= Solution
Using the given constants, the mass is $2\times10^{41}\,\mathrm{kg}$ and the <velocity dispersion> is $2\times10^5\,\mathrm{m\,s^{-1}}$. The nominal <galactic dynamical time> is
$$
\frac{GM}{\sigma^3}=\frac{(7\times10^{-11})(2\times10^{41})}{(2\times10^5)^3}
=1.75\times10^{15}\,\mathrm s\simeq6\times10^7\,\mathrm{yr}.
$$
Thus \b[the collapse clock is a few tens of millions of years, of order $10^8$ years], with $R/\sigma\simeq3\times10^7$ years in the interior <singular isothermal sphere> convention. The nominal radius $GM/\sigma^2\simeq3.5\times10^{20}\,\mathrm m\simeq12\,\mathrm{kpc}$ 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.
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