在原行星盘内不断生长的低质量行星会与盘交换力矩，并相应改变其半长轴。这一过程被称为I型迁移，且强烈依赖于底层盘结构。因此，行星迁移总体上存在许多不确定性。在许多模拟中，当前的I型迁移速率导致行星在盘寿命内到达盘的内边缘。一种新的行星与盘之间的力矩交换，即热力矩，旨在通过加热力矩减缓向内迁移。如果行星光度足够大，加热力矩甚至可能导致行星向外迁移。在此，我们研究在先前的I型模型基础上，热力矩对行星迁移的影响。我们发现，在大多数配置下，Paardekooper et al. (2011)的公式比Jiménez & Masset (2017)的公式允许更多的向外迁移，但我们也发现，在包含卵石和气体吸积的单一行星形成情景中，使用这两个公式时行星演化出非常相似的质量和最终轨道半径。如果行星上的吸积率对应于遵循卵石吸积情景的典型固体吸积率，则添加热力矩可以引入新的但很小的向外迁移区域。如果行星上的吸积率变得非常大，例如在具有大卵石通量的环境中（例如，高金属丰度环境），热力矩可以允许更高效的向外迁移。然而，即便如此，在我们的行星形成情景中，最终质量和轨道位置的变化也相当小。这意味着对于单一行星演化情景，加热力矩的影响可能可以忽略不计。
英文摘要：
Low-mass planets that are in the process of growing larger within protoplanetary disks exchange torques with the disk and change their semi-major axis accordingly. This process is called type I migration and is strongly dependent on the underlying disk structure. As a result, there are many uncertainties about planetary migration in general. In a number of simulations, the current type I migration rates lead to planets reaching the inner edge of the disk within the disk lifetime. A new kind of torque exchange between planet and disk, the thermal torque, aims to slow down inward migration via the heating torque. The heating torque may even cause planets to migrate outwards, if the planetary luminosity is large enough. Here, we study the influence on planetary migration of the thermal torque on top of previous type I models. We find that the formula of Paardekooper et al. (2011) allows for more outward migration than that of Jiménez & Masset (2017) in most configurations, but we also find that planets evolve to very similar mass and final orbital radius using both formulae in a single planet-formation scenario, including pebble and gas accretion. Adding the thermal torque can introduce new, but small, regions of outwards migration if the accretion rates onto the planet correspond to typical solid accretion rates following the pebble accretion scenario. If the accretion rates onto the planets become very large, as could be the case in environments with large pebble fluxes (e.g., high-metallicity environments), the thermal torque can allow more efficient outward migration. However, even then, the changes for the final mass and orbital positions in our planet formation scenario are quite small. This implies that for single planet evolution scenarios, the influence of the heating torque is probably negligible.
Giant planets migrate though the protoplanetary disc as they grow. We investigate how the formation of planetary systems depends on the radial flux of pebbles through the protoplanetary disc and on the planet migration rate. Our N-body simulations confirm previous findings that Jupiter-like planets in orbits outside the water ice line originate from embryos starting out at 20-40 AU when using nominal type-I and type-II migration rates and a pebble flux of 100-200 Earth masses per million years, enough to grow Jupiter within the lifetime of the solar nebula. The planetary embryos placed up to 3…
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…
We study the effect of a migrating planet ($10<M_p<20$ Earth mass) on the dynamics of pebbles in a radiative disk using 2D two-fluid simulations carried out with the RoSSBi code. The combined action of the waves induced by the migrating planet and drag back-reaction on the gas produces a complex evolution of the flux of pebbles. The waves excite zonal flows which act as dust traps, and accumulate pebbles. Owing to drag back-reaction, a Kelvin-Helmholtz instability develops, generating turbulent dust rings containing several Earth masses of solids, where planetesimal formation is likely. The mo…
Planetary migration is a major challenge for planet formation theories. The speed of Type I migration is proportional to the mass of a protoplanet, while the final decade of growth of a pebble-accreting planetary core takes place at a rate that scales with the mass to the two-thirds power. This results in planetary growth tracks (i.e., the evolution of a protoplanet's mass versus its distance from the star) that become increasingly horizontal (migration-dominated) with rising mass of the protoplanet. It has been shown recently that the migration torque on a protoplanet is reduced proportional …
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 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…
Simulating the interplay between the snowline pebble flux and ongoing planet formation and migration
University of Exeter（埃克塞特大学）；Space Telescope Science Institute（太空望远镜科学研究所）；Max Planck Institute for Astronomy（马克斯·普朗克天文学研究所）；University College Cork（科克大学学院）
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
Recent observations have found a valley in the size distribution of close-in super-Earths that is interpreted as a signpost that close-in super-Earths are mostly rocky in composition. However, new models predict that planetesimals should first form at the water ice line such that close-in planets are expected to have a significant water ice component. Here we investigate the water contents of super-Earths by studying the interplay between pebble accretion, planet migration and disc evolution. Planets' compositions are determined by their position relative to different condensation fronts (ice …
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…
The formation of planets depends on the underlying protoplanetary disc structure, which influences both the accretion and migration rates of embedded planets. The disc itself evolves on time-scales of several Myr during which both temperature and density profiles change as matter accretes onto the central star. Here we use a detailed model of an evolving disc to determine the growth of planets by pebble accretion and their migration through the disc. Cores that reach their pebble isolation mass accrete gas to finally form giant planets with extensive gas envelopes, while planets that do not re…
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…
