The Chaotic Solar Cycle [1st ed.] 9789811598203, 9789811598210

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Atmosphere, Earth, Ocean & Space

Arnold Hanslmeier

The Chaotic Solar Cycle

Atmosphere, Earth, Ocean & Space Editor-in-Chief Wing-Huen Ip, Institute of Astronomy, National Central University, Zhongli, Taoyuan, Taiwan Series Editors Masataka Ando, Center for Integrated Research and Education of Natural Hazards, Shizuoka University, Shizuoka, Japan Chen-Tung Arthur Chen, Department of Oceanography, National Sun Yat-Sen University, Kaohsiung, Taiwan Kaichang Di, Institute of Remote Sensing and Digital Earth, Chinese Academy of Sciences, Beijing, China Jianping Gan, Hong Kong University of Science and Technology, Hong Kong, China Philip L.-F. Liu, Department of Civil and Environmental Engineering, National University of Singapore, Singapore Ching-Hua Lo, Department of Geosciences, National Taiwan University, Taipei, Taiwan James A. Slavin, Department of Atmospheric, Oceanic and Space Sciences, University of Michigan–Ann Arbor, USA Keke Zhang, Space Science Institute, Macau University of Science and Technology, Macau, China R. D. Deshpande, Geosciences Division, Physical Research Laboratory, Gujarat, India A. J. Timothy Jull, Geosciences and Physics, University of Arizona AMS Laboratory, Tucson, AZ, USA

The series Atmosphere, Earth, Ocean & Space (AEONS) publishes state-of-art studies spanning all areas of Earth and Space Sciences. It aims to provide the academic communities with new theories, observations, analytical and experimental methods, and technical advances in related fields. The series includes monographs, edited volumes, lecture notes and professional books with high quality. The key topics in AEONS include but are not limited to: Aeronomy and ionospheric physics, Atmospheric sciences, Biogeosciences, Cryosphere sciences, Geochemistry, Geodesy, Geomagnetism, Environmental informatics, Hydrological sciences, Magnetospheric physics, Mineral physics, Natural hazards, Nonlinear geophysics, Ocean sciences, Seismology, Solar-terrestrial sciences, Tectonics and Volcanology.

More information about this series at http://www.springer.com/series/16015

Arnold Hanslmeier

The Chaotic Solar Cycle

123

Arnold Hanslmeier Institute of Physics University of Graz Graz, Austria

ISSN 2524-440X ISSN 2524-4418 (electronic) Atmosphere, Earth, Ocean & Space ISBN 978-981-15-9820-3 ISBN 978-981-15-9821-0 (eBook) https://doi.org/10.1007/978-981-15-9821-0 © Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

I dedicate this monograph to my partner Anita, my children Roland, Christina and Alina and to all my colleagues and friends with whom I can share my enthusiasm and passion for science, especially for astrophysics.

Foreword

The natural laws rule our universe. They lead to the formation of stars and define the processes that can take place on them. One of those stars is the Sun. Due to its proximity to us, it allows us to study stellar processes in detail and investigate the natural laws that influence our lives. Probably, the most remarkable feature of the Sun is its variable magnetic activity. This mirrors in the emergence of sunspots, which are strong magnetic field concentrations. These sunspots appear almost regularly every 11 years with increased frequency. In addition, the changing magnetic activity is accompanied by sometimes vigorous variations of the solar radiation output and particle emission. This space weather can have massive consequences on the solar system including Earth. Having a closer look, one notices, however, that the period of the sunspot cycle is not constant. It can be somewhat longer or shorter, and the numbers of sunspots and the magnetic field strength in the spots vary from cycle to cycle. Historic records of the sunspot cycle indicate that there are further trends and long-term variations present in the Sun’s magnetic field. There are even intermittent periods when the Sun did not show any sunspots for decades. Trying to obtain a theoretical description of the solar dynamo leads to a nonlinear system of differential equations that describe the changing magnetic field. Prof. Dr. Arnold Hanslmeier has studied this solar dynamo extensively during his scientific career and found that solar activity shows signs of chaotic behavior. In this book, he provides a hands-on introduction to understand chaos and how to describe and characterize it. The reader profits from the author’s long-standing experience in studying and finding the natural laws that rule the Sun, and trying to make solar activity predictable. Chapter by chapter, one obtains a comprehensive introduction to historic and modern research and learns a lot about today’s state of knowledge in Solar Physics. Freiburg, Germany September 2020

Markus Roth

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Preface

The Sun is the only star where surface details can be observed directly. It is also the host star of our planetary system and there is a small region around each star where there is a habitable zone that means water can exist in liquid state on a planet there. Of course, the Earth is in the habitable zone around the Sun and Venus and Mars are at the borders or even outside of it. So the Sun, the solar radiation is essential for life on Earth. But is the solar radiation input on Earth constant? The Sun has a total lifetime of about 9 billion years and we will show how the solar radiation behaves over such long timescales. Since the discovery of sunspots already more than 2000 years ago, it became clear that the Sun might be variable on much shorter timescales. In 1610, the first telescopes were used to observe the Sun and 150 years ago it was detected that the Sun has an 11-year activity cycle. The variation of the solar total irradiation over a solar activity cycle is low but there seem to exist longer lasting cycles and intermittent periods where there has been no solar activity at all. Since the solar radiation input is the driver of weather and climate on Earth, it is very important to know whether such variations of the solar output are periodic, what could be the extreme values and the amplitudes of the cycles. Finally, we would like to have a prediction of solar activity because we know that, during extreme solar outbursts, satellites could be damaged by surface charging effects, the enhanced solar UV input changes the composition of the upper Earth’s atmosphere etc. So there are many good reasons to study the behavior of solar activity. The theory to model solar activity is based on dynamos. Dynamo action is described by a set of nonlinear complex equations and methods of nonlinear dynamics are therefore a very helpful tool to understand solar activity behavior. The book starts with some very simple and basic concepts of nonlinear dynamics. How can a simple dynamical system become chaotic? What does chaos mean? Can we quantify chaos? Then we give an overview on chaotic motions in the solar system and describe solar and stellar activity. The basic equations of the solar dynamo are discussed briefly and then we give some application of chaos theory to a simplified solar dynamo model. All examples are written in the PYTHON language and the ix

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program listings are also given so that the reader can make own experiments with the parameters. Solar activity is also compared with stellar activity and the last chapter is about the important problem of solar activity forecasting where several methods are mentioned. The conclusion is that our understanding is still far from being complete, some trends, however, can be forecasted. The situation is comparable to forecasting weather and climate on Earth. It is extremely hard to make precise weather forecasting for longer than 3 weeks, climate variation on timescales of decades are easier to be done. For a deeper penetration into the subject of chaotic solar cycle, the reader can study the long list of cited literature and is encouraged to work with the given simple computer algorithms. Graz, Austria September 2020

Arnold Hanslmeier

Acknowledgements

I would like to thank all my teachers, colleagues, and students with whom I had the good fortune to do research, have discussions and inspiration on the topic of this book. In particular, I am grateful to those who helped to improve the book: proofreading was made by Elizabeth Elliot. The text was read and commented by Dr. Roman Brajsa, Dr. Peter Leitner, and Dr. Markus Roth. I would also like to thank my partner Anita and my children for their love and support.

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Contents

1 What Is Chaos? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Chaos and Causality . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Chaos: Examples from Religion . . . . . . . . . 1.1.2 Chaos in Philosophy . . . . . . . . . . . . . . . . . 1.1.3 Chaos in Modern Science . . . . . . . . . . . . . . 1.1.4 The End of Predictability . . . . . . . . . . . . . . 1.1.5 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 From Simple Equations to Chaos . . . . . . . . . . . . . 1.2.1 The Logistic Equation . . . . . . . . . . . . . . . . 1.2.2 The Tent Map . . . . . . . . . . . . . . . . . . . . . . 1.2.3 A Numerical Example . . . . . . . . . . . . . . . . 1.2.4 Sensitive Dependence on Initial Conditions . 1.2.5 Astrophysical Application: Star Formation . 1.2.6 Autocorrelation Function . . . . . . . . . . . . . . 1.2.7 Powerspectrum: The Fourier Transform . . . 1.3 Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Mathematical Description . . . . . . . . . . . . . . 1.3.2 The Koch Curve . . . . . . . . . . . . . . . . . . . . 1.3.3 The Fractal Dimension . . . . . . . . . . . . . . . . 1.4 The Lorentz System . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Formulation of a Lorentz System . . . . . . . . 1.4.2 Some Experiments with the Lorentz System 1.5 Measuring Chaos . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 The Phase Space . . . . . . . . . . . . . . . . . . . . 1.5.2 Lyapunov Exponent . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Chaos in the Solar System . . . . . . . . . . . . . . . . . . . 2.1 The Solar System . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Solar System Formation . . . . . . . . . . . . . 2.2 Orbital Motions . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Equations of the N-Body Problem . . 2.2.2 Chaos and the N-Body Problem . . . . . . . 2.2.3 Motion and General Relativity . . . . . . . . 2.2.4 Non Gravitational Forces and Chaos . . . . 2.2.5 Chaotic Resonances in the Solar System . 2.2.6 Chaotic Spin-Orbit Resonances . . . . . . . . 2.2.7 Three-Body Resonances . . . . . . . . . . . . . 2.2.8 Chaos Among the Giant Planets . . . . . . . 2.2.9 Chaos Among the Terrestrial Planets . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 The Sun: An Active Star . . . . . . . . . . . . . 3.1 The Structure of the Sun . . . . . . . . . . 3.1.1 The Solar Interior . . . . . . . . . 3.1.2 The Solar Atmosphere . . . . . . 3.2 Solar Activity—An Overview . . . . . . 3.2.1 Sunspots . . . . . . . . . . . . . . . . 3.2.2 Faculae . . . . . . . . . . . . . . . . . 3.2.3 Flares and CMEs . . . . . . . . . . 3.2.4 Solar Wind . . . . . . . . . . . . . . 3.2.5 The Heliosphere . . . . . . . . . . 3.3 The Solar Activity Cycle . . . . . . . . . 3.3.1 The Sunspot Cycle . . . . . . . . 3.3.2 Other Effects of Solar Activity References . . . . . . . . . . . . . . . . . . . . . . . .

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4 MHD and the Solar Dynamo . . . . . . . . . . . . . . 4.1 Solar Magnetohydrodynamics . . . . . . . . . . . 4.1.1 Basic Equations . . . . . . . . . . . . . . . . 4.1.2 Some Important MHD Effects . . . . . 4.1.3 Fluid Equations . . . . . . . . . . . . . . . . 4.1.4 Equation of State . . . . . . . . . . . . . . . 4.1.5 Structured Magnetic Fields . . . . . . . . 4.1.6 Potential Fields . . . . . . . . . . . . . . . . 4.1.7 Charged Particles in Magnetic Fields 4.1.8 MHD Waves . . . . . . . . . . . . . . . . . . 4.1.9 Magnetic Fields and Convection . . . .

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4.2 The Solar Dynamo . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 The Solar Dynamo and Observational Features 4.2.2 The a x Dynamo . . . . . . . . . . . . . . . . . . . . 4.2.3 Differential Solar Rotation . . . . . . . . . . . . . . . 4.2.4 Mathematical Description . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Long Term Solar Activity . . . . . . . . . . . . . . . . . . . . . . . 5.1 Sunspot Recordings . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 The Wolf Number . . . . . . . . . . . . . . . . . . . . 5.1.2 Historical Sunspot Numbers . . . . . . . . . . . . . 5.2 Proxies of Solar Activity . . . . . . . . . . . . . . . . . . . . . 5.2.1 Aurorae . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Cosmogenic Isotopes . . . . . . . . . . . . . . . . . . 5.2.3 Other Proxies . . . . . . . . . . . . . . . . . . . . . . . 5.3 Long Term Solar Activity . . . . . . . . . . . . . . . . . . . . 5.3.1 Solar Activity Since Telescopic Observations 5.3.2 The Maunder Minimum . . . . . . . . . . . . . . . . 5.3.3 Solar Activity Derived from Proxies . . . . . . . 5.3.4 The Total Solar Irradiance, TSI . . . . . . . . . . 5.4 Other Minima and Maxima of Solar Activity . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Stellar Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Hertzsprung–Russell Diagram . . . . . . . . . . 6.1.1 The Basic Parameters . . . . . . . . . . . . . . 6.1.2 The Main Sequence . . . . . . . . . . . . . . . 6.1.3 Giants, Supergiants . . . . . . . . . . . . . . . 6.1.4 Evolution of Stars in the H–R Diagram 6.2 How to Measure Stellar Activity . . . . . . . . . . . 6.2.1 H-K Activity . . . . . . . . . . . . . . . . . . . . 6.2.2 Star Spots . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Stellar Winds . . . . . . . . . . . . . . . . . . . . 6.3 Stellar Activity Cycles . . . . . . . . . . . . . . . . . . 6.3.1 The Mount Wilson Survey . . . . . . . . . . 6.3.2 Stellar Activity Versus Stellar Age . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Solar and Stellar Activity: Cycles or Chaotic Behavior 7.1 Solar Activity Cycles . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 The 11 Year and Other Cycles . . . . . . . . . . . 7.1.2 The Cycle 23 and 24 . . . . . . . . . . . . . . . . . . 7.2 Chaotic Behavior of Solar Activity . . . . . . . . . . . . . 7.2.1 Lyapunov Exponent . . . . . . . . . . . . . . . . . . . 7.2.2 Estimation of Dimension . . . . . . . . . . . . . . .

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7.2.3 Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.2.4 Hurst Exponent Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 151 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 . . . . . . . . . . . . . . . . . . . .

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9 Solar Cycle Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Short Term Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Space Weather . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Space Weather Forecasting . . . . . . . . . . . . . . . . . . . . 9.2 Long Term Solar Activity Prediction . . . . . . . . . . . . . . . . . . 9.2.1 Different Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Stellar Chromospheric Activity Cycles . . . . . . . . . . . 9.2.3 The Faint Young Sun . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 The Basic Methods of Cycle Predicton . . . . . . . . . . . 9.2.5 Summary of Solar Activity Cycle Prediction Methods References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 Chaotic Dynamo Models . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The Solar Tachocline and Convective Zone . . . . . . . . . 8.1.1 The Solar Interior . . . . . . . . . . . . . . . . . . . . . . 8.1.2 The Tachocline . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 The Solar Convection . . . . . . . . . . . . . . . . . . . 8.1.4 Solar Differential Rotation . . . . . . . . . . . . . . . . 8.2 The Solar Dynamo . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 The Rikitake Model . . . . . . . . . . . . . . . . . . . . . 8.2.2 Coupled Disc Dynamos . . . . . . . . . . . . . . . . . . 8.2.3 A Simple Rikitake Model . . . . . . . . . . . . . . . . . 8.2.4 The Babcock Leighton Solar Dynamo . . . . . . . . 8.2.5 A Simple Solar Dynamo Model . . . . . . . . . . . . 8.3 Mean Field Electrodynamics . . . . . . . . . . . . . . . . . . . . 8.3.1 Interface Dynamo . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Surface Dynamo, Flux Transport Dynamo . . . . . 8.3.3 Torsional Oscillations . . . . . . . . . . . . . . . . . . . 8.4 Solar Activity in the Past and Chaotic Dynamo Action . 8.4.1 Reconstruction of Solar Activity in the Past . . . 8.4.2 Grand Minima in Non Linear Dynamos . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Acronyms

ACF ACRIM ARMA AU CIR CME DNA EMD ENSO ESA EUV GCR HF H-R HST IMF IR Ly MHD NASA NOAA NSO pc SDO SEP SOHO TC TSI UV

Autocorrelation function Active cavity radiometer irradiance monitor Autoregressive moving average Astronomical unit Corotating interactive region Coronal mass ejection Deoxyribonucleic acid Empirical mode decomposition El Ni~no Southern Oscillation European Space Agency Extreme ultraviolet Galactic cosmic rays High frequency Hertzsprung Russell Hubble space telescope Interplanetary magnetic field Infrared Light year Magnetohydrodynamics National Aeronautics and Space Administration National Oceanic and Atmospheric Administration National Solar observatory Parsec Solar dynamics observatory Solar energetic particles (protons) Solar and Heliospheric Observatory Total electron content Total solar irradiation Ultraviolet

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Chapter 1

What Is Chaos?

Before discussing the chaotic solar cycle behavior it is adequate to define chaos in general. What characterizes chaotic behavior, what is the difference between chaotic and non chaotic behavior? Could regular predictable motion suddenly transform into chaos? Let us consider the evolution of a physical system e.g. the solar activity as a function of time. Chaos does not imply a complete uncertainness about the future behavior of auch a system. It is shown that besides knowledge about possible onset of chaos in a physical problem it is still possible to make some predictions about the system’s evolution itself. Chaos can be quantized. We start from well known and analyzed mathematical models that show chaotic behavior under certain circumstances. A few mathematical examples of chaos are shortly reviewed demonstrating the principles and the route to the onset of chaos. Certainly, chaos exists in many astrophysical problems such as chaotic motions in our planetary system.

1.1 Chaos and Causality In this section we give a definition of chaos. What is the difference between a deterministic system and a chaotic system? What does deterministic chaos describe? We start with the definition of chaos in philosophy before going to mathematical definitions.

© Springer Nature Singapore Pte Ltd. 2020 A. Hanslmeier, The Chaotic Solar Cycle, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-15-9821-0_1

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1 What Is Chaos?

1.1.1 Chaos: Examples from Religion The word CHAOS comes from the Greek word χ αoσ and the translation into English means a state of disorder or confusion. In ancient Greek it also refers to the void state. According to the ancient Greeks (as well as in many other old religions) this state occurred before the creation of the cosmos, the universe. Sometimes it is also used for the original gap created when heaven and Earth separated. There exist several translations of this word: emptiness, vast void, chasm, abyss etc. The Greek philosopher Pherecydes of Syros (6th century BC) defined CHAOS as something without any definite form, similar to water. Hesiod and other philosophers before Socrates (470–399 BC) believed that Earth and the Sky formed a unity. Later on Earth became separated from the Sky, and CHAOS was formed. The dominant religion in the Indian subcontinent is Hinduism. In the Hindu trinity there exist Brahma, the creator, Shiva the destroyer and Vishnu the preserver. These three have female counterparts: Saraswati, the wife of Brahma, Lakshmi, the wife of Vishnu and Parvati the wife of Shiva. The struggle against chaos (in German: Chaoskampf) appears quite often in myth and legend. In a battle a hero deity fights with a chaos monster often in the shape of a serpent or a dragon. A christianized version of a Chaoskampf is shown in Fig. 1.1. The sentence Quis ut deus means: who is (like) god. This is a literal translation of Michael from the Hebrew language. The archangel Michael is represented as an angelic warrior who fights against a dragon (Fig. 1.2). Fig. 1.1 Chaos by Wenceslaus Hollar (1607–1677)

1.1 Chaos and Causality

3

Fig. 1.2 Depiction of the Christianized Chaoskampf: statue of Archangel Michael slaying a dragon. The inscription on the shield reads: Quis ut Deus?

Hesiod is generally regarded as the first written poet in the Western tradition and he lived around 750–650 BC. Three of his works survived among them the Theogony. Chaos was the first thing to exist but next, out of Chaos, appeared Gaia, Tartarus and Eros. So Chaos lead to structures and forms and the ancient Greeks also believed that Chaos is affected by Zeus like the Earth or the ocean or the air by thunderbolt. Chaos was located below Earth and above Tartarus. The Greeks believed that the universe was formed out of what they called a primal unity, of divine origin. Anaximander (c. 610 BC–547 BC) learned the teachings of his master Thales, Pythagoras was amongst his pupils. He belongs to the era of presocratic philosophy. Anaximander introduced the term apeiron which is an unlimited, divine substance of divine origin. They believed that the lower limit of Earth reaches down to apeiron, and the other limits of Earth are given by the sea, the sky and

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1 What Is Chaos?

Fig. 1.3 Possible rendering of Anaximander’s world map

Tartarus. The latter are located in Chaos. Let us see what he wrote about the origin of humankind: Anaximander of Miletus considered that from warmed up water and earth emerged either fish or entirely fishlike animals. Inside these animals, men took form and embryos were held prisoners until puberty; only then, after these animals burst open, could men and women come out, now able to feed themselves. The world map of Anaximander is shown in Fig. 1.3. Aristophanes (c. 446–c. 386 BC) was a comic playwright of ancient Athens. He wrote forty plays in total, and eleven survived. In his comedy, The Birds there was Chaos, Night, Erebus and Tartarus. From night came Eros, and from Eros and Chaos came the birds. The roman philosopher Ovid (1st century BC) defined Chaos in his Metamorphoses as an unformed mass that contained all elements in a shapeless heap: Ante mare et terras et quod tegit omnia caelum unus erat toto naturae vultus in orbe, quem dixere chaos:… In the Bible Chaos is linked to the term abyss/ tohu wa-boho (Gen. 1:2): In the beginning God created the heavens and the earth. Now the earth was formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters. Abyss denotes a bottomless pit or a chasm that may lead to the underworld or hell. It refers to a state of non-being before creation. It describes a formless state. Some other groups (Church Fathers) spoke about a creation ex nihilo (out of nothing, creation out of nothing) by an omnipotent god.

1.1 Chaos and Causality

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All of these examples are similar in that they describe chaos as a state before the creation. The complete formless primordial waters or primordial darkness was separated by a creator (Demiurge) and Earth and Sky were formed. Figure 1.1 shows a painting of an artist from the 17th century.

1.1.2 Chaos in Philosophy In this section we discuss some philosophical aspects of chaos. In modern physics Chaos also implies that the future evolution of a system is not predictable any more. The physicist Niels Bohr (1885–1962) and the baseball manager Yogi Berra made the following statement: Prediction is difficult, especially about the future. Chaos appears in many philosophical studies. In modern science, chaos means sensitive dependence on initial conditions. This was already mentioned by Aristotle who wrote that the least initial deviation from the truth is multiplied later a thousandfold.1 Let us give an example from the Medieval Period. Paracelsus was physician, alchemist and astrologer of the German Renaissance (1493, Switzerland, Einsiedeln 1541, Austria, Salzburg) (Fig. 1.4). He contributed to the new view that observation in combination with received wisdom leads to progress in medicine. Paracelsus uses chaos synonymously with classical element. The primeval chaos is imagined as a formless congestion of all elements. Earth is the chaos of the gnomi. The element of the gnomes through which these spirits move unobstructed as fish do through water or birds do through air. Paracelsus also wrote that by the light of the soul, or the will of god, all earthly things appear from the primal Chaos. In a book about alchemy, printed in 1612, Martin Ruland the Younger states: ‘A crude mixture of matter or another name for Materia Prima is Chaos, as it is in the Beginning.’ Therefore, as in the examples of religion given in the previous chapter, Chaos is regarded as the state of the universe at the beginning. In the modern view however, chaos theory is about the future states of complex dynamical systems. Some examples of these systems are: • • • •

the weather, the human brain, the stock market, and history.

The application of chaos to understand the human mind was reviewed e.g. by [1]. It is argued that human choice is related to a probabilistic or chaos-based model. Our choices are not random but are chaotic: they are deterministic, but hard to predict due to internal complexity.

1 On the Heavens; in. Barnes J (ed) The complete works of aristotle: the revised Oxford translation,

vol 1. Princeton University Press, Princeton.

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1 What Is Chaos?

Fig. 1.4 Aureolus Theophrastus Bombastus von Hohenheim [Paracelsus]. Reproduction, 1927, of etching by A. Hirschvogel, 1538. CC BY 4.0

1.1.3 Chaos in Modern Science In science Chaos theory states that complex systems follow laws and yet their future states remain unpredictable. Therefore, the future is not predictable based on past events as it was once thought to be. The philosopher George Carlin (1937–2008) said: No one knows what’s next, but everybody does it. The future is not independent of us; it does not follow a straight line from the past. The future is influenced by what we and everybody and everything else does. There is a mixture of human actions and biological and physical happenings and all these interfere with each other. What actually comes next is not predictable. According to J. A. Scott Kelso (1995) there is a circular causation. The complex systems evolve, and the new outputs provide new conditions that affect the system. A typical example is climate change which does not progress in a linear fashion. Each new iteration sets the context for the next iteration. This enables the creation of new unexpected phenomena. Events anywhere in the brain are connected to, and potentially have consequences for, other regions, which may respond to, propagate, enhance or develop that initial event, or alternatively redress it in some way, inhibit it, or strive to re-establish equilibrium. There are no bits, only networks, an almost infinite array of pathways (The Master and His Emissary, 2010, by Iain McGilchrist).

1.1 Chaos and Causality

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Another statement concerning breathing was made by Enrico Coen, 2012: Our ability to breathe depends on the interplay between our nervous system, muscles, skeleton, and lungs. The function of our lungs depends on the composition of the mucus that lines its walls. The composition of the mucus depends on the proteins that transport negatively charged chloride ions. Changes in just one element of the integrated system can have disastrous consequences. Patients with cystic fibrosis have difficulty breathing because they carry a mutation in the gene needed for chloride transport. It only takes one change out of the three billion base pairs in our genome to cause the disease. The functioning of every individual depends on the integration of many different components.

1.1.4 The End of Predictability In the previous chapter we mentioned the concept of an intelligent designer or creator of a regular universe. It was assumed that if we live in a clockwork universe, the existence of a clock-worker is required. Also in science one tends to assume that in those fields of research where regularity and predictability do not seem to exist, this nonexistence can be explained by non exact measurements, non adequate mathematics etc. so that in the end, predictability will be seen in those fields as well. Predictive limitations have been often explained as data or processing inadequacies, omissions, bias or randomness. Einstein said that God does not play dice. In quantum mechanics however, this does not seem to be the case. Electrons appear to distributed over every possible path between two points, they are not simply localized on circular orbits around an atomic nucleus. The path of an electron can be only hypothesized, there is no deterministic certainty. Therefore, in contradiction to Einstein, subatomic particles do play dice. Quantum mechanics is the first element of absolute unpredictability to spoil our confidence in the world being predictable. So what about the philosophical implications of chaos theory? We have seen that on subatomic scales unpredictability dominates over deterministic behavior. Chaos theory applies this statement to our everyday experience. Chaos theory creates a crack in the notion of a regular predictable universe. This is also outlined in Fig. 1.5. Consider the evolution of a system starting from two slightly different initial conditions. • In the case of non chaotic behavior the evolutionary paths will remain close together, and slightly different initial conditions will lead to a slightly different evolution. • In the case of chaos however, the two curves diverge exponentially. After a short time of evolution there is no similarity between them.

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Fig. 1.5 Predictable versus unpredictable universe. Consider the evolution of a physical system. Two neighboring paths showing the evolution of a system stay nearby in a predictable system, however they diverge exponentially for a chaotic system

1.1.5 Causality Causality tells us how the universe progresses. We speak of cause and effect and this principle is well known from everyday life experience. The cause is connected with the following state. The previous process is responsible for the subsequent one, and complex processes have several causes or factors. The nature of cause and effect is a concern of metaphysics. The question is what kind of entity can be a cause and what kind of entity can be an effect. For example one can assume that cause and effect are of one and the same kind of entity; causality is an asymmetric relation between them. We can say in principle: • A is the cause and B is the effect or • B is the cause and A is the effect. However, only one of these two sentences can be true. In the past we expected causation to result in repeatability. Repeatability results in predictability. How does chaos theory disrupt this argumentation? The point is, complex dynamic systems are sensitive to initial conditions. However, we will never be able to set up the same identical set of initial conditions of any system. Even the smallest differences in the initial conditions will lead to completely different results. Certainly causality remains to be true. One thing is following another, but the results will not be the same. This was also stated by Henry Ford: We can no longer view history as merely one damn fact after another. The past effects are themselves the result of chaotic behavior. As the poet Paul Valery puts it: the difficulty of reconstructing the past, even the recent past, is alltogether comparable to that of constructing the future, even

1.1 Chaos and Causality

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the near future; or rather, they are the same difficulty. The prophet is in the same boat as the historian? (Crisis of the Mind, First Letter, 1919). On the other hand, chaos also provides a certain creativity. The sensitivity provides space for unique non repeatable natural processes. S. J Gould argued that historical contingency in this sense plays as great a part in evolution as does natural selection. In Wonderful Life (1989) he said that if we could turn back evolution on Earth to its beginnings and restart the process with slightly different initial conditions, the organisms on our planet would look radically different.2 So chaos leads to historical contingency. But could this also play a role in the very laws of nature? The physicist P. Dirac (1902–1984) notes: At the beginning of time the laws of Nature were probably very different from what they are now. Thus, we should consider the laws of Nature as continually changing with the epoch, instead of holding uniformly throughout space-time. R. Feynman (1918–1988) observes that The only field which has not admitted any evolutionary question is physics. Here are the laws, we say but how did they get that way, in time? So, it might turn out that they are not the same [laws] all the time and that there is a historical, evolutionary, question. The universe could be an evolving chaotic system, even down to its laws.

1.2 From Simple Equations to Chaos In this section we discuss some simple mathematical equations and demonstrate how they lead to chaotic behavior when a certain set of parameters is chosen.

1.2.1 The Logistic Equation The logistic equation or logistic map is a very often cited example of how complex, chaotic behavior can evolve from very simple non-linear equations. The term non linear means that the change of output is not proportional to the change of input. Most systems in nature are non-linear. The logistic map was first mentioned in a paper by R. May in 1976. The logistic equation was first introduced in a demographic model in 1837 by P. F. Verhulst. The mathematical formula is: xt+1 = r xt (1 − xt ).

(1.1)

In this equation, xt is a number between 0 and 1 that gives the ratio of existing population to the maximum possible population. xt+1 describes that population after one timestep. r is a free parameter. The values of interest for r are in the interval

2 Gould

SJ (1989) Wonderful life. Norton, New York.

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[0, 4]. This non-linear equation describes the growth of a given population and two effects can occur: • reproduction: the population will increase at a rate proportional to the current population, when the population size is small. • starvation: the growth rate will decrease. Depending on the value of the parameter r we find: • 0 < r < 1: the population will eventually die, independent of the initial population. • 1 < r < 2 the population will quickly approach the value r −1 r

(1.2)

independent of the initial population. • 2 < r < 3 after some fluctuations the population will again converge to r −1 r

• • •

•

•

(1.3)

The rate of convergence is linear, except for r = 3. In that case, the convergence becomes extremely slow. √ 3 < r < 1 + 6 = 3.44949: from almost all initial conditions the population will approach permanent oscillations among two values. These values are dependent on r . 3.44949 < r < 3.5409: from almost all initial conditions the population will approach permanent oscillations among four values. 3.54409 < r < 3.56995: from almost all initial conditions the population will approach oscillations among 8, 16, 32 etc. values. The lengths of the parameter intervals that yield oscillations of a given length decrease rapidly. We call such a behavior a period-doubling cascade. The ratio between the lengths of two successive bifurcation intervals approaches the Feigenbaum constant δ ∼ 4.66920. r ∼ 3.56995: chaos starts at the end of the period-doubling cascade. From almost all initial conditions, we no longer find oscillations of finite periods. Moreover, slight variations in the initial population yield to dramatically different results. This is a prime characteristic of chaos. beyond r ∼ 3.56995 we see chaotic behavior, but some isolated ranges that show non chaotic behavior also√ exist. These are called islands of stability. This occurs, for example, for r = 1 + 8 = 3.82843.

The behavior of chaotic behavior as the parameter varies from approximately 3.56995 to approximately 3.82843 is called the Pomeau–Manneville scenario. Such as scenario is characterized by the following: The systems evolves periodic, however, the periodic (laminar) phases are interrupted by bursts of aperiodic behavior. The Pomeau–Manneville scenario has an

1.2 From Simple Equations to Chaos

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Fig. 1.6 The logistic map. This diagram is also called a bifurcation diagram

important application in semiconductor devices. In Fig. 1.6 we give the behavior of the logistic map as a function of the parameter r The logistic map describes population growth. The population growth rate in a given time period t2 − t1 can be written as follows: g=

P(t2 ) − P(t1 ) P(t1 )(t2 − t1 )

(1.4)

In this equation P(t1 ) is the population at a time t1 and P(t2 ) denotes the population at a time t2 , t2 > t1 . • g > 0, positive growth rate, the population is increasing, • g < 0, negative growth rate, the population is decreasing. • g = 0: the same number of individuals at the beginning (t1 ) and end (t2 ) of the period. For population dynamics the following equation is used:   P dP = kP 1 − dt K

(1.5)

P(t) is the population after time t, k is the relative growth coefficient and K the carrying capacity of the population. In ecology this means the maximum population size that a particular environment can sustain. This differential equation can be solved: P(t) =

P0 is the initial population at time 0.

K 1 + A exp(−r t) K − P0 A= P0

(1.6) (1.7)

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1.2.2 The Tent Map Another simple example for chaotic series is the tent map. It is defined as:    1   = μ 1 − 2 xn −  2 

xn+1

Let us calculate the first few iterations:    x1 = −μ 2 x0 −

(1.8)

  1  −1 2

The next value x2 :    x2 = −μ 2 x1 −

  1  − 1 2

and by substituting for x1 :        x2 = −μ 2 −μ 2 x0 −

  1  −1 − 2

  1  −1 2

The tent map can also be defined as: f μ := μ min{x, |1 − x|}

(1.9)

For the values of the parameter μ within 0 and 2 f μ maps the unit interval [0, 1] onto itself  xn+1 = f μ (xn ) =

xn < 21 μxn , μ(1 − xn ), 21 ≤ xn

 (1.10)

μ is a positive real constant. Let us assume μ = 2. Then f μ is an operation that is folding the unit interval in two, and we get the interval [0, 1/2]. The next operation gives the interval [0, 1] again. Any point x0 of the interval leads to a sequence xn in the interval [0, 1]. The μ = 2 case for the tent map is a non-linear transformation of • the bit shift map, • the r = 4 case of the logistic map. Mathematicians say that the tent map with μ = 4 and the logistic map with r = 4 are topologically conjugate. Let us now discuss the behavior of the tent map depending on the value of μ:

1.2 From Simple Equations to Chaos

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• if μ < 1, the point x = 0 is an attractive fixed point. All initial values of x converge towards x = 0. • μ = 1: all values of x ≤ 1/2 are fixed points of the system. μ . Both fixed points are • μ > 1: the system has two fixed points at 0 and at μ+1 unstable. If one takes a value close to either of these two fixed points, the results will move away from it. As an example, if μ = 1.5 there is a fixed point at x = 0.6, (1.5/(1 − 0.6)) but if we start at x = 0.61 we get the sequence 0.61 → 0.585 → 0.6225 → 0.56625 → 0.650625 . . . √ • 1 < μ < 2: the system maps a set of intervals between μ − μ2 /2 and μ/2 to themselves. This is the so-called Julia set of the map. It is the smallest invariant sub-set √ of the real line under the map. • μ > 2 these intervals merge, and the Julia set is the whole interval from μ − μ2 /2 and μ/2 (bifurcation diagram). • 1 < μ < 2: the interval [μ − μ2 /2, μ/2] contains both periodic and non-periodic points; the orbits are unstable. Nearby points move away from the orbits. As μ increases longer orbits appear: Let us consider some examples again: μ=1: μ μ μ → 2 → 2 μ2 + 1 μ +1 μ +1 2

or: √ 1+ 5 μ= 2 μ2 μ3 μ μ → 3 → 3 → 3 3 μ +1 μ +1 μ +1 μ +1 or: μ ≈ 1.8393

μ4

μ2 μ3 μ4 μ μ → 4 → 4 → 4 → 4 +1 μ +1 μ +1 μ +1 μ +1

• μ = 2 the system maps [0, 1] onto itself. There are now periodic orbits as well as non-periodic orbits. The periodic orbits in [0, 1] are dense. The map has become chaotic. The dynamics will become aperiodic if and only if x0 is irrational. The autocorrelation function for a sufficient long sequence {xn } will show the value zero, there is no autocorrelation at any non-zero lags. xn cannot be distinguished from white noise using the autocorrelation function. Mathematicians say that the r = 4 case of the logistic map and the μ = 2 case of the tent map are homeomorphic to each other. • μ > 2: the map’s Julia set becomes disconnected. It breaks up into a Cantor set within the interval [0, 1].

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1.2.3 A Numerical Example In this section we give a very simple numerical example that demonstrates how an algorithm is always converging to 1 regardless of the staring point. The algorithm produces a series of integers in a very simple way, and its unfolding is still not completely understood. The formulation was first made by Lothar Collatz: 1. 2. 3. 4.

Choose an arbitrary integer A if A = 1 then STOP if A is even, then replace A by A/2 and go to the previous step. if A is odd, then replace A by 3A + 1 and go to step 2.

It is very interesting that this always seems to end in a STOP. Let us give some examples: • A = 3 leads to the sequence: 10, 5, 16, 8, 4, 2, 1 and STOP. • A = 30 leads to the sequence: 30, 15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1 STOP. • A = 50 leads to the sequence: 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20 10, 5, 16, 8, 4, 2, 1 and STOP In the examples, we see that at the end of the series the numbers become similar. The algorithm works with any complicated number. Let us try with A = 3330. The first numbers are: 3330, 1665, 4996 . . . and the result is shown in Fig. 1.7. The simple python program producing this figure is given below: import matplotlib.pyplot as plt A=3330 x=np.zeros(1500) x[0]=A i1=0

Fig. 1.7 The Lothar Collatz algorithm always leads to 1. Here is the example for A = 3330

1.2 From Simple Equations to Chaos

15

while i11: if A\%2==0: A=A/2 else: if A\%2==1: A=3*A+1 print(A) i1=i1+1 print(’i1’,i1, ’A=’, A) x[i1]=A print(’end’) print(x) plt.plot(x[0:i1]) plt.title(’A=3330’) plt.show()

1.2.4 Sensitive Dependence on Initial Conditions As it was shown in the preceding sections, a typical criterion for chaos is sensitive dependence on initial conditions. To illustrate this in more detail we give a short example: Let us consider a special form of the logistic Verhulst model: pn+1 = pn + r × pn (1 − pn )

(1.11)

Starting with p0 = 0.01 and take r = 3. The resulting values are given in Fig. 1.8 in the uppermost part. Now let us slightly modify the initial condition. Let p0 = p0 ,  = 0.0000001. The resulting values are shown in Fig. 1.8 in the middle panel. The difference between the two series is shown in the figure at the lower panel. It is seen that after a few iterations both curves completely diverge.

1.2.5 Astrophysical Application: Star Formation Let us give a simple short application of the logistic equation. We know that stars are formed by a fragmented contraction of a large interstellar gas cloud. Let us assume that stars are born all identical with mass m as a result of a supernova-triggered event. Assume that all stars end their lives in a supernova explosion. n i is the number of stars where i refers to the ith generation. The number of stars in the next generation,

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1 What Is Chaos?

Fig. 1.8 The logistic map. Starting with slightly different initial conditions (upper and middle panel) leads to totally different series; the difference of the values is given in the lower panel

n i+1 is proportional to n i . M denotes the total amount of gas available. The number n i+1 depends on the available gas in the ith generation, which is M − mn i . Therefore, the stellar population evolves following the law: n i+1 = cn i (M − mn i )

(1.12)

xi+1 = 4r xi (1 − xi )

(1.13)

This can be rewritten as: where xi = mn i /M, and r = cM/4 which is the well known logistic map.

1.2.6 Autocorrelation Function The definition of the autocorrelation function, ACF, which is sometimes also called serial correlation, is as follows. We consider the correlation of a signal with a delayed copy of itself as a function of delay. It measures the similarity between observations as a function of time lag between them. The ACF is a very powerful tool in the analysis of time series. With the help of the ACF we can find repeating patterns in time series. However, in many cases a periodic signal in a measurement can be obscured by noise.

