The relativistic equations of motion for a satellite in orbit about a finite-size, rotating Earth (original) (raw)
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General-relativistic celestial mechanics. IV. Theory of satellite motion
Physical Review D, 1994
The basic equations needed for developing a complete relativistic theory of artificial Earth satellites are explicitly written down. These equations are given both in a local, geocentric frame and in the global, barycentric one. They are derived within our recently introduced general-relativistic celestial mechanics framework. Our approach is more satisfactory than previous ones, especially with regard to its consistency, completeness, and flexibility. In particular, the problem of representing the relativistic gravitational effects associated with the quadrupole and higher multipole moments of the moving Earth, which caused difficulties in several other approaches, is easily dealt with in our approach thanks to the use of previously developed tools: the definition of relativistic multipole moments and transformation theory between reference frames. With this last paper in a series we hope to indicate the way of using our formalism in specific problems in applied celestial mechanics and astrometry.
Relativistic Perturbations of an Earth Satellite
Physical Review Letters, 1984
The inertial frame of reference in the neighborhood of a test body provided by the construction of Fermi normal coordinates is generalized to include the effect of the body's gravitational field. The metric obtained provides a simple physica1 description of relativistic corrections to the orbital motion of a satellite of the Earth. The main correction is the nonlinear Schwarzschild field of the Earth; in these coordinates there are also three much smaller terms arising from the solar tidal influence.
Orbital tests of relativistic gravity using artificial satellites
Physical Review D, 1994
We reexamine non-Einsteinian effects observable in the orbital motion of low-orbit artificial Earth satellites. The motivations for doing so are twofold: (i) recent theoretical studies suggest that the correct theory of gravity might contain a scalar contribution which has been reduced to a small value by the effect of the cosmological expansion; (ii) presently developed space technologies should soon give access to a new generation of satellites endowed with drag-free systems and tracked in three dimensions at the centimeter level. Our analysis suggests that such data could measure two independent combinations of the Eddington parameters β ≡ β − 1 and γ ≡ γ − 1 at the 10 −4 level and probe the time variability of Newton's "constant" at theĠ/G ∼ 10 −13 yr −1 level. These tests would provide well-needed complements to the results of the Lunar Laser Ranging experiment, and of the presently planned experiments aiming at measuring γ. In view of the strong demands they make on the level of non-gravitational perturbations, these tests might require a dedicated mission consisting of an optimized passive drag-free satellite.
Advances in Space Research, 2016
We study the post-Newtonian perturbations in the orbit of a near-Earth satellite by integrating them with a high-fidelity orbit propagation software KASIOP. The perturbations of the orbital elements are evaluated for various cases from a low-Earth orbit to a geostationary one, and from an equatorial to a polar orbit. In particular, the numerical simulation is applied to the LARES-like satellite under a realistic orbital configuration. The relativistic perturbations include the Schwarzschild term, the effects of Lense-Thirring precession, and the post-Newtonian term due to the quadrupole moment of the Earth as well as the post-Newtonian gravitoelectric and gravitomagnetic forces, which are produced by the tidal potential of the solar system bodies, are also modeled. The latter three terms are usually ignored in most orbit-propagation software. The secular variations of the orbital elements are evaluated from the orbital positions propagated for a half year. For a medium altitude orbit like that of the LARES mission, the magnitude of the relativistic perturbations ranges from the order of 10 À7 m/s 2 by the Schwarzschild effect to 10 À15 m/s 2 by the relativistic tidal effects. The orbital integration shows that the secular variations in three orbital elements-the ascending node, the argument of perigee, and the mean anomaly at epoch-are larger than the systematic error as results of the relativistic perturbations. The magnitudes of the secular variation are investigated in terms of the orbital altitude, inclination, and the size of each perturbation force. The numerical simulation rendered in this study shows that the secular post-Newtonian perturbations with the magnitude lying beyond the Schwarzschild and the Lense-Thirring effects need to be taken into account in current and upcoming space geodesy missions.
obs-azur.fr
The "Newton plus relativistic corrections" orbitography software now in wide use faces three major problems. First of all, they ignore 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, corrections 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 native relativistic approach is suggested. In this relativistic approach, the relativistic equations of motion are directly numerically integrated for a chosen metric. Our prototype software, that takes into account non-gravitational forces, is named SCRMI (Semi-Classical Relativistic Motion Integrator).
Relative motion of orbiting bodies
American Journal of Physics, 2001
A problem of relative motion of orbiting bodies is investigated on the example of the free motion of any body ejected from the orbital station that stays in a circular orbit around the earth. An elementary approach is illustrated by a simulation computer program and supported by a mathematical treatment based on approximate differential equations of the relative orbital motion.
Motion and rotation of celestial bodies in the post-Newtonian approximation
Celestial Mechanics, 1987
Consistent post-Newtonian description of motion and precession in a system of N extended slowly rotating bodies is developed in the framework of the post-Newtonian approximation scheme (PNA). The solution of Einstein equations for the metric in the local reference system related to a body of the system is obtained. This metric is used to derive the equations of motion and precession of the considered body on the basis of some relativistic generalization of the model of rigid body. These equations are solved in order to find the first order corrections to nutation theory and to the osculating orbital elements of the body. Another important application of such local metric, concerning the motion of a test particle (e.g., artificial satellite) orbiting the body, is also investigated in this paper.
Relativistic equations of motion of celestial bodies
Symposium - International Astronomical Union, 1996
The problem of relativistic equations of motion for extended celestial bodies in the first post-Newtonian approximation is reviewed. It is argued that the problems dealing with kinematical aspects have been solved in a satisfactory way, but more work has to be done on the dynamical side. Concepts like angular velocity, moments of inertia, Tisserand axes etc. still have to be introduced in a rigorous manner at the 1PN level.
The celestial mechanics approach: theoretical foundations
Journal of Geodesy, 2010
Gravity field determination using the measurements of Global Positioning receivers onboard low Earth orbiters and inter-satellite measurements in a constellation of satellites is a generalized orbit determination problem involving all satellites of the constellation. The celestial mechanics approach (CMA) is comprehensive in the sense that it encompasses many different methods currently in use, in particular so-called short-arc methods, reduced-dynamic methods, and pure dynamic methods. The method is very flexible because the actual solution type may be selected just prior to the combination of the satellite-, arc-and technique-specific normal equation systems. It is thus possible to generate ensembles of substantially different solutions-essentially at the cost of generating one particular solution. The article outlines the general aspects of orbit and gravity field determination. Then the focus is put on the particularities of the CMA, in particular on the way to use accelerometer data and the statistical information associated with it. Keywords Celestial mechanics • Orbit determination • Global gravity field modeling • CHAMP • GRACE 1 Problem description and overview This article has the focus on the theoretical foundations of the so-called celestial mechanics approach (CMA). Applications