Space Science Research Developments [1 ed.] 9781621000495, 9781612090863

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Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

SPACE SCIENCE, EXPLORATION AND POLICIES

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

SPACE SCIENCE RESEARCH DEVELOPMENTS

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Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

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Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

SPACE SCIENCE, EXPLORATION AND POLICIES

SPACE SCIENCE RESEARCH DEVELOPMENTS

JONATHAN C. HENDERSON AND Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

JENNIFER M. BRADLEY EDITORS

Nova Science Publishers, Inc. New York

Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

Copyright © 2011 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Space science research developments / editors, Jonathan C. Henderson and Jennifer M. Bradley. p. cm. Includes index.

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1. Space sciences--Research. I. Henderson, Jonathan C. II. Bradley, Jennifer M. QB500.262.S637 2011 520--dc22 2010047631

Published by Nova Science Publishers, Inc. † New York

Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

CONTENTS Preface Chapter 1

Research on Aerodynamics of Large Bolides V.P. Stulov

Chapter 2

Solar Dynamics and Solar-Terrestrial Influences Katya Georgieva

Chapter 3

Geology of the Terrestrial Planets with Implications to Astrobiology and Mission Design Dirk Schulze-Makuch, James M. Dohm, Alberto G. Fairén, Victor R. Baker, Wolfgang Fink, Robert G. Strom

Chapter 4

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vii

The Origin of the Magnetic Field of Pulsars and the Gravitomagnetic Theory Jacob Biemond

1 19

67

101

Chapter 5

Paleoshorelines and the Evolution of the Lithosphere of Mars Javier Ruiz, Rosa Tejero, David Gómez-Ortiz, and Valle López

131

Chapter 6

Dealing with Potentially Hazardous Asteroids Eric W. Elst

155

Chapter 7

Thermal Properties and Temperature Variations in Martian Soil Analogues F. Gori and S. Corasaniti

Chapter 8

Chapter 9

On The 5D Extra-Force According to Basini-Capozziello-Ponce De Leon Formalism and the Experimental Research of Extra Dimensions On-Board International Space Station(ISS) Using Laser Beams Fernando Loup Origin of the Saturn Rings: Electromagnetic Model of the Sombrero Rings Formation Vladimir V. Tchernyi

Index Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

203

235

267 281

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

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PREFACE This book presents current research in the field of space science. Topics discussed include research on the aerodynamics of large bolides; solar dynamics and solar-terrestrial influences; the geology of terrestrial planets; the origin of the magnetic field of pulsars; paleoshorelines and the evolution of the lithosphere of Mars; thermal properties and temperature variations in Martian soil and the origin of the Saturn rings. Chapter 1 - Crushing and evaporation are basic physical processes at the entrance of natural space bodies of size 0.01-100 m into atmospheres of planets. The relative role of evaporation is characterized by value of mass loss parameter, which is proportional to the relation of kinetic energy of a body mass unit to effective enthalpy of evaporation. Examples of real large bolides for various values of this parameter are given. The offered general approach helps to understand a plenty of observant data of a different level of reliability, and also allows for separation of preliminary real hypotheses from fantastic. Chapter 2 - Much research has been done on the influences of solar activity on processes on Earth and in near-Earth space. While there is a general consensus about the reality of such influences, the results are often contradictory even if highly statistically significant, in many cases the physical mechanism of the influence is unclear, and the mechanism of solar activity itself is not yet fully understood. Obviously there are additional factors affecting solar activity and its influences on the Earth which are not accounted for in the present models. In this paper we are investigating one of these possible factors – the dynamics of the Sun and its significance for the solar-terrestrial influences. One of the most controversial problems in solar-terrestrial physics is the relation between solar activity and climate. Different authors have reported positive, negative or missing correlations between solar activity and surface air temperature in the 11-year sunspot cycle. We demonstrate that the sign of the correlation changes regularly in consecutive secular (Gleissberg) solar cycles, and depends on the long-term changes of North-South solar activity asymmetry which in turn is determined by the movement of the Sun about the barycenter of the Solar system. This movement of the Sun about the barycenter of the Solar system is also related to the dynamics of both the Sun itself and of the Earth. The asymmetry in the rotation of the Northern and Southern solar hemispheres correlates very well with the variations in the Earth rotation rate, and their dominant common periodicity is the periodicity of the rotation of the Sun about the Solar system barycenter. The correlation of the solar dynamics and the Earth dynamics is mediated by the solar wind carrying momentum and magnetic fields and

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viii

Jonathan C. Henderson and Jennifer M. Bradley

modulating the electromagnetic core-mantle coupling torques responsible for the variations in the Earth’s rotation. The variable solar differential rotation affects also the way in which the solar drivers interact with the Earth’s magnetosphere. The periodicities in the interplanetary magnetic field at the Earth’s orbit in any period coincide with the periodicities in the latitudinal gradient of the differential rotation in the more active solar hemisphere, and the azimuthal components of the interplanetary magnetic field are proportional to the solar equatorial rotation rate. The most geoeffective solar driver are the magnetic clouds – coronal mass ejections with smooth rotation over a wide angle of the magnetic field inside the structure. The portion of coronal mass ejections which are magnetic clouds varies throughout the solar cycle, and is determined by the amount of helicity transferred from the solar interior to the surface, and by the surface differential rotation. We demonstrate the influence of the surface differential rotation on the helicity and geoeffectiveness of magnetic clouds. Chapter 3 - Although Mercury, Venus, Earth/Moon, and Mars originated from similar material accreted in close regions of the planetary nebula, the worlds that developed after 4.6 billion years of Solar System evolution are not much alike. Mercury and the Moon, for example, are without a protecting atmosphere and were not exposed to liquid water on their surface for at least 4 billion years. The surfaces of Venus and Mars are desiccated and are presently not a suitable habitat for life, but reservoirs of liquid water remain in the atmosphere of Venus and the subsurface of Mars, and microbial organisms may have adapted to survive in these ecological niches. The search for water and life on any of the terrestrial planets is intrinsically connected to their geological history. Missions should be designed to explore any potential past and present habitats. In addition, ices on Mercury and the Moon should be explored for remnant biogenic material from the early evolution of life on Earth (and elsewhere). Especially desirable are sample return missions and missions that include hierarchical architectures with scalable degrees of mission operation autonomy, which will allow optimal reconnaissance of planetary environments, including surface and subsurface environments. Chapter 4 - Many authors consider a gravitational origin of the magnetic field of celestial bodies. In this chapter the validity of the so-called Wilson-Blackett formula for pulsars is investigated. This formula predicting a dipolar magnetic field for all rotating bodies has previously been deduced from general relativity, e.g., by application of a special interpretation of the gravitomagnetic theory. Other consequences from this theory will also be considered in this chapter. First, the standard quadrupolar charge density and a monopolar charge for pulsars are derived from the gravitomagnetic theory. In addition, contributions to the total magnetic dipole moment from moving charge are deduced. It appears that these contributions may enforce or weaken the basic magnetic dipole moment from gravitomagnetic origin. Many of these results also result from the so-called “dome and torus” model for pulsars. A tentative extension of this model is given, in order to ensure charge neutrality of the system pulsar plus surroundings. The observed magnetic field for fourteen slowly rotating, binary, accreting X-ray pulsars resembles more the gravitomagnetic prediction than the value calculated from the standard magnetic dipole radiation model. At this moment, for five isolated X-ray pulsars such a comparison is difficult, owing to uncertainty in the assignment of the cyclotron lines determining the observed magnetic field.

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Preface

ix

For a sample of 100 pulsars the averaged (gravito)magnetic field, extracted from the magnetic dipole spin-down model, may be compatible with the gravitomagnetic prediction. Unfortunately, the (gravito)magnetic field itself has not yet directly been measured. Finally, first and second order braking indices are discussed. It appears that they do not depend on the magnetic field from gravitomagnetic origin. Chapter 5 - The existence of features indicative of shorelines of ancient oceans on Mars has been proposed for several authors. In this chapter we revise the topography of possible Martian paleoshorelines, and their consequences for the amount of water infilling ocean basins when water load is considered. The authors show that a re-evaluation of paleoshorelines is need. For example, the putative Meridiani shoreline could be the same feature as some portions of the Arabia shoreline. Indeed, elevations in the Meridiani shoreline are roughly similar to that of the Arabia shoreline in northeast Arabia, Utopia (not taken into account the Isidis basin), Elysium, and Amazonis regions. This is still far of an equipotential surface, but a paleoshoreline through these regions and the Meridiani shoreline would be better candidate to represent a paleoequipotential surface than the Arabia shoreline sensu strito. Moreover, the elevation of the Arabia shoreline in northern Arabia Terra after is intriguingly close to the mean elevation of the Deuteronilus shoreline, and it cannot be discarded a “mixed” Arabia/Deuteronilus shoreline, which would include the Arabia shoreline in northern Arabia Terra, and the Deuteronilus shoreline elsewhere. If reality of global shorelines is accepted, as increasing evidence suggests, then presentday topographic variations in these features postdate shorelines formation. So, their topographic range should provide information on large-scale vertical movement of the lithosphere, which in turn provides information on the thermal evolution of Mars. We describe the application of thermal isostasy concept to constraint the ancient thermal state of the lithosphere from present-day paleoshoreline topography. For the ~1.1 km total elevation range of the Deuteronilus shoreline, the relative amplitude of heat flow variations (the ratio between maximum and minimum heat flow) is ≤1.6. This value is clearly lower than that presently observed on continental areas on the Earth. If heat flow variations on Mars are currently greatly disappeared, then the obtained heat flow variations upper limits must be mostly related to the paleoshoreline formation time: the present-day elevation range along Deuteronilus shoreline suggests that differences in the thermal state of the lithosphere in regions along this putative paleoshoreline have been relatively small since the feature was formed, and therefore the absence of lithospheric tectonothermal events by the latest ~3 Gyr, at least. If the Deuteronilus shoreline is a combination of portions of several paleoshorelines, then the total elevation range, and the implied heat flow variations, would be lower, and the lithosphere stability higher. Chapter 6 - An excurse is made through the main belt of asteroids and beyond. Starting with the discovery of the asteroid Ceres, during the first night of January 1801, by Piazzi in Palermo (Sicily), the discipline of asteroid-discovery evolved quickly into large searches of similar bodies in the solar system. At this very moment, more than130.000 new objects have been numbered and the more of them classified into families. One of these groups of asteroids, the NEO’s (Near Earth Objects), subdivided in three substantially different subfamilies (Aten-, Amor- and Apollo-objects) have presently become search objects of primary importance; since they may collide with Earth, resulting in disastrous situations on our planet and (posing) serious risks for the preservation of life on its surface.

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Jonathan C. Henderson and Jennifer M. Bradley

Chapter 7 - The present work reviews the information on mineralogy and chemistry of Martian soils from the data of Landers and remote sensing. Compositional information from Mars Pathfinder Landers indicates a mixture of weathered local rocks of mafic composition. If the Martian soil is composed by sediments only, porosity can be 50% at the surface. In other areas of interest a surface porosity of 20% can be present for lavas. Martian soil analogues, in dry and frozen conditions, are investigated in the present work as far as thermal conductivity and temperature variation, along its depth, are concerned. The thermal conductivity is predicted theoretically with the cubic cell model, which considers the dry soil as a cubic cell, with a solid cubic particle at the center, and gas around it. The thermal conductivity of the porous medium is evaluated from the solution of the one-dimensional Fourier conduction equation with the assumption of isothermal lines. The model needs to know the thermal conductivities of the solid particle and of the materials present, i.e. atmospheric gas and/or frozen ice, and the porosity of the soil analogue. Martian soil analogue is simulated with a kind of olivine tested in laboratory. The soil mineral composition allows to evaluate experimentally the thermal conductivity of the olivine particle, with the help of the theoretical method, which results equal to Ks = 2.94 W/m K. The thermal conductivity of dry Martian soil analogue, evaluated with the theoretical model in the porosity range p = 0 – 1 and at several temperatures, increases with the temperature and decreases with the porosity. The Martian atmospheric pressure is about 6 mbar and the Martian atmospheric gas is composed of CO2 (95%), N2 (3%), Ar (1.5%) and traces of water vapour. Heat capacity of soil analogue is evaluated with the knowledge of its physical properties, the porosity and the specific heats of the materials present. Thermal diffusivity, calculated as the ratio of thermal conductivity and heat capacity, is a function of porosity and ice mass content of the soil analogue. The temperature of Mars surface is assumed variable during the day in the range: 148 K – 298 K. Temperature variations in dry and partially frozen soil analogues are predicted during a Martian day. The temperature variation at different depth is attenuated, as compared to the surface variation, and a phase delay is present depending on the soil thermal properties. The temperature variation, as well as the derivative of the temperature variation with the depth, is dependent on the thermal diffusivity of the soil analogue. In conclusion, the temperature measurement along the depth of a Martian soil analogue can be used to verify its physical status, i.e. dry or partially frozen. Chapter 8 - We analyze the possibility of Experimental Research of Extra Dimensions On-Board International Space Station (ISS) by using a Satellite carrying a Laser device (optical Laser) on the other side of Earth Orbit targeted towards ISS.The Sun will be between the Satellite and the ISS so the Laser will pass the neighborhoods of the Sun at a distance R in order to reach ISS. The Laser beam will be Gravitationally Bent according to Classical General Relativity and the Extra Terms predicted by Kar-Sinha in the Gravitational Bending Of Light due to the presence of Extra Dimensions can perhaps be measured with precision equipment.By computing the Gravitational Bending according to Einstein we know the exact position where the Laser will reach the target on-board ISS.However if the Laser arrives at ISS with a Bending different than the one predicted by Einstein and if this difference is equal to the Extra Terms predicted by Kar-Sinha then this experience would proof that we live in a Universe of more than 4 Dimensions.We demonstrate in this work that ISS have the needed precision to detect these Extra Terms. Such experience would resemble the measures of the Gravitational Bending of Light by Sir Arthur Stanley Eddington in the Sun Eclipse of 1919

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xi

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that helped to proof the correctness of General Relativity although in ISS case would have more degrees ofaccuracy because we would be free from the interference of earth’s atmosphere. Chapter 9 - For the first time the role of superconductivity of the space objects within the Solar system located behind a belt of asteroids is considered. Observation of experimental data for the Saturn’s rings shows that the rings particles may have superconductivity. Theoretical electromagnetic modeling demonstrates that superconductivity can be the physical reason of the origin of the rings of Saturn from the frozen particles of the protoplanetary cloud. The rings appear during some time after magnetic field of planet appears. It happened as a result of interaction of the superconducting iced particles of the protoplanetary cloud with the nonuniform magnetic field of Saturn. Finally, all the Kepler’s orbits of the superconducting particles are localizing as a sombrero disk of rings in the magnetic equator plane, where the energy of particles in the magnetic field of Saturn has a minimum value. Within the sombrero disc all iced particles redistributing by the rings (strips) like it is happened for the iron particles nearby the magnet. Electromagnetism and superconductivity allow us to understand why planetary rings in the solar system appear only for the planet with the magnetic field after the belt of asteroids where the temperature is low enough and why there are no rings for the Earth, and many other phenomena. Versions of these chapters were also published in Journal of Magnetohydrodynamics, Plasma and Space Reserch, Volume 14, Numbers 1-4, edited by Frank Columbus, published by Nova Science Publishers, Inc. They were submitted for appropriate modifications in an effort to encourage wider dissemination of research.

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Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

In: Space Science Research Developments Editors: J.C. Henderson and J. M. Bradley

ISBN 978-1-61209-086-3 © 2011 Nova Science Publishers, Inc.

Chapter 1

RESEARCH ON AERODYNAMICS OF LARGE BOLIDES V.P. Stulov Institute of Mechanics Moscow Lomonosov State University, Moscow, Russia

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ABSTRACT Crushing and evaporation are basic physical processes at the entrance of natural space bodies of size 0.01-100 m into atmospheres of planets. The relative role of evaporation is characterized by value of mass loss parameter, which is proportional to the relation of kinetic energy of a body mass unit to effective enthalpy of evaporation. Examples of real large bolides for various values of this parameter are given. The offered general approach helps to understand a plenty of observant data of a different level of reliability, and also allows us to separate preliminary real hypotheses from fantastic.

INTRODUCTION Modern researches of meteors and bolides reveal two limiting cases of these phenomena. In the first case the question is the physics of meteors generated by particles of the small sizes, less than 1 mm. This case is investigated most in details. The entrance of such particles is accompanied by their full combustion in the high layers of atmosphere, at altitudes of 80100 and more kilometers. These events occur often enough. There is an extensive observant material and numerous analytical researches. The last are reduced, basically, to attempts to reproduce a curve of luminosity of a meteor along its trajectory to determine observable parameters: mass of a meteoric particle, structure and other physical and chemical parameters. There is a significant progress in this area. However some authors mark{celebrate} essential divergences of the theory and observations (see, for example, Lebedinets, 1980). In other limiting case of very much meteoric body sizes, presence of atmosphere is not considered as essential. There is a collision of a body with a planet, giving numerous consequences, elementary of which is formation of craters on the solid surface. Recently the

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V.P. Stulov

similar phenomena are investigated in connection with their probable influence on formation of planets at their early stages. In the given work the intermediate situation, i.e. an entrance into atmosphere of planets of natural space bodies in the size in a range of 0.01-100 m, is considered. These phenomena occur essentially less often, than micrometeors. However the significant observant material recently has collected. There is a hope for its updating in connection with development of observant technical equipment. As stimulus to studying this material serves not only cleanly scientific interest to natural phenomena, in particular, to properties of material bodies in a space, but also improvement of estimations of so-called asteroid hazard which reality increases in process of development of the civilization. We study bolides using methods and results of hypersonic aerodynamics as a meteoroid moves in atmospheric gas in the conditions corresponding{meeting} the fluid dynamics. It has appeared that not only the body size, but also speed of entrance and thermo-physical properties of the body can determine substantially the type of interaction with an atmosphere and the basic consequences. The elementary theory of meteors shows that the solution for trajectories depends on two dimensionless parameters α and β (Stulov, Mirskii, Vislyi, 1995)

1 2

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α = cd

ρ 0 h0 S e c V2 ,β = h e ∗ . M e sin γ 2cd H

(1)

In the solution of the trajectory equations a corner of trajectory γ, coefficients of drag cd and heat exchange ch, and also effective enthalpy of destructions H* are constant. In formulas (1), speed Ve, mass of body Me and the area of middle section Se at the entrance into atmosphere (index «е»), and also height of a homogeneous atmosphere h0 and density of gas on sea level ρ0 are present. Parameter α characterizes intensity of braking as it is equal to the relation of mass of the atmosphere column with cross-section section Se and heght h0 to the body mass. Parameter β is proportional to the relation of kinetic energy of a body mass unit to effective enthalpy of evaporation. The obvious formula for a trajectory containing α and β allows to estimate easily a relative role of evaporation and braking. Also it is easy to determine at what stage of body movement in atmosphere its crushing takes place. It has been made earlier in work of the author (Stulov, 1998а). It has been shown that for large bodies (α ≤ 1) in a wide range of parameter β values (β ≥ 1) process of the body evaporation precedes its braking. For stone bodies crushing accompanies evaporation; more fragile ice bodies at enough big speed of entrance can start to be splitted up even prior to the beginning of heating and evaporation. The relative role of crushing and evaporation can be characterized in value of parameter β. Small β correspond to entrance enough heat-resistant objects with rather small speeds. If solidity of the body is not large there is its crushing on set of fragments, at rather small role of mass loss due to blowing awat a liquid film and evaporation of the body material on its frontal surface. As a result this case corresponds to fall of numerous fragments on the planet surface with formation of meteoric and crater fields. At moderate β, roles of crushing and evaporation during meteoroid interaction with atmosphere are comparable. Depending on morphological properties of a body, crushing can

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Research on Aerodynamics of Large Bolides

3

occur or on set of fragments in one or several stages, or gradually by consecutive separation of small fragments from the parental body. Fine fragments are braked more quickly and lag behind from large. Because of it a role of evaporation on fine fragments fast becomes secondary. At last, great values β correspond to primary gasification of the body, or as single object, or in the form of a cloud of fragments. Thus evaporation (and crushing) takes place at rather early stage of entrance into atmosphere, so braking of meteoroid as a solid body is insignificant. Asymptotic solution (Stulov, 1994) of the trajectory equations for β >> 1 shows that evaporation comes to the end practically at the speed equal to speed of entrance. Below examples of widely known real falling, corresponding three described above ranges of parameter β, are given. They are the widely known Sikhote-Aline meteoric rain (S.A.m., 12.02.1947, Fesenkov, 1951), the Beneshov bolide, recorded by the stations of Czech Republic which are the part of the European bolide network (B., 07.05.1991, Spurný, 1994) and, at last, falling the Tunguska space body (TSB, 30.06.1908, Korobeinikov, Tshushkin, Shurshalov, 1991), investigated up and down, however, till now causing serious disagreements and discussions, including messages on detection of fragments of an other planet spacecraft (the program of telechannel TVC, August, 2004) . In brackets after the name of each event, there are specified: an abbreviation of the name used further, date of event and the reference. Under the reference we shall give here a following explanation. Scientific (and pseudo-scientific) literature on each event is rather extensive. Here there are those papers from which the author took the numerical data used below. In table 1, values of β for each of three events are given. Below for some of initial numbers, corresponding sources are specified. Calculated values of ch and β are received by means of the approximate method, in detail described in the monography (Stulov, Mirskii, Vislyi, 1995). As already it was specified above (after formulas (1)), the coefficient of heat exchange ch in the table is considered to be constant. It was estimated on parameters of entrance into atmosphere and characteristic height of real event (destruction, flash, etc.). Calculated values of ch depend on speed of entrance Ve, the initial characteristic size of body R and density of air. Table 1. Event

Ve, km/s

S.-A.m. B. TSB

14.51 21.182 353

H*, 103J/g 84 25 25

сh, estimated

сh, calculated5

β, estimated

0.010 0.0122 0.100

0.018 0.018 0.090

0.13 1.35 30.63

β, calculated5 0.23 2.03 27.57

In the table there are used the following references: 1 (Fesenkov, 1951), 2 (Spurný, 1994), 3 (Korobeinikov, Tshushkin, Shurshalov, 1991), 4 (Chyba Ch. F., Thomas P.J., Zahnle K.J., 1993), 5 (Stulov, Mirskii, Vislyi, 1995). In all cases it is accepted cd = 1. From the table follows, that calculated and estimated values of β are close, anyway, both calculated, and estimated values of β for three real events differ approximately on the order of value. Thus, if parameter α is responsible for intensity of braking at passage of atmosphere, parameter β characterizes a role of thermochemical destruction and considerably influences

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4

V.P. Stulov

the basic consequences of the meteoroid entrance into atmosphere. Mechanical crushing takes place almost always. Therefore, stated above judgement about parameters α and β concern both to initial meteoroid, and to its fragments.

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CRUSHING OF METEORIC BODIES There are various approaches to the solution of the problem of meteoric bodies crushing under action of aerodynamic loading. In the elementary cases semi-empirical rules of separation of an initial parental body on fragments are offered. More complex approach assumes the solution of the problem of the theory of solidity at variable external loading. We shall discuss more in detail these two points of view. Earlier three models for calculation of destruction and movement of fragments in an atmosphere have been offered: slow consecutive crushing (Baldwin, Sheaffer, 1971), instant destruction (Grigoryan, 1979), fast consecutive crushing (Ivanov, Рыжанский, 1997). We had been constructed analytical solutions for calculation of trajectories according to these models (Stulov, 1998b; Stulov, Stulov, 1999; Stulov, Titova, 2001a.). Then comparison of calculated trajectories on these models for specific variants of falling (Stulov, Titova, 2001b) has been given. The ideas incorporated in the models, only partially reflect the real phenomena. So, analysis of Sikhote-Aline falling has shown (Krinov, 1975; Krinov, Tsvetkov, 1979) that destruction of the space body occured in three discrete stages. Besides, small flat fragments were separated from large fragments during movement. Apparently, this small fragments were observed by eyewitnesses in the form of sparks. Further, the description of in detail documentary Beneshov bolide (Borovicka, Spurný, 1996) testifies that in the beginning on the certain site of its trajectory (24 < h < 42 km) from a parental body were consistently separated rather fine fragments; in the end of this site has occured the explosion-like destruction, accompanied almost instant going out the bolide, i.e. fast braking of products of destruction. Apparently from this brief description, analytical models can be used for the description of separate stages of real falling. Attempts to calculate destruction of a meteoric body in atmosphere by statement and solution of solidity problems have been undertaken. The equations of separation dynamics for meteoric body have been constructed and solved in the work of Grigoryan, 1979. It was shown that the braked and collapsing object transfers its energy to atmospheric gas volume with cross-section sizes considerably exceeding the initial size of the meteoric body. Korobeinikov and co-authors (1994) have calculated the intense condition of a meteoric body in process of introduction into atmosphere. It has appeared that the point of the maximal pressure appears inside of the body. Gradually the area of breaking points extends down to achievement of a surface of the body, then its disintegration begins. Authors emphasize necessity of the account of initial heterogeneity and casual defects for a meteoric body. The rests of meteoroid after destruction move in the form of a swarm of fragments. Below some results of numerical modeling the initial stage of this movement (Zhdan, Stulov, Stulov, 2004а; Zhdan, Stulov, Stulov, 2004b) are given. The stage is characterized by intensive aerodynamic interaction of elements in the swarm.

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Research on Aerodynamics of Large Bolides

5

To study this question the model of a supersonic flow around a system of small number of spheres has been used. The purpose of calculations is definition of aerodynamic coefficients of spheres in the swarm and studying of changes of shock wave configurations depending on distance between bodies. These data promote understanding mechanics of fragments scattering after destruction of a body in a supersonic stream. Here as an example the supersonic flow of three configurations is considered: 1 - two spheres with a line of the centers across a stream, 2 - three spheres in apexes of an equilateral triangle, 3 - six spheres in apexes and the middle of the sides of an equilateral triangle; in cases 2 и 3 the side of thr triangle is perpendicular to the stream. The centers of spheres lay in one plane containing a vector of initial flow velocity. As the size of configurations the size h, equal to half of distance between the nearest points of two next spheres, related to radius of sphere R is used. Comparison of aerodynamic properties of three simple configurations probably will allow to draw the general conclusions about systems of bodies in more complex cases, including both for an irregular arrangement of bodies, and in case of some changes of their form. All the calculations presented here are lead for systems of bodies in a stream of the perfect gas with parameters of M = 6, γ = 1.4. Aerodynamic coefficient of drag cx and cross-section force cy were calculated under following formulas:

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(2) Here S = πR2 is the area of middle sections of the sphere, ρ∞V∞2/p∞ = γM2, pressure р in the integral is related to pressure in the initial stream p∞, corner θ is counted in the meridional plane from a forward point of sphere

Figure 1. Cross-force coefficient for two spheres with the line of centers across the stream (Mach number M = 6) and two semi-cylinders (Shuvalov, Artemieva,Trubetskaja, 2000).

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The supersonic flow of system 1 was studied earlier as the elementary model of fragments scattering a meteoric body after its destruction. In work (Passey, Melosh, 1980) it was offered a rather rough model of a divergence of two fragments in cross direction to a stream. According to this model a constant force acts on two fragments. The force is equal to product of stagnation pressure and middle section, i.e. one has cy = 2. From formulas of uniformly accelerated movements, value of cross-section speed on distance of the order of the characteristic size of a fragment is received. Unfortunately, this model has affirmed in the literature, in the further over and over was used, and even has entered into the review (Nemchinov, Popova, Teterev, 1999). The early experimental and approximated researches (Dubrovina, Stulov, 1989) have shown that, really, aerodynamic interaction of spheres in a configuration 1 stops on distances of the order of the fragment characteristic size when the head shock wave reflected in Mach configuration is tangent the surface of sphere. However numerical solutions for system 1 have found out significant reduction of cross-section force in comparison with the value accepted in work (Passey, Melosh, 1980). It has appeared that the value of cy not too strongly depends on the fragment form. On Figure 1 dependence cy(h) for system 1 (a continuous line) and for two half-cylinders, received by cutting of the cylinder with height R and radius of basis R along a plane passing through its axis and containing the velocity of initial stream is shown. Data for half-cylinders are published earlier (Artemieva, Shuvalov, 1996) and taken by us from later publication (Shuvalov, Artemyeva, Trubetskaja, 2000). The satisfactory consent as on value of max cy ≈ 0.27, and on the greatest distance of interaction h ≈ 0.5 is visible.

Figure 2. Coefficients of drag and cross force in systems 1 (shaped lines), 2 (continuous lines) and 3 (badges:о - cxfm, x - cxfa, cyfa,  - cxsm, cysm,  - cxb).

Results of calculations of the flow around system 1 are given also on Figure 2 by shaped lines. Certainly, during the initial moments after destruction, replacement of fragments with spheres is not quite correct and cross-section force will be defined by stagnation pressure,

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Research on Aerodynamics of Large Bolides

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however development of flow between fragments at once will lead to pressure drop between them. Aerodynamics of system 2 was investigated in comparison to results of a flow around system 1. We shall designate parameters of forward sphere in system 2 by index f, and parameters of back sphere by index b. Coefficient of drag cxf and cross-section force cyf of the forward sphere, and also coefficient of drag cxb of the back sphere are shown on Figure 2 by continuous lines. It is visible that presence of the third sphere only slightly changes conditions of the flow around the forward spheres. This little change is expressed in some reduction both cx and cy of the forward sphere due to increase in ground pressure because of presence of the third sphere. The form of the head shock wave before two forward spheres almost does not change. Let's discuss the flow about the back sphere. At small distances h before it the shock wave limited on scope because of non-uniformity of the initial stream is formed. With growth of h scope of a shock wave increases and drag cxb monotonously increases. The situation does not change and at distances when interaction of forward spheres stops, i.e. at h > 0.5. With increase of h on a frontal part of back sphere the reflected shock wave falls, so the value of cxb continues to increase. At the further increase of h, the point of reflection, and then a lateral part of head shock waves of forward spheres fall on a head shock wave of back sphere. During all period of interaction of shock waves of forward spheres with back sphere 3, the value of cxb exceeds the coefficient of drag of a single sphere in an infinite stream. It is shown on Figure 2 where at 2h ≥ 1.0 the scale on an absciss axis is changed on logarithmic one. Here a new factor of interaction of bodies in system in a supersonic stream is noted: interaction of the reflected shock waves with bodies results not only in occurrence of crosssection force, but also in substantial growth of drag. Properties of the flow around system 3 in the certain degree repeat the tendency noted above. In system 3, the following system of designations is accepted: parameters of spheres in the front on the stream row accept an index f, the second row accept an index s, and the back sphere accept an index b. Further, angular spheres are supplied with an additional index a, the spheres located in the middle of the triangle sides are supplied with an index m. Coefficients of drag and cross-section force of spheres fa and sm and coefficient of drag of spheres fm and b are given on Figure 2. These values are shown by badges. Decoding of badges is given in the signature to Figure 2. We shall note here more essential decrease in drag of sphere fm in comparison with sphere fa for which this decrease practically repeats reduction cxf in system 2. It occurs owing to greater increase in ground pressure at sphere fm because of presence of the third layer made by sphere b in a configuration 3. At the same time, drag of sphere b increases with growth h more slowly, than resistance of sphere f in system 2 or spheres sm in system 3. The form of a head shock wave with increase h changes practically the same as it takes place in configurations 1 and 2. There is a transition from the common shock wave before system to individual shock waves before each sphere in a head part of system. Obviously factor of interaction of the reflected shock waves noted above with head shock waves before spheres of back lines also will take place and lead to appreciable increase in drag of these spheres. The picture of a flow around system 3 is shown on Figure 3 for case at h = 0.5.

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Figure 3. The flow around system 3, h = 0.5; there are shown lines of Mach constant.

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The basic conclusion of research consists in the fact that at a supersonic flow around system of bodies, distortion of the stream due to «collectivity» of obstacles as though is carried downwards on a stream. Small reduction of drag of head spheres arises due to increase in ground pressure. The last, however, practically does not affect reorganization of the head shock wave which with growth of distance between spheres monotonously breaks up to individual shock waves. The characteristic distance between bodies for this disintegration also does not change and quite corresponds to a simple case of a flow of pair spheres with a line of the centers across the stream.

BOLIDES WITH THE MODERATE VALUES OF MASS LOSS PARAMETER The majority of bolides of Prairie and European bolide networks correspond to this range of value of parameter β. Trajectories of the largest bolides of Prairie networks, USA (MacCrosky, Shao, Posen, 1978; MacCrosky, Shao, Posen, 1979) have been processed by the least squares method in which as trial functions solution for a single body with ablation were used. Values of β for eighteen bolides Tab. 7.1 (Stulov, Mirskii, Vislyi, 1995) (excepting bolide 39470*) lay in a range 0.48 < β < 2.76. As it is known, in all cases, except for bolide Lost City, there was not revealed, any traces of falling meteoric substance, despite of careful searches in the areas predicted by extrapolation to the Earth of observable sites of trajectories. Among bolides of the European network the significant attention of researchers was given to the bolide Beneshov, recorded in Czech Republic on May 7, 1991. According to opinion of one of observers (Spurný, 1994), it was one of the brightest and in details documentary bolide. The big observant material on results of the Canadian camera network during 1974-85 was published in the work Halliday, Griffin, Blackwell, 1996. One of fundamental problems of meteoric physics is the definition of preatmospheric mass of bodies which form bolides. Intensity of the meteoric phenomenon serves is defined

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Research on Aerodynamics of Large Bolides

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by kinetic energy of a body at the approach to a planet atmosphere. As it is known, speed of bodies at the entrance into the Earth’s atmosphere lays in rather narrow range 11 < Ve < 72 km/s, so the range of values of the speed contribution to kinetic energy does not exceed 50 times. At the same time value of a meteoric body mass can change in essential wider range, from shares of gram (micrometeors) up to hundreds thousand tons (the Tunguska space body), i. e. on 12-14 orders. Besides the entrance speed is rather simply determined in observation of the initial site of the atmospheric trajectory. On the contrary, reliable ways of definition of the entrance mass, containing an estimation of accuracy of result, now are absent. Approaches known from the literature to an estimation of the entrance mass can be divided conditionally on two groups. The so-called photometric methods using luminosity of bolides enter into the first. More often a method of definition of photometric mass Mph on the basis of the formula tb

I dt. 2 tt τV

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M ph = −2 ∫

(3)

Here I is intensity of a luminescence along an observable site of a trajectory, tb, tt are initial and final times for this site, τ is a factor of proportionality. The formula (3) is based on the assumption that defining contribution to luminosity is given by vapors of the body material. Detailed calculations show that it is impossible to consider it is proved, especially for large bodies. In work of Halliday, Griffin, Blackwell, 1996 attempt is made to modernize a photometric method of definition of preatmospheric meteoric masses. Firstly as the luminous efficiency τ it is taken the variable, proved (calibrated) in early works of authors. It is considered that factor τ depends on speed of bolide movement: τ = 0, V ≤ 10 km/s; τ = 0.04, 10 < V < 36 km/s; τ = 0.069(36/V)2, 36 km/s ≤ V. Secondly, for large bolides the value of the possible mass which remained after going out the bolide, and fell as possible meteorites, is calculated. This mass is determined with cleanly dynamic way by means of the local value of deceleration received by numerical differentiation of the bolide speed distribution on time in the end of the light site of the trajectory. If this dynamic mass appeared to be significant, the conclusion about possible falling meteorites was made. The full preatmospheric mass of the meteoroid was estimated as the sum of photometric mass and final dynamic mass. The hypothesis about falling, however, has not proved to be true in actual searches of meteorites. We have made calculations of preatmospheric mass for bolides of the Canadian camera network by means of the least squares method with use of observant data about the light site of the trajectory. As trial function the form of a trajectory for the limited values of mass loss parameter β (the formula (6), see below) was used. Earlier this method was applied to bolides of the Prairie network, USA; results are given in monography Stulov, Mirskii, Vislyi, 1995. In case of the Canadian camera network, the elementary variant of a method consisting in calculation of parameters α and β under obvious formulas (7.2) of monography Stulov, Mirskii, Vislyi, 1995 was used. Results of calculations for four cases are given in table 2. According to discussion of the elementary variant of the least squares method, given in the monography, two bases for approximation of an observable

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V.P. Stulov

trajectory (the second column of the table) here were used. As it is specified in the monography, the basis of the seven points, not containing an initial site of the trajectory with small braking where v ≥ 0.9 is preferable. Table 2.

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No. of bolide (Tables 4 Basis for least squares method and 6) 018 7 points (p.) t, s: 3.00 and 6 next points (n.p.) 10 p. t, s: 1.00; 2.00 and 8 n.p. 223 7 p. t, s: 4.51 and 6 n.p. 10 p. t, s: 2.07; 3.51 and 8 n.p. 567 7 p. t, s: 12.08 and 6 n.p. 10 p. t, s: 7.08; 9.08; 11.08 and 7 n.p. 925 7 p. t, s: 17.50 and 6 n.p. 10 p. t, s: 10.00; 14.00 and 8 n.p.

α

β

Me, g

MI, g MT, g (Table 4) (Table 4)

30.60

0.9391

6160

6000

39.55 19.78 11.67 111.62 98.66

0.3119 1.6461 2.9458 1.3430 1.6415

2850 80250 240000 390780 6210 42000 8990

44.49 48.36

1.3885 1.6415

724290 1300000 454650

450

8000 210

40000

As one would expect, comparison to calculations of work Halliday, Griffin, Blackwell, 1996 shows various tendencies. We shall note first of all that here there is no so obvious divergence between photometric and dynamic masses as it was at processing of the Prairie network data. It is consequence of new calibration of the parameter τ. At the same time use of the photometric formula still does not guarantee a correct estimation of true preatmospheric meteoroid mass. Addition of the dynamic mass which have remained after going out of the bolide, does not change the situation essentially. This value practically in all cases examined by us is essentially less than photometric mass. Besides, this value is determined in the work of Halliday, Griffin, Blackwell, 1996 by means of local values of deceleration which have small accuracy of definition. Data processing supervision of the Canadian network will be continued by us. Other approach based on value of luminosity is developed by I.V.Nemchinovym with employees and named by a method of radiating radius (references to original works are available in the work of Borovicka, Popova et al., 1998). The authors have calculated the table of luminosity depending on radius of a body, speed and altitude of flight using the method of non-stationary analogy. The radius of an observable body is determined then under this table on value of observable luminosity at the same values of speed and altitude of flight. The second group of methods should be considered dynamic as the body mass is determined on the basis of braking in the atmosphere. Parameter α defines an area of the maximal braking. As it contains the body size twice (Se и Me), it is convenient to rewrite it so

1 2

α = cd

ρ 0 h0 Ae , 23 sin γ ρ m M e1 3

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(4)

Research on Aerodynamics of Large Bolides

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Here Ae is factor of the body form. In dynamic methods, parameter α (4) is defined by comparison of a calculated trajectory with observable one. After that at given Ae and density of body ρm it is determined Me under the formula (4). The lack of such approach is necessity of the a priori task of values Ae and ρm which are not determined in supervision. The dynamic approach based on the solution for a trajectory in variables v, y (dimensionless speed and height) and the least squares method, was used by us earlier (Stulov, Mirskii, Vislyi, 1995). Other approach considering crushing, was offered by authors of work (Ceplecha et al., 1993). It was named by them the method of gross-fragmentation. Below the detailed analysis of results of the Beneshov bolide observation is given. certain The values of parameters α and β are determined by the least squares method. Losses of the bolide mass both due to observable separation of fragments, and by means of evaporation are calculated. Observation and registration of the Beneshov bolide are made by stations of the Czech part of European bolide network on May 7, 1991.. Speed of almost vertical entrance into atmosphere has made over 21 km/s. The shone site of the trajectory reached from 97 up to 16 km on height. Time of movement has made 5.2 s. The estimation of photometric mass of the has given value of 15 t. Since height of 42 km, the separation of fragments from the basic body was observed. In total before achievement of height of 24 km, 4 fragments were separated. The first fragment, in turn, was divided in two pieces during movement. The last observable crushing the basic body on three large and set of fine splinters have occured at h = 24 km. Final fading the bolide took place at height of 19 km at speed of still compact plenty V = 5.2 km/s (Borovicka, Spurný, 1996). The inverse problem for the trajectory of the Beneshov bolide was solved by the least squares method. As trial functions, it was chosen solutions for a trajectory of a uniform body without ablation

y = ln α − ln(− ln v ),

(5)

and for a trajectory of a uniform body in view of ablation

y = ln α − ln(− ln v ) + 0.83β (1 − v ),

(6)

and also a trajectory for model of slow consecutive destruction. Last is described by following system of formulas

αd 2

e

4 − y 3

−

αd 2

e

4 − y0 3

13 ~ βv02 3 t ⎛ ρ 0Ve2 ⎞ ~ e −β 3 Δ 0 e dt ⎟⎟ ,Δ 0 = ∫ 4 3 . ,α d = α ⎜⎜ = 13 t (3 β ) 3 ⎝ σt ⎠ βv 2 3 (7)

Here σt is meteoroid solidity on a stretching. The height and speed in a point of crushing y0 and v0 are determined from a condition

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V.P. Stulov

ρ 0Ve2 exp(− y 0 )v02 ( y 0 ) = σ t ,v0 ( y 0 ) = exp(− αe − y ). 0

(8)

~ For simplification of calculations following approximation of integral Δ0 is used: e −β

3

(3 β )1 3

~ Δ0 = v − 2 3 − v0− 2 3 exp − β ∗ (v0 − v ) . 3

(

) [

]

(9)

Parameter β* differs from β in the numerical factor, giving the best approximation of

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integral

~ Δ0 .

The formula for a trajectory (7) is applied to the analysis of an observable trajectory of the Beneshov bolide under condition of y < y0, i.e. below height of the first crushing. At calculations it was accepted σt = 1.6×107 dyn/cm2. If to accept that appreciable braking begins below height of the first crushing, i.e. v0 = 1, so chosen value of σt according to (8) provides valueof y0, close to observable, i.e. h = 42 km. The assumption of the beginning braking after the beginning crushing also proves to be true in supervision: v = 0.98 at h = 42 km. Thus, formulas (7) - (9) describe the trial function containing two free parameters α and β*. Application of the least squares method to the observable trajectory of the Beneshov bolide has shown, that the best approximation is provided with the formula (6) at following values of free parameters: α = 7.8, β = 1.5. We note that the received value of mass loss parameter β quite corresponds to the calculations and the estimations given in Table 1. These values allow to estimate the entry mass of the bolide and the average ablation factor. Accepting the numerical values offered by observers: cdAe = 1.21, ρm = 3.7 g/сm3, γ = 80.6o, Ve = 21.12 km/s, we shall receive Me = 28 kg, σ = 0.0067 s2/km2.

(10)

The second value (10) corresponds to data of Spurný (1994). As to initial mass Me, this value depends on the assumption of the spherical form of the body. For example, suppose that the body has the form of a plate 1х1х0.25. Then Ae = 2.52, cd = 2, and Ме = 2023 kg. Thus, the best approximation of an observable trajectory turns out by means of model of a uniform body with ablation. This conclusion received by the least squares method, at first sight seems paradoxical. It is possible to explain it only by the assumption that the lump sum of the observable fragments which have separated from the basic body on a greater part of its trajectory (for example, at 42 > h> 24 km), does not make the main share of full meteoroid mass. Observant data about parameters of the fragments, given by Borovicka and Spurný (1996), allow to lead corresponding estimations. According to observation, the final mass of four observable fragments was Mt = 1.82 kg. On the basis of the data adduced by observers about ablation factors for fragments, calculations of their total initial mass, i.e. mass in the moment of their separation from the parental body, have been lead (Barry, Stulov, 2003)). It was received the following value of total initial mass of these fragments: Mb = 7.19 kg.

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Research on Aerodynamics of Large Bolides

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As the bolide velocity in each point of the trajectory is known from observation, it is possible to determine the total mass loss on any site only due to ablation on the basis of known integral of the equations of movement and mass loss. Calculations show that before final destruction at h = 24 km on all trajectory at β = 1.5 the mass equal to 15.81 kg is carried away, i.e. it is more than twice exceeding total mass of the individual fragments. This circumstance explains that the trajectory in variables "speed-height" is described by model of a single uniform body with ablation. Results of estimation of entry mass of the Beneshov bolide by photometric and dynamic methods are given in Table 3. Table 3.

2

Method of radiating radius 3

Method of a grossfragmentation 4

13000

3000-4000

82

Method

Observations, Mph

Observations, Mph

References

1 15000

Me, kg

The least squares method This work Sphere: 28; a plate 1х1х0.5: 523

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Sources of the results shown in the table: 1 (Spurný, 1994), 2 (Borovicka, Spurný, 1996), 3 (Borovicka, Popova et al., 1998), 4 (Ceplecha, etc., 1993). From data of the table it follows that the divergence of estimations of the bolide preatmospheric mass by various methods is still great. Rapproachement is possible as on a way of specification of physicomechanical properties of a meteoric body, in particular, its form, and by means of perfection of methods of the calculation based on an estimation of bolide luminosity.

ENTRANCE INTO ATMOSPHERE OF THE TUNGUSKA SPACE BODY Significant excess of the specific (on a mass unit) kinetic energy brought into atmosphere by the Tunguska object, above the quantity of energy necessary for evaporation of a mass unit of its substance, was a reason of rather unusual consequences of Tunguska falling on June 30, 1908. We mean full absence of craters, and also the fall and burn of the taiga in huge territory which characteristic size approximately on four orders surpasses the probable geometrical size of the Tunguska space body. These features show that influence of the Tunguska space body rests on the Earth’s surface, most likely had purely gas dynamical character. In work (Stulov, 1994) for the first time it has been shown that asymptotic soluyion of the equations of meteoric physics at great values of parameter β has a following appearance v = 1, m = 1 − 2αβe−y

(11)

In other words, at big β fast evaporation of a meteoric body takes place for the lack of braking. After full evaporation there should be a fast braking of products in a mix with air. Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

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This idea has been developed in the subsequent works of the author (Stulov, 1997; Stulov, 1998a; Stulov, 1998b). In second half of the 90th experts discussed in details a new grandiose bolide phenomenon: falling of fragments of comet Shoemaker-Levy 9 on the Jupiter in the summer of 1994. It has appeared that some natural observations of this phenomenon quite keep within the offered hypothesis of bolides behaviour at big values of β. In publications of the American authors (see, for example, Hammel, Beebe et al., 1995), photos of huge gas emissions out of the Jovian atmosphere, received by means of the Hubble telescope, after the entrance of large fragments of the comet into atmosphere are given. Apparently, it were clots of hot evaporation products of fragments in a mix with the atmospheric gases, thrown out of atmosphere under action of buoyancy forces. In June, 2003 at the conference «95 years to the Tunguska problem» in Moscow, V.V.Shuvalov has made the report on the basis of work published before (Shuvalov, Artemieva, 2002) on numerical modeling an large bolide entrance into atmosphere of planets. The numerical solution has completely confirmed the basic features of the asymptotic soluyion (11).

Figure 4. Relative mass of the solid meteoroid (a) and its velocity (b) as functions of flight altitude; continuous lines - numerical results (Shuvalov, Artemieva, 2002), shaped lines - asymptotic solution (11) at α = 0.077 (ρm = 1.5 g/cm3), β = 30.

By the numerical solution of the nonviscous equations of gas dynamics, calculation of movement of a spherical body in radius of 30 m in the Earth’s atmosphere perpendicular to its surface is made. Entry conditions: Ve = 30 km/s at h = 30 km. Full evaporation of a firm phase comes to the end at height of h = 11 km, thus «visible speed of a meteor», i.e. the meteoroid rests and products of evaporation surrounding it, makes 25-26 km/s. At the subsequent stage of calculation «the air-vapor jet» (« air-vapor jet ») is braked till zero speed at height of 4 km then there is a rise of hot gas volume in atmosphere due to buoyancy forces. The characteristic cross-section radius of a jet during the moment of braking exceeds 2 km. From this brief abstract follows that in work (Shuvalov, Artemieva, 2002) for the first time, the basic properties of the asymptotic form of a bolide trajectory, received on the basis of the meteoric physics equations (Stulov, 1994), are shown by calculatios. It is a question about

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Research on Aerodynamics of Large Bolides

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practically full ablation of meteoroid in conditions of insignificant braking. These properties are shown on Figure 4, where change of relative mass m = M/Me (a) and speed of body V (b) depending on height of flight H are given. Continuous lines show the numerical solution (Shuvalov, Artemieva, 2002), shaped lines show the solution (11) constructed on bolide parameters at meteoroid density ρm = 1.5 g/cm3. The stroke-dashed line on Figure 4 (b) shows change of gas volume speed which is not defined in the asymptotic solution. Let's note that this calculation does not reproduce all major details of entrance of the Tunguska space body into the Earth’s atmosphere. However, it is possible to assume that change of entry conditions, for example, up to values of Ve = 35 km/s, Re= 38 m (Korobeinikov, Tshushkin, Shurshalov, 1991) will allow to reproduce falling “an air-vapor jet” on the taiga and it{her} растекание, accompanied by burn and fall of the wood.

CONCLUSION

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The certain dependence of a bolide character from value of mass loss parameter β is noted. This dependence allows to interpret more confidently large falling, excepting, in particular, fantastic or semi- fantastic scripts. As a result of numerical experiment, coefficients of drag and cross-section force of fragments at the initial stage after crushing a meteoric body are received. It has appeared that for two fragments with a line of the centers across a stream the coefficient of cross-section force in some times less of the value used earlier in the literature. In case of big values of parameter β prevailing process is evaporation of fragments of the destroyed meteoric body. It occurs for the lack of significant braking. Therefore interaction of the bolide with the planet surface can have purely gas dynamical character. The author thanks Dr. M.A.Smirnov for discussion of work.

REFERENCES Artemieva N.A., Shuvalov V.V. Interaction of shock waves during the passage of disrupted meteoroid through atmosphere // Shock waves. 1996. V. 5. No.6, pp. 359-367. Baldwin B., Sheaffer Y. Ablation and breakup of large meteoroids during atmospheric entry // J. Geophys. Res. 1971. V. 76. No. 19, pp. 4653-4668. Barry N.G., Stulov V.P. Features of crushing the Beneshov bolide // Astron. vestn. 2003. V. 37. No. 4, pp. 332-335 (in Russian). Borovicka J., Spurný P. Radiation study of two very bright terrestrial bolides and an application to the comet S-L 9 collision with Jupiter // Icarus. 1996. V. 121, pp. 484-510. Borovicka J., Popova O.P., Nemtchinov I.V., Spurný P., Ceplecha Z. Bolides produced by impacts of large meteoroids into the Earth’s atmosphere: comparison of theory with observations I. Benešov bolide dynamics and fragmentation // Astron. and Astrophys. 1998. V. 334, pp. 713-728. Ceplecha Z., Spurný P., Borovicka J., Kecliková J. Atmospheric fragmentation of meteoroids // Astron. and Astrophys. 1993. V. 279, pp. 615-626. Chyba Ch. F., Thomas P.J., Zahnle K.J. The 1908 Tunguska explosion: atmospheric disruption of a stony asteroid // Nature. 1993. V. 361, pp. 40-44.

