II i e NATIONAL AERONAUTIC . AND SPACE ADMINISTRATION Technical Memorandum 33-451 Design and Implementation of Models for the Double Precision Trajectory Program (DPTRAJ,; Gerd W. Spier r N71-25152 tCOD_) (NASA CR OR TN_X _ AD NUMBER) (CAIEG-ORY) JET PROPULSION LABORATORY CALIFOINIA INSTITUTE OF TECNNOLOG-If PASADENA, CALIFORNIA,. April 15, 1971 ':- ........ :,_L." _L :.. i_,%'_''. ' NATIONAL AERONAUTICS AND SPACE ADMINISTRATION Technical Memorandum 33-451 Design and Implementation of Models for the Double Precision Trajectory Program (DPTRAJ) eerd W. Spier 1 _r JET PROPULSION LABORATORY CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENAp CALIFORNIA April 15, 1971 V_}_C_bIG pAC,_B'_K _IOT_'ILLI_ Preface The work described in this report was performed by the Systems Division of the Jet Propulsion Laboratory. % JPL TECHNICAL MEMORANr_UM 33-451 iii Acknowledgmen_ The author wishes to express his appreciation to Mr. Francis Sturms, group supervisor of the Trajectories and Performance Group, and to Mr. Ahmad Khstib, cognizant engineer of DPTRA], both of the JPL Systems Analysis Section (Section 892), for their efforts and contributions to this paper. In addition, he would like to extend appreciation to Mr. Fred Lesh, supervisor of the Traiectory Group, to Mr. Alva Joseph, cognizant programmer, to Mr. Joseph Witt and Dr. Thomas Talbot, all of the JPL Flight Operations and Deep Space Network Progr._mming Section (Section 315), and to Mr. John Strand of Informatics for e- providing valuable insights into the computer program DPTRA]. _) .,..' Jv dPL TECHNICAL MEMORANDUM 33-451 t Contents h Introduction 1 Ih Time and Coordinate Transformations in General 2 A. Coordinate Conversions 2 B. Rotations 2 1. Space-fixed 2 2. Body-fixed 2 IIh Systemsof Time 2 A. Tropical Year and EphemerisTime 3 B. Atomic Time 3 C. Universal Time 3 D. Transformation Between Time Scales 3 IV. Coordinate Type Transformations 4 A. Spherical to Cartesian Cocrd!nate=- Transformation 4 B. Cartesian to Spherical Coordinates Transformation 5 C. Classical to Cartesian Coordinates Transformation 6 1. Elliptic orbit 7 2. Hyperbolic orbit B 3. Parabolic orbit 9 D. Cartesian to Classical Coordinat,;s Transformation 10 E. Pseudo-Asymptote and Asymptote Coordinates to Cartesian Coordinates Transformation 14 V. Rotations of Coordinate Systems. 19 A. Earth-Related Transformations 19 B. Rotation From Mean Earth Equator of 1950.0 to Mean Equator of Date 19 C. Mean Obliquity of Eclipticand its Time Derivative . 22 D. Earth Mean or True Equatorial Coordinates to EclipticCoordinates Rotation 22 E, Mean Earth Equatorand Equinoxof Date Coordinates to True Earth Equator and Equinox of Date Coordinates Rotations , , 23 F. Rotation Transforming Earth.CenteredTrue Equatorial of Date, Space.Fixed Coordinates to Earth-Fixed Coordinates . 2s i JPL TECHNICAL MEMORANDUM 33.451 V /: Contents (contd) . G. Mars-Related Transformations 27 1. Rotation matrices for position vectors 29 2. Rotation matrices for velocity vectors . 30 H. Moon-Related Transformations 31 I. Rotation From Earth Mean Ecliptic to Moon True Equator Coordinate System 33 J. Rotation From Moon True Equator and Equinox of Date Coordinates to Moon-Fixed Coordinates . 34 VI. Translation of Centers 35 VII. Equations of Motion of a Spacecraft. 35 A. Newtonian Point.Mass Acceleration . 36 1. Center of mass and invariable plane . 36 2. Force function . 38 3. Transfer of origin and perturbing forces . 38 B0 Acceleration Caused by an Oblate Body . 39 C. Acceleration Caused by Solar Radiation Pressure and Operation of Attitude-Control System. 42 D° Acceleration Caused by Finite Motor Burns 45 E. Acceleration Caused by Indirect Oblateness 47 1. Basic equations 47 2. Derivation of_erm__2 48 F. Acceleration Caused by General Relativity 52 1. Relativistic equations of motion 53 2. Heliocentric ephemeris of a planet Cother than earth) 55 3. Heliocentric ephemeris of earth-moon barycente_. 55 4. Geocentric ephemeris of moon 55 5. Equations of motion for generation of a spacecraft ephemeris 56 ! VIII. Numerical Integration of Equations of Motion of a Spacecraft . 56 A. Solution Method 56 B. Backward Differences 56 C. Derivation of Predictor-Corrector Formulas 58 D. Computation of Coefficients a_Cs), b_Cs), c_Cs), d_Cs) . 6O _T vi •If_L TECHNICAL MEMORANDUM 33.451 i Contents (contd) E. Starting Procedures 64 1. Start by Taylor series expansion 64 2. Computation of Taylor series coefficients 66 3. Extrapolative start 69 F. Control of Step Size h 70 1. Step-size control by range list 70 2. Automatic step-size control 71 G. Reverse Integration 72 IX. Differential Correction Process 72 A. Interpolation and Differential Correction of Basic Planetary Ephemerides 73 1. Interpolation ._ 73 2. Differential correction o_ basic ephemerides . 74 B. Partial Derivatives Formulation 75 C. Correction to Ephemeris . 77 X. Summary of Results Obtained From Integrating the Equat!ons of Motion . 77 A. Body Group 77 B. Conic Group . 82 ] C. Angle Group . 94 l.