When considering the migration of Jupiter and Saturn, a classical result is to find the planets migrating outwards and locked in the 3:2 mean motion resonance (MMR). These results were obtained in the framework of viscously accreting discs, in which the observed stellar accretion rates constrained the viscosity values. However, it has recently been shown observationally and theoretically that discs are probably less viscous than previously thought. Therefore, in this paper, we explore the dynamics of pairs of giant planets in low-viscosity discs. We performed two-dimensional hydrodynamical simulations using the grid-based code FARGOCA. In contrast to classical viscous discs, we find that the outer planet never crosses the 2:1 resonance and the pair does not migrate outwards. After a wide parameter exploration, including the mass of the outer planet, we find that the planets are primarily locked in the 2:1 MMR and in some cases in the 5:2 MMR. We explain semi-analytically why it is not possible for the outer planet to cross the 2:1 MMR in a low-viscosity disc. We find that pairs of giant planets migrate inwards in low-viscosity discs. Although, in some cases, having a pair of giant planets can slow down the migration speed with respect to a single planet. Such pairs of slowly migrating planets may be located, at the end of the disc phase, in the population of exoplanets of 'warm Jupiters'. However, the planets never migrate outwards. These results could have strong implications on the Solar System's formation scenarios if the Sun's protoplanetary disc had a low viscosity.
Type-II migration of giant planets has a speed proportional to the disc's viscosity for values of the alpha viscosity parameter larger than 1.e-4 . At lower viscosities previous studies, based on 2D simulations have shown that migration can be very chaotic and often characterized by phases of fast migration. The reason is that in low-viscosity discs vortices appear due to the Rossby-wave instability at the edges of the gap opened by the planet. Migration is then determined by vortex-planet interactions. Our aim is to study migration in low viscosity 3D discs. We performed numerical simulations…
The strength and direction of migration of low mass embedded planets depends on the disk's thermodynamic state, where the internal dissipation is balanced by radiative transport, and the migration can be directed outwards, a process which extends the lifetime of growing embryos. Very important parameters determining the structure of disks, and hence the direction of migration, are the viscosity and the adiabatic index. In this paper we investigate the influence of different viscosity prescriptions (alpha-type and constant) and adiabatic indices on disk structures and how this affects the migra…
The migration strength and direction of embedded low-mass planets depends on the disc structure. In discs with an efficient radiative transport, the migration can be directed outwards for planets with more than 3-5 Earth masses. This is due to the entropy driven corotation torque, a process that extends the lifetimes of growing planetary embryos. We investigate the influence on the disc structure caused by a jump in the alpha parameter of the viscosity to model a dead-zone structure in the disc. We focus on M-dot discs, which have a constant net mass flux. Using the resulting disc structure, w…
We present the highest resolution study to date of super-Earths migrating in inviscid and low-viscosity discs, motivated by the connection to laminar, wind-driven models of protoplanetary discs. Our models unveil the critical role of vortices in determining the migration behaviour for partial gap-opening planets. Vortices form in pressure maxima at gap edges, and prevent the disc-feedback stopping of migration for intermediate planets in low-viscosity and inviscid discs, contrary to the concept of the `inertial limit' or `disc feedback' halting predicted from analytical models. Vortices may al…
We study the type III migration of a Saturn mass planet in low viscosity discs. The planet is found to experience cyclic episodes of rapid decay in orbital radius, each amounting to a few Hill radii. We find this to be due to the scattering of large- scale vortices present in the disc. The origin and role of vortices in the context of type III migration is explored. It is shown through numerical simulations and semi- analytical modelling that spiral shocks induced by a sufficiently massive planet will extend close to the planet orbital radius. The production of vortensity across shock tips res…
