 Apsidal evolution under Poynting–Robertson drag

For a spherical grain with constant radiation-pressure coefficient $\beta$, describe its orbit using the reduced central attraction $G{M}_{\star }(1-\beta )$ and treat Poynting–Robertson drag to first order in $v/c$. In that approximation the orbit-averaged apsidal precession isThe 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.^[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.