Relativistic Celestial Mechanics of the Solar System

of the Solar System
This authoritative book presents the theoretical development of gravitational physics as it applies to the dynamics of celestial bodies and the analysis of precise astronomical observations. The authors work at the University of Missouri and the United States Naval Observatory, which is one of the premier institutions in the world for expertise in astrometry, celestial mechanics, and timekeeping. The initial chapters review the fundamental principles of celestial mechanics and of special and general relativity. This background material is the foundation for understanding relativistic celestial mechanics, astrometry, and geodesy which is treated in the main part of the book. The text is based on recent rec- ommendations from the International Astronomical Union.
From the contents: Newtonian celestial mechanics Introduction to Special Relativity General Relativity Relativistic Reference Frames Post-Newtonian Coordinate Transformations Relativistic Celestial Mechanics Relativistic Astrometry Relativistic Geodesy Relativity in IAU Resolutions
Sergei Kopeikin studied general relativity at the Department of Astronomy of Moscow State University, Russia. He obtained his PhD in relativistic astrophysics from Moscow State Uni- versity in 1986, where he was then employed as an associate professor. In 1993, he moved to Japan to teach astronomy at Hitotsubashi University, Tokyo. He was an adjunct staff member and thereafter visiting professor at the National Astronomical Observatory of Japan. In 1997, Professor Kopeikin moved to Germany and worked at the Institute of Theoretical Physics of the Friedrich Schiller University, Jena. Three years later he accepted the position of a professor of physics at the University of Missouri, Columbia, USA.
Michael Efroimsky received his PhD from the University of Oxford in 1995. He then worked at Tufts, Harvard, and the University of Minnesota. Since 2002, he has been working as a staff astronomer at the US Naval Observatory in Washington, D.C. His current research interests are centered around celestial mechanics of the solar system. Dr. Efroimsky served as the Chair of the Division on Dynamical Astronomy of the American Astronomical Society, and is currently a member of several commissions of the International Astronomical Union.
George Kaplan was a staff astronomer at the U.S. Naval Observatory in Washington, D.C., from 1971 to 2007, and now works as an independent consultant. He received his PhD degree from the University of Maryland, USA, in 1985. His professional interests focus on the fi eld of positional astronomy, both its observational and theoretical aspects. His work includes pub- lications in astrometry, celestial reference systems, solar system ephemerides, Earth rotation, navigation algorithms, and astronomical software. Dr. Kaplan is currently the president of Commission 4 (Ephemerides) of the International Astronomical Union. The minor planet 16074 is named in his honor.
Sergei Kopeikin, Michael Efroimsky, George Kaplan
Relativistic Celestial Mechanics of the Solar System
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The Authors
Prof. Sergei Kopeikin University of Missouri Department of Physics and Astronomy Columbia, Missouri, USA [email protected]
Dr. Michael Efroimsky US Naval Observatory 3450 Massachusetts Ave NW Washington, DC, USA [email protected]
Dr. George Kaplan Consultant to US Naval Observatory 3450 Massachusetts Ave NW Washington, DC, USA [email protected]
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Pioneer 10 artwork, Ames Research Center/Nasa Center, 2006
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V
Contents
Symbols and Abbreviations XXIII References XXXI
1 Newtonian Celestial Mechanics 1 1.1 Prolegomena – Classical Mechanics in a Nutshell 1 1.1.1 Kepler’s Laws 1 1.1.2 Fundamental Laws of Motion – from Descartes, Newton, and Leibniz to
Poincaré and Einstein 2 1.1.3 Newton’s Law of Gravity 7 1.2 The N -body Problem 10 1.2.1 Gravitational Potential 11 1.2.2 Gravitational Multipoles 13 1.2.3 Equations of Motion 15 1.2.4 The Integrals of Motion 19 1.2.5 The Equations of Relative Motion with Perturbing Potential 21 1.2.6 The Tidal Potential and Force 22 1.3 The Reduced Two-Body Problem 24 1.3.1 Integrals of Motion and Kepler’s Second Law 24 1.3.2 The Equations of Motion and Kepler’s First Law 27 1.3.3 The Mean and Eccentric Anomalies – Kepler’s Third Law 31 1.3.4 The Laplace–Runge–Lenz Vector 35 1.3.5 Parameterizations of the Reduced Two-Body Problem 37 1.3.5.1 A Keplerian Orbit in the Euclidean Space 37 1.3.5.2 A Keplerian Orbit in the Projective Space 39 1.3.6 The Freedom of Choice of the Anomaly 43 1.4 A Perturbed Two-Body Problem 45 1.4.1 Prefatory Notes 45 1.4.2 Variation of Constants – Osculating Conics 47 1.4.3 The Lagrange and Poisson Brackets 49 1.4.4 Equations of Perturbed Motion for Osculating Elements 51 1.4.5 Equations for Osculating Elements in the Euler–Gauss Form 53 1.4.6 The Planetary Equations in the Form of Lagrange 55
VI Contents
1.4.7 The Planetary Equations in the Form of Delaunay 56 1.4.8 Marking a Minefield 57 1.5 Re-examining the Obvious 58 1.5.1 Why Did Lagrange Impose His Constraint? Can It Be Relaxed? 58 1.5.2 Example – the Gauge Freedom of a Harmonic Oscillator 59 1.5.3 Relaxing the Lagrange Constraint in Celestial Mechanics 62 1.5.3.1 The Gauge Freedom 62 1.5.3.2 The Gauge Transformations 64 1.5.4 The Gauge-Invariant Perturbation Equation in Terms of the Disturbing
