Chaotic phase space structure for a test-particle subject to external perturbers of three different masses. Lines of constant closest approach distance, ﹩a^{\prime} -a(1+e)﹩, in units of Hill radii, ﹩{R}_{\mathrm{Hill}}=a^{\prime} {(\mu ^{\prime} /3)}^{1/3}﹩ are shown in blue. Various stability criteria are also plotted: our resonance overlap boundary (Equation (10)) in red, the Hill stability boundary in orange, and Wisdom (1980)’s resonance overlap boundary, interior to which chaos is predicted for all eccentricity values, indicated by a purple arrow. The semimajor axis separation, ﹩a^{\prime} -a﹩, is shown by ticks on the top of each panel in units of ﹩{R}_{\mathrm{Hill}}﹩. The background gray-scale is the same as in Figure 6 with the addition that initial conditions leading to a close encounters (within a Hill radius) are marked in red. The Hill stability boundary separates initial conditions that could experience such an encounter from those that cannot based on conservation of the Jacobi constant.