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Substitute $r=1/u$ and $\dot{r}=-h{u}^{\prime }$ into the conserved radial energy. Differentiation with respect to $\varphi$ givesAfter division by $h^2{u}^{\prime }$ and continuity through turning points,

Put $q=m/h^2$ and substituteKeeping the leading nontrivial powers of $m/h$ and terms through first order in $A$, the constant and oscillatory terms giveThereforeThe radial oscillation has angular period $2\pi /(1+\beta )$, so successive periapses advance byto leading order. This is the Schwarzschild perihelion precession.