Autarkic Triples in Stellar Dynamics

• Autarkic triples are hierarchical three-body stellar systems defined by a clear inner binary and outer tertiary that remain distinct over gigayear timescales.
• Their evolution is governed by secular Hamiltonian dynamics and the Mardling–Aarseth stability criterion, which prevent transitions to chaotic resonances.
• Stellar evolution processes such as wind mass loss and conservative mass transfer modulate their orbital parameters without breaching the critical stability threshold.

Autarkic triples are hierarchical three-body stellar systems that, despite the combined influence of secular perturbations and moderate evolutionary changes in stellar structure or mass, never violate the critical dynamical stability boundary that would lead to chaotic, democratic three-body resonances. Throughout their evolution, autarkic triples retain a strict distinction between an inner binary and a distant tertiary companion, with their long-term behavior governed predominantly by secular (Hamiltonian) dynamics rather than by direct, resonant three-body interactions. The defining feature of an autarkic triple is that the dimensionless hierarchy parameter, the ratio of outer to inner semi-major axes ($a_\text{out}/a_\text{in}$), always remains above the analytically defined critical threshold for stability, preventing the system from entering chaotic phases or dissolving into a binary plus single configuration (Toonen et al., 2021).

1. Formal Definition and Hierarchical Criteria

A triple system is classified as autarkic if, at every instant during secular orbital evolution and slow changes induced by processes such as wind mass loss or stable mass transfer, its configuration remains hierarchically ordered—the inner and outer orbits never lose their respective identifications. Formally, autarkicity requires

$$\frac{a_\mathrm{out}}{a_\mathrm{in}} \geq \left( \frac{a_\mathrm{out}}{a_\mathrm{in}} \right)_\mathrm{nst}$$

where $(a_\mathrm{out}/a_\mathrm{in})_\mathrm{nst}$ denotes the critical stability ratio determined by the criterion of resonance overlap. Such systems preserve their original structure over gigayear timescales. As a consequence, the secular evolution of autarkic triples is accurately described by double-averaged Hamiltonian dynamics, with no transitions to chaotic "democratic" resonance regimes or loss of hierarchy.

2. Stability Boundaries and the Mardling–Aarseth Criterion

The boundary between autarkic and non-autarkic (potentially chaotic) behavior is quantitatively set by the Mardling–Aarseth overlap-of-resonances stability criterion, with inclination and mass ratio corrections:

$$\left( \frac{a_\mathrm{out}}{a_\mathrm{in}} \right)_\mathrm{nst} \simeq \frac{2.8}{(1-e_\mathrm{out})[1-0.3(i/\pi)]} \left[(1+q_\mathrm{out})(1+e_\mathrm{out})\right]^{2/5} (1-e_\mathrm{out})^{-3/5}$$

where $q_\mathrm{out} = m_3/(m_1+m_2)$ is the outer mass ratio, $e_\mathrm{out}$ is the outer eccentricity, and $i$ is the mutual inclination. Triples for which $$a_\text{out}/a_\text{in} \gg (a_\text{out}/a_\text{in})_\text{nst}$$ invariably remain hierarchical and autarkic. Empirical results show that when the ratio drops below the stability line, eventually the system undergoes chaotic evolution, typically dissolving into a binary and single within $10^2$–$10^3$ crossing times.

3. Secular Hamiltonian Dynamics Versus Chaotic Evolution

The orbital dynamics of autarkic triples are governed by a secular Hamiltonian expansion in powers of $a_\text{in}/a_\text{out}$. At quadrupole order, the classic Lidov–Kozai Hamiltonian dominates, producing periodic oscillations in eccentricity and inclination. Higher-order (octupole) corrections introduce additional angular dependencies and richer modulation, with timescales characterized by the Lidov–Kozai time

$$t_\mathrm{LK} \simeq \frac{P_\mathrm{out}^2}{P_\mathrm{in}} \frac{m_1+m_2+m_3}{m_3}(1-e_\mathrm{out}^2)^{3/2}$$

and octupole modulation timescale

$$\tau_\mathrm{oct} \sim [t_\mathrm{LK}/\epsilon_\mathrm{oct}]^{1/2}$$

where

$$\epsilon_\mathrm{oct} \equiv \frac{m_1-m_2}{m_1+m_2} \frac{a_\mathrm{in}}{a_\mathrm{out}} \frac{e_\mathrm{out}}{1-e_\mathrm{out}^2}$$

Autarkic triples never reach eccentricities or semi-major axis ratios that breach the dynamical stability boundary, so secular oscillations proceed predictably, and transitions to chaotic random walks or resonance-driven exchanges are avoided.

