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 results in 3D show that the dynamical corotation torque has the complete opposite effect and speeds up inward migration. Our numerical experiments implicate buoyancy resonances as the cause. These have two effects: (i) they exert a direct torque on the planet, whose magnitude relative to the Lindblad torque is measured in our simulations to be small; (ii) they torque the gas librating on horseshoe orbits in the corotation region and drive evolution of its vortensity, leading to the negative dynamical corotation torque. This indicates that at low turbulent viscosity, the detailed vertical thermal structure of the protoplanetary disc plays an important role in determining the migration behaviour of embedded planets. If this result holds up under a more refined treatment of disc thermal evolution, then it has important implications for understanding the formation and early evolution of planetary systems.
The dynamical corotation torque arising from the deformation of the horseshoe orbits, along with the vortensity gradient in the background disk, is important for determining orbital migration rate and direction of low-mass planets. Previous two-dimensional studies predicted that the dynamical corotation torque is positive, decelerating the inward planet migration. In contrast, recent three-dimensional studies have shown that buoyancy resonance makes the dynamical corotation torque negative, accelerating the inward migration. In this paper, we study the dependence of the dynamical corotation to…
The importance of the dynamical corotation torque for the migration of low-mass planets -- 1D analytical prescriptions verified by 2D hydrodynamical simulations
Planets in their formative years can migrate due to the influence of gravitational torques in the protoplanetary disk they inhabit. For low-mass planets in an isothermal disk, it is known that there is a strong negative torque on the planet due to its linear perturbation to the disk, causing fast inward migration. The current investigation demonstrates that in these same isothermal disks, for intermediate-mass planets, there is a strong positive nonlinear corotation torque due to the effects of gas being pulled through a gap on horseshoe orbits. For intermediate-mass planets, this positive tor…
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…
We study torques on migrating low-mass planets in locally isothermal discs. Previous work on low-mass planets generally kept the planet on a fixed orbit, after which the torque on the planet was measured. In addition to these static torques, when the planet is allowed to migrate it experiences dynamical torques, which are proportional to the migration rate and whose sign depends on the background vortensity gradient. We show that in discs a few times more massive than the Minimum Mass Solar Nebula, these dynamical torques can have a profound impact on planet migration. Inward migration can be …
I derive a fully analytic expression for the linear corotation torque to first order in eccentricity for planets in non-barotropic protoplanetary disks, taking into account the effect of disk entropy gradients. This torque formula is applicable to both the co-orbital corotation torque and the non co-orbital corotation torques -- for planets in orbits with non-zero eccentricity -- in disks where the thermal diffusivity and viscosity are sufficient to maintain linearity of these interactions. While the co-orbital corotation torque is important for migration of planets in Type I migration, the no…
We study the torque on low mass protoplanets on fixed circular orbits, embedded in a protoplanetary disc in the isothermal limit. For low mass protoplanets and large viscosity the corotation torque behaves as expected from linear theory. However, when the viscosity becomes small enough to enable horseshoe turns to occur, the linear corotation torque exists only temporarily after insertion of a planet into the disc, being replaced by the horseshoe drag first discussed by Ward. This happens after a time that is equal to the horseshoe libration period reduced by a factor amounting to about twice …
We study the effects of diffusion on the non-linear corotation torque, or horseshoe drag, in the two-dimensional limit, focusing on low-mass planets for which the width of the horseshoe region is much smaller than the scale height of the disc. In the absence of diffusion, the non-linear corotation torque saturates, leaving only the Lindblad torque. Diffusion of heat and momentum can act to sustain the corotation torque. In the limit of very strong diffusion, the linear corotation torque is recovered. For the case of thermal diffusion, this limit corresponds to having a locally isothermal equat…
Planetary migration is the process by which a forming planet undergoes a drift of its semi-major axis caused by the tidal interaction with its parent protoplanetary disc. One of the key quantities to assess the migration of embedded planets is the tidal torque between the disc and planet, which has two components: the Lindblad torque and the corotation torque. We review the latest results on both torque components for planets on circular orbits, with a special emphasis on the various processes that give rise to additional, large components of the corotation torque, and those contributing to th…
