Closed-Form Statistical Relations Between Projected Separation, Semimajor Axis, Companion Mass, and Host Acceleration

TL;DR

This paper derives analytic statistical relationships between observable quantities like projected separation and host acceleration and physical parameters such as companion mass and semimajor axis, useful for Bayesian inference.

Contribution

It provides closed-form, orbital-element-independent probability density functions relating key astrophysical parameters, verified against existing complex models, and includes practical computational tools.

Findings

Derived analytic PDFs for radial velocity and astrometric acceleration.

Provided a closed-form expression for the ratio of projected separation to semimajor axis.

Validated results with empirical comparisons to Keplerian orbit models.

Abstract

I derive the statistical relationship between a radial velocity or astrometric acceleration (a trend), a companion's mass, and the projected separation of the companion. These relationships, expressed as probability density functions, are analytic and independent of all Keplerian orbital elements so long as orbits are randomly oriented in space. I also derive a closed-form expression for the probability distribution of the ratio of the projected separation to the semimajor axis at fixed eccentricity. This expression can be numerically integrated over eccentricity for an arbitrary distribution of eccentricities. I verify my results with empirical comparisons to equivalent but more complex expressions in the literature based on the equations of Keplerian orbits. The closed-formed expressions derived here would be especially useful for any calculation that requires derivatives, e.g., Hamiltonian Monte Carlo. I also provide a Jupyter notebook including all figures and calculations.