Kuzmin reflection method ($\Sigma (R)=-{\partial }_z\psi (R,0^{+})/(2\pi G)$)

Reflect a vacuum gravitational potential in the upper half-space evenly across $z=0$. For the convention in which gravitational acceleration is $\nabla \psi$, the derivative jump is $2{\partial }_z\psi (R,0^{+})=-4\pi G\Sigma (R)$. Thus a point mass $m$ below the plane produces a Kuzmin disk, and superposing point masses with weight $\mu (b)db$ produces $\Sigma (R)=(2\pi {)}^{-1}{\int }_0^{\infty }b\mu (b)(R^2+b^2{)}^{-3/2}db$. Positive weights give a nonnegative surface density wherever the integral converges.

For $B>0$ and $0<\beta <1$, the unreferenced relative potential of the Kuzmin reflection method diverges at large $b$. Referencing each component to the origin before integrating gives the finite potential differenceAt small $b$ the subtraction is integrable because $\beta >0$; at large $b$ the difference is $O(b^{-2})$, integrable because $\beta <1$. Its plane force is ${\partial }_R\psi =-C{R}^{\beta -1}$, where $C=(GB/2)B((1+\beta )/2,1-\beta /2)$. A circular speed $V=K{R}^{\beta /2}$ fixes $C=K^2$, givingThe beta function substitutions for the force and surface density have different exponents. Its central mass is locally finite, but its total mass diverges at large radius; a potential zero at infinity cannot be imposed. The force and density approach those of the Mestel disk as $\beta \downarrow 0$.