JuanCastillo29/space-mission-intelligence-data huggingface dataset:
component masses m0 and m1 and semimajor axis a, perturbed by a distant mass m2 on an outer orbit with semimajor axis a2
ϕSA = ϕSA quad + ϕSRF, (6)
where
2 This conservative, single-averaging effect is distinct from the relativistic “precession resonance” of Kuntz (2022), in which gravitational-wave radiation shrinks the inner orbit until its apsidal precession rate is commensurate with the outer orbit.

That mechanism requires both dissipation and strong apsidal precession, whereas our jumps invoke neither.

See also §4.
ϕSA quad = 3a3 2(1 −e2 2)3/2
� 1 −2e2 −(j · ˆr2)2 + 5(e · ˆr2)2� ,
4r3
(7)

SRF Catalysis of Extreme Orbital Evolution 5
and
where
�1 2j2 z + e2 −5
� . (13)
�1 j3 −1
�
�1 j −1
�
+ εRot
2e2 z −1
ϕDA quad = 3
ϕSRF = εGR
j3
j0
�7 24
� −1
�
�1 j9 −1
�1 j7 −1
The potential ϕDA is axisymmetric about the z-axis whether or not SRFs are included, so jz is always conserved in our DA problems.

Without SRFs (ϕSRF = 0) we have ϕDA = ϕDA quad.
+ εTide
j9
j7
�1 j5 −1
�� . (8)
+ 1
j5
This quantity is an integral of motion, but it is more convenient to strip away the constant and the j2 z piece and work instead with the ‘Kozai constant’
Here the dimensionless SRF potential combines contributions from GR, rotational bulges, and tides, following Fabrycky & Tremaine (2007) and Liu et al. (2015), with j0 ≡j(t = 0) the initial value of j.

The terms proportional to j0 have no effect on the dynamics because only the gradients of the potential enter the equations of motion (2)–(3), but we find this parameterization convenient because it means we always have ϕSRF(t = 0) = 0.

The dimensionless coefficients εi are given by
CK ≡4
3ϕDA quad −1
2j2 z + 1
6 (14)
= e2 −5
2 sin2 i sin2 ω � , (15)
2e2 z = e2 � 1 −5
where i is the mutual inclination and ω is the argument of pericenter.

The two integrals of motion jz, CK dictate the entire no-SRF DA dynamics.

When SRFs are included (ϕSRF ̸= 0), the analogous integral of motion is
εGR = 3 m0 + m1
�a2 a
�3
m2
�G(m0 + m1)/c2
� , (9)
2)3/2
× (1 −e2
3ϕDA −1
6 = CK + 4
D ≡4
a
2j2 z + 1
3ϕSRF. (16)
�a2 a
�3
εRot = kq,1 m0 + m1
The integrable dynamics is then dictated entirely by D and jz.

Formally the range of allowed D values is much larger than for CK, because at very high eccentricity (low j), the additional term ϕSRF can become extremely large (see equation (8)).

However, away from peak eccentricity ϕSRF is usually very small3, so D ≃CK to an excellent approximation.

Thus CK remains a useful label of the integrals of motion in the DA problem even with SRFs included.

In either case, the value of CK is important because (i) its sign determines the topology of the phase space motion, separating librating cycles (CK < 0), where ω oscillates, from rotating cycles (CK > 0), where ω circulates; (ii) it changes the secular period PK, often by a large factor near separatrices; (iii) it plays the key role in determining the minimum eccentricity emin; (iv) it modulates the maximum eccentricity emax (though the key role here is played by jz).

The range of allowed CK values depends on jz.
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.

SRF Catalysis of Extreme Orbital Evolution 9
Cumulative residence time T(> e) as a fraction of the total integration time Ttot, versus log10(1 −e), for the BBH–SMBH system of Figures 2–3, run 50 times with random initial outer mean anomaly M2 ∈[0, 2π). Red and blue colors correspond to SRFs (in this case GR precession) being switched on and off respectively. The solid lines show the median SA results and the shaded bands show their 5–95% range. Dashed lines show the DA results; they asymptote towards zero when they reach their predicted maximum eccentricities eDA max (vertical dotted lines).

[Figure: Figure 4.]

[Figure: Figure 6. As in panel (b) of Figure 3, except for the Sun– Neptune system of §3.2, for which a tidal bulge provides the only SRF.]

3.2. Example 2: Tidal precession Next we consider a Sun–Neptune pair (m0 = 1 M⊙, m1 = mNep = 5.1 × 10−5 M⊙), perturbed by an m2 = 0.64 M⊙outer companion on a circular orbit at a2 ≃240 AU (giving an outer period Pout ≃2900 yr). For the inner orbit we choose a ≃12 AU (implying a characteristic secular time Psec ≃8 × 104 yr), e = 0, ω = 0.0◦, i = 88.6◦, Ω= 180◦(precise values are in Appendix A.1). We include apsidal precession due to Neptune’s tidal bulge as our only SRF (so that εGR =

[Figure: Figure 5. Time evolution of log10(1 −e), CK and jz for the Sun–Neptune system of §3.2, and choosing the initial outer mean anomaly to be M2 = 117.3◦. Precession due to the tidal bulge is the only SRF. Only the 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.]
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.

