The sun moves at a speed \(v_{S}=220 \mathrm{~km} / \mathrm{s}\) in a circular orbit of radius \(r_{S}=30,000\) light years around the center of the Milky Way galaxy. The earth requires \(T_{E}=1\) year to orbit the sun, at a radius of \(1.50 \times 10^{11} \mathrm{~m}\).

(a) Using this information, find a formula for the total mass responsible for keeping the sun in its orbit, as a multiple of the sun's mass \(M_{0}\), in terms also of the parameters \(v_{S}, r_{S}, T_{E}\), and \(r_{E}\). Note that \(G\) is not needed here!

(b) Find this mass numerically.