First-order Mean-motion Resonances Research Articles

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.

Kepler-discovered Multiple-planet Systems near Period Ratios Suggestive of Mean-motion Resonances Are Young

Before the launch of the Kepler Space Telescope, models of low-mass planet formation predicted that convergent type I migration would often produce systems of low-mass planets in low-order mean-motion resonances. Instead, Kepler discovered that systems of small planets frequently have period ratios larger than those associated with mean-motion resonances and rarely have period ratios smaller than those associated with mean-motion resonances. Both short-timescale processes related to the formation or early evolution of planetary systems and long-timescale secular processes have been proposed as explanations for these observations. Using a thin disk stellar population’s Galactic velocity dispersion as a relative age proxy, we find that Kepler-discovered multiple-planet systems with at least one planet pair near a period ratio suggestive of a second-order mean-motion resonance have a colder Galactic velocity dispersion and are therefore younger than both single-transiting and multiple-planet systems that lack planet pairs consistent with mean-motion resonances. We argue that a nontidal secular process with a characteristic timescale no less than a few hundred Myr is responsible for moving systems of low-mass planets away from second-order mean-motion resonances. Among systems with at least one planet pair near a period ratio suggestive of a first-order mean-motion resonance, only the population of systems likely affected by tidal dissipation inside their innermost planets has a small Galactic velocity dispersion and is therefore young. We predict that period ratios suggestive of mean-motion resonances are more common in young systems with 10 Myr ≲ τ ≲ 100 Myr and become less common as planetary systems age.

The Global 2:1 Mean-motion Resonance in HD 110067 is Not Vacant!

Extensive modeling of observations obtained by two space telescopes and 14 ground-based observatories revealed the presence of at least six closely packed planets orbiting HD 110067. Their orbital periods form a nearly perfect resonant chain, in which adjacent periods are in a geometric sequence with a ratio of r = 3/2 (planets b to e) that switches to r = 4/3 (planets e to g). The latter part of the sequence is based on single transits of planets f and g and their periods are model-dependent, as only first-order mean-motion resonances (MMRs) were so far investigated; and there is no physical justification for this assumption besides the simplicity imprinted on to the resonant chain itself. Furthermore, the global 2:1 MMR is apparently occupied, whereas this orbit is vacant in all other multibody (extra) solar systems. We summarize the physical arguments for/against the above r = 4/3 period ratios in anticipation of future observations to resolve the issue.

Dissipative Capture of Planets into First-order Mean-motion Resonances

The emergence of orbital resonances among planets is a natural consequence of the early dynamical evolution of planetary systems. While it is well established that convergent migration is necessary for mean-motion commensurabilities to emerge, recent numerical experiments have shown that the existing adiabatic theory of resonant capture provides an incomplete description of the relevant physics, leading to an erroneous mass scaling in the regime of strong dissipation. In this work, we develop a new model for resonance capture that self-consistently accounts for migration and circularization of planetary orbits, and derive an analytic criterion based upon stability analysis that describes the conditions necessary for the formation of mean-motion resonances. We subsequently test our results against numerical simulations and find satisfactory agreement. Our results elucidate the critical role played by adiabaticity and resonant stability in shaping the orbital architectures of planetary systems during the nebular epoch, and provide a valuable tool for understanding their primordial dynamical evolution.

TOI-1136 is a Young, Coplanar, Aligned Planetary System in a Pristine Resonant Chain

