= Solution
{c}
Let $1<S<S_c$ be the ambient <saturation ratio>. There are two equilibrium radii: $r_-<r_c<r_+$. <Condensation> occurs if $S>S_{\rm eq}(r)$; <evaporation> occurs if $S<S_{\rm eq}(r)$. Therefore droplets with $r<r_-$ grow towards $r_-$, while those with $r_-<r<r_c$ shrink towards $r_-$. The smaller equilibrium is stable.
Above $r_c$, droplets with $r_c<r<r_+$ still undergo <evaporation> and ultimately approach $r_-$. Droplets with $r>r_+$ grow, and the larger equilibrium $r_+$ is unstable. Merely exceeding $r_c$ is therefore \b[not sufficient] for continued growth at subcritical ambient <atmospheric supersaturation>. For example, if $(S-1)/(S_c-1)=0.6$, the two intersections have approximately $r_-/r_c=0.676$ and $r_+/r_c=2.349$; a droplet of radius $1.5r_c$ undergoes <evaporation>. If instead $S>S_c$, there is no equilibrium barrier in this approximation and droplets activate into growing <cloud droplets>.
<Köhler theory> explains activation of <cloud condensation nuclei>, rather than rapid production of <atmospheric precipitation>. <Diffusion-controlled droplet growth> is limited by <diffusion> of <water> <vapor> and removal of released <latent heat>; approximately $d(r^2)/dt$ is proportional to the ambient <atmospheric supersaturation>. It slows in radius and tends to narrow a droplet size distribution. The available <water> <vapor>, changing <temperature> and shared depletion of <atmospheric supersaturation> also invalidate unlimited growth at prescribed $S$. Formation of rain-sized drops normally requires <collision-coalescence>, while colder <clouds> can additionally use <cloud ice nucleation versus ice growth>.
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