Intensity of Spherical Shell of Stars

[Kelli Van Brunt]

TL;DR Summary

What is the intensity of radiation at the center of a spherical shell of stars?

Given that L is the luminosity of a single star and there are n stars evenly distributed throughout this thin spherical shell of radius r with thickness dr, what is the total intensity from this shell of stars?
My calculations were as follows: Intensity is the power per unit area per steradian of the sky; the power per unit area is $\frac {nL}{4πr^2}$; the whole sky covers a solid angle of 4π steradians; and so the intensity should be equal to $\frac{nL}{16π^2r^2} dr$. However, my book says that the total intensity is $\frac{nL}{4π} dr$. Can anyone help explain my mistake here? I suspect the error has to do with my lack of familiarity with a solid angle, so if that could be explained, that would be very helpful.

 

[Kelli Van Brunt]

I think I can make a correction here, actually. My book did not define $n$ at all, but looking through later chapters it seems as though $n$ is more commonly used as a number density, rather than just the number of stars. In this case, setting n = # of stars / 4πr^2 dr would give me the same result that the book had. Sorry to inconvenience anyone with this thread, it was a misunderstanding of the variables on my part.

 

[Vanadium 50]

Why is there a dr in it at all? You know what it is for one star, and there are n of them.

 

[Kelli Van Brunt]

Why is there a dr in it at all? You know what it is for one star, and there are n of them.

For my original assumption where I defined n to be the number of stars, the dr was unnecessary, I agree. Looking back in the book, n referred to the density of stars instead (which was not explicitly stated, and that was what caused my confusion) which added a 1/dr factor that later had to be offset by tacking a dr to the end.

 

[hutchphd]

Your book is, I hope and trust, setting the stage for Olber's Paradox...