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The characterization of Super-Earth-to-Neptune sized exoplanets relies heavily on our understanding of their formation and evolution. In this study, we link a model of planet formation by pebble accretion to the planets' long-term observational properties by calculating the interior evolution, starting from the dissipation of the protoplanetary disk. We investigate the evolution of the interior structure in 5-20 Earth masses planets, accounting for silicate redistribution caused by convective mixing, rainout (condensation and settling), and mass loss. Specifically, we have followed the fate of the hot silicate vapor that remained in the planet's envelope after planet formation, as the planet cools. We find that disk dissipation is followed by a rapid contraction of the envelope within 10 Myr. Subsequent cooling leads to substantial growth of the planetary core through silicate rainout, accompanied by inflated radii, in comparison to the standard models of planets that formed with core-envelope structure. We examine the dependence of rainout on the planet's envelope mass, distance from its host star, its silicate mass, and the atmospheric opacity. We find that the population of planets formed with polluted envelopes can be roughly divided in three groups, based on the mass of their gas envelopes: bare rocky cores that have shed their envelopes, super-Earth planets with a core-envelope structure, and Neptune-like planets with diluted cores that undergo gradual rainout. For polluted planets formed with envelope masses below 0.4 Earth mass, we anticipate that the inflation of the planet's radius caused by rainout will enhance mass loss by a factor of 2-8 compared to planets with non-polluted envelopes. Our model provides an explanation for bridging the gap between the predicted composition gradients in massive planets and the core-envelope structure in smaller planets., Comment: Accepted for publication in A&A

Binary solar system objects are common and range from satellite systems with very large mass ratios $M_1/M_2$ to mass ratios very close to unity. A well-known example of a binary is the Pluto-Charon system. With Charon only eight times less massive than Pluto the question arises as for many other systems, why the mass-ratio is still close to unity. There is much evidence that (binary) planet(esimal) formation happened early, when the protoplanetary gas disk was still around. It is likely that (some of) these binaries grew up together subject to pebble accretion. Here we focus on the question of how the mass arriving in the gravitational influence zone of the binary during pebble accretion, is distributed over the binary components. Does the accretion through time lead to a converging mass ratio, or to a diverging mass ratio? We numerically integrate pebble paths in the same well-known fashion as for a single mass subject to pebble accretion and track what the efficiency of accretion is for the two separate binary components, compared to a single body with the same mass. These numerical simulations are done for a range of binary mass-ratios, mutual separations, Stokes numbers and two orbital distances, 2.5 and 39 au. We find that in the limit where pebbles start to spiral around the primary (this holds for relatively large pebbles), the pebble preferentially collides with the secondary, causing the mass ratio to converge towards unity on Myr timescales. In this regime the total sweep-up efficiency can lower to half that of a pebble-accreting single body because pebbles that are thrown out of the system, after close encounters with the system. The results show that systems such as Pluto-Charon and other larger equal mass binaries could well have co-accreted by means of pebble accretion in the disk phase without producing binaries with highly diverging mass-ratios., Comment: 10 pages, 8 figures, submitted to Astronomy and Astrophysics

Recent observations found close-in planets with significant atmospheres of hydrogen and helium in great abundance. These are the so-called super-Earths and mini-Neptunes. Their atmospheric composition suggests that they formed early during the gas-rich phase of the circumstellar disk and were able to avoid becoming hot Jupiters. As a possible explanation, recent studies explored the recycling hypothesis and showed that atmosphere-disk recycling is able to fully compensate for radiative cooling and thereby halt Kelvin-Helmholtz contraction to prevent runaway gas accretion. To understand the parameters that determine the efficiency of atmospheric recycling, we extend our earlier studies by exploring the effects of the core mass, the effect of circumstellar gas on sub-Keplerian orbits (headwind), and the optical depth of the surrounding gas on the recycling timescale. Additionally, we analyze their effects on the size and mass of the forming atmosphere. For the explored parameter space, all simulations eventually reach an equilibrium where heating due to hydrodynamic recycling fully compensates radiative cooling. In this equilibrium, the atmosphere-to-core mass ratio stays well below $10 \, \%$, preventing the atmosphere from becoming self-gravitating and entering runaway gas accretion. Higher core masses cause the atmosphere to become turbulent, which further enhances recycling. Even for our highest core mass of $10 \, M_\mathrm{Earth}$, atmosphere-disk recycling is efficient enough to fully compensate for radiative cooling and prevent the atmosphere from becoming self-gravitating. Hence, in-situ formation of hot Jupiters is very unlikely, and migration of gas giants is a key process required to explain their existence. Our findings imply that atmosphere-disk recycling is the most natural explanation for the prevalence of close-in super-Earths and mini-Neptunes., Comment: 18 pages, 13 figures, Accepted for publication by A&A on 2/7/2022