1.2 From Simple Equations to Chaos

17

Let us consider a continuous signal f (t). The continuous autocorrelation R f f (τ ) is defined as the cross-correlation integral of f (t) with itself at lag τ :  R f f (τ ) =

∞

−∞

f (u + τ ) f ∗ (u)du =



∞

−∞

f (u) f ∗ (u − τ )du

(1.14)

f ∗ is the complex conjugate. For a real function f ∗ = f . The continuous ACF can be also written as a convolution: R f f (τ ) = ( f ∗ g−1 ( f¯))(τ )

(1.15)

The function g−1 is defined as: g−1 ( f (u)) = f (−u)

(1.16)

When the mean values are subtracted before computing the ACF the resulting function is called the auto-covariance function. In practical application there will be always a discrete time series. The discrete autocorrelation R at lag l for a discrete time signal y(n) is given by: R yy (l) =



y(n)y(n − l)

(1.17)

n∈Z

A stochastic process is said to be ergodic if its statistical properties can be deduced from a single random sample of the process. The restriction is that the sample must be sufficiently long. We can also say that the birds-eye view of the collection of samples must represent the whole process. A wide-sense stationary process X (t) has constant mean: μ X = E[X (t)]

(1.18)

The autocovariance function can be written as: r X (τ ) = E[X (t) − μ X )(X (t + τ ) − μ X )]

(1.19)

The process X (t) is said to be mean-ergodic if the time average estimate 1 μˆ X = T



T

X (t)dt

0

converges in squared mean to the ensemble average μ X as T → ∞. For ergodic processes, the ACF definition becomes:

(1.20)

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1 What Is Chaos?

1 R f f (τ ) = lim T →∞ T



T

f (t + τ ) f¯(t)dt

(1.21)

N1 1  y(n) y(n ¯− l) N n=0

(1.22)

0

R yy (l) = lim

N →∞

Finally, let us give some properties of the ACF: • symmetry: R f f (τ ) = R f f (−τ )

(1.23)

Let us consider the continuous case. When f is a real function then R f f (−τ ) = R f f (τ ). Therefore, the ACF is an even function. When f is a complex function the ACF is a Hermitian function: R f f (−τ ) = R ∗f f (τ ). • The continuous ACF has a peak at the origin, for any delay τ : |R f f (τ )| ≤ R f f (0)

(1.24)

This is also valid for the discrete case. • The ACF of the sum of two completely uncorrelated functions is the sum of the ACF of each function separately. We can easily define what is meant by two completely uncorrelated functions: the crosscorrelation is zero for all values of lag τ. • The ACF is a specific type of cross-correlation. • The ACF of a continuous white noise signal has a strong peak at τ = 0 and becomes exactly zero for all other τ . • Wiener–Khinchin Theorem: The ACF of a wide-sense-stationary random process has a spectral decomposition given by the power spectrum of that process. Let us consider a function with discrete values [xn ]. Then the power spectral density function is: ∞ 

r x x[k] exp−i(2π f )k

(1.25)

r x x [k] = E[x[n]∗ x[n − k]]

(1.26)

S( f ) =

k=−∞

We can also define multi-dimensional autocorrelation functions. Let us consider the three dimensional case:  xn,q,r xn− j,q−k,r −l (1.27) R( j, k, l) = n,q,r

When the mean values are subtracted from signals before computing an ACF, the resulting function is called an auto-covariance function.

1.2 From Simple Equations to Chaos

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Fig. 1.9 The autocorrelation function for a periodic sine function (upper panel)

Let us consider a simple example. In Fig. 1.9 we plot a simple sine function in the top panel. In the panel below we show the corresponding ACF. It is evident that the ACF has a maximum at zero shift and then the values become smaller. Because of the periodicity of the sine function several additional maxima appear. The simple python program code is: import matplotlib.pyplot as plt import numpy as np def autocorr(x): result = numpy.correlate(x, x, mode=’full’) return result x=np.arange(1000) x=np.sin(x/57.3) plt.figure(1) plt.subplot(211) plt.plot(x) result=autocorr(x) plt.subplot(212) plt.plot(result[1000:]) plt.show()

We can also test the ACF for random numbers. As it was stated, in such a case the ACF should decline to zero without any peaks. The maximum occurs for lag 0. The result is shown in Fig. 1.10. The ACF program has to be slightly modified by adding the following lines:

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1 What Is Chaos?

Fig. 1.10 The autocorrelation function for an array of random numbers

x1=np.random.rand(1000) print(x1) plt.subplot(211) plt.plot(x1) result=autocorr(x1) plt.subplot(212) plt.plot(result[1000:])

This program simulates the ACF for a white noise signal. In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The final example shows an overlay of a periodic signal and random noise. We define again the sine function as a periodic signal and overlay a random signal with a factor 5 by simple addition to the periodic signal. The plots are given in Fig. 1.11. The corresponding program lines are: x2=5*x1+x figure(3) result=autocorr(x2) plt.subplot(211) plt.plot(x2) plt.subplot(212) plt.plot(result[1000:])

1.2 From Simple Equations to Chaos

21

Fig. 1.11 The autocorrelation function for a periodic signal overlaid by random numbers which were added to the values of the periodic signal (weighting factor 5)

1.2.7 Powerspectrum: The Fourier Transform The Fourier transform is a very powerful tool for analyzing time signals. There are many textbooks about this method, so we will recall only some very basic principles and give one example. The idea is that any time signal can be represented by a superposition of many periodic time signals. Let us consider a very complex signal. The Fourier transform is a method to decompose such a signal and the power spectrum exhibits wether discrete frequencies exist in the signal that seems, at the first inspection very irregular. This method can be used to detect any periods in a time signal. Let us assume a function f (x). The Fourier Transform is defined as:  F(ξ ) =

∞

∞

f (x) exp−2iπ xξ d x

(1.28)

The power spectrum is calculated as the magnitude squared of the Fourier transform (or Fourier series) of the waveform of interest: P( f ) = |X ( f )|2

(1.29)

In Fig. 1.12 we give a simple example: a sine and a cosine function are added. The corresponding power spectrum is shown in the panel below. It exhibits two peaks according to the different frequencies that were given to the two initial functions.

22

1 What Is Chaos?

Fig. 1.12 The power spectrum of the sum of two functions: a sine and a cosine function

import numpy as np import matplotlib.pyplot as plt from scipy import fftpack x=np.arange(1024) xx=x x=np.sin(x/57.3)+np.cos(2*x/57.3) plt.subplot(211) plt.plot(x) # now we computer the power spectral density xfft = fftpack.fft(x) # And the power power = np.abs(xfft) print(xx[1:20]) print(power[1:20]) plt.subplot(212) plt.plot(xx[1:20],power[1:20])

Finally, let us consider a random signal. The python code is listed below and in Fig. 1.13 the results are shown, in the panel above the random signal, in the panel below the corresponding power spectrum is shown. import numpy as np import matplotlib.pyplot as plt from scipy import fftpack xrand=np.random.rand(1024) print(xrand) plt.subplot(211)

1.2 From Simple Equations to Chaos

23

Fig. 1.13 The power spectrum of a random signal does not show any peak

plt.plot(xrand[1:100]) xrandfft=fftpack.fft(xrand) power=np.abs(xrandfft) plt.subplot(212) plt.plot(power[1:100])

The figure shows no specific peaks, so there is white noise. For our study of chaotic motions we can state that by Fourier analysis we can easily depict any periods in a time series. There will be no peaks in the power spectrum of a completely random signal.

1.3 Self-similarity Simply expressed self similarity means that the whole has the same shape as one or more of the parts. This section shows that self similar processes occur quite often in nature. This leads finally to the concept of a fractal or fractal geometry. The dimension has a non integer number.

1.3.1 Mathematical Description Two objects are similar if they have the same shape, regardless to their size. Corresponding angles, however, must be equal, and corresponding line segments must all have the same factor of proportionality. Similarity transformations involve a • scaling • rotation

24

1 What Is Chaos?

• translation Let us describe this mathematically: A scaling operation on any points P(x, y) yields a new point P  = (x  , y  ): x  = sx y  = sy

(1.30) (1.31)

A scaling reduction occurs, if s < 1, an enlargement if s > 1. If a rotation is applied to a point P  = (x  , y  ) it yields P  = (x  , y  ): x  = cos θ x  − sin θ y  y  = sin θ x  + cos θ y 

(1.32) (1.33)

This describes a counterclockwise rotation (mathematically positive) of P  around the origin of the coordinate system by an angle θ . Finally, a translation is described by a displacement Tx , Ty of the point (x  , y  ): x  = x  + Tx y  = y  + Ty

(1.34) (1.35)

The similarity transform can then be written as: P  = T (P  ) = R(R(P  )) = T (R(S(P)))

(1.36)

W (P) = T (R(S(P)))  x = s cos θ x − s sin θ y + Tx

(1.37) (1.38)

y  = s sin θ x + s cos θ y + Ty

(1.39)

This concept was introduced for two dimensions, x, y but can be extended to three dimensions, where an additional process, the tension, can be introduced.

1.3.2 The Koch Curve In this subsection we give some simple examples for scaling and self similarity. The Koch curve: this is a mathematical curve and one of the earliest fractal curves that has been described. It was introduced by the Swedish mathematician Helge von Koch (1904) . The construction is simple. We start with an equilateral triangle, then recursively change each segment according to the following rules: 1. divide the line segment intro three segments of equal length 2. draw an equilateral triangle that has the middle segment from step 1 as its base 3. remove the line segment that is the base of the triangle from step 2.

1.3 Self-similarity

25

Fig. 1.14 The first four iterations to construct a Koch curve

In Fig. 1.14 we give the first four iterations for the Koch curve. Let us give the perimeter of the Koch snowflake. After each iteration, the number of sides of the Koch snowflake increases by a factor of 4. Therefore, after n iterations we have: Nn = Nn−1 4 = 3 × 4n

(1.40)

If the original equilateral triangle has sides of length s: Sn =

s Sn−1 = n 3 3

(1.41)

The perimeter of the snowflake after n iterations becomes: Pn = Nn Sn = 3s

 n 4 3

(1.42)

After each iteration, the length of the curve increases by a factor of 4/3. Each iteration creates 4 times as many line segments as in the previous iteration. As the number of iterations tends to infinity, the limit of the perimeter is infinite. Let us now calculate the area. In each iteration, a new triangle is added on each side of the previous iteration. The number of new triangles added in iteration n is:

26

1 What Is Chaos?

3 Tn = Nn−1 3 × 4n−1 4n 4

(1.43)

The area of each new triangle added in an iteration is 1/9 of the area of each triangle added in the previous iteration: an =

an − 1 a0 9 9n

(1.44)

a0 is the area of the original triangle. The total new area added in iteration n is: bn = Tn an =

3 4

 n 4 a0 9

(1.45)

Thus, after n iterations A n = a0

n  k=1

 a0

   n   n−1   3 4 k 1 4 k 1+ = a0 1 + 4 k=1 9 3 k=0 9

a0 An = 5

  n  4 8−3 9

(1.46)

Therefore, the limit of the area is: lim An =

n→∞

8 a0 5

(1.47)

If the side length of the original triangle is s: √ 2s 2 3 lim An = n→∞ 5

(1.48)

So we found that the Koch curve has following properties: • self replicating • infinite perimeter • finite area.

1.3.3 The Fractal Dimension Curves, surfaces, and volumes can be extremely complex. Ordinary measurements become meaningless. Let us give a simple example. Think about the coast of Great Britain. The smaller the scale we look at it, the longer it would become. So the length

1.3 Self-similarity

27

of the coast of Great Britain depends on which scale we are measuring it. Ordinary measurements thus become meaningless. In this chapter we introduce a method to measure the degree of complexity by evaluating how fast length, surface or volume increases when determined with increasingly smaller scales. The fractal dimension measures how detail in a pattern changes with the scale at which it is measured. Unlike topological dimension, the fractal index can have non integer values. A curve with a fractal dimension very near to 1, say 1.10, behaves quite like an ordinary line, but a curve with fractal dimension 1.9 winds convolutedly through space very nearly like a surface. Let us assume that N denotes the number of sticks and  the scaling factor. The fractal dimension can be given as: N ∝  −D

(1.49)

where D is the fractal dimension. Let us now calculate the fractal dimension. There are several definitions of the fractal dimension. Fractal means, that there is no integer dimension. The Koch curve consists of lines of dimension 1, but the dimension of the fractal is larger than 1 and less than 2. According to the rules described above a Koch curve is obtained by transforming each line line into 4. Each line has become 4 self-similar copies with scaling factor of 3. Therefore N = 4,  = 1/3 and the fractal dimension becomes: d=

log 4 log N = ≈ 1.26 log  log 3

(1.50)

Different types of dimension definitions exist that can be the same value for a given problem, but this is not always the case. For more detail, the reader is referred to textbooks, as we give only the basic definitions here: • Box counting dimension: D is estimated from: log N () →0 log 1 

D0 = lim

(1.51)

In Fig. 1.15 the principle is illustrated. Assume a fractal S. This is lying on an evenly spaced grid. Now we count how many boxes are required to cover the set. N () is the number of boxes of side length  required to cover the set. The box dimension is also defined using balls. Let us consider balls with radius . The covering number NC () is the minimal number of open balls of radius  required to cover the fractal, such that their union contains the fractal. In Euclidean n-space, an (open) n-ball of radius r and center x is the set of all points of distance less than r from x. A closed n-ball of radius r is the set of all points of distance less than or equal to r away from x. The packing number Npack (ε) is the maximal number of disjoint open balls of radius one can situate such that their centers would be inside the fractal. Examples are shown in Fig. 1.16.

28

1 What Is Chaos?

Fig. 1.15 A map of Great Britain. More detail is seen if the box size gets smaller

Fig. 1.16 A map of Great Britain. Ball packing, ball covering, box covering

• Information dimension: D considers how the average information needed to identify an occupied box scales with the box size; p is a probability. D1 = lim+ →0

N  pi () ln pi () ln() i=1

(1.52)

1.3 Self-similarity

29

• Correlation dimension: D is based on M as the number of points used to generate a representation of a fractal g , the number of pairs of points close than  to each other. log(g /M 2 ) M→∞ →0 log 

D2 = lim lim

(1.53)

1.4 The Lorentz System The Lorentz system was one of the first models that were analyzed in Chaos Theory. Several real physics problems such as lasers, electric circuits, osmosis can be described by it. For solar physics, solar convection3 and the solar dynamo4 can be especially simplified to a Lorentz system. Lorentz established these equations to make long term prediction in the Earth’s atmosphere but he realized that such predictions are limited.

1.4.1 Formulation of a Lorentz System The Lorentz system is a set of ordinary differential equations first studied by E. N. Lorentz (1917–2008) in 1963. We consider a two-dimensional fluid layer heated from below and cooled from above. The set of equations is: dx = σ (y − x) dt dy = x(ρ − z) − y dt dz = x y − βz dt

(1.54) (1.55) (1.56)

The three quantities x, y, z are defined as follows: • x is proportional to the rate of convection, • y is the horizontal temperature variation, and • z is the vertical temperature variation. The system is further characterized by: • σ , the Prandtl number, which denotes the viscous diffusion rate over the thermal diffusion rate: 3 The

energy transport in the layers below the solar surface. dynamo models are necessary to understand the solar activity cycles.

4 Solar

30

1 What Is Chaos?

σ =

ν α

(1.57)

If the Prandtl number is small it means that thermal diffusivity dominates. Prandtl numbers for gases are about 1. This means that both momentum and heat dissipate through the fluid at about the same rate. • ρ is the Rayleigh number, which denotes the timescale for thermal transport via diffusion over the timescale for thermal transport via flow at speed u. Let us consider a fluid of length scale l, mass difference ρ, gravitational force ρl 3 g, and viscosity of the fluid η. The Stokes equation gives the force of viscosity on a small sphere, with radius R moving through a viscous fluid: Fd = 6π η Ru

(1.58)

Fg is also called a drag force. If the volume of fluid is falling at speed u the viscous drag is of order Fd = ηu and the speed becomes u ∼ ρl 2 g/η. The Rayleigh number then becomes: ρ=

l 2 /α η/ρlg

(1.59)

α is the thermal diffusivity. • β physical dimensions of the system in the set of Lorentz equations.

1.4.2 Some Experiments with the Lorentz System Let us consider a simple python program as it is shown below: in this listing the variables correspond to s = σ , r = ρ and b = β as it was defined in the above equations. import numpy as np import matplotlib.pyplot as plt from mpl\_toolkits.mplot3d import Axes3D

def lorenz(x, y, z, s=10, r=28, b=2.667): xdot = s*(y - x) ydot = r*x - y - x*z zdot = x*y - b*z return x_dot, y_dot, z_dot

dt = 0.01

1.4 The Lorentz System

31

stepCnt = 10000 # Need one more for the initial values xs = np.empty((stepCnt + 1,)) ys = np.empty((stepCnt + 1,)) zs = np.empty((stepCnt + 1,)) # Setting initial values xs[0], ys[0], zs[0] = (0., 1., 1.05) # Stepping through "time". for i in range(stepCnt): # Derivatives of the X, Y, Z state x\_dot, y\_dot, z\_dot = lorenz(xs[i], ys[i], zs[i]) xs[i + 1] = xs[i] + (x\_dot * dt) ys[i + 1] = ys[i] + (y\_dot * dt) zs[i + 1] = zs[i] + (z\_dot * dt) fig = plt.figure() ax = fig.gca(projection=’3d’) ax.plot(xs, ys, zs, lw=0.5) ax.set\_xlabel("X Axis") ax.set\_ylabel("Y Axis") ax.set\_zlabel("Z Axis") ax.set\_title("Lorenz Attractor") plt.show()

With the parameter set s = 10, r = 28, and b = 2.667 the following plots are obtained: Figures 1.17 (number of integration steps is 100), 1.18 (number of integration steps is 1000), and 1.19 (number of integration steps is 10000). We see the convergence to the strange attractor occurring as the number of integration steps increases.

1.5 Measuring Chaos 1.5.1 The Phase Space A phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. Let

32

1 What Is Chaos?

Fig. 1.17 The Lorentz attractor for s = 10, r = 28, b = 2.667. In this example the number of steps for the integration of the system was chosen as 100

Fig. 1.18 The Lorentz attractor for s = 10, r = 28, b = 2.667. In this example the number of steps for the integration of the system was chosen as 1000

1.5 Measuring Chaos

33

Fig. 1.19 The Lorentz attractor for s = 10, r = 28 and b = 2.667. In this example the number of steps for the integration of the system was chosen as 10000

us consider a mechanical system. For such systems, the phase space consists of all possible values of position and momentum values. Let us also consider a system with one degree of freedom. The equation describing this system is an autonomous ordinary differential equation with a single variable: dy = f (y) dt

(1.60)

A simple example of such a system is the exponential growth model. The phase space is a phase line. For a two-dimensional system there is a phase plane. Such a phase plane occurs when considering a single particle moving in one dimension; here, there are two variables: • position • velocity The horizontal axis gives the position and the vertical axis gives the velocity. As the system evolves the state follows the trajectories on the phase diagram.

34

1 What Is Chaos?

1.5.2 Lyapunov Exponent Chaotic systems can be characterized by the butterfly effect: they are highly sensitive to initial conditions. Even the smallest variations of initial conditions lead to completely different results. The presence of chaos is proven if sensitivity to initial conditions can be demonstrated. Is it possible to quantify this attribute? The Lyapunov exponent measures to the base 2, in bits, the rate at which neighboring points on the attractor diverge as they are moved forward in time. More generally we can say that the Lyapunov exponents measure how a volume of space on the attractor dilates over time. Trajectories on the attractor are embedded in a multi-dimensional space. The divergence is represented as the difference between 2 n-tuples. Let us consider two neighboring trajectories in phase space. The initial separation is denoted by: δZ0

(1.61)

|δZ(t)| ≈ eλt |δZ0 |

(1.62)

Then the divergence can be written as:

where λ is the Lyapunov exponent. However, the phase space has dimension n ≥ 1, therefore there exist n Lyapunov exponents. In chaos theory one is interested in the maximal Lyapunov exponent, because this exponent determines the predictability of the system. If the maximum Lyapunov exponent is positive, this means that the system is chaotic. For a dynamical system given by x˙i = f i (x)

(1.63)

in an n-dimensional space, the spectrum of Lyapunov exponents is λ1 , λ2 , . . . λn

(1.64)

The maximal Lyapunov exponent is 1 |δZ(t)| ln t→∞ δZ0 →0 t |δZ0 |

λ = lim lim

(1.65)

For discrete time series with xn+1 = f (xn ) and an orbit starting with x0 we get:

(1.66)

1.5 Measuring Chaos

35 n−1 1 ln | f  (xi )| n→∞ n i=0

λ(x0 ) = lim

(1.67)

The spectrum of Lyapunov exponents depends on the starting point x0 . We are interested in the attractor of the systems. There will normally be one set of exponents associated with each attractor. The Lyapunov exponents describe the behavior of vectors in the tangent space of the phase space and are given by the Jacobian matrix: Ji j (t) =

d f i (x) |x(t) dx j

(1.68)

The evolution of the system is given by: Y˙ = J Y

(1.69)

The matrix  is given by 1 log(Y (t)Y T (t)) t→∞ 2t

 = lim

(1.70)

and the Lyapunov exponents λi are the eigenvalues of . The system is ergodic if the set of Lyapunov exponents will be the same for almost all starting points. A system is conservative if a volume element in phase space stays constant. Therefore, the sum of all Lyapunov exponents must be zero. The system is dissipative, if the sum of Lyapunov exponents is negative. From the Lyapunov spectrum the rate of entropy production and the fractal dimension can be deduced. The Kaplan–Yorke dimension D K Y is defined as DK Y = k +

k  i=1

λi |λk+1 |

(1.71)

where k is the maximum integer such that the sum of the k largest exponents is still non-negative. The Kaplan Yorke dimension is an upper bound of the information dimension of the system. The sum of all the positive Lyapunov exponents gives an estimate of the Kolmogorov–Sinai entropy.

Reference 1. Ostrowick John M (2015) What is chaos and how is it relevant for philosophy of mind? South Afr J Philos 34(3):323–335

Chapter 2

Chaos in the Solar System

In this section we discuss our planetary system. First we give an overview then we discuss chaotic behavior. Chaotic behavior of solar system bodies can be found in their orbits, rotations, formation processes etc. This chapter finishes with a short outlook to exoplanet systems.

2.1 The Solar System 2.1.1 Overview The Sun is the dominant object in the solar system. It contains about 99.8% of the total mass of the solar system. Astrophysical quantities are often given in solar units: > Astrophysical units Mass of the Sun: 1 M = 2 × 1030 kg Radius of the Sun: 1 R = 7 × 105 km

The objects in the solar system are classified into the following groups: • Planets: there are 8 planets; Mercury, Venus, Earth and Mars are the terrestrial planets with solid surfaces, Jupiter, Saturn, Uranus and Neptune are the gas planets which are larger than the terrestrial planets and have no solid surface. • Dwarf Planets: these are smaller than planets with sizes between 1000 and 2500 km in diameter; they have not cleared up yet their orbit from smaller bodies. Prominent objects belonging to this category are Pluto and Ceres (when detected in 1801 Ceres was classified as an asteroid). © Springer Nature Singapore Pte Ltd. 2020 A. Hanslmeier, The Chaotic Solar Cycle, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-15-9821-0_2

37

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2 Chaos in the Solar System

• Asteroids: these are smaller than the dwarf planets, and only few of them are larger than 100 km. There are several 105 asteroids. They are concentrated in belts: the main asteroid belt is located between the orbits of Mars and Jupiter, and the Kuiper belt is located outside the orbit of Neptune, Pluto being one of its most prominent members. Most asteroids do not have a spherical shape. • Comets: their small nucleus, several 10 km of diameter, consists of volatile elements. Most comets have highly eccentric orbits that are not coplanar with the ecliptic.1 When they approach the Sun (i.e. inside the orbit of Mars) the volatile components vaporize and a cometary coma that extends several thousand kilometers is formed. However, their most prominent feature is a long tail, which can extend over several million kilometers. • Debris: these small bodies (often several centimeters in size) can collide with Earth and when penetrating through the earth’s atmosphere they become heated and appear as bright meteors. Some of these objects are debris from comets or asteroids. • The Oort cloud: this consists of several billions of small objects (mostly cometary) and forms an envelope of the planetary system. It extends up to 105 astronomical units (AUs). 1 AU is the mean distance between the Earth and Sun. > The astronomical unit 1 AU = 149 × 106 km.

Long periodic comets or comets on very high eccentric or inclined orbits are supposed to have originated from the Oort cloud by perturbations.

2.1.2 Solar System Formation There are several observational facts that provide hints about the solar system formation. The orbits of the planets are almost coplanar, with the exception of Mercury and some dwarf planets. The orbit of the other planets and many solar system objects are close to the ecliptic. All planets revolve around the Sun in the same sense. Furthermore, most planets rotate about their axis in the same sense; Venus and Uranus and some small satellites of large planets are exceptions. The Sun and the planets formed when an interstellar gas cloud became gravitationally unstable, starting to contract because of its own gravity. The pressure of the gas particles in an interstellar cloud is given by: P = N kT 1 The

ecliptic is the plane of the Earth’s orbit about the Sun.

(2.1)

2.1 The Solar System

39

where N is the number of particles, k = 1.38 × 10−23 J/K (Boltzmann constant), T is the temperature (several tens of Kelvin). If the mass of the cloud exceeds the Jeans mass, the cloud starts to contract because gravity is not balanced by the gas pressure. The Jeans mass can be derived from the mean free fall time tff . Let us consider a spherically symmetric distribution of mass. The volume of the sphere is V = 4/3π R 3 and the density is ρ=

3M . 4π R 3

Then the mean free fall time becomes:  1 3π 1 ≈ 0.57 √ tff = = 66430 √ 32Gρ ρ Gρ

(2.2)

(2.3)

For a star with a mean density of 1 g/cm3 the free fall time would be 35 min. The lower the density, the longer the free fall time. The typical extension of a giant molecular cloud that started to collapse is about 20 parsec (pc). > Definition parsec 1 pc = 3.26 light years, 1 light year = 1013 km

Such a cloud fragments into 1 pc clouds that again fragment into a dense core of 0.01–1 pc. From such a fragment the Solar System was formed. The cloud mainly consisted of hydrogen and helium (98% of its mass). This scenario also implies that the Sun was formed within a cluster consisting of 100–1000 stars. The diameter of the cluster was between 6.5 and 20 Ly, and the cluster disintegrated between 135 and 535 million years after formation. This early phase of solar system evolution was characterized by strong interactions of the young Sun with close passing stars and can explain anomalies found especially in the outer Solar System. The young Sun had a violent T-Tauri phase. Star and planet formation can be observed in gaseous nebulae like the Orion Nebula (Fig. 2.1). This star forming region is at a distance of about 1500 Ly from us and has an extension of about 12 Ly. The total mass of the Orion Nebula (also called M 42) is about 2000 solar masses. Detailed images show protoplanetary discs around stars (Fig. 2.2). After 50 million years, the Sun became a stable star i.e. an equilibrium was established between the gravitational forces that act inwards and the gas pressure that depends on temperature. The required temperature to explain a high gas pressure was no longer provided by the contraction but by thermonuclear fusion of hydrogen into helium. Since pressure is defined as force per unit area P = F/A the force becomes

40

2 Chaos in the Solar System

Fig. 2.1 The Orion nebula is one of the closest star forming regions. This image is a composite of visual and IR observation. ESA/NASA/HST

F = P A = P4πr 2

(2.4)

where r denotes the distance from the center. Stars are considered as spherically symmetric non rotating non magnetic objects for simplification. The above mentioned force is balanced by gravity: d FG = dmg = ρd V g = ρ4πr 2 drg

(2.5)

In hydrostatic equilibrium both forces are the same: dP = −gρ dr

(2.6)

and since g = G M/r 2 the final form for the condition of hydrostatic equilibrium becomes: dP GM =− 2 ρ dr r

(2.7)

2.1 The Solar System

41

Fig. 2.2 Protoplanetary discs (proplyds) in the Orion nebula. The stars are surrounded by gas and dust. In such protoplanetary discs planets are formed. The proplyds which are closest to the hottest stars of the parent star cluster are seen as bright objects, while the object farthest from the hottest stars is seen as a dark object. The field of view is only 0.14 light-years across. The Orion Nebula star-birth region is 1500 light-years away, in the direction of the constellation Orion. NASA/HST

The Sun will be in hydrostatic equilibrium for about 9 × 109 years. Its luminosity will not change considerably. The long lifetime of the Sun is important for the evolution of life on Earth, because it took at least 109 years for the first primitive forms of life to appear.

2.2 Orbital Motions In this section we discuss the orbital motions of planets. We will focus on the question of whether these motions can turn into chaotic motions.

2.2.1 The Equations of the N-Body Problem The equations of the N-body problem are based on Newton’s equations. We consider n bodies of masses m i , position vectors ri and velocities vi . In Fig. 2.3 this is illustrated with three masses m 1 , m 2 , m 3 . Let us consider the forces exerted by all masses m j on the mass m i . The equations of motion are

42

2 Chaos in the Solar System

Fig. 2.3 The three body problem. Three masses m 1 , m 2 , and m 3 have position vectors r 1 , r 2 , and r3

m i r¨ i =

n  j=1

G

mi m j (ri − r j ) ri3j

i = j

(2.8)

In order to calculate the forces acting on mass m i we have to sum all the other masses indicated by index j, where i = j. The distance between two bodies i and j is given by ri j = (xi − x j )2 + (yi − y j )2 + (z i − z j )2 . Equation (2.8) is a set of non linear differential equations. The equations can be solved by several numerical integration techniques. In order to check the validity of the results conservation laws like energy, linear momentum or angular momentum can be calculated after each integration step. For a two body problem (e.g. the Sun and one planet) there is a simple solution: a conic section (circle, ellipse, parabola or hyperbola). As soon as a third body is involved there is no simple solution. Perturbation theory is one method that can be used to solve the problem. One starts with an unperturbed orbit (an ellipse about the sun). Then the influence of the other bodies is treated as a perturbation.

2.2.2 Chaos and the N-Body Problem As it was pointed out in the previous chapter any errors will propagate with each integration step. Furthermore, the initial conditions such as position and velocity vectors of the bodies considered are not known with full precision. With respect to our discussion about chaos which implies sensitivity on initial conditions there are two sources of uncertainty: • Initial conditions of the planets are not known with infinite precision. • Numerical integration techniques do not work with infinite precision.

2.2 Orbital Motions

43

The butterfly effect demonstrates how such uncertainties can grow exponentially in a chaotic system. Several questions arise immediately: Do the above mentioned uncertainties influence the evolution of an N-body system? Do they cause only small changes in the nearly circular orbits that are coplanar? Might they grow exponentially and become large over the evolution of the solar system? Could such perturbations influence on the habitability of our planet? One well known example is climate variations that are caused by changes in the orbit or by impact of asteroids or comets. Accumulations of interplanetary dust particles that are generated by asteroids and comets may also influence the climate. Long term orbital dynamics of planets and small solar system bodies (SSBs) are therefore very important. Another question that may be of big interest is: How typical is our solar system in the galaxy? What characterizes a stable planetary system? Since the evolution of life on Earth required stability of the planetary system in the range of several billion years we need a so-called continuous habitable zone when searching for life elsewhere in the universe. Let us stress again the meaning of chaos here: > Chaos… describes the irregular behavior that can occur in deterministic dynamical systems. These systems are described by ordinary differential equations; there is no external random influence.

Chaotic motion can be characterized by the Lyapunov time, an e-folding time scale. Another important time scale is the escape time. Chaos in the solar system is connected to resonances, especially concerning the orbits of the objects. Simple cases of resonances occur when the orbital periods of two objects are in the ratio of two small integers. Let us consider a 1:2 resonance between the orbital periods of object A and object B. This means that when object A completes one orbital period, object B completes two. Gravitational resonances may lead to very large orbital changes but could also have only minor effects. In the following we give some examples.

2.2.3 Motion and General Relativity General relativity theory was published in 1915 by A. Einstein. Einstein revolutionized gravitational theory, but Newtonian gravity still provides an excellent approximation of the gravitational forces. According to the general relativity theory of Einstein gravitation can be expressed by a curvature of space-time. In Fig. 2.4 it is illustrated how a single mass causes a curvature of space time. This is summarized in Einstein’s field equations. Particles move on a geodesics, that is the shortest possible

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2 Chaos in the Solar System

Fig. 2.4 According to Einstein’s general relativity theory the space-time curvature is related to mass

distance between two points. Relativistic effects in general become important for strong fields. This is the case in the vicinity of a black hole, a neutron star or even a white dwarf. The distance between any two objects in space-time can be described by the line element ds 2 = d x 2 + dy 2 + dz 2 − c2 dt 2

(2.9)

In the case of the curved space time around a single mass these metrics have to be replaced by the Schwarzschild metrics:  rs  2 1 dt + ds 2 = −c2 1 − r 1−

rs r

dr 2 + r 2 d2 + r 2 sin2 dφ 2

(2.10)

Here, a spherical coordinate system has been used with coordinates (r, , φ) instead of the coordinates (x, y, z). rs is the Schwarzschild radius: rs =

2G M c2

(2.11)

where G is the gravitational constant, and c is the speed of light. For most cases, the ratio of rs /r is very small. In case of the Earth rs = 8.9 mm. The Sun has 330 000 times the mass of the Earth, hence its Schwarzschild radius is about 3 km. On the surface of Earth, corrections to the Newtonian gravity are only one part in a billion. The ratio rs /r becomes large however, in the proximity of ultra dense objects such as black holes or neutron stars. When neglecting relativistic effects to our equations of motion in the solar system still other effects are to be taken into account such as asymmetries in the gravitational fields of the dominant forcing bodies or perturbations by other bodies. A very well known fact is the precession of Mercury’s periapse due to general relativity theory

2.2 Orbital Motions

45

Fig. 2.5 Radiation pressure acting on a particle

though this relativistic effect is only 1/10 times as important as effects by planetary perturbations. However, if the solar system is a chaotic system even the smallest perturbations can lead to completely different results, deterministic chaos. Small particles are affected by radiation forces. This is illustrated in Fig. 2.5.

2.2.4 Non Gravitational Forces and Chaos Let us discuss the effect of radiation pressure on a moving particle. The equation of motion for a particle of mass m and geometrical cross section A, moving with velocity v through a radiation field of energy flux density S is found to be: m v˙ =

SA Q pr c



r˙ ˆ v 1− S− c c

(2.12)

where Sˆ is a unit vector in the direction of the incident radiation, r˙ is the particle’s radial velocity, and c is the speed of light. The radiation pressure factor Q pr = Q abs + Q sca (1− < cos α >), Q abs is the efficiency factor for absorption, and Q sca is the efficiency factor for scattering. This formula was derived in the paper by [1]. Charged small particles are affected by planetary magnetic fields. The trajectories of comets are influenced by the asymmetric loss of volatile material from their surface. Therefore it is impossible to predict their exact trajectories. Non gravitational forces in this case can e.g. be described according to the paper given by [2]: Fi = G i f (r ) The i indices denote:

i = 1, 2, 3

(2.13)

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2 Chaos in the Solar System

• i = 1 radial component (positive outward from the Sun) • i = 2 transverse component (in the orbit plane and positive toward the direction 90◦ ahead of the comet in true anomaly) • i = 3 component perpendicular to the orbit plane. G i is written as G i = Ai exp(−Bi τ )

(2.14)

Ai , and Bi are constants, and τ is the time from some initial epoch. Generally A3 = B3 = 0. In addition, f (r ) = r −3 exp(−r 2 /2)

(2.15)

where r is measured in AU. The variation of forces as a function of heliocentric distance can be calculated. Let us consider comets. The vaporization rate of water snow can be modeled, which influences the trajectories of comets. Another effect that changes orbits is collision. Such impacts are rare except in planetary ring systems. They are also important for moving asteroids and Kuiper belt objects. The Kuiper belt contains many asteroid or dwarf planet sized objects exterior to the orbit of Neptune. Gas drag effects encounter planetary atmospheres, but collisions and gas drag effects were more important during the formation of the planets. How can we obtain analytical results from such problems? The mutual forces between the objects can be expanded into series. The terms of the series contain increasing powers of mass, inclination, and eccentricities. The perturbation expansions are usually done in powers of small parameters. However, in the case of resonances, small divisors are introduced. These small divisors make some high order terms in the power series unexpectedly large. Therefore, the series diverge and have validity only over small time spans. What defines a stable system? We assume that in a stable system, from the astrophysical point of view, no ejections and no mergers between two objects will occur. The result should be robust to sufficiently small perturbations. Chaotic motions mean that for some initial conditions, numerical experiments show that the trajectories are regular (quasiperiodic) while for other initial conditions the trajectories are chaotic and not confined in their motion. The evolution of a chaotic systems strongly depends on the system’s precise initial conditions. Let us consider two trajectories that begin arbitrarily close in phase space: • regular region: these two trajectories diverge from one another as a power (mostly linear) of the elapsed time. • chaotic region: these two trajectories diverge exponentially in time. As we have seen in the previous chapter if d(t) is the distance between two particles having an initially small separation d(0) one finds that

2.2 Orbital Motions

47

• for regular orbits: d(t) − d(0) grows as a power of time (usually linearly), and • for chaotic orbits: d(t) ≈ d(0)eδt

(2.16)

where δ is the Lyapunov characteristic exponent (in the limit of infinitesimal initial separation, t → ∞) and δ −1 is the Lyapunov time scale. Note that this criterion is misleading if bodies are ejected out of the system. In such a case there is no exponential growth but of course it is not a stable system. The Lyapunov time tells us that even a perturbation as small as 10−10 in the initial conditions will result in a 100% discrepancy in ∼25 Lyapunov times. Short Lyapunov times are indicative of large scale chaos. Solar system simulation shows that the time scale for large changes in the principal orbital elements (semi major axis, eccentricity, inclination) is often orders of magnitudes larger than the Lyapunov time scale.

2.2.5 Chaotic Resonances in the Solar System Since the establishment of Newtonian Gravity and its application to calculate the orbits of solar system objects, it was believed that the solar system is a stable system after an initial phase of formation. The number of objects seemed to be fixed. No object should escape the solar system or collide with another object. The discovery of chaos destroyed this picture. A review on chaotic behavior of orbits and rotations of solar system bodies was given in the paper [3]. Let us consider asteroids in the Kirkwood belts (see Fig. 2.6), which occur between the orbits of Mars and Jupiter. Daniel Kirkwood was the first who identified them. He noted that on some locations the orbital period T which depends on the semimajor axis would be of the form T =

p TJ q

(2.17)

where T is the orbital period of the object, TJ is the orbital period of Jupiter, and p and q are integers. In terms of orbital frequencies n = 2π T,

qn J − pn ∼ 0

(2.18)

This is called a resonance. How can such a resonance produce a gap? Jupiter with a mass of μ J ∼ 0.001M 2 exerts a gravitational pull on the objects in the Kuiper belt. The resonance force is zero if the orbits are circular. If the orbits are non-circular, having an eccentricity e and e J is the eccentricity for the orbit of Jupiter, the force is proportional to 2M



= 2 × 1030 kg, the mass of the Sun.

