Corotation Resonance Research Articles

The complex interplay between tidal inertial waves and zonal flows in differentially rotating stellar and planetary convective regions

Context.Quantifying tidal interactions in close-in two-body systems is of prime interest since they have a crucial impact on the architecture and the rotational history of the bodies. Various studies have shown that the dissipation of tides in either body is very sensitive to its structure and to its dynamics. Furthermore, solar-like stars and giant gaseous planets in our Solar System experience differential rotation in their outer convective envelopes. In this respect, numerical simulations of tidal interactions in these objects have shown that the propagation and dissipation properties of tidally excited inertial waves can be strongly modified in the presence of differential rotation.Aims.In particular, tidal inertial waves may strongly interact with zonal flows at the so-called co-rotation resonances, where the wave’s Doppler-shifted frequency is cancelled out. The energy dissipation at such resonances could deeply modify the orbital and spin evolutions of tidally interacting systems. In this context, we aim to provide a deep physical understanding of the dynamics of tidal waves at co-rotation resonances in the presence of differential rotation profiles that are typical of low-mass stars and giant planets.Methods.In this work, we have developed an analytical local model of an inclined shearing box that describes a small patch of the differentially rotating convective zone of a star or a planet. We investigate the propagation and the transmission of free inertial waves at co-rotation, and more generally at critical levels, which are singularities in the governing wave differential equation. Through the construction of an invariant called the wave action flux, we identify different regimes of wave transmission at critical levels, which are confirmed with a one-dimensional three-layer numerical model.Results.We find that inertial waves can be fully transmitted, strongly damped, or even amplified after crossing a critical level. The occurrence of these regimes depends on the assumed profile of differential rotation, on the nature as well as the latitude of the critical level, and on wave parameters such as the inertial frequency and the longitudinal and vertical wavenumbers. Waves can thus either deposit their action flux in the fluid when damped at critical levels, or they can extract action flux from the fluid when amplified at critical levels. Both situations can lead to significant angular momentum exchange between the tidally interacting bodies.

Migration of Jupiter-mass planets in low-viscosity discs

Context. Type-II migration of giant planets has a speed proportional to the disc’s viscosity for values of the α viscosity parameter larger than 10−4. Previous studies based on two-dimensional simulations, have shown that, at even lower viscosities, migration can be very chaotic and is often characterised by phases of fast migration. The reason is that vortices appear in low-viscosity discs due to the Rossby-wave instability at the edges of the gap opened by the planet. Migration is then determined by vortex-planet interactions. Aims. Our goal is to study giant planet migration in low-viscosity discs with 3D simulations. In 3D, vortices are more complex than the simple vertical extension of their 2D counterparts; their impact on planet migration is therefore not obvious. Methods. We performed numerical simulations using two grid-based codes: FARGOCA for three-dimensional simulations and FARGO-ADSG for the two dimensional case. Two-dimensional simulations were used mainly for preliminary tests to check the impact of self-gravity on vortex formation and on vortex-disc dynamics. After selecting disc masses for which self-gravity is not important at the planet location, three-dimensional simulations without self-gravity can be safely used. We have considered an adiabatic equation of state with exponential damping of temperature perturbations in order to avoid the development of the vertical shear instability. In our nominal simulation, we set α = 0 so that only numerical viscosity is present. We then performed simulations with non-zero α values to assess the threshold of prescribed viscosity below which the new migration processes appear. Results. We show that for α ≲ 10−5 two migration modes are possible, which differ from classical Type-II migration in the sense that they are not proportional to the disc’s viscosity. The first occurs when the gap opened by the planet is not very deep. This occurs in 3D simulations and/or when a big vortex forms at the outer edge of the planetary gap, diffusing material into the gap. The de-saturation of co-orbital and co-rotation resonances keeps the planet’s eccentricity low. Inward planet migration then occurs as long as the disc can refill the gap left behind by the migrating planet, either due to diffusion caused by the presence of the vortex or to the inward migration of the vortex itself due to its interaction with the disc. We call this type of migration ‘vortex-driven migration’, which differs from ‘vortex-induced’ migration described in Lin & Papaloizou (2010, MNRAS, 405, 1473, and 2011a, MNRAS, 415, 1445). This migration is very slow and cannot continue indefinitely because eventually the vortex dissolves. The second migration mode occurs when the gap is deep so that the planet’s eccentricity grows to a value e ~ 0.2 due to inefficient eccentricity damping by co-rotation resonances. Once the planet is on an eccentric orbit, gas can pass through the gap and planet migration unlocks from the disc’s viscous evolution. This second, faster migration mode appears to be typical of two-dimensional models in discs with slower damping of temperature perturbations. Conclusions. Vortex-driven migration in low-viscosity discs can be very slow and eventually reverses and stops, offering an interesting mechanism to explain the existence of the cold-Jupiter population, even if these planets originally started growing at the disc’s snowline.