The amount of nebular gas that a planet can bind is limited by its cooling rate, which is set by the opacity of its envelope. Accreting dust and pebbles contribute to the envelope opacity and, thus, influence the outcome of planet formation. Our aim is to model the size evolution and opacity contribution of solids inside planetary envelopes. We then use the resultant opacity relations to study emergent trends in planet formation. We design a model for the opacity of solids in planetary envelopes that accounts for the growth, fragmentation and erosion of pebbles during their sedimentation. We formulate analytical expressions for the opacity of pebbles and dust and map out their trends as a function of depth, planet mass, distance and accretion rate. We find that the accretion of pebbles rather than planetesimals can produce fully convective envelopes, but only in lower-mass planets that reside in the outer disk or in those that are accreting pebbles at a high rate. In these conditions, pebble sizes are limited by fragmentation and erosion, allowing them to pile up in the envelope. At higher planetary masses or reduced accretion rates, a different regime applies where the sizes of sedimenting pebbles are only limited by their rate of growth. The opacity in this growth-limited regime is much lower, steeply declines with depth and planet mass but is invariant with the pebble mass flux. Our results imply that the opacity of a forming planetary envelope can not be approximated by a value that is constant with either depth or planet mass. When applied to the Solar System, we argue that Uranus and Neptune could not have maintained a sufficiently high opacity to avoid runaway gas accretion unless they both experienced sufficiently rapid accretion of solids and formed late., Comment: Accepted to A&A. Comments are welcome

In recent years, space missions such as Kepler and TESS have discovered many close-in planets with significant atmospheres consisting of hydrogen and helium: mini-Neptunes. This indicates that these planets formed early in gas-rich disks while avoiding the runaway gas accretion that would otherwise have turned them into hot-Jupiters. A solution is to invoke a long Kelvin-Helmholtz contraction (or cooling) timescale, but it has also been suggested that thermodynamical cooling can be prevented by hydrodynamical planet atmosphere-disk recycling. We investigate the efficacy of the recycling hypothesis in preventing the collapse of the atmosphere, check for the existence of a steady state configuration, and determine the final atmospheric mass to core mass ratio. We use three-dimensional radiation-hydrodynamic simulations to model the formation of planetary proto-atmospheres. Equations are solved in a local frame centered on the planet. Ignoring small oscillations that average to zero over time, the simulations converge to a steady state where the velocity field of the gas becomes constant in time. In a steady state, the energy loss by radiative cooling is fully compensated by the recycling of the low entropy gas in the planetary atmosphere with high entropy gas from the circumstellar disk. For close-in planets, recycling naturally halts the cooling of planetary proto-atmospheres, preventing them from contracting toward the runaway regime and collapsing into gas giants., Comment: 5 pages, 4 figures, accepted to A&A

The detection of Earth-size exoplanets around low-mass stars -- in stars such as Proxima Centauri and TRAPPIST-1 -- provide an exceptional chance to improve our understanding of the formation of planets around M stars and brown dwarfs. We explore the formation of such planets with a population synthesis code based on a planetesimal-driven model previously used to study the formation of the Jovian satellites. Because the discs have low mass and the stars are cool, the formation is an inefficient process that happens at short periods, generating compact planetary systems. Planets can be trapped in resonances and we follow the evolution of the planets after the gas has dissipated and they undergo orbit crossings and possible mergers. We find that formation of planets above Mars mass and in the planetesimal accretion scenario, is only possible around stars with masses $M_{\star} \ge 0.07 M_{sun}$ and discs of $M_{disc} \ge 10^{-2}~M_{sun} $. We find that planets above Earth-mass form around stars with masses larger than $0.15 M_{\oplus}$ while planets larger than $5 M_{\oplus}$do not form in our model, even not under the most optimal conditions (massive disc), showing that planets such as GJ 3512b form with another, more efficient mechanism. Our results show that the majority of planets form with a significant water fraction; that most of our synthetic planetary systems have 1, 2, or 3 planets, but those with 4, 5, 6, and 7 planets are also common, confirming that compact planetary systems with many planets should be a relatively common outcome of planet formation around small stars., Comment: published in MNRAS