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2 Chaos in the Solar System

Fig. 2.6 Histogram showing the four most prominent Kirkwood gaps and a possible division into inner, middle and outer main-belt asteroids: blue: inner main-belt (a < 2.5 AU); orange: intermediate main-belt (2.5 AU< a < 2.82 AU); green: outer main-belt (a > 2.82 AU)

e| p−q| − se2J

(2.19)

where s is an integer between zero and | p − q|. For a 3:1 resonance p = 3 and q = 1. In that case three force terms are possible [4]: 1. e2 , 2. ee J , and 3. e2J . Assuming e ∼ e J ∼ 0.05, the force exerted on the asteroid by resonance is smaller by a factor of 5 × 10−5 than the force exerted by the Sun.3 So the resonant forces are small. The numerical integration of Wisdom showed that the eccentricity of small bodies placed in the 3:1 gap alternates chaotically between period of low and moderate eccentricity as the result of perturbation from Jupiter. The chaotic motion is very sensitive to initial conditions. The Lyapunov time can be estimated when an asteroid is placed artificially in the 3:1 mean motion resonance. Large scale chaos appears when the libration and precession periods of a resonant asteroid are similar. The Lyapunov time must be equal to the libration period and is some 1.4 × 104 years. It can be shown that the changes of e are of the order of: δe2 ∼

TL M J 2 e T J M

where TL is the Lyapunov time. This is valid for a 3:1 resonance. 3 In

the best possible case of a first order resonance | p − q| = 1.

(2.20)

2.2 Orbital Motions

49

The asteroid Helga is in a 12:7 resonance. The motion is chaotic with a very short Lyapunov time. The diffusion time is comparable or even larger than the age of the solar system. Therefore, Helga may stay in the solar system for about 8 billion years [5]. In the 1:1 resonance there are the Trojan asteroids. Also most meteorites come from the 3:1 resonance or the so-called v6 secular resonance. This is a resonance between the precession frequency of the apsidal line of the asteroid and the sixth fundamental secular frequency of the solar system. The sixth fundamental frequency of the solar system is roughly Saturn’s precession frequency. Cosmic ray exposure of meteorites are typically 20 Myr. The delivery time from a 3:1 resonance is however only 1 Myr as noted before. However, one can assume that most meteorites that are produced by collision between larger bodies in the asteroid belt are not injected directly into either 3:1 or v6 resonance. They are placed in the vicinity of the resonance. So there is a slow dynamical precursor that explains the discrepancies found between the cosmic exposure age and the 3:1 delivery time. Dynamical lifetimes of objects injected into the asteroid belt are studied by [6].

2.2.6 Chaotic Spin-Orbit Resonances In the case of a not completely spherical planet, the asphericity couples to the nonaxisymmetric perturbation produced by orbital eccentricity or inclination. A prominent example is the Moon. In most cases the final result of such perturbations is that the satellites of planets are in 1:1 spin-orbit resonance. Thus the motion is regular. Saturn’s satellite Hyperion shows a chaotic motion [7]. Another prominent example is the variation of the angle between the spin and orbital axes of Mars. This angle varies by approximately ±14◦ around its average which is about 24◦ over millions of years. However, later calculations showed that this variation can even amount to ±20◦ around the present day value The obliquity of Mars seem to evolve chaotically, which has profound implications for climate variation. Most probably the spin axis of Mercury and Venus underwent chaotic variations in the past [8]. Mercury and Venus have been stabilized by tidal dissipation, and the earth may have been stabilized by the moon.

2.2.7 Three-Body Resonances A substantial fraction of main belt asteroids have chaotic orbits with Lyapunov times ≈105 years. The interacting three bodies are: • Inner asteroid belt object: Jupiter and Mars, asteroid. • Outer asteroid belt object: Jupiter and Saturn, asteroid.

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2 Chaos in the Solar System

Three body resonances are proportional to the product of the masses of the two perturbing bodies, for example MJupiter , MMars .

2.2.8 Chaos Among the Giant Planets In 1989 Laskar [9] performed long time integrations of the solar system. The orbits of the planets were averaged, and Pluto was ignored. The result of these studies was that the solar system is chaotic. Laskar suggested that secular resonances were responsible for the chaotic motion. Integrations performed by Sussman and Wisdom [10] confirmed that the full solar system was chaotic with Lyapunov time TL ∼ 5 Myr. The three planets Jupiter, Saturn and Uranus show a three-body resonance. The orbital period of Uranus is nearly seven times that of Jupiter. The orbital frequency n = 2π/T of the three planets is n J (Jupiter), n S (Saturn) and n U (Uranus): n J − 7n U = 5n S − 2n J

(2.21)

The resonant terms in the potential experienced by Uranus are proportional to M J MS 2 − 5n S /n J This appears as a small denominator. The predicted Lyapunov time is TL ∼ 107 years. One can estimate the diffusion time is 1018 years—so the amount of time before Uranus is ejected is larger than the age of the universe. However, detailed studies showed the presence of other chaotic zones near the present orbit of Uranus. A review on the origin of chaos in the outer solar system was given in the paper [11].

2.2.9 Chaos Among the Terrestrial Planets The integration of the full solar system (including also the inner planets) shows evidence of chaos with TL ∼ 5 × 106 years. However, this is not fully understood. Laskar explained it with secular resonances. Several resonances are suggested: σ1 = (ω¯ 1 − ω¯ 5 ) − (1 − 2 )

(2.22)

σ2 = 2(ω¯ 4 − ω¯ 3 ) − (4 − 3 ) σ3 = (ω¯ 4 − ω¯ 3 ) − (4 − 3 )

(2.23) (2.24)

The indices refer to the planets (1 for Mercury, 2 for Venus,…) and ω¯ is the orientation of a planet’s apsidal line. This is the line from the sun to the point of the planet’s orbit

2.2 Orbital Motions

51

closest to the sun.  refers to the orientation of the planet’s nodal line. The nodal line is defined by the intersection of the orbital plane of the planet with the orbital plane of the Earth.

References 1. Burns JA, Lamy PL, Soter S (1979) Radiation forces on small particles in the solar system. 40:1–48 2. Marsden BG, Sekanina Z, Yeomans DK (1973) Comets and nongravitational forces. 78:211 3. Murray N, Holman M (2001) The role of chaotic resonances in the solar system. Nature 410:773–779 4. Wisdom J (1983) Chaotic behavior and the origin of the 3/1 Kirkwood gap. 56:51–74 5. Holman MJ, Murray NW (1996) Chaos in high-order mean resonances in the outer Asteroid belt. 112:1278 6. Gladman BJ, Migliorini F, Morbidelli A, Zappala V, Michel P, Cellino A, Froeschle C, Levison HF, Bailey M, Duncan M (1997) Dynamical lifetimes of objects injected into asteroid belt resonances. Science 277:197–201 7. Wisdom J, Peale SJ, Mignard F (1984) The chaotic rotation of hyperion. 58:137–152 8. Laskar J, Robutel P (1993) The chaotic obliquity of the planets. Nature 361(6413):608–612 9. Laskar J (1989) A numerical experiment on the chaotic behaviour of the solar system. Nature 338:237 10. Sussman GJ, Wisdom J (1992) Chaotic evolution of the solar system. Science 257:56–62 11. Murray N, Holman M (1999) The origin of chaos in the outer solar system. Science 283:1877

Chapter 3

The Sun: An Active Star

The Sun is the star closest to us and the only star where details can be directly observed. By comparison of the parameters describing the Sun with other stars, we understand the physics of stars in more detail and by comparing e.g. our present Sun with stars that are younger or older than the Sun, we can learn about the evolution of the Sun, its formation, and its ultimate fate. The Sun shows an activity cycle and violent solar outbursts, such as flares or Coronal mass ejections, can have a strong influence on the Earth and the space around Earth. Before applying methods of non linear dynamics to describe solar activity we give an overview of these phenomena and describe the structure of the Sun.

3.1 The Structure of the Sun In this section we give a short overview on the structure and basic physics of the Sun. We discuss only the basic features here, for more details the reader is encouraged to read the many textbooks available.

3.1.1 The Solar Interior The interior structure of the Sun can be modeled by a set of simple differential equations. The first equation is the hydrostatic equilibrium equation. For a star to be stable, at every volume element inside it, the inward acting gravitational force given by −gdm (g is the gravitational acceleration, dm a mass element) must be balanced by the net outwards directed pressure force (given by the gas pressure Fg = d Pd A, where A denotes an area element). Therefore: © Springer Nature Singapore Pte Ltd. 2020 A. Hanslmeier, The Chaotic Solar Cycle, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-15-9821-0_3

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3 The Sun: An Active Star

d F = d Pd A = −gdm

(3.1)

The mass element dm can be written as dm = ρd V = ρ4πr 2 dr and from that the equation of hydrostatic equilibrium becomes: dP GM = −gρ = −ρ 2 dr r

(3.2)

where M is the mass of the star. The second equation for stellar structure is trivial: dm(r ) = 4πr 2 ρdr dr

(3.3)

The next equation takes into account the nuclear energy production rate given by  that contributes to the luminosity L: d Lr = 4πr 2 ρ dr

(3.4)

We need one further equation to complete the basic set: energy transport. In the case of radiative energy transport which is appropriate for the interior of solar-type stars, there is a temperature gradient given by: 3κρ L r dT =− dr 64πr 2 σ T 3

(3.5)

where κ is the opacity of the matter. In the solar core, which occupies about 1/3 of the solar radius, the energy is produced by thermonuclear reaction, namely by the fusion of 4 protons into a helium nucleus, also called pp-process: 4 p →4 He

(3.6)

Two protons are converted into neutrons and the resulting He nucleus consisting of 2 neutrons and 2 protons has less mass than the 4 protons that created it. The mass difference m is only about 0.7% and is converted into energy according to E = mc2

(3.7)

This process is sensitive to temperature: pp ∼ T 4...5

(3.8)

Stars with masses larger than the Sun have thermonuclear fusion processes that are even stronger dependent on the temperature. Thus slight variations of temperature

3.1 The Structure of the Sun

55

lead to strong changes in energy production, and the stellar core becomes convectively unstable. The interior structure of the Sun is given by: 1. The innermost zone is the solar core, were energy is produced by thermonuclear reactions. This zone extends to about 0.3 solar radii (R ). 2. In the layer between 0.3 R and 0.6 R the energy is transported outwards by radiation. The high energy gamma-photons are scattered many times in this radiative zone. 3. The outer layers up to the solar surface are convective. The energy is transported by convective motions, hot gas elements rise upwards, cool and move downwards, are heated again and move upwards. This zone is called the convection zone. In the convection zone the temperature gradient becomes:   1 T dP dT = 1− dr γ P dr

(3.9)

where γ is the adiabatic index given by ratio of the specific heats in the gas. For a fully ionized ideal gas γ = 5/3. Between the radiative zone and the convection zone there exists a small transition layer that is called the tachocline.

3.1.2 The Solar Atmosphere The deepest layer visible is the solar photosphere. The temperature is between 4500 and 6000 K, and the density varies between 10−3 and 10−6 kg/m3 . The solar photosphere shows a grainy structure, and this phenomenon is called granulation. In the bright granules hot plasma rises, cools down and sinks back to the hotter regions in the dark intergranular lanes. The thickness of the photosphere is about 400 km. Above the photosphere there is the chromosphere, where the temperature increases from about 4000 K to several 104 K. The chromosphere has a density of about 10−8 times that of the atmosphere of Earth at sea level. Because of its low density it is normally invisible but it can be observed in specific circumstances (Fig. 3.1): • during a total solar eclipse (see Fig. 3.2). • in several spectral lines that are formed in the chromosphere like the Ca H- and K-lines, or the H-α line. The solar corona is the outmost atmospheric layer of the Sun extended over several solar radii (Fig. 3.2). The temperature increases sharply at the transition region between the chromosphere and the corona. In the corona the temperature exceeds several 106 K. The K-corona exhibits a continuum in the spectrum. Because of the high temperature, the free electrons are in larger thermal motions and the spectrum that is reflected and scattered is smeared and no spectral line features can be observed.

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3 The Sun: An Active Star

Fig. 3.1 The interior structure of the Sun: core, radiative zone and convective zone. Above the very thin photosphere there is the chromosphere and the corona. SOHO/NASA/ESA

In the F-Korona, we observe the Fraunhofer spectrum containing spectral lines scattered by dust particles farther away from the Sun and in the E-corona one observes emission lines; these lines are produced by highly ionized elements (e.g. FeXIV).1

3.2 Solar Activity—An Overview In this section we give an overview on different solar activity phenomena. The reader who wants to go into more detail is referred to the many textbooks and review articles on this topic.

1 This

means that Fe has lost 13 electrons.

3.2 Solar Activity—An Overview

57

Fig. 3.2 The solar chromosphere with prominences during the total solar eclipse in 1999. Above the thin chromosphere the white light corona which extends up to several solar radii, is seen. NASA

3.2.1 Sunspots Large sunspots can be seen on the solar disk with the naked eye when the Sun is for example just above the horizon. The earliest sunspot observations date back to the ancient Chinese. The Greek scholar Theophrastus described sunspot observations around 300 BC. He was a student of Plato and Aristotle. On March 17 807 AD the monk Adelmus observed a large sunspot that was visible for eight days. Adelmus interpreted his observations as a transit of Mercury. A large sunspot was observed at the time of Charlemagne’s death in AD 813. In 1129 sunspot activity was described by John of Worcester and Averroes. However, they misinterpreted their observation like Theophrastus and others as planetary transits. An example of typical sunspot drawings from that epoch is seen in Fig. 3.3. Sunspots were first observed telescopically by Harriot and Fabricius. Fabricius used a camera obscura. G. Galilei showed sunspots to his colleagues in Rome in the same year that Ch. Scheiner invented the helioscope to observe them. In 1613 G. Galilei wrote the Letters on Sunspots, in which he showed that sunspots were surface features on the Sun and not transiting planets (Fig. 3.4).

58

Fig. 3.3 Sunspot in the drawings of John of Worcester Fig. 3.4 With his helioscope, Ch. Scheiner could observe the solar disk being projected on a screen behind a telescope. Scheiner, Rosa Ursina

3 The Sun: An Active Star

3.2 Solar Activity—An Overview

59

Sunspots may last from a few days to even a few months. They appear usually in groups. The diameters range from less than 1,000 km to more than 50,000 km.2 Why are sunspots dark? Stars radiate like a black body. The black body radiation can be described by Planck’s law. If Bν (T ) is the spectral radiance (power per unit solid angle and per unit area normal to the propagation), and ν is the frequency of radiation at thermal equilibrium at temperature T then: Bν (T ) =

1 2hν 3 c2 ehν/kT − 1

(3.10)

where h = 6.626 × 10−34 Js is the Planck constant, c is the speed of light in a vacuum, and k = 1.38 × 10−23 J/K. By integrating Planck’s law over all wavelengths we obtain the Stefan–Boltzmann law, which gives the total energy radiated per unit surface area of a black body across all wavelengths per unit time: E = σ Te4

(3.11)

σ = 5.67 × 10−8 Wm−2 K −4 is the Stefan–Boltzmann constant. The luminosity L of an object also depends on its surface area: L = 4π R 4 σ Te4

(3.12)

where Te is the effective temperature. The luminosity of the Sun L  is given by 2 σ T4 L  = 4π R

(3.13)

The distance between the earth and the Sun is a0 . The solar irradiance we receive on Earth (power per unit area) is: E Earth =

L 4πa02

(3.14)

Sunspots appear dark since they emit less energy, therefore they must be cooler than the surrounding solar plasma. The sunspots consist of a dark umbra which is often surrounded by a filamentary penumbra (Fig. 3.5). The temperature of the photosphere is about 5780 K, the umbra is 1000–1900 K cooler than the quiet Sun and the penumbra is 250–400 K cooler. Why are sunspots cooler than the surrounding photosphere? It was detected that certain spectral lines can get split into two or three components when the spectrograph slit is laid across a sunspot. The splitting of spectral lines due to strong magnetic fields was detected 1896 by P. Zeeman. The splitting of spectral lines in the presence of a magnetic field is analogous to the splitting into several components in the presence 2 Can

be seen without a telescope.

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3 The Sun: An Active Star

Fig. 3.5 A big sunspot group showing the dark umbra that is surrounded by a filamentary penumbra. NASA/JAXA

of an electric field (Stark effect). The splitting λ is given by λ = λ2 H g

(3.15)

where H denotes the magnetic field strength, and g is a factor that depends on the energy levels of the atoms involved. G. E. Hale was the first who discovered the Zeeman effect in the solar spectra. The magnetic field strength in the photosphere averaged over a sunspot is approximately 1000–1500 G. It varies from 1800–3700 G in the darkest part of the umbra to 700–1000 G at the outer edge of the penumbra. The field fans out very rapidly. In sunspots one sees the Evershed effect, a horizontal outflow in the photospheric layers of penumbrae. In the chromosphere and transition region it reverses into an inflow region.3 The many phenomena that can be observed within sunspots are e.g. summarized in the review [3] (Fig. 3.6).

3.2.2 Faculae Whereas sunspots are the most prominent features seen in the solar photosphere, there are also other features like faculae. They are associated with sunspots and are 3 The

two layers are above the photosphere.

3.2 Solar Activity—An Overview

61

Fig. 3.6 The entrance slit of a spectrograph is laid across a sunspot. Note the splitting of spectral lines in this area due to the sunspot’s strong magnetic field. NASA/JAXA

seen in white light when the active region appears near the solar limb. Therefore, they are also associated with larger heights in the solar photosphere. When looking at the solar disk in white light, one sees a center to limb variation. The Sun appears to be brighter near the disk center than closer to the limb. The explanation for this center to limb variation which is wavelength dependent, is simple. • Near the solar disk center we see into deeper layers. Temperature increases with depth and so these layers become brighter. • Near the solar limb the optical depth unity is reached at larger geometrical heights. Since the temperature in the photosphere decreases up to a height of approximately 500 km (which is called the temperature minimum) the limb appears darker because less radiation is emitted. Faculae can also be seen very well in the Ca H- and Ca K -lines. Bright faculae observed in these spectral lines are also called Ca plages. The wavelength of these spectral lines are 393.3 nm and 396.8 nm. They were first observed in 1814 by J. Fraunhofer and designated as K and H lines. They are very useful to probe the layer above the solar photosphere, the chromosphere, since their line cores are completely formed there. Another important spectral line to study the chromosphere is the H-α line of hydrogen. This line has a central wavelength of 656.3 nm and it is produced by a transition of the electron in the hydrogen atom from the level n = 2 → n = 3 (Absorption) or n = 3 → n = 2 (Emission) (Fig. 3.7).

3.2.3 Flares and CMEs Flares and CMEs (coronal mass ejections) are phenomena that occur in the chromosphere and corona (especially CMEs). In both cases an enormous amount of energy is released by magnetic reconnection.

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Fig. 3.7 Images of the Sun in different wavelengths: left: H-α, middle: Ca K line, right: white light. NASA/SOHO Table 3.1 X-ray classification of flares Class A B C M X

Flux between 0.1 and 0.8 nm 10−4

Solar flares were first observed by Carrington in 1859. He observed a so called white light flare, a brightening in the photosphere in a strongly active region. Normally flares are related to processes in the chromosphere and can be observed, for example in the spectral line of Hα. Extremely strong flares reach down to the photosphere. Particles are accelerated (mainly electrons), and these interact with the plasma. The acceleration of these particles mainly occurs because of magnetic reconnection. We will briefly discuss these processes. Flares are classified according to the peak flux range at 0.1–0.8 nm (W/m2 ) (see Table 3.1). An X2 flare has a peak flux range of 2 × 10−4 W/m2 in the range 0.1–0.8 nm. Solar flares produce streams of highly energetic particles in the solar wind and short wavelength radiation. Massive solar flares are accompanied by coronal mass ejections, CMEs. The soft X-ray flux of X-class flares increases the ionization of the upper Earth’s atmosphere. Because of heating, the atmosphere expands and the drag on low orbiting satellites is increased. Therefore, their orbits decay. The energetic particles also produce aurorae. Hard X-rays can damage spacecraft electronics. Energetic protons (called SEP, solar energetic protons) can pass through the human body. Therefore the radiation risks posed by solar flares are a major concern of manned space missions, especially of long duration manned space missions (e.g. mission to Mars). Flares produce bremsstrahlung in X-rays and synchrotron radiation in radio. In the visible part of the spectrum they are normally not seen. In Fig. 3.8 simultaneous

3.2 Solar Activity—An Overview

63

Fig. 3.8 Flares observed at different wavelength (given in Angstrom, 1 A = 0.1 nm). Observing the Sun at different wavelengths provides a tomography of the outer solar atmosphere. All wavelength are in the UV. The names of the instruments used to obtain these images are also given. NASA/GOES

images of flares at different UV-wavelengths are shown. These images were taken with satellites since the UV is absorbed by the Earth’s atmosphere. The energy release during a flare is on the order of 1027 erg/s. Large flares can emit up to 1032 erg/s. Note that the latter value corresponds to only 1/10 of the total energy emitted by the Sun every second. In flares, the temperature reaches 10–20 million K, and for the most intense flares this can go up to 100 million K. In order to understand the energy release in flares and CMEs, we have to discuss magnetic reconnection. Magnetic reconnection occurs in highly conducting plasmas. Magnetic energy is converted to kinetic energy, thermal energy and particle acceleration. Typical timescales involved in magnetic reconnection are between the slow resistive diffusion of the magnetic field and fast Alfvénic timescales. Let us consider the Maxwell equation: ∇ × B = μJ + μ

∂E ∂t

(3.16)

There is a resistivity of the current layer. Magnetic flux from either side of the current layer can diffuse through it. Thus, flux from the other side of the boundary

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3 The Sun: An Active Star

is cancelled. The plasma is pulled out by the magnetic tension along the direction of the magnetic field lines. A drop of pressure results and more plasma is pulled which leads to more magnetic flux. Therefore, this process becomes enhanced and self-sustaining. However, this simple model of magnetic reconnection cannot explain the timescales at which solar flares are observed. The flare timescales are 13–14 orders of magnitude faster than from the theory predicted above. One explanation for this discrepancy can come from electromagnetic turbulence in the boundary layer. Electrons are scattered and the plasma’s local resistivity is enhanced. Therefore, magnetic flux can diffuse much more quickly. For slow reconnection the Sweet–Parker model can be used. Sweet demonstrated in 1956 that by pushing together two plasmas with oppositely oriented magnetic fields, resistive diffusion occurs. Let us assume the characteristic inflow velocity; E y is the out-of-plane electric field, and Bin is the upstream magnetic field strength. The initial velocity is ∝ E × B and E y = vin Bin

(3.17)

Neglecting the displacement current in Ampere’s law we can write: J=

∇ ×B μ0

(3.18)

Let δ be the current sheet half-thickness. We can write: Jy ∼

Bin μ0 δ

(3.19)

Note that this relation reverses over a distance ∼2δ. Using Ohm’s law we can write 1 J σ

(3.20)

Ey 1 η ∼ Bin μ0 σ δ δ

(3.21)

E= Then the velocity vin becomes: vin =

where η is the magnetic diffusivity. When the inflow intensity is comparable to the outflow density: vin L ∼ vout δ

(3.22)

This is because of mass conservation. L is the half length of the current sheet, and vout is the outflow velocity. The upstream pressure is given by

3.2 Solar Activity—An Overview

65 2 ρvout 2

(3.23)

Bin2 2μ0

(3.24)

The magnetic pressure is given by

ρ is the mass density of the plasma. The outflow velocity becomes: Bin = vA vout ∼ √ μo ρ

(3.25)

where v A is the Alfvén velocity. From these equation we can write the dimensionless reconnection rate R: R=

δ vin η ∼ ∼ vout vAδ L

(3.26)

The Lundquist number S is given by: S=

Lv A η

(3.27)

The reconnection rate R becomes:  R∼

1 η = 1/2 vA L S

(3.28)

This Sweet–Parker reconnection model allows for reconnection rates faster than global diffusion. However, the reconnection rates observed in solar flares, the Earth’s magnetosphere etc. are still much faster and several factors are neglected such as three dimensional effects, time-dependent effects, viscosity, compressibility and downstream pressure. The fast reconnection is simulated by the Petschek model proposed in 1964 by Harry Petschek. Inflow and outflow regions are separated by stationary slow mode shocks. The aspect ratio of the diffusion region is then of order unity, and the maximum reconnection rate becomes: π vin ∼ vA 8 ln S

(3.29)

This allows for fast reconnection. The two models are sketched in Fig. 3.9. Figure 3.10 summarizes the detailed picture about flares. As it was mentioned already powerful solar flares are often associated with coronal mass ejections. CMEs

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3 The Sun: An Active Star

Fig. 3.9 The two basic fluid models of reconnection. Left: Sweet–Parker’s long thin current sheet model with allow plasma flow across the sheet. Center: Petschek’s small (unspecified) diffusion region/slow shock model. The proposed slow shocks are the four separatrices emanating from the corners of the diffusion region of lengths l ∗ in this figure. Right: The semianalytical solutions of Petschek’s and Sonnerup’s extension of Petschek’s model which includes another discontinuity (data from Vasyliunas, courtesy American Geophysical Union)

Fig. 3.10 Particle acceleration in a solar flare and generation of hard X-rays in a solar flare during reconnection. The hard X-ray emission is from an accelerated electron, and the soft X-ray emission from plasma heating. Credit: https://hesperia.gsfc.nasa/

3.2 Solar Activity—An Overview

67

Fig. 3.11 A coronal mass ejection. NASA/ESA SOHO satellite

throw a huge amount of material out from the Sun. Solar flares normally occur in three stages: • precursor stage: release of magnetic energy is triggered; soft x-ray emission. • impulsive stage: protons and electrons are accelerated to energies exceeding 1 MeV. Radio waves, hard X-rays and gamma rays are emitted. • decay stage: decay of soft X-rays. These stages may last from a few seconds to 1 h. A typical coronal mass ejection is shown in Fig. 3.11 taken from a satellite. To enhance the contrast, the solar disk was obscured artificially. CMEs can be directed towards Earth and reach it as an interplanetary CME. This can be regarded as a shock wave of traveling mass. When it collides with the Earth’s magnetosphere it compresses it on the day side and the magnetosphere extends on the night side as a long tail. The typical velocities of CMEs range from 20 to 3200 km/s, and the average speed is 489 km/s. Therefore, the transit times from Sun to the Earth ranges from 13 h to more than 50 days. The average mass ejected is 1.6 × 1012 kg.

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3 The Sun: An Active Star

Fig. 3.12 The interaction between solar wind and Earth’s magnetosphere. Credit: NASA, William Crochot

3.2.4 Solar Wind The existence of a plasma flow outward from the Sun, partly in the direction to Earth was first suggested by R. C. Carrington in 1859. He was the first observer of a white light flare and he found that about 12 h after the observation of the solar event, the Earth’s magnetic field became strongly disturbed. Later K. Birkeland suggested that ejected solar ions and electrons could be responsible for aurorae. In the 1950 L. Biermann noted that tails of comets always point away from the Sun. Biermann explained this by assuming a stream of particles that pushes cometary tails away from the Sun. E. Parker showed the solar corona is very hot and is a good heat conductor. Since solar gravity decreases as distance from the Sun increases the outer solar corona could escape into interstellar space. In January 1959 the Soviet spacecraft Luna 1 first directly measured the solar wind. Solar wind is observed in two states: • Slow solar wind originates from a region around the Sun’s equatorial belt, also known as streamer belt. Near Earth space: velocity 300–500 km/s, temperature 1.4−1.6 × 106 K.

3.2 Solar Activity—An Overview

69

• Fast solar wind originates from coronal holes; these are funnel-like regions of open magnetic field lines where particles can escape. Near Earth: v = 750 km/s, T ∼ 8 × 105 K. The ram pressure exerted by the solar wind is given by: P = m p nv2 = 1.67 × 10−6 nv2

(3.30)

where m p is the proton mass, P is the pressure in nPa and n is the density in particles/cm3 . The speed is given in km/s. The solar wind intersects with the magnetic field of the planets (this can be clearly seen in the case of Earth, Jupiter and Saturn). The particles are deflected by the Lorentz force F: F = qE + qv × B

(3.31)

Therefore, the particles are forced to travel around the planet and do not arrive at the surface. This is illustrated in Fig. 3.12. Due to solar wind and CMEs the Sun loses mass, the total mass loss each year is about (2 − 3) × 10−14 M which corresponds to (1.3−1.9) × 106 t/s. Since its formation, the Sun has lost about 0.01% of its initial mass.

Fig. 3.13 Voyager 1 and 2 speed and distance from Sun. HORIZONS System, JPL, NASA

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3 The Sun: An Active Star

3.2.5 The Heliosphere The solar magnetic field and the solar wind give rise to a bubble that engulfs the whole planetary system. We can define the heliosphere as the extension of this bubble where the solar influence is larger than the interstellar pressure. Up to now five spacecraft have provided data from the outer borders of this region: Pioneer 10 (data up to 67 AU), Pioneer 11 (data up to 44 AU), Voyager 1, 2 (in 2020 more than 120 AU) and New Horizons. Voyager 1 and Voyager 2 passed the termination shock as well as the Heliopause (see Fig. 3.13). The termination shock is the point in the heliosphere where the solar wind slows down to subsonic speed because of interactions with the local interstellar medium. This causes two effects: • compression of the magnetic field • change of the magnetic field. The distance of the termination shock is in the range of 75–90 AU from the Sun. Voyager 1 crossed it in 2004, Voyager 2 in 2007. Beyond the termination shock there is the heliosheath. At the heliopause the solar wind is stopped by the interstellar

Fig. 3.14 The heliosphere provides also a protection of the planetary system against cosmic rays

3.2 Solar Activity—An Overview

71

medium. At this boundary the solar wind pressure equals the interstellar medium pressure, and a sharp drop in temperature can be expected as well as a rapid increase in cosmic rays. In the case of a strong heliosphere fewer cosmic ray particles can penetrate into the planetary system (Fig. 3.14).

3.3 The Solar Activity Cycle Up to now we have discussed the so called quiet Sun phenomena; these phenomena are always present on the Sun. There is always the granulation and a hot corona emitting solar wind. The number of flares, CMEs, and sunspots varies periodically which is called the solar activity cycle.

3.3.1 The Sunspot Cycle The fact that the Sun is periodically variable was detected by S. H. Schwabe in 1843. He wanted to detect transiting planets that he assumed to exist inside the orbit of Mercury. Therefore, Schwabe carefully made drawings of the sunspots and after 17 years of observation he noticed a periodic variation of about 11 years in the number of sunspots. R. Wolf (1816–1893) reconstructed the solar cycles back to 1745. Therefore, the cycle 1755–1766 is numbered cycle 1. Wolf also created the sunspot number index which still serves as a crude measure for the solar activity. If g denotes the number of observed sunspot groups and f is the total number of spots seen on the solar disk, the sunspot number is defined as: R = k(10g + f )

(3.32)

where k is a factor that takes the quality of observations that are influenced by atmospheric seeing (caused by turbulence in the Earth’s atmosphere) and the quality of the optical system used into account. In small telescopes small sunspots may not be detected. R is also called the Wolf number. The sunspot number is determined from several observing stations as an average value to eliminate the above mentioned uncertainties. In 1908 G. E. Hale detected that the magnetic polarity of sunspot pairs shows peculiarities: • The polarity is constant throughout a cycle. • The polarity is opposite across the equator throughout a cycle. • The polarity reverses itself from one cycle to the next. Therefore, the magnetic cycle is 22 years. The Gnevyshev-Ohl rule states that the 11year cycles alternate between higher and lower sums of the Wolf’s sunspot numbers. The value of 11 years is a mean value. Cycles can last from 9 years up to 14 years,

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3 The Sun: An Active Star

Fig. 3.15 Time versus solar latitude diagram of the radial component of the solar magnetic field, averaged over successive solar rotations. The “butterfly” signature of sunspots is clearly visible at low latitudes. Diagram constructed (and regularly updated) by the solar group at NASA Marshall Space Flight Center. D. Hathaway

and the average is 10.7 years. There are also significant amplitude variations between the cycles. Cycle 24 began in December 2008 and ended in May 2020. The activity remained extremely low until 2010. It is the cycle with the lowest recorded sunspot activity since 1750. At the beginning of a new cycle, spots appear at mid latitudes. In the course of the cycle, they move closer to the solar equator until the minimum is reached. This yields the so-called butterfly diagram (Fig. 3.15). In this diagram the magnetic field (radial component) of sunspots is displayed in colors and their position is plotted as a function of time. At the beginning of a new cycle, they clearly appear at higher latitudes and as the cycle evolves they approach the solar equator. Note also the opposite polarities in the northern and southern solar hemisphere. Cycles with larger maximum amplitudes reach their maximum in less time than cycles with smaller amplitudes. Maximum amplitudes are negatively correlated to the lengths of earlier cycles.4 The Schwabe cycle (11 years) is not the only cycle. There seems to exist an 87 year Gleissberg cycle (ranging between 70 and 100 years). Other periods may be the 210 year Suess cycle and the Hallstatt cycle (2,400 years). A 6,000 year cycle may even exist. The determination of cycles of longer periods than several decades becomes extremely difficult because of lack of exact observations of sunspot numbers. There could even exist a 1000 year period [4].

4 Sometimes

called the Waldmeier effect.

3.3 The Solar Activity Cycle

73

3.3.2 Other Effects of Solar Activity The sunspot number can be easily determined using photospheric solar disk observations. However, there are some subjective effects: • bad atmospheric conditions during the observations, called seeing. In the case of bad seeing the image appears blurred and small sunspots (also called pores) might not be detectable. • imperfect telescope optics: the spatial resolution of a telescope depends on the wavelength at which the observations are made and on the diameter of the telescope. As a crude approximation we can state that a telescope of 10 cm diameter (of lens or main mirror) will yield a spatial resolution of about 1 arcsec in the visible light. Such an angle corresponds to about 720 km on the solar surface. If we want to observe structures or details that have an extension of only 350 km we need a 20-cm telescope. However, the atmospheric seeing blurs the image, and only under good seeing conditions can details below 1 arcsec be resolved. To minimize seeing effects, modern solar telescopes are therefore located on mountains and equipped with adaptive optics that correct the distorted wavefronts induced by the turbulent Earth’s atmosphere. It was attempted to introduce other measures of solar activity e.g. the 10.7 cm radio flux. It is a measure of diffuse, nonradiative coronal plasma heating. This radio flux is very well correlated with the sunspot number. However, since we have recorded sunspot numbers since 1750, the Wolf number is still kept as the main indicator for solar activity. In addition to the number of sunspots, the number of flares also varies over a solar activity cycle. In Fig. 3.16 it is shown how the number of flares increases with decreasing energy. This diagram shows that one superflare having an energy above 1035 erg may occur in 1000 years.

Fig. 3.16 Comparison between the occurrence frequency of superflares on G-type stars and those of solar flares. Adapted from Shibata et al. [2]

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3 The Sun: An Active Star

Using the Kepler satellite,5 observations of Sun-like stars, similar to our Sun in their surface temperature (5600–6000 K) and slow rotation (rotational period > 10 d) [1] have revealed the existence of superflares (with energy of 1033 − 1035 erg). From statistical analyses of these superflares, it was found that superflares with energy of 1034 erg occur once in 800 yr, and superflares with 1035 erg occur once in 5000 yr. In [2], it was examined whether superflares could occur on the present Sun.

References 1. Maehara H, Shibayama T, Notsu S, Notsu Y, Nagao T, Kusaba S, Honda S, Nogami D, Shibata K (2012) Superflares on solar-type stars. Nature 485:478–481 2. Shibata K, Isobe H, Hillier A, Choudhuri AR, Maehara H, Ishii TT, Shibayama T, Notsu S, Notsu Y, Nagao T, Honda S, Nogami D (2013) Can superflares occur on our sun? Astron Soc Jpn 65:49 3. Solanki SK (2003) Sunspots: an overview. Astron Astrophy Rev 11:153–286 ˇ 4. Hanslmeier A, Brajša R, Calogovi´ c J, Vršnak B, Ruždjak D, Steinhilber F, MacLeod CL, Ivezi´c Ž, Skoki´c I (2013) The chaotic solar cycle. II. Analysis of cosmogenic 10 Be data. Astron Astrophys 550:A6

5 Launched

2009, end of mission Oct. 2018.

Chapter 4

MHD and the Solar Dynamo

In this chapter we will explain the basic MHD equations which are needed to understand solar active phenomena such as spots, prominences, flares etc. The solar dynamo is needed to maintain the solar activity cycle.1

4.1 Solar Magnetohydrodynamics 4.1.1 Basic Equations To understand solar activity and the solar cycle it is necessary to briefly outline the principles of magnetohydrodynamics, or MHD. The properties of electromagnetic fields are described by Maxwell’s equations: ∂D ∂t ∂B ∇ ×E = − ∂t divB = 0 divD = ρE

∇ ×H = j+

(4.1) (4.2) (4.3) (4.4)

Here H, B, D, E, j, and ρE are the magnetic field, magnetic induction, electric displacement, electric field, electric current density and electric charge density respectively.

1 See

also, for example, Advances in Solar System Magnetohydrodynamics, E.R. Priest, A. W. Hood, Cambridge, 1991. © Springer Nature Singapore Pte Ltd. 2020 A. Hanslmeier, The Chaotic Solar Cycle, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-15-9821-0_4

75

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4 MHD and the Solar Dynamo

Equation 4.1 is Ampère’s law which states that a spatially varying magnetic field (given by ∇ × H) produces currents j; in MHD often the variation of the E fields is → 0. We can also state that a current j induces a magnetic neglected, thus the term ∂D ∂t field that is in a direction opposite to it. Equation 4.2 is the Faraday law: a time varying magnetic field produces an electric field. If μ0 , and 0 are the permeability and permittivity of free space, then for most gaseous media in the universe: B = μ0 H

D = 0 E

(4.5)

The following equation relates the electric current density to the fields producing it (generalized Ohm’s law): j = σ (E + u × B) (4.6) where σ is the electrical conductivity and u is the bulk velocity of the matter. The final equations depend on the state of plasma; if it consists of electrons and one type of ion: ρE = n i Z i e − n e e (4.7) j = n i Z i eui − n e eue n i , ui , n e , and ue are the number density and velocity of the ions and electrons, respectively, and Z i e, −e are the charges of the ion and the electron. In astrophysics two simplifications are applied: – magnetic fields are treated as permanent, – electric fields are regarded as transient. The third Maxwell equation (4.3) states that there are no magnetic monopoles.2 Electric fields can be produced by separating positive and negative charges through the fourth Maxwell equation (4.4). However, the attraction between these charges is so strong that charge separation is usually canceled out very quickly. Through the second Maxwell equation electric fields can be produced by time varying magnetic fields. Such fields are only significant, if there are rapid changes by time varying magnetic fields. Magnetic fields produced by the displacement current ∂D/∂t are usually insignificant in astrophysical problems because electric fields are unimportant; however they can be produced by a conduction current j, if the electrical conductivity is high enough. Such magnetic fields may be slowly variable in time and space. By neglecting ∂D/∂t, and combining the equations ∇ ×H =j we obtain

2 This

∇ × E = −∂B/∂t

B =μ0 H

1 ∂B + ∇ ×∇ ×B=0 ∂t μo σ

j = σ E,

(4.8)

(4.9)

is a common experience: a division of a permanent magnet into two does not separate north and south poles.

