Growing dust grains in protoplanetary discs - II. The Radial drift barrier problem

We aim to study the migration of growing dust grains in protoplanetary discs, where growth and migration are tightly coupled. This includes the crucial issue of the radial-drift barrier for growing dust grains. We therefore extend the study performed in Paper I, considering models for grain growth and grain dynamics where both the migration and growth rate depend on the grain size and the location in the disc. The parameter space of disc profiles and growth models is exhaustively explored. In doing so, interpretations for the grain motion found in numerical simulations are also provided. We find that a large number of cases is required to characterise entirely the grains radial motion, providing a large number of possible outcomes. Some of them lead dust particles to be accreted onto the central star and some of them don't. We find then that q<1 is required for discs to retain their growing particles, where q is the exponent of the radial temperature profile T(R) proportional to R^-q. Additionally, the initial dust-to gas ratio has to exceed a critical value for grains to pile up efficiently, thus avoiding being accreted onto the central star. Discs are also found to retain efficiently small dust grains regenerated by fragmentation. We show how those results are sensitive to the turbulent model considered. Even though some physical processes have been neglected, this study allows to sketch a scenario in which grains can survive the radial-drift barrier in protoplanetary discs as they grow.Comment: 18 pages, 10 figures. Accepted for publication in MNRAS. v2: typos correcte

Linear growth of streaming instability in pressure bumps

Streaming instability is a powerful mechanism which concentrates dust grains in pro- toplanetary discs, eventually up to the stage where they collapse gravitationally and form planetesimals. Previous studies inferred that it should be ineffective in viscous discs, too efficient in inviscid discs, and may not operate in local pressure maxima where solids accumulate. From a linear analysis of stability, we show that streaming instability behaves differently inside local pressure maxima. Under the action of the strong differential advection imposed by the bump, a novel unstable mode develops and grows even when gas viscosity is large. Hence, pressure bumps are found to be the only places where streaming instability occurs in viscous discs. This offers a promising way to conciliate models of planet formation with recent observations of young discs.Comment: 11 pages, 17 figures, accepted for publication in MNRA

A solution to the overdamping problem when simulating dust-gas mixtures with smoothed particle hydrodynamics

We present a fix to the overdamping problem found by Laibe & Price (2012) when simulating strongly coupled dust-gas mixtures using two different sets of particles using smoothed particle hydrodynamics. Our solution is to compute the drag at the barycentre between gas and dust particle pairs when computing the drag force by reconstructing the velocity field, similar to the procedure in Godunov-type solvers. This fixes the overdamping problem at negligible computational cost, but with additional memory required to store velocity derivatives. We employ slope limiters to avoid spurious oscillations at shocks, finding the van Leer Monotonized Central limiter most effective.Comment: 6 pages, 5 figures, accepted to MNRA

Diversity in the outcome of dust radial drift in protoplanetary discs

The growth of dust particles into planet embryos needs to circumvent the radial-drift barrier, i.e. the accretion of dust particles onto the central star by radial migration. The outcome of the dust radial migration is governed by simple criteria between the dust-to-gas ratio and the exponents p and q of the surface density and temperature power laws. The transfer of radiation provides an additional constraint between these quantities because the disc thermal structure is fixed by the dust spatial distribution. To assess which discs are primarily affected by the radial-drift barrier, we used the radiative transfer code MCFOST to compute the temperature structure of a wide range of disc models, stressing the particular effects of grain size distributions and vertical settling. We find that the outcome of the dust migration process is very sensitive to the physical conditions within the disc. For high dust-to-gas ratios (> 0.01) or flattened disc structures (H/R < 0.05), growing dust grains can efficiently decouple from the gas, leading to a high concentration of grains at a critical radius of a few AU. Decoupling of grains can occur at a large fraction (> 0.1) of the initial radius, for a dust-to-gas ratio greater than ~ 0.05. The exact value of the required dust-to-gas ratio for dust to stop its migration is strongly dependent on the disc temperature structure. Non growing dust grains are accreted for discs with flat surface density profiles (p<0.7) while they always remain in the disc if the surface density is steep enough (p>1.2). Both the presence of large grains and vertical settling tend to favour the accretion of non growing dust grains onto the central object, but it slows down the migration of growing dust grains. All the disc configurations are found to have favourable temperature profiles over most of the disc to retain their planetesimals.Comment: 9 pages, 8 figures, accepted for publications in A&A, corrected typo

