General-Relativistic Effects in Astrometry

1. General-Relativistic Effects in Astrometry S.A.Klioner, M.H.Soffel Lohrmann Observatory, Dresden Technical University 2005 Michelson Summer Workshop, Pasadena, 26 July 2005

2. Newtonian astrometry • Why relativistic astrometry? • Coordinates, observables and the principles of relativistic modelling • Metric tensor and reference systems • BCRS, GCRS and local reference system of an observer • Principal general-relativistic effects • The standard relativistic model for positional observations • Celestial reference frame • Beyond the standard model General-relativistic astrometry

3. Modelling of positional observations in Newtonian physics M. C. Escher Cubic space division, 1952

4. Astronomical observation physically preferred global inertial coordinates observables are directly related to the inertial coordinates

5. Scheme: • aberration • parallax • proper motion • All parameters of the model are defined • in the preferred global coordinates: Modelling of positional observations in Newtonian physics

6. 0 1400 1500 1600 1700 1800 1900 2000 2100 Hipparchus Ulugh Beg 1000 1000 Hevelius 100 Wilhelm IV 100 Flamsteed Tycho Brahe 10 Bradley-Bessel 10 1 as 1 as GC 100 100 FK5 10 10 Hipparcos 1 mas ICRF 1 mas 100 100 Gaia 10 10 SIM 1 µas 1 µas 0 1400 1500 1600 1700 1800 1900 2000 2100 naked eye telescopes space Accuracy of astrometric observations • Accuracy-implied changes of astrometry: • underlying physics: general relativity vs. Newtonian physics • goals: astrophysical picture rather than a kinematical description

7. Newtonian models cannot describe high-accuracy • observations: • many relativistic effects are many orders of • magnitude larger than the observational • accuracy • space astrometry missions would not work • without relativistic modelling • The simplest theory which successfully describes all • available observational data: • APPLIED GENERAL RELATIVITY Why general relativity? 

8. Several general-relativistic effects are confirmed with the following precisions: • VLBI ± 0.0003 • HIPPARCOS ± 0.003 • Viking radar ranging ± 0.002 • Cassini radar ranging ± 0.000023 • Planetary radar ranging ± 0.0001 • Lunar laser ranging I ± 0.0005 • Lunar laser ranging II ± 0.007 • Other tests: • Ranging (Moon and planets) • Pulsar timing: indirect evidence for gravitational radiation Testing general relativity

9. Astronomical observation physically preferred global inertial coordinates observables are directly related to the inertial coordinates

10. Astronomical observation no physically preferred coordinates observables have to be computed as coordinate independent quantities

11. Relativistic reference system(s) Relativistic equations of motion Equations of signal propagation Definition of observables Relativistic models of observables Coordinate-dependent parameters Observational data Astronomical reference frames General relativity for space astrometry

12. Pythagorean theorem in 2-dimensional Euclidean space • length of a curve in Metric tensor

13. special relativity, inertial coordinates Metric tensor: special relativity • The constancy of the velocity of light in inertial coordinates can be expressed as where

14. In relativistic astrometry the • BCRS (Barycentric Celestial Reference System) • GCRS (Geocentric Celestial Reference System) • Local reference system of an observer • play an important role. • All these reference systems are defined by • the form of the corresponding metric tensor. BCRS Metric tensor and reference systems GCRS Local RS of an observer

15. The BCRS: • adopted by the International Astronomical Union (2000) • suitable to model high-accuracy astronomical observations Barycentric Celestial Reference System relativistic gravitational potentials

16. The BCRS is a particular reference system in the curved space-time of the Solar system Barycentric Celestial Reference System • One can • use any • but one • should • fix one

17. The GCRS is adopted by the International Astronomical Union (2000) to model physical processes in the vicinity of the Earth: A: The gravitational field of external bodies is represented only in the form of a relativistic tidal potential. B: The internal gravitational field of the Earth coincides with the gravitational field of a corresponding isolated Earth. Geocentric Celestial Reference System

