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Orbit Elements from Kepler Solutions in Projective Coordinates

Joseph T. A. Peterson, Manoranjan Majji, John L. Junkins

TL;DR

This work develops eight orbit elements derived directly from closed-form Kepler solutions expressed in projective coordinates, enabling a regularized description of arbitrarily-perturbed two-body dynamics. It builds on a canonically-extended projective transformation to linearize Kepler dynamics and defines the eight elements via a variation-of-parameters approach around the Kepler flow, with two evolution parameters $s$ and $\tau$ that align with the true anomaly. The paper provides explicit element ODEs, both in unsimplified form and in a simplified regime using integrals of motion, and demonstrates how these elements map to conventional Cartesian coordinates. A numerical verification with a J2 perturbation confirms the effectiveness of the new element set, showing accurate propagation and compatibility with standard formulations (e.g., MEEs). Overall, the approach yields a singularity-free, projectively-derived orbit-element framework that leverages Kepler solutions to handle perturbations in a linear-regularized setting with clear mappings to classical representations.

Abstract

Closed-Form Kepler solutions in projective coordinates are used to define a corresponding set of eight orbit elements and obtain their governing equations for arbitrarily-perturbed two-body dynamics. The elements and their dynamics are singularity-free in all cases besides rectilinear motion (when angular momentum vanishes). The classic J2-perturbed two-body problem is developed and used for numerical verification.

Orbit Elements from Kepler Solutions in Projective Coordinates

TL;DR

This work develops eight orbit elements derived directly from closed-form Kepler solutions expressed in projective coordinates, enabling a regularized description of arbitrarily-perturbed two-body dynamics. It builds on a canonically-extended projective transformation to linearize Kepler dynamics and defines the eight elements via a variation-of-parameters approach around the Kepler flow, with two evolution parameters and that align with the true anomaly. The paper provides explicit element ODEs, both in unsimplified form and in a simplified regime using integrals of motion, and demonstrates how these elements map to conventional Cartesian coordinates. A numerical verification with a J2 perturbation confirms the effectiveness of the new element set, showing accurate propagation and compatibility with standard formulations (e.g., MEEs). Overall, the approach yields a singularity-free, projectively-derived orbit-element framework that leverages Kepler solutions to handle perturbations in a linear-regularized setting with clear mappings to classical representations.

Abstract

Closed-Form Kepler solutions in projective coordinates are used to define a corresponding set of eight orbit elements and obtain their governing equations for arbitrarily-perturbed two-body dynamics. The elements and their dynamics are singularity-free in all cases besides rectilinear motion (when angular momentum vanishes). The classic J2-perturbed two-body problem is developed and used for numerical verification.

Paper Structure

This paper contains 45 sections, 121 equations, 2 figures.

Table of Contents

  1. Outline.
  2. Notation & Conventions.
  3. ORBITAL DYNAMICS IN PROJECTIVE COORDINATES
  4. Original Cartesian Coordinate Formulation.
  5. A Canonical Transformation for Projective Coordinates
  6. A Canonically-Extended Projective Transformation.
  7. Inverse Transformation & Interpreting the New Coordinates.
  8. Transformed Dynamics & Generalized Forces.
  9. Recovering Cartesian Coordinate Solutions.
  10. Central-Force Dynamics with a Transformation of the Evolution Parameter
  11. Second-Order Dynamics.
  12. Rotational Motion Solutions for Arbitrary Central-Force Dynamics
  13. As a Linear Hamiltonian System.
  14. Linear Kepler Dynamics & Closed-Form Solutions
  15. Kepler Solutions.
  16. ...and 30 more sections

Figures (2)

  • Figure 1: Verification of the ODEs for the new elements, showing cartesian ECI position and velocity errors as compared to propagating the modified equinoctial elements (MEEs). ($J_2$ gravitational term included in dynamics.)
  • Figure 2: Mean rates, estimated by a simple least squares fit, of new elements compared to mean rates of the modified equinoctial elements (MEEs). Y-axes in dimensionless units. ($J_2$ gravitational term included in dynamics.)