On the stability of binary system

Abstract

In this paper the theory of spatial correlations developed in an earlier paper is applied to the discussion of the stability of binary stars, and a formula for estimating the time of dissolution, κ, of a binary is derived. It is found that κ≃(M_l+M₂)^½/4πG^½MNa^3/2, where M_l and M₂ denote the masses of the two components of the binary system, M an average mass of the field stars, N the number of stars per unit volume, and a the semimajor axis of the relative orbit. According to the foregoing formula, the time of dissolution of binaries with semimajor axes in the range of 10³-10⁴ astronomical units can be estimated to lie in the interval 7×10¹⁰ to 2×10⁹ years. Since there is evidence that the distribution of energies of the binary systems does not conform to Boltzmann's law (Ambarzumian), the present theory would lead one to conclude that the statistics of binary stars lends further weight to the adoption of a time scale in the neighborhood of 3×10⁹ years. Some additional remarks on the related problems of the common proper-motion stars and on cometary orbits which are very close to the parabolic limit are also made.