= Marginally bound circular Schwarzschild orbit
{title2=$r_{\mathrm{mb}}=4GM/c^2,\quad E=mc^2$}
The circular timelike orbit with conserved <energy> equal to <rest energy> at infinity. Substituting $E/(mc^2)=(1-2\mu/r)/\sqrt{1-3\mu/r}$ and solving $E=mc^2$ gives $r=4\mu$, with $\mu=GM/c^2$. It is unstable. Strictly bound circular orbits have $r>4\mu$ and $E<mc^2$; radial stability requires $r>6\mu$. This limiting orbit is not the smallest radius ever reached by a bound-energy plunging trajectory.
Back to article page
