the fundamental astronomical reference systems for space missions and the expansion of the universe...
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The fundamental astronomical reference systems for space missions and the expansion of the universe
Michael Soffel & Sergei Klioner
TU Dresden
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
IAU-2000 Resolution B1.3
...cw2
1g
...wc4
g
...cw2
cw2
1g
2ijij
i3i0
4
2
200
IAU -2000 Resolutions: BCRS (t, x) with metric tensor
Equations of translational motion
• The equations of translational motion (e.g. of a satellite) in the BCRS
200
0
24
3
2
2( , )
4( , )
2(
1 ,
,
1 ,
( , )
.)
2
i
ij ij
i
w tc
w tc
w
w tc
g
g
g
tc
x
x
x
x
• The equations coincide with the well-known Einstein-Infeld-Hoffmann (EIH) equations in the corresponding point-mass limit
23
1)
|(
|A
AB
BB A A B
tGMc
x x
Fx
xx
LeVerrier
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.
200 2 4
0 3
2
2 21 ( , ) ( , ) ,
4( , ) ,
21 ( , ) .
aa
ab ab
G W T W Tc c
G W Tc
G W Tc
X X
X
X
, :aW W internal + inertial + tidal external potentials
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.
• Modelling of any local phenomena: observation, attitude, local physics (if necessary)
, :aW W internal + inertial + tidal external potentials
observer
BCRS-metric is asymptotically flat;ignores cosmological effects,fine for the solar-system dynamics and local geometrical optics
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 tosystem below; dynamical time scales grow if we godown the list -> renormalization of constants (sec- aber)
BUT: expansion of the universe has to be taken into account
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 1 pc
221.56° 285.11°
-60.92° 13.91°
40 AUaX 17 24 10 /aX m s
The cosmological principle (CP):
on very large scales the universe is homogeneousand isotropic
The Robertson-Walker metric follows from the CP
Consequences of the RW-metricfor astrometry:
- cosmic redshift
- various distances that differ from each other:
parallax distance luminosity distance angular diameter distance proper motion distance
Is the CP valid?
• Clearly for the dark (vacuum) energy
• For ordinary matter: likely on very large scales
solar-system: 2 x 10 Mpc :
our galaxy: 0.03 Mpc
the local group: 1 - 3 Mpc
-10
The localsupercluster:20 - 30 Mpc
dimensions of great wall:
150 x 70 x 5 Mpc
distance 100 Mpc
Anisotropies in the CMBR
WMAP-data
/ < 10
for
R > 1000 (Mpc/h)
-4
(O.Lahav, 2000)
The CP for ordinary matter seems to be valid for scales
R > R
with R 400 h Mpc
inhom
inhom
-1
The WMAP-data leads to the present(cosmological) standard model:
Age(universe) = 13.7 billion years
Lum = 0.04dark = 0.23 = 0.73 (dark vacuum energy)
H0 = (71 +/- 4) km/s/Mpc
In a first step we considered only the effect of thevacuum energy (the cosmological constant )
...c
'w21g
...wc4
g
...cw2
cw2
1g
2ijij
i3i0
4
2
200
!
(localSchwarzschild-de Sitter)
The -terms lead to a cosmic tidal accelerationin the BCRS proportial to barycentric distance r
effects for the solar-system: completely negligible
only at cosmic distances, i.e. for objectswith non-vanishing cosmic redshift they play a role
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
According to the Equivalence Principlelocal Minkowski coordinates exist everywhere
take x = 0 (geodesic) as origin of a localMinkowskian system
without terms from local physics we can transformthe RW-metric to:
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
• 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
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)
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
Validity of the local expansion hypothesis: unclear
The Einstein-Straussolution ( = 0)
LEH might be wrong
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
THE END