We investigate the tidal interaction between a low-mass planet and a self-gravitating protoplanetary disk, by means of two-dimensional hydrodynamic simulations. We first show that considering a planet freely migrating in a disk without self-gravity leads to a significant overestimate of the migration rate. The overestimate can reach a factor of two for a disk having three times the surface density of the minimum mass solar nebula. Unbiased drift rates may be obtained only by considering a planet and a disk orbiting within the same gravitational potential. In a second part, the disk self-gravity is taken into account. We confirm that the disk gravity enhances the differential Lindblad torque with respect to the situation where neither the planet nor the disk feels the disk gravity. This enhancement only depends on the Toomre parameter at the planet location. It is typically one order of magnitude smaller than the spurious one induced by assuming a planet migrating in a disk without self-gravity. We confirm that the torque enhancement due to the disk gravity can be entirely accounted for by a shift of Lindblad resonances, and can be reproduced by the use of an anisotropic pressure tensor. We do not find any significant impact of the disk gravity on the corotation torque.
We simulate planet migration caused by interactions between planets and a planetesimal disk. We use an N-body integrator optimized for near-Keplerian motion that runs in parallel on a video graphics card, and that computes all pair-wise gravitational interactions. We find that the fraction of planetesimals found in mean motion resonances is reduced and planetary migration rates are on average about 50% slower when gravitational interactions between the planetesimals are computed than when planetesimal self-gravity is neglected. This is likely due to gravitational stirring of the planetesimal d…
We report on the results of novel global high-resolution three-dimensional simulations of disk-planet interaction which incorporate simultaneously realistic radiation physics and the self-gravity of the gas, as well as allowing the planet to move. We find that thermodynamics and radiative physics have a remarkable effect on both migration and accretion of Jupiter mass planets. In simulations with radiative transfer adopting flux-limited diffusion, inward migration can be decreased by about 30% relative to the isothermal case, while in adiabatic runs migration nearly shuts off after a few tens …
Planet formation is accompanied by the formation of comet clouds. In systems where planets migrate on rapid timescales, the diffusive evolution of comet orbits may stall, resulting in a comet cloud intermediate between a flattened Kuiper Belt and a spherical Oort cloud. These `failed Oort clouds' may provide a `smoking gun', indicating that planetary migration has taken place. If some fraction of the scattered component consists of planetary embryos, it may be possible to observe transits of such bodies even when the planetary system is not edge-on to the line of sight.
The tidal interaction between a disk and a planet leads to the planet's migration. A long-standing question regarding this mechanism is how to stop the migration before planets plunge into their central stars. In this paper, we propose a new, simple mechanism to significantly slow down planet migration, and test the possibility by using a hybrid numerical integrator to simulate the disk-planet interaction. The key component of the scenario is the role of low viscosity regions in protostellar disks known as dead zones, which affect planetary migration in two ways. First of all, it allows a smal…
The spatial distribution of Kuiper belt objects (KBOs) in 2:1 exterior resonance with Neptune constrains that planet's migration history. Numerical simulations demonstrate that fast planetary migration generates a larger population of KBOs trailing rather than leading Neptune in orbital longitude. This asymmetry corresponds to a greater proportion of objects caught into asymmetric resonance such that their resonance angles librate about values > pi (trailing) as opposed to < pi (leading). We provide, for the first time, an explanation of this phenomenon, using physical, analytic, and semi-anal…
Recent observations show a clear correlation between the probability of hosting a planet and the metallicity of the parent star. Since radial velocity surveys are biased, however, towards detecting planets with short orbital periods, the probability-metallicity correlation could merely reflect a dependence of migration rates on metallicity. We investigated the possibility, but find no basis to suggest that the migration process is sensitive to the metallicity. The indication is, therefore, that a higher metallicity results in a higher probability for planet formation.
