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

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

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

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

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

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

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

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

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

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

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 3.10-11 m/s2. As its orbital acceleration around the Sun is about 6.10-3 m/s2 , the relative effect is of order 5.10-9 . (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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 1850-2050 range are the most accurate. June 27-29, 2007 Astrocon 2007

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

A Sketchy History of DE Versions This was the best available planetary ephemeris as of 1983, spanning the 1850-2050 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 1974. 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

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 1967. 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 1974. 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 1988. 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

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