Static compression of fluffy dust aggregates in ... - Lund Observatory

Static compression of fluffy dust aggregates
in protoplanetary disks
Akimasa Kataoka
(National Astronomical Observatory of Japan / SOKENDAI)
H.Tanaka (Hokkaido Univ.), S.Okuzumi (Nagoya Univ.), K.Wada (Chiba-tech)

Porosity in dust coagulation
constant
density
coagulation
unrealistic
porosity
evolution
coagulation
realistic
Recent studies have shown that dust grains
grow to fluffy aggregates
cf). Wada et al. 2007, 2009, 2011, Suyama et al. 2008,2012, Okuzumi et al. 2009,2012
Akimasa Kataoka, Ice and Planet Formation @ Lund, May 16th, 2013
2

Planetesimal formation with fluffy aggregates
internal density
0.1μm
1g/cm 3
ISM
1m 1km 10 2-4 km radius
Planetesimals
Other compression
mechanisms are required
cf). Wada et al. 2009,
Okuzumi et al. 2009,2012 ,
Suyama et al. 2008,2012
10 -5 g/cm 3
Collisional Compression
Akimasa Kataoka, Ice and Planet Formation @ Lund, May 16th, 2013 3

Aim of this work
internal density
1μm
1g/cm 3
ISM
Static compression
by gas pressure
gas
10 -5 g/cm 3
1m 1km 10 2-4 km radius
Planetesimals
Static compression
by Self-gravity
gravity
r
We investigate static compression of
highly porous aggregates
Akimasa Kataoka, Ice and Planet Formation @ Lund, May 16th, 2013 4

Strategy
1. we drive static compressive strength by using N-body
simulations (Kataoka et al. 2013, A&A accepted, arXiv:1303.3351)
2. we apply the compressive Physical Model strength to protoplanetary disks
with a given pressure = ram pressure of gas, self gravity.
Particle-Particle Interaction
(Kataoka et al. in prep)
(a) Repulsion/Adhesion (Johnson
et al., 1971)
(b) Rolling
(Dominik & Tielens, 1995)
(c) Sliding
(Dominik & Tielens, 1996)
(d) Twisting
(Dominik & Tielens, 1996)
Direct-interaction model
cf).Dominik & Tielens 1997, Wada et al. 2007
(figure: Seizinger et al. 2012)
Akimasa Kataoka, Ice and Planet Formation @ Lund, May 16th, 2013 5

Monomer radius r [µm] 0.1 0.6
Surface energy
0
[mJ m 2 ] 100 20
Young’s modulus E [GPa] 7.0 2.65
Poisson’s ratio ⌫ 0.25 0.17
Material density ⇢ 0 [g cm 3 ] 1.0 2.65
critical rolling displacement ⇠ crit [Å] 8 20
Periodic boundary
gure 2 in Wada et al. (2007)). We introduce a damping
etween contact particles in normal direction, defined as
k n
m 0
t 0
n c · v r , (4)
n is the damping coe
cient in normal direction and m 0 is
nomer mass. The adopted value of k n is an order of 0.01.
that the result is independent of the normal oscillation
g, we perform N-body simulations with the damping facs
a parameter.
L timescale of damping is
t 0
k n
⇠ 10 2 t 0 , (5)
= 0.01, is much shorter than the simulation timescale,
is typically ⇠ 10 7 t 0 . We show that the obtained comn
strength is independent of the artificial normal damping
our simulations (see Section 3.4).
L
Boundary condition
L
a part of a large aggregate
iform Compression by Moving Boundaries
Fig. 1. Schematic drawing of the periodic boundary condition. Each
pt the periodic boundary The condition dust in our aggregates simulations. ofare the box compressed illustrates a boundary box by with themselves
a side length L for all direction.
When the boundary starts to get closer, the aggregate sticks to the
gregate in the computational region is surrounded by its
as shown in Figure 1. Initially, over wethe set aperiodic cubic box whose boundaries
neighboring aggregates over the boundary and compressed by them. It
re periodic boundaries withL
should be noted that this picture is illustrated in 2D direction, but our
a size of L to be larger than
simulations are performed in 3D.
regate. Thus, the initial →natural BCCA cluster isand detached isotropic from compression
hboring copies over the periodic boundaries. In our sims,
we gradually move the boundaries to the center of the The computational cubic region has length L and the coordinates
Planet inFormation x, y, and @ z directions Lund, May are 16th, set2013 to be L/2 < x < L/2, te to become closer to each other. As Akimasa a result, Kataoka, the aggre- Ice and 6
L

