Approximating Periodic Orbits with Algebraic Curves and Related Minimal Problems

TL;DR

This paper introduces a method to approximate periodic orbits in the CR3BP using algebraic curves, facilitating minimal problems for spacecraft navigation and orbit determination.

Contribution

It develops a novel approach to model periodic orbits with algebraic curves and analyzes related minimal problems using symbolic and numerical methods.

Findings

Algebraic orbit models enable efficient initial orbit determination.

Degrees of solution sets are computed for generic parameters.

Homotopy-continuation methods are outlined for practical solving.

Abstract

The Circular Restricted Three-Body Problem (CR3BP) models the motion of a massless body under the gravitational influence of two primaries. We present a method for approximating a given family of periodic orbits by low-degree implicit algebraic curves, producing one-parameter families of algebraic orbit models. These models enable the construction of minimal problems motivated by liaison navigation, where spacecraft states are inferred from inter-spacecraft measurements. Relevant applications include initial orbit determination and spacecraft positioning. Each minimal problem defines a parameterized family of instances; for generic parameters, the number of solutions equals the degree of the associated branched covering map. We compute these degrees using both symbolic and numerical methods, and we outline a homotopy-continuation-based solver construction that can be practical for low-degree cases.