Q5-84GP The Sun revolves around the cent... [FREE SOLUTION]

Chapter 5: Q5-84GP (page 109)

The Sun revolves around the centre of the Milky Way Galaxy (Fig- 5-49) at a distance of about 30,000 light-years from the centre (\({\bf{1}}\;{\bf{ly}} = {\bf{9}}{\bf{.5}} \times {\bf{1}}{{\bf{0}}^{{\bf{15}}}}\;{\bf{m}}\)). If it takes about 200 million years to make one revolution, estimate the mass of our Galaxy. Assume that the mass distribution of our Galaxy is concentrated mostly in a central uniform sphere. If all the stars had about the mass of our Sun (\({\bf{2}} \times {\bf{1}}{{\bf{0}}^{{\bf{30}}}}\;{\bf{kg}}\)), how many stars would there be in our Galaxy?
Figure 5-49 Edge -on view of our galaxy

Short Answer

Expert verified

The mass of galaxy is \(3 \times {10^{41}}\;{\rm{kg}}\).

Step by step solution

Determination of mass of galaxy

The mass of the galaxy is dependent on the speed of rotation of galaxy and the radius of the Sun orbit. Using the equation of speed of galaxy, the equation for mass of galaxy can be determined.

Given Information

The radius of the Sun orbit is \({r_{{\rm{sun}} - {\rm{orbit}}}} = 30,000\;{\rm{light years}}\)

The time taken in one revolution is \(T = 200\;{\rm{million years}}\)

Determination of the mass of the Galaxy

The speed of revolution is given by,

\(v = \frac{{2\pi {r_{{\rm{sun}} - {\rm{orbit}}}}}}{T}\)

Using the expression for velocity, find equation for mass of galaxy.

\(\begin{aligned}v &= \sqrt {G\frac{{{M_{{\rm{galaxy}}}}}}{{{r_{{\rm{sun}} - {\rm{orbit}}}}}}} \\{M_{{\rm{galaxy}}}} &= \frac{{{r_{{\rm{sun}} - {\rm{orbit}}}}{v^2}}}{G}\\ &= \frac{{4{\pi ^2}{{\left( {{r_{{\rm{sun}} - {\rm{orbit}}}}} \right)}^3}}}{{G{T^2}}}\end{aligned}\)

Substitute values in the above equation,

\(\begin{aligned}{M_{galaxy}} = \frac{{4{\pi ^2}{{\left( {\left( {30000\;{\rm{light years}}} \right)\left( {\frac{{9.5 \times {{10}^{15}}\;{\rm{m}}}}{{1\;{\rm{light years}}}}} \right)} \right)}^3}}}{{\left( {6.67 \times {{10}^{ - 11}}\;{\rm{N}}{{\rm{m}}^2}/{\rm{k}}{{\rm{g}}^2}} \right){{\left( {\left( {200 \times {{10}^6}\;{\rm{yr}}} \right)\left( {\frac{{3.15 \times {{10}^7}}}{{1\;{\rm{yr}}}}} \right)} \right)}^2}}}\\ \approx 3 \times {10^{41}}\;{\rm{kg}}\end{aligned}\)

Thus, the mass of galaxy is \(3 \times {10^{41}}\;{\rm{kg}}\).

Determination of the number of stars in the Galaxy

The number of stars can be calculated as,

\({\rm{n}} = \frac{{{M_{galaxy}}}}{{{M_{Sun}}}}\)

Substiuute values in the equation, to find the number of stars.

\(\begin{aligned}{\rm{n}} &= \frac{{3.452 \times {{10}^{41}}\;{\rm{kg}}}}{{2.0 \times {{10}^{30}}\;{\rm{kg}}}}\\ &= 1.762 \times {10^{11}}\;{\rm{stars}}\end{aligned}\)

Thus, there are \(1.762 \times {10^{11}}\;{\rm{stars}}\) in the Galaxy.