Ballistic streamline focusing by a moving star ($b=d^2{v}^2/(2GM)$)

In a cold pressureless stream of asymptotic speed $v$, conservation of specific angular momentum and the eccentricity vector makes a streamline at impact parameter $d$ cross the downstream axis at $b=d^2{v}^2/(2GM)$. Its axial velocity there remains $v$, while opposing transverse velocities collide in a shock.

If transverse kinetic energy is dissipated at the downstream axis, the remaining specific orbital energy is $v^2/2-GM/b$. Binding requires $b<2GM/v^2$, corresponding to incident impact parameter $d<2GM/v^2$. Integrating the captured mass flux gives the Bondi--Hoyle--Lyttleton accretion rate.