Question

The Sun, which is$\mathbf{2}\mathbf{.2}\mathbf{\times }{\mathbf{10}}^{\mathbf{20}}\mathbf{}\mathbf{\text{m}}$from the center of the Milky Way galaxy, revolves around that center once every $\mathbf{2.5}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{}\mathbf{\text{years}}$. Assuming each star in the Galaxy has a mass equal to the Sun’s mass of $\mathbf{2.0}\mathbf{\times }{\mathbf{10}}^{\mathbf{30}}\mathbf{}\mathbf{\text{kg}}$, the stars are distributed uniformly in a sphere about the galactic center, and the Sun is at the edge of that sphere, estimate the number of stars in the Galaxy.

Short answer

The number of stars in the galaxy are $5.1\times {10}^{10}.$

Step-by-step solution

Step 1: Given

The sun is $2.2\times {10}^{20}\text{m}$far from the center of the Milky Way galaxy

The sun revolves around that center once every ${\text{2.5×10}}^{\text{8}}\text{years}$

Each star in the galaxy has a mass equal to the sun’s mass of $2.0\times {10}^{30}$

Determining the concept

Using the formula of gravitational force, centripetal force and period, find the acceleration of the sun’s motion around the galactic center. Then, using Newton’s second law, find the number of the stars.

Formulae are as follow:

${\mathrm{F}}_{\mathrm{g}}=\frac{\mathrm{GMm}}{{\mathrm{R}}^2}$

${\mathrm{F}}_{\mathrm{C}}=\frac{{\mathrm{Mv}}^2}{\mathrm{R}}$

$\mathrm{V}=\frac{2\mathrm{\pi R}}{\mathrm{T}}$

where, M, m are masses, R is radius, T is time, G is gravitational constant, v, V are velocities and F is corresponding force.

Determining the number of stars in the galaxy

Consider the total mass in the galaxy as $\mathrm{m}=\mathrm{NM}$ ,

where, N is the number of stars in the galaxy and M is the mass of the sun.

Now,

${\mathrm{F}}_{\mathrm{g}}=\frac{\mathrm{GMm}}{{\mathrm{R}}^2}$

Therefore,

${\mathrm{F}}_{\mathrm{g}}=\frac{\mathrm{GMNM}}{{\mathrm{R}}^2}$

${\mathrm{F}}_{\mathrm{g}}=\frac{{\mathrm{GNM}}^2}{{\mathrm{R}}^2}$

The centripetal force on the sun is pointing towards the galactic center,

${\mathrm{F}}_{\mathrm{C}}=\frac{{\mathrm{Mv}}^2}{\mathrm{R}}={\mathrm{F}}_{\mathrm{g}}$

$\frac{{\mathrm{Mv}}^2}{\mathrm{R}}=\frac{{\mathrm{GNM}}^2}{{\mathrm{R}}^2}$

If$\mathrm{T}$is the period of the sun’s motion around the galactic center then,

$\mathrm{T}=\frac{2\mathrm{\pi R}}{\mathrm{V}}$

$\mathrm{V}=\frac{2\mathrm{\pi R}}{\mathrm{T}}$

But,

$\mathrm{a}=\frac{{\mathrm{V}}^2}{\mathrm{R}}$

$\mathrm{a}=\frac{4{\mathrm{\pi }}^2\mathrm{R}}{{\mathrm{T}}^2}$

Therefore, according to Newton’s second law,

$\frac{{\mathrm{GNM}}^2}{{\mathrm{R}}^2}=\frac{4{\mathrm{\pi }}^2\mathrm{MR}}{{\mathrm{T}}^2}$

$\mathrm{N}=\frac{4{\mathrm{\pi }}^2{\mathrm{R}}^3}{{\mathrm{GT}}^2\mathrm{M}}$

As,

$\text{T\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}2.5}\times {\text{10}}^{\text{8}}\text{years}=\text{7.88}\times {\text{10}}^{\text{15}}\text{\hspace{0.17em}s}$

$\mathrm{N}=\frac{{\text{4π}}^{\text{2}}{\left({\text{2.2×10}}^{\text{20}}\text{\hspace{0.17em}m}\right)}^{\text{3}}}{({\text{6.67×10}}^{\text{-11}}{\text{\hspace{0.17em}m}}^{\text{3}}{\text{/s}}^{\text{2}}\cdot \text{kg}){\left({\text{7.88×10}}^{\text{15}}\text{\hspace{0.17em}s}\right)}^{\text{2}}\left({\text{2.0×10}}^{\text{30}}\mathrm{kg}\right)}$

$\mathrm{N}=5.1\times {10}^{10}$

Hence, the number of starts in the galaxy are $5.1\times {10}^{10}.$

Therefore, using the formula of gravitational force, centripetal force and period, the acceleration of the sun around the galactic center can be found. Using Newton’s second law, the number of stars in the galaxy can be found.