Orbital stability of compact three-planet systems III. The role of three-body resonances

TL;DR

This study investigates the stability of extremely compact three-planet systems, revealing that anomalously long-lived configurations arise when the planetary trio is captured into isolated zeroth-order three-body resonances (3BRs). Using high-resolution numerical integrations of coplanar, equal-mass Earth-mass planets around a solar-mass star, the authors map lifetimes in the period-ratio plane and identify five distinct spikes (SPK1–SPK5) associated with specific 3BRs and resonant angles $\phi$. They show that stability is governed not only by two-body resonances but also by the isolation and density of the 3BR network, with the most isolated resonances (notably certain $\alpha = q/p$ values) providing protection from chaotic diffusion even near the Hill and 3BR overlap limits. The findings connect to observed resonant chains in Kepler and TRAPPIST-1, suggesting that very tightly packed resonant triplets preferentially populate the main isolated 3BRs and that 3BR topology should be incorporated into stability criteria and planetary-system formation models.

Abstract

Observational surveys show that at least ~ 30% of short-period multiplanetary systems host tightly packed planets, some of which are locked in stable chains of mean-motion resonances. Despite recent progress, the dynamical stability of these systems remains only partially understood. Numerical simulations have established a general exponential increase in system lifetime with orbital separation, with mean-motion resonances playing a key role in regulating stability. Tightly packed three-planet systems exhibit a distinctive behavior not seen in higher-multiplicity systems: a small yet significant region of phase space is anomalously stable. This study investigates the dynamics of extremely compact three-planet systems, focusing on anomalously long-lived configurations and their connection to resonant chains observed in exoplanetary systems. We perform numerical integrations of coplanar, initially circular, equal-mass three-planet systems over stellar-lifetime timescales and at high resolution in orbital separation, and interpret the results in the context of recent analytical work. We identify regions of phase space hosting anomalously stable orbits, including systems surviving multiple orders of magnitude longer than predicted by the exponential trend. We demonstrate a clear link between stability and isolated three-body mean-motion resonances, showing that extremely compact systems can remain stable when captured into a small subset of isolated zeroth-order resonances. Stability further depends on the initial orbital longitudes and on the interplay between the three-body and two-body resonance networks.

Figures (23)

• Figure 1: Lifetime, $t_c$, of three-planet systems as a function of the initial separation of the orbits of neighboring planets in units of $\beta$ for the set Random (initial longitudes of the middle and outer planets selected randomly). Panel (a) shows the full view with a density of $5\times10^{6}$ per unit $\beta$ within the range [2$\sqrt{3}$, 5.00], and a density of $2\times10^{4}$ within the range (5.00, 5.60]. Systems with a lifetime greater than 5$\sigma$ above the exponential fit are shown with larger size dots that are navy blue in color. The upward-pointing triangles represent systems that remained stable for the entire elapsed time, 10$^{10}$ years. The upper horizontal axis shows the initial period ratios of neighboring planets as defined by Eq. \ref{['eq:periodratio_beta']}. The solid (dotted) red vertical lines denote the locations of first-order (second-order) 2BRs between adjacent pairs of planets, and the solid gold vertical lines denote first-order 2BRs between the innermost and outermost planets. The solid gray line segment represents the exponential fit to all points with a density of 10,000 per unit $\beta$ within the range [3.4645, 5.6000]. The terms SPKi denote the ith spike with $i \in$ [1,2,3,4,5]. Panel (b) shows a zoomed-in view with a density of $10^{7}$ within the range $\beta \in/$ [3.81, 3.85].
• Figure 2: Initial longitude of the middle planet versus the outer planet of systems from the Random set whose lifetimes are greater than 5$\sigma$ above the exponential fit (anomalous systems), i.e., systems in SPK1, SPK2, SPK3, SPK4, and SPK5. For SPK1 we include the systems from the higher resolution batch seen on Fig. \ref{['fig:drawingB']}. The solid black diagonal curve marks the locus of points where the outer pair of planets start in conjunction.
• Figure 3: Lifetime, $t_c$, of three-planet systems as a function of the initial separation of the orbits of neighboring planets for the RR set (see Table \ref{['tab:sets']}) with a fixed density of $2\times10^5$ per unit of $\beta$ over the range [2$\sqrt{3}$, 4.9000].
• Figure 4: Left: lifetime, $t_c/t_0$, of three-planet systems as a function of the initial separation of the orbits of neighboring planets for the set S1 ($\lambda_\mathrm{2,init}$ = 166.273784$^\circ$ and $\lambda_\mathrm{3,init}$ = 165.110014$^\circ$). The triangles represent systems that remained stable for the entire time interval, 10$^{10}$ years. There are five systems that survived for ten billion years; these systems have initial orbital separations of $\beta$ = 3.81783, 3.81921, 3.81922, 3.81926, and 3.82086. The resolution is 10$^5$ per unit of $\beta$. Right: zoom into the spike in the range [3.80, 3.87].
• Figure 5: Lifetime, $t_c/t_0$, of three-planet systems as a function of the initial separation of the orbits of neighboring planets for the set S2 ($\lambda_\mathrm{2,init}$ = 207.16381$^\circ$ and $\lambda_\mathrm{3,init}$ = 208.66160$^\circ$). The resolution is 10$^6$ per unit $\beta$ in the range [3.47643, 3.47645] and $10^4$ elsewhere over the range [3.4645, 4.1300].