System Analysis of Space Missions

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THE MINISTRY of EDUCATION and SCIENCE of RUSSIA FEDERATION SAMARA STATE AEROSPACE UNIVERSITY

V. M. Belokonov, I. V. Belokonov System Analysis of Space Missions

Electronic Lecture Notes

SAMARA 2011

УДК 629.78 ББК 22.62 Authors: Belokonov Vitaliy Mikhailovitch, Belokonov Igor Vitalyevitch

Editorial processing: I. V. Belokonov Computer imposition: I. V. Belokonov Belokonov, V. M. System Analysis of Space Missions = Системный анализ космических миссий[Electronic resource]: Electronic Lecture Notes /V. M. Belokonov, I. V. Belokonov; The Ministry of Education and Science of Russia Federation, Samara State Aerospace University. - Electronic text and graphic data (18,3 Mb). - Samara, 2011. - 1 CD-ROM. The subject of discipline, "System Analysis of Space Missions," includes a set of interrelated problems, a comprehensive solution that allows us to assess the feasibility of a mission to prove the main technical characteristics and design parameters that achieve the goals. Since the central element of any space mission is the flight of the spacecraft, the fundamentals of the theory of flight, engineering models for the design of spacecraft parameters, the methods of motion control with criterial basis of the space mission is a prerequisite. This lecture course includes the information for a coherent assessment of the feasibility of missions in near-Earth space, and for missions within the solar system. Interuniversity Space Research Department, Master Program Educational Content “Space Information Systems and Nanosatellites. Navigation and Remote Sensing” for education direction 010900.68 «Applied Mathematics and Physics»

© Samara State Aerospace University, 2011

CONTENT Introduction

4

1. Mathematical modeling of motion - the main tool for system analysis of space missions

5

1.1. The basis and classification of mathematical motion models

5

1.2. Coordinate systems used in system analysis of space missions

5

1.3. The basis spacecraft motion equations

15

2. The spacecraft motion equations

22

2.1. The motion equations in central force field

22

2.2. The basic integrals of motion

23

2.3. The equations of the orbit and the velocity in polar coordinates

27

2.4. The motion on elliptical orbits

28

2.5. The motion on hyperbolic trajectories

32

2.6. Typical space velocities

35

3. Elements of the space orbits

38

4. Analysis of the possibility of payload orbiting by carrier rocket

42

4.1. Design characteristics of carrier rocket

42

4.2. An estimation of the orbiting possibility

47

4.3. Design of nominal trajectory for the carrier rocket first stage

49

4.4. Design of nominal trajectory for the non-atmoshere carrier rocket stages

49

5. Analysis of the disturbunsed motion of spacecraft

57

5.1. Orbit disturbunses, caused by non-central gravity field

57

5.2. Orbit disturbunses, caused by Earth atmosphere

58

6. Orbiting maneuvers of spacecraft

65

6.1. Orbiting transfers

65

6.2. Maneuvers for orbit corrections

75

6.3. The relative motion of two spacecraft

77

7. Analysis of re-entry vehicles movement

81

7.1. Deorbiting maneuvers

81

7.2. Analysis of movement in the atmosphere

83

8. Analysis of interplanetary missions

85

8.1. An approximate method for design trajectories of interplanetary transfers

85

8.2. Analysis of the transfer from Earth orbit to the orbits of near planets satellites (Mars, Venus)

86

8.3. Analysis of the disturbing maneuvers

91

8.4. Analysis of the spacecraft motion in the Earth-Moon mission

98

2

INTRODUCTION The subject of discipline, "System Analysis of Space Missions," includes a set of interrelated problems, a comprehensive solution that allows us to assess the feasibility of a mission to prove the main technical characteristics and design parameters that achieve the goals. Since the central element of any space mission is the flight of the spacecraft, the fundamentals of the theory of flight, engineering models for the design of spacecraft parameters, the methods of motion control with criterial basis of the space mission is a prerequisite. This lecture course includes the information for a coherent assessment of the feasibility of missions in near-Earth space, and for missions within the solar system. For missions in near-Earth space training materials to evaluate the rationality of the chosen scheme, the duration and the required power consumption of each phase of the mission: elimination spacecraft (SC) on an intermediate orbit, the transition from the orbit of the target launch into orbit and / or maneuvering associated with the need for periodic changes in the orbit if it is a part of the mission, the main disturbance experienced by the spacecraft to the target orbit, maneuvered to rendezvous and docking, such as the orbital station (if required by program mission), transfer the spacecraft into orbit or to reduce disposal of spacecraft from orbit, after the completion of the operation and / or return from orbit space station crew, evaluation of unexpected situations that arise at any stage of the mission. Lectures integrated with the course "Physics of near-Earth space", which reinforces its system component.