Proto-planets embedded in their natal disks acquire hot envelopes as they grow and accrete solids. This ensures that the material they accrete - pebbles, as well as (small) planetesimals - will vaporize to enrich their atmospheres. Enrichment modifies an envelope's structure and significantly alters its further evolution. Our aim is to describe the formation of planets with polluted envelopes from the moment that impactors begin to sublimate to beyond the disk's eventual dissipation. We constructed an analytical interior structure model, characterized by a hot and uniformly mixed high-Z vapor layer surrounding the core, located below the usual unpolluted radiative-convective regions. The evolution of planets with uniformly mixed polluted envelopes follows four potential phases. Initially, the central core grows directly through impacts and rainout until the envelope becomes hot enough to vaporize and absorb all incoming solids. We find that a planet reaches runaway accretion when the sum of its core and vapor mass exceeds a value that we refer to as the critical metal mass - a criterion that supersedes the traditional critical core mass. It scales positively with both the pollutant's evaporation temperature and with the planet's core mass. Hence, planets at shorter orbital separations require the accretion of more solids to reach runaway as they accrete less volatile materials. If the solids accretion rate dries up, we identify the decline of the mean molecular weight - dilution - as a mechanism to limit gas accretion during a polluted planet's embedded cooling phase. When the disk ultimately dissipates, the envelope's inner temperature declines and its vapor eventually rains out, augmenting the mass of the core. The energy release that accompanies this does not result in significant mass-loss, as it only occurs after the planet has substantially contracted., Comment: 18 pages, 11 figures. Published in A&A

Most major planetary bodies in the solar system rotate in the same direction as their orbital motion: their spin is prograde. Theoretical studies to explain the direction as well as the magnitude of the spin vector have had mixed success. When the accreting building blocks are $\sim$ km-size planetesimals -- as predicted by the classical model -- the accretion process is so symmetric that it cancels out prograde with retrograde spin contributions, rendering the net spin minute. For this reason, the currently-favored model for the origin of planetary rotation is the giant impact model, in which a single collision suffices to deliver a spin, which magnitude is close to the breakup rotation rate. However, the giant impact model does not naturally explain the preference for prograde spin. Similarly, an increasing number of spin-vector measurement of asteroids also shows that the spin vector of large (primordial) asteroids is not isotropic. Here, we re-assess the viability of smaller particles to bestow planetary bodies with a net spin, focusing on the pebble accretion model in which gas drag and gravity join forces to accrete small particles at a large cross section. Similar to the classical calculation for planetesimals, we integrate the pebble equation of motion and measure the angular momentum transfer at impact. We consider a variety of disk conditions and pebble properties and conduct our calculations in the limits of 2D (planar) and 3D (homogeneous) pebble distributions. We find that in certain regions of the parameter space the angular momentum transfer is significant, much larger than with planetesimals and on par with or exceeding the current spin of planetary bodies., Comment: Accepted for publication in Icarus

Sub-Neptune planets are a very common type of planets. They are inferred to harbour a primordial (H/He) envelope, on top of a (rocky) core, which dominates the mass. Here, we investigate the long-term consequences of the core properties on the planet mass-radius relation. We consider the role of various core energy sources resulting from core formation, its differentiation, its solidification (latent heat), core contraction and radioactive decay. We divide the evolution of the rocky core into three phases: the formation phase, which sets the initial conditions, the magma ocean phase, characterized by rapid heat transport, and the solid state phase, where cooling is inefficient. We find that for typical sub-Neptune planets of ~2-10 Earth masses and envelope mass fractions of 0.5-10% the magma ocean phase lasts several Gyrs, much longer than for terrestrial planets. The magma ocean phase effectively erases any signs of the initial core thermodynamic state. After solidification, the reduced heat flux from the rocky core causes a significant drop in the rocky core surface temperature, but its effect on the planet radius is limited. In the long run, radioactive heating is the most significant core energy source in our model. Overall, the long term radius uncertainty by core thermal effects is up to 15%., Comment: ApJ Published

