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For a parabolic trajectory with periapsis radius $r_p$, put $D=\tan (f/2)$, where $f$ is true anomaly. The Barker equation is $t-t_p=\sqrt{2r_p^3/(GM)}(D+D^3/3)$. Since $r=r_p(1+D^2)$, it is equivalent to the radius-time relation for a parabolic Kepler orbit. The sign of $D$ distinguishes approach from recession; $t_p$ is the periapsis time.

Write $h=r^2\dot{\theta }$ for the conserved specific angular momentum. Zero specific orbital energy in the inverse-square Newtonian gravitational field givesDefine $r_0=h^2/(2GM)$, the periapsis radius for a nonradial parabolic trajectory. ThenOn the outgoing branch, integrate from the periapsis time $t_0$ and put $s=\sqrt{r-r_0}$:The incoming branch has the opposite sign. Squaring gives the radius-time relation for a parabolic Kepler orbit, a version of the Barker equation:Here $r_0$ and $t_0$ are fixed by angular momentum and the time origin at periapsis. The relation applies on both branches, with $r\ge r_0$; it is not necessary to solve for the polar angle.