We present a numerical study of rapid, so called type III migration for Jupiter-sized planets embedded in a protoplanetary disc. We limit ourselves to the case of outward migration, and study in detail its evolution and physics, concentrating on the structure of the co-rotation and circumplanetary regions, and processes for stopping migration. We also consider the dependence of the migration behaviour on several key parameters. We perform this study using global, two-dimensional hydrodynamical simulations with adaptive mesh refinement. We find that the outward directed type III migration can b…
According to current theories, tidal interactions between a disk and an embedded planet may lead to the rapid migration of the protoplanet on a timescale shorter than the disk lifetime or estimated planetary formation timescales. Therefore, planets can form only if there is a mechanism to hold at least some of the cores back on their way in. Once a giant planet has assembled, there also has to be a mechanism to prevent it from migrating down to the disk center. This paper reviews the different mechanisms that have been proposed to stop or slow down migration.
We analyze the possible relationship between the current orbital elements fits of known exoplanets in the 2/1 mean-motion resonance and the expected orbital configuration due to migration. It is found that, as long as the orbital decay was sufficiently slow to be approximated by an adiabatic process, all captured planets should be in apsidal corotations. In other words, they should show a simultaneous libration of both the resonant angle and the difference in longitudes of pericenter. We present a complete set of corotational solutions for the 2/1 commensurability, including previously known s…
We present a numerical study of rapid, so called type III migration for Jupitersized planets embedded in a protoplanetary disc. We limit ourselves to the case of inward migration, and study in detail its evolution and physics, concentrating on the structure of the corotation and circumplanetary regions, and processes for stopping migration. We also consider the dependence of the migration behaviour on several key parameters. We perform this study using the results of global, two-dimensional hydrodynamical simulations with adaptive mesh refinement. The initial conditions are chosen to satisfy t…
The formation of resonant planets pairs in exoplanetary systems involves planetary migration inside the protoplanetary disc : an inwards migrating outer planet captures in Mean Motion Resonance an inner planet. During the migration of the resonant pair of planets, the eccentricities are expected to rise excessively, if no damping mechanism is applied on the inner planet. We express the required damping action to match the observations, and we show that the inner disk can play this role. This result applies for instance to the system GJ 876 : we reproduce the observed orbital elements through a…
This paper examines how type I planet migration is affected by the presence of turbulent density fluctuations in the circumstellar disk. For type I migration, the planet does not clear a gap in the disk and its secular motion is driven by torques generated by the wakes it creates in the surrounding disk fluid. MHD turbulence creates additional density perturbations that gravitationally interact with the planet and can dominate the torques produced by the migration mechanism itself. This paper shows that conventional type I migration can be readily overwhelmed by turbulent perturbations and hen…
We present results from three-dimensional, self-gravitating radiation hydrodynamical models of gas accretion by planetary cores. In some cases, the accretion flow is resolved down to the surface of the solid core -- the first time such simulations have been performed. We investigate the dependence of the gas accretion rate upon the planetary core mass, and the surface density and opacity of the encompassing protoplanetary disc. Accretion of planetesimals is neglected. We find that high-mass protoplanets are surrounded by thick circumplanetary discs during their gas accretion phase but, contrar…
We present a thorough study of the impact of a migrating planet on a planetesimal disk, by exploring a broad range of masses and eccentricities for the planet. We discuss the sensitivity of the structures generated in debris disks to the basic planet parameters. We perform many N-body numerical simulations, using the symplectic integrator SWIFT, taking into account the gravitational influence of the star and the planet on massless test particles. A constant migration rate is assumed for the planet. The effect of planetary migration on the trapping of particles in mean motion resonances is foun…
There is evidence for the existence of massive planets at orbital radii of several hundred AU from their parent stars where the timescale for planet formation by core accretion is longer than the disc lifetime. These planets could have formed close to their star and then migrated outwards. We consider how the transfer of angular momentum by viscous disc interactions from a massive inner planet could cause significant outward migration of a smaller outer planet. We find that it is in principle possible for planets to migrate to large radii. We note, however, a number of effects which may render…
