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 equation of state. We present some simple models that are able to capture the dependence of the torque on diffusive processes to within 20% of the numerical simulations.
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…
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…
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…
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…
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…
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 …
We determine an expression for the Type I planet migration torque involving a locally isothermal disk, with moderate turbulent viscosity (~0.0005 < alpha < ~0.05), based on three-dimensional nonlinear hydrodynamical simulations. The radial gradients (in a dimensionless logarithmic form) of density and temperature are assumed to be constant near the planet. We find that the torque is roughly equally sensitive to the surface density and temperature radial gradients. Both gradients contribute to inward migration when they are negative. Our results indicate that two-dimensional calculations with a…
Population synthesis studies into planet formation have suggested that distributions consistent with observations can only be reproduced if the actual Type I migration timescale is at least an order of magnitude longer than that deduced from linear theories. Although past studies considered the effect of the Type I migration of protoplanetary embryos, in most cases they used a conventional formula based on static torques in isothermal disks, and employed a reduction factor to account for uncertainty in the mechanism details. However, in addition to static torques, a migrating planet experience…
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 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…
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 importance of the dynamical corotation torque for the migration of low-mass planets -- 1D analytical prescriptions verified by 2D hydrodynamical simulations
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…
Fast inward migration of planetary cores is a common problem in the current planet formation paradigm. Even though dust is ubiquitous in protoplanetary disks, its dynamical role in the migration history of planetary embryos has not been assessed. In this Letter, we show that the scattered pebble-flow induced by a low-mass planetary embryo leads to an asymmetric dust-density distribution that is able to exert a net torque. By analyzing a large suite of multifluid hydrodynamical simulations addressing the interaction between the disk and a low-mass planet on a fixed circular orbit, and neglectin…
We provide torque formulae for low mass planets undergoing type I migration in gaseous disks. These torque formulae put special emphasis on the horseshoe drag, which is prone to saturation: the asymptotic value reached by the horseshoe drag depends on a balance between coorbital dynamics (which tends to cancel out or saturate the torque) and diffusive processes (which tend to restore the unperturbed disk profiles, thereby desaturating the torque). We entertain here the question of this asymptotic value, and we derive torque formulae which give the total torque as a function of the disk's visco…
Quantifying the Impact of the Dust Torque on the Migration of Low-mass Planets II: The Role of Pebble Accretion in Planet Growth within a Global Planet Formation Model
Planets migrate due to the recoil they experience from scattering solid (planetesimal) bodies. To first order, the torques exerted by the interior and exterior disks cancel, analogous to the cancellation of the torques from the gravitational interaction with the gas (type I migration). Assuming the dispersion-dominated regime and power-laws characterized by indices α and β for the surface density and eccentricity profiles, we calculate the net torque on the planet. We consider both distant encounters and close (orbit-crossing) encounters. We find that the close and distant encounter torques ha…
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…
As planets grow the exchange of angular momentum with the gaseous component of the protoplanetary disc produces a net torque resulting in a variation of the semi-major axis of the planet. For low-mass planets not able to open a gap in the gaseous disc this regime is known as type I migration. Pioneer works studied this mechanism in isothermal discs finding fast inward type I migration rates that were unable to reproduce the observed properties of extrasolar planets. In the last years, several improvements have been made in order to extend the study of type I migration rates to non-isothermal d…
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 present the results of hydrodynamical simulations of the orbital evolution of planets undergoing runaway gas accretion in radiative discs. We consider accreting disc models with constant mass flux through the disc, and where radiative cooling balances the effect of viscous heating and stellar irradiation. We assume that 20-30 $M_\oplus$ giant planet cores are formed in the region where viscous heating dominates and migrate outward under the action of a strong corotation torque. In the case where gas accretion is neglected, we find evidence for strong dynamical torques in accreting discs wit…
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…
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…