4.1 Solar Magnetohydrodynamics

77

and using ∇ × ∇ × B = grad divB − ∇ 2 B and divB = 0: ∂B 1 = ∇2B ∂t μ0 σ

(4.10)

This is also called the induction equation for the static case. In cartesian coordinates this equation for the x coordinate is: 1 ∂ Bx = ∂t μ0 σ



∂ 2 By ∂ 2 Bx ∂ 2 Bz + + ∂x2 ∂ y2 ∂z 2

 (4.11)

The solution of these equations shows that magnetic fields decay together with the current producing them. We can derive an approximate decay time: let us assume that currents vary significantly in distance L, then from (4.10) the decay time becomes τD = μ0 σ L 2

(4.12)

If at time t = 0 there exists a sinusoidal field Bx = B0 exp(iky)

(4.13)

Bx = B0 exp(iky)exp(−k 2 t/μ0 σ )

(4.14)

the solution at a later time t is:

Given the wavelength λ of the spatial variation of the field λ = 2π k, the original field decays by a factor e in the time μ0 σ λ2 /4π 2 . Let us consider typical fields of stars: the dimension of the star L and the electrical conductivity are both high (if the gas is fully ionized). Therefore, the lifetime of a magnetic field could exceed the main sequence lifetime, such a field is called a fossil field. The same is not true for the Earth. Its field is produced by currents in a liquid conducting core and continuously regenerated by a dynamo mechanism. The electrical conductivity of an ionized gas is ∼T 3/2 . That means that the characteristic time for decay of currents in the outer layers of the Sun is much less than the solar lifetime, whereas the decay near the center exceeds the lifetime (since the temperature near the surface is about 6 000 K and near the center about 1.5 × 107 K). If the field in the solar interior were a fossil field extending throughout the Sun, the field in the outer layers would now be current free—similar to the field of a dipole. However we don’t observe this. The surface field is very complex and therefore it must be also regenerated by a dynamo. It is conceivable that a fossil field of the Sun was destroyed during the very early evolution of the Sun, when it was fully convective before reaching the main sequence. Helioseismology argues against a strong field.

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4 MHD and the Solar Dynamo

4.1.2 Some Important MHD Effects A magnetic field in a conducting fluid exerts a force per unit volume which is Fmag = j × B = (curlB × B)/μ0 = −grad(B 2 /2μ0 ) + B · ∇B/μ0

(4.15)

This can be interpreted as: – grad(B 2 /2μ0 ) isotropic pressure, – B · ∇B/μ0 tension along the lines of magnetic induction. The isotropic pressure must be added to the gas pressure: let us assume a magnetic tube, in which Pout denotes the pressure outside and Pin the pressure inside, then an equilibrium between outside and inside the fluxtube requires: Pout = Pin + B 2 /2μ0

(4.16)

The ideal gas pressure can be written as ρT /μ

(4.17)

where  is the gas constant and μ is the mean molecular weight. Assuming that Tin = Tout it follows that: (4.18) ρin < ρout Therefore, we have shown that a magnetic flux tube is lighter than its surroundings and will start to rise. This effect is called magnetic buoyancy. There is a strong interaction between motions of the fluid and magnetic field lines. For the Sun two extremes occur: – photosphere: because of the relatively high density the fluid motions drag the magnetic field lines around (the magnetic field is frozen in); – corona: here the density is extremely low and the magnetic force constrains the motion of the fluid. This can be directly observed by loop like plasma structures, the loops are caused by the magnetic field lines that connect two opposite polarities. The plasma follows the magnetic field lines. The tying of the fluid to the magnetic field lines also permits the propagation of MHD waves, which have some similarity to sound waves but a characteristic speed (Alfvén speed):  (4.19) cH = B 2 /μ0 ρ The sound speed is given by cs =



γ P/ρ

(4.20)

This can also be seen from the induction equation. Let us consider again the Maxwell equations. From j = σ E + u × B we can extract E:

4.1 Solar Magnetohydrodynamics

79

E=

1 (j − u × B) σ

(4.21)

This is substituted into the Maxwell equation (4.2) yielding:  ∇×

 j ∂B −u×B =− σ ∂t

(4.22)

We have already argued that the displacement current can be neglected in the first Maxwell equation and therefore ∇ × B = μj, from which j = 1/μ∇ × B and  ∇×

 ∂B 1 ∇ ×B−u×B =− μσ ∂t

(4.23)

using the formula ∇ × (∇ × A) = ∇(∇A) − ∇ 2 A

(4.24)

From vector analysis, we get:  ∇×

  1  1 ∇ ×B = ∇(∇B) − ∇ 2 B μσ μσ

This gives us the final form of the so called induction equation: ∂B = ∇ × (u × B) + η∇ 2 B ∂t

(4.25)

Here η = 1/μσ is the magnetic diffusivity. The case where the plasma is stationary was already discussed above. Stationary plasma means that u = 0; in that case, the field decays with the ohmic decay time: ∂B = η∇ 2 B ∂t B B =η 2 τ L τ = L 2 /η Let us discuss the case when η = 0. Then, the field B is completely determined by the plasma motions u and the induction equation is the equivalent to the vorticity equation for an inviscid fluid. The magnetic flux  through a material surface S which is a surface that moves with the field, is: (4.26)  = B.dS S

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4 MHD and the Solar Dynamo

If v0 , l0 are typical velocity and length-scale values for our system, then the ratio of the two terms on the right hand side of the induction equation gives the Magnetic Reynolds Number (4.27) Rm = l0 v0 /η0 In an active solar surface region one has η0 = 1 m−2 s−1 , l0 = 700 km ∼ 1 arcsec and v0 = 104 m/s we find Rm = 7 × 109  1. Thus the field is frozen to the plasma and the electric field does not drive the plasma but is simply E = −u × B. However, if the length-scales of the system are reduced the diffusion term η∇ 2 B becomes important. Then the field lines are allowed to diffuse through the plasma and this yields to magnetic braking and changing the global topology of the field (magnetic reconnection).

4.1.3 Fluid Equations The continuity or mass equation for a fluid is: Dρ + ρ∇.u = 0 Dt

(4.28)

and the total time derivative means denoted by D/Dt here: ∂ D = + u.∇ Dt ∂t

(4.29)

(See any textbook on fluid dynamics for a derivation of this formula). Now let us consider the equation of motion in a plasma with velocity u: the momentum equation includes the Lorentz force term j × B and other forces F, such as gravity and viscous forces: Du = −∇ P + j × B + F (4.30) ρ Dt Here P is the plasma pressure. Let us assume a Newtonian fluid with isotropic viscosity, then F may be written as: r F = −ρg(r ) + ρν∇ 2 u r

(4.31)

where g(r ) is the local gravity acting in the radial direction and ν the kinematic viscosity. Let us make things more complicated: Consider a frame of reference with angular velocity at a displacement r from the rotation axis:   Du d 1 2 ρ = −∇ P + j × B + F + ρ 2u × + r × + ∇| × r| Dt dt 2

(4.32)

4.1 Solar Magnetohydrodynamics

81

The three terms in [ ] denote: Coriolis force, change of rotation and centrifugal force. Stars rotate more rapidly when they are young. Under most circumstances the latter two terms are small compared with the Coriolis term u × .

4.1.4 Equation of State The perfect gas law P=

kρT = nkT m

(4.33)

determines the constitution of stars, k = 1.38 × 10−23 J/K the Boltzmann constant, m is the mean particle mass and n is the number of particles per unit volume. If s denotes the entropy per unit mass of the plasma, then the flux of energy (heat) through a star becomes: Ds = −L (4.34) ρT Dt where L is the energy loss function. This function describes the net effect of all the sinks and sources of energy. For MHD applications this becomes: ργ D γ − 1 Dt



P ργ

 = −∇.q + κr ∇ 2 T +

j2 +H σ

(4.35)

In this equation we have: – – – – –

q: heat flux due to conduction κr : coefficient of radiative conductivity T temperature j 2 /σ ohmic dissipation (Joule heating) H all other sources.

4.1.5 Structured Magnetic Fields √ If the plasma velocity is small compared with the sound √ speed ( γ P/ρ), the Alfvén √ speed ( B/μρ) and the gravitational free fall speed ( 2gl), the inertial and viscous terms in Eq. 4.30 may be neglected yielding: 0 = −∇ P + j × B + F

(4.36)

This equation must then be solved using ∇ × B = . . ., ∇.B = 0 from the Maxwell equations and the ideal gas law as well as a simplified form of the energy equation. Let us introduce the concept of scale height. Let

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4 MHD and the Solar Dynamo

0=−

dP − ρg dz

(4.37)

Substitute ρ = Pm/kT (ideal gas) in the above equation and integrate:  P = P0 exp −

z 0

dz H p (z)

 (4.38)

(where P0 is the pressure at z = 0). This defines the local pressure scale height H p : H P = kT /mg = P/ρg

(4.39)

At solar photospheric temperatures (T ∼ 6000 K) we find H p = 0.150 Mm, whereas at coronal temperatures T ∼ 106 K we find H P ∼ 30 Mm. That concept can also be applied to MHD in the case of magnetostatic balance discussed above. Assume that gravity acts along the negative z direction and s measures the distance along the field lines inclined at angle θ to this direction, then the component of Eq. (4.36) in the z-direction becomes: 0=−

dP − ρg cos θ ds

dz = ds cos θ

(4.40)

Therefore, the pressure along a given field line decreases with height, the rate of decrease depends on the temperature structure (given by the energy equation). If the height of a structure is much less than the pressure scale height, gravity may be neglected. The ratio β is given by gas pressure P0 to magnetic pressure B02 /2μ. If β  1, any pressure gradient is dominated by the Lorentz force and (4.36) reduces to: j×B=0 (4.41) In this case the magnetic field is said to be force free. In order to satisfy (4.41) either the current must be parallel to B (Beltrami fields) or j = ∇ × B = 0. In the latter case the field is a current free or potential field. ˆ then If β is not negligible and the field is strictly vertical of the form B = B(x) k, (4.36) becomes:   B2 ∂ P+ (4.42) 0= ∂x 2μ

4.1.6 Potential Fields Potential fields result when B vanishes. We can write B = ∇ × A so that ∇ × B = 0; with ∇ · B = 0 one obtains Laplace’s equation:

4.1 Solar Magnetohydrodynamics

83

∇2A = 0

(4.43)

If the normal field component Bn is imposed on the boundary S of a volume V , then the solution within V is unique. Also if Bn is imposed on the boundary S, then the potential field is the one with the minimum magnetic energy. These two statements have many implications for the dynamics of the solar atmosphere. During a solar flare the normal field component through the photosphere remains unchanged. However, since enormous amounts of energy are released during the eruptive phase, the magnetic configuration cannot be potential. The excess magnetic energy could arise from a sheared force-free field. Let us consider an example of a potential field in two dimensions: Consider the solutions A(x, z) = X (x)Z (z) such that ∇ 2 A = 0 gives: 1 d2 Z 1 d2 X = − = −n 2 X dx2 Z dz 2

(4.44)

where n = const. A solution to (4.44) would be:  A=

B0 n



sin(nx)e−nz

(4.45)

For the field components, this gives: ∂A = B0 cos(nx)e−nz ∂x ∂A = −B0 sin(nx)e−nz Bz = ∂z

Bx =

(4.46) (4.47)

The result is a two dimensional model of a potential arcade.

4.1.7 Charged Particles in Magnetic Fields In this section we consider first the motion of a single charged particle in a given electromagnetic field. The particle has charge q and the equation of motion is: m Let us write:

du = q(E + u × B) dt

(4.48)

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4 MHD and the Solar Dynamo

B0 = B0 b E0 = E b + E⊥ v = v b + v⊥

where b is a unit vector and v × B0 = B0 (v⊥ × b) is perpendicular to b. Equation 4.48 splits into a parallel and a perpendicular component (e.g. parallel means parallel to the magnetic field lines): dv = q E dt dv⊥ = q[E ⊥ + B0 (v⊥ × b)] m dt m

(4.49) (4.50)

Equation 4.49 has the solution v = (q E /m)t + v 0

(4.51)

Here v 0 is the velocity component at t = 0 in the direction of the magnetic field line. We see that particles of opposite sign of charge q move in opposite directions, they move along an electric field parallel to a magnetic field which destroys E . Now let us solve Eq. 4.50 by writing:  + E⊥ × b/B0 v⊥ = v⊥

(4.52)

 dv⊥  × b) = q B0 (v⊥ dt

(4.53)

and Eq. 4.50 becomes: m Summarizing we arrive at:

– Motion in which acceleration is ⊥ to the velocity, – constant acceleration to the velocity. This means a motion frequency |q|B0 /m in a circle around the direction of b. The magnitude of the velocity is v⊥0 ; the radius of the orbit, the gyration radius r g is r g = mv⊥0 /|q|B

(4.54)

For an electron the gyration frequency is 1.8 × 1011 (B/Tesla)Hz. The corresponding gyration radius is 6 × 10−9 (v⊥0 /km/s)(B/Tesla)m. The difference in mass between electrons and protons is about 1800. What follows for the radius and frequency of gyration?

4.1 Solar Magnetohydrodynamics

85

The gyration follows the simple statement, that in the absence of other forces, the Lorentz force balances the centripetal force of the particle’s motion around the field line. Let α be the pitch angle, which is the angle between the direction of motion and the local field line. Then for the gyro radius (Larmor radius):



mcv × B

cp⊥



=

RL = qB q B2

(4.55)

The Larmor radius for a 100 keV electron (which is typical for electrons in the inner radiation belt of the Earth) is about 100 m. Let us assume, that the magnetic flux through a particle’s orbit is constant- this is certainly the case when changes of the magnetic field are small over the gyro radius and one gyro period. From the condition that d B /dt = 0 the so called first adiabatic invariant follows: p2 (4.56) μB = ⊥ 2m B or in terms of the particle’s energy: μB =

E sin2 α B

(4.57)

From the conservation of the first adiabatic invariant it follows, that the pitch angle increases, when the particle moves to larger field strength, until α = 90◦ at the mirror point. Summarizing the motion of a particle: – – – –

accelerated motion along the field lines, circular motion around the field, drift velocity E⊥ × b/B0 perpendicular to both electric and magnetic fields, the sense of the accelerated and the circular motions depends on the sign of the electric charge, – the drift velocity is the same for all particles, – in the absence of electric fields, a particle moves with a constant velocity in the direction of the magnetic field and with a velocity of constant magnitude around the field, thus it moves along a helical path, – In all this discussion we have neglected one important effect. Accelerated charged particles radiate, for non relativistically moving particles this radiation is known as cyclotron radiation and for relativistic particles as synchrotron radiation. Finally, if there is a constant non magnetic force F perpendicular to B, there is a drift velocity: cF × B (4.58) vDF = q B2 Please note that again vDF is charge dependent. Let us give some examples for drift motions:

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– Gradient B drift: the field strength in planetary magnetospheres decreases with increasing distance from the planet and the gradient in the field strength induces a force that can be written as: F = −μ B ∇ B The drift velocity is: vB =

μ B cB × ∇ B q B2

(4.59)

(4.60)

Therefore, the particles move perpendicular to the lines of force and perpendicular to the magnetic gradient. Protons and electrons drift in opposite direction. In the Earth’s magnetic field, electrons drift in the eastward direction. This drift motion causes a current system known as ring current. The ring current strengthens the field on its outside, helping expand the size of the magnetosphere but weakens the magnetic field in its interior. The ring current plasma population is enhanced in a magnetic storm. A southward oriented IMF (interplanetary magnetic field) (Bz negative) leads to reconnection at the side of the magnetosphere to the sun and the magnetopause is pushed closer to the Earth. Also injection of particles from the tail occurs which enhances the number of particles. For a typical 100 keV electron or proton it takes about 5–6 h to complete one orbit drift. – field line curvature drift: particles move along curved field lines. The guiding center follows the curved field line and the resulting centripetal force is equal to: Fc =

mv 2 Rc

n

(4.61)

where n is a unit vector outwards. The drift motion is perpendicular to the field line’s radius of curvature and the field line itself. – gravitational field drift. Let us assume that F is the gravitational force F = mg, then (4.62) vDF = mg × B/q B 2 Thus the drift velocity depends on the mass/charge ratio, the ion drift is much larger than the electron drift; the particles drift in opposite directions, and a current is produced. – electric field drift: here the force is F = qE The drift velocity is vE =

cE × B B2

(4.63)

(4.64)

4.1 Solar Magnetohydrodynamics

87

Thus, charged particles move in a direction perpendicular to (a) E and (b) B. Protons and electrons move in the same direction. We stress here, that the ∇ B and E × B drift and curvature forces dominate the drift motions of particles in a magnetosphere. Let us consider a large assembly of particles; these particles interact which is called collision. If τc is the characteristic time between collisions the collision frequency is νc = 1/τc . If νc is large, the particle motions will be disordered and decoupled from the magnetic field, and the fluid will not be tied to the field. If collisions are relatively rare, not only individual particles but the whole fluid will be tied to the field. The collisions provide the electrical resistivity of matter; in a fully ionized gas a good approximation of the value of the electrical conductivity is: σ = n e e2 τc /m e

(4.65)

where τc ∼ T 3/2 .

4.1.8 MHD Waves The equation that describes the connections between the force exerted by the magnetic field and the fluid motions is ρ

dv = −grad P + j × B + ρ grad φ dt

(4.66)

The forces on the gas are the gas pressure P, the gravitational potential φ and the magnetic force j × B. For a full description of the system we write down two additional equations: (a) equation of continuity (conservation of mass)3 dρ + ρ divv = 0 dt

(4.67)

(b) The relation between P and ρ e.g. in the adiabatic form γ dρ 1 dP = P dt ρ dt

(4.68)

Consider the simplest case: a medium with uniform density ρ0 , and pressure P0 , containing a uniform magnetic field B0 . We ignore the influence of the gravitational field and assume that σ is so large that E + v × B = 0. Now let us assume a perturbation for any variable in the form of: 3 Note that d/dt is the d ∂ dt = ∂t + v.grad.

rate of change with time following a fluid element moving with velocity v:

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4 MHD and the Solar Dynamo

f 1 ∼ expi(k.r − ωt) = expi(k x x + k y y + k z z − ωt)

(4.69)

where k is the wave vector, and ω is the wave frequency. The dispersion relation between ω and k in the absence of a magnetic field is: ω2 = k 2 cs2

(4.70)

Therefore, in that case only one type of waves can propagate—sound waves. The wave propagates through the fluid at the wave speed cs = ω/k, k = |k|, which is called the phase velocity of the wave. If there is a magnetic field, the force j × B couples to the equation and also the Maxwell equations must be taken into account. It is very important to note that the magnetic field introduces a preferred direction into the system. In a uniform medium, sound waves travel equally strongly in all directions from its source, this is not true for MHD waves. If we write the magnetic field again in the form B 0 = B0 b, then we find three types of MHD waves: – Alfvén waves : the dispersion relation is given by ω2 = (k.b)2 cH2

(4.71)

– fast and slow magnetosonic waves; their dispersion relation is given by: ω4 − ω2 k 2 (cs2 + cH2 ) + k 2 (k.b)2 cs2 cH2 = 0

(4.72)

Let us consider two special cases: if the waves propagate along the field k.b = k, there are two waves with ω2 = k 2 cH2 and the sound wave ω2 = k 2 cs2 unaffected by the field. For wave propagation perpendicular to the field only one wave survives with ω2 = k 2 (cs2 + cH2 ). When waves propagate anisotropically, it is necessary to introduce another wave velocity in addition to the phase velocity, the group velocity, given by ∂ω/∂k with which the wave carries energy or information. The group velocity for Alfvén waves is always cH b. What does that mean? Regardless of the direction in which it propagates, energy always travels along the field lines with speed cH . Alfvén waves propagate in the direction of the magnetic field lines, the Alfvén velocity is: B (4.73) vA = √ μ0 ρ The Alfvén time is given by (l is a length scale of the system): τA =

l vA

(4.74)

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89

4.1.9 Magnetic Fields and Convection Let L be the length of a box, v a typical velocity. The magnetic diffusivity is 1/μ0 σ , and the eddy turnover time L/v. The resistive decay time is then μ0 σ L 2 and the resistive decay time/eddy turnover time is denoted as magnetic Reynolds number for the flow: (4.75) Rm = Lvμ0 σ If the magnetic Reynolds number Rm is very low, the field is unaffected by the motions, if it is high, it is wound up many times before dissipation occurs. For an intermediate value of Rm the magnetic field is carried from the center of the eddy becoming concentrated in flux ropes at the edge. This buoyant flux ropes rise towards the surface and this leads to the appearance of sunspots. However we must also take into account that convection involves different length scales. Large eddies affect the overall structure of the magnetic field as it has been just described. Also the solar granulation is influenced. It is well known that granulation is suppressed in a sunspot. As it was shown earlier, in the absence of magnetic field convection occurs in a gas, if the ratio of the temperature gradient to the pressure gradient satisfies the relation: γ −1 P dT > T dP γ

(4.76)

If a vertical magnetic field of strength B threads the fluid, then this has to be modified to: P dT γ −1 B2 > + 2 (4.77) T dP γ B +γP Thus a strong magnetic field can prevent convection and a weaker field can interfere with convection. Note also that the magnetic field cannot prevent motions that are oscillatory up and down the field lines but these are likely to be less efficient at carrying energy.

4.2 The Solar Dynamo So far we have discussed the different aspects of solar activity. In the section on MHD it was shown that due to dissipation, such recurrent phenomena on the solar surface and atmosphere cannot be explained by just assuming a fossil magnetic field of the Sun. Therefore, many attempts were made in order to explain the recurrent solar activity phenomena such as sunspots, their migration toward the equator in the course of an activity cycle etc. In the first section of this paragraph we give a general description of the basic dynamo mechanism, in the following chapter some formulas are given.

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4.2.1 The Solar Dynamo and Observational Features Let us briefly recall the observational facts that a successful model for the solar dynamo must explain: – 11 year period of the sunspot cycle; not only does the number of sunspots vary over that period but also other phenomena such as the occurrence of flares, prominences, etc. – the equator-ward drift of active latitudes which is known as Spörers law and can be best seen in the butterfly diagram. At the beginning of a cycle, active regions appear at high latitudes and toward the end they occur near the equator. – Hale’s law: as we have mentioned, the leader and the follower spot have opposite polarities. This reverses after 11 years for each hemisphere so that the magnetic cycle is in fact 22 years. – Sunspot groups have a tilt towards the equator (this is sometimes also called Joy’s law). – Reversal of the polar magnetic fields near the time of the cycle maximum. As we know from fundamental physics, magnetic fields are produced by electric currents. How are these currents generated in the Sun? The solar plasma is ionized and it is not at rest. There are flows on the solar surface as well as in the solar interior producing magnetic fields that contribute to the solar dynamo.

4.2.2 The α − ω Dynamo There are three basic requisites for dynamo theory to work: – an electrically conductive fluid medium; in the Sun most state of the gas occurs as a plasma. – kinetic energy; this is provided by rotation; the Sun rotates differentially; zones near the equator rotate more quickly than zones at higher heliographic latitudes. – an internal energy source to drive motions; this is provided by the core, where energy is generated by thermonuclear fusion. The induction or the creation of the magnetic field is described by the induction equation: ∂B = η∇ 2 B + ∇ × (u × B) ∂t

(4.78)

u is the velocity, B is the magnetic field, η = 1/(σ μ) is the magnetic diffusivity, σ is the electric conductivity, and μ is the permeability.

4.2 The Solar Dynamo

91

Fig. 4.1 The sidereal differential solar rotation at different heliographic latitudes. NASA

4.2.3 Differential Solar Rotation G. Galilei observed sunspots around 1610. He already assumed that sunspots are related to the solar surface. In 1630 Ch. Scheiner mentioned two different rotational periods at the poles and at the equator. This was the first time when differential solar rotation was mentioned. The differential rotation rate is usually described by ω = A + B sin2 (φ) + C sin4 (φ)

(4.79)

where ω is the angular velocity in degrees per day, φ is the solar latitude, and A, B, C are constants. Values often cited in the literature for these values are: A = 14.713 ± 0.0491o /d

(4.80)

B = −2.396 ± 0.188o /d C = −1.787 ± 0.253o /d

(4.81) (4.82)

At the equator the solar rotation is 24.47 days. This is the sidereal rotation. The synodic rotation period is longer, 26.24 days. The synodic rotation is the time for a fixed feature on the Sun to rotate to the same apparent position as seen from Earth. Since the rotation of the Sun and the revolution of Earth around the Sun are in the same sense, the synodic period must be longer. In solar physics the Carrington rotation is usually used: a synodic rotation of 27.2753 days. The sidereal Carrington rotation is 25.38 days. This rotation corresponds to a solar rotation at. 26◦ north or south of the equator. See also Fig. 4.1 for further details.

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4 MHD and the Solar Dynamo

Fig. 4.2 From helioseismology we know about the interior solar rotation. It is differential at the surface and in the convective zone. At about 0.66 r/R, R is the solar radius in the tachocline the rotation changes to a solid body rotation. NASA/NSO

As it was already explained sunspots can be used as tracer for solar rotation. The Sun rotates slightly differently for the two hemispheres. Observations with magnetograph data have shown a rotation period (synodic) of – 26.24 days at the equator – 38 days at the poles. Using helioseismology it is possible to probe the solar interior and to derive the rotation rate at various depths. The differential rotation remains stable throughout the whole convection zone, and at the tachocline there is a rapid transition from differential to rigid body rotation. Therefore the solar radiative zone and the core rotate at constant rotation rate (Fig. 4.2).

4.2.3.1

The ω Effect

Let us consider magnetic fields inside the Sun. Because of the higher density, the field lines are driven by the motion of the plasma. Therefore, magnetic fields within the Sun are stretched out and wound around the Sun by differential rotation (the Sun rotates faster at the equator than near the poles). Let us consider a north-south oriented magnetic field line. Such a field line will be wrapped once around the Sun in about 8 months because of the Sun’s differential rotation (Fig. 4.3).

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93

Fig. 4.3 Illustration of the ω effect. The field lines are wrapped around because of the differential rotation of the Sun

4.2.3.2

The α Effect

The field lines are not only wrapped around the Sun but also twisted by the Sun’s rotation. This effect is caused by the Coriolis force. Because the field lines become twisted loops, this effect was called α effect. Early models of the dynamo assumed that the twisting is produced by the effects of the Sun’s rotation on very large convective flows that transport heat to the Sun’s surface. The main problem of that assumption was, that the expected twisting is too much and would produce magnetic cycles of only a couple of years. More recent dynamo models assume that the twisting is due to the effect of the Sun’s rotation on rising flux tubes. These flux tubes are produced deep within the Sun.

4.2.3.3

The Interface Between Radiation Zone and Convection Zone

If dynamo activity occurs throughout the entire convection zone the magnetic fields within that zone would rapidly rise to the surface and would not have enough time to experience either the alpha or the omega effect. This can be explained as follows: a magnetic field exerts a pressure on its surroundings (∼B 2 , proportional to its strength). Therefore, regions of magnetic fields will push aside the surrounding gas. This produces a bubble that rises continuously to the surface. However, such a buoyancy is not produced in the radiation zone below the convection zone. Here, the magnetic bubble would rise only a short distance before it would find itself as dense

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4 MHD and the Solar Dynamo

as its surroundings. Consequently, it is assumed that magnetic fields are produced at this interface layer between the radiation zone and the convection zone. Helioseismology has established the existence of a layer of strong gradients of angular velocity at the base of the solar convection zone. This layer, having a thickness of about 0.019 R , the tachocline, separates the convection zone and exhibits a strong latitudinal differential rotation from the radiative interior that rotates almost rigidly. Turbulence generated in the tachocline is likely to mix material in the upper radiative zone resulting in the observed deficit of Li and Be. Gilman [1] wrote a summary about the tachoclyne stressing its importance for in situ generation of poloidal fields as well as creating magnetic patterns that are seen on the surface.

4.2.3.4

The Meridional Flow

The solar meridional flow is a flow of material along meridional lines from the equator toward the poles at the surface and from the poles to the equator deep inside. At the surface this flow is in the order of 20 m/s, but the return flow toward the equator deep inside the Sun must be much slower since the density is much higher there—maybe between 1 and 2 m/s. This slow plasma flow carries material from the polar region to the equator in about 20 years. Thus the energy that drives the solar dynamo comes from (a) rotational kinetic energy, (b) another part in the form of small-scale, turbulent fluid motions, pervading the outer 30% in radius of the solar interior (the convection zone).

4.2.4 Mathematical Description Let us discuss some basic mathematics. In the magnetohydrodynamic limit the dynamo process is described by the induction equation: ∂B = ∇ × (u × B) − ∇ × (ηe ∇ × B) ∂t

(4.83)

The flow u is a turbulent flow. In the mean-field electrodynamics one makes the following assumptions: magnetic and flow fields are expressed in terms of a largescale mean component and a small scale fluctuating (turbulent) component. If we average over a suitably chosen scale we obtain an equation that governs the evolution of the mean field. This is identical to the original induction equation but there appears to be a mean electromotive force term associated with the (averaged) correlation between the fluctuation velocity and magnetic field components. The basic principles of mean field electrodynamics were given by Krause and Rädler [2]. The velocity and the field are expressed as: u =< u > +u

B =< B > +B

(4.84)

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95

Fig. 4.4 The MHD relation between flows and magnetic fields

< u >, < B > represent slowly varying mean components and u , B non axisymmetric fluctuating components. The turbulent motion u is assumed to have a correlation time τ and a correlation length λ which are small compared to the scale time t0 and scale length l0 of the variations of < u > and < B >. In other words, τ is a mean time after which the correlation between u (t = τ ) and u (t = 0) is zero and λ is comparable to the mean eddy size. We assume that < u >, < B >= 0 (Fig. 4.4). This is substituted into the induction equation and subtracted from the complete equation: ∂B = ∇ × (< u > ×B + u × < B > +G) − ∇ × (η∇ × B ) ∂t where

E =< u × B >

G = u × B − < u × B >

(4.85)

(4.86)

E is a mean electric field that arises from the interaction of the turbulent motion and the magnetic field. This field must be determined by solving the equation for B and here several assumptions are made. First of all we stressed that < v >= 0. This may be a good assumption when considering a fully turbulent velocity field. However in the Sun we are dealing with a sufficiently ordered convective field where the Coriolis force plays an important role. The other approximation is a first order smoothing: G ∼ 0. That is valid only if B  < B >. Then our equation reduces to: ∂B + ∇ × (η∇ × B ) = ∇ × (u × < B >) ∂t

(4.87)

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4 MHD and the Solar Dynamo

We want to determine E. Thus only B the component of B which is correlated with u must be considered. By definition τ, B(t + τ ) is not correlated with B(t) for any t. B (t) may be determined by integration of the above equation from t − τ to t. Note also, that the order of the convective turn over time τ ∼ λ/v and thus both u and < B > may be regarded as independent of t. Thus the integration yields: E i = αi j < Bi j > +βi jk

∂ < Bj > ∂ xk

(4.88)

where αi j , βi jk depend on the local structure of the velocity field and on τ . If the turbulent field is isotropic, then αi j = αδi j , βi j = βi jk , and E = α < B > −β∇× < B >

(4.89)

If τ is small compared to the decay time λ2 /η, the diffusive term may be neglected and from Eq. 4.87 we get 1 α = − τ < u · ∇ × u >, 3 And finally:

β=

1 2 τv 3

∂B = ∇ × (αB + u × B) − ∇ × [(η + β)∇ × B] ∂t

(4.90)

(4.91)

Compared to the normal induction equation, this contains the term αB and the eddydiffusivity coefficient β. In the mean field dynamo, the magnetic diffusivity η is replaced by a total diffusivity η = η + β and the equation becomes: ∂B = ∇ × (αB + u × B) + η ∇ 2 B ∂t

(4.92)

Please note that most often the prime is dropped on η; however, in the presence of α it is implied to use the turbulent diffusivity. It is assumed that B is axisymmetric. Then it can be represented by its poloidal and toroidal components A(x, z, t) and B(x, z, t) and u = u(x, z, t)j. Neglecting the advection terms: 

 ∂ − η∇ 2 B = [∇u × ∇ A].j − α∇ 2 A ∂t   ∂ − η∇ 2 A = α B ∂t

(4.93) (4.94)

Note that the dynamo action is possible because we have a regeneration of both toroidal and poloidal fields. Let us consider the source term in the first of the two above equations. ∇u describes a non uniform rotation. It can be argued that this term is larger than the next term involving α. This set of equations then describes the so called α − ω-dynamo. The equations describe:

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97

– ω effect: the poloidal field is sheared by non uniform rotation to generate the toroidal field. – α effect: this is the essential feedback. The helicity vc .∇ × vc of the non axisymmetric cyclonic convection generates an azimuthal electromotive force E which is proportional to the helicity and to Bφ . Let us define a characteristic length scale l0 , a decay time t0 = l02 /η and u = s0 ω, where s0 is of the order of the local radius of rotation and ω the local angular velocity. We may rewrite the above equations in terms of the non dimensional variables t  = t/t0 and r  = r/l0 . By an elimination of B and neglecting the α 2 terms we arrive at 

∂ − ∇ 2 ∂t 

2 A=

αl02 s0  [∇ ω × ∇  A].j η2

(4.95)

If α0 and ω0 are scale factors giving the orders of magnitudes of α and |∇  ω| then 

∂ − ∇ 2 ∂t 

2

α A=D α0



 ∇ ω  × ∇ A .j ω0

(4.96)

In that equation the non dimensional dynamo number D is D=

αω0 l02 s0 2η2

(4.97)

It is extremely important to note that the onset of a dynamo action depends on D. Only in case that D for a given system exceeds some critical value will there be dynamo action. Examining our set of equations we may also note that dynamo action is possible when ∇u is negligible compared to α. Such dynamos are called α 2 dynamos. If both terms of the source term are comparable then we speak of an α 2 ω dynamo. Solar like stars have well developed and structured convection zones. Thus, the α − ω dynamo is the most likely dynamo mode. Reviews on the solar dynamo and the emergence of magnetic flux at the surface can be found in Ossendrijver [3], Fisher et al. [4] and Moreno-Insertis [5]. So far we have discussed large dynamos that explain the origin of the solar cycle and of the large scale component of the solar magnetic field. We should add here that the origin of small scale magnetic fields can also be understood in terms of dynamo processes. Recent advances in the theory of dynamo operating in fluids with high electrical conductivity—fast dynamos, indicate that most sufficiently complicated chaotic flows should act as dynamos (Cattaneo [6]). The existence of a large scale dynamo is related to the breaking of symmetries in the underlying field of turbulence (Cattaneo [7]). Steiner and Ferriz-Mas [8], showed how solar radiance variability might be connected to a deeply seated flux-tube dynamo.

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References 1. Gilman PA (2005) The tachocline and the solar dynamo. Astron Nachr 326:208–217 2. Krause F, Raedler KH (1980) Mean-field magnetohydrodynamics and dynamo theory. Pergamon Press, Oxford 3. Ossendrijver M (2003) The solar dynamo. Astron Astrophys Rev 11:287–367 4. Fisher GH, Fan Y, Longcope DW, Linton MG, Pevtsov AA (March 2000) The solar dynamo and emerging flux - (Invited review). Sol Phys 192:119–139 5. Moreno-Insertis F (1994) The magnetic field in the convection zone as a link between the active regions on the surface and the field in the solar interior. In: Solar magnetic fields, pp 117–135 6. Cattaneo F (1999) Dynamo theory and the origin of small scale magnetic fields. In: Motions in the solar atmosphere. ASSL, vol 239, pp 119–137 7. Cattaneo F (1997) The solar dynamo problem. In: SCORe’96: solar convection and oscillations and their relationship. ASSL, vol 225, pp 201–222 8. Steiner O, Ferriz-Mas A (2005) Connecting solar radiance variability to the solar dynamo with the virial theorem. Astron Nachr 326:190–193

Chapter 5

Long Term Solar Activity

In this chapter we discuss long term solar activity behavior. We first address the problem of how we can get longer time series of solar activity since telescopic observations of sunspots reach back only to 1610, when Galileo and others first observed them.

5.1 Sunspot Recordings 5.1.1 The Wolf Number As it was discussed already, the Wolf number or sunspot relative index is given by the total number of sunspots visible, f and the total number of sunspot groups g. The number of groups has a weighting factor of 10, the appearance of a new sunspot group is regarded as a sign of higher solar activity than the appearance of just a small new spot. The smallest spots are called pores and have diameters of a few 100 km. The sunspot relative number is R = k(10g + f )

(5.1)

k takes instrumental effects into account as well as different observing conditions. To minimize these effects, the sunspot relative number is derived from observations of different solar observatories distributed worldwide. This averaged sunspot number takes into account weather, researcher and instrument quality. In 1848 R. Wolf introduced that number in Zurich, so it is therefore also called Zurich number. The advantage of combining the number of groups and individual spots is to compensate for variations in detecting small sunspots. The sunspot cycle was detected in 1843 by S. Schwabe. © Springer Nature Singapore Pte Ltd. 2020 A. Hanslmeier, The Chaotic Solar Cycle, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-15-9821-0_5

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Fig. 5.1 The international sunspot number of the past cycle. SILSO, Royal Obs. Belgium

Since July 1 2015 a revised and updated list of the sunspot numbers has been available. There is an overall increase by a factor of 1.6 to the entire series. This revision includes three major changes. This is a scale change and leads to an increase of about 45% of the most recent part of the series after 1947. – Instead of R. Wolf a new reference observer, A. Wolfer was chosen. This is a scale change and leads to an increase of about 45% of the most recent part of the series after 1947. – Error values: the new values come with the value of the standard deviation. Therefore, there is a better statistical uncertainty measure. – new version numbers. For further details see the reference [1]. Uncertainties in sunspot numbers are discussed in [2]. The international sunspot number of the past cycle is shown in Fig. 5.1. These data are collected at the SIDC (solar index data center) at the Royal Observatory, Brussels. The yearly mean and monthly smoothed sunspot numbers are given in Fig. 5.2. As it is evident from this figure, we do not have sunspot recordings at a regular basis before 1700. In order to study the long term variation of solar activity it is therefore necessary to search for proxies of solar activity that extend our knowledge on solar activity over a much longer time span.

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Fig. 5.2 The yearly mean and monthly smoothed sunspot numbers. SILSO, Royal Obs. Belgium

5.1.2 Historical Sunspot Numbers When the spot is sufficiently large and there are certain atmospheric conditions (mist, dust, or smoke which reduce the intensity of sunlight) large spots can be seen with the naked eye. In ancient China, Korea and Japan there are many historical sources with sunspot observations made with naked eye. In Europe, some medieval reports of Venus and Mercury transits could be interpreted as large sunspots. There are also some catalogues of naked-eye sunspot observations. Examples are found in [3] in which a catalogue of sunspot observations from 165 BC to AD 1684 is given. As was mentioned already, S. H. Schwabe detected the 11 year sunspot cycle. He made notes and drawings of sunspots from 1825–1867. These records are preserved in the manuscript archives of the Royal Astronomical Society, London. For preservation and measurements, these drawing were digitized. In total Schwabe made 8486 drawings and also reports about possible aurorae that are associated with periods of strong solar activity [4]. The original paper of these observations is found in [5]. Other observations in the 19th century were made for example by Spörer (1874–1884). During 1749 and 1796 the amateur astronomer J. C. Staudacher made sunspot observations. The drawings are stored in the library of the Astrophysikalisches Institut Potsdam, Germany. They were digitized [6]. Historical drawings of sunspots are very valuable, because they can be used for determining the solar rotation rate. During 1645 and 1715 there was the so-called Maunder Minimum, a period of extremely low solar activity. Some of the studies suggest that the solar rotation was more differential during the Maunder Minimum. The solar dynamo is influenced by differential rotation. In [7] it was found that the Sun rotated faster near the equator by 3–4% and the differential rotation between zero and ±20◦ latitude was enhanced by a factor 3.

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In [8] a review on historical sunspot observations is given.

5.2 Proxies of Solar Activity Since direct sunspot observation exist only for about 400 years (with telescopes) proxies of the solar activity are important to study its long term behavior.

