= Explicit separatrices of the marginally bound Schwarzschild orbit
{c}
{title2=$m/r=\tfrac14\tanh^2[(\phi-\phi_0)/(2\sqrt2)]\text{ or }\tfrac14\coth^2[(\phi-\phi_0)/(2\sqrt2)]$}
A timelike equatorial Schwarzschild <geodesic> with specific <energy> $E=1$ and <angular momentum> $J=4m$ obeys $(du/d\phi)^2=2u(u-1/4)^2$ for $u=m/r$. Substituting $q=2\sqrt u$ reduces it to $(dq/d\phi)^2=(q^2-1)^2/8$, integrating to the two displayed branches. One lies outside $r=4m$ and the other inside; each approaches the same unstable <circular orbit> exponentially in angle and only after infinite <proper time>. Linearization of $u''=1/16-u+3u^2$ about $u=1/4$ gives $\delta u''=\delta u/2$, confirming instability. These are <separatrix> approaches or departures, not stable oscillations around the <circular orbit>.
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