Resonance sweeping by a decelerating Galactic bar

ABSTRACT We provide the first quantitative evidence for the deceleration of the Galactic bar from local stellar kinematics in agreement with dynamical friction by a typical dark matter halo. The kinematic response of the stellar disc to a decelerating bar is studied using secular perturbation theory and test particle simulations. We show that the velocity distribution at any point in the disc affected by a naturally slowing bar is qualitatively different from that perturbed by a steadily rotating bar with the same current pattern speed Ωp and amplitude. When the bar slows down, its resonances sweep through phase space, trapping, and dragging along a portion of previously free orbits. This enhances occupation on resonances, but also changes the distribution of stars within the resonance. Due to the accumulation of orbits near the boundary of the resonance, the decelerating bar model reproduces with its corotation resonance the offset and strength of the Hercules stream in the local vR-vφ plane and the double-peaked structure of mean vR in the Lz–φ plane. At resonances other than the corotation, resonant dragging by a slowing bar is associated with a continuing increase in radial action, leading to multiple resonance ridges in the action plane as identified in the Gaia data. This work shows models using a constant bar pattern speed likely lead to qualitatively wrong conclusions. Most importantly we provide a quantitative estimate of the current slowing rate of the bar $\dot{\Omega }_{\rm p}= (-4.5 \pm 1.4) \, {\rm km}\, {\rm s}^{-1}\, {\rm kpc}^{-1}\, {\rm Gyr}^{-1}$ with additional systematic uncertainty arising from unmodelled impacts of e.g. spiral arms.

Trimodal structure of Hercules stream explained by originating from bar resonances

ABSTRACT Gaia Data Release 2 revealed detailed structures of nearby stars in phase space. These include the Hercules stream, whose origin is still debated. Most of the previous numerical studies conjectured that the observed structures originate from orbits in resonance with the bar, based on static potential models for the Milky Way. We, in contrast, approach the problem via a self-consistent, dynamic, and morphologically well-resolved model, namely a full N-body simulation of the Milky Way. Our simulation comprises about 5.1 billion particles in the galactic stellar bulge, bar, disc, and dark-matter halo and is evolved to 10 Gyr. Our model’s disc component is composed of 200 million particles, and its simulation snapshots are stored every 10 Myr, enabling us to resolve and classify resonant orbits of representative samples of stars. After choosing the Sun’s position in the simulation, we compare the distribution of stars in its neighbourhood with Gaia’s astrometric data, thereby establishing the role of identified resonantly trapped stars in the formation of Hercules-like structures. From our orbital spectral-analysis, we identify multiple, especially higher order resonances. Our results suggest that the Hercules stream is dominated by the 4:1 and 5:1 outer Lindblad and corotation resonances. In total, this yields a trimodal structure of the Hercules stream. From the relation between resonances and ridges in phase space, our model favoured a slow pattern speed of the Milky-Way bar (40–45 $\mathrm{km \, s^{-1} \, kpc^{-1}}$).

