Dust size distributions in coagulation/fragmentation equilibrium: numerical solutions and analytical fits

By T. Birnstiel, C. Ormel, C. Dullemond. 2010

199 citations recorded by the source corpus.

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.

Subfields: Astronomy and Astrophysics, Atomic and Molecular Physics, and Optics, Instrumentation

View Paper: https://www.semanticscholar.org/paper/f0249fcf74278e5fd60ab58fb5ca6e0cfa2158b8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Astronomy and Astrophysics

In Atomic and Molecular Physics, and Optics

In Instrumentation

Citations in the corpus

Breaking through: The effects of a velocity distribution on barriers to dust growth (2012, background): “It is now well-known known that the gas and dust mass of the solar nebula might have been much higher (Desch 2007), which would increase the coagulation rates and the particle size at the fragmentation barrier (Birnstiel et al. 2011).”
THE MASS DEPENDENCE BETWEEN PROTOPLANETARY DISKS AND THEIR STELLAR HOSTS (2013, background): “Models of this material evolution predict that growth timescales increase like 1/ζ √ M∗ up to grain sizes (a) that are limited by fragmentation (amax ∝ Md; Birnstiel et al. 2011) or drift (amax ∝ M∗Md ζ; Birnstiel et al. 2012).”
Close-in planetesimal formation by pile-up of drifting pebbles (2016, methodology): “For this plot, we used a power-law size (and Stokes number) distribution that is an effect of the coagulation-fragmentation equilibrium (Birnstiel et al. 2011) n(St) · m dSt ∝ St−1/2 dSt (A.8) for seven equally spaced logarithmic bins between St = 10−4 and St = 10−1.”
RESOLVED IMAGES OF LARGE CAVITIES IN PROTOPLANETARY TRANSITION DISKS (2011, background): “Even in a large dead zone without dust in the surface layers, a collisional fragmentation process might still produce a substantial reservoir of small grains and limit growth beyond ∼cm-scales (Brauer et al. 2008; Birnstiel et al. 2010, 2011).”
Global Simulations of the Inner Regions of Protoplanetary Disks with Comprehensive Disk Microphysics (2017, methodology): “We choose grain size m=a 0.1 m and mass fraction of = -f 10 4, where the total surface area is comparable to those obtained by more realistic grain coagulation/fragmentation calculations (Birnstiel et al. 2011).”
Dust Evolution and the Formation of Planetesimals (2016, background): “…2015) – although if this interpretation is true, then that poses a challenge for coagulation theory and experiments to explain how the dominant pebble sizes can be more than one meter, in contrast to drift dominated models that predict sizes closer to 1 cm in the outer disc (Birnstiel et al. 2011).”
The Disk Substructures at High Angular Resolution Project (DSHARP). V. Interpreting ALMA Maps of Protoplanetary Disks in Terms of a Dust Model (2018, methodology): “This is what is expected from models of dust coagulation which include fragmentation (Weidenschilling 1984; Dullemond & Dominik 2005; Brauer et al. 2008; Birnstiel et al. 2010).”
A simple model for the evolution of the dust population in protoplanetary disks (2012, methodology): “In the case where the fragmentation of dust grains due to turbulent relative motion is the limiting effect, analytical models of the grain size distribution have been derived in Birnstiel et al. (2011).”
DUST FILTRATION BY PLANET-INDUCED GAP EDGES: IMPLICATIONS FOR TRANSITIONAL DISKS (2012, background): “The complexity of the problem is illustrated by, for example, Birnstiel et al. (2011) and references therein.”
Pebble-isolation mass: Scaling law and implications for the formation of super-Earths and gas giants (2018, methodology): “The final pebble sizes in our model were determined by radial drift alone, where we did not take the effects of fragmentation (Brauer et al. 2008; Birnstiel et al. 2011), bouncing (Zsom et al. 2010), or condensation (Ros & Johansen 2013; Schoonenberg & Ormel 2017) into account.”
Chemical enrichment of giant planets and discs due to pebble drift (2017, background): “We note that although the dust mass is dominated by large grains, the surface area of the grains is dominated by small grains for both protoplanetary disc dust populations (Birnstiel et al. 2011) and the interstellar medium distribution of grain sizes (Mathis et al. 1977)2.”
Survival of molecular gas in cavities of transition disks - I. CO (2013, methodology): “In a forthcoming paper, we plan to couple the present model with a model for dust evolution (Birnstiel et al. 2010, 2011, 2012b) which will allow us to study the combined use of lines and the continuum in order to better constrain the gap formation mechanism.”