Force 66 1.5.5 The Gauge-Invariant Perturbation Equation in Terms of the Disturbing
Function 67 1.5.6 The Delaunay Equations without the Lagrange Constraint 69 1.5.7 Contact Orbital Elements 72 1.5.8 Osculation and Nonosculation in Rotational Dynamics 75 1.6 Epilogue to the Chapter 76
References 77
2 Introduction to Special Relativity 81 2.1 From Newtonian Mechanics to Special Relativity 81 2.1.1 The Newtonian Spacetime 81 2.1.2 The Newtonian Transformations 84 2.1.3 The Galilean Transformations 85 2.1.4 Form-Invariance of the Newtonian Equations of Motion 88 2.1.5 The Maxwell Equations and the Lorentz Transformations 89 2.2 Building the Special Relativity 94 2.2.1 Basic Requirements to a New Theory of Space and Time 94 2.2.2 On the “Single-Postulate” Approach to Special Relativity 96 2.2.3 The Difference in the Interpretation of Special Relativity by Einstein,
Poincaré and Lorentz 97 2.2.4 From Einstein’s Postulates to Minkowski’s Spacetime of Events 99 2.2.4.1 Dimension of the Minkowski Spacetime 99 2.2.4.2 Homogeneity and Isotropy of the Minkowski Spacetime 99 2.2.4.3 Coordinates and Reference Frames 100 2.2.4.4 Spacetime Interval 100 2.2.4.5 The Null Cone 101 2.2.4.6 The Proper Time 102 2.2.4.7 The Proper Distance 103 2.2.4.8 Causal Relationship 103 2.3 Minkowski Spacetime as a Pseudo-Euclidean Vector Space 103 2.3.1 Axioms of Vector Space 103 2.3.2 Dot-Products and Norms 105 2.3.2.1 Euclidean Space 106 2.3.2.2 Pseudo-Euclidean Space 107 2.3.3 The Vector Basis 108
Contents VII
2.3.4 The Metric Tensor 111 2.3.5 The Lorentz Group 113 2.3.5.1 General Properties 113 2.3.5.2 Parametrization of the Lorentz Group 115 2.3.6 The Poincaré Group 118 2.4 Tensor Algebra 120 2.4.1 Warming up in Three Dimensions – Scalars, Vectors, What Next? 120 2.4.2 Covectors 123 2.4.2.1 Axioms of Covector Space 123 2.4.2.2 The Basis in the Covector Space 125 2.4.2.3 Duality of Covectors and Vectors 126 2.4.2.4 The Transformation Law of Covectors 127 2.4.3 Bilinear Forms 128 2.4.4 Tensors 129 2.4.4.1 Definition of Tensors as Linear Mappings 129 2.4.4.2 Transformations of Tensors Under a Change of the Basis 130 2.4.4.3 Rising and Lowering Indices of Tensors 131 2.4.4.4 Contraction of Tensor Indices 132 2.4.4.5 Tensor Equations 133 2.5 Kinematics 134 2.5.1 The Proper Frame of Observer 134 2.5.2 Four-Velocity and Four-Acceleration 136 2.5.3 Transformation of Velocity 138 2.5.4 Transformation of Acceleration 140 2.5.5 Dilation of Time 142 2.5.6 Simultaneity and Synchronization of Clocks 143 2.5.7 Contraction of Length 146 2.5.8 Aberration of Light 148 2.5.9 The Doppler Effect 150 2.6 Accelerated Frames 152 2.6.1 Worldline of a Uniformly-Accelerated Observer 155 2.6.2 A Tetrad Comoving with a Uniformly-Accelerated Observer 157 2.6.3 The Rindler Coordinates 158 2.6.4 The Radar Coordinates 162 2.7 Relativistic Dynamics 166 2.7.1 Linear Momentum and Energy 166 2.7.2 Relativistic Force and Equations of Motion 169 2.7.3 The Relativistic Transformation of the Minkowski Force 172 2.7.4 The Lorentz Force and Transformation of Electromagnetic Field 174 2.7.5 The Aberration of the Minkowski Force 176 2.7.6 The Center-of-Momentum Frame 178 2.7.7 The Center-of-Mass Frame 182 2.8 Energy-Momentum Tensor 184 2.8.1 Noninteracting Particles 184 2.8.2 Perfect Fluid 188
VIII Contents
2.8.3 Nonperfect Fluid and Solids 189 2.8.4 Electromagnetic Field 190 2.8.5 Scalar Field 191
References 194
3 General Relativity 199 3.1 The Principle of Equivalence 199 3.1.1 The Inertial and Gravitational Masses 199 3.1.2 The Weak Equivalence Principle 201 3.1.3 The Einstein Equivalence Principle 202 3.1.4 The Strong Equivalence Principle 203 3.1.5 The Mach Principle 204 3.2 The Principle of Covariance 207 3.2.1 Lorentz Covariance in Special Relativity 208 3.2.2 Lorentz Covariance in Arbitrary Coordinates 209 3.2.2.1 Covariant Derivative and the Christoffel Symbols in Special
Relativity 211 3.2.2.2 Relationship Between the Christoffel Symbols and the Metric
Tensor 212 3.2.2.3 Covariant Derivative of the Metric Tensor 213 3.2.3 From Lorentz to General Covariance 214 3.2.4 Two Approaches to Gravitation in General Relativity 215 3.3 A Differentiable Manifold 217 3.3.1 Topology of Manifold 217 3.3.2 Local Charts and Atlas 218 3.3.3 Functions 218 3.3.4 Tangent Vectors 219 3.3.5 Tangent Space 220 3.3.6 Covectors and Cotangent Space 222 3.3.7 Tensors 224 3.3.8 The Metric Tensor 224 3.3.8.1 Operation of Rising and Lowering Indices 225 3.3.8.2 Magnitude of a Vector and an Angle Between Vectors 226 3.3.8.3 The Riemann Normal Coordinates 226 3.4 Affine Connection on Manifold 229 3.4.1 Axiomatic Definition of the Affine Connection 230 3.4.2 Components of the Connection 232 3.4.3 Covariant Derivative of Tensors 233 3.4.4 Parallel Transport of Tensors 234 3.4.4.1 Equation of the Parallel Transport 234 3.4.4.2 Geodesics 235 3.4.5 Transformation Law for Connection Components 237 3.5 The Levi-Civita Connection 238 3.5.1 Commutator of Two Vector Fields 238 3.5.2 Torsion Tensor 240
Contents IX