The Earth is strongly depleted in carbon compared to the dust in the ISM, implying efficient removal of refractory carbon before parent body formation. It has been argued that grains get rid of their carbon through oxidation and photolysis in the exposed upper disk layers. We assess the efficacy of these C-removal mechanisms accounting for the vertical and radial transport of grains. We obtain the carbon and carbon free mass budget of solids by solving two 1D advection-diffusion equations, accounting for the dust grain size distribution and radial transport. The carbon removal acts on the fraction of the grains that are in the exposed layer and requires efficient vertical transport. In models without radial transport, oxidation and photolysis can destroy most of the refractory carbon in terrestrial planet formation region. But it only reaches the observed depletion levels for extreme parameter combinations and requires that parent body formation was delayed by 1 Myr. Adding radial transport of solids prevents the depletion entirely, leaving refractory carbon equally distributed throughout the disk. It is unlikely that the observed carbon depletion can ultimately be attributed to mechanisms operating on small grains in the disk surface layers. Other mechanisms need to be studied, for example flash heating events or FU Ori outbursts in order to remove carbon quickly and deeply. However, a sustained drift barrier or strongly reduced radial grain mobility are necessary to prevent replenishment of carbon from the outer disk., Comment: 8 pages, 3 figures, 2 tables, accepted for publication at A&A

Sub-Neptune planets are very common in our galaxy and show a large diversity in their mass-radius relation. In sub-Neptunes most of the planet mass is in the rocky part (hereafter core) which is surrounded by a modest hydrogen-helium envelope. As a result, the total initial heat content of such a planet is dominated by that of the core. Nonetheless, most studies contend that the core cooling will only have a minor effect on the radius evolution of the gaseous envelope, because the core's cooling is in sync with the envelope, i.e., most of the initial heat is released early on timescales of about 10-100 Myr. In this Letter we examine the importance of the core cooling rate for the thermal evolution of the envelope. Thus, we relax the early core cooling assumption and present a model where the core is characterized by two parameters: the initial temperature and the cooling time. We find that core cooling can significantly enhance the radius of the planet when it operates on a timescale similar to the observed age, i.e. several Gyr. Consequently, the interpretation of sub-Neptunes' mass-radius observations depends on the assumed core thermal properties and the uncertainty therein. The degeneracy of composition and core thermal properties can be reduced by obtaining better estimates of the planet ages (in addition to their radii and masses) as envisioned by future observations., Comment: Accepted for publication in A&A Letters

Context: Planet formation by pebble accretion is an alternative to planetesimal-driven core accretion. In this scenario, planets grow by accreting cm-to-m-sized pebbles instead of km-sized planetesimals. One of the main differences with planetesimal-driven core accretion is the increased thermal ablation experienced by pebbles. This provides early enrichment to the planet's envelope, which changes the process of core growth. Aims: We aim to predict core masses and envelope compositions of planets that form by pebble accretion and compare mass deposition of pebbles to planetesimals. Methods: We model the early growth of a proto-planet by calculating the structure of its envelope, taking into account the fate of impacting pebbles or planetesimals. The region where high-Z material can exist in vapor form is determined by the vapor pressure. We include enrichment effects by locally modifying the mean molecular weight. Results: In the pebble case, three phases of core growth can be identified. In the first phase, pebbles impact the core without significant ablation. During the second phase, ablation becomes increasingly severe. A layer of high-Z vapor starts to form around the core that absorbs a small fraction of the ablated mass. The rest either rains out to the core or mixes outwards instead, slowing core growth. In the third phase, the vapor inner region expands outwards, absorbing an increasing fraction of the ablated material as vapor. Rainout ends before the core mass reaches 0.6 M_Earth, terminating direct core growth. In the case of icy H2O pebbles, this happens before 0.1 M_Earth. Conclusions: Our results indicate that pebble accretion can directly form rocky cores up to only 0.6 M_Earth, and is unable to form similarly sized icy cores. Subsequent core growth can proceed indirectly when the planet cools, provided it is able to retain its high-Z material., Comment: 12 pages, 6 figures. Published in A&A. Includes corrigendum to Sect. 4.2

