Dust coagulation with porosity evolution;eﬀects on planetesimal formation andopacity evolution

Akimasa Kataoka (ITA, Heidelberg University)
H.Tanaka (Hokkaido Univ.), S.Okuzumi (Tokyo Tech), C.P. Dullemond (ITA)F. Brauer et al.: Coagulatio

Planet formation in size evolution
Barriers in dust coagulation• Radial drift barrier for planetesimalformation (e.g., Adachi et al. 1976)• Dust grains have to "jump" the barrier• Radial drift barrier for millimeter-wave
observations (e.g., Beckwith & Sargent 1991)• Dust grains has to "jump" the barrier
and we should "keep" thembarrier (e.g., Blum & Münch• Fragmentation
1993)• Dust should overcome high-speed
collisions such as ~ 50 m/s• Bouncing barrier (e.g., Zsom et al. 2010)
Brauer et al. 2008

Fig. 7. The particle size distribution at diﬀerent radii in the disk at diﬀerAkimasa Kataoka (Heidelberg University)2Possible solution
Porous dust aggregates• Radial drift barrier for planetesimalformation→ Rapid coagulation by large cross section(Okuzumi et al. 2012)• Radial drift barrier for millimeter-wave
observations→ Do they account for millimeter-wave
emission? Opacity?

• Fragmentation barrier→ Ice is more sticky than silicate (Wada et al.
2009)

Wada etGray
al. 2009
s consisting of 8000 particles (left) and BCCA clusters consisting of 8192 particles (right).
rings ide with eachother verticallyvelocity
ucol . Here, the cases ofaggregate
Bouncing
barrier at an impact parameter b with a relativeDust•→ Highly porous dust aggregates do notbounce (Wada et al. 2011)

ng and twisting are given bypectively (Wada et al. 2007).

radius : a, mass: minternal density: ρbmax = rc,1 + rc,2 ,3and the normalized impact parameter B is defi

Akimasa Kataoka (Heidelberg University)icantly the critical collision velocity ucol,crit fo
We show that ucol,crit results in around 80 m s−pendent on the size and the mass ratio. We ob
relation!
ucol,crit ≃ 20 Ebreak /mPossible solution
Porous dust aggregates• Radial drift barrier for planetesimalwith Ebreak being the breaking energy for a pairformation
contact
andsimulations
m the monomer mass. From this sc
Numerical→ Rapid coagulation by large cross section
we obtain(Okuzumi et al. 2012)"−5/6
for ice(r/0.1µm)
80
for ice• Radial drift barrier for millimeter-wave ucol,crit = 8 (r/0.1µm)−5/6 [m/s][m/s]
for silicate
for silicate
observations→ Do they account for millimeter-wave(Wada et al. 2009, 2013)– It should be noted that ucol,crit for head-on collemission? Opacity?
Laboratory
experiments
morethan twicethat given in the above equatio
Quarz (Colwell et al. 2003), SiO2, MgSiO3 (Blum andif therealistic averaging over the impact param
Fragmentation barrier
Wurm 2000), Graphite, Al2O3 (Reisshaus et al. 2006), Ice→ Ice is more sticky than silicate (Wada etoﬀset
al. (Gundlach
collisions
dominantandc
and Blumare2015),
CO2 (Musiolik
et al.head-on
2016)

•

2009)

notconsisting
contributeofto8192
the particles
increase(right).
inucol,crit
Wada
etGrayal. .2009s consisting of 8000 particles (left) and BCCA clusters
rings i– The
totala relativeejecta mass
Mejucol=. Here,
Ntargetthe+cases
Nprojof−de with eachotherverticallyatanimpactparameterb
withvelocity
Dustaggregate
Bouncing barrier•

examinedin
oursimulations.
We
obtainascali
radius : a, mass: m→ Highly porous dust aggregates do not
the ejecta mass Mejinternal
averagedover ρthe impact p
density:bounce (Wada et al. 2011)
relatively high ucol > 20 m s−1 asng and twisting are given bypectively (Wada et al. 2007).

