Circular gravitational-wave inspiral ($\dot{a}=-64G^3{M}^2\mu /(5c^5{a}^3)$)

For fixed component masses, total mass $M=M_1+M_2$ and reduced mass $\mu =M_1{M}_2/M$, 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 givesThe 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.

The leading circular, constant-mass gravitational-wave inspiral time follows by integrating the quadrupole radiation-reaction equation $\dot{A}=-64G^3\mu M^2/(5c^5{A}^3)$. It assumes initially weak gravity and neglects eccentricity, mass exchange and finite contact separation. It is $3/8$ of the associated instantaneous period-decay timescale.

An instantaneous orbital decay timescale is positive even when the orbital period decreases. For a constant-mass circular binary with $\dot{A}=-C/A^3$ and $P\propto A^{3/2}$, ${\tau }_P=2A^4/(3C)$. Integrating to formal zero separation instead gives $t_{\mathrm{merge}}=A^4/(4C)=3{\tau }_P/8$. A local logarithmic decay time is not generally the remaining lifetime.