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

question 15

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

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Definitions:

Intercept

A term in mathematics and economics indicating the value where a function or line crosses an axis on a graph.

Budget Line

A budget line represents the combinations of two goods that a consumer can purchase with a given income and prices, illustrating the trade-off between the two goods.

Risk-free Asset

A Risk-free Asset is an investment with a certain return, with no risk of financial loss, often exemplified by government bonds.

Risky Asset

An investment with a significant degree of uncertainty in its future payoffs, including the potential for losing some or all of the original investment.