Recent exoplanet observations reported a large number of multiple-planet systems, in which some of the planets are in a chain of resonances. The fraction of resonant systems to non-resonant systems provides clues about their formation history. We investigated the orbital stability of planets in resonant chains by considering the long-term evolution of planetary mass and stellar mass and using orbital calculations. We found that while resonant chains were stable, they can be destabilized by a change of $\sim$10% in planetary mass. Such a mass evolution can occur by atmospheric escape due to photoevaporation. We also found that resonant chains can be broken by a stellar mass loss of $\lesssim1$%, which would be explained by stellar winds or coronal mass ejections. The long-term mass change of planets and stars plays an important role in the orbital evolutions of planetary systems including super-Earths.
In radial velocity observations, a pair of extrasolar planets near a 2:1 orbital resonance can be misinterpreted as a single eccentric planet, if data are sparse and measurement precision insufficient to distinguish between these models. We determine the fraction of alleged single-planet RV detected systems for which a 2:1 resonant pair of planets is also a viable model and address the question of how the models can be disentangled. By simulation we quantified the mismatch arising from applying the wrong model. Model alternatives are illustrated using the supposed single-planet system HD 27894…
The dynamical excitation of asteroids due to mean motion resonant interactions with planets is enhanced when their parent star leaves the main sequence. However, numerical investigation of resonant outcomes within post-main-sequence simulations is computationally expensive, limiting the extent to which detailed resonant analyses have been performed. Here, we combine the use of a high-performance computer cluster and the general semianalytical libration width formulation of Gallardo et al. (2021) in order to quantify resonant stability, strength and variation instigated by stellar evolution for…
ALMA 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 au to 62 au. In this work we propose that both the crescent and the dust rings are caused by a compact pair (period ratio $\simeq 4:3$) 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 …
The observed deficit and excess of adjacent planet pairs with period ratios narrow and wide of 3:2 and 2:1, the nominal values for the corresponding mean motion resonances (MMRs), have intrigued many. Previously, using a suite of simulations, Chatterjee & Ford (2015) showed that the excess above the 2:1 MMR can be naturally explained if planet pairs, initially trapped in the 2:1 MMR, dynamically interact with nearby planetesimals in a disk. We build on this work by: a) updating the census of discovered planet pairs, b) extending the study to initially non-resonant as well as resonant planet pa…
Transit timing variations (TTVs) are useful to constrain the existence of perturbing planets, especially in resonant systems where the variations are strongly enhanced. Here we focus on Laplace-resonant three-planet systems, and assume the inner planet transits the star. A dynamical study is performed for different masses of the three bodies, with a special attention to terrestrial planets. We consider a maximal time-span of ~ 100 years and discuss the shape of the inner planet TTVs curve. Using frequency analysis, we highlight the three periods related to the evolution of the system: two peri…
Migration of planetary systems caused by the action of dissipative forces may lead the planets to be trapped in a resonance. In this work we study the conditions and the dynamics of such resonant trapping. Particularly, we are interested in finding out whether resonant capture ends up in a long-term stable planetary configuration. For two planet systems we associate the evolution of migration with the existence of families of periodic orbits in the phase space of the three-body problem. The family of circular periodic orbits exhibits a gap at the 2:1 resonance and an instability and bifurcatio…
Early dynamical evolution of close-in planetary systems is shaped by an intricate combination of planetary gravitational interactions, orbital migration, and dissipative effects. While the process of convergent orbital migration is expected to routinely yield resonant planetary systems, previous analyses have shown that the semi-major axes of initially resonant pairs of planets will gradually diverge under the influence of long-term energy damping, producing an overabundance of planetary period ratios in slight excess of exact commensurability. While this feature is clearly evident in the orbi…
The multi-planetary system HD128311 hosts at least two planets. Its dynamical formation history has been studied extensively in the literature. We reanalyse the latest radial velocity data for this system with the affine-invariant Markov chain Monte Carlo sampler EMCEE. Using the high order integrator IAS15, we perform a fully dynamical fit, allowing the planets to interact during the sampling process. A stability analysis using the MEGNO indicator reveals that the system is located in a stable island of the parameter space. In contrast to a previous study, we find that the system is locked in…