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Dubrovina I.A., Stulov V.P. Determination of the greatest distance of aerodynamic interaction for pair spheres and cylinders in a supersonic stream // Vestn. of the Moscow State University. Ser. 1. Mathematics, mechanics. 1989. No. 5, pp. 46-49 (in Russian). Fesenkov V.G. Orbit of the Sikhote-Aline meteorite // Meteoritics. Publishing house AS of the USSR. 1951. Iss. IX, pp. 27-31 (in Russian). Grigoryan S.S. On movement and destruction of meteorites in atmospheres of planets // Kosmich. issled. 1979. V. 17. No. 6, pp. 875-893 (in Russian).. Halliday I., Griffin A.A., Blackwell A.T. Detailed data for 259 fireballs from the Canada camera network and inferences concerning the influx of large meteoroids // Meteoritics and Planetary Science. 1996. V. 31, pp. 185-217. Hammel H.B., Beebe R.F., Ingersoll A.P., Orton G.S., Mills J.R., Simon A.A., Chodas P., Clarke J.T., De Jong E., Dowling T.E., Harrington J., Huber L.F., Karkoschka E., Santon C.M., Toigo A., Yeomans D., West R.A. HST Imaging of Atmospheric Phenomena Created by the Impact of Comet Shoemaker-Levy 9 // Science. 1995. V. 267, pp. 12881296. Ivanov A.G., Ryzhanskii V.A. Fragmentation of a small celestial body at its interaction with atmosphere of a planet // DAN. 1997. V. 353. No.3, pp. 334-337 (in Russian). Korobejnikov V.P., Chushkin P.I., Shurshalov L.V. Complex modeling flight and explosion of a meteoric body in atmosphere // Astron. vestn. 1991. V. 25. No.3, pp 327-343 (in Russian). Korobejnikov V.P., Vlasov V.I., Volkov D.B. Modeling destruction of space bodies at movement in atmospheres of planets // Маth. modeling. 1994. V. 6. No.9, pp. 61-75 (in Russian). Krinov E.L. Crushing Sikhote-Aline meteoric body // Meteoritics. Publishing house AS of the USSR. 1975. Iss. 34, pp. 3-14 (in Russian). Krinov E.L., Tsvetkov V.I. Sikhote-Aline meteoric rain as a classical meteoric falling // Meteoritics. Publishing house AS of the USSR. 1979. Iss. 38, pp. 19-26 (in Russian). Lebedinets V.N. Dust in the top atmosphere and space. Meteors. L.: Hydrometeopub. 1980. 250 p. (in Russian). Mak-Kroski R.E., Shao T.I., Pozen A. Bolides of the Prairie network. 1. The general data and orbits // Meteoritics. Publishing house AS of the USSR. 1978. Iss. 37, pp. 44-59 (in Russian). Mak-Kroski R.E., Shao T.I., Pozen A. Bolides of the Prairie network. 2. Trajectories and curves of shine // Meteoritics. Publishing house AS of the USSR. 1979. Iss. 38, pp. 106156 (in Russian). Nemchinov I.V., Popova O.P., Teterev A.V. Entrance of large meteoroids into atmosphere: the theory and observations // Eng.-phys. J. 1999. V. 72. No.6, pp. 1233-1266 (in Russian). Passey Q.R., Melosh H.J. Effects of Atmospheric Breakup on Crater Field Formation // Icarus. 1980. V. 42, pp. 211-233. Shuvalov V.V., Artemeva N.A., Trubetskaja I.A. Modelling movement of a destroyed meteoroid with evaporation // Astron. vestn. 2000. V. 34. No.1, pp. 55-67 (in Russian). Shuvalov V.V., Artemieva N.A. Numerical modeling of Tunguska-like impacts // Planet. and Space Sci. 2002. V. 50, pp. 181−192.

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Spurný P. Recent fireballs photographed in central Europe // Planet. and Space Sci. 1994. V. 42. No. 2, pp. 157−162. Stulov V.P. Interaction of a comet nuclei with atmosphere of planet // Int. Conf. “Modern problems of the theoretical astronomy”. Abstracts. St-Petersburg. 1994. V. 3, p. 62 (in Russian). Stulov V.P., Mirskij V.N., Vislyi A.I. Fairball aerodynamics. М.: Nauka, 1995, 236 p. (in Russian). Stulov V.P. Movement of a comet nuclei in the top atmosphere of a planet // Vestn. of the Moscow State University. Ser. 1. Mathematics, mechanics. 1997. No. 1, pp. 63-65 (in Russian). Stulov V.P. Gasdynamical model of the Tunguska fall // Planet. and Space Sci. 1998a. V. 46. No. 2/3, pp. 253−260. Stulov V.P. Gasdynamical model of large bolides // Vestn. of the Moscow State University. Ser. 1. Mathematics, mechanics. 1998b. No. 5, pp. 39-45 (in Russian). Stulov V.P. An analytical model of consecutive crushing and ablation of a meteoric body in atmosphere // Astron. vestn. 1998c. V. 32. No.5, pp. 455-458 (in Russian). Stulov V.P., Stulov P.V. Braking and ablation of a meteoroid after its destruction in atmosphere // Astron. vestn. 1999. V. 33. No.1, pp. 45-49 (in Russian). Stulov V.P., Titova L.Ju. A model of meteoroid fragmentation in atmosphere // DAN. 2001a. V. 376. No.1, pp. 53, 54 (in Russian). Stulov V.P., Titova L.Ju. The comparative analysis of crushing models for meteoric bodies // Astron. vestn. 2001b. V. 35. No.4, pp. 345-349 (in Russian). Zhdan I.A., Stulov V.P., Stulov P.V. Aerodynamic interaction of two bodies in a supersonic stream // DAN. 2004а. V. 396. No.2, pp. 191-193 (in Russian). Zhdan I.A., Stulov V.P., Stulov P.V. Characteristic elements of a destroyed body in a supersonic stream // DAN. 2004b. V. 399. No.2, pp. 199-201 (in Russian).

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In: Space Science Research Developments Editors: J.C. Henderson and J. M. Bradley

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Chapter 2

SOLAR DYNAMICS AND SOLAR-TERRESTRIAL INFLUENCES Katya Georgieva* Solar-Terrestrial Influences Laboratory at the Bulgarian Academy of Sciences, Bl.3 Academy G. Bonchev str., 1113 Sofia, Bulgaria

ABSTRACT

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Much research has been done on the influences of solar activity on processes on Earth and in near-Earth space. While there is a general consensus about the reality of such influences, the results are often contradictory even if highly statistically significant, in many cases the physical mechanism of the influence is unclear, and the mechanism of solar activity itself is not yet fully understood. Obviously there are additional factors affecting solar activity and its influences on the Earth which are not accounted for in the present models. In this paper we are investigating one of these possible factors – the dynamics of the Sun and its significance for the solar-terrestrial influences. One of the most controversial problems in solar-terrestrial physics is the relation between solar activity and climate. Different authors have reported positive, negative or missing correlations between solar activity and surface air temperature in the 11-year sunspot cycle. We demonstrate that the sign of the correlation changes regularly in consecutive secular (Gleissberg) solar cycles, and depends on the long-term changes of North-South solar activity asymmetry which in turn is determined by the movement of the Sun about the barycenter of the Solar system. This movement of the Sun about the barycenter of the Solar system is also related to the dynamics of both the Sun itself and of the Earth. The asymmetry in the rotation of the Northern and Southern solar hemispheres correlates very well with the variations in the Earth rotation rate, and their dominant common periodicity is the periodicity of the rotation of the Sun about the Solar system barycenter. The correlation of the solar dynamics and the Earth dynamics is mediated by the solar wind carrying momentum and magnetic fields and modulating the electromagnetic core-mantle coupling torques responsible for the variations in the Earth’s rotation.

*

E-mail address: [email protected]

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Katya Georgieva The variable solar differential rotation affects also the way in which the solar drivers interact with the Earth’s magnetosphere. The periodicities in the interplanetary magnetic field at the Earth’s orbit in any period coincide with the periodicities in the latitudinal gradient of the differential rotation in the more active solar hemisphere, and the azimuthal components of the interplanetary magnetic field are proportional to the solar equatorial rotation rate. The most geoeffective solar driver are the magnetic clouds – coronal mass ejections with smooth rotation over a wide angle of the magnetic field inside the structure. The portion of coronal mass ejections which are magnetic clouds varies throughout the solar cycle, and is determined by the amount of helicity transferred from the solar interior to the surface, and by the surface differential rotation. We demonstrate the influence of the surface differential rotation on the helicity and geoeffectiveness of magnetic clouds.

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INTRODUCTION Sun, the star closest to us, like many other stars is a variable star. Apart from satisfying the purely scientific curiosity about what a variable star looks like from a close distance, we have much more practical reasons to study solar variability – actually, we live in the corona of this star, our Sun, so everything that happens on and around the Earth is if not determined, then at least influenced by solar activity. A long way has been gone toward accepting the reality of this influence, even in areas where earlier it was violently rejected, like climate, earthquakes, or human health, however the ways in which the variations in solar activity affect the Earth, as well as the mechanism of solar activity itself are not quite clear yet. Probably there are factors which present theories don’t account for. Here we are studying one such possible factor – the solar system dynamics and its influence on the solar activity and on the solar influences on the Earth. The study is far from complete, and many of the facts are still waiting for an explanation. We hope that by presenting them we will provoke further investigations in this interesting and controversial area.

A BRIEF HISTORICAL OVERVIEW Sunspots and Solar Rotation By the term “solar activity” usually any type of variation in the appearance or energy output of the Sun is understood. Elements of solar activity are sunspots, solar flares, coronal mass ejections, coronal holes, total and spectral solar irradiance, etc. The most prominent evidence of solar activity and with the longest data record, though not geoeffective themselves but related to geoeffective active regions, are sunspots. Very big sunspots can be seen with naked eye, and old chronicles testify that they have been indeed observed even in ancient times. There is evidence that the Greeks knew of them at least by the 4th century BC, and the earliest records of sunspots observed by Chinese astronomers are from 28 BC. However, systematic observations of sunspots began only early in the 17th century after the telescope was invented. Like many facts about solar activity, the discovery of sunspots is still a matter of controversy. Galileo Galilei and Christoph Scheiner, a German Jesuit astronomer, never

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stopped arguing about the priority in the discovery of sunspots, but neither or them was aware of the earlier observations of Harriot and Fabricius. The English scholar Thomas Harriot was probably the first to observe telescopically sunspots as evidenced by a drawing in his notebook dated December 8, 1610, and the first published observations (June 1611) entitled “De Maculis in Sole Observatis, et Apparente earum cum Sole Conversione Narratio” ("Narration on Spots Observed on the Sun and their Apparent Rotation with the Sun") were by Johannes Fabricius who had been observing systematically the spots for a few months and, as stated in the title of his paper, had noted also their movement across the solar disc. This can be considered the first observational evidence of the solar rotation. Christopher Scheiner (“Rosa Ursine sive solis”, book 4, part 2, 1630) was the first who measured the equatorial rotation rate of the Sun and noticed that the rotation at higher latitudes is slower, so Scheiner can be considered the discoverer of solar differential rotation, and Galileo was the one who found that the solar equator is inclined to the plane of the ecliptic (Abetti, 1965). St.John (1918) was maybe the first one who established that the solar rotation is not constant. He summarized the published solar rotation rates, and concluded that the differences in series measured in different years can hardly be attributed to personal equation or to local disturbances on the Sun, and are probably due to time variations in the rate of rotation. Hubrecht (1915) first found that the two solar hemispheres rotate differently or, as he remarked, “with regard to the equator symmetry is suspiciously absent”. Though he called these differences suspicious, Hubrecht nevertheless found them consistent with Emden’s (1902) solar theory.

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Sunspot Cycle The sunspot cycle was discovered by Heinrich Schwabe, a German pharmacist and amateur astronomer. Schwabe was trying to discover a new planet inside the orbit of Mercury which was tentatively called Vulcan. Because of the close proximity to the Sun, it would have been very difficult to observe Vulcan, and Schwabe believed one possibility to detect the planet might be to see it as a dark spot when passing in front of the Sun. For 17 years, from 1826 to 1843, on every clear day, Schwabe would scan the sun and record its spots trying to detect Vulcan among them. He didn’t find the planet but noticed the regular variation in the number of sunspots and published his findings in a short article entitled “Solar Observations during 1843”. It is worth noting that Schwabe determined very accurately the 11-year sunspot cycle from observations covering only 17 years. This paper at first attracted little attention, but Rudolf Wolf who was at that time the director of Bern observatory, was impressed so he began regular observations of sunspots and studied historical records to make a reconstruction of the number of sunspots as far back in the past as possible. He was the first to note the possible existence in the sunspot record of a longer modulation period of about 55 years – again in quite a short data record. In 1848 Wolf devised the famous relative sunspot number, also known as the "Zürich sunspot number" which is being used ever since: R=k (10g+s),

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Katya Georgieva

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where R is the (“Zürich” or “International”) sunspot number; g is the number of sunspot groups on the solar disk; s is the total number of individual spots in all the groups; and k is a variable scaling factor (usually 0.4 m (for details see Schulze-Makuch et al., 2005a). The blimp/balloon would have the capability of hovering at an altitude of about 50 km or to descend to lower altitudes and to collect samples of cloud particles with aerogels similar to STARDUST and GENESIS (e.g. Knollenberg and Hunten (1980) report a cloud particle density of 10-100 particles per cm3 at about 50 km altitude). These cloud particles, once obtained by the blimp/balloon, could be transported into orbit, and from there to the International Space Station or Earth for analysis. As technology for a sample return mission to Venus exists, the mission could be undertaken at present.

CONCLUSIONS Three of the four terrestrial planets of our Solar System (Venus, Earth, and Mars) may have had liquid water bodies on their surface early in their history. Any liquid water on the Venusian surface evaporated, but significant amounts of water remain in the atmosphere. On Earth liquid water is abundant and stable at the surface, as well as in the subsurface and the atmosphere. Due to the thin martian atmosphere today, liquid water in most locations is not stable at the surface of Mars. Large liquid and frozen water reservoirs, however, are likely to exist in the martian subsurface. The search for life on the inner terrestrial planets should follow the presence of liquid water. Clearly, the geological history of the planetary body in

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question is key to astrobiological considerations. Life is present in the subsurface of Earth, on its surface and, at least transiently, in its atmosphere. It may be present in the venusian atmosphere and the martian subsurface. In addition, ices on Mercury and the Moon should be explored for remnant biogenic material from the early evolution of life on Earth (and elsewhere). Sample return missions and missions that use multi-tier and multi-agent hierarchical architectures with scalable degrees of mission operation autonomy, which will allow optimal reconnaissance of planetary environments, are most suitable for exploring these potential habitats.

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Van Zuilen, M.A., Lepland, A., Arrhenius, G., 2002. Reassessing the evidence for the earliest traces of life. Nature 418, 627-630. Vestal, J.R., 1988. Carbon metabolism of the cryptoendolithic microbiota from the Antarctic desert. Applied and Environmental Microbiology 54, 960-965. Volk, T., 1987. Feedbacks between weathering and atmospheric CO2 over the last 100 million years. Amer. J. Sci. 287, 763-779. Wagner, R.J., Wolf, U., Ivanov, B.A., Neukum, G., 2001. Application of an updated impact cratering chronology model to mercury’s time-stratigraphic system. Lunar Planet. Sci. Conf., XXXII, abstract #8049. Walker, J.C.G., Hays, P.B., and Kasting, J.F., 1981. A negative feedback mechanism for the long-term stabilization of Earth's surface temperature, J. Geophys. Res. 86 (C10), 97769782. Watson, E.B, Harrison, T.M., 2005. Zircon thermometer reveals minimum melting conditions on earliest Earth, Science 308, 841-844. White, R.S., McKenzie, D.P., 1989. Magmatism at rift zones: the generation of volcanic continental margins and flood basalts. J. Geophys.Res. 94, 7685-7729. Wilde, S.A., Valley, J.W., Peck, W.H., Graham, C.M., 2001. Evidence from detrital zircons for the existence of continental crust and oceans on the Earth 4.4 Gyr ago. Nature 409, 175-178. Wildt, R., 1940. Note on the surface temperature of Venus. Astrophys. J. 91, 266-268. Wilhelms, D.E., 1987. The geologic history of the Moon. USGS Prof. Paper 1348. Wurz, P., Lammer, H., 2003. Monte-Carlo simulation of Mercury’s exosphere. Icarus 164, 113. Yung, Y.L., 1988. HDO in the martian atmosphere: implications for the abundance of crustal water. Icarus 76, 146-159.

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In: Space Science Research Developments Editors: J.C. Henderson and J. M. Bradley

ISBN 978-1-61209-086-3 © 2011 Nova Science Publishers, Inc.

Chapter 4

THE ORIGIN OF THE MAGNETIC FIELD OF PULSARS AND THE GRAVITOMAGNETIC THEORY Jacob Biemond* Vrije Universiteit, Amsterdam

ABSTRACT

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Many authors consider a gravitational origin of the magnetic field of celestial bodies. In this chapter the validity of the so-called Wilson-Blackett formula for pulsars is investigated. This formula predicting a dipolar magnetic field for all rotating bodies has previously been deduced from general relativity, e.g., by application of a special interpretation of the gravitomagnetic theory. Other consequences from this theory will also be considered in this chapter. First, the standard quadrupolar charge density and a monopolar charge for pulsars are derived from the gravitomagnetic theory. In addition, contributions to the total magnetic dipole moment from moving charge are deduced. It appears that these contributions may enforce or weaken the basic magnetic dipole moment from gravitomagnetic origin. Many of these results also result from the so-called “dome and torus” model for pulsars. A tentative extension of this model is given, in order to ensure charge neutrality of the system pulsar plus surroundings. The observed magnetic field for fourteen slowly rotating, binary, accreting X-ray pulsars resembles more the gravitomagnetic prediction than the value calculated from the standard magnetic dipole radiation model. At this moment, for five isolated X-ray pulsars such a comparison is difficult, owing to uncertainty in the assignment of the cyclotron lines determining the observed magnetic field. For a sample of 100 pulsars the averaged (gravito)magnetic field, extracted from the magnetic dipole spin-down model, may be compatible with the gravitomagnetic prediction. Unfortunately, the (gravito)magnetic field itself has not yet directly been measured. Finally, first and second order braking indices are discussed. It appears that they do not depend on the magnetic field from gravitomagnetic origin.

*

E-mail address: [email protected], Website: http://www.gewis.nl/~pieterb/gravi/, Postal address: Sansovinostraat 28, 5624 JX Eindhoven, The Netherlands

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INTRODUCTION Since 1891 many authors have discussed a gravitational origin of the magnetic field of rotating celestial bodies. Particularly, the so-called Wilson-Blackett formula has often been considered [1–10] M = – ½ β c–1 G½ S.

(1)

Here M is the magnetic dipole moment of the massive body with angular momentum S, c is the velocity of light in vacuum, G is the gravitational constant and β is a dimensionless constant of order unity. Attempts to derive equation (1) from a more general theory have been made by several authors [8 –16]. A number of them [8 –10, 14–16] have tried to explain equation (1) as a consequence of general relativity. For example, it appeared possible to explain (1) in terms of a special version of the gravitomagnetic theory [8–10]: in this approach the so-called “magnetic-type” gravitational field is identified as a common magnetic field, resulting into the (gravito)magnetic dipole moment M(gm) = M of (1). Experiments on rotating masses in the laboratory in order to test (1) have been performed by Blackett [17] and others [18 –20]. Available observations and theoretical considerations with respect to the relation (1) and other explanations of the origin of the magnetic field of celestial bodies have been reviewed by Biemond [9]. Following Woodward [21], the validity of the Wilson-Blackett (or Schuster-Blackett) formula for pulsars has again been investigated in this work, but now from the gravitomagnetic point of view. The angular momentum S for a sphere of radius R can be calculated from the relation

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S = I Ω, or S = I Ω = 2/5 f m R 2 Ω,

(2)

where m is the mass of the sphere, Ω = 2π P –1 its angular velocity (P is the rotational period), I represents its moment of inertia and f is a dimensionless factor depending on the homo-geneity of the mass density in the sphere (for a homogeneous mass density f = 1). The value of the magnetic dipole moment M may be calculated from M = ½ r 3 Bp , or M = ½ r 3 Bp .

(3)

Here Bp is the magnetic induction field at, say, the north pole of the sphere and r is the distance from the centre of the sphere to the field point where Bp is measured. The magnetic moment M has been derived for r > R, but in order to calculate M values of Bp measured at r ≈ R are mostly substituted into (3). Combination of (1)– (3) yields the following gravitomagnetic prediction for Bp B p(gm) = – β c–1 G½I R –3 Ω, or Bp(gm) = – 5.414×1013 P –1 for β = +1,

(4)

where the minus sign reflects that the vectors Bp(gm) and Ω possess opposite directions for β = +1. Neither the sign nor the value of β follows from the gravitomagnetic theory. The representative values I = 1045 g.cm2 and r ≈ R = 10 km have been inserted in (4) as well as in other formulas in this chapter. It is noticed that (4) imposes no restriction on the maximum of

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Bp(gm), i.e., Bp(gm) may be larger than the critical field strength me2 c3/e ħ = 4.414×1013 Gauss, at which electron-positron pair creation processes may become probable. As pointed out earlier [9, pp. 12, 14, 20 and 49], moving electric charge in the magnetic field from gravitomagnetic origin may cause an additional magnetic field from electromagnetic origin. The latter field may partly or completely compensate the magnetic field from gravitomagnetic origin. It is stressed that the magnetic field generated by rotating neutral mass is generally much smaller than the magnetic field generated by moving charge. For example, for an electron with mass me one may compare the following magnetic moment to angular momentum ratios: (M/S)electromagnetic = – ½ e (me c)–1 and (M/S)gravitomagnetic = – ½ β c–1 G½. Thus, apart from the electric charge – e, the electron may possess a small gravitomagnetic charge e* = – β G½ me. The latter charge is no real electric charge, but it does generate a magnetic field when rotation occurs. For a rotating pulsar with mass m the gravitomagnetic charge Q* can then be written as Q* = – β G½ m.

(5)

Calculation shows that for an electron the ratio (M/S)gravitomagnetic /(M/S)electromagnetic is equal to G½ me e–1 = 4.899×10–22 for β = +1. Therefore, proposed magnetic fields from gravitomagnetic origin are mostly extremely small and difficult to isolate from fields due to electric charges. When both a magnetic induction field Bp(gm) due to gravitomagnetism and a field B p(em) from electromagnetic origin are present at the north pole of the sphere, the total magnetic induction field Bp(tot) is given by

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B p(tot) = B p(gm) + B p(em).

(6)

Since Bp(gm) is antiparallel to Ω for β = +1 (see (4)), it is helpful to define the following dimensionless quantities β* and β′ B p⎟⎟ (tot) ≡ β* B p(gm) and |Bp⊥ (tot)| = |Bp⊥ (em)| ≡ β′ |Bp(gm)|.

(7)

Mostly, the direction of B p⎟⎟ (em) cannot be determined for pulsars, whereas the direction of Bp(gm) has not yet been established. Therefore, absolute values of β* will be given below. Using (4), combination of (6) and (7) yields β* = 1 ± |Bp⎟⎟ (em)|/|Bp(gm)|.

(8)

When the total field B(tot) is only due to gravitomagnetic origin, B(em) = 0 and β* and β′ reduce to β* = 1 and β′ = 0, respectively. As a rule, measurements only yield Bp(tot), so that β* will be approximated by β* = |Bp(tot)|/|Bp(gm)| (compare with (4) and (7a)). Since charges may move in different ways in pulsars and other rotating bodies, one can hardly expect that β* is a constant. Indeed, different results for β* have been found for about fourteen rotating bodies, ranging from metallic cylinders in the laboratory, moons, planets, stars to the Galaxy [9, ch. 1]. On the other hand, for this series a linear regression analysis showed an almost linear relationship between the observed magnetic moment |M|

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and the angular momentum |S| (|M| and |S| vary over an interval of sixty decades!), whereas an average value of |β*| = 0.076 was obtained. Although this result is distinctly different from the gravitomagnetic prediction β* = 1, the correct order of magnitude of β* for so many, strongly different, rotating bodies is amazing (the values of the largely independent parameters m, R, Ω and Bp occurring in (4) (see (2b) and (7a)) also differ by many decades). All these findings may reflect the validity of the gravitomagnetic hypothesis. In general, many properties of pulsars are determined by the emission model used. Therefore, we will first pay attention to some proposed emission models in section 2. In section 3 a theoretical treatment of the influence of the magnetic field on first and second order braking indices is given. In section 6 the obtained predictions are compared with observations. Apart from the Wilson-Blackett formula, other consequences of the gravitomagnetic theory for pulsars are deduced in section 4. In section 5 observational data for different kinds of pulsars are summarized and compared with theoretical predictions. A summary of the conclusions is given in section 7.

EMISSION MODELS OF PULSARS The emission model of pulsars is an important factor in the prediction of the magnetic field of pulsars. Unfortunately, no generally accepted model is available. Therefore, we will discuss some alternative formulas for the emission of pulsars. The rotational energy E = ½ I Ω 2 of the pulsar may change by magnetic dipole radiation, accretion or other mechanisms according to

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Ė = I Ω Ωֹ = 4π 2 I P – 3 Pֹ,

(9)

ֹ and Pֹ are the time derivatives of E, Ω and P, respectively (It will be assumed where Ė, Ω in this chapter that the moment of inertia I is no function of time). The contribution of the (electro)magnetic dipole radiation in vacuum to Ė of (9) is given by (see, e.g., [22, pp. 188– 189] and [23, pp. 176–177]) Ė = – ⅔ c–3 M(em)2 sin2θ Ω 4 = – (32π4/3) c–3 M(em)2 sin2θ P – 4,

(10)

where M(em) is the (electro)magnetic dipole moment of the pulsar and θ is the angle between M(em) and S. Note that Ė does not depend on the component M⎟⎟ (em) (M ⎟⎟ (em) ≡ M(em) cosθ) parallel to S, but only on the component M ⊥ (em) (M ⊥ (em) ≡ M(em) sin θ) perpendicular to S. Moreover, it is stressed that Ė does not depend on the (gravito)magnetic dipole moment M(gm), for (gravito)magnetic dipole radiation does not exist. On the other hand, gravitomagnetic quadrupole radiation may be present, but its influence is not considered in this chapter. For two point masses its energy formula coinciding with the familiar expression for the gravitational quadrupole radiation (see, e.g., [9, ch. 3] and [22]) has, however, been calculated. When Ė in (9b) is taken equal to Ė in (10b), it follows that Pֹ is positive. So, magnetic dipole radiation leads to spin-down. If sinθ ≠ 0, combination of (9b) and (10b) further yields for M(em)

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1

⎛ 3 c3 I ⎞ 2 1 M (em) = ⎜ 2 2 ⎟ ( P ⋅ P ) 2 . ⎝ 8 π sin θ ⎠

(11)

Note that the component M ⊥ (em) ≡ M(em) sin θ can directly be calculated from (11) without knowledge of the angle θ. Usually, the value of θ for pulsars is unknown, so that only an estimate for M(em) can be calculated from (11). The magnetic induction field at the north pole of the pulsar, Bp(sd) = Bp⊥ (em) = M ⊥ (em)/R 3, due to spin-down radiation is then equal to 1

19 ⎛ 3 c3 I ⎞ 1 1 Bp (sd) = ⎜ 2 6 ⎟ ( P ⋅ P ) 2 = 3.200 ×10 ( P ⋅ P ) 2 . ⎝ 8π R ⎠ 2

(12)

Note that the sign of B p(sd) does not follow from the derivation of (12). In addition, from (7b) follows for β′ β′ = |Bp(sd)|/|Bp(gm)|.

(13)

Values for β′ have been given in all tables below, when data were available. Assuming other mechanisms like “deformation of magnetic field lines”, alternative expressions for the (electro)magnetic field Bp(em) at the north pole of the pulsar have been proposed (see, e.g., [23, pp. 188–189]). For example, approximations to Ė were obtained by taking the product of the energy density of the equatorial (electro)magnetic field at the light cylinder BL(em), the effective area of the light cylinder 4π RL2 and the velocity of light c

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E=−

BL (em) 2 4π R L 2 c, 8π

(14)

where RL = c/Ω is the radius of the light cylinder. BL(em) was taken equal to BL(em) = ½ Bp(em) (R/RL) p,

(15)

where Bp(em) is an estimate for the magnetic field Bp⎟⎟ (em) at the north pole of the pulsar. Using (3b), combination of (14) and (15) leads to Ė = – ⅛ c3 – 2 p R 2 p Bp(em)2 Ω 2 p – 2 = – ½ c3 – 2 p R 2 p – 6 M(em) 2 Ω 2 p – 2.

(16)

For example, p = 3 yields Ė = – ½ c – 3 M(em)2 Ω 4.

(17)

Although the derivations of (10a) and (17) start from different assumptions, both equations resemble each other. Since the derivation of (10) is far more rigorous than that of (17), the former expression is considered as the standard formula for the determination of the (electro)-magnetic field of pulsars.

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106

Jacob Biemond When p = 2, equation (16) yields for Ė Ė = – ½ c– 1 R – 2 M(em)2 Ω 2 = – ⅛ c– 1 R 4 Bp(em)2 Ω 2.

(18)

Combination of (9) and (18b) then leads to the following expression for the “deformed” magnetic induction field, denoted by Bp(def) 1

1

1 2 2  2 ⎛ 8 c I ⎞ ⎛ P ⎞ 16 ⎛ P ⎞ Bp (def ) = ⎜ 4 ⎟ ⎜ ⎟ = 1.549 ×10 ⎜ ⎟ . ⎝ R ⎠ ⎝P⎠ ⎝P⎠

(19)

The calculated values of Bp(gm) of (4), Bp(sd) of (12) and Bp(def) of (19) will be compared with each other and with available observational data below.

MAGNETIC FIELD DEPENDENCE OF BRAKING INDICES In order to calculate so-called braking indices, (9a) and (16a) are combined to ֹ = – ⅛ c3 – 2 p I –1 R 2 p Bp(em)2 Ω 2 p – 3. Ω

(20)

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For p = 3 this relation can be compared with the more rigorous result following from (9a) and (10a): Ωֹ = – ⅔ c– 3 I –1 R 6 Bp⊥ (em)2 Ω 3. Contrary to (20), the latter expression clearly shows its dependence on Bp⊥ (em). On the other hand, (20) has the advantage that values p ≠ 3 can be investigated. In addition, (20) may further be generalized to ֹ = – kp c3 – 2 p I –1 R 2 p Bp(em)2 Ω 2 p – 3. Ω

(21)

In this chapter it will be assumed that the quantities kp and p do not depend on time. Instead of using (20) and (21), Kaspi et al. [24] and Johnston and Galloway [25] followed Manchester and Taylor [23] and considered the generalized relation ֹ = – K Ω n0 = – k Bp(em)2 Ω n0, Ω

(22)

where the quantities k and n0 are assumed not to depend on time. ֹֹ can be calculated by differentiation (compare with the From (20)–(22) the quantity Ω integration method discussed in section 6). Subsequently, the so-called first order braking index n can be calculated from (20) or (21)

n≡

 Bp (em) Ω ΩΩ = − + 2 p 3 2 . 2  Ω Bp (em) Ω

(23)

Note that the first order braking index n is defined in terms of the observable ֹ and Ω ֹֹ . In deriving (23) it has been assumed that none of the following quantities Ω, Ω Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

The Origin of the Magnetic Field of Pulsars and the Gravitomagnetic Theory

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parameters depend on time: the moment of inertia I, the radius R, the mass M, or the angle θ of the pulsar. For convenience sake, these assumptions will also be used in the sequel of this chapter. It is noticed that for a fixed value of p the braking index n will become smaller for ֹ < 0. Note that the sign of Bp(em) does Bp(em) > 0 and Bֹ p(em) > 0, since Ω > 0 and usually Ω not follow from (20) or (21). From (22b) follows, analogously to the derivation of (23)

n≡

 B p (em) Ω ΩΩ n 2 . = + 0 2  Bp (em) Ω Ω

(24)

Comparison of (23) and (24) shows that n0 = 2p – 3. Note that, in principle, n and n0 need not to be integers. It is stressed that the braking indices n of (23) and (24) do not directly depend on the gravitomagnetic field Bp(gm) or on Bֹ p(gm). Therefore, the validity of the gravitomagnetic hypothesis cannot been tested by considering the braking index n. Moreover, this result illustrates that the field Bp(gm), assumed to be a magnetic induction field, is not always compatible with the magnetic induction field Bp(em). In the evaluation of (20)– (22) it is often assumed that Bp(em) is no function of time. In that case n in (23) reduces to n = 2p – 3, whereas (24) simplifies to n = n0. Note that, n, p and n0 need not to have integer values. Moreover, if Bp(em) does not depend on time, integration of (20) or (21) yields the following expression for the so-called true age t of the pulsar (compare with, e.g., [23, p. 111])

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t=

n −1 2τ − Ω ⎪⎧ ⎛ Ω ⎞ ⎪⎫ −Ω − ≈ = c , n ≠1 1 ⎨ ⎬ ⎜ ⎟   (n − 1) Ω ⎪⎩ ⎝ Ωi ⎠ ⎪⎭ (n − 1) Ω n − 1

(25)

where Ωi is the initial value of Ω at time t = 0 and τc ≡ – ½ Ω/ Ωֹ = ½ P/Pֹ . Often, the quantity t is an important parameter in evolutionary and other studies of pulsars. From (25b) it can be seen that the characteristic time τc may be used as an estimate for t. Therefore, τc has been added to tables 2 through 5 when P and Pֹ are known. In addition, one may use the field Bp(sd) from (12) depending on the quantities Ω and ֹ Ω as an estimate for the field Bp(em) in (24). Differentiation of Bp(sd) from (12) with respect to time followed by evaluation yields for the ratio Bֹ p(sd) to Bp(sd)

Bp (sd) Bp (sd)

= 12 (n − 3)

 1 Ω = 14 (3 − n) , Ω τc

(26)

where τc is again defined by τc ≡ – ½ Ω/Ωֹ = ½ P/Pֹ. When n = 3, it follows from (26) that Bp(sd) is independent of time, but for n ≠ 3 Bֹp(sd) ≠ 0. When n < 3 and Bp(sd) > 0, Bp(sd) increases in time, whereas for n > 3 and Bp(sd) > 0 Bp(sd) decreases in time. It is noticed that (26a) can rigorously be derived from (9a) and the standard result (10a) (see (12) and compare with comment following (20)).

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When Bp(em) in (24) and Bp(sd) in (26) are taken equal, combination of (24) and (26a) yields n0 = 3. Therefore, the quantity n0 has been called the true braking index (see, e.g., [24]). Moreover, from the relation n0 = 2p – 3 follows p = 3 in this case. Summing up, (electro)magnetic dipole radiation may imply n0 = 3 and p = 3, but according to (26) the value ֹֹ is difficult to measure, only of n also depends on the ratio Bֹp(sd)/Bp(sd). Since the quantity Ω ֹֹ ֹ 2. Note that n need not to be relatively few braking indices n can be calculated from n = Ω Ω /Ω an integer. When Bp(em) is identified with Bp(sd), it is appears possible to deduce an expression for the factor β′ in (13) by combining (4) (taking β = +1) (12) and (13) ֹ )½ β′ = (– 3/2 c5 G –1 I –1 Ω – 5 Ω

(27)

ֹ , and on the particular choice This result shows that β′ depends on the quantities Ω and Ω Bp(em) = Bp(sd). Another possibility is to identify Bp(em) with Bp(def) from (19). Differentiation of Bp(em) = Bp(def) with respect to time yields

B p (def )

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Bp (def )

= 12 (n − 1)

 Ω = Ω

1

4

(1 − n)

1

τc

(28)

.

When in this case n = 1, it follows from (28) that Bp(def) is independent of time. When n < 1 and Bp(def) > 0, Bp(def) increases in time, whereas for n > 1 and Bp(def) > 0 Bp(def) decreases. Combination of (24) and (28a) shows that now n0 = 1. From the relation n0 = 2p – 3 then follows that p = 2. Summing up, in the case of “deformation of magnetic field lines”, n0 and p may have the values n0 = 1 and p = 2. Note again, that the observed braking index n need not to be an integer. In table 1 some results for n and n0 from (23) and (24) with respect to the time dependence of Bp(em) are summarized. See the text for further information. Table 1. Braking indices n and n0 depending on the emission model (p-value) and the field Bp(em) Bp(em) ≠ f(t) n = 2p – 3 n = n0

Bp(em) = Bp(sd)

Bp(em) = Bp(def)

n0 = 2 p – 3

n0 = 2 p – 3

General case

p=p

Magnetic dipole radiation

p=3

n = n0 = 3

n0 = 3

–

Deformation of field lines

p=2

n = n0 = 1

–

n0 = 1

Following Kaspi et al. [24], a second order braking index m can be calculated from (22b). Taking the time dependence of Bp(em) into account, one finds

Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

The Origin of the Magnetic Field of Pulsars and the Gravitomagnetic Theory  (em) Ω 2  B p (em) Ω B p (em) 2 Ω 2 B Ω2 Ω 2 m≡ . = 2n0 − n0 + 6n0 +2 +2 p 3 2  2   2 Bp (em) Ω Bp (em) Ω Bp (em) Ω Ω

109 (29)

ֹ and Ω ֹֹֹ and The quantity m has been defined in terms of the observable quantities Ω, Ω need not to be an integer. When Bp(em) is no function of time, m in (29) reduces to m0 = 2n02 – n0 .

(30)

If Bp(em) in (29) is again identified with Bp(sd) from (12) and if it assumed that Bp(sd) is independent of time, n = n0 = 3 (see comment following (26)) and m0 obtains the value m0 = 15.

(31)

If Bp(em) in (29) is identified with a time dependent Bp(sd) from (12), n ≠ 3 and n0 = 3 (see again comment following (26)). Substitution of Bp(sd) of (12) and its time derivatives ֹֹp (sd) calculated from (12) into (29) yields an identity and no additional expressions Bֹp (sd) and B ֹֹ and ֹΩ ֹֹ are known, n and m can be calculated for m or n. When the observable quantities Ω, Ωֹ , Ω from (23a) and (29a), respectively. When the value of n is known the ratio Bֹp(sd)/Bp(sd) can be calculated from (26). Substitution of the calculated values of m, n0 = 3 and Bֹp(sd)/Bp(sd) ֹֹp(sd)/Bp(sd). Since the quantity Ω ֹֹֹ has been measured for PSR into (29) yields the ratio B B1509–58 (see, [24, 26]), results for this pulsar will be discussed in section 6.

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MAGNETIC FIELD FROM GRAVITO-ELECTROMAGNETIC ORIGIN Pulsars mainly consist of electrically neutral matter, probably neutrons, whereas charged particles may also be present. In this section the charge density in pulsars induced by the magnetic field from gravitomagnetic origin will be investigated. Therefore, some elements of the gravitomagnetic theory will be given here. In the stationary case the magnetic induction field B can be calculated from the simplified gravitomagnetic equations [9, ch. 2] and [10] ∇ × B = – 4π β c– 1 G½ ρ v and ∇ · B = 0,

(32)

where ρ is the homogeneous mass density of a sphere of radius R moving with a velocity v = Ω × r (0 ≤ r ≤ R). A dipolar magnetic field B at distance r > R can be calculated from (32)

B=

3M ⋅ r M r− 3 . 5 r r

(33)

Here M = M(gm) is the gravitomagnetic dipole moment given by (1) (see also (2)). It will be assumed throughout the sequel of this chapter that β = +1, so that M and S possess opposite directions. To my knowledge the origin of the basic dipolar magnetic field B of pulsars has never been explained. The prediction of a such a field B, i.e., B(gm) of (33) from (32) may be considered as a first important merit of the gravitomagnetic theory. Usually, it is assumed that

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the magnetic field of pulsars, due to their ancestors, is effectively frozen into the plasma of the pulsar. Essentially the only way of changing the magnetic field is by “deforming the circuit”, i.e., by macroscopic displacements of the plasma (see, for recent discussions of the origin and evolution of the magnetic fields of pulsars, e.g., Reisenegger [27] and Ruderman [28]). In that case, the field B and the magnetic dipole moment M in (33) can be identified with the (electro)magnetic field B(em) and with the magnetic dipole moment M(em), respectively. So far, no generally accepted model for the calculation of the field B(em) has, however, emerged. The components of B of (33) in spherical coordinates (r, θ and φ) are given by

Br =

2 M cos θ M sin θ er , Bθ = eθ and Bϕ = 0, 3 r r3

(34)

where er and eθ are unit vectors. The field B in (33) has been calculated assuming r > R, but below we also need knowledge of the field B inside the sphere. The field B(gm) inside a sphere has not yet been calculated from (32), but for a sphere with homogeneous mass density ρ the magnetic field at the centre, Bc (gm), can be shown to be [9, app. A]

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Bc(gm) = 5 R – 3 M(gm).

(35)

However, the field B(gm) inside the sphere and its components may approximately be calculated from (33) and (34), respectively, for all values of 0 < r ≤ R, if the magnetic dipole moment M(gm) of the sphere with homogeneous mass density ρ shrinks to an ideal magnetic dipole moment located at the centre of the sphere. Contrary to the full solution Bc(gm) of (35) obtained from (32), the results (33) and (34) for the central ideal dipole model show the flaw of a singularity at r = 0. For the latter model the electric field E inside the sphere can now be calculated from (34), using the following well-known relation from magnetohydrodynamics E = – c– 1 v × B(tot) = – c– 1 (Ω × r) × {B(gm) + B(em)},

(36)

where v is the rotational velocity of the sphere and B(tot) is the total magnetic induction field inside the sphere (compare with (6)). Note that the assumed equivalence of gravitomagnetic and electromagnetic fields has been used here. Equation (36) can be deduced from j = σ {E + c–1 v × B(tot)},

(37)

where j and σ are the current density and the electrical conductivity in the pulsar, respectively. The left hand side of (37) is negligible, if |j| > 1. When in a first approximation B(em) = 0 in (36) is chosen, combination of (34) for 0 < r ≤ R and (36) yields the following components of E in spherical coordinates

Er =

− Ω M sin 2θ 2 Ω M sin θ cos θ e r , Eθ = eθ and Eϕ = 0. 2 cr cr 2

(40)

Then, the charge density ρe inside the pulsar can be calculated from the Maxwell equation ∇ · E = 4π ρe.

(41)

Combination of (40) and (41) yields for the quadrupolar charge density ρe inside the sphere

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ρe =

ΩM Ω⋅B (3cos 2θ − 1) = − , 3 2π c r 2π c

(42)

where M is still equal to M(gm). A related derivation of the standard result ρe related to (42) for the so-called “aligned rotator model” has been given by, e.g., Michel and Li [29]. Note that for θ = 0º the charge density ρe is positive and Ω and Bp are antiparallel in (42) (see (4)). Generally, a negative value for ρe and parallel directions for Ω and Bp are adopted (see, e.g., [29]). The usual result will be obtained, however, if β = +1 in (1) is replaced by β = – 1 (see (1)– (4)). Summing up, the prediction of the much applied quadrupolar charge density ρe of (42) may be considered as a second merit of the gravitomagnetic theory. Integration of the quadrupolar charge density (42) over the whole volume V of the sphere yields a positive and a negative charge Q+ and Q– , respectively Q+ = ∫ ρ e dV =

4 3ΩM R ln = − Q− , ( R0 ≤ r ≤ R ) 9c R0

(43)

where M = M(gm) and R0 is a chosen radius much smaller than R. For R0 = 0 Q+ and Q– show a singularity like B from (33), but the net internal charge from quadrupolar origin, Qquadrupole, satisfies Qquadrupole = Q+ + Q– = 0 for R0 > 0. Note that in our case the chargeseparation of available plasma is caused by the (gravito)magnetic field B = B(gm) of (36). Usually (see, e.g., [29]), this charge-separation is induced by an (electro)magnetic field B(em) of unspecified origin of (36). Analogous to the derivation given by Michel and Li [29], the present treatment leads to the conclusion that the pulsar also bears an intrinsic monopolar charge. Combination of Er

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Jacob Biemond

from (40) and (41) followed by integration over a sphere using Gauss’ formula yields for the magnitude of the charge of this monopole

Qmonopole = ∫ ρe dV = 1/(4π ) ∫ (∇ ⋅ E) dV = 1/(4π ) ∫ E ⋅ dS = −

2Ω M . 3c

(44)

Note that for every r between 0 < r ≤ R the same result for Qmonopole is obtained. In the limiting case of r → 0 Qmonopole can be denoted as a point charge (see discussion in [29, § 4]). Probably, the ambiguity in the integration in (44) can be attributed to the approximate character of the central ideal dipole model. By utilizing (1) (taking β = 1) and (2) (choosing f = 1), the right hand side of (44) can further be rewritten as

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Qmonopole = −

2 Ω 2 R 2 12 G m. 15 c 2

(45)

Mostly, the magnitude of Qmonopole is small compared with that of the gravitomagnetic charge Q* of (5) for pulsars. For example, for a typical pulsar with P = 0.5 s and R = 10 km Qmonopole reduces to – 2.3×10–8 G½ m, but when Ω R approaches c an upper limit Qmonopole = – 2/15 G½ m = – 0.13 G½ m is reached. Some remarks with respect to the electric field Er in (40) may be clarifying. Equation (40) implies that the field Er < 0 at the equator for r = R, so that additional charge outside the pulsar may be present to maintain a stationary charge equilibrium. Therefore, Michel and Li [29, § 4] and Smith et al. [30] adopted varying surface charges and a quadrupolar charge density outside the pulsar, corotating with the star. From their simulations stable stationary charge distributions were found for a pulsar charge Qmonopole corresponding to (45) and for other values of the total charge Qtot of the system, i.e., star plus surrounding plasma. Additional stable charge configurations with Qtot = 0 up to Qtot = 3 Qmonopole were found by Pétri et al. [31]. Using our sign convention with β = +1, combination of this last result and (45) yields Qtot = – 3×(2/15)G½ m = – 0.40 G½ m as an upper limit for Qtot. In our sign convention all the simulations from [29]–[31] result into the model of a dome of positive charge over each magnetic pole and a torus of negative charge around the equatorial zone. The latter topology may be in line of the well-known X-ray image of the Crab nebula from Chandra, as has been discussed by, e.g., Michel [32]. Another consequence of the quadrupolar charge density ρe of (42) deserves special attention. The following electromagnetic dipole moment M(em) can be calculated from ρe of (42) by integration over the whole sphere (0 ≤ r ≤ R) (compare with, e.g., [22, § 44])

M (em) = 1/(2c ) ∫ ρ e r × v dV =

2 Ω2 R2 M (gm). 15 c 2

(46)

Contrary to the results for Q+ and Q– of (43), no singularity problem occurs for M(em). It is noticed that no direct contribution to M(em) from Qmonopole of (45) becomes manifest. This effect may again be attributed to the approximate character of the central ideal dipole model. Nevertheless, in a more full treatment the order of magnitude and the sign of M(em) of (46) are probably correct. When β = +1 in (1) is replaced by β = – 1, the signs on the right hand

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The Origin of the Magnetic Field of Pulsars and the Gravitomagnetic Theory

113

sides of (42) change, but the sign in (46) remains the same. So, M(em) in (46) always enforces M(gm). The resulting field B(em) corresponding to M(em) of (46) is given by an expression analogous to (33). According to (46), the total magnetic moment M(tot) of the pulsar can now be written as

⎛ 2 Ω2 R 2 ⎞ M (tot) = M (gm) + M (em) = ⎜ 1 + ⎟ M (gm). 15 c 2 ⎠ ⎝

(47)

Analogous to the derivation of (42), by using (47), the following higher order quadrupolar charge density ρe′ can be deduced for 0 ≤ r ≤ R

ρ e′ =

ΩM′ Ω ⋅ B′ (3cos 2θ − 1) = − , 3 2π cr 2π c

(48)

where M′ = M′(tot). Moreover, analogous to the derivation of (46), the following higher order magnetic dipole moment M′(em) can be deduced

M′(em) =

2 Ω2 R 2 ⎛ 2 Ω2 R 2 ⎞ ⎜1 + ⎟ M (gm). 15 c 2 ⎝ 15 c 2 ⎠

(49)

Note that combination of (1) (β = 1), (3a), (7a) and (49) then yields for β*

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β* = 1 + 2/15(Ω R/c)2 {1 + 2/15 (Ω R/c)2}.

(50)

An upper limit of β* = 1 + 2/15 (1 + 2/15) = + 1.1511 is obtained from (50), when the ratio Ω R/c approaches unity value for pulsars with a very short period. It is noticed that still higher order values for β* can be calculated, but the series converges quickly to β* = + 1.1538. From the comment following (46) it follows that the sign of β* is positive, regardless the sign of β. It should be noted that the isolated pulsar in the dome-torus model discussed in [29]–[31] bears a non-zero total charge Qtot. Although the calculated charge configurations were found to be stable, the net charge of one kind may somehow be compensated by an opposite charge – Qtot elsewhere. In order to ensure charge neutrality in the system, we tentatively propose an extension of the dome-torus model. Apart from a negative charge Qin, in the pulsar and in the equatorial belt close to the star, we assume that a compensating positive charge Qout will be present in the equatorial belt far from the star, hence Qtot = Qin + Qout = 0 (Note that Qmonopole from (45) contributes to Qin and that Qquadrupole = 0). In addition, no net charge loss will be adopted for the system. In an aligned rotator model the total magnetic moment of the system parallel to Ω, M⎟⎟ (tot), may then be written as M⎟⎟ (tot) = M(gm) + Min⎟⎟ (em) + Mout⎟⎟ (em),

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(51)

114

Jacob Biemond

where the electromagnetic moment contributions Min⎟⎟ (em) and Mout⎟⎟ (em) are due to Qin and Qout, respectively. Substituting (1) (β = +1) into (51) and using crude approximations for the other two terms on the right hand side of (51), this equation changes into M⎟⎟ (tot) = – ½ c–1 G½ S + ½ (m c)–1 Qin S + ½ (mdisk c)–1 Qout Sdisk,

(52)

where mdisk and Sdisk are the mass and the angular momentum of the far disk, respectively. Substitution of S = 2/5 m R 2 Ω and Sdisk = ½ mdisk rout2 Ωdisk (where rout and Ωdisk are the outer radius and the angular velocity of the far disk, respectively) transforms (52) into M⎟⎟ (tot) = 1/5 c–1 (– G½ m R 2 Ω + Qin R 2 Ω + 5/4 Qout rout2 Ωdisk).

(53)

From M⎟⎟ (tot) a crude approximation for the total magnetic field at the north pole of the pulsar Bp⎟⎟ (tot) can be obtained by taking B p⎟⎟ (tot) ≈ 2 R –3 M⎟⎟ (tot). Combination of (4a), (7a) and (53) then leads to the following expression for β*

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β* = 1−

2 Qin 5 Qout rout2 Ωdisk * ) = 1 − Qin + 5 Qin rout Ωdisk (+ β * ). (54) − + ( β 1 1 1 1 current current G 2 m 4 G 2 m R 2Ω G 2 m 4 G 2 m R 2Ω

It is stressed that equation (54) can only be regarded as rough estimate for β*, in view of the crude approximations involved. Two limiting cases may now be distinguished. Firstly, the value of β* may become larger than unity value for a narrow, “subrotating” far disk (i.e., rout < R and Ωdisk < Ω) or for retrograde accretion (in that case the directions of Ωdisk and Ω become antiparallel: compare with (53)). Depending on the magnitude of Qin , the value of β* from (54) may then be larger than the related result β* = 1.15 from (50). Secondly, the value of β* may become smaller than unity value for a broad (rout > R), and/or “superrotating” far equatorial disk (Ωdisk > Ω; as a result of accelerating accretion, for example). In the latter case the contribution to β* from the far disk may (partly) compensate the contributions from gravitomagnetic origin and from Qin. Although (54) illustrates the importance of possible charges in a pulsar (Qin) and in a disk or torus (Qout), another contribution may also explain low values of β*. For example, a toroidal current generated by accretion in the highly conductive pulsar may largely compensate the magnetic field from gravitomagnetic origin. Owing to the slow Ohmic decay, such a current may be retained for a long time. As a result, the right hand side of (54) may contain a contribution β*current . As an example, by (previous) accretion millisecond pulsars may obtain a contribution β*current ≈ – 1 resulting into a value of β* ≈ 0. In the next section we will compare the predictions of β* of (54) with observations on accreting pulsars with an exceptionally long period and with data for millisecond pulsars.

