How to calculate the greenhouse effect?

Question

I tried many equations (Stefan-Boltzmann constant) that gave me a temperature of 249.769 Kelvin (-23.38°C) assuming no atmosphere and 297 Kelvin (24°C) with perfectly absorbing one, but my planet is neither, and SpaceEngine is hopeless (Although it may be right) it gave me a temperature with greenhouse effect of 267.6 Kelvin (-5.5°C (The greenhouse effect is 16.373° Celsius/Kelvin.)).

So how can I calculate it?

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My planet is orbiting a K type Star/Orange dwarf and doesn't have life on it.

Planet:(Unit is Earth = 1) Mass: 1.8 | Radius: 1.2 | Atmosphere: 88% N2 / 9% CO2 / 1.5% Argon / 1.5 other gasses. with Presure at sea level of 1 atm.

Semi-major axis(Changeable): 0.5 AU (Note that ~0.52 AU is the distance where the heat and energy received from the star is the same as Earth does from the sun), I don't know if it would be tidally-locked to the star, so just in case I made it have a very big moon with 0.2 Mass and 0.6251 Radius, so it can break the lock and make it locked to the moon instead, with a semi-major axis of 84500 km.

With ~25% land and ~75% Sea of H2CO3(Carbon Cycle caused Low CO2 on the atmosphere + Sea being H2CO3) Of course, it does have a magnetic-field and plates tectonic.

Star: (Unit is Sol/Sun = 1) Mass: 0.72 | Radius: 0.78 | Luminosity: 0.2687 | Surface Temperature: 4710 Kelvin

Answer 1

Not a simple calculation. If you want to get serious, try the book "Principles of Planetary Climate" by Raymond Pierrehumbert: https://www.cambridge.org/core/books/principles-of-planetary-climate/5B5EEF0534CB6F69FB2E395DD21D3476

For something simpler, I think the EdGCM program will let you plug in various values. Unfortunately, it seems to run only on Windows or Mac, so I've never had a chance to play with it: http://edgcm.columbia.edu/ There's also an Open Climate Workbench, though I've never used it: https://climate.apache.org/

Answer 2

A previous answer said "not simple." Yes, very seriously not simple. Here are just a few considerations.

Clouds are a huge effect. And, quite disconcertingly, they can be positive or negative in different circumstances. Big white fluffy clouds reflect sunlight and cool things. But, big white fluffy clouds at night tend to reflect infrared from the ground. And so a cloudy night can be warmer. Different altitude clouds can also produce different effects. Those thin wispy little clouds at very high altitude don't block much light but may reflect a lot of infrared. So a tiny change in cloud cover, even of cloud type, could produce a substantial effect.

Weather is also a huge factor. Rain moves heat around. A lot of heat. Consider a 1 cm rainfall falling a distance of 1 km. 1 cm over 1 square meter is 10 kg. Falling 1 km is 98,000 Joules of energy per square meter. Evaporation thus carries all of that heat from the surface to the clouds. When the water condenses to rain, it leaves that heat in the clouds, bypassing a bunch of the atmosphere. It's like a vertical heat pump.

Other aspects of weather involve massive movements of air up or down, depending where you are. So warm air can be moved upward in huge amounts at quite rapid speed. Locally and temporarily this can overwhelm any possible greenhouse effect.

Weather related ground cover can make a very large difference. Snow cover reflects a lot of sunlight, as may sand in deserts. Certain types of plant cover reflect more light than others. Different ocean conditions will reflect more or less sunlight.

So calculating the greenhouse effect is a monumental task. Unless you are climate research institute, you probably can't do the full calculation, or even good approximation.

If you are telling a story I suggest you put your world in the "habitable" zone around a star, then just hand-wave it. You've got clouds and trees and oceans, and it all just works.