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…
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 …
Earth-mass bodies are expected to undergo Type I migration directed either inward or outward depending on the thermodynamical state of the protoplanetary disc. Zones of convergent migration exist where the Type I torque cancels out. We study the evolution of multiple protoplanets of a few Earth masses embedded in a non-isothermal protoplanetary disc. The protoplanets are located in the vicinity of a convergence zone located at the transition between two different opacity regimes. Inside the convergence zone, Type I migration is directed outward and outside the zone migration is directed inward…
We 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 …
The stochastic nature of migration of disc instability protoplanets in three-dimensional hydrodynamical and MHD simulations of fragmenting discs
University of Zurich（苏黎世大学）；Shanghai Astronomical Observatory, Chinese academy of Science（中国科学院上海天文台）；Department of Astronomy and Astrophysics, University of California, Santa Cruz（加州大学圣克鲁兹分校天体物理学系）；Institute for Advanced Studies, Tsinghua University（清华大学高等研究院）
In order to reproduce the statistical properties of the observed exoplanets, population synthesis models have shown that the migration of protoplanets should be significantly slowed down, and that processes stalling migration should be at work. Much current theoretical efforts have thus been dedicated to find physical effects that slow down, halt or even reverse migration. Many of these studies rely on the horseshoe drag, whose long term-evolution (saturated or not) is intimately related to the disk viscosity in laminar disk models. We investigate how the horseshoe drag exerted on a low-mass p…
We calculate radial migration rates of protoplanets in laminar minimum mass solar nebula discs using three-dimensional self-gravitating radiation hydrodynamical (RHD) models. The protoplanets are free to migrate, whereupon their migration rates are measured. For low mass protoplanets (10-50 M_\oplus) we find increases in the migration timescales of up to an order of magnitude between locally-isothermal and RHD models. In the high-mass regime the migration rates are changed very little. These results are arrived at by calculating migration rates in locally-isothermal models, before sequentially…
It has recently been shown that the terrestrial planets and asteroid belt can be reproduced if the giant planets underwent an inward-then-outward migration (the "Grand Tack"; Walsh et al 2011). Inward migration occurs when Jupiter opens a gap and type II migrates inward. The planets "tack" and migrate outward when Saturn reaches the gap-opening mass and is caught in the 3:2 resonance with Jupiter. The aim is to test the viability of the Grand Tack model and to study the dynamical evolution of Jupiter and Saturn during their growth from 10 Earth masses cores. We have performed numerical simulat…
Context. The dynamics of a low-mass protoplanet accreting solids is influenced by the heating torque, which was found to suppress inward migration in protoplanetary disks with constant opacities. Aims. We investigate the differences of the heating torque between disks with constant and temperature-dependent opacities. Methods. Interactions of a super-Earth-sized protoplanet with the gas disk are explored using 3D radiation hydrodynamic simulations. Results. Accretion heating of the protoplanet creates a hot underdense region in the surrounding gas, leading to misalignment of the local density …
Concurrent Accretion and Migration of Giant Planets in their Natal Disks with Consistent Accretion Torque (II): Parameter Survey and Condition for Outward Migration
(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…
The torques exerted by a locally isothermal disk on an embedded planet lead to rapid inward migration. Recent work has shown that modeling the thermodynamics without the assumption of local isothermality reveals regions where the net torque on an embedded planet is positive, leading to outward migration of the planet. When a region with negative torque lies directly exterior to this, planets in the inner region migrate outwards and planets in the outer region migrate inwards, converging where the torque is zero. We incorporate the torques from an evolving non-isothermal disk into an N-body sim…
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…
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 study the effects of a large-scale, ordered magnetic field in protoplanetary disks on Type I planet migration using a linear perturbation analysis in the ideal-MHD limit. We focus on wind-driving disks, in which a magnetic torque $\propto B_{0z} \partial B_{0φ}/\partial z$ (where $B_{0z}$ and $B_{0φ}$ are the equilibrium vertical and azimuthal field components) induces vertical angular momentum transport. We derive the governing differential equation for the disk response and identify its resonances and turning points. For a disk containing a slightly subthermal, pure-$B_{0z}$ field, the to…
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…
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…
