Planets In Mean Motion Resonance Research Articles

When, where, and how many planets end up in first-order resonances?

ABSTRACTThe theory of Type I migration has been widely used in many studies. Transiting multiplanet systems offer us the opportunity to examine the consistency between observation and theory, especially for those systems harbouring planets in Mean Motion Resonance (MMR). The displacement these resonant pairs show from exact commensurability provides us with information on their migration and eccentricity-damping histories. Here, we adopt a probabilistic approach, characterized by two distributions – appropriate for either the resonant or non-resonant planets – to fit the observed planet period ratio distribution. With the Markov chain Monte Carlo (MCMC) method, we find that ${\approx }15{{\ \rm per\ cent}}$ of exoplanets are in first-order (j + 1: j) MMRs, the ratio of eccentricity-to-semimajor axis damping is too high to allow overstable librations, and that the results are by-and-large consistent with Type-I migration theory. In addition, our modelling finds that a small fraction of resonant pairs is captured into resonance during migration, implying late planet formation (gas-poor). Most of the resonant pairs park themselves at the migration barrier, indicating early planet formation (gas-rich). Furthermore, after improving the criterion on two-body resonant trapping, we obtain an upper limit of the disc surface density at the time the planets are locked in resonance.

Apsidal alignment and anti-alignment of planets in mean-motion resonance: disc-driven migration and eccentricity driving

ABSTRACT Planets migrating in their natal discs can be captured into mean-motion resonance (MMR), in which the planets’ periods are related by integer ratios. Recent observations indicate that planets in MMR can be either apsidally aligned or anti-aligned. How these different configurations arise is unclear. In this paper, we study the MMR capture process of migrating planets, focusing on the property of the apsidal angles of the captured planets. We show that the standard picture of MMR capture, in which the planets undergo convergent migration and experience eccentricity damping due to planet–disc interactions, always leads to apsidal anti-alignment of the captured planets. However, when the planets experience eccentricity driving from the disc, apsidally aligned configuration in MMR can be produced. In this configuration, both planets’ resonance angles circulate, but a ‘mixed’ resonance angle librates and traps the planets near the nominal resonance location. The MMR capture process in the presence of disc eccentricity driving is generally complex and irregular, and can lead to various outcomes, including apsidal alignment and anti-alignment, as well as the disruption of the resonance. We suggest that the two resonant planets in the K2-19 system, with their moderate eccentricities and aligned apsides, have experienced eccentricity driving from their natal disc in the past.

Inferred Properties of Planets in Mean-motion Resonances are Biased by Measurement Noise

Planetary systems with mean-motion resonances hold special value in terms of their dynamical complexity and their capacity to constrain planet formation and migration histories. The key toward making these connections, however, is to have a reliable characterization of the resonant dynamics, especially the so-called “libration amplitude,” which qualitatively measures how deep the system is into the resonance. In this work, we identify an important complication with the interpretation of libration amplitude estimates from observational data of resonant systems. Specifically, we show that measurement noise causes inferences of the libration amplitude to be systematically biased to larger values, with noisier data yielding a larger bias. We demonstrated this through multiple approaches, including using dynamical fits of synthetic radial velocity data to explore how the libration amplitude distribution inferred from the posterior parameter distribution varies with the degree of measurement noise. We find that even modest levels of noise still result in a slight bias. The origin of the bias stems from the topology of the resonant phase space and the fact that the available phase-space volume increases nonuniformly with increasing libration amplitude. We highlight strategies for mitigating the bias through the usage of particular priors. Our results imply that many known resonant systems are likely deeper in resonance than previously appreciated.

