How would I calculate the greenhouse effect based on atmospheric composition?

Question

Problem: For a known, but hypothetical planet orbiting a celestial body, how would I calculate the surface temperature after greenhouse gases?

What I know about the planet as given:

• Planetary diameter
• Percentage breakdown of atmospheric gases:
  • CO2, N2, O2, Ar, CH4 (Methane), Na, H2, He, "other"
• Distance from star
• Luminosity of star
• Surface albedo
• Total atmospheric mass

Assumptions I am making:

• "All other variables remain constant"
• The planet is completely homogenous

I've been trying to implement a simplistic model, and have been able to calculate the equilibrium temperature but not the temperature after greenhouse gases.

Ideally what I'm looking for is a function using the parameters I have that'd give me the equilibrium temperature with an atmosphere. I've reviewed several articles, but I haven't been able to make sense of their formulas to be able to create a working model.

The equation from the wiki that I found was this: $ T_{eff} = T_V*0.7062 + T_V*0.7062*21*d_{NO}*0.099 + T_V*0.7062*d_{He}*0.218 + (T_V*\frac{0.7062}{262})*m_{CO_2} + (T_V*\frac{0.7062}{262})*m_{GH}*C $

but I've been averse to using this one, since the 0.7062 and other magic numbers are unexplained, and given the formatting issues I'm dubious.

Formula I'm hoping to be able to create:

$ T_{eq} = \left(\frac{L*(1-A_B)}{16 \sigma \pi \alpha^2}\right)^\frac{1}{4} + \Delta T_{CO_2} + \Delta T_{N_2} + ... \\ \text{where} \\ M_{gas} = \text{mass of gas} \\ D = \text{planet diameter, in terms of earths}\\ L = \text{stellar luminosity}\\ G_c = \text{gas temperature constant}\\ \Delta T_{gas} = f(G_c, M_{gas}, D, L, A_b, \alpha, ...) $

Links I've reviewed:

• Greenhouse effect
• Ammonia world
• Terrestrial moon
• Calculate greenhouse
• Science-based calculation (the link to the stepwise process gave me a 403)
• Greenhouse calculator
• The hockey schtick Note: This one I deemed excessively poor because the model cannot calculate for 0 atmosphere, in addition to some very poor notation.

Answer 1

I've found a way to calculate a surface temperature, but it relies on a constant, $\epsilon$, that doesn't appear to have an easy way to generate it.

The function is: $$ \frac{S(1-A)}{4} = \sigma \epsilon T_e^4 + (1 - \epsilon) \sigma T_s^4 $$

For calculating $\epsilon$, my research led me to Dr. Scott Denning at CSU. The answer for calculating $\epsilon$ is..."it's complicated and there's a ton of things that impact it."

But in general, $\epsilon$ has a logarithmic relationship to atmospheric pressure, where higher pressures will net higher $\epsilon$ values.