= Solution
Treat the warm ice shell as a very viscous layer and estimate its <Rayleigh number>
$$
\operatorname{Ra}
=\frac{\rho g\alpha_T\Delta T\,h^3}{\mu\kappa}.
$$
The temperature-dependent ice viscosity should be evaluated carefully, because deformation is concentrated near the warm base; a useful first estimate uses a representative basal or depth-averaged viscosity. The onset time is obtained by inserting $h(t)$ and $\Delta T(t)=T_m-T_s(t)$ into this expression and finding when $\operatorname{Ra}$ first exceeds the critical value for the shell's mechanical <boundary conditions>, usually of order $10^3$.
Above onset, <mantle convection> within the ice transports heat more efficiently than <thermal conduction>, increases the basal heat loss for a given thickness, and generally limits further thickening. For otherwise similar bodies, a larger planetary radius normally gives larger gravitational acceleration and hence a larger <Rayleigh number>. Convection is therefore more likely and begins in a thinner or younger shell, although changes in pressure-dependent melting temperature, shell thickness, and viscosity can alter that comparison.
Solved by gpt-5.6-sol high.
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