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 found to be very sensitive to the initial eccentricity of the planet and of the planetesimals. A planetary eccentricity as low as 0.05 is enough to smear out all the resonant structures, except for the most massive planets. The planetesimals also initially have to be on orbits with a mean eccentricity of less than than 0.1 in order to keep the resonant clumps visible. This numerical work extends previous analytical studies and provides a collection of disk images that may help in interpreting the observations of structures in debris disks. Overall, it shows that stringent conditions must be fulfilled to obtain observable resonant structures in debris disks. Theoretical models of the origin of planetary migration will therefore have to explain how planetary systems remain in a suitable configuration to reproduce the observed structures.
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…
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…
Planets orbiting a planetesimal circumstellar disc can migrate inward from their initial positions because of dynamical friction between planets and planetesimals. The migration rate depends on the disc mass and on its time evolution. Planets that are embedded in long-lived planetesimal discs, having total mass of $10^{-4}-0.01 M_{\odot}$, can migrate inward a large distance and can survive only if the inner disc is truncated or because of tidal interaction with the star. In this case the semi-major axis, a, of the planetary orbit is less than 0.1 AU. Orbits with larger $a$ are obtained for sm…
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…
This paper describes a model which can explain the observed clumpy structures of debris disks. Clumps arise because after a planetary system forms its planets migrate due to angular momentum exchange with the remaining planetesimals. Outward migration of the outermost planet traps planetesimals outside its orbit into its resonances and resonant forces cause azimuthal structure in their distribution. The model is based on numerical simulations of planets of different masses, Mpl, migrating at different rates, dapl/dt, through a dynamically cold (e<0.01) planetesimal disk initially at a semimajo…
In this paper, we further develop the model for the migration of planets introduced in Del Popolo et al. (2001). We first model the protoplanetary nebula as a time-dependent accretion disc and find self-similar solutions to the equations of the accretion disc that give to us explicit formulas for the spatial structure and the temporal evolution of the nebula. These equations are then used to obtain the migration rate of the planet in the planetesimal disc and to study how the migration rate depends on the disc mass, on its time evolution and on some values of the dimensionless viscosity parame…
In this paper, we further develop the model for the migration of planets introduced in Del Popolo, Gambera and Ercan, and extended to time-dependent planetesimal accretion disks in Del Popolo and Eksi. More precisely, the assumption of Del Popolo and Eksi that the surface density in planetesimals is proportional to that of gas is released. Indeed, the evolution of the radial distribution of solids is governed by many processes: gas-solid coupling, coagulation, sedimentation, evaporation/condensation, so that the distribution of planetesimals emerging from a turbulent disk does not necessarily …
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…
Substantial orbital migration of massive planets may occur in most extrasolar planetary systems. Since migration is likely to occur after a significant fraction of the dust has been locked up into planetesimals, ubiquitous migration could reduce the probability of forming terrestrial planets at radii of the order of 1 au. Using a simple time dependent model for the evolution of gas and solids in the disk, I show that replenishment of solid material in the inner disk, following the inward passage of a giant planet, is generally inefficient. Unless the timescale for diffusion of dust is much sho…
The photospheres of stars hosting planets have larger metallicity than stars lacking planets. In the present work we study the possibility of an earlier metal enrichment of the photospheres by means of impacting planetesimals during the first 20-30Myr. Here we explore this contamination process by simulating the interactions of an inward migrating planet with a disc of planetesimal interior to its orbit. The results show the percentage of planetesimals that fall on the star. We identified the dependence of the planet's eccentricity ($e_p$) and time scale of migration ($τ$) on the rate of infal…
The existence of the Oort Comet Cloud, the Kuiper Belt, and plausible inefficiencies in planetary core formation, all suggest that there was once a residual planetesimal disk of mass 10-100 Earth-masses in the vicinity of the giant planets following their formation. Since removal of this disk requires an exchange of orbital energy and angular momentum with the planets, significant planetary migration can ensue. The planet migration phenomenon is examined numerically by evolving the orbits of the giant planets while they are embedded in a planetesimal disk having a mass of M_d=10 to 200 Earth-m…
We examine the effect of a rapidly migrating protoplanet on a ring of planetesimals. The eccentricities of the planetesimals are usually increased by $Δe \in (0.01, 0.1)$, with the exact increase being proportional to the protoplanet's mass, and inversely proportional to its migration rate. The eccentricity distribution is also substantially changed from a Rayleigh distribution. We discuss the possible implications for further planet formation, and suggest that the rapid passage of a protoplanet may not prevent the planetesimal disc from forming further planets.
