I'm looking for a formula, either precise or an approximation (it's for a video game so it's ok to have somewhat rough numbers), that determines heat retained from the sun by an atmosphere, based on its density and composition (specifically CO2 composition, but bonus points if it factors other things in too).
Does such a formula exist? Or if not, are there any rule-of-thumb sort of things I could use for estimating "If an atmosphere doubles in density, its heat will rise by X percent" or "If you replace half the CO2 in the atmosphere with O2, your heat will drop by Y percent", etc?