Resonance-overlap encounter estimate ($j_c\sim {\mu }^{-2/7}$)

A local weak gravitational assist estimate for a nonradiating circular grain gives $\delta ϵ\simeq 32{\mu }^2{ϵ}^{-5}$, where $\mu =M_p/M_{\star }$. Conjunction-longitude sensitivity to an encounter then gives $|\delta \Lambda |\simeq 128\pi {\mu }^2/(3{ϵ}^7)$. Requiring the change to exceed one full turn and using $ϵ\simeq 2/(3j)$ gives $j_c\simeq (2/729{)}^{1/7}{\mu }^{-2/7}$. This motivates loss of coherent resonant conjunctions, rather than proving that every initial phase escapes. The coefficient depends on the encounter convention and the approximations; near-planet Hill equations require a consistent rotating-frame shear instead of mixing local inertial and rotating velocities.