type-I Migration Research Articles

Evolution of protoplanetary discs with magnetically driven disc winds

Aims: We investigate the evolution of protoplanetary discs (PPDs hereafter) with magnetically driven disc winds and viscous heating. Methods: We consider an initially massive disc with ~0.1 Msun to track the evolution from the early stage of PPDs. We solve the time evolution of surface density and temperature by taking into account viscous heating and the loss of the mass and the angular momentum by the disc winds within the framework of a standard alpha model for accretion discs. Our model parameters, turbulent viscosity, disc wind mass loss, and disc wind torque, which are adopted from local magnetohydrodynamical simulations and constrained by the global energetics of the gravitational accretion, largely depends on the physical condition of PPDs, particularly on the evolution of the vertical magnetic flux in weakly ionized PPDs. Results: Although there are still uncertainties concerning the evolution of the vertical magnetic flux remaining, surface densities show a large variety, depending on the combination of these three parameters, some of which are very different from the surface density expected from the standard accretion. When a PPD is in a "wind-driven accretion" state with the preserved vertical magnetic field, the radial dependence of the surface density can be positive in the inner region <1-10 au. The mass accretion rates are consistent with observations, even in the very low level of magnetohydrodynamical turbulence. Such a positive radial slope of the surface density gives a great impact on planet formation because (i)it inhibits the inward drift or even results in the outward drift of pebble/boulder-sized solid bodies, and (ii) it also makes the inward type-I migration of proto-planets slower or even reversed. Conclusions: The variety of our calculated PPDs should yield a wide variety of exoplanet systems.

Runaway gas accretion and gap opening versus type I migration

Growing planets interact with their natal protoplanetary disc, which exerts a torque onto them allowing them to migrate in the disc. Small mass planets do not affect the gas profile and migrate in the fast type-I migration. Although type-I migration can be directed outwards for planets smaller than 20−30M⊕ in some regions of the disc, planets above this mass should be lost into the central star long before the disc disperses. Massive planets push away material from their orbit and open a gap. They subsequently migrate in the slower, type II migration, which could save them from migrating all the way to the star. Hence, growing giant planets can be saved if and only if they can reach the gap opening mass, because this extends their migration timescale, allowing them to eventually survive at large orbits until the disc itself disperses.However, most of the previous studies only measured the torques on planets with fixed masses and orbits to determine the migration rate. Additionally, the transition between type-I and type-II migration itself is not well studied, especially when taking the growth mechanism of rapid gas accretion from the surrounding disc into account. Here we use isothermal 2D disc simulations with FARGO-2D1D to study the migration behaviour of gas accreting protoplanets in discs. We find that migrating giant planets always open gaps in the disc. We further show analytically and numerically that in the runaway gas accretion regime, the growth time-scale is comparable to the type-I migration time-scale, indicating that growing planets will reach gap opening masses before migrating all the way to the central star in type-I migration if the disc is not extremely viscous and/or thick. An accretion rate limited to the radial gas flow in the disc, in contrast, is not fast enough. When gas accretion by the planet is taken into account, the gap opening process is accelerated because the planet accretes material originating from its horseshoe region. This allows an accreting planet to transition to type-II migration before being lost even if gas fails to be provided for a rapid enough growth and the classical gap opening mass is not reached.

On the formation and chemical composition of super Earths

Super Earths are the largest population of exoplanets and are seen to exhibit a rich diversity of compositions as inferred through their mean densities. Here we present a model that combines equilibrium chemistry in evolving disks with core accretion that tracks materials accreted onto planets during their formation. In doing so, we aim to explain why super Earths form so frequently and how they acquire such a diverse range of compositions. A key feature of our model is disk inhomogeneities, or planet traps, that act as barriers to rapid type-I migration. The traps we include are the dead zone, which can be caused by either cosmic ray or X-ray ionization, the ice line, and the heat transition. We find that in disks with sufficiently long lifetimes ($\gtrsim$ 4 Myr), all traps produce Jovian planets. In these disks, planet formation in the heat transition and X-ray dead zone produces hot Jupiters while the ice line and cosmic ray dead zones produce Jupiters at roughly 1 AU. Super Earth formation takes place within short-lived disks ($\lesssim$ 2 Myr), whereby the disks are photoevaporated while planets are in a slow phase of gas accretion. We find that super Earth compositions range from dry and rocky ($<$ 6 % ice by mass) to those with substantial water contents ($>$ 30 % ice by mass). The traps play a crucial role in our results, as they dictate where in the disk particular planets can accrete from, and what compositions they are able to acquire.