Recent observations have revealed the existence of multiple-planet systems composed of Earth-mass planets around late M dwarfs. Most of their orbits are close to commensurabilities, which suggests that planets were commonly trapped in resonant chains in their formation around low-mass stars. We investigate the formation of multiple-planet systems in resonant chains around low-mass stars. A time-evolution model of the multiple-planet formation via pebble accretion in the early phase of the disk evolution is constructed based on the formation model for the TRAPPIST-1 system by Ormel et al. (2017…
The Kepler mission has discovered a plethora of multiple transiting planet candidate exosystems, many of which feature putative pairs of planets near mean motion resonance commensurabilities. Identifying potentially resonant systems could help guide future observations and enhance our understanding of planetary formation scenarios. We develop and apply an algebraic method to determine which Kepler 2-planet systems cannot be in a 1st-4th order resonance, given the current, publicly available data. This method identifies when any potentially resonant angle of a system must circulate. We identify…
Despite the discovery of thousands of exoplanets in recent years, the number of known exoplanets in star clusters remains tiny. This may be a consequence of close stellar encounters perturbing the dynamical evolution of planetary systems in these clusters. Here, we present the results from direct $N$-body simulations of multiplanetary systems embedded in star clusters containing $N = 8k, 16k, 32k$, and $64k$ stars. The planetary systems, which consist of the four Solar system giant planets Jupiter, Saturn, Uranus, and Neptune, are initialized in different orbital configurations, to study the e…
Planetary systems discovered by the Kepler space telescope exhibit an intriguing feature. While the period ratios of adjacent low-mass planets appear largely random, there is a significant excess of pairs that lie just wide of resonances and a deficit on the near side. We demonstrate that this feature naturally arises when two near-resonant planets interact in the presence of weak dissipation that damps eccentricities. The two planets repel each other as orbital energy is lost to heat. This moves near-resonant pairs just beyond resonance, by a distance that reflects the integrated dissipation …
Many exo-solar systems discovered in the last decade consist of planets orbiting in resonant configurations and consequently, their evolution should show long-term stability. However, due to the mutual planetary interactions a multi-planet system shows complicated dynamics with mostly chaotic trajectories. We can determine possible stable configurations by computing resonant periodic trajectories of the general planar three body problem, which can be used for modeling a two-planet system. In this work, we review our model for both the planar and the spatial case. We present families of symmetr…
Since the launch of the Kepler space telescope in 2009 and the subsequent K2 mission, hundreds of multi-planet systems have been discovered. The study of such systems, both as individual systems and as a population, leads to a better understanding of planetary formation and evolution. Kepler-80, a K-dwarf hosting six super-Earths, was the first system known to have four planets in a chain of resonances, a repeated geometric configuration. Transiting planets in resonant chains can enable us to estimate not only the planets' orbits and sizes but also their masses. Since the original resonance an…
Resonant chains are groups of planets for which each pair is in resonance, with an orbital period ratio locked at a rational value (2/1, 3/2, etc.). Such chains naturally form as a result of convergent migration of the planets in the proto-planetary disk. In this article, I present an analytical model of resonant chains of any number of planets. Using this model, I show that a system captured in a resonant chain can librate around several possible equilibrium configurations. The probability of capture around each equilibrium depends on how the chain formed, and especially on the order in which…
A fraction of multiple planet candidate systems discovered from transits by the Kepler mission contain pairs of planet candidates that are in orbital resonance or are spaced slightly too far apart to be in resonance. We focus here on the four planet system, KOI 730, that has planet periods satisfying the ratios 8:6:4:3. By numerically integrating four planets initially in this resonant configuration in proximity to an initially exterior cold planetesimal disk, we find that of the order of a Mars mass of planet-orbit-crossing planetesimals is sufficient to pull this system out of resonance. App…
Resonant locking of two planets is an expected outcome of convergent disc migration. The planets subsequently migrate together as a resonant pair. In the context of circumbinary planets, the disc is truncated internally by the binary. If there were only a single planet, then this inner disc edge would provide a natural parking location. However, for two planets migrating together in resonance there will be a tension between the inner planet stopping at the disc edge, and the outer planet continuing to be torqued inwards. In this paper we study this effect, showing that the outcome is a functio…
