Orbital Resonance Research Articles

Exponentially localized interface eigenmodes in finite chains of resonators

AbstractThis paper studies wave localization in chains of finitely many resonators. There is an extensive theory predicting the existence of localized modes induced by defects in infinitely periodic systems. This work extends these principles to finite‐sized systems. We consider one‐dimensional, finite systems of subwavelength resonators arranged in dimers that have a geometric defect in the structure. This is a classical wave analog of the Su–Schrieffer–Heeger model. We prove the existence of a spectral gap for defectless finite dimer structures and find a direct relationship between eigenvalues being within the spectral gap and the localization of their associated eigenmode. Then, for sufficiently large‐size systems, we show the existence and uniqueness of an eigenvalue in the gap in the defect structure, proving the existence of a unique localized interface mode. To the best of our knowledge, our method, based on Chebyshev polynomials, is the first to characterize quantitatively the localized interface modes in systems of finitely many resonators.

Infiltrative Optic Neuropathy

Infiltrative optic neuropathies represent a subset of optic nerve disorders characterized by the infiltration of cells derived from inflammatory, infectious, or tumor metastasis sources. These conditions pose significant challenges in terms of diagnosis, management, and prognosis due to their diverse clinical presentations and potential for rapid progression. Detailed ophthalmic examinations such as visual field testing and optical coherence tomography (OCT) are necessary to provide valuable insights into the structure and function of the optic nerve. Additionally, brain and orbital magnetic resonance imaging (MRI) are essential for detecting infiltrative lesions and assessing the extent of optic nerve involvement.

Spin orbit resonance cascade via core shell model: application to Mercury and Ganymede

We discuss a model describing the spin orbit resonance cascade. We assume that the body has a two-layer (core–shell) structure; it is composed of a thin external shell and an inner and heavier solid core that are interacting due to the presence of a viscous friction. We assume two sources of dissipation: a viscous one, depending on the relative angular velocity between core and shell and a tidal one, smaller than the first, due to the viscoelastic structure of the core. We show how these two sources of dissipation are needed for the capture in spin–orbit resonance. The shell and the core fall in resonance with different time scales if the viscous coupling between them is big enough. Finally, the tidal dissipation of the viscoelastic core, decreasing the eccentricity, brings the system out of the resonance in a third very long time scale. This mechanism of entry and exit from resonance ends in the 1 : 1 stable state.

Mean-motion resonances with interfering density waves

ABSTRACT In this work, we study the dynamics of two less massive objects moving around a central massive object, which are all embedded within a thin accretion disc. In addition to the gravitational interaction between these objects, the disc–object interaction is also crucial for describing the long-term dynamics of the multibody system, especially in the regime of mean-motion resonances. We point out that near the resonance the density waves generated by the two moving objects generally coherently interfere with each other, giving rise to extra angular momentum fluxes. The resulting backreaction on the objects is derived within the thin-disc scenario, which explicitly depends on the resonant angle and sensitively depends on the smoothing scheme used in the two-dimensional theory. We have performed hydrodynamical simulations with planets embedded within a thin accretion disc and have found qualitatively agreement on the signatures of interfering density waves by measuring the torques on the embedded objects, for the cases of $2:1$ and $3:2$ resonance. By including in interference torque and the migration torques in the evolution of a pair of planets, we show that the chance of resonance trapping depends on the sign of the interference torque. For negative interference torques the pairs are more likely located at off-resonance regimes. The negative interference torques may also explain the $1~{{\ \rm per\ cent}}-2~{{\ \rm per\ cent}}$ offset (for the period ratios) from the exact resonance values as observed in Kepler multiplanet systems.