Simulation setting
BCCA (Ballistic Cluster-Cluster Aggregation)
•Initial aggregate: BCCA
= No compression, suitable for initial
condition
•the filling factor φ increases with time
•We measure φ and P at each time
→to derive P=P(φ)
•The pressure measuring method is
the same as molecular dynamics
cf)
= / 0
movie: H.Tanaka
: internal desnity
0: material density(=1 g/cc)
Purpose: we derive P=P(φ)
Akimasa Kataoka, Ice and Planet Formation @ Lund, May 16th, 2013 7

Initial condition : BCCA
Particle number : 6×10 4
Monomer : 0.1μm, ice
http://www.youtube.com/watch?
feature=player_embedded&v=AY6eq_S6uKE

sumed to be a BCCA clu
5. Summary
Fig. 3. Time evolution of
Results
static compression in the
:
case
compression
of N = 16384. The three figures have
strength
model is based on Domin
0.1
the same scale with di↵erent time epoch. The
white particles are inside a box enclosed by the periodic boundaries. The yellow particles are in neighboring boxes to the box of white particles.
For Fig. visualization, 13. we Same do notas draw Figure the copies 12, in back butand plotted front side with of thelinear boundaries scale but only of 8 copies andWe (2007). investigate We the introduce static comp a ne
of the white particles across the boundaries.
reversal of xy axis to compare with previous studies (see Figure 4 industthe aggregates, periodic boundary whose filling cond
0.0 Individual runs
Average
Seizinger et al.
10 5 (2012)). The dotted (a) line is the result of numerical
10 5 simulations
in the
(b) gregate of ten uniformly runs
10 4 10 3 10 2 10 1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 perform numerical N-body and natu sim
10 4 high density region
P [Pa]
( & 0.1) in Seizinger et al.
10 4 average (2012) highly ary
of ten runs porous condition, dustthe aggregates dust and the thin solid line is the fitting formula proposed by GüttlerE roll et
10 3
10 3
⇥al.
sumed resents to bea part a BCCA of a large cluster. ag
3
r0
(2009). Our results consistently connect to the previous simulations 3 inmodel theis compression based on Dominik of a large &
the high density
10 2 region.
10 2 (2007). moveWe toward introduce the center a newan
m
. 13. Same as Figure 12,
10 1 time
but plotted with linear scale of and
10 1
becomes small. To measu
ersal of xy axis to compare with previous studies (see Figure 4 in
the timeperiodic
boundary conditio
zinger by cluster-cluster et al. (2012)).
10 0 aggregation. The dotted lineA islarge the result voidofexists numerical 10 0 between simtions
gregate adopt uniformly a similarand manner naturall use
the
two
in
smaller
the high
10 clusters 1 density
and
region
they
(
are
&
connected
0.1) in Seizinger
with10 one
et 1 al.
connection
(2012) ary As condition, a result the of the dust numeric aggreg
the thin solid line is the fitting formula proposed by Güttler et al. resents as follows. a part of a large aggre
of monomers
10 2 in contact, represented by dashed10 line 2 in the right
09). Our results consistently connect to the previous simulations in the compression of a large ag
panel of Figure
10 3 14. The compression of the BCCA
10 3 cluster occurs
by crashing
– The compression stren
high density region.
move toward the center and th
10 4 the large void, which requires only rolling of
the monomers at the connection.
10 4 becomes small.
P = E roll To 3 measure
Now, 10 let us 5 estimate the compression strength. 10 In 5 adopt a similar
static compression,
10 the aggregate
r 3 manner , used in
cluster-cluster aggregation. A large void exists between the
0
10 6
As a result of the numerical s
smaller clusters 4 and they is
10 3 compressed are connected
10 2 by
10 1 with external one
10 0 10connection
pressure.
10 6 4 Each
10 3 as follows. 10 2 where 10 1 E 10
BCCA cluster feels a similar pressure, P. Using the pressure, the
0 roll is the rollin
monomers in contact, represented by dashed line in the right
(density)
(density) the monomer radius, a
el force of Figure on the BCCA 14. Thecluster compression is approximately of the BCCA givencluster by ocs
by crashing
– The
the
compression
filling factor
strength
as
Fig. 4. (a) Pressure P in [Pa] against the filling factor . The ten thin solid lines show the results for the initial BCCA clusters with di↵erent
initial
F ⇠random P · r 2 numbers BCCA . the large void, which requires only rolling of
(30) of the whole aggregate
monomers at the and thick We connection.
solid derive line showsthe arithmetic compressive average of the tenstrength runs. (b) Same as the thick P solid = Equation E roll
line in (a) 3 , plotted (35) . withisa dotted indep
line Now, of Equation let us (25). estimate The parameters theare compression N = 16384, C v = strength. 3 ⇥ 10 7 , k n = In 0.01, static and ⇠ crit comssion,
Sincethe aggregate crashing of isthe compressed large voidbyisexternal accompanied pressure. by rolling Each of
= 8 Å.
r
the 3 0number of particles
the boundary speed, th
CA a When pair cluster of the monomers compression feels a similar proceeds in contact, Akimasa pressure, and Kataoka, the the density P. Ice work Using and becomes Planet required the Formation e↵ective pressure, for @ sound the Lund, the speed crash-
where E
May 16th, can be 2013 estimated as roll is the rolling e
9
P [Pa]

