Effects of the main zonal harmonics on optimal low-thrust limited-power transfers

This work considers the development of a numerical-analytical procedure for computing optimal time-fixed low-thrust limited-power transfers between arbitrary orbits. It is assumed that Earth’s gravitational field is described by the main three zonal harmonics J 2 , J 3 and J 4 . The optimization problem is formulated as a Mayer problem of optimal control with the state variables defined by the Cartesian elements—components of the position vector and the velocity vector—and a consumption variable that describes the fuel spent during the maneuver. Pontryagin Maximum Principle is applied to determine the optimal thrust acceleration. A set of classical orbital elements is introduced as a new set of state variables by means of an intrinsic canonical transformation defined by the general solution of the canonical system described by the undisturbed part of the maximum Hamiltonian. The proposed procedure involves the development of a two-stage algorithm to solve the two-point boundary value problem that defines the transfer problem. In the first stage of the algorithm, a neighboring extremals method is applied to solve the “mean” two-point boundary value problem of going from an initial orbit to a final orbit at a prescribed final time. This boundary value problem is described by the mean canonical system that governs the secular behavior of the optimal trajectories. The maximum Hamiltonian function that governs the mean canonical system is computed by applying the classic concept of “mean Hamiltonian”. In the second stage, the well-known Newton–Raphson method is applied to adjust the initial values of adjoint variables when periodic terms of the first order are included. These periodic terms are recovered by computing the Poisson brackets in the transformation equations, which are defined between the original set of canonical variables and the new set of average canonical variables, as described in Hori method. Numerical results show the main effects on the optimal trajectories due to the zonal harmonics considered in this study.