ucol bmax = rc,1 + rc,2 ,

M
=
M
ej
proj
Akimasa Kataoka(Heidelberg
University)4
and the normalized
u impact parameter B is defi

2011 August 10

m/s

Wada ePossible solution
ucol = 2.2 m/s

Porous dust aggregatesucol = 22number
m/s : nc
Coordination
nc=11• Radial drift barrier for planetesimalformation→ Rapid coagulation by large cross section(Okuzumi et al. 2012)• Radial drift barrier for millimeter-wave
observations
E break→ Do they account for millimeter-wave
emission? Opacity?

nc=3.7• Fragmentation barrier→ Ice is more sticky than silicate (Wada et al.
2009)

Wada etGray
al. 2009
s consisting of 8000 particles (left) and BCCA clusters consisting of 8192 particles (right).
rings ide with eachother verticallyvelocity
ucol . Here, the cases ofaggregate
Bouncing
barrier at an impact parameter b with a relativeDust•→ Highly porous dust aggregates do notbounce (Wada et al. 2011)

radius : a, mass: minternal density: ρng and twisting are given by2011= rWada, panel,bmax
c,1 +etc,2
y CPE aggregates for various nc and ucol . Panels are arranged by nc (row) anducol (column).
Inral.
each
pectively(Wadaet Particles
al. 2007).
its cross-sectionalview.
in the cross-sectional views are painted, depending on the amount of diss
Akimasa Kataoka(Heidelberg
University)
5
and thenormalized
impact parameter B is defiStructure evolution of dust aggregates
A&A 557, L4 (2013)(a) Hit-and-stickpressure as an equilibrium density ρeq . Using Eq. (1), we
obtain ρeq as(b) Collisional compression(c) Gas compression

gas flow⎛ 3 ⎞1/3⎜⎜⎜ r0 ⎟⎟⎟ρeq = ⎜⎝P⎟⎠ ρ0 .
Eroll

(2)

We consider a source of the pressure to be ram pressure of thedisk gas or self-gravity of the aggregate.
We obtain ram pressure of the disk gas as follows. We consider a dust aggregate of mass m and radius r, which is moving
in the disk gas with velocity v against the gas. The pressure Pgas
against the aggregate can be defined as the gas drag force divided by the geometrical cross section: Pgas ≡ Fdrag /A, where
Fdrag = mv/ts , A = πr2 , and ts is the stopping time of the aggregate. While the pressure has both compressive and tensile components, we assume that the pressure is compressive. Thus, we
obtain the pressure as
Pgas =

mv 1
·
2
t
πr s

(3)
Suyama et al. 2008

The typical gas drag law is adopted to obtain ts and v. The gas(d) Self-gravitational compression
drag law is the Epstein regime, when the dust radius is less than
4/9 times the mean free path of gas. On the other hand, it is
the Stokes regime if the Reynolds number is less than unity (see→ Growth with fractal dimension of
Eq. (4) in Okuzumi et al. 2012, for example). When the Reynolds~2 dueunity,
to low-velocity
number exceeds
the gas drag lawcollisionschanges as a functionof Reynolds number (see Eqs. (8a) to (8c) in Weidenschillinggravitational force
Kataoka et al. 2013b 1977). The drag force is determined by the relative velocity of
the gas and dust. The relative velocity is induced by Brownian
Fig. 1. Schematic drawing to illustrate dust growth via fluﬀy aggregates. motion, radial drift, azimuthal drift, and turbulence. We use the
Akimasa Kataoka (Heidelberg University)
6a) The dust aggregate hits another aggregate to be stick. This reduces

Structure evolution of dust aggregates
A&A 557, L4 (2013)(a) Hit-and-stickpressure as an equilibrium density ρeq . Using Eq. (1), we
obtain ρeq asfilling⎛ 3
factor⎜⎜⎜ r0ρeq = ⎜⎝(b) Collisional compression

Eroll

⎞1/3
⎟⎟⎟
P⎟⎠ ρ0 .