3.5.3 Nonmetricity Tensor 242 3.5.4 Linking the Connection with the Metric Structure 243 3.6 Lie Derivative 245 3.6.1 A Vector Flow 245 3.6.2 The Directional Derivative of a Function 246 3.6.3 Geometric Interpretation of the Commutator of Two Vector Fields 247 3.6.4 Definition of the Lie Derivative 249 3.6.5 Lie Transport of Tensors 251 3.7 The Riemann Tensor and Curvature of Manifold 253 3.7.1 Noncommutation of Covariant Derivatives 253 3.7.2 The Dependence of the Parallel Transport on the Path 255 3.7.3 The Holonomy of a Connection 256 3.7.4 The Riemann Tensor as a Measure of Flatness 258 3.7.5 The Jacobi Equation and the Geodesics Deviation 261 3.7.6 Properties of the Riemann Tensor 262 3.7.6.1 Algebraic Symmetries 262 3.7.6.2 The Weyl Tensor and the Ricci Decomposition 264 3.7.6.3 The Bianchi Identities 265 3.8 Mathematical and Physical Foundations of General Relativity 266 3.8.1 General Covariance on Curved Manifolds 267 3.8.2 General Relativity Principle Links Gravity to Geometry 269 3.8.3 The Equations of Motion of Test Particles 273 3.8.4 The Correspondence Principle – the Interaction of Matter and
Geometry 277 3.8.4.1 The Newtonian Gravitational Potential and the Metric Tensor 277 3.8.4.2 The Newtonian Gravity and the Einstein Field Equations 279 3.8.5 The Principle of the Gauge Invariance 282 3.8.6 Principles of Measurement of Gravitational Field 286 3.8.6.1 Clocks and Rulers 286 3.8.6.2 Time Measurements 289 3.8.6.3 Space Measurements 290 3.8.6.4 Are Coordinates Measurable? 294 3.8.7 Experimental Testing of General Relativity 297 3.9 Variational Principle in General Relativity 300 3.9.1 The Action Functional 300 3.9.2 Variational Equations 303 3.9.2.1 Variational Equations for Matter 303 3.9.2.2 Variational Equations for Gravitational Field 307 3.9.3 The Hilbert Action and the Einstein Equations 307 3.9.3.1 The Hilbert Lagrangian 307 3.9.3.2 The Einstein Lagrangian 309 3.9.3.3 The Einstein Tensor 310 3.9.3.4 The Generalizations of the Hilbert Lagrangian 313 3.9.4 The Noether Theorem and Conserved Currents 316 3.9.4.1 The Anatomy of the Infinitesimal Variation 316
X Contents
3.9.4.2 Examples of the Gauge Transformations 319 3.9.4.3 Proof of the Noether Theorem 320 3.9.5 The Metrical Energy-Momentum Tensor 322 3.9.5.1 Hardcore of the Metrical Energy-Momentum Tensor 322 3.9.5.2 Gauge Invariance of the Metrical Energy Momentum Tensor 324 3.9.5.3 Electromagnetic Energy-Momentum Tensor 325 3.9.5.4 Energy-Momentum Tensor of a Perfect Fluid 326 3.9.5.5 Energy-Momentum Tensor of a Scalar Field 329 3.9.6 The Canonical Energy-Momentum Tensor 329 3.9.6.1 Definition 329 3.9.6.2 Relationship to the Metrical Energy-Momentum Tensor 331 3.9.6.3 Killing Vectors and the Global Laws of Conservation 332 3.9.6.4 The Canonical Energy-Momentum Tensor for Electromagnetic
Field 333 3.9.6.5 The Canonical Energy-Momentum Tensor for Perfect Fluid 334 3.9.7 Pseudotensor of Landau and Lifshitz 336 3.10 Gravitational Waves 339 3.10.1 The Post-Minkowskian Approximations 340 3.10.2 Multipolar Expansion of a Retarded Potential 344 3.10.3 Multipolar Expansion of Gravitational Field 345 3.10.4 Gravitational Field in Transverse-Traceless Gauge 350 3.10.5 Gravitational Radiation and Detection of Gravitational Waves 352
References 358
4 Relativistic Reference Frames 371 4.1 Historical Background 371 4.2 Isolated Astronomical Systems 378 4.2.1 Field Equations in the Scalar-Tensor Theory of Gravity 378 4.2.2 The Energy-Momentum Tensor 380 4.2.3 Basic Principles of the Post-Newtonian Approximations 382 4.2.4 Gauge Conditions and Residual Gauge Freedom 387 4.2.5 The Reduced Field Equations 389 4.3 Global Astronomical Coordinates 391 4.3.1 Dynamic and Kinematic Properties of the Global Coordinates 391 4.3.2 The Metric Tensor and Scalar Field in the Global Coordinates 395 4.4 Gravitational Multipoles in the Global Coordinates 396 4.4.1 General Description of Multipole Moments 396 4.4.2 Active Multipole Moments 399 4.4.3 Scalar Multipole Moments 401 4.4.4 Conformal Multipole Moments 402 4.4.5 Post-Newtonian Conservation Laws 404 4.5 Local Astronomical Coordinates 406 4.5.1 Dynamic and Kinematic Properties of the Local Coordinates 406 4.5.2 The Metric Tensor and Scalar Field in the Local Coordinates 409 4.5.2.1 The Scalar Field: Internal and External Solutions 410
Contents XI
4.5.2.2 The Metric Tensor: Internal Solution 411 4.5.2.3 The Metric Tensor: External Solution 412 4.5.2.4 The Metric Tensor: The Coupling Terms 419 4.5.3 Multipolar Expansion of Gravitational Field in the Local
Coordinates 420 References 423
5 Post-Newtonian Coordinate Transformations 429 5.1 The Transformation from the Local to Global Coordinates 429 5.1.1 Preliminaries 429 5.1.2 General Structure of the Coordinate Transformation 431 5.1.3 Transformation of the Coordinate Basis 434 5.2 Matching Transformation of the Metric Tensor and Scalar Field 436 5.2.1 Historical Background 436 5.2.2 Method of the Matched Asymptotic Expansions in the PPN
Formalism 439 5.2.3 Transformation of Gravitational Potentials from the Local to Global
Coordinates 442 5.2.3.1 Transformation of the Internal Potentials 442 5.2.3.2 Transformation of the External Potentials 446 5.2.4 Matching for the Scalar Field 447 5.2.5 Matching for the Metric Tensor 447 5.2.5.1 Matching g00(t, x ) and Ogα(u, w ) in the Newtonian Approximation 447 5.2.5.2 Matching gi j (t, x) and Ogα(u, w ) 450 5.2.5.3 Matching g0 i(t, x ) and Ogα(u, w ) 451 5.2.5.4 Matching g00(t, x ) and Ogα(u, w ) in the Post-Newtonian
Approximation 453 5.2.6 Final Form of the PPN Coordinate Transformation 457
References 458
