Question

The sun, which is $2.2\times {10}^{20}$ m from the center of the Milky Way galaxy, revolves around that center once every $2.5\times {10}^8$ years. Assuming each star in the Galaxy has a mass equal to the Sun's mass of $2.0\times {10}^{30}$ 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.

Answer 1

The centripetal force on the Sun is due to the gravitational attraction between the Sun and the stars at the center of the Galaxy.

The magnitude of the Sun's acceleration is $a=v^2/R,$ where $v$ is its speed. If $T$ is the period of the Sun's motion around the galactic center then $v=2\pi R/T$ and $a=4{\pi }^{2}R/T^2$ Newton's second law yields

$GN{M}^2/R^2=4{\pi }^{2}MR/T^2$

The solution for $N$ is

$N=\frac{4{\pi }^2{R}^3}{G{T}^{2}M}$

The period is $2.5\times {10}^{8}y$, which is $7.88\times {10}^{15}s$, so

$N=\frac{4{\pi }^2{(2.2\times {10}^{20}m)}^3}{(6.67\times {10}^{-11}{m}^3/s^2\cdot kg){(7.88\times {10}^{15}s)}^2(2.0\times {10}^{30}kg)}=5.1\times {10}^{10}$

The number of stars in the Milky Way is between ${10}^{11}$ to $4\times {10}^{11}$. Our simplified model provides a reasonable estimate.