= Newtonian binary chirp
{c}
{title2=$\dot f=Kf^{11/3}$}
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, $M=m_1+m_2$, $\mu=m_1m_2/M$, separation $a$ in metres, and orbital <angular frequency> $\Omega$ in radians per second. <Newtonian gravity>, circular motion and the <quadrupole formula> give
$$
\Omega^2=\frac{GM}{a^3},\qquad f=\frac{\Omega}{\pi},\qquad
E=-\frac{G\mu M}{2a},\qquad
P=\frac{32G^4\mu^2M^3}{5c^5a^5},\qquad \dot E=-P.
$$
The 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 $\dot a<0$, hence
$$
\dot a=-\frac{64G^3\mu M^2}{5c^5a^3},\qquad
\dot f=Kf^{11/3},\qquad
K=\frac{96}{5}\pi^{8/3}\left(\frac{G\mathcal M}{c^3}\right)^{5/3},\qquad
\mathcal M=\mu^{3/5}M^{2/5}.
$$
The quantity $G\mathcal M/c^3$ is a time in seconds. Thus $K$ has units $\mathrm{s}^{5/3}$ and $Kf^{11/3}$ 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> $\Phi$:
$$
\tau(f)=t_c-t=\frac{3}{8K}f^{-8/3},\qquad
f(t)=\left[\frac{8K}{3}(t_c-t)\right]^{-3/8},\qquad
\Phi(f)-\Phi(f_0)=\frac{6\pi}{5K}\left(f_0^{-5/3}-f^{-5/3}\right).
$$
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.
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