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The Origin of the Magnetic Field of Pulsars and the Gravitomagnetic Theory

115

OBSERVATIONAL DATA In table 2 the values for the observed magnetic field Bp(tot) of a number of binary, accretion-powered X-ray pulsars are taken from Coburn et al. [33] and Heindl et al. [34], whereas the corresponding values of P and Pֹ are taken from the same references or other ones [35–42]. Values of Bp(tot) are deduced from so-called cyclotron resonance spectral features (CRSFs) that are attributed to resonant scattering of photons by electrons. Unfortunately, no generally accepted model is available for the shape of the continuum part of the X-ray spectrum, whereas such a model is necessary to isolate the line-like spectral features. In references [33] and [34], however, the same continuum model (modified power-formula with high-energy cutoff) was applied to fourteen pulsars. Therefore, their results have been used in table 2. More explicitly, in the presence of a gravitational field the field Bp(tot) has been deduced from the fundamental energy Ecycl 1

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Ecycl

⎛ eh ⎞ ⎛ 2 Gm ⎞ 2 −12 =⎜ ⎟ ⎜ 1 − 2 ⎟ B = 11.58 × 10 0.7658 B(G) keV, c R ⎠ ⎝ me c ⎠ ⎝

(55)

where me is the mass of the electron, m = 1.4 m☼ is the mass of the pulsar and R = 10 km. It is noticed that the location where the photons are scattered by electrons is not exactly known. Therefore, the place where the field B(G) (expressed in Gaussian units) is measured is not precisely known, too. Moreover, the field B(G) appears to be a function of the rotation phase of the pulsar in some cases [34], but in table 2 only rotation averaged values of B(G) have been used. It has been assumed that these fields B(G) are equal to the corresponding fields at the pole of the pulsar Bp(tot). Of course, the latter equalization need not to be valid. Firstly, the obtained magnetic field Bp(tot) and the calculated field Bp(gm) from (4) for β = 1 in table 2 may be compared. Only limited similarity between these fields is found. If the field Bp⎟⎟ (tot) would be equal to Bp(gm), then the quantity β* from (7a) would yield unity value. Particularly, the values of β* for the long-period pulsars are too large, ranging from about 55 to 1. In section 4 an extension of the dome-torus model [29]–[31] is proposed, from which equation (54) for β* follows. According to (54), values β* larger than unity value may be possible for retarding accretion (Ωdisk < Ω) (see text for further information) or retrograde accretion (in that case the directions of Ωdisk and Ω become antiparallel: compare with (53)). Moreover, the contribution β*current from (54) may also contribute. Secondly, if no gravitomagnetic contribution to the field Bp(tot) would be present at all, the spin-down magnetic field Bp(sd) calculated from (12) might coincide with the field Bp(tot). Table 2 shows, however, that in general the field Bp(tot) is much smaller than Bp(sd). Since spin-up episodes for many pulsars in table 2 are observed, the mechanism of magnetic dipole radiation, explicitly leading to spin-down and yielding equation (12), cannot satisfactorily explain the found values of Bp(tot).

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Jacob Biemond

Table 2. Observed and calculated magnetic fields of a number of binary, slowly rotating accretion-powered X-ray emitting pulsars Name (Type) {eclipsing} [references]

Period P (s)

Bp(def) (G)

β*

β′

Log τc (yr)

~ + 4×10–9 since 1981 (– 4×10–9 from 1976 – 1981)

3.2×1012 6.5×1010 5.9×1016 3.4×1010

49

9×105

3.5

681

Spin-up/ spin-down (irregular)

4.2×1012 8.0×1010

53

528.8

Spin-up/ spin-down (irregular)

2.3×1012 1.0×1011

23

440.6

+ 6.9×10–9 (steady spin2.1×1012 1.2×1011 5.6×1016 6.1×1010 down during 15 years)

18

4.7×105

3.01

283.2

spin-down/ spin-up (irregular)

2.8×1012 1.9×1011

15

160

4.1×1012 3.4×1011

12

103.5

1.2×1013 5.2×1011

23

66.3

3.2×1012 8.2×1011

3.9

15.83

3.9×1012 3.4×1012

1.1

4U 0352+309 1

(HMXBa) {no} [33, 35]

837.7

GX 301–2 2

3

4

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5

6

7

8

9

(HMXB) {nearly} [33, 34, 36] 4U 1538–52

(HMXB) {yes} [33, 36] 4U 1907+09 (HMXB) {nearly} [33, 37] Vela X-1 (HMXB) {yes} [33, 36] MX0656–072 (HMXB) {no} [34] A 0535+26b (HMXB) {no} [38] Cep X-4 (HMXB) {no} [34] XTE J1946+ 274 (HMXB) {no} [33]

4U 1626–67 (LMXBc) 10 {no} [33, 36, 39] Cen X-3 (HMXB) 11 {yes} [33, 36, 40]

P (s.s–1)

Bp(tot) (G)

Bp(gm) (G)

Bp(sd) (G)

7.67

+ 4.2×10–11 since 1990 (– 5.0×10–11 from 1977 – 1990)

4.4×1012 7.1×1012 5.7×1014 3.6×1010 0.62

80

3.46

4.82

+ 9×10–11 (long term: – 2×10–11)

3.4×1012 1.1×1013 7×1014

60

2.9

7×1010

0.31

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117

The Origin of the Magnetic Field of Pulsars and the Gravitomagnetic Theory Name (Type) {eclipsing} [references] V 0332+53 (HMXB) 12 {no} [34, 41] 4U 0115+63 13 (HMXB) {no} [33, 34] Her X-1 (LMXB) 14 {yes} [33, 36, 42] a

Period P (s)

P (s.s–1)

Bp(tot) (G)

Bp(gm) (G)

Bp(sd) (G)

Bp(def) (G)

β*

4.37

3.0×1012 1.2×1013

0.25

3.61

1.4×1012 1.5×1013

0.093

1.238

~ + 10–12 (long term: ~ – 10–13)

4.6×1012 4.4×1013

4×1013

1010

0.10

β′

Log τc (yr)

0.9

4.3

HMXB = High mass X-ray binary. b When Ecycl = 110 keV is attributed to the second harmonic instead of the first, Bp(tot) is half as large. c LMXB = Low mass X- ray binary.

Moreover, the accretion process of the pulsars in table 2 may also influence the quantity Ė of (9). In that case, the derivation of Bp(sd) of (12) from a combination of (9) and (10) does not apply and calculated values of Bp(sd) from (12) are not reliable. The discrepancy between Bp(tot) and Bp(sd) may, however, also partly be attributed to electron scattering, not at the polar cap, but at higher altitudes above the surface of the pulsar. One may try to relate β* from (8) with β′ from (13). If |Bp (em)| and |Bp (em)| would be of the same order of magnitude, combination of (8) and (13) (and (54)) would yield

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β* ≈ 1 ± β′ + (β*current).

(56)

Does this relation apply for the pulsars in table 2? Comparison of β* en β′ shows that β′ is much larger than β* in many cases. It should be kept in mind, however, that many pulsars in table 2 show irregular spin-down and spin-down regimes, so that calculation of Bp(sd) and β′ is difficult or impossible. For example, 4U 1626–67 has the peculiarity of showing a steady spin-up from 1977 until 1990 and a steady spin-down afterwards. Only pulsar 4U 1907–09 shows a steady spin-down up to now, consistent with the magnetic dipole radiation model. Perhaps, the very high values of P are mainly caused by a retarding accretion instead of magnetic dipole radiation, so that the values for β′ become much higher (see (12) and (13)). Then, relation (56) is not valid. Thirdly, the field Bp(tot) may be compared with the field Bp(def) of (19) deduced for the mechanism of “deformed magnetic field lines”, if data are available. Similarity between Bp(tot) and Bp(def) in table 2 is better than between Bp(tot) and Bp(sd) in most cases, but Bp(tot) and Bp(gm) compare more favourably. Summing up, the emission mechanism leading to (19) might contribute in a number of pulsars in table 2. In table 3 data and results are given for five isolated, X-ray emitting pulsars with reported values for Bp(tot), analogously to table 2. Data for the soft gamma repeater SGR 1806–20 are taken from Ibrahim et al. [43]. Soft gamma repeaters resemble in their behaviour to the binary, slowly rotating, accretion-powered X-ray pulsars of table 2, but so far no conclusive evidence for a companion or for accretion has been found for these objects. They show, however, gamma ray bursts from time to time. Since the observed absorption cyclotron

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118

Jacob Biemond

features for the soft gamma repeater SGR 1806–20 are attributed to electron or proton resonance, two values for the observed magnetic field Bp(tot) are given in table 3 (see a discussion on this subject in ref. [43]). Further, data for the anomalous X-ray pulsar (= AXP) 1RXS J1708–4009 from Rea et al. [44] and Kaspi and Gavriil [45] are presented in table 3. AXPs are related to SGRs, but they show no gamma ray bursts up to now. Haberl et al. [46] reported data for RX J0720.4–3125. Note that the value of P is rather uncertain. RX J0720.4–3125 is a member of a group of so-called X-ray dim isolated neutron stars (XDINs), which share the following properties: soft blackbody-like X-ray spectrum, radio-quiet and no known association with a super-nova remnant. Data for the radio-quiet pulsar 1E 1207.4–5209 also showing blackbody X-ray radiation and probably situated in the centre of supernova remnant PKS 1209–51/52 were taken from Bignami et al. [47]. These authors observed three, perhaps four, distinct features in the X-ray spectrum of this star and attributed them to electron cyclotron resonance absorptions. Sanwal et al. [48], however, earlier observed two features in the X-ray spectrum and attributed them to atomic transitions of once-ionised helium. Results of all suggested interpretations are given in table 3. Becker et al. [49] recently reported the (marginal) detection of an emission line in the Xray spectrum of the isolated millisecond pulsar B1821–24. Other data for this pulsar in the globular cluster M28 are also given in table 3. When the emission feature is interpreted as an electron cyclotron line, a value of β* = 2.1×10 –5 follows from (7a). This value can be compared with the value β* = 1, when the field Bp(tot) would exclusively be due to gravitomagnetic origin. The extension of the dome-torus model [29]–[31] might explain the low value of β*. According to (54), values β* smaller than unity value may be possible for a broad, “superrotating” far disk (rout > R and Ωdisk > Ω) (see section 4 for detailed information). Apart from that, the found low value of β* may also partly be attributed to electron-scattering, not at the polar cap, but at higher altitudes above the surface of the pulsar. Alternatively, another contribution may explain the low value of β* for a millisecond pulsar like B1821–24. As has been discussed in section 4, a toroidal current in the pulsar caused by previous accretion may result into a contribution β*current ≈ – 1 to (54), yielding a value of β* ≈ 0. From the standard dome-torus model [29]–[31] expression β* of (50) was deduced, as has been shown in section 4. Choosing R = 10 km, from this equation follows a value β* = 1 + 6.3×10 – 4 for B1821–24. In view of the value β* = 2.1×10 –5 deduced from observations, the latter model is unsatisfactory in this case. Finally, in view of the uncertainties in the interpretation of Ecycl it is difficult to decide whether Bp(gm) or Bp(sd) is the best prediction for Bp(tot) in table 3. Much depends on the correct interpretation of the cyclotron features. Do they originate from electron resonance, proton resonance, or from some ion?

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119

The Origin of the Magnetic Field of Pulsars and the Gravitomagnetic Theory Table 3. Observed and calculated magnetic fields of some isolated, X-ray pulsars Name (type) {origin signal} [references]

Period P (s)

P (s.s–1)

7.47

8.2×10–11

Bp(tot) (G)

Bp(gm) (G)

Bp(sd) (G)

Bp(def) (G)

β*

β′

Log τc (yr)

SGR 1806–20 1

{electron} {protona}

5.6×1011 7.8×10–2 12 7.9×1014 5.1×1010 1.1×102 3.16 15 7.2×10 1.0×10 1.4×102

[43] 1RXS J1708– 2

4009 (AXP) {electron} {protona} [44, 45]

11.00 1.9×10–11 9.1×1011 4.9×1012 4.6×1014 2.0×1010 0.19 1.7×1015 3.5×102

94

3.96

3

6.4

RX J0720.4– 3

3125 (XDIN) {electron} {protona}

8.39

5×10–14

[46] 1E 1207.4 – 5209 4

{electron}

{protona} {He+-line} [47, 48]

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5

B1821–24 (INS) {electron}

–14

0.424 1.4×10

3.1×1010 6.5×1012 2×1013 5.6×1013

109

4.8×10–3 8.6

7.9×1010 6.2×10–4 14 14 12 9 1.5×10 1.3×10 2.5×10 2.8×10 1.2 1.9×10–2 14 1.5×10 1.2

5.7

0.00305 1.6×10–18 3.7×1011 1.8×1016 2.2×109 3.5×108 2.1×10–5 1.2×10–7 7.48

[49] a

Bp(tot) is calculated from: Ecycl = 6.305×10–15 0.7658 B(G) keV. Compare with (55): the electron mass me has been replaced by the proton mass mp. Again, for these pulsars m = 1.4 m and R = 10 km have been used.

Furthermore, most values of Bp(def) in table 3 are smaller/much smaller than those of Bp(tot) calculated from the electron/proton cyclotron interpretation, respectively. In the latter case this mechanism seems to be relatively unimportant (compare with table 2). In table 4 the Parkes multi-beam pulsar survey–I [50] has been chosen as a representative sample of 100 pulsars. Earlier, Woodward [21] calculated the values of β′ from β′ = |M(sd)|/|M(WB)| = |Bp(sd)|/|Bp(WB)|, where WB denotes Wilson-Blackett (i.e., WB refers to formula (1) with β = +1), for another sample of more than 100 pulsars and concluded that β′ is no universal constant. In order to explain this non-constancy, he suggested a secondary mechanism of magnetic field generation. Note that (27) offers such an expression for β′ = |Bp(sd)|/|Bp(gm)| depending on the quantities Ω and Ω, but in fact this equation is equivalent to equation (7b). In table 4 the values of β′ from (13) for the pulsars of the Parkes pulsar survey-I [50] are added. From these values a mean value of β′ = 0.060 has been calculated.

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Table 4. Calculated magnetic fields Bp(gm) of (4) for β = 1, Bp(sd) of (12) and Bp(def) of (19), respectively, of the 100 pulsars of the Parkes multi-beam pulsar survey – I

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

PSR J

Period P (s)

(s.s–1)

Bp(gm) (G)

Bp(sd) (G)

Bp(def) (G)

β′

Log τC (yr)

1307–6318 1444–5941 1601–5244 1312–6400 1303–6305 1245–6238 1049–5833 1632–4621 0838–3947 1001–5559 1609–5158 1345–6115 1724–3505 1616–5109 1726–3530 1621–5039 1622–4944 1725–3546 1616–5208 1605–5215 1144–6146 1513–5739 1434–6029 0922–4949 1348–6307 1536–5433 1649–4349 1628–4804 1144–6217 1252–6314 1632–4818 1220–6318 1715–3700 1002–5559 1429–5935 1623–4949 1407–6153 1130–5925 1056–5709 1611–4949

4.962 2.760 2.559 2.437 2.307 2.283 2.202 1.709 1.704 1.661 1.279 1.253 1.222 1.220 1.110 1.084 1.073 1.032 1.026 1.014 0.9878 0.9735 0.9633 0.9503 0.9278 0.8814 0.8707 0.8660 0.8507 0.8233 0.8134 0.7892 0.7796 0.7775 0.7639 0.7257 0.7016 0.6810 0.6761 0.6664

2.11×10–14 8.2×10–15 7.2×10–16 6.8×10–16 2.18×10–15 1.09×10–14 4.41×10–15 7.60×10–14 8×10–16 8.60×10–16 1.30×10–14 3.25×10–15 2.11×10–14 1.91×10–14 1.22×10–12 1.30×10–14 1.71×10–14 1.50×10–14 2.89×10–14 4.75×10–15 – 4×10–17 2.76×1 1.03×10–15 9.76×10–14 3.79×10–15 1.91×10–15 4.4×10–17 1.24×10–15 3.08×10–14 1.1×10–16 6.51×10–13 8×10–17 1.5×10–16 1.57×10–15 4.28×10–14 4.21×10–14 8.85×10–15 9.52×10–16 5.76×10–16 5.4×10–16

1.1×1013 2.0×1013 2.1×1013 2.2×1013 2.3×1013 2.4×1013 2.5×1013 3.2×1013 3.2×1013 3.3×1013 4.2×1013 4.3×1013 4.4×1013 4.4×1013 4.9×1013 5.0×1013 5.0×1013 5.2×1013 5.3×1013 5.3×1013 5.5×1013 5.6×1013 5.6×1013 5.7×1013 5.8×1013 6.1×1013 6.2×1013 6.3×1013 6.4×1013 6.6×1013 6.7×1013 6.9×1013 6.9×1013 7.0×1013 7.1×1013 7.5×1013 7.7×1013 8.0×1013 8.0×1013 8.1×1013

1.0×1013 4.8×1012 1.4×1012 1.3×1012 2.3×1012 5.0×1012 3.2×1012 1.2×1013 1.2×1012 1.2×1012 4.1×1012 2.0×1012 5.1×1012 4.9×1012 3.7×1013 3.8×1012 4.3×1012 4.0×1012 5.5×1012 2.2×1012

1.0×109 8.4×108 2.6×108 2.6×108 4.8×108 1.1×109 6.9×108 3.3×109 3.4×108 3.5×108 1.6×109 7.9×108 2.0×109 1.9×109 1.6×1010 1.7×109 2.0×109 1.9×109 2.6×109 1.1×109

0.91 0.24 6.7×10–2 5.9×10–2 0.10 0.21 0.13 0.38 3.8×10–2 3.6×10–2 9.8×10–2 4.7×10–2 0.12 0.11 0.76 7.6×10–2 8.6×10–2 7.7×10–2 0.10 4.2×10–2

6.57 6.73 7.75 7.75 7.22 6.52 6.90 5.55 7.53 7.49 6.19 6.79 5.96 6.01 4.16 6.12 6.00 6.04 5.75 6.53

a

a

a

a

12

5.2×10 1.0×1012 9.7×1012 1.9×1012 1.3×1012 2.0×1011 1.0×1012 5.2×1012 3.0×1011 2.3×1013 2.5×1011 3.5×1011 1.1×1012 5.8×1012 5.6×1012 2.5×1012 8.1×1011 6.3×1011 6.1×1011

9

2.6×10 5.1×108 5.0×109 9.9×108 7.2×108 1.1×108 5.9×108 2.9×109 1.8×108 1.4×1010 1.6×108 2.1×108 7.0×108 3.7×109 3.7×109 1.7×109 5.8×108 4.5×108 4.4×108

–2

9.3×10 1.8×10–2 0.17 3.3×10–2 2.1×10–2 3.2×10–3 1.6×10–2 8.1×10–2 4.5×10–3 0.34 3.6×10–3 5.1×10–3 1.6×10–2 8.2×10–2 7.5×10–2 3.2×10–2 1.0×10–2 7.9×10–3 7.5×10–3

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5.75 7.17 5.19 6.59 6.86 8.50 7.04 5.64 8.07 4.30 8.19 7.92 6.89 5.45 5.44 6.10 7.05 7.27 7.29

The Origin of the Magnetic Field of Pulsars and the Gravitomagnetic Theory

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

1613–5234 1123–6102 1716–3720 1341–6023 1309–6415 1728–3733 1540–5736 1653–4249 1347–5947 1558–5419 1709–3841 1224–6208 1142–6230 0835–3707 1412–6111

Period P (s) 0.6552 0.6402 0.6303 0.6273 0.6195 0.6155 0.6129 0.6126 0.6100 0.5946 0.5870 0.5858 0.5584 0.5414 0.5292

56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83

1322–6241 1425–6210 1407–6048 1610–5006 1305–6256 0954–5430 1625–4904 1613–5211 0901–4624 1537–5645 1305–6203 1119–6127 1452–5851 1650–4502 1543–5459 1216–6223 1720–3659 1607–5140 1412–6145 1649–4729 1416–6037 1626–4807 1413–6222 1601–5335 1726–3635 1327–6400 1530–5327 1538–5438

0.5061 0.5017 0.4923 0.4811 0.4782 0.4728 0.4603 0.4575 0.4420 0.4305 0.4278 0.4077 0.3866 0.3809 0.3771 0.3740 0.3511 0.3427 0.3152 0.2977 0.2956 0.2939 0.2924 0.2885 0.2874 0.2807 0.2790 0.2767

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121

(s.s–1) 6.63×10–15 6.46×10–15 1.80×10–14 1.95×10–14 8.79×10–15 7×10–17 4.2×10–16 4.81×10–15 1.42×10–14 6.04×10–15 7.86×10–15 2.02×10–14 8×10–17 9.78×10–15 1.91×10–15

Bp(gm) (G) 8.3×1013 8.5×1013 8.6×1013 8.6×1013 8.7×1013 8.8×1013 8.8×1013 8.8×1013 8.9×1013 9.1×1013 9.2×1013 9.2×1013 9.7×1013 1.0×1014 1.0×1014

Bp(sd) (G) 2.1×1012 2.1×1012 3.4×1012 3.5×1012 2.4×1012 2.1×1011 5.1×1011 1.7×1012 3.0×1012 1.9×1012 2.2×1012 3.5×1012 2.1×1011 2.3×1012 1.0×1012

Bp(def) (G) 1.6×109 1.6×109 2.6×109 2.7×109 1.8×109 1.7×108 4.1×108 1.4×109 2.4×109 1.6×109 1.8×109 2.9×109 1.9×108 2.1×109 9.3×108

2.5×10–2 2.5×10–2 4.0×10–2 4.1×10–2 2.8×10–2 2.4×10–3 5.8×10–3 1.9×10–2 3.4×10–2 2.1×10–2 2.4×10–2 3.8×10–2 2.2×10–3 2.3×10–2 1.0×10–2

Log τC (yr) 6.19 6.20 5.74 5.71 6.05 8.14 7.36 6.30 5.83 6.19 6.07 5.66 8.04 5.94 6.64

2.59×10–15 4.8×10–16 3.16×10–15 1.36×10–14 2.11×10–15 4.39×10–14 1.68×10–14 1.92×10–14 8.75×10–14 2.78×10–15 3.21×10–14 4.02×10–12 5.07×10–14 1.61×10–14 5.20×10–14 1.68×10–14 3.27×10–17 2.54×10–15 9.87×10–14 6.55×10–15 4.28×10–15 1.75×10–14 2.23×10–15 6.24×10–14 1.44×10–15 3.12×10–14 4.68×10–15 1.42×10–15

1.1×1014 1.1×1014 1.1×1014 1.1×1014 1.1×1014 1.1×1014 1.2×1014 1.2×1014 1.2×1014 1.3×1014 1.3×1014 1.3×1014 1.4×1014 1.4×1014 1.4×1014 1.4×1014 1.5×1014 1.6×1014 1.7×1014 1.8×1014 1.8×1014 1.8×1014 1.9×1014 1.9×1014 1.9×1014 1.9×1014 1.9×1014 2.0×1014

1.2×1012 5.0×1011 1.3×1012 2.6×1012 1.0×1012 4.6×1012 2.8×1012 3.0×1012 6.3×1012 1.1×1012 3.7×1012 4.1×1013 4.5×1012 2.5×1012 4.5×1012 2.5×1012 1.1×1011 9.4×1011 5.6×1012 1.4×1012 1.1×1012 2.3×1012 8.2×1011 4.3×1012 6.5×1011 3.0×1012 1.2×1012 6.3×1011

1.1×109 4.8×108 1.2×109 2.6×109 1.0×109 4.7×109 3.0×109 3.2×109 6.9×109 1.2×109 4.2×109 4.9×1010 5.6×109 3.2×109 5.8×109 3.3×109 1.5×108 1.3×109 8.7×109 2.3×109 1.9×109 3.8×109 1.4×109 7.2×109 1.1×109 5.2×109 2.0×109 1.1×109

1.1×10–2 4.5×10–3 1.2×10–2 2.4×10–2 9.1×10–3 4.2×10–2 2.3×10–2 2.5×10–2 5.3×10–2 8.5×10–3 2.8×10–2 0.32 3.2×10–2 1.8×10–2 3.2×10–2 1.8×10–2 7.3×10–4 5.9×10–3 3.3×10–2 7.8×10–3 6.1×10–3 1.3×10–2 4.3×10–3 2.3×10–2 3.4×10–3 1.6×10–2 6.3×10–3 3.2×10–3

6.49 7.22 6.39 5.75 6.56 5.23 5.64 5.58 4.90 6.39 5.32 3.21 5.08 5.57 5.06 5.55 8.23 6.33 4.70 5.86 6.04 5.43 6.32 4.86 6.50 5.15 5.98 6.49

β′

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Jacob Biemond

PSR J 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

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a

1622–4802 1317–6302 1115–6052 1349–6130 1406–6121 0957–5432 1723–3659 1301–6305 1548–5607 1138–6207 1232–6501a 1016–5819 0940–5428 1718–3825 1112–6103 1454–5846a 1435–6100a

Period P (s) 0.2651 0.2613 0.2598 0.2594 0.2131 0.2036 0.2027 0.1845 0.1709 0.1176 8.828×10–2 8.783×10–2 8.755×10–2 7.467×10–2 6.496×10–2 4.525×10–2 9.348×10–3

(s.s–1) 3.07×10–16 1.02×10–16 7.24×10–15 5.13×10–15 5.47×10–14 1.95×10–15 8.01×10–15 2.67×10–13 1.07×10–14 1.25×10–14 8.1×10–19 6.98×10–16 3.29×10–14 1.32×10–14 3.15×10–14 8.16×10–19 2.45×10–20

Bp(gm) (G) 2.0×1014 2.1×1014 2.1×1014 2.1×1014 2.5×1014 2.7×1014 2.7×1014 2.9×1014 3.2×1014 4.6×1014 6.1×1014 6.2×1014 6.2×1014 7.3×1014 8.3×1014 1.2×1015 5.8×1015

Bp(sd) (G) 2.9×1011 1.7×1011 1.4×1012 1.2×1012 3.5×1012 6.4×1011 1.3×1012 7.1×1012 1.4×1012 1.2×1012 8.6×109 2.5×1011 1.7×1012 1.0×1012 1.4×1012 6.1×109 4.8×108

Bp(def) (G) 5.3×108 3.1×108 2.6×109 2.2×109 7.8×109 1.5×109 3.1×109 1.9×1010 3.9×109 5.1×109 4.7×107 1.4×109 9.5×109 6.5×109 1.1×1010 6.6×107 2.5×107

β′ 1.5×10–3 8.1×10–4 6.7×10–3 5.7×10–3 1.4×10–2 2.4×10–3 4.8×10–3 2.4×10–2 4.4×10–3 2.6×10–3 1.4×10–5 4.0×10–4 2.7×10–3 1.4×10–3 1.7×10–3 5.1×10–6 8.3×10–8

Log τC (yr) 7.14 7.61 5.75 5.90 4.79 6.22 5.60 4.04 5.40 5.17 9.24 6.30 4.62 4.95 4.51 8.94 9.78

Binary pulsar.

Since the field Bp(tot) has not yet directly been observed for these pulsars, Bp(sd) of (12) may be used to estimate the value of β* using the approximated relation β* ≈ 1 ± β′ of (56). When the mean value 0.060 of β′ of 99 pulsars in table 4 is inserted into the simplified form of (56) the quantity |β*| may lie between 1.06 and 0.94. According to (7a), |Bp (tot)| = |Bp(gm)| implies β* ≈ 1. The deduced results of β* between 0.94 and 1.06 show thus limited similarity with the gravitomagnetic prediction of (4) and with the Wilson-Blackett formula (1). This result, however, depends on the validity of the relation β* ≈ 1 ± β′ from (56), which depends on the assumptions |Bp (em)| = |Bp (em)| and β*current = 0. Furthermore, one may compare the obtained high value β* = 1.06 with the result for β* from (50). The latter expression was deduced from the standard dome-torus model [29]–[31], discussed in section 4. The lower limit of β* from (50) reduces to β* = 1 + 2/15(Ω R/c)2 (for example, an representative choice like P = 0.5 s and R = 10 km yields β* = 1 + 2.3×10 –8), whereas the upper limit of β* is given by β* = + 1.15. Note that the value of β* from (50) is always larger than unity value: the magnetic field Bp(gm) from gravitomagnetic origin will be enforced by the magnetic field from electromagnetic origin embodied in M′(em) from (49). Therefore, the high value β* ≈ 1 + β′ = 1.06 from (56) may lie in the range of values for β* predicted by (50). The validity of the relation β* ≈ 1 ± β′ is uncertain, however. Alternatively, the extended dome-torus model [29]–[31], discussed in section 4 might explain the low value β* = 0.94. According to (54), values β* smaller than unity value may be possible for a broad, “superrotating” far disk (rout > R and Ωdisk > Ω) (see section 4 for detailed information). According to the latter model, the basic magnetic field from gravitomagnetic origin would be weakened by the magnetic field from moving charge. Particularly, the three short-period, binary pulsars in table 4 possess exceptionally low values for β′ (β′ ≈ 10–5–10–7), much lower than the mean value β′ = 0.060 of the pulsars in

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The Origin of the Magnetic Field of Pulsars and the Gravitomagnetic Theory

123

table 4. Perhaps, the low values of P for these binary pulsars are partly caused by prograde accretion (where Ωdisk > Ω), instead of magnetic dipole radiation. As a consequence, the calculated values of Bp(sd) from (12) and β′ from (13) may be too small. Alternatively, if a contribution β*current ≈ – 1 is present, the relation for β* from (56) reduces to β* ≈ ± β′. The values of Bp(def) in table 4 are orders of magnitude smaller than Bp(sd), so that for most pulsars of the list this mechanism seems to be unimportant.

BRAKING INDICES FOR SOME YOUNG PULSARS Data from several authors [23, 24, 26, 51–53] of five isolated, young pulsars have been summarized in table 5. The values of the braking indices have all been calculated from n = Ω Ω /Ω 2. As can be seen from (23) and (24), the braking index n does not directly depend on the gravitomagnetic field Bp(gm) or on Bp(gm). Therefore, the validity of the gravitomagnetic hypothesis cannot be tested by considering n. Table 5. Braking index and other data of some isolated, young pulsars. Name PSR Period P [references] (s)

Bp(sd) (G)

β′

τc (kyr)

1.3×1014

4.1×1013

0.32

1.61

n n0 = 3

0.4076

4.02×10–12

2.91 ± 0.05

0.1507 0.1507

1.537×10–12 1.536×10–12

2.837 ± 0.001 2.839 ± 0.003

0.0893

1.25×10–13

1.4 ± 0.2

6.1×1014

3.4×1012

0.0056

11.3

B0540–69 0.05046

4.79×10–13

2.125 ± 0.001

1.1×1015

5.0×1012

0.0045

1.67

4.23×10–13

2.515 ± 0.005

1.6×1015

3.8×1012

0.0024

1.24

J1119– 6127

m m0 = 15

Bp(gm) (G)

(s.s–1)

[51]

B1509–58

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[24] [26]

B0833–45 Vela pulsar [52] [53]

B0531+21 Crab pulsar [23]

0.0331

14.5 ± 3.6 3.59×1014 1.54×1013 0.0429 1.554 18.3 ± 2.9

Data for PSR B1509–58 from radio timing during 11 years are given by Kaspi et al. [24] in table 5, together with data from 21 years of timing gathered by Livingstone et al. [26]. The calculated values for the first order braking index n and the second order braking index m can be compared with the corresponding braking indices n0 = 3 (see comment following (26)) and m0 = 15 from (31), when the magnetic field Bp(em) is no function of time. Alternatively, if it is assumed that Bp(em) = Bp(sd) is time dependent, substitution of n = 2.839 into (26) yields a value of Bp(sd)/Bp (sd) = + 2.59×10 –5 yr –1. Substitution of this ratio of Bp(sd)/Bp(sd) into (29), together with m = 18.3 and n0 = 3, then yields a value of Bp (sd)/Bp(sd) = + 2.45×10 –7 yr –2. Instead of differentiation, Johnston and Galloway [25] used integration to derive an alternative braking index n0 from Ω = – K Ω n0 = – k Bp (em)2 Ω n0 (see (22): K, n0, k and Bp (em)

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Jacob Biemond

are assumed not to depend on time). From their formula they found n0 = 2.502 for the Crab pulsar compared with n = 2.509 directly calculated from n = Ω Ω/Ω2, so n – n0 = + 0.007. Likewise, for PSR B1509–58 they found n0 = 2.80 compared with n = 2.837; so n – n0 = + 0.037. Thus, n and n0 nearly coincide for both pulsars, when K does not depend on time. On the other hand, if Bp(em) = Bp(sd) from (12) is time dependent, K will be time dependent and combination of (24) and (26a) yields n0 = 3. Then, the values of n and n0 become different and a ratio B p(sd)/Bp(sd) different from zero can be calculated from (26). Taking n = 2.509 for the Crab pulsar, a value of Bp(sd)/Bp(sd) = + 9.90×10 –5 yr –1 can be calculated. Likewise, for PSR B1509–58 a value Bp(sd)/Bp(sd) = + 2.59×10 –5 yr –1 has already been found above. Thus, the absolute value of Bp(sd) of both young pulsars increases in time. Since the true age of the Crab pulsar is known (920 years in 1974 from [23]) and the characteristic time τc = 1240 years, Ωi can be calculated from (25). One finds Ωi = 1.7 Ω, or Pi = 19 ms. Note that the value n = 2.515 has been introduced into (25), instead of the canonical value n0 = 3. In the latter case, Ωi = 2.0 Ω, or Pi = 17 ms (compare with [23, pp. 111–112])]

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CONCLUSION Following Woodward [21], magnetic fields of pulsars deduced from observations are compared with the so-called Wilson-Blackett formula (1). This relation predicting a dipolar magnetic field suggests a general, gravitational origin for the magnetic field of rotating, celestial bodies. Several attempts have been made to derive (1) from general relativity [8 –16], e.g., in the gravitomagnetic approach [8 –10]. The gravitomagnetic prediction for the magnetic induction field at the north pole of the pulsar, Bp(gm), is given in (4). In this chapter the latter field is compared with the total observed field, Bp(tot), when available, or otherwise with the magnetic field, Bp(sd) of (12), deduced from the standard magnetic dipole radiation model. The validity of (1) is tested in table 2 by comparison of the field Bp(gm) from (4) with the observed field Bp(tot) of fourteen young, X-ray emitting, binary pulsars. For these pulsars the ratio β* ≡ |Bp (tot)|/|Bp(gm)| from (7a) has been compared with the gravitomagnetic prediction β* = 1. It appears that an approximation for β*, i.e., β* ≈ |Bp(tot)|/|Bp(gm)| yields values for β* varying from about 55 to 0.1. The found similarity between Bp(tot) and Bp(gm) is only limited but much better than between Bp(tot) and Bp(sd) of (12). The discrepancy between Bp(tot) and Bp(gm) is discussed in section 5. For comparison, a linear regression analysis showed an almost linear relationship between the observed magnetic moment |M| and the angular momentum |S| for a series of about fourteen rotating bodies, ranging from metallic cylinders in the laboratory, moons, planets, stars to the Galaxy (|M| and |S| vary over an interval of sixty decades!). Moreover, from a weighted least-squares fit to the data an average value of | β*| = 0.076 was found [9]. Although this result is distinctly different from the gravitomagnetic prediction β* = 1, the correct order of magnitude of β* for so many, strongly different, rotating bodies is amazing. The discrepancy between Bp(tot) and Bp(gm) may be attributed to magnetic fields from electromagnetic origin [9, ch. 1]. In table 3 values of Bp(gm) of (4) and Bp(sd) of (12) are compared with the corresponding values of Bp(tot) of five isolated pulsars (one soft gamma repeater, one anomalous X-ray

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pulsar and three isolated neutron stars). It is stressed, however, that in view of the uncertainties in the identification of the measured values of Bp(tot), it is difficult to decide whether Bp(gm) or Bp(sd) is the best prediction in table 3. In particular, the observational value β* = 2.1×10 –5 of the millisecond pulsar B1821–24 deserves special attention and has been compared with the gravitomagnetic prediction β* = 1 more in detail. The low value of β* might be explained by an extension of the dome-torus model [29]–[31], proposed in section 4. According to (54), values for β* smaller than unity value are predicted by this model. Alternatively, however, a toroidal current generated by previous accretion in the highly conductive pulsar may largely compensate the magnetic field from gravitomagnetic origin. Owing to the slow Ohmic decay, (54) may then contain a contribution β*current ≈ – 1 resulting into a value of β* ≈ 0. In table 4 data of a sample of 100 pulsars of the Parkes multi-beam survey – I [50] have been given. Since the field Bp(tot) has not yet directly been observed for these pulsars, Bp(sd) of (12) may be used to estimate the value of β* using the approximated relation β* ≈ 1 ± β′ of (56). A mean value β′ = 0.060 has been calculated from 99 pulsars in table 4, so that a lower limit β* = 0.94 and an upper limit β* = 1.06 may be extracted from this value. These results may be compatible with the prediction β* = 1 from the Wilson-Blackett formula (1). The relation β* ≈ 1 ± β′ of (56) depends, however, on the unverified assumption that components of the magnetic field from electromagnetic origin at the north pole, parallel and perpendicular to S, | Bp (em)| and | Bp (em)|, respectively, are of comparable magnitude. However, if a contribution β*current ≈ – 1 is present (for millisecond pulsars, for example), the relation for β* of (56) reduces to β* ≈ β′. Apart from the Wilson-Blackett formula (1), more consequences from the gravitomagnetic theory for pulsars are considered in section 4. From this theory the standard quadrupolar charge density (42) for pulsars is deduced. Analogous to the dome-torus model, (see, e.g., Michel and Li [29] and [30]–[31]), a negative electric monopole charge (45) might also be adopted in the pulsar (using our sign convention). Moreover, a new (electro)magnetic dipole moment M ′(em) of (49), generated by circulating charge in the basic magnetic field from gravitomagnetic origin is deduced. It is found that this contribution to the magnetic field from electromagnetic origin always enforces the magnetic field from gravitomagnetic origin. In addition, a relation (50) for β* is calculated, which always yields values for β* larger than unity value. Furthermore, a tentative extension of the dome-torus model is proposed in section 4. In this model the pulsar and its inner equatorial belt does not only bear a negative charge Qin (in our sign convention), but also an additional positive charge Qout in the outer disk. Charge neutrality of the system is then ensured. As a result, the basic magnetic field from gravitomagnetic origin may then be weakened. According to (54), values for β* smaller than unity value are possible in this model. More observational evidence is necessary, however, to prove the validity of this extension. Special attention has been given to the influence of the magnetic field on first order braking index n (equations (23) and (24)) and second order braking index m (equations (29)– (31)). In deriving these equations, it has been assumed that the moment of inertia I, radius r and mass m of the pulsar and angle θ between magnetic moment M and angular momentum S of the pulsar do not depend on time. Of course, these assumptions need not to be true. Furthermore, the values of n of five young pulsars and one value for m of PSR B1509–58 calculated from observations have been compared with theoretical predictions. In addition, it

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is found that n does not directly depend on the gravitomagnetic field Bp(gm). Therefore, the validity of the gravitomagnetic hypothesis cannot be tested by considering n. In conclusion, one may say that the so-called Wilson-Blackett formula (1) predicts the correct order of magnitude of the magnetic field of many pulsars and in fact of many rotating, electrically neutral bodies. Several effects from electromagnetic origin may account for discrepancies [9]. It has previously been shown [8–10] that relation (1) can be deduced from general relativity, e.g., by application of a special interpretation of the gravitomagnetic theory. Several new consequences of this theory for pulsars are obtained. So far, the proposed gravitomagnetic hypothesis seems to be compatible with known observational and theoretical evidence. In the near future, however, the gravitomagnetic precession rate presently measured by the Gravity Probe B will probably yield more information about the true nature of the gravitomagnetic field [10]. In particular, the equivalence of the gravitomagnetic field and the magnetic field from electromagnetic origin will be tested. Finally, it is noticed that the gravitomagnetic theory is important in view of further unification of existing physical theories [54].

ACKNOWLEDGMENT I would like to thank my son Pieter for technical help in the realization of this chapter.

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REFERENCES [1] Schuster, A., "A critical examination of the possible causes of terrestrial magnetism." Proc. Phys. Soc. Lond. 24, 121-137 (1912). [2] Wilson, H. A., "An experiment on the origin of the Earth's magnetic field." Proc. R. Soc. Lond. A 104, 451-455 (1923). [3] Blackett, P. M. S., "The magnetic field of massive rotating bodies." Nature 159, 658-666 (1947). [4] Russell, C. T., "Re-evaluating Bode's formula of planetary magnetism." Nature 272, 147-148 (1978). [5] Ahluwalia, D. V. and Wu, T.-Y., "On the magnetic field of cosmological bodies." Lett. Nuovo Cimento 23, 406-408 (1978). [6] Sirag, S-P., "Gravitational magnetism." Nature 278, 535-538 (1979). [7] Barut, A. O., "Gravitation and electromagnetism." In: Proceedings of the second Marcel Grossmann meeting on general relativity (Ed. Ruffini R.), North-Holland Publishing Company, Amsterdam, pp. 163-165 (1982). [8] Biemond, J., Gravi-magnetism, 1st ed. (1984). Postal address: Sansovinostraat 28, 5624 JX Eindhoven, The Netherlands. E-mail: [email protected] Website: http://www.gewis.nl/~pieterb/gravi/ [9] Biemond, J., Gravito-magnetism, 2nd ed. (1999). See also ref. [8]. [10] Biemond, J., "Which gravitomagnetic precession rate will be measured by Gravity Probe B?", Physics/0411129.

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[11] Bennett, J. G., Brown, R. L. and Thring, M. W., "Unified field theory in a curvature-free five-dimensional manifold." Proc. R. Soc. Lond. A 198, 39-61(1949). [12] Luchak, G., "A fundamental theory of the magnetism of massive rotating bodies." Can. J. Phys. 29, 470-479 (1951). [13] Barut, A. O. and Gornitz, T. "On the gyromagnetic ratio in the Kaluza-Klein theories and the Schuster-Blackett formula." Found. Phys. 15, 433-437 (1985). [14] Batakis, N. A., "On the puzzling intrinsic magnetic moments of large spinning objects." Class. Quantum Grav. 3, L49-L54 (1986). [15] Widom, A. and Ahluwalia, D. V., "Magnetic moments of astrophysical objects as a consequence of general relativity." Chin. J. Phys. 25, 23-26 (1987). [16] Nieto J. A. “Gravitational magnetism and general relativity. Rev. Mex. Fis. (Mexico) 34, 571-576 (1989). [17] Blackett, P. M. S., "A negative experiment relating to magnetism and the Earth's rotation." Phil. Trans. R. Soc. A 245, 309-370 (1952). [18] Surdin, M., "Magnetic field of rotating bodies." J. Franklin Inst. 303, 493-510 (1977). [19] Surdin, M., "Le champ magnétique des corps tournants." Ann. Fond. L. de Broglie 5, 127-145 (1980). [20] Woodward, J. F., "Electrogravitational induction and rotation." Found. Phys. 12, 467478 (1982). [21] Woodward, J. F., "On nonminimal coupling of the electromagnetic and gravitational fields: The astrophysical evidence for the Schuster-Blackett conjecture and its implications." Found. Phys. 19, 1345-1361 (1989). [22] Landau, L. D. and Lifshitz, E. M., The classical theory of fields, 4th ed., Pergamon Press, Oxford, (1975). [23] Manchester, R. N. and Taylor, J. H., Pulsars, W. H. Freeman and Company, San Francisco, 1977 and references therein. [24] Kaspi, V. M., Manchester, R. N., Siegman, B., Johnston, S. and Lyne, A. G., "On the spin-down of PSR B1509–58." Ap. J. 422, L83-L86 (1994). [25] Johnston, S. and Galloway, D. "Pulsar braking indices revisited." Mon. Not. R. Astron. Soc. 306, L50-L54 (1999). [26] Livingstone, M. A., Kaspi, V. M., Gavriil F. P. and Manchester R. N., "21 years of timing PSR B1509–58." Ap. J. 619 1046-1053 (2005). [27] Reisenegger, A., "Origin and evolution of neutron star magnetic fields." Astroph/0307133. [28] Ruderman, M., "A biography of the magnetic field of a neutron star." Astro-ph/0410607. [29] Michel, F. C. and Li, H., "Electrodynamics of neutron stars." Phys. Rep. 318, 227-297 (1999). [30] Smith, I. A., Michel, F. C. and Thacker, P. D., "Numerical simulations of aligned neutron star magnetospheres." Mon. Not. R. Astron. Soc. 322, 209-217 (2001). [31] Pétri, J., Heyvaerts, J. and Bonazzola, S., "Global static electrospheres of charged pulsars." Astron. Astrophys. 384, 414-432 (2002). [32] Michel, F. C., "The state of pulsar theory." Adv. Space Research 33, 542-551 (2004) and references therein. [33] Coburn, W., Heindl, W. A., Rothschild, R. E., Gruber, D. E., Kreykenbohm, I, Wilms, J., Kretschmar, P. and Staubert, R., "Magnetic fields of accreting X-ray pulsars with the Rossi X-ray timing explorer." Ap. J. 580, 394-412 (2002) and references therein.

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[34] Heindl, W. A., Rothschild, R. E., Coburn, W., Staubert, R., Wilms, J., Kreykenbohm, I. and Kretschmar, P., "Timing and spectroscopy of accreting X-ray pulsars: the state of cyclotron line studies." Astro-ph/0403197 and references therein. [35] Delgado-Martí, H., Levine, A. M., Pfahl, E. and Rappaport, S. A., "The orbit of X Persei and its neutron star companion." Ap. J. 546, 455-468 (2001). [36] BATSE Pulsar team, Pulsar studies. Website: http://www.batse.msfc.nasa.gov/batse/ pulsar/index.html [37] Baykal, A., İnam, Ç., Alpar, M. A., in ’t Zand, J. and Strohmayer, T., "The steady spindown rate of 4U 1907+09." Mon. Not. R. Astron. Soc. 327, 1269-1272 (2001). [38] Negueruela, I., Reig, P., Finger, M. H. and Roche, P., "Detection of X-ray pulsations from the Be/X-ray transient A 0535+26 during a disc loss phase of the primary." Astron. Astrophys. 356, 1003-1009 (2000) and references therein. [39] van Kerkwijk, M. H., Chakrabarty, D., Pringle, J. E. and Wijers, R. A. M. J., "Warped disks as a possible origin of torque reversals in accretion-powered pulsars." Ap. J. 499, L27-L30 (1998). [40] Nelson, R. W., Bildsten, L., Chakrabarty, D., Finger, M. H., Koh, D. T., Prince, T. A., Rubin, B.C., Mathew Scott, D., Vaughan, B. A. and Wilson, R. B., "On the dramatic spin-up/spin-down torque reversals in accreting pulsars." Ap. J. 488, L117-L120 (1997). [41] Shirakawa, A. and Lai, D., "Magnetically driven precession of warped disks and milliHertz variabilities in accreting X-ray pulsars." Ap. J. 565, 1134-1140 (2002) and references therein. [42] Oosterbroek, T., Parmar, A. N., Orlandini, M., Segreto, A., Santangelo, A. and Del Sordo, S., "A BeppoSAX observation of Her X-1 during the first main-on after an anomalous low-state: evidence for rapid spin-down." Astron. Astrophys. 375, 922-926 (2001). [43] Ibrahim, A. I., Safi-Harb, S., Swank, J. H., Parke, W., Zane, S. and Turolla, R., "Discovery of cyclotron resonance features in the soft gamma repeater SGR 1806-20." Ap. J. 574, L51-L55 (2002) and references therein. [44] Rea, N., Israel, G. L., Stella, L., Oosterbroek, T., Mereghetti, S., Angelini, L., Campana, S. and Covino, S., "Evidence of a cyclotron feature in the spectrum of the anomalous Xray pulsar 1RXS J170849-400910." Ap. J. 586, L65-L69 (2003) and references therein. [45] Kaspi, V. M. and Gavriil, F. P., "A second glitch from the “anomalous” X-ray pulsar 1RXS J17049.0-4000910." Ap. J. 596, L71-L74 (2003). [46] Haberl, F., Zavlin, V. E., Trümper J. and Burwitz V. "A phase-dependent absorption line in the spectrum of the X-ray pulsar RX J0720.4-3125." Astron. Astrophys. 419, 10771085 (2004) and references therein. [47] Bignami, G. F., Caraveo, P. A, De Luca, A. and Mereghetti, S., "The magnetic field of an isolated neutron star from X-ray cyclotron absorption lines." Nature 423, 725-727 (2003). [48] Sanwal, D., Pavlov, G. G., Zavlin, V. E. and Teter, M. A., "Discovery of absorption features in the X-ray spectrum of an isolated neutron star." Ap. J. 574, L61-L64 (2002) and references therein. [49] Becker, W., Swartz, D. A., Pavlov, G. G., Elsner, R. F., Grindlay, J., Mignani, R., Tennant, A. F., Backer, D., Pulone, L., Testa, V. and Weisskopf, M. C., "Chandra X-ray observatory observations of the globular cluster M28 and its millisecond pulsar B182124." Ap. J. 594, 798-811 (2003).

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[50] Manchester, R. N., Lyne, A. G., Camilo, F., Bell, J. F., Kaspi, V. M., D’Amico, N, McKay, N. P. F., Crawford, F., Stairs, I. H., Possenti, A., Kramer, M. and Sheppard, D. C., "The Parkes multibeam pulsar survey – I. Observing and data analysis systems, discovery and timing of 100 pulsars." Mon. Not. R. Astron. Soc. 328, 17-35 (2001). [51] Camilio, F., Kaspi, V. M., Lyne, A. G., Manchester, R. N., Bell, J. F., D’Amico, N., McKay, N. P. F. and Crawford, F. "Discovery of two high-magnetic-field radio pulsars." Ap. J. 541, 367-373 (2000). [52] Lyne, A. G., Pritchard, R. S., Graham-Smith, F. and Camilio, F. "Very low braking index for the Vela pulsar." Nature 381, 497-498 (1996). [53] Cusumano, G., Massaro, E. and Mineo, T. "Timing noise, glitches and the braking index of PSR B0540-69." Astron. Astrophys. 402, 647-652 (2003). [54] Biemond, J., "Are electromagnetic phenomena derivable from extended Einstein equations?", Physics/0104009.

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Chapter 5

PALEOSHORELINES AND THE EVOLUTION OF THE LITHOSPHERE OF MARS Javier Ruiz Departamento de Geodinámica, Facultad de Ciencias Geológicas, Universidad Complutense de Madrid, 28040 Madrid, Spain

Rosa Tejero Departamento de Geodinámica, Facultad de Ciencias Geológicas, Universidad Complutense de Madrid, 28040 Madrid, Spain

David Gómez-Ortiz

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Escuela Superior de Ciencias Experimentales y Técnicas-Área de Geología, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain

Valle López Servicio de Sistemas de Información Geográfica, Instituto Geológico y Minero de España, Ríos Rosas 23, 28003 Madrid, Spain

ABSTRACT The existence of features indicative of shorelines of ancient oceans on Mars has been proposed for several authors. In this chapter we revise the topography of possible Martian paleoshorelines, and their consequences for the amount of water infilling ocean basins when water load is considered. We show that a re-evaluation of paleoshorelines is need. For example, the putative Meridiani shoreline could be the same feature as some portions of the Arabia shoreline. Indeed, elevations in the Meridiani shoreline are roughly similar to that of the Arabia shoreline in northeast Arabia, Utopia (not taken into account the Isidis basin), Elysium, and Amazonis regions. This is still far of an equipotential surface, but a paleoshoreline through these regions and the Meridiani shoreline would be better candidate to represent a paleoequipotential surface than the Arabia shoreline sensu strito. Moreover, the elevation of the Arabia shoreline in northern Arabia Terra after is intriguingly close to the mean elevation of the Deuteronilus shoreline, and it cannot be discarded a “mixed” Arabia/Deuteronilus shoreline, which would include the Arabia shoreline in northern Arabia Terra, and the Deuteronilus shoreline elsewhere.

