Height-evaluation gravitational softening (${\Phi }_{ϵ}(\mathbf{R})=-G\int \Sigma ({\mathbf{R}}^{\prime })(|\mathbf{R}-{\mathbf{R}}^{\prime }{|}^2+{ϵ}^2{)}^{-1/2}{d}^2{R}^{\prime }$)

Evaluating the potential of a razor-thin disk at height $ϵ$ gives a softened horizontal force. Its Fourier multiplier is $e^{-kϵ}$ times the razor-thin disk Poisson kernel. This suppresses short wavelengths while retaining the long-wave potential. A real finite-thickness profile generally has a different full reduction factor.

For a vertically separable perturbation with normalized height profile $h(z)$, the midplane gravity reduction factor is $\int h(z)e^{-k|z|}dz$. Its small-$k$ expansion is $1-k⟨|z|⟩+o(k)$ when this first moment is finite. Match it to $e^{-kϵ}$ to get the displayed choice. For $h=e^{-|z|/H}/(2H)$ the exact factor is $(1+kH{)}^{-1}$, demonstrating why no single exponential matches every wavelength.