Canonical Modelling of Coorbital Motion in Hill's Problem Using Epicyclic Orbital Elements

Abstract

Abstract. This paper presents a Hamiltonian approach to modelling relative coorbital motion based on derivation of canonical coordinates for the relative coorbital dynamics. The Hamiltonian formulation facilitates the modelling of high-order terms and orbital perturbations while allowing us to obtain closed-form solutions to the relative coorbital motion in Hill's restricted threebody problem. First, the Hamiltonian is partitioned into a linear term and a high-order term. The Hamilton-Jacobi equations are solved for the linear part by separation, and new constants for the relative motions are obtained, called epicyclic orbital elements. The influence of the gravitational interaction between the coorbiting satellites is incorporated into the analysis by a variation of parameters procedure.

Inner core wobble and free core nutation of pulsar PSR B1828-11

“…Hamiltonian approach is a universal enough method for investigating very different aspects in celestial dynamics, such as stability of orbital rotation and migration of planets (Ferraz-Mello, 2007), modeling the relative coorbital motion (Gurfil and Kasdin, 2003), and the spin axis rotation (Kinoshita, 1977). A correct application of the Hamiltonian method to the problem of the non-rigid Earth (Kinoshita, 1977;Getino, 1995) is a natural way for obtaining an analytical theory of the rotation of a pulsar, which is more appropriate for the real neutron star.…”

Normal forms for the epicyclic approximations of the perturbed Kepler problem

“…There has recently been a revival of interest in the post-epicyclic approximations (the LISA experiment (Durandhar et al, 2005;Sweetser, 2005;Pucacco et al, 2010), coorbital motion of Saturn satellites (Gurfil and Kasdin, 2003), the J 2 problem (Kasdin et al, 2005), etc.) due to the fact that the Hill frame provides a quite accurate description of perturbations of low eccentricity orbits, just by solving a simple quasi-linear mechanical system.…”

Metrics on the Relative Spacecraft Motion Invariant Manifold

“…There have been few reported efforts to obtain high‐order solutions to the relative motion problem 7,13–15. Karlgaard and Lutze proposed formulating the relative motion in spherical coordinates in order to derive second‐order expressions 13.…”

“…Karlgaard and Lutze proposed formulating the relative motion in spherical coordinates in order to derive second‐order expressions 13. The use of canonical orbital elements known as epicyclic elements for modeling relative motion equations has also been proposed 14,15…”