Three-body Resonances Research Articles

The Instability Mechanism of Compact Multiplanet Systems

To improve our understanding of orbital instabilities in compact planetary systems, we compare suites of N-body simulations against numerical integrations of simplified dynamical models. We show that, surprisingly, dynamical models that account for small sets of resonant interactions between the planets can accurately recover N-body instability times. This points toward a simple physical picture in which a handful of three-body resonances, generated by interactions between nearby two-body mean motion resonances, overlap and drive chaotic diffusion, leading to instability. Motivated by this, we show that instability times are well described by a power law relating instability time to planet separations, measured in units of fractional semimajor axis difference divided by the planet-to-star mass ratio to the 1/4 power, rather than the frequently adopted 1/3 power implied by measuring separations in units of mutual Hill radii. For idealized systems, the parameters of this power-law relationship depend only on the ratio of the planets’ orbital eccentricities to the orbit-crossing value, and we report an empirical fit to enable quick instability time predictions. This relationship predicts that observed systems comprised of three or more sub-Neptune-mass planets must be spaced with period ratios P≳1.35 and that tightly spaced systems ( P≲1.5 ) must possess very low eccentricities (e ≲ 0.05) to be stable for more than 109 orbits.

Resonant chains in triple-planet systems

The mean motion resonance is the most important mechanism that may dominate the dynamics of a planetary system. In a multi-planetary system consisting of $N planets, the planets may form a resonant chain when the ratios of orbital periods of planets can be expressed as the ratios of small integers $T_1: T_2: :T_N=k_1: k_2: k_N$. Due to the high degree of freedom, the motion in such systems could be complex and difficult to depict. In this paper, we investigate the dynamics and possible formation of the resonant chain in a triple-planet system. We defined the appropriate Hamiltonian for a three-planet resonant chain and numerically averaged it over the synodic period. The stable stationary solutions -- apsidal corotational resonances (ACRs) -- of this averaged system, corresponding to the local extrema of the Hamiltonian function, can be searched out numerically. The topology of the Hamiltonian around these ACRs reveals their stabilities. We further constructed the dynamical maps on different representative planes to study the dynamics around the stable ACRs, and we calculated the deviation ($ of the resonant angle in the evolution from the uniformly distributed values, by which we distinguished the behaviour of critical angles. Finally, the formation of the resonant chain via convergent planetary migration was simulated and the stable configurations associated with ACRs were verified. We find that the stable ACR families arising from circular orbits always exist for any resonant chain, and they may extend to a high eccentricity region. Around these ACR solutions, regular motion can be found, typically in two types of resonant configurations. One is characterised by libration of both the two-body resonant angles and the three-body Laplace resonant angle, and the other by libration of only two-body resonant angles. The three-body Laplace resonance does not seem to contribute to the stability of the resonant chain much. The resonant chain can be formed via convergent migration, and the resonant configuration evolves along the ACR families to eccentric orbits once the planets are captured into the chain. Ideally, our methods introduced in this paper can be applied to any resonant chain of any number of planets at any eccentricity.

The nature of the Laplace resonance between the Galilean moons

The Laplace resonance is a mean-motion resonance that involves the three inner Galilean moons of Jupiter. However, its true nature is in part unclear; in particular, different views can be found in the literature on whether the Laplace resonance is a pure three-body resonance or a mere superposition of two-body resonances. To settle this question, we conduct a thorough analysis of the many resonances involved, starting from the two-body 2:1 commensurabilities of the couples Io–Europa and Europa–Ganymede, and ending with the three-body 4:2:1 commensurability between the three moons. By artificially varying the parameters of the system and monitoring its fundamental frequencies, we cartography all resonances involved and their interactions. From the analysis of the individual 2:1 commensurabilities, we find that despite the oscillation of the resonant angles they are not genuine resonances, as the trajectory of the system in the phase space is not enclosed by separatrices. On the contrary, as suggested by previous works, we show that the only current true mean-motion resonance is the pure three-body resonance between all three satellites. Moreover, we find that the current values of the moons’ orbital elements make the Laplace resonance sufficiently separated from the individual two-body 2:1 resonances, preventing chaotic effects from appearing.

