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.
We calculate global $m=1$ modes with low pattern speed corresponding to introducing a finite eccentricity into a protoplanetary disc. We consider disc models which are either isolated or contain one or two protoplanets orbiting in an inner cavity. Global modes that are strongly coupled to inner protoplanets are found to have disc orbits which tend to have apsidal lines antialigned with respect to those of the inner protoplanets. Other modes corresponding to free disc modes may be global over a large range of length scales and accordingly be long lived. We consider the motion of a protoplanet i…
We present a mechanism related to the migration of giant protoplanets embedded in a protoplanetary disc whereby a giant protoplanet is caught up, before having migrated all the way to the central star, by a lighter outer giant protoplanet. This outer protoplanet may get captured into the 2:3 resonance with the more massive one, in which case the gaps that the two planets open in the disc overlap. Two effects arise, namely a squared mass weighted torque imbalance and an increased mass flow through the overlapping gaps from the outer disc to the inner disc, which both play in favour of an outwar…
We review results about protoplanetary disk models, protoplanet migration and formation of giant planets with migrating cores. We first model the protoplanetary nebula as an α-accretion disk and present steady state calculations for different values of αand gas accretion rate through the disk. We then review the current theories of protoplanet migration in the context of these models, focusing on the gaseous disk-protoplanet tidal interaction. According to these theories, the migration timescale may be shorter than the planetary formation timescale. Therefore we investigate planet formation in…
We calculate rates of Type I migration of protoplanets in a non-isothermal three-dimensional protoplanetary disk, building upon planet-disk models developed in previous work. We find that including the vertical thickness of the disk results in a decrease in the Type I migration rate by a factor of ~2 from a two-dimensional disk. The vertical temperature variation has only a modest effect on migration rates since the torques at the midplane are weighted heavily both because the density and the perturbing potential are maximized at the midplane. However, temperature perturbations resulting from …
We investigate the gravitational interaction of a Jovian mass protoplanet with a gaseous disc with aspect ratio and kinematic viscosity expected for the protoplanetary disc from which it formed. Different disc surface density distributions have been investigated. We focus on the tidal interaction with the disc with the consequent gap formation and orbital migration of the protoplanet. Nonlinear hydrodynamic simulations are employed using three independent numerical codes. A principal result is that the direction of the orbital migration is always inwards and such that the protoplanet reaches t…
Planet evolution is tightly connected to the dynamics of both distant and close disk material. Hence, an appropriate description of disk-planet interaction requires global and high resolution computations, which we accomplish by applying a Nested-Grid method. Through simulations in two and three dimensions, we investigate how migration and accretion are affected by long and short range interactions. For small mass objects, 3D models provide longer growth and migration time scales than 2D ones do, whereas time lengths are comparable for large mass planets.
We present the results of hydrodynamical simulations of low mass protoplanets embedded in circumbinary accretion disks. The aim is to examine the migration and long term orbital evolution of the protoplanets, in order to establish the stability properties of planets that form in circumbinary disks. Simulations were performed using a grid--based hydrodynamics code. First we present a set of calculations that study how a binary interacts with a circumbinary disk. We evolve the system for 10^5 binary orbits, which is the time needed for the system to reach a quasi-equilibrium state. From this tim…
Precise Doppler searches for extrasolar planets find a surfeit of planets with orbital periods of 3-4 days, and no planets with orbital periods less than 3 days. The circumstellar distance, R_0, where small grains in a protoplanetary disk reach sublimation temperature (~1500 K) corresponds to a period of ~6 days. Interior to R_0, turbulent accretion due to magneto-rotational instability may evacuate the disk center. We suggest that planets with orbital periods of 3-4 days are so common because migrating planets halt once this evacuated region contains the sites of their exterior 2:1 Lindblad r…