Chains of Planets in Mean Motion Resonances Arising from Oligarchic Growth

Abstract Exoplanet systems with multiple planets in mean motion resonances have often been hailed as a signpost of disk-driven migration. Resonant chains like Kepler-223 and Kepler-80 consist of a trio of planets with the three-body resonant angle librating and/or with a two-body resonant angle librating for each pair. Here we investigate whether close-in super-Earths and mini-Neptunes forming in situ can lock into resonant chains due to dissipation from a depleted gas disk. We simulate the giant impact phase of planet formation, including eccentricity damping from a gaseous disk, followed by subsequent dynamical evolution over tens of millions of years. In a fraction of simulated systems, we find that planets naturally lock into resonant chains. These planets achieve a chain of near-integer period ratios during the gas-disk stage, experience eccentricity damping that captures them into resonance, stay in resonance as the gas disk dissipates, and avoid subsequent giant impacts, eccentricity excitation, and chaotic diffusion that would dislodge the planets from resonance. Disk conditions that enable planets to complete their formation during the gas-disk stage enable those planets to achieve tight period ratios ≤2 and, if they happen to be near-integer period ratios, lock into resonance. Using the weighting of different disk conditions deduced by MacDonald et al. and forward modeling Kepler selection effects, we find that our simulations of in situ formation via oligarchic growth lead to a rate of observable trios with integer period ratios and librating resonant angles comparable to observed Kepler systems.

Dust distribution around low-mass planets on converging orbits

Context. Super-Earths can form at large orbital radii and migrate inward due to tidal interactions with the circumstellar disk. In this scenario, convergent migration may occur and lead to the formation of resonant pairs of planets. Aims. We explore the conditions under which convergent migration and resonance capture take place, and what dynamical consequences can be expected on the dust distribution surrounding the resonant pair. Methods. We combine hydrodynamic planet–disk interaction models with dust evolution calculations to investigate the signatures produced in the dust distribution by a pair of planets in mean-motion resonances. Results. We find that convergent migration takes place when the outer planet is the more massive. However, convergent migration also depends on the local properties of the disk, and divergent migration may result as well. For similar disk parameters, the capture in low degree resonances (e.g., 2:1 or 3:2) is preferred close to the star where the resonance strength can more easily overcome the tidal torques exerted by the gaseous disk. Farther away from the star, convergent migration may result in capture in high degree resonances. The dust distribution shows potentially observable features typically when the planets are trapped in a 2:1 resonance. In other cases, with higher degree resonances (e.g., 5:4 or 6:5) dust features may not be sufficiently pronounced to be easily observable. Conclusions. The degree of resonance established by a pair of super-Earths may be indicative of the location in the disk where capture occurred. There can be significant differences in the dust distribution around a single super-Earth and a pair of super-Earths in resonance.

Exploiting periodic orbits as dynamical clues for Kepler and K2 systems

Aims. Many extrasolar systems possessing planets in mean-motion resonance or resonant chain have been discovered to date. The transit method coupled with transit timing variation analysis provides an insight into the physical and orbital parameters of the systems, but suffers from observational limitations. When a (near-)resonant planetary system resides in the dynamical neighbourhood of a stable periodic orbit, its long-term stability, and thus survival, can be guaranteed. We use the intrinsic property of the periodic orbits, namely their linear horizontal and vertical stability, to validate or further constrain the orbital elements of detected two-planet systems. Methods. We computed the families of periodic orbits in the general three-body problem for several two-planet Kepler and K2 systems. The dynamical neighbourhood of the systems is unveiled with maps of dynamical stability. Results. Additional validations or constraints on the orbital elements of K2-21, K2-24, Kepler-9, and (non-coplanar) Kepler-108 near-resonant systems were achieved. While a mean-motion resonance locking protects the long-term evolution of the systems K2-21 and K2-24, such a resonant evolution is not possible for the Kepler-9 system, whose stability is maintained through an apsidal anti-alignment. For the Kepler-108 system, we find that the stability of its mutually inclined planets could be justified either solely by a mean-motion resonance, or in tandem with an inclination-type resonance. Going forward, dynamical analyses based on periodic orbits could yield better constrained orbital elements of near-resonant extrasolar systems when performed in parallel to the fitting of the observational data.