Convergent disk migration has long been suspected to be responsible for forming planetary systems with a chain of mean-motion resonances (MMRs). Dynamical evolution over time could disrupt the delicate resonant configuration. We present TOI-1136, a 700 ± 150 Myr old G star hosting at least six transiting planets between ∼2 and 5 R ⊕. The orbital period ratios deviate from exact commensurability by only 10−4, smaller than the ∼10−2 deviations seen in typical Kepler near-resonant systems. A transit-timing analysis measured the masses of the planets (3–8M ⊕) and demonstrated that the planets in TOI-1136 are in true resonances with librating resonant angles. Based on a Rossiter–McLaughlin measurement of planet d, the star’s rotation appears to be aligned with the planetary orbital planes. The well-aligned planetary system and the lack of a detected binary companion together suggest that TOI-1136's resonant chain formed in an isolated, quiescent disk with no stellar flyby, disk warp, or significant axial asymmetry. With period ratios near 3:2, 2:1, 3:2, 7:5, and 3:2, TOI-1136 is the first known resonant chain involving a second-order MMR (7:5) between two first-order MMRs. The formation of the delicate 7:5 resonance places strong constraints on the system’s migration history. Short-scale (starting from ∼0.1 au) Type-I migration with an inner disk edge is most consistent with the formation of TOI-1136. A low disk surface density (Σ1 au ≲ 103g cm−2; lower than the minimum-mass solar nebula) and the resultant slower migration rate likely facilitated the formation of the 7:5 second-order MMR.

Dynamics of co-orbital exoplanets in a first-order resonance chain with tidal dissipation

Co-orbital planets (in a 1: 1 mean motion resonance) can be formed within a Laplace resonance chain. We develop a secular model tc study the dynamics of the resonance chain p: p : p + 1, where the co-orbital pair is in a first-order mean motion resonance with the outermost third planet. Our model takes into account tidal dissipation through the use of a Hamiltonian version of the constant time-lag model, which extends the Hamiltonian formalism of the point-mass case. We show the existence of several families of equilibria, anc how these equilibria extend to the complete system. In one family, which we call the main branch, a secular resonance between the libration frequency of the co-orbitals and the precession frequency of the pericentres has unexpected dynamical consequences when tidal dissipation is added. We report the existence of two distinct mechanisms that make co-orbital planets much more stable within the p : p : p + 1 resonance chain rather than outside it. The first is due to negative real parts of the eigenvalues of the linearised system with tides, in the region of the secular resonance mentioned above. The second comes from non-linear contributions of the vector fielt and is due to eccentricity damping. These two stabilising mechanisms increase the chances of a future detection of exoplanets in the co-orbital configuration.

Proximity of exoplanets to first-order mean-motion resonances

ABSTRACT Planetary formation theories and, more specifically, migration models predict that planets can be captured in mean-motion resonances (MMRs) during the disc phase. The distribution of period ratios between adjacent planets shows an accumulation in the vicinity of the resonance, which is not centred on the nominal resonance but instead presents an offset slightly exterior to it. Here, we extend on previous works by thoroughly exploring the effect of different disc and planet parameters on the resonance offset during the disc migration phase. The dynamical study is carried out for several first-order MMRs and for both low-mass Earth-like planets undergoing type-I migration and giant planets evolving under type-II migration. We find that the offset varies with time during the migration of the two-planet system along the apsidal corotation resonance family. The departure from the nominal resonance increases for higher planetary masses and stronger eccentricity damping. In the Earth to super-Earth regime, we find offset values in agreement with the observations when using a sophisticated modelling for the planet–disc interactions, where the damping time-scale depends on the eccentricity. This dependence causes a feedback that induces an increase of the resonance offsets. Regarding giant planets, the offsets of detected planet pairs are well reproduced with a classical K-factor prescription for the planet–disc interactions when the eccentricity damping rate remains low to moderate. In both regimes, eccentricities are in agreement with the observations too. As a result, planet–disc interactions provide a generic channel to generate the offsets found in the observations.