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Javier Ruiz, Rosa Tejero, David Gómez-Ortiz et al. If reality of global shorelines is accepted, as increasing evidence suggests, then present-day topographic variations in these features postdate shorelines formation. So, their topographic range should provide information on large-scale vertical movement of the lithosphere, which in turn provides information on the thermal evolution of Mars. We describe the application of thermal isostasy concept to constraint the ancient thermal state of the lithosphere from present-day paleoshoreline topography. For the ~1.1 km total elevation range of the Deuteronilus shoreline, the relative amplitude of heat flow variations (the ratio between maximum and minimum heat flow) is ≤1.6. This value is clearly lower than that presently observed on continental areas on the Earth. If heat flow variations on Mars are currently greatly disappeared, then the obtained heat flow variations upper limits must be mostly related to the paleoshoreline formation time: the present-day elevation range along Deuteronilus shoreline suggests that differences in the thermal state of the lithosphere in regions along this putative paleoshoreline have been relatively small since the feature was formed, and therefore the absence of lithospheric tectonothermal events by the latest ~3 Gyr, at least. If the Deuteronilus shoreline is a combination of portions of several paleoshorelines, then the total elevation range, and the implied heat flow variations, would be lower, and the lithosphere stability higher.

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INTRODUCTION A plenty of studies about the characteristics, thermal state, and evolution of the lithosphere of Mars has been previously performed from several lines of work. On a hand, numerous efforts have focused on the mechanical properties of the lithosphere, which serve to calculate its effective elastic thickness and thermal structure (e.g., Comer et al., 1985; Solomon and Head, 1990; Anderson and Grimm, 1998; Zuber et al., 2000; Nimmo, 2002; McGovern et al., 2002, 2004; McKenzie et al., 2002; Kieffer, 2004; Ruiz et al., 2006), and the topography and geometry or greath faults, and hence the brittleductile transition depth (e.g., Schultz and Lin, 2001; Schultz and Watters, 2001; Vidal et al., 2005). These studies inform about the thermal structure and heat flow in the time when the structures were formed. On the other hand, thermal history models make predictions about the evolution of surface heat flow and lithospheric thickness, or even crustal growth (e.g., Stevenson et al., 1983; Schubert and Spohn, 1990; Schubert et al., 1992; Grasset and Parmentier, 1998; Nimmo and Stevenson, 2000; Choblet and Sotin, 2001; Spohn et al., 2001; Hauck and Phillips, 2002; Breuer and Spohn, 2003). Obviously, results from thermal history models should be consistent with constraints imposed by the studies of the mechanical properties of the lithosphere. A third line of information about the evolution of the Martian lithosphere is the analysis of present-day topography of features interpreted as paleoshorelines. Indeed, the presence of features indicative of shorelines of ancient oceans on Mars has been proposed (Parker et al., 1989, 1993; Edgett and Parker, 1997; Clifford and Parker, 2001). If reality of global paleoshorelines is accepted, then present-day topographic variations in these features postdate shorelines formation. So, their topographic range should provide information on large-scale vertical movement of the lithosphere, which in turn would provide information on the thermal evolution of Mars (Ruiz, 2003): total elevation differences along paleoshorelines impose constraints to the differences in the evolution of the thermal structure of the lithosphere in shoreline-crossed regions. In fact, it is possible to make an approximate calculation of the

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amplitude of the ancient heat flow variations necessary to compensate, through thermal isostasy, elevation differences, and transform the paleoshorelines into equipotential surfaces. Like other geological processes could have produced vertical movements, the results so obtained suppose an upper limit. Diverse efforts have been carried out to test reality of the proposed paleoshorelines. Specifically targeted MOC images have been interpreted as not supporting the shoreline hypothesis (Malin and Edgett, 1999, 2001), although these results have been disputed (Parker et al., 2001; Clifford and Parker, 2001; Fairén et al., 2003). Moreover, recent works using high resolution images have found clear evidences of erosion in places located in putative paleoshorelines (Webb and McGill, 2003; Webb, 2004), which support its formation in relation to coastal processes. The observed present-day Martian topography (Smith et al., 1999, 2001) has also been used to analyze elevations along of the main proposed paleoshorelines, in order to test their chance to represent true paleoshorelines (Head et al., 1998, 1999; Carr and Head, 2003, Webb, 2004). These analyses obtain that the Late Hesperian Deuteronilus shoreline is a viable paleoshoreline, since this feature slightly deviates from an equipotential surface. Otherwise, the putative older and higher-standing Arabia shoreline deviates substantially from an equipotential surface, indicating that it may not be representative of a true shoreline. The evaluation of possible paleoshorelines through assessment of present-day topography must be made cautiously, because it is not necessarily true that a paleoequipotential surface must fit well a present-day equipotential surface. Lithosphere rebound due to water unloading associated with the disappearance of an ocean with irregularly shaped margins could result in deviations of equipotentiality of up to several hundreds of meters (Leverington et al., 2003; Leverington and Ghent, 2004). Different thermal histories among regions may have appreciably contributed to the deformation of the original large wavelength topography of putative paleoshorelines: variations in Martian heat flow, similar in relative amplitude to those observed in terrestrial continental tectonothermally stable areas, could result in large wavelength elevation differences of kilometric scale through differential thermal isostasy (Ruiz, 2003, Ruiz et al., 2003, 2004), an important amount of deformation for any possible paleoshoreline. Tectonic processes could also produce vertical movements, and erosional or sedimentary activity could affect the original paleoshoreline signatures (Clifford and Parker, 2001). Moreover, lateral continuity of paleoshorelines is not well established, and diverse division and mixing of the originally proposed features are likely required (Ruiz et al, 2003; Webb, 2004; Ruiz, 2005). In this chapter we revise and re-evaluate the topography of possible Martian paleoshorelines, and the consequences of the water load for the amount of water infilling ocean basins. We also describe the application of thermal isostasy concept to constraint the ancient thermal state of the lithosphere from present-day paleoshoreline topography, and we present the results obtained and their implications for the evolution of the lithosphere of Mars.

PALEOSHORELINES In this section we first revise elevation ranges in the paleoshorelines originally proposed by Parker et al. (1989, 1993) and Edgett and Parker (1997), which were revised, redrawn and

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renamed (as Deuteronilus, Arabia and Meridiani) by Clifford and Parker (2001) (Figure 1 shows a map of Mars with some of the geographical features mentioned in the text, and Figure 2 shows the paleoshorelines after Clifford and Parker (2001) represented on the Martian topography). Later, we use MOLA topography (Smith et al., 1999, 2001) to refine calculations of the water volume infilling the ancient oceanic basins, by taking into account both present-day topography and the maximum possible effect caused by the water weight on the oceanic floors topography. Finally, we re-evaluate these features through of comparing their respective elevations (geomorphologic revisions or re-evaluations are beyond the scope of this chapter), using the mapping of Clifford and Parker (2001) and MOLA topography. In this point it is necessary recall that Martian elevations are given with respect to an arbitrary zero elevation level, defined as the equipotential surface (gravitational + rotational) whose average value at the equator is equal to the mean radius (see Smith et al., 2001). Table 1. Martian chronology after Hartmann and Neukum (2001) and Hartmann (2005) Period Late Amazonian

Age (Gyr)

Middle Amazonian

0.2-0.6

Early Amazonian Late Hesperian Early Hesperian

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Late Noachian Middle Noachian Early Noachian

1.4-2.1 2.9-3.2 3.2-3.6 3.5-3.7 3.6-3.9 3.8-4.1

Figure 1. Map of Mars with some of the geographical features mentioned in the text, and other representative features indicated on the Martian topography. Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

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The better evidence for a Martian paleoshoreline (dating from the Late Hesperian; temporal equivalences, as obtained from crater counts statistical, are given in Table 1) is the presence in the northern lowlands of a mapping contact (originally named Contact 2) marking the outer boundary of the northern plains, which was interpreted to be the shoreline of an ancient Martian ocean (Parker et al., 1989, 1993). This “contact” was later redrawn and renamed Deuteronilus shoreline by Clifford and Parker (2001). Head et al. (1998, 1999) and Carr and Head (2003), using MOLA data, have shown that this putative shoreline represents a relatively good approximation to an equipotential surface: its mean altitude is −3.792 ± 0.236 km, and its whole topographic range is ~1.1 km, from −3.2 to −4.3 km (Carr and Head, 2003). The elevation of the base levels of the Chryse outflow channels, −3.742 ± 0.153 km (Ivanov and Head, 2001), is close to the mean level of Deuteronilus shoreline, could also indicate that they debouched into a large standing body of water (Head et al., 1999; Ivanov and Head, 2001); this seem also be the case for other channels termini (Salamuniccar, 2004). Additionally, there is clear evidence for erosion along the Deuteronilus shoreline (Webb, 2004). Alternatively (or complementarily), Carr and Head (2003) considered that the Late Hesperian Vastitas Borealis Formation, which extends for a great part of the northern lowlands, represents better support for the past existence of a large standing body of water on Mars. The Vastitas Borealis Formation has been interpreted as a sedimentary veneer at least 100 m thick on the East Hesperian ridged plains (Head et al., 2002), which could have originated as a sublimation residue from a large (probably frozen) water body (Kreslavsky and Head, 2002). The outer contact of the Vastitas Borealis Formation is coincident with the trace of the Deuteronilus shoreline in the Deuteronilus, Nilosyrtis, Isidis, Tempe, and Chryse regions, but not in Elysium or the Olympus Mons aureole. If the outer contact of the Vastitas Borealis Formation in the Utopia basin is ignored (where it is covered by younger Amazonian Elysium materials, and therefore, the original contact trace is not visible), the outer contact of the Vastitas Borealis Formation has a mean altitude of −3.658 ± 0.282 km, with a whole elevation range of ~1.0 km, from −3.3 to −4.3 km (Carr and Head, 2003; see their Figure 12). 60 Arcadia Planitia

Utopia Planitia

Acidalia Planitia

Alba Patera

Cassini

Isidis Basin

Mo nt es

Elysium Mons

Olympus Mons

0

Arabia Terra

Th ar sis

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Elevations Range along Deuteronilus, Arabia, and Meridiani Shorelines

0

60

120

180

240

300

360

Figure 2. The Deuteronilus (yellow), Arabia (green) and Meridiani (red) shorelines after Clifford and Parker (2001), represented on the Martian topography.

Parker et al. (1989, 1993) also proposed an older, higher-standing Contact 1, later on renamed Arabia shoreline (Clifford and Parker, 2001). This shoreline, which would be of Noachian age (see Clifford and Parker, 2001), is roughly coincident with the Martian dichotomy separating the lowlands from the highlands, and the elevation along its outline

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Figure 3. Constant elevation contours in km (black) crossed by the Meridiani shoreline (red). Elevation contours are spaced 0.5 km. 25

20

Basin volume (km-3)

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highly deviates from an equipotential surface (Head et al., 1998, 1999), and thus it is not a good candidate to paleoshoreline. The topography along the Arabia shoreline is characterized (Carr and Head, 2003) by a mean altitude of −2.090 ± 1.400 km, and a total elevation range of ~5.6 km (from 1.6 to −4.0 km). Finally, the existence of a pre-Arabia shoreline has been proposed in northern Sinus Meridiani and western Arabia Terra (Edgett and Parker, 1997; Clifford and Parker, 2001); precisely, the MER Opportunity has recently found evidences for an aqueous, maybe searelated, environment at Meridiani Planum, close to this possible paleoshoreline (Squyres et al., 2004). The mean elevation of this Meridiani shoreline (as named by Clifford and Parker, 2001), would be about −1.5 km (Parker et al., 2000), although its topography has not been examined in previous works. Figure 3 shows constant elevation contours (0.5 km spaced) crossed by the Meridiani shoreline. It can be seen that elevations along the mapped paleoshoreline mostly range between 0 and −2 km. If Hesperian chaos materials (see Tanaka et al., 1992), impact craters, and an isolated peak are not taken into account (see also Figure 6), then total elevation range is ~1 km, from −0.5 to −1.5 km, a reasonable amount for a very old paleoshoreline.

water loaded topography

15 present-day topography

10

5

0 -4.5

-3.5

-2.5

-1.5

-0.5

Elevation (km)

Figure 4. Basin volume as a function of the elevation level, for both, present-day topography and water loaded, Airy compensated, topography (below the -4.35 km level volume includes the contribution from both North Polar and Utopia basins). Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

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Water Volumes in the Ancient Oceans MOLA topography and mean elevations of proposed paleoshorelines has been previously used to calculate the water volumes contained in the basin related to these putative coastal limits (Head et al., 1998, 1999; Carr and Head, 2003; Öner et al., 2004). Similar calculations are presented in Table 2, and Figure 4 shows basin volume as a function of the elevation level. For the Meridiani shoreline Results in Table 2 are very lower than that in Carr and Head (2003) due to the fact that these authors used a value of 0 for the mean paleoshoreline elevation, whereas we use a mean elevation of −1.5 km following Parker et al. (2000). Table 2. Basin characteristics at the mean elevations of the proposed Deuteronilus, Arabia and Meridiani shorelines. North polar cap contribution is included in every case.

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Deuteronilus Arabia Meridiani shoreline shoreline shoreline Mean elevation (km) a −3.792 −2.090 −1.5 Basin area enclosed 2.47 4.66 5.34 1.93 8.66 11.88 Present-day basin volume b (107 km3) Total water volume c,d (107 km3) 2.00-2.80 8.77-12.28 11.99-16.79 GEL (km) c,d 0.14-0.19 0.61-0.85 0.83-1.16 Mean depth (km) c,d 0.81-1.13 1.88-2.63 2.25-3.14 1.46-2.04 3.16-4.42 3.75-5.25 Maximum depth (km) c a Mean elevations for Deuteronilus and Arabia shoreline after Carr and Head (2003), and for Meridiani Shoreline after Parker et al. (2000), b North polar cap contribution not included. c Lower limit: present-day topography; Upper limit: water loaded, Airy compensated, topography. d North polar cap contribution included.

These calculations assume that the topography of the northern plains has changed little since the putative shorelines were formed. Irrespective of the accuracy of this assumption, estimations based on observed present-day topography can only provide lower limits to the basins volume. Indeed, the existence of oceans in the northern lowlands would imply that the water column provided an additional load over the lithosphere in the regions covered by water (Leverington et al., 2003; Leverington and Ghent, 2004). The effects of variations of the water load on topography are well known for Earth (for a review see Watts, 2001). Therefore, the weight of the water column would have produced the subsidence of the sea floor during the possible periods in which an ocean occupied the lowlands, increasing the basins volume; the subsequent desiccation of the ocean would results in the opposite effect (Hiesinger and Head, 2000; Thomson and Head, 2001; Kreslavsky and Head, 2002; Leverington et al., 2003; Leverington and Ghent, 2004). We take into account the effect of water load by assuming Airy compensation to calculate upper limits to the basins volume for Meridiani, Arabia and Deuteronilus shoreline. To assume Airy compensation achieved in the mantle implies that elevation variation in the sea floor due to changes in the height of the overlying water column is

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Δy floor =

yw ρ w ρm

,

where yw is the height of the water column (i.e., the ocean depth), and ρw and ρm are the ocean and mantle densities, respectively. If yw is taken as the depth of an ancient ocean, and y is the altitude difference between the ocean basin floor for the empty basin and the sea level when the basin was filled, obviously Δy floor = yw − y, and the isostasy principle requires

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yw ρm = y (ρ m − ρ w )

.

So, if y is taken as the present-day depth of the basin under the sea level, then this implies that estimations for the water volume enclosed in the Martian oceans should be increase by a factor yw/y (with higher ocean density and lower mantle density increasing this factor). If we assume ρw = 1000 kg m−3 and ρm = 3500 kg m−3, it is obtained yw/y = 1.4. Figure 4 shows basins volume as a function of the elevation level for water loaded, Airy-compensated, topography. It is important to note that, although water load of putative ancient oceans would result in a substantial increase in the basin volume with respect to calculations based on present-day topography, the assumption of Airy isostasy would imply that the lithosphere has not rigidity, and therefore the increasing factor yw/y is an upper limit. So, values presented in Table 2 for water volume, GEL (the Global Equivalent Layer if the water is homogeneously distributed on the surface of Mars), mean depth and maximum depth are given as intervals between estimated results for present-day topography and results for water load compensated by Airy isostasy. In the calculations of total water volumes the north polar cap volume under the appropriate mean shoreline elevation is taken into account (the total volume of the north polar cap is 1.14 × 106 km3; Smith et al., 2001), since this contributes to the present-day topography. Lithospheric flexure due to north polar cap loading is not considered here, since that the assumption of not flexure represents the lower limit for north polar contribution to the basins volumes, and upper limits are here calculate assuming Airy compensation.

Re-evaluation of Paleoshorelines The original global mapping of the putative paleoshorelines was limited by resolution of Viking images. Besides this, it is fairly evident that diverse degradational processes could have affected the original morphology (and topography) of any putative paleoshoreline. Thus, reevaluations of paleoshorelines mapping are probably guaranteed. These revaluations would be of great interest to improve the knowledge the hydrogeological history, but also for the tectonothermal history of Mars, because they could affect the elevation range attributed to a given paleoshoreline, and hence the information derived from elevation ranges.

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The possibility that the putative Meridiani shoreline could be the same feature as some portions of the Arabia shoreline was first suggested by Ruiz et al. (2003) related to preliminary work about thermal isostasy applied to Mars. Indeed, the mean elevation in the Meridiani shoreline (−1.5 km following Parker et al. (2000)) is roughly similar to that of the Arabia shoreline in northeastern Arabia, Utopia (not taken into account the Isidis impact basin), Elysium, and Amazonis regions. Thus, a possible paleoshoreline might follow the outline of the Arabia shoreline in these regions, but including the outline of Meridiani Shoreline in western Arabia Terra and Sinus Meridiani. The elevation range of this “mixed” Meridiani/Arabia shoreline, although not examined, would be mostly about 2 km, from −1 to −3 km after Ruiz et al. (2004) on the basis of the topography analysis of Arabia shoreline in Carr and Head (2003). This is still far of an equipotential surface, but this Meridiani/Arabia shoreline would be better candidate to represent a paleoequipotential surface than the Arabia shoreline sensu strito: for that reason, it was incorporated to the hypothesis for the Martian hydrogeological history suggested by Fairén et al. (2003), in order to represent the boundary of a putative Noachian ocean.

Figure 5. Meridiani (red) and Arabia (green) shorelines, and constant elevation contours in km (black) crossed by the Meridiani shoreline, which are spaced each 0.5 km..

Figure 5 shows Meridiani and Arabia shorelines, as mapped by Clifford and Parker (2001), and constant elevation contours (0.5 km spaced) crossed by the Meridiani shoreline. It can be seen that elevations of the Arabia shoreline at northeastern Arabia, Utopia, and Elysium regions are similar to these along the Meridiani Shoreline, which support the “mixed” Meridiani/Arabia shoreline as a true paleoshoreline. Also is evident that elevations in the Arabia shoreline are much lower in northwestern Arabia Terra, as well as further to the west. Figure 6a shows Meridiani and Arabia shorelines and the contour of the −1.5 km elevation level (the mean level of the Meridiani shoreline after Parker et al. (2000)) superimposed on MOLA topography. The Arabia shoreline at northeastern Arabia, Utopia, Elysium, and Amazonis regions is well close to the −1.5 km elevation level at the majority of places, further supporting the “mixed” Meridiani/Arabia shoreline as a true paleoshoreline. Figure 6b is similar, but showing the contour of the −2.09 kn elevation, corresponding to the mean elevation of the Arabia shoreline sensu strito: the Arabia shoreline at Arabia, Utopia, Elysium, and Amazonis regions is at, or generally above, the −2.09 kn elevation level (with

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Javier Ruiz, R Rosa Tejjero, David Gómez-Ortiz ett al.

thhe exception of o Isidis impaact basin, whhich probably postdates paaleoshoreline formation). f Thus, the who ole topographiic range in thhe Meridiani/A Arabia shorelline would bee ~1.6 km, beetween −0.5 and −2.1 km m. Moreover, if the Arabiaa shoreline seensu strito is not a true paaleoshoreline, then areas, volumes, v meann depths and GELs G obtainedd for this “feaature” from M MOLA topograaphy are not representative r e of any Martiian oceanic sttage, but estim mations for thhe −1.5 km eleevation level would w be roughly appropriatte for the Meridiani/Arabia shoreline.

a

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b

c Fiigure 6. The Deuteronilus (yellow), Arabia (green) ( and Meeridiani (red) shhorelines after Clifford C and Paarker (2001), reepresented on the t Martian toppography (scalee in km). Also represented r (bllack) are the coontour of the (a)) -1.5 km, (b) -22.09 km, and (cc) -3.792 km eleevation levels.

The elevattions along puutative shorellines on northhern Arabia Terra T has beeen recently annalyzed in hig gher resolutioon (Webb, 20004), finding a elevation of 3707 ± 21 m, for the A Arabia shorelin ne, and two different d elevvations, 4000 ± 14 m and 4200 ± 12 m, m for two seeparate portion ns of the Deuuteronilus shorreline, which could therefore represent tw wo distinct shhorelines. Eleevation differrence betweenn the two seeparate portioons of the Deuteronilus

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shoreline is difficultly due to post-formation processes, because elevation in the two portions is nearly constant along distances of about 500 km, and to the clear bimodality in the elevation values. On the other hand, the elevation of the Arabia shoreline in northern Arabia Terra after (Webb, 2004) is intriguingly close to the mean elevation of the Deuteronilus shoreline, and it the possibility of a “mixed” Arabia/Deuteronilus shoreline, which would include the Arabia shoreline in northern Arabia Terra and the Deuteronilus shoreline elsewhere, has been mentioned (Ruiz, 2005). Figure 6c shows the Arabia and Deuteronilus shorelines, and the contour of the −3.792 km elevation level, the mean elevation of the Deuteronilus shoreline, superimposed on MOLA topography. It is obvious that the Arabia shoreline is very close to −3.792 km elevation level at northwestern Arabia Terra and northeastern Tempe Terra. In turn, Deuteronilus shoreline elevation at Tempe Terra is close to −4 km (see also Carr and Head, 2003), a similar elevation to that found for this feature by Webb (2004) for northern Arabia Terra. Thus, locally at northern Arabia Terra and northeastern Tempe Terra putative shorelines fit well equipotential surfaces, but they are suggesting a complex scenario for the possible evolution of Martian oceans.

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Elevation Ranges along Paleoshorelines and Vertical Movements of the Lithosphere Implications of paleoshoreline reevaluations are evident: the lower the true elevation range of a paleoshoreline the lower the magnitude of vertical movements postdating its formation. For example, elevation range of 1.1 km along the Deuteronilus shoreline is an upper limit, because this range probably includes portions of several different paleoshorelines. Vertical movements deduced from the differences of this feature with respect to an equipotential surface should also be an upper limit. The fit, although rough, of paleoshorelines to equipotential surfaces would imply a relatively calmed history for the Martian lithosphere, at least since these features were formed. This is clearly the case for the Deuteronilus shoreline sensu strito, if this feature is a true paleoshoreline, and moreover for the “revised” Deuteronilus shorelines, which are dating from the Late Hesperian, ~3 Gyr ago (see Table 1). The evidences for a reasonably equipotential Noachian paleoshoreline are attractive, but they must be taken more carefully. If the Meridiani/Arabia shoreline represents a true paleoshoreline, then its whole elevation range of 1.6 km would imply a quite stable lithosphere since 3.5 Gyr ago at least. By contrast, Gratton et al. (2003) find that the Meridiani, Arabia and Deuteronilus shorelines could fit equipotential surfaces if the reference ellipsoid for the planet has changed with the time, maybe due to the dichotomy formation or Tharsis volcanism. Although this possibility is attractive, it is based on the interpretation of the Meridiani, Arabia, and Deuteronilus sensu strito shorelines as true paleoshorelines, which is unlikely (see above).

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THERMAL ISOSTASY AND THERMAL EVOLUTION OF THE LITHOSPHERE A significant relation exists between Earth’s surface elevation and the thermal state of the lithosphere: the warmer the lithosphere, the lower its mean density, and the higher its buoyancy with respect to the underlying fluid materials. This principle (known as thermal isostasy) has been broadly applied to the thermal subsidence of the cooling oceanic lithosphere (e.g., Turcotte and Schubert, 2002), but it can be also applied in a general way to the continental lithosphere (Lachenbruch and Morgan, 1990), including a tectonothermally stable one. This allows the use of topography and surface heat flow data to constrain Earth’s continental lithospheric thermal structure (e.g., Tejero and Ruiz, 2002; Lewis et al., 2003). This relation between surface elevation and thermal state of the lithosphere has been also applied to Mars (Ruiz, 2003; Ruiz et al., 2003, 2004). In this section we present a complete description of the application to Mars of the thermal isostasy concept. We also revise and extend the results obtained about the evolution of the Martian lithosphere.

Thermal Isostasy Because of thermal expansion and contraction, the elevation of the surface, referenced to the free height of the asthenosphere, depends on the thermal state of the lithosphere and has contributions from the lithospheric mantle and crust,

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H = Hm + Hc,

(1)

where Hm and Hc are the lithospheric mantle and crust contribution to the elevation of the surface respectively. (The term lithosphere is used here to define a thermally conductive layer, in which base isostatic compensation can be achieved.) The contribution due to the lithospheric mantle is given by (Lachenbruch and Morgan, 1990)

H m = α(Tm − Ta )bm ,

(2)

where α is the volumetric thermal expansion coefficient, Ta is the temperature of the asthenosphere, Tm is the mean temperature of the lithospheric mantle, and bm is the lithospheric mantle thickness; density differences between asthenospheric and lithospheric mantle are taken as solely due to temperature differences, which is a very reasonable approximation. A similar equation can be written to describe the crustal contribution, but taking into account a correction factor for the lesser crustal density,

Hc =

αρ c (Tc − T )bc ρa

,

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(3)

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where ρc and ρa are respectively a reference crustal density and the density of the asthenosphere, and bc is the crustal thickness. In turn, the mean temperatures of each lithospheric layer is given by

T =

1 z 2 − z1

∫

z2

T ( z )dz

z1

. (4) where z, z1 and z2 are the depth, the depth in the layer top, and the depth at the layer base, respectively. The component of the topography due to thermal isostasy, expressed in terms of heat flow, is (Ruiz, 2003) h = Hm (Fh) + Hc (Fh) – Hm (Fo) – Hc (Fo) ,

(5)

where h is the local elevation with respect to a reference elevation, Fh is the local surface heat flow, and Fo is the heat flow for the reference elevation.

Temperature Profiles Temperature profiles in this chapter are calculated by assuming radioactive heat sources homogeneously distributed in the crust, and linear thermal gradients for the lithospheric mantle mantle. The temperature at a depth z within the crust is given by (Roy et al., 1968)

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Fz Az 2 Tz = Ts + − kc 2k c ,

(6)

where Ts is the surface temperature, F is the surface heat flow, kc is the thermal conductivity of the crust, and A is the volumetric heat production rate. Ruiz (2003) calculate

T

bm, Tm , and c in terms of the surface heat flow, and the proportion f of the heat flow originated from crustal heat sources, assuming that crustal heat sources are homogeneously distributed. The factor f can be formally defined as

f =

Abc F ;

(7)

so, within the crust, the temperature at a depth z is

Tz = Ts +

Fz ⎛ fz ⎜⎜1 − k c ⎝ 2bc

⎞ ⎟⎟ ⎠.

(8)

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layer, and within the lithospheric mantle, the heat flow can be assumed constant (e.g., Turcotte and Schubert, 2002); so, within the mantle lithosphere, the temperature at a depth z is

F (1 − f )( z − bc ) km

Tz = Tc +

, (9) where Tc is the temperature at the crust base, calculated taking z = bc in equation (8); in turn, bm is calculated from

bm = bc +

k m (Ta − Tc ) F (1 − f ) .

(10)

Mean lithospheric mantle and crust temperatures are respectively given by

Tm =

Ta + Tc 2 ,

(11)

and

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Tc = Ts +

Fbc (1 − f / 3) 2k c

.

(12)

Parameter Values The calculations have been performed using α = 3 × 10-5 ºC-1, kc = 2.5 W m-1 ºC-1, km = 3.5 W m-1 ºC-1, ρc = 2900 kg m-3, and ρa = 3500 kg m-3 for material properties. The surface temperature is taken as 0ºC, maybe more appropriate for a time in which an ocean is assumed than the current mean surface temperature of about −50ºC (although the results are relatively insensitive to the election of this parameter). The asthenosphere temperature is taken as 1300ºC, which is a value typically used for the Earth’s asthenosphere (e.g., Ranalli, 1997). Crustal thickness is assumed to be 40 km, in accordance with the typical mean crustal thickness below the northern lowlands derived from topography and gravity data (Zuber et al., 2000). Two possibilities have been taken for the value of f in equation (8), although we note that this value could locally vary: f = 0 (corresponding to a linear thermal gradient through the crust) and f = 0.5. The latter value is in accordance with the proposal (made from geochemical arguments drawn from the materials on Mars’ surface) that perhaps over 50% (or even 75%) of radioactive heat sources in this planet are placed in its crust (McLennan, 2001, 2003); similarly, in the Earth, the 40-60% of the heat flow lost in continental areas originates from crustal heat sources (Pollack and Chapman, 1977; Turcotte and Schubert, 2002).

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Deformation of Paleoshorelines Ruiz et al. (2004) have used the thermal isostasy concept to show that it is not necessarily true that a paleoequipotential surface implies a good fit to a present-day equipotential surface. A similar conclusion has been obtained taking into account the lithosphere rebound due to water uploading associated to the disappearance of an ocean of irregularly shaped margins (Leverington et al., 2003; Leverington and Ghent, 2004). Here we present the first-order calculations of the magnitude of the relation between possible variations in the thermal state of the lithosphere and elevation differences, which could cause concrete deviations in equipotentiallity along the proposed paleoshorelines. Note that for the purposes of this chapter, only large wavelength topographic differences are relevant, since that the rigidity of the Martian lithosphere could prevent small-scale isostatic adjustment.

Surface elevation (km)

0

-2 f =0 -4

-6

f = 0.5

-8

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-10 50

40

30

20

10

Heat flow (mW m-2)

Figure 7. Elevation over the free height of the asthenosphere in terms of surface heat flow for f = 0 and f = 0.5. Surface heat flow is represented in reverse order (describing the topographic evolution of a cooling region isostatically compensated). Adapted from Ruiz et al. (2004).

Figure 7 shows H in terms of surface heat flow for f = 0 and f = 0.5. Calculations have been made for a range of F values between 10 and 50 mW m-2, which roughly correspond to the whole range of surface heat flows proposed for diverse regions and times using estimates of the elastic thickness of the lithosphere (McGovern et al., 2002, 2004). It is important to note that the f value can change along a possible paleoshoreline (for example, due to local variations in crustal heat sources, mantle heat flow, or both), or with time (due to waning of radiogenic dissipation intensity or to changes in the efficiency of convective heat transfer). In any case, these possibilities are not important for the purpose of this first-order calculation, which is show the feasibility of differential thermal isostasy histories to affect the large wavelength topography of possible Martian paleoshorelines. For the purposes of this analysis the interesting point is the relative differences of H, and not the absolute values obtained for this parameter (planetary topographies are referred to arbitrary datum). Figure 7 indicate that

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variations in thermal state of the lithosphere can result in differential thermal isostasy, which in turn can result in important elevation differences, even of kilometric scale as occur on the Earth (Lewis et al., 2003).

Relative amplitude of surface heat flow variations

2.2 2 1.8 f =0 1.6 1.4

f = 0.5

1.2 1 10

20

30

40

50

Reference heat flow (mW m-2)

a

Relative amplitude of surface heat flow variations

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1.5

1.4

1.3

f =0

1.2 f = 0.5 1.1

1 10

20

30

40

50

-2

Reference heat flow (mW m )

b Figure 8. Relative amplitude of surface heat flow variations that can produce elevation ranges of 1 (a) and 0.5 (b) km centered on the H value corresponding to a reference heat flow in the range from 10 to 50 mW m-2. As in Figure 1 surface heat flow is represented in reverse order. Adapted from Ruiz et al. (2004).

Assuming a Martian paleoshoreline, the posterior attenuation, disappearance (as is expected with the waning of internal heat sources), or formation (if reheating of the lithosphere postdating the shoreline formation occurred) of heat flow variations must result in the deformation of the paleoshoreline topography, deviating it from an equipotential surface. Figure 8 shows the relative amplitude of surface heat flow variations that can produce elevation ranges of 1 and 0.5 km centered on the H value corresponding to a reference heat flow in the range from 10 to 50 mW m-2. The relative amplitude of heat flow variations is obtained as the quotient between the maximum and minimum heat flow that can produce

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positive and negative elevations, respectively, of 0.5 and 0.25 km with respect to the reference H value. In Figure 8 it can be seen that ancient surface heat flow variations less than a factor ~2 may account for differences of elevation of 1 km. An elevation range of 0.5 km could be produced by surface heat flow variations less than a factor of ~1.5. These values are further lowered if a substantial amount of the Martian heat sources are located within the crust. On the Earth, present-day surface heat flow variations expand by more than one order of magnitude (e.g. Pollack et al., 1993). The major part of these variations is due to plate tectonics, but for Mars an early phase of plate tectonics is controversial. Variations in surface heat flow in Earth’s continental regions, from contoured maps (Cermak, 1993; Pollack et al., 1993), can be higher than a factor of 2 or 3, sufficient for important deformation of paleoshorelines (see Figure 7). Those areas include terrains of different ages, and it is known for continental areas that an inverse relation exists between surface heat flow and age of the last tectonothermal stabilization (e.g., Hamza, 1979; Vitorello and Pollack, 1980; Cermak, 1993). Moreover, continental heat flow depends also on a wide array of factors, as for example radioactive heat production in the crust, local mantle heat flow, or tectonic or erosive redistribution of crustal heat producing elements (for reviews see Beardsmore and Cull, 2001; Sandiford and McLaren, 2002). In any case, heat flow variations on old and tectonothermally stable terrestrial continental areas can be as high as a factor ~1.5-2 (e.g., Cermak, 1993; Roy and Rao, 2000; Rolandone et al., 2002). If local variations of surface heat flow of at least similar amplitude existed in Mars during any moment of its history, then these results indicate that differential thermal isostasy should result in important deformation, and deviation of equipotentiallity, along putative shorelines. Moreover, it is significant that if the half of the surface heat flow was originated from crustal heat sources (as it is the case on the Earth) when the paleoshorelines were formed, then heat flow variations lower than a modest factor of ~1.2-1.4 may account for present-day elevation ranges of 0.5-1 km (if, as it seem reasonable, these heat flow variations are currently greatly attenuated). These elevation ranges are respectively similar to the ±1 standard deviation and whole elevation ranges in the Deuteronilus shoreline, but they represent an important amount of deformation along any possible paleoshoreline. It is important remind that thermal isostasy is only a contributor to the topography. Though it is not our intentions to discuss the many other factors that may have contributed to the modification of an equipotential surface, we highlight that other possibilities may be degradation by wind, water, tectonic, and volcanic modification (e.g., Clifford and Parker, 2001; Fairén et al., 2003), rebound of the lithosphere due to dissipation of a water body (Leverington et al., 2003; Leverington and Ghent, 2004; for a review of isostatic and flexural effects related to changes in sea level see Watts, 2001), flexure (non-thermal) isostasy due to surface loading, erosion, or subsurface magmatic intrusions. In fact, endogenic-driven geologic activity (probably implying vertical movements) and exogenic activity clearly postdates the possible paleoshorelines in Arabia Terra and Tharsis and Elysium (e.g., Head et al., 1999; Anderson et al., 2001). Together, all those possibilities make more pressing the main argument in this section: any paleoequipotential surface dating of the ancient Mars must be importantly deformed at present, even in a range of elevations of a kilometric scale.

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If the existence of paleoshorelines is accepted, then elevation differences along these paleoshorelines impose constraints to the differences in the evolution of the thermal structure of the lithosphere in shoreline-crossed regions (Ruiz, 2003). Indeed, it is possible to calculate the amplitude of heat flow variations necessary to compensate present-day topography and transform a paleoshoreline into an equipotential surface. If surface heat flow variations on Mars are currently almost disappeared (as it would expectable for an efficiently cooled planet), then the heat flow variations deduced from shoreline topography must be mostly related to the time when this feature was formed. Like other than thermal isostasy processes could have produced vertical movements in the shoreline-crossed regions, the results obtained in this way suppose an upper limit to the amplitude of heat flow variations. Diverse processes, including geomorphologic evolution, can affect small-scale topographic variations, and the rigidity of the Martian lithosphere could also prevent small-scale isostatic adjustment; therefore, large wavelength topographic variations, in which isostatic adjustment can work, are more relevant again for the purposes of this argument. The calculations have been performed for the Deuteronilus shoreline, but the argument and results are also valid for the outer contact of the Vastitas Borealis Formation (if this is considered to be an ancient oceanic limit) since the ranges of elevations are similar in both features. So, h is taken as ±0.55 km, for a total elevation range of 1.1 km for the Deuteronilus shoreline. This corresponds to a reference elevation (for which h = 0 and Fh = Fo) of –3.75 km, close to the mean elevation of the Deuteronilus shoreline and to the mean elevation of the termini of the Chryse outflow channels. Like elevation differences along the shoreline must be compensated by heat flow variations, h should have sign minus in equation (5) for mathematical consistency. The Fo value is not known, and for that reason, the calculations have been performed for a range of Fo values between 15 and 35 mW m-2, a range based on estimates (uncertainty included) of the late Hesperian/early Amazonian elastic lithosphere thickness (Zuber et al., 2000; McGovern et al., 2002, 2004). 50

Surface heat flow (mW m-2)

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Ancient Heat Flow Variations

h = −0.55 km 40

30

h = 0.55 km

20

10 15

20

25

30

35

Reference surface heat flow (mW m-2)

Figure 9. Fh values for h = 550 m (lower curves) and h = –550 m (upper curves) in terms of Fo. Gray and black lines indicate f = 0 and f = 0.5, respectively. In each case, the difference between the upper and lower curves gives the maximum surface heat flow variations permitted assuming the Deuteronilus shoreline as a paleo-equipotential surface. Adapted from Ruiz (2003).

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Figure 9 shows Fh values for h = ±0.55 km in terms of Fo. In each case, the difference between the upper and lower curves gives the maximum heat flow variations allowed, taking into account the Deuteronilus shoreline topography. Figure 10 shows upper limits to the relative amplitude of surface heat flow variations on Deuteronilus shoreline locations; these relative amplitude upper limits are the ratio between the maximum and minimum values shown in Figure 9.

Relative amplitude of surface heat flow variations

1.8

1.6 f =0 1.4 f = 0.5 1.2

1 15

20

25

30

35

-2

Reference heat flow (mW m )

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Figure 10. Upper limits to the relative amplitude of surface heat flow variations on the Deuteronilus shoreline locations. These relative amplitude upper limits are given by the ratio between upper and lower values. Adapted from Ruiz (2003).

It can be seen in Figures 9 and 10 that variations in heat flow in regions through the Deuteronilus shoreline were small in the Late Hesperian. In fact, the obtained upper limits for the relative amplitude of these variations are, at most, a factor of 1.6. If crustal heat sources are taken into account, the magnitude of the relative amplitude variations decreases for each given value of Fo. For the Fo range used here, the depth to the 1300ºC isotherm is ~100-300 km for f = 0 and ~200-600 km for f = 0.5. Since the small radius of Mars the calculations for the case f = 0.5 (at low heat flows values) should take in account the spherical shape of Mars. In addition, the range of Fo values used here is based on works assuming linear thermal gradients. Calculation of surface heat flows from elastic thicknesses would result in higher values if heat sources are present in the crust (Solomon and Head, 1990; Ruiz et al., 2005). This, in turn, decreases the depth to the 1300ºC isotherm and also increases the relative amplitude of variations in Fh for the case f = 0.5. As the relative amplitude of Fh variations is clearly lower in the f = 0.5 case than in the f = 0 one, the main conclusions so obtained are not altered. The upper limits for the relative amplitude of heat flow variations obtained here are clearly lower than those presently observed on Earth. On our planet, the higher heat flows are associated with sea floor spreading centers, but there is not clear evidence for a phase of plate tectonics in the Mars’s history (and, in any case, not for the late Hesperian or later on), and so, those very high heat flows are not relevant for the purposed of is chapter. As above

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mentioned, heat flow show variations in continental areas can be higher than a factor 2-3, and as high as a factor ~1.5-2 in tectonothermally stable terrestrial continental areas. Those areas include terrains of different ages, and it is known for continental areas that an inverse relation exists between surface heat flow and age of the last tectonothermal stabilization (e.g., Hamza, 1979; Vitorello and Pollack, 1980; Cermak, 1993).

Relative amplitude of surface heat flow variations

1.6 1.5 1.4

f =0

1.3 1.2

f = 0.5

1.1 1 0

0.2

0.4

0.6

0.8

1

2

Elevation range (km )

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Figure 11. Upper limits to the relative amplitude of surface heat flow variations in terms of the total elevation range of a paleoshoreline, calculated for Fo = 35 mW m-2 (the upper limits in the range of heat flows assumed for analyze the Deuteronilus shoreline case) at the H value corresponding to the central value in the elevation range.

If surface heat flow variations on Mars are currently almost disappeared, then the upper limits to the heat flows variations deduced of Deuteronilus shoreline topography are related to the time when this feature was formed (i.e., the Late Hesperian, ~3 Gyr ago; Hartmann and Neukum, 2001). In this case, the present-day elevation range along Deuteronilus shoreline suggests that differences in the thermal state of the lithosphere in the "Deuteronilus shoreline regions" have been relatively small since the feature was formed, and therefore, that very large areas of the Martian lithosphere has been tectonothermally stable since the Late Hesperian. This is consistent with near complete building of the Tharsis rise by the end of the Noachian (Phillips et al., 2001), with a significant decrease in volcanic resurfacing rates following the Hesperian’s end (e.g., Hartmann and Neukum, 2001) and with the localization of the waning Amazonian magmatic and tectonic activity at areas in Tharsis and Elysium (e.g., Anderson et al., 2001; Dohm et al., 2001; Head et al., 2001). Alternatively, reheating of the lithosphere postdating the Deuteronilus shoreline could have caused, or contributed to, the distortion of the topography. In this case, the reheating should have been maintained (at least partially) until the present time, since the dissipation of the thermal anomalies should lead to the disappearance of their effect on the topography. But, as above mentioned, Amazonian geological activity represents the waning and localized magmatic and tectonic activity on Mars, and for that reason, the obtained upper limits to the

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heat flow variations more probably refer to the thermal state of the Martian lithosphere when the Deuteronilus shoreline was formed. In summary, if the Deuteronilus shoreline (or equivalently, the outer contact of the Vastitas Borealis Formation) represents a Late Hesperian paleo-equipotential surface, then three conclusions can be deduced from these calculations. First, the relative variations in Late Hesperian surface heat flow in shoreline regions were lower than relative variations in present-day surface heat flow in continental areas on Earth. Second, if substantial amounts of radiogenic heat sources are located in the Martian crust, those relative variations are likely lower. Finally, very large areas of the Martian lithosphere have been tectonothermally stable since (at least) the Late Hesperian. If the Deuteronilus shoreline is a feature combining portions of several paleoshorelines, then the total elevation range would be lower, and the arguments above more pressing, as clearly illustrated by Figure 11. This figure shows upper limits to the relative amplitude of heat variations, for f = 0 and f = 0.5, as functions of the total elevation range of a paleoshoreline. As only upper limits for a given elevation range are represented, the calculations have been performed for a Fo value of 35 mW m-2 (the upper limits in the range of heat flows assumed for analyze the Deuteronilus shoreline case) at the H value corresponding to the central value in the elevation range, and therefore h = ±ΔH/2.

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CONCLUSION The range of elevations of the Deuteronilus shoreline is strongly indicating the great tectonothermal stability of the Martian lithosphere since the Late Hesperian, at least. If an important proportion of radioactive heat sources are located in the crust, as geochemical evidences suggest, the lithosphere should have been greatly thermally homogeneus in “Deuteronilus shoreline” regions when coastal processes creating this paleoshoreline were working. There are evidences suggesting that the lateral continuity of the originally proposed paleoshorelines is not well established: diverse division and mixing of the originally proposed paleoshorelines seem be required. These revaluations are of great interest for the understanding of the evolution of the Martian lithosphere, because they would modify the elevation range attributed to a given paleoshoreline. Thus, it is necessary a careful geomorphologic reassessment of the diverse features interpreted as paleoshorelines and of the relation among them. The Deuteronilus shoreline (sensu strito) seems integrate portions of several separate paleoshorelines, which would imply a still higher thermal stability and homogeneity of the lithosphere by the latest ~3 Gyr. The confirmation of the Meridiani/Arabia shoreline would be indicative of a longer lithospheric stability.

ACKNOWLEDGMENTS The authors thank Frank Columbus for your invitation to prepare this chapter. JR was supported by a grant of the Spanish Secretaría de Estado de Educación y Universidades.

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REFERENCES Anderson, S.; Grimm, R. E. J. Geophys. Res. 1998, 103, 11,113-11,124. Anderson, R. C.; Dohm, J. M.; Golombek, M. P.; Haldemann, A. F. C.; Franklin, B. J.; Tanaka, K. L.; Lias, J.; Peer, B. J. Geophys. Res. 2001, 106, 20,563-20,585. Beardsmore, G. R.; Cull, J. P. Crustal Heat Flow. A Guide to Measurement and Modelling; University Press: Cambridge, UK, 2001; pp 1-324. Breuer, D.; Spohn, T. J. Geophys. Res. 2003, 108 (E7), 8.1-8.13. Carr, M. H.; Head, J.W. J. Geophys. Res. 2003, 108 (E5), 8.1-8.28. Cermak, V. Phys. Earth Planet. Inter.1993, 79, 179-193. Choblet, G.; Sotin, C. Geophys. Res. Lett. 2001, 28, 3035-3038. Clifford, S. M.; Parker, T. J. Icarus 2001, 154, 40–79. Comer, R. P.; Solomon, S. C.; Head, J. W. Rev. Geophys. 1985, 23, 61-92. Dohm, J. M.; Ferris, J. C.; Baker, V. R.; Anderson, R. C.; Hare, T. M.; Strom R. G.; Barlow N. G.; Tanaka, K. L.; Klemaszewski, J. E.; Scott D. H. J. Geophys. Res. 2001, 106, 32,943-32,958. Edgett, K. S.; Parker, T. J. Geophys. Res. Lett. 1997, 24, 2897-2900. Fairén, A. G.; Dohm, J. M., Baker V. R.; de Pablo M. A.; Ruiz J.; Ferris J. C.; Anderson, R. C. Icarus 2003, 165, 53-67, 2003. Grasset, O.; Parmentier, E. M. J. Geophys. Res. 1998, 103, 18,171-18.181. Gratton, R.; Mareschal, J.; Sotin, C. Eos Trans. AGU Fall Meet. Suppl. 2003, 84(46), abstract G41B-0033. Hamza, V. M. Pure Appl. Geophys. 1979, 117, 65-74. Hartmann, W. K. Icarus 2005, 174, 294-320. Hartmann, W. K.; Neukum, G. Space Sci. Rev. 2001, 96, 165-194. Hauck, S. A.; Phillips, R. J. J. Geophys. Res. 2002, 107 (E7), 6.1-6.19. Head, J. W.; Kreslavsky, M.; Hiesinger, H.; Ivanov, M. A.; Pratt, S.; Seibert, N.; Smith, D. E.; Zuber, M. T. Geophys. Res. Lett. 1998, 25, 4401-4404. Head, J. W.; Hiesinger, H.; Ivanov, M. A.; Kreslavsky, M. A.; Pratt, S.; Thomson, B. J. Science 1999, 286, 2134-2137. Head, J. W.; Greeley R.; Golombek, M. P.; Hartmann, W. K.; Hauer, E.; Jaumann, R.; Masson, P.; Neukum, G.; Nyquist, L. E.; Carr, M. H. Space Sci. Rev. 2001, 96, 263-292. Head, J. W.; Kreslavsky, M. A.; Pratt, S. J. Geophys. Res. 2002, 106 (E1), 3.1-3.29. Hiesinger, H.; Head, J. W. J. Geophys. Res. 2000, 105, 11,999-12,022. Ivanov, M. A.; Head, J. W. J. Geophys. Res., 106, 3275-3295, 2001. Kiefer, W. S. Earth Planet. Sci. Lett. 2004, 222, 349-361. Kreslavsky, M. A.; Head, J. W. J. Geophys. Res. 2002, 107 (E12), 4.1-4.25. Lachenbruch, A. H.; Morgan, P. Tectonophysics 1990, 174, 39–62. Leverington, D. W.; Ghent R. R. J. Geophys. Res. 2004, 109 (E01005), 1-10. Leverington, D. W.; Ghent, R. R.; Irwin R. P.; Craddock R. A.; Maxwell T. A. Lunar Planet. Sci. 2003, 34, abstract 1282. Lewis, T. J.; Hyndman, R. D.; Flück P. J. Geophys. Res. 2003, 108 (B6), 16.1-16.18. Malin, M. C.; Edgett, K.C. Geophys. Res. Lett. 1999, 26, 3049-3052. Malin, M. C.; Edgett, K.C. J. Geophys. Res. 2001, 106, 23,429-23,570.

