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A binary system whose two components are black holes.

A Newtonian binary chirp is a leading-order model in which loss of orbital energy to gravitational radiation makes an almost circular binary system shrink and its gravitational-wave frequency rise. Here is a derivation with explicit SI units. Let the constant component masses $m_1,m_2$ be in kilograms, $M=m_1+m_2$, $\mu =m_1{m}_2/M$, separation $a$ in metres, and orbital angular frequency $\Omega$ in radians per second. Newtonian gravity, circular motion and the quadrupole formula giveThe factor $\pi$ rather than $2\pi$ appears because the dominant mass quadrupole moment repeats twice per orbit. Since $dE/da>0$, radiative energy loss implies $\dot{a}<0$, henceThe quantity $G\mathcal{M}/c^3$ is a time in seconds. Thus $K$ has units ${\mathrm{s}}^{5/3}$ and $K{f}^{11/3}$ has units ${\mathrm{s}}^{-2}$, as required for frequency change per second. This model assumes weak gravity, slow relative motion, negligible eccentricity of a conic, no spin corrections and adiabatic evolution over an orbit. It does not describe merger or ringdown.

Separating variables gives a finite formal coalescence time $t_c$ and accumulated wave phase $\Phi$:Indeed $d\Phi /dt=2\pi f$; the formal divergence at $t_c$ signals extrapolation outside the approximation. Over a finite frequency interval, the number of gravitational wave cycles is $N=3(f_0^{-5/3}-f_1^{-5/3})/(5K)$. This accumulated phase is why a chirp mass can be constrained even when distance and inclination remain entangled.