Question

Calculate the approximate mass of the Milky Way Galaxy from the fact that the Sun orbits the galactic center every 230 million years at a distance of 27,000 light-years. (As dis-cussed in Chapter 19, this calculation tells us only the mass of the galaxy within the Sun’s orbit.)

Answer 1

Answer:

The approximate mass of the Milky Way Galaxy within the Sun's orbit is $4.73x10^{39}Kg$.

Explanation:

The Universal law of gravitation shows the interaction of gravity between two bodies:

$F = G\frac{Mm}{r^{2}}$  (1)

Where G is the gravitational constant, M and m are the masses of the two objects and r is the distance between them.

For this particular case, M is the mass of the Milky Way Galaxy and m is the mass of the Sun. Since it is a circular motion the centripetal acceleration will be:

$a = \frac{v^{2}}{r}$  (2)

Then Newton's second law ($F = ma$) will be replaced in equation (1):

$ma = G\frac{Mm}{r^{2}}$

By replacing (2) in equation (1) it is gotten:

$m\frac{v^{2}}{r} = G\frac{Mm}{r^{2}}$ (3)

Therefore, the mass of the Milky Way Galaxy can be determined if M is isolated from equation (3):

$M = \frac{rv^{2}}{G}$ (4)

But r is 27000 light years (r = 27000 ly), Notice that it is necessary to express r in units of meters:

$r = 27000ly \cdot \frac{9.461x10^{15}m}{1ly}$ ⇒ $2.55x10^{20}m$

Before using equation 4 it is necessary to find the velocity:

$v = \frac{d}{t}$ (5)

Where d is the distance and t is the time, for this case t = 230000000 years.

$t = 230000000years \cdot \frac{31536000s}{1year}$ ⇒ $7.25x10^{15}s$

Then, the values of d and t can be replaced in equation 5:

$v = \frac{2.55x10^{20}m}{7.25x10^{15}s}$

$v = 35172m/s$

Finally, equation 4 can be used:

$M = \frac{rv^{2}}{G}$

$M = \frac{(2.55x10^{20}m)(35172m/s)^{2}}{6.67x10^{-11}kg.m/s^{2}.m^{2}/kg^{2}}$

$M = 4.73x10^{39}Kg$

Hence, the approximate mass of the Milky Way Galaxy within the Sun's orbit is $4.73x10^{39}Kg$.

Appendix:

The value of a light year in meters can be determined by means of equation 5:

$v = \frac{d}{t}$

$d = v.t$

$d = (3x10^{8}m/s)(31536000s)$

$d = 9.461x10^{15}m$