(Abridged) We present the results of N-body simulations of planetary systems formation in radiatively-inefficient disc models, where positive corotation torques may counter the rapid inward migration of low mass planets driven by Lindblad torques. The aim of this work is to examine the nature of planetary systems that arise from oligarchic growth in such discs. We adapt the commonly-used Mercury-6 symplectic integrator by including simple prescriptions for planetary migration (types I and II), planetary atmospheres that enhance the probability of planetesimal accretion by protoplanets, gas acc…
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…
Earth-mass planets embedded in gaseous protoplanetary disks undergo Type I orbital migration. In radiative disks an additional component of the corotation torque scaling with the entropy gradient across the horseshoe region can counteract the general inward migration, Type I migration can then be directed either inward or outward depending on the local disk properties. Thus, special locations exist in the disk toward which planets migrate in a convergent way. Here we present N-body simulations of the convergent migration of systems of low-mass (M=1-10 m_earth) planets. We show that planets do …
Recent developments in non-ideal magnetohydrodynamic simulations of protoplanetary disks suggest that instead of being traditional turbulent (viscous) accretion disks, they have a largely laminar flow with accretion driven by large-scale wind torques. These disks are possibly threaded by Hall-effect generated large-scale horizontal magnetic fields. We have examined the dynamics of the corotation region of a low mass planet embedded in such a disk and the evolution of the associated migration torque. These disks lack strong turbulence and associated turbulent diffusion, and the presence of a ma…
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…
Low-mass objects embedded in isothermal protoplanetary discs are known to suffer rapid inward Type I migration. In non-isothermal discs, recent work has shown that a decreasing radial profile of the disc entropy can lead to a strong positive corotation torque which can significantly slow down or reverse Type I migration in laminar viscous disc models. It is not clear however how this picture changes in turbulent disc models. The aim of this study is to examine the impact of turbulence on the torque experienced by a low-mass planet embedded in a non-isothermal protoplanetary disc. We particular…
The migration of planets on nearly circular, non-inclined orbits in protoplanetary discs is entirely described by the disc's torque. This torque is a complex function of the disc parameters, and essentially amounts to the sum of two components: the Lindblad torque and the corotation torque. Known torque formulae do not reproduce accurately the torque actually experienced in numerical simulations by low- and intermediate- mass planets in radiative discs. One of the main reasons for this inaccuracy is that these formulae have been worked out in two-dimensional analyses. Here we revisit the torqu…
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…
The migration of low-mass planets is driven by the differential Lindblad torque and the corotation torque in non-magnetic viscous models of protoplanetary discs. The corotation torque has recently received detailed attention as it may slow down, stall, or reverse migration. In laminar viscous disc models, the long-term evolution of the corotation torque is intimately related to viscous and thermal diffusion processes in the planet's horseshoe region. This paper examines the properties of the corotation torque in discs where MHD turbulence develops as a result of the magnetorotational instabili…
We study Type I migration of a planet in a radiatively efficient disk using global two dimensional hydrodynamic simulations. The large positive corotation torque is exerted on a planet by an adiabatic disk at early times when the disk has the steep negative entropy gradient. The gas on the horseshoe orbit of the planet is compressed adiabatically during the change of the orbit from the slow orbit to the fast orbit, increasing its density and exerting the positive torque on the planet. The planet would migrate outward in the adiabatic disk before saturation sets in. We further study the effect …
We present N-body simulations of planetary system formation in thermally-evolving, viscous disc models. The simulations incorporate type I migration (including corotation torques and their saturation), gap formation, type II migration, gas accretion onto planetary cores, and gas disc dispersal through photoevaporation. The aim is to examine whether or not the oligarchic growth scenario, when combined with self-consistent disc models and up-to-date prescriptions for disc-driven migration, can produce planetary systems similar to those that have been observed. The results correlate with the init…
Disc-driven planet migration is integral to the formation of planetary systems. In standard, gas-dominated protoplanetary discs, low-mass planets or planetary cores undergo rapid inwards migration and are lost to the central star. However, several recent studies indicate that the solid component in protoplanetary discs can have a significant dynamical effect on disc-planet interaction, especially when the solid-to-gas mass ratio approaches unity or larger and the dust-on-gas drag forces become significant. As there are several ways to raise the solid abundance in protoplanetary discs, for exam…
