# Relativistic satellite orbits: central body with higher zonal harmonics

Relativistic satellite orbits: central body with higher zonal harmonics Satellite orbits around a central body with arbitrary zonal harmonics are considered in a relativistic framework. Our starting point is the relativistic Celestial Mechanics based upon the first post-Newtonian approximation to Einstein’s theory of gravity as it has been formulated by Damour et al. (Phys Rev D 43:3273–3307, 1991; 45:1017–1044, 1992; 47:3124–3135, 1993; 49:618–635, 1994). Since effects of order $$(\mathrm{GM}/c^2R) \times J_k$$ ( GM / c 2 R ) × J k with $$k \ge 2$$ k ≥ 2 for the Earth are very small (of order $$7 \times 10^{-10}\,\times \,J_k$$ 7 × 10 - 10 × J k ) we consider an axially symmetric body with arbitrary zonal harmonics and a static external gravitational field. In such a field the explicit $$J_k/c^2$$ J k / c 2 -terms (direct terms) in the equations of motion for the coordinate acceleration of a satellite are treated first with first-order perturbation theory. The derived perturbation theoretical results of first order have been checked by purely numerical integrations of the equations of motion. Additional terms of the same order result from the interaction of the Newtonian $$J_k$$ J k -terms with the post-Newtonian Schwarzschild terms (relativistic terms related to the mass of the central body). These ‘mixed terms’ are treated by means of second-order perturbation theory based on the Lie-series method (Hori–Deprit method). Here we concentrate on the secular drifts of the ascending node $$<\!{\dot{\Omega }}\!>$$ < Ω ˙ > and argument of the pericenter $$<\!{\dot{\omega }}\!>$$ < ω ˙ > . Finally orders of magnitude are given and discussed. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Celestial Mechanics and Dynamical Astronomy Springer Journals

# Relativistic satellite orbits: central body with higher zonal harmonics

, Volume 130 (6) – May 29, 2018
24 pages

/lp/springer_journal/relativistic-satellite-orbits-central-body-with-higher-zonal-harmonics-epi0RcFdqu
Publisher
Springer Journals
Subject
Physics; Astrophysics and Astroparticles; Dynamical Systems and Ergodic Theory; Aerospace Technology and Astronautics; Geophysics/Geodesy; Classical Mechanics
ISSN
0923-2958
eISSN
1572-9478
D.O.I.
10.1007/s10569-018-9836-6
Publisher site
See Article on Publisher Site

### Abstract

Satellite orbits around a central body with arbitrary zonal harmonics are considered in a relativistic framework. Our starting point is the relativistic Celestial Mechanics based upon the first post-Newtonian approximation to Einstein’s theory of gravity as it has been formulated by Damour et al. (Phys Rev D 43:3273–3307, 1991; 45:1017–1044, 1992; 47:3124–3135, 1993; 49:618–635, 1994). Since effects of order $$(\mathrm{GM}/c^2R) \times J_k$$ ( GM / c 2 R ) × J k with $$k \ge 2$$ k ≥ 2 for the Earth are very small (of order $$7 \times 10^{-10}\,\times \,J_k$$ 7 × 10 - 10 × J k ) we consider an axially symmetric body with arbitrary zonal harmonics and a static external gravitational field. In such a field the explicit $$J_k/c^2$$ J k / c 2 -terms (direct terms) in the equations of motion for the coordinate acceleration of a satellite are treated first with first-order perturbation theory. The derived perturbation theoretical results of first order have been checked by purely numerical integrations of the equations of motion. Additional terms of the same order result from the interaction of the Newtonian $$J_k$$ J k -terms with the post-Newtonian Schwarzschild terms (relativistic terms related to the mass of the central body). These ‘mixed terms’ are treated by means of second-order perturbation theory based on the Lie-series method (Hori–Deprit method). Here we concentrate on the secular drifts of the ascending node $$<\!{\dot{\Omega }}\!>$$ < Ω ˙ > and argument of the pericenter $$<\!{\dot{\omega }}\!>$$ < ω ˙ > . Finally orders of magnitude are given and discussed.

### Journal

Celestial Mechanics and Dynamical AstronomySpringer Journals

Published: May 29, 2018

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