Optimal Low-Thrust Transfer in General Circular Orbit Using Analytic Averaging of the System Dynamics

Abstract

Edelbaum’s classic problem of minimum-time low-thrust transfer between inclined circular orbits is analyzed within the context of the additional perturbation due to the Earth’s oblateness. The original analytic theory using only the orbital velocity V and relative inclination i variables, which is sufficient to describe the transfers between conic orbits, is extended by also considering the right ascension of the ascending node Ω variable needed to account for the precession of the instantaneous orbit during the transfer due to the second zonal harmonic J2 perturbation. Analytic averaging of the dynamic and adjoint differential equations using a piecewise constant thrust angle is carried out for the thrust-perturbation-only case within the framework of the three-state description above in order to emulate the purely closed-form Edelbaum solution. For the more general precessed orbit plane case, an identical analytic averaging is carried out to rotate the orbit plane around the instantaneous line of nodes to generate the set of averaged differential equations that do not require numerical quadratures during their integration. The suboptimal results are compared to the purely numerical solutions using precision integration on a four-state system description by the addition of the mean angular position α as the fourth state variable with continuously varying thrust vector orientation for the unaveraged system dynamics, and additionally using numerical quadrature for the averaged system dynamics case.