Two-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.

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.

Reshaped three-body interactions and the observation of an Efimov state in the continuum

Efimov trimers are exotic three-body quantum states that emerge from the different types of three-body continua in the vicinity of two-atom Feshbach resonances. In particular, as the strength of the interaction is decreased to a critical point, an Efimov state merges into the atom-dimer threshold and eventually dissociates into an unbound atom-dimer pair. Here we explore the Efimov state in the vicinity of this critical point using coherent few-body spectroscopy in 7Li atoms using a narrow two-body Feshbach resonance. Contrary to the expectation, we find that the 7Li Efimov trimer does not immediately dissociate when passing the threshold, and survives as a metastable state embedded in the atom-dimer continuum. We identify this behavior with a universal phenomenon related to the emergence of a repulsive interaction in the atom-dimer channel which reshapes the three-body interactions in any system characterized by a narrow Feshbach resonance. Specifically, our results shed light on the nature of 7Li Efimov states and provide a path to understand various puzzling phenomena associated with them.

Study of the e+e−→ π+π−ω process at center-of-mass energies between 4.0 and 4.6 GeV

Using 15.6 fb−1 of e+e− collision data collected at twenty-four center-of-mass energies from 4.0 to 4.6 GeV with the BESIII detector, the helicity amplitudes of the process e+e− → π+π−ω are analyzed for the first time. Born cross section measurements of two-body intermediate resonance states with statistical significance greater than 5σ are presented, such as f0(500), f0(980), f2(1270), f0(1370), b1(1235)±, and ρ(1450)±. In addition, evidence of a resonance state in e+e− → π+π−ω production is found. The mass of this state obtained by line shape fitting is about 4.2 GeV/c2, which is consistent with the production of ψ(4160) or Y(4220).

A Six-planet Resonance Chain in K2-138?

The K2-138 system hosts six planets and presents an interesting case study due to its distinctive dynamical structure. Its five inner planets are near a chain of 3/2 two-body mean-motion resonances, while the outermost body (planet g) is significantly detached, having a mean-motion ratio of n f/n g ∼ 3.3 with its closest neighbor. We show that the orbit of m g is actually consistent with the first-order three-planet resonance (3P-MMR) characterized by the relation 2n e − 4n f + 3n g = 0 and is the first time a pure first-order 3P-MMR has been found in a multiplanet system and tied to its current dynamical structure. Adequate values for the masses allow one to trace the dynamical history of the system from an initial capture in a six-planet chain (with n f/n g in a 3/1 resonance) up to its current configuration due to tidal interactions over the age of the star. The increase in resonance offset with semimajor axis, as well as its large value for n f/n g, can be explained by the slopes of the pure three-planet resonances in the mean-motion ratio plane. The triplets slide outward over these curves when the innermost pair is pulled apart by tidal effects, in a pantograph-like manner. The capture into the 3P-MMR is found to be surprisingly robust given similar masses for m g and m f, and it is possible that the same effect may also be found in other compact planetary systems.

ExoMMR: A New Python Package to Confirm and Characterize Mean Motion Resonances

The study of orbital resonances allows for the constraint of planetary properties of compact systems. We can predict a system’s resonances by observing the orbital periods of the planets, as planets in or near mean motion resonance (MMR) have period ratios that reduce to a ratio of small numbers. However, a period ratio near commensurability does not guarantee a resonance; we must study the system’s dynamics and resonant angles to confirm resonance. Because resonances require in-depth study to confirm, and because two-body resonances require a measurement of the eccentricity vector which is quite challenging, very few resonant pairs or chains have been confirmed. We thus remain in the era of small-number statistics, not yet able to perform large population synthesis or informatics studies. To address this problem, we build a python package to find, confirm, and analyze MMRs, primarily through N-body simulations. We then analyze all near-resonant planets in the Kepler/K2 and TESS catalogs, confirming over 60 new resonant pairs and various new resonant chains. We additionally demonstrate the package’s functionality and potential by characterizing the mass–eccentricity degeneracy of Kepler-80g, exploring the likelihood of an exterior giant planet in Kepler-80, and constraining the masses of planets in Kepler-305. We find that our methods overestimate the libration amplitudes of the resonant angles and struggle to confirm resonances in systems with more than three planets. We identify various systems that are likely resonant chains but that we are unable to confirm, and highlight next steps for exoplanetary resonances.