18. The GCRS is adopted by the International Astronomical Union (2000) to model physical processes in the vicinity of the Earth: A: The gravitational field of external bodies is represented only in the form of a relativistic tidal potential. B: The internal gravitational field of the Earth coincides with the gravitational field of a corresponding isolated Earth. Geocentric Celestial Reference System

19. The GCRS is adopted by the International Astronomical Union (2000) to model physical processes in the vicinity of the Earth: A: The gravitational field of external bodies is represented only in the form of a relativistic tidal potential. B: The internal gravitational field of the Earth coincides with the gravitational field of a corresponding isolated Earth. Geocentric Celestial Reference System internal + inertial + tidal external potentials

20. internal + inertial + tidal external potentials The version of the GCRS for a massless observer: A: The gravitational field of external bodies is represented only in the form of a relativistic tidal potential. Local reference system of an observer observer • Modelling of any local phenomena: • observation, • attitude, • local physics (if necessary)

21. The equations of translational motion • (e.g. of a satellite) in the BCRS Equations of translational motion • The equations coincide with the well-known Einstein-Infeld-Hoffmann (EIH) • equations in the corresponding limit

22. The equations of light propagation • in the BCRS Equations of light propagation • Relativistic corrections to • the “Newtonian” straight line:

23. Proper timeof an observer can be related • to the BCRS coordinate time t=TCB using • the BCRS metric tensor • the observer’s trajectory xio(t) in the BCRS Observables I: proper time

24. To describe observed directions (angles) one should introduce spatial • reference vectors moving with the observer explicitly into the formalism Observables II: proper direction • Observed angles between incident light rays and a spatial reference vector • can be computed with the metric of the local reference system of the observer

25. s the observed direction • n tangential to the light ray • at the moment of observation •  tangential to the light ray • at • k the coordinate direction • from the source to the observer • l the coordinate direction • from the barycentre to the source •  the parallax of the source • in the BCRS observed related to the light ray defined in BCRS coordinates The standard astrometric model

26. Stars: Sequences of transformations • Solar system objects: (1) aberration (2) gravitational deflection (3) coupling to finite distance (4) parallax (5) proper motion, etc. (6) orbit determination

27. Lorentz transformation with the scaled velocity of the observer: Aberration: s n • For an observer on the Earth or on a typical satellite: • Newtonian aberration 20 • relativistic aberration  4 mas • second-order relativistic aberration 1 as • Requirement for the accuracy of the orbit:

28. with Sun without Sun Gravitational light deflection: n  k • Several kinds of gravitational fields deflecting light • monopole field • quadrupole field • gravitomagnetic field due to translational motion • gravitomagnetic field due to rotational motion • post-post-Newtonian corrections (ppN)

29. The principal effects due to the major bodies of the solar system in as • The maximal angular distance to the bodies where the effect is still >1 as Gravitational light deflection: n  k

30. A body of mean density  produces a light deflection not less than  • if its radius: Gravitational light deflection: n  k Pluto 7 Charon 4 Titania 3 Oberon 3 Iapetus 2 Rea 2 Dione 1 Ariel 1 Umbriel 1 Ceres 1 Ganymede 35 Titan 32 Io 30 Callisto 28 Triton 20 Europe 19

31. Gravitational light deflection: n  k Jos de Bruijne, 2002

32. All formulas here are formally Euclidean: Parallax and proper motion: k  l  l0, 0, 0 • Expansion in powers of several small parameters:

33. All astrometrical parameters of sources obtained from astrometric • observations are defined in BCRS coordinates: • positions • proper motions • parallaxes • radial velocities • orbits of minor planets, etc. • orbits of binaries, etc. • These parameters represent a realization (materialization) of the BCRS • This materialization is „the goal of astrometry“ and is called • Celestial Reference Frame Celestial Reference Frame

34. Gravitational light deflection caused by the gravitational fields • generated outside the solar system • microlensing on stars of the Galaxy, • gravitational waves from compact sources, • primordial (cosmological) gravitational waves, • binary companions, … • Microlensing noise could be • a crucial problem • for going well below 1 microarcsecond… • Cosmological effects Beyond the standard model