1. The Science of Relativistic Celestial Mechanics
The Science of Relativistic Celestial Mechanics. Introduction for a layman.
Sergei Kopeikin
June 27-29, 2007
Astrocon 2007

2. The Founders Albert Einstein Hendric A. Lorentz Karl Schwarzschild
Willem de Sitter
Vladimir A. Fock
Tullio Levi-Civita
Leopold Infeld
Arthur S. Eddington
Hans Thirring
Lev D. Landau

3. The Solar System: Hierarchy of Celestial Frames
Barycentric Frame
Heliocentric Frame
Geocentric
Frame
Lunocentric Frame
Earth-Moon
Barycentric Frame
June 27-29, 2007
Astrocon 2007

4. Newtonian Gravity Field Equations
Gravitational potential
(a scalar function)
Density of matter
(a scalar function)
June 27-29, 2007
Astrocon 2007

5. Boundary Conditions and Reference Frames
Body’s Frame
Barycentric Frame r M
Field point m
June 27-29, 2007
Astrocon 2007

6. Multipolar Fields in Body’s Frame
mass
dipole
intrinsic
quadrupole
intrinsic
octupole
monopole
acceleration
tidal
quadrupole
tidal
octupole
June 27-29, 2007
Astrocon 2007

7. Multipolar Fields in Global Frame
June 27-29, 2007
Astrocon 2007

8. The Frame Matching Technique
Matching Coordinate Transformations:
June 27-29, 2007
Astrocon 2007

9. Microscopic Equations of Motion
June 27-29, 2007
Astrocon 2007

10. Equations of Translational Motion in the Local Frame
June 27-29, 2007
Astrocon 2007

11. Picture Moon Earth The center of mass moves with
The center of mass of a massive body having non-zero intrinsic multipoles moves with acceleration with respect to a spherically-symmetric body because of the coupling of the intrinsic and external multipoles
Picture
The center of mass moves with
acceleration a with respect to
the world line of a spherically-symmetric
body.
For the Earth this acceleration
amounts to m/s2. As its
orbital acceleration around the
Sun is about m/s2 , the relative
effect is of order (taken into
account in JPL ephemerides) S
Moon
Earth
The world-line of Earth’s
center of mass
World-line of a sperically-symmetric body
June 27-29, 2007
Astrocon 2007

12. Equations of Rotational Motion
June 27-29, 2007
Astrocon 2007

13. Equations of Orbitalal Motion in the Barycentric Frame
June 27-29, 2007
Astrocon 2007

14. Einstein’s Definition of Relativity
"Put your hand on a hot stove for a minute, and it seems like an hour. Sit with a pretty girl for an hour, and it seems like a minute. THAT's relativity."
A. Einstein.
June 27-29, 2007
Astrocon 2007

15. Gravitational Field is not a Scalar!
June 27-29, 2007
Astrocon 2007

16. Building Blocks of General Relativity
June 27-29, 2007
Astrocon 2007

17. Field Equations and Gauge Freedom
June 27-29, 2007
Astrocon 2007

18. Solving Einstein’s Equations
June 27-29, 2007
Astrocon 2007

19. Residual Gauge Freedom and Coordinates
June 27-29, 2007
Astrocon 2007

20. Form-invariance of the Metric Tensor
June 27-29, 2007
Astrocon 2007

21. Reference Frames and Boundary Conditions
Global Coordinates
Local Coordinates
June 27-29, 2007
Astrocon 2007

22. Global and Local Frames
June 27-29, 2007
Astrocon 2007

23. Mathematical Techniques for Deriving Equations of Motion
Einstein-Infeld-Hoffmann
Fock-Papapetrou
Dixon-Synge
Asymptotic Matching (D’Eath)
June 27-29, 2007
Astrocon 2007

24. Derivation of equations of motion
Derivation of equations of motion. The internal-structure effacing principle
Lagrangian-based theory of gravity
Field equations: tensor, vector, scalar
Boundary and initial conditions:
External problem - global frame
Boundary and initial conditions:
Internal problem - local frame(s)
External solution of the field equations:
metric tensor + other fields in entire space
Internal solution of the field equations:
metric tensor + other fields in a local domain;
external and internal multipole moments
Matching of external and internal solutions
Coordinate transformations
between the global and local frames
External multipole moments in terms of
external gravitational potentials
Laws of motion: external
Laws of transformation of the
internal and external moments
Laws of motion: internal;
Fixing the origin of the local frame
Equations of motion: external
Equations of motion: internal
Effacing principle: equations of motion of spherical and non-rotating bodies depend only on
their relativistic masses
June 27-29, 2007
Astrocon 2007

25. Equations of Motion of Spherically-Symmetric Bodies
Spherical symmetry of a moving body is ill-defined in the global frame because of the Lorentz (special-relativistic) and Einstein (general-relativistic) contractions. Spherical symmetry can be physically defined only in the body’s local frame (tides are neglected) 1 2 3 4
June 27-29, 2007
Astrocon 2007

26. Matching Global and Local Coordinates
Geocentric coordinates (u,w) cover
interior of the world tube bounded by
radius of the lunar orbit. Metric tensor
Barycentric coordinates (t,x ) cover
the entire space-time. Metric tensor
. The two coordinate systems
overlaps admitting the matching
transformation:
Matching Global and Local Coordinates
Moon
Sun
Earth
June 27-29, 2007
Astrocon 2007

