Michael Soffel & Sergei Klioner TU Dresden

1. The fundamental astronomical reference systems for space missions and the expansion of the universe Michael Soffel & Sergei Klioner TU Dresden

2. IAU-2000 Resolution B1.3 Definition of BCRS (t, x) with t = x0 = TCB, spatial coordinates x and metric tensor g • post-Newtonian metric in harmonic coordinates determined by potentials w, w i

3. IAU -2000 Resolutions: BCRS (t, x) with metric tensor

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

5. 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

6. 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)

7. BCRS-metric is asymptotically flat; ignores cosmological effects, fine for the solar-system dynamics and local geometrical optics

8. One might continue with a hierarchy of systems • GCRS (geocentric celestial reference system) • BCRS (barycentric) • GaCRS (galactic) • LoGrCRS (local group) etc. • each systems contains tidal forces due to • system below; dynamical time scales grow if we go • down the list -> renormalization of constants (sec- aber) • BUT: • expansion of the universe has to be taken into account

9. Tidal forces from the next 100 stars: their quadrupole moment can be represented by two fictitious bodies: BCRS for a non-isolated system

10. The cosmological principle (CP): on very large scales the universe is homogeneous and isotropic The Robertson-Walker metric follows from the CP

11. Consequences of the RW-metric for astrometry: - cosmic redshift - various distances that differ from each other: parallax distance luminosity distance angular diameter distance proper motion distance

12. Is the CP valid? • Clearly for the dark (vacuum) energy • For ordinary matter: likely on very large scales

13. -10 solar-system: 2 x 10 Mpc : our galaxy: 0.03 Mpc the local group: 1 - 3 Mpc

14. The local supercluster: 20 - 30 Mpc

15. dimensions of great wall: 150 x 70 x 5 Mpc distance 100 Mpc

16. Anisotropies in the CMBR WMAP-data

17. -4 / < 10 for R > 1000 (Mpc/h) (O.Lahav, 2000)

18. The CP for ordinary matter seems to be valid for scales R > R with R  400 h Mpc inhom -1 inhom

19. The WMAP-data leads to the present (cosmological) standard model: Age(universe) = 13.7 billion years Lum = 0.04 dark = 0.23  = 0.73 (dark vacuum energy) H0 = (71 +/- 4) km/s/Mpc

21. In a first step we considered only the effect of the vacuum energy (the cosmological constant ) !

23. (local Schwarzschild-de Sitter)

24. The -terms lead to a cosmic tidal acceleration in the BCRS proportial to barycentric distance r effects for the solar-system: completely negligible only at cosmic distances, i.e. for objects with non-vanishing cosmic redshift they play a role

25. Further studies: • transformation of the RW-metric to ‚local • coordinates‘ • construction of a local metric for a barycenter in motion • w.r.t. the cosmic energy distribution • - cosmic effects: orders of magnitude

26. According to the Equivalence Principle local Minkowski coordinates exist everywhere take x = 0 (geodesic) as origin of a local Minkowskian system without terms from local physics we can transform the RW-metric to:

27. Transformation of the RW-metric to ‚local coordinates‘

28. ‘ Construction of a local metric for a barycenter in motion w.r.t. the cosmic energy distribution

29. Cosmic effects: orders of magnitude • Quasi-Newtonian cosmic tidal acceleration at Pluto‘s orbit • 2 x 10**(-23) m/s**2away from Sun • (Pioneer anomaly: 8.7 x 10**(-10) m/s**2 towards Sun) • perturbations of planetary osculating elements: e.g., • perihelion prec of Pluto‘s orbit: 10**(-5) microas/cen • 4-acceleration of barycenter due to motion of • solar-system in the g-field of -Cen • solar-system in the g-field of the Milky-Way • Milky-Way in the g-field of the Virgo cluster • < 10**(-19) m/s**2

30. The problem of ‚ordinary cosmic matter‘ The local expansion hypothesis: the cosmic expansion occurs on all length scales, i.e., also locally If true: how does the expansion influence local physics ? question has a very long history (McVittie 1933; Järnefelt 1940, 1942; Dicke et al., 1964; Gautreau 1984; Cooperstock et al., 1998)

31. The local expansion hypothesis: the cosmic expansion induced by ordinary (visible and dark) matter occurs on all length scales, i.e., also locally Is that true? Obviously this is true for the -part

32. Validity of the local expansion hypothesis: unclear The Einstein-Straus solution ( = 0) LEH might be wrong

33. Conclusions If one is interested in cosmology, position vectors or radial coordinates of remote objects (e.g., quasars) the present BCRS is not sufficient  the expansion of the universe has to be considered  modification of the BCRS and matching to the cosmic R-W metric becomes necessary

34. THE END