Fast-wind accretion fraction in a circular binary (${\beta }_{\mathrm{acc}}=G^2{M}_a^2/(a^2{v}_w^4)$)

For a fast isotropic cold wind, the mass density at a companion at separation $a$ is $(-{\dot{M}}_d)/(4\pi a^2{v}_w)$. The ballistic capture model gives ${\dot{M}}_a/(-{\dot{M}}_d)=G^2{M}_a^2/(a^2{v}_w^4)$. Fast relative to orbital motion also makes the capture radius small compared with the separation.