Retention of Long-period Gas Giant Planets: Type II Migration Revisited

Abstract During their formation, emerging protoplanets tidally interact with their natal disks. Proto–gas giant planets, with Hill radii larger than the disk thickness, open gaps and quench gas flow in the vicinity of their orbits. It is usually assumed that their type II migration is coupled to the viscous evolution of the disk. Although this hypothesis provides an explanation for the origin of close-in planets, it also encounters a predicament on the retention of long-period orbits for most gas giant planets. Moreover, numerical simulations indicate that the planets’ migrations are not solely determined by the viscous diffusion of their natal disk. Here we carry out a series of hydrodynamic simulations combined with analytic studies to examine the transition between different paradigms of type II migration. We find a range of planetary mass for which gas continues to flow through a severely depleted gap so that the surface density distribution in the disk region beyond the gap is maintained in a quasi-steady state. The associated gap profile modifies the location of corotation and Lindblad resonances. In the proximity of the planet’s orbit, high-order Lindblad and corotation torque are weakened by the gas depletion in the gap, while low-order Lindblad torques near the gap walls preserve their magnitude. Consequently, the intrinsic surface density distribution of the disk delicately determines both the pace and direction of the planets’ type II migration. We show that this effect might stall the inward migration of giant planets and preserve them in disk regions where the surface density is steep.

Modeling Radial Velocity Data of Resonant Planets to Infer Migration Histories

Abstract A number of giant-planet pairs with period ratios ≲2 discovered by the radial velocity (RV) method 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 protoplanetary disk should not only be captured into mean motion resonances but also end up in a specific dynamical configuration within the resonance, sometimes referred to as apsidal corotation resonance (ACR). Here we develop a method for testing the hypothesis that a planet pair resides in an ACR by directly fitting RV data. The ACR hypothesis strongly restricts the number of free parameters describing the RV signal, and we compare fits using this highly restricted model to fits using a more conventional two-planet RV model by using nested sampling simulations. We apply our method to HD 45364 and HD 33844, two systems hosting giant-planet pairs in 3:2 and 5:3 resonances, respectively. The observations of both systems are consistent with ACR configurations, which are formally preferred based on the Bayes factors computed from nested sampling simulations. We use the results of our ACR model fits to constrain the possible migration histories of these systems.

Escapees from the bar resonances

Understanding radial migration is a crucial point for building relevant chemical and dynamical evolution models of the Milky Way disk. In this paper we analyze a high-resolution N-body simulation of a Milky Way-type galaxy to study the role that the slowing down of a stellar bar has in generating migration from the inner to the outer disk. Stellar particles are trapped by the main resonances (corotation and outer Lindblad resonance, OLR) which then propagate outward across the disk due to the bar slowing down. Once the bar strength reaches its maximum amplitude, some of the stars delivered to the outer disk escape the resonances and some of them settle on nearly circular orbits. The number of escaped stars gradually increases, also due to the decrease in the bar strength when the boxy/peanut bulge forms. We show that this mechanism is not limited to stars on nearly circular orbits; stars initially on more eccentric orbits can also be transferred outward (out to the OLR location) and can end up on nearly circular orbits. Therefore, the propagation of the bar resonances outward can induce the circularization of the orbits of some of the migrating stars. The mechanism investigated in this paper can explain the presence of metal-rich stars at the solar vicinity and more generally in the outer Galactic disk. Our dynamical model predicts that up to 3% of stars between corotation and the OLR can be formed in the innermost region of the Milky Way. The epoch of the Milky Way bar formation can be potentially constrained by analyzing the age distribution of the most metal-rich stars at the solar vicinity.

