July 2014
Example of a calculation with the new structure equation for grain growth. The left panel shows the normalized temperature and density while the right panel plots the grain properties: their size, abundance, and opacity. Results of two simulations are presented: (i) grains only enter at the top of the atmosphere (solid lines); (ii) grains are additionally produced by ablating planetesimals within the atmosphere (dashed lines). In the latter grains coagulate quickly, but the bigger particles suppress the grain opacity. As a result, the temperature and density profiles are very similar.


Early atmospheres around planets — those still embedded in gaseous disks — are voluminous, but not necessarily very massive. How massive such atmosphere can become depends to a large extent on the thermodynamic properties of the atmospheric gas and its ability to cool.
This means that it depends on the opacity of the gas or rather the opacity of the grains within the gas. There are many uncertainties here: how many grains are there; how well do they grow (make planetesimals) in disks; what are their optical properties. Grains can also change their abundance, size and optical properties in such atmospheres. In fact, they will do so very rapidly. To take these effects into account one may solve for the grain size distribution as function of depth by combining the Smoluchowski equation and the transport equation. This is accurate but computationally intensive.
However, the problem can be simplified by assuming that there is a single dominant grain size and solve for this grain size and its corresponding abundance and optical properties as function of depth. What I have done is, in fact, added an additional stellar structure equation for the grains.



C.W. Ormel
An Atmospheric Structure Equation for Grain Growth
The Astrophysical Journal Letters, Vol. 789...L18 (2014) [ADS] [arXiv]
May 2013
Comparison of turbulent motions (black dashed), strength curves (red solid and dashed) and the escape velocity (blue)


The classical idea in planetformation theory is the following: A) planetesimals (~kmsize bodies) form out of dust; B) they quickly produce a few protoplanetary seeds; C) these protoplanets sweepup the leftover planetesimals; D) (proto)planets then accrete gas, migrate (possibly), and experience dynamical interactions before settling into a stable configuration.
We wondered whether phase B is compatible with a turbulent protoplanetary disks, for which nature there are ample indications. The short answer is: not really. The underlying reason is that turbulence causes density fluctuations in the gas that torque bodies gravitationally; and this while phase B relies on a phenomenon of runaway growth, which requires very quiescent conditions. The figure illustrates this point graphically. Here, we plot the turbulent excitation as function of the size of bodies. For bodies until 30 km the relative velocity lies above the escape velocity (v_{esc}). In that case there is no runaway growth and no rapid assembly of protoplanets. Bodies have to growth beyond this threshold (denoted by the blue dot in the figure) to initiate phase B.
Actually, the figure represents one of the more positive scenarios, because of the presence of a so called deadzone — a region in the disk where the ionization is very low and the turbulent instability is suppressed. Unfortunately, we find that due to dust coagulation the small grains and the deadzone to disappear. These calculations imply that turbulent disks are not a conducive environment for planet assembly.



S. Okuzumi & C.W. Ormel
The fate of planetesimals in turbulent disks with dead zones. I. The turbulent stirring recipe
The Astrophysical Journal, Vol. 771...43 (2013) [ADS] [arXiv]
C.W. Ormel & S. Okuzumi
The fate of planetesimals in turbulent disks with dead zones. II. The viability of runaway growth
The Astrophysical Journal, Vol. 771...44 (2013) [ADS] [arXiv]
January 2013
Pantha rhei ("Everything flows"—Heraclitus, ~500 BC); but how do gravitating bodies like planets affect the flow pattern of the gas that attempts to stream past? When protoplanetary embryos made from accumulation of solid particles exceed a certain threshold mass (corresponding to ~1000 km in size), they can start to bind the gas from the nebula disk. The atmospheres of these young, low mass planets are (presumably) hot; generally, the gas does not collapse onto the planet and the atmosphere is in pressureequilibrium with the disk. In fact, the boundary between planet and nebular disks has to be determined from the velocity of the flow.
We have calculated the steadystate flow pattern which emerges in the vicinity of the planet (see figure). Left and right one notes the background shearing flow, which is almost unperturbed. Towards the top and the bottom, one sees, very clearly, the horseshoe region where streamlines make a "Uturn". At the very center the flow curls around the planet in the prograde (counterclockwise) direction. This is the atmosphere of the protoplanet.
Implications concern the migration behavior of protoplanets (for which the width of the horseshoe region is a key parameter), the thermodynamical structure of the protoplanet atmosphere, circumplanetary disks formation, and the accretion behavior of small particles (for which the gas drag is important).


