= Poynting–Robertson decay of a circumstellar orbit
{c}
{title2=$\langle\dot a\rangle_{\rm cs}$}
With $A=\beta GM_\star/c$ constant, average the drag <work> $\dot\varepsilon=-A(2\dot r^2+v_\theta^2)/r^2$ over a bound <elliptic orbit>. Using $dt=r^2df/h$ gives the integral $\pi(2+3e^2)\mu^2/h^2$, where $\mu$ is the conservative central parameter. Combining with $\dot a=2a^2\dot\varepsilon/\mu$ gives
$$
\langle\dot a\rangle=-\frac A a\frac{2+3e^2}{(1-e^2)^{3/2}}.
$$
This is a weak-drag secular average. It also holds when the conservative parameter is $(1-\beta)GM_\star$, provided a bound <orbit> exists; the conservative parameter cancels but the drag amplitude remains $A$. For a <circular Kepler orbit>, $a^2(t)=a_0^2-4At$.
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