In the current paper, we further improved the model for the migration of planets introduced in Del Popolo et al. (2001) and extended to time-dependent planetesimal accretion disks in Del Popolo and Eksi (2002). In the current study, the assumption of Del Popolo and Eksi (2002), that the surface density in planetesimals is proportional to that of gas, is released. In order to obtain the evolution of planetesimal density, we use a method developed in Stepinski and Valageas (1997) which is able to simultaneously follow the evolution of gas and solid particles for up to 10^7 yrs. Then, the disk mo…
Gravitational torques between a planet and gas in the protoplanetary disk result in orbital migration of the planet, and are likely to play an important role in the formation and early evolution of planetary systems. For masses comparable to those of observed giant extrasolar planets, the interaction with the disk is strong enough to form a gap, leading to coupled evolution of the planet and disk on a viscous time scale (Type II migration). Both the existence of hot Jupiters, and the statistical distribution of observed orbital radii, are consistent with an important role for Type II migration…
We study the effect of poloidal magnetic field on type I planetary migration by linear perturbation analysis in the shearing-sheet approximation and the analytic results are compared with numerical calculations. In contrast to the unmagnetized case, the basic equations that describe the wake due to the planet in the disk allow magnetic resonances at which density perturbation diverges. In order to simplify the problem, we consider the case without magneto-rotational instability. We perform two sets of analyses: two-dimensional and three-dimensional. In two-dimensional analysis, we find the gen…
In this letter, we briefly describe the evolution of a variety of self-gravitating protoplanetary disk models that contain annular grooves (e.g. gaps) in their surface density. These grooves are inspired by the density gaps that are presumed to open in response to the formation of a giant planet. Our work provides an extension of the previously studied groove modes that are known in the context of stellar disks. The emergence of spiral gravitational instabilities (GI) is predicted via a generalized eigenvalue code that performs a linear analysis, and confirmed with hydrodynamical simulations. …
In the current paper, we further develop the model for the migration of planets introduced in Del Popolo et al. (2001) and extended to time-dependent accretion discs in Del Popolo and Eksi (2002). We use a method developed by Stepinski and Valageas (1996, 1997), that is able to simultaneously follow the evolution of gas and solid particles for up to $10^7 {\rm yr}$. The disc model is coupled to the migration model introduced in Del Popolo et al. (2001) in order to obtain the migration rate of the planet in the planetesimal disc. We find that in the case of discs having total mass of $10^{-3}-0…
Using 2D MHD numerical simulations performed with two different finite difference Eulerian codes, we analyze the effect that a toroidal magnetic field has on low mass planet migration in nonturbulent protoplanetary discs. The presence of the magnetic field modifies the waves that can propagate in the disc. In agreement with a recent linear analysis (Terquem 2003), we find that two magnetic resonances develop on both sides of the planet orbit, which contribute to a significant global torque. In order to measure the torque exerted by the disc on the planet, we perform simulations in which the la…
A significant number of the known multiple exoplanetary systems are containing a pair of giant planets engaged in a low order mean motion resonance. Such a resonant condition protects the dynamics of these planets resulting in very stable orbits. According to recent studies the capture into a resonance is the result of a planetary migration process induced by the interaction of the planets with a protoplanetary disk. If the migration is slow enough (adiabatic) next to a mean motion resonance, the two planets will also be in apsidal corotation. The recently refined orbital parameters of the sys…
We develop a simple model for computing planetary formation based on the core instability model for the gas accretion and the oligarchic growth regime for the accretion of the solid core. In this model several planets can form simultaneously in the disc, a fact that has important implications specially for the changes in the dynamic of the planetesimals and the growth of the cores since we consider the collision between them as a source of potential growth. The type I and II migration of the embryos and the migration of the planetesimals due to the interaction with the disc of gas are also tak…