Migration of protoplanets in their gaseous host disks may be largely responsible for the observed orbital distribution of extrasolar planets. Recent simulations have shown that the magnetorotational turbulence thought to drive accretion in protoplanetary disks can affect migration by turning it into an orbital random walk. However, these simulations neglected the disk's ionization structure. Low ionization fraction near the midplane of the disk can decouple the magnetic field from the gas, forming a dead zone with reduced or no turbulence. Here, to understand the effect of dead zones on protop…
We present new results related to the coupled evolution of a two giant planet system embedded in a protoplanetary disk, in which a Saturn mass protoplanet is trapped in an outer mean motion resonance with a Jupiter mass protoplanet. The gaps opened in the disk by the two planets overlap, therefore the two planet system exchanges angular momentum with the disk at the inner planet's Inner Lindblad Resonances (ILRs) and at the outer planet Outer Lindblad Resonances (OLRs). Since the torques are proportional to the square of the planet masses, and since the inner planet is about three times more m…
We consider two protoplanets gravitationally interacting with each other and a protoplanetary disc. The two planets orbit interior to a tidally maintained disc cavity while the disc interaction indices inward migration. When the migration is slow enough, the more rapidly migrating outer protoplanet approaches and becomes locked in a 2:1 commensurability with the inner one. This is maintained in subsequent evolution. We study this evolution using a simple anaytic model, full hydrodynamic 2D simulations of the disc planet system and longer time N body integrations incorporating simple prescripti…
(Abridged) We consider models of gas giant planets forming in protoplanetary disks consisting of solid cores with gaseous envelopes in contact with their critical Hill spheres while accreting gas from the surrounding disk.We suppose the luminosity derives from gas accretion alone.We label such models as type A and follow their evolution which may occur on a time scale similar to the protostellar disk lifetime until rapid gas accretion. We consider another set of models, we label type B, with a free surface, powered by gravitational contraction, while accreting through a disk.We find these mode…
More than 30 extra-solar Jupiter-like planets have shorter periods than the planet Mercury. It is generally accepted that they formed further out, and migrated inwards. In order to be driven by tidal torques from the gaseous disc, the disc exterior to the planet had to contain about a planetary mass. The fact that the planets stopped migrating means that their outer disc was removed. We suggest that the outer disc was accreted by the planet. In this scenario, the endgame is a race. The planet survives if it accretes its outer disc before being accreted by the star. The winner is determined sol…
We investigate the migration rates of high-mass protoplanets embedded in accretion discs via two and three-dimensional hydrodynamical simulations. The simulations follow the planet's radial motion and employ a nested-grid code that allows for high resolution close to the planet. We concentrate on the possible role of the coorbital torques in affecting migration rates. We analyse two cases: (a) a Jupiter-mass planet in a low-mass disc and (b) a Saturn-mass planet in a high-mass disc. Planet migration in case (b) is much more susceptible to coorbital torques than in case (a). We find that the co…
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…
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…
Protoplanet eccentricities of e >~ H/r can slow or reverse migration, but previous 2D studies have shown that gravitational scattering cannot maintain significant planet eccentricities against disc-induced damping. We simulate the evolution of low-mass protoplanetary swarms in three dimensions. The aim is to examine both protoplanet survival rates and the dynamical structure of the resulting planetary systems, and to compare them with 2D simulations. We present results from a 3D hydrodynamic simulation of eight protoplanets embedded in a protoplanetary disc. We also present a suite of simulati…
(Abridged).We present the results of MHD simulations of low mass protoplanets interacting with turbulent disks. We calculate the orbital evolution of `planetesimals' and protoplanets with masses in the range 0 < m_p < 30 M_Earth. Planetesimals and protoplanets undergo stochastic migration due to interaction with turbulent density fluctuations. Over run times of ~ 150 planet orbits, stochastic migration dominates over type I migration for many models. Fourier analysis of the torques experienced by planets indicates that the torque fluctuations contain components with significant power whose tim…