AN INCREASE IN THE MASS OF PLANETARY SYSTEMS AROUND LOWER-MASS STARS

Trends in the planet population with host star mass provide an avenue to constrain planet formation theories. We derive the planet radius distribution function for Kepler stars of different spectral types, sampling a range in host star masses. We find that M dwarf stars have 3.5 times more small planets (1.0-2.8 R_Earth) than main-sequence FGK stars, but two times fewer Neptune-sized and larger planets (>2.8 R_Earth). We find no systematic trend in the planet size distribution between spectral types F, G, and K to explain the increasing occurrence rates. Taking into account the mass-radius relationship and heavy-element mass of observed exoplanets, and assuming those are independent of spectral type, we derive the inventory of the heavy-element mass locked up in exoplanets at short orbits. The overall higher planet occurrence rates around M stars are not consistent with the redistribution of the same mass into more, smaller planets. At the orbital periods and planet radii where Kepler observations are complete for all spectral types, the average heavy-element mass locked up in exoplanets increases roughly inversely with stellar mass from 4 M_Earth in F stars to 5 M_Earth in G and K stars to 7 M_Earth in M stars. This trend stands in stark contrast with observed protoplanetary disk masses that decrease towards lower mass stars, and provides a challenge for current planet formation models. Neither models of in situ formation nor migration of fully-formed planets are consistent with these results. Instead, these results are indicative of large-scale inward migration of planetary building blocks --- either through type-I migration or radial drift of dust grains --- that is more efficient for lower mass stars, but does not result in significantly larger or smaller planets.

3D Simulations of Type-I Migration in Nearly Laminar Disks

We report results from 3D hydrodynamic disk simulations with an embedded planet. We examine the planet’s migration dependence on the disk viscosity. We find that the Type I migration can be slowed down significantly in low viscosity disks. The density feedback effect that was studied in 2D simulations still exists in 3D disks and causes the slowing down of Type I. Future work will address further the quantitative differences between 3D and 2D in slowing down the migration.

MIGRATION RATES OF PLANETS DUE TO SCATTERING OF PLANETESIMALS

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 will 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 have opposite signs with respect to α and β; and that the torque is especially sensitive to the eccentricity gradient β. Compared to Type-I migration due to excitation of density waves, the planetesimal-driven migration rate is generally lower due to the lower surface density of solids in gas-rich disk, although this may be partially or fully offset when their eccentricity and inclinaton are small. Allowing for the feedback of the planet on the planetesimal disk through viscous stirring, we find that under certain conditions a self-regulated migration scenario emerges, in which the planet migrates at a steady pace that approaches the rate corresponding to the one-sided torque. If the ratio of the local disk mass in planetesimals to planet mass is low, however, migration will stall. We quantify the boundaries separating the three accretion regimes.

On type-I migration near opacity transitions. A generalized Lindblad torque formula for planetary population synthesis

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 simplicity of the standard Lindblad torque formula and gives results that accurately reproduce those of numerical simulations, even at locations where the disk temperature undergoes abrupt changes. Our study is conducted by means of customized numerical simulations in the low-mass regime, in locally isothermal disks, and compared to linear torque estimates obtained by summing fully analytic torque estimates at each Lindblad resonance. The functional dependence of our modified Lindblad torque expression is suggested by an estimate of the shift of the Lindblad resonances that mostly contribute to the torque, in a disk with sharp gradients of temperature or surface density, while the numerical coefficients of the new terms are adjusted to seek agreement with numerics. As side results, we find that the vortensity related corotation torque undergoes a boost at an opacity transition that can counteract migration, and we find evidence from numerical simulations that the linear corotation torque has a non-negligible dependency upon the temperature gradient, in a locally isothermal disk.