Dynamics of the 11:10 Corotation and Lindblad resonances with Mimas, and application to Anthe

The short-term, resonant and long-term dynamics of Anthe (S/2007 S4), are investigated with the most recent published ephemeris of the Saturnian system. We identify and explain many components in the time variations of the elements of Anthe as being caused by the joined effects of the Corotation resonance, the secular precessions of the orbits of Anthe and Mimas due to the effects of the oblateness of Saturn and long-term mutual interactions. Firstly, it is shown that the current resonant state of the pair Mimas-Anthe is characterized uniquely by the stable libration of the corotation angle ϕ1 ≡ 11λA − 10λM − ϖM around zero, where A, M, λ, ϖ refer to Anthe, Mimas, mean longitude and longitude of the pericenter, respectively. We find that only the corotation angle ϕ1 is stably librating, while all other arguments for the other sub-resonances (including the angular combination 11λA − 10λM − ϖA − ΩA + ΩM, where Ω is the longitude of the ascending node) are circulating on long time scales. This is consistent with the absence of chaos in Anthe's orbit.In the second part of this work, we map the 11:10 Mimas-Anthe resonance in the eccentricity-semi-major axis phase space. We identify non-chaotic regions around both the Corotation and Lindblad resonances, which are separated by chaotic zones. The orbit of Anthe is located deeply inside the stable regions of the Corotation resonance.

Effects of immersed moonlets in the ring arc particles of Saturn

Ring arcs are the result of particles in corotation resonances with nearby satellites. Arcs are present in Saturn and Neptune systems, in Saturn they are also associated with small satellites immersed on them. The satellite Aegaeon is immersed in the G ring arc, and the satellites Anthe and Methone are embedded in arcs named after them. Since most of the population of the arcs is formed by μm-sized particles the dissipative effects, such as the plasma drag and the solar radiation force, decrease the lifetime of the arcs. We analysed the effects of the immersed satellites on these arcs by computing the mass production rate and the perturbation caused by them in the arc particles. By comparing the lifetime of the particles and the mass production rate we concluded that Aegaeon, Anthe and Methone did not act as sources for their arcs. We took a step further by analysing a hypothetical scenario formed by an immersed moonlet of different sizes. As a result we found that regardless of the size of the hypothetical moonlet (from about 0.10 km–4.0 km) these moonlets will not act as a source. These arcs are temporary structures and they will disappear in a very short period of time.

Trojans in the Solar Neighborhood

Abstract About 20% of stars in the solar vicinity are in the Hercules stream, a bundle of stars that move together with a velocity distinct from the Sun. Its origin is still uncertain. Here, we explore the possibility that Hercules is made of trojans, stars captured at L4, one of the Lagrangian points of the stellar bar. Using GALAKOS–a high-resolution N-body simulation of the Galactic disk–we follow the motions of stars in the corotating frame of the bar and confirm previous studies on Hercules being formed by stars in corotation resonance with the bar. Unlike previous work, we demonstrate that the retrograde nature of trojan orbits causes the asymmetry in the radial velocity distribution, typical of Hercules in the solar vicinity. We show that trojans remain at capture for only a finite amount of time, before escaping L4 without being captured again. We anticipate that in the kinematic plane the Hercules stream will depopulate along the bar’s major axis and be visible at azimuthal angles behind the solar vicinity with a peak toward L4. This test can exclude the outer Lindblad resonance origin of the Hercules stream and be validated by Gaia DR3 and DR4.

Exploring the Origin of Moving Groups and Diagonal Ridges by Simulations of Stellar Orbits and Birthplaces

