# What is the mass of the Milky Way?

We can consider the Sun to be a satellite of the Milky Way. The Sun revolves around the center of the Milky Way with a radius of 8 kpc. The period of revolution is 230 million years. What is the mass of the Milky Way?

###### Solution.

The gravitational force between the center of the Milky Way and the Sun is in the radial direction and its magnitude is given by the Newton’s equation

F = GMm/r2    (1)

where G is the gravitational constant, M and m are the masses of the Milky Way and the Sun respectively and r is the radius of the orbit.

In case of the circular motion, the net force equals mass times centripetal acceleration, where the centripetal acceleration can be calculated by ω2r, where ω is the angular rate of rotation also known as angular velocity.

Thereby,

F = mac = mω2r.   (2)

The angular velocity is given by

ω = 2π/T,   (3)

where T is the period of revolution.

We substitute (3) into the equation (2) and we get

F = m4π2r/T2.   (4)

We can equate equations (4) and (1) to obtain

m4π2r/T2 = GMm/r2.

We divide the last equation by m and we get

2r/T2 = GM/r2.   (5)

We can rewrite the equation (5) as

M = 4π2r3/(GT2)   (6)

1 kpc (kiloparsec) equals 3.086×1019 m. So, 8 kiloparsecs equal to 2.469×1020 m.

1 year equals 3.154×107 s and 230 million years equal to 7.253×1015 s.

We substitute the values into the equation (6) and solve

M = 4π2×(2.469×1020)3/(6.67408×10−11×(7.253×1015)2) ≈ 1.69×1041 kg.