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) $\int_{0}^{120} 4 \pi r\left(1+\left(\frac{r}{120}\right)^{3}\right)^{-10} d r$

Question

Quizplus · Accepted Answer

To find the total number of stars in the cluster, we integrate the density function over the volume of the sphere. The volume element in spherical coordinates is $4\pi r^2 dr$, and when multiplied by the density function $\rho(r)$, it gives the integrand in option A.

The Globular Cluster M13 Is a Spherical Distribution of Stars ρ(r)=(1+(r120)3)−5\rho(r)=\left(1+\left(\frac{r}{120}\right)^{3}\right)^{-5}ρ(r)=(1+(120r​)3)−5

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