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I have a question regarding tidally locked planets, regarding the length it takes to reach a certain temperature.

How would one calculate the estimated temperature after a certain amount of time being tidally locked. For example, how hot would a planet be after 10,000 years of tidally locking as compared to millions or billions? As I understand it it would actually be far cooler, but I am curious as to how exactly one would calculate this in regards to distance to the source, luminosity, and the size and albedo of the planet. If such a formula exists.

In short, how to find the surface temperature of a tidally-locked planet in regards to time. I am capable of doing the math myself, I simply require help finding a formula that describes what I am looking for. I only need an approximation, as long as it remains in the correct order of magnitude.

Your help would be much appreciated.

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    $\begingroup$ What is the surface temperature of a tidally locked body in the first place? Or varies a lot, depending on distance to the terminator. And it depends a lot on the atmosphere (see e. g. Venus) $\endgroup$ Commented Sep 18, 2023 at 6:15
  • $\begingroup$ While Venus is not tidally locked, it's an interesting pseudo case study for this context. Venus, despite having a huge greenhouse effect, is roughly at temperature equilibrium - that is, heat in ~= heat out - after it began heating up ~700mya. Other factors to consider are the thermodynamics of atmosphere exchange between the bright and dark sides, the composition of the surface and its heat retention characteristics, any tectonic activity, and magnetic field mitigation of incoming radiation. I am not sure you're going to find any one unified equation to describe all of this. $\endgroup$ Commented Sep 19, 2023 at 18:11
  • $\begingroup$ As an elaboration, the case I am considering is a planet that has a relatively Earth-like atmosphere. So you wouldn't have to worry about absurd levels of the greenhouse effect. Also, yeah, I know Emmisivity and the level of "trade" between sides is important. $\endgroup$ Commented Sep 20, 2023 at 1:34

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