Spin–Orbit Coupling of the Ellipsoidal Secondary in a Binary Asteroid System

In our solar system, spin–orbit coupling is a common phenomenon in binary asteroid systems, where the mutual orbits are no longer invariant due to exchange of angular momentum between translation and rotation. In this work, dynamical structures in phase space are explored for the problem of spin–orbit coupling by taking advantage of analytical and numerical methods. In particular, the technique of Poincaré sections is adopted to reveal numerical structures, which are dependent on the total angular momentum, the Hamiltonian, mass ratio, and asphericity parameter. Analytical study based on perturbative treatments shows that high-order and/or secondary spin–orbit resonances are responsible for numerical structures arising in Poincaré sections. Analytical solutions are applied to (65803) Didymos, (80218) VO123 and (4383) Suruga to reveal their phase-space structures, showing that there is a high possibility for them to locate inside secondary 1:1 spin–orbit resonance.

Family of 2:1 resonant quasi-periodic distant retrograde orbits in cislunar space

Given the current enthusiasm for lunar exploration, the 2:1 resonant distant retrograde orbit (DRO) in Earth-Moon space is of interest. To gain an in-depth understanding of the complex dynamic environment in cislunar space and provide more options for parking orbits, this paper investigates the existence of quasi-periodic orbits near the 2:1 resonant DRO in the circular restricted three-body problem (CR3BP). Firstly, the numerical computation approach, continuation strategy, and stability analysis method of quasi-periodic orbits are introduced. Then, addressing the primary challenges in the continuation progress, we have developed an adaptive continuation algorithm with automatic adjustment of the step size and the number of discrete points that characterize the invariant torus. Finally, two types of 2D quasi-DROs and their linear stability properties are explored. Using Poincaré sections, we investigated the boundaries of the maximum extent attainable by both 2D quasi-DRO families in the CR3BP at a specific Jacobi energy, confirming that both types of quasi-periodic families have reached their respective boundaries. The algorithm described in this paper is beneficial for facilitating the computation of quasi-periodic families and aids in discovering additional potential dynamical structures.

A perturbative treatment of the Yarkovsky-driven drifts in the 2:1 mean motion resonance

Aims. Our aim is to gain a qualitative understanding as well as to perform a quantitative analysis of the interplay between the Yarkovsky effect and the Jovian 2:1 mean motion resonance under the planar elliptic restricted three-body problem. Methods. We adopted the semi-analytical perturbation method valid for arbitrary eccentricity to obtain the resonance structures inside the Jovian 2:1 resonance. We averaged the Yarkovsky force so it could be applied to the integrable approximations for the 2:1 resonance and the ν5 secular resonance. The rates of Yarkovsky-driven drifts in the action space were derived from the quasi-integrable approximations perturbed by the averaged Yarkovsky force. Pseudo-proper elements of test particles inside the 2:1 resonance were computed using N-body simulations incorporated with the Yarkovsky effect to verify the semi-analytical results. Results. In the planar elliptic restricted model, we identified two main types of systematic drifts in the action space: (Type I) for orbits not trapped in the ν5 resonance, the footprints are parallel to the resonance curve of the stable center of the 2:1 resonance; (Type II) for orbits trapped in the ν5 resonance, the footprints are parallel to the resonance curve of the stable center of the ν5 resonance. Using the semi-analytical perturbation method, a vector field in the action space corresponding to the two types of systematic drifts was derived. The Type I drift with small eccentricities and small libration amplitudes of 2:1 resonance can be modeled by a harmonic oscillator with a slowly varying parameter, for which an analytical treatment using the adiabatic invariant theory was carried out.

