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. Geocentric Celestial Reference System
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.
internal + inertial + tidal external potentials

6. Local reference system of an observer
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.
observer
internal + inertial + tidal external potentials
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. BCRS for a non-isolated system
Tidal forces from the next 100 stars:
their quadrupole moment can be represented by two fictitious bodies:
Body 1
Body 2
Mass
1.67 Msun
0.19 MSun
Distance
1 pc 
221.56°
285.11° 
-60.92°
13.91°

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

14. The local
supercluster:
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**2 away 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