Abstract The present paper is the culmination of a series aimed at contributing to the understanding of the kinematic structures of the solar neighborhood (SN), explaining the origin of the Local Arm, and relating the moving groups with the spiral-arms resonances in the disk. With a model for the Galactic potential, with the Sun inside the spiral corotation resonance (CR), we integrate the 2D orbits of test particles distributed in birthplaces along the main spiral arms, the Local Arm, and in the axisymmetric disk. A comparison of the resulting U–V plane of the SN with that provided by Gaia DR2 confirms our previous conclusion that the moving groups of Coma Berenices, Pleiades, and Hyades are associated with the CR, and that the Hercules stream is formed by the bulk of high-order inner Lindblad resonances. The kinematic structures result from stellar orbits trapped by the spiral resonances in a timespan of ∼1 Gyr, indicating the long-lived nature of the spiral structure and challenging recent arguments in favor of short-lived structures originated from incomplete phase mixing in the Galactic disk. As a byproduct, our simulations give some insight into the birthplaces of the stars presently located in the SN; the majority of stars of the main moving groups and the Hercules stream were likely born in the Local Arm, while stars of the Sirius group possibly originated from the outer segment of the Sagittarius–Carina arm. We also propose the spiral resonances as the dynamical origin for the diagonal ridges in the Galactic distribution of rotation velocities.

Tracing Hercules in Galactic azimuth with Gaia DR2

The second data release of the Gaia mission has revealed, in stellar velocity and action space, multiple ridges, the exact origin of which is still debated. Recently, we demonstrated that a large Galactic bar with pattern speed 39 km s−1 kpc−1 creates most of the observed ridges. Among these ridges, the Hercules moving group would then be associated with orbits trapped at the co-rotation resonance of the bar. Here we show that a distinctive prediction of such a model is that the angular momentum of Hercules at the Sun’s radius must significantly decrease with increasing Galactocentric azimuth (i.e. when getting closer to the major axis of the bar). We show that this dependence of the angular momentum of trapped orbits on the azimuth on the other hand does not happen close to the outer Lindblad resonance of a faster bar, unless the orbital distribution is still far from phase-mixed, namely for a bar perturbation younger than ∼2 Gyr. Using Gaia DR2 and Bayesian distances from the StarHorse code, and tracing the average Galactocentric radial velocity as a function of angular momentum and azimuth, we show that the Hercules angular momentum changes significantly with azimuth as expected for the co-rotation resonance of a dynamically old large bar.

Renewal of Transient Spiral Modes in Disk Galaxies

Abstract Spiral structure in disk galaxies could arise from transient modes that create conditions conducive for their regeneration; this is the proposal of Sellwood and Carlberg, based on simulations of stellar disks. The linear response of an axisymmetric stellar disk, to an adiabatic nonaxisymmetric transient mode, gives a final distribution function (DF) that is equal to the initial DF everywhere in phase space, except at the Lindblad and corotation resonances where the final DF is singular. We use the nonlinear theory of adiabatic capture into resonance to resolve the singularities and calculate the finite changes in the DF. These take the form of axisymmetric “scars” concentrated around resonances, whose DFs have simple general forms. Global changes in the physical properties are explored for a cool Mestel disk: we calculate the DFs of scars and estimate the changes in the disk angular momentum, surface density, and orbital frequencies leading to shifts in resonances. Resonant torques between disk stars and any new linear nonaxisymmetric mode are suppressed within a scar, as is epicyclic heating. Because all resonances of a linear mode with the same angular wavenumber and pattern speed as its precursor lie inside the scars of the precursor, it suffers less damping. Hence, scars filter the spectrum of noise-generated modes, promoting the renewal of a few select modes. Relic scars sustained by a galaxy disk, due to past tidal interaction with a passing companion, may still be active enablers of nonaxisymmetric modes, such as the two-armed “grand design” spiral patterns.

Secular dynamics in extrasolar systems with two planets in mutually inclined orbits

AbstractWe revisit the problem of the secular dynamics in two-planet systems in which the planetary orbits exhibit a high value of the mutual inclination. We propose a ‘basic hamiltonian model’ for secular dynamics, parameterized in terms of the system’s Angular Momentum Deficit (AMD). The secular Hamiltonian can be obtained in closed form, using multipole expansions in powers of the distance ratio between the planets, or in the usual Laplace-Lagrange form. The main features of the phase space (number and stability of periodic orbits, bifurcations from the main apsidal corotation resonances, Kozai resonance etc.) can all be recovered by choosing the corresponding terms in the ‘basic Hamiltonian’. Applications include the semi-analytical determination of the actual orbital state of the system using Hamiltonian normalization techniques. An example is discussed referring to the system of two outermost planets of the ν-Andromedae system.