Planetesimal-driven planet migration in the presence of a gas disk

We report here on an extension of a previous study by Kirsh et al. (Kirsh, D.R., Duncan, M., Brasser, R., Levison, H.F. [2009]. Icarus 199, 197–209) of planetesimal-driven migration using our N-body code SyMBA (Duncan, M.J., Levison, H.F., Lee, M.H. [1998]. Astron. J. 116, 2067–2077). The previous work focused on the case of a single planet of mass M em, immersed in a planetesimal disk with a power-law surface density distribution and Rayleigh distributed eccentricities and inclinations. Typically 10 4–10 5 equal-mass planetesimals were used, where the gravitational force (and the back-reaction) on each planetesimal by the Sun and planet were included, while planetesimal–planetesimal interactions were neglected. The runs reported on here incorporate the dynamical effects of a gas disk, where the Adachi et al. (Adachi, I., Hayashi, C., Nakazawa, K. [1976]. Prog. Theor. Phys. 56, 1756–1771) prescription of aerodynamic gas drag is implemented for all bodies. In some cases the Papaloizou and Larwood (Papaloizou, J.C.B., Larwood, J.D. [2000]. Mon. Not. R. Astron. Soc. 315, 823–833) prescription of Type-I migration for the planet are implemented, as well as a mass distribution. In the gas-free cases, rapid planet migration was observed – at a rate independent of the planet’s mass – provided the planet’s mass was not large compared to the mass in planetesimals capable of entering its Hill sphere. In such cases, both inward and outward migrations can be self-sustaining, but there is a strong propensity for inward migration. When a gas disk is present, aerodynamic drag can substantially modify the dynamics of scattered planetesimals. For sufficiently large or small mono-dispersed planetesimals, the planet typically migrates inward. However, for a range of plausible planetesimal sizes (i.e. 0.5–5.0 km at 5.0 AU in a minimum mass Hayashi disk) outward migration is usually triggered, often accompanied by substantial planetary mass accretion. The origins of this behaviour are explained in terms of a toy model. The effects of including a size distribution and torques associated with Type-I migration are also discussed.

Planet migration in three-dimensional radiative discs

The migration of growing protoplanets depends on the thermodynamics of the ambient disc. Standard modelling, using locally isothermal discs, indicate in the low planet mass regime an inward (type-I) migration. Taking into account non-isothermal effects, recent studies have shown that the direction of the type-I migration can change from inward to outward. In this paper we extend previous two-dimensional studies, and investigate the planet-disc interaction in viscous, radiative discs using fully three-dimensional radiation hydrodynamical simulations of protoplanetary accretion discs with embedded planets, for a range of planetary masses. We use an explicit three-dimensional (3D) hydrodynamical code NIRVANA that includes full tensor viscosity. We have added implicit radiation transport in the flux-limited diffusion approximation, and to speed up the simulations significantly we have newly adapted and implemented the FARGO-algorithm in a 3D context. First, we present results of test simulations that demonstrate the accuracy of the newly implemented FARGO-method in 3D. For a planet mass of 20 M_earth we then show that the inclusion of radiative effects yields a torque reversal also in full 3D. For the same opacity law used the effect is even stronger in 3D than in the corresponding 2D simulations, due to a slightly thinner disc. Finally, we demonstrate the extent of the torque reversal by calculating a sequence of planet masses. Through full 3D simulations of embedded planets in viscous, radiative discs we confirm that the migration can be directed outwards up to planet masses of about 33 M_earth. Hence, the effect may help to resolve the problem of too rapid inward migration of planets during their type-I phase.

N-BODY SIMULATIONS OF PLANETARY ACCRETION AROUND M DWARF STARS

We have investigated planetary accretion from planetesimals in terrestrial planet regions inside the ice line around M dwarf stars through N-body simulations including tidal interactions with disk gas. Because of low luminosity of M dwarfs, habitable zones (HZs) are located in inner regions. In the close-in HZ, type-I migration and the orbital decay induced by eccentricity damping are efficient according to the high disk gas density in the small orbital radii. In the case of full efficiency of type-I migration predicted by the linear theory, we found that protoplanets that migrate to the vicinity of the host star undergo close scatterings and collisions, and 4 to 6 planets eventually remain in mutual mean motion resonances and their orbits have small eccentricities and they are stable both before and after disk gas decays. In the case of slow migration, the resonant capture is so efficient that densely-packed ~ 40 small protoplanets remain in mutual mean motion resonances. In this case, they start orbit crossing, after the disk gas decays and eccentricity damping due to tidal interaction with gas is no more effective. Through merging of the protoplanets, several planets in widely-separated non-resonant orbits with relatively large eccentricities are formed. Thus, the final orbital configurations of the terrestrial planets around M dwarfs sensitively depend on strength of type-I migration. We also found that large amount of water-ice is delivered by type-I migration from outer regions and final planets near the inner disk edge around M dwarfs are generally abundant in water-ice except for the innermost one that is shielded by the outer planets, unless type-I migration speed is reduced by a factor of more than 100 from that predicted by the linear theory.

