Retrograde orbits perturbed by a third-body

Abstract

This paper develops a semi-analytical study of the perturbation caused to a spacecraft by a third body involved in the dynamics. There are several important applications for this research, such as to calculate the effect of lunar and solar perturbations on high-altitude Earth satellites. In the present research the goal is to study the evolution of retrograde orbits around the Earth. There is a special interest to see under which conditions a near-circular orbit remains near-circular. The existence of circular, equatorial and frozen orbits are also considered, The results are valid for any system of primaries by making a time transformation that depends on the masses of the bodies involved. Several plots will show the time-histories of the Keplerian elements of the orbits involved

Interesting dynamics at high mutual inclination in the framework of the Kozai problem with an eccentric perturber

“…Practically, the term can be expressed in the following way – see Prado (2005) for a similar 2D calculation: where 1 ( i 1 , ω 1 , Ω 1 ) and 2 ( i 2 , ω 2 , Ω 2 ) are Eulerian rotations of the orbital reference frames of the masses m 1 and m 2 , respectively, and and have rather simple expressions: R i , j being the element in the i th row and j th column of the matrix . The averaging of the Hamiltonian () over the short‐period terms, is realized, in practice, over the eccentric anomaly E 1 of the perturbed body and over the true anomaly f 2 of the outer perturbing body.…”

“…Practically, the term cos S = r 1 · r 2 /(r 1 r 2 ) can be expressed in the following way -see Prado (2005) for a similar 2D calculation:…”

“…For the sake of completeness, we present hereafter the technical details of the calculations. Practically, the term cos S = r 1 • r 2 /(r 1 r 2 ) can be expressed in the following way -see Prado (2005) for a similar 2-D calculation:…”