JohnRC
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chill_factor said:yep that's true I hadn't thought of the carbon dioxide and water separating from the atmosphere.
would it be possible to pump huge amounts of methane into the atmosphere in amounts great enough to slow down the cooling?
I have checked up on the geothermal energy flux through the surface, and at present it amounts to about 0.1 watt per square metre. For that particular energy flux, the steady state emission temperature would probably be in the range of about 50-60 K if the emissivity of an ice-covered Earth were in the range 10-25%. The flux would increase with the geothermal gradient if the Earth's surface were to cool, but probably not by very much.
The steady state emission temperature is a firm number for a particular level of emissivity. Any remaining uncertainty and discrepancy between the picture that I have set out and that of chill_factor is down to the altitude and the "blackness" of the surface or atmospheric layer that the emission is effectively coming from.
There are two reasons, though, why pumping large amounts of methane might be unnecessary. The first is that the Earth is a huge thermal reservoir that would probably cool very slowly to reach this steady state. It should be possible to calculate how slowly; I have not yet done that. The second is that all of the plants are going to die in the darkness. They and the remaining microbiota will continue for awhile to metabolize and produce carbon dioxide, (and methane in some cases), until their environment gets too cool for them to continue. (Likewise the animals, but they are a much smaller part of the equation than plants and microbiota, which are roughly equal). This will prolong the period of high atmospheric carbon dioxide, and high level atmospheric emission from a layer much cooler than the surface.