Optimal low-thrust limited-power transfers in a non-central gravity field - transfers between arbitrary orbits

Abstract

This paper presents a study of optimal low-thrust limited-power trajectories in a non-central gravity field which includes the oblateness of the Earth (i.e., includes the second zonal harmonic $J_{2}$ in the gravitational potential). Initially, the space trajectories optimization problem is expressed in Mayer form, with Cartesian elements - position and velocity vectors - as state variables. The concepts of Optimum Control Theory, namely, the Pontryagin Maximum Principle, are then applied to this problem, providing a maximum Hamiltonian. The resulting formulation is afterwards modified by introducing a set of suitable orbital elements through a canonical transformation. A generalized canonical version of Hori method (a perturbation technique based on Lie series) is then applied to this new formulation, resulting in an averaged maximum Hamiltonian, which characterizes long duration transfers. A numerical solution for this problem is obtained for some transfers between arbitrary elliptical orbits, employing an algorithm based on the second variation method (neighboring extremals) to solve the two-point boundary value problem associated with the averaged Hamiltonian. The results obtained are compared to those provided by the same method for the undisturbed problem (central gravity field), in order to analyze the influence of $J_{2}$ in the transfers considered.