Super-Earths are found in tighter orbits than the Earth's around more than one third of main sequence stars. It has been proposed that super-Earths are scaled-up terrestrial planets that formed similarly, through mutual accretion of planetary embryos, but in discs much denser than the solar protoplanetary disc. We argue instead that terrestrial planets and super-Earths have two distinct formation pathways that are regulated by the disc's pebble reservoir. Through numerical integrations, which combine pebble accretion and N-body gravity between embryos, we show that a difference of a factor of …
Current models of (exo)planet formation often rely on a large influx of so-called `pebbles' from the outer disk into the planet formation region. In this paper, we investigate how the formation of pebbles in the cold outer regions of protoplanetary disks and their subsequent migration to the inner disk can alter the gas-phase CO distribution both interior and exterior to the midplane CO snowline. By simulating the resulting CO abundances in the midplane as well as the warm surface layer, we identify observable signatures of large-scale pebble formation and migration that can be used as `smokin…
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 …
We propose a pebble-driven planet formation scenario to form giant planets with high multiplicity and large orbital distances in the early gas disk phase. We perform N-body simulations to investigate the growth and migration of low-mass protoplanets in the disk with inner viscously heated and outer stellar irradiated regions. The key feature of this model is that the giant planet cores grow rapidly by a combination of pebble accretion and planet-planet collisions. This consequently speeds up their gas accretion. Because of efficient growth, the planet transitions from rapid type I migration to…
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 …
At least 30\% of main sequence stars host planets with sizes of between 1 and 4 Earth radii and orbital periods of less than 100 days. We use N-body simulations including a model for gas-assisted pebble accretion and disk--planet tidal interaction to study the formation of super-Earth systems. We show that the integrated pebble mass reservoir creates a bifurcation between hot super-Earths or hot-Neptunes ($\lesssim15M_{\oplus}$) and super-massive planetary cores potentially able to become gas giant planets ($\gtrsim15M_{\oplus}$). Simulations with moderate pebble fluxes grow multiple super-Ear…
It is generally accepted that the four major (Galilean) satellites formed out of the gas disk that accompanied Jupiter's formation. However, understanding the specifics of the formation process is challenging as both small particles (pebbles) as well as the satellites are subject to fast migration processes. Here, we hypothesize a new scenario for the origin of the Galilean system, based on the capture of several planetesimal seeds and subsequent slow accretion of pebbles. To halt migration, we invoke an inner disk truncation radius, and other parameters are tuned for the model to match physic…
Context: Planetary embryos can continue to grow by pebble accretion until they become giant planet cores. Simultaneously, these embryos mutually interact and also migrate due to torques arising from the protoplanetary disk. Aims: Our aim is to investigate how pebble accretion alters the orbital evolution of embryos undergoing the Type-I migration. In particular, we study whether they establish resonant chains, whether these chains are prone to instabilities and if giant planet cores form through embryo merging, thus occurring more rapidly than by pebble accretion alone. Methods: For the first …
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…
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…
Pebble accretion has become a popular component to core accretion models of planet formation, and is especially relevant to the formation of compact, resonant terrestrial planetary systems. Pebbles initially form in the inner protoplanetary disc, sweeping outwards in a radially expanding front, potentially forming planetesimals and planetary cores via migration and the streaming instability. This pebble front appears at early times, in what is typically assumed to be a low mass disc. We argue this picture is in conflict with the reality of young circumstellar discs, which are massive and self-…
We present a synergic study of protoplanetary disks to investigate links between inner disk gas molecules and the large-scale migration of solid pebbles. The sample includes 63 disks where two types of measurements are available: i) spatially-resolved disk images revealing the radial distribution of disk pebbles (mm-cm dust grains), from millimeter observations with ALMA or the SMA, and ii) infrared molecular emission spectra as observed with Spitzer. The line flux ratios of H2O with HCN, C2H2, and CO2 all anti-correlate with the dust disk radius R$_{dust}$, expanding previous results found by…
The origins of the Galilean satellites - namely Io, Europa, Ganymede, and Callisto - is not fully understood yet. Here we use N-body numerical simulations to study the formation of Galilean satellites in a gaseous circumplanetary disk around Jupiter. Our model includes the effects of pebble accretion, gas-driven migration, and gas tidal damping and drag. Satellitesimals in our simulations first grow via pebble accretion and start to migrate inwards. When they reach the trap at the disk inner edge, scattering events and collisions take place promoting additional growth. Growing satellites event…
In the past decade, ALMA observations have revealed that a large fraction of protoplanetary discs contains rings in the dust continuum. These rings are the locations where pebbles accumulate, which is beneficial for planetesimal formation and subsequent planet assembly. We investigate the viability of planet formation inside ALMA rings in which pebbles are trapped by either a Gaussian-shape pressure bump or by the strong dust backreaction. Planetesimals form at the midplane of the ring via streaming instability. By conducting N-body simulations, we study the growth of these planetesimals by co…