ticles, which is equivalent toWe the investigate di↵erentthe sizes static of compression the inil
dust aggregates. Figure 6 shows dependence on the num-
10highly 0 porou
strength of
rmula proposed by
r of particles of the Results
dust Güttler
initial BCCA: cluster. dependence
aggregates, et whose al. resents a part of a l
filling
The initial numbers on
factor
size
is lower than
10 1 0.1. W
1
ct10 2 to10the 3 4 previous
10 5 10 6
simulations
perform numerical
in
N-body
the
simulations
compression
of static compression
of
o
a]
highly porous dust aggregates. The initial dust aggregate
10 5
10 2 is a
sumed to be a BCCA cluster. The particle-particle interactio
10 4 N=16384 model is based on Dominik & Larger Tielens N (1997) 10and 3 Wada et a
N=4096 (2007). We introduce a newbecomes method → well forfitted compression. small. to lower
10 3
10 4 We φTo
adop
tted with linear scale of and
evious studies (see Figure N=1024 4 in
the periodic boundary condition in order to compress the dust ag
e is the result
10 2 of numerical E roll
⇥sim-
gregate uniformly and naturally. adopt 10 5
3 →The
Because a similar compression
of the periodic man bound
large
r0
0.1) in Seizinger voidet al. exists 3 (2012) ary
between
condition, the
the
dust aggregate in computational
10 1
strength formula 10
10 6 region rep
As a result ofis
mula proposed by Güttler et al.
valid to lower φ
4 the n
resents a part of a large aggregate, and thus we can investiga
10
onnected t to the previous
10 0
simulations with one in time theconnection
compression of a large
as
aggregate.
follows.
The periodic boundarie
move toward the center and the distance between the boundarie
ed by dashed
10 1 linebecomes the small. right To measure the pressure of the aggregate, w
arge void
10
exists 2
adopt a similar manner used in molecular dynamics simulation
ion of the between BCCA the cluster oc-
by dashed requires line in the right only
– The Fig. 7. compressio
As a result of the numerical simulations, our Dependence main findings onar
th
nnected with
10 3 one connection
ten initial conditions varyi
dhich
as follows.
rolling of
k n = 10 2 , and k n = 10 1 .
on of the BCCA
10 4 cluster ocich
requires
– The compression strength can beother written parameters as are N = 1
P = E roll 3
10 5 only rolling of
P = E roll
ssion strength. In static 3 , and ⇠ crit = 8 Å. Each(35
l
ssion strength.
10 6 In static comd
by external 10 pressure. 4 10Each
3 10 2 0
r 3 comed
by external pressure. where E 10 Each
1 10 0 that the
r 3 ,
k n = 0, 10 0 2 , and 10 1 , r
(
normal

ht
c-
of
as follows.
The compressive strength
– The compression strength can be writte
Eroll
P = E roll
3 ,
m-
ch
he
cf)
r 3 0
where E roll is the rolling energy of mon
the monomer radius, and is the fillin
the filling factor
= / 0
as
Energy required
= ⇢/⇢
of monomers
0 , where ⇢
Eroll: rolling energy
r0: monomer radius
φ: filling factor ( )
in contact to rotate 90°
0)
of
h-
of the whole aggregate, and ⇢ 0 is the m
Equation (35) is independent of the nu
the number of particles, the size of the
the boundary speed, the normal dampin
In application:
We use the compressive strength formula as φ=φ(P), to obtain
the φ in a given pressure (=ram pressure of gas, self gravity)
Akimasa Kataoka, Ice and Planet Formation @ Lund, May 16th, 2013 11

sal of xy axis to compare with previous studies (see Figure 4 in
nger et al. (2012)). The dotted line is the result of numerical simns
in the high density region ( & Conclusions
0.1) in Seizinger et al. (2012)
he thin solid line is the fitting formula proposed by Güttler et al.
). Our results consistently connect to the previous simulations in
igh density region.
• We investigate the the static compressive strength of highly
porous aggregates (φ