A satellite, the sun and the satellites heat protector

[Olly_price]

Homework Statement

A spacecraft is protected from the Sun’s radiation by a planar baffle whose size is much greater than that of the spacecraft itself. The baffle is aligned perpendicular to the direction of the Sun. Show that the equilibrium temperature of the baffle is $$T_b=\Big(\frac{\alpha_d^2}{8}\Big)^{\frac{1}{4}}T_s$$ where T_s is the heat of the sun and is 5800K, $\alpha_d$ is the angular diameter of the Sun as seen from the spacecraft .

Homework Equations

Flux of the sun = L / $2\pi d^2$ where $d$ is the distance of the Sun from the baffle and L is the luminosity of the Sun.
Any classical equations involving thermal equilibriums etc.

The Attempt at a Solution

I was thinking of using the flux of the Sun stated above, and then the flux of the Sun's radiation on the baffle, considering the Sun's rays projected onto the baffle. Some form of ratios may help, but I didn't get anywhere.

I also tried using some geometries involving the angular diameter but could not successfully isolate $\alpha$. Perhaps we need to take a small angle approximation?

 

[mfb]

I was thinking of using the flux of the Sun stated above, and then the flux of the Sun's radiation on the baffle

You can use that approach. It is probably useful to introduce some variable for its area. It will cancel out later but it makes the formulas easier to follow.

Perhaps we need to take a small angle approximation?

The final result looks like it uses that approximation, yes, but it should not be necessary to get a result.