= Adiabatic cloud droplet number retrieval
{title2=$N\propto c_w^{1/2}\tau_c^3W^{-5/2}$}
A shallow liquid cloud with $w=c_wh$ has <liquid-water path> $W=c_wH^2/2$. If $N$, extinction efficiency $Q$ and shape factor $k$ are constant with height, integrating droplet extinction gives $\tau_c=(3/5)\pi Qk^{1/3}[3c_w/(4\pi\rho_w)]^{2/3}N^{1/3}H^{5/3}$. Eliminating $H$ yields $N=[250\sqrt2/(243\pi Q^3k)]\rho_w^2c_w^{1/2}\tau_c^3/W^{5/2}$. Nonadiabatic water profiles and biased water paths strongly affect the retrieval.
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