Radiative equilibrium temperature of a satellite

[guitarstorm]

Homework Statement

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 $T_{E}$ = 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.

Homework Equations

$F=\sigma T_{E}^{4}=\pi I$

$\int dE=\int \int I\, d\omega \, dA$

The Attempt at a Solution

First, I set up the integral of dE as:

$\int dE=\int_{A=\pi R_{E}^{2}}\int_{2.21}I\, d\omega \, dA$

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 $4\pi R_{E}^{2}$)?

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

$E=2.21\pi IR_{E}^{2}$

Substituting in for I and then $F_{E}$, it becomes:

$E=2.21R_{E}^{2}\sigma T_{E}^{4}$

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

$E = 2.15 * 10^{16}W$, 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:

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

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

Plugging in the values and calculating, I get $T_{s} = 165 K$.

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

 

[mark726]

any feedback would be appreciated!Your approach and calculations look correct to me. As for the limit of integration for dA, it should be over the entire surface area of the Earth, so 4πR_E^2. This is because the satellite is receiving radiation from all parts of the Earth's surface, not just the disk that is facing it.

As for the final temperature, it does seem a bit low, but it is possible for the satellite to have a lower equilibrium temperature in the Earth's shadow compared to when it is facing the sun. This is because the satellite is not only receiving radiation from the Earth, but it is also radiating its own energy out into space. This balance between incoming and outgoing radiation leads to a lower equilibrium temperature.

Overall, your solution looks good and your calculations seem correct. Great job!