Question

The Sun orbits the center of the Galaxy in 225 million years at a distance of 26,000 lightyears. Given that a^3 = (M1 +M2) x P^2, where a is the semimajor axis in AU and P is the orbital period in years, what is the mass of the Galaxy within the Sun’s orbit?

Answer 1

According to Kepler third law the relation between orbital period and radio of matter in the Galaxy is given by

$a^3 = (M_1+M_2)p^2$

Where,

a = Radius of start orbit

p = Orbital Period

M = Total mass in a sphere of radius centered on galactic center

Our values are given as

$a = 26000ly (\frac{9.461*10^{15}m}{1LY}) = 2.45*10^{20}m$

$p = 225million year = 225*10^6 year$

Replacing we have,

$M = \frac{2.45*10^{20}}{225*10^6}$

$M = 1.088*10^{12}M_{sun}$

$M = 108Billion M_{sun}$

Therefore the mass of the galaxy within the sun's orbit is 108Billion the mass of the sun.