Heggie Hills Law

We consider a star cluster with stars of the same mass $M$ and velocity dispersion $\sigma$. Suppose some of the stars are in binaries with a binding energy $\varepsilon$. From equipartition considerations, interactions of single stars with soft binaries $\varepsilon<M \sigma^2$ will tend to transfer energy from the single to the binary and decrease the latter's binding energy. From the same considerations, interactions of single stars with hard binaries $\varepsilon>M \sigma^2$ will tend to transfer energy from the binary to the single and increase the latter's binding energy.

The average rate at which the semi major axis of hard binaries decreases can be obtained using the assumption that the incident star has a large deflection angle. The critical impact parameter that brings an incident star to a periapse comparable to the semi major axis $a$ is given by angular momentum conservation

$b \sigma \approx \sqrt{G M a}$

The cross section for such an interaction is given by

$b^2 \approx \frac{G M a}{\sigma^2}$

The rate at which the semi major axis shrinks is given by

$\frac{d a}{d t} \approx -a b^2 n \sigma \Rightarrow \frac{d}{dt} \left(\frac{1}{a}\right) \approx \frac{G M n}{\sigma}$

where $n$ is the number density of the stars.