Aurélien Crida

On the width and shape of gaps in protoplanetary disks

Although it is well known that a massive planet opens a gap in a protoplanetary gaseous disk, there is no analytic description of the surface density profile in and near the gap. The simplest approach, which is based upon the balance between the torques due to the viscosity and the gravity of the planet and assumes local damping, leads to gaps with overestimated width, especially at low viscosity. Here, we take into account the fraction of the gravity torque that is evacuated by pressure supported waves. With a novel approach, which consists of following the fluid elements along their trajectories, we show that the flux of angular momentum carried by the waves corresponds to a pressure torque. The equilibrium profile of the disk is then set by the balance between gravity, viscous and pressure torques. We check that this balance is satisfied in numerical simulations, with a planet on a fixed circular orbit. We then use a reference numerical simulation to get an ansatz for the pressure torque, that yields gap profiles for any value of the disk viscosity, pressure scale height and planet to primary mass ratio. Those are in good agreement with profiles obtained in numerical simulations over a wide range of parameters. Finally, we provide a gap opening criterion that simultaneously involves the planet mass, the disk viscosity and the aspect ratio.

DISK SURFACE DENSITY TRANSITIONS AS PROTOPLANET TRAPS

The tidal torque exerted by a protoplanetary disk with power-law surface density and temperature profiles onto anembedded protoplanetary embryois generally anegative quantity that leads to the embryoinward migration. Here we investigate how the tidal torque balance is affected at a disk surface density radial jump. The jump has two consequences:(1)Itaffects thedifferential Lindbladtorque.Inparticular,ifthediskismerelyemptyontheinnerside,the differential Lindblad torque almost amounts to the large negative outer Lindblad torque. (2) It affects the corotation torque, which is a quantity very sensitive to the local gradient of the disk surface density. In particular, if the disk is depleted on the inside and the jump occurs radially over a few pressure scale heights, the corotation torque is a positive quantity that ismuchlargerthan inapower-lawdisk.Weshow bymeans ofcustomized numerical simulations of low-massplanetsembedded inprotoplanetarynebulaewithasurfacedensity jump that thesecond effect isdominant; that is, that the corotation torque largely dominates the differential Lindblad torque on the edge of a central depletion, even a shallow one. Namely, a disk surface density jump of about 50% over 3–5 disk thicknesses suffices to cancel out the total torque. As a consequence, the type I migration of low-mass objects reaching the jump should be halted, and all these objects should be trapped there provided some amount of dissipation is present in the disk to prevent the corotation torque saturation. As dissipation is provided by turbulence, which induces a jitter of the planet semimajor axis, we investigate under which conditions the trapping process overcomes the trend of turbulence to induce stochastic migration across the disk. We show that a cavity with a large outer to inner surface density ratio efficiently traps embryos from 1 to 15 M� , at any radius up to 5 AU from the central object, in a disk that has same surface density profile as the minimum mass solar nebula (MMSN). Shallow surface density transitions require light disks to efficiently trap embryos. In the case of the MMSN, this could happen in the very central parts (r < 0:03 AU). We discusswhereinaprotoplanetarydiskonecanexpectasurfacedensityjump.Thiseffectcouldconstituteasolutionto the well-known problem that the buildup of the first protogiant solid core in a disk takes much longer than its type I migration toward the central object. Subject headings: accretion, accretion disks — hydrodynamics — methods: numerical — planetary systems: formation — planetary systems: protoplanetary disks

Dynamics of the Giant Planets of the Solar System in the Gaseous Protoplanetary Disk and Their Relationship to the Current Orbital Architecture

We study the orbital evolution of the four giant planets of our solar system in a gas disk. Our investigation extends the previous works by Masset & Snellgrove and Morbidelli & Crida, which focused on the dynamics of the Jupiter-Saturn system. The only systems we found to reach a steady state are those in which the planets are locked in a quadruple mean-motion resonance (i.e., each planet is in resonance with its neighbor). In total, we found six such configurations. For the gas-disk parameters found in Morbidelli & Crida, these configurations are characterized by a negligible migration rate. After the disappearance of the gas, and in the absence of planetesimals, only two of these six configurations (the least compact ones) are stable for a time of hundreds of millions of years or more. The others become unstable on a timescale of a few Myr. Our preliminary simulations show that, when a planetesimal disk is added beyond the orbit of the outermost planet, the planets can evolve from the most stable of these configurations to their current orbits in a fashion qualitatively similar to that described in Tsiganis et al.