An integrable model for first-order three-planet mean motion resonances

Recent works on three-planet mean motion resonances (MMRs) have highlighted their importance for understanding the details of the dynamics of planet formation and evolution. While the dynamics of two-planet MMRs are well understood and approximately described by a one-degree-of-freedom Hamiltonian, little is known of the exact dynamics of three-body resonances besides the cases of zeroth-order MMRs or when one of the bodies is a test particle. In this work, I propose the first general integrable model for first-order three-planet mean motion resonances. I show that one can generalize the strategy proposed in the two-planet case to obtain a one-degree-of-freedom Hamiltonian. The dynamics of these resonances are governed by the second fundamental model of resonance. The model is valid for any mass ratio between the planets and for every first-order resonance. I show the agreement of the analytical model with numerical simulations. As examples of application, I show how this model could improve our understanding of the capture into MMRs as well as their role in the stability of planetary systems.

Bridges and gaps at low-eccentricity first-order resonances

ABSTRACT Previous works on the divergence of first-order mean-motion resonances (MMRs) have studied in detail the extent of the pericentric and apocentric libration zones of adjacent first-order MMRs, highlighting possible bridges between them in the low eccentricity circular restricted three-body problem. Here, we describe the previous results in the context of periodic orbits and show that the so-called circular family of periodic orbits is the path that can drive the passage between neighbouring resonances under dissipative effects. We illustrate that the circular family can bridge first-order and higher order resonances, while its gaps at first-order MMRs can serve as boundaries that stop transitions between resonances. In particular, for the Sun–asteroid–Jupiter problem, we show that, during the migration of Jupiter in the protoplanetary disc, a system initially evolving below the apocentric branch of a first-order MMR follows the circular family and can either be captured into the pericentric branch of an adjacent first-order MMR if the orbital migration is rapid or in a higher order MMR in case of slow migration. Radial transport via the circular family can be extended to many small body and planetary system configurations undergoing dissipative effects (e.g. tidal dissipation, solar mass-loss, and gas drag).

Orbital evolution of Saturn’s satellites due to the interaction between the moons and the massive rings

Context. Saturn’s mid-sized moons (satellites) have a puzzling orbital configuration with trapping in mean-motion resonances with every-other pairs (Mimas-Tethys 4:2 and Enceladus-Dione 2:1). To reproduce their current orbital configuration on the basis of a recent model of satellite formation from a hypothetical ancient massive ring, adjacent pairs must pass first-order mean-motion resonances without being trapped. Aims. The trapping could be avoided by fast orbital migration and/or excitation of the satellite’s eccentricity caused by gravitational interactions between the satellites and the rings (the disk), which are still unknown. In our research we investigate the satellite orbital evolution due to interactions with the disk through full N-body simulations. Methods. We performed global high-resolution N-body simulations of a self-gravitating particle disk interacting with a single satellite. We used N ∼ 105 particles for the disk. Gravitational forces of all the particles and their inelastic collisions are taken into account. Results. Dense short-wavelength wake structure is created by the disk self-gravity and a few global spiral arms are induced by the satellite. The self-gravity wakes regulate the orbital evolution of the satellite, which has been considered as a disk spreading mechanism, but not as a driver for the orbital evolution. Conclusions. The self-gravity wake torque to the satellite is so effective that the satellite migration is much faster than was predicted with the spiral arm torque. It provides a possible model to avoid the resonance capture of adjacent satellite pairs and establish the current orbital configuration of Saturn’s mid-sized satellites.

On the divergence of first-order resonance widths at low eccentricities

ABSTRACT Orbital resonances play an important role in the dynamics of planetary systems. Classical theoretical analyses found in textbooks report that libration widths of first-order mean motion resonances diverge for nearly circular orbits. Here, we examine the nature of this divergence with a non-perturbative analysis of a few first-order resonances interior to a Jupiter-mass planet. We show that a first-order resonance has two branches, the pericentric and the apocentric resonance zone. As the eccentricity approaches zero, the centres of these zones diverge away from the nominal resonance location but their widths shrink. We also report a novel finding of ‘bridges’ between adjacent first-order resonances: at low eccentricities, the apocentric libration zone of a first-order resonance smoothly connects with the pericentric libration zone of the neighbouring first-order resonance. These bridges may facilitate resonant migration across large radial distances in planetary systems, entirely in the low-eccentricity regime.