SRF Catalysis of Extreme Orbital Evolution 11 (so εGR = εTide = 0).7 Then we have jz = 0.010, (31) CK = 1.0 × 10−6, (32) Pout/Psec = 0.021, (33) εRot = 1.0 × 10−6, (34) jDA min = 0.014. (35) In Figure 8 we show the resulting SA evolution: again the no-SRF run (blue) tracks the regular DA cycle, while the run with the SRF included (red) jumps nonadiabatically, with CK wandering across much of its allowed range. In Figure 9 we show the corresponding emax versus PK scatter plot, which exhibits the same catalysis as in the previous examples. In fact, just as in the tidal example (Figure 6), there are two families of eccentricity maxima: an upper family (reaching 1 −emax ∼10−4) that could be reasonably well-described by the semi-analytic prescription ±δe, and a separate lower family (clustered around 1 −emax ∼10−5) that can only be the result of nonadiabatic jumps. Finally, in Figure 10 we plot the cumulative residencetime fraction for 50 realizations of this system with randomized initial M2; as in the previous examples, the SA+SRF median result (red) sits to the left of and above the no-SRF median (blue), confirming that the rotational bulge too catalyzes extreme eccentricity evolution.

[Figure: Figure 8. As in Figure 5, but for the Sun–Jupiter-like system described in §3.3 where the only SRF included is due to a rotational bulge.]

4. DISCUSSION Hierarchical triples are remarkably rich dynamical systems whose behavior is still not fully understood. Here we have uncovered yet further layers of this richness. Namely, as the tertiary orbits in a hierarchical triple, it exerts a torque on the inner binary that fluctuates on the outer orbital timescale, and when these fluctuations act in concert with SRF-induced apsidal precession — i.e., when the system lies in the fourth quadrant of Figure 1 — the inner binary can suffer nonadiabatic jumps at high eccentricity. These jumps occur because the rapid apsidal precession when e →1 swings the argument of pericenter ω through an O(1) angle on the outer orbital timescale. Then the system has no time to oscillate around its original secular (DA) trajectory, and instead emerges near a new secular trajectory, pinned to a new
7 We do this only to show that the rotational bulge alone can drive nonadiabatic jumps. In reality the other SRFs are not negligible in this system; in particular εTide ≃10−13 turns out to be the dominant effect. This example is thus a controlled demonstration, not a fully realistic system.

[Figure: Figure 9. As in Figure 6, but for the Sun–Jupiter-like system described in §3.3 where the only SRF included is due to a rotational bulge.]

12 Klein & Hamilton

[Figure: Figure 10. As in Figures 4 and 7, but for the example from §3.3 with the rotational bulge providing the only SRF.]
how ZLK cycles subject to nonadiabatic jumps become even more extreme when octupolar effects are important.
• Systems that break the double averaging (DA) approximation (so the validity criterion (17) is violated) and are subject to apsidal precession from SRFs — i.e., those sitting in the fourth quadrant
C.H. was partly supported by the John N.

Bahcall Fellowship Fund at the Institute for Advanced Study.

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2015; Basha et al. 2025; Kinoshita & Nakai 2007; Vashkov’yak 1999) and start every integration there, e = emin, so that each run begins at a well-defined phase of its cycle.

Second, we fix the apsidal branch from the sign of CK: we take ω = π/2 for librating cycles (CK < 0) and ω = 0 for rotating cycles (CK > 0).

Third, we orient the orbit in the outer frame with Ω.

These three choices determine the initial j and e.

We then integrate the Milankovitch equations (2)–(3) with the dimensionless potential ϕ = ϕquad + ϕSRF, taking the quadrupole part to be either the single-averaged potential ϕSA quad (6) or the double-averaged potential ϕDA quad (12) as appropriate, forward in dimensionless time τ = t/Psec.

Finally, to display a system in physical units we attach a single dimensional scale—equivalently, the component
B.

VALIDITY OF THE SINGLE-AVERAGING AND TEST-PARTICLE APPROXIMATIONS Throughout the main text we relied on two simplifications of the underlying few-body dynamics: (i) singleaveraging (SA) of the inner orbit (ignoring all fluctuations on the timescale of the inner orbital period Pin), and (ii) the test-particle approximation (so the outer orbit is perfectly Keplerian and does not feel any backreaction from the inner binary dynamics).

The latter approximation is what allowed the problem to be cast in fully dimensionless form (Appendix A).

Both of these approximations can break down in certain circumstances (Antonini et al. 2014; Anderson et al. 2016).

However, we checked the qualitative phenomenology established in §3 — jumps in CK, the SRF-induced secular evolution of the adiabatic invariants CK and jz, and the resulting SRF-catalyzed excursions to extreme eccentricity, all
8 In DA, Ωis a gauge angle, since the potential is axisymmetric about the outer angular-momentum axis; the same holds in SA when e2 = 0.