Migration of protoplanets in radiative discs

Context. In isothermal discs, the migration of protoplanets is directed inwards. For small planetary masses, the standard type I migration rates are so high that this can result in an unrealistic loss of planets into the stars. Aims. We investigate the planet-disc interaction in non-isothermal discs and analyse the magnitude and direction of migration for an extended range of planet masses. Methods. We performed detailed two-dimensional numerical simulations of embedded planets including heating/cooling effects as well as radiative diffusion for realistic opacities. Results. In radiative discs, small planets with M planet < 50 M Earth migrate outwards at a rate comparable to the absolute magnitude of standard type I migration. For larger masses, the migration is inwards and approaches the isothermal, type II migration rate. Conclusions. Our findings are particularly important for the first growth phase of planets and ease the problem of too rapid inward type I migration.

Cavity opening by a giant planet in a protoplanetary disc and effects on planetary migration

We study the effect of a Jovian planet on the gas distribution of a protoplanetary disc, using a new numerical scheme that allows us to take into consideration the global evolution of the disc, down to an arbitrarily small inner physical radius. We find that Jovian planets do not open cavities in the inner part of the disc (i.e. interior to their orbits) unless (a) the inner physical edge of the disc is close to the planet’s location or (b) the planet is much more massive than the disc. In all other cases the planet simply opens a gap in the gas density distribution, whose global profile is essentially unchanged relative to the one that it would have if the planet were absent. We recognize, though, that the dust distribution can be significantly different from the gas distribution and that dust cavities might be opened in some situations, even if the gas is still present in the inner part of the disc. Concerning the migration of the planet, we find that classical type II migration (with speed proportional to the viscosity of the disc) occurs only if the gap opened by the planet is deep and clean. If there is still a significant amount of gas in the gap, the migration of the planet is generally slower than the theoretical type II migration rate. In some situations, migration can be stopped or even reversed. We develop a simple model that reproduces satisfactorily the migration rate observed in the simulations, for a wide range of disc viscosities and planet masses and locations relative to the inner disc edge. Our results are relevant for extrasolar planetary systems, as they explain (a) why some hot Jupiters did not migrate all the way down to their parent stars and (b) why the outermost of a pair of resonant planets is typically the most massive one.

Stellar irradiated discs and implications on migration of embedded planets - I. Equilibrium discs

The strength and direction of migration of low mass planets depends on the discs thermodynamics. In discs where the viscous heating is balanced by radiative transport, the migration can be directed outwards, a process which extends the lifetime of growing planetary embryos. We investigate the influence of opacity and stellar irradiation on the disc thermodynamics. Utilizing the resulting disc structure, we determine the regions of outward migration. We perform two-dimensional numerical simulations of equilibrium discs with viscous heating, radiative cooling and stellar irradiation. We use the hydrodynamical code NIRVANA that includes a full tensor viscosity and stellar irradiation, as well as a two temperature solver that includes radiation transport in the flux-limited diffusion approximation. The migration is studied by using torque formulae. In the constant opacity case, we reproduce the analytical results of a black-body disc: the stellar irradiation dominates in the outer regions -- leading to flaring -- while the viscous heating dominates close to the star. We find that the inner edge of the disc should not be significantly puffed-up by the stellar irradiation. If the opacity depends on the local density and temperature, the structure of the disc is different, and several bumps in the aspect ratio H/r appear, due to transitions between different opacity regimes. The bumps in the disc can shield the outer disc from stellar irradiation. Stellar irradiation is an important factor for determining the disc structure and has dramatic consequences for the migration of embedded planets. Compared to discs with only viscous heating, a stellar irradiated disc features a much smaller region of outward migration for a range of planetary masses. This suggests that the region where the formation of giant planet cores takes place is smaller, which in turn might lead to a shorter growth phase.

Accretion of Saturn's mid-sized moons during the viscous spreading of young massive rings: Solving the paradox of silicate-poor rings versus silicate-rich moons

Abstract The origin of Saturn’s inner mid-sized moons (Mimas, Enceladus, Tethys, Dione and Rhea) and Saturn’s rings is debated. Charnoz et al. [Charnoz, S., Salmon J., Crida A., 2010. Nature 465, 752–754] introduced the idea that the smallest inner moons could form from the spreading of the rings’ edge while Salmon et al. [Salmon, J., Charnoz, S., Crida, A., Brahic, A., 2010. Icarus 209, 771–785] showed that the rings could have been initially massive, and so was the ring’s progenitor itself. One may wonder if the mid-sized moons may have formed also from the debris of a massive ring progenitor, as also suggested by Canup [Canup, R., 2010. Nature 468, 943–946]. However, the process driving mid-sized moon accretion from the icy debris disks has not been investigated in details. In particular, Canup’s (2010) model does not seem able to explain the varying silicate contents of the mid-sized moons (from 6% to 57% in mass). Here, we explore the formation of large objects from a massive ice-rich ring (a few times Rhea’s mass) and describe the fundamental properties and implications of this new process. Using a hybrid computer model, we show that accretion within massive icy rings can form all mid-sized moons from Mimas to Rhea. However in order to explain their current locations, intense dissipation within Saturn (with Q p