(Abridged) We present global disc and local shearing box simulations of planets interacting with a MHD turbulent disc. We examine the torque exerted by the disc on the embedded planets as a function of planet mass, and thus make a first study of orbital migration of planets due to interaction with turbulent discs. Global simulations were performed for a disc with H/R=0.07 and planet masses M_p=3,10,30 Earth masses, and 3 Jupiter masses. Shearing box runs were performed for values of (M_p/M_*)/(H/R)^3=0.1,0.3,1.0 and 2.0, M_* being the central mass. These allow embedded and gap forming planets …
Young planets embedded in their protoplanetary disk interact gravitationally with it leading to energy and angular momentum exchange. This interaction determines the evolution of the planet through changes to the orbital parameters. We investigate changes in the orbital elements of a 20 Earth--mass planet due to the torques from the disk. We focus on the non-linear evolution of initially non-vanishing eccentricity $e$ and/or inclination $i$. We treat the disk as a two- or three-dimensional viscous fluid and perform hydrodynamical simulations with an embedded planet. We find rapid exponential d…
The evolution of a system consisting of a protoplanetary disc with two embedded Jupiter sized planets is studied numerically. The disc is assumed to be flat and non-self gravitating, which is modeled by the planar (two-dimensional) Navier-Stokes equations. The mutual gravitational interaction of the planets and the star, and the gravitational torques of the disc acting on the planets and the central star are included. The planets have an initial mass of one Jupiter mass $M_{Jup}$ each and the radial distances from the star are one and two semi-major axis of Jupiter, respectively. During the ev…
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…
We performed a series of hydro-dynamic simulations to investigate the orbital migration of a Jovian planet embedded in a proto-stellar disk. In order to take into account of the effect of the disk's self gravity, we developed and adopted an \textbf{Antares} code which is based on a 2-D Godunov scheme to obtain the exact Reimann solution for isothermal or polytropic gas, with non-reflecting boundary conditions. Our simulations indicate that in the study of the runaway (type III) migration, it is important to carry out a fully self consistent treatment of the gravitational interaction between th…
We present a new computation of the linear tidal interaction of a protoplanetary core with a thin gaseous disc in which it is fully embedded. For the first time a discussion of the orbital evolution of cores with eccentricity (e) significantly larger than the gas-disc scale height to radius ratio (H/r) is given. We find that the direction of orbital migration reverses for e>1.1H/r. This occurs as a result of the orbital crossing of resonances in the disc that do not overlap the orbit when the eccentricity is very small. Simple expressions giving approximate fits to the eccentricity damping rat…
Aims: We investigate the effect of including a proper energy balance on the interaction of a low-mass planet with a protoplanetary disk. Methods: We use a three-dimensional version of the RODEO method to perform hydrodynamical simulations including the energy equation. Radiation is included in the flux-limited diffusion approach. Results: The sign of the torque depends sensitively on the ability of the disk to radiate away the energy generated in the immediate surroundings of the planet. In the case of high opacity, corresponding to the dense inner regions of protoplanetary disks, migration is…
The growth of Jovian mass planets during migration in their protoplanetary disks is one of the most important problems that needs to be solved in light of observations of the exosolar planets. Studies of the migration of planets in standard gas disk models routinely show that migration is too fast to form Jovian planets, and that such migrating planetary cores generally plunge into the central stars in less than a Myr. In previous work, we have shown that a poorly ionized, less viscous region in a protoplanetary disk called a dead zone slows down the migration of fixed-mass planets. In this pa…
We present the results of a three dimensional, locally isothermal, non-self-gravitating SPH code which models protoplanetary disks with two fluids: gas and dust. We ran simulations of a 1 Msun star surrounded by a 0.01 Msun disk comprising 99% gas and 1% dust in mass and extending from 0.5 to ~300 AU. The grain size ranges from 0.001 mm to 10 m for the low resolution (~25 000 SPH particles) simulations and from 0.1 mm to 10 cm for the high resolution (~160 000 SPH particles) simulations. Dust grains are slowed down by the sub-Keplerian gas and lose angular momentum, forcing them to migrate tow…
We present a new 3D continuum radiative transfer code, MCFOST, based on a Monte-Carlo method. The reliability and efficiency of the code is tested by comparison with five different radiative transfer codes previously tested by Pascucci et al., 2004, using a 2D disk configuration. When tested against the same disk configuration, no significant difference is found between the temperature and SED calculated with MCFOST and with the other codes. The computed values are well within the range of values computed by the other codes. The code-to-code differences are small, they rarely exceed 10% and ar…