The 𝒯ℛ𝒪𝒴 project

Context. Co-orbital objects, also known as trojans, are frequently found in simulations of planetary system formation. In these configurations, a planet shares its orbit with other massive bodies. It is still unclear why there have not been any co-orbitals discovered thus far in exoplanetary systems (exotrojans) or even pairs of planets found in such a 1:1 mean motion resonance. Reconciling observations and theory is an open subject in the field. Aims. The main objective of the 𝒯ℛ𝒪𝒴 project is to conduct an exhaustive search for exotrojans using diverse observational techniques. In this work, we analyze the radial velocity time series informed by transits, focusing the search around low-mass stars. Methods. We employed the α-test method on confirmed planets searching for shifts between spectral and photometric mid-transit times. This technique is sensitive to mass imbalances within the planetary orbit, allowing us to identify non-negligible co-orbital masses. Results. Among the 95 transiting planets examined, we find one robust exotrojan candidate with a significant 3-σ detection. Additionally, 25 exoplanets show compatibility with the presence of exotrojan companions at a 1-σ level, requiring further observations to better constrain their presence. For two of those weak candidates, we find dimmings in their light curves within the predicted Lagrangian region. We established upper limits on the co-orbital masses for either the candidates and null detections. Conclusions. Our analysis reveals that current high-resolution spectrographs effectively rule out co-orbitals more massive than Saturn around low-mass stars. This work points out to dozens of targets that have the potential to better constraint their exotrojan upper mass limit with dedicated radial velocity observations. We also explored the potential of observing the secondary eclipses of the confirmed exoplanets in our sample to enhance the exotrojan search, ultimately leading to a more accurate estimation of the occurrence rate of exotrojans.

TESS Investigation—Demographics of Young Exoplanets (TI-DYE). II. A Second Giant Planet in the 17 Myr System HIP 67522

The youngest (<50 Myr) planets are vital to understand planet formation and early evolution. The 17 Myr system HIP 67522 is already known to host a giant (≃10R ⊕) planet on a tight orbit. In their discovery paper, Rizzuto et al. reported a tentative single-transit detection of an additional planet in the system using TESS. Here, we report the discovery of HIP 67522c, a 7.9 R ⊕ planet that matches with that single-transit event. We confirm the signal with ground-based multiwavelength photometry from Sinistro and MuSCAT4. At a period of 14.33 days, planet c is close to a 2:1 mean-motion resonance with b (6.96 days or 2.06:1). The light curve shows distortions during many of the transits, which are consistent with spot-crossing events and/or flares. Fewer stellar activity events are seen in the transits of planet b, suggesting that planet c is crossing a more active latitude. Such distortions, combined with systematics in the TESS light-curve extraction, likely explain why planet c was previously missed.

Low-Energy Endgame Trajectory Design for Callisto Orbiter

Callisto is a promising target for future Jovian system exploration due to its high scientific value and low-radiation environment. This paper proposes a new low-energy endgame that can achieve a low-Δv transfer with short flight times for Callisto orbiting and landing missions. In contrast to the endgames designed for other Galilean moons, the proposed Callisto endgame would perform interior transfers from Ganymede in the low-energy region rather than exterior transfers. The definition and analysis of the Callisto endgame are completed using the Tisserand–Poincaré graph. Tisserand-leveraging transfers and isolating blocks techniques are used to search for different trajectory arcs in the endgame. The developed trajectory solver is able to find an optimal complete endgame trajectory with constraints on the initial state and the orbit around Callisto. In the designed scenarios, the spacecraft can complete the transfer from the Ganymede resonance orbit to the polar science orbit around Callisto with a minimum Δv of 712.3 m/s and a transfer time of about 140 days, which is a significant improvement compared to the high-energy endgame.

Dynamical evolution of the Uranian satellite system I.: From the 5/3 Ariel–Umbriel mean motion resonance to the present

Mutual gravitational interactions between the five major Uranian satellites raise small quasi-periodic fluctuations on their orbital elements. At the same time, tidal interactions between the satellites and the planet induce a slow outward drift of the orbits, while damping the eccentricities and the inclinations. In this paper, we revisit the current and near past evolution of this system using a N−body integrator, including spin evolution and tidal dissipation with the weak friction model. We update the secular eigenmodes of the system and show that it is unlikely that any of the main satellites were recently captured into a high obliquity Cassini state. We rather expect that the Uranian satellites are in a low obliquity Cassini state and compute their values. We also estimate the current variations in the eccentricities and inclinations, and show that they are not fully damped. We constrain the modified quality factor of Uranus to be QU′=(1.2±0.4)×105, and that of Ariel to be QA′=(7±3)×104. We find that the system most likely encountered the 5/3 mean motion resonance between Ariel and Umbriel in the past, at about (0.7±0.2) Gyr ago. We additionally determine the eccentricities and inclinations of all satellites just after the resonance passage that comply with the current system. We finally show that, from the crossing of the 5/3 MMR to the present, the evolution of the system is mostly peaceful and dominated by tides raised on Uranus by the satellites.

