adiabatic jumps can place these systems on entirely different DA cycles, with a generally different
jz = 0.010, (21) CK = 1.0 × 10−6, (22) Pout/Psec = 0.021, (23)
4 Strictly we should use the Kozai period PK in the criterion (17) rather than the characteristic timescale Psec.

However, we nearly always have PK ∼Psec, and the latter is much easier to evaluate quickly.

The exception is near the separatrix CK = 0 where we can have PK ≫Psec.

The main thing to keep in mind is that criteria like (17) are only heuristics guides and should not be trusted blindly.
εGR = 0.0451, (24) jDA min = 0.042. (25) Note that one can scale, e.g., the binary component masses and the two semimajor axes while keeping the SRF Catalysis of Extreme Orbital Evolution 7

Time evolution of log10(1 −e), CK and jz for the BBH–SMBH triple of §3.1, for initial outer mean anomaly M2 = 278.6◦. GR apsidal precession is the only SRF. The left (right) column shows DA (SA) results. Red (blue) curves show results from runs with GR precession switched on (off). The two dashed horizontal lines in panel (a) mark the DA-predicted eccentricity maximum 1 −eDA max. The dashed black lines in panels (e) and (f) mark jz = 0.

[Figure: Figure 2.]

[Figure: Figure 3. Scatter plot of maximum eccentricity 1 −emax and secular period PK measured at the end of each secular cycle. The system under study and the plotting conventions are the same as in Figure 2. The red ‘error bars’ in each panel mark the semi-analytic prediction eDA max ±δe that is supposed to bracket the SA fluctuations in eccentricity; here δe is obtained by inserting the jz-oscillation amplitude of Luo et al. (2016, their Eqs. 32–33) for an eccentric outer orbit into the fluctuating-eccentricity prescription of Grishin et al. (2018, their Eq. 36).]

8 Klein & Hamilton dimensionless numbers (25) fixed, so that the results we are about to show apply equally to, e.g., lower-mass BBHs and binary neutron stars (BNSs); precise parameters for three such realizations (including the 30+30 M⊙ system shown here) are listed in Appendix A.1.

In Figure 2 we show the evolution of log10(1 −e), CK, and jz as a function of time for this system.

The panels on the left show the DA results, while those on the right are the SA results for a single initial outer mean anomaly M2 = 278.6◦.

In each panel the result of the no-SRF control experiment is shown in blue, while the result including the SRF is shown in red.

We see from the left column in this Figure that in the DA approximation all trajectories are regular, jz is constant, and the primary role of GR precession is to lower the maximum eccentricity reached, as expected (Fabrycky & Tremaine 2007; quadrant 2 of Figure 1).

As for the SA runs (right column), the results of the non-SRF control case (blue) look similar to those in the DA runs, apart from some extra fluctuations in the maximum eccentricity and in jz around the DA predicted value (Ivanov et al. 2005; quadrant 3 of Figure 1).

But the run including GR precession (red) looks completely different: the minimum eccentricity, maximum eccentricity, and secular period all change from one cycle to the next, reflecting the fact that the ‘adiabatic invariants’ CK and jz are now not even approximately conserved.

Panels (d) and (f) make clear that jumps in CK towards the edges of its allowed range coincide with decrements in the magnitude of jz (§2.1).

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).