An analytically treatable model of long-term dynamics in a mean motion resonance with coexisting resonant modes

We examine a 2-DOF Hamiltonian system modeling some previously unexplored dynamical effects in first-order mean motion resonances in the spatial circular restricted three-body problem “star-planet-asteroid.” In distinction from the well-known integrable model of Sessin and Ferraz-Mello (Celest Mech Dyn Astron 32:307–332, 1984) and Wisdom (Celest Mech Dyn Astron 38:175–180, 1986), in our analysis, we kept more terms in the expansion for the disturbing function. This allowed us to study the phenomena caused by coexisting resonant modes, that were lost in the integrable model. In addition, we identified the areas of dynamical chaos in the phase space of the considered system. Despite the non-integrability of our model, we were able to compute analytically many quantities characterizing its dynamical behavior. Finally, we illustrate some of our results through the examples of the known resonant Kuiper belt objects.

Tidally Induced Radius Inflation of Sub-Neptunes

Abstract Recent work suggests that many short-period super-Earth and sub-Neptune planets may have significant spin axis tilts (“obliquities”). When planets are locked in high-obliquity states, the tidal dissipation rate may increase by several orders of magnitude. This intensified heat deposition within the planets’ interiors should generate significant structural consequences, including atmospheric inflation leading to larger transit radii. Using up-to-date radius estimates from Gaia Data Release 2, we show evidence for ∼50% larger average radii of planets wide of first-order mean-motion resonances, a population of planets with a theorized frequent occurrence of high obliquities. We investigate whether this radius trend could be a signature of obliquity tides. Using an adaptation of the Modules for Experiments in Stellar Astrophysics (MESA) stellar evolution toolkit, we model the atmospheric evolution of sub-Neptune-mass planets in response to additional internal heat from obliquity tides. The degree of radius inflation predicted by the models is ∼10%–100% for tidal luminosities ≳10−5 of the incident stellar power; this degree of inflation is broadly consistent with the observations and can approximately be described by power-law relationships. We present a few case studies of very low density “super-puff” planets—Kepler-79 d, Kepler-31 c, and Kepler-27 b—and show that they are strong candidates for potentially having undergone tidally induced radius inflation. We also discuss how the discrepancy between the two populations of planets with masses derived from radial velocities and transit timing variations is connected to the radius distribution features we have identified. Altogether, the calculations in this work confirm that tidal dissipation has nonnegligible consequences for the structural properties of short-period sub-Neptunes.

Dynamical interactions in the planetary system GJ4276

ABSTRACT GJ4276 is an M4.0 dwarf star with an inferred Neptune mass planet from radial velocity (RV) observations. We re-analyse the RV data for this system and focus on the possibility of a second, super-Earth mass, planet. We compute the time-scale for fast resonant librations in the eccentricity to be $\sim \!2000 \, \mathrm{d}$. Given that the observations were taken over $700\, \mathrm{d}$, we expect to see the effect of these librations in the observations. We perform a fully dynamical fit to test this hypothesis. Similar to previous results, we determine that the data could be fit by two planets in a 2:1 mean motion resonance. However, we also find solutions near the 5:4 mean motion resonance that are not present when planet–planet interactions are ignored. Using the mean exponential growth of nearby orbits indicator, we analyse the stability of the system and find that our solutions lie in a stable region of parameter space. We also find that though out-of-resonance solutions are possible, the system favours a configuration that is in a first-order mean motion resonance. The existence of mean motion resonances has important implications in many planet formation theories. Although we do not attempt to distinguish between the one- and two-planet models in this work, in either case, the predicted orbital parameters are interesting enough to merit further study. Future observations should be able to distinguish between the different scenarios within the next 5 yr.

