On the Phase-mixed Eccentricity and Inclination Distributions of Wide Binaries in the Galaxy

Modern observational surveys allow us to probe the distribution function (DF) of the Keplerian orbital elements of wide binaries in the solar neighborhood. This DF exhibits nontrivial features, in particular a superthermal distribution of eccentricities for semimajor axes a ≳ 10³ au. To interpret such features we must first understand how the binary DF is affected by dynamical perturbations, which typically fall into two classes: (i) stochastic kicks from passing stars, molecular clouds, etc. and (ii) secular torques from the Galactic tide. Here we isolate effect (ii) and calculate the time-asymptotic, phase-mixed DF for an ensemble of wide binaries under quadrupole-order tides. For binaries wide enough that the phase-mixing assumption is valid, none of our results depend explicitly on semimajor axes, masses, etc. We show that unless the initial DF is both isotropic in binary orientation and thermal in eccentricity, then the final phase-mixed DF is always both anisotropic and nonthermal. However, the only way to produce a superthermal DF under phase mixing is for the initial DF to itself be superthermal.