Dust tail

A dust tail is an extended distribution of grains released from a localized moving source. Radiation pressure changes the grains' orbits; their release times, grain properties and survival probabilities determine the spatial profile.

For a single-stream dust tail with injection rate $\dot{N}$, monotone angular distance $s(t)$, and survival probability $P(t)$, the number per unit angle is $n(s)=\dot{N}P(t(s))/\dot{s}$. A vanishing angular drift creates an idealized density enhancement; velocity dispersion rounds it.

A zero of the angular drift in a cold single-grain-size dust tail produces a formally divergent number density per unit angle. The divergence is integrable when the exact local angle-to-age map is retained. A spread of launch velocities or radiation-pressure coefficients removes the singularity.

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