Young planets interact with their parent gas disks through tidal torques. An imbalance between inner and outer torques causes bodies of mass $\ga 0.1$ Earth masses to lose angular momentum and migrate inward rapidly relative to the disk; this is known as ``Type I'' migration. However, protoplanets that grow to gas giant mass, O($10^2) M_\oplus$, open a gap in the disk and are subsequently constrained to migrate more slowly, locked into the disk's viscous evolution in what is called "Type II" migration. In a young planetary system, both Type I and Type II bodies likely coexist; if so, different…
We study and review disk protoplanet interactions using local shearing box simulations. These suffer the disadvantage of having potential artefacts arising from periodic boundary conditions but the advantage, when compared to global simulations, of being able to capture much of the dynamics close to the protoplanet at high resolution for low computational cost. Cases with and without self sustained MHD turbulence are considered. The conditions for gap formation and the transition from type I migration are investigated and found to depend on whether the single parameter M_p R^3/(M_* H^3), with …
We analyse the non-linear, three-dimensional response of a gaseous, viscous protoplanetary disc to the presence of a planet of mass ranging from one Earth mass (1 M$_e$) to one Jupiter mass (1 M$_J$) by using the ZEUS hydrodynamics code. We determine the gas flow pattern, and the accretion and migration rates of the planet. The planet is assumed to be in a fixed circular orbit about the central star. It is also assumed to be able to accrete gas without expansion on the scale of its Roche radius. Only planets with masses $M \gsim 0.1$ M$_J$ produce significant perturbations in the disc's surfac…
We present a method for calculating the radiative tranfer on a protoplanetary disk perturbed by a protoplanet. We apply this method to determine the effect on the temperature structure within the photosphere of a passive circumstellar disk in the vicinity of a small protoplanet of up to 20 Earth masses. The gravitational potential of a protoplanet induces a compression of the disk material near it, resulting in a decrement in the density at the disk's surface. Thus, an isodensity contour at the height of the photosphere takes on the shape of a well. When such a well is illuminated by stellar i…
Context: Our previous models of a giant planet migrating through an inner protoplanet/planetesimal disk find that the giant shepherds a portion of the material it encounters into interior orbits, whilst scattering the rest into external orbits. Scattering tends to dominate, leaving behind abundant material that can accrete into terrestrial planets. Aims: We add to the possible realism of our model by simulating type I migration forces which cause an inward drift, and strong eccentricity and inclination damping of protoplanetary bodies. This extra dissipation might be expected to enhance shephe…
We investigate the evolution of two-planet systems embedded in a protoplanetary disc, which are composed of a Jupiter-mass planet plus another body located further out in the disc. We consider outermost planets with masses ranging from 10 earth masses to 1 M_J. We also examine the case of outermost bodies with masses < 10 earth masses (M_E). Differential migration of the planets due to disc torques leads to different evolution outcomes depending on the mass of the outer protoplanet. For planets with mass < 3.5 M_E the type II migration rate of the giant exceeds the type I migration rate of the…
We analyze the line radiative transfer in protoplanetary disks using several approximate methods and a well-tested Accelerated Monte Carlo code. A low-mass flaring disk model with uniform as well as stratified molecular abundances is adopted. Radiative transfer in low and high rotational lines of CO, C18O, HCO+, DCO+, HCN, CS, and H2CO is simulated. The corresponding excitation temperatures, synthetic spectra, and channel maps are derived and compared to the results of the Monte Carlo calculations. A simple scheme that describes the conditions of the line excitation for a chosen molecular tran…
Aims: In order to understand the first stages of planet formation, when tiny grains aggregate to form planetesimals, one needs to simultaneously model grain growth, vertical settling and radial migration of dust in protoplanetary disks. In this study, we implement an analytical prescription for grain growth into a 3D two-phase hydrodynamics code to understand its effects on the dust distribution in disks. Methods: Following the analytic derivation of Stepinski & Valageas (1997), which assumes that grains stick perfectly upon collision, we implement a convenient and fast method of following gra…
