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I encountered this problem in my textbooK:

"The Sun has the radius of 700,000km and a surface temperature of 5800K. The Earth is 150,000,000km far from the Sun and it has the radius of 6400km. In the simplified model the Sun transfers heat through radiation, and part of it (proportional to the Earth cross-section area) is absorbed by Earth.In the equilibrium the Earth radiates the same amount of heat through its surface. The emissivity is 1 for both Sun and Earth. Based on this model calculate the surface temperature of the Earth in◦C."

I don't know where to start. Should I first find the heat of radiation of the sun at the distance of the earth? How do I use it to find out the radiation absorbed by the earth in order to find its temperature?

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  • $\begingroup$ Hint: A blackbody (such as the Earth in this problem) absorbs all energy incident on it, and radiates such that the power absorbed is equal to the power emitted. $\endgroup$
    – probably_someone
    Commented Mar 2, 2018 at 4:49
  • $\begingroup$ It's a matter of ratios: there's a ratio of the earth-surface and sun-surface temperatures, that you have enough info to compute. $\endgroup$
    – Whit3rd
    Commented Mar 2, 2018 at 7:45
  • $\begingroup$ Please note that this site is not a place to obtain solutions to worked problems. Please see this Meta post on asking homework-like questions and this Meta post for "check my work problems". $\endgroup$
    – Kyle Kanos
    Commented Mar 2, 2018 at 11:08

1 Answer 1

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From the radius and temperature of the Sun, you should be able to (given the Sun is a black body) calculate the total Luminosity (power) radiated by the Sun. Given the total Luminosity and the distance between the Sun and Earth, you should be able to get a Flux (power/area) of radiation at Earth which you can combine with the radius of the Earth to calculate the total power that the Earth absorbs from the Sun. Given that in equilibrium power absorbed must equal power radiated, you can then get the Earth's surface Temperature.

Hints: See the Stefan-Boltzmann Law.

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  • $\begingroup$ Or one uses the solid angle subtended by the Sun. And realizes that Earth would have the same temperature as the Sun if that would be $4\pi$. And uses $T^4$. $\endgroup$
    – user137289
    Commented Mar 2, 2018 at 6:57
  • $\begingroup$ I've never heard of a flux radiation formula and everything I search on google about flux is related to electric fields or calculus. How do I adapt the formulas there for radiation? $\endgroup$
    – A.boj
    Commented Mar 2, 2018 at 19:53
  • $\begingroup$ See here: en.wikipedia.org/wiki/Radiative_flux and here: en.wikipedia.org/wiki/Spectral_flux_density The first article is the "flux" I'm talking about, but the second article goes into a bit more detail. The Flux I'm talking about is basically the integral over all wavelengths of the spectral flux density talked about in the second article. $\endgroup$
    – enumaris
    Commented Mar 4, 2018 at 20:50

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