What is density of dark matter as a function of distance from the galactic core?

[Lotto]

Homework Statement

Because of dark matter, our stars in the Galaxy with core of radius ##r_1##, whose matter is distributed spherically symetrically, are moving with speed ##v_0## that is not dependent on distance from the Galaxy's centre. This is true for distances from the centre smaller than ##r_2##. Consider the distribution of dark matter to be spherically symetric around the centre. What is denisty of dark matter dependent on distance ##r## from the Galaxy's centre in interval from ##r_1## to ##r_2##?

Relevant Equations

##G\frac{mM}{r^2}=m\frac {{v_0}^2}{r}##
##M_1=\frac{{v_0}^2r_1}{G}##

This problem builds on my previous post, where we calculated that core's mass is ##M_1=\frac{{v_0}^2r_1}{G}##. So if we consider mass of dark matter dependent on distance ##r## to be ##M_2(r)##, we can calculated it from

##G\frac{(M_2(r)+M_1)m}{r^2}=m\frac{{v_0}^2}{r}.##

So ##M_2(r)=\frac{{v_0}^2}{G}(r-r_1)##.

An average density in interval from ##r## to ##r+\Delta r## is ##\frac{\frac{{v_0}^2}{G}(r+\Delta r-r_1)-\frac{{v_0}^2}{G}(r-r_1)}{\frac 43 \pi [(r+\Delta r)^3-{r_1}^3]-\frac 43 \pi (r^3-{r_1}^3)}##. If we make a limit of it for ##\Delta r \rightarrow 0##, we get

##\rho(r)=\frac{{v_0}^2}{4\pi r^2G}##.

Is it correct? If not, where are my thoughts wrong?

 

[mfb]

Looks right.