C. P. Dullemond

Dust coagulation in protoplanetary disks: A rapid depletion of small grains

We model the process of dust coagulation in protoplanetary disks and calculate how it affects their observational appearance. Our model involves the detailed solution of the coagulation equation at every location in the disk. At regular time intervals we feed the resulting 3D dust distribution functions into a continuum radiative transfer code to obtain spectral energy distributions. We find that, even if only the very basic - and well understood - coagulation mechanisms are included, the process of grain growth is much too quick to be consistent with infrared observations of T Tauri disks. Small grains are removed so efficiently that, long before the disk reaches an age of 10 6 years typical of T Tauri stars, the SED shows only very weak infrared excess. This is inconsistent with observed SEDs of most classical T Tauri stars. Small grains must be replenished, for instance by aggregate fragmentation through high-speed collisions. A very simplified calculation shows that when aggregate fragmentation is included, a quasi-stationary grain size distribution is obtained in which growth and fragmentation are in equilibrium. This quasi-stationary state may last 10 6 years or even longer, depending on the circumstances in the disk, and may bring the time scales into the right regime. If this is indeed the case, or if other processes are responsible for the replenishment of small grains, then the typical grain sizes inferred from infrared spectral features of T Tauri disks do not necessarily reflect the age of the system (small grains → young, larger grains → older), as is often proposed. Indeed, there is evidence reported in the literature that the typical inferred grain sizes do not correlate with the age of the star. Instead, it is more likely that the typical grain sizes found in T Tauri star (and Herbig Ae/Be star and Brown Dwarf) disks reflect the state of the disk in some more complicated way, e.g. the strength of the turbulence, the amount of dust mass transformed into planetesimals, the amount of gas lost via evaporation etc. A simple evolutionary scenario in which grains slowly grow from pristine 0.1 µm grains to larger grains over a period of a few Myr is most likely incorrect.

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

Context. The evolution of dust particles in protoplanetary disks determines many observable and structural properties of the disk, such as the spectral energy distribution (SED), appearance of disks, temperature profile, and chemistry. Dust coagulation is also the first step towards planet formation. Aims. We investigate dust growth due to settling in a 1D vertical column of a disk. It is known from the ten 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 the particle properties change as a function of height above the midplane. Methods. We used a Monte Carlo code to follow the mass and porosity evolution of the particles in time. We gradually build up the complexity of the models by considering the effects of porosity, different collision models, turbulence, and different gas models, respectively. This way we can distinguish the effects of these physical processes on particle growth and motion. The collision model used is based on laboratory experiments performed on dust aggregates. As the experiments cannot cover all possible collision scenarios, the largest uncertainty of our model comes from the necessary extrapolations we had to perform. We simultaneously solved for the particle growth and motion. Particles can move vertically due to settling and turbulent mixing. We assumed that the vertical profile of the gas density is fixed in time and that only the solid component evolves. Results. 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 α = 10−2 is needed to keep the dust atmospheres dusty, but such strong turbulence can produce only small particles at the midplane, which does not favor 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 ten micron feature of disk SEDs. Turbulence may therefore help keep small dust particles in the disk atmosphere.

Gas- and dust evolution in protoplanetary disks

Context. Current models of the size- and radial evolution of dust in protoplanetary disks generally oversimplify either the radial evolution of the disk (by focussing at one single radius or by using steady state disk models) or they assume particle growth to proceed monodispersely or without fragmentation. Further studies of protoplanetary disks - such as observations, disk chemistry and structure calculations or planet population synthesis models - depend on the distribution of dust as a function of grain size and radial position in the disk. Aims. We attempt to improve upon current models to be able to investigate how the initial conditions, the build-up phase, and the evolution of the protoplanetary disk influence growth and transport of dust. Methods. We introduce a new model similar to Brauer et al. (2008, A&A, 480, 859) in which we now include the time-dependent viscous evolution of the gas disk, and in which more advanced input physics and numerical integration methods are implemented. Results. We show that grain properties, the gas pressure gradient, and the amount of turbulence are much more influencing the evolution of dust than the initial conditions or the build-up phase of the protoplanetary disk. We quantify which conditions or environments are favorable for growth beyond the meter size barrier. High gas surface densities or zonal flows may help to overcome the problem of radial drift, however already a small amount of turbulence poses a much stronger obstacle for grain growth.