Confirming Resonance in Three Transiting Systems

Although resonant planets have orbital periods near commensurability, resonance is also dictated by other factors, such as the planets’ eccentricities and masses, and therefore must be confirmed through a study of the system’s dynamics. Here, we perform such a study for five multiplanet systems: Kepler-226, Kepler-254, Kepler-363, Kepler-1542, and K2-32. For each system, we run a suite of N-body simulations that span the full parameter space that is consistent with the constrained orbital and planetary properties. We study the stability of each system and look for resonances based on the libration of the critical resonant angles. We find strong evidence for a two-body resonance in each system; we confirm a 3:2 resonance between Kepler-226c and Kepler-226d, confirm a 3:2 resonance between Kepler-254c and Kepler-254d, and confirm a three-body 1:2:3 resonant chain between the three planets of Kepler-363. We explore the dynamical history of two of these systems and find that these resonances most likely formed without migration. Migration leads to the libration of the three-body resonant angle, but these angles circulate in both Kepler-254 and Kepler-363. Applying our methods to additional near-resonant systems could help us identify which systems are truly resonant or nonresonant and which systems require additional follow-up analysis.

Eigenvector continuation for emulating and extrapolating two-body resonances

Eigenvector continuation for emulating and extrapolating two-body resonances

Analysis of rescattering effects in 3pi final states

Decays into three particles are often described in terms of two-body resonances and a non-interacting spectator particle. To go beyond this simplest isobar model, crossed-channel rescattering effects need to be accounted for. We quantify the importance of these rescattering effects in three-pion systems for different decay masses and angular-momentum quantum numbers. We provide amplitude decompositions for four decay processes with total J^{PC} = 0^{--}, 1^{--}, 1^{-+}, and 2^{++}, all of which decay predominantly as rho pi states. Two-pion rescattering is described in terms of an Omnès function, which incorporates the rho resonance. Inclusion of crossed-channel effects is achieved by solving the Khuri–Treiman integral equations. The unbinned log-likelihood estimator is used to determine the significance of the rescattering effects beyond two-body resonances; we compute the minimum number of events necessary to unambiguously find these in future Dalitz-plot analyses. Kinematic effects that enhance or dilute the rescattering are identified for the selected set of quantum numbers and various masses.

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.

High-partial-wave separable potential method for investigating Feshbach resonances of ultracold atoms

We present a high-partial-wave (non-zero partial-wave) separable potential method (SPM) for investigating magnetically induced Feshbach resonances in collisions of ultracold atoms, then derive the Lippmann–Schwinger equation for two ultracold alkali-metal atoms in an external magnetic field and calculate the generalized scattering length and weakly bound state energy. We apply this method to the 6Li system and calculate the weakly bound state energy and the scattering volume near three typical p-wave Feshbach resonances. The calculated results by using the SPM agree well with those by using the coupled channel method and with experimental measurements. The SPM is expected to simplify computation of the three-body collision Faddeev equation in the vicinity of two-body high partial-wave Feshbach resonances.

Toffoli gate based on a three-body fine-structure-state-changing Förster resonance in Rydberg atoms

We have developed an improved scheme of a three-qubit Toffoli gate based on fine structure state changing three-body Stark-tuned Rydberg interaction. This scheme is a substantial improvement of our previous proposal [I.I.Beterov et al., Physical Review A 98, 042704 (2018)]. Due to the use of a different type of three-body F\"orster resonance we substantially simplified the scheme of laser excitation and phase dynamics of collective three-body states. This type of F\"orster resonance exists only in systems with more than two atoms, while the two-body resonance is absent. We reduced the sensitivity of the gate fidelity to fluctuations of external electric field and eliminated the necessity to use external magnetic field for fine tuning of the resonant electric field value, compared to the previous scheme of Toffoli gate based on Rydberg atoms. A gate fidelity of >99% was demonstrated in the calculations.