Oligarchic growth with migration and fragmentation

In the core-accretion model, giant-planet cores form by oligarchic growth from a population of planetesimals prior to the dispersal of the disk gas. Once a core reaches a critical mass of roughly 10 Earth masses, it begins to accrete a gaseous envelope, forming a giant planet. Collisions between planetesimals cause fragmentation. Planetesimal fragments are more easily captured by cores, speeding up growth, but fragments are also lost by radial drift, reducing the total solid mass in the disk. Interaction with the gas causes cores to undergo inward type-I migration. Migration allows a core to accrete planetesimals from a larger region, but migrating cores may be lost if they reach the star. Thus, migration and fragmentation have both a positive and a negative impact on core formation. Here we describe results of new simulations of oligarchic growth that include fragmentation and/or migration. In the absence of migration, cores grow until they reach their isolation mass, which increases with distance from the star, or until the disk gas disperses. Fragmentation increases the maximum core mass by increasing growth rates in the outer disk, allowing objects to reach their isolation mass during the disk lifetime. When migration is present, cores migrate inwards rapidly when they approach 1 Earth mass. Most migrating cores are lost. Migrating cores gain little extra mass since they are passing through regions that have been depleted by earlier generations of cores. For a disk viscosity parameter alpha = 1 e − 3 and planetesimal radius = 10 km, the maximum core mass is roughly 4 and 0.5 Earth masses with/without fragmentation, respectively, with little dependence on the disk mass. Formation and survival of 10-Earth-mass cores, in the presence of migration, requires large alpha ( 1 e − 2 ) and a massive disk (0.1 solar masses). When alpha is large, type-I migration rates decrease rapidly with time, allowing large, late-forming cores to survive. The addition of a stochastic (random-walk) migration component makes little difference to the outcome, provided that stochastic migration affects only cores larger than 0.01 Earth masses. Stochastic migration becomes increasingly important if it also affects lower-mass objects.

Migration of protoplanets in radiative discs

In isothermal disks the migration of protoplanets is directed inward. For small planetary masses the standard type-I migration rates are so fast that this may result in an unrealistic loss of planets into the stars. We investigate the planet-disk interaction in non-isothermal disks and analyze the magnitude and direction of migration for an extended range of planet masses. We have performed detailed two-dimensional numerical simulations of embedded planets including heating/cooling effects as well as radiative diffusion for realistic opacities. In radiative disks, small planets with M_planet < 50 M_Earth do migrate outward with a rate comparable to absolute magnitude of standard type-I migration. For larger masses the migration is inward and approaches the isothermal, type-II migration rate. Our findings are particularly important for the first growth phase of planets and ease the problem of too rapid inward type-I migration.

Orbital migration and mass-semimajor axis distributions of extrasolar planets

AbstractHere we discuss the effects of type-I migration of protoplanetary embryos on mass and semimajor axis distributions of extrasolar planets. We summarize the results of Ida &amp; Lin (2008a, 2008b), in which Monte Carlo simulations with a deterministic planet-formation model were carried out. The strength of type-I migration regulates the distribution of extrasolar gas giant planets as well as terrestrial planets. To be consistent with the existing observational data of extrasolar gas giants, the type-I migration speed has to be an order of magnitude slower than that given by the linear theory. The introduction of type-I migration inhibits in situ formation of gas giants in habitable zones (HZs) and reduces the probability of passage of gas giants through HZs, both of which facilitate retention of terrestrial planets in HZs. We also point out that the effect of magneto-rotational instability (MRI) could lead to trapping of migrating protoplanetary embryos in the regions near an ice line in the disk and it significantly enhances formation/retention probability of gas giants against type-I migration.

