Chapter 10, Problem 3P

To determine

The distance between the stars in an open cluster and in a globular cluster.

Answer to Problem 3P

The distance between the stars in the open cluster ranges from 1.25 pc to 5.8 pc.

The distance between the stars in the globular cluster ranges from 0.015 pc to 0.11 pc.

Explanation of Solution

Calculation:

The volume of a Sphere is $\frac{4}{3}\pi r^3$ . The share of the cluster’s volume belongs to a single star.

The volume of a cluster of stars, $V_{cluster}$ is related to its diameter $d_{cluster}$

  $V_{cluster}=\frac{4\pi }{3}{\left(\frac{d_{cluster}}{2}\right)}^3$

The volume of space between each star in an open cluster is the volume of the open cluster divide by the number of stars in the cluster, N .

  $V_{star}=\frac{V_{cluster}}{N_{cluster}}$

Substitute the volume of the cluster in volume of star.

  $V_{star}=\frac{\frac{4\pi }{3}{\left(\frac{d_{cluster}}{2}\right)}^3}{N_{cluster}}$

The distance between the stars, r is the radius of the sphere of empty space around each star.

  $\begin{array}{c}r={\left(\frac{3V_{star}}{4\pi }\right)}^{\frac{1}{3}}\\ ={\left(\frac{3}{4\pi }\left(\frac{\frac{4\pi }{3}{\left(\frac{d_{cluster}}{2}\right)}^3}{N_{cluster}}\right)\right)}^{\frac{1}{3}}\\ =\frac{d_{cluster}}{2{\left(N_{cluster}\right)}^{\frac{1}{3}}}\end{array}$

Because there is a range of star numbers, there is a range of distances between the stars. To calculate the maximum space between the stars, use the smallest number of stars in the cluster.

An open cluster has a diameter of 25 pc and a minimum number of 10 stars.

  $\begin{array}{c}{r}_{oc}=\frac{d_{oc}}{2{\left(N_{\min }\right)}^{\frac{1}{3}}}\\ =\frac{25\text{pc}}{2{\left(10\right)}^{\frac{1}{3}}}\\ =5.8\text{pc}\end{array}$

To calculate the minimum space between the stars, use the largest number of stars in the cluster. The maximum number of stars in an open cluster is 1000 stars.

  $\begin{array}{c}{r}_{oc}=\frac{d_{oc}}{2{\left(N_{\max }\right)}^{\frac{1}{3}}}\\ =\frac{25\text{pc}}{2{\left(1000\right)}^{\frac{1}{3}}}\\ =1.25\text{pc}\end{array}$

The distance between the stars in the open cluster ranges from 1.25 pc to 5.8 pc.

Global clusters have a range of diameters between 10 pc and 30 pc and contain between 100,000 and 1,000,000.

A globular cluster with the maximum space between the stars would have the largest diameter and the smallest number of stars.

  $\begin{array}{c}{r}_{oc}=\frac{d_{oc}}{2{\left(N_{\min }\right)}^{\frac{1}{3}}}\\ =\frac{30\text{pc}}{2{\left({10}^6\right)}^{\frac{1}{3}}}\\ =1.5\times {10}^{-2}\text{pc}\end{array}$

A globular cluster with the minimum space between the stars would have the smallest diameter and the greatest number of stars.

  $\begin{array}{c}{r}_{oc}=\frac{d_{oc}}{2{\left(N_{\max }\right)}^{\frac{1}{3}}}\\ =\frac{10\text{pc}}{2{\left({10}^5\right)}^{\frac{1}{3}}}\\ =0.1077\\ \approx 0.11\text{pc}\end{array}$

The distance between the stars in the globular cluster ranges from 0.015 pc to 0.11 pc.