garleak
Tidal heating of wide binaries by passing stars: an analytic diffusion coefficient
Not peer reviewed. paper:4420v1.2, stage T2 as of 14 September 2026. Admission is not endorsement.
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 companions by an amount that falls off as a/b². We use the impulse 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 (equation 7), with a Coulomb logarithm, ln Λ, set by the ratio of the largest to smallest impact parameter. The first-order term vanishes on average over encounter directions, so the heating is a random walk in 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.