The relativistic equations of motion for a satellite in orbit about a finite-size, rotating Earth (original) (raw)
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).
General relativistic effects acting on the orbits of Galileo satellites
Celestial Mechanics and Dynamical Astronomy, 2021
The first pair of satellites belonging to the European Global Navigation Satellite System (GNSS)-Galileo-has been accidentally launched into highly eccentric, instead of circular, orbits. The final height of these two satellites varies between 17,180 and 26,020 km, making these satellites very suitable for the verification of the effects emerging from general relativity. We employ the post-Newtonian parameterization (PPN) for describing the perturbations acting on Keplerian orbit parameters of artificial Earth satellites caused by the Schwarzschild, Lense-Thirring, and de Sitter general relativity effects. The values emerging from PPN numerical simulations are compared with the approximations based on the Gaussian perturbations for the temporal variations of the Keplerian elements of Galileo satellites in nominal, near-circular orbits, as well as in the highly elliptical orbits. We discuss what kinds of perturbations are detectable using the current accuracy of precise orbit determination of artificial Earth satellites, including the expected secular and periodic variations, as well as the constant offsets of Keplerian parameters. We found that not only secular but also periodic variations of orbit parameters caused by general relativity effects exceed the value of 1 cm within 24 h; thus, they should be fully detectable using the current GNSS precise orbit determination methods. Many of the 1-PPN effects are detectable using the Galileo satellite system, but the Lense-Thirring effect is not. Gravitation • Artificial Earth satellites • General relativity • Orbit perturbations • Osculating Keplerian elements • GNSS B K. Sośnica
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.
Satellites: The Differential Equations of Motion of the System
International journal of scientific and research publications, 2018
This paper deals with the analyses of the effect of earth's oblateness and magnetic force on the motion of a system of two particles .The relative motion of the particles ,for case of circular motion of the center of mass of the system; The satellites may oscillate about some equilibrium position of the system in elliptical orbit. In three dimensional spaces, the particle is bound to attain the sphere .At a certain moment and after that the motion will take place with tight string and the particle will start moving on the surface of the sphere.
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
On Six DOF Relative Orbital Motion of Satellites
In this chapter, we reveal a dual-tensor-based procedure to obtain exact expressions for the six degree of freedom (6-DOF) relative orbital law of motion in the specific case of two Keplerian confocal orbits. The result is achieved by pure analytical methods in the general case of any leader and deputy motion, without singularities or implying any secular terms. Orthogonal dual tensors play a very important role, with the representation of the solution being, to the authors' knowledge, the shortest approach for describing the complete onboard solution of the 6-DOF orbital motion problem. The solution does not depend on the local-vertical–local-horizontal (LVLH) properties involves that is true in any reference frame of the leader with the origin in its mass center. A representation theorem is provided for the full-body initial value problem. Furthermore, the representation theorems for rotation part and translation part of the relative motion are obtained.
A Theory of Low Eccentricity Earth Satellite Motion
Journal of The Astronautical Sciences, 2012
Earth satellite motion is considered from the point of view of periodic orbits and Floquet theory in the Earth's zonal potential field. Periodic orbits in the zonal potential are nearly circular, except near the critical inclination. The local linear solution near the periodic orbit includes two degenerate modes that locally mirror the global invariance to time and nodal rotation, at least in the zonal potential. Since the Earth's oblateness is included in the periodic orbit, perturbations generally begin at one part in 10 5 , not one part in 10 3. Perturbations to the periodic orbit are calculated for sectoral and tesseral potential terms, for air drag, and for third body effects. The one free oscillatory mode of the periodic orbit is the eccentricity / argument of perigee analogues, and this can be extended past the first order in small quantities. There results a compact, purely numerical set of algorithms that may rival numerical integration in their accuracy, but have the usual "general perturbations" advantage of calculation directly at the time of interest, without having to perform a long propagation.
Satellite motion in a non-singular gravitational potential
Abstract We study the effects of a non-singular gravitational potential on satellite orbits by deriving the corresponding time rates of change of its orbital elements. This is achieved by expanding the non-singular potential into power series up to second order. This series contains three terms, the first been the Newtonian potential and the other two, here R1 (first order term) and R2 (second order term), express deviationsof the singular potential from the Newtonian. These deviations from the Newtonian potential are taken as disturbing potential terms in the Lagrange planetary equations that provide the time rates of change of the orbital elements of a satellite in a non-singular gravitational field. We split these effects into secular, low and high frequency components and we evaluate them numerically using the low Earth orbiting mission Gravity Recovery and Climate Experiment (GRACE). We show that the secular effect of the secondorder disturbing term R2 on the perigee and the mean anomaly are 4.307 × 10−9/a, and −2.533 × 10−15/a, respectively.These effects are far too small and most likely cannot easily be observed with today’s technology. Numerical evaluation of the low and high frequency effects of the disturbing term R2 on low Earth orbiters like GRACE are very small and undetectable by current observational means.
Simple algorithms for relative motion of satellites
New Astronomy, 2015
We derive in a simple way a basic set of solutions for the variational equations of the two-body motion. This is shown to be a useful approximation for the relative motion of two-satellites even in the so called J 2 problem, provided one uses the secular J 2-theory to obtain the orbit precession. We also present an accurate simple-to-implement numerical method for the computation of the relative motion of two-satellites. This is presented for both the linearized approximation and in an exact formulation. The numerical results are compared with approximation produced by the two-body variational but precessed approximation. We find very good agreement for quasi-circular orbits with same inclination.
General Relativistic Satellite Astrometry
1998
The non-perturbative general relativistic approach to global astrometry introduced by de Felice et al. (1998) is here extended to account for the star motions on the Schwarzschild celestial sphere. A new expression of the observables, i.e. angular distances among stars, is provided, which takes into account the effects of parallax and proper motions. This dynamical model is then tested on an end-to-end simulation of the global astrometry mission GAIA. The results confirm the findings of our earlier work, which applied to the case of a static (angular coordinates only) sphere. In particular, measurements of large arcs among stars (each measurement good to ∼100 µarcsec, as expected for V ∼ 17 mag stars) repeated over an observing period comparable to the mission lifetime foreseen for GAIA, can be modeled to yield estimates of positions, parallaxes, and annual proper motions good to ∼15 µarcsec. This second round of experiments confirms, within the limitations of the simulation and the assumptions of the current relativistic model, that the space-born global astrometry initiated with Hipparcos can be pushed down to the 10 −5 arcsec accuracy level proposed with the GAIA mission. Finally, the simplified case we have solved can be used as reference for testing the limiting behavior of more realistic models as they become available.
Some Aspects of Relativistic Astrometry from Within the Solar System
Celestial Mechanics & Dynamical Astronomy, 2003
In this article we outline the structure of a general relativistic astrometric model which has been developed to deduce the position and proper motion of stars from 1 µarcsecond optical observations made by an astrometric satellite orbiting around the Sun. The basic assumption of our model is that the Solar System is the only source of gravity, hence we show how we modeled the satellite observations in a many-body perturbative approach limiting ourselves to the order of accuracy of (v/c)2. The microarcsecond observing scenario outlined is that for the GAIA astrometric mission.