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The ratio of the third to the second radius moment determines projected extinction area per liquid mass. For size-independent extinction efficiency $Q$, the local particle extinction coefficient is $3Qw/(4{\rho }_w{r}_e)$. It is not generally the mean radius or the mean-volume radius. In adiabatic cloud droplet moments, the shape factor gives $r_e=r_v/k^{1/3}$.

Using the adiabatic cloud droplet moments, define $N=\int n(r)dr$ and the mean volume radius by $r_v^3=N^{-1}\int r^{3}n(r)dr$. Summing the liquid mass of spherical droplets givesTherefore, with the shallow-cloud liquid-water content $w=c_{w}h$,The growth is $h^{1/3}$ at fixed number concentration, and the radius scales as $N^{-1/3}$ at fixed water content. The dimensions are consistent: $c_w$ has units $\mathrm{kg}{\mathrm{m}}^{-4}$, so $c_{w}h/({\rho }_{w}N)$ has units ${\mathrm{m}}^3$. The formula is a third-moment radius; a monodisperse distribution is not required. Fixed $N$ per volume is the shallow-cloud approximation, not exact conservation of number per parcel volume during an arbitrarily deep expansion.

Let $r_s^2=N^{-1}\int r^{2}n(r)dr$. The effective droplet radius and shape factor give $r_e=r_v^3/r_s^2$ and $k^{1/3}=r_v/r_e$, so $r_s^2=k^{1/3}{r}_v^2$. If the extinction efficiency is approximately independent of droplet size over the distribution,For visible wavelengths and droplets large compared with the wavelength, geometrical optics makes $Q_{\mathrm{ext}}\simeq 2$ a useful approximation. Retaining it as the specified constant, and taking $N,k,c_w$ constant over the cloud,The $3/5$ comes from integrating $h^{2/3}$; replacing the whole cloud by its cloud-top extinction would incorrectly omit it. The effective-radius relation treats the distribution shape through $k$, without assuming all droplets have the same size. Liquid phase, negligible vertical variation in that shape factor, and negligible departures from the adiabatic water profile are essential assumptions.

In the adiabatic cloud droplet number retrieval, the independently retrieved liquid-water path fixes the geometrical thickness in the adiabatic model:Define $C_c=(3/5)\pi Q_{\mathrm{ext}}{k}^{1/3}[3c_w/(4\pi {\rho }_w){]}^{2/3}$. Since ${\tau }_c=C_c{N}^{1/3}{H}^{5/3}$, eliminating $H$ givesUse SI units to obtain $N$ in ${\mathrm{m}}^{-3}$, then divide by $10^6$ for $\mathrm{c}{\mathrm{m}}^{-3}$. The sensitivity is substantial: $d\log N=3d\log {\tau }_c-(5/2)d\log W+(1/2)d\log c_w-d\log k-3d\log Q_{\mathrm{ext}}$. Thus a numerical retrieval additionally needs the adiabatic condensation coefficient, shape factor and extinction efficiency. A nonadiabatic water profile, mixed phase, horizontal inhomogeneity or a biased water-path retrieval can produce a large number-concentration bias.