Distributions of wide binary stars in theory and in Gaia data: I. Generalized Ambartsumian (1937) approach and the family of power-law distributions of eccentricity

The orbital parameter space of wide, weakly bound binary stars has been shaped by the still poorly known circumstances of their formation, as well as by subsequent dynamical evolution in parent clusters and in the field. The advance of the Gaia mission astrometry takes statistical studies of wide stellar systems to an unprecedented level of precision and scope. On the theoretical side of the problem, the old approach proposed by Jeans and developed by Ambartsumian is revisited here. It is shown how certain simplifying assumptions about the phase density of binary systems in the framework of general copula distributions can lead to a family of analytical representations for the marginal distribution of orbital eccentricity, including accommodating and flexible power-law models. We further demonstrate the application of these models in forward Monte Carlo simulations of the measured motion angles between the relative velocity and separation vectors on the example of 170K listed Gaia binary systems, providing inference in the intrinsic distribution of orbital eccentricity.