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McGovern, P. J.; Solomon, S. C.; Smith, D. E.; Zuber, M. T.; Simons, M.; Wieczorek, M. A.; Phillips, R. J.; Neumann, G. A.; Aharonson, O.; Head, J. W. J. Geophys. Res. 2002, 107 (E12), 19.1-19.25. McGovern, P. J.; Solomon, S. C.; Smith, D. E.; Zuber, M. T.; Simons, M.; Wieczorek, M. A.; Phillips, R. J.; Neumann, G. A.; Aharonson, O.; Head, J. W. J. Geophys. Res. 2004, 109 (E07007), 1-5. McKenzie, D.; Barnett, D. N.; Yuan, D. L. Earth Planet. Sci. Lett. 2002, 195, 1-16. McLennan, S. M. Geophys. Res. Lett. 2001, 28, 4019-4022. McLennan, S. M. Sixth International Conference on Mars 2003, abstract 3099. Nimmo, F.; J. Geophys. Res. 2002, 107 (E11), 27.1-27.16. Nimmo, F.; Stevenson, D. J. J. Geophys. Res. 2000, 105, 11,969-11.979. Öner, A. T.; Ruiz, J.; Fairén, A. G.; Tejero, R.; Dohm J. M. Lunar Planet. Sci. 2004, 35, abstract 1319. Parker, T. J.; Saunder, R. S.; Schneeberger, D. M. Icarus 1989, 82, 111-145. Parker, T. J.; Gorsline, D. S.: Saunders, R. S.; Pieri, D. C.; Schneeberger, D. M. J. Geophys. Res. 1993, 98, 11,061-11,078. Parker, T. J.; Clifford, S. M.; Banerdt, W. B. Lunar Planet. Sci. 2000, 31, abstract 2033. Parker, T. J.; Grant, J. A.; Franklin B. J.; Rice J. W. Lunar Planet. Sci. 2001, 32, abstract 2051. Phillips R. J. et al. Science 2001, 291, 2587-2591. Pollack, H. N.; Chapman, D. S. Earth Planet. Sci. Lett. 1977, 34, 174-184. Pollack, H. N.; Hunter, S. J.; Johnson, J. R. Rev. Geophys. 1993, 31, 267-280. Ranalli, G., Geol. Soc. Spec. Publ. 1997, 121, 19-37. Rolandone, F.; Jaupart, C.; Mareschal, J. C.; Gariépy, C.; Bienfait, G.; Carbonne, C., Lapointe, R. J. Geophys. Res. 2002, 107 (B12), 7.1-7.19. Roy, R. F.; Blackwell, D. D.; Birch, F. Earth Planet. Sci. Lett. 1968, 5, 1-12. Roy, S.; Rao, R. U. M. J. Geophys. Res. 2000, 105, 25,587-25,604. Ruiz, J. J. Geophys. Res. 2003, 108 (E11), 8.1-8.5. Ruiz, J. Lunar Planet. Sci. 2005, 36, abstract 1135. Ruiz, J.; Fairén, A. G.; de Pablo, M. A. Lunar Planet. Sci. 2003, 34, abstract 1090. Ruiz, J.; Fairén, A. G.; Dohm, J. M.; Tejero, R. Planet. Space Sci. 2004, 52, 1297-1301. Ruiz, J.; McGovern, P. J.; Tejero, R. Earth Planet. Sci. Lett. 2006, 241, 2-10. Salamuniccar, G. Lunar Planet. Sci. 2004, 35, abstract 1992. Sandiford, M.; McLaren, S. Earth Planet. Sci. Lett. 2002, 204, 133-150. Schultz, R. A.; Lin, J. J. Geophys. Res. 2001, 106, 16,549-16,566. Schultz, R. A.; Watters, T. R. Geophys. Res. Lett. 2001, 28, 4659-4662. Solomon, S. C.; Head, J. W. J. Geophys. Res. 1990, 95, 11,073-11,083. Schubert, G.; Spohn, T. J. Geophys. Res. 1990, 95, 14,095-14,104. Schubert, G.; Solomon, S. C., Turcotte, D. L.; Drake, M. J., Sleep, N. H. In Mars; Kieffer H.; Jakosky, B. M.; Snyder, C. W.; Matthews, M. S.; Eds.; University of Arizona Press: Tucson, AZ, 1992; pp 249-297. Smith, D. E. et al. Science 1999, 284, 1495-1503. Smith, D. E. et al. J. Geophys. Res. 2001, 106, 23,689-23,722. Spohn, T.; Acuña, M. H.; Breuer, D.; Golombek, M.; Greeley, R., Halliday, A.; Hauber, E.; Jaumann, R.; Sohl, F. Space Sci. Rev. 2001, 96, 231-262. Squyres, S. W. et al. Science 2004, 306, 1709-1714.

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Stevenson, D. J.; Spohn, T.; Schubert, G. Icarus 1983, 54, 466-489. Tanaka, K. N.; Scott, D.H.; Greeley, R. 1992. In Mars; Kieffer H.; Jakosky, B. M.; Snyder, C. W.; Matthews, M. S.; Eds.; University of Arizona Press: Tucson, AZ, 1992; pp 345-383 Tejero, R.; Ruiz, J. Tectonophysics 2002, 350, 49-62. Thomson, B. J.; Head, J. W. J. Geophys. Res. 2001, 106, 23,209-23,230. Turcotte, D. L.; Schubert, G. Geodynamics. Second edition; Cambridge University Press: Cambridge, UK, 2002; pp 1-456. Vidal, A.; Mueller, K. M.; Golombek, M. P. Lunar Planet. Sci. 2005, 36, abstract 2333. Vitorello, I.; Pollack, H. N. J. Geophys. Res. 1980, 85, 983-995. Watts, A. B. Isostasy and flexure of the lithosphere; Cambridge University Press: Cambridge, UK, 2001; pp 1-458. Webb, V. E. J. Geophys. Res. 2004, 109 (E09010), 1-12. Webb, V. E.; McGill, G. E. Lunar Planet. Sci. 2003, 34, abstract 1132. Zuber, M. T. et al. Science 2000, 287, 1788-1793.

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Chapter 6

DEALING WITH POTENTIALLY HAZARDOUS ASTEROIDS Eric W. Elst Royal Observatory at Uccle & Holbach-Foundation Mortsel, Belgium

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ABSTRACT An excurse is made through the main belt of asteroids and beyond. Starting with the discovery of the asteroid Ceres, during the first night of January 1801, by Piazzi in Palermo (Sicily), the discipline of asteroid-discovery evolved quickly into large searches of similar bodies in the solar system. At this very moment, more than130.000 new objects have been numbered and the more of them classified into families. One of these groups of asteroids, the NEO’s (Near Earth Objects), subdivided in three substantially different subfamilies (Aten-, Amor- and Apollo-objects) have presently become search objects of primary importance; since they may collide with Earth, resulting in disastrous situations on our planet and (posing) serious risks for the preservation of life on its surface.

INTRODUCTION Ceres, the first and largest asteroid of the main belt of asteroids, was discovered visually at the 5 feet vertical circle by G. Piazzi (1746-1826), during the first night of 1801 at the Observatory of Palermo (Sicily). J. Palisa (1848-1925), with the 15cm astrograph at the Pola Observatory (Yougoslavia) and afterwards with the 62cm astrograph at the Vienna Observatory (Austria) found visually 122 asteroids during the period of 1874-1923, promoting himself with these discoveries to the most succesful visual discoverer of asteroids. By introducing the photographical method at the Bruce-twin 40cm astrograph for hunting asteroids, M. Wolf (1863-1932) at the Heidelberg Observatory was honoured with 248 discoveries. One of his discoveries was the earth approaching asteroid (887) Alinda, in 1920. Before that remarkable event he had already discovered the first Trojan (588) Achilles, in 1906.

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At the same observatory K. Reinmuth (1892-1979) would even discover 395 asteroids, holding for a long time the record of asteroid discovery. He was removed from the first place by the team of the van Houten’s at Leiden and T. Gehrels (1970) at Palomar, with the Palomar-Leiden surveys during 1960-1977, with 3428 discoveries, and individually by E.W. Elst, during 1986-1999, with 3161 discoveries (current situation). However Reinmuth was the first to discover a PHA (Potentially Hazardous Asteroid), on April 24th of 1932. It was named (1862) Apollo and it became the prototype of the Apollos from which several objects have been classified as potentially hazardous. With the discovery of the prototype of the Amor-asteroids, (1221) Amor by E. Delporte in 1932, at Uccle Observatory and in 1976 of the first Aten asteroid (2062) Aten by E. Helin at Palomar, more PHAs have been added to the list of potentially hazardous objects. Introducing CCDs (charge-coupled devices) into astronomical imaging in 1987, CCD-discovery by larger teams, such as LINEAR, LONEOS, Spacewatch end NEAT asteroid discovery, went far beyond all earlier visual and photographical discoveries. Above mentioned teams account for 65.113 CCD-discoveries from a total of 99947 numbered asteroids (by all means; July 2005). Among these discoveries we distinguish 286 Aten-, 1663 Apollo- and 1472 Amor asteroids, including 663 PHAs, of which 120 objects have so far been definitively numbered and 48 of them have received an appropriate (mythological) name (MPC, July 2005). Table 1 lists the total amount of discoveries by individuals and by teams. The heading of the table speaks for itself including the symbols Co for codiscoveries and * for +I. van Houten-Groeneveld, T. Gehrels. Table 2 lists the most prolific sites where asteroids have been observed very intensively. Table 3 deserves our special attention, since it reviews the almost incredible effort that has been put into observation of asteroids, during the last 5 years. The table lists the evolution of the observation of numbered and unnumbered asteroids (cumulative numbers). Under Grand total all individual observations of comets have also been included. It is interesting to note from this table, that in the beginning of 2000 the larger part of the observations still deal with the unnumbered objects (17995549 observations for unnumbered objects versus against 1509331 observations for numbered asteroids). Around May 2001 we notice a turning point, where the numbered objects take over from the unnumbered objects, pointing to the fact that statistically more numbered objects than unnumbered objects show up in the sky. This evolution is undoubtly confirmed after June 2001, where the recovery of the numbered objects is steadily moving higher. At present (21 July 2005) a total of 21609020 observations of numbered objects have been collected (recovery of numbered objects) against 9254137 observations of unnumbered objects (discovery of new objects and recovery of unnumbered objects). Are we approaching the end of asteroid discovery? This question will be answered in next paragraph. Table 4_a,b,c lists the Apollos and numbered PHAs. For the meaning of the symbols in these tables we refer to Annex1.

1. THE END OF ASTEROID DISCOVERY? Table 3 suggests that in a near future the only observations may be of numbered asteroids. Elst (2001) has made a statistical investigation, in order to elucidate the problem. From the enormous increase of recent asteroid discoveries by LINEAR, LONEOS, NEAT, Catalina Sky Survey and Spacewatch, there is indeed an indication that the number of truly new discoveries is slowly decreasing. What is the reason for this? Is there a cutoff in the

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distribution of fainter and faintest asteroids or is it the consequence of the fact that most instruments/devices are not reaching far enough? In the past many searches for new asteroids have been performed. Among the photographical searches we notice the survey done during 1951-1952 at the Observatory of McDonald, the Palomar-Leiden Asteroid Surveys (1960-1973) and the Palomar-CometAsteroid Survey (1970-1995). At the European Southern Observatory (ESO) succesful asteroid-searches were performed during 1976-1998. Intensive CCD-searches for asteroids were initiated in 1997 by LINEAR at Socorro (New Mexico), taking very quickly a leading role in the discovery of asteroids (and comets), with the discovery of 50.113 new asteroids (Table 1, MPC, July 2005). However, all these discoveries remain below visual magnitude v=19.5. Only the CCD-surveys done at Spacewatch (Arizona, 1m reflector) and NEAT (Palomar,1m Schmidt) are reaching further and have discovered several thousands of new and fainter asteroids (Table 1, MPC, July 2005), including all photographical discoveries during the past. In 2001, in the Annual Report (1999) of the Minor Planet Center, Brian Marsden, director of MPC wrote the following: "As anticipated, the year 1999 handsomely broke all previous records for Minor Planet Center activity. The number of pages of actual observations in the Minor Planet Circulars Supplement (MPSs), 4998, each exceeded the total for the previous year by more than 65 percent. The 3145 new numberings of minor planets were 85 percent more than in 1998 and more that four times as many as in 1997. A milestone was reached in March 1999 with the numbering of minor planet (10000). Although it had taken more than 198 years, following the discovery of Ceres, to attain this milestone, the milestone rather loses significance when one realizes that it will less than two years more to reach the numbering of minor planet (20000)"! From this quote we could get the impression that discovering and numbering minor planets will continue for ever. But the figures from table 3 remain and therefore it is necessary that we investigate the problem more thoroughly. During September/October 1960 Gehrels took 130 photographical plates at Palomar, with the 48 inch Schmidt telescope, which were afterwards measured by van Houten et al (1970). More than 2000 faint asteroids were discovered. Statistical investigation made clear that, at least for asteroids as faint as magnitude v=19.5, there was surely no cutoff to expect for asteroids beyond the limiting magnitude of the instrument.

2: ASTEROID FAMILIES AND COMPLETENESS OF SEARCHES In 1866 Kirkwood discovered the presence of gaps in the main belt of asteroids and found that for particular distances from the sun corresponding to the value of the semimajor axis of the objects involved- asteroids were accumulating (Figure1), revealing the existence of families, i.e. asteroids with almost the same orbital elements (Kirkwood, 1866). We want to mention here that if we want to study families in a more appropriate way, then it will be necessary to use proper elements (orbital elements, freed from perturbation)(Knezevic and Milani, 1993). Flora asteroids are accumulating at 2.3 A.U., Themis asteroids at 3.15 A.U., Hilda asteroids at 3.9 A.U. and Trojans at 5.2 A.U. At present we deal with about 20 well distinguished families, but there seems to exist a lot more (e.g. Hirayama asteroid families).

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Itt is clear that the t height of a peak may bee interpreted as a a measure for f relative completeness off a particular family, assum ming that beloow the instrum mental limitinng magnitude v all their m members havee been discoovered. Extennding the lim miting magniitude, by ussing larger innstruments and d more sensitiive CCDs, wiill result in higgher peaks, siince we are reeaching the faainter asteroids of that particcular family. Since S we know w the distancee a of the familly from the suun, it is possib ble to calculatee the absolute magnitude H of the newly discovered assteroids, by ussing the follow wing approxim mate formula (Cunningham, ( 1988) H = v – 5*llog(a(a-1))

(1)

where v is thee limiting maagnitude, H the w t absolute magnitude m off the asteroidd and a its seemimajor axiss. With v=19.55 and a=2.3 we w obtain a vaalue H=17.1. By B means of the t formula (M Meeus J., 1991 1) = 3.6 –0.2*H logD(km)=

(2)

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w obtain the metric we m dimensiion of the asteroid (i.e. D=1.5km).

Fiigure 1. Numbeer of asteroids versus v semimajoor axis.

We may no ow plot the nuumber of all discovered d asteroids (interm mixing all famiilies), for a paarticular limitiing magnitudee (i.e. v=19.5), versus absoluute magnitudee H (Figure2).

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Figure 2 prrovides a reliaable measure for completenness, assuminng that full skyy coverage haas been performed for obseervations up to t the limitingg magnitude v. v In the interrval H=15155.5 (step=0.25 5 magnitude) about 150000 numbered asteroids a havee been discovvered. This im mplies also th hat no more asteroids a withh H=15-15.5 will be discoovered in thatt particular innterval. Only, by pushing the observatioons beyond thhe limiting magnitude m v new n fainter obbjects will be sighted and eventually e num mbered, resultting in a higheer peak whichh will move att the same tim me to fainter values v of the absolute a magnnitude. It goess without sayiing that we caan construct an nalogue plots for all familiees separately (e.g.: Figure 3)).

Fiigure 2. Total number n of asterooids versus absoolute magnitudee H.

Fiigure 3. Total number n of Near Earth Asteroidss (July 2005). Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

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3. HAZARDOUS ASTEROIDS What is a hazardous asteroid? It is a celestial body that during its revolution around the sun crosses the orbit of the earth (e.g. Alinda) and may therefore collide (impact) with our home planet, with all possible disastrous consequences. Figure 3 shows the total number of numbered and unnumbered NEAs _Near Earth Asteroids_ (Apollos, Amors and Atens, including all PHAs) versus absolute magnitude. From inspection of this figure it follows that we have nearly completeness for an absolute magnitude of H=20.0 (corresponding with a metric dimension of about 400 meter), to be compared with an absolute magnitude of 18.5 (850 meter) in 2000. It is clear that NEAs, and therefore also PHAs, with the same absolute magnitude may be still discovered in the near future; but they will generally show large inclinations, a signature for objects that are discovered more seldomly, since they are moving for the largest part of their existence far outside the ecliptic, where during the past searches for new asteroids have been more scarce or even completely been neglected. Only by pushing the limiting instrumental magnitude v to fainter absolute magnitudes new asteroids will be discovered. It is interesting to note that with completeness of H=20.0, and even with H=18.5, we have reached the objects with dimension smaller than 1km, and therefore the theoretical limit for non global destruction effects, if one of these objects should hit the earth. The main belt remains the principal reservoir of new objects that, due to internal collisions could find their way towards the inner parts of the solar system (Ipatov, 1995). However, the collisions do not lead directly to the generation of new potentially hazardous asteroids, but they may supply new objects that are partially injected into resonant, chaotic orbits undergoing large variations of the eccentricity and becoming therefore amenable to planetary encounters (Menichella et al.,1996). Farinella et al. (1993) estimated the fraction of main belt fragments that may reach the 1:3 mean motion resonance with Jupiter or the v6 secular resonance, which are the two most effective dynamical routes from the main belt to the inner planet zone (Elst, 2001)

4. CONSEQUENCES OF AN IMPACT OF A CELESTIAL BODY WITH EARTH The impact of a cosmic body with Earth essentially causes an explosion: an instantaneous conversion of kinetic energy of the impactor into fragmentation, comminution and cratering of the substrate; heating, melting and vaporisation of the projectile and target materials; kinetic energy of cratering ejecta and seismic shock waves penetrating the planet. A significant fraction of the energy is dissipated in the ecosphere, the thin shell of air, water and surface rocks and soils, whose constancy sustains and nurtures life. While the greatest damage is obviously at ground level, the stratosphere is badly polluted with dust on a global scale from impacts of objects exceeding 1 km diameter. Smaller impacts (e.g. by a 200m impactor), if into the ocean, can cause devastation thousands of kilometers away due to the efficient transmission of energy to great distances by tsunamis (Chapman, 2002, Ward and Aspaug, 2002). Examination of the cratering record on the Earth and terrestrial planets _and

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especially on the moon_, has demonstrated the continuity through the past 3.5 billion years of impact processes.

Figure 4. Almost extinction of the ferns.

The disaster of an impact at the level of fauna and flora is impressive. Below are two figures which were taken from the book La botanique rédécouverte (1999) from Aline Raynal, professor at the Jardin des Plantes in Paris, France. Figure4 shows the evolution of the ferns slowing down abruptly, 225 million years ago, epoch of the Yucatán impact. Figure5 is still more impressive and shows the almost complete extinction of the gymnospermia (conifers) and at the same time the overdeveloping of the angiospermia (fruit-bearing trees). The extinction of the dinosaurs falls at the same epoch (65 millions year ago).

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Figure 5. Overdeveloping of the angiospermia.

5. SOURCES OF THE PHAS NEAs originate from five sources: asteroids clustering near the resonances v6 and 1:3 (Alindas), asteroids from the outer zones of the belt, Mars-crossers -asteroids that may cross the orbit of Mars- and comets perturbed by Jupiter and originating from the Kuiper Belt. NEAs _and with them the PHAs_ are sightet at the entire sky. From inspection of the orbital elements of the NEAs (Table 5) we may get the impression, that some preferable directions are preferred by the possible impactors. Let us therefore look at the inclinations. Figure6a,b shows us a swarm of NEAs (the first 50 items from table 5). There seems hardly to exist a relation between semimajor axis and inclination for the involved objects (Fig6a, rectangles). However, the excentricity declines slowly with increasing inclination (Fig6b, stars)).

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In c lin a t io n v e r s u s e x c e n t r ic it y / s e m im a jo r a x is

3 2 .5

e/a

2 1 .5 1 0 .5 0 0

10

20

30

40

i Figure 6. Semimajor axis/excentricity versus inclination.

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Let us now plot semimajor axis versus inclination (Figure7) for the objects of Figure6. Here we notice clustering of the objects with a tendency to move upwards with increasing eccentricity. This is a consequence of the fact that earth-crossing asteroids are distinguished on the basis of semimajor axis and eccentricity. The Jacobi constant has the following form (Klose, 1928): C= 1/a+2*Sqr(a(1-e2))*cosi.

Figure 7. Clustering of a swarm of NEAs.

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(3)

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Eric W. Elst

6. A STRANGE OBJECT: COMET/ASTEROID ELST-PIZARRO.

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Let us give an example. In September 1987 an Apollo-asteroid (1987 SB) has been discovered at the Bulgarian National Observatory (Elst et all, 1989). It is characterized by an eccentricity of 0.65, an inclination of 3° and a semimajor axis of 2.16. Apollo-Amor objects have C-values in the range of 2.7 and 3.0. With a value of C= 2.7 the object 1987 SB fits nicely into the Apollo-region. This means that if we are not considering the inclination, by putting semimajor axis versus eccentricity, then the eccentricity (semimajor axis) in equation (3) becomes dependent of the semimajor axis (eccentricity), explaining the particular behavior of the objects in Figure7. Comets may also be considered as potentially hazardous if their orbits allow them to approach Earth. A special case is the object Elst-Pizarro that has been classified as an asteroid and as a comet. The object was discovered by E. W. Elst and G. Pizarro, during a routine search for new asteroids with the 1m ESO-Schmidt at the European Southern Observatory. Preliminary orbit calculation by Marsden (1996) showed that the orbit could be linked up with the Themis asteroids. However, a narrow dust tail remained for more than four months. Therefore an explanation had to be found for the persistence of the tail. Figure8 shows 133P/Elst-Pizarro, imaged on 2002 September 07 by the University of Hawaii 2.2m telescope on Mauna Kea. Nucleus and tail are indicated by white arrows. The tail extends about 3.5 arcminutes, corresponding to about 400.000 km.

Figure 8. Comet/asteroid Elst-Pizarro, 2.2m telescope on Mauna Kea (Hawaii).

The identification of 133P/Elst-Pizarro as a comet is not in doubt, but the origin of the mass loss is unclear. In 1996 the tail had been previously explained as debris released by an impact, but the discovery in 2002 of recurrent activity renders this interpretation implausible. The ejection of particles may be consistenly explained if the object is a barely active Jupiterfamily comet, that has evolved into an asteroid-like orbit. A poor understanding of the precise effects of nongravitational forces on the long-term dynamical evolution of such bodies, results in uncertain knowledge of the fraction of inactive Jupiter-family comets among bona

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fide asteroids. 133P/Elst-Pizarro could be a true Themis family member upon which subsurface ice has been recently exposed, but would then still blur the asteroid-comet boundary by showing that small main-belt asteroids can preserve ice over astronomically long timescales. Such a scenario would greatly complicate the determination of source regions for near-Earth objects, since the presence of significant ice in a body could no longer be considered a diagnostic of origin in the outer solar system and not in the main asteroid belt (Hsieh, 2002).

7. RESONANCE OBJECTS (ALINDA ASTEROIDS)

Alin d a -a s te ro id s 2,58 semimajor axis

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Alinda asteroids have semimajor axes of about 2.5 AU and eccentricities between 0.4 and 0.65. The objects are in a 1:3 orbital resonance with Jupiter and therefor in a 1:4 resonance with Earth. An object in this resonance has its orbital eccentricity steadily increased by gravitational interactions with Jupiter, until it approaches an inner planet that disrupts the resonance. Deorbited, the asteroids eventually come in collision with Earth or some other planet of the solar system. Several Alindas have their perihelia very close to Earth, resulting in a series of close encounters at almost exactly four-year intervals, due to the 1:4 resonance. One consequence of this is that if an Alinda asteroid happens to be in an unfavorable position for observing it, at the time of its close approach to Earth, this situation can persist for decades. We should not wonder that we find back a few PHAs among the Alinda-asteroids (Table 7). Figure9 shows a plot with the semimajor axis versus eccentricity for these objects. There is a tendency of declining semimajor axis with increasing eccentricity, but this appearance is inherent with the classification-scheme used for the Alinda-asteroids (Jacobi constant, Klose 1928). The apparent difference between Figure7 and Figure9 can be explained by the fact that in Figure7 the NEAs have been intermixed with each other, while in Figure9 (less scattering) we are looking at a single family of asteroids.

2,56 2,54 2,52 2,5 2,48 2,46 2,44 0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

e x ce n tricity

Figure 9. Alinda-asteroids: semimajor axis versus exentricity.

8. HUNGARIA- AND PHOCAEA ASTEROIDS Fig 10 shows the distributions of the minor planets, semimajor axis versus inclination. The center of the cluster of Hungaria asteroids is at =1.85 (semimajor axis between 1.78 and 2

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E W. Elst Eric

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AU) and =2 A 23° (inclinatioon between 166° and 34°). The T Hungariass have eccentrricities less thhan 0.18 and move m just outsside the orbit of Mars. The cluster of Phocaeas lies at = 2.35 (ssemimajor axiis between 2.225 and 2.5 AU U), = 24°(inclination beetween 18° annd 32°) and ecccentricities greater g than 0.1. Hungaria asteroids a dwelll near the 8:99 resonance with w Jupiter, reeason why wee suspect that they are reguularly removedd from this loocation in the main belt. The eccentricity increases steadily untiil the asteroiid escapes frrom its clustter-position w whereafter it may m eventuallyy be injected innto the orbit of Earth. The problem p remaiins why the H Hungaria asterroids were originally residing at this particular pllace. The vallue of the innclination of th he earth axis may (23°.5) may m point to a liaison of thhese asteroids with Earth itself. But the intriguing i asppect of the em mpty place bellow the clusteers of the Hunngaria- and mperatively bee investigated. A substantiaal part of the NEAs N may Phhocaea asteroids should im haave found theere their origiin, flattening at the same time the low wer part of thee Hungaria assteroids. Let us u look upon thhe NEA, 20033 NO4, that haas been discovvered some tim me ago and thhat could be probably p linkeed with the cluuster of Hunggaria asteroidss (Table 11). The T orbital ellements (i,a) of 2003 NO44 correspond nicely with the elementss of a normall Hungaria assteroid, with exception e of its eccentricity,, which is in thhis case largerr than 0.18. If we assume thhat these NEA As have their origin o in the Hungaria H clusteer, than of couurse the asterooids have to find a way to leave l the clustter, which is done d by meanns of Jupiters gravitational field. 2005 Y16 has been discovered more m recently. We notice herre an elevated eccentricity and a a lower PY innclination.

Fiigure 10. Minorr Planets: Inclinnation versus semimajor axis. Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

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Table 11. Orbital elements: 2003 NO4 n a e P

0.43884482 Peri. 171.04428 1.7149919 Node 135.44363 0.3127116 Incl. 22.6940 2.25 H 18.2 G 0.15

Orbital elements: 2005 PY16 n a e P

0.35485324 Peri. 193.40426 1.9759235 Node 159.45300 0.5247886 Incl. 6.41602 2.78 H 19.3 G 0.15

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Let us look now at a plot which shows the Hungaria- and Phocaea asteroids, inclination versus semimajor axis (Figure11).

Figure 11. Hungaria and Phocaea asteroids.

The clusters are isolated from each other -and from other asteroids- by resonances. v5 and v6 are the eccentricity terms and v16 the inclination term of the perturbation by Jupiter (and Saturn). It is certainly no coincidence that below the cluster of the Hungaria asteroids space is almost empty (to be compared with Figure10). All asteroids once residing there have been removed by resonances during the past. A larger part of them hasmost probably found its way to Earth. The same applies to the Phocaea-asteroids. There seems indeed to exist a continuous transition from Phocaeas to Mars-crossers, which differ from the Phocaeas only by larger eccentricities which enable them to cross the orbit of Mars. We note additionaly that the part that has been removed by resonances follows neatly the shape of the resonances. However, we must consider the possibility that at the indicated spaces (E1 and E2) no asteroids have ever been present, actually due to the combined action of the resonances v6 and v16.

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9. CONCLUSIONS Collisions between asteroids and the Yarkovsky Effect -a very weak thermal impulse when the asteroid surface is heated by the Sun, affecting asteroids less than 10 km in diameter- may feed the resonances capable of injecting new NEAs over several million years into the inner part of the Solar System. Bottke et al. (19xx) estimate that 61% of NEAs of H 1, is presented in Figure 8d. T1

g as

ic e water

T5 T9 T2

lt lw l wf ls

s o lid T3 T1 T6 T4

Figure 8d. Cubic cell model for a soil analogue with ice disposed among adjacent particles, for γf > 1 [38].

The effective thermal conductivity, keff , is then given by: β ⋅γ − β ⋅γ f β ⋅γ f − β 1 β 2⋅ − β ⋅ γ + = + + 2 2 2 2 2 2 2 keff katm ⋅ β − γ f + ki ⋅ γ f katm ⋅ β − γ + ki ⋅ γ A + ki ⋅ γ + 2 ⋅ β ⋅ γ f − 2 ⋅ γ ⋅ γ f Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

(

)

(

)

(

)

β ks + ki ⋅ (γ − 1 + 2 ⋅ β ⋅ γ f − 2 ⋅ γ ⋅ γ f ) + A 2

(9)

where A is given by Eq. (8). The first term of Eq. (9) is the thermal resistance of the materials gas and ice in the section ( lt 2 ), of length (lt – lw). The second term is the thermal resistance of the materials gas and ice in the section ( lt 2 ), of length (lw–lwf). The third term is the thermal resistance of the materials gas and ice in the section ( lt2 ), of length ( lw f - ls ). The fourth term is the thermal resistance of the materials gas, ice and solid in the section ( lt 2 ), of length ( ls )

2. Martian Analogues The Martian soil analogue investigated in the present work is the olivine tested in the laboratory [21]. The thermal conductivity of the solid particle has been evaluated experimentally resulting equal to ks= 2.94 W/m/K [21]. The evaluation of the thermal conductivity of the solid particle is based on two experimental measurements, i.e. in dry and

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Thermal Properties and Temperature Variations…

215

water saturated conditions. The two experimental values are then used in Eq. (1) as keff to give two equations which are then solved to calculate the values of β and ks . The Martian atmospheric pressure is about 6 mbar and the atmospheric gas is composed of CO2 (95%), N2 (3%), Ar (1.5%) and traces of water vapour. The thermal conductivity of the Martian atmospheric gas is evaluated as 92.8% of the value at 1 bar and as a weighed average. Table 1 reports the values at several temperatures. Table 1. Thermal conductivity of the Martian atmospheric gas, katm [38]. T (K)

10 3(W/m/K)

148 173 223 273 298

6.86 8.39 11.87 14.49 16.01

A relation between ice volume content, f = Vi / Vv , ice mass content, a = Mi / Mt and soil porosity, p, can be found, where Mi is the mass of ice in the porous medium, Mt the total mass of the porous medium, Vi the ice volume in the porous medium, and Vv the void volume of the cubic cell. The ice mass content, a, is given by:

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a=

1 ρ 1 − f ρ s ⋅ (1 − p ) 1 + atm ⋅ + ρ ice f ρ ice ⋅ f ⋅ p

(10)

For a given soil porosity, p, the ice mass content a = Mi / Mt, corresponding to saturated frozen soil, is reported in Table 2. Table 2. Porosity and ice mass content, a = Mi / Mt, at T=223 K, for saturated soil [38]. p 0.20 0.30 0.40 0.50

a 0.074 0.12 0.17 0.24

3. Numerical Evaluation of Two Phase Martian Soil Analogue Figure 9 presents the thermal conductivity of a dry Martian soil analogue in the porosity range p = 0 – 1 at several temperatures. The thermal conductivity increases with the temperature and decreases with the increase of porosity. At p = 0 the thermal conductivity is equal to the thermal conductivity of the solid particle (2.94 W/m/K) and at p = 1 is equal to the thermal conductivity of the Martian atmospheric gas.

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Figure 10 presents the thermal diffusivity of a dry Martian soil analogue, in the porosity range p = 0 - 1 and at several temperatures, evaluated assuming the thermal conductivity given by Figure 9 and the value of (ρ·cp)d measured experimentally in [21]. 3,00 T = 148 K

k eff , W/(mK)

2,50

T = 173 K T= 223 K

2,00

T = 273 K T = 298 K

1,50 1,00 0,50 0,00 0

0,2

0,4

0,6

0,8

1

p

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Figure 9. Thermal conductivity of dry Martian soil analogue versus porosity [29].

Figure 10. Thermal diffusivity of dry Martian soil analogue versus porosity [29].

4. Numerical Evaluation of Three Phase Partially Frozen Martian Soil Analogue Figure 11 presents the thermal conductivity of a partially frozen Martian soil analogue versus ice volume content at several porosities and T = 223 K. The abrupt increase in the theoretical effective thermal conductivity is due to the step change from the adsorbed ice configuration

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Thermal Properties and Temperature Variations…

217

of Figure 8b to the configuration of Figure 8c, where ice is disposed among adjacent soil particles. The experimental step increase is smooth as confirmed by unpublished results. 3,00

keff , W/(mK)

2,50 2,00 1,50

( p = 0.20 ) ( p =0.40 )

1,00

( p =0.60 ) ( p = 0.80 )

0,50

( p = 0.98 )

0,00 0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

f

Figure 12 presents the thermal conductivity of a partially frozen Martian soil analogue versus the ice mass content, a, at several porosities and T = 223 K. For a porosity equal to 0.2, the thermal conductivity spans from the minimum value in dry condition, 0.144 W/m/K, to the maximum one in saturated conditions, 2.7 W/m/K. For a porosity equal to 0.5, the thermal conductivity spans from the minimum value in dry conditions, 0.05 W/m/K, to the maximum one, in saturated conditions, 2.7 W/m/K. Figure 12 reports also the lines with the same ice volume content f. The thermal conductivity, at the same f, decreases with the increase of the ice mass content, because the thermal conductivity of the solid particle is higher than the ice value. 3,00

( p = 0.20 ) ( p =0.40 ) ( p = 0.50 ) ( p =0.60 ) ( p =0.70 ) ( p = 0.80 ) ( p = 0.90 ) ( p = 0.98 ) f = 0.084 f = 0.20 f = 0.30 f = 0.40 f = 0.50 f = 0.60 f = 0.70 f = 0.80 f = 0.90

2,50 k eff ,W/(mK)

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Figure 11. Thermal conductivity versus ice volume content at 223 K.

2,00 1,50 1,00 0,50 0,00 0

0,2

0,4

0,6

0,8

1

a

Figure 12. Thermal conductivity of Martian soil analogue versus ice mass content, a, at 223K [38]. Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

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Figure 13 presents the thermal conductivity of the Martian soil analogue versus porosity at several ice mass contents, a, and 223 K. In dry condition (a = 0) the thermal conductivity decreases from the value of the solid particle (2.94 W/m/K), p=0, to the atmospheric gas (0.012 W/m/K), p=1. At each ice mass content, a = Mi/Mt, the thermal conductivity decreases from the maximum value, predicted at low porosities, when the cubic cell is full of ice, to the minimum ones, at high porosity, when the void space inside the cell is occupied by the atmospheric gas. It can be noted that the maximum thermal conductivity is around 2.7 W/m/K, almost independently of the ice mass content. 3,50 T = 223 K

keff , W/ (mK)

3,00

a = 0,05 a = 0,10 a = 0,15 a = 0,20 a = 0,30 a = 0,50 a = 0,00

2,50 2,00 1,50 1,00 0,50 0,00 0

0,2

0,4

0,6

0,8

1

p

Figure 14 present the thermal conductivity of the Martian soil analogue versus porosity at several ice mass contents and 273 K. 3,50 T = 273 K

3,00 k eff , W/ (mK)

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Figure 13. Thermal conductivity of Martian soil analogue versus porosity, at 223 K [38].

a = 0,05 a = 0,10 a = 0,15 a = 0,20 a = 0,30 a = 0,50 a = 0,00

2,50 2,00 1,50 1,00 0,50 0,00 0

0,2

0,4

0,6

0,8

1

p

Figure 14. Thermal conductivity of Martian soil analogue versus porosity, at 273 K, [38]. Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

Thermal Properties and Temperature Variations…

219

The heat capacity of the Martian soil analogue is evaluated with the following Eq.11, and the values are reported in Figure 15. The heat capacity increases, for a fixed porosity, with the ice volume content, with a higher increase at higher porosities.

(ρ ⋅ c )

p eff

= (ρ ⋅ c p )s ⋅ (1 − p) + (ρ ⋅ c p )ice ⋅ f ⋅ p + (ρ ⋅ c p )atm ⋅ (1 − f ) ⋅ p

(11)

2,50E+06

3

(ρ c p) , J / (m K)

2,00E+06

p = 0.20 p = 0.40 p = 0.50 p = 0.60 p = 0.70 p = 0.80 p = 0.90 p = 0.98

1,50E+06 1,00E+06 5,00E+05 0,00E+00 0

0,2

0,4

0,6

0,8

1

f

Figure 15. Heat capacity of Martian soil analogue versus ice volume content, a, at 223 K [38]. 1,80E-06 1,60E-06 ( p = 0.20 ( p =0.40 ( p = 0.50 ( p =0.60 ( p =0.70 ( p = 0.80 ( p = 0.90 ( p = 0.98

1,00E-06 8,00E-07

2

, m /s

1,20E-06

eff

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1,40E-06

6,00E-07 4,00E-07

) ) ) ) ) ) ) )

2,00E-07 0,00E+00 0

0,2

0,4

0,6

0,8

1

f

Figure 16. Thermal diffusivity of Martian soil analogue versus ice volume content, f, at 223 K [38].

The thermal diffusivity of the Martian soil analogue is evaluated by the ratio of keff and ρ cp. Figure 16 presents the thermal diffusivity versus the volume ice content, f, at several porosities and 223 K. The theoretical thermal diffusivity has a step increase from the configuration of ice adsorbed around the solid particle, Figure 8b, for f < 0.083, to the configuration of ice disposed among the solid particles, for f > 0.083, Figure 8c, because of

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F. Gori and S. Corasaniti

the ice bridges present among the solid particles. The thermal diffusivity increases with f with a higher increase at higher porosities. Figures 17-19 report the thermal diffusivity of the Martian soil analogue, versus porosity, at several ice mass contents, a, and three temperatures, i.e. 148 K, 173 K and 223 K. The results show that the thermal diffusivity of the Martian soil analogue changes of more than an order of magnitude from dry to partially frozen conditions. 3,00E-06 T = 148 K

2,50E-06

a = 0,05 a = 0,10 a = 0,15

2

, m /s

2,00E-06 1,50E-06

eff

a = 0,20 a = 0,30

1,00E-06

a = 0,50 a = 0,00

5,00E-07 0,00E+00 0

0,2

0,4

0,6

0,8

1

p

Figure 17. Thermal diffusivity of Martian soil analogue versus porosity, p, at 148 K [38].

T = 173 K

2

α eff , m /s

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2,50E-06 2,00E-06

a = 0,05

1,50E-06

a = 0,15

a = 0,10

a = 0,20

1,00E-06

a = 0,30 a = 0,50

5,00E-07

a = 0,00

0,00E+00 0

0,2

0,4

0,6

0,8

1

p Figure 18. Thermal diffusivity of Martian soil analogue versus porosity, p, at 173 K [29].

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Thermal Properties and Temperature Variations…

221

2,50E-06 T = 223 K

1,00E-06

α

2

, m /s

1,50E-06

eff

2,00E-06

a = 0,05 a = 0,10 a = 0,15 a = 0,20 a = 0,30 a = 0,50

5,00E-07

a = 0,00

0,00E+00 0

0,2

0,4

0,6

0,8

1

p

Figure 19. Thermal diffusivity of Martian soil analogue versus porosity, p, at 223 K [29].

Figure 20 presents the thermal diffusivity versus the ice mass contents, a, along to the lines of constant ice volume content, f. 1,80E-06

( p = 0.20 ) ( p =0.40 ) ( p = 0.50 ) ( p =0.60 ) ( p =0.70 ) ( p = 0.80 ) ( p = 0.90 ) ( p = 0.98 ) f = 0.084 f = 0.20 f = 0.30 f = 0.40 f = 0.50 f = 0.60 f = 0.70 f = 0.80 f = 0.90

1,60E-06

1,20E-06

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2

α eff , m /s

1,40E-06

1,00E-06 8,00E-07 6,00E-07 4,00E-07 2,00E-07 0,00E+00 0

0,2

0,4

0,6

0,8

1

a Figure 20. Thermal diffusivity of Martian soil analogue versus ice mass content, a, at 223 K [29].

III. TEMPERATURE VARIATION IN MARTIAN SOIL ANALOGUES 1. Temperature Variation The differential equation of heat conduction in the soil ∂ ∂z

∂ϑ ⎛ ∂ϑ ⎞ ⎜⎜ k ⎟⎟ = C ⋅ z ∂t ∂ ⎝ ⎠

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(12)

222

F. Gori and S. Corasaniti

has the solution z ⎛ z⎞ ⎛ ⎞ ϑ (z, t ) = ϑa + ϑs exp⎜ − ⎟ ⋅ sin⎜ ωt − + ϕ ⎟ D ⎝ D⎠ ⎝ ⎠

(13)

under the following boundary conditions: - on the surface, z = 0, the temperature variation is

ϑ (0, t ) = ϑ a + ϑs ⋅ sin (ωt + ϕ )

(14)

- at an infinite depth the temperature variation is zero lim ϑ = ϑa

(15)

z →∞

where ϑa is the average temperature in the soil, ϑs the amplitude of the temperature variation on the surface, t the time, ω is the radial frequency ω=

2π 2π = = 7.27 ⋅ 10 −5 sec −1 T 86400

(16)

The Martian day is assumed equal to 24 h and ϕ is the phase delay. Further on:

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D=

2⋅k C ⋅ω

,

(17)

where k is the thermal conductivity of the soil, C the heat capacity and α = k/ C the thermal diffusivity.

1.1. Martian Soil Analogues, Type A The first Martian soil analogue investigated, type A, has the porosity p = 0.20. The temperature variation on the surface is variable from the minimum value of 148 K to the maximum one of 298 K. 1.1.1. Dry Martian Soil Analogue, Type AD The dry soil analogue, AD, has a thermal conductivity equal to k = 0.144 W/m/K, as evaluated by the theoretical model at 223 K, Figure 13, and a thermal diffusivity equal α = 10-6 m2/s, as suggested by the literature review. Figures 21-22 show the daily temperature variations (24 hours) in the soil AD. Figure 21 reports the temperatures of the soil up to a depth of 0.60 m and Figure 22 below 0.60 m. At 0.30 m the total temperature variation, during the day, is about 25 K and at 0.60 m is 4 K, Figure 21. At 1 m the temperature variation is 0.4 K, while at 1.30 m is 0.06 K, Figure 22.

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Thermal Properties and Temperature Variations…

223

303 z= 0 m z = 0.2 m z = 0.4 m z = 0.6 m

283 263

z = 0.1 m z = 0.3 m z = 0.5 m

T (K)

243 223 203

0

5

10

15

20

183 163 143 t (h)

Figure 21. Temperature variations in the dry soil analogue, type AD, up to 0.60 m [29].

223,8 z = 0.8 m z=2m z=4m z = 1.3 m

223,6 223,4

z=1m z=3m z=5m

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T (K)

223,2 223 222,8

0

5

10

15

20

222,6 222,4 222,2 t (h)

Figure 22. Temperature variations in the dry soil analogue, type AD, below 0.60 m [29].

The temperature distributions are presented versus the depth of the soil analogue, in Figure 23, at several hours during the day. Figure 23 shows that the temperature variations are detectable inside the dry soil analogue, type AD, up to a depth of about 1 m.

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224

F. Gori and S. Corasaniti 300 t=8h

t = 12 h

t = 16 h

t = 20 h

t = 24 h

T (K)

250

t=4h

200

0

1

2

3

150 z (m)

Figure 23. Temperature distribution in the dry soil analogue, type AD, versus depth, during the Martian day, [38].

1.1.2. Frozen Martian Soil Analogues, Type AF The frozen Martian soil analogue, type AF, has the following characteristics: p = 0.20, f = 1, a = M ice / M t = 0.0728, thermal conductivity equal to k = 2.70 W/m/K according to the theoretical model at 223 K, Figure 13, and thermal diffusivity equal to α = 10-5 m2/s, one order of magnitude higher than the dry soil, as suggested by the literature review. The temperature variation on the surface is variable from the minimum value of 148 K to the maximum one 298 K, with an average value of 223 K. 303 z= 0 m z = 0.2 m z = 0.4 m z = 0.6 m

263

z = 0.1 m z = 0.3 m z = 0.5 m

243 T (K)

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283

223 203

0

5

10

15

20

183 163 143 t (h)

Figure 24. Temperature variations in the frozen soil analogue, type AF, up to 0.60 m [29].

Figures 24-25 show the daily temperature variations in the soil analogue, type AF. Figure 24 reports the temperature of the soil up to a depth of 0.60 m. At 0.30 m, the temperature variation, during the day, is about 85 K and at 0.60 m is about 48 K, Figure 24. At 1 m the temperature variation is about 22 K, at 1.30 m about 13 K and at 2 m about 3.3 K, Figure 25.

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Thermal Properties and Temperature Variations…

225

245 240

T (K)

235 230 225 220 0

5

10

15 z = 0.8 m z= 2 m z= 4 m z = 1.3 m

215 210

20

z= 1 m z= 3 m z= 5 m

205 t (h)

Figure 25. Temperature variations in the frozen soil analogue, type AF, below 0.60 m [29].

The temperature distributions are presented versus the depth of the soil analogue, type AF, in Figure 26, at several hours during the Martian day. Figure 26 shows that the temperature variations are detectable, inside the frozen soil analogue, up to a depth of about 3 m. In conclusion the temperature variation is more penetrating in the frozen soil, because the thermal diffusivity is one order of magnitude higher than in the dry soil.

290

250 T (K)

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270

t=4h

t=8h

t = 12 h

t = 16 h

t = 20 h

t = 24 h

230 210 0

1

2

3

4

190 170 150 z (m)

Figure 26. Temperature distribution in the frozen soil analogue, type AF, versus depth, during the Martian day [38].

1.2. Martian Soil Analogue, Type B The second Martian soil analogue investigated, type B, has a porosity equal to p = 0.50. The temperature variation on the surface is variable from the minimum value of 148 K to the maximum one of 298 K.

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226

F. Gori and S. Corasaniti

1.2.1. Dry Martian Soil Analogue, Type BD The dry soil analogue, type BD, has a thermal conductivity equal to k = 0.05 W/m/K, as evaluated by the theoretical model at 223 K, Figure 13, and a thermal diffusivity equal to α = 0.45·10-7 m2/s, as evaluated by the theoretical model, Figure 19. Figures 27-28 show the daily temperature variations (24 hours) in the soil BD. Figure 27 reports the temperatures of the soil up to a depth of 0.60 m and Figure 28 below 0.60 m. At 0.30 m the total temperature variation, during the day, is about 0.03 K and at 0.60 m is negligible, Figure 27. At 0.80 m and at 1 m the temperature variation is negligible, Figure 28. 303 z=0m z = 0.2 m z = 0.4 m z = 0.6 m

283 263

z = 0.1 m z = 0.3 m z = 0.5 m

T (K)

243 223 203

0

5

10

15

20

183 163 143 t (h)

Figure 27. Temperature variations in the dry soil analogue, type BD, up to 0.60 m [29]. z = 0.8 m z=2m z=4m z = 1.3 m

223,000000010 223,000000005

T (K)

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223,000000015

z=1m z=3m z=5m

223,000000000 222,999999995

0

5

10

15

20

222,999999990 222,999999985

t (h) Figure 28. Temperature variations in the dry soil analogue, type BD, below 0.60 m [29].

The temperature distributions are presented versus the depth of the soil analogue, type BD, in Figure 29, at several hours during the Martian day. Figure 29 shows that the temperature variations are detectable, inside the dry soil analogue, up to a depth of about 0.20 m.

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Thermal Properties and Temperature Variations…

227

300 t=4h t = 12 h t = 20 h

T (K)

250

t=8h t = 16 h t = 24 h

200

150 0

0,5

1

1,5

2

2,5 z (m)

3

3,5

4

4,5

5

Figure 29. Temperature distribution in the dry soil analogue, type BD, versus depth, during the day [29].

303

z=0m z = 0.2 m z = 0.4 m z = 0.6 m

283 263

z = 0.1 m z = 0.3 m z = 0.5 m

243 T (K)

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1.2.2. Frozen Martian Soil Analogue, Type BF The frozen Martian soil analogue, type BF, has the following characteristics: p = 0.50, f = 1, a = M ice / M t = 0.2413, thermal conductivity equal to k = 2.70 W/m/K, according to the evaluation of the theoretical model at 223 K, Figure 13, and thermal diffusivity equal to α = 1.36·10-6 m2/s, as evaluated by the theoretical model, Figure 19. The temperature variation on the surface is variable from the minimum value 148 K to the maximum one 298 K, with an average value of 223 K.

223 203 0

5

10

15

20

183 163 143 t (h)

Figure 30. Temperature variations in the frozen soil analogue, type BF, up to 0.60 m [29].

Figures 30-31 show the daily temperature variations in the soil analogue, type BF. Figure 30 reports the temperature of the soil up to a depth of 0.60 m. At 0.30 m the temperature variation, during the day, is 31.8 K and at 0.60 m is 6.8 K, Figure 30. Figure 31 reports the

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228

F. Gori and S. Corasaniti

temperature variations below 0.60 m. At 1 m the temperature variation is 0.8 K, at 1. 30 m is 0.2 K while is negligible at 2 m, Figure 31. 224,50 z = 0.8 m z=2m z=4m z = 1.3 m

224,00

T (K)

223,50

z=1m z=3m z=5m

223,00 0

5

10

15

20

222,50 222,00 221,50 t (h)

Figure 31. Temperature variations in the frozen soil analogue, type BF, below 0.60 m [29].

The temperature variations are presented versus the depth of the soil analogue, type BF, in Figure 32, at several hours during the Martian day. Figure 32 shows that the temperature variations are detectable, inside the frozen soil analogue, up to a depth of about 1.3 0 m. In conclusion, the temperature variation is more penetrating in the frozen soil because the thermal diffusivity is more than one order of magnitude higher than in the dry soil.

t=4h t = 12 h t = 20 h

250

t=8h t = 16 h t = 24 h

T (K)

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300

200

0

1

2

3

4

5

150 z (m)

Figure 32. Temperature distribution in the frozen soil analogue, type BF, versus depth at several hours [29].

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Thermal Properties and Temperature Variations…

229

2. Relation between Temperature Distribution and Physical Properties The temperature variations in the soil analogues, type A, are reported in Table 3, at several depths. Table 3. Temperature variation, ΔT = Tmax – T min, in the soil analogues, type A, at several depths [38].

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Depth z(m) 0.10 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.30 2.00 3.00 4.00 5.00

Type AF ΔT (K) 123.7 102.4 84.6 69.9 57.8 47.7 32.6 22.2 12.6 3.3 0.5 0.1 0.0

Type AD ΔT (K) 82.0 44.9 24.6 13.4 7.4 4.0 1.2 0.4 0.1 0.0 0.0 0.0 0.0

Table 3 shows the comparison between the temperature variations in the dry and frozen soil analogues, type AD and AF. At small depths, i.e. up to 0.50 m, the temperature variation is of the same order of magnitude, although higher in the frozen soil, but, at larger depths, i.e. below 0.60 m, Table 3 shows that the temperature variations are higher of one or two orders of magnitudes, in the frozen soil as compared to the dry soil. Table 4. Temperature variation, ΔT= Tmax – T min, in the soil analogues, type B, at several depths [29]. Depth z(m) 0.10 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.30 2.00 3.00 4.00 5.00

Type BF ΔT (K) 89.5 53.4 31.9 19.0 11.4 6.8 2.4 0.9 0.2 0.0 0.0 0.0 0.0

Type BD ΔT (K) 8.7 0.5 0.03 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

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230

F. Gori and S. Corasaniti

Table 4 reports the temperature variations in the soil analogue, type B, at several depths. The temperature variations are lower than those of Table 3 because of the lower thermal diffusivity but they are detectable in the frozen soil analogue, type BF, up to 1.30 m while in the type BD they are detectable only up to 0.20 m. Table 5. Derivative of the temperature variation, ΔT/Δz (K/m), for the soil analogues, type A, at several depths [38]. Depth

Type AF

Type AD

z(m)

ΔT/Δz (K/m)

ΔT/Δz (K/m)

0.15

213.0

371.0

0.25

178.0

203.0

0.35

147.0

112.0

0.45

121.0

60.0

0.55

101.0

34.0

0.70

75.5

14.0

0.90

52.0

4.0

1.15

32.0

1.1

1.65

13.3

0.1

2.50

2.8

0.0

3.50

0.4

0.0

4.50

0.1

0.0

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The data of Table 3 have been used to calculate the variation of ΔT = (Tmax – Tmin) with the depth or, in other words, the derivative of ΔT with z, ∂T/∂z=ΔT/Δz (K/m). The results for the soil analogues, types A and B, are reported in Table 5-6. Table 6. Derivative of the temperature variation, ΔT/Δz (K/m), for the soil analogues, type B, at several depths [29]. Depth

Type BF

Type BD

z(m)

ΔT/Δz (K/m)

ΔT/Δz (K/m)

0.05

604.9

1412.8

0.15

360.9

82.1

0.25

215.5

4.8

0.35

128.5

0.3

0.45

76.8

0.02

0.55

45.8

0.70

21.8

0.0 0.0

0.90

7.8

0.0

1.15

2.2

0.0

1.65

0.3

0.0

2.50

0.0

0.0

3.50

0.0

0.0

4.50

0.0

0.0

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Thermal Properties and Temperature Variations…

231

Table 5 shows the comparison between the derivative of the temperature variation, ΔT/Δz, in the two soil analogues, types AF and AD. At small depths, i.e. up to 0.25 m, ΔT/Δz is higher for the dry soil than for the frozen soil. At greater depths, i.e. below 0.35 m, Table 5 shows that ΔT/Δz are higher for the frozen soil. Below 0.70 m, ΔT/Δz are higher for the frozen soil of one order of magnitude and are detectable up to a depth of 3.5 m while in the type AD they are detectable only up to 1.15 m. Table 6 reports the derivative of the temperature variation, ΔT/Δz, in the soil analogue, type B, at several depths. Table 6 shows that at very small depths, i.e. up to 0.05 m, ΔT/Δz is higher for the dry soil than for the frozen soil. At greater depths, i.e. below 0.15 m, ΔT/Δz is higher for the frozen soil. For the type BF ΔT/Δz are detectable up to a depth of 1.65 m while in the type AD they are detectable only up to 0.35 m.

