Approximating Periodic Orbits with Algebraic Curves and Related Minimal Problems

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.