Satellite motion in a non-singular gravitational potential (original) (raw)
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Satellite Motion in a Non-Singular Potential
2010
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 R 1 (first order term) and R 2 (second order term), express deviations of 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 R 2 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 R 2 on low Earth orbiters like GRACE are very small and undetectable by current observational means.
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.
Analysis of orbital perturbations due to gravitational field harmonics
This document details the procedure of calculation of orbital perturbation due to gravitational harmonics by simulation of various orbits at different altitudes and inclinations to the earth's equatorial plane. The orbits were first simulated and data of position and velocity was collected. The data was then analysed by another program to calculate the perturbations. Runge – Kutta fourth order method was used to simulate the orbits. The orbits were assumed to start from the perigee position which lied on the longitudinal plane containing the point of vernal equinox. The perturbations obtained were analysed w.r.t inclination of orbits and eccentricity of the orbits. The order of magnitude of the perturbations obtained were analysed w.r.t altitude. The earth was assumed to be symmetrical in all other terms.
The impact of common versus separate estimation of orbit parameters on GRACE gravity field solutions
Journal of Geodesy, 2015
Gravity field parameters are usually determined from observations of the GRACE satellite mission together with arc-specific parameters in a generalized orbit determination process. When separating the estimation of gravity field parameters from the determination of the satellites' orbits, correlations between orbit parameters and gravity field coefficients are ignored and the latter parameters are biased towards the a priori force model. We are thus confronted with a kind of hidden regularization. To decipher the underlying mechanisms, the Celestial Mechanics Approach is complemented by tools to modify the impact of the pseudo-stochastic arc-specific parameters on the normal equations level and to efficiently generate ensembles of solutions. By introducing a time variable a priori model and solving for hourly pseudo-stochastic accelerations, a significant reduction of noisy striping in the monthly solutions can be achieved. Setting up more frequent pseudo-stochastic parameters results in a further reduction of the noise, but also in a notable damping of the observed geophysical signals. To quantify the effect of the a priori model on the monthly solutions, the process of fixing the orbit parameters is replaced by an equivalent introduction of special pseudo-observations, i.e., by explicit regularization. The contribution of the thereby introduced a priori information is determined by a contribution analysis. The presented mechanism is valid universally. It may be used to separate any subset of parameters by pseudo-observations of a special design and to quantify the damage imposed on the solution.
2021
This paper presents a collection of analytical formulae that can be used in the long-term propagation of the motion of a spacecraft subject to low-thrust acceleration and orbital perturbations. The paper considers accelerations due to: a low-thrust profile following an inverse square law, gravity perturbations due to the central body gravity field and the third-body gravitational perturbation. The analytical formulae are expressed in terms of non-singular equinoctial elements. The formulae for the third-body gravitational perturbation have been obtained starting from equations for the third-body potential already available in the literature. However, the final analytical formulae for the variation of the equinoctial orbital elements are a novel derivation. The results are validated, for different orbital regimes, using high-precision numerical orbit propagators.
We numerically investigate the impact of the General Theory of Relativity (GTR) on the orbital part of the satellite-to-satellite range q and range-rate q_ of the twin GRACE A/B spacecrafts through their post-Newtonian (PN) dynamical equations of motion integrated in an Earth-centered frame over a time span DP ¼ 1 d. The present-day accuracies in measuring the GRACE biased range and range-rate are rq 1 10 lm; rq_ 0:1 1 lm s1. The GTR range and range-rate effects turn out to be Dq ¼ 80 lm and Dq_ ¼ 0:012 lm s1 (1PN gravitomagnetic), and Dq ¼ 6000 lm and Dq_ ¼ 10 lm s1 (1PN gravitoelectric). It turns out that the range shifts Dq corresponding to the GTR-induced time delays Dt on the propagation of the electromagnetic waves linking the GRACE spacecrafts are either negligible (1PN gravitomagnetic) or smaller (1PN gravitoelectric) than the orbital effects by about 1 order of magnitude over DP ¼ 1 d. We also compute the dynamical range and range-rate perturbations caused by the first six zonal harmonic coefficients J‘; ‘ ¼ 2; 3; 4; 5; 6; 7 of the classical multipolar expansion of the geopotential to evaluate their aliasing impact on the relativistic effects. Conversely, we quantitatively, and preliminarily, assess the possible a-priori “imprinting” of GTR itself, not solved-for in all the GRACE-based Earth’s gravity models produced so far, on the low degree zonals of the geopotential. The present sensitivity analysis can also be extended, in principle, to different orbital configurations in order to design a suitable dedicated mission able to accurately measure GTR. Moreover, it may be the starting point for more refined numerical investigations concerning the actual measurability of the relativistic effects involving, e.g., a simulation of full GRACE data, including GTR itself, and the consequent parameters’ estimation. Finally, also other non-classical dynamical features of motion, caused by, e.g., modified models of gravity, may be considered in further studies.
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.
An analytical treatment of resonance effects on satellite orbits
Celestial Mechanics, 1987
A first order analytical approximation of the tesseral harmonic resonance perturbations of the Keplerian elements is presented, and the mean elements (the Keplerian elements with the long period portions averaged out) will also be given in closed form. The results of a numerical test, which compares the analytical solution against a numeric.'l integration of the Lagrange equations of motion, will be summarized, and the implementation of the solution in the analytical orbit determination routine ANODE
On the third-body perturbations of high-altitude orbits
Celestial Mechanics and Dynamical Astronomy, 2012
The long-term effects of a distant third-body on a massless satellite that is orbiting an oblate body are studied for a high order expansion of the third-body disturbing function. This high order may be required, for instance, for Earth artificial satellites in the so-called MEO region. After filtering analytically the short-period angles via averaging, the evolution of the orbital elements is efficiently integrated numerically with very long step-sizes. The necessity of retaining higher orders in the expansion of the third-body disturbing function becomes apparent when recovering the short-periodic effects required in the computation of reliable osculating elements. Keywords Third-body perturbation • Lie transforms • Averaging • High-altitude orbits 1 Introduction Semi-analytical integration is the common approach used by aerospace engineers in the mathematical problem of investigating the long-term dynamics of uncontrolled man-made Earth satellites, like non-operational orbits and debris. Short-periodic angles are filtered analytically M.
The celestial mechanics approach: application to data of the GRACE mission
Journal of Geodesy, 2010
The celestial mechanics approach (CMA) has its roots in the Bernese GPS software and was extensively used for determining the orbits of high-orbiting satellites. The CMA was extended to determine the orbits of Low Earth Orbiting satellites (LEOs) equipped with GPS receivers and of constellations of LEOs equipped in addition with intersatellite links. In recent years the CMA was further developed and used for gravity field determination. The CMA was developed by the Astronomical Institute of the University of Bern (AIUB). The CMA is presented from the theoretical perspective in (Beutler et al. 2010). The key elements of the CMA are illustrated here using data from 50 days of GPS, K-Band, and accelerometer observations gathered by the Gravity Recovery And Climate Experiment (GRACE) mission in 2007. We study in particular the impact of (1) analyzing different observables [Global Positioning System (GPS) observations only, inter-satellite measurements only], (2) analyzing a combination of observations of different types on the level of the normal equation systems (NEQs), (3) using accelerometer data, (4) different orbit parametrizations (short-arc, reduced-dynamic) by imposing different constraints on the stochastic orbit parameters, and (5) using either the