A typical flow pattern past a low mass planet, where it is in steady state. Arrows indicate the velocity and gray contours are streamlines of the flow. Red/Pink circles are isodensity contours.


C.W. Ormel
The flow pattern past gravitating bodies
Monthly Notices of the Royal Astronomical Society, Vol. 428...3526 (2013) [ADS] [arXiv]
August 2012
If you ever shot a gun, you probably noticed that the gun recoiled back on you. This is due to the conservation of momentum. In disks, a migrating planet likewise reflects that its (angular) momentum is changing. The gravitational interaction with the gaseous disks is a wellknown effect (TypeI migration).
In a disk solid bodies (planetesimals) act as the bullets. The gravitational interaction with the much more massive planet will slingshot them to different orbits (scatterings). The planet feels the recoil, which causes it to migrate. Overall, this planetesimaldriven migration is analogous to the (more wellknown) TypeI migration; but both can be understood as a consequence of dynamical friction.
A complication is that, to first order, interactions with the interior disk cancel interactions with the exterior disk. Therefore, higher order effects must be included. This means that the migration depends on the gradient in the surface density and eccentricity. We investigate the effects of an eccentricity gradient and find a strong dependence. In addition, we find a regime where the migration is selfsustained.


Mapping the three migration regimes


C.W. Ormel, S. Ida, & H. Tanaka
Migration rates due to scattering of planetesimals
The Astrophysical Journal, Vol 758... 80 (2012) [ADS] [arXiv]
July 2012
It is already difficult to make one planet — let alone two. Therefore, we have investigated the idea for triggered planet formation. Applied to the solar system this means: form Jupiter first, then Saturn.
To make life a bit easier, we have assumed that Jupiter did already form (without specifying how) and carved a wide gap in the primordial gasrich protoplanetary disk. This gap causes a pressure maxima, whose location could coincide with Saturn's for plausible parameters. This is important because debris ('small stuff') will pile up at these pressure maxima. The debris originates from collisions among planetesimals from the outer disks.
The figure on the right tells the story: without a pressure maxima (dashed lines) Saturn will not grow big — it doesn't even get to the Earth! With a pressure maxima created by Jupiter (solid line), it easily jumps over the Earth in terms of mass and become a gas giant next to Jupiter. Note that this mechanism works better when the outer disk contains smalle bodies, because these are weaker and collide more frequent.


Evolution of protoplanet mass for several models


H. Kobayashi, C.W. Ormel, & S. Ida
Rapid Formation of Saturn after Jupiter Completion
The Astrophysical Journal, Vol. 756...70 (2012) [ADS] [arXiv]
December 2011
A flowchart of the toy model


Protoplanet growth is complex. There are a multitude of physical processes — for instance, planetesimal fragmentation, radial drift, turbulent diffusion, gas drag — that determine its efficiency. Catching all these mechanisms in one selfconsistent model is virtually impossible. Let alone to perform a statistically viable parameter study.
Here, a toy model serves as a useful tool to quickly explore the parameter space. We construct a toy model for the protoplanet growth, emphasizing simplicity and versatility:
 Speed  a single run takes ~seconds
 Completeness  include many physical processes
 Transparency  keep the physics simple and make the model publicly available for the community
This means that we follow only three components — embryos, planetesimals, and fragments — include many physical processes (some of them are named in the sketch), and that we opt for a modular nature of the code, i.e., the features can be turned on or off at the user's discretion.
Compared to previous toy models, we have especially focused on a more realistic treatment of the interaction of small particles (fragments) with the gas.



