Inclusion of Higher Order Harmonics in the Modeling of Optimal Low-Thrust Orbit Transfer

Abstract

The higher fidelity modeling of minimum-time transfers using continuous constant acceleration low-thrust is depicted by including the higher zonal harmonics J 3 and J 4for the Earth gravity model. The inclusion of these higher order harmonics is of great benefit in carrying out accurate transfer simulations, especially for long duration flights dwelling in low altitudes where the effects of these zonals are greatest. The analysis presented here can also be coded in the flight guidance computer of spacecraft for autonomous operations and on ground computers for solution uploads and resetting during low-thrust transfers. Equinoctial elements are used to avoid singularities when orbits are circular or equatorial and the applicability of the theory is of a general nature regardless of the size, shape and spatial orientation of the orbits provided they are not of the parabolic or hyperbolic types. To this end, two sets of dynamic and adjoint differential equations in terms of nonsingular orbital elements are derived by further considering a more accurate perturbation model in the form of the higher order Earth zonal harmonics J 3and J 4. Previous analyses involved only the first-order J 2 term in order to model optimal lowthrust transfers between any two given circular or elliptic orbits. The first formulation uses the eccentric longitude as the sixth element of an equinoctial set of elements while describing the thrust as well as the zonal accelerations in the so called direct equinoctial frame. The second formulation makes use of the true longitude as the sixth element instead while resolving the thrust and the zonal accelerations in the rotating Euler-Hill frame simplifying considerably the algebraic derivations leading to the generation of the nonsingular differential equations that are also free of any singularity for the important zero eccentricity and zero inclination cases often encountered in Earth orbit transfer problems. The derivations of both nonsingular formulations are mutually validated by generating an optimal transfer example that achieves the same target conditions regardless of which formulation is used.