1. Mathematical modeling of motion - the main tool for system analysis of space missions 1.1. The basis and classification of mathematical motion models A mathematical model for the spacecraft motion is a set of differential and functional operators, graphical and tabular dependences that uniquely determine the spacecraft trajectory. The mathematical model consists of five main elements: – motion equations (differential equations, reduced to the Cauchy form and functional relationships); – methods for solving differential equations (the most common method is the numerical integration of Runge-Kutta of fourth order); – characteristics of the gravitational and other fields, as well as the atmosphere; –parameters of the spacecraft, the characteristics of the propulsion system, aerodynamic characteristics; – laws of motion control. The first two elements form the basic mathematical model that is universal. The remaining elements of the model form its binding to a particular appearance of the spacecraft and the intended purpose. There are two types of mathematical models: deterministic and stochastic. In a deterministic mathematical model all the original characteristics are completely defined by the specified values or functions. In the stochastic mathematical model the baseline characteristics describe the probabilistic models (are random parameters or random functions). 1.2. Coordinate systems used in systems analysis of space missions Coordinate system used for recording movement patterns are uniquely characterized by three elements: the origin, the base line and the focus. Coordinate systems can be used as reference systems for describing the spatial position of the spacecraft and for the design of the vector equations of motion. Classification of coordinate systems:  heliocentric (origin at the Sun mass center);  planetocentric (origin at the planet mass center);  topocentric (origin at this point on the surface of the planet);  aircraftcentric (origin at the aircraft mass center).

1.2.1 Heliocentric ecliptic coordinate system Heliocentric ecliptic coordinate system (OE XE, YE, ZE) is shown in figure 1.

Figure 1. Heliocentric ecliptic coordinate system.

It is used the following definitions and notations: the plane of the ecliptic (the main plane) is the plane motion of the Earth around the Sun; γ – vernal equinox; ХE – major axis directed toward the vernal equinox (the vernal equinox - the point is the celestial sphere in which the Sun on the vernal equinox); ZE – axis directed to the north world pole; YE – axis directed to the point of the winter solstice (WSS); FPC - first point of cancer; SS - solstice. Heliocentric ecliptic coordinate system is an inertial coordinate system and is used for calculating of interplanetary trajectories. Tropical year – the time duration for one complete turn of Earth around the Sun ТTROP = 365, 2422 days Mean solar day (time of one Earth turn) - the time between two consecutive lower transit of the Sun ТMS = 24 hours = 86400 sec. Sidereal day - the time of one revolution with respect to the fixed stars ТSDD = 86164 sec.

1.2.2 Planetocentric coordinate systems Geocentric equatorial inertial coordinate system (0e, Xi, Yi, Zi,) is shown in Figure 2, where - the angle of right ascension LA, δ - declination angle of the spacecraft, ω3 angular velocity of Earth rotation, S - sidereal time at Greenwich, the local star time S 0  ( S   ) λ - Geographic longitude of spacecraft; φ - geographic latitude of spacecraft, P - subsatellite point.

Figure 2. Geocentric equatorial coordinate system

Greenwich geocentric coordinate system (0e, XG, YG, ZG) is a non inertial rotating system.

1.2.3 Topocentric coordinate system Earth's geographic coordinate system (0s, X0, Y0, Z0) is a non-inertial coordinate system and shown in Figure 3

Figure 3. Earth's geographical coordinate system.

Here the origin of coordinate system is starting point C or starting position (i.e., spaceport); X0C axis (pillar) - tangential to the meridian; Z0C axis - tangential to the parallel; Y0C axis - perpendicular to the plane Z0-X0 and the angle A is the azimuth of the launch. Earthly starting coordinate system (0c, XC, Yc, Zc) is a non-inertial coordinate system and shown in Figure 3.

1.2.4 Spacecraftcentric coordinate system The normal system of coordinates (0, Xd, Yd, Zd) is shown in Figure 4.

Figure 4. The normal system of coordinates.

Here OXg axis parallel to the tangent to the meridian, the axis parallel to the tangent to OZg parallels; OYg axis perpendicular to the plane Zg OXg;  - angular velocity due to the change in longitude of the spacecraft;  - angular velocity due to the change in latitude 

of the spacecraft;  д     - kinematic equation of the angular velocity of rotation of the normal coordinate system.  To operate the spacecraft: thrust P , gravitational attractive force G , aerodynamic force Ra .

In a normal coordinate system, the force of gravity is given by the highest accuracy GÃ  xä   0   GÃ   GÃ  xä     mg  - vector of the Earth's gravity.       GÃ  zä   0 

Coupled system of coordinates (0, X, Y, Z) is shown in Figure 5. This coordinate system is commonly used in studying the movement of spacecraft around the mass center.

Figure 5. Coupled system of coordinates.

