1. measured by accelerometers Newtonian contributions from the Moon, Sun and Planets A semi-classical relativistic motion integrator (SCRMI), to model the orbits of space probes around the Earth and other planets. Sophie PIREAUX, Jean-Pierre BARRIOT and Pascal ROSENBLATT, UMR 5562, Observatoire Midi-Pyrénées, Toulouse, France sophie.pireaux@cnes.fr - tel. +33 (0)5 61 33 28 44 – Fax +33 (0)5 61 33 29 00sophie.pireaux@cnes.fr Royal Observatory of Belgium, Brussels, Belgium. ABSTRACT Today, the motion of spacecrafts is still described according to the classical Newtonian equations plus the so-called « relativistic corrections », computed with the required precision using the Post-(Post-…) Newtonian formalism. The current approach, with the increase of tracking precision (Ka-Band Doppler, interplanetary lasers) and clock stabilities (atomic fountains) is reaching its limits in terms of complexity, and is furthermore error prone. Those problems are especially relevant when modeling tiny geophysical effects through orbitography, like for example the waxing-waning mechanism of the Martian polar caps, with temporal frequencies in the same band as relativistic effects. In the more appropriate frame of General Relativity, we study a method to numerically integrate the native relativistic equations of motion for a weak gravitational field, taking into account not only gravitational forces, but also non gravitational forces (atmospheric drag, solar radiation pressure, albedo pressure, thermal emission…). The latter are treated as perturbations, in the sense that we assume that both the local structure of space-time ( ) is not modified by these forces, and that the unperturbed satellite motion follows the geodesics of the local space-time. We also advice the use of a symplectic integrator to compute the unperturbed geodesic motion, in order to insure the constancy of the norm of the proper velocity quadrivector ( ). Figure 1Figure 2Figure 3 THE (SEMI CLASSICAL) RELATIVISTIC APPROACH: (SC)RMI Method: GINS provides template orbits to validate the RMI orbits: - simulations with 1) Schwarzschild metric => validate Schwarzschild correction 2) (Schwarzschild + GRIM4-S4) metric => validate harmonic contributions 3) Kerr metric => validate Lens-Thirring correction 4) GCRS metric with(out) Sun, Moon, Planets => validate geodetic precession (other bodies contributions) RMI will go beyond GINS capabilities: - will include 1) IAU 2000 standard GCRS metric [6,7] 2) IAU 2000 time transformation prescriptions [6,7] 3) IAU 2000/IERS 2003 new standards on Earth rotation [5,6] 4) Post Newtonian parameters in the metric and space-time transformations - separate modules allow to easily update for metric, potential model (EGM96)… prescriptions. REFERENCES: [1] GRGS. Descriptif modèle de forces: logiciel GINS. Note technique du Groupe de Recherche en Géodesie Spatiale (GRGS), (2001). [2] X. Moisson. Intégration du mouvement des planètes dans le cadre de la relativité générale (thèse). Observatoire de Paris (2000). [3] A. W. Irwin and T. Fukushima. A numerical time ephemeris of the Earth. Astronomy and Astrophysics, 338, 642-652 (1999). [4] SOFA homepage. The SOFA libraries. IAU Division 1: Fundamental Astronomy. ICRS Working Group Task 5: Computation Tools. Standards of Fundamental Astronomy Review Board. ( 2003) http://www.iau-sofa.rl.ac.uk/product.html.http://www.iau-sofa.rl.ac.uk/product.html Observatoire Midi-Pyrenees The classical Newton plus relativistic corrections method briefly described below faces three major problems. First of all, it ignores that in General Relativity time and space are intimately related, as in the classical approach, time and space are separate entities. Secondly, a (complete) review of all the corrections is needed in case of a change in conventions (metric adopted), or if precision is gained in measurements. Thirdly, with such a method, one correction can sometimes be counted twice (for example, the reference frequency provided by the GPS satellites is already corrected for the main relativistic effect), if not forgotten. For those reasons, a new approach was suggested. The prototype is called (Semi-Classical) Relativistic Motion Integrator (whether non gravitational forces are considered). In this relativistic approach, the relativistic equations of motion are directly numerically integrated for a chosen metric. 1 1 Gravitational potential model for the Earth THE CLASSICAL APPROACH: GINS * * Géodésie par Intégrations Numériques Simultanées is a software developed by CNES Relativistic corrections on the forces Figure 5Figure 6 Figure 7 Figure 4 Financial support provided through: European Community's Improving Human Potential Program under contract RTN2-2002-00217, MAGE; where Newton’s 2nd law of motion with [1] is due to satellite colliding with residual gas molecules (hyp: free molecular flux);is due to change in satellite momentum owing to solar photon flux;is the Earth tide potential due to the Sun and Moon, corrected for Love number frequencies, ellipticity and polar tide;is the ocean tide potential (single layer model); and is a gravitational acceleration induced by the redistribution of atmospheric masses (single layer model). Figure 1-10: The following graphs were plotted by selecting only certain gravitational contributions. Those examples show the correction to tangential/ normal/ radial directions on the trajectory of LAGEOS 1 satellite, due to the selected effect, after one day. Integration is carried out during one day, from JD17000 to JD17001. The corresponding induced acceleration on JD17000 is given below each graph. We clearly see the orbital periodicity of 6. 39 revolution/day in each figure, as well as the additional periodicities due to J2 (Fig.2) and higher orders in the gravitational potential (Fig.3). Capital letters are used for geocentric coordinates and velocities; while lower cases are used for barycentric quantities. Name of planets/Sun are shown by indices; no index is used in case of the satellite. Figure 8Figure 9 For the appropriate metric, the relativistic equation of motion contains all needed gravitational effects (blue terms). Non gravitational forces [8] are encoded in measured by accelerometers. When, this equation reduces to the geodesic equation of motion. withand (…) withthe normalizedsemi-major axis of the Earth, harmonic coefficients, given in GRIM4-S4 model [5] D. D. McCarthy and G. Petit. IERS conventions (2003). IERS technical note 200?. (2003). http://maia.usno.navy.mil/conv2000.html.http://maia.usno.navy.mil/conv2000.html [6] IAU 2000 resolutions. IAU Information Bulletin, 88 (2001). Erratum on resolution B1.3. Information Bulletin, 89 (2001). [7] M. Soffel et al. The IAU 2000 resolutions for astrometry, celestial mechanics and metrology in the relativistic framework: explanatory supplement. astro-ph/0303376v1 ( 2003). [8] A. Lichnerowicz. Théories relativistes de la gravitation et de l’électromagnétisme. Masson & Cie Editeurs 1955. TAI J2000 (“inertial”) INTEGRATOR TAI J2000 (“inertial”) ORBIT PLANET EPHEMERIS DE403 for in and TDB GRAVITATIONAL POTENTIAL MODEL FOR EARTH GRIM4-S4 Earth rotation model ITRS (non inertial) GCRS (“inertial”) INTEGRATOR TCG GCRS (“inertial”) ORBIT PLANET EPHEMERIS DE403 for in TDB GRAVITATIONAL POTENTIAL MODEL FOR EARTH GRIM4-S4 Earth rotation model METRIC MODEL IAU2000 GCRS metric ITRS (non inertial) [5] 1 2 2 with quadri-”force” + first integral Need for symplectic integrator Earth tides with l=2,3 in the Earth gravitational potential, due to Sun and Moon. Figure 10 Royal Observatory of Belgium 1 2 Belgian ESA/PRODEX 7 contract.