Mass Ratio Dependence of Three-Body Resonance Lifetimes in 1D and 3D

Mass Ratio Dependence of Three-Body Resonance Lifetimes in 1D and 3D

Titania's Heat Fluxes Revealed by Messina Chasmata

Messina Chasmata is a relatively young tectonic structure on Titania based on cross-cutting relationships, although an absolute age has not been estimated. We investigated lithospheric flexure bounding Messina and found that the terrain along both rims reflects Titania’s thermal properties. We estimate Titania’s heat fluxes to have been 5–12 mW m−2 in this region, assuming that the lithosphere is composed of pure H2O ice without porosity. These estimates are lower if lithospheric porosity and/or NH3–H2O are also present. If Messina is ancient, forming as a result of freeze expansion, then the reflected heat fluxes are consistent with radiogenic heating. However, if Messina is instead young, then an additional heat source is required. In this scenario, perhaps tidal heating associated with the past three-body resonance shared between Titania, Ariel, and Umbriel generated this heat. However, this scenario is unlikely because the amount of tidal heating produced on Titania would have been minimal. Titania’s heat fluxes are notably lower than estimates for Miranda or Ariel, and future work is needed to investigate Umbriel and Oberon to gain a fuller understanding of Uranian moon thermal and orbital histories. Additionally, further constraints on Titania’s more ancient heat fluxes could be obtained by investigating relatively older features, such as some viscously relaxed impact craters.

A new python package for identifying celestial bodies trapped in mean-motion resonances

In this paper, a new open-source package ‘resonances’ written in python is introduced. It allows to find, analyse, and plot two-body and three-body mean-motion eccentricity-type resonances in the Solar and other planetary systems. The package has a better accuracy of the automatic identification procedure for resonant objects compared to previous studies. Furthermore, it has built-in integrations with AstDyS and NASA JPL catalogues. The code is extensively documented and tested with automatic tests. The package is available on GitHub under MIT Licence.

A Gap-sharing Planet Pair Shaping the Crescent in HD 163296: A Disk Sculpted by a Resonant Chain

The Atacama Large Millimeter Array observations of the disk around HD 163296 have resolved a crescent-shape substructure at around 55 au, inside and off-center from a gap in the dust that extends from 38 to 62 au. In this work we propose that both the crescent and the dust rings are caused by a compact pair (period ratio ≃4:3) of sub-Saturn-mass planets inside the gap, with the crescent corresponding to dust trapped at the L 5 Lagrange point of the outer planet. This interpretation also reproduces well the gap in the gas recently measured from the CO observations, which is shallower than what is expected in a model where the gap is carved by a single planet. Building on previous works arguing for outer planets at ≈86 and ≈137 au, we provide a global model of the disk that best reproduces the data and shows that all four planets may fall into a long resonant chain, with the outer three planets in a 1:2:4 Laplace resonance. We show that this configuration is not only an expected outcome from disk–planet interaction in this system, but it can also help constrain the radial and angular position of the planet candidates using three-body resonances.

Three-body resonances in the φ4 theory

We study the properties of three-body resonances using a lattice complex scalar φ4 theory with two scalars, with parameters chosen such that one heavy particle can decay into three light ones. We determine the two- and three-body spectra for several lattice volumes using variational techniques, and then analyze them with two versions of the three-particle finite-volume formalism: the Relativistic Field Theory approach and the Finite-Volume Unitarity approach. We find that both methods provide an equivalent description of the energy levels, and we are able to fit the spectra using simple parametrizations of the scattering quantities. By solving the integral equations of the corresponding three-particle formalisms, we determine the pole position of the resonance in the complex energy plane and thereby its mass and width. We find very good agreement between the two methods at different values of the coupling of the theory.

Kepler-80 Revisited: Assessing the Participation of a Newly Discovered Planet in the Resonant Chain

