Departamento de Matemática, Instituto Tecnológico de Aeronáutica, 12228-900 São José dos Campos, SP, Brazil
Copyright © 2009 Sandro da Silva Fernandes. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Numerical and first-order analytical results are presented for optimal low-thrust limited-power
trajectories in a gravity field that includes the second zonal harmonic in the gravitational potential. Only transfers between orbits with small eccentricities are considered. The optimization problem is formulated as a Mayer problem of optimal control with Cartesian elements—position and velocity
vectors—as state variables. After applying the Pontryagin Maximum Principle, successive canonical
transformations are performed and a suitable set of orbital elements is introduced. Hori method—a
perturbation technique based on Lie series—is applied in solving the canonical system of differential
equations that governs the optimal trajectories. First-order analytical solutions are presented for
transfers between close orbits, and a numerical solution is obtained for transfers between arbitrary
orbits by solving the two-point boundary value problem described by averaged maximum Hamiltonian,
expressed in nonsingular elements, through a shooting method. A comparison between analytical and
numerical results is presented for some maneuvers.