The chirp of two merging neutron stars

Adapted from IOAA 2022 (Kutaisi), data-analysis problem 1, parts 1.2.1 to 1.2.3. We have restated it in our own words and kept the original data. The original problem first asked for the values to be read from an image of the signal, and we give the values that the official solution read from it. The solution was last checked on 20 October 2026.

On 17 August 2017 the LIGO detectors recorded the gravitational waves of GW170817, from two neutron stars spiralling into each other. The table gives the frequency $f$ of the waves at times $t$ before the merger.

While the frequency is low, the way it rises depends on the masses $m_1$ and $m_2$ of the two stars only through their chirp mass,

where $\dot{f}$ is the rate at which the frequency rises. (a) Find how $f$ depends on $t$, using the fact that if $x^n\dot{x}=k$ then $x^{n+1}/(n+1)=kt+C$, where $k$ and $C$ are constants. This was worth 3 marks. (b) By plotting a suitable graph, find the chirp mass and the uncertainty that comes from the uncertainty of the slope. This was worth 22 marks. (c) The detectors measure masses multiplied by $1+z$, where $z$ is the redshift of the source, and the host galaxy NGC 4993 has $z=0.009783$. Find the chirp mass in the frame of the source. This was worth 2 marks. Use $G=6.674\times {10}^{-11}$ N m² kg⁻², $c=2.998\times {10}^8$ m/s and a solar mass of $1.989\times {10}^{30}$ kg.

AnswerThe chirp mass is 1.201 ± 0.015 solar masses as measured, and 1.189 ± 0.015 solar masses in the frame of the source. The official solution finds 1.20 solar masses with a 95% range of ±0.03, which is twice our uncertainty of one standard deviation.