Secular phase approximation for a dust tail

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 $n(s)\propto s^{-2/3}$.