New Transient Co-orbital Asteroids of Venus

Venus has no known natural satellites but has 5 known co-orbitals. These are objects trapped in a 1:1 mean-motion resonance with Venus. Co-orbital configurations include retrograde satellite orbits (RS), tadpole orbits (T) around the Lagrangian equilibrium points L4 or L5, and horseshoe orbits around both L4 and L5 (H). At high eccentricity or inclination, co-orbital configurations may also involve compounds of T and RS (T-RS, T-RS-T), H and RS (H-RS) orbits, or transitions between different co-orbital modes. Here we identify asteroids in 2 RS, 1 L4-tadpole, 2 H-RS, and 2 T-RS orbits, as well as 8 additional asteroids in possible temporary co-orbital status. Although the majority of these objects do not yet have well-characterized orbits, 2020 CL1, and 2020 SB do and are very likely to be new co-orbital asteroids. With the new candidates, Venus would have a population of 20 co-orbital asteroids, comparable to those of Mars and Earth.

Cosmic‐ray exposure age accumulated in near‐Earth space: A carbonaceous chondrite case study

AbstractThis study investigates the expected cosmic‐ray exposure (CRE) of meteorites if they were to be ejected by a near‐Earth object, that is, from an object already transferred to an Earth‐crossing orbit by an orbital resonance. Specifically, we examine the CRE ages of CI and CM carbonaceous chondrites (CCs), which have some of the shortest measured CRE ages of any meteorite type. A steady‐state near‐Earth carbonaceous meteoroid probability density function is estimated based on the low‐albedo near‐Earth asteroid population, including parameters such as the near‐Earth dynamic lifetime, the impact probability with the Earth, and the orbital parameters. This model was then compared to the orbits and CRE ages of the five CC falls with precisely measured orbits: Tagish Lake, Maribo, Sutter's Mill, Flensburg, and Winchcombe. The study examined two meteoroid ejection scenarios for CI/CM meteoroids: Main Belt collisions and ejections in near‐Earth space. The results indicated that applying a maximum physical lifetime in near‐Earth space of 2–10 Myr to meteoroids and eliminating events evolving onto orbits entirely detached from the Main Belt (Q &lt; 1.78 au) significantly improved the agreement with the observed orbits of carbonaceous falls. Additionally, the CRE ages of three of the five carbonaceous falls have measured CRE ages one to three orders of magnitude shorter than expected for an object originating from the Main Belt with the corresponding semi‐major axis value. This discrepancy between the expected CRE ages from the model and the measured ages of three of the carbonaceous falls indicates that some CI/CM meteoroids are being ejected in near‐Earth space. This study proposes a nuanced hypothesis involving meteoroid impacts and tidal disruptions as significant contributors to the ejection and subsequent CRE age accumulation of CI/CM chondrites in near‐Earth space.

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 and Ultra-short-period Planet Systems Are at Opposite Ends of the Exoplanet Age Distribution

