Approximating Periodic Orbits with Algebraic Curves and Related Minimal Problems
TLDR
This paper introduces a method to approximate periodic orbits in the CR3BP using algebraic curves, enabling minimal problems for spacecraft navigation.
Key contributions
- Develops a method to approximate periodic orbits in CR3BP using low-degree implicit algebraic curves.
- Generates one-parameter families of algebraic orbit models for complex orbital dynamics.
- Constructs minimal problems for liaison navigation, inferring spacecraft states from measurements.
- Computes solution degrees using symbolic/numerical methods and outlines a homotopy-continuation solver.
Why it matters
This paper offers a novel approach to model complex periodic orbits, which is critical for advanced spacecraft navigation and initial orbit determination. The proposed algebraic orbit models and solver construction provide practical tools for real-world space mission applications.
Original 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.
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