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…
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 analyze the physical processes of gap formation in an inviscid protoplanetary disk with an embedded protoplanet using two-dimensional local shearing-sheet model. Spiral density wave launched by the planet shocks and the angular momentum carried by the wave is transferred to the background flow. The exchange of the angular momentum can affect the mass flux in the vicinity of the planet to form an underdense region, or gap, around the planetary orbit. We first perform weakly non-linear analyses to show that the specific vorticity formed by shock dissipation of density wave can be a source of …
In the innermost regions of protoplanerary discs, the solid-to-gas ratio can be increased considerably by a number of processes, including photoevaporative and particle drift. MHD disc models also suggest the existence of a dead-zone at $R\lesssim 10$ AU, where the regions close to the midplane remain laminar. In this context, we use two-fluid hydrodynamical simulations to study the interaction between a low-mass planet ($\sim 1.7 \;{\rm M_\oplus}$) on a fixed orbit and an inviscid pebble-rich disc with solid-to-gas ratio $ε\ge 0.5$. For pebbles with Stokes numbers St=0.1, 0.5, multiple dusty …
[Abridged] The migration of low mass planets has been studied in hydrodynamical disc models for more than three decades, but the impact of a magnetic field in the protoplanetary disc is less known. When the disc's magnetic field is strong enough to prevent horseshoe motion, the corotation torque is replaced by a torque arising from magnetic resonances. For weak enough magnetic fields, horseshoe motion and a corotation torque exist, and recent turbulent MHD simulations have reported the existence of a new component of the corotation torque in the presence of a mean toroidal field. The aim of th…
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…
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 …
During their formation, emerging protoplanets tidally interact with their natal disks. Proto-gas-giant planets, with Hills radius 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 encounter predicament on the retention of long-period orbits for most gas giant planets. Moreover, numerical simulations indicate that planets migrations are not solely determined by…
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…
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…
Magnetically-driven disk winds would alter the surface density slope of gas in the inner region of a protoplanetary disk $(r \lesssim 1 {\rm au})$. This in turn affects planet formation. Recently, the effect of disk wind torque has been considered with the suggestion that it would carve out the surface density of the disk from inside and would induce global gas flows (wind-driven accretion). We aim to investigate effects of global gas flows on type I migration and also examine planet formation. A simplified approach was taken to address this issue, and N-body simulations with isolation-mass pl…
We evaluate the horseshoe drag exerted on a low-mass planet embedded in a gaseous disk, assuming the disk's flow in the coorbital region to be adiabatic. We restrict this analysis to the case of a planet on a circular orbit, and we assume a steady flow in the corotating frame. We also assume that the corotational flow upstream of the U-turns is unperturbed, so that we discard saturation effects. In addition to the classical expression for the horseshoe drag in barotropic disks, which features the vortensity gradient across corotation, we find an additional term which scales with the entropy gr…
We present the results of high resolution 2D simulations of low mass planets on fixed eccentric orbits embedded in protoplanetary discs. The aim of this study is to determine how the strength of the sustained, non-linear corotation torque experienced by embedded planets varies as a function of orbital eccentricity, disc parameters, and planetary mass. In agreement with previous work we find that the corotation torque diminishes as orbital eccentricity, $e$, increases. Analysis of the time-averaged streamlines in the disc demonstrates that the width of the horseshoe region narrows as the eccent…
We study the torque on low-mass planets embedded in protoplanetary discs in the two-dimensional approximation, incorporating non-isothermal effects. We couple linear estimates of the Lindblad (or wave) torque to a simple, but non-linear, model of adiabatic corotation torques (or horseshoe drag), resulting in a simple formula that governs Type I migration in non-isothermal discs. This formula should apply in optically thick regions of the disc, where viscous and thermal diffusion act to keep the horseshoe drag unsaturated. We check this formula against numerical hydrodynamical simulations, usin…
Inward migration of low-mass planets and embryos of giant planets can be stopped at the disc-cavity boundaries due to co-orbital corotation torque. We performed the first global three-dimensional (3D) simulations of planet migration at the disc-cavity boundary, and have shown that the boundary is a robust trap for low-mass planets and embryos. A protoplanetary disc may have several such trapping regions at various distances from the star, such as at the edge of the stellar magnetosphere, the inner edge of the dead zone, the dust-sublimation radius and the snow lines. Corotation traps located a…
