## 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.