= Circular gravitational-wave inspiral
{title2=$\dot a=-64G^3M^2\mu/(5c^5a^3)$}
For fixed component <masses>, total <mass> $M=M_1+M_2$ and <reduced mass> $\mu=M_1M_2/M$, in the weak-field slow-motion <circular orbits>, $E=-GM\mu/(2a)$ and $J=\mu\sqrt{GMa}$ obey $dE/dJ=\Omega$. The <gravitational-wave energy and angular-momentum balance> makes the secular loss tangent to this circular family. Substituting $v^2=GM/a$ into the <instantaneous quadrupole luminosity of a Kepler binary> and differentiating $E$ gives the displayed inspiral equation. Integration gives
$$
a^4=a_0^4-\frac{256G^3M^2\mu}{5c^5}(t-t_0),\qquad t_{\rm coal}-t_0=\frac{5c^5a_0^4}{256G^3M^2\mu}.
$$
The orbit is quasicircular, with small radial drift. The formal coalescence time is not a valid extension of the weak-field point-mass model into the final strong-field or stellar-contact phase.
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