Chapter 6 Problem 60

Problem

The Sun rotates about the center of the Milky Way Galaxy at a distance of about 30,000 light-years from the center (1 ly = $9.5\times 10^{35}$ m). If it takes about 200 million years to make one rotation, estimate the mass of our Galaxy. Assume that the mass distribution of our Galaxy is concentrated mostly in a central uniform sphere. If all the stars had about the mass of our Sun ($2\times 10^{30}$ kg), how many stars would there be in our Galaxy?

Solution

Like the previous question

$G{\frac {m_{\text{Sun}}M_{\text{galaxy}}}{R^{2}}}=m_{\text{Sun}}{\frac {v^{2}}{R}}$ $\rightarrow v={\sqrt {G{\frac {M_{\text{galaxy}}}{R}}}}$

$M_{\text{galaxy}}={\frac {Rv^{2}}{G}}$

also

$v={\frac {2\pi R}{T}}$

thus

$M_{\text{galaxy}}={\frac {4\pi ^{2}R^{3}}{GT^{2}}}={\frac {4\pi ^{2}\left(30000\times 9.5\times 10^{35}\right)^{3}{\text{m}}^{3}}{\left(6.67\times 10^{-11}{\text{Nm}}^{2}/{\text{kg}}^{2}\right)\left[\left(200\times 10^{6}{\text{y}}\right)\left(3.14\times 10^{7}{\text{s/y}}\right)\right]^{2}}}$ $=3.452\times 10^{41}{\text{kg}}\approx 3\times 10^{41}{\text{kg}}$

Number of stars $={\frac {M_{\text{galaxy}}}{M_{\text{sun}}}}={\frac {3.452\times 10^{41}{\text{kg}}}{2\times 10^{30}{\text{kg}}}}$ $=1.726\times 10^{11}\approx 2\times 10^{11}$

see Fermi Paradox