Linearization in Special Cases of the Perturbed Two-Body Problem

Abstract

In the last years, a lot of analytical methods have been developed to improve accuracy of numerical integrations of orbital problems. Most of them may be envisaged as special tecniques of approximate linearization because, by using them, we can linearize the Keplerian problem, but not the perturbed problem of two bodies.

Looking for a more exact linearization is the purpose of this paper, i.e. linearizing exactly a perturbed problem closer to the original than the Keplerian. We use the method of linearization by means of time transformations studied by Szebehely (1976), and applied by Belen’kii (1981), and Cid, Elipe and Ferrer (1983) in central force fields.

If the Hamiltonian of a perturbed two problem is conservative, we replace it by a more simple spatial problem, defined by a radial intermediary which is independent of the argument of latitude θ, when it is expressed in Hill’s variables. The existence and uniqueness of linearizing functions is proved for the latter, and their explicit forms are given in the most frequent cases.

Then, an application to the Earth’s artificial satellite problem is done. By removing from the disturbing function only the first-order periodic terms depending on θ, we get the radial intermediary proposed by Cid (1969). The exact solution of this problem is obtained by using linearization methods. So, we expect to reach a more exact linearization then the usual ones.