= Apsidal evolution under Poynting–Robertson drag
For a spherical grain with constant <radiation-pressure coefficient> $\beta$, describe its orbit using the reduced central attraction $GM_\star(1-\beta)$ and treat <Poynting–Robertson drag> to first order in $v/c$. In that approximation the orbit-averaged <apsidal precession> is
$$
\langle\dot\varpi\rangle=0.
$$
The dissipative force has a radial component proportional to $-2\dot r/r^2$ and a transverse component proportional to $-v_\theta/r^2$, as obtained from the first-order radiation-force expression.<Radiation Forces on Small Particles in the Solar System by Joseph A. Burns and others>[Abstract: equation of motion retaining terms of order v/c]{ref}
The cancellation follows by <orbit averaging>: radial <velocity> is odd in <true anomaly>, whereas radius, transverse <speed> and the time-weighting factor are even. The terms contributing to the change of apsidal angle integrate to zero over a full orbit. <Semi-major axis> and <orbital eccentricity> can still decrease. Instantaneous osculating angles can oscillate, and nonspherical forces or additional planetary perturbations need not share this cancellation; a circular orbit has no defined apsidal direction.
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