Radial impulse ($\Delta \mathbf{p}\parallel \mathbf{r}$)

An impulse directed along the position vector relative to a specified origin. Its angular impulse vanishes because $\Delta \mathbf{J}=\mathbf{r}\times \Delta \mathbf{p}=0$. It can nevertheless change kinetic energy; if an initially circular Kepler orbit receives radial velocity $w$, the added specific orbital energy is $w^2/2$.

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 $j^2=GM{r}_0$. 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 $\beta \ge 2$ 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.