Multiplanet systems in inviscid discs can avoid forming resonant chains

ABSTRACT Convergent migration involving multiple planets embedded in a viscous protoplanetary disc is expected to produce a chain of planets in mean motion resonances (MMRs), 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 MMRs, 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 disc surface density and drive vortex formation, the smooth capture of planets into MMRs is disrupted. As planets pile up in close orbits, not protected by resonances, close encounters increase the probability of planet–planet collisions, even while the gas disc is still present. While inviscid discs often produce unstable non-resonant systems, stable, closely packed, non-resonant systems can also be formed. Thus, when examining the expectation for planet migration to produce planetary systems in MMRs, the effective turbulent viscosity of the protoplanetary disc is a key parameter.

Instabilities in multiplanet circumbinary systems

ABSTRACT The majority of the discovered transiting circumbinary planets are located very near the innermost stable orbits permitted, raising questions about the origins of planets in such perturbed environments. Most favoured formation scenarios invoke formation at larger distances and subsequent migration to their current locations. Disc-driven planet migration in multiplanet systems is likely to trap planets in mean motion resonances and drive planets inwards into regions of larger dynamical perturbations from the binary. We demonstrate how planet–planet resonances can interact with the binary through secular forcing and mean-motion resonances, driving chaos in the system. We show how this chaos will shape the architecture of circumbinary systems, with specific applications to Kepler 47 and the Pluto–Charon system, limiting maximum possible stable eccentricities and indicating what resonances are likely to exist. We are also able to constrain the minimum migration rates of resonant circumbinary planets.

Breaking mean-motion resonances during Type I planet migration

AbstractWe present 2D hydrodynamical simulations of pairs of planets migrating simultaneously in the Type I regime in a protoplanetary disc. Convergent migration naturally leads to the trapping of these planets in mean-motion resonances. Once in resonance the planets’ eccentricity grows rapidly, and disc-planet torques cause the planets to escape resonance on a time-scale of a few hundred orbits. The effect is more pronounced in highly viscous discs, but operates efficiently even in inviscid discs. We attribute this resonance-breaking to overstable librations driven by moderate eccentricity damping, but find that this mechanism operates differently in hydrodynamic simulations than in previous analytic calculations. Planets escaping resonance in this manner can potentially explain the observed paucity of resonances in Kepler multitransiting systems, and we suggest that simultaneous disc-driven migration remains the most plausible means of assembling tightly packed planetary systems.

On the spin states of habitable zone exoplanets around M dwarfs: the effect of a near-resonant companion

One longstanding problem for the potential habitability of planets within M dwarf systems is their likelihood to be tidally locked in a synchronously rotating spin state. This problem thus far has largely been addressed only by considering two objects: the star and the planet itself. However, many systems have been found to harbor multiple planets, with some in or very near to mean-motion resonances. The presence of a planetary companion near a mean-motion resonance can induce oscillatory variations in the mean-motion of the planet, which we demonstrate can have significant effects on the spin-state of an otherwise synchronously rotating planet. In particular, we find that a planetary companion near a mean-motion resonance can excite the spin states of planets in the habitable zone of small, cool stars, pushing otherwise synchronously rotating planets into higher amplitude librations of the spin state, or even complete circulation resulting in effective stellar days with full surface coverage on the order of years or decades. This increase in illuminated area can have potentially dramatic influences on climate, and thus on habitability. We also find that the resultant spin state can be very sensitive to initial conditions due to the chaotic nature of the spin state at early times within certain regimes. We apply our model to two hypothetical planetary systems inspired by the K00255 and TRAPPIST-1 systems, which both have Earth-sized planets in mean-motion resonances orbiting cool stars.