5.2.1 Aurorae In high latitude regions around the Arctic and Antarctic aurorae, also called polar lights are produced when the Earth’s magnetosphere is disturbed by the solar wind. Mainly electrons and protons enter the upper atmosphere. They cause ionization and excitation of molecules in the upper atmosphere and the different compounds produce the different colors of the aurorae. Optical emissions are usually produced by incident hydrogen atoms after gaining electrons, and proton aurorae are observed even at lower latitudes. An example of the aurora oval over the Earth’ south pole is given in Fig. 5.3. The auroral zone is 3–6◦ wide in latitude and between 10 and 20◦ from the geomagnetic poles. Aurorae in the northern latitudes are called aurorae borealis, northern lights, and aurorae in the southern latitudes are called aurorae australis or southern lights. During a geomagnetic storm they are seen below the auroral zone. Large geomagnetic storms are most common during the peak of the 11-year sunspot cycle but may be also remain spectacular during the first 3 years after the solar maximum. Typical colors are: – red: excited atomic oxygen emits at 630 nm. The concentration of oxygen at larger atmospheric heights is low, therefore these emission lines are faint and they are only seen when solar activity is at a high level. – green: at lower latitudes. The emission is around 557.7 nm, and they are emitted by atomic oxygen (which is not in an excited state, therefore it is more frequent and the intensity is larger). – blue caused by molecular nitrogen and ionized molecular nitrogen. Dominant at 428 nm. There are also emissions in the UV and IR. UV aurorae were identified on Mars, Jupiter and Saturn as well. As an example an aurora over Jupiter’s pole is shown in Fig. 5.5. The Earth is constantly hit by solar wind particles. The solar wind reaches the Earth at a typical velocity of 400 km/s, the ion number density is about 5/cm3 . The magnetic field intensity is around 2–5 T. The Earth’s surface field for comparison, is between 30,000–50,000 nT. During magnetic storms the values for the velocity can

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Fig. 5.3 Aurora australis (11 September 2005) as captured by NASA’s IMAGE satellite

be much higher. The solar wind collides with the Earth’s magnetosphere and it is reshaped. At about 11 Earth radii (R E ) or a distance of 70,000 km a bow shock is produced, at 1.0–2.4 R E (12,000–15,000 km) a further upstream. The width of the magnetosphere is up to 30 R E and on the night side a magnetotail forms that extends to distances >200 RE . The high latitude magnetosphere is filled with plasma as the solar wind passes. If the solar wind has additional turbulence, density and speed (in case of high solar activity) the flow of plasma into the magnetosphere is enhanced. In Fig. 5.4 a schematic of the Earth’s magnetosphere is given. Note that at the dayside the magnetosphere is strongly deformed and compressed, and at the nightside a long tail is seen. Solar wind particles are deflected and by reconnection in the magnetotail they can approach the Earth over the magnetic poles. Also the Van Allen radiation belts are shown, which is where particles are trapped (Fig. 5.5). Historical recordings of aurorae are extremely valuable for solar activity reconstruction. A Japanese dictionary made in 1770 indicates a sighting of aurorae over Kyoto. That storm may have been larger by 7% than the Carrington event.1 The storms during the Carrington event on August 28 and September 2 1859 are probably the most spectacular storms in recent history. The white light solar flare on September 1, 1859 produced aurorae that were reported worldwide throughout the US, Europe, 1 This

event affected telegraph networks.

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Fig. 5.4 Schematic of Earth’s magnetosphere. NASA, A. Kaase

Japan and Australia. The New York times reported that in Boston on Sep. 2 1859, the aurora was so brilliant, that at one o’clock in the night a newspaper could be read without any other artificial light. It was possible to operate an American Telegraph line between Boston and Portland without any additional power from a battery. In ancient Greece Pytheas reported about an aurora (4th century BC). The Aboriginal Australians associated aurorae with fire. Also Seneca wrote about aurorae and their different colors. Halley noted that before the aurora of 1716, no such phenomenon had been recorded for more than 80 years. No appearance of an aurora is recorded in the Transaction of the French Academy of Sciences between 1666 and 1716. Only one is recorded in Berlin Miscellany 1797 and this was called a rare event. In 1723 one was observed at Bologna. During the night after the Battle of Fredericksburg (1862), an aurora was seen from the battlefield. The Confederate Army took this as a sign that God was on their side.

5.2.2 Cosmogenic Isotopes Cosmogenic isotopes are produced when cosmic rays collide with atmospheric molecules. The production rate of the cosmogenic isotopes depends on the strength

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Fig. 5.5 Aurora on Jupiter. NASA/ESA, J. T. Clarke Table 5.1 Commonly used cosmogenic isotopes Beryllium 10 1,387,000 Exposure dating of rocks, soils, ice cores Aluminium 26 720,000 Exposure dating of rocks, sediment Chlorine 36 308,000 Exposure dating of rocks, groundwater tracer Calcium 41 103,000 Exposure dating of carbonate rocks Iodine 129 15,700,000 Groundwater tracer Carbon 14 5730 Radiocarbon dating Sulfur 35 0.24 Water residence times Sodium 22 2.6 Water residence times Tritium 3 12.32 Water residence times Argon 39 269 Groundwater tracer Krypton 81 229,000 Groundwater tracer

of the cosmic radiation. However, the strength of the cosmic radiation varies with the strength of the earth’s magnetic field and solar activity. Therefore, studies of the abundances of cosmogenic isotopes provide a clue for understanding variations of – Earth’s magnetic field variations, and – Solar activity proxy. In Table 5.1 we give a list of commonly used cosmogenic isotopes. 14 C is produced in the upper layers of the troposphere and the stratosphere. Thermal neutrons are absorbed by nitrogen atoms.

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Fig. 5.6 The 10 Be production and deposit. GFZ

14 n +14 7 N →6 C + p

(5.2)

The highest rate of carbon-14 production takes place at altitudes between 9 and 15 km. Therefore, it takes some time before this isotope reaches the ground. After the production the carbon-14 atoms oxidize to 14 CO2 which dilutes all over the atmosphere and is also absorbed in the oceans. The atmospheric half-life of 14 C is between 12 and 16 years. The production changes with solar activity because the cosmic ray flux is modulated by the strength of the heliosphere and the Earth’s magnetic field. Time series show a spike from 774–775. This can be explained by an extreme solar particle event. 10 Be is formed by cosmic ray spallation of nitrogen and oxygen in the Earth’s atmosphere. It is transported to the surface via precipitation and exists in solutions below pH 5.5 (Fig. 5.6). Again, there is an anticorrelation between 10 Be production and solar activity. During episodes of high solar activity, less 10 Be is produced because the number of cosmic ray particles that enter the heliosphere is reduced. There is, however, a lack of direct high-resolution atmospheric time series on 10 Be that enable estimating atmospheric modulation on the production signal. The paper of [9] indicates intrusion of stratosphere/upper troposphere air masses that can

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Fig. 5.7 Tree rings can also be used to derive the climate in the past. NASA

modulate the isotopes production signal, and may induce relative peaks in the natural 10 Be archives (i.e., ice and sediment). The atmospheric impact on the Be-isotopes can disturb the production signals and consequently the estimate of past solar activity amplitude.

5.2.3 Other Proxies Other proxies of measuring solar activity exist, however, it has to be stressed that these proxies cannot provide exact datings of individual solar cycles. The analysis of tree rings is dependent on the 14 C production. Cosmogenic 14 C is generated in the Earth’s stratosphere and upper troposphere by energetic cosmic rays. It is then oxidized to 14 CO2 which takes part in a number geochemical and geophysical processes. It becomes stored in tree rings. Therefore, the concentration of 14 C in tree rings is a further proxy of solar activity. In [10] New Zealand tree ring-index data as a proxy of 11-year solar activity are discussed. Tree rings can also be used to reconstruct past climate (Fig. 5.7). Another proxy for solar activity can be the variation of the solar diameter. Is there a variation of the solar diameter as a function of the solar activity cycle? Initiated in the XVIIth century, the diameter measurements were carried out primarily for astrometric purposes, in particular, for determining precisely the instant in which the center of the Sun crossed the local meridian. Nowadays, these measurements are relevant to solar physics.

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Several possibilities to precisely measure the solar diameter exist: – Ground based measurements: telescopic measurements of the solar limb shape, transits of Mercury, solar eclipses. – Balloon based measurements; an advantage over ground-based methods due to less perturbations by the Earth’s atmosphere. – Measurements from space: For example the MDI instrument of SOHO could make such measurements; the problem is the varying thermal condition that influences on the focal length of the optical system. – Helioseismology: gives different values for the solar diameter than the other techniques. It is evident that in order to avoid effects of the Earth’s atmosphere, measurements from space should be made, however they are not available for long time series [11]. The widths of total solar eclipse paths depends on the diameter of the Sun, so if observations are obtained near both the northern and southern limits of the eclipse path, in principle, the angular diameter of the Sun can be measured. Concerted efforts have been made to obtain contact timings from locations near total solar eclipse path edges since the mid 19th century, and Edmund Halley organized a rather successful first effort in 1715 [12]. The result of this study is that although the observations seem to show small variations from eclipse to eclipse, they are only a little larger than the assessed accuracies. On the other hand these measurements are extremely important because small variations of the solar diameter also mean a variation of the total solar irradiation [13]. The authors limit variations of the solar radius between 1850 and 1937 to about 0.25 arc second; modeling of the sun indicates that the solar constant did not vary by more than 0.3% during that time. Models predict the following relation between solar diameter variation and solar output [14]: W =

D R DL R L

(5.3)

where D is the photospheric radius, L is the luminosity, and W is a dimensionless parameter. In [15] a decrease in the solar radius is determined using the technique of [16], in which timed observations are made just inside the path edges. When the method is applied to the solar eclipses of 1715, 1976, and 1979, the solar radius for 1715 is 0.34 ± 0.2 arc second larger than the recent values, with no significant change between 1976 and 1979. The duration of totality is examined as a function of distance from the edges of the path.

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Fig. 5.8 Sunspot drawing made by Ch. Scheiner in his work Rosa Ursina

5.3 Long Term Solar Activity 5.3.1 Solar Activity Since Telescopic Observations The first telescopic observations of the Sun were made by Galilei, Fabricius and Scheiner around 1610. They projected the solar disk image on a screen behind the telescope and could clearly see sunspots. Scheiner (1573–1650) made drawings of sunspots in the Rosa Ursina. Before that in 1612 he published several letters in which he claimed that sunspots are satellites of the Sun. An example of such a sunspot drawing is given in Fig. 5.8. The quality of the drawings had a remarkably high level and can be compared to drawings made 2 centuries later. However, certainly the first instruments to observe the sunspots were not of high quality compared to our modern instruments and since the sunspot index depends on the quality of the instrument this has to be taken into account when combining such observations with modern data for a time series. The instrument that was used for these observations was the helioscope. Scheiner was one of the first to observe sunspots by telescope, in March 1611, and in 1612 he published his findings anonymously. This led to a famous controversy

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with Galileo, who claimed to have observed sunspots earlier, involving the exchange of several letters. In Galileo’s letters to Wesler, published in 1613, he identified sunspots correctly as markings on the sun, confirming that the sun rotates monthly, as the position of the spots move. Early Observations of Sunspots by Scheiner and Galileo were discussed in [17]. In Fig. 5.9 we give the number of sunspot groups since 1610 instead of individual sunspot numbers. The Maunder Minimum, between 1645 and 1715, when sunspots were scarce is clearly visible. The modulations of the 11-year solar cycle is clearly seen, as well as the 70–100-year Gleissberg cycle. The Maunder Minimum will be discussed in the next chapter. Sunspot positions and areas from observations by Galileo Galilei are reported, for example, in [18]. In this paper further literature is cited.

5.3.2 The Maunder Minimum As it has been shown, solar activity can be reconstructed using cosmogenic isotopes like 14 C, 10 Be or 44 Ti. These data show that the 11 year Schwabe cycle is modulated by much longer-term variations. Deep minima and grand maxima of activity occurred. One famous example is the Maunder minimum that occurred during 1645–1715; this term was introduced by J. A. Eddy [19]. During that period only few sunspots were observed. That was not because of lack of observations or bad telescopic optics. G. D. Cassini (1625–1712) who was director of the Paris observatory carried out systematic solar observations. Before Cassini moved to France in 1669 he was in San Petronio, Bologna. There he was able to convince church officials to create an improved sundial meridian line at the San Petronio Basilica, moving the pinhole gnomon that projected the Sun’s image up into the church’s vaults 66.8 m (219 ft) away from the meridian inscribed in the floor. The much larger image of the Sun’s disk projected by the camera obscura effect allowed him to measure the change in diameter of the Sun’s disk over the year as the Earth moved toward and then away from the Sun. He concluded the changes in size he measured were consistent with Johannes Kepler’s 1609 heliocentric theory, in which the Earth was moving around the Sun in an elliptical orbit instead of the Ptolemaic system where the Sun orbited the Earth. The arrangement of how to get a solar disk image is shown in Fig. 5.10. He reported on a huge sunspot observed in 1680. There were maxima of small amplitudes during the Maunder Minimum as well as minima: – – – –

Minimum: 1676–1677 Minimum: 1684 Minimum: 1695 Minimum: 1705.

From the drawings of the sunspots it was found that the reduced solar activity was concentrated in the southern solar hemisphere except for the last cycle.

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Fig. 5.9 The number of sunspot groups since 1610. Image credit WDC-SILSO

Fig. 5.10 The pinhole-projected image of the Sun on the floor at Florence Cathedral. Cassini measured a similar image over a year at San Petronio Basilica to try to prove the Earth orbited the Sun. I. Sailko

The Maunder Minimum coincided with a period of cooler temperatures that were recorded in several countries. This period of a colder climate is known as little ice age. Europe and North America experienced colder than average temperatures. Very cold winters in Europe, Canada or Greenland were recorded for the periods 1683–84, 1694–95 and 1708–09. However, a unique correlation between sunspot activity and climate on Earth is still a matter of big debate. Also during the little ice age there were exceptional warm and mild winters. The drop in global average temperatures at the start of the little ice age was between 1560 and 1600, but the Maunder minimum began almost 50 years later. The temperature change between 1680 and 1780 is shown in Fig. 5.11. The temperature difference between 1680, a year at the center of the Maunder Minimum, and 1780, a year of normal solar activity, was calculated by a general circulation model. Deep blue across eastern and central North America and northern Eurasia illustrates where the drop in temperature was the greatest. Nearly all other land areas were

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Fig. 5.11 The temperature change during the Maunder minimum (1680) and 1780, at a time of normal solar activity from global circulation models. NASA

Fig. 5.12 The Maunder Minimum and the Dalton Minimum are clearly seen in sunspot recordings

also cooler in 1680, as indicated by the varying shades of blue. The few regions that appear to have been warmer in 1680 are Alaska and the eastern Pacific Ocean (left), the North Atlantic Ocean south of Greenland (left of center), and north of Iceland (top center). In sunspot recordings it is clearly seen that there were only few sunspots visible during the Maunder minimum (Fig. 5.12). Using 14 C and 14 Be recordings, solar activity can be also recorded. Both curves indicate a lower solar activity during the Maunder Minimum. A very valuable source of data is the Flamsteed drawing. These data suggest that the Sun’s surface rotation slowed in the deep Maunder Minimum. However, also controversial data exist. During the Maunder Minimum aurorae had been observed at normal frequencies. In the work of [20] it is shown that many catalogues of historical aurorae suffer from various problems and shortcomings: – East Asian records are often available only in later compilations (out of context), – original source texts from Europe are often not given (it is time-consuming to find the original), – observational reports from Europe are mostly in Latin, – in particular in the 17th century, there are double entries due to confusion between the Julian and Gregorian calendar.

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Fig. 5.13 Frost fair on the Thames

– Furthermore, we need clear criteria to distinguish aurorae from other meteorological and astronomical phenomena. Reports exist , e.g., war armies or dragons on sky at night can refer to auroral features or lunar halo effects. The authors show that many of the published presumable aurorae in the Maunder Minimum are non-auroral in origin. Even in the collection of sightings of aurora borealis in the area of Hungary by Botley [21], considered the most homogeneous sample in and around the Maunder Minimum, almost all reports during its deep phase are most certainly halo displays including elaborated narratives. Aurorae would allow a reconstruction of the geomagnetic field independent from archaeological and geophysical samples; in contrast to aurorae, solar activity reconstructions with radiocarbon samples suffer from inhomogeneities due to human intervention (e.g. Suess effect) and depend on the reconstructed geo-magnetic field.

5.3.3 Solar Activity Derived from Proxies “Frost fairs” on the Thames in London from 1684, which were during the Maunder minimum are shown in Fig. 5.13. The first use of the term ‘frost fair’ appears to be 1608, but there are numerous surviving records of the Thames freezing over before then, including in AD 250, 695, 923, 1150 and 1309, although these records are likely to be incomplete.

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Fig. 5.14 Comparing records of solar activity (top) with monthly minimum winter & maximum summer temperatures in Central England (CET, middle rows) and volcanic activity (bottom). The vertical lines represent years when there are records of the Thames freezing (yellow) and of Frost Fairs (purple). The black vertical lines indicate 1825, when the old London Bridge was demolished, and the completion of the Thames embankments in 1870, both of which changed the flow characteristics of the river. From: Ed Hawkins, Climate Lab

In Fig. 5.14 a comparison between sunspot recordings, radiocarbon data and minimum winter and maximum summer temperature in central England is given together with frost fairs on the Thames river (purple vertical lines) and Thames freezing (yellow lines). There was another period of low solar activity, the Spörer minimum: AD 1400– 1550. All these studies depend on long solar activity recordings and many attempts were made to reconstruct the sunspot number into the pre telescopic times. For the study of the influence of solar activity on climate on Earth, the total solar irradiance is an important parameter. In the paper of [22] two approaches were made: – Reconstruction of the short term trend by using sunspot numbers – Reconstruction of the long term trend by using radionuclide data. The reconstructed total solar irradiance, TSI, correlates with r = 0.84 with space based instruments. With such methods the TSI can be reconstructed back to the Maunder minimum and beyond.

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What was the amplitude of the solar irradiance variations during the Maunder minimum? The near and mid UV variation were studied by reconstruction methods using an extension of the Monte Carlo Solar Spectral Irradiance Model [23]. The spectral range of that model extends to 150 nm and it goes back to the year 1610. The following values for the variation were found: – 300–400 nm range: ∼1% – below 200 nm: 3–5%. The reduction of solar UV radiation during the Maunder minimum led to less production of ozone in the higher Earth’s atmosphere between 10 and 50 km. This influenced the planetary waves (jet streams) and the North Atlantic oscillation (NAO), the balance between a permanent low-pressure system near Greenland and a permanent high-pressure system to its south. This oscillatory pattern came into a negative phase and both pressure systems became relatively weak. The effects of a positive or negative phase are: – negative phase: Under these conditions, winter storms crossing the Atlantic generally head eastward toward Europe, which experiences a more severe winter. – positive phase: When the NAO is positive, winter storms track farther north, making winters in Europe milder. In the paper of [24] the total solar irradiance and Ca H and K fluxes (HK) for the Maunder minimum are estimated from scaling laws for solar-type stars using historical solar rotation rates and solar diameters. It was found that the irradiance may have been lower than modern solar minimum values by 1.23% in 1683 and by 0.37% in 1715 (Dalton Minimum). The estimate for 1683 is substantially lower than previously reported. Analysis of cosmogenic isotope records in ice cores and tree rings shows continuation of the Sun’s magnetic cycle through the Maunder minimum; therefore, the HK fluxes were found to be 0.161 for 1683 and 0.163 for 1715, compared with the modern solar minimum flux value of ∼0.164. This suggests that the Sun never reached a noncycling state. More recent studies, however, suggest a more moderate TSI difference of less than 1 Wm−2 , possibly even less (0–0.3 Wm−2 ) [25]. This is in contrast to previous published values which are about 5.8 Wm−2 . In the paper of [26] it is stressed again that the total solar irradiance change during the Maunder minimum ranges from 0.5–1.5 Wm−2 . One criticism of this is that this value is mainly estimated from sunspot recordings and by comparison with solar like stars. It is discussed that the sunspot number can go to zero, however, irradiance variations could be larger than the above cited values. Therefore, another proxy was used to derive the irradiance variations during the Maunder minimum, the 14 C data. During the transition from cycle 23–24 there was a long period of very low solar activity. However, concerning the UV output, this emission was still higher than during the Maunder minimum.

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5.3.4 The Total Solar Irradiance, TSI The total solar irradiance is key input factor into the climate, it is the main external driver of the Earth’s climate. The definition of the TSI is power per unit area received from the Sun in the form of electromagnetic radiation (mostly within a certain wavelength range). Measurements on the Earth’s surface are influenced by absorption and scattering in the Earth’s atmosphere. Therefore, exact data can only be achieved from space measurements. The value depends on several factors. Since the orbit of Earth around the Sun is elliptical, there is a small variation depending on the number of a day n of the year:   Q = S0 1 + 0.034 cos 2π

n  365.25

(5.4)

S0 is the solar constant. If R0 is the mean distance Earth-Sun, and R E the actual Sun-Earth distance, then:  Q = S0

R0 RE

2 cos(θ )

(5.5)

This formula describes the solar flux density (insolation) onto a plane tangent to the sphere of the Earth above the atmosphere (an elevation of 2 to n = 2. This means that an electron jumps between these discrete energy levels in a hydrogen atom. For example the Hα line is formed in

Table 6.1 The photometric UBV system and R, I, and J filters Filter Effective wavelength midpoint Full width half maximum (nm) (nm) U B V R I J

365 445 551 658 896 1229

66 94 88 138 149 213

6.1 The Hertzsprung–Russell Diagram

125

– emission: n = 3 → n = 2. – absorption: n = 2 → n = 3. n is the quantum number of the electron. On the ordinate of the H–R diagram the luminosity of stars is plotted. The luminosity of a star depends on (i) radiative energy output, and (ii) the surface area. The total radiative output is obtained by integration of the Planck function over all wavelengths λ: 4 E = σ Teff

(6.3)

This is also known as Stefan Boltzmann law. σ = 5.67 × 10−8 Wm−2 K −4 . Note that if the temperature of a star increases by a factor of 2, its total energy output increases by a factor of 24 = 16! Stars with a large surface radiate more than stars with a smaller surface area. Doubling the radius means that the radiative output increases by a factor of 4 because the surface area is given by A = 4π R 2 . The luminosity of a star is therefore: 4 L = 4π R 2 σ Teff

(6.4)

The temperature defined by the Stefan-Boltzmann law is also called effective temperature. Normally the brightness of a star is measured by its apparent magnitude m. The relation between m and intensity I is given by: m = −2.5 log I

(6.5)

In ancient Greece, the brightest stars in the sky were said to be of first magnitude m = 1, and the faintest stars visible to the naked eye have m = 6. This historical definition was extended. A star of magnitude m is about 2.512 times as bright as a star of magnitude m + 1. The apparent magnitude of a star depends on its true luminosity and its distance. Very luminous stars could appear very faint in the sky if they are at great distances. Therefore, the absolute magnitude M was introduced. The absolute magnitude is the apparent magnitude if the stars would be at a distance of 10 parsec.1 Thus, the difference m − M is related to distance: m − M = 5 log r − 5 In the H–R diagram the absolute magnitude of a star M is usually plotted.

11

parsec =1 pc =3.26 Light years, Ly. 1 Ly = 1013 km.

(6.6)

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6.1.2 The Main Sequence If we plot the spectral type, the temperature or the color versus the absolute magnitude of stars in a diagram, we obtain an H–R diagram. In Fig. 6.1 the H–R diagram is shown. On the x-axis the spectral type and the temperature is given, and also the color of stars is indicated. On the y-axis, the luminosity is given in solar units. In this diagram the Sun has spectral type G2. More than 80% of stars are found on a diagonal. This is called the main sequence.

Fig. 6.1 The H–R diagram. The main features seen are: main sequence, giants, supergiants, white dwarfs. ESO

6.1 The Hertzsprung–Russell Diagram

127

6.1.3 Giants, Supergiants The luminosity of a star depends on (i) temperature, and (ii) surface area. As seen in Fig. 6.1 there are stars that have the same temperature as main sequence stars but appear above the main sequence having a higher luminosity. This implies that they must have a larger surface area, hence a larger radius. Therefore, these stars are called giants or supergiants. The classification with O-B-A-F-G-K-M is not unique. For example a G-star can be a main sequence star or a giant or even a supergiant. Luminosity classes were introduced to overcome this ambiguity: – – – – – –

0, Ia hypergiants I supergiants II bright giants III regular giants IV sub giants V main sequence stars.

In this classification scheme the Sun is a G2V star. The scheme is shown in Fig. 6.2.

Fig. 6.2 The H–R diagram and luminosity classes. Note that the Sun has absolute magnitude M = 4.5; seen from a distance of 10 pc it would belong to the group of faint stars and visible to the naked only under clear dark sky conditions. Rursus

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6.1.4 Evolution of Stars in the H–R Diagram Why do we find more than 80% of stars on the main sequence? The answer is simple. Stars spend most of their life time on the main sequence and often in astrophysics, when one talks about stellar age one means the main sequence lifetime. The luminosity of a star determines its energy output. The higher the luminosity L, the larger the energy output which means that the nuclear fusion processes must become more efficient and more mass is converted into energy. Therefore, stars of high luminosities have a shorter main sequence lifetime than stars with lower luminosity. For main sequence stars a simple relation between L and their masses M exists : L ∼ M 3.5

(6.7)

From this equation the main sequence lifetime can be estimated: 

τMS

M∗ ∼ 10 years M

−2.5

10

(6.8)

This relationship is valid for main-sequence stars in the range of 0.1 − 50 M . In Table 6.2 some parameters for stars of different spectral type are given. In Fig. 6.3 the different nuclear fusion processes are shown for stars of different temperatures. For the Sun and cool stars the p-p reaction is the dominant process, for hotter stars the CNO cycle and the triple α process become relevant. The latter two are strongly temperature dependent and therefore the core where these reactions take place becomes convective. As soon as the central hydrogen fuel is consumed up, hydrogen fusion occurs within a shell around the core. This hydrogen burning shell moves slowly outwards. The core, for solar like stars consisting of helium shrinks and heats up. Due to the outwards moving hydrogen burning shell, the star gets bigger, the diameter increases,

Table 6.2 Some parameters for main-sequence stars Class Radius Mass O6 B0 A0 F0 G0 G2 K0 M0

18 7.4 2.5 1.4 1.05 1 0.85 0.63

40 18 3.2 1.7 1.10 1 0.78 0.47

Luminosity

Temperature

500,000 20,000 80 6 1.26 1 0.40 0.063

38,000 30,000 10,800 7,240 5902 5780 5240 3920

6.1 The Hertzsprung–Russell Diagram

129

Fig. 6.3 This plot shows the rate of nuclear energy generation  for a main sequence star as a function of temperature (T). The green curve shows the proton-proton cycle, blue is the CNO cycle and red is the triple-α process. The brown vertical line represents the core temperature of the Sun, demonstrating that the p-p chain is the primary source of energy generation. The dashed line joining the p-p curve to the CNO cycle represents the net energy generation rate of the combined nuclear hydrogen burning cycles. The slope of the curves shows the greater temperature sensitivity of the CNO cycle and triple-α process. RJHALL

and the temperature decreases only by a few 100 K. Thus the star gets larger and leaves the main sequence moving upwards to the region of giants or even supergiants. As soon as the central temperature gets large enough, helium burning starts, i.e. helium is converted into carbon by the triple α reaction. The star has reached the highest point in the H–R diagram. Since it has now two energy producing zones, the core with helium burning and a still outwards moving hydrogen burning shell, it becomes unstable. The further evolution depends on the mass of the star. Massive stars burn the carbon into O, Si etc. The process ends with the formation of the iron isotope 56 Fe. Elements heavier than 56 Fe cannot be formed by thermonuclear fusion. As soon as the iron core exceeds 1.44 M it collapses.2 This leads to the formation of a neutron star with a typical diameter of only few tens of kilometers. If the mass exceeds 4 to 5 solar masses, the final stage of stellar evolution is a black hole. For stars with masses below 1.44 M the final state is that of a white dwarf with a size similar to that of Earth. Since stellar evolution depends on mass, the Sun will become a white dwarf in about 4.5 billion years, but will leave several hundreds of millions of years before that the main sequence and become a red giant. At this stage its radius will include the orbit of Earth. The evolution of a star from a protostellar cloud to the main sequence is shown in Fig. 6.4. In Fig. 6.5 the post main sequence evolution of the Sun is shown. 2 This

mass is also called Chandrasekhar mass.

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Fig. 6.4 Tracks are plotted on the H–R diagram to show how stars of different masses change during the early parts of their lives. The number next to each dark point on a track is the rough number of years it takes an embryo star to reach that stage. http://cnx.org/contents/[email protected] Fig. 6.5 Post main sequence evolution of the Sun. At the end of is evolution, the Sun will become a white dwarf

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The H–R diagram of star clusters can be used to determine the age of such a cluster. Assuming that all stars in a cluster were born at nearly the same time the turn off point in the H–R diagram gives the age. The turn off point is the location where stars in a cluster are no longer found on the main sequence. As it was pointed out, the age of a star depends on its mass. Massive stars evolve faster than low mass stars. Therefore, if in the H–R-diagram, all stars are still found on the main sequence, it must be a young cluster since even massive stars that have a lifetime of only a few million years, are still found on the main sequence.

6.2 How to Measure Stellar Activity Even through the largest telescopes stars appear point like and cannot be resolved into disks. We cannot directly observe features such as star spots, prominences, flares etc. However, it is still possible to detect such phenomena on stars.

6.2.1 H-K Activity The H- and K-lines are prominent absorption lines in the spectra of solar like stars or cooler stars. They are caused by singly ionized calcium (Ca II) at wavelengths 396.9 and 393.4 nm. In their line cores emission features are visible, especially when there is strong stellar activity. In Fig. 6.7 the CaII H-profile near the line center is shown for different objects. Note the enhancement of the Ca II H-emission for the active star 36 Oph A.

Fig. 6.6 Prominent spectral lines in the spectrum of solar like stars. The H- and K-lines of Ca II are at the blue edge of the visible spectrum

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Fig. 6.7 The line core of CaII K line for (i) the Sun, (ii) two components of HD 219542 (iii) the active star 36 Oph A. [1]

6.2.2 Star Spots Spots on stars can only be detected when they are much larger than sunspots. They can be detected from periodic variations in a star’s integrated light because the spots move across the stellar disk due to the star’s rotation. RS Canum Venaticorum and BY Draconis stars are well known examples of spotted stars. A first review of starspots was given by [2]. Typical starspots cover up to 30% of the stellar surface and they may be 100 times larger than those on the Sun. The observed starspots have temperatures 500–2000 K cooler than the surrounding stellar photosphere. The corresponding brightness variation can reach values up to 0.6 magnitudes (between the spot and the surface). In general, four different methods are used to detect them: – – – –

Rapidly rotating stars: Doppler imaging and Zeeman-Doppler imaging. Slowly rotating stars: line depth ratio. Eclipsing binary stars: eclipse mappings. Stars with transiting exoplanets: light curve variations.

Doppler imaging was first used by S. Vogt and D. Penrod in 1983 [3]. They examined the variable star HR 1099 (V711 Tau). In the spectrum of a rapidly rotating star, a

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Fig. 6.8 Spectral line profiles for a model fast rotating star with no spots (dashed line) and with a spot moving across the disk as the star rotates (solid line). S. Berdyugina

one-to-one correspondence between wavelength position across a spectral line profile and spatial position across the stellar disk exists. Any dark or bright region on the stellar surface will produce an associated bump or depression at the corresponding location in the line profile. Because of the stellar rotation, the associated spectral feature (bump or depression) propagates across the line profile. This effect can be detected only on fast rotating stars because in such a case the line profile becomes broadened (see also the illustration in [4]). The principle of Doppler imaging is illustrated in Fig. 6.8. Inversion of a time series of the stellar line profiles results in a map, or image, of the stellar surface. The ZeemanDoppler imaging is based on the analysis of high-resolution spectropolarimetric data. The magnetic field distribution on the stellar surface due to different Doppler shifts of Zeeman-split local line profiles in the spectrum of a rotating star is obtained. If there is no rotation, the net circular polarization signal in spectral lines would be zero due to mutual cancellations of contributions regions of opposite field polarity. This technique was first introduced by [7]. A general review of starspots as a key to the stellar dynamo is given in the Springer Living Reviews Series by Svetlana Berdyugina, 2005.

6.2.3 Stellar Winds A stellar wind is a flow of gas ejected from the upper atmosphere of a star, similar to solar wind. Most stars show continuous outflows of gas from their surfaces. Stellar winds are generally symmetric, not bipolar outflows, which are characteristic for young stars. Massive O- and B-stars have stellar winds associated with mass loss rates M˙ < 10−6 M /year. The velocities are high and range from 1000–2000 km/s.

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Fig. 6.9 This image shows the wind from the star LL Orionis generating a bow shock (the bright arc) as it collides with material in the surrounding Orion Nebula. NASA

The physical mechanisms driving these winds is radiation pressure. For solar like stars (G-type) the winds are mainly driven by their hot magnetized corona. These winds are not stellar wind bubbles but consist mainly of high energy electrons and protons (1 keV). For low mass stars the influence of stellar winds on their evolution is negligible, for massive stars stellar winds can transport up to 50% of the initial stellar mass. Figure 6.9 shows how stellar wind from the star LL Orionis generates a bow shock when the particles collides with the plasma in the Orion nebula where the star is embedded in. Stars are often binary systems.3 The companion star has a strong influence in shaping the wind, especially in the case of cool luminous stars in close binary systems. Such companion stars could be an explanation for the often complex structure of planetary nebulae that are formed from expanding stellar envelopes when the stars has reached the red giant branch in the H–R diagram. Stellar wind can be detected for example by P Cygni line profiles. In fact the first P Cygni line profiles were detected in the late 1960s in the UV-spectra of O-stars. The typical characteristics of a P Cygni profile are: – strong emission lines; these are produced from other parts of the expanding shell. – blueshifted absorption lines; these are produced by matter moving away from the star toward the observer.

3 About

50% of all stars may be binary systems or even multiple systems.

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The intensity of stellar winds also depends on stellar age. It is most intense at the beginning and end of a star’s lifetime. T-Tauri stars are stars that are still contracting from their protostellar cloud and have not reached yet the main sequence where they would remain in hydrostatic equilibrium. During their main sequence phase, the mass loss by stellar wind is normally about 10−14 M /yr. A semi-empirical relation exists: 1.1 M˙ ∝ Z 0.7 L 2.2 M −1.3 (v∞ /vesc )−1.4 Teff

(6.9)

More details can be found e.g., in [5]. Stellar winds also influence on the habitability of planets. The stellar wind interaction with the planet depends on the magnetic field and atmosphere of the planet. Mass, energy, and momentum capture from stellar winds by magnetized and unmagnetized planets and the implications for atmospheric erosion and habitability were investigated by [6]. A basic parameter for such studies is the so-called standoff distance. Let us assume a planet with a dipole magnetic field and a stellar wind dominated by ram pressure. The standoff distance ds = rs , where rs is determined by where the magnetic pressure of the planet equals the ram pressure of the impinging stellar wind: rs3 = r 3p



B 2p

1/2

8πρw vw2

(6.10)

where r p is the radius of the planet, B p the magnetic field strength at the planetary surface, ρw is the stellar wind density, and vw is the stellar wind speed at rs .

6.3 Stellar Activity Cycles Stellar activity can only be detected at levels above solar activity. Moreover, long time series of stellar activity recordings are available only for few objects.

6.3.1 The Mount Wilson Survey After about 30 years of Ca II K observations of selected stars this survey ended in the year 2000. It was the first observational proof of stellar activity cycles. An index called S was defined. This index is given by the flux of singly ionized calcium H& K line core. The index is a composition of four channels: SMWO = α

NH + NK N R + NV

(6.11)

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The filters are centered around the H and K line with a FWHM bandpass of 0.109 nm center in the CaII H+K line cores. N H , N K denotes the number of counts. N R , N V are in the 2 nm continuum bandpasses R, V centered at 390.107 nm and 400.107 nm. The factor α is a calibrations factor. The Mt Wilson HK project measured a sample of over 100 stars from early F to early M spectral type. For comparing solar activity to that of Sun-like stars it is important to place the Sun on this S scale. This can be done e.g., by measurements of the reflected sunlight from the Moon. Several important relations could be derived: – Stellar activity decreases with slower rotation rate. – Stellar activity is influenced by stellar convection. Deeper convection zones produce higher stellar activity. – Stars with high activity seem to be spot dominated. – Stars with lower activity seem to be faculae dominated.

6.3.2 Stellar Activity Versus Stellar Age The first idea of a simple relation between stellar activity and stellar age was given by the work of [6]. The Ca H-K emission after correction for spectral-type effects for several samples of stars was investigated for the following objects: – – – –

Pleiades, Ursa Major, Hyades and the the Sun.

These objects are at different ages. The Pleiades4 are an open stellar cluster and the age of the stars is about 100 million years (Fig. 6.10). The cluster will survive another 250 million years before it will disperse due to gravitational interactions. The distance is about 136 pc. How can the age of this cluster be determined? One way is with stellar model calculations. Another way is to look at the lowest-mass objects. In normal main sequence stars, lithium is rapidly destroyed by nuclear fusion reactions. Brown dwarfs can contain their Li, which has a low ignition temperature (2.5 ×106 K), the highest-mass brown dwarfs will burn it eventually. Therefore, the determination of the highest mass of brown dwarfs found in a cluster still containing Li provides an estimate for the age of the cluster. With this Li depletion method, an age of 115 million years was estimated. The total mass of the cluster is about 800 solar masses and it contains about 1000 members. The Hyades is the nearest open cluster, and the distance is about 47 pc. The members of the cluster (total mass estimated about 400 solar masses) are about 625 million years old. The five brightest member stars of the Hyades have consumed the hydrogen fuel at their cores and are now evolving to giant stars. From our perspective, 4 Also

called seven sisters.

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Fig. 6.10 The Pleiades cluster. NASA./ESA/AURA

the bright star Aldebaran (αTau) is in the center of the cluster, but it is not even a member of the cluster. It is at a distance of only 20 pc. When plotting the Ca-Index, the rotation velocity and the Li abundance and the age one recognizes the following trend: with increasing age, these values decrease, the Li abundance decreases more quickly because of the reasons mentioned above. This is illustrated in Fig. 6.11 adopted from [8]. Rotation plays an important role in stellar activity. In Table 6.3 we give stellar rotational velocities for different spectral types. Note that hotter stars tend to rotate more quickly than cooler stars. The decline in rotation for main sequence stars can be written as: e ∝ t −1/2

(6.12)

e is the angular velocity at the equator. t denotes the stellar age. This relation is also named Skumanich’s law. Note also that at the end of the main sequence ultracool and brown dwarfs experience faster rotation as they age. This is caused by conservation of angular momentum because they contract. Rotation leads to an equatorial bulge. An extreme example is the star Regulus (α Leo). The star has a measured equatorial velocity of 317 ± 3 km/s. This corresponds to a rotation period of 15.9 h. The dramatic high equatorial rotation rate of Regulus can be seen by noting that the equatorial rotational velocity corresponds to 86% of the velocity at which the star would break apart. The equatorial radius is 32% larger than the polar radius. Other rapidly rotating stars include Pleione and Vega. The break-up velocity of a star follows from the case where the centrifugal force at the equator is equal to the gravitational force.

