Orbit crossing

Orbit crossing occurs when two geometrical orbital paths intersect. It permits collisions or close encounters when bodies also reach the intersection at compatible times; crossing alone does not imply a collision. For nearly circular coplanar nested orbits, a useful local test follows by differentiating their radial positions at fixed longitude.

For a cold coplanar disk with complex eccentricity $z(a,t)$, the first-order radial shape is $r=a[1-\mathrm{Re}(z{e}^{-i\theta })]$. 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 $t_{\mathrm{cross}}\simeq 2/(3Z_{f}A)$. The finite-neighbor version requires the accumulated phase difference to remain small when replacing a difference by a derivative.