Dynamics of Multiple Bodies in a Corotation Resonance: Conserved Quantities and Relevance to Ring Arcs

Abstract The interactions among objects in a mean motion resonance are important for the orbital evolution of satellites and rings, especially Saturn’s ring arcs and associated moons. In this work, we examine interactions among massive bodies in the same corotation eccentricity resonance site that affect the orbital evolution of those bodies using numerical simulations. During these simulations, the bodies exchange angular momentum and energy during close encounters, altering their orbits. This energy exchange, however, does not mean that one body necessarily moves closer to exact corotation when the other moves away from it. Indeed, if one object moves toward one of these sites, the other object is equally likely to move toward or away from it. This happens because the timescale of these close encounters is short compared to the synodic period between these particles and the secondary mass (i.e., the timescale where corotation sites can be treated as potential maxima). Because the timescale of a gravitational encounter is comparable to the timescale of a collision, we could expect energy to be exchanged in a similar way for collisional interactions. In that case, these findings could be relevant for denser systems like the arcs in Neptune’s Adams ring and how they can be maintained in the face of frequent inelastic collisions.

2D chemical evolution model: The impact of Galactic disc asymmetries on azimuthal chemical abundance variations

Context. Galactic disc chemical evolution models generally ignore azimuthal surface density variation that can introduce chemical abundance azimuthal gradients. Recent observations, however, have revealed chemical abundance changes with azimuth in the gas and stellar components of both the Milky Way and external galaxies.Aims. Our aim is to quantify the effects of spiral arm density fluctuations on the azimuthal variations of the oxygen and iron abundances in disc galaxies.Methods.We developed a new 2D Galactic disc chemical evolution model that is capable of following not just radial but also azimuthal inhomogeneities.Results. The density fluctuations resulting from a Milky Way-like N-body disc formation simulation produce azimuthal variations in the oxygen abundance gradients of the order of 0.1 dex. Moreover, the azimuthal variations are more evident in the outer Galactic regions, which is in agreement with the most recent observations in external galaxies. Using a simple analytical model, we show that the largest fluctuations with azimuth result near the spiral structure co-rotation resonance where the relative speed between the spiral and gaseous disc is the slowest.Conclusion. We provide a new 2D chemical evolution model capable of following azimuthal density variations. Density fluctuations extracted from a Milky Way-like dynamical model lead to a scatter in the azimuthal variations of the oxygen abundance gradient, which is in agreement with observations in external galaxies. We interpret the presence of azimuthal scatter at all radii by the presence of multiple spiral modes moving at different pattern speeds, as found in both observations and numerical simulations.

Signatures of the resonances of a large Galactic bar in local velocity space

The second data release of the Gaia mission has revealed a very rich structure in local velocity space. In terms of in-plane motions, this rich structure is also seen as multiple ridges in the actions of the axisymmetric background potential of the Galaxy. These ridges are probably related to a combination of effects from ongoing phase-mixing and resonances from the spiral arms and the bar. We have recently developed a method for capturing the behaviour of the stellar phase-space distribution function at a resonance by re-expressing it in terms of a new set of canonical actions and angles variables valid in the resonant region. Here, by properly treating the distribution function at resonances, and by using a realistic model for a slowly rotating large Galactic bar with pattern speed Ωb = 39 km s−1 kpc−1, we show that no fewer than six ridges in local action space can be related to resonances with the bar. Two of these ridges at low angular momentum correspond to the corotation resonance, and can be associated with the Hercules moving group in local velocity space. Another ridge at high angular momentum corresponds to the outer Lindblad resonance, and can tentatively be associated with the velocity structure seen as an arch at high azimuthal velocities in Gaia data. The other ridges are associated with the 3:1, 4:1, and 6:1 resonances. The last can be associated with the so-called “horn” of the local velocity distribution. While it is clear that effects from spiral arms and incomplete phase-mixing related to external perturbations also play a role in shaping the complex kinematics revealed by Gaia data, the present work demonstrates that, contrary to common misconceptions, the bar alone can create multiple prominent ridges in velocity and action space.