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Fig. 6.11 The relation between stellar age and Ca-H-K activity, stellar rotation and Li abundance. Adapted from [8]

Table 6.3 Some parameters for main-sequence stars Spectral type Rotational velocity (km/s) O5 B0 B5 A0 A5 F0 G0

190 200 210 190 160 95 12

For stellar dynamos the differential rotation is also important. As we have shown for the Sun, differential rotation causes a concentration of magnetic field lines near the equator because the magnetic field is frozen in the photospheric plasma. Differential rotation means that the angular velocity decreases with increasing distance from the equator i.e. with increasing latitude. However, the reverse has also been observed, e.g., the star HD 31993. The star AB Doradus is the first star where differential rotation has been mapped in detail. AB Doradus is a young rapidly rotating star, and its period of rotation is 12 h (cf. Sun 25–34 days). The equatorial rotation velocity is > 80 km/s (cf. Sun 2 km/s). The Einstein X-ray satellite found X-rays. Rotational modulation in visible light was also found. This suggests strong stellar activity, starspots. Moreover, variable absorption features in the Hα profile were measured.

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These can be attributed to huge prominences. The age was estimated between 100 and 150 Myr. It is believed that turbulent convection is the mechanism that causes differential rotation. Through convection, energy is transported toward the surface. The rising plasma bubbles transports angular velocity of the star. When turbulence occurs through shear and rotation, the angular momentum can become redistributed to different latitudes through meridional flow. There are interfaces with sharp rotational gradients (like the tachocline in the Sun); at these interfaces the dynamo process works.

References 1. Desidera S et al (2003) A search for planets the metal-enriched binary HD 219542. Astron Astrophys 05. https://doi.org/10.1051/0004-6361:20030531 2. Vogt SS (1983), Spots, spot-cycles, and magnetic fields of late-type dwarfs. In: Byrne PB, Rodono, M (eds), IAU colloquium 71: activity in red-dwarf stars, volume 102 of astrophysics and space science library, pp 137–155 3. Vogt SS, Penrod GD (1983) Doppler imaging of spotted stars—application to the RS Canum Venaticorum star HR 1099. Publ Astr Soc Pacific 95:565–576 4. Vogt SS, Penrod GD (1983) Doppler imaging of starspots. In: Byrne PB, Rodono M (eds), IAU colloquium 71: activity in red-dwarf stars, volume 102 of astrophysics and space science library, pp 379–385 5. Vink JS (2011) The theory of stellar winds. Astrophys Space Sci 336:163–167 6. Blackman EG, Tarduno JA (2018) Mass, energy, and momentum capture from stellar winds by magnetized and unmagnetized planets: implications for atmospheric erosion and habitability. Monthly Not 481:5146–5155 7. Semel M (1989) Zeeman–Doppler imaging of active stars. I—basic principles. Astron Astrophys 225:456–466 8. Skumanich A (1972) Time scales for CA II emission decay, rotational braking, and lithium depletion. Astrophys J 171:565

Chapter 7

Solar and Stellar Activity: Cycles or Chaotic Behavior

In this chapter we discuss solar and stellar activity cycles. Are they really periodic? Do several periods exist and work simultaneously leading to intermittency effects of extremely low and high activity? Can such a behavior be derived from the dynamo equations?

7.1 Solar Activity Cycles In this section we review the long term behavior of solar activity.

7.1.1 The 11 Year and Other Cycles We have discussed in detail the 11-year solar activity cycle that was detected by Schwabe when comparing his sunspot recordings. However this cycle is not strictly 11-years. It is believed that there were 28 cycles between 1699 and 2008. This gives a mean cycle length of 11.04 Years. It was claimed however, that the longest cycle (1784–1799) consisted indeed of two cycles. Therefore, now the average length of a cycle is assumed to be 10.7 years. This was found by [1] using newly recorded solar drawings by the 18–19th century observers Staudacher and Hamilton. A sudden systematic occurrence of sunspots at high solar latitudes in 1793–1796 indicated the onset of a new cycle in 1793. This cycle was lost in the traditional Wolf sunspot series. Wolf observed sunspots from 1848 to 1893. He constructed the monthly sunspot numbers starting in 1749 using archival records and proxy data. Cycle 1 started in 1755. In the 1790s there were few sunspots so Wolf claimed that cycle number 4 was the longest ever observed. © Springer Nature Singapore Pte Ltd. 2020 A. Hanslmeier, The Chaotic Solar Cycle, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-15-9821-0_7

141

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7 Solar and Stellar Activity: Cycles or Chaotic Behavior

Fig. 7.1 Radiocarbon data indicate solar activity. Credit: Stuiver, 1979

During the above mentioned time interval (1699–2009), the cycle length varied from as short as 9 years to as long as 14 years. Also the amplitudes varied considerably as it was previously discussed. Direct observations of sunspots are available for the past four centuries, but we need longer time series to get a better insight into the complex solar dynamo and its periodic/aperiodic behavior as well as to identify the possible influence of solar activity on climate. As it was pointed out already dendrochronology can be used as a proxy. Radiocarbon concentrations allow a reconstruction of solar activity over the last 11,400 years. The data indicate that: • over the past 70 years solar activity has been at an exceptionally high level. • the last period of equal high solar activity was about 8,000 years ago. • over that last 11,400 years the Sun spent about 10% of the time at a similarly high activity level. • almost all of the earlier high-activity periods were shorter than the present episode. This was reviewed by [2] (Fig. 7.1). In additions to the 11 years cycle, there are cycles of longer lengths: • Gleissberg cycle: approximately 87 years • Suess cycle: approximately 210 years • Hallstatt cycle: named after a cool and wet period in Europe when glaciers advanced; 2,400 years. This is illustrated in Fig. 7.2. • a possible cycle cycle of 6,400 years, • during the upper Permian, 240 million years ago, a cycle of length 2,500 years was proposed from mineral layers.

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Fig. 7.2 Records of radiocarbon production. Residual (detrended; dashed lines) record of 14 C in which the long-term secular variation trend has been removed. The heavy red curve is a weighted smoothing function showing Hallstatt Cycles with periodicities of about 2,000 years. Credit: [3]

We stress that the magnetic cycle behaves differently. Reversal in the magnetic field polarity occurs every 11 years among the maximum phase of each 11 year cycle.1 • The Sun’s polar magnetic field attains a maximum at solar minimum. • The Sun’s polar magnetic field attains a minimum at solar maximum.

7.1.2 The Cycle 23 and 24 Solar cycle 23 started in August 1996. The maximum was in November 2001 and it ended in December 2008. The maximum sunspot number was 180.3, and the minimum count was 11.2. It was very strange to observe the transition from cycle 23 to cycle 24. During the minimum transit between these two cycles there was a total of 817 days with no sunspots. Compared to the preceding cycle, the activity level of cycle 23 was also fairly average. Large solar flares occurred on September 7 (class X17) and April 2 and April 15 2001 (X20+, X 14.4) as well on October 29 2003 (X10). On July 14 2000 there was a coronal mass ejection (CME) that produced an extreme (so-called G5 level) geomagnetic storm. This is known in the literature as the Bastille day event. The storm caused damage to GPS systems and power companies. Aurorae were visible as far south as Texas. This event started with an X 5.7-class flare in 1 Reversals

of planetary magnetic field polarities are also known, for Earth the timescale of such reversals is on the scale of milenia.

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7 Solar and Stellar Activity: Cycles or Chaotic Behavior

Fig. 7.3 Composite image of an aurora seen by the DMSP satellite on 30 October 2003. Source http://www.af.mil/photos/mediasearch.asp?q=aurorabtnG.x=0btnG.y=0

an active sunspot region and fifteen minutes later energetic protons bombarded the Earth’s ionosphere. The geomagnetic superstorm occurred on July 15–16. (level G5). Between October 28 and November 4 2003 there was the Halloween event. Also during this event Aurorae could be seen in low geographic latitudes (see example in Fig. 7.3). Astronauts aboard the International Space Station, ISS, had to stay inside more shielded parts because they had protect themselves against the high radiation levels. Higher radiation levels were also measured by the spacecrafts Ulysses (which was near Jupiter at that time) and Cassini (which was near Saturn).

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7.2 Chaotic Behavior of Solar Activity As we discussed in the first chapter there are several methods in non linear dynamics to study chaotic behavior of a system. In this section we will review the application of these methods to the study of the long term behavior of solar activity. In the 21st century technological progress has made human life increasingly dependent on satellites. Solar coronal mass ejections (CMEs) solar flares, and high speed solar wind streams often cause disturbances of the Earth’s magnetosphere, atmosphere and even surface. Therefore, the time-varying Sun as the main source of space weather is of extreme interest for our modern technological society. The solar cycle as a strange attractor was first introduced by [4]. The author stresses that the Maunder minimum and other periods of low or extremely high solar activity are strong indicators of a non linear acting dynamo mechanism. This observational facts are regarded as hints for a stochastic nature of solar activity.

7.2.1 Lyapunov Exponent Lyapunov exponents are the most basic and useful dynamical diagnostic for deterministic chaotic systems. They yield a qualitative measure to characterize the stability and instability of systems with respect to initial conditions. A system with one or more positive Lyapunov exponents is defined to be chaotic. Let us consider a continuous dynamical system in an n-dimensional phase space. We monitor the long-term evolution of an infinitesimal n-sphere of initial conditions. The sphere will be deformed during the system’s evolution. It will become an nellipsoid due to the locally deforming nature of the flow. The i th one-dimensional Lyapunov exponent is then defined in terms of the length of the ellipsoidal principal axis pi (t) 1 pi (t) log2 t→∞ t pi (0)

λi = lim

(7.1)

The λi are ordered from largest to smallest. The Lyapunov exponents are related to the expanding or contracting nature of different directions in phase space. The exponents measure the rate at which system processes create or destroy information. The exponents are expressed in bits of information/s or bits/orbit for a continuous system and bits/iteration for a discrete system. Let us consider an example: The Lorentz attractor has a positive Lyapunov exponent of 2.16 bits/s. If an initial point were specified with an accuracy of one part per million (20 bits, 220 ∼ 106 ), the future behavior could not be predicted after about 9 s because: 20 bits 2.16 bits s−1

(7.2)

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7 Solar and Stellar Activity: Cycles or Chaotic Behavior

After this time the small initial uncertainty will essentially cover the entire attractor. This is explained in the paper [5]. Two general approaches for their estimation exist: • geometrical approach, • Jacobian approach. In both cases a long time series is important. Let us consider the calculation of the Lyapunov exponents with the Jacobian approach. We consider a discrete dynamical system: xk+1 = f (xk ), k = 0, 1, ..., (7.3) xk is the state vector in the R m space and f is a continuously differentiable nonlinear function. The linearized system for a small range about the trajectory in the phase space can be written as: xk+1 = Jk xk , with Jk =

k = 0, 1, ..., ∂f | x∈R m ∂x

(7.4)

(7.5)

the Jacobian matrix in point k. The Lyapunov exponents are defined as: let Y k = . Jk−1 Jk−2 . . J0 :  T  2k1  = lim Y k Y k (7.6) k→∞

This is a positive symmetric definite matrix. The logarithms of its eigenvalues are called the Lyapunov exponents. The practical calculation can become extremely complex for large values of k. In this case the so-called QR factorization algorithm is used. Given an orthogonal Q 0 such that Q 0T Q 0 = I

(7.7)

Then solve Z k+1 = Jk Q k

k = 0, 1, ...

(7.8)

and obtain the decomposition Z k+1 = Q k+1 Rk+1 with Q k an orthogonal matrix and Rk+1 an upper triangular matrix with positive diagonal elements, solve:   1 1 log((Rk )ii ...(Rk )ii ) = lim log (R j )ii , k→∞ k k→∞ k j=1 k

λk = lim

i = 1, ..., m

(7.9) We briefly discuss the polynomial methods. The dynamical behavior of a system is described with the following non linear differential equation:

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147

y(k + 1) = f (y(k))

(7.10)

where f is a continuously differentiable function and y(k) is delayed vector: y(k) = [y(k − m + 1)y(k − m + 2)...y(k)]

(7.11)

m is the minimum embedding dimension of time series. Further details can be found in the literature. Finally, we mention the adaptive method for estimation of Lyapunov exponents. The most suitable structure (embedding dimension and degree of non linearity) should be estimated, dopt , n p . The state vector is defined as the following delay vector: ⎞ ⎛ ⎛ ⎞ y(k − (dopt − 1)) x1 (k) ⎜ x2 (k) ⎟ ⎜ y(k − (dopt − 2)) ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ . ⎜ ⎟ ⎜ ⎟ (7.12) x(k) = ⎜ ⎟=⎜ ⎟ . . ⎟ ⎜ ⎜ ⎟ ⎝ . ⎠ ⎝ ⎠ . xopt (k) y(k) ⎛

This can be written as:

⎜ ⎜ ⎜ x(k + 1) = ⎜ ⎜ ⎜ ⎝

⎞ x2 (k) x3 (k) ⎟ ⎟ ⎟ . ⎟ ⎟ . ⎟ ⎠ . f (x(k))

(7.13)

The Jacobian matrix Jk in each point k of the typical trajectory for this canoncial representation is: ⎤ ⎡ 0 1 0 ... 0 ⎢ 0 0 1 ... 0 ⎥ ⎥ ⎢ (7.14) Jk = ⎢ . . . .. ⎥ . .. .. .. ⎣ .. . ⎦ D f 1 D f 2 . . . D f d−1 D f d The Lyapunov exponents follow from: λk+ =

1 (kλi (k) + log(R(k + 1)ii )) k+1

i = 1, . . . , m

(7.15)

In the paper of [6] several cases studies were made: sunspot number, disturbance storm time and proton temperature. These values were considered during coronal mass ejections, CMEs. The Lyapunov exponents were estimated from the Jacobian approach. The method was proposed by [7].

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7.2.2 Estimation of Dimension In this section we outline the estimation of the dimension which characterizes an attractor and its stochastic behavior. This value is close to the minimum number of independent variables needed to describe the system. Suppose we have a time series such as the number of sunspots as a function of time x(t). We transform this series into an n-dimensional phase space with vectors: Xk = X(tk ) = [x(tk ), . . . , x(tk + (n − 1)τ )]

(7.16)

where τ is the delay time. According to Grassberger and Procaccia [8], the dimension d can be calculated as a slop of the curves log2 C(r ) − log2 r : d(n) =  log2 C(r )/ log2 r

(7.17)

r is the hypersphere diameter C(r ) is the correlation sum: C(r ) =

1  (r − ||Xi − X j ||) N 2 i= j

(7.18)

 is the heaviside function. One way to describe this function is: (n) = 0 (n) = 1

n 2ds + 1. Considering the Wolf numbers two values are found in the literature, i.e. two plateaus around the values • d ∼ 4.3 characterizes the basic period, the 11-year cycle; the local features of the attractor. • d ∼ 1.6 characterizes a global modulation. These values are cited in [9]. These authors analyzed the following time series: 1. Wolf number average over one year: dimension of the attractor was 3.3 2. Wolf number average over a month: dimension of the attractor was 4.3 3. radiocarbon data: dimension of the attractor was 4.73.

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They found positive values of the Kolmogorov entropy and the first Lyapunov exponent. This indicates a stochastic behavior of the solar activity. Long time series of solar activity proxies can also be analyzed with nonlinear dynamics methods. In [10] 10 Be-data were used and in [11] 14 C-data. The time span of the data covers about 10 000 years. The solar activity cycle is found to be on the edge of chaotic behavior. This can explain the observed intermittent period of longer lasting solar activity minima.

7.2.3 Wavelet Analysis In [12] a wavelet analysis method was used to derive a measure of the disorder content of solar activity. The method is based on wavelet entropy. The Fourier analysis is suitable for detecting and quantifying constant periodic fluctuations in a time series. However, for intermittent and transient multiscale phenomena, the wavelet transform is more adequate. With the wavelet transform we are able to detect time evolutions of the frequency distribution. As we have discussed, the long term behavior of solar activity occurs in such a way. Let us consider a time series x(t) given as a function of both time t and frequency scale a: Then W (a, t) =



1 a 1/2

∞

x(t) ∗

−∞



t −τ a

 dt

(7.22)

where  is called an analyzing wavelet. It must fulfill the following conditions: 

∞

c = 0

1 2 ˆ |(ω)| dω < ∞ ω

(7.23)

where ˆ (ω) =



∞ −∞

(t) exp(−iωt)dt

(7.24)

is the related Fourier transform. The parameters a and t denote: • a dilation scale factor • t time shift parameter. The local wavelet spectrum is defined as: Pω (k, t) =

1 2c k0

 |W

 k0 , t |2 k

k≥0

(7.25)

k0 denotes the peak frequency of the analyzing wavelet . From the local wavelet spectrum we can derive a mean or global wavelet spectrum Pω (k):

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7 Solar and Stellar Activity: Cycles or Chaotic Behavior

 Pω (k) =

∞ −∞

Pω (k, t)dt

(7.26)

The relationship between the ordinary Fourier spectrum PF (ω) and the mean wavelet spectrum Pω (k) is given by: Pω (k) =

1 c k



∞

ˆ PF (ω)|

0



 k0 ω 2 | dω k

(7.27)

The mean wavelet spectrum is the average of the Fourier spectrum weighted by the square of the Fourier transform of the analysing wavelet  shifted at frequency k. The Morlet wavelet is given by (η) = π −1/4 exp(iω0 η) exp(−η2 /2)

(7.28)

ω0 is a non dimensional frequency and equals to 6 in order to satisfy the conditions described above. A wavelet entropy is defined by: W St = −



pt,l log2 ( pt,k )

(7.29)

Pω (k, t) Pω (k, t)dω

(7.30)

k

pt,k = 

The last equation is the energy probability function for each level. How can we interpret a wavelet analysis? • An ordered activity corresponds to a narrow frequency distribution of energy, low wavelet entropy • A random activity corresponds to a broad frequency distribution with high wavelet entropy. • Higher values for wavelet entropy mean higher complexity, higher irregular behavior, and lower predictability. The wavelet procedure was applied in [12] to the record of solar activity given by the mean monthly number of sunspots. The data are available in the SIDC archive. In total 3013 observations were used covering the time span from 1749-2000.04. The time evolution of the wavelet entropy was also calculated. It was found that the degree of disorder reaches maximum local values at times of high solar activity. The recorded maximum entropy value has been reached during the maximum of solar cycle 4 (1789), the entropy results quite low for longer periods such as solar cycles 5–7 (1798–1828) and cycle 9–14 (1848–1912). For the cycles 21, 22 and 23 a clear average increase of the wavelet entropy was obtained. This suggests a more complex dynamic with a higher level of disorder due to broad frequency energy distribution. This is also consistent with a lower value for Lyapunov’s predictability time.

7.2 Chaotic Behavior of Solar Activity

151

An important side effect of such an analysis is that with the use of wavelet entropy we can quantify the intermittent degree in solar activity. This is of particular interest for possible stochastic fluctuations in meridional circulation of plasma.

7.2.4 Hurst Exponent Analysis In the work of [13] it is examined whether or not non-periodic variations in solar activity are caused by a white-noise, random process. A white noise signal is continuous and has no distinct pattern. For that purpose the Hurst exponent was calculated. Let us assume a time series. White noise means that amplitude of the variations at different times are independent of one another. The Hurst exponent describes the coherence or persistence in a long time series. The first application of this method to solar physics was made already in 1969 ([14]. ). The authors found that the continuum variation of sunspot numbers in the time interval range of from 1 to 200 years was not random. The Hurst exponent was significantly larger than 21 . The problem of whether the solar dynamo is quasi-periodic or chaotic is addressed in [15] by examining 1500 years of sunspot, geomagnetic, and auroral activity cycles. The authors discuss sub-harmonics of the fundamental solar cycle period during the years preceding the Maunder minimum and investigate the loss of phase of the subharmonic on emergence from it. These phenomena are indicative of chaos. Therefore, solar dynamo is chaotic and operates in a region close to the transition between period doubling and chaos. Since Maunder-type minima reoccur irregularly for millennia, it appears that the sun remains close to this transition to and from chaos. This is postulated to be a universal characteristic of solar type stars caused by feedback in the dynamo number.

References 1. Usoskin IG, Mursula K, Arlt R, Kovaltsov GA (2009) A solar cycle lost in 1793–1800: early sunspot observations resolve the old mystery. Astrophys J Lett 700:L154–L157 2. Solanki SK, Usoskin IG, Kromer B, Schüssler M, Beer J (2004) Unusual activity of the Sun during recent decades compared to the previous 11,000 years. Nature 431:1084–1087 3. Damon PE, Sonett CP (1991) Solar and terrestrial components of the atmospheric C-14 variation spectrum. In: Sonett CP, Giampapa MS, Matthews MS (eds) The sun in time, pp 360–388 4. Ruzmaikin AA (1981) The solar cycle as a strange attractor. Comments Astrophys 9:85–93 5. Wolf A, Swift JB, Swinney HL, Vastano JA (1985) Determining Lyapunov exponents from a time series. Phys D Nonlinear Phenom 16:285–317 6. Mirmomeni M, Lucas C (2009) Analyzing the variation of Lyapunov exponents of solar and geomagnetic activity indices during coronal mass ejections. Space Weather 7:S07002 7. Khaki-Sedigh A, Ataei M, Lohmann B, Lucas C (2004) Adaptive calculation of lyapunov exponents from time series observations of chaotic time varying dynamical systems. Nonlinear Dyn Syst Theory 4:145–159

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8. Grassberger P, Procaccia I (1983) Measuring the strangeness of strange attractors. Phys D Nonlinear Phenom 9(1–2):189–208 9. Ostriakov VM, Usoskin IG (1990) On the dimension of solar attractor. Solar Phys 127:405–412 ˇ 10. Hanslmeier A, Brajša R, Calogovi´ c J, Vršnak B, Ruždjak D, Steinhilber F, MacLeod CL, Ivezi´c Ž, Skoki´c I (2013) The chaotic solar cycle. II. Analysis of cosmogenic 10 Be data. Astron Astrophys 550:A6 11. Hanslmeier A, Brajša R (2010) The chaotic solar cycle. I. Analysis of cosmogenic 14 C-data. Astron Astrophys 509:A5 (2010) 12. Sello S (2000) Wavelet entropy as a measure of solar cycle complexity. Astron Astrophys 363:311–315 13. Ruzmaikin A, Feynman J, Robinson P (1994) Long-term persistence of solar activity. Solar Phys 149:395–403 14. Mandelbrot BB, Wallis JR (1969) Some long-run properties of geophysical records. Water Resour Res 5:321–340 15. Feynman J, Gabriel SB (1990) Period and phase of the 88-year solar cycle and the Maunder minimum - evidence for a chaotic sun. Solar Phys 127:393–403

Chapter 8

Chaotic Dynamo Models

In this section we discuss dynamo models that are used to explain the solar activity cycle. Any solar dynamo model should explain several observational features of solar activity: such as different solar cycles, polarity law, meridional flow, 22 year magnetic cycle etc. The basic ideas about an α − ω dynamo were already discussed in preceding chapters.

8.1 The Solar Tachocline and Convective Zone In this section we examine the region where magnetic flux is produced. The interior of the Sun is structured into different layers by the mode of energy generation and energy transport from the interior to the solar surface.

8.1.1 The Solar Interior The inner 20% by solar radius is the region where nuclear fusion powers the solar energy trough the p-p cycle . There are two important byproducts of these reactions: photons and neutrinos. This is shown in Fig. 8.1. Near the center of the Sun the temperature is about 15.7 million K and the density is about 150 g/cm3 . High energy photons are released in the fusion reactions. These photons travel outwards but due to the high density of the plasma they are absorbed within a few millimeters and then re-emitted in random directions at slightly lower energies. This is a free-free scattering process. As a result of these scattering processes, the photons diffuse outwards from the core to the Sun’s surface through a slow random walk process. The transit time is about 100,000 years. The neutrinos © Springer Nature Singapore Pte Ltd. 2020 A. Hanslmeier, The Chaotic Solar Cycle, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-15-9821-0_8

153

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8 Chaotic Dynamo Models

Fig. 8.1 The p-p cycle. 4 protons are converted into a He nucleus, 2 of the protons decay to neutrons, a neutrino and a positron is emitted. Sarang

are much faster, they reach the solar surface within a few seconds, because they have an extremely low scattering coefficient and they do not interact with the plasma. Observing solar neutrinos therefore gives valuable insight in the nuclear fusion processes. Below ∼0.7R the temperature gradient becomes sufficiently small and radiative transfer by photons is the primary means of energy transport from the core upwards. At about 0.7 R the temperature is approximately 2 million K. Here, radiation becomes less efficient because the opacity increases. Much of the energy transport occurs by thermal convection. Therefore, the internal structure of the Sun is given by: • Solar core: here the energy is generated by nuclear fusion. • Radiative zone: the energy is transported outward by radiative processes • Convective zone: reaches up to the surface, energy transport by convection.

8.1.2 The Tachocline The radiative zone and the convective zone are separated by a thin layer. The thickness is about 0.05 R [1]. This layer is called the tachocline. The existence of the tachocline was first proven from helioseismic inversions (e.g. [2]). The solar rotation behaves differently below and above the tachocline: • Below the tachocline: the Sun rotates like a rigid body, • Above the tachocline: the Sun rotates differentially, i.e. faster at the equator than at higher latitudes.

8.1 The Solar Tachocline and Convective Zone

155

Fig. 8.2 Solar differential rotation as a function of distance from the center. At the tachocline the differential rotation ceases and the Sun rotates like a rigid body. Image courtesy of GONG: http://gong.nso.edu/

Therefore, the tachocline that separates the upper lying convection zone from the inner part that rotates like a rigid body, is a region where strong shears are to be expected. The transition between the two rotation laws occurs in a thin, unresolved layer as shown in Fig. 8.2: the differential rotation of the Sun (curves at different heliographic latitudes 00 ,150 ..). These curves converge at about 0.66 R . The tachocline is the origin of the observed features of solar activity like coherent magnetic flux tubes that form sunspots and magnetic prominences. This was pointed out in varous studies, for example [3]. However, this layer also influences the dynamics of the convection zone. The latitudinal entropy gradient in the tachocline explains the differential rotation in the convection zone and the tachocline. From helioseismology the thickness was determined only 4% of the solar radius [4]. Therefore, the angular momentum transport in the tachocline must be predominantly horizontal and frictional—a down gradient in angular momentum velocity. Such a transport could be explained by • Horizontal turbulence [5]. • Maxwell stresses from a primordial magnetic field [6]. The Maxwell stress tensor represents the interaction between electromagnetic forces and mechanical momentum. If we consider for example a point charge moving freely in a homogeneous magnetic field, the forces on the charge are given by the Lorentz force law: F = q(E + v × B)

(8.1)

The Maxwell stress tensor is part of the electromagnetic stress-energy tensor:

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8 Chaotic Dynamo Models

T μν =

  1 1 F μα Fαν − ημν Fαβ F αβ μ0 4

(8.2)

where ημν is the Minkowski metric. The Maxwell stress tensor becomes: σi j = 0 E i E j +

  1 1 1 2 0 E 2 + Bi B j − B δi j μ0 2 μ0

(8.3)

where 0 , and μ0 are the electric and magnetic constants and δi j is Kronecker’s delta. The explanation by this process has several advantages over the turbulent model: it explains (i) the thickness of the tachocline, and (ii) uniform rotation of the radiation zone. The only condition is that the field remains confined below the convection zone. The field lines should not cross the tachocline into the convection zone. This is also called the magnetic confinement problem. One can estimate the time scale of ohmic decay in the radiation zone: τ=

4π σ R R c2

(8.4)

σ is the electrical conductivity, R = 1 R and R ≈ 0.3 R . The electrical conductivity for plasma is very high and increases with T 3/2 . For example σCu = 107 . In many cases of stellar plasma physics σ → ∞. Therefore, we see, that the ohmic decay time for a magnetic field in the radiative zone is at least in the order of the age of the Sun. A fully nonlinear 3D numerical model of the solar tachocline was given by [7]. In their simulation a tachocline forms self-consistently at the interface between the convective zone and the radiation zone and remains for a long time. Uniform rotation is maintained in the radiation zone confined by a primordial magnetic field.

8.1.3 The Solar Convection The convection is essential to explain solar activity. Here, we must make a distinction between small scales and large scales. • Small scales: Convection directly sustains magnetic fields, this explains the Sun’s small-scale dynamo field. • Large scales: The turbulent convection in the rotating Sun produces Reynolds stresses which drive large scale (global) motions. The velocity field of a flow can be given as (8.5) vi =< vi > +vi

8.1 The Solar Tachocline and Convective Zone

157

This is also called Reynolds decomposition: the velocity field of a flow can be decomposed into 1. mean part v¯ i 2. fluctuating part vi vi is the velocity component in the xi direction. The Reynolds stresses are given by (in the case of constant density) Ri j =< vi , vj >

(8.6)

The Reynolds stress tensor is also given by: R¯ i j = ρ Ri j

(8.7)

These motions amplify the global magnetic field. Bipolar active regions, such as large sunspot groups occur, and these are stressed by convective motions and produces flares and all other space weather relevant phenomena. On long time scales, magnetic fields also transport angular momentum from the Sun, which is known as magnetic braking. Convection also leads to mixing of plasma inside the stars. Such a mixing can significantly affect the lifetimes of main sequence stars. To understand convection we must take into account that solar convection occurs on extreme values. For example the density contrast between the top (corresponds to the solar surface) and the bottom (corresponds to the tachocline) is about 105 . The typical regime that describes solar convection is Re ∼ 1014 , Ra ∼ 1020 and Pr ∼ 10−6 , where Re is the Reynolds number, Ra the Rayleigh number and Pr is the Prantl number.

8.1.4 Solar Differential Rotation The basic idea to explain solar activity at large scale is a self sustaining dynamo. An originally poloidal field is concentrated by the action of differential rotation to a toroidal field. This field then as transformed into a poloidal field again by the influence of the coriolis force and convection, both induce a poloidal component. Here we give some basic equations following the fundamental paper by [5]. The basic equations are the equation of motion with respect to a differentially rotating frame with angular velocity (r, t), r = |r| denotes the radial coordinate. The absolute velocity is given by ×r+V

(8.8)

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8 Chaotic Dynamo Models

where V is the local velocity in the differentially rotating frame. The following equations are needed to describe the motions and radiative transport: • Equation of the conservation of mass: ∂ρ + ∇.(ρV) = 0 ∂t • conservation of momentum:   ∂V ˙ ×r =− + (V .∇)V + 2 × V +  ρ ∂t ∇ P − ρ∇ + ∇.|τ |

(8.9)

(8.10) (8.11)

• Conservation of heat (or entropy) ρT

∂S + ρT V.∇ S = ∇.(χ ∇T ) ∂t

(8.12)

∂ ∂ +u ∂t ∂r

(8.13)

Please note that ˙ = 

which is called substantial derivative. u denotes the radial component of the velocity. ρ is the density, P is the pressure, T is the temperature and S is the specific entropy. |τ | is the viscous stress (turbulent) stress tensor, χ the thermal conductivity and  is the gravitational potential. At the distances we are interested in, nuclear energy production can be neglected,  is prescribed and no magnetic field is assumed. In addition to the given set of equations some simplifications are assumed: • Axisymmetric flow field, V = V(r, , t). • No oblateness of the Sun due to centrifugal force. • The flow occurs on a long time scale, acoustic modes may be filtered out. ∂ρ/∂ T = 0 anelastic approximation. • Thermodynamic fluctuations that appear as quadratic terms can be neglected. • Small Rossby Number. The Rossby number is given by the ratio of the coriolis acceleration to the advection term.  r     1 (8.14) R0 = h 2  is the differential rotation, h is the vertical scale of variation of the flow. • The tachocline is very thin h r . The scale height of any function describing the structure of the layer will be of order h. This also implies neglecting the vertical velocity when compared with the horizontal velocity in the meridional plane. • Geostrophic flow: below the convection zone, the turbulent viscous forces are much less important than the Coriolis forces. This also means that the so-called

8.1 The Solar Tachocline and Convective Zone

159

Eckman numbers must be small: νV 2h 2 νH = 2 2R

EV = EH

(8.15) (8.16)

Using these assumption simplifies the set of equations considerably. We introduce spherical coordinates (r, , ) Each dependent variable is separated into • mean value, and • perturbation. The temperature is:

and

T (r, t) + T¯ (r, , t) 

π

 sin  = 0

(8.17)

(8.18)

0

The linearized equation of state is:

The velocity field is

ρ¯ T¯ P¯ = + P ρ T

(8.19)

¯ sin ) V = (u, v, r 

(8.20)

¯ is the differential rotation with respect to the rotation of .  Introducing a stream function for the meridional flow we can write: ∂ ∂x ∂ rρ sin v = ∂r r 2 ρu =

(8.21) (8.22)

with x = cos . This leads to a complex set of equations. The first two equations express hydrostatic and geostrophic balance: T¯ 1 ∂ P¯ +g =0 ρ ∂r T ¯ ¯ = 1 ∂P −2r ×  ρr ∂ x −

The third equation describes the advection

(8.23) (8.24)

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8 Chaotic Dynamo Models

¯ ∂ ∂ + 2 × = ∂t ∂r



¯ ¯ ρνV r 4 ∂∂r + ρ ∂∂x ν H (1 − x 2 )2 ∂∂x

ρr 2 (1 − x 2 ) (1 − x 2 )

r 2 ∂r∂

(8.25) (8.26)

The fourth equation gives the diffusion of angular momentum and heat: ∂ T¯ N 2 T ∂ 1 ∂ + = ∂t g ρr 2 ∂ x ρC p r 2 ∂r

 χr

¯

2 ∂T

∂r

(8.27)

where the buoyancy frequency is given by N2 =

g (∇ad − ∇) Hp

(8.28)

where H P is the pressure scale height and ∇ = ∂ ln T /∂ ln P the logarithmic temperature gradient. The turbulent stresses may be anisotropic, therefore we have vertical (νV ) and horizontal (ν H ) components. These equations describe the evolution of differential rotation. A further simplification is made: the dependence on latitude at a certain distance from the center, r = r0 is imposed from above the convection zone, there is no feedback of the flows in the radiative zone. Observations of the differential rotation are usually expressed as polynomials x = cos θ : ¯ 0 , x) = 0 (1 − ax 2 − bx 4 ) zc =  + (r

(8.29)

Form helioseismology a rotation law at the base of the convection zone was found (see [8]): zc = (462 − 64x 2 − 73x 4 ) nHz 2π

(8.30)

Several boundary conditions must be assumed. Since we are only covering the basic ideas, the reader is referred to the above cited paper. One interesting result of such a model is an estimate for the thickness of the tachocline: h=

3π r0 2μ4



 N

1/2 

κ νH

1/4 (8.31)

where μ4 = 4.933; the horizontal turbulence enforces a stationary state, in which the advection of angular momentum is balanced by the Reynolds stresses acting on the horizontal shear. The thickness of the tachocline in the present Sun is approximately: h ∼ 20000σ −1/4

(8.32)

8.1 The Solar Tachocline and Convective Zone

161

Here σ is the Prandtl number σ = ν H /κ. The Prandtl number is the ratio of momentum diffusivity to thermal diffusivity. Large values of σ mean that the momentum diffusivity dominates the behavior.

8.2 The Solar Dynamo In this section we discuss solar dynamo models and examine them whether they show chaotic behavior. We start with a very simple model.

8.2.1 The Rikitake Model The dynamics of MHD dynamos is extremely complex. A first insight can be gained by extremely simplified models. Even these models show that very simple systems can behave in an extremely complex manner.

8.2.2 Coupled Disc Dynamos Rikitake studied two dynamos. Each dynamo consists of a metallic disc driven by a certain couple. If a magnetic field crosses either rotating disc it induces an electromotive force that can be used to drive a current through a coil which produces a magnetic field that crosses the other disc (Fig. 8.3). What is the analogy of this model to for example the Earth’s magnetic field?

Fig. 8.3 The Rikitake model for dynamo action

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8 Chaotic Dynamo Models

• The Earth’s magnetic poloidal field is the magnetic field through one of the disc dynamos. • The Earth’s toroidal field is the other dynamo. The poloidal magnetic field of the Earth drives the currents that produce the toroidal field either by (i) differential rotation or (ii) the α effect. The toroidal field of the Earth drives the currents that produce the poloidal field (α effect). Such a system can be written as a simple system of ordinary differential equations. I1 , I2 are the currents, L is the self inductance, R the resistance associated with each dynamo, M the mutual inductance, J the moment of inertia of either dynamo about its axis. dI1 1 2 1 + R11 I 1 = M23 I ω dt dI2 2 2 2 + R22 I 2 = M14 L 22 I ω dt dω1 3 1 2 = F 3 − M12 J33 I I dt dω2 = F 4 − M12 4 I 1 I 2 J44 dt L 11

(8.33) (8.34) (8.35) (8.36)

Let us fix the difference ω1 − ω2 of the angular velocities. The systems has two steady solutions: • First solution: (I10 , I20 , ω10 , ω20 ) • Second solution: (−I10 , −I20 , ω10 , ω20 ) However, the steady solutions are unstable. If we start extremely close to solution (i) it will systematically depart through oscillations of increasing amplitude to the state (ii) and vice versa. Random occurrence of reversals is an intrinsic property of such a simple system. The equations for the Rikitake system are equivalent to equations of motion in the case of a magnetohydrodynamic dynamo. By combining Faraday’s and Ohm’s law the induction equation is obtained 1 ∂B = − ∇ × (∇ × B) + ∇ × (v × B) ∂t σ

(8.37)

The right hand side of this equation includes: j

• diffusion term; this corresponds to the expressions Ri I j in the Rikitake system (a resistance) • the electromotive force M jk I j ωk generated by the rotating disc is equivalent to the electromotive force due to the interaction between the fluid motion and magnetic flux (second term). The current I 1 is a toroidal current, ω2 is the mean differential rotation in the core. 2 1 2 This differential rotation is generated by the driving force F 4 . Therefore, M14 I ω

8.2 The Solar Dynamo

163

Fig. 8.4 A simple Rikitake model. The obtained time series are plotted for the coordinates

represents the production of toroidal magnetic field due to the ω-effect. I 2 is the total poloidal current. ω1 is a measure of the intensity of the α-effect, which is generated 1 2 1 I ω is the production of the poloidal magnetic by the driving force F 3 . The term M23 field due to the α effect. Both ω- and α-effects reinforce the original magnetic field.

8.2.3 A Simple Rikitake Model In this section we give a simple Python program that models a Rikitake system. The set of differential equations is: dx = −μx + zy dt dy = −μyy + x(z − a) dt dz = 1 − xy dt

(8.38) (8.39) (8.40)

In Fig. 8.4 the results for the parameters μ = 1, a = 5 are given. We give three plots in which the obtained time series for the coordinates are plotted (a) x(t) versus y(t), (b) y(t) versus z(t) and (c) x(t) versus y(t). Figure 8.5 is a 3-D plot for all coordinates that clearly shows the attractor.

