I've been struggling to find equations that express how many degrees of warming greenhouse gases contribute, given the composition of an atmosphere (and solar insolation).
What I did find was the Stefan–Boltzmann Law, and that wiki article has good equations about black body radiation. It shows how to get a value of $279 K = 6 °C = 43 F$ for the plain Earth with no atmosphere. Then to account for the albedo of Earth, the effective temperature is calculated at $255 K = −18 °C = -0.4 F$.
Then the article basically states what the real values are without presenting any more equations or explaining how to calculate it.
However, long-wave radiation from the surface of the earth is partially absorbed and re-radiated back down by greenhouse gases, namely water vapor, carbon dioxide and methane. Since the emissivity with greenhouse effect (weighted more in the longer wavelengths where the Earth radiates) is reduced more than the absorptivity (weighted more in the shorter wavelengths of the Sun's radiation) is reduced, the equilibrium temperature is higher than the simple black-body calculation estimates. As a result, the Earth's actual average surface temperature is about 288 K (15 °C), which is higher than the 255 K effective temperature, and even higher than the 279 K temperature that a black body would have.
And there were only 2 cites that only cited the fact that H2O, CO2, and CH4 are greenhouse gases.
I've been searching for how to calculate the warming due to GHGs, but no luck. I'm really hoping someone can shed some light on this. I'm looking for an equation that takes into account atmospheric composition and probably the rotation rate of Earth (or whatever body), and of course the solar insolation.
Please note I'm not really asking about climate change calculations. Climate change is more about how the atmospheric composition is changing (namely GHG increases). I'm simply looking for equations that dictate the temperature based on a given/constant composition of the atmosphere.