World function and as astrometry

1. World function and as astrometry Christophe Le Poncin-Lafitte and Pierre Teyssandier Observatory of Paris, SYRTE CNRS/UMR8630

2. 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

3. The world function • 1. Definition SAB= geodesic distance between xA and xB e = 1, 0, -1 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 - W(xA,xB) satisfies equations of the Hamilton-Jacobi type at xA and xB : • - gAB is a light ray • Deduction of the time transfer function

4. Post-Minkowskian expansion of W(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

5. (Cf Synge) Using Hamilton-Jacobi equations, we find • and the general form of W(2) where and the straight line connecting xA and xB

6. 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 :

7. 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.)

8. 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

9. 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

10. Post-Post-Minkowskian contribution of a static spherically symmetric body • Consider the following metric (John 1975, Richter & Matzner 1983) • We obtain for W(xA,xB) with and

11. Time transfer and vector tangent at xB up to the order G²/c4 We deduce the time transfer (for a different method in GR, see Brumberg 1987 Vector tangent at xB is obtained where

12. 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.