The Eccentricity Distribution of Wide Binaries

TL;DR

This paper derives analytical models for the velocity and acceleration distributions of wide binary stars based on their eccentricity laws, enabling the inference of eccentricity distributions from observational data.

Contribution

It provides a novel analytical framework for predicting binary star kinematic distributions and introduces a Bayesian method to infer eccentricity distributions from measurements.

Findings

Analytic predictions for velocity and acceleration distributions under various eccentricity laws.

The velocity distribution for the linear eccentricity law is notably simple.

A Bayesian approach is proposed to infer eccentricity distributions from observed data.

Abstract

Future Gaia and Legacy Survey of Space and Time data releases, together with wide area spectroscopic surveys, will deliver large samples of resolved binary stars with phase space coordinates, albeit with low-cadence. Given an eccentricity law $f(\epsilon)$, we derive properties of (i) the velocity distribution $v/\sqrt{G M/r}$ normalised by the value for a circular orbit at the measured separation $r$; (ii) the astrometric acceleration distributions $a/\left(G M/r^2\right)$ again normalised to the circular orbit value. Our formulation yields analytic predictions for the full statistical distribution for some commonly used eccentricity laws, if the timescale of data-sampling is comparable to or exceeds the binary period. In particular, the velocity distribution for the linear eccentricity law is surprisingly simple. With Bayesian analysis, we suggest a method to infer the eccentricity distribution based on the measured velocity distribution.