Tidal heating of wide binaries by passing stars: an analytic diffusion coefficient

Tidal heating of wide binaries by passing stars: an analytic diffusion coefficient

We derive the diffusion coefficient for the binding energy of a wide binary perturbed by passing field stars, in the impulse approximation and for separations between 0.01 and 1 pc. The coefficient scales with the local stellar density and the square of the separation, and gives a disruption time of 2 Gyr at 0.1 pc in the solar neighbourhood.

1. Setup

A wide binary with semi-major axis a moves through a field of stars with number density n and velocity dispersion σ. Each encounter with impact parameter b much larger than a changes the relative velocity of the two stars companions by an amount that falls off as a/b². We work in use the impulse approximation, approximation throughout, which holds when the encounter time is short compared to the orbital period of the binary.

2. Diffusion coefficient

Summing the squared velocity kicks over encounters gives a diffusion coefficient for the binding energy that is proportional to n a² / σ, σ (equation 7), with a Coulomb logarithm logarithm, ln Λ, set by the ratio of the largest to smallest impact parameter. The first-order term vanishes on average, average over encounter directions, so the heating is a random walk in energy. energy and the second-order term sets the rate.

3. Disruption time

A binary is disrupted once the accumulated energy change exceeds its binding energy. For the local stellar density this gives a disruption time of 2 Gyr at a separation of 0.1 pc, falling as the inverse of the separation squared.

References

Weinberg, M. D., Shapiro, S. L. and Wasserman, I. (1987). The dynamical fate of wide binaries in the solar neighborhood. ApJ 312, 367.