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