Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 54 / 2 / a

For a razor-thin disk, integrate the delta function in height to getThe corresponding horizontal force kernel is proportional to $(\mathbf{R}-{\mathbf{R}}^{\prime })/(|\mathbf{R}-{\mathbf{R}}^{\prime }{|}^2+{ϵ}^2{)}^{3/2}$. It removes the short-distance point-force singularity and smooths forces on separations of order $ϵ$. This is height-evaluation gravitational softening.

To represent finite vertical thickness, choose $ϵ$ of order the disc disk scale height $H$. There is no universal exact numerical choice: a true vertical mass density profile produces the Fourier reduction factor ${\Sigma }^{-1}\int \rho (z)e^{-k|z|}dz$, which is not generally $e^{-kϵ}$. Matching the long-wave term gives $ϵ=⟨|z|{⟩}_{\rho }$, the vertical-profile softening match. For an exponential vertical profile this mean is $H$, while its full reduction factor is $(1+kH{)}^{-1}$.