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…
We investigate the process of an inhomogeneous planetesimal disk evolution caused by the planetesimal-planetesimal gravitational scattering. We develop a rather general approach based on the kinetic theory which self-consistently describes the evolution in time and space of both the disk surface density and its kinematic properties -- dispersions of eccentricity and inclination. The gravitational scattering of planetesimals is assumed to be in the dispersion-dominated regime which considerably simplifies analytical treatment. The resultant equations are of advection-diffusion type. Distance de…
(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…
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…
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 investigate how the statistical distribution of extrasolar planets may be combined with knowledge of the host stars' metallicity to yield constraints on the migration histories of gas giant planets. At any radius, planets that barely manage to form around the lowest metallicity stars accrete their envelopes just as the gas disk is being dissipated, so the lower envelope of planets in a plot of metallicity vs semi-major axis defines a sample of non-migratory planets that will have suffered less than average migration subsequent to gap opening. Under the assumption that metallicity largely co…
We study the migration and resonant capture of planetesimals in a planetary system consisting of a gaseous disc analogous to the primordial solar nebula and a Neptune-like planet. Using a simple treatment of the drag force we find that planetesimals are mainly trapped in the 3:2 and 2:1 resonances and that the resonant populations are correlated with the gaseous drag strength in a sense that the 3:2 resonant population increases with the stronger gaseous drag, but the 2:1 resonant population does not. Since planetesimals can lead to the formation of larger bodies similar to asteroids and Kuipe…
We use the model for the migration of planets introduced in Del Popolo, Yesilyurt & Ercan (2003) to calculate the observed mass and semimajor axis distribution of extra-solar planets. The assumption that the surface density in planetesimals is proportional to that of gas is relaxed, and in order to describe disc evolution we use a method which, using a series of simplifying assumptions, is able to simultaneously follow the evolution of gas and solid particles for up to $10^7 {\rm yr}$. The distribution of planetesimals obtained after $10^7 {\rm yr}$ is used to study the migration rate of a gia…
We present the results of N--body simulations which examine the effect that gas giant planet migration has on the formation of terrestrial planets. The models incorporate a 0.5 Jupiter mass planet undergoing type II migration through an inner protoplanet--planetesimal disk, with gas drag included. Each model is initiated with the inner disk being at successively increased levels of maturity, so that it is undergoing either oligarchic or giant impact style growth as the gas giant migrates. In all cases, a large fraction of the disk mass survives the passage of the giant, either by accreting int…
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…
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 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.
Some large trans-Neptunian objects could be formed by the compression of rarefied dust condensations, but not by the accumulation of smaller planetesimals. A considerable portion of near-Earth objects could have come from the trans-Neptunian region. Our runs of the evolution of thousands orbits of Jupiter-family comets under the gravitational influence of planets showed that former Jupiter-family comets collide with the terrestrial planets mostly from orbits with aphelia located deep inside Jupiter's orbit.
