).

Over the full run these jumps carry CK across almost the entire interval [−3/2, 1], so the system alternates between circulating and librating ZLK trajectories, while jz repeatedly changes sign — behavior entirely absent from the no-SRF and DA runs.5
ation’ obtained by inserting the jz-oscillation amplitude of Luo et al. (2016, their Eqs. 32–33) for eccentric outer orbits (here e2 = 0.8) into the fluctuating-eccentricity prescription of Grishin et al. (2018, their Eq. 36).

We see from panel (a) of Figure 3 that without SRFs, the SA fluctuations drive essentially no change in secular period and only a modest spread in emax values, consistent with equations (19)-(20).

But in panel (b), with SRFs included, the story is completely different.

The value of 1 −emax in these runs varies by ∼4 orders of magnitude, dwarfing the expected semi-analytic correction δe.

The secular period PK varies by almost an order of magnitude across cycles, contrary to the expectation that it should be roughly constant.

Figures 2–3 were for the choice of outer mean anomaly M2 = 278.6◦.

Different choices of M2 generally produce different evolution.

To characterize the spread of possible outcomes, we ran an ensemble of 50 realizations with M2 drawn uniformly from [0, 2π) and, for each realization, computed the amount of time T(> e) the inner binary spent above each eccentricity e.

In Figure 4 we plot the resulting cumulative ‘residence time’ T(> e)/Ttot (where Ttot = 400Psec is the total integration time) as a function of log10(1−e).

The dashed curves show the DA results (which are realization-independent), the solid curves show the median of the SA ensemble, and the shaded bands show their 5–95% ranges.

We see that the two DA results are nearly identical at all eccentricities in the range log10(1 −e) ≳−2, which is what we expect since SRFs are irrelevant there.

As we increase e past this threshold (i.e., move further left in Figure 4), the red dashed curve (DA with SRFs) cuts off before the blue dashed curve (DA without SRFs) does, which again is expected since in DA theory the SRFs suppress extreme eccentricity behavior (Fabrycky & Tremaine 2007; Liu et al. 2015).

Importantly, though, the median SA results (solid lines) show the opposite ordering, with red sitting to the left of blue.

This is the consequence of SRFs catalyzing, rather than suppressing, extreme eccentricity behavior when SA fluctuations are included.

As a result, for any eccentricity smaller than e ≃0.9999, the cumulative time spent above that eccentricity is about an order of magnitude larger with SRFs than without!

These are two distinct effects: the leftward shift of the SA curve reflects a higher maximum eccentricity emax, while the upward shift — the extra time spent above a fixed eccentricity — reflects not only emax but also the cycle-to-cycle changes in the minimum eccentricity emin and the secular period PK.

The residence history is therefore sensitive to the whole post-jump sequence of cycles, not just the single deepest eccentricity peak.

How do these jumps affect (i) the maximum eccentricity emax and (ii) the secular period PK?

To find out, in Figure 3 we scatter plot the peak eccentricity reached during each secular ZLK cycle versus the secular period of that cycle (defined as the time since the previous eccentricity peak).

The DA calculations produce single, analytic values for emax and PK, and we again show these results with stars, while the SA results are shown with dots, colored by the time of the measurement.

Finally with the red error bars in each panel we show the naive semi-analytic prediction for the spread in eccentricity maxima, eDA max ±δe, with δe ≃3.7×10−4 the ‘SA fluctu-
5 Strictly, with SRFs the conserved DA invariant is D = CK + (4/3)ϕSRF (equation (16)) rather than CK itself, so in the DA run CK undergoes short excursions during each high-e episode (where ϕSRF becomes large) while it is D that stays fixed.