 Solution

For the isotropic Jeans-Abel mass inversion, let $J(R)=I(R){\sigma }_{\mathrm{los}}^2(R)$ and $p_{\ast }(r)=\nu (r){\sigma }_r^2(r)$. The two preceding projections have the same Abel integral equation kernel, givingTheir ratio recovers ${\sigma }_r^2=p_{\ast }/\nu$, and the isotropic Jeans equation givesEquivalently put $A(r)={\int }_r^{\infty }{I}^{\prime }(R)/\sqrt{R^2-r^2}dR$ and $B(r)={\int }_r^{\infty }{J}^{\prime }(R)/\sqrt{R^2-r^2}dR$. Then $M=-r^2{B}^{\prime }(r)/(GA(r))$: the common stellar luminosity cancels from the inferred mass. Differentiating these singular-kernel integrals is understood through suitable regularity or a nonsingular variable change, not by naively keeping an infinite lower-end boundary term.

The inversion assumes adequate radial coverage, convergence and a smooth spherical equilibrium. In practice derivatives amplify noise and the unobserved outer tail needs treatment or a model. Unknown anisotropy reintroduces a mass–anisotropy degeneracy; the uniqueness just derived is conditional on the explicitly isotropic velocity distribution.