= Vertical-profile softening match
{title2=$\epsilon_{\rm match}=\langle|z|\rangle_\rho$}
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 $1-k\langle|z|\rangle+o(k)$ when this first moment is finite. Match it to $e^{-k\epsilon}$ 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>.
Back to article page