27. Einstein-Infeld-Hoffmann Force in the Global Reference Frame
The JPL Solar System Ephemeris specifies the past and future positions of the Sun, Moon, and nine planets in three-dimensional space. Many versions of this ephemeris have been produced to include improved measurements of the positions of the Moon and planets and to conform to new and improved coordinate system definitions.
June 27-29, 2007
Astrocon 2007

28. JPL Development Ephemeris (DE) E. M. Standish, X. X. Newhall, J. G
JPL Development Ephemeris (DE) E. M. Standish, X.X. Newhall, J.G. Williams
The DE100-series ephemeris is in the B1950 coordinate system
The DE200 series is in the J2000 system
The DE400 series is in the reference frame defined by the International Earth Rotation Service (IERS).
June 27-29, 2007
Astrocon 2007

29. Planetary positions are generated by a computer integration fit to the best available observations of the positions of the Sun, Moon, planets, and five largest asteroids. The computer integration involves stepwise computation of the position of each planet as determined by the gravitation of all of the other objects in the solar system.
June 27-29, 2007
Astrocon 2007

30. The observation are mainly from: transit circles since 1911,
planetary radar ranging since 1964,
lunar laser ranging since 1969,
distances to the Viking lander on Mars since 1976,
Very Long Baseline Interferometry since 1987.
The computer calculations have been extended as far as 3000 BC to 3000 AD, but positions for the range are the most accurate.
June 27-29, 2007
Astrocon 2007

31. Subtle differences exist between:
the best ephemeris model coordinates and the standard definitions of B1950 and J2000,
the coordinate systems defined by star positions and the B1950 and J2000 standards,
the coordinate systems defined by stars and radio sources.
These differences, which start at the level of a couple of milliseconds and a few tenths of an arcsecond, are very important to pulsar timing and radio interferometry. With care and consistency, all-sky accuracies of a few hundred nanoseconds and a few milliarcseconds are currently being achieved
June 27-29, 2007
Astrocon 2007

32. A Sketchy History of DE Versions
This was the best available planetary ephemeris as of 1983, spanning the time range, based on transit circle measurements since 1911, planetary radar since 1964, lunar laser ranging since 1969, and Viking spacecraft ranging on Mars since Its larger time span companion was DE102, which covered 1411 BC to 3002 AD. The major ephemerides leading to DE118 were DE96, DE102, DE108, and DE111. All of these ephemerides, including DE118 are in the B1950 coordinate system (FK4 catalogue)
DE200 : (includes nutations but not librations)
This is DE118 rotated into the J2000 coordinate system. DE200 has been the basis for the calculation of Astronomical Almanac planetary tables since 1984.
DE125
Created in July 1985 for the Voyager encounter with Uranus.
DE130
Created in October 1987 for the Voyager encounter with Neptune.
DE202
This is DE130 rotated into the J2000 coordinate system. DE202 is more accurate for the outer planets than is DE200.
DE403 : (includes both nutations and librations)
A new ephemeris aligned with the (J2000) reference frame of the Radio Source Catalog of the International Earth Rotation Service (IERS). It it based on planetary and reference frame data available in 1995.
DE405 : (includes both nutations and librations)
It is based upon the International Celestial Reference Frame (ICRF). (DE200 is within 0.01 arcseconds of the frame of the ICRF). DE405 was created in May-June 1997.
DE406 : the New "JPL Long Ephemeris" (includes neither nutations nor librations)
This is the same ephemeris as DE405, though the accuracy of the interpolating polynomials has been lessened. For DE406, the interpolating accuracy is no worse than 25 meters for any planet and no worse than 1 meter for the moon.
June 27-29, 2007
Astrocon 2007

33. Other Ephemeris Programs
Planetary Ephemeris Program (PEP)
This is the MIT Harvard Smithsonian Astrophysics Center ephemeris. Originally generated by I. Shapiro, M. Ash, R. King in Significantly improved in the spring of 1975 by Bob Goldstein. John Chandler has maintained PEP since the middle of 80th. PEP has the same accuracy as DE.
Ephemerides of Planets and the Moon (EPM)
This is the Institute of Applied Astronomy, St. Petersburg ephemeris code. Created by Geogre Krasinsky in Major contributions and improvements by Elena Pitjeva, Michael Sveshnikov. Previous versions: EPM87, EPM98, EPM2000. Current version EPM2006 has the same acuracy as DE405/414, and it is maintained by G. Krasinsky and E. Pitjeva. There are ephemeris programs in the Institute of Applied Mathematics and the Space Flight Control Center.
Variations Seculaires des Orbites Planetaires (VSOP)
Institute de Mechanique Celeste et de Calcul des Ephémérides (IMCCE). Created by P. Bretagnon and G. Francou in Recent developments by A. Fienga and J.-L. Simon (VSOP2002) which includes the Moon, 300 asteroids, solar oblateness, and relativity. Diverges from DE405 up to 100 meters over 30 years.
June 27-29, 2007
Astrocon 2007

34. THANK YOU !!!
June 27-29, 2007
Astrocon 2007