Extended planetary chaotic zones

ABSTRACT We consider the chaotic motion of low-mass bodies in two-body high-order mean-motion resonances with planets in model planetary systems, and analytically estimate the Lyapunov and diffusion time-scales of the motion in multiplets of interacting subresonances corresponding to the mean-motion resonances. We show that the densely distributed (though not overlapping) high-order mean-motion resonances, when certain conditions on the planetary system parameters are satisfied, may produce extended planetary chaotic zones – ‘zones of weak chaotization,’ – much broader than the well-known planetary connected chaotic zone, the Wisdom gap. This extended planetary chaotic zone covers the orbital range between the 2/1 and 1/1 resonances with the planet. On the other hand, the orbital space inner (closer to the host star) with respect to the 2/1 resonance location is essentially long-term stable. This difference arises because the adiabaticity parameter of subresonance multiplets specifically depends on the particle’s orbit size. The revealed effect may control the structure of planetesimal discs in planetary systems: the orbital zone between the 2/1 and 1/1 resonances with a planet should be normally free from low-mass material (only that occasionally captured in the first-order 3/2 or 4/3 resonances may survive); whereas any low-mass population inner to the 2/1 resonance location should be normally long-lived (if not perturbed by secular resonances, which we do not consider in this study).

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.

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.

Three-body Resonances in the Saturnian System

Abstract Saturn has a dynamically rich satellite system, which includes at least three orbital resonances between three pairs of moons: Mimas–Tethys 4:2, Enceladus–Dione 2:1, and Titan–Hyperion 4:3 mean-motion resonances. Studies of the orbital history of Saturn’s moons usually assume that their past dynamics was also dominated solely by two-body resonances. Using direct numerical integrations, we find that three-body resonances among Saturnian satellites were quite common in the past, and could result in a relatively long-term, but finite capture time (10 Myr or longer). We find that these three-body resonances are invariably of the eccentricity type and do not appear to affect the moons’ inclinations. While some three-body resonances are located close to two-body resonances (but involve the orbital precession of the third body), others are isolated, with no two-body arguments being near resonance. We conclude that future studies of the system’s past must take full account of three-body resonances, which have been overlooked in the past work.

A Criterion for the Onset of Chaos in Compact, Eccentric Multiplanet Systems

Abstract We derive a semianalytic criterion for the presence of chaos in compact, eccentric multiplanet systems. Beyond a minimum semimajor axis separation, below which the dynamics are chaotic at all eccentricities, we show that (i) the onset of chaos is determined by the overlap of two-body mean motion resonances (MMRs), like it is in two-planet systems; and (ii) secular evolution causes the MMR widths to expand and contract adiabatically, so that the chaotic boundary is established where MMRs overlap at their greatest width. For closely spaced two-planet systems, a near symmetry strongly suppresses this secular modulation, explaining why the chaotic boundaries for two-planet systems are qualitatively different from cases with more than two planets. We use these results to derive an improved angular momentum deficit (AMD) stability criterion, i.e., the critical system AMD below which stability should be guaranteed. This introduces an additional factor to the expression from Laskar and Petit that is exponential in the interplanetary separations, which corrects the AMD threshold toward lower eccentricities by a factor of several for tightly packed configurations. We make routines for evaluating the chaotic boundary available to the community through the open-source SPOCK package.