We give an expression for the Lindblad torque acting on a low-mass planet embedded in a protoplanetary disk that is valid even at locations where the surface density or temperature profile cannot be approximated by a power law, such as an opacity transition. At such locations, the Lindblad torque is known to suffer strong deviation from its standard value, with potentially important implications for type I migration, but the full treatment of the tidal interaction is cumbersome and not well suited to models of planetary population synthesis. The expression that we propose retains the simplicit…
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 …
The majority of gas giants (planets of masses $\gtrsim10^2 M_\oplus$) are found to reside at distances beyond $\sim1$ au from their host stars. Within 1 au, the planetary population is dominated by super-Earths of $2-20 M_\oplus$. We show that this dichotomy between inner super-Earths and outer gas giants can be naturally explained should they form in nearly inviscid disks. In laminar disks, a planet can more easily repel disk gas away from its orbit. The feedback torque from the pile-up of gas inside the planet's orbit slows down and eventually halts migration. A pressure bump outside the pla…
We investigate the unsaturated horseshoe drag exerted on a low-mass planet by an isothermal gaseous disk. In the globally isothermal case, we use a formal- ism, based on the use of a Bernoulli invariant, that takes into account pressure effects, and that extends the torque estimate to a region wider than the horse- shoe region. We find a result that is strictly identical to the standard horseshoe drag. This shows that the horseshoe drag accounts for the torque of the whole corotation region, and not only of the horseshoe region, thereby deserving to be called corotation torque. We find that ev…
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 …
We present two-dimensional hydrodynamical simulations of pairs of planets migrating simultaneously in the Type I regime in a protoplanetary disc. Convergent migration naturally leads to the trapping of these planets in mean-motion resonances. Once in resonance the planets' eccentricity grows rapidly, and disc-planet torques cause the planets to escape resonance on a time-scale of a few hundred orbits. The effect is more pronounced in highly viscous discs, but operates efficiently even in inviscid discs. We attribute this resonance-breaking to overstable librations driven by moderate eccentrici…
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…
A number of giant planet pairs discovered by the radial velocity method with period ratios $\lesssim 2$ 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 proto-planetary disk should not only capture into mean motion resonances but also end up in a specific dynamical configuration within the resonance, sometimes referred to as apsidal corotation resonance (ACR). H…
As planets form and grow within gaseous protoplanetary disks, the mutual gravitational interaction between the disk and planet leads to the exchange of angular momentum, and migration of the planet. We review current understanding of disk-planet interactions, focussing in particular on physical processes that determine the speed and direction of migration. We describe the evolution of low mass planets embedded in protoplanetary disks, and examine the influence of Lindblad and corotation torques as a function of the disk properties. The role of the disk in causing the evolution of eccentricitie…
3D MHD simulations of planet migration in cavities and inner discs of magnetized stars
Cornell University（康奈尔大学）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Keldysh Institute for Applied Mathematics（凯尔迪什应用数学研究所）；Boston University（波士顿大学）
As planets form they tidally interact with their natal disks. Though the tidal perturbation induced by Earth and super-Earth mass planets is generally too weak to significantly modify the structure of the disk, the interaction is potentially strong enough to cause the planets to undergo rapid type I migration. This physical process may provide a source of short-period super-Earths, though it may also pose a challenge to the emergence and retention of cores on long-period orbits with sufficient mass to evolve into gas giants. Previous numerical simulations have shown that the type I migration r…
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…
Context: Several recent studies have found that planet migration in adiabatic discs differs significantly from migration in isothermal discs. Depending on the thermodynamic conditions, i.e., the effectiveness of radiative cooling, and the radial surface density profile, planets migrate inward or outward. Clearly, this will influence the semimajor axis - mass distribution of planets as predicted by population synthesis simulations. Aims: Our goal is to study the global effects of radiative cooling, viscous torque desaturation and gap opening as well as stellar irradiation on the tidal migration…
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…
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 obs…
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 …