Influence of an inner disc on the orbital evolution of massive planets migrating in resonance

Context. The formation of resonant planet pairs in exoplanetary systems involves planetary migration inside the protoplanetary disc. After a resonant capture, subsequent migration leads to a large increase in planetary eccentricities, if no damping mechanism is applied. This has led to the conclusion that the migration of resonant planetary systems cannot take place across large radial distances, but must be terminated rapidly by disc dissipation. Aims. We investigate if the presence of an inner disc could supply eccentricity damping to the inner planet, and if this effect could explain observed eccentricities in some planetary systems. Methods. We compute hydrodynamic simulations of giant planets, in orbits of a given eccentricity about an inner gas disc, and measure the effect of the gas disc on the planetary orbital parameters. We perform detailed long-term calculations of the GJ 876 system. We complete N-body simulations, which include artificial forces on the planets that recreate the effect of the inner and outer discs. Results. We find that we cannot neglect the influence of the inner disc, and that the disc could explain the observed eccentricities. In particular, we reproduce the orbital parameters of a few systems engaged in 2:1 mean motion resonances: GJ 876, HD 73 526, HD 82 943, and HD 128 311. Analytically, we derive the effect that the inner disc should have on the inner planet to reach a specific orbital configuration, for any given damping effect of the outer disc on the outer planet. Conclusions. We conclude that an inner disc, even though difficult to model properly in hydrodynamical simulations, should be taken into account because of its damping effect on the eccentricity of the inner planet. By including this effect, we can explain quite naturally the observed orbital elements of the pairs of known resonant exoplanets.

Building giant-planet cores at a planet trap

Context. A well-known bottleneck for the core-accretion model of giant-planet formation is the loss of the cores into the star by type I migration, due to the tidal interactions with the gas disk. It has been shown that a steep surface-density gradient in the disk, such as the one expected at the boundary between an active and a dead zone, acts as a planet trap and prevents isolated cores from migrating down to the central star. Aims. We study the relevance of the planet trap concept for the accretion and evolution of systems of multiple planetary embryos/cores. Methods. We performed hydrodynamical simulations of the evolution of systems of multiple massive objects in the vicinity of a planet trap. The planetary embryos evolve in 3 dimensions, whereas the disk is modeled with a 2D grid. Synthetic forces are applied onto the embryos to mimic the damping effect that the disk has on their inclinations. Results. Systems with two embryos tend to acquire stable, separated and non-migrating orbits, with the more massive embryo placed at the planet trap and the lighter one farther out in the disk. Systems of multiple embryos are intrinsically unstable. Consequently, a long phase of mutual scattering can lead to accreting collisions among embryos; some embryos are injected into the inner part of the disk, where they can be evacuated into the star by type I migration. The system can resume a stable, non-migrating configuration only when the number of surviving embryos decreases to a small value (∼2-4). This can explain the limited number of giant planets in our solar system. These results should apply in general to any case in which the type I migration of the inner embryo is prevented by some mechanism, and not solely to the planet trap scenario.

Stellar irradiated discs and implications on migration of embedded planets - II. Accreting-discs

The strength and direction of migration of embedded low mass planets depends on the discs structure. It has been shown that, in discs with viscous heating and radiative transport, the migration can be directed outwards. In this paper we investigate the influence of a constant dM/dt-flux through the disc, as well as the influence of the discs metallicity on the discs thermodynamics. We focus on dM/dt discs, which have a net mass flux through them. Utilizing the resulting disc structure, we determine the regions of outward migration in the disc. We perform numerical hydrosimulations of dM/dt discs with viscous heating, radiative cooling and stellar irradiation in 2D in the r-z-plane. We use the explicit/implicit hydrodynamical code FARGOCA that includes a full tensor viscosity and stellar irradiation, as well as a two temperature solver that includes radiation transport in the flux-limited diffusion approximation. The migration of embedded planets is studied by using torque formulae. For a disc of gas surface density and viscosity, we find that the discs thermal structure depends on the product Sigma_G \nu and the amount of heavy elements, while the migration of planets additionally to the mentioned quantities, depends on the amount of viscosity itself. As a result of this, the disc structure can not be approximated by simple power laws. During the lifetime of the disc, the structure of the disc changes significantly in a non-linear way in the inner parts. In the late stages of the discs evolution, outward migration is only possible if the metallicity of the disc is high. For low metallicity, planets would migrate inwards and could potentially be lost. The presented disc structures and migration maps have important consequences on the formation of planets, as they can give hints on the different formation mechanisms for different types of planets as a function of metallicity.