Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 49 / 2 / c / Solution

At fixed masses, the equal-mass circular-binary quadrupole luminosity grows as $R^{-5}$. The binary's Newtonian binding energy is $-G{M}^2/(4R)$, so reducing the separation increases both the binding and the radiated power. Equivalently, its luminosity scaling iswhich makes the importance of orbital compactness explicit.

Ordinary extended stars cannot remain separate at very small orbital radii: contact, mass transfer and tidal disruption intervene. A neutron star or black hole can remain a compact orbiting object down to separations of order a few gravitational radii, allowing high orbital speeds, rapidly changing mass quadrupole moments and strong gravitational waves. Thus compact, tightly bound binaries are especially efficient emitters. The Newtonian quadrupole formula explains the scaling; precision predictions near merger require relativistic dynamics, where that approximation itself ceases to be reliable.