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A Theoretical Foundation https://ntrs.nasa.gov/search.jsp?R=19970010384 2020-06-16T02:40:11+00:00Z NASA-CR-203^22 JPL Publication 96-13 < . Relativistic Navigation: A Theoretical Foundation Slava G. Turyshev July 15, 1996 National Aeronautics and Space Administration Jet Propulsion Laboratory California Institute of Technology Pasadena, California JPL Publication 96-13 Relativistic Navigation: A Theoretical Foundation Slava G. Turyshev July 15, 1996 National Aeronautics and Space Administration Jet Propulsion Laboratory California Institute of Technology Pasadena, California The research described in this publication was performed while the author held an NRC-NASA Resident Research Associateship award at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. The author was on leave from the Bogolyubov Institute for Theoretical Microphysics, Moscow State University, Moscow, Russia. Abstract We present a theoretical foundation for relativistic astronomical measurements in curved space-time. In particular, we discuss a new iterative approach for describing the dynamics of an isolated astronomical N-body system in metric theories of gravity. To do this, we generalize the Fock-Chandrasekhar method of the weak-field and slow-motion approximation (WFSMA) and develop a theory of relativistic reference frames (RFs) for a gravitationally bounded many-extended-body problem. In any proper RF constructed in the immediate vicinity of an arbitrary body, the N-body solutions of the gravitational field equations are formally presented as a sum of the Riemann-flat inertial space-time, the gravitational field generated by the body itself, the unperturbed solutions for each body in the system transformed to the coordinates of this proper RF, and the gravitational interaction term. We develop the basic concept of a general WFSMA theory of the celestial RFs applicable to a wide class of metric theories of gravity and an arbitrary model of matter distribution. We apply the proposed method to general relativity. Celestial bodies are described us- ing a perfect fluid model; as such, they possess any number of internal mass and current multipole moments that explicitly characterize their internal structures. The obtained rel- ativistic corrections to the geodetic equations of motion arise because of a coupling of the bodies' multiple moments to the surrounding gravitational field. The resulting relativistic transformations between the different RFs extend the Poincare group to the motion of de- formable self-gravitating bodies. Within the present accuracy of astronomical measurements we discuss the properties of the Fermi-normal-like proper RF that is defined in the immediate vicinity of the extended compact bodies. We further generalize the proposed approximation method and include two Eddington parameters (7, /?). This generalized approach was used to derive the relativistic equations of satellite motion in the vicinity of the extended bodies. Anticipating improvements in radio and laser tracking technologies over the next few decades, we apply this method to spacecraft orbit determination. We emphasize the number of feasi- ble relativistic gravity tests that may be performed within the context of the parameterized WFSMA. Based on the planeto-centric equations of motion of a spacecraft around the planet, we suggested a new null test of the Strong Equivalence Principle (SEP). The experiment to measure the corresponding SEP violation effect could be performed with the future Mercury Orbiter mission. We discuss other relativistic effects, including the perihelion advance and the redshift and geodetic precession of the or biter's orbital plane about Mercury, as well as the possible future implementation of the proposed formalism in software codes developed for solar-system orbit determination. All the important calculations are completely documented, and the references contain an extensive list of cited literature. Ill "PAGE MISSING FROM AVAILABLE VERSION" Contents 0 Notations and Definitions 1 1 Introduction and Overview 1 1.1 The Motivation and the Structure of the Report 1 1.2 The Problem of Relativistic Astronomical Measurements 7 1.3 The Qualitative Description of the Astronomical Systems of Interest 18 1.4 Different Methods of Constructing the Proper RF 21 2 Parametrized Post-Newtonian Metric Gravity 27 2.1 An Isolated One-Body Problem 27 2.2 The Limitations of the Standard PPN Formalism 29 2.2.1 The Simplified Lagrangian Function of an Isolated N-Body System ... 30 2.2.2 The Simplified Barycentric Equations of Motion 32 3 WFSMA for an Isolated Astronomical N-Body System 35 3.1 The General Form of the N-Body Solution <. 35 3.2 The Post-Newtonian KLQ Parameterization 39 3.2.1 The Properties of the Coordinate Transformations in the WFSMA ... 