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Consider a circular circumplanetary orbit well inside the Hill sphere, with dust speed $u^2=G{M}_p/a$ and unit plane normal $\hat{\mathbf{k}}$. During one dust orbital period, take the stellar direction $\hat{\mathbf{R}}$ and flux at the planet's stellar radius $a_p$ as constant. The velocity-dependent Poynting–Robertson drag has relative workThe constant terms have zero work over the closed circular orbit. Since $⟨u_i{u}_j⟩=(u^2/2)({\delta }_{ij}-k_i{k}_j)$,The coefficient is three for coplanar stellar and dust orbits; it is two for a dust plane normal to the instantaneous stellar direction. Averaging over a circular planetary year at fixed dust orbital inclination $I$ gives $3-{\sin }^{2}I/2$. Conservative perturbations must remain small enough for this circular-orbit average.

The orbit-averaged Poynting–Robertson drag equations preserve $C=a(1-e^2)e^{-4/5}$ for $0<e<1$. This relates shrinking semi-major axis to decreasing orbital eccentricity; a circular orbit is treated by its separate limiting solution.

The zero-kick circular-parent threshold is $\beta =1/2$: below it grains are initially bound, at it their conservative release orbit is parabolic, and above it they are initially unbound. Poynting–Robertson drag can modify a marginal orbit, so the threshold is a statement about the instantaneous release orbit, not an exact capture criterion with dissipation included.

A planet can block stellar radiation during part of a circumplanetary orbit. Then the full-orbit constant-flux averages for radiation pressure and Poynting–Robertson drag require modification. The detailed shadow duty cycle depends on the orbital inclination, stellar angular extent and dust radius.