Dust-tail angular lag ($\Delta \theta ={\theta }_d-{\theta }_p$)

For unwrapped grain and parent azimuths, define a signed lag $\Delta \theta ={\theta }_d-{\theta }_p$ and a positive trailing distance $s=-\Delta \theta$. A grain released without a velocity kick from a circular parent orbit has $\Delta \theta =2\beta (\sin {\theta }_d-{\theta }_d)+O({\beta }^2{\theta }_d)$. Keeping track of this sign avoids mistaking trailing motion for leading motion.

The replacement $s=2\beta (\theta -\sin \theta )\mapsto 2\beta \theta$ neglects a bounded orbital ripple in accumulated lag. Substituting the resulting angle into the instantaneous periodic drift gives a convenient ${\csc }^2(s/(4\beta ))$ profile, but is not a uniform small-$\beta$ approximation. The consistent first-order profile retains the implicit angle-to-lag map; near release it gives $n(s)\propto s^{-2/3}$.