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Homework Help: Radiative equilibrium temperature of a satellite

  1. Oct 2, 2011 #1
    1. The problem statement, all variables and given/known data

    A small, perfectly black, spherical satellite is in orbit around the Earth. If the Earth radiates as a black body at an equivalent blackbody temperature [itex]T_{E}[/itex] = 255 K, calculate the radiative equilibrium temperature of the satellite when it is in the Earth’s shadow. Start by setting dQ as function of solid angle dω and let integration over the arc of solid angle be 2.21.

    2. Relevant equations

    [itex]F=\sigma T_{E}^{4}=\pi I[/itex]

    [itex]\int dE=\int \int I\, d\omega \, dA[/itex]

    3. The attempt at a solution

    First, I set up the integral of dE as:

    [itex]\int dE=\int_{A=\pi R_{E}^{2}}\int_{2.21}I\, d\omega \, dA[/itex]

    My only question here is my limit of integration for dA... Should it be over the area of Earth's disk or the entire surface area (which would be [itex]4\pi R_{E}^{2}[/itex])?

    Assuming the way I have it is right, I get:

    [itex]E=2.21\pi IR_{E}^{2}[/itex]

    Substituting in for I and then [itex]F_{E}[/itex], it becomes:

    [itex]E=2.21R_{E}^{2}\sigma T_{E}^{4}[/itex]

    And plugging in the values [itex]R_{E} = 6.37 * 10^{6}m[/itex], [itex]\sigma = 5.67 * 10^{-8} Wm^{-2}K^{-4}[/itex], and [itex]T_{E} = 255 K [/itex],

    [itex]E = 2.15 * 10^{16}W[/itex], which is the energy transfer to the satellite per unit time.

    Using the Steffan-Boltzmann Law again, I set up the equation for the temperature of the satellite as:

    [itex]F_{s}=\sigma T_{s}^{4}[/itex], which I believe is the same as [itex]\frac{E}{4\pi R_{E}^{2}}=\sigma T_{s}^{4}[/itex].

    Rearranging, [itex]T_{s}=\frac{E}{4\pi \sigma R_{E}^{2}}^{1/4}[/itex].

    Plugging in the values and calculating, I get [itex]T_{s} = 165 K [/itex].

    I was a bit uncertain about whether I did this last step correctly, and my answer seems a bit low...
  2. jcsd
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