Metallicity effect and planet mass function in pebble-based planet formation models

One of the main scenarios of planet formation is the core accretion model where a massive core forms first and then accretes a gaseous envelope. This core forms by accreting solids, either planetesimals, or pebbles. A key constraint in this model is that the accretion of gas must proceed before the dissipation of the gas disc. Classical planetesimal accretion scenario predicts that the time needed to form a giant planets core is much longer than the time needed to dissipate the disc. This difficulty led to the development of another accretion scenario, in which cores grow by accretion of pebbles, which are much smaller and thus more easily accreted, leading to a more rapid formation. The aim of this paper is to compare our updated pebble-based planet formation model with observations, in particular the well studied metallicity effect. We adopt the Bitsch et al. 2015a disc model and the Bitsch et al. 2015b pebble model and use a population synthesis approach to compare the formed planets with observations. We find that keeping the same parameters as in Bitsch et al. 2015b leads to no planet growth due to a computation mistake in the pebble flux (Bitsch et al. 2017). Indeed a large fraction of the heavy elements should be put into pebbles (Zpeb/Ztot = 0.9) in order to form massive planets using this approach. The resulting mass functions show a huge amount of giants and a lack of Neptune mass planets, which are abundant according to observations. To overcome this issue we include the computation of the internal structure for the planetary atmosphere to our model. This leads to the formation of Neptune mass planets but no observable giants. Reducing the opacity of the planetary envelope finally matches observations better. We conclude that modeling the internal structure for the planetary atmosphere is necessary to reproduce observations.Comment: 13 pages, 22 figure

Stability of the co-orbital resonance under dissipation: Application to its evolution in protoplanetary discs

Despite the existence of co-orbital bodies in the solar system, and the prediction of the formation of co-orbital planets by planetary system formation models, no co-orbital exoplanets (also called trojans) have been detected thus far. In this paper we investigate how a pair of co-orbital exoplanets would fare during their migration in a protoplanetary disc. To this end, we computed a stability criterion of the Lagrangian equilibria L4 and L5 under generic dissipation and slow mass evolution. Depending on the strength and shape of these perturbations, the system can either evolve towards the Lagrangian equilibrium, or tend to increase its amplitude of libration, possibly all the way to horseshoe orbits or even exiting the resonance. We estimated the various terms of our criterion using a set of hydrodynamical simulations, and show that the dynamical coupling between the disc perturbations and both planets have a significant impact on the stability: the structures induced by each planet in the disc perturb the dissipative forces applied on the other planets over each libration cycle. Amongst our results on the stability of co-orbitals, several are of interest to constrain the observability of such configurations: long-distance inward migration and smaller leading planets tend to increase the libration amplitude around the Lagrangian equilibria, while leading massive planets and belonging to a resonant chain tend to stabilise it. We also show that, depending on the strength of the dissipative forces, both the inclination and the eccentricity of the smaller of the two co-orbitals can be significantly increased during the inward migration of the co-orbital pair, which can have a significant impact on the detectability by transit of such configurations

The formation of Jupiter by hybrid pebble-planetesimal accretion

The standard model for giant planet formation is based on the accretion of solids by a growing planetary embryo, followed by rapid gas accretion once the planet exceeds a so-called critical mass. The dominant size of the accreted solids (cm-size particles named pebbles or km to hundred km-size bodies named planetesimals) is, however, unknown. Recently, high-precision measurements of isotopes in meteorites provided evidence for the existence of two reservoirs in the early Solar System. These reservoirs remained separated from ~1 until ~ 3 Myr after the beginning of the Solar System's formation. This separation is interpreted as resulting from Jupiter growing and becoming a barrier for material transport. In this framework, Jupiter reached ~20 Earth masses within ~1 Myr and slowly grew to ~50 Earth masses in the subsequent 2 Myr before reaching its present-day mass. The evidence that Jupiter slowed down its growth after reaching 20 Earth masses for at least 2 Myr is puzzling because a planet of this mass is expected to trigger fast runaway gas accretion. Here, we use theoretical models to describe the conditions allowing for such a slow accretion and show that Jupiter grew in three distinct phases. First, rapid pebble accretion brought the major part of Jupiter's core mass. Second, slow planetesimal accretion provided the energy required to hinder runaway gas accretion during 2 Myr. Third, runaway gas accretion proceeded. Both pebbles and planetesimals therefore have an important role in Jupiter's formation.Comment: Published in Nature Astronomy on August 27, 201

