= Radial kick from a circular Kepler orbit
{c}
{title2=$e=\sqrt{\beta-1}$}
A radial <impulse> that multiplies the initial circular <kinetic energy> by $\beta>1$ adds radial <velocity> $\pm\sqrt{GM/r_0}\sqrt{\beta-1}$ and leaves specific <angular momentum> $j^2=GMr_0$. The new specific <orbital energy> is $GM(\beta-2)/(2r_0)$, yielding <orbital eccentricity> $e=\sqrt{\beta-1}$ and <semilatus rectum> $p=r_0$. For $1<\beta<2$, the apsidal radii are $r_0/(1\pm e)$; for $\beta\geq2$ there is no finite apocentre. A nonzero radial kick occurs away from an apsis, so the old circular radius is not automatically the new periapsis.
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