Mass Density of the Galaxy: Blog 6, Worksheet 3.1, Problem 5

We know something suspicious is happening with the mass of the galaxy in our approximations. A good next step is to check related variables and see if they are suspicious too. A good example is density:

M(r < r) is related to the mass density ρ(r) by the integral: $$M(< r) = \int_{0}^{r}  4π'^{2} ρ(r') dr'$$ (Recall that the $4πr^{2}$ comes from the surface area of each spherical shell, and the dr' is the thickness of each thin shell; talk to a TF if this is not clear.) The fundamental theorem of calculus then implies that 4πr^2ρ(r) = dM(< r)/dr. For the case in question 4, what is ρ(r)? Is the density finite as r --> 0 in the case of a flat rotation curve?

The Milky Way's rotation curve 

Source

This calculation is simply a matter of using what we already know. We have from question 4: $$M(r) = \frac{ V_c^2 r }{G}$$ Taking the derivative gives us: $$\frac{dM}{dr} = \frac{ V_c^2}{G}$$ which we can set equal to $$\frac{dM}{dr} = 4 \pi r^2 \rho (r)$$ And solve for density. $$4 \pi r^2 \rho (r)  = \frac{ V_c^2}{G}$$ $$\rho (r)  = \frac{ V_c^2}{4 \pi r^2 G}$$

This calculated density would be infinite in the case of a flat rotation curve.