We develop an idealized dynamical model to predict the typical properties of outer extrasolar planetary systems, at radii beyond 5 AU. Our hypothesis is that dynamical evolution in outer planetary systems is controlled by a combination of planet-planet scattering and planetary interactions with an exterior disk of small bodies ("planetesimals"). Using 5,000 long duration N-body simulations, we follow the evolution of three planets surrounded by a 50 Earth mass primordial planetesimal disk. For large planet masses (above that of Saturn) the influence of the disk is modest, and we recover the ob…
Stellar spin-orbit misalignments (obliquities) in hot Jupiter systems have been extensively probed. Such obliquities may reveal clues about hot Jupiter dynamical histories. Common explanations for generating obliquities include high-eccentricity migration and primordial disk misalignment. This paper investigates another mechanism for producing stellar spin-orbit misalignments in systems hosting a close-in planet with an external, modestly inclined companion. Spin-orbit misalignment may be excited due to a secular resonance, occurring when the precession rate of the stellar spin axis (driven by…
A diversity of resonance configurations may be formed under different migration of two giant planets. And the researchers show that the HD 128311 and HD 73526 planetary systems are involved in a 2:1 mean motion resonance but not in apsidal corotation, because one of the resonance argument circulates over the dynamical evolution. In this paper, we investigate potential mechanisms to form the 2:1 librating-circulating resonance configuration. In the late stage of planetary formation, scattering or colliding among planetesimals and planetary embryos can frequently occur. Hence, in our model, we c…
We study the stability of mean-motion resonances (MMR) between two planets during their migration in a protoplanetary disk. We use an analytical model of resonances, and describe the effect of the disk by a migration timescale (T_{m,i}) and an eccentricity damping timescale (T_{e,i}) for each planet (i=1,2 respectively for the inner and outer planet). We show that the resonant configuration is stable if T_{e,1}/T_{e,2} > (e_1/e_2)^2. This general result can be used to put constraints on specific models of disk-planet interactions. For instance, using classical prescriptions for type I migratio…
The disc surrounding PDS 70, with two directly imaged embedded giant planets, is an ideal laboratory to study planet-disc interaction. We present three-dimensional smoothed particle hydrodynamics simulations of the system. In our simulations, planets, which are free to migrate and accrete mass, end up in a locked resonant configuration that is dynamically stable. We show that features observed at infrared (scattered light) and millimetre (thermal continuum) wavelengths are naturally explained by the accretion stream onto the outer planet, without requiring a circumplanetary disc around planet …
Uncovering the formation process that reproduces the distinct properties of compact super-Earth exoplanet systems is a major goal of planet formation theory. The most successful model argues that non-resonant systems begin as resonant chains of planets that later experience a dynamical instability. However, both the boundary of stability in resonant chains and the mechanism of the instability itself are poorly understood. Previous work postulated that a secondary resonance between the fastest libration frequency and a difference in synodic frequencies destabilizes the system. Here, we use that…
The Kepler mission is dramatically increasing the number of planets known in multi-planetary systems. Many adjacent planets have orbital period ratios near resonant values, with a tendency to be larger than required for exact first-order mean-motion resonances. This intriguing feature has been shown to be a natural outcome of orbital circularization of resonant planetary pairs due to star-planet tidal interactions. However, this feature holds in multi-planetary systems with periods longer than ten days, for which tidal circularization is unlikely to provide efficient divergent evolution of the…
Most stars form in star clusters and stellar associated. To understand the roles of star cluster environments in shaping the dynamical evolution of planetary systems, we carry out direct $N$-body simulations of four planetary systems models in three different star cluster environments with respectively N=2k, 8k and 32k stars. In each cluster, an ensemble of initially identical planetary systems are assigned to solar-type stars with $\sim 1 M_{\odot}$ and evolved for 50~Myr. We found that following the depletion of protoplanetary disks, external perturbations and planet-planet interactions are …
The four longest period Kuiper belt objects have orbital periods close to integer ratios with each other. A hypothetical planet with orbital period $\sim$17,117 years, semimajor axis $\sim$665 AU, would have N/1 and N/2 period ratios with these four objects. The orbital geometries and dynamics of resonant orbits constrain the orbital plane, the orbital eccentricity and the mass of such a planet, as well as its current location in its orbital path.