Pushing planets into an inner cavity by a resonant chain

Context. The orbital distribution of exoplanets indicates an accumulation of super-Earth sized planets close to their host stars in compact systems. When an inward disc-driven migration scenario is assumed for their formation, these planets could have been stopped and might have been parked at an inner edge of the disc, or be pushed through the inner disc cavity by a resonant chain. This topic has not been properly and extensively studied. Using numerical simulations, we investigate the possibility that the inner planets in a resonant chain can be pushed into the disc inner cavity by outer planets. We performed hydrodynamical and N-body simulations of planetary systems embedded in their nascent disc. The inner edge of the disc was represented in two different ways, resembling either a dead zone inner edge (DZ) or a disc inner boundary (IB). The main difference lies in the steepness of the surface density profile. The innermost planet always has a mass of 10 M Earth , with additional outer planets of equal or higher mass. A steeper profile is able to stop a chain of planets more efficiently than a shallower profile. The final configurations in our DZ models are usually tighter than in their IB counterparts, and therefore more prone to instability. We derive analytical expressions for the stopping conditions based on power equilibrium, and show that the final eccentricities result from torque equilibrium. For planets in thinner discs, we found, for the first time, clear signs for over-stable librations in the hydrodynamical simulations, leading to very compact systems. We also found that the popular N-body simulations may overestimate the number of planets in the disc inner cavity.Comment: 22 pages (including appendices), Accepted for publication in A &

The role of disc torques in forming resonant planetary systems

Context. The most accurate method for modelling planetary migration and hence the formation of resonant systems is using hydrodynamical simulations. Usually, the force (torque) acting on a planet is calculated using the forces from the gas disc and the star, while the gas accelerations are computed using the pressure gradient, the star, and the planet’s gravity, ignoring its own gravity. For a non-migrating planet the neglect of the disc gravity results in a consistent torque calculation while for a migrating case it is inconsistent. Aims. We aim to study how much this inconsistent torque calculation can affect the final configuration of a two-planet system. We focus on low-mass planets because most of the multi-planetary systems, discovered by the Kepler survey, have masses around ten Earth masses. Methods. Performing hydrodynamical simulations of planet–disc interaction, we measured the torques on non-migrating and migrating planets for various disc masses as well as density and temperature slopes with and without considering the self-gravity of the disc. Using this data, we found a relation that quantifies the inconsistency, used this relation in an N-body code, and performed an extended parameter study modelling the migration of a planetary system with different planet mass ratios and disc surface densities, to investigate the impact of the torque inconsistency on the architecture of the planetary system. Results. Not considering disc self-gravity produces an artificially larger torque on the migrating planet that can result in tighter planetary systems. The deviation of this torque from the correct value is larger in discs with steeper surface density profiles. Conclusions. In hydrodynamical modelling of multi-planetary systems, it is crucial to account for the torque correction, otherwise the results favour more packed systems. We examine two simple correction methods existing in the literature and show that they properly correct this problem

1:1 orbital resonance of circumbinary planets

The recent detection of the third planet in Kepler-47 has shown that binary stars can host several planets in circumbinary orbits. To understand the evolution of these systems we have performed two-dimensional hydrodynamic simulations of the circumbinary disc with two embedded planets for several Kepler systems. In two cases, Kepler-47 and -413, the planets are captured in a 1:1 mean-motion resonance at the planet “parking position” near the inner edge of the disc. The orbits are fully aligned and have mean eccentricities of about 0.25 to 0.30; the planets are entangled in a horseshoe-type motion. Subsequent n-body simulations without the disc show that the configurations are stable. Our results point to the existence of a new class of stable resonant orbits around binary stars. It remains to be seen if such orbits exist in reality

The formation of Jupiter by hybrid pebble–planetesimal accretion

The standard model for giant planet formation is based on the accretion of solids by a growing planetary embryo, followed by rapid gas accretion once the planet exceeds a so-called critical mass1. However, the dominant size of the accreted solids (‘pebbles’ of the order of centimetres or ‘planetesimals’ of the order of kilometres to hundreds of kilometres) is unknown1,2. Recently, high-precision measurements of isotopes in meteorites have provided evidence for the existence of two reservoirs of small bodies in the early Solar System3. These reservoirs remained separated from ~1 Myr until ~3 Myr after the Solar System started to form. This separation is interpreted as resulting from Jupiter growing and becoming a barrier for material transport. In this framework, Jupiter reached ~20 Earth masses (M⊕) within ~1 Myr and slowly grew to ~50 M⊕ in the subsequent 2 Myr before reaching its present-day mass3. The evidence that Jupiter’s growth slowed after reaching 20 M⊕ for at least 2 Myr is puzzling because a planet of this mass is expected to trigger fast runaway gas accretion4,5. Here, we use theoretical models to describe the conditions allowing for such a slow accretion and show that Jupiter grew in three distinct phases. First, rapid pebble accretion supplied the major part of Jupiter’s core mass. Second, slow planetesimal accretion provided the energy required to hinder runaway gas accretion during the 2 Myr. Third, runaway gas accretion proceeded. Both pebbles and planetesimals therefore play an important role in Jupiter’s formation