On the orbital evolution of a pair of giant planets in mean motion resonance

Pairs of extrasolar giant planets in a mean motion commensurability are common with 2:1 resonance occurring most frequently. Disc-planet interaction provides a mechanism for their origin. However, the time scale on which this could operate in particular cases is unclear. We perform 2D and 3D numerical simulations of pairs of giant planets in a protoplanetary disc as they form and maintain a mean motion commensurability. We consider systems with current parameters similar to those of HD 155358, 24 Sextantis and HD 60532, and disc models of varying mass, decreasing mass corresponding to increasing age. For the lowest mass discs, systems with planets in the Jovian mass range migrate inwards maintaining a 2:1 commensurability. Systems with the inner planet currently at around 1 au from the central star could have originated at a few au and migrated inwards on a time scale comparable to protoplanetary disc lifetimes. Systems of larger mass planets such as HD 60532 attain 3:1 resonance as observed. For a given mass accretion rate, results are insensitive to the disc model for the range of viscosity prescriptions adopted, there being good agreement between 2D and 3D simulations. However, in a higher mass disc a pair of Jovian mass planets passes through 2:1 resonance before attaining a temporary phase lasting a few thousand orbits in an unstable 5:3 resonance prior to undergoing a scattering. Thus finding systems in this commensurability is unlikely.

A Survey of Multiple Planet Systems

As of August 2008, over 30 multiple exoplanet systems are known, and 28% of stars with planets show significant evidence of a second companion. I briefly review these 30 systems individually, broadly grouping them into five categories: 1) systems with 3 or more giant (Msini > 0.2 M_Jup) planets, 2) systems with two giant planets in mean motion resonance (MMR), 3) systems with two giant planets not in MMR but whose dynamical evolution is affected by planet-planet interactions, 4) highly hierarchical systems, having two giant planets with very large period ratios (> 30:1), and 5) systems of ``Super-Earths'', containing only planets with (Msini < 20 M_Earth). It now appears that eccentricities are not markedly higher among planets in known multiple planet systems, and that planets with Msini < 1 M_Jup have lower eccentricities than more massive planets. The distribution of semimajor axes for planets in multiplanet systems does not show the 3-day pile-up or the 1 AU "jump" of the apparently-single planet distribution.

LONG RANGE OUTWARD MIGRATION OF GIANT PLANETS, WITH APPLICATION TO FOMALHAUT b

Recent observations of exoplanets by direct imaging, reveal that giant planets orbit at a few dozens to more than a hundred of AU from their central star. The question of the origin of these planets challenges the standard theories of planet formation. We propose a new way of obtaining such far planets, by outward migration of a pair of planets formed in the 10 AU region. Two giant planets in mean motion resonance in a common gap in the protoplanetary disk migrate outwards, if the inner one is significantly more massive than the outer one. Using hydrodynamical simulations, we show that their semi major axes can increase by almost one order of magnitude. In a flared disk, the pair of planets should reach an asymptotic radius. This mechanism could account for the presence of Fomalhaut b ; then, a second, more massive planet, should be orbiting Fomalhaut at about 75 AU.

Mean Motion Resonances from Planet-Planet Scattering

Planet-planet scattering is the leading mechanism to explain the large eccentricities of the observed exoplanet population. However, scattering has not been considered important to the production of pairs of planets in mean motion resonances (MMRs). We present results from a large number of numerical simulations of dynamical instabilities in 3-planet systems. We show that MMRs arise naturally in about five percent of cases. The most common resonances we populate are the 2:1 and 3:1 MMRs, although a wide variety of MMRs can occur, including high-order MMRs (up to eleventh order). MMRs are generated preferentially in systems with uneven mass distributions: the smallest planet is typically ejected after a series of close encounters, leaving the remaining, more massive planets in resonance. The distribution of resonant planets is consistent with the phase-space density of resonant orbits, meaning that planets are randomly thrown into MMRs rather than being slowly pulled into them. It may be possible to distinguish between MMRs created by scattering vs. convergent migration in a gaseous disk by considering planetary mass ratios: resonant pairs of planets beyond ~1 AU with more massive outer planets are likely to have formed by scattering. In addition, scattering may be responsible for pairs of planets in high-order MMRs (3:1 and higher) that are not easily populated by migration. The frequency of MMRs from scattering is comparable to the expected survival rate of MMRs in turbulent disks. Thus, planet-planet scattering is likely to be a major contributor to the population of resonant planets.

