single-averaged results are shown, since here the DA results with and without SRFs are indistinguishable. Red curves include the tide (+SRF), blue curves do not, and the dashed line in panel (c) marks jz = 0.]

10 Klein & Hamilton

[Figure: Figure 7. As in Figure 4, except for the Sun–Neptune system of §3.2, for which a tidal bulge provides the only SRF. Unlike in Figure 4 the no-SRF and +SRF DA curves now coincide, because at the DA level the tidal precession does nothing.]

(20), and another around 1−emax ≃10−4 that can only be produced by the nonadiabatic jumps. Meanwhile the secular period can be shorter than the DA prediction by almost an order of magnitude. Finally, in Figure 7 we plot the cumulative residence time fraction for over 50 realizations of this system, each initialized with random outer mean anomaly. The red and blue dashed DA curves coincide in this case — at the DA level, the tidal precession does nothing. Yet the SA ensemble again behaves as it did in Figure 4: the median curve with the SRF included (red) sits to the left of and above the no-SRF median (blue). The same two effects operate here: the leftward shift reflects a higher emax, while the upward shift reflects the cycle-to-cycle changes in emin and the secular period PK. This example confirms that — just as in the GR case of §3.1 — apsidal precession due to tides may catalyze, rather than suppress, extreme eccentricity evolution when SA fluctuations are included.
εRot = 0)6, using Love number k2 = 0.127 (Gavrilov & Zharkov 1977) and radius RNep ≃2.5 × 104 km. These choices give jz = 0.025, (26) CK = 1.0 × 10−6, (27) Pout/Psec = 0.035, (28) εTide = 3.08 × 10−16, (29) jDA min = 0.032. (30) In Figure 5 we show the SA evolution of log10(1 −e), CK and jz for initial outer mean anomaly M2 = 117.3◦. As in the GR example of §3.1, the no-SRF SA run (blue) tracks the regular DA cycle with only small fluctuations, whereas the run including the SRF (red) shows jumping behavior near eccentricity maxima. The value of CK wanders across much of its allowed range [−3/2, 1] and the secular period and peak eccentricity change from one cycle to the next. The striking point is that the SRF is fantastically weak (εTide = 3.08 × 10−16) — so small in fact, that in the DA approximation it does nothing at all. And yet, in the SA case, it drives evolution much more extreme than could have been predicted from DA theory. In Figure 6 we show a scatter plot of maximum eccentricity versus secular period, only for the cases with the SRF switched on (cf. panel (b) of Figure 3). In this case one can think of the maximum eccentricity measurements as being split into two ‘families’ — one centered around 1−emax ≃2.5×10−4, which are reasonably welldescribed by the standard semi-analytic approximation
3.3. Example 3: Rotational bulge For our final example we consider a Sun-like star (m0 = 1 M⊙) hosting a Jupiter-mass planet (m1 = 1 MJ), perturbed by an m2 = 0.20 M⊙M-dwarf on a circular orbit at a2 = 100 AU (so the outer period is Pout ≃900 yr). We choose the inner orbital elements a = 7 AU (so the secular time is Psec ≃4.3 × 104 yr), e = 0, ω = 0.0◦, i = 89.4◦, Ω= 180◦(precise values in Appendix A.1). We let the planet spin near its primordial rate with period (Pspin = 9.93 hr; Batygin 2018), and include rotational bulges as the only SRF
6 Although εGR is not strictly zero for this system (in fact εGR ∼ 3 × 10−5), including it does not change our results.