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I'm looking for an algorithmic model which will gives rough estimate of the average temperature of a planet's atmosphere (good enough to say whether there will be liquid water anywhere, or if metals and plastics will melt/ignite) based on the luminosity of its star, the radius of its orbit, the constituents of its atmosphere, the mass of the planet, and the surface area of the planet.

Specifically, I figure I need two mathematical models. The first would simulate the incoming thermal energy from the star, and the second would calculate how much is lost as blackbody radiation by the planet, taking into account the effects of the 'greenhouse effect'.

Does anyone know how to model these things?

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It sounds like what you're looking for is related to "habitable zone" calculations. Check out planetarybiology.com/calculating_habitable_zone.htm and arxiv.org/abs/1301.6674 –  Brandon Enright Jun 1 '13 at 17:26
Another subtlety you may or may not be interested in is the idea that the initial heat of formation and/or heat from radioactive decay could be considered. In cases of dense atmospheres, Dave Stevenson has argued (this article, preprint here) that planets can retain this heat even without any input from a star. –  Chris White Jun 1 '13 at 18:16
Habitable zone calculations are useful, but not what I need here. I'm trying to find the temperature of the atmosphere, in order to calculate what state certain materials can be in on that planet, and what requirements a craft must meet to make a landing there. It's for a video game. –  Schilcote Jun 1 '13 at 18:23
@Schilcote Ah yes, I remember your other post now. Good luck with the game :) My advice here parallels something I said on that other question: don't fret too much about getting the answer, since this is very much an active field of research. Hopefully you get a somewhat accurate, yet simple-to-implement result. –  Chris White Jun 2 '13 at 2:36

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I recently wrote a tutorial for my astronomy club which seems to address your question. It includes numerical examples that specifically covers planetary temperatures and greenhouse effects.

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Ah, yes, I think that's what I need. I assume that a planet without any clouds will have an average albedo somewhere zero? And how do I determine the value of the greenhouse factor? –  Schilcote Jun 2 '13 at 18:49
Clouds are not the only albedo factors. Checkout albedo on WIKI. The greenhouse factor is mainly determined by water on Earth, but other planets like Venus and Pluto have carbon dioxide and methane factors that dominate. –  Michael Luciuk Jun 3 '13 at 11:25

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