So much for back of the envelope...
You should integrate along the atmosphere how much of the IR radiation gets absorbed by the atmosphere, taking into consideration the cross section given by the percentage (or per-millage) ) of atmospheric CO2.
Considering that O2 and N2 are mostly transparent to IR, while water vapor and CO2 are not. Actually, H2O is far more absorbent, but that one falls back down as rain, while CO2 mostly stays there.
At 100% CO2 concentration, you should absorb something like 10~15% of all IR, judging by this plot (I'm just eyeballing it here):
You can then assume linearity, where 1% of CO2 in the atmosphere, leads to ~0.1% of IR being absorbed.... and just let it go on a feedback loop.
The hotter the surface gets, the more IR it emits.
You should integrate along the atmosphere how much of the IR radiation gets absorbed by the atmosphere, taking into consideration the cross section given by the percentage (or per-millage) ) of atmospheric CO2.
Considering that O2 and N2 are mostly transparent to IR, while water vapor and CO2 are not. Actually, H2O is far more absorbent, but that one falls back down as rain, while CO2 mostly stays there.
At 100% CO2 concentration, you should absorb something like 10~15% of all IR, judging by this plot (I'm just eyeballing it here):
You can then assume linearity, where 1% of CO2 in the atmosphere, leads to ~0.1% of IR being absorbed.... and just let it go on a feedback loop.
The hotter the surface gets, the more IR it emits.