Growth of eccentricity in planet-disc interactions

The origin and wide distribution of eccentricities in planetary systems remains to be explained, in particular in the context of planet-disc interactions. Here we present a set of linear equations that describe the behavior of small eccentricities in a protoplanetary system consisting of a gaseous disc and a planet. Eccentricity propagates through the disc by means of pressure, and is exchanged with the planet via secular interactions. Excitation and damping of eccentricity can occur through Lindblad and corotation resonances, as well as viscosity. Three-dimensional effects allow for an eccentric mode to be trapped in the inner parts of the disc. This eccentric mode can easily grow within the disc's lifetime. An eccentric mode dominated by the planet can also grow, although less rapidly. Application to a hot Jupiter surrounded by a gaseous disc suggests that the eccentricity of the planet can grow. Finally, the linear theory is compared to hydrodynamical simulations, and a very good agreement is found.

Ring dynamics around non-axisymmetric bodies with application to Chariklo and Haumea

Dense and narrow rings have been discovered recently around the small Centaur object Chariklo and the dwarf planet Haumea, while being suspected around the Centaur Chiron. They are the first rings observed in the Solar System elsewhere than around giant planets. Contrarily to the latters, gravitational fields of small bodies may exhibit large non-axisymmetric terms that create strong resonances between the spin of the object and the mean motion of rings particles. Here we show that modest topographic features or elongations of Chariklo and Haumea explain why their rings are relatively far away from the central body, when scaled to those of the giant planets. Lindblad-type resonances actually clear on decadal time-scales an initial collisional disk that straddles the corotation resonance (where the particles mean motion matches the spin rate of the body). The disk material inside the corotation radius migrates onto the body, while the material outside the corotation radius is pushed outside the 1/2 resonance, where the particles complete one revolution while the body completes two rotations. Consequently, the existence of rings around non-axisymmetric bodies requires that the 1/2 resonance resides inside the Roche limit of the body, favoring fast rotators for being surrounded by rings.

Forming Gliese 876 through Smooth Disk Migration

We run a suite of dissipative N-body simulations to determine which regions of phase space for smooth disk migration are consistent with the GJ876 system, an M-dwarf hosting three planets orbiting in a chaotic 4:2:1 Laplace resonance. We adopt adaptive mesh refinement (AMR) methods which are commonly used in hydrodynamical simulations to efficiently explore the parameter space defined by the semi-major axis and eccentricity damping timescales. We find that there is a large region of phase space which produces systems in the chaotic Laplace resonance and a smaller region consistent with the observed eccentricities and libration amplitudes for the resonant angles. Under the assumptions of Type I migration for the outer planet, we translate these damping timescales into constraints on the protoplanetary disk surface density and thickness. When we strongly (weakly) damp the eccentricities of the inner two Laplace planets, these timescales correspond to disk surface densities around ten thousand (a few hundred) grams per square centimeter and disk aspect ratios between 1-10%. Additionally, smooth migration produces systems with a range of chaotic timescales, from decades and centuries to upwards of thousands of years. In agreement with previous studies, the less chaotic regions of phase space coincide with the system being in a low energy double apsidal corotation resonance. Our detailed modeling of multi-planetary systems coupled with our AMR exploration method enhances our ability to map out the parameter space of planet formation models, and is well suited to study other resonant chain systems such as Trappist-1, Kepler-60, and others.