The Globular Cluster M13 Is a Spherical Distribution of Stars $\rho(r)=\left(1+\left(\frac{r}{120}\right)^{3}\right)^{-5}$

Multiple Choice

The globular cluster M13 is a spherical distribution of stars which orbits our galaxy.Suppose that the density of stars in the cluster is purely a function of distance r from the center of the cluster and is given as $\rho(r) =\left(1+\left(\frac{r}{120}\right) ^{3}\right) ^{-5}$ stars/(ly) ³, where r is measured in light-years, and 0 $\le$ r $\le$ 120 ly.(One light-year is the distance light travels in one year; "light-year" is abbreviated as "ly".) Set up an integral whose value is the exact number of stars in M13.

A) $\int_{0}^{120} 4 \pi r^{2}\left(1+\left(\frac{r}{120}\right) ^{3}\right) ^{-5} d r$
B) $\int_{0}^{120} 2 \pi r\left(1+\left(\frac{r}{120}\right) ^{3}\right) ^{-5} d r$
C) $\int_{0}^{120} \pi\left(1+\left(\frac{r}{120}\right) ^{3}\right) ^{-10} d r$
D) $\int_{0}^{120} 4 \pi r\left(1+\left(\frac{r}{120}\right) ^{3}\right) ^{-10} d r$