Planetary migration in evolving planetesimals discs

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 model is coupled to migration model introduced in Del Popolo et al. (2001) in order to obtain the migration rate of the planet in the planetesimal. We find that the properties of solids known to exist in protoplanetary systems, together with reasonable density profiles for the disk, lead to a characteristic radius in the range 0.03-0.2 AU for the final semi-major axis of the giant planet.Comment: IJMP A in prin

Protoplanetary Disk Turbulence Driven by the Streaming Instability: Non-Linear Saturation and Particle Concentration

We present simulations of the non-linear evolution of streaming instabilities in protoplanetary disks. The two components of the disk, gas treated with grid hydrodynamics and solids treated as superparticles, are mutually coupled by drag forces. We find that the initially laminar equilibrium flow spontaneously develops into turbulence in our unstratified local model. Marginally coupled solids (that couple to the gas on a Keplerian time-scale) trigger an upward cascade to large particle clumps with peak overdensities above 100. The clumps evolve dynamically by losing material downstream to the radial drift flow while receiving recycled material from upstream. Smaller, more tightly coupled solids produce weaker turbulence with more transient overdensities on smaller length scales. The net inward radial drift is decreased for marginally coupled particles, whereas the tightly coupled particles migrate faster in the saturated turbulent state. The turbulent diffusion of solid particles, measured by their random walk, depends strongly on their stopping time and on the solids-to-gas ratio of the background state, but diffusion is generally modest, particularly for tightly coupled solids. Angular momentum transport is too weak and of the wrong sign to influence stellar accretion. Self-gravity and collisions will be needed to determine the relevance of particle overdensities for planetesimal formation.Comment: Accepted for publication in ApJ (17 pages). Movies of the simulations can be downloaded at http://www.mpia.de/~johansen/research_en.ph

Dusty gas with SPH - II. Implicit timestepping and astrophysical drag regimes

In a companion paper (Laibe & Price 2011b), we have presented an algorithm for simulating two-fluid gas and dust mixtures in Smoothed Particle Hydrodynamics (SPH). In this paper, we develop an implicit timestepping method that preserves the exact conservation of the both linear and angular momentum in the underlying SPH algorithm, but unlike previous schemes, allows the iterations to converge to arbitrary accuracy and is suited to the treatment of non- linear drag regimes. The algorithm presented in Paper I is also extended to deal with realistic astrophysical drag regimes, including both linear and non-linear Epstein and Stokes drag. The scheme is benchmarked against the test suite presented in Paper I, including i) the analytic solutions of the dustybox problem and ii) solutions of the dustywave, dustyshock, dustysedov and dustydisc obtained with explicit timestepping. We find that the implicit method is 1- 10 times faster than the explicit temporal integration when the ratio r between the the timestep and the drag stopping time is 1 < r < 1000.Comment: Accepted for publication in MNRA

Growth and migration of solids in evolving protostellar disks I: Methods and Analytical tests

This series of papers investigates the early stages of planet formation by modeling the evolution of the gas and solid content of protostellar disks from the early T Tauri phase until complete dispersal of the gas. In this first paper, I present a new set of simplified equations modeling the growth and migration of various species of grains in a gaseous protostellar disk evolving as a result of the combined effects of viscous accretion and photo-evaporation from the central star. Using the assumption that the grain size distribution function always maintains a power-law structure approximating the average outcome of the exact coagulation/shattering equation, the model focuses on the calculation of the growth rate of the largest grains only. The coupled evolution equations for the maximum grain size, the surface density of the gas and the surface density of solids are then presented and solved self-consistently using a standard 1+1 dimensional formalism. I show that the global evolution of solids is controlled by a leaky reservoir of small grains at large radii, and propose an empirically derived evolution equation for the total mass of solids, which can be used to estimate the total heavy element retention efficiency in the planet formation paradigm. Consistency with observation of the total mass of solids in the Minimum Solar Nebula augmented with the mass of the Oort cloud sets strong upper limit on the initial grain size distribution, as well as on the turbulent parameter \alphat. Detailed comparisons with SED observations are presented in a following paper.Comment: Submitted to ApJ. 23 pages and 13 figure

Dust Size Growth and Settling in a Protoplanetary Disk

We have studied dust evolution in a quiescent or turbulent protoplanetary disk by numerically solving coagulation equation for settling dust particles, using the minimum mass solar nebular model. As a result, if we assume an ideally quiescent disk, the dust particles settle toward the disk midplane to form a gravitationally unstable layer within 2x10^3 - 4x10^4 yr at 1 - 30 AU, which is in good agreement with an analytic calculation by Nakagawa, Sekiya, & Hayashi (1986) although they did not take into account the particle size distribution explicitly. In an opposite extreme case of a globally turbulent disk, on the other hand, the dust particles fluctuate owing to turbulent motion of the gas and most particles become large enough to move inward very rapidly within 70 - 3x10^4 yr at 1 - 30 AU, depending on the strength of turbulence. Our result suggests that global turbulent motion should cease for the planetesimal formation in protoplanetary disks.Comment: 27 pages, 8 figures, accepted for publication in the Ap

The Eccentricity-Mass Distribution of Exoplanets: Signatures of Different Formation Mechanisms?