Context. Giant planets open gaps in their protoplanetary and subsequently suffer so-called type II migration. Schematically, planets are thought to be tightly locked within their surrounding disks, and forced to follow the viscous advection of gas onto the central star. This fundamental principle however has recently been questioned, as migrating planets were shown to decouple from the gas' radial drift. Aims. In this framework, we question whether the traditionally used linear scaling of migration rate of a giant planet with the disk's viscosity still holds. Additionally, we assess the role o…
Overcoming type I migration and preventing low-mass planets from spiralling into the central star is a long-studied topic. It is well known that outward migration is possible in viscous-heated discs relatively close to the central star because the entropy gradient can be sufficiently steep that the positive corotation torque overcomes the negative Lindblad torque. Yet efficiently trapping planets in this region remains elusive. Here we study disc conditions that yield outward migration for low-mass planets under specific planet migration prescriptions. Using a steady-state disc model with a co…
Towards a global model for planet formation in layered MHD wind-driven discs: A population synthesis approach to investigate the impact of low viscosity and accretion layer thickness
Center for Space and Habitability, University of Bern（伯尔尼大学太空与宜居性中心）；Division of Space Research and Planetary Sciences, Physics Institute, University of Bern（伯尔尼大学物理研究所空间研究与行星科学系）
Migration of giant planets in discs with low viscosity has been studied recently. The proportionality between migration speed and the disc's viscosity is broken by the presence of vortices that appear at the edges of the planet-induced gap. Our goal is to investigate vortex-driven migration in low-viscosity discs in the presence of radial advection of gas, as expected from angular momentum removal by magnetised disc winds. We performed three dimensional simulations using the grid-based code FARGOCA. We mimicked the effects of a disc wind by applying a synthetic torque on a surface layer of the…
Motivated by models suggesting that the inner planet forming regions of protoplanetary discs are predominantly lacking in viscosity-inducing turbulence, and are possibly threaded by Hall-effect generated large-scale horizontal magnetic fields, we examine the dynamics of the corotation region of a low-mass planet in such an environment. The corotation torque in an inviscid, isothermal, dead zone ought to saturate, with the libration region becoming both symmetrical and of a uniform vortensity, leading to fast inward migration driven by the Lindblad torques alone. However, in such a low viscosit…
The importance of the dynamical corotation torque for the migration of low-mass planets -- 1D analytical prescriptions verified by 2D hydrodynamical simulations
The strength and direction of migration of embedded low mass planets depends on the disc's 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 disc's metallicity on the disc's 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 …
The migration of planetary cores embedded in a protoplanetary disk is an important mechanism within planet-formation theory, relevant for the architecture of planetary systems. Consequently, planet migration is actively discussed, yet often results of independent theoretical or numerical studies are unconstrained due to the lack of observational diagnostics designed in light of planet migration. In this work we follow the idea of inferring the migration behavior of embedded planets by means of the characteristic radial structures that they imprint in the disk's dust density distribution. We ru…
ALMA observations of dust ring/gap structures in a minority but growing sample of protoplanetary disks can be explained by the presence of planets at large disk radii - yet the origins of these planets remains debated. We perform planet formation simulations using a semi-analytic model of the HL Tau disk to follow the growth and migration of hundreds of planetary embryos initially distributed throughout the disk, assuming either a high or low turbulent $α$ viscosity. We have discovered that there is a bifurcation in the migration history of forming planets as a consequence of varying the disk …
We carry out 2-D high resolution numerical simulations of type I planet migration with different disk viscosities. We find that the planet migration is strongly dependent on disk viscosities. Two kinds of density wave damping mechanisms are discussed. Accordingly, the angular momentum transport can be either viscosity dominated or shock dominated, depending on the disk viscosities. The long term migration behavior is different as well. Influences of the Rossby vortex instability on planet migration are also discussed. In addition, we investigate very weak shock generation in inviscid disks by …