At least 30\% of main sequence stars host planets with sizes of between 1 and 4 Earth radii and orbital periods of less than 100 days. We use N-body simulations including a model for gas-assisted pebble accretion and disk--planet tidal interaction to study the formation of super-Earth systems. We show that the integrated pebble mass reservoir creates a bifurcation between hot super-Earths or hot-Neptunes ($\lesssim15M_{\oplus}$) and super-massive planetary cores potentially able to become gas giant planets ($\gtrsim15M_{\oplus}$). Simulations with moderate pebble fluxes grow multiple super-Ear…
Close-in planetary systems detected by the Kepler mission present an excess of periods ratio that are just slightly larger than some low order resonant values. This feature occurs naturally when resonant couples undergo dissipation that damps the eccentricities. However, the resonant angles appear to librate at the end of the migration process, which is often believed to be an evidence that the systems remain in resonance. Here we provide an analytical model for the dissipation in resonant planetary systems valid for low eccentricities. We confirm that dissipation accounts for an excess of pai…
Multi-planetary systems detected by the Kepler mission present an excess of planets close to first-order mean-motion resonances (2:1 and 3:2) but with a period ratio slightly higher than the resonant value. Several mechanisms have been proposed to explain this observation. Here we provide some clues that these near-resonant systems were initially in resonance and reached their current configuration through tidal dissipation. The argument that has been opposed to this scenario is that it only applies to the close-in systems and not to the farthest ones for which the tidal effect is too weak. Us…
Surveys have revealed many multi-planet systems containing super-Earths and Neptunes in orbits of a few days to a few months. There is debate whether in situ assembly or inward migration is the dominant mechanism of the formation of such planetary systems. Simulations suggest that migration creates tightly packed systems with planets whose orbital periods may be expressed as ratios of small integers (resonances), often in a many-planet series (chain). In the hundreds of multi-planet systems of sub-Neptunes, more planet pairs are observed near resonances than would generally be expected, but no…
Convergent migration involving multiple planets embedded in a viscous protoplanetary disc is expected to produce a chain of planets in mean motion resonances, but the multiplanet systems observed by the Kepler spacecraft are generally not in resonance. We demonstrate that under equivalent conditions, where in a viscous disc convergent migration will form a long-term stable system of planets in a chain of mean motion resonances, migration in an inviscid disc often produces a system which is highly dynamically unstable. In particular, if planets are massive enough to significantly perturb the di…
Context. Findings by the Helios and STEREO mission have indicated the presence of a resonant circumsolar ring of dust associated with Venus. Attempts to model this phenomenon as an analogue to the resonant ring of Earth - as a result of migrating dust trapped in external mean-motion resonances (MMRs) - have so far been unable to reproduce the observed dust feature. Other theories of origin have recently been put forward. However, the reason for the low trapping efficiency of Venus's external MMRs remains unclear. Aims. Here we look into the nature of the dust trapping resonant phenomena that a…