We present a new continuum 3D radiative transfer code, MCFOST, based on a Monte-Carlo method. MCFOST can be used to calculate (i) monochromatic images in scattered light and/or thermal emission, (ii) polarisation maps, (iii) interferometric visibilities, (iv) spectral energy distributions and (v) dust temperature distributions of protoplanetary disks. Several improvements to the standard Monte Carlo method are implemented in MCFOST to increase efficiency and reduce convergence time, including wavelength distribution adjustments, mean intensity calculations and an adaptive sampling of the radia…
This paper presents a fully three-dimensional radiative hydrodymanics simulation with realistic opacities for a gravitationally unstable 0.07 Msun disk around a 0.5 Msun star. We address the following aspects of disk evolution: the strength of gravitational instabilities under realistic cooling, mass transport in the disk that arises from GIs, comparisons between the gravitational and Reynolds stresses measured in the disk and those expected in an alpha-disk, and comparisons between the SED derived for the disk and SEDs derived from observationally determined parameters. The mass transport in …
In this paper, we study a simple model of an orbiting protoplanet in a central magnetospheric cavity, the entry into such a cavity having been proposed as a mechanism for halting inward orbital migration. We have calculated the gravitational interaction of the protoplanet with the magnetosphere using a local model and determined the rate of evolution of the orbit. The interaction is found to be determined by the outward flux of MHD waves and thus the possibility of the existence of such waves in the cavity is significant. The estimated orbital evolution rates due to gravitational and other int…
We investigate the fast (type III) migration regime of high-mass protoplanets orbiting in protoplanetary disks. This type of migration is dominated by corotational torques. We study the details of flow structure in the planet's vicinity, the dependence of migration rate on the adopted disc model, and the numerical convergence of models (independence of certain numerical parameters such as gravitational softening). We use two-dimensional hydrodynamical simulations with adaptive mesh refinement,based on the FLASH code with improved time-stepping scheme. We perform global disk simulations with su…
We have performed Smoothed Particle Hydrodynamics (SPH) simulations to study the time evolution of one and two protoplanets embedded in a protoplanetary accretion disc. We investigate accretion and migration rates of a single protoplanet depending on several parameters of the protoplanetary disc, mainly viscosity and scale height. Additionally, we consider the influence of a second protoplanet in a long time simulation and examine the migration of the two planets in the disc, especially the growth of eccentricity and chaotic behaviour. One aim of this work is to establish the feasibility of SP…
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.
[Abridged] We evaluate the coorbital corotation torque on a migrating protoplanet. The coorbital torque is assumed to come from orbit crossing fluid elements which exchange angular momentum with the planet when they execute a U-turn at the end of horseshoe streamlines. When the planet migrates inward, the fluid elements of the inner disk undergo one such exchange as they pass to the outer disk. The angular momentum they gain is removed from the planet, and this corresponds to a negative contribution to the corotation torque, which scales with the drift rate. In addition, the material trapped i…
In the core accretion model, gas giant formation is a race between growth and migration; for a core to become a jovian planet, it must accrete its envelope before it spirals into the host star. We use a multizone numerical model to extend our previous investigation of the "window of opportunity" for gas giant formation within a disk. When the collision cross-section enhancement due to core atmospheres is taken into account, we find that a broad range of protoplanetary disks posses such a window.
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 …
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 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…
We study the interaction of a low-mass planet with a protoplanetary disk with a realistic treatment of the energy balance by doing radiation-hydrodynamical simulations. We look at accretion and migration rates and compare them to isothermal studies. We used a three-dimensional version of the hydrodynamical method RODEO, together with radiative transport in the flux-limited diffusion approach. The accretion rate, as well as the torque on the planet, depend critically on the ability of the disk to cool efficiently. For densities appropriate to 5 AU in the solar nebula, the accretion rate drops b…