Tsiganis et al. (2005) have proposed that the current orbital architecture of the outer solar system could have been established if it was initially compact and Jupiter and Saturn crossed the 2:1 orbital resonance by divergent migration. The crossing led to close encounters among the giant planets, but the orbital eccentricities and inclinations were damped to their current values by interactions with planetesimals. Brunini (2006) has presented widely publicized numerical results showing that the close encounters led to the current obliquities of the giant planets. We present a simple analytic…
The process of gap formation by a growing planetary embryo embedded in a planetesimal disk is considered. It is shown that there exists a single parameter characterizing this process, which represents the competition between the gravitational influence of the embryo and planetesimal-planetesimal scattering. For realistic assumptions about the properties of the planetesimal disk and the planetary embryo, a gap is opened long before the embryo can accrete all the bodies within its region of influence. The implication of this result is that the embryo stops growing and, thus, large bodies formed …
This pedagogical review covers an unsolved problem in the theory of protoplanetary disks: the growth of dust grains into planetesimals, solids at least a kilometer in size. I summarize timescale constraints imposed on planetesimal formation by circumstellar disk observations, analysis of meteorites, and aerodynamic radial migration. The infall of ~meter-sized solids in a hundred years is the most stringent constraint. I review proposed mechanisms for planetesimal formation. Collisional coagulation models are informed by laboratory studies of microgravity collisions. The gravitational collapse …
We present a series of calculations aimed at examining how an inner system of planetesimals/protoplanets, undergoing terrestrial planet formation, evolves under the influence of a giant planet undergoing inward type II migration through the region bounded between 5 - 0.1 AU. We find that > 60% of the solids disk survives by being scattered by the giant planet into external orbits. Planetesimals are scattered outward almost as efficiently as protoplanets, resulting in the regeneration of a solids disk where dynamical friction is strong and terrestrial planet formation is able to resume. A simul…
We investigate the formation of planetesimals via the gravitational instability of solids that have settled to the midplane of a circumstellar disk. Vertical shear between the gas and a subdisk of solids induces turbulent mixing which inhibits gravitational instability. Working in the limit of small, well-coupled particles, we find that the mixing becomes ineffective when the surface density ratio of solids to gas exceeds a critical value. Solids in excess of this precipitation limit can undergo midplane gravitational instability and form planetesimals. However, this saturation effect typicall…
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 …
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 numerically explore planet formation around alpha Cen A by focusing on the crucial planetesimals-to-embryos phase. Our code computes the relative velocity distribution, and thus the accretion vs. fragmentation trend, of planetesimal populations having any given size distribution. This is a critical aspect of planet formation in binaries since the pericenter alignment of planetesimal orbits due to the gravitational perturbations of the companion star and to gas friction strongly depends on size. We find that, for the nominal case of a MMSN gas disc, the region beyond 0.5AU from the primary i…
The first challenge in the formation of both terrestrial planets and the cores of gas giants is the retention of grains in protoplanetary disks. In most regions of these disks, gas attains sub-Keplerian speeds as a consequence of a negative pressure gradient. Hydrodynamic drag leads to orbital decay and depletion of the solid material in the disk, with characteristic timescales as short as only a few hundred years for meter-sized objects at 1 AU. In this paper, we suggest a particle retention mechanism which promotes the accumulation of grains and the formation of planetesimals near the water …
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…
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 have investigated effects of early stellar encounters on a protoplanetary disk (planetesimal disk) and found that they can explain the high eccentricities and inclinations observed in the outer part ($>42$AU) of the Edgeworth-Kuiper Belt (EKB). The proto-sun is considered as a member of a stellar aggregation that undergoes dissolution on a timescale $\sim 10^8$yrs, such that a planetesimal disk experiences a flyby encounter at pericenter distance ($q$) on the order of 100AU. We simulated dynamical evolution of a planetesimal (test particle) disk perturbed by a passing star. We show that the…
Planetary embryos embedded in a gas disc suffer a decay in semimajor axis -- type I migration -- due to the asymmetric torques produced by the interior and exterior wakes raised by the body (Goldreich & Tremaine 1980; Ward 1986). This presents a challenge for standard oligarchic approaches to forming the terrestrial planets (Kokubo & Ida 1998) as the timescale to grow the progenitor objects near 1 AU is longer than that for them to decay into the Sun. In this paper we investigate the middle and late stages of oligarchic growth using both semi-analytic methods (based upon Thommes et al. 2003) a…