Planetesimals in gaseous protoplanetary disks may grow by collecting dust particles. Hydrodynamical studies show that small particles generally avoid collisions with the planetesimals because they are entrained by the flow around them. This occurs when $St$, the Stokes number, defined as the ratio of the dust stopping time to the planetesimal crossing time, becomes much smaller than unity. However, these studies have been limited to the laminar case, whereas these disks are believed to be turbulent. We want to estimate the influence of gas turbulence on the dust-planetesimal collision rate and on the impact speeds. We used three-dimensional direct numerical simulations of a fixed sphere (planetesimal) facing a laminar and turbulent flow seeded with small inertial particles (dust) subject to a Stokes drag. A no-slip boundary condition on the planetesimal surface is modeled via a penalty method. We find that turbulence can significantly increase the collision rate of dust particles with planetesimals. For a high turbulence case (when the amplitude of turbulent fluctuations is similar to the headwind velocity), we find that the collision probability remains equal to the geometrical rate or even higher for $St\geq 0.1$, i.e., for dust sizes an order of magnitude smaller than in the laminar case. We derive expressions to calculate impact probabilities as a function of dust and planetesimal size and turbulent intensity.

The recently discovered coreshine effect can aid in exploring the core properties and in probing the large grain population of the ISM. We discuss the implications of the coreshine detected from the molecular cloud core L1506C in the Taurus filament for the history of the core and the existence of a primitive ISM component of large grains becoming visible in cores. The coreshine surface brightness of L1506C is determined from IRAC Spitzer images at 3.6 micron. We perform grain growth calculations to estimate the grain size distribution in model cores similar in gas density, radius, and turbulent velocity to L1506C. Scattered light intensities at 3.6 micron are calculated for a variety of MRN and grain growth distributions to compare with the observed coreshine. For a core with the overall physical properties of L1506C, no detectable coreshine is predicted for an MRN size distribution. Extending the distribution to grain radii of about 0.65 $\mu$m allows to reproduce the observed surface brightness level in scattered light. Assuming the properties of L1506C to be preserved, models for the growth of grains in cores do not yield sufficient scattered light to account for the coreshine within the lifetime of the Taurus complex. Only increasing the core density and the turbulence amplifies the scattered light intensity to a level consistent with the observed coreshine brightness. The grains could be part of primitive omni-present large grain population becoming visible in the densest part of the ISM, could grow under the turbulent dense conditions of former cores, or in L1506C itself. In the later case, L1506C must have passed through a period of larger density and stronger turbulence. This would be consistent with the surprisingly strong depletion usually attributed to high column densities, and with the large-scale outward motion of the core envelope observed today., Comment: 6 pages, 6 figures, accepted for publication in Astronomy & Astrophysics

Gravitating bodies significantly alter the flow pattern (density and velocity) of the gas that attempts to stream past. Still, small protoplanets in the Mars--super-Earth range can only bind limited amounts of nebular gas; until the so-called critical core mass has been reached (~1-10 Earth masses) this gas is in near hydrostatic equilibrium with the nebula. Here we aim for a general description of the flow pattern surrounding these low-mass, embedded planets. Using various simplifying assumptions (subsonic, 2D, inviscid flow, etc), we reduce the problem to a partial differential equation that we solve numerically as well as approximate analytically. It is found that the boundary between the atmosphere and the nebula gas strongly depends on the value of the disc headwind (deviation from Keplerian rotation). With increasing headwind the atmosphere decreases in size and also becomes more asymmetrical. Using the derived flow pattern for the gas, trajectories of small solid particles, which experience both gas drag and gravitational forces, are integrated numerically. Accretion rates for small particles (dust) are found to be low, as they closely follow the streamlines, which curl away from the planet. However, pebble-size particles achieve large accretion rates, in agreement with previous numerical and analytical works., Comment: 19 pages, 12 figures, accepted for publication in MNRAS

We investigate dust growth due to settling in a 1D vertical column of a protoplanetary disk. It is known from the observed 10 micron feature in disk SEDs, that small micron-sized grains are present at the disk atmosphere throughout the lifetime of the disk. We hope to explain such questions as what process can keep the disk atmospheres dusty for the lifetime of the disk and how does the particle properties change as a function of height above the midplane. We use a Monte Carlo code to follow the mass and porosity evolution of the particles in time. The used collision model is based on laboratory experiments performed on dust aggregates. As the experiments cannot cover all possible collision scenarios, the largest uncertainty of our model is the necessary extrapolations we had to perform. We simultaneously solve for the particle growth and motion. Particles can move vertically due to settling and turbulent mixing. We assume that the vertical profile of the gas density is fixed in time and only the solid component evolves. We find that the used collision model strongly influences the masses and sizes of the particles. The laboratory experiment based collision model greatly reduces the particle sizes compared to models that assume sticking at all collision velocities. We find that a turbulence parameter of alpha = 10^-2 is needed to keep the dust atmospheres dusty, but such strong turbulence can produce only small particles at the midplane which is not favorable for planetesimal formation models. We also see that the particles are larger at the midplane and smaller at the upper layers of the disk. At 3-4 pressure scale heights micron-sized particles are produced. These particle sizes are needed to explain the 10 micron feature of disk SEDs. Turbulence may therefore help to keep small dust particles in the disk atmosphere., Comment: accepted for publication in A&A

