Planar isotropic stellar distribution function ($F(E),E=v^2/2+\Phi$)

For a stationary collisionless stellar system confined to a plane, velocity isotropy makes the galactic distribution function depend on speed through $u=v^2/2$. The Collisionless Boltzmann equation implies ${\nabla }_{x}f=f_u{\nabla }_x\Phi$, hence $f=F(E)$ with specific orbital energy $E=u+\Phi$. This conclusion requires stationarity as well as isotropy.

The surface density of a disk associated with a planar isotropic distribution is $S(\Phi )=2\pi {\int }_{\Phi }^{\infty }F(E)dE$. Differentiation gives $F(E)=-(2\pi {)}^{-1}dS/d\Phi {|}_{\Phi =E}$. A vanishing upper-limit boundary value recovers the original integral; monotonic decrease of $S(\Phi )$ ensures a nonnegative distribution.