Here ψ , υ , γ – respectively, the angles of yaw, pitch, roll;  - angular velocity of pitch;      - angular velocity of yaw;  - angular velocity of roll;        - the kinematic equation of spacecraft rotational motion. In a body system of coordinates is given thrust with the highest accuracy P  P   0  - vector of thrust.    0 



Speed (aerodynamic) coordinate system (0, Xa, Ya, Za) is shown in Figure 6.  Here α - angle of attack; β - slip angle; V - airspeed; Ya - aerodynamic lift force; X a aerodynamic drag force; Z a - lateral aerodynamic force. In the high-speed aerodynamic force coordinate system is defined with the highest accuracy     Ra   X a  Ya  Z a

 X a   Ra   Ya  - vector of the aerodynamic force    Z a 

 

Figure 6. Speed (aerodynamic) coordinate system.

Orbital coordinate system (0, Xk, Yk., Zk.) is shown in Figure 7 (in this course is the primary coordinate system).

  Vk - speed of the spacecraft trajectory; W - wind speed;  a - aerodynamic roll angle; ψ к –   corner of the course; Θ – angle of the trajectory;  к - course angular velocity;   trajectory slope angular velocity;  кg   к   - vector kinematic equation of motion.

Figure 7. Orbital coordinate system.

1.2.5 Transfer from one coordinate system to another The challenge is in terms of vector projections from one coordinate system to another. Look at an example of transfer from converting a normal coordinate system to coupled coordinatecsystem (Fig. 8) a  a xg  iä  a yg  jä  a zg  k ä 

Figure 8. vector a a xg , a yg , a zg 

 a ñâ

 ax      ay  ; a   z

 a g

 axä      a yä  a   zä 

 a ñâ   A XXg    a g  a x  11  12 a     22  y   21  a z   31  32

 13   a xg     23   a yg  ,   33   a zg 

where 11  cos  x?; xg  ,  21  cos  y ?; xg  ,  31  cos  z ?; xg  12  cos  x ?; yg  ,  22  cos  y ?; yg  ,  32  cos  z ?; y g  13  cos  x ?; z g  ,  23  cos  y ?; zg  ,  33  cos  z ?; z g 

The transition matrix from one coordinate system to another is determined by multiplying the matrices of elementary rotations, taken in sequence, the opposite sequence of turns (Fig. 9).

Figure 9. The transition from one coordinate system to another one.

Carried out a sequence of elementary rotations: the first rotation – on the angle ψ, the second rotation - on the angle υ, the third rotation – on the angle γ.

A   A  A  A  - the transition matrix XX g



 cos  A    0  sin 



0 1 0



 sin   0  - rotation by angle ψ  cos  

 cos sin  0 A    sin  cos 0 - rotation by angle υ  0 0 1 0 1  A   0 cos  0  sin  0 1  A  A   0 cos 0  sin 

0  sin   - rotation by angle γ cos   0   cos sin 0  cos sin   sin cos 0   sin  cos    cos   0 0 1  sin  sin

The resulting transition matrix takes the form

sin cos  cos   cos  sin 

0  sin    cos 

cos sin  0  cos 0  sin     A    sin   cos  cos  cos  sin    0 1 0    sin   sin   cos  sin  cos    sin  0 cos  cos  cos sin   sin   cos      sin   cos   cos cos  cos  sin   cos   sin   sin   cos  sin   sin   cos  sin   cos   cos  sin   sin   sin   sin   cos   cos 



XX g



1.3 The basis spacecraft motion equations Assume that S – solid (not deformable) surface (Fig. 10). The equations of spacecraft motion (variable composition) can be written as the equation of rigid body motion, which include the mass of the hardened body, if the forces acting on the spacecraft, added the variation of force and Coriolis forces of inertia and moment of this force, a purely reactive force and moment.

Рис.10. The body of variable composition

   dQ  e   F  Fp  Fвар  Fкор dt

     dK 0  M e  M Fp  M вар  M кор dt

where

    F p , M Fр - reactive force and moment by joining and dropping masses; Fвар , M вар -

variational forces and moments due to unsteady motion of the particles in the channel of   the propulsion system (due to the smallness of neglect); Fкор , M кор - Coriolis forces and moments arising from the relative motion of particles inside the channel by rotating the propulsion of spacecraft (due to the smallness of neglect). When testing the engine is measured purely reactive force of pressure and static forces due to external pressure, which in summation give equation    P  F p  Fст.д.