6 Relativistic Celestial Mechanics 463 6.1 Post-Newtonian Equations of Orbital Motion 463 6.1.1 Introduction 463 6.1.2 Macroscopic Post-Newtonian Equations of Motion 467 6.1.3 Mass and the Linear Momentum of a Self-Gravitating Body 468 6.1.4 Translational Equation of Motion in the Local Coordinates 473 6.1.5 Orbital Equation of Motion in the Global Coordinates 477 6.2 Rotational Equations of Motion of Extended Bodies 479 6.2.1 The Angular Momentum of a Self-Gravitating Body 479 6.2.2 Equations of Rotational Motion in the Local Coordinates 480 6.3 Motion of Spherically-Symmetric and Rigidly-Rotating Bodies 483 6.3.1 Definition of a Spherically-Symmetric and Rigidly-Rotating Body 483 6.3.2 Coordinate Transformation of the Multipole Moments 487 6.3.3 Gravitational Multipoles in the Global Coordinates 490 6.3.4 Orbital Post-Newtonian Equations of Motion 492 6.3.5 Rotational Equations of Motion 500
XII Contents
6.4 Post-Newtonian Two-Body Problem 501 6.4.1 Introduction 501 6.4.2 Perturbing Post-Newtonian Force 503 6.4.3 Orbital Solution in the Two-Body Problem 505 6.4.3.1 Osculating Elements Parametrization 505 6.4.3.2 The Damour–Deruelle Parametrization 508 6.4.3.3 The Epstein–Haugan Parametrization 511 6.4.3.4 The Brumberg Parametrization 512
References 513
7 Relativistic Astrometry 519 7.1 Introduction 519 7.2 Gravitational Liénard–Wiechert Potentials 524 7.3 Mathematical Technique for Integrating Equations of Propagation of
Photons 529 7.4 Gravitational Perturbations of Photon’s Trajectory 538 7.5 Observable Relativistic Effects 541 7.5.1 Gravitational Time Delay 541 7.5.2 Gravitational Bending and the Deflection Angle of Light 547 7.5.3 Gravitational Shift of Electromagnetic-Wave Frequency 552 7.6 Applications to Relativistic Astrophysics and Astrometry 557 7.6.1 Gravitational Time Delay in Binary Pulsars 557 7.6.1.1 Pulsars – Rotating Radio Beacons 557 7.6.1.2 The Approximation Scheme 560 7.6.1.3 Post-Newtonian Versus Post-Minkowski Calculations of Time Delay in
Binary Systems 565 7.6.1.4 Time Delay in the Parameterized Post-Keplerian Formalism 567 7.6.2 Moving Gravitational Lenses 572 7.6.2.1 Gravitational Lens Equation 572 7.6.2.2 Gravitational Shift of Frequency by Moving Bodies 580 7.7 Relativistic Astrometry in the Solar System 584 7.7.1 Near-Zone and Far-Zone Astrometry 584 7.7.2 Pulsar Timing 590 7.7.3 Very Long Baseline Interferometry 593 7.7.4 Relativistic Space Astrometry 600 7.8 Doppler Tracking of Interplanetary Spacecrafts 604 7.8.1 Definition and Calculation of the Doppler Shift 607 7.8.2 The Null Cone Partial Derivatives 609 7.8.3 Doppler Effect in Spacecraft-Planetary Conjunctions 611 7.8.4 The Doppler Effect Revisited 613 7.8.5 The Explicit Doppler Tracking Formula 617 7.9 Astrometric Experiments with the Solar System Planets 619 7.9.1 Motivations 619 7.9.2 The Unperturbed Light-Ray Trajectory 624 7.9.3 The Gravitational Field 626
Contents XIII
7.9.3.1 The Field Equations 626 7.9.3.2 The Planet’s Gravitational Multipoles 628 7.9.4 The Light-Ray Gravitational Perturbations 631 7.9.4.1 The Light-Ray Propagation Equation 631 7.9.4.2 The Null Cone Integration Technique 632 7.9.4.3 The Speed of Gravity, Causality, and the Principle of Equivalence 636 7.9.5 Light-Ray Deflection Patterns 640 7.9.5.1 The Deflection Angle 640 7.9.5.2 Snapshot Patterns 642 7.9.5.3 Dynamic Patterns of the Light Deflection 646 7.9.6 Testing Relativity and Reference Frames 650 7.9.6.1 The Monopolar Deflection 652 7.9.6.2 The Dipolar Deflection 653 7.9.6.3 The Quadrupolar Deflection 655
References 656
8 Relativistic Geodesy 671 8.1 Introduction 671 8.2 Basic Equations 676 8.3 Geocentric Reference Frame 681 8.4 Topocentric Reference Frame 684 8.5 Relationship Between the Geocentric and Topocentric Frames 687 8.6 Post-Newtonian Gravimetry 689 8.7 Post-Newtonian Gradiometry 694 8.8 Relativistic Geoid 703 8.8.1 Definition of a Geoid in the Post-Newtonian Gravity 703 8.8.2 Post-Newtonian u-Geoid 704 8.8.3 Post-Newtonian a-Geoid 705 8.8.4 Post-Newtonian Level Surface 706 8.8.5 Post-Newtonian Clairaut’s Equation 707
References 709
9 Relativity in IAU Resolutions 715 9.1 Introduction 715 9.1.1 Overview of the Resolutions 716 9.1.2 About this Chapter 718 9.1.3 Other Resources 719 9.2 Relativity 720 9.2.1 Background 720 9.2.2 The BCRS and the GCRS 722 9.2.3 Computing Observables 724 9.2.4 Other Considerations 727 9.3 Time Scales 728 9.3.1 Different Flavors of Time 729 9.3.2 Time Scales Based on the SI Second 730 9.3.3 Time Scales Based on the Rotation of the Earth 733
XIV Contents
9.3.4 Coordinated Universal Time (UTC) 735 9.3.5 To Leap or not to Leap 735 9.3.6 Formulas 737 9.3.6.1 Formulas for Time Scales Based on the SI Second 737 9.3.6.2 Formulas for Time Scales Based on the Rotation of the Earth 740 9.4 The Fundamental Celestial Reference System 743 9.4.1 The ICRS, ICRF, and the HCRF 744 9.4.2 Background: Reference Systems and Reference Frames 746 9.4.3 The Effect of Catalogue Errors on Reference Frames 748 9.4.4 Late Twentieth Century Developments 750 9.4.5 ICRS Implementation 752 9.4.5.1 The Defining Extragalactic Frame 752 9.4.5.2 The Frame at Optical Wavelengths 753 9.4.6 Standard Algorithms 753 9.4.7 Relationship to Other Systems 754 9.4.8 Data in the ICRS 755 9.4.9 Formulas 757 9.5 Ephemerides of the Major Solar System Bodies 758 9.5.1 The JPL Ephemerides 759 9.5.2 DE405 760 9.5.3 Recent Ephemeris Development 761 9.5.4 Sizes, Shapes, and Rotational Data 762 9.6 Precession and Nutation 763 9.6.1 Aspects of Earth Rotation 764 9.6.2 Which Pole? 765 9.6.3 The New Models 768 9.6.4 Formulas 771 9.6.5 Formulas for Precession 774 9.6.6 Formulas for Nutation 778 9.6.7 Alternative Combined Transformation 781 9.6.8 Observational Corrections to Precession-Nutation 782 9.6.9 Sample Nutation Terms 783 9.7 Modeling the Earth’s Rotation 786 9.7.1 A Messy Business 786 9.7.2 Nonrotating Origins 788 9.7.3 The Path of the CIO on the Sky 790 9.7.4 Transforming Vectors Between Reference Systems 791 9.7.5 Formulas 794 9.7.5.1 Location of Cardinal Points 795 9.7.5.2 CIO Location Relative to the Equinox 795 9.7.5.3 CIO Location from Numerical Integration 797 9.7.5.4 CIO Location from the Arc-Difference s 798 9.7.5.5 Geodetic Position Vectors and Polar Motion 799 9.7.5.6 Complete Terrestrial to Celestial Transformation 801 9.7.5.7 Hour Angle 802
