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The upward arrows at the surface are reflected short-wave flux $\alpha F_r$, emitted long-wave flux $ϵ\sigma T_s^4$ from the Stefan–Boltzmann law, and, when positive, conductive heat arriving from the planetary ice shell. The downward arrow is the incident solar flux $F_r$. The planetary surface energy balance is thereforeThe relevant planetary albedo is the Bond albedo. In a steady one-dimensional shell, Fourier's law and $k(T)=a/T$ make the upward conductive fluxwhere $z$ points downward and $T(h)=T_m$. Substitution gives the requested implicit equation

A steady planetary surface energy balance equates absorbed incoming radiation plus upward internal heat flux with outgoing thermal radiation. For local incident stellar radiative flux $F_r$, surface albedo $\alpha$, upward conductive heat flux $F_c$ and thermal emissivity $ϵ$,The reflected part $\alpha F_r$ has already been subtracted. Equivalently, incident plus internal inputs equal reflected plus emitted outputs: $F_r+F_c=\alpha F_r+ϵ\sigma T_s^4$. Including reflected radiation again alongside an already absorbed input would count it twice. This simple surface equation omits downward atmospheric long-wave radiation and turbulent or latent heat fluxes; those require additional terms when present.

For an airless spherical planet with uniform temperature, intercepted stellar power is $\pi R^2{F}_{\star }$, while the emitting area is $4\pi R^2$. A whole-planet balance therefore uses mean incident flux $F_{\star }/4$ and Bond albedo ${\alpha }_B$:when internal heating is negligible. The factor four comes from geometry, not an extra loss mechanism. A nonuniform temperature field requires averaging $T_s^4$, rather than raising the mean temperature to the fourth power. In a planetary ice shell, a nonzero basal conductive contribution can keep the surface warmer than stellar heating alone predicts.