In the classical core-accretion planet formation scenario, rapid inward migration and accretion timescales of kilometer size planetesimals may not favor the formation of massive cores of giant planets before the dissipation of protoplanetary disks. On the other hand, the existence of pressure maxima in the disk could act as migration traps and locations for solid material accumulation, favoring the formation of massive cores. We aim to study the radial drift of pebbles and planetesimals and planet migration at pressure maxima in a protoplanetary disk and their implications for the formation of…
In this paper we analyse giant gap-opening planet migration in protoplanetary discs, focusing on the type II migration regime. According to standard type II theory, planets migrate at the same rate as the gas in the disc, as they are coupled to the disc viscous evolution; however, recent studies questioned this paradigm, suggesting that planets migrate faster than the disc material. We study the problem through 2D long-time simulations of systems consistent with type II regime, using the hydrodynamical grid code FARGO3D. Even though our simulations confirm the presence of an initial phase char…
We investigate the migration of a low-mass ($\lesssim 10 M_\oplus$) planet near the inner edge of a protoplanetary disc using two-dimensional viscous hydrodynamics simulations. We employ an inner boundary condition representing the truncation of the disc at the stellar corotation radius. As described by Tsang (2011), wave reflection at the inner disc boundary modifies the Type I migration torque on the planet, allowing migration to be halted before the planet reaches the inner edge of the disc. For low-viscosity discs ($α\lesssim 10^{-3}$), planets may be trapped with semi-major axes as large …
Concurrent Accretion and Migration of Giant Planets in their Natal Disks with Consistent Accretion Torque (II): Parameter Survey and Condition for Outward Migration
The outward migration of a pair of resonant-orbit planets, driven by tidal interactions with a gas-dominated disk, is studied in the context of evolved solar nebula models. The planets' masses, M1 and M2, correspond to those of Jupiter and Saturn. Hydrodynamical calculations in two and three dimensions are used to quantify the migration rates and analyze the conditions under which the outward migration mechanism may operate. The planets are taken to be fully formed after 1e6 and before 3e6 years. The orbital evolution of the planets in an evolving disk is then calculated until the disk's gas i…
We performed 3D MHD simulations of planet migration in stratified disks using the Godunov code PLUTO, where the disk is turbulent due to the magnetorotational instability. We study the migration for planets with different planet-star mass ratios $q=M_{p}/M_{s}$. In agreement with previous studies, for the low-mass planet cases ($q=5\times10^{-6}$ and $10^{-5}$), migration is dominated by random fluctuations in the torque. For a Jupiter-mass planet $(q=M_{p}/M_{s}=10^{-3}$ for $M_{s}=1M_{\odot})$, we find a reduction of the magnetic stress inside the orbit of the planet and around the gap regio…
Two longstanding problems in planet formation include (1) understanding how planets survive migration, and (2) articulating the process by which protoplanetary disks disperse---and in particular how they accrete onto their central stars. We can go a long way toward solving both problems if the disk gas surrounding planets has no intrinsic diffusivity ("viscosity"). In inviscid, laminar disks, a planet readily repels gas away from its orbit. On short timescales, zero viscosity gas accumulates inside a planet's orbit to slow Type I migration by orders of magnitude. On longer timescales, multiple…
[Abridged] The occurrence rate of observed sub-Neptunes has a break at 0.1 au, which is often attributed to a migration trap at the inner rim of protoplanetary disks where a positive co-rotation torque prevents inward migration. We argue that conditions in inner disk regions are such that sub-Neptunes are likely to open gaps, lose the support of the co-rotation torque as their co-rotation regions become depleted, and the trapping efficiency then becomes uncertain. We study what it takes to trap such gap-opening planets at the inner disk rim. We performed 2D locally isothermal and non-isotherma…
We examine the migration of low mass planets in laminar protoplanetary discs, threaded by large scale magnetic fields in the dead zone that drive radial gas flows. As shown in Paper I, a dynamical corotation torque arises due to the flow-induced asymmetric distortion of the corotation region and the evolving vortensity contrast between the librating horseshoe material and background disc flow. Using simulations of laminar torqued discs containing migrating planets, we demonstrate the existence of the four distinct migration regimes predicted in Paper I. In two regimes, the migration is approxi…
