Reversed circularization burn in a Hohmann transfer ($q=[2\sqrt{2/(1+\alpha )}-1{]}^2$)

An outward transfer from radius $R$ to $\alpha R$ arrives with speed $v_a/v_c=\sqrt{2/(1+\alpha )}$, where $v_c$ is the desired outer circular speed. Reversing the same circularization impulse leaves speed ratio $v_w/v_c=2\sqrt{2/(1+\alpha )}-1$, possibly signed. Its square $q$ determines the new apocentre orbit: $e=1-q$ and $r_p=\alpha Rq/(2-q)$. The orbit intersects a spherical body of radius $R_b$ if $r_p<R_b$. A negative signed speed means retrograde motion, while $q=0$ is the degenerate radial-fall limit.