References 805
Contents XV
Appendix A Fundamental Solution of the Laplace Equation 813 References 817
Appendix B Astronomical Constants 819 References 823
Appendix C Text of IAU Resolutions 825 C.1 Text of IAU Resolutions of 1997 Adopted at the XXIIIrd General
Assembly, Kyoto 825 C.2 Text of IAU Resolutions of 2000 Adopted at the XXIVth General
Assembly, Manchester 829 C.3 Text of IAU Resolutions of 2006 Adopted at the XXVIth General
Assembly, Prague 841 C.4 Text of IAU Resolutions of 2009 Adopted at the XXVIIth General
Assembly, Rio de Janeiro 847
Index 851
XVII
Preface
The general theory of relativity was developed by Einstein a century ago. Since then, it has become the standard theory of gravity, especially important to the fields of fundamental astronomy, astrophysics, cosmology, and experimental gravitational physics. Today, the application of general relativity is also essential for many practical purposes involving astrometry, navigation, geodesy, and time synchronization. Numerous experiments have successfully tested general relativity to a remarkable level of precision. Exploring relativistic gravity in the solar system now involves a variety of high-accuracy techniques, for example, very long baseline radio interferometry, pulsar timing, spacecraft Doppler tracking, planetary radio ranging, lunar laser ranging, the global positioning system (GPS), torsion balances and atomic clocks.
Over the last few decades, various groups within the International Astronomical Union have been active in exploring the application of the general theory of relativity to the modeling and interpretation of high-accuracy astronomical observations in the solar system and beyond. A Working Group on Relativity in Celestial Me- chanics and Astrometry was formed in 1994 to define and implement a relativistic theory of reference frames and time scales. This task was successfully completed with the adoption of a series of resolutions on astronomical reference systems, time scales, and Earth rotation models by the 24th General Assembly of the IAU, held in Manchester, UK, in 2000. However, these resolutions only form a framework for the practical application of relativity theory, and there have been continuing questions on the details of the proper application of relativity theory to many common astronomical problems. To ensure that these questions are properly addressed, the 26th General Assembly of the IAU, held in Prague in August 2006, established IAU Commission 52, “Relativity in Fundamental Astronomy”. The general scientific goals of the new commission are to:
clarify the geometrical and dynamical concepts of fundamental astronomy within a relativistic framework,
provide adequate mathematical and physical formulations to be used in fundamental astronomy,
deepen the understanding of relativity among astronomers and students of astronomy, and
promote research needed to accomplish these tasks.
XVIII Preface
The present book is intended to make a theoretical contribution to the efforts undertaken by this commission. The first three chapters of the book review the foundations of celestial mechanics as well as those of special and general relativity. Subsequent chapters discuss the theoretical and experimental principles of applied relativity in the solar system. The book is written for graduate students and researchers working in the area of gravitational physics and its applications in modern astronomy. Chapters 1 to 3 were written by Michael Efroimsky and Sergei Kopeikin, Chapters 4 to 8 by Sergei Kopeikin, and Chapter 9 by George Kaplan. Sergei Kopeikin also edited the overall text.
It hardly needs to be said that Newtonian celestial mechanics is a very broad area. In Chapter 1, we have concentrated on derivation of the basic equations, on explanation of the perturbed two-body problem in terms of osculating and nonosculating elements, and on discussion of the gauge freedom in the six-dimensional configuration space of the orbital parameters. The gauge freedom of the configuration space has many similarities to the gauge freedom of solutions of the Einstein field equations in general theory of relativity. It is an important element of the New- tonian theory of gravity, which is often ignored in the books on classic celestial mechanics.
Special relativity is discussed in Chapter 2. While our treatment is in many aspects similar to the other books on special relativity, we have carefully emphasized the explanation of the Lorentz and Poincaré transformations, and the appropriate transformation properties of geometric objects like vectors and tensors, for example, the velocity, acceleration, force, electromagnetic field, and so on.
Chapter 3 is devoted to general relativity. It explains the main ideas of the tensor calculus on curved manifolds, the theory of the affine connection and parallel transport, and the mathematical and physical foundations of Einstein’s approach to gravity. Within this chapter, we have also included topics which are not well- covered in standard books on general relativity: namely, the variational analysis on manifolds and the multipolar expansion of gravitational radiation.
