 Surface density of a size-sorted dust burst

For initial dust mass $m_d$ in a ring at $r_0$ with a fixed-diameter power-law size distribution of index $\alpha <4$, burst transport under inverse-size radial drag maps radius to diameter $D_r=2\gamma t/(r_0^2-r^2)$. Set $T_r=t_r(D_{\max })$ and $\eta =D_{\min }/D_{\max }$. The surface density is the azimuthal average: $2\pi r\Sigma dr=(\pi \rho K/6)D_r^{3-\alpha }d{D}_r$. Since $d{D}_r/dr=2r{D}_r/(r_0^2-r^2)$, mass normalization of a power-law size distribution gives the displayed expression. It is supported where $\eta \le t/T_r\le 1$. It applies at $t>0$ before removal at the radius in question; the initial state is a delta-function ring. A local angular density needs additional phase information.