= Reversed circularization burn in a Hohmann transfer
{title2=$q=[2\sqrt{2/(1+\alpha)}-1]^2$}
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.
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