Answer to Question #109181 in Mechanics | Relativity for Tanya Saini

Question

The sun is approximately 25,000 light years away from the centre of milky way and moves around it,in an approximately circularly path,in roughly 170 million years.given the sunlight takes approximately 8 mins to reach the Earth ,what is the ratio of masses of galaxy and sun?

Answer 1

Given:

$a_S$ - distance between the Sun and galaxy;

$T_S$ - period of rotation of the sun around the galaxy;

$a_E$ - distance between the earth and sun;

$T_E$ - 1 year, earth's sidereal period.

Solution:

If we treat the sun and the milky way as point masses, and if we assume that the solar mass in infinitesimal as compared to the mass of the galaxy, and the earth's mass is infinitesimal relative to the Sun, applying Kepler's third law, we get the following result.

For the galaxy-sun system:

$$\frac{a_S^3}{T_S^2}=\frac{GM_G}{4\pi^2}$$

For the earth-sun system:

$$\frac{a_E^3}{T_E^2}=\frac{GM_S}{4\pi^2}.$$

Therefore, to find the ratio, we only need to divide galaxy-sun equation by sun-earth 1.53equation:

$$\frac{M_G}{M_S}=\frac{T_E^2a_S^3}{T_S^2a_E^3}=\\ \space\\ =\frac{1^2(25000)^3}{170000000^2(8/(60\cdot24\cdot365))^3}=1.53\cdot10^{11}.$$