Arnaud Pierens

A comparative study of disc–planet interaction

We perform numerical simulations of a disc-planet system using various grid-based and smoothed particle hydrodynamics (SPH) codes. The tests are run for a simple setup where Jupiter and Neptune mass planets on a circular orbit open a gap in a protoplanetary disc during a few hundred orbital periods. We compare the surface density contours, potential vorticity and smoothed radial profiles at several times. The disc mass and gravitational torque time evolution are analysed with high temporal resolution. There is overall consistency between the codes. The density profiles agree within about 5 per cent for the Eulerian simulations. The SPH results predict the correct shape of the gap although have less resolution in the low-density regions and weaker planetary wakes. The disc masses after 200 orbital periods agree within 10 per cent. The spread is larger in the tidal torques acting on the planet which agree within a factor of 2 at the end of the simulation. In the Neptune case, the dispersion in the torques is greater than for Jupiter, possibly owing to the contribution from the not completely cleared region close to the planet.

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.

Hot super-Earths and giant planet cores from different migration histories

Planetary embryos embedded in gaseous protoplanetary disks undergo Type I orbital migration. Migration can be inward or outward depending on the local disk properties but, in general, only planets more massive than several

Migration and gas accretion scenarios for the Kepler 16, 34, and 35 circumbinary planets

M_\oplus

Possible planet formation in the young, low-mass, multiple stellar system GG Tau A.

can migrate outward. Here we propose that an embryos migration history determines whether it becomes a hot super-Earth or the core of a giant planet. Systems of hot super-Earths (or mini-Neptunes) form when embryos migrate inward and pile up at the inner edge of the disk. Giant planet cores form when inward-migrating embryos become massive enough to switch direction and migrate outward. We present simulations of this process using a modified N-body code, starting from a swarm of planetary embryos. Systems of hot super-Earths form in resonant chains with the innermost planet at or interior to the disk inner edge. Resonant chains are disrupted by late dynamical instabilities triggered by the dispersal of the gaseous disk. Giant planet cores migrate outward toward zero-torque zones, which move inward and eventually disappear as the disk disperses. Giant planet cores migrate inward with these zones and are stranded at ~1-5 AU. Our model reproduces several properties of the observed extra-solar planet populations. The frequency of giant planet cores increases strongly when the mass in solids is increased, consistent with the observed giant exoplanet - stellar metallicity correlation. The frequency of hot super-Earths is not a function of stellar metallicity, also in agreement with observations. Our simulations can reproduce the broad characteristics of the observed super-Earth population.

On the evolution of multiple low mass planets embedded in a circumbinary disc

Context. Several circumbinary planets have been detected by the Kepler mission. Recent work has emphasized the difficulty of forming these planets at their observed locations due to perturbations by the binary. It has been suggested that these planets formed further out in their discs in more quiescent environments and migrated in to locations where they are observed. Aims. We examine the orbital evolution of planets embedded in circumbinary disc models for the three systems Kepler-16, Kepler-34 and Kepler-35. The aims are: to explore the plausibility of a formation scenario in which cores form at large distances from the binaries and undergo inward migration and gas accretion as the gas disc disperses; to determine which sets of disc parameters lead to planets whose final orbits provide reasonable fits to the observed systems. Methods. Using a grid-based hydrodynamics code we performed simulations of a close binary system interacting with circumbinary discs with differing aspect ratios, h, and viscous stress parameters α. Once the binary+disc system reaches quasi-equilibrium we embed a planet in the disc and examine its evolution under the action of binary and disc forces. We consider fully-formed planets with masses equal to those inferred from Kepler data, and low-mass cores that migrate and accrete gas while the gas disc is being dispersed. Results. A typical outcome for all systems is stalling of inward migration as the planet enters the tidally-truncated inner cavity formed by the binary system. The circumbinary disc becomes eccentric through interaction with the binary, and the disc eccentricity forces the planet into a non-circular orbit. For each of the Kepler-16b, Kepler-34b and Kepler-35b systems we obtain planets whose parameters agree reasonably well with the observational data, but none of our simulations are able to produce highly accurate fits for all orbital parameters. Conclusions. The final orbital configuration of a circumbinary planet is determined by a delicate interplay between the detailed stucture of the circumbinary disc and the orbital parameters of the planet as it migrates into the inner disc cavity. Simplified simulations such as those presented here provide support for a formation scenario in which a core forms, migrates inward and accretes gas, but accurate fitting of the observed Kepler systems is likely to require disc models that are significantly more sophisticated in terms of their input physics.

Making giant planet cores: convergent migration and growth of planetary embryos in non-isothermal discs

The formation of planets around binary stars may be more difficult than around single stars. In a close binary star (with a separation of less than a hundred astronomical units), theory predicts the presence of circumstellar disks around each star, and an outer circumbinary disk surrounding a gravitationally cleared inner cavity around the stars. Given that the inner disks are depleted by accretion onto the stars on timescales of a few thousand years, any replenishing material must be transferred from the outer reservoir to fuel planet formation (which occurs on timescales of about one million years). Gas flowing through disk cavities has been detected in single star systems. A circumbinary disk was discovered around the young low-mass binary system GG Tau A (ref. 7), which has recently been shown to be a hierarchical triple system. It has one large inner disk around the single star, GG Tau Aa, and shows small amounts of shocked hydrogen gas residing within the central cavity, but other than a single weak detection, the distribution of cold gas in this cavity or in any other binary or multiple star system has not hitherto been determined. Here we report imaging of gas fragments emitting radiation characteristic of carbon monoxide within the GG Tau A cavity. From the kinematics we conclude that the flow appears capable of sustaining the inner disk (around GG Tau Aa) beyond the accretion lifetime, leaving time for planet formation to occur there. These results show the complexity of planet formation around multiple stars and confirm the general picture predicted by numerical simulations.