Application of the Shannon entropy in the planar (non-restricted) four-body problem: the long-term stability of the Kepler-60 exoplanetary system

ABSTRACT In this paper, we present an application of the Shannon entropy in the case of the planar (non-restricted) four-body problem. Specifically, the Kepler-60 extrasolar system is being investigated with a primary interest in the resonant configuration of the planets that exhibit a chain of mean-motion commensurabilities with the ratios 5:4:3. In the dynamical maps provided, the Shannon entropy is utilized to explore the general structure of the phase space, while, based on the time evolution of the entropy, we also determine the extent and rate of the chaotic diffusion as well as the characteristic times of stability for the planets. Two cases are considered: (i) the pure Laplace resonance when the critical angles of the two-body resonances circulate and that of the three-body resonance librates; and (ii) the chain of two two-body resonances when all the critical angles librate. Our results suggest that case (ii) is the more favourable configuration, but we state too that, in either case, the relevant resonance plays an important role in stabilizing the system. The derived stability times are no shorter than 108 yr in the central parts of the resonances.

Direct and sequential four-body recombination rates at low temperatures

We investigate four-body nuclear reactions in stellar environments contributing to creation of light nuclei, exemplified by $^9$Be and $^{12}$C. The originally assumed process is radiative capture, where nuclear clusters combine into the excited final nucleus and photon emission populates the stable nuclear ground states. Instead, we consider nuclear four-body recombination reactions where a spectator nuclear particle replaces the photon. We first develop the elaborate formalism for both, direct and sequential capture processes, where the decaying three-body resonance is formed without and with population of an intermediate two-body resonance, respectively. To facilitate both calculations and practical applications we parameterize the involved cross sections as done successfully in previous computations of reaction rates. We consider the lowest-lying nuclear states with their dominant contributions at low stellar temperatures. We calculate and compare reaction and production rates for different processes. The direct reaction mechanism dominates by many orders of magnitude at low temperature, where the sequential stepping stones are energetically too expensive to use. At somewhat higher temperatures these two different nuclear four-body mechanisms become comparable. Comparison to radiative three-body capture reveals already formally, but also numerically, that four-body nuclear recombination must dominate for sufficiently high nuclear densities. Numerical values are given for all these rates as function of temperature and density. The relative importance is exhibited.

Evidence against the Efimov effect in C12 from spectroscopy and astrophysics

Background: The Efimov effect is a universal phenomenon in physics whereby three-body systems are stabilized via the interaction of an unbound two-body sub-systems. A hypothetical state in $^{12}\mathrm{C}$ at 7.458 MeV excitation energy, comprising of a loose structure of three $\alpha$-particles in mutual two-body resonance, has been suggested in the literature to correspond to an Efimov state in nuclear physics. The existence of such a state has not been demonstrated experimentally. Method: Using the combined data sets from two recent experiments, one with the TexAT TPC to measure $\alpha$-decay and the other with Gammasphere to measure $\gamma$-decay of states in $^{12}\mathrm{C}$ populated by $^{12}\mathrm{N}$ and $^{12}\mathrm{B}$ $\beta$-decay respectively, we achieve high sensitivity to states in close-proximity to the $\alpha$-threshold in $^{12}\mathrm{C}$. Results: No evidence of a state at 7.458 MeV is seen in either data set. Using a likelihood method, the 95\% C.L. $\gamma$-decay branching ratio is determined as a function of the $\beta$-decay feeding strength relative to the Hoyle state. In parallel, calculations of the triple-alpha reaction rate show the inclusion of the Efimov corresponds to a large increase in the reaction rate around $5 \times 10^{7}$ K. Conclusion: From decay spectroscopy - at the 95\% C.L., the Efimov state cannot exist at 7.458 MeV with any $\gamma$-decay branching ratio unless the $\beta$-strength is less than 0.7\% of the Hoyle state. This limit is evaluated for a range of different excitation energies and the results are not favorable for existence of the hypothetical Efimov state in $^{12}\mathrm{C}$. Furthermore, the triple-alpha reaction rate with the inclusion of a state between 7.43 and 7.53 MeV exceeds the rate required for stars to undergo the red giant phase.