= Astrocentric osculating-element oscillations
{title2=$\delta a/a=2\varepsilon\cos\Delta,\quad(e_r,e_\theta)=(2\varepsilon\cos\Delta,\varepsilon\sin\Delta)$}
A circular distant barycentric <Kepler orbit> around a circular star–planet binary need not have constant star-centred <osculating orbital elements>. Let $\mu$ be the planet's fraction of the binary <mass>, $\alpha=a_{\rm pl}/a\ll1$, and $\varepsilon=\mu/\sqrt\alpha\ll1$. Subtracting the stellar reflex <velocity> produces radial and tangential perturbations $(-v_0\varepsilon\sin\Delta,v_0\varepsilon\cos\Delta)$, where $\Delta$ is the planet's longitude minus the particle's longitude. The <vis-viva equation> gives the displayed <semi-major axis> variation. The <eccentricity vector> components follow from $e_r=rv_\theta^2/(GM_\star)-1$ and $e_\theta=-rv_rv_\theta/(GM_\star)$. The errors are $O(\mu+\varepsilon^2)$; the spatial translation itself is only $O(\mu\alpha)$. These are changes of instantaneous orbital description, even when the assumed barycentric trajectory is fixed.
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