The interaction of a satellite with a gaseous disk results in the excitation of spiral density waves which remove angular momentum from the orbit. In addition, if the orbit is not coplanar with the disk, three-dimensional effects will excite bending and eccentricity waves. We perform three-dimensional hydrodynamic simulations to study nonlinear disk-satellite interaction in inviscid protoplanetary disks for a variety of orbital inclinations from $0^\circ$ to $180^\circ$. It is well known that three-dimensional effects are important even for zero inclination. In this work we (1) show that for p…
Recent studies on the planet-dominated regime of Type II migration showed that, contrary to the conventional wisdom, massive planets can migrate outwards. Using `fixed-planet' simulations these studies found a correlation between the sign of the torques acting on the planet and the parameter $K'$ (which describes the depth of the gap carved by the planet in the disc). We perform `live-planet' simulations exploring a range of $K'$ and disc mass values to test and extend these results. The excitation of planet eccentricity in live-planet simulations breaks the direct dependence of migration rate…
Previous models of the combined growth and migration of protoplanets needed large ad hoc reduction factors for the type I migration rate as found in the isothermal approximation. In order to eliminate these factors, a simple semi-analytical model is presented that incorporates recent results on the migration of low mass planets in non-isothermal disks. It allows for outward migration. The model is used to conduct planetary populations synthesis calculations. Two points with zero torque are found in the disks. Planets migrate both in- and outward towards these convergence zones. They could be i…
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…
Using N-body simulations with planet-disk interactions, we present a mechanism capable of forming compact systems of hot super Earths such as Kepler 11. Recent studies show that outward migration is common in the inner parts of radiative disks. However we show that two processes naturally tip the balance in favor of inward migration. First the corotation torque is too weak to generate outward migration for planetary embryos less massive than $4\mearth$. Second, system of multiple embryos generate sustained non-zero eccentricities that damp the corotation torque and again favor inward migration…
We aim to develop a simple prescription for migration and accretion in 1D disc models, calibrated with results of 3D hydrodynamic simulations. Our focus lies on non-self-gravitating discs, but we also discuss to what degree our prescription could be applied when the discs are self-gravitating. We study migration using torque densities. Our model for the torque density is based on existing fitting formulas, which we subsequently modify to prevent premature gap-opening. At higher planetary masses, we also apply torque densities from hydrodynamic simulations directly to our 1D model. These torque…
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 …
In this paper we investigate the possibility of a migration-induced resonance locking in systems containing three planets, namely an Earth analog, a super-Earth and a gas giant. The planets have been listed in order of increasing orbital periods. All three bodies are embedded in a locally isothermal gaseous disc and orbit around a solar mass star. We are interested in answering the following question: Will the low-mass planets form the same resonant structures with each other in the vicinity of the gas giant as in the case when the gas giant is absent? When there is no gas giant in the system,…
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…
Hundred meter sized objects have been identified by the Cassini spacecraft in Saturn's A ring through the so-called "propeller" features they create in the ring. These moonlets should migrate, due to their gravitational interaction with the ring ; in fact, some orbital variation have been detected. The standard theory of type I migration of planets in protoplanetary disks can't be applied to the ring system, as it is pressureless. Thus, we compute the differential torque felt by a moonlet embedded in a two-dimensional disk of solid particles, with flat surface density profile, both analyticall…
Massive cores of the giant planets are thought to have formed in a gas disk by accretion of pebble-size particles whose accretional cross-section is enhanced by aerodynamic gas drag [1][2]. A commonly held view is that the terrestrial planet system formed later (30-200 Myr after the dispersal of the gas disk) by giant collisions of tens of roughly Mars-size protoplanets [3]. Here we propose, instead, that the terrestrial planets formed earlier by gas-driven convergent migration of protoplanets toward $\sim\!1\,{\rm au}$ (related ref. [4] invoked a different process to concentrate planetesimals…
We examine the migration of luminous low-mass cores in laminar protoplanetary discs where accretion occurs mainly because of disc winds and where the planet luminosity is generated by pebble accretion. Using 2D hydrodynamical simulations, we determine the eccentricities induced by thermal forces as a function of gas and pebble accretion rates, and also evaluate the importance of the torque exerted by the solid component relative to the gas torque. For a gas accretion rate $\dot M= 2\times 10^{-8}$ $M_\odot/$yr and pebble flux $\dot M_{peb}=170$ $M_\oplus$/Myr, we find that embryo eccentricitie…
Planet-Disk Interactions and the Convective Overstability. I. Low Mass Planets
Iowa State University（爱荷华州立大学）；Institute of Astronomy and Astrophysics, Academia Sinica（中央研究院天文及天文物理研究所）；National Center for Theoretical Sciences（国家理论科学中心）
