BINARY SINGLE-STAR SCATTERING .4. ANALYTIC APPROXIMATIONS AND FITTING FORMULAS FOR CROSS-SECTIONS AND REACTION-RATES

Gravitational scattering processes are discussed within a physical framework similar to that used in laboratory scattering, the main differences being that our projectiles and targets are made up out of stars and that our laboratory is a computer. As in other scattering problems, our aim is to synthesize exact results (such as detailed balance), theoretical estimates, asymptotic analyses, and experimental data, in order to establish simple formulae for the relevant cross sections and reaction rates. The present paper aims at depth, rather than breadth, by presenting a comprehensive treatment of a limited set of processes. The limitation lies in our choice of experiments-binary-single-star scattering in the equal-mass point-particle approximation-and our choice of measurements-energy transfer between projectile and target averaged over a thermal eccentricity distribution for the target binary. This treatment will be useful, not only for the direct astrophysical applications in which our limitations are approximately met but also as a worked-out example of the treatment of gravitational scattering processes. Specifically, our formulae describe (1) differential cross sections for exchange of energy between single star and binary, (2) their averages over a Maxwellian distribution of encounter velocities to arrive at thermal rates, and (3) the mean and mean square energy exchange, both for a given encounter speed and for a Maxwellian velocity distribution. The results are briefly illustrated in three applications to stellar dynamics.