The equations of sapacecraft motion in the inertial coordinate system have the form  dV à  e   m  F  P  dt  . e   dK 0  M Mp  dt

In drawing up the system equations of motion for the non-inertial coordinate system, added to the portable and Coriolis forces of inertia caused by rotation of the reference system.       dV m  F e  P  Fине  Fинкор    dt      dK 0 e е кор   M  M p  M ин  M ин  dt

Projection of the vector on the axis of rotation system coordinate can be written as:      da x  da y  da z    da d di dj dk da '    a x i  a y j  az k  i j k  ax  ay  az  a dt dt dt dt dt dt dt dt dt    di  dt    i        da da     d j   j  a dt  dt  dt  dk     k   dt 





Then we can write the vector equation for the velocity and the angular momentum vector K in the rotating coordinate system   dV d V      V dt dt  dK d K       K0 dt dt

Differential motion equations of the mass center in the projections on the axes of an rotating moving frame of reference will take the form:



  dV  m x   yV z   zV y    Fix   dt    dV y   m   zV x   xVz    Fiy   dt     dV z  m   xV y   yVx    Fiz   dt  

where   x ,  y ,  z  - angular velocity of the moving coordinate system. Differential motion equations around the mass center in terms of projections on the principal central inertia axes can be written as:     K 0  I x  x iгл  I y  y j гл  I z  z k гл

d x  I z  I y    y   z   M ix dt d y Iy  I x  I z    x   z   M iy dt dz Iz  I y  I x   y   x   M iz dt Ix

X, Y, Z – principal central axes of inertia. Vector motion equation of of spacecraft mass center the relative to an inertial coordinate system of Greenwich are given        dV m  P  Ra  mg  Fине  Fинкор   Fi    dt сила тяжести mg

Let us write the motion equations of the spacecraft trajectory in projections on the axes of the orbital coordinate system (Fig. 12)

Figure 12. Orbital coordinate system

  dV d 'V      V dt dt

Projecting the vector motion equation, it can obtain  dV  m xk   ykVzk   zkV yk    Fixk dt  

 dV  m yk   zkVxk   xkVzk    Fiyk  dt   dV  m zk   xkV yk   ykVxk    Fizk  dt 

because Vxk  V , V yk  Vzk  0

, then dV    Fixk  dt  mV zk   Fiyk   mV yk   Fizk    m

Let us form the kinematic equations (Fig. 12).          ä     ; êä    k ; k  ä  õä or      k      k  

Figure 11. A normal coordinate system

 

To determine the projections of the angular velocity  ,  use conversion matrix method           xk     x x      A k д   yk            zk 





 cos        sin  ;  0   

where

         

  

  xk x д yk   A  zk 

xk





 0   0       

        cos  sin   cos  cos  sin        sin  cos       sin  cos k k k   yk ,   yk ,   zk ,   yk  0  zk   ,      sin  k ,   zk ,







    

k yk

  к cos

k zk

0

 yk   cos k  sin   cos  cos  sin     sin  k sin    k cos   zk    sin k cos    cos k  

, After transformations, we write the kinematic motion equations V V  cos   cos k    xд   r r  Vzд V cos  sin k     r cos  r cos  

Substituting V cos sin k V cos sin k V cos sin k  yk  cos k sin  cos   cos sin   sin k sin   k cos r cos r cos r cos V cos2  sin k tg   yk   k cos  , r

 zk  

V cos   r

are written the dynamic motion equations mV   Fixk

V 2 cos  mV   Fiyk  m r mV 2 cos2  sin k tg  mV cos k   Fizk  r

We carry out the design of the external forces acting on the aircraft at the orbital axis of the coordinate system кор.  xa   0   Fин. xk   .    y a   A xk xд   mg    Fинкор . yk  .   z a   0   Fинкор . zk     ik jk kk       mWk  2m 3  V  2m  3 xk  3 yk  3 zk  2mV 3 zk j  2mV3 ykVk V 0 0

  Fxk    xk x  Fyk   A   Fzk   

P   0   A xk xa  0 

 

Fинкор .













Finally, the motion equations in the trajectory coordinate system take the form - dynamic motion equations mV  P cos  cos   X a  mg sin 

mV  Pcos sin  sin  a  sin  cos a   Ya cos a  Z a sin  a  mg cos  2mV3 zk  mV cos  k  P   cos  sin  cos  a  sin  sin  a   Ya sin  a  Z a cos  a  2mV 3 yk  m V 2  sin k  cos 2   tan   r

- kinematic motion equations

mV 2 cos 3 r

 

V cos cos k V cos  sin k  ,  H  V sin  r r ,

Overload vector is the sum of external forces except gravity (the thrust forces and aerodynamic forces), the force of gravity    P  Ra n - overload mg

The projections of the overload vector - on the axis of the orbital coordinate system P cos  cos   X a n xk  n yk n zk

mg P cos  sin  sin  a  sin  cos  a   Ya cos  a  Z a sin  a  mg P  cos  sin  cos  a  sin  sin  a   Ya sin  a  Z a cos  a  mg

- on the axis of the velocity coordinate system: P cos  cos   X a n xa  n ya n za

mg P sin   Ya  mg  P cos  sin   Z a  mg

- related to the e-axis coordinate system: n x   n xa   n   A ХХ к   n   y  ya   n za   n z 





The equations of motion in the g-forces (in dimensionless form) can be written as 1  V  n xk  sin  g V  V V 2 cos   n yk  cos  2  zk  g g gr 

V 2 sin  k cos 2   tg V cos 2V  k  n zk   yk  g g gr

2. The spacecraft motion equations The motion of the spacecraft is considered as a motion of material point in the Earth gravitational field. Following assumptions are accepted: - attracting body is spherical and the spherical density distribution (Fig. 12); - spacecraft mass is negligible compared to the mass of the Earth (LA does not affect the motion of the Earth); - neglects the influence of perturbing forces: gravitational perturbations, the resistance of the the atmosphere, influence of radiation pressure, electromagnetic forces;

- used a spherical coordinate system .

