Relativistic Celestial Mechanics of the Solar System E-Book

Sergei Kopeikin

• Herausgeber: Wiley-VCH
• Kategorie: Lebensstil
• Sprache: Englisch
• Veröffentlichungsjahr: 2011

Seitenzahl: 1687

Beschreibung

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. In so doing, it fills the need for a textbook that teaches modern dynamical astronomy with a strong emphasis on the relativistic aspects of the subject produced by the curved geometry of four-dimensional spacetime. The first three chapters review the fundamental principles of celestial mechanics and of special and general relativity. This background material forms the basis for understanding relativistic reference frames, the celestial mechanics of N-body systems, and high-precision astrometry, navigation, and geodesy, which are then treated in the following five chapters. The final chapter provides an overview of the new field of applied relativity, based on recent recommendations from the International Astronomical Union. The book is suitable for teaching advanced undergraduate honors programs and graduate courses, while equally serving as a reference for professional research scientists working in relativity and dynamical astronomy. The authors bring their extensive theoretical and practical experience to the subject. Sergei Kopeikin is a professor at the University of Missouri, while Michael Efroimsky and George Kaplan work at the United States Naval Observatory, one of the world?s premier institutions for expertise in astrometry, celestial mechanics, and timekeeping.

Leseprobe

Contents

Cover

Half Title page

Related Titles

Title page

Copyright page

Preface

Symbols and Abbreviations

General Notations

Mathematical Symbols Used in the Book

Abbreviations and Symbols Frequently Used in Astronomy

References

Chapter 1: Newtonian Celestial Mechanics

1.1 Prolegomena – Classical Mechanics in a Nutshell

1.2 The N-body Problem

1.3 The Reduced Two-Body Problem

1.4 A Perturbed Two-Body Problem

1.5 Re-examining the Obvious

1.6 Epilogue to the Chapter

References

Chapter 2: Introduction to Special Relativity

2.1 From Newtonian Mechanics to Special Relativity

2.2 Building the Special Relativity

2.3 Minkowski Spacetime as a Pseudo-Euclidean Vector Space

2.4 Tensor Algebra

2.5 Kinematics

2.6 Accelerated Frames

2.7 Relativistic Dynamics

2.8 Energy-Momentum Tensor

References

Chapter 3: General Relativity

3.1 The Principle of Equivalence

3.2 The Principle of Covariance

3.3 A Differentiable Manifold

3.4 Affine Connection on Manifold

3.5 The Levi-Civita Connection

3.6 Lie Derivative

3.7 The Riemann Tensor and Curvature of Manifold

3.8 Mathematical and Physical Foundations of General Relativity

3.9 Variational Principle in General Relativity

3.10 Gravitational Waves

References

Chapter 4: Relativistic Reference Frames

4.1 Historical Background

4.2 Isolated Astronomical Systems

4.3 Global Astronomical Coordinates

4.4 Gravitational Multipoles in the Global Coordinates

4.5 Local Astronomical Coordinates

References

Chapter 5: Post-Newtonian coordinate transformation

5.1 The Transformation from the Local to Global Coordinates

5.2 Matching Transformation of the Metric Tensor and Scalar Field

References

Chapter 6: Relativistic Celestial Mechanics

6.1 Post-Newtonian Equations of Orbital Motion

6.2 Rotational Equations of Motion of Extended Bodies

6.3 Motion of Spherically-Symmetric and Rigidly-Rotating Bodies

6.4 Post-Newtonian Two-Body Problem

References

Chapter 7: Relativistic Astrometry

7.1 Introduction

7.2 Gravitational Liénard–Wiechert Potentials

7.3 Mathematical Technique for Integrating Equations of Propagation of Photons

7.4 Gravitational Perturbations of Photon’s Trajectory

7.5 Observable Relativistic Effects

7.6 Applications to Relativistic Astrophysics and Astrometry

7.7 Relativistic Astrometry in the Solar System

7.8 Doppler Tracking of Interplanetary Spacecrafts

7.9 Astrometric Experiments with the Solar System Planets

References

Chapter 8: Relativistic Geodesy

8.1 Introduction

8.2 Basic Equations

8.3 Geocentric Reference Frame

8.4 Topocentric Reference Frame

8.5 Relationship Between the Geocentric and Topocentric Frames

8.6 Post-Newtonian Gravimetry

8.7 Post-Newtonian Gradiometry

8.8 Relativistic Geoid

References

Chapter 9: Relativity in IAU Resolutions

9.1 Introduction

9.2 Relativity

9.3 Time Scales

9.4 The Fundamental Celestial Reference System

9.5 Ephemerides of the Major Solar System Bodies

9.6 Precession and Nutation

9.7 Modeling the Earth’s Rotation

References

Appendix A: Fundamental Solution of the Laplace Equation

References

Appendix B: Astronomical Constants

References

Appendix C: Text of IAU Resolutions

C.1 Text of IAU Resolutions of 1997 Adopted at the XXIIIrd General Assembly, Kyoto

C.2 Text of IAU Resolutions of 2000 Adopted at the XXIVth General Assembly, Manchester

C.3 Text of IAU Resolutions of 2006 Adopted at the XXVIth General Assembly, Prague

C.4 Text of IAU Resolutions of 2009 Adopted at the XXVIIth General Assembly, Rio de Janeiro

Index

Sergei Kopeikin,Michael Efroimsky,and George Kaplan

Relativistic Celestial Mechanicsof the Solar System

Related Titles

Dvorak, R. (ed.)Extrasolar PlanetsFormation, Detection and Dynamics2008ISBN: 978-3-527-40671-5

Stahler, S. W., Palla, F.The Formation of Stars2004ISBN: 978-3-527-40559-6

Fortescue, P. W., Stark, J. P. W., Swinerd, G. (eds.)Spacecraft Systems Engineering2003ISBN: 978-0-471-61951-2

Shore, S. N.The Tapestry of Modern Astrophysics2003ISBN: 978-0-471-16816-4

Shapiro, S. L., Teukolsky, S. A.Black Holes, White Dwarfs and Neutron StarsThe Physics of Compact Objects1983ISBN: 978-0-471-87317-4

The Authors

Prof. Sergei Kopeikin University of Missouri Department of Physics and Astronomy Columbia, Missouri, [e-mail]

Dr. Michael Efroimsky US Naval Observatory 3450 Massachusetts Ave NW Washington, DC, [e-mail]

Dr. George Kaplan Consultant to US Naval Observatory 3450 Massachusetts Ave NW Washington, DC, [e-mail]

Cover Picture

Pioneer 10 artwork, Ames Research Center/Nasa Center, 2006

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ISBN Print 978-3-527-40856-6

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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 Mechanics 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, andpromote research needed to accomplish these tasks.

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 Newtonian 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 applicability 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, 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. Simplification 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 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 generalization 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 Nanjing 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); Ignazio 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 (Massachusetts, 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 University (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 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 responsibility.

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 assistance 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 managing 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 US Naval Observatory, Washington, DC US Naval Observatory, Washington, DC June 2011

Sergei Kopeikin Michael Efroimsky George Kaplan