= Vertical-energy Abel inversion of a stellar distribution
{title2=$F(E)=-\frac1{\pi\sqrt2}\int_E^\infty\frac{d\nu/d\psi}{\sqrt{\psi-E}}\,d\psi$}
For a phase-mixed one-dimensional <collisionless stellar system>, $E=v_z^2/2+\psi(z)$ is an <integral of motion> and $f_z=F(E)$. Integrating both signs of velocity gives $\nu(\psi)=\sqrt2\int_\psi^\infty F(E)(E-\psi)^{-1/2}\,dE$. Applying the <Abel transform> a second time gives $\int_E^\infty\nu(\psi)(\psi-E)^{-1/2}\,d\psi=\pi\sqrt2\int_E^\infty F(t)\,dt$; differentiating proves the inverse. Sufficient convergence, regularity, and correct escape-energy boundary conditions are essential. An arbitrary tracer-density/potential pair is physical only if the reconstructed <phase-space distribution function> is nonnegative. Applying this inversion to a <galactic disk> requires a locally separable vertical dynamics or a justified approximation to it, and control of radial mixing and the tilted <velocity ellipsoid>.
Back to article page