C.W. Ormel & H. Kobayashi
Understanding how planets become massive? I. Description and validation of a new toy model
The Astrophysical Journal, Vol. 747...115 (2012) [ADS] [arXiv]
June 2011
Coagulation will affect the dust size distribution in dense molecular clouds. This, in turn, affects the dust opacity — a critical quantity required for any interpretation of observational data sets. Following previous work (below) where we computed the dust size distribution as function of time, we now present the corresponding massweighted opacities for infrared wavelengths. The figure shows that the opacities change on timescales of ~Myr (or less if the cloud's density is higher than the assumed n=10^{5} cm^{3}). At visible and nearIR wavelengths the opacitiy decreases, but at longer wavelengths it will increase with time. We have quantified this evolutionary trend in terms of the strength in the 9.7μm silicate feature vs nearIR color excess and in terms of the submm slope β.


Opacity changes with time/coagulation state


C.W. Ormel, M. Min, X. Tielens, C. Dominik, & D. Paszun
Dust coagulation and fragmentation in molecular clouds. II. The opacity of the dust aggregate size distribution
Astronomy & Astrophysics, Volume 532, A43 (2011) [ADS] [arXiv]
July 2010
Examples of orbits for particles experiencing various amounts of gas drag and gravitational forces


Protoplanets can sweepup particles effectively due to their gravitational focusing effect which allows particles to be accreted with a collision cross section much larger than the geometrical cross section of the protoplanets. The amount of focusing depends to a large degree on the velocity at which the bodies approach, with the largest focusing being achieved at low relative velocities. This effects is well described in the literature; however the effects of gas drag —a force that becomes especially important for small particles— on this process are less clear. In this project we determined how gas drag affects the gravitational focusing of particles.
The figure shows some examples of particle trajectories under the influence of varying levels of gas drag. The protoplanet is in the center of the coordinate system.



C.W. Ormel & H.H. Klahr
The effect of gas drag on the growth of protoplanets—Analytical expressions for the accretion of small bodies in laminar disks
Astronomy and Astrophysics, Volume 520, A43 (2010) [ADS] [arXiv] [A&A Highlights]
January 2011
It is well known that a collisional cascade produces a powerlaw size distribution. The most notably example being the dust size distribution in the interstellar medium, the so called MRN distribution. This situation represents a steadystate: collisions among particles of a certain size deplete their number at the same rate as the replenishment rate by collisions among larger particles. A collisional cascade is applicable to situations where collisions results in fragmentation. However, in a protoplanetary disks the gas damps the motions of the smaller particles. As a result, the small particles coagulate whereas only collisions with highermass particles are energetic enough to produce fragments. In this situation a steadystate emerges but the size distribution is now determined by the characteristics of both the fragmentation and the coagulation.
We have considered a coagulationfragmentation steady state for the dust in protoplanetary disks. Two key parameters are: i) the velocity field (Brownian motion or turbulence); and ii) the properties of the fragmentation event (the size distribution of particles within a collision). In case of a power law dependence on mass of these quantities we have derived the ensuing exponent for the size distribution.


Size distribution for different fragmentation states


T. Birnstiel, C.W. Ormel, & C.P. Dullemond
Dust size distributions in coagulation/fragmentation equilibrium: numerical solutions and analytical fits
Astronomy and Astrophysics, Volume 525, A11 (2011) [ADS] [arXiv]
March 2010
When bodies are large enough to attract each other gravitationally the effective cross section for collisions becomes larger than the geometrical cross section. This enhancement factor is called the gravitational focusing factor (GFF) and is approximately given by the square of the ratio of the escape velocity of the protoplanet and the relative velocity at wich the bodies approach. The nature of this phenomenon is such that the largest bodies grow faster than other bodies, a situation referred as runaway growth (RG). RG indicates a positive feedback effect: due to the growth, the gravitational focusing increases. RG or, generally, a large GFF is required in order to grow protoplanets in a sufficiently short time span.
However, the positive feedback of RG is counteracted by viscous stirring, the (longrange) deflection of bodies' trajectories that causes the mean relative motion to increase. The growth of the protoplanet then slows down, switching to the much slower oligarch growth stage.
In this study we have investigated the conditions at which runaway growth turns into oligarchic growth. In the figure the GFF is plotted as function of the (evolving) radius of the protoplanet, which indicates time. First GFF increase (the RGstage) but decrease after the transition size R_{1}=R_{tr} (the oligarchic growth stage). We provide a criterion for the start of the oligarchic growth phase (i.e. for R_{tr}) in terms of environmental conditions (radius and surface density of planetsimals, semimajor axis, etc.)


