Exploring Olbers' Paradox in Matt Roots' Introduction to Cosmology

[alejandromeira]

I'm beginning to study the Matt Roots book Introduction to Cosmology and in the section 1.3 Olbers' Paradox he writes:
"If the surface area of an average star is A, then its brightness is B=L/A. The sun may be taken to be such an average star, mainly because we know it so well.
The number of stars in a spherical shell of radius r and thickness dr is then $4\pi r²ndr$. Their total radiation as observed at the origin of a static universe of infinite extent is then found by integrating the spherical shells from 0 to $\infty$:"
$$\int_{0}^\infty 4\pi r^2nBdr = \int_{0}^\infty nLdr = \infty$$
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I suppose that he use $B=\frac{L}{4\pi r^2}$ for obtain the second integral, but r is the radius of the shell not the average radius of the stars. I'm a little bit confused whit that.

Of course if the Universe is infinite and the integration runs from 0 to infinity the total luminosity must be infinity.

My doubt is about the use of r above, in the radius of shell and also the same letter for the radius of a star... and then vanishing... I'm a little bit confused.

 

[Orodruin]

$A$ should be the area of a sphere with radius equal to the distance to the star. This is because the energy flux from the star is assumed to be evenly spread over that sphere.

 

[alejandromeira]

Ok. it is understood. Also just after your answer I was thinking that the energy that we receive from a star a distance r, must be spreaded in a sphere of radius r.
Ok thanks a lot! Thread solved.