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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.

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  • $\begingroup$ Unfortunately Google's not allowing Page8 to be displayed. Do you have another source? $\endgroup$ – Carl Witthoft Dec 9 '13 at 18:21
  • $\begingroup$ The same image is remade for this homework set (NB pdf) $\endgroup$ – Kyle Kanos Dec 9 '13 at 18:27
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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 $\sin^{-1}R/r\equiv\theta_c$.

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  • $\begingroup$ 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. $\endgroup$ – Virgo Dec 9 '13 at 20:25

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