4. Effects of Stellar Evolution on Autarkicity

Stellar evolution processes, particularly slow isotropic wind mass loss and conservative mass transfer within the inner binary, modulate the semi-major axes and total mass of the system:

• Wind mass loss: Orbits widen adiabatically, and in a triple,

$$(a_\mathrm{out}/a_\mathrm{in})' = (a_\mathrm{out}/a_\mathrm{in}) \left[\frac{M_1+M_2+M_3}{M_1'+M_2'+M_3'}\right]\left[\frac{M_1'+M_2'}{M_1+M_2}\right]$$

Post-main-sequence mass loss by the inner pair tends to reduce $a_\text{out}/a_\text{in}$, potentially pushing systems close to or below the stability threshold.

• Conservative mass transfer: The semi-major axis ratio evolves as

$a'/a = \left( \frac{m_d m_a}{m_d' m_a'} \right)^2$

Cumulatively, such changes can reduce $a_\text{out}/a_\text{in}$ by tens of percent over Gyr timescales, causing some originally autarkic triples to approach instability and even cross into the chaotic regime.

5. Population Outcomes and Statistical Properties

Population synthesis using the TRES code and direct N-body integration for $\gtrsim 10^6$ primordial triples reveals the following:

• Autarkic survivors: 54–69% of all destabilisation candidates never lose hierarchy, maintaining autarkicity even after $10^{3}$–$10^{6}$ crossing times.
• Inclination bias: Prograde triples ($i < 90^\circ$) tend to eject the tertiary in a slingshot process, producing a binary and single star without losing hierarchy. Retrograde configurations are more likely to enter democratic resonances.
• Mass ratio dependence: Systems with $q_\text{out} \lesssim 0.3$ are most robustly autarkic. Those with near-equal tertiary masses ($q_\text{out} \gtrsim 0.8$) destabilise more readily.
• Survivor demographics: Autarkic triples cluster at $a_\text{out}/a_\text{in} \approx 2$–$5$ above the critical line, with $a_\text{in} \sim 10^2$–$10^3\,R_\odot$, $a_\text{out} \sim 10^3$–$10^4\,R_\odot$, $e_\text{in} \sim 0.2$–$0.6$, $e_\text{out} \sim 0.1$–$0.5$, and mutual inclinations $i \sim 20^\circ$–$60^\circ$.

6. Observational Signatures and Astrophysical Implications

Autarkic triples are identified by their secular variability, with the following expected diagnostics:

• Predictable eclipse timing variations (ETVs): The period of ETV modulation matches the outer orbital period in compact eclipsing binaries.
• Apsidal motion rates: Consistent with predictions from the quadrupole plus octupole orders in the secular Hamiltonian.
• Stable light-curve modulation: Tidal and relativistic effects show smooth, long-term variations on the Lidov–Kozai timescale $t_\text{LK}$.

In contrast, destabilised (non-autarkic) triples generate:

• Runaway and walkaway stars: Stellar ejections at velocities up to several $\times 10$ km/s, with maxima of a few $100$ km/s for compact systems, potentially explaining the population of field O/B runaways.
• Blue straggler and exotic binary formation: Main-sequence collisions at rates of $\sim 10^{-4}$ yr$^{-1}$ in the Galaxy, with remnants sometimes appearing as non-coeval binaries ("twin stragglers").
• Eccentric, wide binaries: Systems may exhibit sub- or superthermal eccentricity distributions, anomalous component ages, or spins.
• Stellar-collision transients: Intermediate-luminosity optical transients (ILOTs) and luminous novae at rates of $10^{-4}$–$10^{-5}$ yr$^{-1}$, sometimes involving giants or WD–WD supernovae.

Autarkic triples, by never breaching the Mardling–Aarseth boundary, avoid these chaotic outcomes and instead display slow, predictable orbital evolution (Toonen et al., 2021).