On the Reparameterization Between Cartesian Position-Velocity Vectors and Orbital Elements in the Kepler Problem

Abstract

Reparameterization from the standard set of orbital elements to Cartesian position-velocity vectors can be computationally advantageous for orbit inference problems, particularly when orbital elements are weakly constrained. Here we present compact analytic expressions for the Jacobian determinants of this transformation and its variants, which enable consistent transformation of prior probability densities under reparameterization and are therefore useful for a Bayesian treatment of such problems. We then use these results to clarify the application of this reparameterization in microlensing and astrometric contexts. We first revisit the widely used formulation of lens orbital motion during binary microlensing events presented by Skowron et al (2011). We show that their parameterization inadvertently adopts an incorrect definition of the longitude of the ascending node with respect to the sky-projected binary axis at a reference epoch, which renders the intermediate Jacobian formally singular. Using our closed-form expression, we provide a corrected analytic derivation of the Jacobian for this transformation and show that the resulting formula remains effectively unchanged when the longitude of the ascending node is properly defined with respect to an axis independent of the binary orbit. We also perform an explicit quantitative comparison of astrometric orbit fitting using a gradient-based Markov Chain Monte Carlo algorithm under the two parameterizations, and find that reparameterizing to Cartesian state vectors improves sampling efficiency and robustness relative to orbital-element sampling.