Chapter 22, Problem 26Q

(a)

To determine

The orbital period of the cloud, if a gas cloud located in the spiral arm of a distant galaxy has an orbital velocity of 400 km/s while moving in a circular orbit and the distance from the center of the galaxy is 20,000 pc.

Answer to Problem 26Q

Solution:

$3.1\times {10}^8\text{yr}$

Explanation of Solution

Given data:

For a gas cloud located in the spiral arm of a distant galaxy, the orbital velocity is $400\text{km/s}$ and the distance from the center of the galaxy is $20,000\text{pc}$.

Formula used:

The expression for time taken by an object in circular motion to complete an orbit is,

$P=\frac{2\pi r}{v}$

Here, $P$ is the orbital period, $r$ is the distance from the center of the circular orbit and $v$ is the orbital speed.

Conversion from km/s to m/s is performed as,

$1\text{km/s =1000}\text{m/s}$

Conversion from pc to m is performed as,

$1\text{pc }=3.\text{09}\times {\text{10}}^{16}\text{m}$

Conversion from 1 s to 1 year is performed as,

$\begin{array}{c}\text{1}\text{year }=3\text{65}\text{days}\\ \text{1 day }=24\text{hours}\\ \text{1 hour }=6\text{0}\text{minute}\\ \text{1minute}=6\text{0}\text{second}\end{array}$

Hence, $1\text{year }=3.\text{16}\times {\text{10}}^7\text{s}$

Explanation:

Write the expression for the period of the cloud’s orbit around the galactic center.

$P=\frac{2\pi r}{v}$

Substitute $400\text{km/s}$ for $v$ and $20,000\text{pc}$ for $r$.

$\begin{array}{c}P=\frac{2\pi \left(20,000\text{pc}\right)}{400\text{km/s}}\\ =\frac{2\pi \left(20,000\text{pc}\left(\frac{3.09\times {10}^{16}\text{m}}{1\text{pc}}\right)\right)}{\left(400\text{km/s}\left(\frac{1000\text{m/s}}{1\text{m/s}}\right)\right)}\\ =\frac{2\pi \left(6.18\times {10}^{20}\text{m}\right)}{\left(4\times {10}^5\text{m/s}\right)}\\ =9.71\times {10}^{15}\text{s}\end{array}$

Further, solve as,

$\begin{array}{c}P=9.71\times {10}^{15}\text{s}\left(\frac{1\text{yr}}{3.15\times {10}^7\text{s}}\right)\\ \simeq 3.1\times {10}^8\text{ yr}\end{array}$

Conclusion:

Therefore, the orbital period of the cloud is $3.1\times {10}^8\text{yr}$.

(b)

To determine

The mass of the galaxy contained within the cloud’s orbit, if a gas cloud located in the spiral arm of a distant galaxy has an orbital velocity of 400 km/s while moving in a circular orbit and the distance from the center of the galaxy is 20,000 pc.

Answer to Problem 26Q

Solution:

$7.4\times {10}^{11}{\text{ M}}_{\odot }$

Explanation of Solution

Given data:

For a gas cloud located in the spiral arm of a distant galaxy, the orbital velocity is $400\text{km/s}$ and the distance from the center of the galaxy is $20,000\text{pc}$.

Formula used:

The expression for mass of our galaxy within the cloud is,

$M=\frac{r{v}^2}{G}$

Here, $r$ is the distance from the center of the galaxy when the cloud is moving in circular orbit, $v$ is the orbital speed of the cloud and $G$ is the constant of gravitation.

The value of $G$ is $6.673\times {10}^{-11} {\text{m}}^{\text{3}}\text{/kg}\cdot {\text{s}}^{\text{2}}$

Conversion from km/s to m/s is performed as,

$1\text{km/s }=1\text{000}\text{m/s}$

Conversion from pc to m is performed as,

$1\text{pc }=3.\text{09}\times {\text{10}}^{16}\text{m}$

Explanation:

Recall the expression for mass of our galaxy within the cloud.

$M=\frac{r{v}^2}{G}$

Substitute $400\text{km/s}$ for $v$, $20,000\text{pc}$ for $r$ and $6.673\times {10}^{-11} {\text{m}}^{\text{3}}\text{/kg}\cdot {\text{s}}^{\text{2}}$ for $G$.

$\begin{array}{c}M=\frac{\left(20,000\text{pc}\left(\frac{3.09\times {10}^{16}\text{m}}{1\text{pc}}\right)\right){\left(400\text{km/s}\left(\frac{1000\text{m/s}}{1\text{km/s}}\right)\right)}^2}{6.673\times {10}^{-11} {\text{m}}^{\text{3}}\text{/kg}\cdot {\text{s}}^{\text{2}}}\\ =\frac{\left(6.18\times {10}^{20}\text{m}\right){\left(4\times {10}^5\text{m/s}\right)}^2}{6.673\times {10}^{-11} {\text{m}}^{\text{3}}\text{/kg}\cdot {\text{s}}^{\text{2}}}\\ =14.8\times {10}^{41}\text{kg}\left(\frac{1{\text{ M}}_{\odot }}{2\times {10}^{30}\text{ kg}}\right)\\ =7.4\times {10}^{11}{\text{ M}}_{\odot }\end{array}$

Conclusion:

Therefore, the mass of our galaxy within the cloud is $7.4\times {10}^{11}{\text{ M}}_{\odot }$