Marginally bound circular Schwarzschild orbit ($r_{\mathrm{mb}}=4GM/c^2,E=m{c}^2$)

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