Secular phase approximation for a dust tail
= Secular phase approximation for a dust tail
The replacement $s=2\beta(\theta-\sin\theta)\mapsto2\beta\theta$ 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 $n(s)\propto s^{-2/3}$.