Adiabatic cloud droplet moments ($r_v^3=⟨r^3⟩,r_e=⟨r^3⟩/⟨r^2⟩$)

For droplet number density $N$ and radius distribution $n(r)$, let $r_v^3=N^{-1}\int r^{3}n(r)dr$ and $r_s^2=N^{-1}\int r^{2}n(r)dr$. The effective radius is $r_e=r_v^3/r_s^2$. With liquid-water content $w=c_{w}h$, $r_v=[3c_{w}h/(4\pi {\rho }_{w}N){]}^{1/3}$. 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.