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Fig. 8.5 A simple Rikitake model. The 3-D plot of all coordinates

Fig. 8.6 A simple Rikitake model. The obtained time series are plotted for the coordinates. with a different set of parameters

8.2 The Solar Dynamo

165

Fig. 8.7 A simple Rikitake model. The 3-D plot of all coordinates with a different set of parameters

In order to demonstrate the dependence of the results on the input parameters, we give another set of examples in Figs. 8.6 and 8.7. These two Figures show the results for the parameters a = 5, μ = 5. The program listing is as follows:

import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import axes3d

# noqa: F401 unused import

# ##################################################################### # Function declaration # #####################################################################

def rikitake(x, y, z, **kwargs): """ Calculate the next coordinate X, Y, Z for 3rd-order Rikitake system Parameters ---------x, y, z : float Input coordinates Z, Y, Z respectively kwargs : float mu, a - are Rikitake system parameters """ # Default Rikitake parameters: aa = kwargs.get(’a’, 5) mu = kwargs.get(’mu’, 1) # Next step coordinates:

166

8 Chaotic Dynamo Models x_out = -mu * x + z * y y_out = -mu * y + x * (z - aa) z_out = 1 - x * y return x_out, y_out, z_out

# ##################################################################### # Calculate attractor # ##################################################################### NW = 10000 dt = 100

# Number of points # Step for equations (leave default as 100)

# Create zero arrays for coordinates xt = np.zeros(NW) yt = np.zeros(NW) zt = np.zeros(NW) # Set initial values for [X, Y, Z] xt[0], yt[0], zt[0] = 1.0, 0.0, 0.5 # Set system parameters # -->> AH set here the different parameters! original value= a=5, mu=1 params = { ’a’: 5, ’mu’: 5 } # Calculate the next coordinates of system for i in range(NW-1): x_next, y_next, z_next = rikitake(xt[i], yt[i], zt[i], **params) xt[i+1] = xt[i] + (x_next / dt) yt[i+1] = yt[i] + (y_next / dt) zt[i+1] = zt[i] + (z_next / dt) # ##################################################################### # Plot results # ##################################################################### # Plot 3D model

fig = plt.figure(’3D model of chaotic system’) ax = fig.gca(projection=’3d’) ax.plot(xt, yt, zt, ’o-’, linewidth=0.1, markersize=0.3) ax.set_xlabel("X") ax.set_ylabel("Y") ax.set_zlabel("Z") ax.set_title("Rikitake Attractor") plt.show()

# Plot 2D coordinates in time axis lin = plt.figure(’Coordinates evolution in time’)

8.2 The Solar Dynamo

167

plt.subplot(311) plt.plot(xt,yt, linewidth=0.75) plt.grid() plt.xlabel(’X’) plt.ylabel(’Y’) plt.subplot(312) plt.plot(yt,zt, linewidth=0.75) plt.grid() plt.xlabel(’Y’) plt.ylabel(’Z’) plt.subplot(313) plt.plot(xt, zt, linewidth=0.75) plt.grid() plt.xlabel(’X’) plt.ylabel(’Z’) plt.tight_layout()

plt.show()

8.2.4 The Babcock Leighton Solar Dynamo The basic idea of this dynamo model to explain the solar cycle, and other related phenomena such as the migration of active photospheric regions towards the solar equator etc. is as follows. We follow one of the earliest papers that describes these processes [9]. The Sun possesses a polar dipole field. The strength is about 1 gauss. Near the solar equator bands of toroidal magnetic field should exist beneath the photosphere. Each band is about 30 degrees wide in latitude and successive bands and corresponding bands north and south of the equator have opposite signs. The bands appear at the surface first at about ±400 latitude as photospheric bipolar active regions. During the solar cycle of about 11 years they migrate from this latitude to the equator, where they disappear. The axis of the cyclonic convective cells are in the x-direction, and the shearing motion is in the z-direction. Let us assume an initial field given by a magnetic band consisting of a field Bt in the z-direction. The cells move upwards carrying Bt upwards. The field is twisted into a plane perpendicular to Bt and as result, a loop is formed. This is illustrated in Fig. 8.8. The next step is diffusion. A number of magnetic loops in a meridional plane is shown in Fig. 8.9. These loops are formed from a toroidal field by cyclonic convective cells. By interdiffusion and coalescence of loops a large-scale poloidal field is formed (Fig. 8.10). The magnetic field of the loop is represented by a vector potential A0 in the z-direction. The rate of generation of this vector potential depends on

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8 Chaotic Dynamo Models

Fig. 8.8 The upward motion of a magnetic flux bundle through convective motions

Fig. 8.9 The upward motion of a magnetic flux bundle through convective motions

8.2 The Solar Dynamo

169

Fig. 8.10 The formation of a poloidal field by diffusion and merging of loops

• Bt , • size of the cyclonic cells Γ , and • the term ∇ 2 A0 /μσ takes into account the diffusion of the field. σ is the electric resistivity. We can write:

∂A0 1 2 0 = −ex  Bt + ∇ A ∂t μσ

(8.41)

There is also an interaction of the cell with the y-component of ∇ × A0 to generate a new field A1 : 1 2 1 ∂A1 ∂ A0 = ey  + ∇ A (8.42) ∂t ∂x μσ The cells will interact with A1 to form a field A2 : 1 2 2 ∂A2 ∂ A1 = −ez  + ∇ A ∂t ∂x μσ

(8.43)

The final step is that the large scale shearing ez v(x, y) of the fluid will shear the xand y- components of the field to regenerate Bt according to:

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8 Chaotic Dynamo Models



∞ ∂Bt 1 2 2n =∇ × v × ∇ Bt = ∇ ×A + ∂t μσ n=0 ez (∇v) ×

∞ n=0

∇ A2n +

1 2 ∇ Bt μσ

(8.44) (8.45)

The above mentioned set of equations have a simple solution in the form of:        1 ∂v 1/2 k ∂v 1/2 k2 Bt = ez B0 exp ± − t k ∂ x 2 ∂x μσ     k ∂v 1/2 × cos ∓ t + ky 2 ∂x B0 A = ∓ez exp k 0

(8.46)

     k ∂v 1/2 k2 ± − t 2 ∂x μσ

    k ∂v 1/2 π × cos ∓ t + ky + 2 ∂x 4 An = 0

n≥1

(8.47)

(8.48)

We obtain a traveling dynamo wave with velocity:   ∂v 1 1/2  ∂ x 2k

(8.49)

The amplitude of the wave varies exponentially with time. Under the assumption of high electrical conductivity we can neglect the term k 2 /μσ . Cyclonic rotation vanishes at the equator. The sign is opposite in the northern and southern hemispheres. In each hemisphere the dynamo waves migrate toward the equator. We must take also the magnetic buoyancy into account. Inside a toroidal band there is a magnetic pressure in addition to the gas pressure and therefore the gas density inside such a band is less than outside which explains the buoyant force.

8.2.5 A Simple Solar Dynamo Model Let us consider the solar activity cycle. It was found that there is a difference between even and odd cycle. For references see the paper of [10] where further citations are

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found. We give a simplified Babcock–Leighton model that reduces the complex equations to the well known logistic equation. With this simplification we can study the chaotic behavior of the solar dynamo. We follow the paper of [11]. The dynamo equations are reduced to a map. The principle of a Babcock–Leighton dynamo is that: • a large scale polar field exists, • this polar field is sheared by the differential rotation • thus a toroidal field is produced. Let n be a certain solar cycle number. A shear-interface layer enveloping the core produces a toroidal field for the next cycle and the production of that toroidal component T is proportional to the poloidal component P in the following way: Tn+1 = ( )( t)Pn

(8.50)

That means that the toroidal component of cycle n + 1 is proportional to the poloidal component of cycle n. We assume that the poloidal field remains constant during the time interval L1 − L2 (8.51) t = u This is the time interval required to be advected by meridional circulation from the surface down to the equatorial part of the shear layer. During that time the poloidal field is being acted upon by a constant rotational shear . The next step according to the Babcok–Leighton dynamo is that the production of the poloidal field is assumed to be a function of the toroidal field of the current cycle: Pn+1 = f (Tn+1 )Tn+1 (8.52) The function f (Tn+1 ) measures the efficiency of the Babcock–Leighton dynamo as a function of the toroidal field strength in the shear layer. The poloidal and toroidal ¯ T¯ . We can then convert the above equation into field amplitudes are denoted by P, a non-dimensional form: tn+1 = apn  ¯ T f (tn+1 ) pn+1 = P¯ with

(8.53)

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Pn P¯ Tn tn = T¯ P¯ a =  t T¯

pn =

(8.54) (8.55) (8.56)

Without loss of generality we can set a = 1. By substitution we get pn+1 and finally

 ¯ T f ( pn ) pn = P¯

 ¯ P (1 + β(1 − tn+1 )) f (tn + 1; β) = T¯

(8.57)

β>0

(8.58)

The parameter β measures the efficiency of the Babcock–Leighton mechanism for the poloidal field regeneration. This then leads to pn+1 = pn (1 + β(1 − pn ))

(8.59)

or pn+1 = g( pn ; β) with β > 1. We obtain a one-dimensional parametric iterative map. In the paper of [10] it was shown that 2 ≤ β ≤ 2.45 yields a sequence of amplitude iterates pn ’s alternating between higher and lower-than average. This corresponds to the odd-even effect for the sunspot data. There are two simple solutions of the equations, so called fixed points. A fixed point is defined as: (8.60) pn+1 = pn There are two fixed points: • Pn = 0, from

pn+1 = pn (1 + β(1 − pn ))

we get pn+1 = 0 • Pn = 1. We get pn+1 = pn (1 + β(1 − pn )) = 1(1 + β(1 − 1)) = 1 For these two points the value of β is irrelevant. For the analysis let us first start with a very simple form: xn+1 = xn r (1 − xn )

(8.61)

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Fig. 8.11 A simple example of the logistic map xn+1 = r xn (1 − xn ). The successive values of xn are given as a function of n. The blue curve shows the behavior for x0 = 0.1, the orange curve for x0 = 0.2

In Fig. 8.11 a plot of the successive iterations xn is given for increasing values of n. Two initial seed values are considered: (a) x0 = 0.1, (b) x0 = 0.2. It can be seen that the system after some first transient phase in both cases for the seed evolves to a cycle of 2. It takes two iterations to complete a cycle. The system oscillates between the numerical values of about 0.558016 and 0.764566. Now we choose two values close to 1: (a) x0 = 0.99, and (b) x0 = 0.999. The results are given in Fig. 8.12. We see that in both cases the population (since the logistic equation describes population evolution, see Chap. 1 of this book) is reduced almost to zero after the first few steps but then recovers and we obtain the same values as in the case above. The decrease of population after 1 step e.g. for x0 = 0.999 is down to the value of 0.003. The main result of this strongly simplified dynamo is: • The fixed point x0 = 0 is reached independently of the value of r . It has zero measure, which means that it is reached only with exact numerical values. • The fixed point x0 = 1 is an attractor of the system; however, here the condition is, that r must be ≤ 2. At that point it has a period cycle 2. The solution oscillates between two different states. • The period-2 limit cycle looses then stability in a period-4 limit cycle. • This period-4 limit cycle becomes unstable in a period-8 limit cycle and so on. In Fig. 8.13 we give two examples with the following set of values:

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Fig. 8.12 A simple example of the logistic map xn+1 = r xn (1 − xn ). The successive values of xn are given as a function of n. The blue curve shows the behavior for x0 = 0.99, the orange curve for x0 = 0.999

1. x0 = 0.1, r = 1.9, 2. x0 = 0.1, r = 2.7. The red line shows the behavior for the first set of values, and the parameter is 1.9. The system evolves to 1 after about 50 iterations (the number of iterations is given on the x-axis). In the case of parameter r = 2.7 the situation becomes more complex: the blue line in the figure shows no equilibrium at any step, but a kind of oscillatory pattern that is not really periodic in a strict sense. A bifurcation occurs when a small change of the parameter values that describe the system causes a sudden topological change in its behavior. Figure 8.14 shows a bifurcation diagram for the logistic map. The bifurcation parameter is r . We see how the system behaves asymptotically for a large set of values x. In this figure the following range of the parameter r was chosen: r = 1.5....4.0. We see that for example for r = 2 the asymptotic value for xn becomes approximately 2. At the point r = 3 there are two fixed points. These split again near r = 3.5 into 4 points, etc. The Lyapunov Exponent is defined as:    d f (x (r ) )   r i  λ(r ) = lim log   n→∞  dx  i=0 n−1

which becomes for the logistic equation:

(8.62)

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175

Fig. 8.13 The logistic equation solution. In both cases the seed was chosen at 0.1, the parameter is 1.9 (red line) and 2.7 (blue line). xn is plotted as a function of n

Fig. 8.14 Bifurcation diagram for the logistic equation

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8 Chaotic Dynamo Models

Fig. 8.15 The Lyapunov exponents for the logistic equation

fr = r x − r x 2 d fr = r − 2r x dx

(8.63) (8.64)

In Fig. 8.15 we give a plot of the Lyapunov exponents. The negative values are shown as red dots, and the positive values as blue dots. It is clear that for the logistic equation the behavior varies from negative (i.e. non chaotic) to positive values (i.e. chaotic). Let us now come back to the modified logistic equation that gives an approach to a solar dynamo: (8.65) xn+1 = xn (1 + r (1 − xn )) The two figures show the plot for a range of values x0 ∈ {0, ...., 1}. The fixed points 0 and 1 are marked by a dot (Fig. 8.16). Next we examine the bifurcation diagram. A bifurcation diagram is a very useful tool for the study of dynamical systems. It shows the values visited or approached asymptotically (fixed points, periodic orbits) as a function of a bifurcation parameter. There are two types of values. • stable values: represented by a solid line, • unstable values: represented by a dot. In Fig. 8.17 we have plotted successive values of xn (the former pn ) against r (the former β) after avoiding transients associated with the choice of x0 (the former p0 ) .

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Fig. 8.16 A simple solar dynamo model following the logistic map approach

Fig. 8.17 Bifurcation diagram for solar dynamo approximated by the logistic equation

Finally, in analogy to the logistic equation we plot the Lyapunov exponent for our simplified solar dynamo. The Lyapunov Exponent becomes: fr = r x − r x 2 + x d fr = r − 2r x + 1 dx

(8.66) (8.67)

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Fig. 8.18 The Lyapunov exponent a for solar dynamo approximated by the logistic equation

The results are shown in Fig. 8.18. Again, the red points indicate negative numbers, and the blue points indicate positive numbers (including the case where λ = 0). From a value of r > 2.5 the systems is characterized by positive Lyapunov exponents with some short intermittent negative values. Let us now consider a slightly different case according to the above cited paper. The efficiency of poloidal field production decreases as the toroidal field increases in strength. This is a result of the equation f (tn+1 , β) =

P¯ (1 + β(1 − tn+1 )) T¯

β>0

The parameter β gives the efficiency of the Babcock–Leighton mechanism i.e. it describes the poloidal field regeneration. So we can investigate other choices for f (tn+1 ) Numerical calculations that are cited in the paper suggest lower and upper operating thresholds. The poloidal field production decreases faster than tn as the toroidal field falls below 10 kG. This can be simulated by P¯ γ >0 γ tn+1 (1 − tn+1 ), T¯ = γ pn2 (1 − pn ) γ >0

f (tn+1 ; γ ) = pn+1

The corresponding bifurcation diagram is given in Fig. 8.19. As an example, we give also a Python program listing:

(8.68) (8.69)

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import numpy as np import matplotlib.pyplot as plt

def logistic(r, x): return r*x*x*(1-x) n = 10000 r = np.linspace(3.5, 7.0, n) print(r[1:10]) iterations = 5000 last = 50 x=0.7 #x = 1e-5 * np.ones(n) lyapunov = np.zeros(n) # fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(8, 9), # # for i in range(iterations): x = logistic(r, x) # We compute the partial sum of the # Lyapunov exponent. lyapunov += np.log(abs(r - 2 * r * x)) # We display the bifurcation diagram. if i >= (iterations - last): #plt.plot(r, x, ’,r’, alpha=.25) plt.plot(r, x, ’,r’, alpha=.45) # use this parameter for alpha for zoom plt.axis([3.5,7,0,1.0]) plt.xlabel(’Parameter r’) plt.ylabel(’Aysmptotic $x_n$’) plt.title(’Bifurcation diagram modified equation’) plt.show()

A closer look at the values shows that chaotic behavior starts as soon as the parameter r = γ becomes 5.8 as it is shown in the zoom of Fig. 8.20. The onset of chaos is also clearly seen when considering the Lyapunov exponents (Fig. 8.21). As an example, we give a simple Python code: import numpy as np import matplotlib.pyplot as plt

def logistic(r, x): return r * x*x * (1 - x) n = 10000 r = np.linspace(3.5, 7.0, n) iterations = 1000 last = 50 x=0.7 #x = 1e-5 * np.ones(n) lyapunov = np.zeros(n) # fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(8, 9), #

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Fig. 8.19 The modified solar dynamo bifurcation diagram

Fig. 8.20 The modified solar dynamo bifurcation diagram zoomed in to see the onset of chaos

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Fig. 8.21 The modified solar dynamo Lyapunov exponents

# for i in range(iterations): x = logistic(r, x) # We compute the partial sum of the # Lyapunov exponent. lyapunov += np.log(abs(2*r*x-3*r*x*x))

# We display the Lyapunov exponent. plt.plot(r[lyapunov =0]/iterations,’.b’,alpha=.1) plt.xlabel(’Parameter r’) plt.xlim(0,10)

plt.ylabel(’Lyapunov exponent’) plt.title(’Lyapunov exponent’) plt.show()

It can be easily shown that a non trivial solution with xn = 0 exists only for the parameter range 4 < r < 6.5433. But also, inside this range the attractor is confined between the numbers 0.2 and 1.0. Any initial value x0 outside this range converges to zero (i.e. xn → 0, n → ∞). This behavior cannot be found for the first dynamo approximation. The solar activity cycle is characterized by an even-odd effect. This can also be seen in Fig. 8.19 for the interval 16/3 < r < 5.7635. The transition to chaos takes

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place at r = 5.8878. This even-odd effect can also be found in the case for the first dynamo approximation used: here, the parameter range is 2.0 ≤ r ≤ 2.4494. Does that mean that this even-odd effect is not very robust? We know that in the convective zone there are turbulent motions. This could mean that r also varies stochastically over a significant range of numerical values. This could then lead to a range for r in which this even-odd behavior is not found any more. The next step would therefore be to test a dynamo that is described by: pn+1 = γn pn2 (−1 pn ) + n

(8.70)

For the values of γn we can choose the interval [γ1 , γ 2]. The n is taken from [0, ] with the restriction that  = 0

(8.77)

Substituting this into the induction equation and averaging, we obtain an equation for the mean field: ∂B0 = ∇ × (U0 × B0 ) + ∇ ×  + η∇ 2 B0 ∂t

(8.78)

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8 Chaotic Dynamo Models

Here  =< u × b > is the mean electromotive force. The next step is to express this force in terms for the mean quantities alone. We can write: i = αi j B0 j + βi jk

∂ B0 j ∂ xk

(8.79)

The tensors αi j , βi jk depend upon the fluctuation velocity and the diffusivity. All higher derivatives are negligible under the assumption that L is sufficiently large. The simplest case is a homogeneous isotropic turbulence, where the tensors αi j , βi jk are isotropic: βi jk = βδi jk (8.80) αi j = αδi j We obtain: ∂B0 = ∇ × (U0 × B0 ) + ∇ × (αB0 ) + (η + β)∇ 2 B0 ∂t

(8.81)

We see that: • β represents a turbulent enhancement of the magnetic diffusivity • α represents a source for a mean field. Since we need a non-zero α, the velocity filed must be non symmetrical. This can be achieved a fluid motion within a rotating body. The α-effect therefore stands for differential rotation. The kinematic dynamo problem consists of solving the Eq. 8.81 for B0 where a velocity U0 , diffusivity and α are assumed. This is a linear problem, and the solution are either exponentially growing or decaying. Therefore, we need to take into account feedback effects of the magnetic field for the fluid flow and Eq. 8.81 has to be solved together with an equation for the fluid flow. With such feedback effects, the amplitude of the generated field becomes limited. We can also represent the α effect by α0 (8.82) α∼ 1 + ξ B2 α0 is the value in the absence of a magnetic field, and ξ is the quenching parameter. Let us discuss a more realistic situation. The magnetic field can be described by poloidal and toroidal components. The poloidal components are in the meridional plane, and the toroidal components are in the azimuthal plane. A kinematic dynamo requires a continuous transfer to regenerate the toroidal and poloidal components which is achieved by the differential rotation. The plasma is highly conductive, the magnetic field lines appear to be frozen in, they follow the plasma motions. The field lines become stretched along the equator because of differential rotation— the equator rotates faster than higher latitudes. Therefore, a toroidal component evolves, which is known as the ω-effect. The toroidal component can be easily understood by introducing the differential rotation on a poloidal field. But how to regenerate a poloidal field from a toroidal component? In 1955 Parker suggested that a small helical component that results from convective motions could twist a toroidal

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185

field into loops in the meridional plane, which is known as the α-effect. Mean field electrodynamics were developed in [13] by Steenbeck, Krause and Rädler. These authors studied a magnetic field for an electrically conduction fluid and the medium investigated was influenced by turbulent motions due to Coriolis force. The mean velocity of the medium is zero, and the magnetic field is not. For the mean values of the magnetic field, the magnetic flux density and the magnetic field we have: ∇ × E = −B˙ ∇ × H = ˙j

(8.84)

∇B = 0

(8.85)

(8.83)

and ¯ B¯ = μH ¯j = σ (E + v × B)

(8.86) (8.87)

H = H + H

(8.88)

Normally, v × B = v × B. We can write:

where H is a small perturbation. For small values of v we obtain for the perturbed field: 1 2  ∂H − ∇ H = ∇ × (v × H ) ∂t μσ

(8.89)

This leads to: Hi (x, t)

 = (δil δ jm − δim δ jl )

with (x, t) =

∂ (vl (x  , t  )H m (x  , t  ))Γ (x − x  , t − t  )d 3 x  dt  ∂ x j (8.90)  μσ 1/2 4π t

  μσ exp − xi xl δil 4t

(8.91)

Under the assumption of a barometric pressure gradient and Coriolis force the turbulence is simplified and a non vanishing component of the following appears v × B = μv × H

(8.92)

This component is in the direction of H and this means a current i in the direction of H. This gives a self sustaining dynamo as it is sketched in Fig. 8.23. Of course this is not a perpetuum mobile. The energy comes from the turbulent field.

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Fig. 8.23 A simplified self sustaining dynamo. A current i0 in the ring 1 generates a magnetic field H0 . This field in the ring 2 has a component of current along itself, i1 this generates a field H1 in the ring 2 which has a component i2 and so on

Such a type of dynamo explains the many features about active regions of the Sun, the sunspot cycle, etc., but there are also problems. One of these problems is that regions of concentrated magnetic flux are less dense than their surroundings, so buoyancy forces work. This can be easily seen by comparing the pressure inside a flux bundle Pin with the pressure outside. In the case of Pin in addition to the gas pressure also the magnetic pressure Pm ∼ B 2 has to be considered: Pin = n in kT

(8.93)

Pout = n out kT

(8.94)

This buoyant motions occur on a timescale that is short in comparison to the solar cycle, the magnetic flux could not be held long enough within the convection zone to be amplified. Turbulent motion all expel magnetic flux from the convection zone. This leads to a concentration in a thin layer below the base of the convection zone, the dynamo may be located in the region around the base of the convection zone (Galloway and Weiss 1981). This is a stably stratified region not susceptible to magnetic buoyancy instabilities. Stronger fields may develop before becoming unstable due to buoyancy. Another problem with strong fields is however, that they resist convective deformation, therefore the α-effect become less effective. This is also called the α-quenching. This problem becomes less important as soon most of the flux is concentrated below the convective zone.

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187

8.3.1 Interface Dynamo Results of helioseismology have shown that there exists a tachocline [5] that is a transition region between the differentially rotating convection zone and the almost rigidly rotating region below. In the tachocline we find the strongest differential rotation within the Sun. These results led to the formulation of the interface dynamo [3]. The two generation effects are spatially separated: • α-effect: operates in the turbulent convective layer, • ω-effect: operates in the shear layer below the convective zone. There is a diffusive transport of flux between the two regions. Turbulence enhances the diffusion. The diffusion within the tachocline is smaller than in the convection zone. The α-quenching is solved, and stronger toroidal fields can be generated in the tachocline away from the region where the α-effect operates. Also pumping mechanisms have been suggested. From the point of dynamo theory the solar activity cycle can be described as follows: • A toroidal field is generated via shearing of the poloidal field in the tachocline. • The toroidal field becomes unstable because of buoyancy and rise into the convection zone. • In the convection zone the poloidal field is regenerated via the α-effect. • The convection zone acts like a filter, – only the strongest field bundles rise to the surface and appear as active regions. – If the weaker fields are recycled back to the tachocline by turbulent pumping, the cycle repeats.

8.3.2 Surface Dynamo, Flux Transport Dynamo These basic ideas can be modified, for example by representing the differential rotation by an analytic fit based on the results of helioseismology (e.g. [14]). In the Babcock–Leighton dynamo [15, 16] the decay of tilted bipolar active regions that produce poloidal flux operates only at the surface. In the flux transport dynamos the ω-effect operates in the tachocline and a meridional circulation is invoked in order to couple the two regions. A polewards flow was observed (Hathaway 1996). Because of compressibility, the equatorward flow will be very weak, the efficiency of flux transport will be reduced. A surface α-effect requires a strong field, the quenching issue discussed above is less important but how to explain the working of the dynamo in times of grand minima? These models have in common that there is a strong separation between the two regions important for dynamo action. In [14], an additional deep seated α-effect

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calculated from the kinetic helicity profile is used and also a tachocline-based αeffect is discussed.

8.3.3 Torsional Oscillations One of the aims of dynamo theory is to understand and predict long term solar activity behavior including periods of strongly reduced activity like the Maunder minimum. Modifications of the dynamo theory can be made by including torsional oscillations. In 1980 Howard and La Bonte detected variations in the solar differential rotation profiles [17]. They found a surface pattern of alternating band of faster and slower than average local rotation which migrate from mid to low latitudes with an 11-year periodicity. Because of of the coincidence with the 11-year sunspot cycle the underlying mechanism that explains these torsional oscillation is thought to be magnetism. Therefore in the dynamo models an oscillatory part of the total velocity perturbation is introduced. Observations suggest that the strongest part of these oscillations occurs near the surface and maybe even an additional band of oscillations at high latitudes. This is in contraction to dynamo models where the dynamo action is assumed to be concentrated at low latitudes at the base of the convection zone. An explanation of this discrepancy is the strong stratification of the convection zone. A relatively small perturbation to the local angular momentum at the surface would produce a large angular velocity perturbation. The explanation of high latitude oscillations is more difficult. Maybe a weak dynamo action at high latitudes could contribute to that.

8.4 Solar Activity in the Past and Chaotic Dynamo Action 8.4.1 Reconstruction of Solar Activity in the Past In Chap. 5 we have outlined how solar activity in the past can be reconstructed using several proxies: • sunspot numbers: reliable numbers are available for about 300 years. Pre telescopic observations of sunspots have been reported from naked eye observations in the case of the presence of extremely large sunspots. By an analysis of sunspot positions in the recordings, Ribes and Nesme-Ribes [18] indicate a stronger differential rotation at the end of the Maunder minimum. Also, the sunspot distribution seemed to be unusual. Therefore, quadrupolar modes of the magnetic fields may also play an important role in the solar dynamo, at least during certain time intervals. For dynamo theories the coupling between the two different hemispheres is

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189

important. A phase difference of 35–40 years was found and thus the period of a phase lag change is between 70–80 years [19]. • Aurorae, • Galactic cosmic ray records: Galactic cosmic rays are deflected by the presence of magnetic field. The magnetic fields in the heliosphere are subject to modulation by the solar activity cycle and therefore, the cosmic ray flux detected on Earth is modulated. The cosmic rays lead to a production of 14 C and 10 Be. Therefore e.g. measurements from Greenland and Antarctic ice cores the solar activity can be reconstructed over the last 10 000 years. This time series must be filtered by a Principal Component Analysis to filter out climatic effects. A Fourier analysis of the resulting function reveals several periods such as the Gleissberg cycle at 87 years. The whole data series contain periods with strong modulation of solar magnetic activity with the presence of prominent grand maxima and minima. Two examples of minima in the past are the Maunder Minimum (1645–1715) and the Dalton Minimum (1790–1830). Galactic cosmic ray records were analyzed by [20] (10 Be-data) and by [21] (14 Cdata).

8.4.2 Grand Minima in Non Linear Dynamos Grand minima can be simulated with mean-field dynamo models by invoking fluctuations in the main dynamo drivers which can be: • differential rotation, • meridional circulation, • small-scale turbulence effects. The solutions of the equation including one of these effects lead to irregular behavior that can be interpreted as grand-minima type effects. Sunspot number reconstructions (SNRs) based on dendrochronologically dated radiocarbon concentrations have been analyzed through the Empirical Mode Decomposition (EMD) by [22]. This Method is ideally for time series analysis. The output remains in the time spectrum. The fast Fourier transform assumes that the signal is based on simple sine waves, and the EMD is based on an intrinsic mode function. There is no assumption made on the dataset, so it is empirical. The authors came to the conclusion that the grand minima generation can be attributed to the coupling between the Gleissberg (60–120 years) and Suess (2000 years) cycles and that the events occurring around minima of the Suess cycle are the longest and most intense Spörer-like minimum.

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References 1. Hanasoge S, Gizon L, Sreenivasan KR (2016) Seismic sounding of convection in the sun. Ann Rev Fluid Mech 48(1):191–217 2. Brown TM, Christensen-Dalsgaard J, Dziembowski WA, Goode P, Gough DO, Morrow CA (1989) Inferring the sun’s internal angular velocity from observed p-mode frequency splittings. Astrophys J 343:526 3. Parker EN (1993) A solar dynamo surface wave at the interface between convection and nonuniform rotation. Astrophys J 408:707 4. Basu S, Antia HM (2003) Changes in solar dynamics from 1995 to 2002. Astrophys J 585(1):553–565 5. Spiegel EA, Zahn JP (1992) The solar tachocline. Astron Astrophys 265:106–114 6. Rudiger G, Kitchatinov LL (1997) The slender solar tachocline: a magnetic model. Astron Nachricht 318(5):273 7. Wood TS, Brummell NH (2018) A self-consistent model of the solar tachocline. Astrophys J 853(2):97 8. Goode PR, Dziembowski WA, Korzennik SG, Rhodes Jr EJ (1991) What we know about the sun’s internal rotation from solar oscillations. Astrophys J 367:649 9. Parker EN (1957) The solar hydromagnetic dynamo. Proc Natl Acad Sci 43(1):8–14 10. Durney BR (2000) On the differences between odd and even solar cycles. Solar Phys 196(2):421–426 11. Charbonneau P (2001) Multiperiodicity, chaos, and intermittency in a reduced model of the solar cycle. Solar Phys 199(2):385–404 12. Moffatt HK (1978) Magnetic field generation in electrically conducting fluids 13. Steenbeck M, Krause F, Rädler KH (1966) Berechnung der mittleren LORENTZ-Feldstärke für ein elektrisch leitendes Medium in turbulenter, durch CORIOLIS-Kräfte beeinflußter Bewegung. Zeitschrift Naturforschung Teil A 21:369 14. Dikpati M, de Toma G, Gilman PA, Arge CN, White OR (2004) Diagnostics of polar field reversal in solar cycle 23 using a flux transport dynamo model. Astrophys J 601(2):1136–1151 15. Babcock HW (1961) The topology of the sun’s magnetic field and the 22-year cycle. Astrophys J 133:572 16. Leighton RB (1969) A magneto-kinematic model of the solar cycle. Astrophys J 156:1 17. Howard R, Labonte BJ (1980) The sun is observed to be a torsional oscillator with a period of 11 years. Astrophys J Lett 239:L33–L36 18. Ribes JC, Nesme-Ribes E (1993) The solar sunspot cycle in the Maunder minimum AD1645 to AD1715. Astron Astrophys 276:549 19. Zolotova NV, Ponyavin DI, Arlt R, Tuominen I (2010) Secular variation of hemispheric phase differences in the solar cycle. Astron Nachricht 331(8):765 ˇ 20. Hanslmeier A, Brajša R, Calogovi´ c J, Vršnak B, Ruždjak D, Steinhilber F, MacLeod CL, Ivezi´c Ž, Skoki´c I (2013) The chaotic solar cycle. II. Analysis of cosmogenic 10 Be data. Astron Astrophys 550:A6 21. Hanslmeier A, Brajša R (2010) The chaotic solar cycle. I. Analysis of cosmogenic 14 C-data. Astron Astrophys 509:A5 22. Vecchio A, Lepreti F, Laurenza M, Carbone V, Alberti T (2019) Solar activity cycles and grand minima occurrence. Nuovo Cimento C Geophys Space Phys C 42(1):15

Chapter 9

Solar Cycle Forecasting

In this chapter we discuss several methods for forecasting solar activity on different time scales. The solar radiation and particle emission influences on the space around Earth, the Earth’s magnetosphere, the Earth’s atmosphere, and can be harmful for satellites and manned spacecraft missions and disrupt communication systems. In extreme cases even power lines on the Earth’s surface are perturbed or can become disrupted. Solar influences on timescales of days to weeks are summarized as space weather. Space climate denotes influences on longer time scales.1 The amplitude and number of space weather related effects strongly depend on the solar activity cycle, so cycle predictions are essential to estimate space weather effects. On the other hand, the forecasting of space weather events requires forecasting over relatively short time scales in the order of a few days which is at present not available because of lack of data and precise understanding of the physical processes that trigger space weather relevant solar phenomena like flares, CMEs, and strong solar wind.

9.1 Short Term Forecasting 9.1.1 Space Weather According to the National Oceanic and Atmospheric Administration’s (NOAA) Space Weather Prediction Center the “monitoring and forecasting solar outbursts in time to reduce its effect on space-based technologies have become a national 1 There are also other influences that contribute to space climate such as our galactic neighborhood.

© Springer Nature Singapore Pte Ltd. 2020 A. Hanslmeier, The Chaotic Solar Cycle, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-15-9821-0_9

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priority”. The forecasting is made for space weather relevant processes such as flares, CMEs, solar wind, heliosphere strength. In general the Sun impacts • • • • • •

The Earth’s climate, Electric power transmissions, GPS systems, HF communications, Satellite drag, and Satellite communications.

Does the sun influence the climate? It is generally accepted that modern climate variation is driven by greenhouse gases, volcanoes, ENSO (el Ni˜no Southern Oscillation) and solar irradiance. However, the solar irradiance varies over a solar cycle and the amplitude of the variability strongly depends on the wavelength: • • • •

Visible (400–800 nm): Variation is about 0.1%. UV (120–400 nm): Variation changes up to 15%. Extreme UV (EUV): Sun changes by 30–300% at very short timescales (minutes). Infrared (IR, 800–10000 nm) changes below 1%.

The shorter the wavelength range, the stronger the variation. The total solar irradiance, or TSI, denotes the total wavelength-integrated energy from the Sun and can be measured with high precision only by satellites. The variations range from 1365.5 W/m2 at solar minimum to 1366.5 W/m2 at solar maximum. An increase of 0.1% of TSI corresponds to 1.3 W/m2 energy input at the top of the Earths atmosphere. This amount is small but detectable. The sun radiates mostly in the visible and near IR and therefore, the total variation seems to be small. Radiation in the UV and X wavelength range is absorbed in the atmosphere above the troposphere so its influence on weather systems might be negligible. Another impact on climate variation comes from solar energetic particles. When these particles hit the higher atmosphere, its chemical composition, is changed. Energetic particles influence on the amount of NO and ozone. Another effect is the duration of the minimum of solar activity. During a solar minimum a larger amount of cosmic rays penetrates the weaker heliosphere. These rays produce particle showers in our atmosphere which can act as nucleation sites and create cloud formation. The different space weather effects are summarized in Fig. 9.1. Generally, space weather concerns many phenomena (Fig. 9.2): • Aurorae, • Coronal Holes: cooler darker, less dense regions than the surrounding coronal plasma, regions of open unipolar fields. They are more common and persistent mostly around the solar poles during solar minima. Coronal holes are the sources of high speed solar wind streams. Coronal holes near the equator produce a corotating interactive region (CIR). • Coronal Mass Ejections, these travel outward from the Sun at speeds from 250 km/s to 3000 km/s. The fastest Earth-directed CMEs arrive with a delay between 15–18 h.

9.1 Short Term Forecasting

Fig. 9.1 Space weather effects. NOAO Fig. 9.2 Cosmic ray and solar energetic particle environment for aviation radiation. SEP denote solar energetic particles, GCR galactic cosmic rays. Ktobiska

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Table 9.1 NOAA scale for geomagnetic storms Scale Description Kp-Index G5 G4 G3 G2 G1

Extreme Severe Strong Moderate Minor

9 8 7 6 5

Frequency per cycle

Days per cycle

4 100 200 600 1700

4 60 130 380 900

• Earth’s Magnetosphere. • F10.7 cm Radio Emissions; the radio flux at 10.7 cm (2800 MHz) is an indicator for solar activity. This parameter is now available over 6 solar cycles. The radiation is produced from the chromosphere and the corona. • Galactic Cosmic Rays; these originate outside the solar system, and are most probable during supernova explosions. Near solar maximum, when the heliosphere is strongest, the cosmic ray flux is at minimum. At high geographic latitudes and high altitudes aircrew and passengers are exposed to to this flux. • Geomagnetic Storms: the solar wind produces changes in the currents, plasmas and fields in the Earth’s magnetosphere. The strongest effect comes from a southward directed solar magnetic field, the direction is opposite the direction of the Earth’s field and magnetic reconnection occurs. Storms generate currents in the ionosphere, that heat the upper atmosphere and heat is added to the upper atmosphere. As a result it expands causing additional drag on low orbiting satellites. Because of strong variations in the ionosphere the path of radio signals will be affected and, for example, navigation systems can be disrupted. The NOAA scales range from G1 to G5. G5 indicates severe problems on ground based power systems (e.g. voltage control, transformers may receive damage), satellites suffer from extensive surface charging, problems of their orientation; pipeline currents can reach hundreds of amps, HF radio propagation may become impossible for days, aurora can be seen at low geographic latitudes (e.g. Italy, Florida). The number of expected events is about 4 per cycle. It is measured by the so-called Kp-Index. Table 9.1 gives a summary of events that are statistically possible during a solar cycle. • Ionosphere: the extension is from 80 to 600 km and the ionization is caused by the Sun’s EUV and x-ray radiation that ionize atoms and molecules. The ionosphere reflects and modifies radio waves thus affecting communication and navigation. The change of the EUV- and x-ray input during a solar activity cycle varies by a factor of 10. • Ionospheric Scintillation: these are rapid variations in the ionosphere. The variation occurs on scales of several 10 m to kilometers. In Table 9.2 the NOAA scale for radiation storms is given. Note that an S5 storms means high radiation hazard to astronauts on extravehicular activities, and even

9.1 Short Term Forecasting Table 9.2 NOAA scale for solar storms Scale Description S5 S4 S3 S2 S1

• • •

•

•

•

Extreme Severe Strong Moderate Minor

195

Flux level (≤ 10 MeV Frequency per cycle particles) 105 104 103 102 10

Fewer than 1 3 10 25 50

passenger and aircrew are exposed to radiation risk at high latitudes. Complete radio fadeout over polar regions is highly probable. Radiation Belts: these are region of enhanced populations of electrons and protons around the Earth. Whenever satellites pass through these belts they are exposed to high levels of radiation. Solar EUV irradiance: the EUV radiation impacts the ionosphere and correlates well with the 10.7 cm flux. Solar Flares (Radio Blackouts): these eruptions last from minutes to hours (Fig. 9.6). The radiation propagates at the speed of light reaching the Earth after 8 min. In quiet solar activity conditions, radio waves are reflected in higher ionospheric layers. However, during a strong solar flare, more dense layers (D-layer) are produced in the lower ionosphere, the radio waves are reflected there and this results in a radio blackout, mostly at frequencies in the 3 to 30 MHz band. In Table 9.3 a correlation between flare energy (measured in the 0.1 to 0.8 nm band, X-flares denote 10−4 Wm−2 , M-flares 10−5 Wm−2 and so on) and the so-called NOAA scales are given. Solar Radiation Storm: these occur during a large-scale magnetic eruption, often associated with a coronal mass ejection. Protons get accelerated to relativistic velocities, and they can arrive in about 10 min to the space around Earth. These high energetic particles penetrate deep inside the objects with which they collide (e.g. man made satellites, human body), and cause circuits in the case of satellites or DNA damage in the case of living organisms. In the NOAA scheme they are classified as S1...S5. The GOES2 satellite is in a geosynchronous orbit and measures this flux. Solar Wind: continuous outflow of protons and electrons with embedded magnetic field. Coronal holes produce high speed solar wind (500 to 800 km/s). Most of the coronal holes are located around the solar poles. The Earth is in the ecliptic and therefore is mostly affected by the solar wind from an equatorial current sheet. The particle speed is about 400 km/s. The current sheet is nearly flat during solar minima. Sunspots/Solar Cycle

2 Geostationary

Operational Environmental Satellite

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Table 9.3 Solare flares and radio blackouts Radio blackout X-ray flare parameter R1 R2 R3 R4 R5

M1 M5 X1 X10 X20

Flux (W/m)

Severity

0.00001 0.00005 0.0001 0.001 0.002

Minor Moderate Strong Severe Extreme

• Total Electron Content; this is the total number of electrons along a path between a radio transmitter and a receiver. The numbers are given in TEC units. The TEC depends on local time, latitude, longitude, season, geomagnetic conditions, solar cycle, solar activity and conditions in the troposphere. 1 TECU = 1016 electrons/m2

(9.1)

9.1.2 Space Weather Forecasting After having summarized the different aspects of space weather we shortly discuss the possibility of forecasting these events which is of course very important for satellite missions and even power lines on Earth. Several space weather prediction centers exist. Space weather service means that the parameters that drive the space weather should be predicted for the next few days (Fig. 9.3). Solar energetic particles (SEPs) provide a major component to space weather. A large flow of SEPs may disrupt radio frequency communications, and passengers and crew on polar flight routes, as well as astronauts, especially when they are out of low earth orbit, are exposed to increased radiation (Fig. 9.4). Physical simulations and modeling to forecast outbursts of SEPs are not at a level to predict these events. There exist empirical and semi-empirical models to produce such forecasts. An overview of these methods is given in [1]. Coronal mass ejections can be observed from ground stations. These observations are important because they are the main cause of geomagnetic storms. For predicting solar flares and CMEs a continuous 24-h monitoring of the Sun would be ideal (Fig. 9.5). The problem is that solar activity shows chaotic properties, and therefore, a limit for the prediction horizon exists. For Earth related studies, we must also take into account that there is interaction with the rotation of the Earth’s core when studying geomagnetic activity as well as local ionospheric currents. The solid inner core, with a radius of about 1220 km, rotates within a liquid outer core, extending out to 3400 km. The circulation of the outer core is driven by the transfer of heat from the

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Fig. 9.3 A solar flare observed in at 19.3 nm by NASA Solar Dynamics Observatory. NASA/SDO

inner core, to the core-mantle boundary. As a result the geomagnetic indices exhibit nonlinear variability and behave like multifractals. The basic forecasting methods involve: • Regressions, • Neural networks, and • Frequency domain algorithm. The forecasting is based on the following data sets [2]: • sunspots and solar irradiance (from 1820 onward). The solar irradiance data from satellites have been available since 1978 (ACRIM data3 ) but they have been reconstructed as far back as 1610 [3]. • Aa geomagnetic index (from 1868 onward); this index is based on the magnetic activity measured at two antipodal stations: Canberra, Australia and Hartland, England. It is defined as the average of the northern and southern values of magnetic activity, weighted to account for differences in the latitudes of the two stations and local induction effects. • Am index (from 1959 onward). The prediction horizon is between 1–7 days, for irradiance and sunspots persistence is better than for the geomagnetic indices. The geomagnetic indices are less correlated to solar activity because they also depend on properties and variations of the Earth’s magnetic field. Currently the errors estimates are: • 48% for 1 day • 54% at later horizons. 3 Active

Cavity Radiometer Irradiance Monitor (ACRIM) space flight experiment.