Embedded in the gaseous protoplanetary disk, Jupiter and Saturn naturally become trapped in 3:2 resonance and migrate outward. This serves as the basis of the Grand Tack model. However, previous hydrodynamical simulations were restricted to isothermal disks, with moderate aspect ratio and viscosity. Here we simulate the orbital evolution of the gas giants in disks with viscous heating and radiative cooling. We find that Jupiter and Saturn migrate outward in 3:2 resonance in modest-mass ($M_{disk} \approx M_{MMSN}$, where MMSN is the "minimum-mass solar nebula") disks with viscous stress parame…
Recent simulations show that giant planets of about one Jupiter mass migrate inward at a rate that differs from the Type II prediction. Here we show that at higher masses, planets migrate outward. Our result differs from previous ones because of our longer simulation times, lower viscosity, and our boundary conditions that allow the disk to reach viscous steady state. We show that, for planets on circular orbits, the transition from inward to outward migration coincides with the known transition from circular to eccentric disks that occurs for planets more massive than a few Jupiters. In an ec…
The planet migration due to the disk--planet interaction is one of the most important processes to determine the architecture of planetary systems. A sufficiently massive planet forms a density gap and migrates together with the gap. By carrying out two-dimensional and two-fluid (gas and dust grains) hydrodynamic simulations, we investigated the effects of the dust feedback on the migration of the gap-opening planet, which was not considered in previous studies. We found that the gas surface density at the outer edge of the gap becomes smaller due to the dust feedback, and thus the torque exer…
Massive planets that open a gap in the accretion disk are believed to migrate with exactly the viscous speed of the disk, a regime termed type II migration. Population synthesis models indicate that standard type II migration is too rapid to be in agreement with the observations. We study the migration of massive planets between $2\times10^{-4}$ and $2\times10^{-3} M_\odot$ corresponding to 0.2 to 2 Jupiter masses $M_J$. in order to estimate the migration rate in comparison to type II migration. We follow the evolution of planets embedded in two-dimensional, locally isothermal disks with non-z…
We study the torque acting on a planet embedded in an optically thick accretion disc, using global two-dimensional hydrodynamic simulations. The temperature of an optically thick accretion disc is determined by the energy balance between the viscous heating and the radiative cooling. The radiative cooling rate depends on the opacity of the disc. The opacity is expressed as a function of the temperature. We find the disc is divided into three regions that have different temperature distributions. The slope of the entropy distribution becomes steep in the inner region of the disc with the high t…
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 radiat…
In order to study the origin of the architectures of low mass planetary systems, we perform numerical surveys of the evolution of pairs of coplanar planets in the mass range $(1-4)\ \rmn{M}_{\oplus}.$ These evolve for up to $2\times10^7 \rmn{yr}$ under a range of orbital migration torques and circularization rates assumed to arise through interaction with a protoplanetary disc. Near the inner disc boundary, significant variations of viscosity, interaction with density waves or with the stellar magnetic field could occur and halt migration, but allow ircularization to continue. This was modelle…
The final orbital position of growing planets is determined by their migration speed, which is essentially set by the planetary mass. Small mass planets migrate in type I migration, while more massive planets migrate in type II migration, which is thought to depend mostly on the viscous evolution rate of the disc. A planet is most vulnerable to inward migration before it reaches type II migration and can lose a significant fraction of its semi-major axis at this stage. We investigated the influence of different disc viscosities, the dynamical torque and gas accretion from within the horseshoe …
A planet is formed within a protoplanetary disk. Recent observations have revealed substructures such as gaps and rings, which may indicate forming planets within the disk. Due to disk--planet interaction, the planet migrates within the disk, which can affect a shape of the planet-induced gap. In this paper, we investigate effects of fast inward migration of the planet on the gap shape, by carrying out hydrodynamic simulations. We found that when the migration timescale is shorter than the timescale of the gap-opening, the orbital radius is shifted inward as compared to the radial location of …