In this paper, we consider the chain of resonances in the Kepler-80 system and evaluate the impact that the additional member of the resonant chain discovered by Shallue & Vanderburg has on the dynamics of the system and the physical parameters that can be recovered by a fit to the transit timing variations (TTVs). Ultimately, we calculate the mass of Kepler-80 g to be 0.8 ± 0.3M ⊕ when assuming all planets have zero eccentricity, and 1.0 ± 0.3 M ⊕ when relaxing that assumption. We show that the outer five planets are in successive three-body mean-motion resonances (MMRs). We assess the current state of two-body MMRs in the system and find that the planets do not appear to be in two-body MMRs. We find that while the existence of the additional member of the resonant chain does not significantly alter the character of the Kepler-80 three-body MMRs, it can alter the physical parameters derived from the TTVs, suggesting caution should be applied when drawing conclusions from TTVs for potentially incomplete systems. We also compare our results to those of MacDonald et al., who perform a similar analysis on the same system with a different method. Although the results of this work and MacDonald et al. show that different fit methodologies and underlying assumptions can result in different measured orbital parameters, the most secure conclusion is that which holds true across all lines of analysis: Kepler-80 contains a chain of planets in three-body MMRs but not in two-body MMRs.

Efimov-like physics in fraction-dimensional Bose systems with three-body interaction

A few-body properties of spinless Bose particles interacting via the contact three-body potential in geometries with fractional dimensions $1<d<2$ are considered. In the four-body sector at three-body resonance we predict the existence of infinite tower of the Efimov bound states, and a similar behavior is found in the five-body problem. It is shown that a ratio of the high-energy levels in these two sectors is a universal constant. The consequences of emergence of the Efimov physics on the many-body behavior are briefly discussed.

Environment-modified three-body energy transfer

Resonant energy transfer from a donor to an acceptor is one of the most basic interactions between atomic and molecular systems. In real-life situations, the donor and acceptor are not isolated but in fact coupled to their environment and to other atoms and molecules. The presence of a third body can modify the rate of energy transfer between donor and acceptor in distinctive and intricate ways, especially when the three-site system is itself interacting with a larger macroscopic background such as a solvent. The rate can be calculated perturbatively, which ordinarily requires the summation of very large numbers of Feynman-like diagrams. Here we demonstrate a method based on canonical perturbation theory that allows us to reduce the computational effort required, and use this technique to derive a formula for the rate of three-body resonance energy transfer in a background environment. As a proof of principle, we apply this to the situation of a dimer positioned near a dielectric interface, with a distant third molecule controlling the rate, finding both enhancement or suppression of the rate depending on system parameters.

Machine learning applied to asteroid dynamics

Machine learning (ML) is the branch of computer science that studies computer algorithms that can learn from data. It is mainly divided into supervised learning, where the computer is presented with examples of entries, and the goal is to learn a general rule that maps inputs to outputs, and unsupervised learning, where no label is provided to the learning algorithm, leaving it alone to find structures. Deep learning is a branch of machine learning based on numerous layers of artificial neural networks, which are computing systems inspired by the biological neural networks that constitute animal brains. In asteroid dynamics, machine learning methods have been recently used to identify members of asteroid families, small bodies images in astronomical fields, and to identify resonant arguments images of asteroids in three-body resonances, among other applications. Here, we will conduct a full review of available literature in the field and classify it in terms of metrics recently used by other authors to assess the state of the art of applications of machine learning in other astronomical subfields. For comparison, applications of machine learning to Solar System bodies, a larger area that includes imaging and spectrophotometry of small bodies, have already reached a state classified as progressing. Research communities and methodologies are more established, and the use of ML led to the discovery of new celestial objects or features, or new insights in the area. ML applied to asteroid dynamics, however, is still in the emerging phase, with smaller groups, methodologies still not well-established, and fewer papers producing discoveries or insights. Large observational surveys, like those conducted at the Zwicky Transient Facility or at the Vera C. Rubin Observatory, will produce in the next years very substantial datasets of orbital and physical properties for asteroids. Applications of ML for clustering, image identification, and anomaly detection, among others, are currently being developed and are expected of being of great help in the next few years.

Discovery of an asteroid family linked to (22) Kalliope and its moon Linus

Aims. According to adaptive-optics observations, (22) Kalliope is a 150-km-wide, dense, and differentiated body. Here, we interpret (22) Kalliope in the context of the bodies in its surroundings. While there is a known moon, Linus, with a 5:1 size ratio, no family has been reported in the literature, which is in contradiction with the existence of the moon. Methods. Using the hierarchical clustering method along with physical data, we identified the Kalliope family. It had previously been associated with (7481) San Marcello. We then used various models (N-body, Monte Carlo, and SPH) of its orbital and collisional evolution, including the breakup of the parent body, to estimate the dynamical age of the family and address its link to Linus. Results. The best-fit age is (900 ± 100) Myr according to our collisional model; this is in agreement with the position of (22) Kalliope, which was modified by chaotic diffusion due to 4–1–1 three-body resonance with Jupiter and Saturn. It seems possible that Linus and the Kalliope family were created at the same time, although our SPH simulations show a variety of outcomes for both satellite size and the family size-frequency distribution. The shape of (22) Kalliope itself was most likely affected by the gravitational re-accumulation of ‘streams’, which creates the characteristic hills observed on its surface. If the body was differentiated, its internal structure is most likely asymmetric.