Exoplanet systems are thought to evolve on secular timescales over billions of years. This evolution is impossible to directly observe on human timescales in most individual systems. While the availability of accurate and precise age inferences for individual exoplanet host stars with ages τ in the interval 1 Gyr ≲ τ ≲ 10 Gyr would constrain this evolution, accurate and precise age inferences are difficult to obtain for isolated field dwarfs like the host stars of most exoplanets. The Galactic velocity dispersion of a thin-disk stellar population monotonically grows with time, and the relationship between age and velocity dispersion in a given Galactic location can be calibrated by a stellar population for which accurate and precise age inferences are possible. Using a sample of subgiants with precise age inferences, we calibrate the age–velocity dispersion relation in the Kepler field. Applying this relation to the Kepler field’s planet populations, we find that Kepler-discovered systems plausibly in second-order mean-motion resonances have 1 Gyr ≲ τ ≲ 2 Gyr. The same is true for systems plausibly in first-order mean-motion resonances, but only for systems likely affected by tidal dissipation inside their innermost planets. These observations suggest that many planetary systems diffuse away from initially resonant configurations on secular timescales. Our calibrated relation also indicates that ultra-short-period (USP) planet systems have typical ages in the interval 5 Gyr ≲ τ ≲ 6 Gyr. We propose that USP planets tidally migrated from initial periods in the range 1 day ≲ P ≲ 2 days to their observed locations at P < 1 day over billions of years and trillions of cycles of secular eccentricity excitation and inside-planet damping.

The Solar System could have formed in a low-viscosity disc: A dynamical study from giant planet migration to the Nice model

Context. In the context of low-viscosity protoplanetary discs (PPDs), the formation scenarios of the Solar System should be revisited. In particular, the Jupiter-Saturn pair has been shown to lock in the 2:1 mean motion resonance while migrating generally inwards, making the Grand Tack scenario impossible. Aims. We explore what resonant chains of multiple giant planets can form in a low-viscosity disc, and whether these configurations can evolve into forming the Solar System in the post gas disc phase. Methods. We used hydrodynamical simulations with the code FARGOCA to study the migration of the giant planets in a disc with viscosity parameter of α = 10−4. After a transition phase to a gas-less configuration, we studied the stability of the obtained resonant chains through their interactions with a disc of leftover planetesimals by performing N-body simulations using rebound. Results. The gaps opened by giant planets are wider and deeper for lower viscosity, reducing the damping effect of the disc. Thus, when planets enter a resonance, the resonant angle remains closer to circulation, making the chain weaker. Exploring numerous configurations, we found five stable resonant chains of four or five planets. In a thin (cold) PPD, the four giant planets revert their migration and migrate outwards. After disc dispersal, under the influence of a belt of planetesimals, some resonant chains undergo an instability phase while others migrate smoothly over a billion years. For three of our resonant chains, about ~1% of the final configurations pass the four criteria to fit the Solar System. The most successful runs are obtained for systems formed in a cold PPD with a massive planetesimal disc. Conclusions. This work provides a fully consistent study of the dynamical history of the Solar System’s giant planets, from the protoplanetary disc phase up to the giant planet instability. Although building resonant configurations is difficult in low-viscosity discs, we find it possible to reproduce the Solar System from a cold, low-viscosity protoplanetary disc.

Nitrogen Loss from Pluto’s Birth to the Present Day via Atmospheric Escape, Photochemical Destruction, and Impact Erosion

We estimate the loss of nitrogen from Pluto over its lifetime, including the giant planet instability period, which we term the “Wild Years.” We analyze the orbital migration of 53 simulated Plutinos, which are Kuiper Belt Objects (KBOs) captured into 3:2 mean-motion resonance with Neptune during the instability. This orbital migration brought the Plutinos from 20 to 30 au to their present-day orbits near 40 au along a nonlinear path that includes orbits with semimajor axes from 10 to 100 au. We model the thermal history that results from this migration and estimate the volatile loss rates due to the ever-changing thermal environment. Due to the early Sun’s enhanced ultraviolet radiation, the photochemical destruction rate during the Wild Years was a factor of 100 higher than the present-day rate, but this only results in a loss of ∼10 m global equivalent layer (GEL). The enhanced Jeans escape rate varies wildly with time, and a net loss of ∼100 cm GEL is predicted. Additionally, we model the impact history during the migration and find that impacts are a net source, not loss, of N2, contributing ∼100 cm GEL. The 100 cm GEL is 0.1% of the amount of N2 in Sputnik Planitia. We therefore conclude that Pluto did not lose an excessive amount of volatiles during the Wild Years, and its primordial volatile inventory can be approximated as its present-day inventory. However, significant fractions of this small total loss of N2 occurred during the Wild Years, so estimates made using present-day rates will be underestimates.

