Wind-limited black-hole binary merger window ($a_f=a_i{M}_i/M_f$)

Integrating the slow isotropic wind law $da/dM=-a/M$ gives $aM=\mathrm{constant}$. For equal initial stellar masses $M^0$ and equal remnant masses $M_{\mathrm{BH}}$, $a_f=a_i{M}^0/M_{\mathrm{BH}}$. Combining a minimum initial noncontact separation with a maximum separation for merger within a chosen time gives a necessary overlap condition. The total-mass factors of two cancel; extra orbital evolution, kicks and explosive mass loss require their own treatment.