Satellite orbiting the Earth (heat radiation)

[glorfindel1000]

Homework Statement

Satellite is orbiting earth and is heated by sun to 323 K. To what temperature satellite cools down during its 45 min cycle in shadow? Satellite is 1 m radius and 2 mm thick copper. Emissivity is 0,75. Space temperature is assumed to be 0 K.(hint. Write a formula to heat change in function of time, you have to integrate)

Relevant Equations

\begin{align*}
\frac{dQ}{dt} &= e \sigma A T^4 \text{ or } \frac{dQ}{dt} = e * \sigma * A * T^4 \\
\text{(maybe)}E &= cm \Delta T \\
A &= 4 * \pi * r^2 \\
\sigma &= 5,67 \times 10^{-8}{\rm W/(m^2 * K^4)} \\
e &= 0,75
\end{align*}

Mass of a satellite is 750g($m = \rho V = 8,96 \frac{g}{cm^3}\cdot (\frac{4\pi(100cm)^3}{3} -\frac{4\pi(99,9cm)^3}{3}) = \approx 750g =0,75kg$)

I am not sure what to integrate. I solved T there but it seems far stretched
$$T =\sqrt[4]{\frac{dQ}{dt}\frac{1}{e\sigma A}} $$

How to get the function to integrate is basically my problem. I changed that dt to other side

$dQ = e\sigma AT^4dt$

and integrated it from 0 to 2700.

$Q = \bigg/_{\!\!\! 0}^{\,2700}e\sigma AT^4t$

Not really sure if that helps. What to do with Q then?

 

[jim mcnamara]

Use two # characters like # # (minus the space) to delimit an inline latex statement. Use $ $ to delimit a separate block. Below the bottom left of the window you are typing in, there is a link for more latex help. I edited your post to fix latex. Please make sure I did not cause more problems.

 

[vela]

I fixed the LaTeX in the relevant equations section as well. Separate equations were running together, which made them confusing to understand.

In your second relevant equation, $E = mc\Delta T$, what does the $E$ stand for? How is it related to the heat radiated away by the satellite?