Repelling Planet Pairs by Ping-pong Scattering

The Kepler mission reveals a peculiar trough-peak feature in the orbital spacing of close-in planets near mean-motion resonances: a deficit and an excess that are, respectively, a couple of percent interior to and wide of the resonances. This feature has received two main classes of explanations: one involving eccentricity damping and the other scattering with small bodies. Here, we point out a few issues with the damping scenario and study the scattering scenario in more detail. We elucidate why scattering small bodies tends to repel two planets. As the small bodies random-walk in energy and angular momentum space, they tend to absorb fractionally more energy than angular momentum. This, which we call “ping-pong repulsion,” transports angular momentum from the inner to the outer planet and pushes the two planets apart. Such a process, even if ubiquitous, leaves identifiable marks only near first-order resonances: diverging pairs jump across the resonance quickly and produce the mean-motion resonance asymmetry. To explain the observed positions of the trough-peaks, a total scattering mass of order a few percent of the planet masses is required. Moreover, if this mass is dominated by a handful of Mercury-sized bodies, one can also explain the planet eccentricities as inferred from transit-time variations. Last, we suggest how these conditions may have naturally arisen during the late stages of planet formation.

Photo-dynamical characterisation of the TOI-178 resonant chain

Context. The TOI-178 system consists of a nearby, late-K-dwarf with six transiting planets in the super-Earth to mini-Neptune regime, with radii ranging from to 2.9 R⊕ and orbital periods between 1.9 and 20.7 days. All the planets, but the innermost one, form a chain of Laplace resonances. The fine-tuning and fragility of such orbital configurations ensure that no significant scattering or collision event has taken place since the formation and migration of the planets in the protoplanetary disc, thereby providing important anchors for planet formation models. Aims. We aim to improve the characterisation of the architecture of this key system and, in particular, the masses and radii of its planets. In addition, since this system is one of the few resonant chains that can be characterised by both photometry and radial velocities, we propose to use it as a test bench for the robustness of the planetary mass determination with each technique. Methods. We performed a global analysis of all the available photometry from CHEOPS, TESS and NGTS, and radial velocity from ESPRESSO, using a photo-dynamical modelling of the light curve. We also tried different sets of priors on the masses and eccentricity, as well as different stellar activity models, to study their effects on the masses estimated by transit-timing variations (TTVs) and radial velocities (RVs). Results. We demonstrate how stellar activity prevents a robust mass estimation for the three outer planets using radial velocity data alone. We also show that our joint photo-dynamical and radial velocity analysis has resulted in a robust mass determination for planets c to 𝑔, with precision of ~ 12% for the mass of planet c, and better than 10% for planets d to 𝑔. The new precisions on the radii range from 2 to 3%. The understanding of this synergy between photometric and radial velocity measurements will be valuable for the PLATO mission. We also show that TOI-178 is indeed currently locked in the resonant configuration, librating around an equilibrium of the chain.

An Earth Encounter as the Cause of Chaotic Dynamics in Binary Asteroid (35107) 1991VH

Among binary asteroids, (35107) 1991VH stands out as unique given the likely chaotic rotation within its secondary component. The source of this excited dynamical state is unknown. In this work, we demonstrate that a past close encounter with Earth could have provided the necessary perturbation to allow the natural internal dynamics, characterized by spin–orbit coupling, to evolve the system into its current dynamical state. In this hypothesis, the secondary of 1991VH was previously in a classical 1:1 spin–orbit resonance with an orbit period likely between 28 and 35 hr before being perturbed by an Earth encounter within ∼80,000 km. We find that if the energy dissipation within the secondary is relatively inefficient, this excited dynamical state could persist to today and produce the observed ground-based measurements. Coupled with the orbital history of 1991VH, we can then place a constraint on the tidal dissipation parameters of the secondary.