Figure 12. The model of a spherical Earth

2.1. The equation of motion of the spacecraft in a central force field The motion equation in a central gravity field is

  dV  Mm r m  m G r ; Gr   2    3  r , dt r r r

where γ – universal gravitation constant, μ - Earth's gravitational parameter,    g Г   3 r - gravity acceleration vector. r

Differential motion equation of spacecraft in vector form is  dV M   r 0 dt r 3

where      M   3,98602 10 5 км3/с2 Differential motion equations in projections on the axes of the inertial coordinate system are:   x 

x  0 r3   y  3 y  0 r   z  3 z  0   r

where

x  f1 t , C1 ,..., C 6    y  f 2 t , C1 ,..., C6  - called the first integrals; z  f 3 t , C1 ,..., C 6  x  f 4 t , C1 ,..., C 6   y  f 5 t , C1 ,..., C 6  - called the second integral. z  f 6 t , C1 ,..., C 6 

Integrating the differential equation in vector form can be obtained three integrals.

2.2 Basic integrals of equations of motion

Integral of energy Let us differential motion equation in vector form multiply scalar on the velocity vector   dV    dr  V

dt



r 0 r 3  dt 

Given that

   dV 1 dV 2 1 dV 2 dV d  V 2   V   V   dt 2 dt 2 dt dt dt  2    dr dr  dr d    r  r , 3 r    dt dt r dt dt  r  d V 2       0, dt  2 r 

perform the integration and obtain V2  h   , where h - constant energy integral. 2

r

2

V2 

2  h - Integral of energy r

The first term of this formula is the kinetic energy per unit of body mass; the second term is the potential energy. Thus, the energy integral expresses the conservation of total mechanical energy in a central gravity field: the sum of kinetic and potential energy per unit of body mass during the whole period of its motion remains constant. The constant h is the energy integral for the initial conditions: 2 t = 0, r = r0, V = V0 => h  V02  r0

Depending on the sign of the constant energy integral orbit may be closed and unclosed.

because V 2 

2 2 2  h  0 , when, at h ≥ 0 – orbit is open, at h < 0:  h  0, r  r r h

orbit is closed. The area integral 

Radius vector r multiply vector on the left on the differential motion equation:   dV    r d dt d dt

dt



r3

r  r   0

 0   dr   dV r V  V  r  dt dt   r V  0









Integrating this expression is found in the vector form the area integral    r V  C

The area integral expresses the law of conservation of angular momentum in the central gravity field.  It is possible to find the module of the vector constant С of the area integral. In scalar form the area integral can be wtitten (Figure 13) r  V sin   r V cos   C , where θ-angle of the velocity vector to the local horizon. r  V cos   C - scalar form of the area integral. Module of area integral constant is considered over the initial conditions: t = t0, r = r0, V = V0, θ = θ0 C  r0  V0 cos 0

Figure 13 Graphical interpretation of the area integral

The integral area can be written in the coordinate form:  i   r V  x x





  j k y z y z

 yz  y z  C1 (projection on axis Ox)   zx  xz  C 2 (projection on axis 0y) - the area integral in the coordinate form.  xy - x y  C (projection on axis 0z) 3 

Multiplying by the area integral on differential dt, we obtain    r  Vdt  Cdt

which implies 2d  C  dt Then d C2 - Kepler's second law, which reads: dt

in the central gravity field, swept by the radius vector of a moving point per unit time remains constant. Of the area integral also implies that movement of spacecraft in the central gravity field is 



carried out in the constant plane (the two vectors r and V form the motion plane).

Laplace integral Vector differential motion equation multiply on the vector constant of area integral on the right    dV        dr     dt  C   r 3  r   r  dt   0    

Then, after elementary transformations  dV  d   C  V C dt dt       dr    r  r     r 3   dt  r 3 d      V C  r   0 dt  r 







        dr  d r    r  1 dr  1 dr  d    r r   r  r   r         dt  dt   r 2 dt r dt  dt  r    



we obtain the expression, which is called Laplace integral in vector form



    r  (V  C )   f r

where f - vector constant of the Laplace integral (Figure 14).