Inspired by the close-proximity pair of planets in the Kepler-36 system, we consider two effects that may have important ramifications for the development of life in similar systems where a pair of planets may reside entirely in the habitable zone of the hosting star. Specifically, we run numerical simulations to determine whether strong, resonant (or non-resonant) planet-planet interactions can cause large variations in planet obliquity---thereby inducing large variations in climate. We also determine whether or not resonant interactions affect the rate of lithopanspermia between the planet p…
An intriguing trend among Kepler's multi-planet systems is an overabundance of planet pairs with period ratios just wide of mean motion resonances (MMR) and a dearth of systems just narrow of them. In a recently published paper Chatterjee & Ford (2015; henceforth CF15) has proposed that gas-disk migration traps planets in a MMR. After gas dispersal, orbits of these trapped planets are altered through interaction with a residual planetesimal disk. They found that for massive enough disks planet-planetesimal disk interactions can break resonances and naturally create moderate to large positive o…
Hundreds of giant planets have been discovered so far and the quest of exo-Earths in giant planet systems has become intriguing. In this work, we aim to address the question of the possible long-term coexistence of a terrestrial companion on an orbit interior to a giant planet, and explore the extent of the stability regions for both non-resonant and resonant configurations. Our study focuses on the restricted three-body problem, where an inner terrestrial planet (massless body) moves under the gravitational attraction of a star and an outer massive planet on a circular or elliptic orbit. Usin…
I consider the dynamics of mean motion resonances between pairs of co-planar planets and derive a new integrable Hamiltonian model for planets' resonant motion. The new model generalizes previously-derived integrable Hamiltonians for first-order resonances to treat higher-order resonances by exploiting a surprising near-symmetry of the full, non-integrable Hamiltonians of higher-order resonances. Whereas past works have frequently relied on truncated disturbing function expansions to derive integrable approximations to resonant motion, I show that no such expansion is necessary, thus enabling …
Diversity in the properties of exoplanetary systems arises, in part, from dynamical evolution that occurs after planet formation. We use numerical integrations to explore the relative role of secular and resonant dynamics in the long-term evolution of model planetary systems, made up of three equal mass giant planets on initially eccentric orbits. The range of separations studied is dominated by secular processes, but intersects chains of high-order mean-motion resonances. Over time-scales of $10^8$ orbits, the secular evolution of the simulated systems is predominantly regular. High-order res…
It has been suggested that HD 200964 is the first exoplanetary system with two Jovian planets evolving in the 4/3 mean- motion resonance. Previous scenarios to simulate the formation of two giant planets in the stable 4/3 resonance configuration have failed. Moreover, the orbital parameters available in the literature point out an unstable configuration of the planetary pair. The purpose of this paper is i) to determine the orbits of the planets from the RV measurements and update the value of the stellar mass (1.57 M), ii) to analyse the stability of the planetary evolution in the vicinity an…
We consider the dynamics of a small trojan companion of a hypothetical giant exoplanet under the secular perturbations of additional planets. By a suitable choice of action-angle variables, the problem is amenable to the study of the slow modulation, induced by secular perturbations, to the dynamics of an otherwise called `basic' Hamiltonian model of two degrees of freedom (planar case). We present this Hamiltonian decomposition, which implies that the slow chaotic diffusion at resonances is best described by the paradigm of modulational diffusion.
