= Isotropic stellar pressure deprojection
{title2=$p(r)=-\frac1{2\pi r}\frac d{dr}\int_{r^2}^\infty\frac{N(\sqrt u)\sigma_{\rm los}^2(\sqrt u)}{\sqrt{u-r^2}}\,du$}
For a spherical, nonstreaming <stellar population> with isotropic <velocity dispersion>, each component of the random <pressure tensor> equals $p=n\langle v_r^2\rangle$. The observed second <stellar velocity moment> satisfies $N(R)\sigma_{\rm los}^2(R)=2\int_R^\infty p(r)r/\sqrt{r^2-R^2}\,dr$. Composing this <Abel transform> with itself gives the displayed inverse under sufficient decay and regularity. The <Spherical Jeans equation> then gives $\psi'=p'/n$, with <gravitational acceleration> $\nabla\psi$. The <Poisson equation for Newtonian gravity> recovers total gravitating <mass density>, not just the tracer density. Without velocity isotropy the same projected moment also contains anisotropic stresses.
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