We present simulations of low-mass planet--disc interactions in inviscid three-dimensional discs. We show that a wind-driven laminar accretion flow through the surface layers of the disc does not significantly modify the migration torque experienced by embedded planets. More importantly, we find that 3D effects lead to a dramatic change in the behaviour of the dynamical corotation torque compared to earlier 2D theory and simulations. Although it was previously shown that the dynamical corotation torque could act to slow and essentially stall the inward migration of a low-mass planet, our resul…
A key process in planet formation is the exchange of angular momentum between a growing planet and the protoplanetary disc, which makes the planet migrate through the disc. Several works show that in general low-mass and intermediate-mass planets migrate towards the central star, unless corotation torques become dominant. Recently, a new kind of torque, called the thermal torque, was proposed as a new source that can generate outward migration of low-mass planets. While the Lindblad and corotation torques depend mostly on the properties of the protoplanetary disc and on the planet mass, the th…
We study the stability of gaps opened by a giant planet in a self-gravitating protoplanetary disc. We find a linear instability associated with both the self-gravity of the disc and local vortensity maxima which coincide with gap edges. For our models, these edge modes develop and extend to twice the orbital radius of a Saturn mass planet in discs with disc-to-star mass ratio >0.06, corresponding to a Toomre Q < 1.5 at the outer disc boundary. Unlike the local vortex-forming instabilities associated with gap edges in weakly or non-self-gravitating low viscosity discs, the edge modes are global…
Orbital mean motion resonances in planetary systems originate from dissipative processes in disk-planet interactions that lead to orbital migration. In multi-planet systems that host giant planets, the perturbation of the protoplanetary disk strongly affects the migration of companion planets. By studying the well-characterized resonant planetary system around GJ 876 we aim to explore which effects shape disk-driven migration in such a multi-planet system to form resonant chains. We modelled the orbital migration of three planets embedded in a protoplanetary disk using two-dimensional locally …
We investigate the effects of viscosity on disk-planet interaction and discuss how type I migration of planets is modified. We have performed a linear calculation using shearing-sheet approximation and obtained the detailed, high resolution density structure around the planet embedded in a viscous disk with a wide range of viscous coefficients. We use a time-dependent formalism that is useful in investigating the effects of various physical processes on disk-planet interaction. We find that the density structure in the vicinity of the planet is modified and the main contribution to the torque …
We herein develop a new simple model for giant planet formation, which predicts the final mass of a giant planet born in a given disk, by adding the disk mass loss due to photoevaporation and a new type II migration formula to our previous model. The proposed model provides some interesting results. First, it gives universal evolution tracks in the diagram of planetary mass and orbital radius, which clarifies how giant planets migrate at growth in the runaway gas accretion stage. Giant planets with a few Jupiter masses or less suffer only a slight radial migration in the runaway gas accretion …
The strength and direction of migration of low mass planets depends on the disc's 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…
We study the possibility of tidal dissipation in the solid cores of giant planets and its implication for the formation of hot Jupiters through high-eccentricity migration. We present a general framework by which the tidal evolution of planetary systems can be computed for any form of tidal dissipation, characterized by the imaginary part of the complex tidal Love number, ${\rm Im}[{\tilde k}_2(ω)]$, as a function of the forcing frequency $ω$. Using the simplest viscoelastic dissipation model (the Maxwell model) for the rocky core and including the effect of a nondissipative fluid envelope, we…
We investigate how planets interact with viscous accretion disks, in the limit that the disk is sufficiently low mass that the planet migrates more slowly than the disk material. In that case, the disk's surface density profile is determined by the disk being in viscous steady state (VSS), while overflowing the planet's orbit. We compute the VSS profiles with 2D hydrodynamical simulations, and show that disk material piles up behind the planet, with the planet effectively acting as a leaky dam. Previous 2D hydrodynamical simulations missed the pileup effect because of incorrect boundary condit…