-Clock and cone angles 94 2. Angle iASD and limb angles . 96 3. Hinge and swivel angles . 96 , Appendix A. Proof of Kepler's Laws 98 r_ Appendix B. Kepler's Equation 103 Appendix C. Solution of Kepler's Equation 108 -_ Appendix D. The Vis-Viva Integral 111 l Appendix E. Linearized Flight Time 113 i Glossary 117 1 References . 131 * Bibliography 132 Index 123 JPL TECHNICAL MEMORANDUM 33-451 vii Contents (contd) Table 1. Raddii of relativity spheres 54 Figures 1. Spherical coordinates 4 2. Enlargement of the H-plane ° • ° ° 5 3. Orbital elements . 7 4. Eccentric and true anomalies 7 5. A hyperbolic orbit 8 6. A parabolic orbit . 9 7. Orbit of a spacecraft . 11 8. Spacecraft flight plane 15 9. Launch geometry 16 10. The flight-path angle P 17 11. The radius vector Rm_ 17 12. Mean equator of 1950.0 to mean equator of date 20 13. Rotation about X"-axis 20 14. Rotation about Z'-axis 20 15. Mean obliquity of the ecliptic 22 16. True or mean obliquity . 23 17. Mean and true equinox of date 24 18. Prime meridian of the earth . 25 19. Greenwich hour angle . 26 20. Nutation in longitude, true and mean obliquity 26 21. Martian equator and equinox 27 22. Spherical triangle 1 . 28 23. Spherical triangle 2 29 24. Lunar equator and orbit . 31 25, The orbit of the moon relative to earth . 32 26. Earth mean ecliptic and moon true equatorial coordinate systems . 33 27. Space. and body-fixed lunar' coordinates 34 viii JPL TECHNICAL MEMORANDUM 33.451 .... !ii 2 Contents (contd) Figures (contd) Translation of centers 35 A system of bodies in an inertial coordinate frame . 36 Rectangular coordinate system axes x', y', and zp, relative to body-fixed axes Xb, yb, Zb . 40 The angle K . 44 Orientation of a spacecraft relative to sun, earth, and reference body 45 The parameter D . 45 Spacecraft moving into or out of shadow of body 45 Up-north-east coordinate system 52 Simplified schematic representation of gravity in space-time system 53 Step sizes h and hp 69 Right ascension and declination of a spacecraft 79 The vector II_ 79 Angles P and _ . 79 The vector/'* 80 Angles y and 8O The quantity/_ 82 Sun-shadow parameter d 82 3 Sun-shadow parameter d_ 82 Eccentric anomaly E 84 The quantity F 85 L_ Angles p and vm.... 85 Deflection angle between asymptote vectors 86 A A ^ The vectors W, N, and M 88 ^ A The vectors g, V, ._, and _ 89 i Impact parameter vector B 9O Magnitude of vector B 9O Coordinates of vector B . 90 B vector impact radius 91 Incoming and outgoing asymptote vectors 92 The R, S, T target coordinate system 93 JPL TECHNICAL MEMORANDUM 33-451 Ix Contents (contd) Figures (contd) 58. The vectors Jibe and R_o 93 59. The angle ET_ 0 94 60. The angle XYZ 94 61. Vectors _, and _ . 95 62. Clock and cone angles 95 63. The angle iASD 95 64. The angle xPNi 96 ^ 65. The vector $ . 96 66. Hinge. and swivel angles 96 A-1. Two-body gravitational attraction 98 A-2. Radial component of 100 A-3. Velocities in polar reference frame 101 B-I. An elliptic orbit 103 B-2. A hyperbolic orbit 105 B-:3. A parabolic orbit . 106 C-1. The Method of false position 109 D-1. Velocity components of body b 111 E-1. Trajectory approach geometry 113 i JPL TECHNICAL MEMORANDUM 33-451 '_ ............ i--. " -_'-_._.-_- .; --" " ......... : .............. ' ...... - _ -__ ... - Abstract A common requirement for all lunar and planetary missions is the extremely accurate determination of the trajectory of a spacecraft. The Double Precision Trajectory Program (DPTRAJ) developed by JPL has proved to be a very accurate and dependable tool for the computation of interplanetary trajectories during the Mariner missions in 1969. This report describes the mathematical models that are used in DPTRA] at present, with emphasis on the development of the equations. +! ,+." +, 4 JPL TECHNICAL MEMORANDUM 33-4_I x| Design and Implementation of Models for the Double Precision Trajectory Program (DPTRAJ) I. Introductior, Because the total acc.rleration of a spacecraft cannot be integrated in closed form, recourse must be taken to An important factor in determining high-precision numerical methods. At present, the equatiorts of motion interplanetary trajectories is the computation and sub- of a spacecraft are integrated by a so-called second-sum sequent integrahon of the acceleration of a spa.,2ecraft numerical-integration scheme relaHve to some central that is moving in the solar system and is subject to a body (Cowcll method). variety of forces. The forces acting upon the spacecraft determine its acceleration according to Newton's second law; therefore, knowledge of the forces implies knowledge The equations of motion are solved for the spacecraft of the acceleration of the spacecraft. Integration of the ordy, and ignore the negligible perturbations of the space- total acceleration in some convenient frame of reference craft on celestial bodies (i.e., on the sun, moon, and establishes the ephemeris of the spacecraft, and hence planets); hence, it is sufficient to obtain positions and its trajectory.