# Proof of the inverse square law for a uniformly bright sphere

On page 8 of "Radiative processes in Astrophysics" by Rybicki and Lightman they have a proof that the flux of a uniform sphere is inversely proportional to the distance of the sphere from the observer. I don't understand why they claim $\theta_c=\sin^{-1}R/r$, where the geometry is shown on pg 8, figure 1.6. It seems to me that its should rather be $\theta_c=\tan^{-1}R/r$. But then I don't think the proof would work. Any help in clarifying this for me would be greatly appreciated.

• Unfortunately Google's not allowing Page8 to be displayed. Do you have another source? – Carl Witthoft Dec 9 '13 at 18:21
• The same image is remade for this homework set (NB pdf) – Kyle Kanos Dec 9 '13 at 18:27

## 1 Answer

If you look carefully at the figure, you will see that $r$ is the length of the hypotenuse (point $P$ to the center of the source), not the length of point $P$ to the tangent. Thus, we have that $\sin^{-1}R/r\equiv\theta_c$.

• Thanks, I was mistaking the line going from P to the tangent of the sphere as the hypotenuse. But now I can appreciate, the hypotenuse is in fact the horizontal line in the figure. I see now this makes sense as it is ofcourse the tangent which is perpendicular to the radius. – Virgo Dec 9 '13 at 20:25