The dust size distribution in molecular clouds can be strongly affected by ice-mantle formation and (subsequent) grain coagulation. Following previous work where the dust size distribution has been calculated from a state-of-the art collision model for dust aggregates that involves both coagulation and fragmentation (Paper I), the corresponding opacities are presented in this study. The opacities are calculated by applying the effective medium theory assuming that the dust aggregates are a mix of 0.1{\mu}m silicate and graphite grains and vacuum. In particular, we explore how the coagulation affects the near-IR opacities and the opacity in the 9.7{\mu}m silicate feature. We find that as dust aggregates grow to {\mu}m-sizes both the near-IR color excess and the opacity in the 9.7 {\mu}m feature increases. Despite their coagulation, porous aggregates help to prolong the presence of the 9.7{\mu}m feature. We find that the ratio between the opacity in the silicate feature and the near-IR color excess becomes lower with respect to the ISM, in accordance with many observations of dark clouds. However, this trend is primarily a result of ice mantle formation and the mixed material composition of the aggregates, rather than being driven by coagulation. With stronger growth, when most of the dust mass resides in particles of size 10{\mu}m or larger, both the near-IR color excess and the 9.7{\mu}m silicate feature significantly diminish. Observations at additional wavelengths, in particular in the sub-mm range, are essential to provide quantitative constraints on the dust size distribution within dense cores. Our results indicate that the sub-mm index {\beta} will increase appreciably, if aggregates grow to ~100{\mu}m in size., Comment: 10 pages, accepted for publication in A&A

Context. Grains in circumstellar disks are believed to grow by mutual collisions and subsequent sticking due to surface forces. Results of many fields of research involving circumstellar disks, such as radiative transfer calculations, disk chemistry, magneto-hydrodynamic simulations largely depend on the unknown grain size distribution. Aims. As detailed calculations of grain growth and fragmentation are both numerically challenging and computationally expensive, we aim to find simple recipes and analytical solutions for the grain size distribution in circumstellar disks for a scenario in which grain growth is limited by fragmentation and radial drift can be neglected. Methods. We generalize previous analytical work on self-similar steady-state grain distributions. Numerical simulations are carried out to identify under which conditions the grain size distributions can be understood in terms of a combination of power-law distributions. A physically motivated fitting formula for grain size distributions is derived using our analytical predictions and numerical simulations. Results. We find good agreement between analytical results and numerical solutions of the Smoluchowski equation for simple shapes of the kernel function. The results for more complicated and realistic cases can be fitted with a physically motivated "black box" recipe presented in this paper. Our results show that the shape of the dust distribution is mostly dominated by the gas surface density (not the dust-to-gas ratio), the turbulence strength and the temperature and does not obey an MRN type distribution., Comment: 16 pages, 9 figures, accepted for publication in A&A

Planetary bodies form by accretion of smaller bodies. It has been suggested that a very efficient way to grow protoplanets is by accreting particles of size <

(abridged) When preplanetary bodies reach proportions of ~1 km or larger in size, their accretion rate is enhanced due to gravitational focusing (GF). We have developed a new numerical model to calculate the collisional evolution of the gravitationally-enhanced growth stage. We validate our approach against existing N-body and statistical codes. Using the numerical model, we explore the characteristics of the runaway growth and the oligarchic growth accretion phases starting from an initial population of single planetesimal radius R_0. In models where the initial random velocity dispersion (as derived from their eccentricity) starts out below the escape speed of the planetesimal bodies, the system experiences runaway growth. We find that during the runaway growth phase the size distribution remains continuous but evolves into a power-law at the high mass end, consistent with previous studies. Furthermore, we find that the largest body accretes from all mass bins; a simple two component approximation is inapplicable during this stage. However, with growth the runaway body stirs up the random motions of the planetesimal population from which it is accreting. Ultimately, this feedback stops the fast growth and the system passes into oligarchy, where competitor bodies from neighboring zones catch up in terms of mass. Compared to previous estimates, we find that the system leaves the runaway growth phase at a somewhat larger radius. Furthermore, we assess the relevance of small, single-size fragments on the growth process. In classical models, where the initial velocity dispersion of bodies is small, these do not play a critical role during the runaway growth; however, in models that are characterized by large initial relative velocities due to external stirring of their random motions, a situation can emerge where fragments dominate the accretion., Comment: Accepted for publication in Icarus