Context. The origin of giant planets at moderate separations $\simeq$$1$$-$$10$ au is still not fully understood because numerical studies of Type II migration in protoplanetary disks often predict a decay of the semi-major axis that is too fast. According to recent 2D simulations, inward migration of a gap-opening planet can be slowed down or even reversed if the outer gap edge becomes heated by irradiation from the central star, and puffed up. Aims. Here we study how stellar irradiation reduces the disk-driven torque and affects migration in more realistic 3D disks. Methods. Using 3D hydrody…
Convergent migration involving multiple planets embedded in a viscous protoplanetary disc is expected to produce a chain of planets in mean motion resonances, 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 mean motion resonances, 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 di…
Several white dwarfs with atmospheric metal pollution have been found to host small planetary bodies (planetesimals) orbiting near the tidal disruption radius. We study the physical properties and dynamical origin of these bodies under the hypothesis that they underwent high-eccentricity migration from initial distances of several astronomical units. We examine two plausible mechanisms for orbital migration and circularization: tidal friction and ram-pressure drag in a compact disc. For each mechanism, we derive general analytic expressions for the evolution of the orbit that can be rescaled f…
In this paper the migration of a 10 Earth-mass planetary core is investigated at the outer boundary of the dead zone of a protoplanetary disc by means of 2D hydrodynamic simulations done with the graphics processor unit version of the FARGO code. In the dead zone, the effective viscosity is greatly reduced due to the disc self-shielding against stellar UV radiation, X-rays from the stellar magnetosphere and interstellar cosmic rays. As a consequence, mass accumulation occurs near the outer dead zone edge, which is assumed to trap planetary cores enhancing the efficiency of the core-accretion s…
The Kepler space mission discovered about a dozen planets orbiting around binary stars systems. Most of these circumbinary planets lie near their instability boundaries at about 3 to 5 binary separations. Past attempts to match these final locations through an inward migration process were only successful for the Kepler-16 system. Here, we study 10 circumbinary systems and try to match the final parking locations and orbital parameters of the planets with a disc driven migration scenario. We performed 2D locally isothermal hydrodynamical simulations of circumbinary discs with embedded planets …
The migration of planets plays an important role in the early planet-formation process. An important problem has been that standard migration theories predict very rapid inward migration, which poses problems for population synthesis models. However, it has been shown recently that low-mass planets (20-30 Earth Masses) that are still embedded in the protoplanetary disc can migrate outwards under certain conditions. Simulations have been performed mostly for planets at given radii for a particular disc model. Here, we plan to extend previous work and consider different masses of the disc to qua…
We perform three-dimensional self-gravitating radiative transfer simulations of protoplanet migration in circumstellar discs to explore the impact upon migration of the radial temperature profiles in these discs. We model protoplanets with masses ranging between 10-100 M\bigoplus, in discs with surface density profiles of Σ \varpropto r^-1/2, and temperature profiles of the form T \varpropto r^-β, where β ranges 0-2. We find that steep (β > 1) temperature profiles lead to outward migration of low mass protoplanets in interstellar grain opacity discs, but in more optically thin discs the migrat…
We present the results of our recent study on the interactions between a giant planet and a self-gravitating gas disk. We investigate how the disk's self-gravity affects the gap formation process and the migration of the giant planet. Two series of 1-D and 2-D hydrodynamic simulations are performed. We select several surface densities and focus on the gravitationally stable region. To obtain more reliable gravity torques exerted on the planet, a refined treatment of disk's gravity is adopted in the vicinity of the planet. Our results indicate that the net effect of the disk's self-gravity on t…
Origins of Compact Mean-Motion Resonances: Evidence for Long-Range Migration and the Case of Kepler-36
California Institute of Technology（加州理工学院）；Laboratoire Lagrange, Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS（拉格朗日实验室，蔚蓝海岸大学，蔚蓝海岸天文台，法国国家科学研究中心）；Collège de France, Université PSL（法兰西公学院，巴黎文理研究大学）