Fast migration of low-mass planets in radiative discs

Context. Previous work has shown that the tidal interaction between a binary system and a circumbinary disc leads to the formation of a large inner cavity in the disc. Subsequent formation and inward migration of a low mass planet causes it to become trapped at the cavity edge, where it orbits until further mass growth or disc dispersal. The question of how systems of multiple planets in circumbinary discs evolve has not yet been addressed. Aims. We present the results of hydrodynamic simulations of multiple low mass planets embedded in a circumbinary disc. The aim is to examine their long term evolution as they approach and become trapped at the edge of the tidally truncated inner cavity. Methods. A grid-based hydrodynamics code was used to compute simulations of 2D circumbinary disc models with embedded planets. The 3D evolution of the planet orbits was computed, and inclination damping due to the disc was calculated using prescribed forces. We present a suite of simulations which study the evolution of pairs of planets migrating in the disc. We also present the results of hydrodynamic simulations of five-planet systems, and study their long term evolution after disc dispersal using a N-body code. Results. For the two-planet simulations we assume that the innermost planet has migrated to the edge of the inner cavity and remains trapped there, and study the subsequent evolution of the system as the outermost planet migrates inward. We find that the outcomes largely depend on the mass ratio q = m i /m o , where m i (m o ) is the mass of the innermost (outermost) planet. For q 1 the systems reach equilibrium configurations in which the planets are locked into mean motion resonances, and remain trapped at the edge of the inner cavity without further migration. Most simulations of five-planet systems we performed resulted in collisions and scattering events, such that only a single planet remained in orbit about the binary. In one case however, a multiplanet resonant system was found to be dynamically stable over long time scales, suggesting that such systems may be observed in planet searches focussed on close binary systems.

Protoplanetary migration in non-isothermal discs with turbulence driven by stochastic forcing

Earth-mass bodies are expected to undergo Type I migration directed either inward or outward depending on the thermodynamical state of the protoplanetary disc. Zones of convergent migration exist where the Type I torque cancels out. We study the evolution of multiple protoplanets of a few Earth masses embedded in a non-isothermal protoplanetary disc. The protoplanets are located in the vicinity of a convergence zone located at the transition between two different opacity regimes. Inside the convergence zone, Type I migration is directed outward and outside the zone migration is directed inward. We used a grid-based hydrodynamical code that includes radiative effects. We performed simulations varying the initial number of embryos and tested the effect of including stochastic forces to mimic the effects resulting from turbulence. We also performed N-body runs calibrated on hydrodynamical calculations to follow the evolution on Myr timescales. For a small number of initial embryos (N = 5-7) and in the absence of stochastic forcing, the population of protoplanets migrates convergently toward the zero-torque radius and forms a stable resonant chain that protects embryos from close encounters. In systems with a larger initial number of embryos, or in which stochastic forces were included, these resonant configurations are disrupted. This in turn leads to the growth of larger cores via a phase of giant impacts, after which the system settles to a new stable resonant configuration. Giant planets cores with masses of 10 Earth masses formed in about half of the simulations with initial protoplanet masses of m_p = 3 Earth masses but in only 15% of simulations with m_p = 1 Earth mass. This suggests that if ~2-3 Earth mass protoplanets can form in less than ~1 Myr, convergent migration and giant collisions can grow giant planet cores at Type I migration convergence zones.

On the dynamics of resonant super-Earths in disks with turbulence driven by stochastic forcing

Low-mass planets are known to undergo Type I migration and this process must have played a key role during the evolution of planetary systems. Analytical formulae for the disc torque have been derived assuming that the planet evolves on a fixed circular orbit. However, recent work has shown that in isothermal discs, a migrating protoplanet may also experience dynamical corotation torques that scale with the planet drift rate. The aim of this study is to examine whether dynamical corotation torques can also affect the migration of low-mass planets in non-isothermal discs. We performed 2D radiative hydrodynamical simulations to examine the orbital evolution outcome of migrating protoplanets as a function of disc mass. We find that a protoplanet can enter a fast migration regime when it migrates in the direction set by the entropy-related horseshoe drag and when the Toomre stability parameter is less than a threshold value below which the horseshoe region contracts into a tadpole-like region. In that case, an underdense trapped region appears near the planet, with an entropy excess compared to the ambient disc. If the viscosity and thermal diffusivity are small enough so that the entropy excess is conserved during migration, the planet then experiences strong corotation torques arising from the material flowing across the planet orbit. During fast migration, we observe that a protoplanet can pass through the zero-torque line predicted by static torques. We also find that fast migration may help in disrupting the mean-motion resonances that are formed by convergent migration of embryos.