= Adiabatic cloud droplet moments
{title2=$r_v^3=\langle r^3\rangle,\quad r_e=\langle r^3\rangle/\langle r^2\rangle$}
For droplet <number density> $N$ and radius distribution $n(r)$, let $r_v^3=N^{-1}\int r^3n(r)dr$ and $r_s^2=N^{-1}\int r^2n(r)dr$. The effective radius is $r_e=r_v^3/r_s^2$. With liquid-water content $w=c_wh$, $r_v=[3c_wh/(4\pi\rho_wN)]^{1/3}$. The shape factor $k=r_v^3/r_e^3$ gives $r_s^2=k^{1/3}r_v^2$; these are distribution moments, not a monodisperse assumption.
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