Time Evolution of Viscous Circumstellar Disks due to Photoevaporation by Far-Ultraviolet, Extreme-Ultraviolet, and X-ray Radiation from the Central Star

We present the time evolution of viscously accreting circumstellar disks as they are irradiated by ultraviolet and X-ray photons from a low-mass central star. Our model is a hybrid of a one-dimensional (1D) time-dependent viscous disk model coupled to a 1+1D disk vertical structure model used for calculating the disk structure and photoevaporation rates. We find that disks of initial mass 0.1 M ☉ around ~1 M ☉ stars survive for ~4 × 106 yr, assuming a viscosity parameter α = 0.01, a time-dependent FUV luminosity L FUV ~ 10–2-10–3 L ☉ and with X-ray and EUV luminosities LX ~ L EUV ~ 10–3 L ☉. We find that FUV/X-ray-induced photoevaporation and viscous accretion are both important in depleting disk mass. Photoevaporation rates are most significant at ~1-10 AU and at 30 AU. Viscosity spreads the disk which causes mass loss by accretion onto the central star and feeds mass loss by photoevaporation in the outer disk. We find that FUV photons can create gaps in the inner, planet-forming regions of the disk (~1-10 AU) at relatively early epochs in disk evolution while disk masses are still substantial. EUV and X-ray photons are also capable of driving gaps, but EUV can only do so at late, low accretion-rate epochs after the disk mass has already declined substantially. Disks around stars with predominantly soft X-ray fields experience enhanced photoevaporative mass loss. We follow disk evolution around stars of different masses, and find that disk survival time is relatively independent of mass for stars with M * 3 M ☉; for M * 3 M ☉ the disks are short-lived (~105 yr).

The Gas Temperature in the Surface Layers of Protoplanetary Disks

Models for the structure of protoplanetary disks have thus far been based on the assumption that the gas and dust temperatures are equal. The gas temperature, an essential ingredient in the equations of hydrostatic equilibrium of the disk, is then determined from a continuum radiative transfer calculation, in which the continuum opacity is provided by the dust. It has long been debated whether this assumption still holds in the surface layers of the disk, in which the dust infrared emission features are produced. In this paper we compute the temperature of the gas in the surface layers of the disk in a self-consistent manner. The gas temperature is determined from a heating-cooling balance equation in which processes such as photoelectric heating, dissociative heating, dust-gas thermal heat exchange, and line cooling are included. The abundances of the dominant cooling species such as CO, C, C+, and O are determined from a chemical network based on the atomic species H, He, C, O, S, Mg, Si, and Fe. The underlying disk models to our calculations are the models of Dullemond, van Zadelhoff, & Natta. We find that in general the dust and gas temperatures are equal to within 10% for AV 0.1, which is above the location of the superheated surface layer in which the dust emission features are produced. High above the disk surface the gas temperature exceeds the dust temperature and can become—in the presence of polycyclic aromatic hydrocarbons—as high as 600 K at a radius of 100 AU. This is a region in which CO has fully dissociated, but a significant fraction of hydrogen is still in molecular form. The densities are still high enough for nonnegligible H2 emission to be produced. At radii inward of 50 AU, the temperature of the gas above the photosphere can reach up to ~104 K. In the disk surface layers, the gas temperature exceeds the virial temperature of hydrogen. Some of this material could possibly evaporate, but firm conclusions have to await fully self-consistent disk models, in which the disk structure and gas temperature determination will be solved iteratively.

Benchmark problems for continuum radiative transfer. High optical depths, anisotropic scattering, and polarisation

Solving the continuum radiative transfer equation in high opacity media requires sophisticated numerical tools. In order to test the reliability of such tools, we present a benchmark of radiative transfer codes in a 2D disc configuration. We test the accuracy of seven independently developed radiative transfer codes by comparing the temperature structures, spectral energy distributions, scattered light images, and linear polarisation maps that each model predicts for a variety of disc opacities and viewing angles. The test cases have been chosen to be numerically challenging, with midplane optical depths up 10^6, a sharp density transition at the inner edge and complex scattering matrices. We also review recent progress in the implementation of the Monte Carlo method that allow an efficient solution to these kinds of problems and discuss the advantages and limitations of Monte Carlo codes compared to those of discrete ordinate codes. For each of the test cases, the predicted results from the radiative transfer codes are within good agreement. The results indicate that these codes can be confidently used to interpret present and future observations of protoplanetary discs.

