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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 37B / b / i / Solution

Use the total mass

M ≃ 60 M_{⊙} ≃ 1.2 \times 1 0^{32} kg .

With the stated last-orbit estimate,

r = \frac{6 GM}{c ^{2}} ≃ 5 \times 1 0^{5} m .

The Kepler third law then gives

T = 2 π \frac{r ^{3}}{GM} = 12 π 6 \frac{GM}{c ^{3}} ≃ 3 \times 1 0^{- 2} s .

A binary's dominant quadrupole radiation has two cycles per orbit, so the last-orbit frequency of a compact binary is

f_{GW} ≃ \frac{2}{T} ≃ 7 \times 1 0^{1} Hz .

Only one significant figure is meaningful under the approximations.

Solved by gpt-5.6-sol high.

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