Figure 14 Graphical representation of the Laplace integral

From the Laplace motion integrals implies an important property in a central attraction: in the motion plane exists a constant direction defined by the Laplace vector. The line through the attractive center parallel to the vector Laplace, called the axis of apsides is taken as the reference direction of the angular spacecraft motion (the angle of the true anomaly ν). i jk Because (V  C )  x y z  ( y  C3  z  C2 ) i  ( z  C1  x  C3 ) j  ( x  C2  y  C1 ) k , then C1C2C3

Laplace integral in scalar form is

 yC3  zC 2    x 





 f1  r   zC3  xC3   y  f 2  r   xC 2  yC1   z  f 3  r 

Vector f is perpendicular C , it follows that there is a connection between the constants found by the integrals   f  C  0  f 1C1  f 2 C 2  f 3 C 3

2       2          f  f   V  C  r   V 2C 2  r  V  C  2 r 2 ; r  V  C  C  r  V  C 2 r  2r r   2 f 2  V 2C 2  2 C 2   2   2  C 2  (V 2  ) r r 2

2

















Equation relating the nodule of the Laplace constant and module of the area integral constant and the energy integral constant is f 2   2  C 2h because coupling constants between the two ratios of the integrals exists, then the sevenderived scalar integrals are only five independent.

2.3 The equations of the orbit and the velocity in polar coordinates Let us vector Laplace constant multiply scalar on the radius vector             f  r  V  C   r   r  V  C  r  r 2  c 2    r r     f  r  f  r cos c 2    r  f  r cos .

r

Then we obtain the equation of the orbit in polar coordinates c2 c2  p r     f cos 1  f cos 1  e cos  where we use the following notation: c2 p - focal parameter (characterizes geometric size of the orbit);  f e - eccentricity (characterizes the shape of the orbit). 

This result reflects the first law of Kepler for motion in a central gravitational field: the trajectory is conic section, one of the focuses of which is placed in the center of attraction, and the main focal axis coincides with the direction of the Laplace vector. There is the following classification of orbits as a function of eccentricity: e = 0 – circle orbit; 0 < e < 1 – elliptic orbit; e = 1 – parabolic orbit; e > 1 – hyperbolic orbit. Let us find the projection of the velocity in polar coordinates (Figure 15).

Figure 15. Velocity in polar coordinates dr dr d Vr   - radial velocity component - the projection of the velocity vector in the dt d dt

direction of the radius vector; d Vn  r  V cos - transverse velocity component - projection of the vector in the dt

direction perpendicular to the radius - vector. C  r V  cos   r Vn  Vn  r

d C  dt r

c r

d C  dt r 2

hence Vr  Vn 

2

p sin 

1  e cos 

p 1  e cos

2

c 1  e cos   p2

 e sin  p

  p 1  e cos    1  e cos  2 p p

V  Vr2  Vn2 

 1  e 2  2 cos p

The above expressions are used to solve problems connected with the observation of satellites and the measurement of the orbital parameters of the Earth's surface

2.4 Motion on elliptical orbits

The geometry of the elliptical orbit Elliptical orbits are the most common in nature (Fig. 16)

Figure 16. The motion of the spacecraft in an elliptical orbit

The equation of the elliptical orbit

x2 y2   1 -. a2 b2

0 < e 50 km), it is necessary to make choice of the scheme launch. Depending on the structure and purpose support can meet the following options for removing schemas by reference orbit. Option 1. Launch provides an upper (second) stage of a two-stage vehicle with a rocket engine. When choosing a program of motion is solved a two-parameter boundary value problem elimination ( H k  H îðá ,  k   îðá ). Option 2. Launch of the complete two upper level of a three-stage vehicle with a rocket engine. The intermediate stage provides a gradual decrease in the slope of the trajectory to  k 2  1 3...1 4 k 1 . When choosing a program of the movement of this stage is solved oneparameter boundary value problem. The upper stage completes the withdrawal of the orbit. When you select a mission upper stage of a two-parameter boundary value problem is solved ( H îðá ,  îðá ). Option 3. Launch of the complete two upper level with a solid propellant rocket motors. Solid-fuel engines are working short time during which the engine in continuous operation payload does not have time to get up to a given height. In this case, enter the passive leg of the route between the intermediate and upper-stage vehicle. The intermediate stage provides injection at an angle of inclination of the trajectory at the end of the stage to a passive flight along a ballistic trajectory to apogee was reached given the orbit H îðá . At the apogee motor turns the upper stage, which provides acceleration of the payload to orbital velocity, subject to conditions   0 . In determining the trajectories of the upper stages of the carrier must take into account the curvature of the Earth's surface and the inhomogeneity of the gravitational field. Aerodynamic and inertial forces caused by Earth's rotation are neglected. The system of differential equations of motion in projections on the axis start coordinate system (Fig. 30), taking into account the assumptions made after linearization of the projections of gravitational acceleration is given by:

Fig.30. Traffic Pattern upper stage vehicle 2

u  p cos   x,

    p sin   g 0  2 2 y,  x  u, y   ,  n g n g P where p  0 0  0 0 – acceleration traction; m 1   t 1  a  m n g    0 0 – relative flow rate of fuel; m PóäÏ

a  mT m — coefficient of filling the fuel level;   t t k — dimensionless time; mT ; m — mass of fuel and the initial mass of stage; t k — Working time step;  2  g 0 R .

