Answer to Question #169328 in Astronomy | Astrophysics for kaashfi

Question

Given that the Sun takes ῀2 x 108 years to make a circular orbit around the Galactic Centre, staying at a distance of I0 kpc, estimate the mass of the Galaxy contained within the solar orbit, assuming that it is spherical and ignoring any effects of mass lying outside the orbit. (You may use Newtonian dynamics and gravitation.)

Answer 1

According to the third Kepler's law (see https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion#Third_law):

$$\dfrac{a^3}{T^2} \approx \frac{GM}{4\pi^2}$$

where $a = 10kpc \approx 3.086\times 10^{20}m$ is the radius of the orbit, $T = 2\times 10^8 years \approx 6.307\times 10^{15}s$ is the period, $G = 6.674\times 10^{-11}N\cdot m^2/kg^2$ is the gravitational constant, and $M$ is the mass of the Galaxy. The equality is approximate because we do not take into account the mass of the Sun. But it is clear that it is much much less then the mass of the Galaxy, thus, we can neglect it.

For $M$ obtain:

$$M = \dfrac{4\pi^2a^3}{GT^2}\\ M = \dfrac{4\pi^2\cdot (3.086\times 10^{20})^3}{6.674\times 10^{-11}\cdot (6.307\times 10^{15})^2} \approx 4.294\times 10^{43}kg$$

Answer. $4.294\times 10^{43}kg$