Numerical Modeling of the Coagulation and Porosity Evolution of Dust Aggregates

Porosity evolution of dust aggregates is crucial in understanding dust evolution in protoplanetary disks. In this study, we present useful tools to study the coagulation and porosity evolution of dust aggregates. First, we present a new numerical method for simulating dust coagulation and porosity evolution as an extension of the conventional Smoluchowski equation. This method follows the evolution of the mean porosity for each aggregate mass simultaneously with the evolution of the mass distribution function. This method reproduces the results of previous Monte Carlo simulations with much less computational expense. Second, we propose a new collision model for porous dust aggregates on the basis of our N-body experiments on aggregate collisions. We first obtain empirical data on porosity changes between the classical limits of ballistic cluster-cluster and particle-cluster aggregation. Using the data, we construct a recipe for the porosity change due to general hit-and-stick collisions as well as formulae for the aerodynamical and collisional cross sections. Simple coagulation simulations using the extended Smoluchowski method show that our collision model explains the fractal dimensions of porous aggregates observed in a full N-body simulation and a laboratory experiment. Besides, we discover that aggregates at the high-mass end of the distribution can have a considerably small aerodynamical cross section per unit mass compared with aggregates of lower masses. We point out an important implication of this discovery for dust growth in protoplanetary disks.Comment: 17 pages, 15 figures; v2: version to appear in ApJ (typos corrected

Electrostatic Barrier against Dust Growth in Protoplanetary Disks. I. Classifying the Evolution of Size Distribution

Collisional growth of submicron-sized dust grains into macroscopic aggregates is the first step of planet formation in protoplanetary disks. These grains are expected to carry nonzero negative charges in the weakly ionized disks, but its effect on their collisional growth has not been fully understood so far. In this paper, we investigate how the charging affects the evolution of the dust size distribution properly taking into account the charging mechanism in a weakly ionized gas as well as porosity evolution through low-energy collisions. To clarify the role of the size distribution, we divide our analysis into two steps. First, we analyze the collisional growth of charged aggregates assuming a monodisperse (i.e., narrow) size distribution. We show that the monodisperse growth stalls due to the electrostatic repulsion when a certain condition is met, as is already expected in the previous work. Second, we numerically simulate dust coagulation using Smoluchowski's method to see how the outcome changes when the size distribution is allowed to freely evolve. We find that, under certain conditions, the dust undergoes bimodal growth where only a limited number of aggregates continue to grow carrying the major part of the dust mass in the system. This occurs because remaining small aggregates efficiently sweep up free electrons to prevent the larger aggregates from being strongly charged. We obtain a set of simple criteria that allows us to predict how the size distribution evolves for a given condition. In Paper II (arXiv:1009.3101), we apply these criteria to dust growth in protoplanetary disks.Comment: 20 pages, 22 figures, accepted for publication in Ap

The outcome of protoplanetary dust growth: pebbles, boulders, or planetesimals? II. Introducing the bouncing barrier

The sticking of micron sized dust particles due to surface forces in circumstellar disks is the first stage in the production of asteroids and planets. The key ingredients that drive this process are the relative velocity between the dust particles in this environment and the complex physics of dust aggregate collisions. Here we present the results of a collision model, which is based on laboratory experiments of these aggregates. We investigate the maximum aggregate size and mass that can be reached by coagulation in protoplanetary disks. We model the growth of dust aggregates at 1 AU at the midplane at three different gas densities. We find that the evolution of the dust does not follow the previously assumed growth-fragmentation cycles. Catastrophic fragmentation hardly occurs in the three disk models. Furthermore we see long lived, quasi-steady states in the distribution function of the aggregates due to bouncing. We explore how the mass and the porosity change upon varying the turbulence parameter and by varying the critical mass ratio of dust particles. Particles reach Stokes numbers of roughly 10^-4 during the simulations. The particle growth is stopped by bouncing rather than fragmentation in these models. The final Stokes number of the aggregates is rather insensitive to the variations of the gas density and the strength of turbulence. The maximum mass of the particles is limited to approximately 1 gram (chondrule-sized particles). Planetesimal formation can proceed via the turbulent concentration of these aerodynamically size-sorted chondrule-sized particles.Comment: accepted for publication in A&

