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Basis for a native relativistic software integrating Observatoire Midi-Pyrennees the motion of satellites Sophie PIREAUX 1 1 UMR 5562, Observatoire Midi-Pyrénnées, Toulouse, France Financial support provided through the European Community's Improving Human Potential Program under contract RTN2-2002-00217, MAGE [email protected] - tel. +33 (0)5 61 33 28 44 – Fax +33 (0)5 61 33 29 00 INTRODUCTION And finally, a correction on the clock frequency, owing to the presence of a gravitational field, is considered, leading to a relativistic Doppler effect. This is a natural consequence of the distinction between proper time and coordinate time in relativistic theories. We introduce here a relativistically consistent software called RMI (Relativistic Motion Integrator). We compare it with the GINS software [2] as an example of the classical approach consisting in the Newtonian formalism plus relativistic corrections. GINS is a software routinely used to evaluate the gravity potential (GRIM5, EIGENS 1-2) from the determination of orbit perturbations, or for precise orbit determinations around the Earth (CHAMP, GRACE...) and around Mars (MGS, modeling of future missions). It is based upon the usual formalism used in spatial geodesy, i.e. it relies upon the classical Newtonian description of motion, to which relativistic corrections are added. The number/type of corrections needed depends upon the precision in the measurements (Clock precision/stability). The relativistic corrections on the forces already taken into account in GINS are: the Schwarzschild effect, function of the position and speed of the satellite; the Lens-Thirring effect, due to the rotation of the attracting body; the geodetic precession, function of the chosen coordinate system and due to the non inertial motion of the gravitational source in the solar system. Corrections are also applied on the measurements: a datation correction, which stems from the transformation between the Universal Time Coordinate and the International Atomic Time, or the time referred to in the ephemeris; a relativistic time delay correction, due to the curvature of space-time through which light travels. Newton’s 2nd law of motion with [1] THE CLASSICAL APPROACH: GINS The classical Newton plus relativistic corrections method briefly described here 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. In this relativistic approach, the geodesic equations of motion are directly numerically integrated for a chosen metric. A gradU E A Perturbating Bodies E A Radiation Pressure Earth Tides A Atmospheric Pressure Relativistic Ocean Tides where A grad U grad U A A Atmospheric Drag is due to satellite colliding with residual gas molecules (hyp: free molecular flux); A AD is due to change in satellite momentum owing to solar photon flux; is the ocean tide potential (single layer model); and A RP ET U Earth tide potential due to the Sun and Moon, corrected for Love number frequencies, ellipticity and polar tide; is the U OT AP is a gravitational acceleration induced by the redistribution of atmospheric masses (single layer model). Figure 1-9: 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 JD17001 to JD17002. 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. Relativistic corrections on the forces A A The gravitational potential model for the Earth U E GM R lmax E l 0 l m 0 a P (sin )C cos m S sin m R lm lm lm with A a semi-major axis of the Earth, C , S , the normalized lm E lm S harmonic coefficients, given in GRIM4-S4 model GM 4GM V c R R E 2 E 3 2 R 4 V R V GP Figure 2 Figure 4 Figure 3 0.187604 4.524321 - 0.210319 The Newtonian contributions from the Moon, Sun and Planets A A A PB E J2 - Moon coupling body "n " with 3rd body n A J2 - Moon coupling 3 GM 2 R Mo Mo 5 Z 5 C a 5 R 2 20 2 Mo 2 E Mo 0 1 R 2 R 0 1 Mo Mo and A 3rd body n XYZ0 n 3 n S 10 m/s 11 Geodetic(De Sitter) Precession A Lens-Thirring Precession LTP LTP GM cR 2 E 3 3 S R R S R E E 2 Figure 6 0.13 34.83 40.10 2 XYZ0 10 m/s 12 2 XYZ0 ORBIT 3 n 0.245 2.141 0.928 2 RR R GM RR R n 2 E Figure 5 10 m/s 8 LTP 3 v grad U ( x ) 2c E A A 2 V GP A 2 V GP Figure 1 Schwarzschild R l E n X ,Y , Z ;V ,V ,V x Y GRAVITATIONAL POTENTIAL MODEL FOR EARTH z TAI (…) GRIM4-S4 ITRS (non inertial) J2000 (“inertial”) INTEGRATOR A PLANET EPHEMERIS DE403 TAI 10 m/s 6 2 XYZ0 0.5313475 5.5674979 1.3066652 Figure 9 10 m/s 7 2 XYZ0 0.314 0.560 0.257 E J2000 (“inertial”) 10 m/s 11 PBE E TDB 2 XYZ0 For the appropriate metric, the geodesic equation of motion, contains all needed relativistic effects.(For a massive particle, the affine parameter the proper time) Method: GINS provides template orbits to validate the RMI orbits: THE RELATIVISTIC APPROACH: RMI GRAVITATIONAL POTENTIAL MODEL FOR EARTH dU U U d dX U d GRIM4-S4 ITRS (non inertial) ORBIT X ;U - simulations with 1) Schwarzschild metric => validate Schwarzschild correction TCG 2) (Schwarzschild + GRIM4-S4) metric => validate harmonic contributions GCRS (“inertial”) 4) GCRS metric with(out) Sun, Moon, Planets => validate geodetic precession (other bodies contributions) (…) RMI goes beyond GINS capabilities: INTEGRATOR - includes 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] - separate modules allow to easily update for metric, potential model (EGM96)… prescriptions. TCG METRIC MODEL 3) Kerr metric => validate Lens-Thirring correction 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). GP for x , v in A and A [5] 0.58286072 1.02761036 0.34965593 Figure 8 TAI TT TDB Earth rotation model Figure 7 TCG dU ;U d IAU2000 GCRS metric G TCG TDB PLANET EPHEMERIS DE403 for G in GCRS (“inertial”) TDB [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. [5] D. D. McCarthy and G. Petit. IERS conventions (2003). IERS technical note 200?. (2003). 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).