Numerical Simulation ofStructure Evolution of Dust Aggregates
Growing in Protoplanetary Disks

Toru Suyama, Koji Wada, Hidekazu Tanaka
Institute of Low Temperature Science, Hokkaido University1

Abstract
Dust growth: In protoplanetary disks, dust aggregates grow through mutual collisions.
Small dust aggregates have the fluffy structure. However, dust aggregates are compressedas they grow. Thus, large dust aggregates can not keep such fluffy structure. Suchcompression changes the cross section of dust aggregates. Thus, when and how tocompression is important on dust growth.
Method: In this study, we perform the N-body simulation of aggregate collisions of realistic
oblique collisions. We focused the radius of gyration and the cross section as the propertyof the structure of dust aggregates. We examine the evolution of the radius of gyration andthe cross section during aggregate growth, respectively.
Result: We examine the change of the radius of gyration and the cross section at collisions
between aggregates which have a variety of the mass and the density. At oblique collisions,
dust aggregates are elongated. For the radius of gyration, aggregates are not compressed.
However, for the cross section, aggregates are compressed. Applying our compression
model based on numerical simulations of head-on collisions (Suyama et al. 2008) tooblique collisions, we succeed the formulation of the evolution of the cross section. As aresult of this model, the impact compression make low density. Such aggregates have alarge cross section. Thus, dust growth which previous study considered changes
quantitatively.
2

Structure evolution of the dust aggregates
Dust

Planetesimals

Planets

Initial condition of planet formation
Dust grow through mutual collisions

The change of the cross section
and the density causes
the impact velocity change⇒ It affects dust growth.

Impact compression?

Low velocity

No compression
BCCA(Ballistic Cluster-Cluster Aggregation)
The porosity increases and density
decreases.
⇒Dust must be compressed!

It is important to examine the evolutionof density in the course of dust growth.
Simulation of aggregate collision
Dominik and Tielens 1997
Wada et al. 2007, 2008
Suyama et al. 2008
Paszun and Dominik 2008
Only one collision⇒Dust must grow through many collisions.
Head-on collision3⇒Most collisions are oblique ones.

Aim
For oblique collisions,
how does the internal structure change?
What is the difference between
oblique collisions and head-on collisions?

We performed 3D numerical simulation of
aggregate collisions and examine the density changes
through dust growth.
4Interaction model
TensileJohnson et al. (1971)
Dominik and Tielens (1997)
Wada et al. (2007)

Roll

The energy necessary to roll: ErollSlideTwist

We calculate the force for each degree of freedom.

5

N-body simulation of dust collisions
We simulate the collisional growth of dust aggregates which consisted of sphericalparticles (Ice, r=0.1μm).

Method：・ We repeat the collisions between two identical aggregates.
A sequence of collision: 「2 ＋ 2 ⇒ 4 ＋ 4 ⇒…⇒ 16384 ＋ 16384」
・ Oblique collisions (Impact parameter is chosen randomly).
・ The impact velocity is constant.
・ The estimation of the density:
We perform the 10 sequences of collisions with each velocity. We obtain
the averaged value.

48327686

Effects of oblique collisions
At oblique collisions, dust aggregates are elongated.

Head-on

Oblique7

Resultant aggregates
N=16384

N=32768

Vimp=4.4m/s

Vimp=1.1m/s

8

At higher velocity collisions, dust aggregates are compressed for small scale structure.

Evolution of radius of gyration
Nrg

Head-onrg =∑ x −x
i =1i

N

G

2

xi: the positions of particles
xG: the center of the aggregate

Oblique

At oblique collisions, aggregates are not compressed 9
because they are elongated.

rg

rg

Evolution of density ρ(rg)rc

Head-on

5rc =rg
3m1 N
ρ (rg ) = 4π 3
3 rc

Oblique

At oblique collisions, the density decreases asthat of BCCA clusters.

10Evolution of cross section
We calculated the cross section ofthe shadow of dust aggregates (30-directions).

Head-on

Oblique

Even at oblique collisions, aggregates are compressed!

11Evolution of density ρ(S)We evaluated the density fromthe cross section.

Head-onrs =Sπm1 N
ρ ( S ) = 4π 33 rs

ObliqueHigh velocity oblique collisions = Low velocity head-on collisions

12

Internal structure of compressed dust
We count the number of particles N’(r)within a distance r.

Fractal dimension

d ln N in (r )Df ≡d ln r

Head-onHead-on：Df=2.5
Oblique：Df=2.5
Compact：Df=3.0, BCCA：Df=2.0
Fractal dimension is 2.5.
⇒Even at oblique collisions,
dust is compressed.

Oblique
13Compression model:
compressive strength
Compression at collisions between BCCA clusters (Wada et al. 2008).
A relation between the volume V(rg) and the impact energy Eimp .

dEimp = − PdV
1332
ρ(r)
⎛⎞Eroll ρ1g3
Compressive strength: P ≡ −⎜⎟N= 0.5
dVm1 ⎜⎝ ρ1 ⎟⎠dEimpWe obtain the volume change at a collision with Eimp .
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Compression model: ρ(rg) at head-on
Eimp = − ∫V final

Vinitial

d ln ρ (rg )d ln N=−PdV

15⎛ ρ (rg ) ⎞⎟⎟+ 0.23⎜⎜⎝ ρm ⎠−

103⎛M ⎞⎜⎜ ⎟⎟⎝ m1 ⎠−

2
3

Eimp
NEroll1/ 33 ⎛⎜ m1 N ⎞⎟rg =5 ⎜⎝ 43π ρ (rg ) ⎟⎠We obtain the evolution of thedensity and the radius of gyration
for given impact velocity.

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Compression model: ρ(S)We apply density evolution model about ρ(rg) to ρ(S).
⎛ ρ (S ) ⎞d ln ρ ( S )1⎟⎟= − + B⎜⎜d ln N5⎝ ρm ⎠

Head-on:
B=0.055−103⎛M ⎞⎜⎜ ⎟⎟⎝ m1 ⎠−23

EimpNEroll⎛ m0 N ⎞⎟S = π ⎜⎜ 4π⎟S
ρ
()
3
⎝⎠

2/3

Oblique:
B=0.0035

16

We obtain the evolution of the density and the cross section.

Evolution of density in protoplanetary disks
Settling

MidplaneCompression and growthwith settling

R=5AU

Compression and growthat the midplane

Impact compression results invery low density objects！
(ρ~10-5g/cm3)

17Summary• We performed the simulation of oblique collisions
between dust aggregates and examined thecompression process.
• For the radius of gyration, aggregates are notcompressed.
• For the cross section, aggregates are compressed.
• We formulated the evolution of the density and thecross section.
• Impact compression results in very low density
objects. (ρ~10-5g/cm3)18