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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.

Use the usual spherical circular-shell adiabatic contraction approximation: initial dark-matter orbits are circular, slow accumulation keeps them circular, and dark shells do not cross. Angular momentum is conserved in a spherical potential, and the vanishing radial action stays zero as an adiabatic invariant. For a circular shell,Here $r_i$ is the initial shell radius and $r$ its final radius. The final flat circular speed $V=2v_{\mathrm{NFW},\mathrm{c},\max }$ gives $M_f(r)=V^{2}r/G$.

For an initial shell in the inner NFW profile, write $A=2\pi {\rho }_1{r}_1$, so $M_i(r_i)\simeq A{r}_i^2$. The invariant givesConservation of dark shell mass yieldsand differentiation givesTo express the requested ratios, set $K=F(x_{\max })\simeq 0.2162166$ and $B=8K\simeq 1.72973$, so $V^2/(GA)=B{r}_1$. Since the original cusp has ${\rho }_D(r)\simeq A/(2\pi r)$,The total density of the final spherical flat-rotation model is ${\rho }_{\mathrm{tot}}=V^2/(4\pi G{r}^2)$. Its dark fraction isSubtracting dark matter from the total gives the added-matter density, ${\rho }_B={\rho }_{\mathrm{tot}}(1-h)$, and thusThese expressions describe the leading inner approximation, not an extension to radii where the denominator becomes negative or the initial shell is outside the NFW cusp. As $r/r_1\to 0$, dark matter is enhanced relative to its original density while its fraction of the even steeper total density tends to zero. This is inner NFW contraction to a flat rotation curve.

The orbital assumptions matter. An initial density profile does not specify the phase-space distribution function. For eccentric orbits, slow evolution conserves their actions, but $rM(r)$ at an instantaneous radius is not generally their invariant. The above ratios are the intended circular-shell estimate; their numerical coefficients cannot be uniquely inferred from the density profile and slow accumulation alone.

The radial action isWith the variable $x$ from part b this becomesApplying the supplied integral twice, with $c=0$ and $c=2$, and using the root sum and products from parts b and c givesThis is the radial action of the spherical isochrone model. Rearranging,and squaring yields the Hamiltonian in action-angle variables:As a check, differentiating this Hamiltonian gives ${\Omega }_r=(-2E{)}^{3/2}/(GM)$ and the frequency ratio found in part c.

The spherical isochrone model has potentialIts radial period depends only on orbital energy, while its radial action and frequencies are elementary functions of the integrals of motion.