Formation of terrestrial planets from planetesimals around M dwarfs

AbstractWe have investigated accretion of terrestrial planets from planetesimals around M dwarfs through N-body simulations including the effect of tidal interaction with disk gas. Because of low luminosity of M dwarfs, habitable zones around them are located near the disk inner edge. Planetary embryos undergo type-I migration and pile up near the disk inner edge. We found that after repeated close scatterings and occasional collisions, three or four planets eventually remain in stable orbits in their mean motion resonances. Furthermore, large amount of water-rich planetesimals rapidly migrate to the terrestrial planet regions from outside of the snow line, so that formed planets in these regions have much more water contents than those around solar-type stars.

Accretion of terrestrial planets from oligarchs in a turbulent disk

We have investigated the final accretion stage of terrestrial planets from Mars-mass protoplanets that formed through oligarchic growth in a disk comparable to the minimum mass solar nebula (MMSN), through N-body simulation including random torques exerted by disk turbulence due to Magneto-Rotational Instability. For the torques, we used the semi-analytical formula developed by Laughlin et al. [Laughlin, G., Steinacker, A., Adams, F.C., 2004. Astrophys. J. 608, 489–496]. The damping of orbital eccentricities (in all runs) and type-I migration (in some runs) due to the tidal interactions with disk gas is also included. Without any effect of disk gas, Earth-mass planets are formed in terrestrial planet regions in a disk comparable to MMSN but with too large orbital eccentricities to be consistent with the present eccentricities of Earth and Venus in our Solar System. With the eccentricity damping caused by the tidal interaction with a remnant gas disk, Earth-mass planets with eccentricities consistent with those of Earth and Venus are formed in a limited range of disk gas surface density (∼10 −4 times MMSN). However, in this case, on average, too many (≳6) planets remain in terrestrial planet regions, because the damping leads to isolation between the planets. We have carried out a series of N-body simulations including the random torques with different disk surface density and strength of turbulence. We found that the orbital eccentricities pumped up by the turbulent torques and associated random walks in semimajor axes tend to delay isolation of planets, resulting in more coagulation of planets. The eccentricities are still damped after planets become isolated. As a result, the number of final planets decreases with increase in strength of the turbulence, while Earth-mass planets with small eccentricities are still formed. In the case of relatively strong turbulence, the number of final planets are 4–5 at 0.5–2 AU, which is more consistent with Solar System, for relatively wide range of disk gas surface density ( ∼ 10 −4 – 10 −2 times MMSN).

Planet Formation with Migration

In the core-accretion model, gas-giant planets form solid cores which then accrete gaseous envelopes. Tidal interactions with disk gas cause a core to undergo inward type-I migration in 10^4 to 10^5 years. Cores must form faster than this to survive. Giant planets clear a gap in the disk and undergo inward type-II migration in <10^6 years if observed disk accretion rates apply to the disk as a whole. Type-II migration times exceed typical disk lifetimes if viscous accretion occurs mainly in the surface layers of disks. Low turbulent viscosities near the midplane may allow planetesimals to form by coagulation of dust grains. The radius r of such planetesimals is unknown. If r<0.5 km, the core formation time is shorter than the type-I migration timescale and cores will survive. Migration is substantial in most cases, leading to a wide range of planetary orbits, consistent with the observed variety of extrasolar systems. When r is of order 100m and midplane alpha is of order 3 times 10^-5, giant planets similar to those in the Solar System can form.

Orbital evolution and accretion of protoplanets tidally interacting with a gas disk: II. Solid surface density evolution with type-I migration

