Short-Range Forces Can Catalyze Extreme Orbital Evolution in Hierarchical Triples

Technical Summary: Short-Range Forces Can Catalyze Extreme Orbital Evolution in Hierarchical Triples

Problem Statement
Hierarchical triple systems are widely recognized as potential origin sites for exotic astrophysical phenomena, such as black hole mergers and hot Jupiters, driven by the von Zeipel-Lidov-Kozai (ZLK) effect. In the standard theoretical framework, a distant tertiary torques an inner binary to high eccentricity on secular timescales. The dynamics are typically analyzed using the double-averaged (DA) approximation, where the inner and outer orbits are averaged out. In this limit, the system is integrable, governed by two constants of motion: the angular momentum component perpendicular to the outer orbit ($j_z$) and the Kozai constant ($C_K$).

It is well-established that "short-range forces" (SRFs)—including general relativity (GR), equilibrium tidal bulges, and rotational bulges—induce apsidal precession. In the DA approximation, this precession suppresses the maximum eccentricity ($e_{\max}$) achievable by the inner binary but leaves the secular period ($P_K$) unchanged. Furthermore, it has been generally assumed that relaxing the DA approximation to the single-averaged (SA) approximation (which retains fluctuations on the outer orbital timescale) only introduces small, adiabatic corrections to $e_{\max}$ and $P_K$, without fundamentally altering the secular evolution.

This paper challenges the assumption that SRFs merely suppress extreme eccentricity and that the DA approximation (even with SRFs) provides a robust baseline for secular periods. The authors investigate the regime where both SRFs and the breakdown of the DA approximation (due to rapid apsidal precession at high eccentricity) are present simultaneously.

Methodology
The authors analyze the test-particle, quadrupole-order ZLK problem. They employ a dynamical framework based on the Milankovitch equations for the evolution of the eccentricity vector ($\mathbf{e}$) and angular momentum vector ($\mathbf{j}$).

1. Approximations: The study contrasts two levels of averaging:
   • Double-Averaged (DA): Averages over both inner and outer orbits. This yields an integrable system with conserved $j_z$ and a modified constant $D$ (which includes SRF contributions).
   • Single-Averaged (SA): Averages only over the inner orbit, retaining the instantaneous position of the tertiary. This introduces fluctuations in the orbital elements.
2. SRF Inclusion: The authors include SRFs (GR, tides, and rotational bulges) in the SA potential. These forces cause apsidal precession rates ($\dot{\omega}_{\text{SRF}}$) that scale steeply with eccentricity (e.g., $\propto (1-e^2)^{-n}$).
3. Numerical Integration: The authors integrate the SA equations of motion for three distinct astrophysical scenarios (a binary black hole orbiting a supermassive black hole, a Sun-Neptune system, and a white dwarf binary), isolating one SRF at a time. They compare these results against DA predictions and control runs without SRFs.
4. Parameter Space: The analysis focuses on systems capable of reaching very high eccentricities ($|j_z| \ll 1$) where the validity criterion for the DA approximation ($P_{\text{out}}/P_{\text{sec}} \ll j_{\min}^{\text{DA}}$) is marginal or violated.

Key Contributions and Mechanism
The central contribution of this work is the identification of a mechanism where SRFs catalyze, rather than suppress, extreme orbital evolution when the DA approximation is relaxed.

• Nonadiabatic Jumps: The authors demonstrate that during high-eccentricity episodes, the SRF-induced apsidal precession becomes so rapid that the argument of pericenter $\omega$ changes by an order-unity amount on the timescale of the outer orbit. This violates the DA assumption that $\omega$ varies only on the secular timescale.
• Breakdown of Adiabatic Invariance: Consequently, the system does not return to its original "parent" DA trajectory after a high-eccentricity excursion. Instead, it undergoes a discrete, nonadiabatic jump to a different DA trajectory, characterized by a new pair of invariants $(C_K, j_z)$.
• Secular Evolution of $j_z$: Crucially, the authors show that these jumps can alter the magnitude and sign of $j_z$. This represents the first demonstration of secular evolution in $j_z$ driven by a quadrupole-order, three-body mechanism, whereas previous theories treated $j_z$ as strictly conserved in the absence of dissipation or octupole terms.

Results
Numerical simulations reveal that the interaction between SA fluctuations and SRF precession leads to dynamics qualitatively different from the other three regimes (DA/No-SRF, DA+SRF, SA/No-SRF):

1. Variable Maximum Eccentricity: The maximum eccentricity ($e_{\max}$) and minimum eccentricity ($e_{\min}$) vary drastically from one secular cycle to the next. In the BBH-SMBH example, $1-e_{\max}$ varies by approximately four orders of magnitude across cycles, far exceeding the small fluctuations predicted by semi-analytic SA corrections.
2. Variable Secular Period: Contrary to the standard view that SRFs leave the secular period unchanged, the authors find that $P_K$ varies by nearly an order of magnitude across cycles due to the jumps in $C_K$.
3. Catalysis of Extreme Eccentricity: While DA theory predicts that SRFs suppress the reach of extreme eccentricity, the SA+SRF results show the opposite. The cumulative "residence time" spent at very high eccentricities is significantly larger with SRFs than without. The system explores a much broader region of phase space than implied by any (semi-)analytic ZLK theory.
4. Robustness: These effects are observed across different SRF types (GR, tides, rotation) and different astrophysical mass scales, suggesting the phenomenon is generic to hierarchical triples in the relevant parameter regime.

Significance
The paper claims that the standard (semi-)analytic answers to the two fundamental questions of ZLK dynamics—(i) the maximum possible eccentricity and (ii) the secular period—may need dramatic revision for realistic systems.

• Re-evaluation of Population Syntheses: The authors warn that population synthesis studies relying on the "DA+SRF" recipe (which assumes SRFs only suppress $e_{\max}$ and leave $P_K$ fixed) may be highly misleading. The nonadiabatic jumps imply that binaries can explore phase space regions previously thought inaccessible or transient.
• Theoretical Implications: The work highlights a regime where adiabatic invariance is completely broken, leading to stochastic-like evolution of the integrals of motion even in a conservative, quadrupole-order system.
• Future Work: The authors note that this study is conducted at the test-particle quadrupole level. In a companion paper, they indicate that combining these jumps with octupolar ZLK evolution leads to even more extreme behavior.

In summary, the paper argues that short-range forces, often treated as a suppressive correction in hierarchical triples, can act as a catalyst for extreme orbital evolution when the timescale separation required for double-averaging breaks down, leading to discrete jumps in the system's adiabatic invariants and a fundamentally different secular evolution.