# Turbulence-induced relative velocity of dust particles. III. The probability distribution

## Abstract

Motivated by its important role in the collisional growth of dust particles in protoplanetary disks, we investigate the probability distribution function (PDF) of the relative velocity of inertial particles suspended in turbulent flows. Using the simulation from our previous work, we compute the relative velocity PDF as a function of the friction timescales, τ{sub p1} and τ{sub p2}, of two particles of arbitrary sizes. The friction time of the particles included in the simulation ranges from 0.1τ{sub η} to 54T {sub L}, where τ{sub η} and T {sub L} are the Kolmogorov time and the Lagrangian correlation time of the flow, respectively. The relative velocity PDF is generically non-Gaussian, exhibiting fat tails. For a fixed value of τ{sub p1}, the PDF shape is the fattest for equal-size particles (τ{sub p2} = τ{sub p1}), and becomes thinner at both τ{sub p2} < τ{sub p1} and τ{sub p2} > τ{sub p1}. Defining f as the friction time ratio of the smaller particle to the larger one, we find that, at a given f in (1/2) ≲ f ≲ 1, the PDF fatness first increases with the friction time τ{sub p,h} of the larger particle, peaks at τ{sub p,h} ≅ τ{sub η}, andmore »

- Authors:

- Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 (United States)
- ICREA and ICC, University of Barcelona, Marti i Franquès 1, E-08028 Barcelona (Spain)
- Department of Astronomy, University of Texas, Austin, TX 78712 (United States)

- Publication Date:

- OSTI Identifier:
- 22365171

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Astrophysical Journal; Journal Volume: 792; Journal Issue: 1; Other Information: Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; ASTEROIDS; CORRELATIONS; DIFFUSION BARRIERS; DISTRIBUTION; DISTRIBUTION FUNCTIONS; DUSTS; FRAGMENTATION; LAGRANGIAN FUNCTION; PARTICLES; PLANETS; PROBABILITY; PROTOPLANETS; SATELLITES; SIMULATION; TURBULENCE; TURBULENT FLOW; VELOCITY

### Citation Formats

```
Pan, Liubin, Padoan, Paolo, and Scalo, John, E-mail: lpan@cfa.harvard.edu, E-mail: ppadoan@icc.ub.edu, E-mail: parrot@astro.as.utexas.edu.
```*Turbulence-induced relative velocity of dust particles. III. The probability distribution*. United States: N. p., 2014.
Web. doi:10.1088/0004-637X/792/1/69.

```
Pan, Liubin, Padoan, Paolo, & Scalo, John, E-mail: lpan@cfa.harvard.edu, E-mail: ppadoan@icc.ub.edu, E-mail: parrot@astro.as.utexas.edu.
```*Turbulence-induced relative velocity of dust particles. III. The probability distribution*. United States. doi:10.1088/0004-637X/792/1/69.

```
Pan, Liubin, Padoan, Paolo, and Scalo, John, E-mail: lpan@cfa.harvard.edu, E-mail: ppadoan@icc.ub.edu, E-mail: parrot@astro.as.utexas.edu. Mon .
"Turbulence-induced relative velocity of dust particles. III. The probability distribution". United States.
doi:10.1088/0004-637X/792/1/69.
```

```
@article{osti_22365171,
```

title = {Turbulence-induced relative velocity of dust particles. III. The probability distribution},

author = {Pan, Liubin and Padoan, Paolo and Scalo, John, E-mail: lpan@cfa.harvard.edu, E-mail: ppadoan@icc.ub.edu, E-mail: parrot@astro.as.utexas.edu},

abstractNote = {Motivated by its important role in the collisional growth of dust particles in protoplanetary disks, we investigate the probability distribution function (PDF) of the relative velocity of inertial particles suspended in turbulent flows. Using the simulation from our previous work, we compute the relative velocity PDF as a function of the friction timescales, τ{sub p1} and τ{sub p2}, of two particles of arbitrary sizes. The friction time of the particles included in the simulation ranges from 0.1τ{sub η} to 54T {sub L}, where τ{sub η} and T {sub L} are the Kolmogorov time and the Lagrangian correlation time of the flow, respectively. The relative velocity PDF is generically non-Gaussian, exhibiting fat tails. For a fixed value of τ{sub p1}, the PDF shape is the fattest for equal-size particles (τ{sub p2} = τ{sub p1}), and becomes thinner at both τ{sub p2} < τ{sub p1} and τ{sub p2} > τ{sub p1}. Defining f as the friction time ratio of the smaller particle to the larger one, we find that, at a given f in (1/2) ≲ f ≲ 1, the PDF fatness first increases with the friction time τ{sub p,h} of the larger particle, peaks at τ{sub p,h} ≅ τ{sub η}, and then decreases as τ{sub p,h} increases further. For 0 ≤ f ≲ (1/4), the PDF becomes continuously thinner with increasing τ{sub p,h}. The PDF is nearly Gaussian only if τ{sub p,h} is sufficiently large (>>T {sub L}). These features are successfully explained by the Pan and Padoan model. Using our simulation data and some simplifying assumptions, we estimated the fractions of collisions resulting in sticking, bouncing, and fragmentation as a function of the dust size in protoplanetary disks, and argued that accounting for non-Gaussianity of the collision velocity may help further alleviate the bouncing barrier problem.},

doi = {10.1088/0004-637X/792/1/69},

journal = {Astrophysical Journal},

number = 1,

volume = 792,

place = {United States},

year = {Mon Sep 01 00:00:00 EDT 2014},

month = {Mon Sep 01 00:00:00 EDT 2014}

}