= Circular-shell adiabatic contraction
{title2=$r_iM_i(r_i)=r_fM_f(r_f)$}
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.
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