 Circular ash deposit profile

For the monodisperse circular ash runout, integrate the local particle deposition flux ${\rho }_p{w}_{s}c(t)$ from first arrival to infinite time. Here $s=\max (s_0,r^2/R_{\infty }^2)$ for $0\le r<R_{\infty }$ and $s_0=R_0^2/R_{\infty }^2$. The displayed $\Sigma$ is deposited particle mass per horizontal area, and vanishes outside the limiting current. It is constant under the initial disk, then decreases to zero; integrating $2\pi r\Sigma$ gives the initial ash mass ${\rho }_p{c}_0\mathcal{V}$. A deposit thickness additionally requires its packing fraction or bulk deposit density.