Context. The growth processes from protoplanetary dust to planetesimals are not fully understood. Laboratory experiments and theoretical models have shown that collisions among the dust aggregates can lead to sticking, bouncing, and fragmentation. However, no systematic study on the collisional outcome of protoplanetary dust has been performed so far, so that a physical model of the dust evolution in protoplanetary disks is still missing. Aims. We intend to map the parameter space for the collisional interaction of arbitrarily porous dust aggregates. This parameter space encompasses the dust-aggregate masses, their porosities and the collision velocity. With such a complete mapping of the collisional outcomes of protoplanetary dust aggregates, it will be possible to follow the collisional evolution of dust in a protoplanetary disk environment. Methods. We use literature data, perform laboratory experiments, and apply simple physical models to get a complete picture of the collisional interaction of protoplanetary dust aggregates. Results. We found four different kinds of sticking, two kinds of bouncing, and three kinds of fragmentation as possible outcomes in collisions among protoplanetary dust aggregates. Our best collision model distinguishes between porous and compact dust. We also differentiate between collisions among similar-sized and different-sized bodies. All in all, eight combinations of porosity and mass ratio can be discerned. For each of these cases, we present a complete collision model for dust-aggregate masses between 10 −12 and 10 2 g and collision velocities in the range of 10 −4 . . . 10 4 cm s −1 for arbitrary porosities. This model comprises the collisional outcome, the mass(es) of the resulting aggregate(s) and their porosities. Conclusions. We present the first complete collision model for protoplanetary dust. This collision model can be used for the determination of the dust-growth rate in protoplanetary disks.

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 km (e.g., chondrules, boulders, or fragments of larger bodies) as they can be kept dynamically cold. We investigate the effects of gas drag on the impact radii and the accretion rates of these particles. As simplifying assumptions we restrict our analysis to 2D settings, a gas drag law linear in velocity, and a laminar disk characterized by a smooth (global) pressure gradient that causes particles to drift in radially. These approximations, however, enable us to cover an arbitrary large parameter space. The framework of the circularly restricted three body problem is used to numerically integrate particle trajectories and to derive their impact parameters. Three accretion modes can be distinguished: hyperbolic encounters, where the 2-body gravitational focusing enhances the impact parameter; three-body encounters, where gas drag enhances the capture probability; and settling encounters, where particles settle towards the protoplanet. An analysis of the observed behavior is presented; and we provide a recipe to analytically calculate the impact radius, which confirms the numerical findings. We apply our results to the sweepup of fragments by a protoplanet at a distance of 5 AU. Accretion of debris on small protoplanets ( < ∼ 50 km) is found to be slow, because the fragments are distributed over a rather thick layer. However, the newly found settling mechanism, which is characterized by much larger impact radii, becomes relevant for protoplanets of ∼10 3 km in size and provides a much faster channel for growth.

Context. The sticking of micron-sized dust particles caused by surface forces within circumstellar disks is the first stage in the production of asteroids and planets. The key components describing this process are the relative velocity between the dust particles in this environment and the complex physics of dust aggregate collisions. Aims. We present the results of a collision model based on laboratory experiments of these aggregates. We investigate the maximum aggregate size and mass that can be reached by coagulation in protoplanetary disks. Methods. We use the results of laboratory experiments to establish the collision model previously published by Güttler et al. The collision model is based on the assumptions that we model the aggregates as spheres with compact and porous "phases" and that there is a continuous transition between these two. We apply this collision model to the Monte Carlo method developed previously by Zsom & Dullemond and include Brownian motion, radial drift, and turbulence as contributors of relative velocity between dust particles. Results. We model the growth of dust aggregates at 1 AU in the midplane for 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 caused by bouncing. We explore how the mass and the porosity depend on both the turbulence parameter and the critical mass ratio of dust particles. Upon varying the turbulence parameter, the system behaves in a non-linear way, and we find that the critical mass ratio has a strong effect on the particle sizes and masses. Particles reach Stokes numbers of roughly 10 −4 during the simulations. Conclusions. 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 in the gas density and the strength of turbulence. The maximum mass of the particles is limited to ≈1 g (chondrule-sized particles). Planetesimal formation can proceed by the means of the turbulent concentration of these aerodynamically size-sorted, chondrule-sized particles.

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.

Close-in super-Earths having radii 1-4 R ⊕ may possess hydrogen atmospheres comprising a few percent by mass of their rocky cores. We determine the conditions under which such atmospheres can be accreted by cores from their parent circumstellar disks. Accretion from the nebula is problematic because it is too efficient: we find that 10 M ⊕ cores embedded in solar metallicity disks tend to undergo runaway gas accretion and explode into Jupiters, irrespective of orbital location. The threat of runaway is especially dire at ∼0.1 AU, where solids may coagulate on timescales orders of magnitude shorter than gas clearing times; thus nascent atmospheres on close-in orbits are unlikely to be supported against collapse by planetesimal accretion. The time to runaway accretion is well approximated by the cooling time of the atmosphere's innermost convective zone, whose extent is controlled by where H 2 dissociates. Insofar as the temperatures characterizing H 2 dissociation are universal, timescales for core instability tend not to vary with orbital distance -and to be alarmingly short for 10 M ⊕ cores. Nevertheless, in the thicket of parameter space, we identify two scenarios, not mutually exclusive, that can reproduce the preponderance of percent-by-mass atmospheres for super-Earths at ∼0.1 AU, while still ensuring the formation of Jupiters at 1 AU. Scenario (a): planets form in disks with dust-to-gas ratios that range from ∼20× solar at 0.1 AU to ∼2× solar at 5 AU. Scenario (b): the final assembly of super-Earth cores from mergers of proto-cores -a process that completes quickly at ∼0.1 AU once begun -is delayed by gas dynamical friction until just before disk gas dissipates completely. Both scenarios predict that the occurrence rate for super-Earths vs. orbital distance, and the corresponding rate for Jupiters, should trend in opposite directions, as the former population is transformed into the latter: as gas giants become more frequent from ∼1 to 10 AU, super-Earths should become more rare.