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Snow-lines can be thermally unstable

Volatile species in protoplanetary discs can undergo a phase change from vapour to solid. These "snow-lines" can play vital roles in planet formation at all scales, from dust coagulation to planetary migration. In the outer regions of protoplanetary discs, the temperature profile is set by the absorption of reprocessed stellar light by the solids. Further, the temperature profile sets the distribution of solids through sublimation and condensation at various snow-lines. Hence the snow-line position depends on the temperature profile and vice-versa. We show that this coupling can be thermally unstable, such that a patch of the disc at a snow-line will produce either run-away sublimation or condensation. This thermal instability arises at moderate optical depths, where heating by absorption of reprocessed stellar light from the disc's atmosphere is optically thick, yet cooling is optically thin. Since volatiles in the solid phase drift much faster than volatiles in the vapour phase, this thermal instability results in a limit-cycle. The snow-line progressively moves in, condensing volatiles, before receding, as the volatiles sublimate. Using numerical simulations, we study the evolution of the CO snow-line. We find the CO snow-line is thermally unstable under typical disc conditions and evolves inwards from $\sim50$ to $\sim30$~AU on timescales from 1,000-10,000 years. The CO snow-line spends between $\sim 10\%-50\%$ of its time at smaller separations, where the exact value is sensitive to the total opacity and turbulent viscosity. The evolving snow-line also creates ring-like structures in the solid distribution interior to the snow-line. Multiple ring-like structures created by moving snow-lines could potentially explain the sub-structures seen in many {\it ALMA} images.