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Let's assume there is this planet with no atmosphere, no geothermal activity and an average temperature $\ T_p$.

Now, if the distance between the planet and the star is $\ d$ and the radius of the star is $\ r$ , how do you express the temperature $\ T_s$ of the star?

I can't figure out whether the Stefan-Boltzmann law is the appropriate thing to use here.

Thank you!

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    $\begingroup$ Yes, you can use the Stefan-Boltzmann law, but you have to make assumptions about the reflectivity (albedo) of the planet. See this Wikipedia article for more en.wikipedia.org/wiki/Effective_temperature $\endgroup$
    – Thomas
    Dec 3 '21 at 19:25
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    $\begingroup$ It depends on the albedo characteristics of the particular planet. A planet that absorbs all incoming radiation would have maximum temperature (depending only on the distance and the luminosity (i.e. temperature and radius) of the star), a fully reflecting planet would have zero temperature.In reality it would be somewhere in between. $\endgroup$
    – Thomas
    Dec 3 '21 at 20:56
  • $\begingroup$ Thank you very much! $\endgroup$ Dec 3 '21 at 21:06
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this sounds like a fun homework question for upper-division astronomy !!

a couple more points to consider in your solution:

  • small angle approximation (to go from radius r to solid angle subtended by the star's disk as seen from the planet).
  • is the planet rotating? if rotating fast, temperature as a function of longitude will be constant. if rotating slow, it will have a hot side and cold side. this affects how you interpret the average temperature T_p.
  • temperature will vary with latitude. this will be symmetric at equinox and asymmetric if there is an obliquity. this affects how you interpret the average temperature T_p.
  • albedo, as mentioned by @Thomas in the comments. emissivity is usually assumed to be 1 for these kind of problems.

PS. i am a new contributor, so my apologies for "answering" with not a complete answer. i have a low reputation, so i am not able to put these important points as comments. besides, i'm not sure what the ethics are of just giving a straight-up answer to homework questions on here.

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I can't figure out whether the Stefan-Boltzmann law is the appropriate thing to use here.

Yes, that's exactly what to use, and the 1/4 exponent in the equation below suggests to us that that's how it was derived even before we review the derivation in the article.

Wikipedia's Planetary equilibrium temperature; Calculation for extrasolar planets provides this relationship.

$$T_{eq} = T_{star} \sqrt{\frac{R}{2a}} \left(1 - A_B \right)^{1/4}$$

where $R$ is the radius of the star, $a$ is the orbital distance of the planet, and $A_B$ is the Bond albedo of the planet. You will need to read the article and other further sources to understand what Bond albedo is and how one might estimate it for the planet. See also

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