We examine the distributions of eccentricity and host star metallicity of exoplanets as a function of their mass. Planets with M sin i >~ 4 M_J have an eccentricity distribution consistent with that of binary stars, while planets with M sin i <~ 4 M_J are less eccentric than binary stars and more massive planets. In addition, host star metallicities decrease with planet mass. The statistical significance of both of these trends is only marginal with the present sample of exoplanets. To account for these trends, we hypothesize that there are two populations of gaseous planets: the low-mass population forms by gas accretion onto a rock-ice core in a circumstellar disk and is more abundant at high metalliticities, and the high-mass population forms directly by fragmentation of a pre-stellar cloud. Planets of the first population form in initially circular orbits and grow their eccentricities later, and may have a mass upper limit from the total mass of the disk that can be accreted by the core. The second population may have a mass lower limit resulting from opacity-limited fragmentation. This would roughly divide the two populations in mass, although they would likely overlap over some mass range. If most objects in the second population form before the pre-stellar cloud becomes highly opaque, they would have to be initially located in orbits larger than ~30 AU, and would need to migrate to the much smaller orbits in which they are observed. The higher mean orbital eccentricity of the second population might be caused by the larger required intervals of radial migration, and the brown dwarf desert might be due to the inability of high-mass brown dwarfs to migrate inwards sufficiently in radius.Comment: 7 pages, 4 figures. Version with expanded discussion section. Accepted for publication in A&

Predictions for the frequency and orbital radii of massive extrasolar planets

We investigate the migration of massive extrasolar planets due to gravitational interaction with a viscous protoplanetary disc. We show that a model in which planets form at 5 AU at a constant rate, before migrating, leads to a predicted distribution of planets that is a steeply rising function of log (a), where a is the orbital radius. Between 1 AU and 3 AU, the expected number of planets per logarithmic interval in orbital radius roughly doubles. We demonstrate that, once selection effects are accounted for, this is consistent with current data, and then extrapolate the observed planet fraction to masses and radii that are inaccessible to current observations. In total, about 15 percent of stars targeted by existing radial velocity searches are predicted to possess planets with masses 0.3 M_Jupiter < M_p sin (i) < 10 M_Jupiter, and radii 0.1 AU < a < 5 AU. A third of these planets (around 5 percent of the target stars) lie at the radii most amenable to detection via microlensing. A further 5-10 percent of stars could have planets at radii of 5 AU < a < 8 AU that have migrated outwards. We discuss the probability of forming a system (akin to the Solar System) in which significant radial migration of the most massive planet does not occur. About 10-15 percent of systems with a surviving massive planet are estimated to fall into this class. Finally, we note that a smaller fraction of low mass planets than high mass planets is expected to survive without being consumed by the star. The initial mass function for planets is thus predicted to rise more steeply towards small masses than the observed mass function.Comment: MNRAS, in pres

The gas temperature in flaring disks around pre-main sequence stars

A model is presented which calculates the gas temperature and chemistry in the surface layers of flaring circumstellar disks using a code developed for photon-dominated regions. Special attention is given to the influence of dust settling. It is found that the gas temperature exceeds the dust temperature by up to several hundreds of Kelvins in the part of the disk that is optically thin to ultraviolet radiation, indicating that the common assumption that Tgas=Tdust is not valid throughout the disk. In the optically thick part, gas and dust are strongly coupled and the gas temperature equals the dust temperature. Dust settling has little effect on the chemistry in the disk, but increases the amount of hot gas deeper in the disk. The effects of the higher gas temperature on several emission lines arising in the surface layer are examined. The higher gas temperatures increase the intensities of molecular and fine-structure lines by up to an order of magnitude, and can also have an important effect on the line shapes.Comment: 14 pages, 10 figures, accepted for publication in A&

On the accumulation of planetesimals near disc gaps created by protoplanets

We have performed three-dimensional two-fluid (gas-dust) hydrodynamical models of circumstellar discs with embedded protoplanets (3 - 333 M\oplu) and small solid bodies (radii 10cm to 10m). We find that high mass planets (\gtrsim Saturn mass) open sufficiently deep gaps in the gas disc such that the density maximum at the outer edge of the gap can very efficiently trap metre-sized solid bodies. This allows the accumulation of solids at the outer edge of the gap as solids from large radii spiral inwards to the trapping region. This process of accumulation occurs fastest for those bodies that spiral inwards most rapidly, typically metre-sized boulders, whilst smaller and larger objects will not migrate sufficiently rapidly in the discs lifetime to benefit from the process. Around a Jupiter mass planet we find that bound clumps of solid material, as large as several Earth masses, may form, potentially collapsing under self-gravity to form planets or planetesimals. These results are in agreement with Lyra et al. (2009), supporting their finding that the formation of a second generation of planetesimals or of terrestrial mass planets may be triggered by the presence of a high mass planet.Comment: 14 pages, 10 figures. Accepted for publication in MNRA

Ice Lines, Planetesimal Composition and Solid Surface Density in the Solar Nebula

To date, there is no core accretion simulation that can successfully account for the formation of Uranus or Neptune within the observed 2-3 Myr lifetimes of protoplanetary disks. Since solid accretion rate is directly proportional to the available planetesimal surface density, one way to speed up planet formation is to take a full accounting of all the planetesimal-forming solids present in the solar nebula. By combining a viscously evolving protostellar disk with a kinetic model of ice formation, we calculate the solid surface density in the solar nebula as a function of heliocentric distance and time. We find three effects that strongly favor giant planet formation: (1) a decretion flow that brings mass from the inner solar nebula to the giant planet-forming region, (2) recent lab results (Collings et al. 2004) showing that the ammonia and water ice lines should coincide, and (3) the presence of a substantial amount of methane ice in the trans-Saturnian region. Our results show higher solid surface densities than assumed in the core accretion models of Pollack et al. (1996) by a factor of 3 to 4 throughout the trans-Saturnian region. We also discuss the location of ice lines and their movement through the solar nebula, and provide new constraints on the possible initial disk configurations from gravitational stability arguments.Comment: Version 2: reflects lead author's name and affiliation change, contains minor changes to text from version 1. 12 figures, 7 tables, accepted for publication in Icaru