Orbital decay timescale (${\tau }_P=P/|\dot{P}|$)

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.