Chapter 4 introduces a detailed theory of relativistic reference frames and time scales in an N-body system comprised of massive, extended bodies – like our own solar system. Here, we go beyond general relativity and base our analysis on the scalar-tensor theory of gravity. This allows us to extend the domain of applicabil- ity of the IAU resolutions on relativistic reference frames, which in their original form were applicable only in the framework of general relativity. We explain the principles of construction of reference frames, and explore their relationship to the solutions of the gravitational field equations. We also discuss the post-Newtonian multipole moments of the gravitational field from the viewpoint of global and local coordinates.
Chapter 5 discusses the principles of derivation of transformations between reference frames in relativistic celestial mechanics. The standard parameterized post- Newtonian (PPN) formalism by K. Nordtevdt and C. Will operates with a single coordinate frame covering the entire N-body system, but it is insufficient for discussion of more subtle relativistic effects showing up in orbital and rotational motion of extended bodies. Consideration of such effects require, besides the global frame,
Preface XIX
the introduction of a set of local frames needed to properly treat each body and its internal structure and dynamics. The entire set of global and local frames allows us to to discover and eliminate spurious coordinate effects that have no physical meaning. The basic mathematical technique used in our theoretical treatment is based on matching of asymptotic post-Newtonian expansions of the solutions of the gravity field equations.
In Chapter 6, we discuss the principles of relativistic celestial mechanics of massive bodies and particles. We focus on derivation of the post-Newtonian equations of orbital and rotational motion of an extended body possessing multipolar moments. These moments couple with the tidal gravitational fields of other bodies, making the motion of the body under consideration very complicated. Simplifica- tion is possible if the body can be assumed spherically symmetric. We discuss the conditions under which this simplification can be afforded, and derive the equations of motion of spherically-symmetric bodies. These equations are solved in the case of the two-body problem, and we demonstrate the rich nature of the possible coordinate presentations of such a solution.
The relativistic celestial mechanics of light particles (photons) propagating in a time-dependent gravitational field of an N-body system is addressed in Chapter 7. This is a primary subject of relativistic astrometry which became especially important for the analysis of space observations from the Hipparcos satellite in the early 1990s. New astrometric space missions, orders of magnitude more accurate than Hipparcos, for example, Gaia, SIM, JASMINE, and so on, will require even more complete developments. Additionally, relativistic effects play an important role in other areas of modern astronomy, such as, pulsar timing, very long baseline radio interferometry, cosmological gravitational lensing, and so on. High-precision measurements of gravitational light bending in the solar system are among the most crucial experimental tests of the general theory of relativity. Einstein predicted that the amount of light bending by the Sun is twice that given by a Newtonian theory of gravity. This prediction has been confirmed with a relative precision about 0.01%. Measurements of light bending by major planets of the solar system allow us to test the dynamical characteristics of spacetime and draw conclusions about the ultimate speed of gravity as well as to explore the so-called gravitomagnetic phenomena.
Chapter 8 deals with the theoretical principles and methods of the high-precision gravimetry and geodesy, based on the framework of general relativity. A gravitational field and the properties of geocentric and topocentric reference frames are described by the metric tensor obtained from the Einstein equations with the help of post-Newtonian iterations. By matching the asymptotic, post-Newtonian expansions of the metric tensor in geocentric and topocentric coordinates, we derive the relationship between the reference frames, and relativistic corrections to the Earth’s force of gravity and its gradient. Two definitions of a relativistic geoid are discussed, and we prove that these geoids coincide under the condition of a constant rigid-body rotation of the Earth. We consider, as a model of the Earth’s matter, the notion of the relativistic level surface of a self-gravitating perfect fluid. We discover that, under conditions of constant rigid rotation of the fluid and hydrostatic
XX Preface
behavior of tides, the post-Newtonian equation of the level surface is the same as that of the relativistic geoid. In the conclusion of this chapter, a relativistic general- ization of the Clairaut’s equation is obtained.
Chapter 9 is a practical guide to the relativistic resolutions of the IAU, with enough background information to place these resolutions into the context of late twentieth century positional astronomy. These resolutions involve the definitions of reference systems, time scales, and Earth rotation models; and some of the resolutions are quite detailed. Although the recommended Earth rotation models have not been developed ab initio within the relativistic framework presented in the other resolutions (in that regard, there still exist some difficult problems to solve), their relativistic terms are accurate enough for all the current and near-future observational techniques. At that level, the Earth rotation models are consistent with the general relativity framework recommended by the IAU and considered in this book. The chapter presents practical algorithms for implementing the recommended models.
The appendices to the book contain a list of astronomical constants and the original text of the relevant IAU resolutions adopted by the IAU General Assemblies in 1997, 2000, 2006, and 2009.
Numerous colleagues have contributed to this book in one way or or another. It is a pleasure for us to acknowledge the enlightening discussions which one or more of the authors had on different occasions with Victor A. Brumberg of the Institute of Applied Astronomy (St. Petersburg, Russia); Tianyi Huang and Yi Xie of Nan- jing University (China); Edward B. Fomalont of the National Radio Astronomical Observatory (USA); Valeri V. Makarov, William J. Tangren, and James L. Hilton of the US Naval Observatory; Gerhard Schäfer of the Institute of Theoretical Physics (Jena, Germany); Clifford M. Will of Washington University (St. Louis, USA); Ig- nazio Ciufolini of the Universitá del Salento and INFN Sezione di Lecce (Italy); and Patrick Wallace, retired from Her Majesty’s Nautical Almanac Office (UK).
We also would like to thank Richard G. French of Wellesley College (Mas- sachusetts, USA); Michael Soffel and Sergei Klioner of the Technical University of Dresden; Bahram Mashhoon of the University of Missouri-Columbia; John D. Anderson, retired from the Jet Propulsion Laboratory (USA); the late Giacomo Giampieri, also of JPL; Michael Kramer, Axel Jessner, and Norbert Wex of the Max- Planck-Institut für Radioastronomie (Bonn, Germany); Alexander F. Zakharov of the Institute of Theoretical and Experimental Physics (Moscow, Russia); the late Yuri P. Ilyasov from Astro Space Center of Russian Academy of Science; Michael V. Sazhin, Vladimir A. Zharov, and Igor Yu. Vlasov of the Sternberg Astronomical Institute (Moscow, Russia); and Vladimir B. Braginsky of Moscow State Univer- sity (Russia) for their remarks and comments, all of which helped us to properly formulate the theoretical concepts and other material presented in this book.
