The Sun treated as a perfect Black Body

[Keiner Nichts]

Homework Statement

At lunch, the Sun's thermal energy incident on the surface of the Earth is 1.4 kW/m^2. Given the radius of the Sun, R, distance from Earth, r, and treating the Sun like a perfect black body, calculate the total intensity of its radiation and determine its temperature.

Homework Equations

The Attempt at a Solution

I have no idea which formula to actually use. I've attempted applying Rayleigh-Jeans and Planck's derivation of it but I simply cannot see how R and r come into play. Perhaps as integer wavelength conditions?

 

[Drakkith]

You're given the thermal energy incident on the Earth. How does that play into r, the distance from the Sun to the Earth? What happens to the incident power if r changes?

Knowing that the Sun outputs a constant amount of power, what has to happen as R varies?

 

[Keiner Nichts]

I know the given power is proportional to T^4 through the Stefan-Boltzmann law but I was not really taught the formula for how the incident power changes with r. I imagine it would be proportional to (1/r)^2? Also, temperature of the Sun would be proportional to either (1/R)^2 or (1/R)^3 for constant output, I think...

 

[Drakkith]

I imagine it would be proportional to (1/r)^2?

The inverse-square law, that's right. So why is this law important? What's happening as the distance changes?

Also, temperature of the Sun would be proportional to either (1/R)^2 or (1/R)^3 for constant output, I think...

Ah, but why? What's important about the radius of the Sun? Where exactly is the light emitted from?

 

[Keiner Nichts]

Oooh, ok, so it's only emitted from the surface. Now...if I were to consider the output as constant, it means that the given value in the beginning, let's call it P, would satisfy: P×4πr^2=P'×4πR^2 where P' is what I need to find. And then through Stefan Boltzmann I divide it by the constant and raise it to the power of 1/4 to find out the temperature at the photosphere. I hope that is correct...

 

[Drakkith]

What do you get if you do the math?

 

[Keiner Nichts]

Well, for the temperature I got the Sun's temperature at the photosphere (or a very good approximation at least, I got the real value from Wikipedia), so I imagine the total intensity is good as well. Thank you a lot!