When Jupiter's Roche Lobe radius exceeded the scale height of the protoplanetary disk, Jupiter opened a gap in the disk. When the gap was wide enough, tidal torques from the disk interior and exterior to Jupiter were suppressed and migration continued on the accretion time scale. In the 'minimum solar nebula' about two Jupiter masses of gas remained between Jupiter and Saturn and about five Jupiter masses between Jupiter and Uranus. Unless all but a Jupiter mass of the outer disk was removed, Jupiter would have migrated into the Sun on the accretion time scale. So far, no mechanism has been po…
Torque fluctuations due to magnetorotational turbulence in proto-planetary disks may greatly influence the migration patterns and survival probabilities of nascent planets. Provided that the turbulence is a stationary stochastic process with finite amplitude and correlation time, the resulting diffusive migration can be described with a Fokker-Planck equation, which we reduce to an advection-diffusion equation. We calibrate the coefficients with existing turbulent-disk simulations and mean-migration estimates, and solve the equation both analytically and numerically. Diffusion tends to dominat…
We study the effect of a Jovian planet on the gas distribution of a protoplanetary disc, using a new numerical scheme that allows us to take into consideration the global evolution of the disc, down to an arbitrarily small inner physical radius. We find that Jovian planets do not open cavities in the inner part of the disc (i.e. interior to their orbits) unless (a) the inner physical edge of the disc is close to the planet's location or (b) the planet is much more massive than the disc. In all other cases the planet simply opens a gap in the gas density distribution, whose global profile is es…
Gap formation in a gas disk triggered by disk-planet tidal interaction is considered. Density waves launched by the planet are assumed to be damped as a result of their nonlinear evolution leading to shock formation and its subsequent dissipation. As a consequence wave angular momentum is transferred to the disk,leading to evolution of its surface density. Planetary migration is an important ingredient of the theory; effects of the planet-induced surface density perturbations on the migration speed are considered. A gap is assumed to form when a stationary solution for the surface density prof…
(Abridged) A diversity of 2:1 resonance configurations can be expected in extrasolar planetary systems, and their geometry can provide information about the origin of the resonances. Assembly during planet formation by the differential migration of planets due to planet-disk interaction is one scenario for the origin of mean-motion resonances in extrasolar planetary systems. The stable 2:1 resonance configurations that can be reached by differential migration of planets with constant masses and initially coplanar and nearly circular orbits are (1) anti-symmetric configurations with the mean-mo…
In a further development of a deterministic planet-formation model (Ida & Lin 2004), we consider the effect of type-I migration of protoplanetary embryos due to their tidal interaction with their nascent disks. During the early embedded phase of protostellar disks, although embryos rapidly emerge in regions interior to the ice line, uninhibited type-I migration leads to their efficient self-clearing. But, embryos continue to form from residual planetesimals at increasingly large radii, repeatedly migrate inward, and provide a main channel of heavy element accretion onto their host stars. Durin…
Many recent observational studies have concluded that planetary systems commonly exist in multiple-star systems. At least ~20%, and presumably a larger fraction of the known extrasolar planetary systems are associated with one or more stellar companions. These stellar companions normally exist at large distances from the planetary systems (typical projected binary separations are on the orders 100-10000AU) and are often faint (ranging from F to T spectral types). Yet, secular cyclic angular momentum exchange with these distant stellar companions can significantly alter the orbital configuratio…
During orbital migration of a giant extrasolar planet via ejection of planetesimals (Murray et al.~1998), inner mean motion resonances can be strong enough to cause planetesimals to graze or impact the star. We integrate numerically the motions of particles which pass through the 3:1 or 4:1 mean motion resonances of a migrating Jupiter mass planet. We find that many particles can be trapped in the 3:1 or 4:1 resonances and pumped to high enough eccentricities that they impact the star. This implies that for a planet migrating a substantial fraction of its semi major axis, a significant fractio…
The dust disk around Beta Pictoris must be produced by collision or by evaporation of orbiting Kuiper belt-like objects. Here we extend the already proposed Orbiting-Evaporating-Bodies (OEB) scenario in which the disk is a gigantic multi-cometary tail supplied by slowly evaporating bodies like Chiron. We show that the number of these OEBs must be several tens of millions, and that this is consistent with the number of bodies needed to explain the presence of CO and CI in the gaseous disk. We explore some possible origin of the required perturbation on the OEBs. If dust is produced by evaporati…
