The fundamental astronomical reference systems for space missions

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

IAU-2000 Resolution B 1. 3 Definition of BCRS (t, x) with t = x 0 = TCB, spatial coordinates x and metric tensor g Ä post-Newtonian metric in harmonic coordinates determined by potentials w, w i

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 • The equations coincide with the well-known Einstein-Infeld-Hoffmann (EIH) equations in the corresponding point-mass limit Le. Verrier

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. observer internal + inertial + tidal external potentials • Modelling of any local phenomena: observation, attitude, local physics (if necessary)

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) • Ga. CRS (galactic) • Lo. Gr. CRS (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 1 Body 2 Mass 1. 67 Msun 0. 19 MSun Distance 1 pc 221. 56° 285. 11° -60. 92° 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 -10 Mpc : our galaxy: 0. 03 Mpc the local group: 1 - 3 Mpc

The local supercluster: 20 - 30 Mpc

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

Anisotropies in the CMBR WMAP-data

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

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

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 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 (Mc. Vittie 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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