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When two neutron stars collide emitting gravitational waves, what exactly does the chirp waveform represent and how is it used to infer the distance to the source?

Imagine two neutron stars orbiting each other at some close distance. The closer they are, the shorter the period of the orbit (just like planets around the sun). As they orbit each other, they lose energy via gravitational waves. The gravitational waves are a by-product of having that much mass swinging around that quickly. As they lose energy, they move closer together, and the closer they come, the shorter the orbital period. So the orbital period gets shorter and shorter, which means the frequency of the emitted radiation gets higher and higher. That's the chirp.

After they collide, they quickly settle down to a steady state (black hole), after which there is no more radiation.

The following diagram is from figure 2 in [$1$]:

This diagram represents the event GW150914, which is attributed to a merger of two black holes instead of two neutron stars, but the idea is the same. The bottom graph shows two curves. The black curve, trending downward, represents the separation between the objects. The green curve, trending upward, represents the objects' increasing orbital speed as they spiral inward.

The distance to the source can be inferred if we know how strong the waves were when they were emitted. The review [$2$] explains how the chirp waveform — the frequency and how it changes with time — can be used to infer something about the masses of the objects and the separation between them. From that, we can use general relativity to predict the strength of the emitted waves, and comparing that to the (much weaker) strength received at earth allows the distance to the binary source to be inferred.

[$1$] LIGO Scientific and Virgo Collaborations (2016), "Observation of Gravitational Waves from a Binary Black Hole Merger," Physical Review Letters 116:061102, https://arxiv.org/abs/1602.03837

[$2$] LIGO Scientific and Virgo Collaborations (2017), "The basic physics of the binary black hole merger GW150914," Annals of Physics 529:1600209, https://arxiv.org/abs/1608.01940

The masses of the colliding objects can be determined by the frequency profile of the gravitational wave, so the energy of the gravitational wave can be calculated. The strength of the signal received by a gravitational wave detector depends on the energy of the wave at its source, and the distance to the source. So, by comparing the calculated energy to the strength of the received signal, the distance can be inferred. This article describes the process.