The migration of a planet through a gaseous disc causes the locations of their resonant interactions to drift and can alter the torques exerted between the planet and the disc. We analyse the time-dependent dynamics of a non-coorbital corotation resonance under these circumstances. The ratio of the resonant torque in a steady state to the value given by Goldreich & Tremaine (1979) depends essentially on two dimensionless quantities: a dimensionless turbulent diffusion time-scale and a dimensionless radial drift speed. When the drift speed is comparable to the libration speed and the viscosity …
We evaluate the coorbital corotation torque on a planet on a fixed circular orbit embedded in a viscous protoplanetary disk, for the case of a steady flow in the planet frame. This torque can be evaluated just from the flow properties at the separatrix between the librating (horseshoe) and circulating streamlines. A stationary solution is searched for the flow in the librating region. When used to evaluate the torque exerted by the circulating material of the outer and inner disk on the trapped material of the librating region, this solution leads to an expression of the coorbital corotation t…
The two planets about the star GJ 876 appear to have undergone extensive migration from their point of origin in the protoplanetary disk -- both because of their close proximity to the star (30 and 60 day orbital periods) and because of their occupying three stable orbital resonances at the 2:1 mean-motion commensurability. The resonances were most likely established by converging differential migration of the planets leading to capture into the resonances. A problem with this scenario is that continued migration of the system while it is trapped in the resonances leads to orbital eccentriciti…
This paper presents a parametric study of giant planet migration through the combined action of disk torques and planet-planet scattering. The torques exerted on planets during Type II migration in circumstellar disks readily decrease the semi-major axes, whereas scattering between planets increases the orbital eccentricities. This paper presents a parametric exploration of the possible parameter space for this migration scenario using two (initial) planetary mass distributions and a range of values for the time scale of eccentricity damping (due to the disk). For each class of systems, many r…
A residual planetesimal disk of mass 10-100 Earth masses remained in the outer solar system following the birth of the giant planets, as implied by the existence of the Oort cloud, coagulation requirements for Pluto, and inefficiencies in planet formation. Upon gravitationally scattering planetesimal debris, planets migrate. Orbital migration can lead to resonance capture, as evidenced here in the Kuiper and asteroid belts, and abroad in extra-solar systems. Finite sizes of planetesimals render migration stochastic ("noisy"). At fixed disk mass, larger (fewer) planetesimals generate more noise…
We calculate the linear torque exerted by a planet on a circular orbit on a disc containing a toroidal magnetic field. All fluid perturbations are singular at the so--called magnetic resonances, where the Doppler shifted frequency of the perturbation matches that of a slow MHD wave propagating along the field line. These lie on both sides of the corotation radius. Waves propagate outside the Lindblad resonances, and also in a restricted region around the magnetic resonances. The magnetic resonances contribute to a significant global torque which, like the Lindblad torque, is negative (positive…
This paper explores the intermediate-time dynamics of newly formed solar systems with a focus on possible mechanisms for planetary migration. We consider two limiting corners of the available parameter space -- crowded systems containing N=10 giant planets in the outer solar system, and solar systems with N=2 planets that are tidally interacting with a circumstellar disk. For a given set of initial conditions, dynamical relaxation leads to a well-defined distribution of possible solar systems. For each class of initial conditions, we perform large numbers of N-body simulations to obtain a stat…
Among the many new irregular satellites that have been discovered in the last five years, at least six are in the so-called Kozai resonance. Due to solar perturbations, the argument of pericenter of a satellite usually precesses from 0 to 360 degrees. However, at inclinations higher than 39.3 degrees and lower than 140.7 degrees a new kind of behavior occurs for which the argument of pericenter oscillates around +/-90 degrees. In this work we will concentrate on the orbital history of the saturnian satellite S/2000 S5 Kiviuq, one of the satellites currently known to be in such resonance Kiviuq…
A planetary system may undergo significant radial rearrangement during the early part of its lifetime. Planet migration can come about through interaction with the surrounding planetesimal disk and the gas disk--while the latter is still present--as well as through planet-planet interactions. We review the major proposed migration mechanisms in the context of the planet formation process, in our Solar System as well as in others.
