= Inner NFW contraction to a flat rotation curve
{title2=$\rho'_D\propto r^{-5/3}$}
An inner <NFW profile> has $M_i(r_i)=Ar_i^2$, where $A=2\pi\rho_s r_s$. If the final total <circular speed> is $V$, <circular-shell adiabatic contraction> gives $r_i=(V^2/(GA))^{1/3}r^{2/3}$ and $M'_D=A^{1/3}(V^2/G)^{2/3}r^{4/3}$. Thus $\rho'_D=A^{1/3}(V^2/G)^{2/3}r^{-5/3}/(3\pi)$. The total density producing the flat curve is $V^2/(4\pi Gr^2)$, so the dark fraction tends to zero as $r^{1/3}$ despite its increase relative to the original cusp.
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