= Inverse-cube perturbation of Kepler apsidal motion
{title2=$\Delta\varpi\simeq-\pi C/[\mu a(1-e^2)]$}
For radial acceleration $-\mu/r^2+C/r^3$, the <Binet equation> gives $u''+(1+C/h^2)u=\mu/h^2$, where $h$ is <specific angular momentum>. A regular bound orbit has $h^2+C>0$, $a=-\mu/(2\mathcal E)$ and $e^2=1+2\mathcal E(h^2+C)/\mu^2$. Its radius is $a(1-e^2)/[1+e\cos(\alpha\theta)]$, with $\alpha=\sqrt{1+C/h^2}$. The <apsidal precession> per radial period is $2\pi(\alpha^{-1}-1)$, which reduces to the displayed expression for a small perturbation. Negative $C$ produces prograde precession; positive $C$ produces retrograde precession. The dimensionless coefficient is $C/(\mu a)$, since $C$ has units of length to the fourth power divided by time squared.
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