Context. Observations at sub-millimeter and mm wavelengths will in the near future be able to resolve the radial dependence of the mm spectral slope in circumstellar disks with a resolution of around a few AU at the distance of the closest star-forming regions. Aims. We aim to constrain physical models of grain growth and fragmentation by a large sample of (sub-)mm observations of disks around pre-main sequence stars in the Taurus-Auriga and Ophiuchus star-forming regions. Methods. State-of-the-art coagulation/fragmentation and disk-structure codes are coupled to produce steady-state grain size distributions and to predict the spectral slopes at (sub-)mm wavelengths. Results. This work presents the first calculations predicting the mm spectral slope based on a physical model of grain growth. Our models can quite naturally reproduce the observed mm-slopes, but a simultaneous match to the observed range of flux levels can only be reached by a reduction of the dust mass by a factor of a few up to about 30 while keeping the gas mass of the disk the same. This dust reduction can either be due to radial drift at a reduced rate or during an earlier evolutionary time (otherwise the predicted fluxes would become too low) or due to efficient conversion of dust into larger, unseen bodies., Comment: Accepted for publication in A&A Letters. 5 pages, 3 figures

Accretion among macroscopic bodies of ~km size or larger is enhanced significantly due to gravitational focusing. Two regimes can be distinguished. Initially, the system experiences runaway growth, in which the gravitational focusing factors increase, and bodies at the high-mass tail of the distribution grow fastest. However, at some point the runaway body dynamically heats its environment, gravitational focusing factors decrease, and runaway growth passes into oligarchic growth. Based on the results of recent simulations, we reconsider the runaway growth-oligarchy transition. In contrast to oligarchy, we find that runaway growth cannot be approximated with a two component model (of small and large bodies) and that the criterion of Ida & Makino (1993), which is frequently adopted as the start of oligarchy, is not a sufficient condition to signify the transition. Instead, we propose a new criterion based on timescale arguments. We then find a larger value for the runaway growth-oligarchy transition: from several hundreds of km in the inner disk regions up to ~10^3 km for the outer disk. These findings are consistent with the view that runaway growth has been responsible for the size distribution of the present day Kuiper belt objects. Our finding furthermore outlines the proper initial conditions at the start of the oligarchy stage., Comment: 5 pages, 5 figures, accepted for publication in the Astrophysical Journal Letters

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&A

In dense molecular clouds collisions between dust grains alter the ISM-dust size distribution. We study this process by inserting the results from detailed numerical simulations of two colliding dust aggregates into a coagulation model that computes the dust size distribution with time. All collisional outcomes -- sticking, fragmentation (shattering, breakage, and erosion) -- are included and the effects on the internal structure of the aggregates are also tabulated. The dust aggregate evolution model is applied to an homogeneous and static cloud of temperature 10 K and gas densities between 10^3 and 10^7 cm^-3. The coagulation is followed locally on timescales of ~10^7 yr. We find that the growth can be divided into two stages: a growth dominated phase and a fragmentation dominated phase. Initially, the mass distribution is relatively narrow and shifts to larger sizes with time. At a certain point, dependent on the material properties of the grains as well as on the gas density, collision velocities will become sufficiently energetic to fragment particles, halting the growth and replenishing particles of lower mass. Eventually, a steady state is reached, where the mass distribution is characterized by a mass spectrum of approximately equal amount of mass per logarithmic size bin. The amount of growth that is achieved depends on the cloud's lifetime. If clouds exist on free-fall timescales the effects of coagulation on the dust size distribution are very minor. On the other hand, if clouds have long-term support mechanisms, the impact of coagulation is important, resulting in a significant decrease of the opacity on timescales longer than the initial collision timescale between big grains., Comment: 25 pages, 20 figures, accepted for publication in Astronomy & Astrophysics