3. CONCLUSIONS

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The temperature variation along the depth of a soil analogue during a Martian day, and the knowledge of its porosity, can be employed to evaluate the physical conditions, i.e. either in dry or frozen state. The temperature variations in a frozen Martian soil analogue with porosity equal to 0.2 are detectable up to a depth of about 3 m while in a dry soil analogue they are detectable only up to 1 m. The derivative of the temperature variation with the depth is detectable in a frozen soil analogue up to a depth of 3.5m while in a dry soil analogue it is detectable only up to 1.15 m. The temperature variations in a frozen Martian soil analogue, with porosity equal to 0.5, are detectable up to a depth of about 1.3 m while in a dry soil analogue they are detectable only up to 0.2 m. The derivative of the temperature variation with the depth is detectable in a frozen soil analogue up to a depth of 1.65 m while in a dry soil analogue it is detectable only up to 0.35 m.

REFERENCES [1] [2] [3] [4] [5] [6]

[7]

Clifford, S.M. and S.P. Fanale, The Thermal Conductivity Of The Martian Crust, Lunar Planet Sci. Conf. XVI, 144-145, 1985. Clifford, S.M., A Model For The Hydrologic And Climate Behavior Of Water On Mars, Jour. Geophys. Res. 98 (E6), 10973-11016, 1993. Fanale F.P., Martian Volatiles: Their Degassing History And Geochemical Fate, Icarus,28,179-202, 1976. Crescenti G.H., Analysis Of Permafrost Depths On Mars, NASA Tech.Mem. TM 79730, 1984. Fanale, F.P., J.R. Salvail, A.P. Zent, and S.E. Postawko, Global Distribution And Migration Of Subsurface On Mars, Icarus, 67,1-18, 1986. Solomon, S.C. and J. W. Head, Heterogeneities In The Elastic Lithosphere Of Mars: Constraints In The Heat Flow And Internal Dynamics, Jour. Geophys. Res. 95,1107311083, 1990. Gori F., A Theoretical Model for Predicting the Effective Thermal Conductivity of Unsaturated Frozen Soils, The 4th International Conference on Permafrost, Fairbanks, (U.S.A.), 363-368, 1983.

Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

232 [8]

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[14]

[15]

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[16]

[17] [18]

[19]

[20]

[21]

[22] [23]

F. Gori and S. Corasaniti Gori F., Prediction of Effective Thermal Conductivity of Two-Phase Media (including Saturated Soils), XVI International Congress of Refrigeration, Paris, B1-26, 525531, 1983. Gori F., On the Theoretical Prediction of the Effective Thermal Conductivity of Bricks, The Eight International Heat Transfer Conference, S. Francisco (U.S.A.), II, 627632, 1986. Gori F. and Corasaniti S., Considerations On A Theoretical Method For The Prediction Of The Thermal Conductivity Of Porous Media, 17th UIT National Heat Transfer Conference, Ferrara (Italy), vol. II, pp. 683-694, 1999. Pande R.N. and Gori F., Effective Media Formation and Conduction Through Unsaturated Granular Materials, International Journal of Heat and Mass Transfer, 30, 5, 993-1000, 1987. Pande R.N., Chaudhary D.R. and Gori F., Effective Medium in Dispersed Systems, Pramana - J. Phys., 29, 2, 217-223, 1987. Gori F. and Corasaniti S., A Theoretical Model for Combined Radiation Conduction Heat Transfer in Porous Media, 19th UIT National Heat Transfer Conference, Modena (Italy), 371-376, 2001. Gori F. and Corasaniti S., Theoretical Thermal Conductivity of Soils at High Temperatures, IMECE, International Mechanical Engineering Congress and Exposition, HTD-24152, ASME, new York, NY, pp.1-8, 2001. Gori F. and Corasaniti S., Theoretical Prediction of the Thermal Conductivity of Soils at Moderately High Temperatures, Journal of Heat Transfer, ASME,126 (6), 10011008, 2002. Gori F. and Corasaniti S., Theoretical Prediction of the Effective Thermal Conductivity of Particulate Materials in Extraterrestrial Conditions and of Foams at Low Density, Microgravity and Space Station Utilization, Vol. 2, Nr. 2-3-4, pp 23-24, JuneDecember 2001. Gori F. and Corasaniti S., Temperature Variations Inside Dry And Partially Frozen Mars Soils, IEEE Aerospace Conference, March 8-15, Big Sky, Montana,2003. Gori F. and Corasaniti S., Theoretical Prediction Of The Thermal Conductivity And Temperature Variation Inside Mars Soil Analogues, Planetary and Space Science, 52, 91-99, 2004. Gori F., Marino C. and Pietrafesa M., Experimental Measurements and Theoretical Predictions of the Effective Thermal Conductivity of Porous Saturated Two Phases Media, International Communications in Heat and Mass Transfer, Vol. 28, N. 8 (01), pp. 1091-1102, 2001. Gori F. and Corasaniti S., On the Effective Thermal Conductivity of Dry Olivine, 5th World Conference on Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, vol.2,1257-1262, 2001. Gori F. and Corasaniti S., Experimental Measurement and Theoretical Prediction of the Thermal Conductivity of Two and Three Phases Water Olivine Systems, European Conference on Thermophysical Properties, ECTP, London, 2002. Gori F. and Corasaniti S., Experimental Measurement of the thermal conductivity of porous materials, 57 th ATI Congress, pp. II, 93-98, 2002. Gori F. and Corasaniti S., Experimental Measurements and Theoretical Prediction of the Thermal Conductivity of Two-and Three-Phase Water/Olivine Systems,

Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

Thermal Properties and Temperature Variations…

[24]

[25]

[26] [27]

[28]

[29] [30] [31]

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International Journal of Thermophysics, Vol. 24, No. 5, pp. 1339-1353, September 2003. Gori F. and Corasaniti S., The Measurement Of The Thermal Diffusivity To Detect The Amount Of Water In Olivine, Proceedings of the 14th International Symposium on Transport Phenomena, Bali, Indonesia, pp. 581-585, 6-10 July, 2003. Tarnawski V.R., Gori F., Wagner B. and Buchan G.D., Modelling Approaches To Predicting Thermal Conductivity Of Soils At High Temperatures, International Journal of Energy Research, n 24, pp. 403-423, 2000. Tarnawski V.R. and Gori F., Enhancement of the Cubic Cell Soil Thermal Conductivity Model, International Journal of Energy Research, n. 26, 143,157, 2002. Tarnawski V.R., Leong W.H., Gori F., Buchan G.D. and Sundberg J., Inter-Particle Contact Heat Transfer In Soil Systems At Moderate Temperatures, International Journal of Energy Research, 26:1345-1358, 2002. Leong W. H., Tarnawski V. R., Gori F., Buchan G. D. and Sundberg J., Inter-Particle Contact Heat Transfer Models: An Extension To Soils At Elevated Temperatures, International Journal of Energy Research, 2005; 29:131-144. Gori F. and Corasaniti S., Extraterrestrial Soil Thermophysical- Properties, EST-PRO, Final Report, European Space Agency, 2002. Map of Mars and landing sites of Viking and Pathfinder mission (http://mars.jpl.nasa.gov/mgs/target/marsmap1b.jpg) Scott D.H. and M.H. Carr, Geological Map Of Mars, 1:25.000.000, Miscellaneous investigations series, U.S. Geological Survey, 1978. Rieder R., T. Economou, H. Wanke, A. Turkevich, J. Crisp, J. Bruckner, G. Dreibus and H.Y. McSween Jr., The Chemical Composition Of Martian Soil And Rocks Returned By The Mobile Alpha Proton X-Ray Spectrometer: Preliminary Results From The X-Ray Mode, Science, 278, 1771-1774, 1997. (http://mars.jpl.nasa.gov/MPF/ops/NEcolor_annot.jpg). Withers P. and G.A. Neumann, Enigmatic Northern Plains Of Mars, Nature, 410, 651, 2001. Mangold N., P. Allemand, P. Duval, Y. Geraud and P. Thomas, Experimental And Theoretical Deformation Of Ice-Rock Mixtures: Implications On Rheology And Ice Content Of Martian Permafrost, Planetary and Space Science, 50, 385-401, 2002. Britt D.T., R. Anderson, J.F. Bell III, J. Crisp, T. Economou, K.E. Herkenhoff, M.B. Madsen, H.Y. Mc Sween, S. Murchie, R. Reid, R. Rieder, R.B. Singer and L. Soderblom, The Mineralogy Of The Mars Pathfinder Landing Site, 29th Lunar and Planetary Science conference, 1998. (http://mars.jpl.nasa.gov/MPF/ops/hap_5.jpg). Reprinted from [18] with permission from Elsevier. Luikov A.V., Heat And Mass Transfer In Capillary-Porous Bodies. Pergamon Press, 1966.

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In: Space Science Research Developments Editors: J. C. Henderson and J. M. Bradley

ISBN: 978-1-61209-086-3 c 2011 Nova Science Publishers, Inc.

Chapter 8

O N T HE 5D E XTRA -F ORCE ACCORDING TO B ASINI -C APOZZIELLO-P ONCE D E L EON F ORMALISM AND THE E XPERIMENTAL R ESEARCH OF E XTRA D IMENSIONS O N -B OARD I NTERNATIONAL S PACE S TATION (ISS) U SING L ASER B EAMS∗ Fernando Loup† Residencia de Estudantes Universitas Lisboa, Portugal

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Abstract We analyze the possibility of Experimental Research of Extra Dimensions OnBoard International Space Station (ISS) by using a Satellite carrying a Laser device(optical Laser) on the other side of Earth Orbit targeted towards ISS.The Sun will be between the Satellite and the ISS so the Laser will pass the neighborhoods of the Sun at a distance R in order to reach ISS. The Laser beam will be Gravitationally Bent according to Classical General Relativity and the Extra Terms predicted by Kar-Sinha in the Gravitational Bending Of Light due to the presence of Extra Dimensions can perhaps be measured with precision equipment.By computing the Gravitational Bending according to Einstein we know the exact position where the Laser will reach the target on-board ISS.However if the Laser arrives at ISS with a Bending different than the one predicted by Einstein and if this difference is equal to the Extra Terms predicted by Kar-Sinha then this experience would proof that we live in a Universe of more than 4 Dimensions.We demonstrate in this work that ISS have the needed precision to detect these Extra Terms(see eq 137 in this work).Such experience would resemble the measures of the Gravitational Bending Of Light by Sir Arthur Stanley Eddington in the Sun Eclipse of 1919 that helped to proof the correctness of General Relativity although in ISS case would have more degrees of accuracy because we would be free from the interference of Earth Atmosphere. ∗ A version of this chapter also appears in Journal of Magnetohydrodynamics, Plasma and Space Research Volume 14, Number 3/4, published by Nova Science Publishers, Inc. It was submitted for appropriate modifications in an effort to encourage wider dissemination of research. † E-mail address: [email protected] Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

236

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1.

Fernando Loup

Introduction

Much has been said about the so-called Extra Dimentional nature of the Universe.It was first proposed by Theodore Kaluza and Oskar Klein in 1918 1 in an attempt to unify Gravity and Electromagnetism. However the physical nature of the Extra Dimension was not well defined in a clear way. They used a so-called Compactification Mechanism to explain why we cannot see the 5D Extra Dimension but this mechanism was not clearly understood. Later on and with more advanced scientific knowledge other authors appeared with the same idea under the exotic concept of the so-called BraneWorld. In the BraneWorld concept our visible Universe is a 3 + 1 Dimensional sheet of Spacetime :a Brane involved by a Spacetime of Higher Dimensional nature. This idea came mainly from Strings Theory where Gravitational Forces are being represented by an Elementary Particle called Graviton while other interactions are represented by other sets of Elementary Particles:Electromagnetic Interaction is being represented by an Elementary Particle called Photon. According to Strings Theory Gravitons are Closed Loops and can leave easily our 3 + 1 Dimensional Spacetime and escape into the Extra Dimensions while Photons are Open Strings and are ”trapped” in our 3 + 1 Spacetime. To resume: In 1918 Kaluza-Klein tried to unify Gravity and Electromagnetism and Einstein among other scientists were interested in the same thing. But however inside the framework of the so-called Strings Theory how can a Closed Loop be unified with a Open String? How can an Interaction that with some degrees of freedom is allowed to probe the Extra Dimensional Spacetime be unified with Interactions confined to our 3 + 1 Spacetime? A puzzle to solve.So the so-called Strings Theory is trying to unify Gravity with Electromagnetism and other Interactions but the framework is not completed or not well understood. On the other hand the Compactification Mechanism in the original Kaluza-Klein theory explains why we cannot see beyond the 3 + 1 Spacetime because the Extra Dimensions are Compactified or Curled Up but it does not explains why we have 3 + 1 Uncurled or Uncompactified Dimensions while the remaining Extra Dimensions are Curled and what generates this Compactification Mechanism in the first place? We adopt in this work the so-called Basini-Capozziello Ponce De Leon formalism coupled to the formalisms of Mashoon-Wesson-Liu and Overduin-Wesson in which Extra Dimensions are not compactified but opened like the 3 + 1 Spacetime Dimensions we can see.There are small differences between these formalisms but Basini-Capozziello Ponce De Leon admits a non-null 5 RAB Ricci Tensor while the others make the Ricci Tensor 5 RAB = 02 but essentially these formalisms are mathematically equivalent.In the Basini-Capozziello Ponce de Leon the ordinary Spacetime of 3 + 1 Dimensions is embedded into a large Higher Dimensional Spacetime,however in a flat or Minkowsly Spacetime the Spacetime Curvature eg Ricci and Einstein Tensors of the Higher Dimensional Spacetime reduces to the same Ricci and Einstein Tensors of a 3 + 1 Spacetime. This explains without Compactification Mechanisms why we cannot see beyond the 3 + 1 Spacetime:our everyday Spacetime is essentialy Minkowskian or flat 3 and a 5D Ricci Tensor reduces to a Ricci Tensor of a 3 + 1 1 see

[21] for an excellent account on Kaluza-Klein History [21] pg 31 after eq 48 and see [2] eq 20 [11] eq 21 and [20] eq 8. We prefer to assume that exists matter in the 5D due to the last section of [20] about the particle Z 3 we consider our Spacetime as a Schwarzschild Spacetime however at a large distance from the Gravitational Source it reduces to a Minkowsky SR Spacetime due to a large R and the ratio M R tends to zero 2 see

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On The 5D Extra-Force According to Basini-Capozziello-Ponce De Leon ...

237

Spacetime. On the other hand in this formalism all masses,electric charges and spins of all the Elementary Particles seen in 4D are function of a 5D rest-mass coupled with Spacetime Geometry. We can observe in 4D Spacetime a multitude of Elementary Particles with different masses,electric charges or spins but according to the Basini-Capozziello Ponce De Leon formalism eg the 5D to 4D Dimensional Reduction all these different Elementary Particles with all these 4D rest-masses,electric charges or spins are as a matter of fact a small group of 5D Elementary Particles with a 5D rest-mass and is the geometry of the 5D Spacetime coupled with the Dimensional Reduction that generates these apparent differences. Hence two particles with the same 5D rest-mass M5 can be seen in 4D with two different rest-masses m0 making ourselves think that the particles are different but the difference is apparent and is generated by the Dimensional Reduction from 5 D to 4D.Look to the set of equations below:We will explore in this work these equations with details but two particles with the same 5D rest mass M5 can be seen in the 4D with two different rest masses m q0 if the Dimensional Reduction from 5D to 4D or the Spacetime Geometric Coupling 2 1 − Φ2 ( dy ds ) is different for each particle.

All the particles in the Table of

4

Elementary Particles given below with non-zero rest-mass m0 seen in 4D canqas a matter of

2 fact have the same rest-mass M5 in 5D and the Dimensional Reduction term 1 − Φ2 ( dy ds ) generates the apparent different 4D rest-masses.This is very attractive from the point of view of a Unified Physics theory.There exists a small set of particles in 5 D and all the huge number of Elementary Particles in 4D is a geometric projection from the 5D Spacetime into a 4D one([2] eq 20,[11] eq 21 and [20] eq 8)([2] eq 14,[20] eq 1 and eq 2).

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m0 = q

Particle u d s c b t e− µ− τ− νe νµ ντ γ gluon W+ Z graviton 4 extracted

spin (~) B 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1 1 1 1 2

1/3 1/3 1/3 1/3 1/3 1/3 0 0 0 0 0 0 0 0 0 0 0

M5

(1)

2 1 − Φ2 ( dy ds )

dS2 = guv dxu dxv − Φ2 dy2

(2)

dS2 = ds2 − Φ2 dy2

(3)

L

T

T3

S

C

B∗

charge (e)

m0 (MeV)

antipart.

0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0

1/2 1/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1/2 −1/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0

+2/3 −1/3 −1/3 +2/3 −1/3 +2/3 −1 −1 −1 0 0 0 0 0 +1 0 0

5 9 175 1350 4500 173000 0.511 105.658 1777.1 0(?) 0(?) 0(?) 0 0 80220 91187 0

u d s c b t e+ µ+ τ+ νe νµ ντ γ gluon W− Z graviton

from the Formulary Of Physics by J.C.A. Wevers available on Internet

Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

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238

Fernando Loup

We employ in this work the Basini-Capozziello Ponce De Leon Formalism to demonstrate that while in flat or Minkowsky Spacetime the curvature in 5 D reduces to a one in 4D due to the Dimensional Reduction suffered by the Ricci and Einstein Tensors and we cannot tell if we live in a 5D or in a 4D Universe due to the absence of Strong Gravitational Fields but in an environment of Strong Gravity the 5 D Ricci and Einstein Tensors cannot be reduced to similar 4D ones and the Curvature of a 5D Spacetime is different than the one of a 4D because the 5D extra terms in the Ricci and Einstein Tensors have the terms of the Strong Gravitational Field and cannot be reduced to 4 D.Perhaps the study of the conditions of extreme Gravitational Fields in large Black Holes will tell if we live in a 5 D Universe or in a 4D one.We also demonstrate that the International Space Station (ISS) can perhaps be used to study the Experimental Detection Of Extra Dimensions using the Gravitational Bending of Light of the Sun or the similar for large Black Holes.Higher Dimensional Spacetimes affects the Gravitational Bending Of Light adding Extra Terms as predicted by KarSinha(see abstact of [3]).(see also pg 73 in [20]).International Space Station is intended to be a laboratory designated to test the forefront theories of Physics and ISS can provide a better environment for Physical experiences without the interference of Earth secondary effects that will disturb careful measures specially for gravity-related experiments.(see pg 602-603 for the advantage of the free-fall conditions for experiments in [8]) We argue that the Gravitational Bending Of Light first measured by Sir Arthur Stanley Eddington in a Sun Eclipse in 1919 is widely known as the episode that made Einstein famous but had more than 30 percent of error margin. In order to find out if the Extra Dimension exists(or not) we need to measure the factor C2 of Kar-Sinha coupled to a Higher Dimensional definition of m0 with a accurate precision of more than 4ω < 2.8 × 10−4 .( see pg 1783 in [3]).ISS can use the orbit of the Moon to create a ”artificial eclipse” 5 to get precise measurements of the Gravitational Bending Of Light and according to Kar-Sinha detect the existence of the 5D Extra Dimension although in this work we will propose a better idea.ISS will be used to probe the foundations of General Relativity and Gravitational Bending of Light certainly will figure out in the experiences(see pg 626 in [6]).ISS Gravitational shifts are capable −6 to detect measures of 4ω ω with Expected Uncertainly of 12 × 10 (see pg 629 Table I in [6]) smaller than the one predicted by Kar-Sinha for the Extra Dimensions although the ISS shifts are red-shifts and not Bending of Light similar precision can be achieved.Also red-shifts are due to a time delay in signals and the 5 D can also affect this measure as proposed by Kar-Sinha.(see pg 1782 in [3]).The goal of ISS is to achieve a Gravitational Shift Precision of 2.4 × 10−7 (see pg 631 Table II in [6]) by far more than enough to detect the existence of the 5D Extra Dimension.We believe that Gravitational Bending Of Light can clarify the question if we live in a Higher Dimensional Universe or not.A small deviation in a photon path different than the one predicted by Einstein can solve the quest for Higher Dimensional Spacetimes. We propose here the use of a Satellite with a Laser beam in the other side of the Earth Orbit targeted towards ISS. The Laser would pass the neighborhoods of the Sun at a distance R in order to reach ISS and would be Gravitational Bent according to General Relativity and affected by the Kar-Sinha Extra Terms due to the presence of the Higher Dimensional Spacetime.Hence the Gravitational Bending Of Light can be measured with precision equipment of ISS.We demonstrate in this work that ISS have the needed pre5 we

agree that idea is weird but is better than to wait for a Sun Eclipse in the proper conditions

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On The 5D Extra-Force According to Basini-Capozziello-Ponce De Leon ...

239

cision to detect these Extra Terms(see eq 137 in this work).By computing the Gravitational Bending according to Einstein we know where the photons would reach ISS.However if the photons arrives at ISS with a Bending angle different than the one predicted by Einstein and if this difference is equal to the Kar-Sinha Extra Terms then we would have a proof that we live in a Universe of more than 4 Dimensions.We also examine Gravitational Red Shifts affected by the presence of the Extra Dimension.This experience made on-board ISS would have the same impact of the Sir Arthur Stanley Eddington measures of Gravitational Bending Of Light in the Sun Eclipse of 1919 and if the result is ”positive” then the International Space Station ISS would change forever our way to see the Universe.

2.

The Basini-Capozziello Ponce De Leon Formalism and Resemblances with Mashoon-Wesson-Liu and Overduin-Wesson Formalisms

Basini-Capozziello Ponce de Leon argues that our 3 + 1 Dimensional Spacetime we can see is a Dimensional Reduction from a larger 5D one and according to a given Spacetime Geometry we can see(or not) the 5D Extra Dimension .This is also advocated in the almost similar Formalisms of Mashoon-Wesson-Liu and Overduin-Wesson. A 5 D Spacetime metric is defined as([1] eq 32,[5] eq 18,[20] eq 62 and [9] pg 556 Section 2) and contains all the 3 + 1 Spacetime Dimensions of our observable Universe plus the 5 D Extra Dimension.Then A, B = 0, 1, 2, 3, 4 where 0, 1, 2, 3 are the Dimensons of the 4D Spacetime and 4 is the script of the 5D Extra Dimension(see [5] pg 2225 after eq 18 and again [9] pg 556).

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dS2 = gAB dxA dxB

(4)

Note that this equation is common not only to Basini-Capozziello-Ponce De Leon but also to Mashoon-Wesson-Liu and Overduin-Wesson Formalisms.These formalisms advocates the Dimensional Reduction from 5D to 4D6 and we need to separate in this Spacetime Metric both the 3 + 1 Components of our visible Universe and the components of the 5 D Extra Dimension.The resulting equation would then be([1] eq 56,[2] eq 12 and 14,[5] eq 42,[9] eq 32 and 33 without vector potential,[11] eq 10, [12] pg 308,[20] eq 109 and [13] pg 1346)789 dS2 = gAB dxA dxB = gαβ dxα dxβ − Φ2 dy2

(5)

Note that when the Warp Field 10 Φ = 1 the Spacetime Metric becomes: dS2 = gAB dxA dxB = gαβ dxα dxβ − dy2

(6)

6 We will skip a tedious definition and concentrate on the Dimensional Reduction.A unfamiliar reader must study first [1] pg 122 Section 2.2 to pg 127,[5] pg 2225 Section 3 to pg 2229 and [20] pg 1434 Section 4 to pg 1441.see also [21] pg 29 Section 6 to pg 31 7 [12] with spacelike signature 8 see [13] pg 1341 the Campbell-Magaard Theorem 9 see [1] eq 57,[5] eq 43 and [21] eq 47. 10 the term ”Warp” appears in pg 1340 in [2]

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240

Fernando Loup

Writing the 5 Rαβ Ricci Tensor and the 5 R Ricci Scalar according to Basini-Capozziello using these equations:(α, β = 0, 1, 2, 3)([1] eq 58, [5] eq 44 and [20] eq 111.See also [21] eq 48 for 5 Rαβ )11 5

5

gµν gµν,4 gαβ,4 Φ,a;b 1 Φ,4 gαβ,4 λµ − ( − g + g g g − ) αβ,44 αλ,4 βµ,4 Φ 2Φ2 Φ 2

(7)

gµνgµν,4 gαβ,4 Φ,a;b αβ 1 αβ Φ,4 gαβ,4 λµ g − g ( − g + g g g − ) αβ,44 αλ,4 βµ,4 Φ 2Φ2 Φ 2

(8)

Rαβ = Rαβ −

R = R−

Simplifying for diagonalized metrics we should expect for: 5

5

Rαβ = Rαβ −

R = R−

gµν gµν,4 gαβ,4 Φ,a;b 1 Φ,4 gαβ,4 − ( − g + ) αβ,44 Φ 2Φ2 Φ 2

gµν gµν,4 gαβ,4 Φ,a;b αβ 1 αβ Φ,4 gαβ,4 g − g ( − g + ) αβ,44 Φ 2Φ2 Φ 2

(9)

(10)

Note that the term 4 Φ = ∇α Φα = gαβ (Φα );β = gαβ [(Φα)β − ΓKβα ΦK ] corresponds to the D’Alembertian in 4D1213 so we can write for the Ricci Scalar the following expression: gµν gµν,4 gαβ,4 1 αβ Φ,4 gαβ,4 g ( − g + ) (11) αβ,44 Φ 2Φ2 Φ 2 If according to Basini-Capozziello the terms gαβ have no dependance with respect to to the Extra Coordinate y after the Reduction from 5 D to 4D then all the derivatives with respect to y vanish and we are left out with the following expression for the Ricci Scalar:([1] eq 59,[5] eq 45 and [20] eq 116)).We will analyze this in details when studying the 5 D to 4D Dimensional Reduction.

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5

R = R−

4 Φ

−

5

R = R−

4 Φ

Φ

(12)

Writing the remaining Ricci Tensors we should expect for([21] eq 48) 14 :

Rˆ α4 =

∇β (∂α Φ) 1 − Φ 2Φ2



∂4 Φ ∂4 gαβ − ∂4 gαβ Φ ! gγδ ∂4 gγδ ∂4 gαβ γδ + g ∂4 gαγ ∂4 gβδ − , 2

Rˆ αβ = Rαβ −

∂β gβγ ∂4 gγα g44 gβγ (∂4 gβγ ∂α g44 − ∂γ g44 ∂4 gαβ ) + 4 2

the terms α,λ,µ,β and ν are all equal pg 129 in [1] and pg 2230 in [5] 13 see also pg 311 in [12] 14 adapted from the arXiv.org LaTeX file of [21] eq 48.Note the difference between the first term R ˆ αβ between [21] eq 48 ,[1] eq 58,[5] eq 44 and [20] eq 111 11 Working with diagonalized metrics 12 see

Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

On The 5D Extra-Force According to Basini-Capozziello-Ponce De Leon ... gβγ ∂4 (∂β gγα ) ∂α gβγ ∂4 gβγ gβγ ∂4 (∂α gβγ ) − − 2 2 2 gβγ gδε ∂4 gγα ∂β gδε ∂4 gβγ ∂α gβγ + + , 4 4 ∂4 gαβ ∂4 gαβ gαβ ∂4 (∂4 gαβ ) = ΦΦ − − 2 2 ∂4 Φ gαβ ∂4 gαβ gαβ gγδ ∂4 gγβ ∂4 gαδ + − , 2Φ 4

241

+

Rˆ 44

(13)

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where “” is defined as usual (in four dimensions) by Φ ≡ gαβ ∇β (∂α Φ). Note that the Overduin-Wesson definition is exactly equal to the one presented by Basini-Capozziello Ponce De Leon.Both Formalisms are equivalent except that BasiniCapozziello-Ponce De Leon admits a 5 RAB not null. Working with diagonalized Spacetime Metrics of signature (+,-,-,-,-) 15 the Ricci Tensors would be written as:  ∇α (∂α Φ) 1 ∂4 Φ ∂4 gαα ˆ − − ∂4 gαα Rαα = Rαα − 2 Φ 2Φ Φ  gαα ∂4 gαα ∂4 gαα , + gαα ∂4 gαα ∂4 gαα − 2 g44 gαα ∂α gαα ∂4 gαα Rˆ α4 = (∂4 gαα ∂α g44 − ∂α g44 ∂4 gαα) + 4 2 gαα ∂4 (∂α gαα) ∂α gαα ∂4 gαα gαα ∂4 (∂α gαα ) − − + 2 2 2 gαα gαα ∂4 gαα ∂α gαα ∂4 gαα ∂α gαα + + , 4 4 ∂4 gαα ∂4 gαα gαα ∂4 (∂4 gαα ) − Rˆ 44 = ΦΦ − 2 2 αα αα ∂4 Φ g ∂4 gαα g gαα ∂4 gαα ∂4 gαα + − , 2Φ 4

(14)

Compare the first of the Ricci Tensors above with [20] eq 113. Note that the MashoonWesson-Liu Formalism is exactly equal to the Basini-Capozziello-Ponce De Leon and Overduin-Wesson Formalisms. Look to the equations [9] eq 32 and 33 without vector potential.Compare with dS2 = gαβ dxα dxβ − Φ2 dy2

(15)

One can see that we already presented this equation proving without shadows of doubt that the three formalisms are equivalent.Mashoon-Wesson-Liu in [9] pg 557 makes g44 = −Φ2 . They also makes g44 = −1 (see pg 558)giving the equation below: dS2 = gαβ dxα dxβ − dy2 15 α = β =

γ=δ=ε

Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

(16)

242

Fernando Loup

We already presented this equation:is the 5 D Spacetime Geometry without the Warp Field. One thing advocated by Mashoon-Wesson-Liu and Basini-Capozziello Ponce De Leon is the fact that the 5D Extra Dimension generate a 5D Extra Force that can be detected in 4D.(see [9] abstract and pgs 556,562 look to eq 24,pg 563 look to eq 31 and the comment below this equation,pg 565 eq 38,39 and the comments on the de-acceleration,pg 566 definition of β,pg 567 look to the comment of a small force but detectable).(see also [2] abstract and pgs 1336,1337,1341 eq 20, pg 1342 eq 25,pg 1343 eq 30).We will now prove that the 5D Extra Force in both formalisms is equivalent. If we have a Spacetime Geometry defined as: dS2 = gαβ dxα dxβ − Φ2 dy2

(17)

dS2 = gαβ dxα dxβ − [φ(t, x)χ(y)]2dy2

(18)

where we defined the Warp Field Φ according to Basini-Capoziello([1] eq 76,[5] eq 70 and [20] eq 132 ) we have two choices: • M5 the 5D Mass is not zero and we have matter in the 5D Extra Dimension according to one of the Ponce De Leon Options making also 5 RAB the Ricci Tensor in 5D not null.

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

• M5 The 5D Mass is zero and we have no matter in the 5D Extra Dimension according to another of the Ponce De Leon Options making also 5 RAB the Ricci Tensor in 5D null. Overduin-Wesson and Mashoon-Wesson-Liu formalisms agree with the second option of Ponce De Leon.(see [21] pg 31 after eq 48 and [9] pg 557 eq 2) According to Ponce De Leon in option 1 if we have a rest-mass in 5 D M5 this rest-mass will be seen in 4D as a rest-mass m0 as follows([2] eq 20,[11] eq 21 and [20] eq 8): m0 = q m0 = q

M5 2 1 − Φ2 ( dy ds )

M5

(19)

(20)

2 1 − [φ(t, x)χ(y)]2( dy ds )

We have Quantum Chromodynamics for Quarks and a Quantum Electrodynamics for Leptons like Electron but as a matter of fact two particles with the same rest-mass in 5 D M5 can appear in our 4D Spacetime with different rest masses m0 making one appear as a Quark and the other as a Lepton q the Dimensional Reduction from 5 D to 4D q depending on

2 2 dy 2 or the Spacetime Coupling 1 − Φ2 ( dy ds ) , 1 − [φ(t, x)χ(y)] ( ds ) although in 5D both particles are the same. This is a very interesting perspective of Modern Physics.Why Quantum Electrodynamics and Quantum Chromodynamics in 4D while as a matter of fact in 5D both are the same? Look again to the table below 16 : 16 extracted

from the Formulary Of Physics by J.C.A. Wevers available on Internet

Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

On The 5D Extra-Force According to Basini-Capozziello-Ponce De Leon ... Particle u d s c b t e− µ− τ− νe νµ ντ γ gluon W+ Z graviton

spin (~) B 1/2 1/3 1/2 1/3 1/2 1/3 1/2 1/3 1/2 1/3 1/2 1/3 1/2 0 1/2 0 1/2 0 1/2 0 1/2 0 1/2 0 1 0 1 0 1 0 1 0 2 0

L 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0

T 1/2 1/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

T3 1/2 −1/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

S 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

C 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

B∗ 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0

charge (e) +2/3 −1/3 −1/3 +2/3 −1/3 +2/3 −1 −1 −1 0 0 0 0 0 +1 0 0

m0 (MeV) 5 9 175 1350 4500 173000 0.511 105.658 1777.1 0(?) 0(?) 0(?) 0 0 80220 91187 0

243

antipart. u d s c b t e+ µ+ τ+ νe νµ ντ γ gluon W− Z graviton

The Extra Force generated by the 5D seen in 4D for a massive 5D particle M5 seen in 4D as m0 according to Ponce De Leon is defined as follows([2] eq 25 and [20] eq 15): 1 ∂guv dy ∂Φ dy 1 dm0 = − uu uv − Φuu u ( )2 m0 ds 2 ∂y ds ∂x ds

(21)

We have here two choices: • The Warp Field Φ = [φ(t, x)χ(y)]([1] eq 76,[5] eq 70 and [20] eq 132 ) is not null and we have a Warp Field coupled to the 5D Extra Dimension. Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

• The Warp Field Φ = 1 and we have no Warp Field at all. For a 5D Extra Dimension coupled with a Warp Field according to BasiniCapozziello([1] eq 76,[5] eq 70 and [20] eq 132 ) the Extra Force is given by: 1 ∂guv dy ∂φ(t, x)χ(y) dy 2 1 dm0 = − uu uv − φ(t, x)χ(y)uu ( ) m0 ds 2 ∂y ds ∂xu ds

(22)

1 dm0 1 ∂guv dy ∂φ(t, x) dy 2 = − uu uv − φ(t, x)χ(y)2uu ( ) m0 ds 2 ∂y ds ∂xu ds

(23)

If we have no Warp Field at all Φ = 1 the equation is simply: 1 ∂guv dy 1 dm0 = − uu uv m0 ds 2 ∂y ds

(24)

Note that this equation is exactly equal to the 5 D Extra Force equation as defined by Mashoon-Wesson-Liu([9] eq 24 and 38 because they used g44 = −Φ2 = −1 see [9] pg 558).Of course we expected this result because Basini-Capozziello Ponce De Leon and Mashoon-Wesson-Liu formalisms are equivalent. For the case of a null 5D rest-mass M5 the option 2 of Ponce De Leon the equation of the 5D Extra Force seen in 4D is given by([2] eq 30 and [20] eq 19):

Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

244

Fernando Loup 1 dm0 − 1 ∂guv u v uu ∂Φ = u u − m0 ds + 2Φ ∂y Φ ∂xu

(25)

1 dm0 − 1 ∂guv u v uu ∂φ(t, x) =+ u u − m0 ds 2φ(t, x)χ(y) ∂y φ(t, x) ∂xu

(26)

If we have a no Warp Field at all the equation becomes: 1 dm0 − 1 ∂guv u v = u u m0 ds + 2 ∂y

(27)

This is equal to ([9] eq 24 and 38 with dy = ds a Null-Like 5D Spacetime Geometry) According to the following Spacetime Geometry as defined by Basini-Capozziello Ponce De Leon,Mashoon-Wesson-Liu and Overduin-Wesson formalisms([1] eq 56,[2] eq 12 and 14,[5] eq 42 and [20] eq 1) dS2 = guv dxu dxv − Φ2 dy2

(28)

dS2 = ds2 − Φ2 dy2

(29)

ds2 = guv dxu dxv

(30)

We have three different types of Spacetime Geometries: • Timelike 5D Geometry

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dS2 > 0 99K ds2 − Φ2 dy2 > 0 99K ds2 > Φ2 dy2 99K

1 dy > ( )2 99K Timelike5D 2 Φ ds (31)

• Null-Like 5D Geometry dS2 = 0 99K ds2 − Φ2 dy2 = 0 99K ds2 = Φ2 dy2 99K

1 dy = ( )2 99K Nulllike5D Φ2 ds (32)

• Spacelike 5D Geometry dS2 < 0 99K ds2 − Φ2 dy2 < 0 99K ds2 < Φ2 dy2 99K

1 dy < ( )2 99K Spacelike5D 2 Φ ds (33)

Note that for a Null-Like 5D Geometry the equation of the 4D rest-mass m0 in function of the 5D rest-mass M5 is not valid.([2] eq 20,[11] eq 21 and [20] eq 8). m0 = q

M5 2 1 − Φ2 ( dy ds )

(34)

Hence we suppose that for a Null-Like 5D Geodesics the Extra Dimension have no mass at all or all matter in the 5D Extra Dimension obeys Timelike 5D Geometries. Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

On The 5D Extra-Force According to Basini-Capozziello-Ponce De Leon ...

245

Then we can say that the Basini-Capozziello Ponce De Leon 5D formalism is for Timelike 5D Geometries bacause they admit a non-null 5D rest-mass while the formalisms of Mashoon-Wesson-Liu and Overduin-Wesson are valid for a Null-Like 5 D Geometry where we have a M5 = 0 a null 5D Ricci Tensor or a flat 5D Spacetime. •

•

•

dy dy 1 > ( )2 99K 1 > Φ2 ( )2 2 Φ ds ds

(35)

1 dy dy = ( )2 99K 1 = Φ2 ( )2 2 Φ ds ds

(36)

dy dy 1 < ( )2 99K 1 < Φ2 ( )2 2 Φ ds ds

(37)

Note that a small Warp Field 0 < Φ2 < 1 will generate a large Φ12 ideal for a 5D Timelike Geodesics. Although we can have a Null 5D rest-mass M5 the Warp Field in the 5D Extra Dimension can still account for the generation of rest-masses in 4 D. See these Ponce De Leon Equations for the 4D rest-mass m0 ([2] eq 27 and 28,[20] eq 16,17 and 18) dy dλ

(38)

1 ds m0

(39)

m0 =+ −Φ

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dλ =

Combining eqs 45 and 46 we can clearly see that: 17 dy 1 = (40) ds Φ The 5D Extra Force seen in 4D for massless particles in 5D is given by:([2] eq 30,[20] eq 19)1819 1 dm0 − 1 ∂guv u v uu ∂Φ = u u − m0 ds + 2Φ ∂y Φ ∂xu

(41)

This equation although for massless 5 D particles have many resemblance with its similar for massive 5D particles as pointed out by Ponce De Leon and can easily be obtained combining eqs 15 and 18 of [20](see pg 1343 in [2]). According to the Table of Elementary Particles already presented in this work(two times and we think its enough) Photons or Gravitons have a 4 D rest-mass m0 = 0 corresponding to 17 see

pg 1343 in [2] but the result is obvious from [20] eq 10

18 note that like for its analogous 5 D massive counterpart the Warp Field function only of the Extra Coordinate

makes the second term vanish(examine eqs 50 and 52 in [1]) 19 compare this equation with [9] eq 24 and look for the + signal in this equation while [9] eq 24 only have the - sign Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

246

Fernando Loup

a 5D Null-Like Spacetime Geometry or in hence a stationary particle a particle that is at the dy dy dy rest in the 5D Spacetime,a particle with a m0 =+ − Φ dλ 99K m0 = 0 99K dλ = 0 99K ds = 0. But of course we can have a 5D rest-mass M5 = 0 giving a non-null 4D rest-mass m0 6= 0 dy even with a Warp Field Φ = 1 if dλ 6= 0 according to the following equations although we believe that non-null rest-masses m0 in 4D comes from non-null rest-masses M5 in 5D(see sections 8 and 9 about particle Z in [20]) : dy dλ

(42)

1 ds m0

(43)

m0 =+ − dλ =

dy =1 ds

(44)

1 dm0 − 1 ∂guv u v = u u m0 ds + 2 ∂y

(45)

2 Note that if the Warp Field Φ = 1 with dy ds = 1 and dS = 0 the equation of the 5D Extra Force for a massless particle in 5D M5 = 0 becomes equivalent to [9] eq 24 proving that the Ponce De Leon equations are equivalent to the Mashoon-Wesson-Liu ones.

1 dm0 − 1 ∂guv u v = u u m0 ds + 2 ∂y

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3.

(46)

Dimensional Reduction from 5D to 4D According to BasiniCapozziello Ponce De Leon, Mashoon-Wesson-Liu and Overduin-Wesson. Possible Experimental Detection of Extra Dimensions in Strong Gravitational Fields or On-Board the International Space Station (ISS) Using the Gravitational Bending Of Light in Extra Dimensions

The most important thing to keep in mind when we study models of BraneWorlds or Extra Dimensions is to explain why we cannot ”see” directly the presence of the Extra Dimension although we can ”feel” its effects in the 4 D everyday Physics.We avoid here the models with compactification or ”curling-up” of the Extra Dimension because these models don’t explain why we have 3 + 1 Large Dimensions while the remaining ones are small and ”unseen” Extra Dimensions and also these models dont explain what generates the ”Compactification” or ”Curling” mechanism in the first place.Also some of these models develop ”Unphysical” features.An excellent account of the difference between compactified and uncompactified models of Extra Dimensions is given by [21](see pgs 2 to 31).We prefer to adopt the fact that like the 3 + 1 ordinary Large Spacetime Dimensions the Extra Dimensions are Large and uncompactified but due to a Dimensional Reduction from 5 D to 4D we cannot ”see” these Extra Dimensions although we can ”feel” some of its effects.

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On The 5D Extra-Force According to Basini-Capozziello-Ponce De Leon ...

247

(see abs and pg 123 of [1] when Basini-Capozziello mentions the fact that we cannot perceive Time as the fourth Dimension and hence we cannot perceive the Spacelike Nature of the 5D Extra Dimension).(see also pg 1424 and pg 1434 begining of section 4 in [20]).(see also abs pg 2218 and 2219 of [5].Note the comment on Dimensional reduction and a 4 D Spacetime embedded into a larger 5D one).We will now demonstrate how the Dimensional Reduction from 5D to 4D work and why in ordinary conditions we cannot ”see” the 5 D Extra Dimensions but we can ”feel” some of its effects.Also we will see that changing the Spacetime Geometry and the shape of the Warp Field the 5D Extra Dimension will become visible.(Dimensional Reductions from 5D to 4D appears also in pg 2040 of [4]). We know that in ordinary 3 + 1 Spacetime the curvature of the Einstein Tensor is neglectable and Spacetime can be considered as Minkowskian or flat where Special Relativity holds.A Minkowskian 5D Spacetime with a Warp Field can be given by(see eq 325 in [20]): dS2 = dt 2 − dX 2 − Φ2 dy2

(47)

The Warp Field considered here have small values between 0 and 1 nearly close to 0 and we recover the ordinary Special Relativity Ansatz.A Minkowskian 5 D Spacetime with no Warp Field at all would be given by(see eq 326 in [20]): dS2 = dt 2 − dX 2 − dy2

(48)

The Ricci Tensors and Scalars for the Basini-Capozziello 5D Spacetime Fomalism and Ansatz given by dS2 = gµνdxµ dxν − Φ2 dy2 are shown below:(see pg 128 eq 58 in [1],pg 2230 eq 44 in [5] and pg 1442 eqs 111 to 115 in [20])

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5

5

Rαβ = Rαβ −

R = R− 5

gµν gµν,4 gαβ,4 Φ,a;b 1 Φ,4 gαβ,4 − ( − g + ) αβ,44 Φ 2Φ2 Φ 2

(49)

gµν gµν,4 gαβ,4 Φ,a;b αβ 1 αβ Φ,4 gαβ,4 g − g ( − g + ) αβ,44 Φ 2Φ2 Φ 2

(50)

gµν gµν,4 gαβ,4 1 αβ Φ,4 gαβ,4 Φ − − g ) g ( + αβ,44 Φ 2Φ2 Φ 2

(51)

4

R = R−

For a 5D Spacetime Metric without Warp Field defined as dS2 = gµν dxµ dxν − dy2 the Ricci Tensor and Scalar would then be(see eqs 330 and 331 pg 1477 in [20]): 5

gµνgµν,4 gαβ,4 1 Rαβ = Rαβ − (−gαβ,44 + ) 2 2

(52)

gµνgµν,4 gαβ,4 1 R = R − gαβ (−gαβ,44 + ) (53) 2 2 But remember that a Minkowskian 5D Spacetime in which Special Relativity holds have all the 3 + 1 Spacetime Metric Tensor Components defined by gµν = (+1, −1, −1, −1)(see pg 1476 and 1477 in [20]) and the derivatives of the Metric Tensor vanishes and hence we are left with the following results(see eqs 332 and 333 in [20]) 5

5

Rαβ = Rαβ

Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

(54)

248

Fernando Loup 5

R=R

(55)

From the results above in a flat Minkowsky 5D Spacetime the Ricci Tensor in 5D is equal to its counterpart in 4D and since the Spacetime is flat then both are equal to zero.Then its impossible to tell if we live in a 4 D Spacetime or in a larger 5D Extra Dimensional one.(see pg 1477 after eq 333 in [20]).If the Geometry of a flat 5D Extra Dimensional Spacetime is equivalent to the Geometry of a 3 + 1 Spacetime we cannot distinguish if we live in a 4D or a 5D Universe.This is one of the most important things in the Dimensional Reduction from 5D to 4D as proposed by Basini-Capozziello.The 5D Extra Dimension is Large and Uncompactified but the physical reality we see is a Dimensional Reduction from 5D to 4D because we live in a nearly flat Minkowsky Spacetime 20 where Special Relativity holds and the 5D Ricci Tensor is equal to the 3 + 1 counterpart.No Compactification mechanisms needed.The 5D Extra Dimension have a real physical meaning(see pg 2226 in [5] and pg 127 in [1]).(see also pg 2230 in [5] the part of the reduction of the Ricci Tensor from 5D to 4D eqs 44 and 45.if the Warp Field Φ = 1 both 5D and 4D Ricci Tensors from eq 45 are equal.the same can be seen in pg 128 to 129 eqs 58 to 59 in [1].see also pg 1442 eqs 115 to 116 in [20]). If in a flat 5D Minkowsky Spacetime we cannot ”see” the Extra Dimension then we have three choices in order to tell if we live in a 5 D Extra Dimensional Spacetime or a 3 + 1 Ordinary Dimensional one.The choices are:

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

• Making the Warp Field Φ 6= 1 in order to generate a difference between the 5D Extra Dimensional Ricci Tensor and the 3 + 1 Spacetime counterpart according to eq 45 in [5],eq 59 in [1] and eq 116 in [20].This difference can tell the difference between a 5D Universe and a 3 + 1 one.(see pg 1477 in [20]) • Making the 3 + 1 Spacetime Metric Tensor components be a function of the 5D Extra Dimension in order to do not vanish the derivatives of the Metric Tensor with respect to the Extra Dimension generating a difference between the 5 D Ricci Tensor and the 3 + 1 counterpart according to eqs 330 and 331 pg 1477 in [20]. A Strong Gravitational Field of a Large Maartens-Clarkson 5D Schwarzschild Black String have the Spacetime Metric Tensor Components defined in function of the 4 D rest-mass M but the 4D rest-mass is function of the 5D Extra Dimensional Spacetime Geometry according to eq 20 in [2]. • Making both conditions above hold true We will examine all of the items above in this section. We live in a region of Spacetime where the Warp Field Φ = 1 then we cannot see the 5D Extra Dimension.Or we can live in a region of Spacetime where the Warp Field Φ = 0 and this cancels out the term Φ2 dy2 in the 5D Spacetime Ansatz making the Extra Dimension invisible.Or perhaps we can live in a region of spacetime where 0 ≤ Φ ≤ 1 but near to 0 or 1 so its very difficult to detect the presence of the 5D Extra Dimension although we can ”feel” some of its effects.Considering now a Warp Field Φ 6= 1 the Minkowsky 5D Spacetime Ansatz would still have the terms of the 3 + 1 Spacetime Metric Tensor given by gµν = (+1, −1, −1, −1).Hence the 5D Spacetime Ansatz would then be: 20 The

Systen Earth-Sun have a weak Gravitational Field so around Earth the Spacetime is cosidered flat

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On The 5D Extra-Force According to Basini-Capozziello-Ponce De Leon ...

