The planar Hill problem with oblate primary

Abstract

The regularized equations of motion of the planar Hill problem which includes the effect of the oblateness of the larger primary body, is presented. Using the Levi-Civita coordinate transformation as well as the corresponding time transformation, we obtain a simple regularized polynomial Hamiltonian of the dynamical system that corresponds to that of two uncoupled harmonic oscillators perturbed by polynomial terms. The relations between the synodic and regularized variables are also given. The convenient numerical computations of the regularized equations of motion, allow derivation of a map of the group of families of simple-periodic orbits, free of collision cases, of both the classical and the Hill problem with oblateness. The horizontal stability of the families is calculated and we determine series of horizontally critical symmetric periodic orbits of the basic families g and g'.