The initial conditions are expressed in terms of the parameters obtained at the end of the trajectory of the previous stage: u 0  Vk 1 cos  k1 ,  0  Vk1 sin  k 1 , x0  x k1 , y 0  y k1 . The final conditions depend on the version of the scheme launch. As an approximation, the optimal pitch angle of the program adopted the program, obtained by solving the variational problem of motion of the upper stage of a planeparallel gravitational field outside the atmosphere (Fig. 31): tg  tg 0  Bt , где B  b t k After selecting the parameters of the optimal program for the pitch angle finally found the parameters of the final calculated parameters of movement of the upper stage relative to the coordinate system starting from the formulas. Errors are then determined by the final parameters and their comparison with the specified tolerances.

Fig. 31. The optimal program of the movement of upper stages

Note. For the option of launching the spacecraft after the final calculation of motion parameters with the tabulated value of the coefficient of filling the fuel level a  should

make the final conversion of the motion parameters to the actual value of the coefficient a stage by the following algorithm: the first formula Aviation defined increment of velocity and final velocity for a full time job steps: V  u Ï ln z  ln z , z   1 1  a , Vk  Vk  V ; the second formula is determined by Tsiolkovsky increment the path traversed by the last stage of the arc of the orbit over time t k  t k  :  ln z    ln z      S  V0 t k  u Ï t k  1     V0 t k  u Ï t k  1    z  1    z   1   where V0 — initial rate of stage;

is the increment of angular range and level of total angular range:  k  S R  H îðá ,  k   k   k ; calculate the final stage of the motion parameters. x k  R  H îðá   sin  k , y k  R  H îðá   cos  k  R,

u k  Vk cos  k ,  k  Vk sin  k .

The solution of two-parameter boundary value problem of deducing the orbit pa with full fuel burn-up the last stage of maximum possible final speed, which may not coincide with the speed required to travel on a given orbit. To ensure the withdrawal of the payload at the right speed to make the conversion time of movement of the last stage, which is equivalent to the change in fuel supply. Assuming that at the end of the last stage of the movement occurs without changing the angle of inclination of the trajectory (for spacecraft  îðá  0 ) and without the resistance of the atmosphere, to calculate the speed is acceptable to apply a formula Aviation. Located excess (deficiency) of velocity given supplements of the Earth's rotation: V  Vk  Vâð  Vîðá . This excess (deficiency) of velocity is due to excessive (insufficient) supply of fuel last stage. Determined by the need fuel and Working time steps according to the formulas:   sin  îðá , V  u n  ln z n  ln z n   g í t kn  t kn z n  z n exp  V u Ï  ; z n  1 mn ; z n   mTn  m n , t kn   mÒn

Here, n number of the last stage, the quantities with primes correspond to the needs values. With the new time value of t kn the last stage of the media need to repeat the calculation of two-parameter (for SC) boundary value problem for the elimination of the previously cited algorithms. As can be seen from the above, put into the desired orbit requires solving a threeparameter for the SC boundary value problem which is solved by successive approximations. After the conversion should be to find an estimate of the payload mass that can be output to the desired orbit, and if there is an excess (deficiency) of fuel mT , it must be taken into account in the payload:  ; mÏÍ  m ÏÍ  mÒ . mT  mTn  mTn

Recalculation of the final parameters of the starting coordinates in the geocentric inertial coordinate system The calculation of the trajectory coordinates were found breeding, and projection speeds at the end of the active site relative to the coordinate system starting Ox c y c z c . To determine the characteristics of the orbital motion of the need for this same point in time to calculate the coordinates and velocity components relative to the inertial reference system. As an inertial frame of reference we take the geocentric equatorial coordinate system axis OÇ xè meridian which passes through the starting point at the end of the active site. Note that in this way entered the inertial coordinate system is rotated with respect to the geocentric inertial frame stellar angle at S, where S - local sidereal time at the starting point in the moment of the spacecraft to the reference orbit. Starting position of the coordinate system O3 x c y c z c

Fig. 32 The transition from home to an inertial coordinate system.

