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 = x 0 = 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

IAU Resolutions: BCRS (t, x) with metric tensor

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

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

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) 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 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

BCRS for a non-isolated system Tidal forces from the next 100 stars: their quadrupole moment can be represented by two fictitious bodies: Body 1Body 2 Mass1.67 M sun 0.19 M Sun Distance1 pc  °285.11°  °13.91°

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

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

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: Mpc -10

The local supercluster: 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

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) H 0 = (71 +/- 4) km/s/Mpc

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

(local Schwarzschild-de Sitter)

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

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 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:

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-Straus solution (  = 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

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