A Pair of Planets Likely in Mean-motion Resonance From Gravitational Microlensing

Abstract We combine statistical arguments and dynamical analysis to study the orbital configuration of the microlensing planetary system OGLE-2012-BLG-0026L. This system is known to host two massive planets, both projected close to the Einstein ring at the time of the detection. Assuming an isotropic distribution for the orbital orientation, we find that the two planets should also be closely spaced in 3D space and that the ratio of their orbital periods is almost certainly less than two. With N-body numerical integrations, we then identify two types of stable configurations: the two planets can be in first-order mean-motion resonances (MMRs) and have significant (≳0.1) eccentricities, or they stay out of MMRs and have nearly circular orbits. The latter is disfavored, given the absence of similar planet pairs in radial velocity observations as well as the theoretical difficulties in forming such a configuration. Therefore, the two massive planets in OGLE-2012-BLG-0026L are likely in a resonance configuration. Our work shows that the microlensing technique, which usually only measures the projected configurations, can also probe the detailed dynamical state of multi-planet systems. We also discuss theoretical implications of measuring the multiplicity and the orbital architecture of cold planets.

Long-term Hydrodynamic Simulations on the Planetesimals Trapped in the First-order Mean Motion Resonances

Abstract The resonant perturbations from planets are able to halt the drag-induced migration, and capture the inwardly drifting planetesimals into mean motion resonances. The equilibrium eccentricity of planetesimals in resonances, and the minimum size of planetesimal that can trigger resonance trapping, have been analyzed and formulated. However, the analytical works based on the assumption that the disk is axisymmetric, which is violated by the asymmetric structures developed by planets. We perform long-term 2D hydrodynamic simulations to study the dynamics of planetesimals in the j: (j + 1) first-order exterior resonances, and reexamine the theoretical expressions. We find the expression of equilibrium eccentricity underestimates the values for resonances with j &lt; 5, in particular the 1:2 resonance that the underestimation can be 30%–40%. Within the parameter space we explored, we find the equilibrium eccentricity and the minimum size are reduced in an asymmetric disk. The amount of discrepancy in eccentricity depends on the degree of asymmetric structures. For cases of Earth-sized planets, where the disk is less disturbed, the planetesimal’s eccentricity can reach to the values predicted by our modified expression. For gaseous planets, however, the eccentricity can be 0.01–0.02 smaller in value. We find the minimum size is 10 times smaller, and the factor seems to be independent of the planet’s mass. The influences of asymmetric profiles on the eccentricity and the minimum size could affect the outcome of collisions between resonant and nonresonant planetesimals, and the amount of planetesimals migrated into the planet’s feeding zone.

Dynamical instability and its implications for planetary system architecture

We examine the effects that dynamical instability has on shaping the orbital properties of exoplanetary systems. Using N-body simulations of non-EMS (Equal Mutual Separation), multi-planet systems we find that the lower limit of the instability timescale $t$ is determined by the minimal mutual separation $K_{\rm min}$ in units of the mutual Hill radius. Planetary systems showing instability generally include planet pairs with period ratio $<1.33$. Our final period ratio distribution of all adjacent planet pairs shows dip-peak structures near first-order mean motion resonances similar to those observed in the \kepler\ planetary data. Then we compare the probability density function (PDF) of the de-biased \kepler\ period ratios with those in our simulations and find a lack of planet pairs with period ratio $> 2.1$ in the observations---possibly caused either by inward migration before the dissipation of the disk or by planet pairs not forming with period ratios $> 2.1$ with the same frequency they do with smaller period ratios. By comparing the PDF of the period ratio between simulation and observation, we obtain an upper limit of 0.03 on the scale parameter of the Rayleigh distributed eccentricities when the gas disk dissipated. Finally, our results suggest that a viable definition for a `packed' or `compact' planetary system be one that has at least one planet pair with a period ratio less than 1.33. This criterion would imply that 4\% of the \kepler\ systems (or 6\% of the systems with more than two planets) are compact.

