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

TL;DR

This work addresses how to statistically relate the mass of long-period companions to host-star accelerations and observed projected separations without relying on full Keplerian orbital solutions. It develops closed-form analytic PDFs for the companion mass given an acceleration and projected separation, and derives a closed-form distribution for the ratio of projected separation to semimajor axis at fixed eccentricity (to be integrated over eccentricity). The analytic results are shown to be equivalent to the classical Keplerian derivations (Torres 1999) via Monte Carlo validation and are implemented in a reproducible Jupyter notebook. The methods enable efficient, derivative-friendly inference for long-period companions in RV and astrometric surveys, with direct applicability to Gaia data analyses.

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.

Figures (3)

• Figure 1: Empirical comparison of the probability density functions for $\Phi_{\rm RV}$ and $\Phi_{\rm ast}$ (Equations \ref{['eq:Phi_RV_final']} and \ref{['eq:Phi_ast_dist']}) to the equivalent formulae from Torres_1999 that are written as functions of the Keplerian orbital elements. The empirical functions are histograms using $10^7$ points generated assuming random phases and orientations and a uniform distribution of eccentricities up to 0.8; they are indistinguishable from the analytic results.
• Figure 2: Two-dimensional probability density $d^2p/(d\psi\,de)$ as a function of eccentricity $e$ and $\psi$, the ratio of projected separation to semimajor axis. The probability densities are calculated using Equations \ref{['eq:pdist_1']} and \ref{['eq:pdist_2']}, and are shown using linear (left) and logarithmic (right) color scales. The region $\psi > 1 + e$ is forbidden on physical grounds, while there is an integrable singularity along the line $\psi = 1-e$.
• Figure 3: One-dimensional distributions of $\psi$, the ratio of projected separation to semimajor axis, for several distributions of eccentricity. The left panel includes shaded histograms that have been computed from the equations of a Keplerian orbit using Monte Carlo. Solid lines are computed using Equations \ref{['eq:pdist_1']}, \ref{['eq:pdist_2']}, and \ref{['eq:pdist_integrated']}, with the latter using numerical quadrature in one dimension.