The discussions among the members of the IAU Working Group on Relativity in Celestial Mechanics and Astrometry as well as those within the Working Group on Nomenclature for Fundamental Astronomy have also been quite valuable and have contributed to what is presented here. The numerous scientific papers written by Nicole Capitaine of the Paris Observatory and her collaborators have been essential
Preface XXI
references. Victor Slabinski and Dennis D. McCarthy of the US Naval Observatory, P. Kenneth Seidelmann of the University of Virginia, Catherine Y. Hohenkerk of Her Majesty’s Nautical Almanac Office, and E. Myles Standish, retired from the Jet Propulsion Laboratory, reviewed early drafts of the material that became Chapter 9 and made many substantial suggestions for improvement.
We were, of course, influenced by many other textbooks available in this field. We would like to pay particular tribute to:
C.W. Misner, K. S. Thorne and J. A. Wheeler “Gravitation” V.A. Brumberg “Essential Relativistic Celestial Mechanics” B.F. Schutz “Geometrical Methods of Mathematical Physics” M.H. Soffel “Relativity in Celestial Mechanics, Astrometry and Geodesy” C.M. Will “Theory and Experiment in Gravitational Physics”.
There are many other books and influential papers that are important as well which are referenced in the relevant parts of the present book.
Not one of our aforementioned colleagues is responsible for any remaining errors or omissions in this book, for which, of course, the authors bear full responsi- bility.
Last, but by no means least, Michael Efroimsky and George Kaplan wish to thank John A. Bangert of the US Naval Observatory for the administrative support which he so kindly provided to the project during all of its stages. Sergei Kopeikin is sincerely grateful to the Research Council of the University of Missouri-Columbia for the generous financial support (grants RL-08-027, URC-08-062B, SRF-09-012) that was essential for the successful completion of the book.
It is a great pleasure for the Authors to acknowledge the work, support and as- sistance of the Wiley Editors, who have made the publication of this monograph possible. Originally, the monograph was commissioned, on behalf of Wiley, by Dr. Christoph von Friedeburg who presently is an Editorial Director at the scientific publishing house of Walter de Gruyter. A large volume of subsequent manag- ing and technical work was carried out by the Commissioning Editor, Ms. Ulrike Werner, and the team of le-tex publishing services. To all these people the Authors express their sincerest gratitude.
University of Missouri, Columbia Sergei Kopeikin US Naval Observatory, Washington, DC Michael Efroimsky US Naval Observatory, Washington, DC George Kaplan June 2011
XXIII
General Notations
Greek indices α, , γ , . . . run from 0 to 3 and mark spacetime components of four- dimensional objects. Roman indices i, j, k, . . . run from 1 to 3 and denote components of three-dimensional objects (zero component belongs to time). Repeated indices mean the Einstein summation rule with respect to corresponding indices, for instance, Aα Bα D A0B0 C A1B1 C A2B2 C A3B3, T k
k D T 1 1 C T 2
2 C T 3 3 , and so
on. Minkowski metric has signature ηα D diag(1, C1, C1, C1). Kronecker sym-
bol (the unit matrix) is denoted by δ i j D diag(1, 1, 1). The Levi-Civita fully- antisymmetric symbol is ε i j k such that ε123 D C1. The Kronecker symbol is used to rise and lower Roman indices. Complete metric tensor gα is used to rise and lower the Greek indices in exact tensor equations whereas the Minkows- ki metric ηα is employed for rising and lowering indices in equations of the post-Newtonian and post-Minkowskian approximations.
Round brackets surrounding a group of Roman indices mean full symmetriza- tion with a corresponding normalizing coefficient, for example,
A (i j ) 1 2!
A i j C A j i
,
A (i j k ) 1 3!
,
and so on. Square brackets around a group of Roman indices denote antisym- metrization with a corresponding normalizing coefficient, that is,
A [i j ] 1 2!
A i j A j i
,
A [i j k ] 1 3!
,
and so on. Angular brackets surrounding a group of Roman indices denote the symmetric trace-free (STF) part of the corresponding three-dimensional object, for
XXIV Symbols and Abbreviations
A hi j i D A (i j ) 1 3
δ i j A k k ,
A hi j ki D A (i j k ) 1 5
δ i j A k p p 1 5
δ j k A i p p 1 5
δ i k A j p p ,
and the general definition of STF tensor is discussed in Sections 1.2.2 and 3.10.2. We also use multi-index notations, for example, A L A i1 i2...i l , BP1
Bi1 i2... i p1 , DhLi D Dhi1 i2... i l i. Contraction over multi-indices is understood as follows, A L QL D A i1 i2... i l Q
i1 i2...i l , PaL1T bL1 D Pai1 i2...i l1 T bi1 i2...i l1 , and so on. The sign @ in front of indices denotes a partial derivative with respect to a corresponding coordinate which is taken as many times as the number of indices following the @, for example, @α φ D @φ/@x α , @α φ D @2φ/@x α@x b , where @0φ D c1@φ/@t, @i φ D @φ/@x i , and, similarly, r T α denotes a covariant derivative. The partial derivatives will be also denoted sometimes with a comma, for example, F,α @F/@x α , and so on. L-order partial derivative with respect to spatial coordinates is denoted by @L D @i1 i2...i l D @i1 . . . @i l . Other conventions are intro- duced and explained as they appear in the text of the book. Particular symbols for various mathematical objects are given below.
