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Homework Help: Earth Temperature

  1. May 6, 2017 #1
    1. The problem statement, all variables and given/known data
    Calculate the temperature of the Earth assuming that the Sun and the Earth are black bodies. Assume that Earth is in a steady state as far as energy balance is concerned

    2. Relevant equations
    ##\frac{E}{St} = \sigma T^4## - Stefan-Boltzman law

    3. The attempt at a solution
    The power radiated by sun is ##P=\sigma T_{sun}^4 S_{sun} = \sigma T_{sun}^4 4\pi R_{sun}^2##. The amount of this received by earth is proportional to: ##\frac{\pi R_{earth}^2}{4 \pi R_{sun-earth}^2}##, with ##R_{sun-earth}## being the distance from sun to earth. The power radiated by Earth is ##\sigma T_{earth}^4 4\pi R_{earth}##. As the earth is in a steady state we have in the end: ##\sigma T_{sun}^4 4\pi R_{sun}^2 \frac{\pi R_{earth}^2}{4 \pi R_{sun-earth}^2} = \sigma T_{earht}^4 4\pi R_{earth}^2## and from here we can get the temperature of the earth, as all the other constants are considered to be known. Is this correct? I obtained a numerical value of about 900K. I am aware that here we ignore the shielding of the atmosphere, the albedo and other effects that would influence the temperature, but it still seems to be pretty high.
    Last edited: May 6, 2017
  2. jcsd
  3. May 6, 2017 #2


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    Science Advisor
    Gold Member

    You've made an error here. The radius should be squared.
    Correcting that will net you a factor of ~1/9, so if you've got 900K out of the equation, there has to be some other issue elsewhere. You should get a bit below 300K. The equation is otherwise sound, so it has to be either a number crunching error, or bad constants. Check those first.
  4. May 6, 2017 #3
    Hey, sorry I have squared there on paper I just mistyped it here. Ok, I will check the numbers again. Thank you!
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