H3 - the cored isothermal sphere

the velocity profile of a spiral can be well described by combining two components:
- Keplerian component with $v (r) \propto r^{- 1 / 2}$
- dark matter halo component
the dark matter halo component is well described by the cored isothermal sphere
the total mass enclosed can be calculated as:

M_{e n c} = \int_{0}^{r_{e n c}} ρ (r) 4 π r^{2} d r

for the integral to be proportional to $r$ (as necesary for $v (r) \propto M (r) / r =$ constant), $ρ (r) \propto r^{- 2}$ is required for a constant integrand
a common equation satisfying this is:

ρ (r) = \frac{v_{h}^{2}}{4 π G} \frac{1}{r^{2} + a^{2}}

where, $v_{h}$ is the asymptotic rotational velocity, and $a$ is the core radius

eg, for the MW, $M_{e n c} \approx 5 \times 10^{11} M_{⊙}$ within $\sim 50$ Mpc, but as inferred from the visible content, $M_{v i s} \approx 5 \times 10^{10} M_{⊙}$ , which is $\sim 10 %$
over the full virial radius $\sim 200$ kpc, $M \sim 10^{12} M_{⊙}$ , so the dark matter fraction is even larger