Modeling approaches for precise relativistic orbits: Analytical, Lie-series, and pN approximation

authored by

Dennis Philipp, Florian Wöske, Liliane Biskupek, Eva Hackmann, Enrico Mai, Meike List, Claus Lämmerzahl, Benny Rievers

Abstract

Accurate orbit modeling plays a key role in contemporary and future space missions such as GRACE and its successor GRACE-FO, GNSS, and altimetry missions. To fully exploit the technological capabilities and correctly interpret measurements, relativistic orbital effects need to be taken into account. Within the theory of General Relativity, equations of motion for freely falling test objects, such as satellites orbiting the Earth, are given by the geodesic equation. We analyze and compare different solution methods in a spherically symmetric background, i.e. for the Schwarzschild spacetime, as a test bed. We investigate satellite orbits and use direct numerical orbit integration as well as the semi-analytical Lie-series approach. The results are compared to the exact analytical reference solution in terms of elliptic functions. For a set of exemplary orbits, we determine the respective accuracy of the different methods. Within the post-Newtonian approximation of General Relativity, modified orbital equations are obtained by adding relativistic corrections to the Newtonian equations of motion. We analyze the accuracy of this approximation with respect to the general relativistic setting. Therefore, we solve the post-Newtonian equation of motion using the eXtended High Performance Satellite dynamics Simulator. For corresponding initial conditions, we compare orbits in the Schwarzschild spacetime to those in its post-Newtonian approximation. Moreover, we compare the magnitude of relativistic contributions to several typical perturbations of satellite orbits due to, e.g., solar radiation pressure, Earth's albedo, and atmospheric drag. This comparison is done for our test scenarios and for a real GRACE orbit to highlight the importance of relativistic effects in geodetic space missions. For the considered orbits, first-order relativistic contributions give accelerations of about 20 nm/s

² and are dominant in the radial direction.

Organisation(s)

Institute of Geodesy

External Organisation(s)

Center of Applied Space Technology and Microgravity (ZARM)
University of Bremen

Type

Article

Journal

Advances in Space Research

Volume

Pages

921-934

No. of pages

ISSN

0273-1177

Publication date

15.08.2018

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Aerospace Engineering, Astronomy and Astrophysics, Geophysics, Atmospheric Science, Space and Planetary Science, Earth and Planetary Sciences(all)

Electronic version(s)

https://doi.org/10.48550/arXiv.1708.04609 (Access: Open)
https://doi.org/10.1016/j.asr.2018.05.020 (Access: Closed)

Details in the research portal "Research@Leibniz University"

BibTeX

@article{acab027ccc554bc1a8944be5f375b3f0,
title = "Modeling approaches for precise relativistic orbits: Analytical, Lie-series, and pN approximation",
abstract = "Accurate orbit modeling plays a key role in contemporary and future space missions such as GRACE and its successor GRACE-FO, GNSS, and altimetry missions. To fully exploit the technological capabilities and correctly interpret measurements, relativistic orbital effects need to be taken into account. Within the theory of General Relativity, equations of motion for freely falling test objects, such as satellites orbiting the Earth, are given by the geodesic equation. We analyze and compare different solution methods in a spherically symmetric background, i.e. for the Schwarzschild spacetime, as a test bed. We investigate satellite orbits and use direct numerical orbit integration as well as the semi-analytical Lie-series approach. The results are compared to the exact analytical reference solution in terms of elliptic functions. For a set of exemplary orbits, we determine the respective accuracy of the different methods. Within the post-Newtonian approximation of General Relativity, modified orbital equations are obtained by adding relativistic corrections to the Newtonian equations of motion. We analyze the accuracy of this approximation with respect to the general relativistic setting. Therefore, we solve the post-Newtonian equation of motion using the eXtended High Performance Satellite dynamics Simulator. For corresponding initial conditions, we compare orbits in the Schwarzschild spacetime to those in its post-Newtonian approximation. Moreover, we compare the magnitude of relativistic contributions to several typical perturbations of satellite orbits due to, e.g., solar radiation pressure, Earth's albedo, and atmospheric drag. This comparison is done for our test scenarios and for a real GRACE orbit to highlight the importance of relativistic effects in geodetic space missions. For the considered orbits, first-order relativistic contributions give accelerations of about 20 nm/s 2 and are dominant in the radial direction. ",
keywords = "Orbit propagation, Post-Newtonian theory, Relativistic effects, Relativistic geodesy, Satellite orbits",
author = "Dennis Philipp and Florian W{\"o}ske and Liliane Biskupek and Eva Hackmann and Enrico Mai and Meike List and Claus L{\"a}mmerzahl and Benny Rievers",
note = "Funding Information: The present work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the Sonderforschungsbereich (SFB) 1128 Relativistic Geodesy and Gravimetry with Quantum Sensors (geo-Q) and the Research Training Group 1620 Models of Gravity. We also acknowledge support by the German Space Agency DLR with funds provided by the Federal Ministry of Economics and Technology (BMWi) under Grant No. DLR 50WM1547 .",
year = "2018",
month = aug,
day = "15",
doi = "10.48550/arXiv.1708.04609",
language = "English",
volume = "62",
pages = "921--934",
journal = "Advances in Space Research",
issn = "0273-1177",
publisher = "Elsevier Ltd.",
number = "4",
}