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.
