Fig. 4.

Numerical simulations for 2 Gy of a modified three-body model of the Sun, Neptune, and (massless) Pluto in which we include an oblateness for the Sun parameterized by a value of J₂, as indicated at the top of each pair of panels. The top panel in each pair plots Pluto’s critical resonant angle $\varphi$ versus time, and the bottom panel in each pair is a plot of $(e\cos \omega ,e\sin \omega )$. Simulations were carried out for 74 values of J₂ in the range 1 to 10,000; only a small selection of eight cases is shown here. See SI Appendix for more detail. The output interval in the plots is 1 My. The color scheme follows that of Fig. 5.