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I tried to find the habitable planets, thus I suppose to calculate the effective temperatures that incident on the planets from their host stars, I used

$$ T_\mathrm{eff}=\frac{L}{4\pi\sigma_\mathrm{SB}R^2} $$

but I don't know how to calculate the effective temperatures in polar regions?

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    $\begingroup$ Your equation is dimensionally incorrect. Neither is it the correct way to calculate the equilibrium temperature of a planet, which can be found in multiple places if you search for "habitable zone". $\endgroup$
    – ProfRob
    Commented Aug 6, 2017 at 22:46
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    $\begingroup$ You might start at this Wikipedia page on Effective Temperature (and links on that page) and also this page on the Circumstellar habitable zone. Temperature at the poles depends on the details of the planet's atmosphere and ocean, and the wind system which is also shaped by land masses in detail. So very complex. $\endgroup$
    – StephenG - Help Ukraine
    Commented Aug 7, 2017 at 7:10
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    $\begingroup$ You should read this (appeared today). arxiv.org/pdf/1708.01363.pdf $\endgroup$
    – ProfRob
    Commented Aug 7, 2017 at 12:16
  • $\begingroup$ @RobJeffries that link is quite a nice commentary indeed. $\endgroup$
    – uhoh
    Commented Jan 17, 2018 at 2:42

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The formula you have written resembles the Stefan-Boltzmann law, and its correct form is $$T^4 = \frac{L}{4\pi R^2 \sigma_{SB}}$$. It relates the luminosity of a blackbody of a given radius to its effective temperature. Earth as a whole can be assumed to be a black body at an effective temperature. So you can calculate the effective temperature of the "earth". This is very well explained in the example section of this wiki page: https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law

For calculating the effective temperature of the polar region, first assumption is that polar region is a black body. Then the question to ask is: what is the effective teperature of the polar region given that the it reaches equilibrium when the Sun light falls on it. To calculate this further, you would need to calculate the angle at which the sun light fall on the polar region you are interested in and also the distance travelled by the sun light in the atmosphere. These parameters will vary with respect to the time of the year, time of the day and the region you are interested in (North pole, South pole). Some calculated temperatures are given on the same wiki page. I hope that helps.

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