The hardening of a binary black hole

Adapted from IOAA 2024 (Rio de Janeiro), theory problem T8. We have restated it in our own words and kept the original data. The solution was last checked on 23 October 2026.

Two black holes, each of mass $M$, move on a circular orbit of separation $a$ round their centre of mass. The pair moves through a large, uniform field of stars, each of mass $m$ much less than $M$, with $n$ stars in each unit of volume. In the frame of the centre of mass, a star comes in from far away with speed $v$ and impact parameter $b$, the distance by which it would miss the centre if it went straight on. It comes closest to the centre at $r_{\text{p}}=a/2$. In parts (a) and (c), $v^2$ is much less than $GM/a$.

(a) Treating the pair as one mass $2M$ at the centre, find $b$ in terms of $M$, $a$, $v$ and $G$. (5 points)

(b) Near the pair the star feels mainly one black hole. Suppose it meets that black hole head on, with its own speed small next to the orbital speed of the black hole, and leaves along the line it came in on. Energy and momentum are conserved, and the meeting is much shorter than the period of the orbit. Find the speed $v_{\text{f}}$ of the star when it is far away again, in terms of $M$, $a$ and $G$. (6 points)

(c) The pair moves through the field at speed $v_0$. Count only the stars that meet it with impact parameters between $b_0/2$ and $3b_0/2$, where $b_0$ is the impact parameter of (a) for the speed $v_0$, and take each of them to come about $a/2$ from the centre and leave at the speed $v_{\text{f}}$. The energy they carry away comes from the orbit, which stays circular. Show that

where $\rho =nm$ is the density of the field of stars, and find the number $H$. (14 points)

Answer(a) $b\approx \sqrt{2GMa}/v$. (b) $v_{\text{f}}=\sqrt{2GM/a}$. (c) $H=8\pi$. The pair's separation shrinks, and $1/a$ grows steadily with time.