249

dS2 = dt 2 − dX 2 − Φ2 dy2

(56)

The derivatives of the 3 + 1 components of the Spacetime Metric Tensor vanishes but note that the 5D component do not vanish.The Ricci Tensor and Scalar would be given by the following expressions(see eqs 335,336 and 337 in [20]): 5

Φ,a;b Φ

(57)

Φ,a;b αβ g Φ

(58)

Rαβ = Rαβ −

5

R = R− 5

R = R−

4 Φ

(59) Φ Note that now the scenario is different:while with the Warp Field Φ = 1 the 5D Ricci Tensor is equal to its 3 + 1 counterpart and we cannot tell if we live in a 5 D or in a 3 + 1 Universe but when the Warp Field Φ 6= 1 there exists a difference between the Ricci Tensor in 5D and the 3 + 1 one.The Geometrical Properties of Spacetime of the 5D Spacetime are now different than the 3 + 1 equivalent one and this makes the 5 D Extra Dimension visible.(see also pg 1478 after eq 337 in [20]) According to Basini-Capozziello the Warp Field can be decomposed in two parts:one in 3 + 1 ordinary Spacetime and another in the 5D Extra Dimension given by the following equation:([1] eq 76,[5] eq 70 and [20] eq 132 and 338 )

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Φ = φ(t, x)χ(y)

(60)

Note that when we compute the covariant derivative of the Warp Field with respect to the 3 + 1 Spacetime the terms of the 5D Extra Dimension are cancelled out and we are left with derivatives of the 3 + 1 components of the Warp Field(see eq 339 in [20] )

Φ,a;b = χ(y)[φ,a;b] = χ(y)[(φα)β ] − ΓKβα φK ] 99K =

Φ,a;b χ(y)[φ,a;b] = Φ φ(t, x)χ(y)

[(φα)β − ΓKβα φK ] [φ,a;b ] = φ(t, x) φ(t, x)

(61)

The result shown below is very important.It demonstrates that only the 3 + 1 component of the Warp Field φ(t, x) fortunately the component that lies in ”our side of the wall” and its derivatives with(again fortunately) respect to our 3 + 1 Spacetime coordinates can make the 5D Ricci Tensor be different than its 3 + 1 counterpart and since we are considering in this case a flat Minkowsky Spacetime the 4D Ricci Tensor reduces to zero and this means to say that 5 Rαβ = −

φ,a;b φ

or better 5 Rαβ = − 5

[(φα )β −ΓK βα φK ] φ(t,x)

Rαβ = Rαβ −

φ,a;b φ

Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

(62)

250

Fernando Loup

5

Rαβ = Rαβ −

[(φα)β − ΓKβα φK ] φ(t, x)

(63)

Note that if the 4D Ricci Tensor vanishes due to a flat Minkowsky Spacetime and we are left with derivatives of the 3 + 1 Spacetime components of the Warp Field with respect to (again fortunately for the second time) 3 + 1 Spacetime coordinates and we are left with a result in which the 5D Ricci Tensor and our capability to detect the existence of the 5 D Extra Dimension depends on the shape of the 3 + 1 component of the Warp Field.If we can detect the derivatives of the Warp Field we can detect the existence of the 5 D. The other way to make the 5D Extra Dimension visible is to make the derivatives of the 3 + 1 Spacetime Metric Tensor components gµν = (g00, g11 , g22 , g33)non-null with respect to the 5D Extra Dimension. dS2 = dt 2 − gµµ d(X µ )2 − Φ2 dy2

(64)

For our special case of diagonalized metric: dS2 = dt 2 − gµµ d(X µ )2 − Φ2 dy2

(65)

Considering the 3 + 1 Spacetime Metric Tensor Components g00 and g11 (see eqs 353 and 354 in [20]). 5

Φ,0;0 g00 g00,4 g00,4 1 Φ,4 g00,4 − ( − g + ) 00,44 Φ 2Φ2 Φ 2

(66)

1 Φ,4 g11,4 Φ,1;1 g11 g11,4 g11,4 − − g ) (67) ( + 11,44 Φ 2Φ2 Φ 2 Now we can see that if the derivatives of g00 and g11 do not vanish with respect to Φ Φ g g00 g00,4 g00,4 Φ the Extra Coordinate then the terms Φ,0;0 − 2Φ1 2 ( ,4Φ00,4 − g00,44 + ) and Φ,1;1 − 2 5

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

R00 = R00 −

1 Φ,4 g11,4 ( Φ 2Φ2

R11 = R11 −

g11 g

g

11,4 11,4 − g11,44 + ) will generates a difference between the 5D Ricci Tensor 2 and its 3 +1 Ordinary Spapcetime Dimensional counterpart.Remember also that g00 and g11 can be defined as the Spacetime Metric Tensor Components of the Maartens-Clarkson 5D Schwarzschild Black String centerd on a large Black Hole for example in which M is the 4D rest-mass of the Black Hole but M can be defined in function of the 5D Extra Dimensional rest-mass M5 and also defined in function of the 5 D Spacetime Geometry according to Ponce De Leon eq 20 in [2].This can make the 5D Extra Dimension becomes visible. Writing the Maartens-Clarkson 5D Schwarzschild Cosmic Black String as follows:([7] eq 1,[20] eq 380):

dS2 = [(1 −

2GM 2 dR2 )dt − − R2 dη2 ] − Φdy2 R (1 − 2GM ) R

(68)

Where the Spacetime Metric Tensor Components of the Black String are given by:g00 = 2GM −1 (1 − 2GM R ) and g11 = −(1 − R ) .The derivatives with respect to the Extra Coordinate 21 are then : 21 only time

and radial components are considered here.

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On The 5D Extra-Force According to Basini-Capozziello-Ponce De Leon ...

251

∂M ∂g00 ∂(1 − 2GM ∂M ∂R−1 R ) = = −2G R = −2G[ × R−1 + × M] ∂y ∂y ∂y ∂y ∂y

(69)

We know that the 4D rest-mass M of the Maartens-Clarkson 5D Schwarzschild Cosmic Black String can be defined in function of the Ponce De Leon 5 D rest-mass M5 eq 20 in [2].The final result would then be: 1 M5 dy ∂ ds dy ∂Φ 1 ∂R M5 ∂g00 = −2G[ q [Φ2 + ( )2 Φ ] − 2 ( ) q ] ∂y R [ 1 − Φ2 ( dy )2 ]3 ds ∂y ds ∂y R ∂y 1 − Φ2 ( dy )2 dy

ds

ds

(70) ∂g11 ∂y

=

∂g00 ∂y g200

−2G 1 M5 dy ∂ ds dy ∂Φ [ q [Φ2 + ( )2 Φ ] 2 ds ∂y ds ∂y g00 R [ 1 − Φ2 ( dy )2 ]3 dy

=

(71)

ds

1 ∂R M5 − 2 ( )q ] R ∂y 1 − Φ2 ( dy )2 ds

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

Note that in a Strong Gravitational Field these derivatives will have high values and this will make the 5D Ricci Tensor be highly different than its 3 + 1 counterpart making the 5D Extra Dimension be visible but far away from the center of the Black String MR5 ' 0 and the derivatives will vanish due to the Weak Gravitational Field however the term corresponding to the Warp Field will remain as shown below: 5

R00 = R00 −

Φ,0;0 Φ

(72)

5

R11 = R11 −

Φ,1;1 Φ

(73)

Then in a Weak or Null Gravitational Field 22 is the Warp Field that can make the 5D Ricci Tensor be different than its 3 + 1 counterpart and can tell if we live in a 5D Extra Dimensional Spacetime or in a ordinary 3 + 1 one.Remember that at faraway distances from Gravitational Field the Spacetime is flat or Minkowskian and the 3 + 1 Ricci Tensor is zero or nearly zero.Then we could rewrite the two equations above as follows: 5

R00 = −

Φ,0;0 Φ

(74)

Φ,1;1 (75) Φ We already know that when computing derivatives of the Warp Field with respect to 3 + 1 Coordinates the 5D Extra Dimensional terms are cancelled out and we will get these results: 5

22 eg

R11 = −

Earth-Sun System or a Spaceship far away from a Black Hole

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252

Fernando Loup

5

R00 = −

[(φ0 )0 − ΓK00 φK ] φ(t, x)

(76)

5

R11 = −

[(φ1 )1 − ΓK11 φK ] φ(t, x)

(77)

Writing the Ricci Tensors with the derivatives of the 3 + 1 Spacetime Metric Tensor Components of the Warp Field explicitly written we have: 5

[ ∂ φ(t,x) − ΓK00 ∂φ(t,x) ] ∂t 2 ∂xK R00 = − φ(t, x)

(78)

5

[ ∂ φ(t,x) − ΓK11 ∂φ(t,x) ] 2 ∂xK R11 = − ∂R φ(t, x)

(79)

2

2

ΓK00

∂φ(t, x) ∂φ(t, x) ∂φ(t, x) = Γ000 + Γ100 ∂xK ∂t ∂R

(80)

∂φ(t, x) ∂φ(t, x) ∂φ(t, x) = Γ011 + Γ111 (81) ∂xK ∂t ∂R We still don’t know the shape of the Warp Field but remember that the 3 + 1 component of the Warp Field can be coupled to Gravity as defined by Basini-Capozziello in [1] pg 119 and [5] pg 2235. Considering only valid Christoffel Symbols we have 23 :

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ΓK11

ΓK00

∂φ(t, x) ∂φ(t, x) 1 1 ∂g00 ∂φ(t, x) 1 1 ∂g00 ∂φ(t, x) ∂y = Γ000 = = K ∂x ∂t 2 g00 ∂t ∂t 2 g00 ∂y ∂t ∂t

(82)

ΓK11

∂φ(t, x) ∂φ(t, x) 1 1 ∂g11 ∂φ(t, x) 1 1 ∂g11 ∂φ(t, x) ∂y = Γ111 = = K ∂x ∂R 2 g11 ∂R ∂R 2 g11 ∂y ∂R ∂R

(83)

These expressions are valid for Strong or Weak Gravitational Fields .But we are considering here Weak Gravitational Fields where the Gravitational Force almost vanishes and the derivatives of the Spacetime Metric Tensor Components of the Maartens-Clarkson 5 D Schwarzschild Cosmic Black String vanishes due to the term MR5 24 and the final expression for the 5D Ricci Tensors can be given by: 5

5

R00 = −

R11 = −

[

∂2 φ(t,x) ] ∂t 2

φ(t, x) [

∂2 φ(t,x) ] ∂R2

φ(t, x)

The 5D Ricci Scalar would be given by: 23 diagonalized metrics 24 making

g00 = 1 and g11 = 1

Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

(84)

(85)

On The 5D Extra-Force According to Basini-Capozziello-Ponce De Leon ... φ(t,x) ] [ ∂ φ(t,x) [ ∂ ∂R 1 4 2 ] ∂t 2 R = R00 + R11 = − +− =− φ(t, x)2 φ(t, x) φ(t, x) φ(t, x) 2

5

5

253

2

5

(86)

Remarkably we can extract the Ricci Scalar from the 5D to 4D Dimensional Reduction of Basini-Capozziello . If the Warp Field Coupled to Gravity is defined by Basini-Capozziello then this can be regarded as a final proof that the 5D Extra Dimension really exists Considering now the case of the 5D Spacetime Metric with no Warp Field at all Φ = 1 the difference between the 5D and the 4D Ricci Tensors will depend on the derivatives of the Spacetime Metric Tensor Components with respect to the 5D Extra Dimension that will vanish far away from the center of the Maartens Clarkson 5D Schwarzschild Cosmic Black String making the 5D be invisible but in the regions of intense Gravitational Field the 5D Ricci Tensor will be different than its 4 D counterpart making the 5D Extra Dimension becomes visible 5

gµνgµν,4 gαβ,4 1 Rαβ = Rαβ − (−gαβ,44 + ) 2 2

(87)

gµνgµν,4 gαβ,4 1 R = R − gαβ (−gαβ,44 + ) (88) 2 2 Writing the Ricci Tensor for the time and radial components we have(see eqs 343 and 344 pg 1479 in [20]): 5

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5

g00 g00,4g00,4 1 R00 = R00 − (−g00,44 + ) 2 2

(89)

1 g11 g11,4g11,4 R11 = R11 − (−g11,44 + ) (90) 2 2 We we know that the derivatives of the Spacetime Metric Tensor for the 5 D MaartensClarskon Schwarzschild Cosmic Black String are given by: 5

1 M5 dy ∂( ds ) M5 1 ∂R ∂g00 = −2G[ q −q ] ∂y R [ 1 − ( dy )2 ]3 ds ∂y dy 2 R2 ∂y 1−( ) dy

ds

(91)

ds

∂g00

1 1 M5 M5 dy ∂( ds ) 1 ∂R ∂g11 ∂y = 2 = − 2 2G[ q −q ] ∂y R g00 g00 dy 2 3 ds ∂y dy 2 R2 ∂y [ 1−( ) ] 1−( ) dy

ds

(92)

ds

Note that far away from the Black String the ratio MR5 ' 0 and the derivatives of the Spacetime Metric Tensor Components will vanish making the 5D Ricci Tensor equal to its 3 + 1 counterpart and the 5D Extra Dimension will become invisible.But in the neighborhoods of the Black String center Gravity becomes so high that the Extra Terms will make the 5D Ricci Tensor be different than its 3 + 1 counterpart. Another way to measure the presence of the 5D Extra Dimension is to measure how the Extra Dimension affects

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Fernando Loup

the Gravitational Bending of Light in the vicinity of the Black String according to KarSinha(see abstract of [3]).In one of our works([20] pg 1495 section 8) we proposed the use of the International Space Station ISS 2526 to measure the Kar-Sinha Gravitational Bending of Light of the Sun to find out if it can be affected by the presence of the 5 D Extra Dimension.(see also pg 1467 before eq 290 in [20]) While the Sun have a ”weak” Gravitational Field a Black String is a Black Hole in 5 D and in the vicinity of the Black String perhaps the Gravitational Bending Of Light affected by the presence of the 5 D Extra Dimension according to Kar-Sinha would be better noticeable(ISS could still be used in this fashion to measure Gravitational Bending Of Light affected by the presence of Extra Dimensions by observing accretion disks of Black Holes free from the disturbances of Earth Atmosphere.We propose here to use beams of neutrons or photons to measure the Extra Terms On-Board ISS ). Writing the Kar-Sinha Gravitational Bending Of Light affected by the presence of the 5D Extra Dimension in the neighborhoods of a Black String or in the neighborhoods of the Sun to be measured On-Board ISS with a non-null Warp Field as follows(see pg 1467 eq 288 to 291 and pg 1468 eq 295 to 296 in [20])(see also pg 1781 eq 18 in [3])2728 : dy 2 2GM (2 + [Φ ] ) c2 R cdt

(93)

dy 2GM (2 + [φ(t, x)χ(y) ]2 ) c2 R cdt

(94)

M5 dy 2 2G q (2 + [Φ ] ) c2 R 1 − Φ2 ( dy )2 cdt

(95)

dy 2G M5 q (2 + [φ(t, x)χ(y) ]2 ) 2 c R 1 − φ(t, x)2 χ(y)2 ( dy )2 cdt

(96)

4ω = 4ω = 4ω =

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

ds

4ω =

ds

The same expression for a null Warp Field would be given by: dy 2 2GM (2 + [ ]) 2 c R cdt

(97)

M5 dy 2 2G q (2 + [ ] ) 2 c R cdt 2 1 − ( dy ) ds

(98)

4ω = 4ω =

In the above equations M5 and M are the 5D and 4D rest-masses of the Sun or the Black Hole and R the distance between the accretion disk and the Black Hole or the distance 25 more

on General Relativity and ISS in [6],[8],[16],[17] and [18] will also appear in the next section 27 equations written without Warp Factors and with the Gravitational Constant 28 see also eqs 156 to 158 pg 70 section 8 .7 in [21].see also in the same reference the comment on the velocity dy along the 5D Extra Dimension in pg 71 after eq 159 dψ dt similar to our dt .see also between page 70 and 71 the comment that the shift is physically measurable.we will examine photon paths in the 5 D Maartens-Clarkson Schwarzschild Cosmic Black String in this section also but we will use the Ponce De Leon point of view of pg 1343 after eq 30 in [2].look to the Ponce De Leon comment of genuine manifestation of the 5 D Extra Dimension before section 4 26 ISS

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between the photon beam and the Sun. We know that the Warp Field must have values between 0 and 1 so the shift in the Gravitational Bending Of Light must be very small making the value of the expression in 5 D be close to its 4D counterpart.The presence of the 2 Gravitational Constant G = 6, 67 × 10−11 Newtonkgtimesm divided by the square of the Light 2 Speed would make the things even worst.This is the reason why we need a Black String of large rest mass M or M5 to make the shift noticeable.Perhaps in the Sun we would never be able to measure the shift. Note also the comment on [21] pg 71 that the derivative dψ dy dt analogous to our dt is null for photons and we know from Ponce De Leon that the 5D Spacetime Metric dS5 = ds2 − Φ2 dy2 is null for photons making ds2 = Φ2 dy2 M5 = 0 and m0 = 0.Remember that in 4D SR ds2 = 0 for photons and dy ds = 0 making the shift in 5D be equal to its 4D counterpart.Kar-Sinha mentions in pg 1783 [3] the fact that if −4 the photon propagates in 5D the value of dy and the shift 4ω affected by ds < 2, 8 × 10 the 5D Extra Dimension must lie between the error margins of the observed values of pgs 39 to 41 in [19] 29 .Remember also that the Spacetime at a distance R from the Sun where the photon passes by in order to be Bent or Deflected is described by the 5 D Black String centered on the Sun according to [7] eq 1,[20] eq 380.Although dy dt is zero for photons it may be not for the Sun Mass M and then the Extra Terms in the Gravitational Bending Of Light for photons may still be measurable anyway.The Gravitational Bending can be observed for other particles with a non-null M5 and a non-null m0 and perhaps the study of the motion of high-speed relativistic particles from accretion disks of large Black Holes can tell the difference between the 5D 4ω and its 4D counterpart.For a non-null dy ds particle the Gravitational Bending formulas could be given by: dy ds 2 2GM (2 + [Φ ]) 2 c R ds cdt

(99)

dy ds 2 2GM (2 + [φ(t, x)χ(y) ]) 2 c R ds cdt

(100)

M5 dy ds 2 2G q (2 + [Φ ]) 2 c R 1 − Φ2 ( dy )2 ds cdt

(101)

M5 dy ds 2 2G q (2 + [φ(t, x)χ(y) ] ) 2 c R 1 − φ(t, x)2 χ(y)2 ( dy )2 ds cdt

(102)

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4ω = 4ω = 4ω =

ds

4ω =

ds

The same for a Warp Field Φ = 1 dy ds 2 2GM (2 + [ ]) 2 c R ds cdt

(103)

M5 dy ds 2 2G q ]) (2 + [ c2 R 1 − ( dy )2 ds cdt

(104)

4ω = 4ω =

ds

Note that a relativistic beam of neutrons would not suffer the deflection by electromagnetic fields and could be used to measure the Extra Terms in the Gravitational Bending of 29 this

reference contains one of the best explanations for the Gravitational Bending Of Light Geometry and describes even the 30 percent margin of error in the 1919 measurements Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

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Light due to the presence of the Higher Dimensional Spacetime but a beam of photons is more easy to be obtained.Consider a Satellite carrying a small Laser device in the other side of the Earth Orbit targeting the beam towards a target in ISS.The beam in order to reach ISS must pass at a distance R from the Sun.The Extra Terms in 4ω could perhaps be measured in Outer Space On-Board International Space Station ISS free of the interference of the Earth Atmosphere.Computing the Classical Bending of Light as(see pg 1781 eq 18 in [3]): 4GM c2 R If we observe these deviations of the Bending Angle 4ωClassic =

4ωExtraTerms =

2GM dy 2 [Φ ] c2 R cdt

(105)

(106)

2GM dy 2 [ ] (107) c2 R cdt different than the original Classical value then we can demonstrate that we live in a 5 D Higher Dimensional Spacetime. The Laser beam would be affected by the Sun Mass M while passing at a distance R from the Sun but would reach ISS.Computing the Classical Einstein Bending we know where the Laser will reach the target.If the observed Bending is equal to the Classical Einstein then this would mean that there are no Extra Dimensions in the Universe.But if the deviated photon arrives at ISS with an angle different than the one predicted by Einstein and if this difference in the angle matches the Extra-Terms predicted by Kar-Sinha then ISS would proof that we live in a Universe of more than 4 Dimensions.Then this experiment onboard ISS would have the same degree of importance of the measures of the Gravitational Bending of Light by Sir Arthur Stanley Eddington in the Sun Eclipse of 1919 Computing the Classical Einstein Bending Of Light for a Laser beam passing the Sun at distance R = 150.000km 30 we would have: Mass of the Sun(in 4D):

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

4ωExtraTerms =

M = 1, 9891 × 1030kg

(108)

Newton Gravitational Constant (in 4 D): G = 6, 67 × 10−11 Newton × m2 /kg2

(109)

There exists a common factor between the Classical Bending Of Light in 4 D and the Kar-Sinha Extra Terms given by : 4ωCommonFactor =

2GM c2 R

(110)

The common factor would be given by: 30 Remember

that the Sun lies between ISS and the Satellite placed in the other side of the Earth Orbit.The laser must pass in the neighbourhoods of the Sun

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On The 5D Extra-Force According to Basini-Capozziello-Ponce De Leon ...

257

4ωCommomFactor = 2 × 6, 67 × 10−11 × 1, 9 × 1030 /(1, 5 × 108 × 9 × 1016 ) = 1, 877481481481 × 10−5

(111)

2 × 6, 67 × 10−11 × 1, 9 × 1030 = 25, 346 × 1019

(112)

1, 5 × 108 × 9 × 1016 = 13, 5 × 1024

(113)

25, 346 = 1, 877481481481 13, 5

(114)

The Classical Einstein Bending Of Light would be given by:

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4ωClassic = 4 × 6, 67 × 10−11 × 1, 9 × 1030 /(1, 5 × 108 × 9 × 1016 ) = 3, 754962962962 × 10−5

(115)

4 × 6, 67 × 10−11 × 1, 9 × 1030 = 50, 692 × 1019

(116)

50, 692 = 3, 754962962962 13, 5

(117)

We already outlined in the Introduction Section the fact that the goal of ISS is to achieve a Gravitational Shift Precision of 2.4 × 10−7 (see pg 631 Table II in [6]) and the faxt that ISS Gravitational shifts are capable to detect measures of 4ω ω with Expected Uncertainly of 12 × 10−6 (see pg 629 Table I in [6]) The difference between Kar-Sinha and Classical Gravitational Bending Of Light is given by: 2GM 4GM dy 2 (2 + [Φ ] )− 2 2 c R cdt c R

(118)

4GM 2GM dy 2 4GM + 2 [Φ ] − 2 c2 R c R cdt c R

(119)

2GM dy 2 [Φ ] c2 R cdt

(120)

4ωKarSinha − 4ωClassic = 4ωKarSinha − 4ωClassic =

4ωKarSinha − 4ωClassic =

4ωKarSinha − 4ωClassic = 4ωCommonFactor [Φ

dy 2 ] cdt

4ωKarSinha − 4ωClassic = 1, 877481481481 × 10−5[Φ

dy 2 ] cdt

4ωKarSinha − 4ωClassic = 4ωExtraTerms Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

(121) (122) (123)

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Fernando Loup

And this difference is equal to the Common Factor multiplied by the Kar-Sinha Extra Terms due to the presence of the Higher Dimensional Spacetime The Kar-Sinha additional terms depends on the derivative of the 5 D Extra Coordinate dy 2 or the value of the Warp Field Φ due to the Extra Dimensional factors [Φ cdt ] .We know that for a Timelike 5D geodesics according to Ponce De Leon we have 0 < Φ < 1 and 2 0 < [Φ dy ds ] < 1(see eq3 pg 1426 in [20] without the Warp Factors Ω = 1)and for photons we would have a Null-Like Geodesics(see eq4 pg 1426 in [20] without the Warp Factors 2 Ω = 1) giving 0 < [Φ dy ds ] = 1.Assuming also a small derivative of the Extra Coordinate dy with respect to time cdt then we would have very small values for the Extra Dimensional dy 2 Term [Φ cdt ] at least for a Timelike Geodesics. Examining now the case for photons in a Null-Like 5 D Geodesics in the neighborhoods of the Sun as a Maartens-Clarkson Schwarzschild Black String 31 : dS2 = 0 99K ds2 − Φ2 dy2 = 0 99K ds2 = Φ2 dy2 99K 1 = Φ2 (

dy 2 dy 2 cdt 2 dy 2 ds 2 ) 99K 1 = Φ2 ( )( ) 99K Φ2 ( ) =( ) ds cdt ds cdt cdt

4ωExtraTerms = 1, 877481481481 ×10−5[Φ

dS2 = [(1 − Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

1 dy dy = ( )2 99K 1 = Φ2 ( )2 (124) Φ2 ds ds

dy 2 ds 2 ] = 1, 877481481481 ×10−5( ) (126) cdt cdt

2GM dR2 2 )(cdt) − − R2 dη2 ] − Φdy2 2GM c2 R (1 − c2 R )

(127)

2GM dR2 2 )(cdt) − − R2 dη2 ] c2 R (1 − 2GM ) 2 c R

(128)

ds2 = [(1 −

2 dR2 2GM 1 ds2 2 dη = [(1 − ) − − R ] (cdt)2 c2 R (cdt)2 (1 − 2GM ) (cdt)2 c2 R 2 1 dR2 ds2 2 dη = [(1 − 4ω ) − − R ] CommonFactor (cdt)2 (1 − 4ωCommonFactor ) (cdt)2 (cdt)2

1−

(125)

(129)

(130)

2GM = 1 − 4ωCommonFactor = 1 − 1, 877481481481 × 10−5 = 0, 9998122518518519 c2 R (131) 2 1 1 dR2 ds2 2 dη = (1 − 4ω ) − [ − R ] CommonFactor (cdt)2 c2 (1 − 4ωCommonFactor ) (dt)2 (dt)2

31 We

(132)

already outlined the fact that Kar-Sinha mentions in pg 1783 [3] the fact that if the photon propagates −4 and the shift 4ω affected by the 5D Extra Dimension must lie between the in 5D the value of dy ds < 2, 8 × 10 error margins of the observed values of pgs 39 to 41 in [19] Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

On The 5D Extra-Force According to Basini-Capozziello-Ponce De Leon ... Note that the term −16 10 .

1 c2

259

will make the second right term neglectable due to the factor

ds2 = (1 − 4ωCommonFactor ) = 0, 9998122518518519 = 9, 998122518518519 × 10−1 (cdt)2 (133) dy 2 ] cdt

(134)

ds 2 ) cdt

(135)

4ωKarSinha − 4ωClassic = 4ωExtraTerms = 1, 877481481481 × 10−5[Φ

4ωKarSinha − 4ωClassic = 4ωExtraTerms = 1, 877481481481 × 10−5(

4ωKarSinha − 4ωClassic = 4ωExtraTerms = 1, 877481481481 × 10−5

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

×9, 998122518518519 × 10−1

(136)

4ωKarSinha − 4ωClassic = 4ωExtraTerms = 18, 771289878096695909454046639 × 10−6 (137) The result above is the most important of this work and means to say that the Kar-Sinha Extra Terms in the Gravitational Bending of Light due to presence of the Extra Dimensions are in the range of the detection capability of the International Space Station ISS enclosing the Gravitational Shift Precision of 2 .4 × 10−7(see pg 631 Table II in [6]) and the Gravita−6 tional Shifts of 4ω ω with Expected Uncertainly of 12 × 10 (see pg 629 Table I in [6]). If this Extra Term is detected with ”positive” results then the International Space Station ISS can demonstrate for the first time that our Universe have more than 4 D Dimensions making the Physics of Extra Dimensions an Experimental Branch of Modern Physics.

4.

Experimental Detection of Extra Dimensions Using Gravitational Red-Shifts On-Board the International Space Station ISS

From the abstract of [6] we know that Gravitational Red Shifts are also considered to experiments On-Board the International Space Station ISS.We already outlined before the Gravitational Shift precision of ISS.We also know from Kar-Sinha that Extra Dimensions affects the Gravitational Red Shift(see pg 1782 in [3]) generating Extra Terms in a way similar to the ones for the Gravitational Bending Of Light. We will propose in this Section a way to detect these Extra Terms On-Board ISS as a second proof that Extra Dimensions exists(or not). There are two Classical expressions for the Gravitational Red-Shift 4λ(The wavelength displacement in Spectral Lines due to Gravity as seen by a far away observer in free

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Fernando Loup

space).One approximate and one exact. The approximate expression is given by: GM c2 R

4λapproximate = 4λClassical =

(138)

And the exact one by: 1 −1 4λexact = q 1 − 2GM c2 R

(139)

These expressions considering the Kar-Sinha Extra Terms due to the presence of the Extra Dimensions would be given by: 4λapproximate ks =

GM dy ds 2 (1 + [Φ ]) 2 c R ds cdt

1 4λexactks = q −1 dy ds 2 1 − 2GM (2 + [Φ ] ) ds cdt c2 R 4λapproximate ks = 4λexactks = q

GM dy 2 (1 + [Φ ] ) 2 c R cdt 1

dy 2 1 − 2GM (2 + [Φ cdt ]) c2 R

−1

(140) (141)

(142) (143)

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The approximate expression is more than enough to illustrate our point of view. 4λKarSinha =

4λKarSinha = 4λClassical (1 + [Φ

GM dy 2 (1 + [Φ ]) c2 R cdt

dy 2 dy 2 ] ) = 4λClassical + 4λClassical [Φ ]] cdt cdt

(144)

(145)

dy 2 ] = 4λExtraTerms (146) cdt Our idea is to send a second Satellite with another Laser beam to the Venus Orbit but directed towards the ISS.The Satellite would send the Laser beam to ISS with a certain blue wavelength but when arriving at ISS due to the difference of Sun Gravitational Fields between Earth and Venus the beam would be Red-Shifted and the Kar-Sinha Extra Terms due to the presence of Extra Dimensions could be detected. In Venus the Gravitational Red Shift would be given by: 4λKarSinha − 4λClassical = 4λClassical [Φ

4λKarSinhaVenus =

GM c2 DVenus

(1 + [Φ

dy 2 ] ) cdt Venus

From the Previous Section we know that

Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

(147)

On The 5D Extra-Force According to Basini-Capozziello-Ponce De Leon ...

4λKarSinhaVenus =

GM ds 2 (1 + [ ] ) c2 DVenus cdt Venus

261

(148)

With DVenus being the Sun Radius R plus the distance dVenus from Sun to Venus On Earth the Gravitational Red Shift would be given by: 4λKarSinhaEarth =

GM c2 D

4λKarSinhaEarth =

(1 + [Φ

Earth

GM c2 D

Earth

(1 + [

dy 2 ] ) cdt Earth

ds 2 ] ) cdt Earth

(149) (150)

With DEarth being the Sun Radius R plus the distance dEarth from Sun to Earth Since in Venus the Laser is still Blue-Shifted when sent towards ISS the Red Shift detected by ISS would be generated in the Venus-Earth trip. GM c2 D

Earth

−

GM c2 DVenus

=

GM 1 1 GM 1 1 [ − ]∼ − ] = 2 [ 2 c DEarth DVenus c dEarth dVenus

(151)

The Kar-Sinha Gravitational Red Shift in Extra Dimensions epresions from the Venus Earth trip would be given by the following expressions 4λKarSinhaEarthVenus =

GM 1 1 dy 2 ]) [ − ](1 + [Φ 2 c dEarth dVenus cdt

(152)

GM 1 1 ds 2 [ − ](1 + [ ] ) 2 c dEarth dVenus cdt

(153)

4λKarSinhaEarthVenus = Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

And the Extra Terms due to the presence of the Extra Dimensions would be given by: 4λExtraTermsEarthVenus =

GM 1 1 dy 2 [ − ][Φ ] c2 dEarth dVenus cdt

(154)

GM 1 1 ds 2 [ − ][ ] c2 dEarth dVenus cdt

(155)

4λExtraTermsEarthVenus =

From the previous Section we know that ISS can measure these Extra Terms if the Extra Dimensions exists.

5.

Conclusion-Physics of Extra Dimensions as an Experimental Branch of Physics for the First Time

Our approach to the study of Extra Dimensions was centered on the Basini-CapozzielloPonce de Leon Formalism.While other formalisms of Extra Dimensions uses 3 + 1 uncompactified ordinary spacetime dimensions while the Extra Dimensions are compactified bringing the question of why 3 + 1 large ordinary dimensions and the rest of the Extra Dimensions ”curled-up” over themselves and what causes or generates the ”compactification mechanism”? In the Basini-Capozziello-Ponce de Leon Formalism the Extra Dimensions

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Fernando Loup

are large the same size of the 3 + 1 ordinary dimensions avoiding the need of ”exotic” compactification mechanisms but we cannot ”see” these dimensions in normal conditions due to the reasons presented in this work. Also it can explain the multitude of particles seen in 4D as Dimensional Reductions from a small group of particles in 5 D allowing perhaps the ”unification” of Physics from the point of view of the Extra Dimensional Spacetime. This is very attractive from the point of view of a Unified Physics theory.There exists a small set of particles in 5D and all the huge number of Elementary Particles in 4D is a geometric projection from the 5D Spacetime into a 4D one([2] eq 20,[11] eq 21 and [20] eq 8).32

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

m0 = q Particle u d s c b t e− µ− τ− νe νµ ντ γ gluon W+ Z graviton

spin (~) B 1/2 1/3 1/2 1/3 1/2 1/3 1/2 1/3 1/2 1/3 1/2 1/3 1/2 0 1/2 0 1/2 0 1/2 0 1/2 0 1/2 0 1 0 1 0 1 0 1 0 2 0

L 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0

T 1/2 1/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

T3 1/2 −1/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

M5

(156)

2 1 − Φ2 ( dy ds )

S 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

C 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

B∗ 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0

charge (e) +2/3 −1/3 −1/3 +2/3 −1/3 +2/3 −1 −1 −1 0 0 0 0 0 +1 0 0

m0 (MeV) 5 9 175 1350 4500 173000 0.511 105.658 1777.1 0(?) 0(?) 0(?) 0 0 80220 91187 0

antipart. u d s c b t e+ µ+ τ+ νe νµ ντ γ gluon W− Z graviton

Look to the Elementary Particles Table above:we have a multitude of Quarks,Leptons,Muons and Heavy particles with apparently different rest-masses m0 in 4D but these particles can have the same rest-mass M5 in the 5D Extra Dimension and the differences are being generated by the Dimensional Reduction from 5 D to 4D according to the Basini-Capozziello-Ponce De Leon formalism.This can bring new perspectives for the desired dream of the Unification of Physics We proposed here the use of a Satellite 33 with a Laser device placed in the other side of Earth Orbit with the Sun between the Satellite and ISS.The satellite will send the Laser beam towards ISS and the beam must pass in the neighborhoods of the Sun in order to reach ISS. The bemm will be Gravitationally Bent according to Einstein and if Extra Dimensions exists then the Extra Terms in the Gravitational Bending Of Light affected by the presence of the Extra Dimensions predicted by Kar-Sinha will appear. We demonstrated here that ISS have the needed precision to spot the Extra Terms predicted by Kar-Sinha and ISS could answer for the first time the question if the Universe have or not more than 4 Dimensions predicted 32 We

know that we are repeating the table for the third time but the table coupled with the Ponce De Leon equation illustrates the beauty of this point of view 33 Such a Satellite could perfectly be christened as ”Eddington” as a Homage to Sir Arthur Stanley Eddington Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

On The 5D Extra-Force According to Basini-Capozziello-Ponce De Leon ...

263

by many Physics Theories but never seen before. Such an experiment would have the same degree of importance of the measures of Gravitational Bending Of Light made by Sir Arthur Stanley Eddington in the Sun Eclipse of 1919 that proved valid the Einstein General Theory Of Relativity The implications of a ”positive” result would be enormous making the Physics of Extra Dimensions an Experimental Branch of Physics for the first time 34 35 . All the theories of Physics Unification that predicts the existence of Extra Dimensions would be regarded as valid Physical Descriptions of Nature and not only mere Mathematical Models adjusted to ”normalize consistently ” some calculations and this would pose major modifications in Particle Physics since it could be proved that the multitude of apparent different particles we see in 4 Dimensions are Dimensional Reductions or Dimensional Projections from a small group of perhaps the same particles in 5 Dimensions according to the Basini-Capozziello Ponce De Leon formalism If this experiment is performed with ”positive” results then the International Space Station ISS could change drastically and forever our way to understand the Universe.Young Jedi Knight Padawan:Stay Away From The Dark Side And May The Force Be With You36

6.

Epilogue • ”The only way of discovering the limits of the possible is to venture a little way past them into the impossible.”-Arthur C.Clarke 37

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

• ”The supreme task of the physicist is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can reach them”-Albert Einstein 3839

Acknowledgments We would like to express the most profound and sincere gratitude towards Doctor Frank Columbus Editor-Chief of NOVA Scientific Publishers United States of America for the invitation to write a paper for his presentation conference meeting entitled ”Space Stations: Crew, Experiments and Missions.” This arXiv.org paper is our answer to the invitation. We also would like to Acknowledge Professor Doctor Martin Tajmar of University Of Viena, Austria – ESA (European Space Agency) – ESTEC-SV (European Space Technology and Engineering Center - Space Vehicles Division) and Seibesdorf Austria Aerospace Corporation GmBH-ASPS (Advanced Space Propulsion Systems), for his kindness and goodwill 34 An

excellent account of the progresses in this area is given by the link below: physics/1/abs : +AND + dimensions + AND + experimental + extra/0/ 1/0/all/0/1 36 Slightly modified from Frank Oz as Master Yoda in the George Lucas movie Star Wars Episode II The Attack Of The Clones 37 special thanks to Maria Matreno from Residencia de Estudantes Universitas Lisboa Portugal for providing the Second Law Of Arthur C.Clarke 38 ”Ideas And Opinions” Einstein compilation, ISBN 0 − 517 − 88440 − 2, on page 226.”Principles of Research” ([Ideas and Opinions],pp.224-227), described as ”Address delivered in celebration of Max Planck’s sixtieth birthday (1918) before the Physical Society in Berlin” 39 appears also in the Eric Baird book Relativity in Curved Spacetime ISBN 978 − 0 − 9557068 − 0 − 6 35 htt p : //arXiv.org/ f ind/gr p

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Fernando Loup

for being our arXiv.org sponsor and to Acknowledge Paulo Alexandre Santos and Dorabella Martins da Silva Santos from University of Aveiro Portugal for the access to the scientific publication General Relativity and Gravitation(GRG). In closing we would also like to Acknowledge the Administrators and Moderators of arXiv.org at the Cornell University, United States of America for their agreement in accepting this document.

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7.

Remarks

The bulk of the bibliographic sources used in our research came from the refereed scientific publication General Relativity and Gravitation (GRG) from Springer-Verlag GmBH (formerly Kluver/Plenum Academic Publishing Corp)(ISSN:0001-7701 paper)(ISSN:15729532 electronic) under the auspices of the International Comitee on General Relativity and Gravitation and quoted by Deutsche Zentralblatt Math of EMS(European Mathematical Society). The Volume 36 Issue 03 March 2004 under the title:”Fundamental Physics on the ISS” was totally dedicated to test experimentally General Relativity,Quantum Gravity,Extra Dimensions and other physics theories in Outer Space on-board International Space Station(ISS) under the auspices of ESA(European Space Agency) and NASA(National Aeronautics and Space Administration). All the mention to pages of the references in the main text and in the footnotes of this work are for GRG and Liv Rev Rel references originally from the published version since we have access to GRG and Liv Rel Rel although we provide the number of the arXiv.org available GRG and Liv Rev Rel papers but for PhysRpt the page numbers are originally from the arXiv.org version since we cannot access this journal and sometimes exists differences in page numbers between the arXiv.org version and the published version due to different editorial styles preferred by scientific journals. 40 We choose to adopt in our research mainly refereed published papers from these publications not only due to their prestige and reputation among the scientific community but also because we are advocating new points of view in this work but based on the solid ground of certifiable and credible references

8.

Legacy

This work is dedicated to the memory of the British Astronomer Sir Arthur Stanley Eddington that measured for the first time the Gravitational Bending Of Light in the Sun Eclipse of 1919 proving valid the Einstein General Theory Of Relativity. This work is also dedicated with a feeling of gratitude to all the people of NASA(National Aeronautics and Space Administration),ESA(European Space Agency),to all the people of the Space Agencies of Canada,Russia,Japan,China,Brazil,Argentina.Mexico,Israel and India all of these people with major contributions and Manned Space Missions or responsible and involved in one way or another with the project of the International Space Station ISS 40 readers

that can access GRG can compare for example gr-qc/0310078 with [2] or gr-qc/0603106 with [20]

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On The 5D Extra-Force According to Basini-Capozziello-Ponce De Leon ...

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References [1] Basini G. and Capozziello S. (2005).Gen Rel Grav 37 115 . [2] Ponce De Leon J. (2004).Gen Rel Grav 36 1335,gr-qc/0310078. [3] Kar S. and Sinha M. (2003).Gen Rel Grav 35 1775. [4] Loup F. Santos P. and Santos D. (2003).Gen Rel Grav 35 2035. [5] Basini G. and Capozziello S. (2003).Gen Rel Grav 35 2217. [6] C.Lammerzahl;G. Ahlers N. Ashby, M. Barmatz,P. L. Biermann, H. Dittus, V. Dohm, R. Duncan, K. Gibble, J. Lipa, N. Lockerbie, N. Mulders and C. Salomon. (2004). Gen Rel Grav 36 615 [7] Clarkson C. and Maartens R. (2005).Gen Rel Grav 37 1681,astro-ph/0505277 [8] Dittus H. (2004). Gen Rel Grav 36 601 [9] Mashhoon B., Wesson P. and Liu H. (1998). Gen Rel Grav 30 555 [10] Loup F. Santos P. and Santos D. (2003).Gen Rel Grav 35 1849 [11] Ponce De Leon J. (2003).Gen Rel Grav 35 1365,gr-qc/0207108 [12] Wesson P. (2003). Gen Rel Grav 35 307,gr-qc/0302092 [13] Seahra S. and Wesson P. (2005). Gen Rel Grav 37 1339 Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

[14] Seahra S. and Wesson P. (2001). Gen Rel Grav 33 1731,gr-qc/0105041 [15] Billyard A. and Sajko W. (2001).Gen Rel Grav 33 1929,gr-qc/0105074 [16] Paik H. , Moody M. and Strayer D. (2004).Gen Rel Grav 36 523 [17] Dittus H. , Lammerzahl C. and Selig H. (2004).Gen Rel Grav 36 571 [18] Walz J and Hansch T. (2004).Gen Rel Grav 36 561 [19] Will C.M.,(2006).Liv Rev Rel,(9) lrr-2006-3,gr-qc/0103036 [20] Loup F (2006).Gen Rel Grav 38 1423,gr-qc/0603106 [21] Overduin J.M. and Wesson P. (1997). Phys.Rept. 283 303-380,gr-qc/9805018

Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

In: Space Science Research Developments Editors: J.C. Henderson and J. M. Bradley

ISBN 978-1-61209-086-3 © 2011 Nova Science Publishers, Inc.

Chapter 9

ORIGIN OF THE SATURN RINGS: ELECTROMAGNETIC MODEL OF THE SOMBRERO RINGS FORMATION Vladimir V. Tchernyi Modern Science Institute, SAIBR. 20-2-702, Osennii blvd. Moscow 121614, Russia

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ABSTRACT For the first time the role of superconductivity of the space objects within the Solar system located behind a belt of asteroids is considered. Observation of experimental data for the Saturn’s rings shows that the rings particles may have superconductivity. Theoretical electromagnetic modeling demonstrates that superconductivity can be the physical reason of the origin of the rings of Saturn from the frozen particles of the protoplanetary cloud. The rings appear during some time after magnetic field of planet appears. It happened as a result of interaction of the superconducting iced particles of the protoplanetary cloud with the nonuniform magnetic field of Saturn. Finally, all the Kepler’s orbits of the superconducting particles are localizing as a sombrero disk of rings in the magnetic equator plane, where the energy of particles in the magnetic field of Saturn has a minimum value. Within the sombrero disc all iced particles redistributing by the rings (strips) like it is happened for the iron particles nearby the magnet. Electromagnetism and superconductivity allow us to understand why planetary rings in the solar system appear only for the planet with the magnetic field after the belt of asteroids where the temperature is low enough and why there are no rings for the Earth, and many other phenomena.

1. INTRODUCTION There is no yet clear picture of the origin of Saturn’s rings. There are two versions of the rings origin. The most common version saying that the rings originated when an asteroid type body approached Saturn and was destroyed by the gravity and centrifugal forces and then from the debris the rings were created. It looks like this version contains a mysterious fact. Another idea relates to the origin of rings from the particles of the protoplanetary cloud around Saturn, and this problem has not been resolved yet. This paper compensates the lack of this knowledge, demonstrating how rings could originate and form from the frozen

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particles of the protoplanetary cloud after the appearance of the magnetic field of Saturn due to electromagnetic interaction of icy particles with the planetary magnetic field. The founder of the theory of electromagnetic waves J.K. Maxwell in his award winning paper on the subject “On the stability of the motion of Saturn’s rings” (1859), deduces that the rings of Saturn cannot be solid and the rings could be stable only if they consist of “an indefinite number of unconnected particles orbiting Saturn in much the same way as our Moon orbits the Earth” [1]. Otherwise gravitational forces would destroy them. Ground-based experiments and the data of the Pioneer, Voyager 1, 2 and Cassini-Huygens space missions have revealed the rings to be composed of icy particles, and icy particles with impurities. After G. Galileo (1610) many researchers have studied the nature of the rings [for example, 1-16]. From the consideration of the gravity and celestial movements it follows that different ring systems are morphologically quite distinct and are all shaped by a few common processes. This is the outward transport of angular momentum by rings particles and by gravitational interactions between satellites, moons and ring material. Orbital resonances between satellites, moons and ring particles play an important role in enhancing the influence of satellites and forming specific structure of the rings and gaps. At the same time, extensive experimental data confirmed the importance of magnetohydrodynamic plasma phenomena and, particularly, electromagnetism of the rings structure origin. Despite the rich available database, there is no yet physically satisfactory model of the Saturn rings and the mystery of many experimental data has no explanation: origin, evolution and dynamics of the rings; why particles are separated and at the same time they could stick together; considerable flattening and the sharp edges of the rings; thin periodic structure of the rings; deformation of the magnetic field lines nearby ring F; formation of “spokes” in the ring B; high radio-wave reflectivity and low brightness of the rings; anomalous reflection of circularly polarized microwaves (like from magnetic mirror); strong pulse electromagnetic radiation of the rings in the 20.4 kHz - 40.2 MHz range; spectral anomalies of the thermal radiation of the rings; why substance of the rings does not mix, but preserves its small-scale color differences; existence of an atmosphere of unknown origin nearby the rings; why there is existence of the waves of density and bending waves within the rings; why the planetary rings in the solar system appear for the planets which are located only outside the belt of asteroids and why the Earth has no rings, etc. The superconductivity of the ring particles may follow from the fact that the ring particles are relics of the early days of the Solar system and particles were never subject to coalescence and heating. Indeed, the Sun heats the rings weakly, because temperature in the area of the rings is about 70-110 K. It makes possible the existence of the superconducting substance in the space behind the belt of asteroids. The superconductive particles cannot stick together because the magnetic field emanates from them and pushes the particles apart. In 1933 W. Meissner and R. Ochsenfeld found that a superconducting material will repel a magnetic field. The high-temperature superconductivity was discovered by Bednortz and Muller in 1986 [17]. In 1986 superconductivity of ice was experimentally demonstrated by A.N. Babushkin1 et al [18]. It is assumed that superconducting matter of the particles of the rings of Saturn allows extending classical theories of the planetary rings (gravitational, mechanical, 1

Professor Aleksey N. Babushkin is a Dean of the Physics Department at the Ural State University, Yekaterinburg, Russia. Tel: +7(912)243-6892, +7(343)261-1885, +7(343)261-6058; E-mail: [email protected]

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magnetohydrodynamic and plasma interactions) by non-conflicting superconducting model [20-44]. An interesting fact of this paper is that even though its subject concerns problem related to astronomy, the solution may come from the general electromagnetic theory.

2. AN EXPERIMENTAL DATA OBSERVATION Thin Structure and Sharp Edges of the Rings Similar to magnetic particles creating dense and rarefied areas in a nonuniform magnetic field, the superconductive ice particles also form their groups, which from outside look like a system of rings. Superconducting particles will collapse into a system of rings as the result of their replacement in the area with the less density of the magnetic flow, within plane of magnetic equator, with the force: F = - mdH/dz, where m – magnetic moment of a particle, dH/dz – gradient of intensity of a magnetic field along an axis z of the magnetic dipole. The force of diamagnetic push-out is forming the sharp edges of the ring: F= -mdH/dy, where dH/dy - gradient of intensity of a magnetic field along the radius of the ring. The casual break in the ring will be stabilized by the force of diamagnetic push-out F = - mdH/dx, where dH/dx - gradient of intensity of a magnetic field in tangential direction. Measurements of the magnetic field nearby the rings F by the Pioneer mission [5, 6, 12] have registered deformation (distortion) of the magnetic field lines like it is happened for the superconducting disc in laboratory under Meissner - Ochsenfeld state [17].

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Planetcentrical Dust Flow For superconducting particles there is London’s depth λL of penetration of the magnetic field inside superconductor [17]. Influence of penetration of the magnetic flow becomes appreciable for particles with the size comparable with London’s depth of penetration. Smaller particles are not cooperating with the planetary magnetic field, because they lost their superconductivity by size. Dynamics of these particles is different from dynamics of particles with bigger size which is >2λL. These particles will fall down to the planet due to the gravity. Thus, existence of the planetcentrical dust flows of submicron’s size particles related to disappearance of superconductivity of the matter of the rings particles due to reducing their size. It is also possible for the particles to loose their superconductivity by influence of collisions and by fluctuations of magnetic field.

Change of the Azimuth Brightness of the A Ring of Saturn There is a number of theories explaining this fact, based on assumptions of a synchronous rotation of the ring’s particles with their asymmetrical form as extended ellipsoids directed under a small angle to the orbit, or with asymmetrical albedo of the surface [5, 6, 12]. Let’s go to superconducting model. If superconductor is placed in the magnetic field, additional moment directed opposite to the external field arises. The matter is magnetized, not along the

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Vladimir V. Tchernyi

external magnetic field but in the opposite direction. The rod of superdiamagnetic substance of the ring particle tries to locate itself perpendicularly to the magnetic field lines. From the science that studies ice [19] it is a known fact that at the temperature below – 22 0C growing snowflakes take the form of prisms. Thus, the prism of the superconducting iced particle will be oriented perpendicularly to the field lines of the polhoidal and toroidal constituents of the magnetic fields of Saturn. So, it’s clear that variable azimuth brightness of the Saturn’s rings system A relates to orientation of the elongated ellipsoid of superconducting particles to the normal direction to the magnetic field of the planet.

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Spokes in the Ring B of Saturn As well as the spokes of any wheel, are located almost along radiuses. According to the laws of Kepler any radial formation should be distorted and washed out for a few tens of minutes. However, experimental data show that lifetime of separated spoke 103 - 104sec though Kepler washing outs in them are nevertheless proven. The size of spokes themselves is about 104km along the radius and about 103km along the orbit of the ring. The matter of the spokes consists of micron and submicron size particles [11]. There were many attempts to explain the nature of the spokes. Mostly all theories are based on the action of the force of gravity. At the same time, there were some ideas that registration of rotating spokes somehow related to electromagnetic interaction because of its rotation synchronously going along with the magnetosphere of the Saturn [5, 6, 12]. The analysis of spectral radiant power of spokes provides specific periodicity about 640.6±3.5 min, which is almost coincident to the period of rotation of the magnetic field of Saturn, which is 639.4 min. Moreover, the strong correlation of maxima and minima of activity of spokes with the spectral magnetic longitudes is connected to presence or absence of the radiation of Saturn’s Kilometric Radiation (SKR). It confirms the assumption of the dependence of the spokes dynamics on the magnetic field of Saturn and testifies to the presence of large-scale anomalies in the magnetic field of Saturn. Superconducting iced particles of the rings matter are rotating in accordance with Kepler’s law, and at the same time magnetic field is rotating along with the planet and has anomalies itself. Superconducting particles coming to anomalies of the magnetic field positions and the balance of the three forces acting with each particle will change due to change of the electromagnetic force. Then within the anomaly of the planetary magnetic field all particles will try to get another position and observer can see all these chaotic movements of the particles on the picture of the rings as a spokes. After passing position of the planetary magnetic field anomaly all particles will get positions in accordance with Kepler law.