The process of planet conglomeration, which primarily unfolds in a geometrically thin disk of gas and dust, is often accompanied by dynamical excitation of the forming planets and planetesimals. The ensuing orbital crossing can lead to large-scale collisional fragmentation, populating the system with icy and rocky debris. In a gaseous nebula, such leftover solid matter tends to spiral down towards the host star due to aerodynamic drag. Along the way, the inward drifting debris can encounter planets and gravitationally couple to them via mean-motion resonances, sapping them of their orbital ene…
A considerable fraction of multi-planet systems discovered by the observational surveys of extrasolar planets reside in mild proximity to first-order mean motion resonances. However, the relative remoteness of such systems from nominal resonant period ratios (e.g. 2:1, 3:2, 4:3) has been interpreted as evidence for lack of resonant interactions. Here we show that a slow divergence away from exact commensurability is a natural outcome of dissipative evolution and demonstrate that libration of critical angles can be maintained tens of percent away from nominal resonance. We construct an analytic…
Dust grains migrating under Poynting-Robertson drag may be trapped in mean-motion resonances with planets. Such resonantly trapped grains are observed in the solar system. In extrasolar systems, the exozodiacal light produced by dust grains is expected to be a major obstacle to future missions attempting to directly image terrestrial planets. The patterns made by resonantly trapped dust, however, can be used to infer the presence of planets, and the properties of those planets, if the capture and evolution of the grains can be modelled. This has been done with N-body methods, but such methods …
Among multi-planet planetary systems there are a large fraction of resonant systems. Studying the dynamics and formation of these systems can provide valuable informations on processes taking place in protoplanetary disks where the planets are thought have been formed. The recently discovered resonant system HD 60532 is the only confirmed case, in which the central star hosts a pair of giant planets in 3:1 mean motion resonance. We intend to provide a physical scenario for the formation of HD 60532, which is consistent with the orbital solutions derived from the radial velocity measurements. O…
We analyze solutions drawn from the recently published posterior distribution of the TRAPPIST-1 system, which consists of seven Earth-size planets appearing to be in a resonant chain around a red dwarf. We show that all the planets are simultaneously in two-planet and three-planet resonances, apart from the innermost pair for which the two-planet resonant angles circulate. By means of a frequency analysis, we highlight that the transit-timing variation (TTV) signals possess a series of common periods varying from days to decades, which are also present in the variations of the dynamical variab…
Massive planets form within the lifetime of protoplanetary disks and undergo orbital migration due to planet-disk interactions. When the first planet reaches the inner edge of the disk its migration stops and the second planet is locked in resonance. We detail the resonant trapping comparing semi-analytical formulae and numerical simulations in the case of two equal-mass coplanar planets trapped in first order resonances. We describe the family of resonant stable equilibrium points (zero-amplitude libration orbits) using expanded and non-expanded Hamiltonians. We show that during convergent mi…
Many extrasolar planetary systems containing multiple super-Earths have been discovered. N-body simulations taking into account standard type-I planetary migration suggest that protoplanets are captured into mean-motion resonant orbits near the inner disk edge at which the migration is halted. Previous N-body simulations suggested that orbital stability of the resonant systems depends on number of the captured planets. In the unstable case, through close scattering and merging between planets, non-resonant multiple systems are finally formed. In this paper, we investigate the critical number o…
Convergent migration allows pairs of planet to become trapped into mean motion resonances. Once in resonance, the planets' eccentricities grow to an equilibrium value that depends on the ratio of migration time scale to the eccentricity damping timescale, $K=τ_a/τ_e$, with higher values of equilibrium eccentricity for lower values of $K$. For low equilibrium eccentricities, $e_{eq}\propto K^{-1/2}$. The stability of a planet pair depends on eccentricity so the system can become unstable before it reaches its equilibrium eccentricity. Using a resonant overlap criterion that takes into account t…
A number of giant planet pairs discovered by the radial velocity method with period ratios $\lesssim 2$ may reside in mean motion resonances. Convergent orbital migration and resonant capture at the time of formation would naturally explain the present-day resonant orbital configurations of these systems. Planets that experience smooth migration and eccentricity damping forces due to a proto-planetary disk should not only capture into mean motion resonances but also end up in a specific dynamical configuration within the resonance, sometimes referred to as apsidal corotation resonance (ACR). H…