First-order mean motion resonances in two-planet systems: general analysis and observed systems

This paper focuses on two-planet systems in a first-order $(q+1):q$ mean motion resonance and undergoing type-I migration in a disc. We present a detailed analysis of the resonance valid for any value of $q$. Expressions for the equilibrium eccentricities, mean motions and departure from exact resonance are derived in the case of smooth convergent migration. We show that this departure, not assumed to be small, is such that period ratio normally exceeds, but can also be less than, $ (q+1)/q.$ Departure from exact resonance as a function of time for systems starting in resonance and undergoing divergent migration is also calculated. We discuss observed systems in which two low mass planets are close to a first-order resonance. We argue that the data are consistent with only a small fraction of the systems having been captured in resonance. Furthermore, when capture does happen, it is not in general during smooth convergent migration through the disc but after the planets reach the disc inner parts. We show that although resonances may be disrupted when the inner planet enters a central cavity, this alone cannot explain the spread of observed separations. Disruption is found to result in either the system moving interior to the resonance by a few percent, or attaining another resonance. We postulate two populations of low mass planets: a small one for which extensive smooth migration has occurred, and a larger one that formed approximately in-situ with very limited migration.

Migration of planets into and out of mean motion resonances in protoplanetary discs: overstability of capture and non-linear eccentricity damping

A number of multiplanet systems are observed to contain planets very close to mean motion resonances, although there is no significant pileup of precise resonance pairs. We present theoretical and numerical studies on the outcome of capture into first-order mean motion resonances (MMRs) using a parametrized planet migration model that takes into account nonlinear eccentricity damping due to planet-disk interaction. This parametrization is based on numerical hydrodynamical simulations and is more realistic than the simple linear parametrization widely used in previous analytic studies. We find that nonlinear eccentricity damping can significantly influence the stability and outcome of resonance capture. In particular, the equilibrium eccentricity of the planet captured into MMRs become larger, and the captured MMR state tends to be more stable compared to the prediction based on the simple migration model. In addition, when the migration is sufficiently fast or/and the planet mass ratio is sufficiently small, we observe a novel phenomenon of eccentricity overshoot, where the planet's eccentricity becomes very large before settling down to the lower equilibrium value. This can lead to the ejection of the smaller planet if its eccentricity approaches unity during the overshoot. This may help explain the lack of low-mass planet companion of hot Jupiters when compared to warm Jupiters.

Near Mean-motion Resonances in the System Observed by Kepler: Affected by Mass Accretion and Type I Migration

Abstract The Kepler mission has released over 4496 planetary candidates, among which 3483 planets have been confirmed as of 2017 April. The statistical results of the planets show that there are two peaks around 1.5 and 2.0 in the distribution of orbital period ratios. The observations indicate that plenty of planet pairs could have first been captured into mean-motion resonances (MMRs) in planetary formation. Subsequently, these planets depart from exact resonant locations to be near-MMR configurations. Through type I migration, two low-mass planets have a tendency to be trapped in first-order MMRs (2:1 or 3:2 MMRs); however, two scenarios of mass accretion of planets and potential outward migration play important roles in reshaping their final orbital configurations. Under the scenario of mass accretion, the planet pairs can cross 2:1 MMRs and then enter into 3:2 MMRs, especially for the inner pairs. With such a formation scenario, the possibility that two planets are locked into 3:2 MMRs can increase if they are formed in a flat disk. Moreover, the outward migration can make planets have a high likelihood to be trapped into 3:2 MMRs. We perform additional runs to investigate the mass relationship for those planets in three-planet systems, and we show that two peaks near 1.5 and 2.0 for the period ratios of two planets can be easily reproduced through our formation scenario. We further show that the systems in chain resonances (e.g., 4:2:1, 3:2:1, 6:3:2, and 9:6:4 MMRs), have been observed in our simulations. This mechanism can be applicable to understand the formation of systems of Kepler-48, Kepler-53, Kepler-100, Kepler-192, Kepler-297, Kepler-399, and Kepler-450.