Weighing the Invisible

Gas orbits in a galaxy at radius $r = 20\ \text{kpc}$ with orbital speed $v = 220\ \text{km s}^{-1}$. Estimate the enclosed mass using $M(\lt r) = r v^2/G$. Use $1\ \text{kpc} = 3.086 \times 10^{21}\ \text{cm}$, $1\ \text{km s}^{-1} = 10^5\ \text{cm s}^{-1}$, $G = 6.674 \times 10^{-8}\ \text{cm}^3\,\text{g}^{-1}\,\text{s}^{-2}$, and $M_\odot = 2.0 \times 10^{33}\ \text{g}$.

$r = 20\ \text{kpc} \times (3.086 \times 10^{21}\ \text{cm/kpc}) = 6.172 \times 10^{22}\ \text{cm}$, and $v = 220\ \text{km s}^{-1} \times (10^5\ \text{cm s}^{-1}\,\text{per km s}^{-1}) = 2.20 \times 10^7\ \text{cm s}^{-1}$.

$v^2 = (2.20 \times 10^7)^2 = 4.84 \times 10^{14}\ \text{cm}^2\,\text{s}^{-2}$, so the numerator is $r v^2 = (6.172 \times 10^{22})(4.84 \times 10^{14}) = 2.99 \times 10^{37}\ \text{cm}^3\,\text{s}^{-2}$, and dividing by $G$ gives $M(\lt r) \approx 4.48 \times 10^{44}\ \text{g}$.

$\dfrac{\text{cm}\,(\text{cm}^2\,\text{s}^{-2})}{\text{cm}^3\,\text{g}^{-1}\,\text{s}^{-2}} = \text{g}$ — the result has units of mass, as a mass estimate must.

Converting, $M(\lt 20\ \text{kpc}) = 4.48 \times 10^{44}\ \text{g} \times (1\,M_\odot / 2.0 \times 10^{33}\ \text{g}) \approx 2.2 \times 10^{11}\,M_\odot$. This is a dynamical mass: the gravitating matter required to produce the measured motion under the model assumptions. It is not automatically the mass in stars — to separate stars, gas, black holes, and dark matter, astronomers compare this dynamical mass to the mass traced by light and gas emission.