High Reflection and Low Brightness of the Rings Particles in the Radiofrequency Range We can explain also using superconducting model. The discovery of strong radar-tracking reflection from the rings of Saturn in 1973 was surprising [5, 6, 12]. It turned out that the rings of Saturn actually have the greatest radar-tracking section among all bodies of the Solar system. It was explained by the metallic nature of the particles. The data of the Voyager excludes this possibility. The disk of superconducting particles completely reflects radiation

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with frequencies below 1011Hz and poorly reflects radiation with higher frequencies, as in the case of superconductor. The superconductors have practically no resistance up to frequencies of 100MHz. At frequencies about 100GHz there comes a limit, above which the frequent quantum phenomena cause a fast increase of resistance, as it is shown on figure 1. Hence a specific picture of the dependence of brightness.

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Figure 1. Top diagram is the dependence of the brightness temperature of the rings on the wavelength: transition from the radiation of the almost black body to practically complete reflection is observed [5, 6, 12]. Bottom diagram is the dependence of the surface resistance of the superconductor on frequency for Nb at T=4,2K [17].

Own Wide Band Pulse Radiation of the Rings Data from the Voyager have shown that the rings radiate own wide band pulse radiation within the 20 KHz-40,2MHz [5, 6, 12]. These waves, probably, are a result of interaction of charged particles with the particles of ice or destruction and friction of iced particles when costriking occur. These incidental radio discharges are named as Saturn’s Electrostatic Discharges (SED). The average period SED is well determined and was established as 10 hour 10±5 min and 10 hour 11±5 min by Voyager-1, -2. If the ring has a source of SED, the area of this source can be located at the distance of 107,990 – 109,000km from the planet according to measured periodicity. Data of the experiments can help specify electrodynamics coupling between the planetary ring system and the magnetosphere, in which SKR, SED and activity of spokes are subordinated to longitude regulation. In accordance with superconducting model, approaching of superconducting rings’ particles up to distance of about 10-8m, or, the existence of narrowing or dot contact will result in the formation of a weak link (superconducting transition) through which superconducting electrons can be tunneled. When the difference of phases between superconductors under action of the electrical or magnetic field occurs, the weak link will generate electromagnetic radiation with frequency proportional to power failure on this transition (nonstationary Josephson phenomenon) [17, 20-36]. The radiation frequency is proportional to the voltage in the

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transition, ν=2eV/h, where 2e/h= 483,6 MHz/µV, e is a charge of electron, h is Plank constant.

Frequency Anomalies of Thermal Radiation of the Rings in the 100µm - 1cm Range The measured brightness temperature on the short waves is less than true brightness temperature of the rings, and on the longer waves the rings look much colder than in the case when the radiation corresponds to their physical temperature [5, 6, 12]. On the range 100µm 1mm brightness temperature of the ring (figure 2) sharply falls to the meanings smaller than those ones characteristic of an absolutely black body. On the wavelengths longer than 1cm a ring behaves as the diffusion screen, reflecting planetary and cold space radiation. The central part of the spectral range 100µm – 1cm is the most sensitive to the parameter of refraction, and may contain the important determining information of fundamental properties of the substance. In accordance with the model, under the superconducting condition the electrons do not interact with a crystal lattice and do not exchange energy with it, therefore they cannot transfer heat from one part of the body into another. Hence, when the substance passes into a superconducting condition, its heat conductivity is lowered. This effect can be obvious under temperatures much less critical, when there are very few conventional electrons capable of transferring heat [20-36].

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Color Difference of Rings in a Small Scale The balance of three forces determines the position of the superconducting particles in the gravitational and magnetic planetary fields: gravitational force, centrifugal one and magnetic levitation (diamagnetic push out), figure 1. Going along with our model let’s consider the distribution of three particles (a, b, c) with equal weights on close orbits. Let a particle a be wholly superconducting, b – have an impurity clathrate-hydrates of ammonia or methane (NH3; CH4 H2O), c - has an impurity of sulphur and Ferro containing silicates (H2S). Each impurity will give the contribution to reduction of the volume of superconducting phase and will determine the color of the particle. Force of diamagnetic push out - FL depends on the volume of the superconducting phase, therefore for each of considered particles the balance of three forces will be carried out in the orbits with different radiuses [20-36].

Anomalous Inversion Reflection of Microwaves with Circular Polarization Above 1 Cm The research of reflection of radiowaves above 1 cm from the rings was carried out with the use of ground based radio-locators and by the space probes [5, 6, 12]. The reflection appeared rather large, and the geometrical albedo is equal approximately to 0.34 and has no strong functional dependence on the wavelength or on the angle of the inclination of ring’s pitch. The rings are strong depolarizers. Therefore, in order to get any information from the reflection it is necessary to measure separately the intensity of two orthogonal polarized

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reflected signals. It provides information of the factor of the ring’s polarization, which carries information about properties of particles. For the majority of single objects of the Solar system, for example the planets, factor of reflection unobserved polarization (orthogonal to observable) is rather small. As to the rings, the supervision in some range of wavelengths and angles of inclination give reflection factor of unobserved polarization between 0.4 – 1.0 . Let’s go to our superconducting model [20-36]. The superconductors have an essential difference from ideal conductors; besides almost infinite conductivity they also demonstrate an ideal diamagnetism. The falling electromagnetic wave will induce circular currents in superconductor, which will completely compensate action of the magnetic field of the incident wave. So that the absence of the magnetic field in the volume of the superconductor should be carried out. Superconductor will be acting as a magnetic mirror. Thus, if the falling on superconductor electromagnetic wave has a determined direction of a circular polarization (the spirality), the direction of circular polarization (spirality) will be kept in the reflected wave.

An Atmosphere of “Unknown” Origin at the Rings

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The atmosphere of Saturn’s rings can exist as a result of thin balance of forces of gravitational attraction and diamagnetic push-out of gas molecules. The levitation of gas molecules comes true at the expense of forces of diamagnetic push-out induced in superconducting particles by molecular magnetic moment of gas [20-36]. The similar situation can be observed under laboratory conditions when atmospheric water steam is precipitated on substance as whitefrost at the transition moment of substance from a superconducting into a conventional one.

Existence of Wave of Density and Bending Waves within the Rings The existence of waves of density and bending waves in the Saturn’s rings has no complete explanation based only on gravitation phenomena. Let’s use the superconducting model. It is possible to note that the external magnetic field is directed along the free surface of diamagnetic liquid which represents a disk of the rings. In case of periodic deformation of a free surface a normal field under a hollow decreased, and it increased under chamber. Consequently, the ponderomotive force works on the side of restoration of the flat form of the free surface. Thus, the field increases a rigidity of a free surface. The change of rigidity of the free surface in a field gives a chance to excite its parametrical fluctuations. When the quantity of the amplitude of intensity of a variable field is larger than the critical one, the standing wave is occurring on a free surface of superconducting liquid. The constant phase lines of this wave are transverse to the vector of intensity of the field. The wavelength is determined by the condition of the parametric resonance. The constant phase lines begin to be bent when intensity of the field increases, and the excited ripple becomes casual [20-36].

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3. THEORETICAL SOLUTION OF THE SATURN’S RINGS ORIGIN Following from the solution of the electromagnetic problem we will demonstrate how rings of Saturn could be originated from the iced particles located within the protoplanetary cloud. Before appearance of the magnetic field of Saturn all particles within the protoplanetary cloud are located on such an orbit as Kepler’s, where there is a balance of the force of gravity and the centrifugal force. With the occurrence of the magnetic field of the Saturn the superconducting particles of the protoplanetary cloud begin to demonstrate an ideal diamagnetism (Meissner-Ochsenfeld phenomenon). Particles start to interact with the magnetic field and all particles become to be involved in additional azimuth-orbital movement. Let’s estimate the result of this movement [37-38]. If the magnetic field of the planet is equal H, and the planetary magnetic moment is equal G μ , then the magnetic field at any particular point within the protoplanetary cloud, located on

G

the distance r , can be presented as:

H=

3 ⋅ r ⋅ (r, μ) μ − 3 r5 r

(3.1)

Then the superconducting ball with the radius R, which is located within the protoplanetary cloud, gets the magnetic moment equal:

M = −R 3 ⋅ H .

(3.2)

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That energy of a superconductor in a magnetic field gets the following value:

U H = − (M , H ) = R 3 ⋅ H 2

(3.3)

If the beginning of coordinates to place in the center of a planet, and an axis z to direct along the magnetic moment of a planet (orthogonal to equator) magnetic energy thus will be equal:

UH =

R3μ 2 (3 cos 2 θ + 1) . 6 r

(3.4)

G

Here θ - an angle between a vector r and an axis Z. We can see from the expression (3.4), that magnetic energy of the superconducting particle has a minimum value when the radiusG vector r (position of the superconducting particle) appears in a plane of magnetic equator, a

perpendicular to the axis Z ( cos θ = 0 ) . It is clear, that for one particle its trajectory (orbit)

in the azimuth-orbital movement will be only disturbed by the magnetic field. However, in case of a huge amount of particles, like its happened within the protoplanetary cloud, after some time, collisions between particles will compensate their azimuth-orbital movements, and, as a result, eventually, during some time, all orbits of the particles of the protoplanetary cloud should come together to magnetic equator plane and create highly flattening disc around planet. Within the disc of rings all particles will be located on such an orbit as

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Kepler’s, where there is a balance of the force of gravity, the centrifugal and electromagnetic forces. At the same time, orbital resonances (due to gravity force) between satellites, moons and the rings particles play an important role in forming specific structure of the rings and gaps.

4. SEPARATION AND COLLISION OF THE PARTICLES WITHIN THE SOMBRERO OF RINGS We can define the energy of the interaction of two superconducting particles with magnetic moment μ1 и μ2 if they are located on the distance r1 и r2, respectively as:

U = −μ1H 2 ,

(4.1)

Where magnetic field H2 is produced by the magnetic moment μ2 it can be presented as

H2 =

3(r1 − r2 )(μ 2 (r1 − r2 )) r1 − r2

5

−

μ2 r1 − r2

(4.2)

3

If we place the particles with magnetic moment μ2 at the beginning of coordinate (r2=0) then the expression for the energy of the interaction of two particles (4.1) will be the following:

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U =−

3(μ 1r1 )(μ 2 r1 ) r1

5

+

μ 1μ 2 r1

3

.

(4.3)

In the plane of the rings of Saturn the magnetic field of the planet coincides with the rotation axis of the planet. If the axis Z is directed along with the rotation axis of the planet then magnetic moment of the particles also will be directed along the axis Z. Using the cylindrical system of coordinate

( ρ ,ϕ , z ) we can represent expression (4.3) as:

⎛ ⎞ 2 2 3z 2 1 ⎟ μ1z μ2 z = ρ − 2 z μ1z μ 2 z , U = −⎜ − 2 2 52 ⎜ ( ρ 2 + z 2 )5 2 ( ρ 2 + z 2 )3 2 ⎟ ρ z + ( ) ⎝ ⎠

(4.4)

Let us use the expression (4.4) to estimate how two superconducting particles will be interacting in two different cases. The first one is when two particles are located in the same plane within the sombrero of the rings (Z=0), and the second situation is when two particles are located on the different planes but on the same axis

( ρ = 0) .

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Vladimir V. Tchernyi From the expression (4.4) follows that for the particles with magnetic moment

μ2 z

μ1z

and

which are located on the same plane, Z=0, we can get that energy of their interaction is

equal: U

=

μ1z μ2 z ρ3

,

(4.5)

From the expression (4.5) follows that in this situation both particles will be pushing each other and they will be holding separation distance in between them. This result has been confirmed by the data of Cassini mission: the particles are separated2. Then for another situation both particles are located on the same axis but on the different planes, and, as it follows from (4.4) the expression for the interaction energy is:

U =−

μ1z μ 2 z z

3

,

(4.6)

We can see that in this case both particles will be attracting each other, they could even collide or stick together and form bigger pieces or lumps of ice. This fact has an experimental conformation by Cassini mission. As we know from the data of Cassini mission it was registered that particles within the sombrero of the rings can collide or even stick together and form bigger pieces of ice3. Then later the particles with 50 meters or more in diameter can be destroyed into smaller pieces by the common action of gravity and centrifugal force.

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4. CONCLUSION Theoretical electromagnetic model of the origin of the rings of Saturn from superconducting particles of the protoplanetary cloud, which is presented in this paper, is a direct continuation of the paper by J.K. Maxwell (1859). Unfortunately, at his time there was no knowledge about superconductivity (1911) and force of diamagnetic push-out of superconductor (1933). Superconductivity of ice and high temperature superconductivity was discovered just recently, in 1986 [17, 18]. As it follows from above consideration, observation of experimental data and electromagnetic modeling confirmed the suggested model. The model considered here makes possible a “magnetic coupling” between protosun and superconducting particles of its protoplanetary cloud, that in the process of formation of the solar system at an early stage of its development, when the temperature was low enough to have superconductivity, lead to the carrying of the moment of momentum from the Sun to other planets by the electromagnetic means of the superconducting substance of the space environment. Following this we can conclude that the idea of H. Alfen [10-11] that “solar system history as recorded in the Saturn rings structure” becomes to be a physical reality. From the approach presented above, with the possibility of an electromagnetic origin for the rings, it follows as important for the space physics to take into account the natural space 2 3

http://pds-rings.seti.org saturn.jpl.nasa.gov/multimedia/images/index.cfm

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superconductivity for the space substance after the belt of asteroids. It may have a fundamental importance for analyzing the data of the Cassini-Huygens probe, and striking parallels to those that occur in more remote disc systems such as galactic discs and accretion discs around stars and black holes. The force of diamagnetic push-out of superconductor in the magnetic field may be a driving force for propagation of organic molecules within the interstellar space by electromagnetic means, and organic molecules can also be contained within the rings of Saturn, as it is presented in [45-47]. The author would like to express greatest thanks for valuable discussions to A.M. Prokhorov, V.N. Strakhov, V.V. Migulin, Yu.V. Gulyaev, V.I. Pustovoit, A.A. Rukhadze, B.I. Rabinovich, V.G. Kurt, A.Yu. Pospelov, A.N. Malov, O.N. Rzhiga, V.A. Miliaev, E.V. Chensky, V.N. Lugovenko, S.V. Girich, M.V. Belodedov, E.P. Bazhanov, A.V. Zaitsev from the Russian Academy of Sciences; S.V. Vasilyev (SVVTI, CA), V.P. Vasilyev (SOLERC, Ukraine), J.R. Whinnery and T.K. Gustafson (UC Berkeley); J.A. Kong (MIT), D. Osheroff (Stanford), E.A. Marouf (SJSU); J.N. Guzzi, R.B. Hoover, J.F. Spann, R. Sheldon, D. Gallagher, K. Mazuruk and A. Pakhomov (NASA); P. Goldreich (Caltech), L. Spilker (JPL), C.T. Russel and Y. Rahmat-Samii (UCLA), G. Shoemaker (CSUS), L.N. Vanderhoef, P. Rock, R.T. Shelton, R.R. Freeman, A. Albrecht, B. Backer, W. Ko and W. Pickett (UC Davis), R.P. Kudritzki and R.D. Joseph (IFA UH), J.A. Burns (Cornell University, NY), M. Pardavi-Horvath (GTU), P.M. Cincotta (IAFE, Buenos Aires), G. Gerlach (Oriongroup, CA), A. Mendis, L. Peterson, M. Fomenkova (UCSD), N. Castle, R.S. Henderson (San Diego, CA), K. Fischer (Los Gatos, CA), C.B. Vesecky (Palo Alto, CA), A. Pagliere (Sacramento, CA), A.V. Aliaev, V.V. Tsykalo, P.A. Razvin, E.N. Muraviev, O.I. Chernaya (Moscow, Russia) for support. The author is also grateful for fruitful discussions to all participants of the seminars and conferences at the NASA Marshall Space Flight Center and the Huntsville Space Physics Colloquium, the Institute for Astronomy at the University of Hawaii, Astrophysics and the Space Research Center at the University of California in San Diego, CA, the Electrical Engineering and Computer Sciences Department at the University of California in Berkeley, the University of California in Davis, the Institute of Astronomy and Physics at La Plata in Buenos Aires, the Progress In Electromagnetic Research Symposium (PIERS) organized by MIT, the 42nd – 50th SPIE Annual Meetings, the National Bureau of Standard in Washington D.C., the 30th Annual Meeting of the Division of Planetary Sciences of the American Astronomical Society.

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[24] Pospelov A.Yu., Tchernyi V.V., Girich S.V. Possible explanation of the planet’s rings behavior in the radio and mm-wave range via superdiamagnetic model // Kona, HW: SPIE International Symposiumon Astronomical Telescopes and Instrumentation. 20-28 March 1998. № 132. № 73. [25] Pospelov A.Yu., Tchernyi V.V., Girich S.V. Possible explanation of the planet’s rings behavior in the radio and mm-wave range via superdiamagnetic model // International Aerospace Abstracts of American Institute of Aeronautics and Astronautics, Inc. January 1999. № 1. P. A99-10781. [26] Pospelov A.Yu., Tchernyi V.V., Girich S.V. Superdiamagnetic model of planetary rings behavior in the millimeter and submullimeter range // Digest 3465 – 4th International Conference on MM and SMM Waves and Applications. San Diego, CA: Proc. SPIE 43 Annual International Symposium. 20-23 July 1998. San Diego, CA, 1998. P. 172-173. [27] Girich S.V., Pospelov A.Yu., Tchernyi V.V. Radar data explanation via superdiamagnetic model of the Saturn’s rings // Annual Report of the AAS. 30th Meeting Division of Planetary Science. Madison, WI. 11-16 October 1998. Bulletin of the American Astronomical Society. 1998. V. 30. № 3. P. 1043. [28] Pospelov A.Yu., Tchernyi V.V., Girich S.V. Anomalous inversion of polarization of icy satellites and Saturn’s rings: superdiamagnetic model // Proc. 44th SPIE Annual Meeting. Denver, CO. 18-23 July 1999. “Polarization: measurements, analysis and remote sensing II”. 1999. V. 3754. P. 329-333. [29] Pospelov A.Yu., Tchernyi V.V., Girich S.V. Are Saturn’s rings superconducting? // Progress In Electromagnetic Research Symposium (PIERS). 5-14 July 2000. Cambridge, MA: MIT. 2000. P. 1158. [30] Pospelov A.Yu., Tchernyi V.V., Girich S.V. What data could confirm Saturn’s rings superconductivity? // SPIE conference on Astronomical Telescopes and Instrumentation. Munich, Germany. 27-31 March 2000. N. 4015-67. [31] Pospelov A.Yu., Tchernyi V.V. Magnetic levitation of Saturn’s rings. // Progress In Electromagnetic Research Symposium. (PIERS). 5-14 July 2002. MIT. Cambridge, MA. 2002. C. 135. [32] Tchernyi V.V., Pospelov A.Yu. Possible role of space electromagnetism for Saturn’s rings existence // Progress In Electromagnetic research Symposium. (PIERS). 13-16 October 2003. Honolulu, HW. 2003. [33] Tchernyi V.V., Pospelov A.Yu. Possible magnetic levitation of Saturn’s (planetary) rings. Pisa, Italy. // Progress in Electromagnetic Research Symposium. (PIERS). 28-31 March, 2004. N. P08. [34] Pospelov A.Yu., Tchernyi V.V. Space electromagnetism: modeling of magnetic levitation of superconducting Saturn rings // Progress In Electromagnetic Research Symposium (PIERS). Hangzhou, China. 22-26 August 2005. [35] Tchernyi V.V., Pospelov A.Yu. Possible electromagnetic nature of the Saturn’s rings: superconductivity and magnetic levitation // Progress in electromagnetic research (PIER). Cambridge, MA: MIT Press. 2005. V. 52. P. 277-299. [36] Tchernyi V.V., Pospelov A.Yu. About possible electromagnetic nature of the planetary rings: magnetic levitation of superconducting rings of Saturn // Fizika volnovykh protzessov i radiotekhnicheskikh sistem. (In Russian). 2005. Т. 8. № 2. P. 4-16.

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[37] Tchernyi V.V., Chensky E.V. Electromagnetic background for possible magnetic levitation of the superconducting rings of Saturn // Journal of Electromagnetic Waves and Applications. Cambridge, MA: MIT Press. 2005. V. 19. № 15. P. 1997-2006. [38] Tchernyi V.V., Chensky E.V. Movements of the protoplanetary superconding particles in the magnetic field of Saturn lead to the origin of rings // IEEE Geoscience and remote sensing letters. 2005. V. 2. No. 4. P. 445-446. Corrections // IEEE GRSL. 2006. V. 3. No. 2. [39] Tchernyi V.V. (Cherny). About possible role of electromagnetism and superconductivity for the origin of Saturn’s rings. Prikladnaya fizika. (Applied Physics). 2006. N. 5. P. 10-16. (In Russian). [40] Tchernyi V.V. (Cherny). Possible role of superconductivity and electromagnetism for the origin of the rings of Saturn. Proc. Intern. Conf. “Fundamental principles of engineering sciences”. Devoted to 90-years birthday of Nobel prize winner A.M. Prokhorov. Moscow, Oct. 25-27, 2006. P.257-259. (In Russian). [41] Tchernyi V.V. Responsibility of electromagnetism for the origin of the rings of Saturn from superconducting particles of the protoplanetary cloud // Progress in electromagnetic research symposium (PIERS). Tokyo, Japan. 2006. [42] Tchernyi V.V., Pospelov A.Yu. About hypothesis of the superconducting origin of the Saturn’s rings // Astrophysics and space science. Springer. 2007. V. 307. No. 4. P. 347356. [43] Tchernyi V.V. To the glory of G. Galileo and J.K. Maxwell: electromagnetic modelling of the origin of Saturn’s rings from superconducting particles of the protoplanetary cloud // The 23 rd Annual review of progress in applied computational electromagnetics. Verona, Italy. March 19-23, 2007. [44] Tchernyi V.V. Modeling of electromagnetic origin of the rings of Saturn from superconducting particles of the protoplanetary cloud // Progress In Electromagnetic Research Symposium (PIERS). Hangzhou, China. 2007. [45] Tchernyi V.V., Kapranov S.V. Possible role of superconductivity for simplest life propagation within interstellar space by electromagnetic force of magnetic levitation // Journal of Electromagnetic Waves and Applications. Cambridge, MA: MIT Press. 2005. V. 19. № 15. P. 1997-2006. [46] Tchernyi V.V., Kapranov S.V. Is electromagnetic force a possible means for life transmission in the universe? // Progress In Electromagnetic Research Symposium (PIERS). Hangzhou, China. 22-26 August 2005. [47] Tchernyi V.V., Kapranov S.V. Contribution of superconductivity to possible interstellar propagation of organic molecules by electromagnetic way // Proc. 50th SPIE Annual Meeting. July 31-Aug. 4, 2005. San Diego, CA. Hoover R.B. et al. (Eds.). Astrobiology and planetary missions. SPIE, 2005. V. 5906.

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INDEX 2  20th century, 27, 28, 29, 31, 32, 34, 35, 36, 37

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A  absorption, 98, 118, 129 access, 88, 264, 265 accuracy, 9, 10, 33, 137 acid, 75, 83 acidic, 75, 79, 84, 93 acquisitions, 25 adaptation, 82, 86, 99 adjustment, 145, 148 adsorption, 205 aerogels, 89 Africa, 97 age, 71, 74, 75, 107, 124, 135, 147, 150 agriculture, 25 air temperature, vii, 19, 23, 24, 25, 27, 29, 32, 37, 40 airborne particles, 92 alpha proton x-ray spectrometer (APXS, 206 ammonia, 272 amplitude, ix, 31, 65, 132, 133, 146, 147, 148, 149, 150, 151, 222, 273 ancestors, 84, 110 apex, 98 aquifers, 77, 86 architecture, 88 Argentina, 265 assessment, 80, 133 assimilation, 83 asteroids, ix, xi, 74, 155, 156, 157, 158, 159, 160, 162, 163, 164, 165, 167, 168, 173, 174, 176, 201, 267, 268, 277 asymmetry, vii, 19, 27, 28, 29, 35, 37, 39, 40, 42, 46, 49, 65

atmosphere, viii, xi, 1, 2, 3, 4, 9, 10, 11, 13, 14, 15, 16, 17, 22, 24, 27, 30, 32, 33, 34, 37, 38, 41, 43, 45, 64, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 82, 83, 84, 85, 87, 88, 89, 90, 91, 93, 95, 96, 97, 98, 99, 100, 204, 205, 207, 268, 273 atmospheric pressure, x, 31, 71, 76, 203, 215 atoms, 71 ATP, 94 Austria, 155, 264 automation, 88 autonomy, viii, 67, 68, 88, 90

B  bacteria, 77, 82, 92 base, 35, 45, 52, 75, 83, 91, 99, 135, 142, 143, 144, 212 beams, 254 Belgium, 155 bending, 268, 273 benign, 82, 84, 89 biological activity, 82 biosphere, 84, 86, 96 biotic, 99 black hole, 277 body size, 1, 2, 10 bolides, vii, 1, 2, 8, 9, 14, 15, 17 boundary conditions, 222 Brazil, 265 bridges, 220 Bulgaria, 19 burn, 13, 15

C  calibration, 10 candidates, 86 carbon, 70, 75, 77, 78, 83

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282

Index

carbon dioxide (CO2), x, 70, 73, 75, 76, 77, 78, 79, 83, 87, 90, 95, 96, 100, 203, 215 celestial bodies, viii, 101, 102, 124 chaos, 98, 136 charge density, viii, 101, 109, 111, 112, 113, 125 chemical, 1, 69, 82, 84, 206, 207, 210 China, 96, 265, 279, 280 chlorine, 88, 95 chlorophyll, 94 chronology, 94, 100, 134 circulation, 22, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 61 civilization, 2 classification, 94, 165, 210 climate, vii, 19, 20, 23, 24, 25, 27, 28, 29, 31, 68, 85, 91, 94, 95, 204 climate change, 25 climatic factors, 25 clustering, 162, 163 clusters, 166, 167 coherence, 39, 41 collisions, 160, 269, 274 color, 206, 268, 272 combined effect, 49 combustion, 1 communication, 91, 94, 201 communities, 86, 94 community, 91, 94, 265 comparative analysis, 17 compensation, 137, 138, 142 compilation, 24, 264 complement, 88 composition, x, 55, 69, 70, 78, 80, 85, 89, 203, 204, 206, 207, 209 compounds, 77, 80, 83, 84 compression, 73 computation, 33 computing, x, 235, 239, 252 condensation, 85 conduction, x, 203, 205, 211, 221 conductivity, x, 45, 110, 111, 143, 203, 204, 205, 211, 212, 214, 215, 216, 217, 218, 222, 224, 226, 227, 232, 272, 273 conference, 14, 233, 264, 279 configuration, 6, 7, 55, 61, 213, 216, 219 configurations, 5, 7, 61, 112, 113 consensus, vii, 19 conservation, 33, 36, 37 constituents, 204, 270 continuous data, 27 contour, 139, 140, 141 controversial, vii, 19, 20, 69, 76, 147 convention, 112, 126

cooling, 22, 74, 142, 145, 209 Coriolis effect, 30 correlation, vii, 19, 23, 24, 26, 27, 28, 32, 35, 36, 37, 39, 40, 41, 50, 90, 270 correlation coefficient, 50 correlation function, 26, 50 correlations, vii, 19, 23, 24, 27, 29, 35, 96 cosmos, 264 cost, 68 covering, 21, 73 crust, 74, 77, 78, 79, 81, 84, 93, 96, 100, 142, 143, 144, 147, 149, 151, 204, 209 crustal growth, 132 cycles, vii, 19, 22, 27, 28, 29, 36, 37, 40, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 80, 86 cycling, 74 cyclones, 30 Czech Republic, 3, 8

D  data analysis, 65, 68, 129 data availability, 24 data processing, 68 data set, 25, 26, 27 database, 25, 55, 268 decay, 85, 114, 125 defects, 4 deformation, 34, 72, 73, 98, 105, 108, 133, 146, 147, 268, 269, 273 degradation, 147 deposition, 86, 204, 209 deposits, 74, 78, 80, 86, 87, 91, 92, 207, 208 depth, x, 86, 132, 137, 138, 143, 144, 149, 203, 204, 205, 222, 223, 224, 225, 226, 227, 228, 230, 231, 269 derivatives, 104, 109, 240, 248, 249, 250, 251, 252, 253, 254 desiccation, 76, 137 destruction, 3, 4, 5, 6, 11, 13, 16, 17, 160, 271 detectable, 223, 225, 226, 228, 230, 231, 242 detection, 3, 69, 118, 168, 260 deviation, 23, 61, 147, 239 dichotomy, 69, 135, 141 diffusion, 272 diffusivity, x, 204, 205, 216, 219, 220, 221, 222, 224, 225, 226, 227, 228, 230 direct measure, 25 disaster, 161 discharges, 271 discs, 277 disequilibrium, 82, 84 displacement, 260 distortion, 8, 150, 269

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Index distribution, 9, 25, 61, 75, 90, 91, 94, 95, 96, 99, 157, 205, 213, 214, 224, 225, 227, 228, 272 disturbances, 21, 22, 23, 50, 55 divergence, 6, 10, 13 DNA, 83 drainage, 86, 92

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E  earthquakes, 20 election, 144 electric charge, 103, 237 electric field, 41, 110, 112 electrical conductivity, 45, 110 electromagnetic, viii, xi, 19, 41, 103, 109, 110, 112, 114, 123, 125, 126, 127, 129, 267, 268, 269, 270, 271, 273, 274, 275, 276, 279, 280 electromagnetic field, 41, 110 electromagnetic origin, 103, 123, 125, 126, 276, 280 electromagnetic waves, 268 electromagnetism, 127, 268, 279, 280 electron, 55, 103, 115, 117, 118, 119, 272 electron cyclotron resonance, 118 electrons, 115, 271, 272 emission, 23, 85, 104, 108, 117, 118 EMS, 264 energy, vii, xi, 1, 2, 4, 9, 13, 20, 29, 68, 76, 80, 82, 84, 87, 104, 105, 115, 160, 205, 267, 272, 274, 275, 276 energy density, 105 engineering, 68, 280 environment, 63, 68, 69, 70, 77, 84, 86, 87, 95, 97, 99, 136, 238, 276 environmental conditions, 68, 84 environmental crisis, 84 equilibrium, 112, 207 equipment, x, 2, 235, 239 equipotential surface, 78, 134, 136, 141 erosion, 73, 76, 79, 86, 96, 97, 133, 135, 147, 209 EST, 233 eukaryotic cell, 84 Europe, 17, 25, 32, 70, 86 European Southern Observatory (ESO XE "ESO" ), 157, 164 evaporation, vii, 1, 2, 3, 11, 13, 14, 15, 16, 81, 204 evidence, ix, 20, 21, 65, 69, 74, 79, 85, 90, 91, 92, 93, 96, 97, 98, 99, 100, 118, 126, 127, 129, 132, 135, 149 evolution, vii, viii, ix, 28, 45, 67, 71, 73, 74, 76, 77, 80, 84, 85, 90, 91, 93, 94, 95, 96, 97, 99, 110, 128, 132, 133, 141, 142, 145, 148, 151, 156, 161, 164, 268 exploration, 68, 69, 84, 86, 87, 88, 92, 93 extinction, 161

283

F  faults, 72, 73, 74, 132 fauna, 161 feedback, 61, 85 FFT, 39 field theory, 127 filters, 70 flare, 22 flight, 10, 14, 15, 16 flooding, 80 floods, 209 flora, 161 flow field, 72 fluctuations, 41, 269, 273 fluid, 2, 70, 142, 205 fluvial deposits, 86 flybys, 69, 70 force, 5, 6, 7, 15, 30, 33, 41, 61, 242, 269, 270, 272, 273, 274, 275, 276, 277, 280 formation, ix, 1, 2, 34, 72, 74, 75, 76, 77, 79, 80, 81, 90, 94, 98, 132, 133, 140, 141, 146, 208, 268, 270, 271, 276, 277 formula, viii, 2, 9, 10, 11, 12, 49, 50, 52, 53, 101, 102, 104, 105, 112, 115, 119, 122, 123, 124, 125, 126, 127, 158 fossils, 91 foundations, 238 fractures, 75 fragments, 2, 3, 4, 5, 6, 11, 12, 13, 14, 15, 160 France, 161 freedom, 236 freezing, 77, 79 frequencies, 40, 271 friction, 33, 271 frost, 78, 273

G  Galaxy, 103, 125 Galileo, 20, 22, 62, 268, 280 gasification, 3 General Relativity, x, 235, 238, 239, 254, 264, 265 genus, 98 geological history, viii, 68, 89 geology, vii, 68 geometry, 132, 237, 277 Germany, 98, 279 global scale, 74, 76, 160 Gori, 203, 231, 232, 233 gravitation, 273 gravitational constant, 102 gravitational field, 102, 115, 127, 166

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284

Index

gravitational force, 268, 272 gravitomagnetic origin, 103, 109, 114, 123, 126 gravity, 22, 61, 69, 70, 78, 88, 144, 238, 267, 268, 269, 270, 274, 275, 276 Greeks, 20 greenhouse, 75, 76, 77, 82, 95, 98 greenhouse gases, 98 groundwater, 96 growth, 7, 8, 98, 132

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H  habitat, viii, 67, 82, 89 habitats, viii, 68, 80, 86, 90 Hawaii, 164, 277 haze, 77, 97 heat capacity, x, 204, 219, 222 heat conductivity, 272 heat release, 72 heat transfer, 145 height, 2, 3, 6, 11, 12, 13, 14, 137, 138, 142, 145, 158 helicity, viii, 20, 58, 59, 60 helium, 118 hematite, 91 hemisphere, viii, 20, 22, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 42, 44, 45, 47, 49, 53, 54, 58, 59, 70 heterogeneity, 4 highlands, 69, 76, 78, 87, 135 history, viii, 68, 69, 71, 72, 73, 74, 76, 77, 79, 80, 82, 84, 86, 89, 90, 91, 95, 99, 100, 132, 138, 139, 141, 147, 149, 276 homeostasis, 96 homogeneity, 25, 151 hot spots, 72 hot springs, 84 House, 97 human, 20, 68, 85 human health, 20 Hunter, 153 hunting, 155 hydrocarbons, 98 hydrogen, 42, 85, 91, 95, 205 hydrothermal activity, 87, 99 hydrothermal system, 87 hypersonic aerodynamics, 2 hypothesis, 9, 14, 24, 38, 53, 90, 91, 96, 104, 107, 123, 126, 133, 139, 205, 280 hysteresis, 65

I  ice, x, 2, 73, 74, 77, 78, 79, 80, 81, 85, 87, 88, 90, 91, 93, 94, 97, 99, 165, 203, 204, 205, 206, 209, 210, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 224, 227, 268, 269, 270, 271, 276, 278 Iceland, 32 identification, 97, 125, 164 identity, 109 imagery, 70, 209 images, 69, 70, 81, 83, 88, 94, 133, 138, 276 IMF, 28, 29, 45, 46, 47, 49, 50, 51, 52, 53, 54, 57 impacts, 15, 16, 73, 74, 81, 85, 160, 205 impurities, 268 incidental radio XE "radio" discharges, 271 India, 63, 65, 77, 95, 265 indirect measure, 34 individuals, 25, 156 Indonesia, 233 induction, 102, 103, 105, 106, 107, 109, 110, 124, 127 inertia, 41, 69, 102, 104, 107, 126 inferences, 16 initiation, 80 INS, 119 integration, 106, 107, 112, 123 interference, xi, 235, 238, 256 International Space Station, v, x, 88, 89, 235, 238, 247, 254, 256, 260, 264, 265 intrusions, 147 inversion, 279 iron, xi, 70, 71, 78, 79, 206, 267 islands, 209 isotherms, 205, 211 isotope, 97 Israel, 129, 265 Italy, 59, 203, 232, 279, 280

J  Japan, 64, 70, 94, 96, 265, 280 Java, 72, 77, 98

L  lakes, 73, 77, 78 landings, 69 landscape, 87, 90 late heavy bombardment, 74 laws, 264, 270 lead, 5, 7, 12, 23, 34, 38, 41, 45, 49, 53, 150, 160, 276, 280 lifetime, 74, 270 light, 9, 22, 24, 83, 97, 102, 105

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Index lithology, 204 localization, 150 lower prices, 27 luminescence, 9 luminosity, 1, 9, 10, 13

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M  macromolecules, 83 magnesium, 79, 206 magnet, xi, 267 magnetic declination, 22 magnetic field, vii, viii, ix, xi, 19, 20, 22, 28, 33, 40, 41, 44, 45, 46, 48, 49, 50, 52, 53, 55, 56, 58, 59, 60, 61, 70, 73, 101, 102, 103, 104, 105, 108, 109, 110, 111, 114, 115, 116, 117, 118, 119, 120, 123, 124, 125, 126, 127, 128, 129, 267, 268, 269, 270, 271, 273, 274, 275, 277, 278, 280 magnetic moment, 102, 103, 113, 125, 126, 127, 269, 273, 274, 275, 276 magnetic particles, 269 magnetism, 73, 127 magnetosphere, viii, 20, 29, 38, 41, 45, 47, 71, 77, 80, 270, 271 magnitude, 38, 41, 45, 46, 52, 53, 54, 55, 57, 72, 73, 104, 112, 114, 117, 123, 125, 126, 141, 145, 147, 149, 157, 158, 159, 160, 168, 220, 224, 225, 228, 229, 231 main belt, ix, 155, 157, 160, 166, 168, 201 majority, 8, 27, 139, 273 mantle, viii, 19, 33, 41, 45, 72, 77, 78, 79, 94, 96, 97, 99, 137, 138, 142, 143, 144, 145, 147 mapping, 69, 75, 78, 91, 92, 94, 95, 99, 134, 135, 138 Mars, vii, viii, ix, x, 67, 68, 69, 70, 71, 72, 73, 74, 77, 78, 79, 80, 82, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 131, 132, 133, 134, 135, 138, 139, 142, 144, 147, 148, 149, 150, 153, 154, 162, 166, 167, 203, 204, 205, 206, 207, 209, 231, 232, 233 Mars Odyssey, 70, 88, 91, 205 Martian topography, 133, 140 mass, vii, viii, x, 1, 2, 8, 9, 10, 11, 12, 13, 14, 15, 20, 24, 39, 40, 44, 55, 56, 58, 71, 78, 79, 89, 102, 103, 107, 109, 110, 114, 115, 117, 119, 126, 164, 204, 205, 215, 217, 218, 220, 221, 237, 242, 243, 244, 245, 246, 249, 251, 255, 263 mass loss, vii, 1, 2, 9, 12, 13, 15, 164 material bodies, 2 materials, x, 73, 74, 75, 79, 81, 85, 88, 90, 135, 136, 142, 144, 160, 203, 204, 205, 206, 207, 211, 212, 214, 232 matter, 20, 76, 77, 85, 109, 236, 237, 242, 243, 245, 268, 269, 270

285

measurements, 25, 26, 27, 33, 38, 53, 59, 60, 69, 93, 103, 204, 205, 214, 238, 255, 279 mechanical properties, 132 melting, 100, 160 mercury, 24, 92, 100 Mercury, viii, 21, 67, 68, 70, 71, 72, 73, 74, 78, 80, 81, 82, 87, 90, 94, 95, 97, 99, 100 metabolic pathways, 83 metabolism, 69, 87, 100 meteor, 1, 2, 14 meteorites, 9, 16, 82, 209, 210 meter, 160, 204 Mexico, 127, 157, 265 microbial communities, 86, 94 microbiota, 100 microorganisms, 69, 83, 92, 94 microwaves, 268 migration, 72, 86 mineralogy, x, 203, 206 mission, viii, 68, 69, 70, 87, 88, 89, 90, 96, 97, 98, 99, 205, 206, 233, 269, 276, 278 missions, viii, 68, 69, 70, 78, 86, 87, 88, 90, 93, 268, 280 mixing, 31, 133, 151 modeling, xi, 4, 14, 16, 69, 205, 267, 276, 279 modelling, 280 models, vii, 4, 17, 19, 69, 94, 104, 132, 205, 209, 247 modification, 147 modifications, xi, 263 molecules, 273, 277, 280 momentum, vii, 19, 29, 33, 36, 38, 40, 41, 45, 102, 103, 104, 114, 125, 126, 268, 276 Mongolia, 95 monitoring, 25 Montana, 232 Moon, viii, 33, 67, 68, 69, 72, 73, 74, 77, 78, 80, 85, 87, 90, 91, 94, 95, 96, 97, 99, 100, 201, 205, 238, 268 morphology, 95, 138 Moscow, 1, 14, 16, 17, 64, 65, 92, 93, 267, 277, 278, 280 multimedia, 276

N  Netherlands, 101, 127 neutral, 41, 94, 103, 109, 126 neutron stars, 118, 125, 128 neutrons, 93, 109, 254, 256 nitrogen, 77, 78, 83, 94 noise, 129 North America, 25 nuclei, 17

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286

Index

null, 236, 241, 242, 243, 244, 245, 246, 250, 254, 255, 256 nutrient, 82, 86

O  obstacles, 8 oceans, ix, 31, 73, 75, 76, 77, 78, 79, 85, 90, 92, 93, 95, 100, 131, 132, 137, 138, 141 oil, 205, 232 one dimension, 205 opportunities, 86 orbit, viii, 20, 21, 40, 45, 46, 47, 49, 50, 53, 59, 60, 70, 71, 77, 78, 88, 89, 128, 160, 162, 164, 166, 167, 168, 238, 269, 270, 274 organic compounds, 84 organic matter, 77 organism, 84 oscillation, 65 oscillations, 33, 38, 40 oxidation, 95, 97 oxygen, 77, 78, 84, 90, 91

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P  Pacific, 31, 63, 65, 77 paleoequipotential surface, ix, 131, 133, 139, 145, 147 paleoshorelines, vii, ix, 98, 131, 132, 133, 137, 138, 141, 147, 148, 151 paradigm shift, 93 parallel, 30, 104, 111, 113, 125, 205, 211 parity, 40 participants, 277 particles, xi, 1, 75, 82, 89, 109, 164, 205, 213, 237, 243, 246, 256, 262, 263, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 280 pathways, 83 periodicity, vii, 19, 23, 39, 40, 41, 52, 53, 270, 271 perseverance, 80 personal communication, 201 pH, 79, 82, 83, 98 phosphorus, 83 photons, 115, 239, 254, 255, 256, 258 photosynthesis, 77, 87 phototrophic life, 82, 99 physical properties, x, 2, 70, 203 physical theories, 126 physics, vii, 1, 8, 13, 14, 19, 22, 265, 276 phytoplankton, 91 pitch, 272 planetary exploration, 87 planetary missions, 69 planetcentrical dust flows, 269

planets, vii, viii, 1, 2, 14, 16, 22, 40, 65, 68, 69, 78, 80, 87, 89, 92, 95, 98, 103, 125, 157, 160, 165, 168, 169, 201, 205, 268, 273, 276, 277 Plank constant, 272 plausibility, 95 polar, 30, 32, 34, 44, 45, 49, 53, 69, 73, 74, 78, 80, 87, 92, 94, 95, 99, 117, 118, 137, 138 polarity, 22, 28, 42, 43, 44, 45, 46, 48, 49, 51, 52, 53, 65 polarization, 74, 273, 279 porosity, x, 203, 204, 205, 207, 210, 211, 213, 215, 216, 217, 218, 219, 220, 221, 222, 225, 231 porous materials, 232 porous media, 205 Portugal, 235, 264 positive correlation, 23, 27, 35, 37 positive feedback, 76 positron, 103 potassium, 71 precipitation, 96 preservation, ix, 155 probability, 56, 88 probe, 236, 238, 277 productivity, 94 project, 265 prokaryotes, 84 propagation, 50, 277, 280 proportionality, 9 protection, 83 proteins, 83 prototype, 156 pulsars, vii, viii, ix, 101, 102, 103, 104, 105, 107, 109, 112, 113, 114, 115, 116, 117, 118, 119, 120, 122, 123, 124, 125, 126, 128, 129

Q  quantum phenomena, 271 quartz, 207

R  radar, 70, 74, 75, 81, 82, 94, 99, 270 radiation, viii, 15, 71, 83, 85, 98, 101, 104, 105, 108, 115, 117, 118, 123, 125, 205, 232, 268, 270, 271, 272, 277 radio, 118, 123, 129, 268, 271, 272, 279 radius, 5, 6, 10, 13, 14, 40, 49, 89, 102, 105, 107, 109, 111, 114, 126, 134, 149, 269, 270, 274 reactant, 82 reactions, 82, 83, 87, 205 reality, vii, ix, 2, 19, 20, 24, 132, 133, 248, 276 recall, 134 reconstruction, 21, 32, 209

Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

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Index recovery, 156 recycling, 73, 76 redistribution, 43, 147 redundancy, 87, 88 reference system, 47 reflectivity, 74, 268 regression, 103, 125 regression analysis, 103, 125 relativity, viii, 101, 102, 124, 126, 127 relevance, 89, 98 reliability, vii, 1, 25 remote sensing, x, 88, 203, 206, 279, 280 replacement, 6, 269 reputation, 265 researchers, 8, 25, 268 resistance, 7, 211, 212, 214, 271 resolution, 69, 70, 94, 133, 138, 140 restoration, 273 retardation, 38 ring material, 268 ring particle, 268, 270 rings, vii, xi, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280 risks, ix, 155 river systems, 77 rocks, x, 71, 76, 77, 78, 96, 97, 160, 203, 206, 207, 209, 210 rotation axis, 275 rotations, 55, 60 routes, 160 runoff, 96 Russia, 1, 26, 265, 267, 268, 277, 278

S  safety, 68, 87, 88 salts, 79 satellites, 71, 74, 169, 268, 275, 279 saturation, 204 scaling, 22 scattering, 5, 6, 76, 115, 117, 118, 165 science, vii, 68, 69, 85, 87, 88, 270, 280 scientific knowledge, 236 scope, 7, 134 scripts, 15 sea level, 2, 32, 138, 147 seasonal changes, 77 sediment, 207 sedimentation, 79, 96 sediments, x, 77, 87, 97, 203, 206, 207, 210 seminars, 277 sensing, x, 88, 203, 205, 206, 279, 280 sensors, 88 shape, 115, 149, 167, 205, 247, 250, 253

287

shock, 5, 6, 7, 8, 15, 75, 160 shock waves, 7, 8, 15, 75, 160 shoreline, ix, 131, 132, 133, 135, 136, 137, 138, 139, 140, 141, 146, 147, 148, 149, 150, 151 shorelines, ix, 131, 132, 135, 137, 139, 140, 141 Siberia, 95 signals, 74, 239, 273 significance level, 42, 50 signs, 23, 112 silica, 206 silicon, 209 simulations, 79, 100, 112, 128 Sinai, 209 SiO2, 207, 209, 210 smoothing, 48 sodium, 71, 74 software, 65, 88 soil particles, 217 solar activity, vii, 19, 20, 23, 24, 27, 28, 29, 32, 35, 36, 37, 38, 40, 41, 49, 61, 64, 65 solar system, ix, xi, 20, 40, 61, 75, 81, 155, 160, 165, 267, 268, 276 Solar System, viii, 67, 69, 71, 76, 81, 84, 87, 89, 201 solution, x, 2, 3, 4, 8, 11, 14, 110, 203, 211, 222, 269, 274 South Africa, 97 space environment, 276 spacetime, 249, 262 Spain, 67, 131 species, 68 specific heat, x, 203, 204 spectrophotometry, 94 spectroscopy, 128 speculation, 78 spin, ix, 40, 71, 78, 101, 104, 105, 115, 116, 117, 128, 129, 238, 243, 263 stability, ix, 80, 86, 97, 132, 151, 268 stabilization, 100, 147, 150 standard deviation, 52, 147 standard error, 43 Star Wars, 264 stars, 20, 61, 103, 118, 125, 128, 162, 277 state, ix, 80, 96, 128, 129, 132, 133, 142, 145, 146, 150, 151, 231, 269 statistics, 35, 57 steroids, 155, 201 stimulus, 2 storms, 55, 56, 57, 59 stratification, 209, 210 streams, 55 structure, viii, 1, 20, 22, 28, 55, 59, 69, 70, 75, 91, 132, 142, 148, 205, 209, 268, 275, 276, 278 style, 86, 168, 169

Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.

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Index

substitution, 123 substrate, 86, 160 sulfate, 79, 94 sulfur, 83, 91, 93, 94, 97, 99, 210 sulfuric acid, 75, 83 sulphur, 91, 272 sunspot cycle, 21, 22, 44, 45, 47, 51, 56, 57 sunspots, 20, 21, 22, 23, 28, 38, 42, 48 superconductivity, xi, 267, 268, 269, 276, 277, 278, 279, 280 superconductor, 269, 271, 273, 274, 276, 277 superdiamagnetic substance, 270 supervision, 10, 11, 12, 273 surface area, 88 surface modification, 80 surface structure, 69 survey, 81, 84, 119, 120, 125, 129, 157, 201 survival, 86 symmetry, 21, 61 synthesis, 81, 93 Syria, 72

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T  target, x, 160, 206, 233, 235, 256, 257 teams, 156 technology, 70, 89 temperature, vii, x, xi, 19, 23, 24, 25, 26, 27, 29, 32, 33, 34, 35, 36, 37, 40, 44, 55, 56, 61, 75, 77, 80, 84, 89, 100, 142, 143, 144, 203, 204, 205, 215, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 267, 268, 270, 271, 272, 276 tension, 73 terrestrial planets, viii, 68, 80, 89, 160 territory, 13 testing, 84 texture, 209 thermal expansion, 142 thermal history, 132 thermal properties, vii, x, 204, 205 thermal resistance, 211, 212, 214 thermal stability, 97, 151 thermodynamic calculations, 79 thermometer, 24, 25, 100 tides, 33 time periods, 76, 86 time series, 25, 27 time use, 10 topology, 112 torus, viii, 101, 112, 113, 114, 115, 118, 122, 123, 125, 126 trade, 30, 34 trajectory, 1, 2, 3, 4, 9, 10, 11, 12, 13, 14, 274 transformation, 77

transformations, 40 transmission, 160, 280 transport, 34, 168, 201, 268

U  U.S. Geological Survey, 233 Ukraine, 277 unification, 126, 262 United States (USA), 8, 9, 65, 67, 70, 96, 169, 264 universe, 280 USSR, 16, 70 UV, 82, 83, 98, 99 UV radiation, 83

V  vacuum, 102, 104 valleys, 78, 84, 86, 87, 88, 94, 96 vapor, 14, 15, 78, 83, 205 variables, 11, 13, 35 variations, vii, ix, x, 19, 20, 21, 22, 25, 28, 29, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 48, 49, 50, 69, 80, 93, 132, 133, 137, 145, 146, 147, 148, 149, 150, 151, 160, 204, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231 vector, 5, 240, 242, 273, 274 velocity, 5, 6, 13, 14, 22, 28, 32, 33, 34, 38, 41, 43, 44, 49, 50, 52, 55, 59, 74, 102, 105, 109, 110, 111, 114, 254 Venus, viii, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 80, 82, 83, 84, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 261 Viking, 69, 78, 96, 97, 138, 204, 206, 207, 209, 233 viscosity, 76

W  Washington, 62, 63, 67, 91, 277, 278 water, viii, ix, x, 34, 67, 68, 70, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 91, 93, 95, 96, 97, 100, 131, 133, 134, 135, 136, 137, 138, 145, 147, 160, 203, 205, 207, 209, 212, 215, 273 water vapor, 78, 83 wavelengths, 70, 83, 272, 273

X  X-ray, viii, 90, 101, 112, 115, 116, 117, 118, 119, 125, 128, 129

Y  yield, 26, 27, 88, 103, 115, 117, 126

Space Science Research Developments, edited by Jonathan C. Henderson, and Jennifer M. Bradley, Nova Science Publishers, Incorporated, 2011.