We present a method which extends Monte Carlo studies to situations that require a large dynamic range in particle number. The underlying idea is that, in order to calculate the collisional evolution of a system, some particle interactions are more important than others and require more resolution, while the behavior of the less important, usually of smaller mass, particles can be considered collectively. In this approximation groups of identical particles, sharing the same mass and structural parameters, operate as one unit. The amount of grouping is determined by the zoom factor -- a free parameter that determines on which particles the computational effort is focused. Two methods for choosing the zoom factors are discussed: the `equal mass method,' in which the groups trace the mass density of the distribution, and the `distribution method,' which additionally follows fluctuations in the distribution. Both methods achieve excellent correspondence with analytic solutions to the Smoluchowski coagulation equation. The grouping method is furthermore applied to simulations involving runaway kernels, where the particle interaction rate is a strong function of particle mass, and to situations that include catastrophic fragmentation. For the runaway simulations previous predictions for the decrease of the runaway timescale with the initial number of particles ${\cal N}$ are reconfirmed, extending ${\cal N}$ to $10^{160}$. Astrophysical applications include modeling of dust coagulation, planetesimal accretion, and the dynamical evolution of stars in large globular clusters. The proposed method is a powerful tool to compute the evolution of any system where the particles interact through discrete events, with the particle properties characterized by structural parameters., Comment: 18 pages, 10 figures. Re-submitted to ApJ with comments of the referee included

We present a mechanism for chondrules to stick together by means of compaction of a porous dust rim they sweep up as they move through the dusty nebula gas. It is shown that dust aggregates formed out of micron-sized grains stick to chondrules, forming a porous dust rim. When chondrules collide, this dust can be compacted by means of rolling motions within the porous dust layer. This mechanism dissipates the collisional energy, compacting the rim and allowing chondrules to stick. The structure of the obtained chondrule-dust agglomerates (referred to as compounds) then consists of three phases: chondrules, porous dust, and dust that has been compacted by collisions. Subsequently, these compounds accrete their own dust and collide with other compounds. The evolution of the compound size distribution and the relative importance of the phases is calculated by a Monte Carlo code. Growth ends, and a simulation is terminated when all the dust in the compounds has been compacted. Numerous runs are performed, reflecting the uncertainty in the physical conditions at the chondrule formation time. It is found that compounds can grow by 1-2 orders of magnitudes in radius, upto dm-sizes when turbulence levels are low. However, relative velocities associated with radial drift form a barrier for further growth. Earlier findings that the dust sweep-up by chondrules is proportional to their sizes are confirmed. We contrast two scenarios regarding how this dust evolved further towards the densely packed rims seen in chondrites., Comment: 23 pages, accepted for publication in ApJ

In this note we present complete, closed-form expressions for random relative velocities between colliding particles of arbitrary size in nebula turbulence. These results are exact for very small particles (those with stopping times much shorter than the large eddy overturn time) and are also surprisingly accurate in complete generality (that is, also apply for particles with stopping times comparable to, or much longer than, the large eddy overturn time). We note that some previous studies may have adopted previous simple expressions, which we find to be in error regarding the size dependence in the large particle regime., Comment: 8 pages, accepted as Research Note by A&A

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 shortened

This paper presents results from modelling 450 micron and 850 micron continuum and HCO+ line observations of three distinct cores of an infrared dark cloud (IRDC) directed toward the W51 GMC. In the sub-mm continuum these cores appear as bright, isolated emission features. One of them coincides with the peak of 8.3 micron extinction as measured by the Midcourse Space Experiment satellite. Detailed radiative transfer codes are applied to constrain the cores' physical conditions to address the key question: Do these IRDC-cores harbour luminous sources? The results of the continuum model, expressed in the $\chi^2$ quality-of-fit parameter, are also constrained by the absence of 100 micron emission from IRAS. For the sub-mm emission peaks this shows that sources of 300 solar luminosities are embedded within the cores. For the extinction peak, the combination of continuum and HCO+ line modelling indicates that a heating source is present as well. Furthermore, the line model provides constraints on the clumpiness of the medium. All three cores have similar masses of about 70-150 solar masses and similar density structures. The extinction peak differs from the other two cores by hosting a much weaker heating source, and the sub-mm emission core at the edge of the IRDC deviates from the other cores by a higher internal clumpiness., Comment: 13 pages, 13 figures, accepted for publication in A&A