The chemical history of molecules in circumstellar disks - I. Ices

Context. Many chemical changes occur during the collapse of a molecular cloud to form a low-mass star and the surrounding disk. One-dimensional models have been used so far to analyse these chemical processes, but they cannot properly describe the incorporation of material into disks. Aims. The goal of this work is to understand how material changes chemically as it is transported from the cloud to the star and the disk. Of special interest is the chemical history of the material in the disk at the end of the collapse. Methods. A two-dimensional, semi-analytical model is presented that, for the first time, follows the chemical evolution from the pre-stellar core to the protostar and circumstellar disk. The model computes infall trajectories from any point in the cloud and tracks the radial and vertical motion of material in the viscously evolving disk. It includes a full time-dependent radiative transfer treatment of the dust temperature, which controls much of the chemistry. A small parameter grid is explored to understand the effects of the sound speed and the mass and rotation of the cloud. The freeze-out and evaporation of carbon monoxide (CO) and water (H 2 O), as well as the potential for forming complex organic molecules in ices, are considered as important first steps towards illustrating the full chemistry. Results. Both species freeze out towards the centre before the collapse begins. Pure CO ice evaporates during the infall phase and re-adsorbs in those parts of the disk that cool below the CO desorption temperature of ∼ 18 K. Water remains solid almost everywhere during the infall and disk formation phases and evaporates within ∼10 AU of the star. Mixed CO-H 2 O ices are important in keeping some solid CO above 18 K and in explaining the presence of CO in comets. Material that ends up in the planet- and comet-forming zones of the disk (∼5-30 AU from the star) is predicted to spend enough time in a warm zone (several 104 yr at a dust temperature of 20-40 K) during the collapse to form first-generation complex organic species on the grains. The dynamical timescales in the hot inner envelope (hot core or hot corino) are too short for abundant formation of second-generation molecules by high-temperature gas-phase chemistry.

Radiative transfer in very optically thick circumstellar disks

Aims. In this paper we present two efficient implementations of the diffusion approximation to be employed in Monte Carlo computations of radiative transfer in dusty media of massive circumstellar disks. The aim is to improve the accuracy of the computed temperature structure and to decrease the computation time. The accuracy, efficiency, and applicability of the methods in various corners of parameter space are investigated. The effects of using these methods on the vertical structure of the circumstellar disk as obtained from hydrostatic equilibrium computations are also addressed. Methods. Two methods are presented. First, an energy diffusion approximation is used to improve the accuracy of the temperature structure in highly obscured regions of the disk, where photon counts are low. Second, a modified random walk approximation is employed to decrease the computation time. This modified random walk ensures that the photons that end up in the high-density regions can quickly escape to the lower density regions, while the energy deposited by these photons in the disk is still computed accurately. A new radiative transfer code, MCMax, is presented in which both these diffusion approximations are implemented. These can be used simultaneously to increase both computational speed and decrease statistical noise. Results. We conclude that the diffusion approximations allow for fast and accurate computations of the temperature structure, vertical disk structure and observables of very optically thick circumstellar disks.

[O I] 6300 A emission in Herbig Ae/Be systems: Signature of Keplerian rotation

We present high spectral-resolution optical spectra of 49 Herbig Ae/Be stars in a search for the [OI] 6300 A line. The vast majority of the stars in our sample show narrow (FWHM < 100 km s -1 ) emission lines, centered on the stellar radial velocity. In only three sources is the feature much broader (∼400 km s -1 ), and strongly blueshifted (-200 km s -1 ) compared to the stellar radial velocity. Some stars in our sample show double-peaked line profiles, with peak-to-peak separations of ∼10 km s -1 . The presence and strength of the [OI] line emission appears to be correlated with the far-infrared energy distribution of each source: stars with a strong excess at 60 μm have in general stronger [OI] emission than stars with weaker 60 μm excesses. We interpret these narrow [OI] 6300 A line profiles as arising in the surface layers of the protoplanetary disks surrounding Herbig Ae/Be stars. A simple model for [OI] 6300 A line emission due to the photodissociation of OH molecules shows that our results are in quantitative agreement with that expected from the emission of a flared disk if the fractional OH abundance is ∼5 × 10 -7 .

Vertical structure models of T Tauri and Herbig Ae/Be disks

In this paper we present detailed models of the vertical structure (temperature and density) of passive irradiated circumstellar disks around T Tauri and Herbig Ae/Be stars. In contrast to earlier work, we use full frequency- and angle-dependent radiative transfer instead of the usual moment equations. We nd that this improvement of the radiative transfer has a strong influence on the resulting vertical structure of the disk, with dierences in temperature as large as 70%. However, the spectral energy distribution (SED) is only mildly aected by this change. In fact, the SED compares reasonably well with that of improved versions of the Chiang & Goldreich (CG) model. This shows that the latter is a reasonable model for the SED, in spite of its simplicity. It also shows that from the SED alone, little can be learned about the vertical structure of a passive circumstellar disk. The molecular line emission from these disks is more sensitive to the vertical temperature and density structure, and we show as an example how the intensity and proles of various CO lines depend on the adopted disk model. The models presented in this paper can also serve as the basis of theoretical studies of e.g. dust coagulation and settling in disks.