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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.

A rotating gravitational pattern resonates with a stellar orbit when their azimuthal frequencies agree. In action-angle variables, this is the $k_R=0$ resonance of an azimuthal harmonic. Its torque changes azimuthal angular momentum without changing radial action to leading order, allowing stellar radial migration with little heating. The circular-orbit location is the corotation radius.

The $k_R=-1$ orbital resonance of an $m$-fold perturbation of a galactic disk. For near-circular stellar orbits, ${\Omega }_R$ is the radial epicyclic frequency. The harmonic changes radial action and azimuthal angular momentum in the ratio $\Delta J_R/\Delta J_{\phi }=-1/m$, so an orbit losing angular momentum gains random radial motion.

The $k_R=+1$ orbital resonance of an $m$-fold perturbation of a galactic disk. For near-circular stellar orbits, ${\Omega }_R$ is the radial epicyclic frequency. The harmonic changes radial action and azimuthal angular momentum in the ratio $\Delta J_R/\Delta J_{\phi }=1/m$, so an orbit gaining angular momentum gains random radial motion.

Let $⟨\cdot ⟩$ denote $(2\pi {)}^{-2}\int d^2\theta$. Average the Collisionless Boltzmann equation to second order. All first-order angle derivatives average to zero, leavingAn angle integration by parts shows thatThus the angle-averaged galactic distribution function obeys an action-space continuity equation,This sign also follows from Hamilton's equations, ${\dot{J}}_i=-{\partial }_{{\theta }_i}\delta \Phi$: the flux is $⟨\delta f{\dot{J}}_i⟩$.

Reality gives $\delta {\varphi }_{-\mathbf{k}}=\delta {\varphi }_{\mathbf{k}}^{\ast }$ and $\delta f_{-\mathbf{k}}=\delta f_{\mathbf{k}}^{\ast }$. Orthogonality of the Fourier series impliesInsert b to obtainAlthough individual terms are complex, pairing $\mathbf{k}$ with $-\mathbf{k}$ makes the flux real.

Now take the marginalization specified by $F_0=2\pi {\int }_0^{\infty }{f}_{0}d{J}_R$. Assume vanishing radial-action flux at $J_R=0$ and $J_R\to \infty$: the first boundary does not permit negative radial action, and the second has no escaping flux in this calculation. ThenwithIf radial-action boundary flux were present, its explicit boundary term would have to be retained.

For $k_R=\pm 1$, the selected orbital resonances satisfyFor near-circular stellar orbits, ${\Omega }_R$ is the radial epicyclic frequency. Thus $k_R=-1$ is the inner Lindblad resonance, $m({\Omega }_{\phi }-{\Omega }_{\mathrm{p}})={\Omega }_R$, and $k_R=+1$ is the outer Lindblad resonance, $m({\Omega }_{\phi }-{\Omega }_{\mathrm{p}})=-{\Omega }_R$.

The flux is driven by $m{\partial }_{J_{\phi }}{f}_0\pm {\partial }_{J_R}{f}_0$, and each resonant harmonic exchanges both angular momentum and radial action in the ratioThese terms can therefore produce galactic disk heating as well as angular momentum transport. In the usual absorbing response, stars losing angular momentum at the inner Lindblad resonance gain radial action, and stars gaining angular momentum at the outer Lindblad resonance also gain radial action. The actual direction of flux follows from the displayed derivatives of $f_0$; it is not fixed by the resonance condition alone. A finite pulse replaces exact Lindblad resonance selection by a band of detunings of width of order $1/\tau$.

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.

A change in a star's mean orbital radius through exchange of angular momentum, particularly at a corotation resonance with a rotating perturbation. For the $k_R=0$ harmonic in action-angle variables, the leading change of radial action vanishes even while azimuthal angular momentum changes. Migration can therefore rearrange a galactic disk with much less heating than exchange at a Lindblad resonance.