We study the possibility that the mutual interactions between Jupiter and Saturn prevented Type II migration from driving these planets much closer to the Sun. Our work extends previous results by Masset and Snellgrove (2001), by exploring a wider set of initial conditions and disk parameters, and by using a new hydrodynamical code that properly describes for the global viscous evolution of the disk. Initially both planets migrate towards the Sun, and Saturn's migration tends to be faster. As a consequence, they eventually end up locked in a mean motion resonance. If this happens in the 2:3 re…
The kinetic theory is used to study the evolution of the self-gravitating disk of planetesimals. The effects of frequent collisions between planetesimals are taken into account by using a Krook integral in the Boltzmann kinetic equation. It is shown that as a result of an aperiodic collision-dissipative instability of small gravity disturbances the disk is subdivided into numerous dense fragments. These can eventually condense into the planetary sequence.
Close-in giant planets are thought to have formed in the cold outer regions of planetary systems and migrated inward, passing through the orbital parameter space occupied by the terrestrial planets in our own Solar System. We present dynamical simulations of the effects of a migrating giant planet on a disk of protoplanetary material and the subsequent evolution of the planetary system. We numerically investigate the dynamics of post-migration planetary systems over 200 million years using models with a single migrating giant planet, one migrating and one non-migrating giant planet, and exclud…
The modern study of the dynamics of stellar systems requires the use of high-performance computers. Indeed, an accurate modelization of the structure and evolution of self-gravitating systems like planetary systems, open clusters, globular clusters and galaxies imply the evaluation of body-body interaction over the whole size of the structure, a task that is computationally very expensive, in particular when it is performed over long intervals of time. In this report we give a concise overview of the main problems of stellar systems simulations and present some exciting results we obtained abo…
We investigate the evolution of protoplanets with different masses embedded in an accretion disk, via global fully three-dimensional hydrodynamical simulations. We consider a range of planetary masses extending from one and a half Earth's masses up to one Jupiter's mass, and we take into account physically realistic gravitational potentials of forming planets. In order to calculate accurately the gravitational torques exerted by disk material and to investigate the accretion process onto the planet, the flow dynamics has to be thoroughly resolved on long as well as short length scales. We achi…
HD 69830 exhibits radial velocity variations attributed to three planets as well as infrared emission attributed to a warm debris disk. Previous studies have developed models for the planet migration and mass growth (Alibert et al. 2006) and the replenishment of warm grains (Wyatt et al. 2007). We perform n-body integrations in order to explore the implications of these models for: 1) the excitation of planetary eccentricity, 2) the accretion and clearing of a putative planetesimal disk, 3) the distribution of planetesimal orbits following migration, and 4) the implications for the origin of t…
We investigate the survivability of Trojan-type companions of Neptune during primordial radial migration of the giant planets Jupiter, Saturn, Uranus, and Neptune. Loss of Neptune Trojans during planetary migration is not a random diffusion process. Rather, losses occur almost exclusively during discrete episodes when Trojan particles are swept by secondary resonances associated with mean-motion commensurabilities of Uranus with Neptune. The single greatest episode of loss ejects nearly 75% of existing Neptune Trojans and occurs just prior to Neptune reaching its final orbit.
We investigate the orbital evolution of planetesimals in a self-gravitating circumstellar disc in the size regime ($\sim 1-5000$ km) where the planetesimals behave approximately as test particles in the disc's non-axisymmetric potential. We find that the particles respond to the stochastic, regenerative spiral features in the disc by executing large random excursions (up to a factor of two in radius in $\sim 1000$ years), although typical random orbital velocities are of order one tenth of the Keplerian speed. The limited time frame and small number of planetesimals modeled does not permit us …
These notes provide an introduction to the theory of the formation and early evolution of planetary systems. Topics covered include the structure, evolution and dispersal of protoplanetary disks; the formation of planetesimals, terrestrial and gas giant planets; and orbital evolution due to gas disk migration, planetesimal scattering, planet-planet interactions, and tides.
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 <…