Dusty gas with one fluid

In this paper, we show how the two-fluid equations describing the evolution of a dust and gas mixture can be reformulated to describe a single fluid moving with the barycentric velocity of the mixture. This leads to evolution equations for the total density, momentum, the differential velocity between the dust and the gas phases and either the dust-to-gas ratio or the dust fraction. The equations are similar to the usual equations of gas dynamics, providing a convenient way to extend existing codes to simulate two-fluid mixtures without modifying the code architecture. Our approach avoids the inherent difficulties related to the standard approach where the two phases are separate and coupled via a drag term. In particular, the requirements of infinite spatial and temporal resolution as the stopping time tends to zero are no longer necessary. This means that both small and large grains can be straightforwardly treated with the same method, with no need for complicated implicit schemes. Since there is only one resolution scale the method also avoids the problem of unphysical trapping of one fluid (e.g. dust) below the resolution of the other. We also derive a simplified set of equations applicable to the case of strong drag/small grains, consisting of the standard fluid equations with a modified sound speed, plus an advection-diffusion equation for the dust-to-gas ratio. This provides a simple and fast way to evolve the mixture when the stopping time is smaller than the Courant timestep. We present a Smoothed Particle Hydrodynamics implementation in a companion paper.Comment: Accepted for publication in MNRAS (very minor revisions included

An opening criterion for dust gaps in protoplanetary discs

We aim to understand under which conditions a low mass planet can open a gap in viscous dusty protoplanetary discs. For this purpose, we extend the theory of dust radial drift to include the contribution from the tides of an embedded planet and from the gas viscous forces. From this formalism, we derive i) a grain size-dependent criterion for dust gap opening in discs, ii) an estimate of the location of the outer edge of the dust gap and iii) an estimate of the minimum Stokes number above which low-mass planets are able to carve gaps which appear only in the dust disc. These analytical estimates are particularly helpful to appraise the minimum mass of an hypothetical planet carving gaps in discs observed at long wavelengths and high resolution. We validate the theory against 3D SPH simulations of planet-disc interaction in a broad range of dusty protoplanetary discs. We find a remarkable agreement between the theoretical model and the numerical experiments.Comment: 17 pages, 13 figures. Accepted for publication in MNRA

A fast and explicit algorithm for simulating the dynamics of small dust grains with smoothed particle hydrodynamics

We describe a simple method for simulating the dynamics of small grains in a dusty gas, relevant to micron-sized grains in the interstellar medium and grains of centimetre size and smaller in protoplanetary discs. The method involves solving one extra diffusion equation for the dust fraction in addition to the usual equations of hydrodynamics. This "diffusion approximation for dust" is valid when the dust stopping time is smaller than the computational timestep. We present a numerical implementation using Smoothed Particle Hydrodynamics (SPH) that is conservative, accurate and fast. It does not require any implicit timestepping and can be straightforwardly ported into existing 3D codes.Comment: 15 pages, 10 figures, accepted to MNRAS. Code implementation (ndspmhd v2.1) and setup of test problems available at: http://users.monash.edu.au/~dprice/ndspmhd/. v3: sign errors fixed as per erratum to published pape

Dusty gas with one fluid in smoothed particle hydrodynamics

In a companion paper we have shown how the equations describing gas and dust as two fluids coupled by a drag term can be reformulated to describe the system as a single fluid mixture. Here we present a numerical implementation of the one-fluid dusty gas algorithm using Smoothed Particle Hydrodynamics (SPH). The algorithm preserves the conservation properties of the SPH formalism. In particular, the total gas and dust mass, momentum, angular momentum and energy are all exactly conserved. Shock viscosity and conductivity terms are generalised to handle the two-phase mixture accordingly. The algorithm is benchmarked against a comprehensive suit of problems: dustybox, dustywave, dustyshock and dustyoscill, each of them addressing different properties of the method. We compare the performance of the one-fluid algorithm to the standard two-fluid approach. The one-fluid algorithm is found to solve both of the fundamental limitations of the two- fluid algorithm: it is no longer possible to concentrate dust below the resolution of the gas (they have the same resolution by definition), and the spatial resolution criterion h < csts, required in two-fluid codes to avoid over-damping of kinetic energy, is unnecessary. Implicit time stepping is straightforward. As a result, the algorithm is up to ten billion times more efficient for 3D simulations of small grains. Additional benefits include the use of half as many particles, a single kernel and fewer SPH interpolations. The only limitation is that it does not capture multi-streaming of dust in the limit of zero coupling, suggesting that in this case a hybrid approach may be required.Comment: Accepted for publication in MNRAS. Numerical code and input files for dustybox, wave and shock tests available from http://users.monash.edu.au/~dprice/ndspmhd