Relativity and Reference Frame Working group – Nice, November 2003 World function and  as astrometry Christophe Le Poncin-Lafitte and Pierre Teyssandier Observatory of Paris, SYRTE CNRS/UMR8630

Relativity and Reference Frame Working group – Nice, November 2003 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 approachKlioner & 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

Relativity and Reference Frame Working group – Nice, November 2003 The world function 1. Definition S AB = geodesic distance between x A and x B  for timelike, null and spacelike geodesics, respectively 2. Fundamental properties - Given x A and x B, let  be the unique geodesic path joining x A and x B, vectors tangent to  at x A and x B  (x A,x B ) satisfies equations of the Hamilton-Jacobi type at x A and x B :  AB is a light ray   Deduction of the time transfer function

Relativity and Reference Frame Working group – Nice, November 2003 Post-Minkowskian expansion of  (x A,x B ) 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

Relativity and Reference Frame Working group – Nice, November 2003 Using Hamilton-Jacobi equations, we find and the general form of  (2) where and the straight line connecting x A and x B (Cf Synge)

Relativity and Reference Frame Working group – Nice, November 2003 Relativistic astrometric measurement Consider an observer located at x B and moving with an unite 4-velocity u Let k be the vector tangent to the light ray observed at x B. The projection of k obtained from the world function on the associated 3-plane in x B orthogonal to u is => Direction of the light ray :

Relativity and Reference Frame Working group – Nice, November 2003 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/c 4 (2PP-Minkowskian approx.)

Relativity and Reference Frame Working group – Nice, November 2003 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,x A,x B ) is the Shapiro kernel function For a stationary space-time, we have for the tangent vector at x B

Relativity and Reference Frame Working group – Nice, November 2003 As a consequence, the tangent vector at x B is Where With a general definition of the unite 4-velocity => Determination of the observed vector of light direction in the 3-plane in x B

Relativity and Reference Frame Working group – Nice, November 2003 Post-Post-Minkowskian contribution of a static spherically symmetric body Consider the following metric (John 1975, Richter & Matzner 1983) We obtain for  (x A,x B ) with and

Relativity and Reference Frame Working group – Nice, November 2003 Time transfer and vector tangent at x B up to the order G²/c 4 where We deduce the time transfer (for a different method in GR, see Brumberg 1987 Vector tangent at x B is obtained

Relativity and Reference Frame Working group – Nice, November 2003 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.