What to Expect from Transiting Multiplanet Systems

AbstractSo far radial velocity measurements have discovered ~25 stars to host multiple planets. The statistics imply that many of the known hosts of transiting planets should have additional planets, yet none have been solidly detected. They will be soon, via complementary search methods of RV, transit-time variations of the known planet, and transits of the additional planet. When they are found, what can transit measurements add to studies of multiplanet dynamical evolution? First, mutual inclinations become measurable, for comparison to the solar system's disk-like configuration. Such measurements will give important constraints to planet-planet scattering models, just as the radial velocity measurements of eccentricity have done. Second, the Rossiter-McLaughlin effect measures stellar obliquity, which can be modified by two-planet dynamics with a tidally evolving inner planet. Third, transit-time variations are exquisitely sensitive to planets in mean motion resonance. Two planets differentially migrating in the disk can establish such resonances, and tidal evolution of the planets can break them, so the configuration and frequency of these resonances as a function of planetary parameters will constrain these processes.

Constraints on resonant-trapping for two planets embedded in a protoplanetary disc

Context. A number of extrasolar planet systems contain pairs of Jupiter-like planets in mean motion resonances. As yet there are no known resonant systems which consist of a giant planet and a significantly lower-mass body. Aims. We investigate the evolution of two-planet systems embedded in a protoplanetary disc, which are composed of a Jupiter-mass planet plus another body located further out in the disc. The aim is to examine how the long-term evolution of such a system depends on the mass of the outer planet. Methods. We have performed 2D numerical simulations using a grid-based hydrodynamics code. The planets can interact with each other and with the disc in which they are embedded. We consider outermost planets with masses ranging from 10 M⊕ to 1 MJ. Combining the results of these calculations and analytical estimates, we also examine the case of outermost bodies with masses <10 M⊕. Results. Differential migration of the planets due to disc torques leads to different evolution outcomes depending on the mass of the outer protoplanet. For planets with mass <3.5 M⊕ the type II migration rate of the giant exceeds the type I migration rate of the outer body, resulting in divergent migration. Outer bodies with masses in the range 3.5 < mo ≤ 20 M⊕ become trapped at the edge of the gap formed by the giant planet, because of corotation torques. Higher mass planets are captured into resonance with the inner planet. If 30 ≤ mo ≤ 40 M⊕ or mo = 1 MJ, then the 2:1 resonance is established. If 80 ≤ mo ≤ 100 M⊕, the 3:2 resonance is favoured. Simulations of gas-accreting protoplanets of mass mo ≥ 20 M⊕, trapped initially at the edge of the gap, or in the 2:1 resonance, also result in eventual capture in the 3:2 resonance as the planet mass grows to become close to the Saturnian value. Conclusions. Our results suggest that there is a theoretical lower limit to the mass of an outer planet that can be captured into resonance with an inner Jovian planet, which is relevant to observations of extrasolar multiplanet systems. Furthermore, capture of a Saturn-like planet into the 3:2 resonance with a Jupiter-like planet is a very robust outcome of simulations, independent of initial conditions. This result is relevant to recent scenarios of early Solar System evolution which require Saturn to have existed interior to the 2:1 resonance with Jupiter prior to the onset of the Late Heavy Bombardment.

Resonances and stability of extra-solar planetary systems

This paper reviews recent results on the dynamics of multiple-planet extra-solar systems, including main sequence stars and the pulsar PSR B1257+12 and, comparatively, our own Solar System. Taking into account the degree of gravitational interaction of the planets, the known planetary systems may be separated into four main groups: (Ia) Planets in Mean-motion resonance (Ib) Low-eccentricity near-resonant pairs; (II) Non-resonant planets with a significant secular dynamics; and (III) Weakly interacting planet pairs. Different analytical and numerical tools can help to understand the structure of the phase space, to identify stability mechanisms and to categorize different types of motions in the cases of more significant dynamical interaction. The origin of resonant configurations is discussed in the light of the hypothesis of planetary migration. To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html