= Finite-thickness disk relaxation
{title2=$t_{\rm disk}/t_{\rm sphere}\sim\lambda^3(z_0/R)\log(R/b_0)/\log(z_0/b_0)$}
For a disk with thickness $z_0$, random relative speeds $u=\lambda v$ and small-impact cutoff $b_0\ll z_0\ll R$, the leading three-dimensional logarithmic encounter term has upper cutoff of order $z_0$. At fixed <mass> and radius, its number <volume> density exceeds the spherical density by order $R/z_0$. Hence $t_{\rm disk}/t_{\rm sphere}\sim\lambda^3(z_0/R)\log(R/b_0)/\log(z_0/b_0)$ when the same cutoff is used for comparison. Different speed-dependent cutoffs must be distinguished in numerical applications. Distant planar encounters contribute a convergent, non-logarithmic correction. This scaling ceases to apply if thickness is comparable to the cutoff.
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