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Fig. 9.4 One of the GOES satellites. NASA

Fig. 9.5 The D-region absorption prediction model is used as a guide to understand the high frequency (HF) radio degradation and communication interruptions. This is provided by the NOAA space weather prediction center, NOAA-SPWC

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Special tests can be extended to 1–4 weeks. At 1 week the best models reduce the error to about 35%, and for >4 weeks the errors become very large. Let us give a few examples to illustrate the different methods. The moving average method works as follows: let us assume we have a series of values 11, 12, 13, 14, 15, 16, 17. From this we create a moving average series over 5 days, and the first value of series (11 + 12 + 13 + 14 + 15)/5 = 13, the second value (12 + 13 + 14 + 15 + 16)/5 = 14 and the third value (14 + 15 + 16 + 17 + 18)/5 = 15. Let us recall the autocorrelation function AC F of a time series that gives the correlation between a time series and a lagged time series of itself over its entire length. The autocorrelation shows the degree of similarity between the values of the same variables over successive time intervals. A series of numbers in which values can be predicted based on preceding values has an autocorrelation. In a correlogram we plot the ACF as a function of lag. For different lags, the correlation might be different. μ is the mean value, and σ the standard deviation: AC F(k) =

E[(xt − μ)(xt−k − μ)] σ2

(9.2)

When the AC F shows significant peaks, then current values of the time series depend on past values. Here the autoregressive method (AR) can be useful. An autoregressive process has coefficients that quantify the dependence of current value on recent past values: (9.3) xt = a1 xt−1 + a2 xt−2 + ... + a p xt− p + t  is a random error (Gaussian in the ideal case) with zero mean and constant variance. p is the order of the process (how many lags) and the a’s are coefficients for each lag up to the order p. A moving average process (MA) has coefficients that quantify the dependence of current values on recent past random shocks to the system: xt = t + b1 t−1 + b2 t−2 + ... + bq t−q

(9.4)

The combination of the two gives a combined ARMA(p,q) process. More details can be found in [4]. Let us sketch the application to predict the geomagnetic substorm index AL. The impulse response function H is connected with solar wind inputs: AL(t + t) =

T 

H ( jt)[V Bz ](t − jt)

j=0

Also an ARMA filter can be applied to get the Dst Index.

(9.5)

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Fig. 9.6 Scheme of a node. From pathmind.com

Dst (t + δt) =

m−1  i=0

ai Dst (t − it) +

l−1 

b j V Bs (t − jt)

(9.6)

j=0

Another method for space weather forecasting is using neural networks. Neural networks interpret data through a kind of machine perception, they recognize patterns etc. Using neural networks an unknown function f (x) = y between any input x and any output y can be approximated. By learning a correlation can be found between present and future events. Networks are composed of several layers, and the layers are made up of nodes (Fig. 9.6). Solar magnetic field measurements can be used for flare prediction models. Flares produce energetic particles and may generate CMEs that are strong drivers of space weather and therefore their study and forecasting is extremely valuable. Many attempts in this direction have been made and we just give a few examples here. The SHARP data product is a set of key data used to derive physical properties and develop flare prediction models. SHARP is an acronym for Space—Weather HMI Active Region Patches. The HMI (Helioseismic and Magnetic Imager) instrument is on board of the Solar dynamics observatory,4 or SDO [5]. The data include each component of the vector magnetic field, the line-of-sight magnetic field, continuum intensity, Doppler velocity, error maps and bitmaps. The data segments are not full-disk; rather, they are partial-disk, automatically-identified active region patches. Often, there is more than one active region on the solar disk at any given time. The SHARP analysis code calculates the following spaceweather quantities for the region (keyword, quantity, units)5 : • • • • • • •

USFLUX Total unsigned flux in Maxwells MEANGAM Mean inclination angle, gamma, in degrees MEANGBT Mean value of the total field gradient, in Gauss/Mm MEANGBZ Mean value of the vertical field gradient, in Gauss/Mm MEANGBH Mean value of the horizontal field gradient, in Gauss/Mm MEANJZD Mean vertical current density, in mA/m2 TOTUSJZ Total unsigned vertical current, in Amperes

4 Launched 5 Form:

Feb. 11, 2010. jsocs.stanford.edu.

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201

• • • • • •

MEANALP Total twist parameter, alpha, in 1/Mm MEANJZH Mean current helicity in G2/m TOTUSJH Total unsigned current helicity in G2/m ABSNJZH Absolute value of the net current helicity in G2/m SAVNCPP Sum of the Absolute Value of the Net Currents Per Polarity in Amperes MEANPOT Mean photospheric excess magnetic energy density in ergs per cubic centimeter • TOTPOT Total photospheric magnetic energy density in ergs per cubic centimeter • MEANSHR Mean shear angle (measured using Btotal ) in degrees • SHRGT45 Percentage of pixels with a mean shear angle greater than 45 degrees in percent.

9.2 Long Term Solar Activity Prediction The Sun is changing over long time scales. The first question to be answered is, if the changing solar activity over long time scales may correlate with climate effects on Earth. Many attempts to answer this question have been made but there is no definite conclusion. Apart from this question, a long term forecast of solar activity at least over several solar cycles would be highly desirable.

9.2.1 Different Cycles The 11 year cycle was detected by Schwabe and it is called the Schwabe cycle. However, it was detected soon afterwards that longer cycles may be acting as well. By applying a low pass filtering to the recorded sunspot numbers, Gleissberg (e.g. in [6]) detected a cycle of about 80 years. One simple method to detect periodicities in sunspot data is the Fourier analysis. In this technique the record and a sinusoidal signal of a given frequency are compared. The problem is however that the 11-year signal is pseudo-periodic. Its amplitude changes by several orders of magnitude over the time interval under consideration. In that sense the Gleissberg cycle can also be regarded as a pseudo-periodic signal. Also, the length of the Gleissberg cycle is only a few times shorter than the length of the time series (which is approximately 400 years in the cases of sunspots). Therefore more appropriate methods should be considered. In [7] the wavelet analysis was applied to the sunspot data (from 1610 to 1994) and found that cycles of greater duration are connected with periods of weak solar activity. Unlike sinusoidals, wavelets are localized near time t0 . They decay if |t − t0 | > a where a is a characteristic scale.

(9.7)

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The Gleissberg cycle is now assumed to occur in the range of 60–120 years. However, there is also another feature in the solar time series activity record. There are intervals of extended and decreased solar activity that were already mentioned: Maunder minimum ( 1645–1715), Dalton minimum (1790–1845) and modern maximum (1940–1980) to mention a few of them. Apart from these periods the Sun changes on even longer timescales which we discuss in the next paragraph.

9.2.2 Stellar Chromospheric Activity Cycles Stellar coronae are thought be heated by the release of magnetic energy. This energy is produced by a magnetic dynamo that is driven by differential rotation and in the stellar interior. Stellar chromospheric activity can be observed for example by the emission of the Ca II H and K lines. We mention here that the wavelet analysis was also applied to the analysis of stellar chromospheric activity variation. The emission of Ca II H (396.8 nm) and Ca II K(393.4 nm) of stars at the lower main sequence range in the Herzsprung Russell diagram were studied at the Mount Wilson observatory on longer timescales (several decades). For more information see also the paper [8]. About 1/3 of the sample investigated exhibited activity periods comparable to the 11 year sunspot cycle. But it was also found that some stars shows a Maunder minimum like phase and others are found at the transition to a Maunder minimum like phase. For example in [9] the wavelet analysis was applied to the stars HD 3651, HD 10700, HD 10476 and HD 201091. The parameter that defines stellar chromospheric activity was defined as: NH + NK (9.8) S=α N R + NV N denotes the number of photons counted, α is a calibration factor. Besides the Hand K- passbands there where two passbands in the near continuum at R (centered at 400.1 nm) and V (centered at 390.1 nm). In [10] non-dimensional relationships between the magnetic dynamo cycle period Pcyc , the rotational period Prot , the activity level (as observed in Ca II H and Ca II K), and other stellar properties was discussed. The α− effect (see chapter about dynamo) increases with mean magnetic field strength and α and ωcyc decrease with stellar age. Let us consider the following parameters: • • • •

cycle frequency ωcyc „ cycle period Pcyc rotational frequency , rotational period Prot convective turnover time τc , Rossby number: Ro = Prot /4π τc

(9.9)

9.2 Long Term Solar Activity Prediction

• Ca H Ca K emission flux:

  4 = FHK /4π σ Teff RHK

203

(9.10)

A different behavior for young and old stars was found, they are on parallel bands and the old stars show ωcyc / values 6 times larger than young stars. Both stars show  . The ratio ωcyc / decreases in time as a power-law relationship with R0−1 and RHK −0.35 . A strong increase is found around 2–3 Gyr. t When considering the luminosity in the X-ray, a simple relation can be established: L X /L bol ∼ Ro−k

(9.11)

for example k = 2.7 [11]. One obtains for: • young stars: L X /L bol ∼ 10−3 , • old stars L X /L bol ∼ 10−8 ...10−4 The coronal activity rotation relationship is considered to be a proxy for the underlying stellar dynamo responsible for magnetic activity in solar and late-type stars.

9.2.3 The Faint Young Sun Now we discuss the long term behavior of solar activity. According to stellar evolutionary models, the early Sun had about 70% of its present day luminosity. As a result of the fusion of hydrogen into helium in the solar core, the mean molecular weight has increased over long timescales. A higher molecular weight would reduce the gas pressure, therefore the temperature must increase in order to maintain hydrostatic equilibrium. As a result, the thermonuclear fusion rate increases and thus the luminosity. Geologic and fossil evidence indicate that temperatures on Earth have always been similar to present day temperatures. This is called the faint young Sun paradox. By comparing the Sun with solar analogues we can characterize its parameters when it was a young star: • Rotation: the Sun rotated more than 10 times as quickly as today, so one solar rotation was less than three days. • Because of the fast rotation, a strong dynamo was acting. • X-ray and UV emissions were up to several hundred times higher. • There was a strong solar wind particle emission. • There was a high frequency of large CMEs. The explanation of this paradox is that the concentration of the greenhouse gas CO2 was much higher in the past and therefore, the temperature remained constant even when the solar input was less. A minimum partial pressure of 0.1 bar of CO2 could

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Fig. 9.7 Variation of solar luminosity (given by the solar constant S, S0 is the present day value = 1) and the assumed variation of the concentration of carbon dioxide in the Earth’s atmosphere. By Gretashum - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php? curid=79674471

be sufficient to keep the temperatures at the same level as today. This is illustrated in Fig. 9.7. Solar long term variation was first mentioned in the papers of [12] and later reviewed by [13]. Solar activity and the Earth’s climate was also reviewed in [14]. The nonlinear features of the mean sunspot numbers that are a very simply observable for solar activity are documented by many different papers. One of the first papers on the chaotic behavior of the solar activity was given by Rozelot [15]. The times series of solar activity proxies such as sunspots shows both periodic and intermittent phases. Prolonged minima of solar activity over the last 2000 years are: • • • •

Oort Minimum (1010–1050) Wolf (1280–1340) Spörer (1420–2530) and Maunder (1645–1715).

In addition to sunspot data, cosmogenic isotopes such as 14 C, aurora records etc. can also be used.

9.2.4 The Basic Methods of Cycle Predicton Three basic methods for solar cycle prediction exist: 1. Precursor methods 2. Extrapolation methods, and

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205

3. Model-based predictions. In the precursor methods any proxy of solar activity may be used as a precursor. In many cases, sunspots are used as precursors because a data set of more than three centuries is available. All valuable precursors need to be evaluated at around the time of the solar minimum and refer then to the next solar maximum. However, it seems to be difficult to establish a correlation between solar minimum parameters and solar maximum value. Each solar cycle is a consistent unit in itself, but solar activity on the other hand consists of a series of much less coherent individual cycles, which was already suggested by W. Gleissberg [16]. A very simple method is to correlate the amplitudes of sunspot numbers of consecutive cycles. Then a correlation coefficient of about 0.35 is found. This is a weak correlation and the explanation is simple: (i) a secular variation of solar activity (e.g. the Gleissberg cycle and even longer cycles) exists, (ii) strong cycle-to-cycle variations certainly exist. Another method is to correlate the minimum activity level and the amplitude of the next maximum. This yields an almost linear relation with a correlation coefficient of about 0.7. Empirical relations such as Rmax = 126.4 + 5.6Rmin

(9.12)

can be found.6 If for example the value for Rmin = 1.8 for the sunspot number in the recent minimum (monthly averaged smoothed value in Dec. 2019), the next maximum will reach values around 136. The 1σ error is about ±20. The authors claim that it is better to use the activity level before the minimum in order to predict the next maximum. For example using a time lag of 3 years before the minimum, the following formula can be obtained: Rmax = 76.8 + 1.5R(tmin − 3)

(9.13)

For cycle 24 the predictor is 28.5. Therefore, from that formula an amplitude of 120 has to be expected for cycle 25. The problem with this method is however, that the epoch of the minimum of R cannot be known with certainty until about a year after the minimum. Therefore, a prediction will become available 2–3 years before the maximum. Other methods use the correlation Rmax − tcycle;n which means a correlation between cycle length and maximum sunspot number Rmax . The polar fields on the solar surface reach their maximum near the minima of the sunspot cycle. Since 1976 direct measurements of the magnetic field in the polar areas of the Sun have been available. The polar field precursor method is based on the already discussed Babcock–Leighton scenario of the solar dynamo. Instead of direct polar field measurements, Hα synoptic maps can be used. 6 Brajsa,

R., Verbanac, G., Bandic, M., Hanslmeier, A., Skokic, I., and Sudar, D. On the minimum - maximum method for prediction of the solar cycle amplitude, submitted to Astronomy and Astrophysics, 2020.

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Also data from space missions such as Hinode,7 can be used for considering the faculae poleward 50◦ latitude. The faculae are considered as proxies for magnetic field strength. Another relation found is: (n+1) (n) (n) (n) = C1 + C2 Rmax (trev − tmax ), Rmax C1 = 83 ± 11, C2 = 0.009 ± 0.02

(9.14) (9.15) (9.16)

(n) is the epoch of the polarity reversal in cycle n which typically occurs a where trev (n) [17]. year after tmax Also geomagnetic and interplanetary precursors can be used. Flares and CMEs emit particles that hit the Earth’s magnetosphere. The occurrence of the largest flares and CMEs peak some years after the sunspot maximum. Now we discuss extrapolation methods. These methods use time series of solar activity parameters (such as sunspot numbers) and rely on more than one previous point to predict trends. Mathematically speaking these are time series analysis methods; for historical reasons they are also called regression methods. The assumption is that the time series is homogeneous. What does that mean? Mathematical irregularities underlying its variation are the same at any point of time. A forecast of n years ahead has equal chance of success in the maximum, minimum, rising or decaying phase of the solar cycle. The precursor methods consider the solar cycles as individual units, which is not the case for these methods. Let us assume a time series given at time t − t, t − 2t, ..., t − pt., some random error n . Now we want to calculate the value of R in point n:

Rn = R0

p 

cn−i Rn−i + n

(9.17)

i=1

where p is the order of the autoregression, and ci are the weights. The autoregressive moving average model (ARMA) admits a propagation of errors from the previous q points: p q   cn−i Rn−i + n + dn−i n−i (9.18) Rn = R0 + i=1

i=1

This method was used, for example, in [18]. In Fig. 9.8 an example of a forecasting is shown. We follow the example of forecasting that was given in [15]. The authors used the Wolf numbers from 1749–1992 and divided the dataset into • past values • based on the methods applied on the past values the predicted values were calculated and compared with the real values. 7 Launch:

2006.

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207

Fig. 9.8 Solar cycle forecast example. Kind permission of W.D. Pan, B.B. Jason, Vettubu Benson

From the analysis the following conclusions were drawn: (a) The behavior of the Sun’s activity is governed by a chaotic attractor of dimension around 3, which is relatively low. (b) Deterministic chaos takes into account at least 90% of the activity. (c) Solar activity seems to be governed by few periodicities ( around 7). This overall shape of solar activity can therefore be predicted but accurate forecasting is limited to 2–4 years because of a Lyapunov exponent of 0.5. Numerical models of the solar dynamo were also done. The dimensionless dynamo number ν (9.19) D = η2 K 3 2 can be varied for the system of equations that describe such a dynamo. The quantities appearing in the dynamo number are: α regeneration of the field through helicity, η regeneration of the field through turbulent dissipation, ν  average value of the velocity shear, and K wave number of the dynamo wave (see also [19]). The main result is that with increasing dynamo number (>3.84 the period-doubled-chaotic-bifurcation appears. In the chaotic regime intervals of quasi-periodic behavior interrupted by intervals of suppressed activity were found. The time between successive intervals of suppressed activity varies. In the nearby periodic regime the intervals of suppressed activity were evenly distributed. Observations of solar activity parameters show periods of 11, 22, 88-years. There seems to be no 44-year period. There could be a self-regulating dynamo number process to explain the behavior of solar activity, from intermittent phases to quasi periodic doubling phase. During suppressed activity the motions are suppressed, the dynamo number becomes smaller and the system changes out of chaos to the period-doubling regime. The overall impression is that our Sun is near to the region of bifurcation to and from chaos. In [20] a combination of WaveNet and Long Short-Term Memory neural networks were applied to forecast the sunspot number using the years 1749 to 2019 and total sunspot area using the years 1874 to 2019 time series data for the upcoming Solar Cycle 25. The main conclusion in this work was that Solar Cycle 25 will have a maximum sunspot number around 106 ± 19.75, and a maximum total sunspot area

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around 1771 ± 381.17. This indicates that the cycle will be slightly weaker than Solar Cycle 24. An interesting empirical fact about solar cycle behavior is the even-odd rule, also known as the Gnevychev-Ohl rule. Solar cycles follow an alternating pattern of lower and higher maxima. Odd cycles have been typically stronger than even cycles. The last method to characterize the solar cycle comes form non linear dynamics methods. It is based on the fact the (M-1) values of the sunspot number are used to predict the current value. The problem is thus restricted to an M-dimensional phase space. The dimensions are the current value and (M-1) values. Let us assume a time series of length N . We have N − M + 1 points fixed in the phase space. Consecutive points are connected by a line, the phase space trajectory. From this trajectory we can derive the nature of the underlying attractor. How can we find the embedding dimension and the structure of the attractor. One method is the false nearest neighbors method. This algorithm was proposed by [21]. How the number of neighbors of a point along a signal trajectory change with increasing embedding dimension is examined. If the embedding dimension is too low, many of the neighbors will be false. As soon as the correct value of the embedding dimension is found the false neighbors will no longer be neighbors.In short, this method studies how the number of neighbors change as a function of dimension. Another method is based on the correlation dimension [22]. Here it is counted, how the number of neighbors in an embedding space of dimension M >> 1 increases with the distance from a point. The logarithmic steepness d of that function should become constant and this is the correlation dimension. The results on the dimensions are controversial. The problem of determining the dimension has been studied in several papers such as [23, 24] where more references can be found. The last prediction method to be mentioned are neural networks. These are algorithms built up from a large number of small interconnected units, called neurons. Once such a network has been correctly trained it is capable of further predictions. Let us briefly outline this method following the paper of [25]. Let us start with an input layer, which is a group of neurons that are fed by external stimuli represented by a column vector I. The input units send these stimuli to hidden neurons that are not connected to the environment, the units process the information they receive and pass their results to the group of output layer neurons, and the response is represented by O.

9.2.5 Summary of Solar Activity Cycle Prediction Methods Here, we give a short summary of the different methods (the list is not complete, but the methods mentioned here are the most common). Therefore, we can summarize the methods: • compare amplitudes of cycles, • polar fields,

9.2 Long Term Solar Activity Prediction

• • • • • • •

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geomagnetic indices, extrapolation, ARMA model, dynamo waves, neural networks, Gnevychev-Ohl, nonlinear dynamics, chaos theory.

The situation of forecasting solar activity is similar to forecasting Earth’s climate and weather. It is easier to predict long term trends than to predict the exact weather within the next three weeks. Because of the physical nature of both systems, on the onset of chaotic behavior even larger data sampling will not lead to better short term predictions.

References 1. Bain H, Onsager T, Balch C, Steenburgh R, Carr C, Biesecker D, Adamson E (2019) Solar energetic particle forecasting: current capabilities and future directions from a NOAA space weather prediction center perspective. In: Solar heliospheric and interplanetary environment (SHINE 2019), p 239 2. Reikard G (2018) Forecasting space weather over short horizons: revised and updated estimates 62:62–69 3. Krivova NA, Vieira LEA, Solanki SK (2010) Reconstruction of solar spectral irradiance since the Maunder minimum. J Geophys Res (Space Phys) 115(A12):A12112 4. Feigelson ED, Babu GJ, Caceres GA (2018) Autoregressive times series methods for time domain astronomy. Frontiers Phys 6:80 5. Bobra MG, Hoeksema JT, Sun X, HMI Magnetic Field Team (2011) SHARP: space-weather HMI active region patches. In: SDO-3: solar dynamics and magnetism from the interior to the atmosphere, p 17 6. Gleissberg W (1962) Untersuchungen an drei achizigjährigen Zyklen der Sonnentätigkeit. Mit 3 Textabbildungen 55:153 7. Frick P, Galyagin D, Hoyt DV, Nesme-Ribes E, Schatten KH, Sokoloff D, Zakharov V (1997) Wavelet analysis of solar activity recorded by sunspot groups. Astron Astrophys 328:670–681 8. Wilson OC (1978) Chromospheric variations in main-sequence stars. Astrophys J 226:379–396 9. Frick P, Baliunas SL, Galyagin D, Sokoloff D, Soon W (1997) Wavelet analysis of stellar chromospheric activity variations. Astrophys J 483(1):426–434 10. Saar SH, Brandenburg A (1998) Time evolution of the magnetic activity cycle period: results for an expanded stellar sample. In: American astronomical society meeting abstracts. American astronomical society meeting abstracts, vol 193, p 44.04 11. Wright NJ, Newton ER, Williams PKG, Drake JJ, Yadav RK (2018) The stellar rotation-activity relationship in fully convective M dwarfs. Mon Notices 479(2):2351–2360 12. Guinan EF, Ribas I (2002) Our changing sun: the role of solar nuclear evolution and magnetic activity on earth’s atmosphere and climate. In: Montesinos B, Gimenez A, Guinan EF (eds) The evolving sun and its influence on planetary environments. Astronomical society of the pacific conference series, vol 269, p 85 13. Schrijver JC, Siscoe GL (2010) Evolving Solar Activity and the Climates of Space and Earth. Heliophysics 14. Benestad RE (2006) Solar activity and earth’s climate, 2nd ed 15. Rozelot JP (1995) On the chaotic behaviour of the solar activity. Astron Astrophys 297:L45

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16. Gleissberg W (1951) A forecast of solar activity. J Geophys Res 56:292 17. Tlatov AG (2009) The minimum activity epoch as a precursor of the solar activity. Solar Phys 260(2):465–477 18. Brajša R, Wöhl H, Hanslmeier A, Verbanac G, Ruždjak D, Cliver E, Svalgaard L, Roth M (2009) On solar cycle predictions and reconstructions. Astron Astrophys 496(3):855–861 19. Weiss NO, Cattaneo F, Jones CA (1984) Periodic and aperiodic dynamo waves. Geophys Astrophys Fluid Dyn 30(4):305–341 20. Benson B, Pan WD, Prasad A, Gary GA, Hu Q (2020) Forecasting solar cycle 25 using deep neural networks. Solar Phys 295(5):65 21. Kennel MB, Brown R, Abarbanel HDI (1992) Determining embedding dimension for phasespace reconstruction using a geometrical construction 45(6):3403–3411 22. Grassberger P, Procaccia I (1983) Measuring the strangeness of strange attractors. Phys D Nonlinear Phenom 9(1–2):189–208 23. Hanslmeier A, Brajša R (2010) The chaotic solar cycle. I. Analysis of cosmogenic 14 C-data. Astron Astrophys, 509:A5 24. Kurths J, Ruzmaikin AA (1990) On forecasting the sunspot numbers. Solar Phys 126(2):407– 410 25. Calvo RA, Ceccato HA, Piacentini RD (1995) Neural network prediction of solar activity. Astrophys J 444:916

Index

A Aa index, 197 AB Doradus, 138 Absolute magnitude, 125 Active Cavity Radiometer Irradiance Monitor (ACRIM), 197 Adelmus, 57 Adiabatic invariant, 85 Aldebaran, 137 Alfvénic timescale, 63 Alfvén speed, 78, 81, 88 Alfvén time, 88 Alfvén velocity, 65 Alfvén waves, 88 α effect, 93 α − ω-dynamo, 96 Am index, 197 Anaximander, 3, 4 Anelastic approximation, 158 AR autoregressive method, 199 Aristophanes, 4 Asteroids, 38 Astronomical Unit (AU), 37 Aurora, 192 Aurorae, 62, 68, 102 Autocorrelation Function (ACF), 16, 199 Autocovariance, 17 Autoregressive Moving Average (ARMA), 199

B Babcock Leighton Solar Dynamo, 167 Balmer lines, 124 Bastille day event, 143 10 Be, 107 Beltrami fields, 82

Bible, 4 Biermann, L., 68 Bifurcation diagram, 13 Birkeland, K., 68 Blackbody radiation, 59 Boltzmann constant, 124 Box counting dimension, 27 Bremsstrahlung, 63 Butterfly diagram, 72 Butterfly effect, 34 Bz, 86

C 14 C, 105 Ca II H and K, 61 Ca II H, K, 202 Ca II line, 131 Ca plages, 61 Carbon burning, 129 Carlin, G., 6 Carrington event, 103 Carrington, R. C., 62, 68 Carrington rotation, 91 Cassini, 144 Cassini, G. D., 111 Causality, 8 Center to limb variation, 61 Chaos, 2 definition, 1 solar system, 37 Chaoskampf, 2 Charge separation, 76 Chromosphere, 55 CIr, 192 Climate variations, 43 Cloud formation, 192

© Springer Nature Singapore Pte Ltd. 2020 A. Hanslmeier, The Chaotic Solar Cycle, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-15-9821-0

211

212 CNO cycle, 128 Coen, E., 7 Collisions, 46 Comets, 38, 45, 68 Contingency, 9 Convective turnover time, 202 Convective zone, 55 Coriolis force, 157 Coriolis term, 81 Corona, 55, 78 Coronal holes, 192 Coronal Mass Ejection (CME), 61, 65, 192 Correlation dimension, 29, 208 Cosmic rays, 192 Cosmogenic isotopes, 104 Cyclotron radiation, 85

D Dalton, J., 120 Dalton Minimum, 120, 189 Decay time, 77 Demiurge, 5 Dendrochronology, 142 Deoxyribonucleic Acid (DNA), 195 Dirac, P., 9 Dispersion relation, 88 Displacement current, 79 D-layer, 195 Doppler imaging, 132, 133 Drift velocity, 85, 86 Dynamo mechanism, 77 Dynamo number, 97, 207

E Earth magnetosphere, 67, 86 rotation, 197 Earth’s climate, 192 Eckman numbers, 159 Ecliptic, 38, 43 Effective temperature, 125 Einstein, A., 7, 43 Electric power transmissions, 192 El Ni˜no Southern Oscillation (ENSO), 192 Embedding dimension, 208 Empirical mode decomposition, 189 Entropy, 81 Equation of state, 81 Ergodic, 17, 35 Evershed effect, 60 Extrapolation methods, 206

Index F Fabricius, 57 Faculae, 60 Faint young Sun paradox, 203 Faraday law, 76 Fe, 129 Feigenbaum constant, 10 F10.7 emission, 194 Feynman, R., 9 Flare prediction models, 200 Flares, 61, 67, 83, 192 classification, 62 Fluid equations, 80 Force free, 82 Ford, H., 8 Fossil field, 77 Fourier analysis, 203 Fourier transform, 21 Fractal dimension, 27 Fraunhofer, J., 61 Free-free scattering, 153 Frozen field, 80 G Galactic Cosmic Rays (GCR), 189, 193 Galilei, G., 57, 91, 109 Gas drag, 46 General relativity, 45 Geomagnetic storm, 194 Giants, 127 Gleissberg cycle, 72, 110, 142, 203, 205 Gnevychev-Ohl rule, 208 Gnevyshev-Ohl, 71 GOES, 195 Gould, 9 GPS systems, 192 Grad B drift, 86 Granulation, 55 Grassberger, 148 Gyration frequency, 84 Gyration radius, 84 H Hale, G. E., 60, 71 Hale’s law, 90 Halloween event, 144 Hallstatt cycle, 72, 142 H-α, 61 Hα line, 124 Hamilton, 141 Harriot, Th., 57 HD 31993, 138

Index Helga, 49 Helioscope, 109 Helioseismic inversion, 154 Helioseismology, 77, 92, 108 Helioshetath, 70 Heliosphere, 70, 71, 192 Helium burning, 129 Hertzsprung–Russell diagram, 123 Hesiod, 2, 3 HF communications, 192 Hinode, 206 H-K activity, 131 HK flux, 115 Hurst exponent, 151 Hyades, 136 Hydrostatic equilibrium, 40, 53 Hyperion, 49

I Induction equation, 77, 78, 162 Information dimension, 28 Interface dynamo, 187 International Space Station (ISS), 144 Interplanetary CME, 67 Interstellar cloud, 38 Ionosphere, 194

J Jacobian approach, 146 Jeans mass, 39 Joule heating, 81 Joy’s law, 90 Julia set, 13 Jupiter, 102

K Kaplan–Yorke dimension, 35 Kelso, S, J. A., 6 Kirkwood belts, 47 Koch curve, 24 Kolmogorov entropy, 149 Kolmogorov–Sinai entropy, 35 Kp-index, 194 Krause, 185 Kuiper belt, 38, 46

L Larmor radius, 85 Li depletion, 136 Little ice age, 111

213 LL Ori, 134 Logistic map, 9 Long term solar activity, 109 Lorentz attractor, 32 Lorentz, E. N., 29 Lorentz force, 69, 80, 82 Lorentz system, 29, 30 Lothar Collatz example, 14 Luminosity class, 127 Luna 1, 68 Lyapunov exponent, 34, 145 Lyapunov time, 43 Lyapunov time scale, 47

M Magnetic braking, 157 Magnetic buoyancy, 78 Magnetic cycle, 71 Magnetic diffusivity, 79 Magnetic field corona, 78 frozen in, 78 photosphere, 78 Magnetic flux freezing, 78 Magnetic mirror, 85 Magnetic reconnection, 62, 63, 194 Magnetic Reynolds number, 80, 89 Magnetosonic waves, 88 Magnetosphere ring current, 86 Magnitude, 125 Main sequence, 126 Main sequence lifetime, 128 Mars, 102 climate variation, 49 Maunder Minimum, 101, 110, 189 Maxwell equations, 75 Maxwell stress tensor, 155 May, R., 9 Mean field electrodynamics, 94, 182 Mercury precession, 45 Meridional flow, 94 Meteorites, 49 MHD waves, 78, 87 Mirror point, 85 Morlet wavelet, 150 Moving average, 199 Mt Wilson survey, 135

214 N National Oceanic and Atmospheric Administration (NOAA), 194 n-body problem, 41 Neural networks, 200, 208 Neutrinos, 153 Neutron star, 129 New Horizons, 70 NO, 192 NOAA scale, 195 Non gravitational forces, 45 O Ohmic decay, 156 Ohmic decay time, 79 Ohm’s law, 76 ω effect, 92 Oort cloud, 38 Opacity, 54, 154 36 Oph A, 131 Orbital decay, 62 Orion nebula, 39 Ovid, 4 Ozone, 192 P Paracelsus, 2, 5 Parker, 184 Parker, E., 68 Particle collision, 87 Particle motions, 83 P-Cygni, 134 Perfect gas law, 81 Petschek model, 65 Phase space, 33 Pherecydes, 2 Photosphere, 55 Pioneer, 70 Planck constant, 124 Planck law, 59, 124 Planetary nebula, 134 Planets, 37 Pleiades, 136 Pleione, 137 Polar fields, 205 Pomeau–Manneville scenario, 11 Population growth, 11 Pores, 99 Potential field, 82 Powerspectrum, 21 p-p cycle, 153 pp-process, 54

Index p-p reaction, 128 Prandtl number, 30 Precursor methods, 205 Predictability, 7 Prediction horizon, 197 Pressure scale height, 82 Procaccia, 148 Proplyds, 41 Protoplanetary disc, 41 Pytheas, 104

Q Quantum mechanics, 7

R Radiation belts, 195 Radiation pressure, 45 Radiative zone, 55 Radio fadeout, 195 Rädler, 185 Random walk process, 153 Rayleigh number, 30 Regulus, 137 Resonances, 43, 47 Revised sunspot number, 100 Reynolds decomposition, 157 Reynolds stresses, 157 Rikitake model, 161 Ring current, 86 Rossby number, 158, 202 Rotation, 24

S Satellite drag, 192 Saturn, 102 Scale height, 81 Scaling operation, 24 Scheiner, Ch., 57, 91, 109 Schwabe cycle, 72, 203 Schwabe, S. H., 71, 101 Schwarzschild radius, 44 Self similarity, 23 SHARP, 200 Sidereal rotation, 91 Skumanich’s law, 137 Socrates, 2 Solar activity chaotic behavior, 204 Solar activity cycle, 71 Solar activity proxies, 102 Solar convection, 156

Index Solar core, 55 Solar cycle forecasting, 191 Solar diameter variation, 108 Solar differential rotation, 157 Solar Dynamics Observatory (SDO), 200 Solar dynamo, 89, 153 Solar Energetic Particles (SEP), 62, 192, 193, 196 Solar flares, 195 Solar Index Data Center (SIDC), 100 Solar interior, 153 Solar irradiance, 59, 192 Solar magnetohydrodynamics, 75 Solar radiation storm, 195 Solar rotation tracer, 92 Solar system chaos, 43, 50 formation, 38 Solar wind, 62, 68, 102, 195 Sound speed, 78 Sound wave, 88 Space climate, 191 Space weather, 191 Space weather forecasting, 196 Spin-orbit resonances, 49 Spörer G., 101 Spörer Minimum, 114, 120 Spörer’s law, 90 SSB, 43 Stand-off-distance, 135 Star formation, 16 Stark effect, 60 Star spots, 131 Staudacher, 141 Staudacher, J. C., 101 Steenbeck, 185 Stefan Boltzmann law, 59, 125 Stellar activity, 131 Stellar activity cycle, 135 Stellar age, 135 Stellar chromospheric cycles, 202 Stellar rotation, 81 Stellar winds, 133 Streamer belt, 68 Suess cycle, 72, 142 Sun, 37 differential rotation, 91 energy transport, 54 interior, 53 luminosity, 59 mass loss, 69 sidereal rotation, 91

215 spectral type, 127 synodic rotation, 91 Sunspot relative number, 99 Sunspots, 57, 92 Superflare, 73 Supergiants, 127 Sweet–Parker model, 64 Synchrotron radiation, 85 Synodic rotation, 91

T Tachocline, 55, 92, 94, 154 Tambora, 120 TEC, 196 Tent map, 12 Termination shock, 71 Theorphrastus, 57 Thermonuclear fusion, 54 Tohu wa bohu, 4 Torsional oscillations, 188 Total Solar Irradiance (TSI), 116, 192 Translation, 24 Tree rings, 107 Triple α, 128 Trojan asteroids, 49 T Tauri, 135 T-Tauri phase, 39 Turn-off point, 131

U UBV, 124 Ulysses, 144

V Van Allen belts, 103 Vega, 137 Verhulst, P. F., 9 Voyager, 70 V711 Tau, 132

W Wavelet analysis, 149, 203 White dwarf, 129 White light flare, 62 White noise signal, 20 Wiener–Khinchin Theorem, 18 Wien’s law, 124 Wolf, 141 Wolf number, 71, 99 Wolf, R., 71, 99

216 Z Zeeman-Doppler imaging, 133 Zeeman effect, 60

Index Zeeman, P., 60 Zurich number, 99