Mestel disk ($\Sigma (R)=v_M^2/(2\pi GR)$)

A Mestel disk is a thin axisymmetric astrophysical disk with surface density inversely proportional to radius. Its galaxy rotation curve is exactly flat, $v_c=v_M$, and $v_c^2=GM(R)/R$ despite its nonspherical geometry. In the Legendre expansion of thin-disk gravity, all positive-degree interior and exterior terms cancel. Its total mass is infinite, so only potential differences, rather than a potential zero at infinity, are defined.

The planar Mestel disk has $F(E)\propto e^{-E/v_0^2}$, giving independent centered Gaussian distributions for $v_R,v_{\varphi }$ with velocity dispersions ${\sigma }_R={\sigma }_{\varphi }=v_0$. Its streaming velocity is zero. A maximally prograde stellar distribution instead has mean $\sqrt{2/\pi }{v}_0$ and azimuthal variance $v_0^2(1-2/\pi )$, while all second raw moments remain unchanged.

The reflection-symmetric Newtonian gravitational potential $\Phi =v_0^2\log [(\sqrt{R^2+z^2}+|z|)/R_{\ast }]$ is harmonic off the plane. Its normal-derivative jump is $2v_0^2/R$, so the Poisson equation for Newtonian gravity gives surface density of a disk $S=v_0^2/(2\pi GR)$ and volume mass density $\rho =S\delta (z)$. The midplane circular speed is constant. The scale-free model has infinite total mass and does not use a finite-mass boundary condition at infinity.

The midplane galaxy rotation curve fixes the radial derivative of the Newtonian gravitational potential, not every exterior boundary condition. Adding $a|z|$ leaves that radial derivative unchanged but adds a uniform sheet of surface density of a disk $a/(2\pi G)$. The pure scale-free Mestel disk excludes such extra sources.