Evolution of the gravitational focusing during runaway growth


C.W. Ormel, C.P. Dullemond, & M. Spaans
A New Condition for the Transition from Runaway to Oligarchic Growth
The Astrophysical Journal Letters, 714, 103 (2010) [ADS] [arXiv]
October 2010
When bodies reach kmsizes (planetesimals), their collisional and dynamical evolution becomes dominated by gravity. Ideally, one would model the collisional evolution by an Nbody methods; hower the shear number of planetesimals limit these attempts in practise. However, to model the (runaway and oligarchic) growth correctly, its discrete nature must be taken into account. For this reason we have extended our Monte Carlo /superparticle code to treat dynamical interactions.
The figure shows the size distribution at three times. Protoplanets can be seen to separate from the population of (leftover) planetesimals. The color shows the amount of dynamical excitation with respect to the largest body in the simulation: blue colors indicate a dynamical cold system, whereas red colors indicate a dynamical hot system. Click the image for an mpeg movie!


Planetesimal accretion—click on image for animation


C.W. Ormel, C.P. Dullemond, & M. Spaans
Accretion among preplanetary bodies: the many faces of runaway growth
Icarus, Volume 210, Issue 1, p. 507538 (2010) [ADS] [arXiv]
October 2009
Very small, μmsize, dust particles easily stick, setting the first steps in a coagulation process that will eventually form planets. The sticking assumpting becomes less obvious for larger particles, however, with laboratory experiments indicating that equallysized dust particles often bounce off. But in other experiments, particles can still stick, especially if the size ratio of the particles involved is large. Yet in other experiments, fragmentation is observed at larger impact velocities. The porosity of the aggregates is also important to determine the outcome of a collision.
To investigate the implications of these diverse results on the coagulation process, we have calculated the collisional evolution with a Monte Carlo code. Each particle is characterized by two properties —its mass and porosity— and represents a certain share of the total dust's mass budget. The Monte Carlo code calculates the probability of a collision among any combination of particles with the outcome of this collisions being given by (or interpolated from) thelaboratory results.
Initially we find (click image to start movie) that the growth is fractal: the porosity (=enlargement factor) of the bodies increases. However, at some stage the fractal growth stops, as collisions become more energetic. Thereafter, there is a sudden (almost runaway) growth stage that is triggered by a wide size distribution. However, this growth is subsequently negated by fragmentary or masstransfer collisions. In the end, bouncing predominates and little evolution is present. We find that the bouncing result is a nearuniversal outcome, i.e., it seems very difficult to circumvent.
In a followup study we have extended the model to include a vertical dimension to investigate the sedimentation/diffusion behavior of the dust. One of the underlying questions is whether or not the disk atmosphere can be kept dusty (meaning that small particles are present) on long timescales. Dusty disk atmospheres are observationally favored to explain the presence of, among others, the 10μm silicate feature. We found that, provided turbulence is strong, particle fragmentation can indeed replenish small particles. However, the drawback is that particles at the midplane —where planets presumably form— also do not grow large!


