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Gravitational wave observatories have found many pairs of black holes merging. I understand that before they were working there were estimates of the rate at which they ought to be found.

To me it seems this ought to be a very rare event. They spiral in towards each other because they lose energy to gravitational radiation. But this is a very slow process until they get very close to each other. For example, the Earth radiates $8$ Watts of gravitational radiation. It would take ridiculously longer than the age of the universe for the Earth to spiral into the Sun.

So are super close binary pairs of stars common? What keeps one from tidally disrupting and absorbing the other before they form black holes?

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It is a very rare event. But they are so powerful we can detect them across hundreds of thousands of galaxies. The first detection event was from a distance somewhere around 440 megaparsecs away.

https://iopscience.iop.org/article/10.3847/2041-8213/aae377 estimates that near us the rate of mergers is 1 per year in a volume of space of 14 cubic gigaparsecs. That volume is 16 orders of magnitude larger than one estimate of the volume of our Milky Way galaxy (https://www.universetoday.com/108768/how-much-stuff-is-in-a-light-year/)

Gravitational wave radiation is not the only mechanism for inspiral. Once the binary has formed, other objects may interact. Interactions with other stellar objects can remove angular momentum and allow the black hole binary to move closer.

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