Question

Our Sun, with mass 2.00×1030 kg, revolves about the center of the Milky Way galaxy, which is 2.20×1020 m away, once every 2.50×108 years. Assuming that each of the stars in the galaxy has a mass equal to that of our Sun, that the stars are distributed uniformly in a sphere about the galactic center, and that our Sun is essentially at the edge of that sphere, estimate roughly the number of stars in the galaxy.

Answer 1

If the mass of the sun is m, then the total mass of the stars in the galaxy is Nm, where N is the number of stars. Assume the sun is moving in a circle and use the formula for centripetal acceleration. Then, the force of the galaxy on the sun is mv^2 / r , where v is the sun's velocity around the center of the galaxy and r is the galactic radius. From here, use Newton's law of gravity F = G *m_1 * m_2 / r^2 (where the m's are the different masses, and r again will be the galactic radius). Set the two forces equal to each other. In this case, m_2 will be the mass of the galaxy Nm.

F = GNm^2 / r^2 = mv^2 / r --->>> N = v^2 * r / (Gm)

The period of the orbit P = 2*pi*r / v, or, v = 2*pi*r / P. Substituting this in to the previous expression,

N = 4*pi^2*r^3 / (G m P^2) [make sure to convert P into seconds]

time period = P = 2.50×10^8 years = 2.50×10^8×12×30×24×60×60 seconds = 7.776×10^15 seconds

N = 4?^2×(2.20×10^20)/(6.67×10^-11×2×10^30×(7.776×10^15))

N = 5.21×10^10.....Approx.

Please check calculation carefuly.