Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 60 / 3 / ii / Solution

Using the given constants, the mass is $2\times 10^{41}\mathrm{kg}$ and the velocity dispersion is $2\times 10^5\mathrm{m}{\mathrm{s}}^{-1}$. The nominal galactic dynamical time isThus the collapse clock is a few tens of millions of years, of order $10^8$ years, with $R/\sigma \simeq 3\times 10^7$ years in the interior singular isothermal sphere convention. The nominal radius $GM/{\sigma }^2\simeq 3.5\times 10^{20}\mathrm{m}\simeq 12\mathrm{kpc}$ is a plausible galactic scale. A full stellar orbit takes several crossing times, giving order $10^8$ years near a galactic disc radius, whereas the age of the Milky Way is of order $10^{10}$ years. Many orbital passages therefore fit into the lifetime of the galaxy, as expected; a galactic dynamical time need not equal either its full orbital period or its age.