Mathematical Symbols Used in the Book
eα vector basis on manifold Qωα covector basis on manifold Λα
0 , Λα0
the matrix of transformation from one basis on manifold to another
gμν physical (Jordan–Fierz frame) metric tensor Qgμν conformal (Einstein frame) metric tensor g the determinant of gμν
Qg the determinant of Qgμν
ημν the Minkowski (flat) metric tensor G
αγ the affine connection
Kαγ the contortion tensor Rαγ the Ricci rotation coeffieicnts Dαγ the deviation tensor T α
γ the torsion tensor Qα
γ the nonmetricity tensor Γ α
μν the Christoffel symbol Rμν the Ricci tensor R the Ricci scalar QRμν the conformal Ricci tensor
Tμν the energy-momentum tensor of matter T D T α
α the trace of the energy-momentum tensor t α the canonical pseudotensor of gravitational field
Symbols and Abbreviations XXV
t α LL the pseudotensor of Landau and Lifshitz
Λα the effective tensor of matter and gravitational field φ the scalar field φ0 the background value of the scalar field φ the dimensionless perturbation of the scalar field θ (φ) the coupling function of the scalar field g the Laplace–Beltrami operator the D’Alembert operator in the Minkowski
spacetime the density of matter in the comoving frame the invariant (Fock) density of matter Π the internal energy of matter in the comoving
frame πμν the tensor of (anisotropic) stresses of matter uα the four-velocity of matter v i the 3-dimensional velocity of matter in the global
frame ω the asymptotic value of the coupling function θ (φ) ω0 the asymptotic value of the derivative of the
coupling function θ (φ) c the ultimate speed of general and special theories
of relativity a small dimensional parameter, D 1/c hμν the metric tensor perturbation, gμν ημν (n) h μν the metric tensor perturbation of order n in the
post-Newtonian expansion of the metric tensor
N a shorthand notation for (2) h 00
L a shorthand notation for (4) h 00
Ni a shorthand notation for (1) h 0 i
L i a shorthand notation for (3) h 0 i
Hi j a shorthand notation for (2) h i j
H a shorthand notation for (2) h k k
QN , QL shorthand notations for perturbations of conformal metric Qgμν
γ the “space-curvature” PPN parameter the “nonlinearity” PPN parameter η the Nordtvedt parameter, η D 4 γ 3 G the observed value of the universal gravitational
constant G the bare value of the universal gravitational
constant
XXVI Symbols and Abbreviations
x α D (x0, x i ) the global coordinates with x0 D c t and x i x w α D (w0, w i ) the local coordinates with w0 D cu and w i w U the Newtonian gravitational potential in the global
frame U (A) the Newtonian gravitational potential of body A in
the global frame Ui a vector potential in the global frame U (A)
i a vector potential of body A in the global frame , Φ1, . . . , Φ4 various special gravitational potentials in the global
frame V, V i potentials of the physical metric in the global
frame σ, σ i the active mass and current-mass densities in the
global frame IhLi the active mass multipole moments in the global
frame ShLi the active spin multipole moments in the global
frame NV potential of the scalar field in the global frame Nσ scalar mass density in the global frame NIhLi scalar mass multipole moments in the global
frame QV gravitational potential of the conformal metric in
the global frame Qσ the conformal mass density in the global frame QIhLi the conformal mass multipole moments in the
global frame M conserved mass of an isolated system P i conserved linear momentum of an isolated system S i conserved angular momentum of an isolated
system D i integral of the center of mass of an isolated system OA symbols with the hat stand for quantities in the
local frame (B) sub-index referring to the body and standing for
the internal solution in the local frame (E) sub-index referring to the external with respect to
(B) bodies and standing for the external solution in the local frame
(C) sub-index standing for the coupling part of the solution in the local frame
PL external STF multipole moments of the scalar field Q L external STF gravitoelectric multipole moments of
the metric tensor
CL external STF gravitomagnetic multipole moments of the metric tensor
ZL, SL other sets of STF multipole moments entering the general solution for the spacetime part of the external local metric
YL, BL, DL, EL, FL, GL STF multipole moments entering the general solution for the space-space part of the external local metric
Vi , Ωi linear and angular velocities of kinematic motion of the local frame
ν i 3-dimensional velocity of matter in the local frame IL active STF mass multipole moments of the body in
the local frame σB active mass density of body B in the local frame NIL scalar STF mass multipole moments of the body in
the local frame NσB scalar mass density of body B in the local frame QIL conformal STF mass multipole moments of the
body in the local frame QσB conformal mass density of body B in the local
frame σ i
B current mass density of body B in the local frame SL spin STF multipole moments of the body in the
local frame 0, i Relativistic corrections in the post-Newtonian
transformation of time and space coordinates x i
B , v i B , ai
B position, velocity and acceleration of the body’s center of mass with respect to the global frame
R i B x i x i
B (t), that is, the spacial coordinates taken with respect to the center of mass of body B in the global frame
A,BhLi functions appearing in the relativistic transformation of time
DhLi,FhLi, EhLi functions appearing in the relativistic transformation of spacial coordinates
Λ α matrix of transformation between local and global
coordinate bases in the post-Newtonian approximation scheme
Ç α matrix of the inverse transformation between local
and global coordinate bases in the post-Newtonian approximation scheme
B, D, Bi , P i , R i j the terms in the post-Newtonian expansion of the
matrix of transformation Λ α
NU , NU i , etc. external gravitational potentials
XXVIII Symbols and Abbreviations
NU,L(x B ), NU i ,L(x B ) lth spatial derivative of an external potential taken
at the center of mass of body B U (B) PN correction in the formula of matching of the
local Newtonian potential F i k the matrix of relativistic precession of local
coordinates with respect to global coordinates M,J i
,P i Newtonian-type mass, center of mass, and linear
momentum of the body in the local frame M general relativistic PN mass of the body in the local
frame M active mass of the body in the local frame
QM conformal mass of the body in the local frame I(2) rotational moment of inertia of the body in the
local frame N L a set of STF multipole moments P i PN linear momentum of the body in the local
frame Δ PP i scalar-tensor PN correction to PP i
QMi j conformal anisotropic mass of the body in the local frame
F i N , F i
p N , ΔF i p N gravitational forces in the expression (6.20) for Q i
S i the bare post-Newtonian definition of the angular momentum (spin) of a body
T i the post-Newtonian torque in equations of rotational motion
ΔT i the post-Newtonian correction to the torque T i
ΔS i the post-Newtonian correction to the bare spin S i
Ri velocity-dependent multipole moments S i
C the (measured) post-Newtonian spin of the body r radial space coordinate in the body’s local frame,
r D jw j Ω j
B angular velocity of rigid rotation of the body B referred to its local frame
I (2l) B lth rotational moment of inertia of the body B
IL B multipole moments of the multipolar expansion of
the Newtonian potential in the global coordinates RB jRB j, where RB D x x B
R i B C x i
B x i C
E I H , F i S , F i
IG R , F i IS T forces from the equation of motion of

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