= Differential secular orbit-crossing criterion
{title2=$|z+a\partial_a z|=1$}
For a cold coplanar disk with <complex eccentricity> $z(a,t)$, the first-order radial shape is $r=a[1-\operatorname{Re}(ze^{-i\theta})]$. Infinitesimally neighboring orbits first cross when $|z+a\partial_a z|=1$. Under <secular forcing by a distant outer planet>, $z=Z_f(1-e^{iAt})$, $Z_f\propto a$ and $A\propto a^{3/2}$, so the secular phase gradient gives $t_{\rm cross}\simeq2/(3Z_fA)$. The finite-neighbor version requires the accumulated phase difference to remain small when replacing a difference by a derivative.
Back to article page
