= Circular ash deposit profile
{title2=$\Sigma(r)=\rho_pw_sc_0\tau[2/3-s+s^3/3]/(1-s_0^2)^2$}
For the <monodisperse circular ash runout>, integrate the local particle <deposition flux> $\rho_pw_sc(t)$ from first arrival to infinite time. Here $s=\max(s_0,r^2/R_\infty^2)$ for $0\leq r<R_\infty$ 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 $\rho_pc_0\mathcal V$. A deposit thickness additionally requires its packing fraction or bulk deposit density.
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