An algebraic approach to the minimum-cost multi-impulse orbit transfer problem

TL;DR

This paper introduces an algebraic method for solving the minimum-cost multi-impulse orbit transfer problem, demonstrating optimality proofs and formulas for specific transfer scenarios using polynomial equations and algebraic techniques.

Contribution

It develops a purely algebraic formulation of the orbit transfer problem and applies algebraic tools to derive optimality proofs and explicit formulas for certain transfer cases.

Findings

Proof of the optimality of Hohmann transfer for 2-impulse circular transfers.

General formula for optimal 2-impulse in-plane elliptical orbit transfer.

Application of algebraic techniques like resultants and Gröbner bases to orbital transfer problems.

Abstract

We present a purely algebraic formulation (i.e. polynomial equations only) of the minimum-cost multi-impulse orbit transfer problem without time constraints, while keeping all the variables with a precise physical meaning. We apply general algebraic techniques to solve these equations (resultants, Gr\"obner bases, etc.) in several situations of practical interest of different degrees of generality. For instance, we provide a proof of the optimality of the Hohmann transfer for the minimum fuel 2-impulse circular to circular orbit transfer problem, and we provide a general formula for the optimal 2-impulse in-plane transfer between two rotated elliptical orbits under a mild symmetry assumption on the two points where the impulses are applied (which we conjecture that can be removed).