Long-term tidal evolution of the TRAPPIST-1 system

ABSTRACT The ultracool M-dwarf star TRAPPIST-1 is surrounded by seven planets configured in a resonant chain. Transit-timing variations have shown that the planets are caught in multiple three-body resonances and that their orbits are slightly eccentric, probably caused by resonant forcing. The current values of the eccentricities could be a remnant from their formation. Here, we run numerical simulations using fictitious forces of trapping the fully grown planets in resonances as they migrated in the gas disc, followed by numerical simulations detailing their tidal evolution. For a reduced disc scale height h ∼ 0.03–0.05, the eccentricities of the planets upon capture in resonance are higher than their current values by factors of a few. We show that the current eccentricities and spacing of planets d to h are natural outcomes of coupled tidal evolution wherein the planets simultaneously damp their eccentricities and separate due to their resonant interaction. We further show that the planets evolve along a set of equilibrium curves in semimajor axis–eccentricity phase space that are defined by the resonances, and that conserve angular momentum. As such, the current 8:5–5:3–(3:2)2–4:3–3:2 resonant configuration cannot be reproduced from a primordial (3:2)4–4:3–3:2 resonant configuration from tidal dissipation in the planets alone. We use our simulations to constrain the long-term tidal parameters k2/Q for planets b to e, which are in the range of 10−3 to 10−2, and show that these are mostly consistent with those obtained from interior modelling following reasonable assumptions.

Three-body resonances in pionless effective field theory

We investigate the appearance of resonances in three-body systems using pionless effective field theory at leading order. The Faddeev equation is analytically continued to the unphysical sheet adjacent to the positive real energy axis using a contour rotation. We consider both, the three-boson system and the three-neutron system. For the former, we calculate the trajectory of Borromean three-body Efimov states turning into resonances as they cross the three-body threshold. For the latter, we find no sign of three-body resonances or virtual states at leading order. This result is validated by exploring the level structure of three-body states in a finite volume approach.

The Criterion for Chaos in Three-planet Systems

We establish the criterion for chaos in three-planet systems, for systems similar to those discovered by the Kepler spacecraft. Our main results are as follows: (i) The simplest criterion, which is based on overlapping mean motion resonances (MMRs), only agrees with numerical simulations at a very crude level. (ii) Much greater accuracy is attained by considering neighboring MMRs that do not overlap. We work out the widths of the chaotic zones around each of the neighbors, and also provide simple approximate expressions for the widths. (iii) Even greater accuracy is provided by the overlap of three-body resonances (3BRs), which accounts for the fine-grained structure seen in maps from N-body simulations, and also predicts Lyapunov times. From previous studies, it is unclear whether interplanetary chaos should be attributed to the overlap of MMRs or of 3BRs. We show that the two apparently contradictory viewpoints are in fact consistent: both predict the same criterion for chaos. (iv) We compare the predicted criterion with high-resolution maps of chaos from N-body simulations, and show that they agree at a high level of detail.

Periodic orbits in the 1:2:3 resonant chain and their impact on the orbital dynamics of the Kepler-51 planetary system

