Finding the Temp of the Sun given only 3 variables.

[MarcZero]

Homework Statement

This is for my Optics class:

Given the radius of our planet, its temperature, and the distance to the Sun, estimate the temperature of the Sun (hint: use the Stephan-Boltzmann law).

Homework Equations

Radius of Earth $$\widetilde{=}$$ 6,378.1 km
Temperature of the Earth $$\widetilde{=}$$ 14.4 C or 287.55 K
Distance from Sun to Earth $$\widetilde{=}$$ 150,000,000 km

j* = A$$\epsilon\sigma$$T⁴

(I don't know why it is showing the greek symbols as exponents. I can't seem to get them all to be in the normal line. I apologize about this.)

The Attempt at a Solution

I can use what is given and estimate the surface are of the Earth (A_Earth) as:

4$$\pi$$r² = 5.11 X 10⁸ km²

I am also assuming that the Sun is a perfect blackbody radiation source so that $$\epsilon$$ = 1.

So I am guessing that to find T_Sun I need to find the Irradiance (j*) of the Sun somehow using the radius and temperature of the earth. This question is from the professor and is not in our Optics book anywhere that I can see (Fowles - Introduction to Modern Optics). I am struggling to figure out how to relate the Earth's info to the Sun's Irradiance. I can't seem to find a formula to plug things into. I can imagine that somehow the Energy of the sun falls off as the distance increases and we could probably find somehow that this is related to the temperature of the Earth, but I am at a loss to do so. I know the units of J* are [W/s/m²], but I can't see how I'm supposed to relate these to what I have to work with.

Any insight or a point in the right direction would be most appreciated. Thank you for your time.

 

[Redbelly98]

I think the key, at least to get started, is to realize that the "equivalent area" that receives radiation from the sun is πr², while the radiating area of the Earth is its entire surface area.

(I would have thought we'll need the diameter of the sun too, but I haven't work this one out completely yet.)

Oh, here's a copy-and-pastable σ for you, on the house