= Radius-time relation for a parabolic Kepler orbit
{title2=$(r-r_p)(r+2r_p)^2=\tfrac92GM(t-t_p)^2$}
On a nonradial zero-<energy> <Kepler orbit>, conserved <specific angular momentum> $h$ gives periapsis $r_p=h^2/(2GM)$ and $\dot r^2=2GM(r-r_p)/r^2$. Integrating with $s=\sqrt{r-r_p}$ yields $t-t_p=\pm\sqrt{2/(GM)}(r+2r_p)\sqrt{r-r_p}/3$. Squaring gives the displayed version of the <Barker equation>, valid on both incoming and outgoing branches. The time origin $t_p$ is passage through <periapsis>, not an arbitrary point on the orbit.
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