world function and as astrometry

Relativity and Reference Frame Working group – Nice, 27-28 November 2003

World function and as astrometry

Christophe Le Poncin-Lafitte and Pierre Teyssandier

Observatory of Paris, SYRTE CNRS/UMR8630


Modeling light deflection

Shape of bodies (multipolar structure)

• We must take into account Motion of the

bodies

Several models based on integration of geodesic differential equations to obtain the path of the photon :- Post-Newtonian approach Klioner & Kopeikin (1992)

Klioner (2003)

- Post-Minkowskian approach Kopeikin & Schäfer (1999) Kopeikin & Mashhoom (2002)

We propose- Use of the world function spares the trouble of geodesic

determination

- Post-Post-Minkowskian approach for the Sun (spherically symmetric case)

- Post-Minkowskian formulation for other bodies of Solar System


The world function• 1. Definition

SAB= geodesic distance between xA and xB

for timelike, null and spacelike geodesics, respectively

• 2. Fundamental properties

- Given xA and xB, let be the unique geodesic path joining xA and xB , vectors tangent to at xA and xB

(xA,xB) satisfies equations of the Hamilton-Jacobi type at xA and xB :

AB is a light ray

Deduction of the time transfer function


Post-Minkowskian expansion of (xA,xB)

• The post-post-Minkowskian metric may be written as

Field of self-gravitating, slowly moving sources :

• The world function can be written as

where and


Using Hamilton-Jacobi equations, we find

• and the general form of (2)

where

and

the straight line connecting xA and xB

(Cf Synge)


Relativistic astrometric measurement

• Consider an observer located at xB and moving with an unite 4-velocity u

• Let k be the vector tangent to the light ray observed at xB. The projection of k obtained from the world function on the associated 3-plane in xB orthogonal to u is

• => Direction of the light ray :


Applications to the as accuracy

• For the light behaviour in solar system, we must determine :

– The effects of planets with a multipolar structure at 1PN

– The effect of post-post-Minkowskian terms for the Sun (spherically symmetric body)

• We treat the problem for 2 types of stationary field :– Axisymmetric rotating body in the Nordtvedt-Will PPN formalism

– Spherically symmetric body up to the order G²/c4 (2PP-Minkowskian approx.)


Case of a stationary axisymmetric body within the Will-Nordtvedt PPN formalism

• From (1) , it has been shown (Linet & Teyssandier 2002) for a light ray

where F(x,xA,xB) is the Shapiro kernel function

•For a stationary space-time, we have for the tangent vector at xB


• As a consequence, the tangent vector at xB is

Where

With a general definition of the unite 4-velocity

=> Determination of the observed vector of light direction in the 3-plane in xB


Post-Post-Minkowskian contribution of a static spherically symmetric body

• Consider the following metric (John 1975, Richter & Matzner 1983)

• We obtain for (xA,xB)

with

and


Time transfer and vector tangent at xB up to the order G²/c4

where

We deduce the time transfer (for a different method in GR, see Brumberg 1987

Vector tangent at xB is obtained


Conclusions

• Powerful method to describe the light between 2 points located at finite distance without integrating geodesic equations.

• Obtention of time transfer and tangent vector at the reception point with all multipolar contributions in stationary space-time at 1PN approx.

• Obtention of time transfer and tangent vector at the reception point in spherically symmetric space-time at 2PM.

• Possibility to extend the general determination of the world function at any N-post-Minkowskian order (in preparation).

• To consider the problem of parallax in stationary space-time.

• To take into account motion of bodies.

world function and as astrometry

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