How do you model an NFW Dark Matter Halo?

I am trying to reproduce the results of Sofue's 2016 paper (https://arxiv.org/abs/1510.05752). I'm able to use least squares minimization to model the bulge and disk effectively, but I get into trouble with the halo. By simply minimizing the residuals of this equation:

$$M_h(R)=4 \pi \rho_0h^3\left(\ln\left(1+X\right)-\frac{X}{1+X}\right)$$ $$V_h(R)=\sqrt{\frac {G M_h(R)}{R}}$$

I can match the data pretty well, except I get a density that is far less dense than critical, which is impossible (you can't have a halo that is less dense than the surrounding universe). Sofue provides some equations describing $M_{200}$, $R_{200}$ and $X_{200}$, but I can't connect the dots. How do you model an Navarro-Frenk-White (NFW) halo so that it matches the data but has a reasonable density?

Update: Sofue has an $R_{MAX}$ value which limits the size of the halo scale radius, but even using this, the minimization always hits $R_{MAX}$ as a limit for the halo scale radius. It appears that any attempt to simply match a dark matter halo to a velocity curve is going to dilute the dark matter to an unreasonable quantity.

• Model how? Like plotting the distribution? For use in $n$-body simulations? – Kyle Kanos Aug 26 '17 at 19:47
• @KyleKanos - To compare observed velocity against predicted you need to determine the masses for the bulge, disk and halo components of a galaxy. Typically you use properties like the bulge mass, bulge scale radius, disk central density, disk scale radius, halo representative density, halo scale radius to model the total mass at a given radius. This mass tells you the predicted velocity of an object in orbit at that radius. The model velocity can then be compared to the measured velocity. – Quarkly Aug 27 '17 at 19:20

protected by Community♦Sep 21 '17 at 12:17

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