Circular-shell adiabatic contraction ($r_i{M}_i(r_i)=r_f{M}_f(r_f)$)

In a slowly changing spherical gravitational potential, angular momentum is exactly conserved and the radial action is an adiabatic invariant. An initially circular orbit remains circular in the adiabatic limit, so $J_r=0$ and $L^2=GrM(r)$ give the displayed shell relation. If dark-matter shells do not cross, their enclosed dark mass is also conserved. A density profile alone does not specify this orbital model: eccentric-orbit populations require their full action distribution.

An inner NFW profile has $M_i(r_i)=A{r}_i^2$, 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 $M_D^{\prime }=A^{1/3}(V^2/G{)}^{2/3}{r}^{4/3}$. Thus ${\rho }_D^{\prime }=A^{1/3}(V^2/G{)}^{2/3}{r}^{-5/3}/(3\pi )$. The total density producing the flat curve is $V^2/(4\pi G{r}^2)$, so the dark fraction tends to zero as $r^{1/3}$ despite its increase relative to the original cusp.