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 …
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…
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…
Gas-giant planets, such as Jupiter, Saturn and massive exoplanets, were formed via the gas accretion onto the solid cores each with a mass of roughly ten Earth masses. However, rapid radial migration due to disk-planet interaction prevents the formation of such massive cores via planetesimal accretion. Comparably rapid core growth via pebble accretion requires very massive protoplanetary disks because most pebbles fall into the central star. Although planetesimal formation, planetary migration, and gas-giant core formation have been studied with much effort, the full evolution path from dust t…
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…
Context. Planet formation with pebbles has been proposed to solve a couple of long-standing issues in the classical formation model. Some sophisticated simulations have been done to confirm the efficiency of pebble accretion. However, there has not been any global N-body simulations that compare the outcomes of planet formation via pebble accretion with observed extrasolar planetary systems. Aims. In this paper, we study the effects of a range of initial parameters of planet formation via pebble accretion, and present the first results of our simulations. Methods. We incorporate the pebble acc…
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…
Recent detailed observations of protoplanetary discs revealed a lot of sub-structures which are mostly ring-like. One interpretation is that these rings are caused by growing planets. These potential planets are not yet opening very deep gaps in their discs. These planets instead form small gaps in the discs to generate small pressure bumps exterior to their orbits that stop the inflow of the largest dust particles. In the pebble accretion paradigm, this planetary mass corresponds to the pebble isolation mass, where pebble accretion stops and efficient gas accretion starts. We perform planet p…
Atmospheric chemical abundances of giant planets lead to important constraints on planetary formation and migration. Studies have shown that giant planets that migrate through the protoplanetary disk can accrete substantial amounts of oxygen-rich planetesimals, leading to super-solar metallicities in the envelope and solar or sub-solar C/O ratios. Pebble accretion has been demonstrated to play an important role in core accretion and to have growth rates that are consistent with planetary migration. The high pebble accretion rates allow planetary cores to start their growth beyond 10 AU and sub…
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…
Recent observations of protoplanetary disks have revealed ring-like structures that can be associated to pressure maxima. Pressure maxima are known to be dust collectors and planet migration traps. Most of planet formation works are based either on the pebble accretion model or on the planetesimal accretion model. However, recent studies proposed the possible formation of Jupiter by the hybrid accretion of pebbles and planetesimals. We aim to study the full process of planet formation consisting of dust evolution, planetesimal formation and planet growth at a pressure maximum in a protoplaneta…
Models of planetary core growth by either planetesimal or pebble accretion are traditionally disconnected from the models of dust evolution and formation of the first gravitationally-bound planetesimals. The state-of-the-art models typically start with massive planetary cores already present. We aim to study the formation and growth of planetary cores in a pressure bump, motivated by the annular structures observed in protoplanetary disks, starting with sub-micron-sized dust grains. We connect the models of dust coagulation and drift, planetesimal formation in the streaming instability, gravit…
Protoplanets of Super-Earth sizes may get trapped in convergence zones for planetary migration and form gas giants there. These growing planets undergo accretion heating, which triggers a hot-trail effect that can reverse migration directions, increase eccentricities and prevent resonant captures (Chrenko et al. 2017). We study populations of embryos accreting pebbles using Fargo-Thorin 2D hydrocode. We find that embryos in a disk with high surface density ($Σ_0 = 990\,{\rm g}\,{\rm cm}^{-2}$) undergo `unsuccessful' two-body encounters which do not lead to a merger. Only when a 3rd protoplanet…
We develop a pebble-driven model to study the formation and evolution of planets around stars in the mass range of 0.08 and 1 solar mass. The growth and migration of a large number of individual protoplanetary embryos are simulated in a population synthesis manner. We test two hypotheses for the birth locations of embryos: at the water ice line or log-uniformly distributed over entire protoplanetary disks. Two types of disks with different turbulent viscous parameters alpha of 1e-3 and 1e-4 are investigated, to shed light on the role of outward migration of protoplanets. The forming planets ar…
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…
Recent work has shown that aside from the classical view of collisions by increasingly massive planetesimals, the accretion of mm- to m-sized 'pebbles' can also reproduce the mass-orbit distribution of the terrestrial planets. Here, we perform N-body simulations to study the effects of pebble accretion onto growing planetesimals of different diameters located in the inner Solar System. The simulations are run to occur during the lifetime of the gas disc while also simultaneously taking Jupiter's growth into account. We find that pebble accretion can increase the mass in the solid disc by at le…
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…
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…
