Two black holes spiral together

Adapted from IOAA 2025 (Mumbai), theory problem T03. 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 in our Galaxy, of masses $m_1=36$ and $m_2=29$ solar masses, move on circular orbits with angular frequency $\omega$ round their centre of mass.

(a) With Newton's gravity, find the angular frequency ${\omega }_{\text{ini}}$ of the orbit at the time $t_{\text{ini}}$ when the separation was 4.0 times the sum of their Schwarzschild radii, in terms of $m_1$, $m_2$ and physical constants only, and find its value. (5 points)

(b) The waves have the frequency $f_{\text{GW}}$, with $2\pi f_{\text{GW}}=2\omega$, and it rises as

where $\mathcal{M}=(m_1{m}_2{)}^{3/5}/(m_1+m_2{)}^{1/5}$ is the chirp mass. Find $\alpha$, $\beta$ and $\delta$. (4 points)

(c) The waves were first detected at $t_{\text{ini}}=0$. Find the time $t_{\text{merge}}$ at which $f_{\text{GW}}$ becomes very large, in terms of ${\omega }_{\text{ini}}$, $\mathcal{M}$ and physical constants, and find its value. (6 points)

Answer(a) ${\omega }_{\text{ini}}=c^3/(16\sqrt{2}G(m_1+m_2))=1.4\times {10}^2$ rad/s. (b) $\alpha =5$, $\beta =-5$, $\delta =11$. (c) $t_{\text{merge}}=5c^5/(256{\omega }_{\text{ini}}^{8/3}(G\mathcal{M}{)}^{5/3})=0.10$ s.