@blouis79: On the “snowball earth” hypothesis: if we consider the case of an earth with no sun and no ability to radiate. In such conditions, the earth would be isothermal. Isothermal earth would not be a snowball, it would be rather hotter because of the molten core. It is the ability to radiate and lose heat to space (there is no other mode of heat loss to space of note) that results in the real earth surface including a sun warming it being cooler than an isothermal radiatively insulated earth with no sun. Show me a model that can correctly predict these boundary conditions.
You already have one: the one you used to determine what you think will happen.
Furthermore every model that predicts the same outcome as yours would need the same feature as yours: a surface with zero emissivity. Other details would then be largely irrelevant.
As a picky point, from a real-world physics standpoint your question is purely hypothetical. If we take “surface” to mean the top layer of molecules, zero emissivity is not remotely approachable physically. The thickness of such a surface would be 0.0001 of the wavelength of emitted radiation. That layer, and for that matter several thousand such layers deeper, would have essentially unit emissivity. This is because such layers are largely transparent to the thermal radiation they emit. You need a thickness on the order of a wavelength to even begin blocking thermal radiation.
But in that case conductivity kicks in to warm the top molecular layers from below. That is, the outgoing thermal flow is weakly conductive up to the top few microns, and strongly radiative thereafter.
This conductive flow from the core up to the top few microns can be estimated at around 60-80 mW/m2. Without the Sun to keep it warm, the surface of the Earth would lose heat at essentially that rate. As it cooled, practically all of the atmosphere would freeze and settle on (and hence become) the surface.
When equilibrium with the 2.7 K temperature of space was reached, the temperature of the surface would be roughly 35 K, while the core would remain molten, kept warm by the decay of radioactive elements, for the order of their half-life anyway.
I computed the 35 K surface temperature as sqrt(sqrt(.09/sb)) where .09 W/m2 is is a tad more than the current rate of heat flow from the core to the surface, and sb is the Stefan-Boltzmann constant 5.67 x 10^-8. The outgoing heat from the core dwarfs all incoming heat from space whence the latter can be neglected.
35 K under the real-world assumption that a wavelength of thermal radiation is thousands of molecules wide is a very different outcome from a model that assumes exactly zero emissivity.