40 3.2.2 The Inverse Transformations 41 3.2.3 The Coordinate Transformations Between the Two Proper RFs 43 3.2.4 Notes on an Arbitrary Rotation of the Spatial Axes 44 3.3 The Definition of the Proper RF 46 4 General Relativity: Solutions for the Field Equations 51 4.1 The Solution for the Interaction Term 51 4.2 The Solution of the Field Equations in the Proper RF 57 4.3 Decomposition of the Fields in the Proper RF 59 5 General Relativity: Transformations to the Proper RF 63 5.1 Finding the Functions KA and Q^ 64 5.1.1 Equations for the Functions KA and Q^ 64 5.1.2 The Solution for the Function KA 64 5.1.3 The Solution for the Function Q% 66 5.2 Finding the Function LA 68 5.2.1 Equations for the Function LA 68 5.2.2 The Solution for the Function LA 70 5.3 Equations of Motion for the Massive Bodies 72 5.4 The Proper RF of the Small Self-Gravitating Body 73 5.5 The Fermi-Normal-Like Coordinates 77 6 General Relativity: The Proper RF for the Extended Body 81 6.1 The Extended-Body Generalization 81 6.2 Conservation Laws in the Proper RF 88 6.3 The Solution for the Function 6w%0 96 7 Parameterized Proper RF 99 7.1 Parameterized Coordinate Transformations 99 7.2 Equations of the Spacecraft Motion 103 7.3 Gravitational Experiment for Post-2000 Missions 107 7.3.1 Mercury's Perihelion Advance 109 7.3.2 The Redshift Experiment 110 7.3.3 The SEP Violation Effect 110 7.3.4 The Precession Phenomena 112 8 Discussion: Relativistic Astronomical RFs. 115 8.1 The Geocentric Proper RF 116 8.2 The Satellite Proper RF 118 8.3 The Topocentric Proper RF 120 8.4 Discussion 122 Appendix A: Generalized Gravitational Potentials 125 Appendix B: Power Expansions of the General Geometric Quantities 127 B.I. Expansion for the Metric Tensor gmn 127 mn B.2. Expansion for the det [gmn] and g 127 B.3. Expansion for the gmn = J=ggmn 128 B.4. Expansion for the Gauge Conditions 129 B.5. Expansion of the Christoffel Symbols — 129' B.6. Expansion for the Ricci Tensor Rmn 130 B.7. Expansion for an Arbitrary Energy-Momentum Tensor T^n 131 Appendix C: Transformation Laws of the Coordinate Base Vectors 133 C.I. Direct Transformation of the Coordinate Base Vectors 133 C.2. Transformation of the Background Metric 7mn 133 C.3. Inverse Transformation of the Coordinate Base Vectors 135 C.4. Mutual Transformation Between the Two Quasi-Inertial RFs 136 Appendix D: Transformations of Some Physical Quantities and Solutions 137 D.I. Transformation of the Gauge Conditions 137 D.2. Transformation of the Ricci Tensor Rmn 137 D.3. Transformation Law for an Arbitrary Energy-Momentum Tensor Tmn 138 D.4. Transformation of the Unperturbed Solutions hmn 140 D.5. Transformation Rules for the Interaction Term /ijjj£ 141 D.6. Transformation for the Energy-Momentum Tensor of a Perfect Fluid 141 Appendix E: Transformations of the Gravitational Potentials 143 Appendix F: Christoffel Symbols in the Proper RF^ 149 F.I. Christoffel Symbols With Respect to the Background Metric 7^ 149 F.2. Christoffel Symbols With Respect to the Riemann Metric g^n 149 Appendix G: The Component g$Q and the Riemann Tensor 151 G.I. The Form of the Component 7^, 151 G.2. Lemma : 152 G.3. The Form of the 'Inertia! Friction' Term 153 G.4. The Form of the Interaction Term , 154 G.5. The Form of the Riemann Tensor in the Proper RF^ 154 Appendix H: Some Important Identities 157 Appendix I: Astrophysical Parameters Used in the Report 161 Acknowledgments 163 References 163 VI 0 Notations and Definitions. In this report, the notations are the same as in (Landau & Lifshitz, 1988). In particular, the small latin letters n,m,k... run from 0 to 3 and the Greek letters a,/3,7,... run from 1 to 3; the italic capitals A,B,C number the bodies and run from 1 to N; the comma denotes a standard partial derivative and the semicolon denotes a covariant derivative; repeated indices imply an Einstein rule of summation; round brackets surrounding indices denote symmetrization and square brackets denote anti-symmetrization. The geometrical units c — G = I are used throughout the report, where G is the universal gravitational constant and c is the speed of light. We designate 6apg as the fully anti-symmetric Levi-Civita symbol (6123 = 1); the metric convention is accepted to be (H ); ^mn = diag(l,—l,—l,-l) is the Minkowski metric in Cartesian coordinates of the inertial RF; 7^n(2A) *s tne Minkowski metric in the coordinates (z^) of the RF^ that is constructed in the immediate vicinity of an arbitrary body (A); gmn denotes the effective Riemann metric of the curved space-time; and g = det(gmn). To enable one to deal conveniently with sequences of many spatial indices, we shall use an abbreviated notation for 'multi-indices' where an upper-case letter in curly brackets denotes a multi-index, while the corresponding lower-case letter denotes its number of indices, for example: {P} := so Hi^2 • • -P-P-, S{j} := 5MlM2...w.
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