Combined evolution of particle mass and porosity. Click image for animation


C. Güttler, J. Blum, A. Zsom, C.W. Ormel, & C.P. Dullemond
The outcome of protoplanetary dust growth: pebbles, boulders, or planetesimals?. I. Mapping the zoo of laboratory collision experiments
Astronomy and Astrophysics, Volume 513, A56 (2010) [ADS] [arXiv]
A. Zsom, C.W. Ormel, C. Güttler, J. Blum, & C.P. Dullemond
The outcome of protoplanetary dust growth: pebbles, boulders, or planetesimals? II. Introducing the bouncing barrier
Astronomy and Astrophysics, Volume 513, A57 (2010) [ADS] [arXiv]
A. Zsom, C.W. Ormel, C.P. Dullemond, & T. Henning
The outcome of protoplanetary dust growth: pebbles, boulders, or planetesimals? III. Sedimentation driven coagulation inside the snowline
Astronomy and Astrophysics, Volume 534, A73 (2011) [ADS] [arXiv]
June 2009
Cores in molecular clouds are dense enough for grains to collide — a process which has implications for the interpretation of their observations. Not all the collisions are necessarily sticking, however; the relative velocites among dust particles is relatively large and the material properties will likewise influence the growth process. In this study we have investigated the process of dust coagulation and fragmentation by combining an state of the art molecular dynamics model for the outcome of collisions among individual dust aggreagtes and a Monte Carlo collisional evolution code.
We consider two type of coagulation modes: i) among slicatelike grains; and ii) among icecoated grains. It is found that the former state leads (quickly) to steadystate dust distributions, where destructive collisions among highmass aggregates replenish the small grains. However, if the grains are icecoated their growth can become significant (see figure). The material properties of ices are very conducive for grain growth, perhaps up to a factor 10^{3} in size.
Whether or not this strong growth materializes also depends on the lifetime of the molecular cloud (or core). If the condensation is simply a transient phenomenon, e.g., a fluctuation in a turbulent environment, it lifetime — essentially the freefall time — is proberly only ~10^{5} yr. In that case, the imprints of grain growth are probably very minor. However, magnetic fields may ertard the collapse, perhaps by a factor of 10 or so, promoting growth.
Observationally, grain growth manifests itself by the reduction of the opacity. We have calculated the effects of grain growth on the (geometrical) opacity and found that the initial decline can be understood in terms of the initial collision timescale between dust grains. Thus, a higher density compensates a shorter lifetime to give a similar observational signature. At large times, fragmentation and the replenishment of smaller grains stabilizes the opacity. More sophisticated opacity calculations are upcoming.


Evolution of the grain size distribution
Opacity evolution for several models


C.W. Ormel, D. Paszun, C. Dominik, & A.G.G.M. Tielens
Coagulation and fragmentation in molecular clouds: I. How collisions between dust aggregates alters the dust size distribution
Astronomy and Astrophysics, Volume 502...845 (2009) [ADS] [arXiv]
April 2008
Monte Carlo methods simulating the coagulation process suffer from one fundamental shortcoming: their limited dynamic range. Coagulation removes particles from the distribution and the number of collisions that can be followed is necessarily less than the initial number of particles N. The latter is of course very large for astrophysical purposes, but is in practice small due to computational reasons.
In this study we have implemented a new algorithm, introducing the superparticle concept to Monte Carlo simulations; The number of superparticles N_g is then limited, but N can be virtually infinite. Collisions are then between particle groups; rather than between individual particles. The algorithm leaves the user the freedom to choose the relation between superparticles and physical bodies.
In our favorite implementation, we assign the superparticles equally over logarithmic mass space, such that the highmass, exponentially declining tail of the distribution is well resolved too. This means that the algorithm is ideally suited to study runaway growth systems. An aplication is presented in the figure, where the timescale for runaway growth (a.k.a. gelation) is plotted vs. initial partice number N for two runaway kernels. The larger the box size the sooner the gelation proceeds (although the particle density is constant in all cases).


Runaway (gelation) growth time as function of system size


C.W. Ormel & M. Spaans
Monte Carlo Simulation of Particle Interactions at High Dynamic Range: Advancing beyond the Googol
The Astrophysical Journal, Volume 684...1291 (2008) [ADS] [arXiv]