Only for SA with e2 > 0 does the outer eccentricity vector pick out a preferred apsidal direction, which makes Ωa genuine initial condition.

18 Klein & Hamilton Dimensional realizations of the three numerical examples of §3. The three GR columns share one dimensionless parameter set and differ only by an overall scaling. Inner-orbit angles follow from (jz, CK) via Appendix A.

[Table: Table 1.]
8 In DA, Ωis a gauge angle, since the potential is axisymmetric about the outer angular-momentum axis; the same holds in SA when e2 = 0.

Only for SA with e2 > 0 does the outer eccentricity vector pick out a preferred apsidal direction, which makes Ωa genuine initial condition.

18 Klein & Hamilton Dimensional realizations of the three numerical examples of §3. The three GR columns share one dimensionless parameter set and differ only by an overall scaling. Inner-orbit angles follow from (jz, CK) via Appendix A.

[Table: Table 1.]

GR: NS+NS GR: 10+10 M⊙ GR: 30+30 M⊙ Tides: Rotation:
(shown) Sun–Neptune Sun–Jupiter
Bodies m0 [M⊙] 1.4 10 30 1 1 m1 [M⊙] 1.4 10 30 5.150 × 10−5 (MNep) 9.546 × 10−4 (MJup) m2 [M⊙] 107 107 107
0.642 0.20 R1 [AU] · · · · · · · · · 1.646 × 10−4 (RNep) 4.778 × 10−4 (RJup) k2
· · · · · · · · · 0.127 0.37 · · · · · · · · · · · · 9.93 hr
Pspin
Orbit (inner / outer) a [AU] 0.763 5.45 16.34 11.94 7 a2 [AU] 1.449 × 104 5.373 × 104 1.118 × 105 238.9 100 a2 [pc] 0.0703 0.260 0.542 · · · · · · e (= emin) 1.0 × 10−3 1.0 × 10−3 1.0 × 10−3 1.0 × 10−3 1.0 × 10−3
e2
0.8 0.8 0.8 0 0 i [deg] 89.4 89.4 89.4 88.6 89.4 ω [deg] 0 0 0 0 0 Ω[deg] 180 180 180 180 180 M2 [deg] 278.6 278.6 278.6 117.3 349.5
Periods Pin [yr] 0.398 2.845 8.527 41.3 18.51 Pout [yr] 551 3939 1.181 × 104 2881 912 Psec [Myr] 0.0262 0.187 0.562 0.082 0.0430
Dimensionless parameter set
jz
0.010 0.010 0.010 0.025 0.010
CK 10−6 10−6 10−6 10−6 10−6 Pout/Psec
0.021 0.021 0.021 0.035 0.021
εGR
0.0451 0.0451 0.0451 0 0
εTide
0 0 0 3.08 × 10−16 0 εRot
0 0 0 0 1.0 × 10−6
captured by the cumulative residence-time diagnostic — are robust to these approximations. Here we give some concrete examples. In Figure 11 we zoom in on the first eccentricity maximum (τ ≈4.33) of the trajectory shown in Figure 2, which we recall is for a BBH–SMBH triple (see §3.1). We show the results of integrating this system using multiple different methods. The red dashed and black dashed
lines reproduce the SA and DA results from Figure 2 respectively. The blue dashed line shows the SA integration without the test-particle approximation (i.e., including the back-reaction onto the outer orbit), and the thick gray line shows the results of a direct N-body integration (no averaging at all) using SpaceHub (Wang et al. 2021). We see that the two SA integrations and SRF Catalysis of Extreme Orbital Evolution 19 the N-body integrations produce results that are indistinguishable, giving us confidence in our approach.
Jackie Villadsen,1, 2, 3 Carter Russell,2 Luna Guerrero,3 Ethan Harvie,1 Ariana Watson,1 Arjun Anand,1 John Sebastian Pineda,4 Vanessa Moss,5, 6 Daniele d’Antonio,7, 6
Louisa Canepa,8, 6 E. Cappellazzo,9, 10, 6 and Andrew Zic11, 6 1Bucknell University One Dent Drive, Lewisburg, PA 17837, USA 2St. Mary’s College of Maryland 47645 College Dr., St. Mary’s City, MD 20686, USA 3Vassar College 124 Raymond Avenue, Poughkeepsie, New York 12604, USA 4Laboratory for Atmospheric and Space Physics 1234 Innovation Dr, Boulder, CO 80303 5University of Sydney Camperdown NSW 2006, Australia 6ATNF, CSIRO, Space and Astronomy, PO Box 76, Epping, NSW 1710, Australia 7University of Technology, Sydney 8University of New South Wales 9School of Mathematical and Physical Sciences, Macquarie University, Sydney, NSW 2109, Australia 10Astrophysics and Space Technologies Research Centre, Macquarie University, Sydney, NSW 2109, Australia 11Macquarie University