This paper investigates the surface density evolution of a planetesimal disk due to the effect of type-I migration by carrying out N-body simulation and through analytical method, focusing on terrestrial planet formation. The coagulation and the growth of the planetesimals take place in the abundant gas disk except for a final stage. A protoplanet excites density waves in the gas disk, which causes the torque on the protoplanet. The torque imbalance makes the protoplanet suffer radial migration, which is known as type-I migration. Type-I migration time scale derived by the linear theory may be too short for the terrestrial planets to survive, which is one of the major problems in the planet formation scenario. Although the linear theory assumes a protoplanet being in a gas disk alone, Kominami et al. [Kominami, J., Tanaka, H., Ida, S., 2005. Icarus 167, 231–243] showed that the effect of the interaction with the planetesimal disk and the neighboring protoplanets on type-I migration is negligible. The migration becomes pronounced before the planet's mass reaches the isolation mass, and decreases the solid component in the disk. Runaway protoplanets form again in the planetesimal disk with decreased surface density. In this paper, we present the analytical formulas that describe the evolution of the solid surface density of the disk as a function of gas-to-dust ratio, gas depletion time scale and semimajor axis, which agree well with our results of N-body simulations. In general, significant depletion of solid material is likely to take place in inner regions of disks. This might be responsible for the fact that there is no planet inside Mercury's orbit in our Solar System. Our most important result is that the final surface density of solid components ( Σ d ) and mass of surviving planets depend on gas surface density ( Σ g ) and its depletion time scale ( τ dep ) but not on initial Σ d ; they decrease with increase in Σ g and τ dep . For a fixed gas-to-dust ratio and τ dep , larger initial Σ d results in smaller final Σ d and smaller surviving planets, because of larger Σ g . To retain a specific amount of Σ d , the efficient disk condition is not an initially large Σ d but the initial Σ d as small as the specified final one and a smaller gas-to-dust ratio. To retain Σ d comparable to that of the minimum mass solar nebula (MMSN), a disk must have the same Σ d and a gas-to-dust ratio that is smaller than that of MMSN by a factor of 1.3 × ( τ dep / 1 Myr ) at ∼1 AU. (Equivalently, type-I migration speed is slower than that predicted by the linear theory by the same factor.) The surviving planets are Mars-sized ones in this case; in order to form Earth-sized planets, their eccentricities must be pumped up to start orbit crossing and coagulation among them. At ∼5 AU, Σ d of MMSN is retained under the same condition, but to form a core massive enough to start runaway gas accretion, a gas-to-dust ratio must be smaller than that of MMSN by a factor of 3 × τ dep / 1 Myr .

Orbital evolution and accretion of protoplanets tidally interacting with a gas disk: I. Effects of interaction with planetesimals and other protoplanets

We have performed N-body simulations on the stage of protoplanet formation from planetesimals, taking into account so-called “type-I migration,” and damping of orbital eccentricities and inclinations, as a result of tidal interaction with a gas disk without gap formation. One of the most serious problems in formation of terrestrial planets and jovian planet cores is that the migration time scale predicted by the linear theory is shorter than the disk lifetime (10 6–10 7 years). In this paper, we investigate retardation of type-I migration of a protoplanet due to a torque from a planetesimal disk in which a gap is opened up by the protoplanet, and torques from other protoplanets which are formed in inner and outer regions. In the first series of runs, we carried out N-body simulations of the planetesimal disk, which ranges from 0.9 to 1.1 AU, with a protoplanet seed in order to clarify how much retardation can be induced by the planetesimal disk and how long such retardation can last. We simulated six cases with different migration speeds. We found that in all of our simulations, a clear gap is not maintained for more than 10 5 years in the planetesimal disk. For very fast migration, a gap cannot be created in the planetesimal disk. For migration slower than some critical speed, a gap does form. However, because of the growth of the surrounding planetesimals, gravitational perturbation of the planetesimals eventually becomes so strong that the planetesimals diffuse into the vicinity of the protoplanets, resulting in destruction of the gap. After the gap is destroyed, close encounters with the planetesimals rather accelerate the protoplanet migration. In this way, the migration cannot be retarded by the torque from the planetesimal disk, regardless of the migration speed. In the second series of runs, we simulated accretion of planetesimals in wide range of semimajor axis, 0.5 to 2–5 AU, starting with equal mass planetesimals without a protoplanet seed. Since formation of comparable-mass multiple protoplanets (“oligarchic growth”) is expected, the interactions with other protoplanets have a potential to alter the migration speed. However, inner protoplanets migrate before outer ones are formed, so that the migration and the accretion process of a runaway protoplanet are not affected by the other protoplanets placed inner and outer regions of its orbit. From the results of these two series of simulations, we conclude that the existence of planetesimals and multiple protoplanets do not affect type-I migration and therefore the migration shall proceed as the linear theory has suggested.

How does disk gravity really influence type-I migration?

We report an analytical expression for the locations of Lindblad resonances induced by a perturbing protoplanet, including the effect of disk gravity. Inner, outer and differential torques are found to be enhanced compared to situations where a keplerian velocity field for the dynamics of both the disk and the planet is assumed. Inward migration is strongly accelerated when the disk gravity is only accounted for in the planet orbital motion. The addition of disk self-gravity slows down the planet drift but not enough to stop it.