Critical Z as a fraction of ﹩{e}_{\mathrm{cross}}/\sqrt{2}﹩, plotted vs. the quantity ﹩{(a/{\rm{\Delta }}a)}^{4}({\mu }_{1}+{\mu }_{2})﹩ computed from numerical simulations over wide range of planet spacings and masses in the range ﹩[{10}^{-10}\mbox{--}{10}^{-3}]{M}_{* }﹩. Points are colored according to planet mass. The square markers and error bars indicate the median and 1σ range of binned numerical results for planet masses ﹩\gt {10}^{-6}{M}_{* }﹩ (red) and ﹩\lt {10}^{-6}{M}_{* }﹩ (blue). Quantities for the planet pair Kepler-36 b and c are indicated by the black circle (see Section 4.2).