Onset of resonance overlap, and hence chaos, as a function of perturber mass and planet spacing. The red lines show our overlap criterion from setting ﹩{\tau }_{\mathrm{res}}=1﹩ in Equation (10) at three values of critical eccentricity, as labeled. The purple line shows the onset of chaos due to the overlap of first-order mean motion resonances at small eccentricity according to Wisdom (1980)’s ﹩{\mu }^{2/7}﹩ criterion. Our overlap criterion, which predicts the critical eccentricity for the onset of chaos, applies below this line. The black dashed line indicates the 2:1 resonance. Since our result for the critical eccentricity shown in Figure 3 adopts the “close approximation” (see Section 2.2), it becomes quantitatively incorrect beyond the 2:1.