Decimetre dust aggregates in protoplanetary discs

The growth of planetesimals is an essential step in planet formation. Decimetre-size dust agglomerates mark a transition point in this growth process. In laboratory experiments we simulated the formation, evolution, and properties of decimetre-scale dusty bodies in protoplanetary discs. Small sub-mm size dust aggregates consisting of micron-size SiO$_2$ particles randomly interacted with dust targets of varying initial conditions in a continuous sequence of independent collisions. Impact velocities were 7.7 m/s on average and in the range expected for collisions with decimetre bodies in protoplanetary discs. The targets all evolved by forming dust \emph{crusts} with up to several cm thickness and a unique filling factor of 31% $\pm$3%. A part of the projectiles sticks directly. In addition, some projectile fragments slowly return to the target by gravity. All initially porous parts of the surface, i.e. built from the slowly returning fragments, are compacted and firmly attached to the underlying dust layers by the subsequent impacts. Growth is possible at impact angles from 0$^{\circ}$ (central collision) to 70$^{\circ}$. No growth occurs at steeper dust surfaces. We measured the velocity, angle, and size distribution of collision fragments. The average restitution coefficient is 3.8% or 0.29 m/s ejection velocity. Ejecta sizes are comparable to the projectile sizes. The high filling factor is close to the most compact configuration of dust aggregates by local compression ($\sim 33$%). This implies that the history of the surface formation and target growth is completely erased. In view of this, the filling factor of 31% seems to be a universal value in the collision experiments of all self-consistently evolving targets at the given impact velocities. We suggest that decimetre and probably larger bodies can simply be characterised by one single filling factor.Comment: 10 pages, 9 figure

Opacity of fluffy dust aggregates

Context. Dust grains coagulate to form dust aggregates in protoplanetary disks. Their porosity can be extremely high in the disks. Although disk emission may come from fluffy dust aggregates, the emission has been modeled with compact grains. Aims. We aim to reveal the mass opacity of fluffy aggregates from infrared to millimeter wavelengths with the filling factor ranging from 1 down to $10^{-4}$. Methods. We use Mie calculations with an effective medium theory. The monomers are assumed to be 0.1 ${\rm \mu m}$ sized grains, which is much shorter than the wavelengths that we focus on. Results. We find that the absorption mass opacity of fluffy aggregates are characterized by the product $a\times f$, where $a$ is the dust radius and $f$ is the filling factor, except for the interference structure. The scattering mass opacity is also characterized by $af$ at short wavelengths while it is higher in more fluffy aggregates at long wavelengths. We also derive the analytic formula of the mass opacity and find that it reproduces the Mie calculations. We also calculate the expected difference of the emission between compact and fluffy aggregates in protoplanetary disks with a simple dust growth and drift model. We find that compact grains and fluffy aggregates can be distinguished by the radial distribution of the opacity index $\beta$. The previous observation of the radial distribution of $\beta$ is consistent with the fluffy case, but more observations are required to distinguish between fluffy or compact. In addition, we find that the scattered light would be another way to distinguish between compact grains and fluffy aggregates.Comment: 16 pages, 17 figures, published in A&A, 568, A4

Dust coagulation in protoplanetary disks: porosity matters

Context: Sticking of colliding dust particles through van der Waals forces is the first stage in the grain growth process in protoplanetary disks, eventually leading to the formation of comets, asteroids and planets. A key aspect of the collisional evolution is the coupling between dust and gas motions, which depends on the internal structure (porosity) of aggregates. Aims: To quantify the importance of the internal structure on the collisional evolution of particles, and to create a new coagulation model to investigate the difference between porous and compact coagulation in the context of a turbulent protoplanetary disk. Methods: We have developed simple prescriptions for the collisional evolution of porosity of grain-aggregates in grain-grain collisions. Three regimes can then be distinguished: `hit-and-stick' at low velocities, with an increase in porosity; compaction at intermediate velocities, with a decrease of porosity; and fragmentation at high velocities. (..) Results: (..) We can discern three different stages in the particle growth process (..) We find that when compared to standard, compact models of coagulation, porous growth delays the onset of settling, because the surface area-to-mass ratio is higher, a consequence of the build-up of porosity during the initial stages. As a result, particles grow orders of magnitudes larger in mass before they rain-out to the mid-plane. Depending on the turbulent viscosity and on the position in the nebula, aggregates can grow to (porous) sizes of ~ 10 cm in a few thousand years. We also find that collisional energies are higher than in the limited PCA/CCA fractal models, thereby allowing aggregates to restructure. It is concluded that the microphysics of collisions plays a key role in the growth process.Comment: 21 pages, 15 figures. Accepted for publication in A&A. Abstract shortene