Aims. Space missions have discovered a large number of exoplanets evolving in (or close to) mean-motion resonances (MMRs) and resonant chains. Often, the published data exhibit very high uncertainties due to the observational limitations that introduce chaos into the evolution of the system on especially shorter or longer timescales. We propose a study of the dynamics of such systems by exploring particular regions in phase space. Methods. We exemplify our method by studying the long-term orbital stability of the three-planet system Kepler-51 and either favor or constrain its data. It is a dual process which breaks down in two steps: the computation of the families of periodic orbits in the 1:2:3 resonant chain and the visualization of the phase space through maps of dynamical stability. Results. We present novel results for the general four-body problem. Stable periodic orbits were found only in the low-eccentricity regime. We demonstrate three possible scenarios safeguarding Kepler-51, each followed by constraints. Firstly, there are the 2/1 and 3/2 two-body MMRs, in which eb &lt; 0.02, such that these two-body MMRs last for extended time spans. Secondly, there is the 1:2:3 three-body Laplace-like resonance, in which ec &lt; 0.016 and ed &lt; 0.006 are necessary for such a chain to be viable. Thirdly, there is the combination comprising the 1/1 secondary resonance inside the 2/1 MMR for the inner pair of planets and an apsidal difference oscillation for the outer pair of planets in which the observational eccentricities, eb and ec, are favored as long as ed ≈ 0. Conclusions. With the aim to obtain an optimum deduction of the orbital elements, this study showcases the need for dynamical analyses based on periodic orbits performed in parallel to the fitting processes.

Web of resonances and possible path of evolution of the small Uranian satellites

Satellite systems around giant planets are immersed in a region of complex resonant configurations. Understanding the role of satellite resonances contributes to comprehending the dynamical processes in planetary formation and posterior evolution. Our main goal is to analyse the resonant structure of small moons around Uranus and propose different scenarios able to describe the current configuration of these satellites. We focus our study on the external members of the regular satellites interior to Miranda, namely Rosalind, Cupid, Belinda, Perdita, Puck, and Mab, respectively. We use N-body integrations to perform dynamical maps to analyse their dynamics and proximity to two-body and three-body mean-motion resonances (MMR). We found a complicated web of low-order resonances amongst them. Employing analytical prescriptions, we analysed the evolution by gas drag and type-I migration in a circumplanetary disc (CPD) to explain different possible histories for these moons. We also model the tidal evolution of these satellites using some crude approximations and found possible paths that could lead to MMRs crossing between pairs of moons. Finally, our simulations show that each mechanism can generate significant satellite radial drift leading to possible resonant capture, depending on the distances and sizes.

An Updated Ephemeris for K2-138 d

Abstract K2-138 d (EPIC 245950175 d) is one of five planets in a chain of 3:2 mean motion resonances discovered by Christiansen et al. in K2 Campaign 12. An additional planet, confirmed with the Spitzer Space Telescope by Hardegree-Ullman et al., is not in the resonant chain. The near first-order resonances, coupled with the planets being locked in a set of three-body Laplace resonances, make this system a unique target to study for transit timing variations (TTVs). As the predicted transit timing amplitude (∼7 minutes) was below the typical transit timing uncertainty (∼10 minutes) of the K2 data, Christiansen et al. were unable to detect TTVs for K2-138 d in the K2 time series. Here, we describe new observations that allow us to refine the ephemeris for K2-138 d and perform a brief search for TTVs. Our efforts result in a refined orbital period that is 19 times more precise than previously available measurements, but our data are insufficient to confirm a TTV signal for K2-138 d.

The dynamics of the TRAPPIST-1 system in the context of its formation

ABSTRACT TRAPPIST-1 is an 0.09 M⊙ star, which harbours a system of seven Earth-sized planets. Two main features stand out: (i) all planets have similar radii, masses, and compositions; and (ii) all planets are in resonance. Previous works have outlined a pebble-driven formation scenario where planets of similar composition form sequentially at the H2O snowline (∼0.1 au for this low-mass star). It was hypothesized that the subsequent formation and migration led to the current resonant configuration. Here, we investigate whether the sequential planet formation model is indeed capable to produce the present-day resonant configuration, characterized by its two-body and three-body mean motion resonances structure. We carry out N-body simulations, accounting for type-I migration, stellar tidal damping, disc eccentricity-damping, and featuring a migration barrier located at the disc’s inner edge. Due to the sequential migration, planets naturally form a chain of first-order resonances. But to explain the period ratios of the b/c/d-system, which are presently in higher order resonances, we find that planets b and c must have marched across the migration barrier, into the gas-free cavity, before the disc has dispersed. We investigate both an early and late cavity infall scenario and find that the early infall model best matches the constraints, as well as being more probable. After the dispersal of the gaseous disc, stellar tidal torque also contributes towards a modest separation of the inner system. We outline how the insights obtained in this work can be applied to aid the understanding of other compact resonant planet systems.