Power received close to a large black body radiator

Question

I am interested in calculating the power received by an object near a black body radiator. Say, for example, I had a piece of paper perpendicular to the earth's surface normal. If I make assumptions about the earth's temperature, I can calculate its black body radiation. If I had a range of wavelengths I was interested in, I could calculate the power in that spectral range.

Now if I assume my paper is itself a blackbody in the spectral range of interest, then I believe I could say that if it were placed very, very close to the earth, then the power it absorbed would be the power emitted per unit area by the earth multiplied by the area of my paper. This is based on the image below.

My question is, what if I start to move my paper up? Is there anything I can say about the power it receives at different ranges from the surface of the earth? (Assume we are not so far from the earth that it can be treated as a point source). What I would like to be able to say is that the paper receives that maximum possible power when it is very, very close to the earth's surface. I am unsure how to prove this or if it is even true. Any kind of bounding limit on the received power I could derive would be helpful

Answer 1

Thermal radiation of a solid sphere that has the same temperature everywhere, has intensity that is (as a function of position) spherically symmetric - it depends only on distance from the center of the sphere. Also, it falls off with distance as inverse square of this distance - this follows from the assumption that power that goes through any imaginary concentric sphere outside the solid sphere has to be the same, so for any two radii $r_1,r_2$, radiation intensities are such that

$$ I_1 4\pi r_1^2 = I_2 4\pi r_2^2. $$

So indeed the captured power is the largest when the paper piece is on the surface, and decreases as it is pulled up.