adopted with respect to an inertial O3 xè y è z è determined by the breadth of the item starting  0 and azimuth of launch A0 (Fig. 32). The transition from the coordinates of the end of the active site x k , y k , z k in the starting system to the initial coordinates x0 , y 0 , z 0 orbital motion in the geocentric inertial frame (Fig. 16) is satisfied by the formulas: x 0   x k cos A0 sin  0  R  y k  cos  0 ; y 0  x k sin A0 ; z 0  x k cos A0 cos  0  R  y k sin  0 . The value of the radius - vector of the initial point of the orbital motion r0  x02  y 02  z 02 . The projections of the relative velocity Vk on the axis of the geocentric system O3 xè y è z è expressed through the projections of the relative velocity u k ,  k starting on the axis of the analogous formulas. Absolute velocity at the beginning of the orbital motion of the sum of the relative velocity V r  Vk ,. and drive speed, which is defined by: 





iè

jè

è

V e   3 r 0  0

0

x0

y0

 3   3 y 0 i è   3 x 0 j è , z0

















where i è , j è ,  è – unit vectors of the geocentric coordinate system. Thus, the projection of the absolute velocity in the geocentric coordinate axis and the starting point of the orbit are given by: Vx 0  x 0  u k cos A0 sin  0   k cos  0   3 y 0 ; V y 0  y 0  u k sin A0   3 x0 ; Vz 0  z0  u k cos A0 cos  0   k sin  0 . The initial velocity of orbital motion, and angle it to the local horizon, respectively, V0 

x 02  y 02  z 02 ,  0  arcsin

x0 x 0  y 0 y 0  z 0 z 0 . r0V0

5. Analysis of the perturbed motion of spacecraft After reaching orbit the spacecraft experiences a different perturbing forces that affect the nature of assigned tasks. It is therefore necessary for the system analysis of space missions to assess at least the main disturbing factors, which include non-central gravitational field and the orbits below 500 km - inhibition atmosphere. The motion of the spacecraft is in a complex force field, which is characterized by a large number of different physical forces of nature. The main force is the force of gravitational attraction nyutonovckaya corresponding to a spherical Earth model with a uniform radial distribution of mass. The motion of this force is called the unperturbed motion, characterized by the constancy of the orbital elements. When designing a spacecraft moving at low altitudes up to 1,000 km, it is necessary to use more complex movement patterns that take into account additional forces. Among them are a force that appears when using a more complex model of the Earth - Ellipsoid of revolution (usually limited to the second zonal harmonic light of the potential expansion of the field of gravity), as well as the aerodynamic force, which arises from the action of the Earth's atmosphere. Such motion of the spacecraft is perturbed movement. Accounting for the additional forces leads to the appearance of periodic and secular perturbations in the motion of the spacecraft. In the systemic analysis of space missions usually take into account the secular perturbations, which grow monotonically from the coil to the thread. Secular perturbations are calculated by the approximate method for one turn of a spacecraft around the Earth (in the range of variation of the argument of latitude and from 0 to 2 ).

5.1 5.1 Perturbations of the orbit caused by the non-central gravitational field of the Earth Secular perturbations of the inclination i, the focal parameter p, orbital eccentricity e is zero . The secular perturbation (precession) relative to the stars (in the absolute coordinate system) longitude of the ascending node of the orbit for one turn is determined by the ratio of Under the influence of compression of the Earth orbit ascending node moves in the direction opposite to the rotation of the Earth (the West), for direct orbits ( ) and the direction of rotation of the Earth (the East) for the inverse orbit ( ).For polar orbits (i = 90°) precession of the orbital plane is missing ( ). The secular perturbation (precession) of the argument of the perigee in the absolute coordinate system for a loop is determined by the ratio of

Under the influence of compression of the Earth at perigee argument argument moves in the direction of motion of the spacecraft, with argument of the perigee moves in the opposite direction of the spacecraft. At displacement of the perigee is not . Secular perturbations of the N turns :

В формулах (13),(14) р - focal parameter and e - eccentricity of the orbit at the initial time.

5.2 Disturbances of movement caused by braking the atmosphere Secular perturbations of the longitude of ascending node , orbital inclination i (if not included capture the atmosphere of the Earth rotation), the argument of perigee (for the exponential model of the atmospheric density) are equal to zero ( ).Secular perturbations of the focal parameter p and eccentricity e for a loop defined by the following formula for the model of an isothermal atmosphere : - elliptical orbit with a small initial eccentricity

(16)

- elliptical orbit with an average initial eccentricity

- elliptical orbit with a high initial eccentricity

Здесь

, - height of the atmosphere (equal to the height of a fictitious column of

the atmosphere, whose density is equal to , and which is at a height the same pressure as the atmosphere is considered), - atmospheric density at perigee, - the average density of the atmosphere for near-circular orbits. Secular perturbations of the focal parameter p and eccentricity e for N turns are calculated by the formulas. Under the influence of atmospheric elliptical orbit of the spacecraft over time more and more close to the circular. The period of revolution is monotonically decreasing, and the average speed increases. The maximum speed of lowering the orbit to account for peak area and the minimum - at the district's perigee. For a circular (or near-circular e