(2)

We consider a source of the pressure to be ram pressure of thecompression
disk gas or self-gravity of theCollisional
aggregate.
We obtain ram pressure of the disk gas as follows. (D~2.5)
We consider a dust aggregate of mass m and radius r, which is moving
in the disk gas with velocity v against the gas. The pressure Pgas
against the aggregate can be defined as the gas drag force divided by the geometrical cross section: Pgas ≡ Fdrag /A, where
Fdrag = mv/ts , A = πr2 , and ts is the stopping time of the aggregrowthgate. While theFractalpressure hasboth compressive and tensile components, we assume that the(D~2)
pressure is compressive. Thus, we
obtain the pressure as(c) Gas compression

Pgas =

gas flow

mv 1
·
2
t
πr s

(3)

numberoftoparticles
The typical gas drag law
is adopted
obtain t and v. The gas
Suyamaetlessal. 2008(d) Self-gravitational compression
drag law is the Epstein regime, when the dust
radius isthan
4/9 times
the aggregatesmean freein pathof gas. forOnvarious
the other
it isand c
Fig. 5.— Density evolutionof growing
the simulations
impacthand,velocities
regimeif theReynolds
numberless thanunity (see2, 8,Stokes
16 8. The
solid lines
show
the densitiesof theisresultant
aggregatesin our si
and 4.4 m s"1 and !crit ¼ the
Eq. (4)
Okuzumiet ofal.particles
2012, for
When theasReynolds
to start compression,
estimated froclusters. Filled circles indicate
theincriticalnumberNcritexample).
Collisional
compression:
ineffective
number exceeds unity, the gas drag law changes as a function
to numbermake (seeaggregatesof Reynolds
Eqs. (8a) tocompact(8c) in Weidenschillinggravitational force
Kataoka et al. 2013b 1977). The drag force is determined by the relative velocity of
the gas and dust. The relative velocity is induced by Brownian
Fig. 1. Schematic drawing to illustrate dust growth via fluﬀy aggregates. motion, radial drift, azimuthal drift, and turbulence. We use the
Akimasa Kataoka (Heidelberg University)
7
Fig. 5.— Density evolution of growing aggregates in the simulations for various impact velocities and critical rolling displacements. We set vimp ¼ 0:27, 0.54, 1.1, 2.2,
and 4.4 m s"1 and !crit ¼ 2, 8, 16 8. The solid lines show the densities of the resultant aggregates in our simulations, and the dashed lines indicate the densities of BCCA
clusters. Filled circles indicate the critical number of particles Ncrit to start compression, as estimated from eq. (17) with b ¼ 0:5. sa) The dust aggregate hits another aggregate to be stick. This reduces

P [Pa]sumed to be a BCCA cluster. The
model
is basedon Dominik & Tiele
Structure evolutionof
dust
aggregates
A&A 557, L4 (2013)(2007). We introduce a new methodure(a)12,but
plottedwithlinearscaleofandHit-and-stick
Using Eq. (1), in
we or
an equilibrium
density ρeq . condition
theas periodic
boundary
ompare with previous studies (see Figure 4 inpressure
obtain ρeq as
uniformly and naturally. Bec
The dotted line is the result of numerical sim- ⎛gregate⎞1/3
r03 condition,⎜⎜⎜arythe dust aggregate(2) insity region ( & 0.1) in Seizinger et al. (2012)ρeq = ⎜⎝ P⎟⎟⎟⎟⎠ ρ0 .
Erollresentsa part of a large aggregate, a
the fitting formula proposed by Güttler et al.
(b) Collisional compression
a source of the pressure
toabelarge
ram pressureof theistentlyconnect to the previous simulations inWe consider
thecompressionofaggregatdisk gas or self-gravity of the aggregate.
towardthecenterthe
Wemoveobtain ram
pressure of
the disk
gas as andfollows.
We distaconsider a dust aggregate of mass m and radius r, which is moving
becomessmall.
Tothemeasure
thePgaspre
in the disk
gas with velocity
v against
gas. The pressure
againstadopt
the aggregate
can be definedas the gas
draginforce
diasimilarmannerused
molec
by the geometrical cross section: Pgas ≡ Fdrag /A, where(c) Gas compression
regation.
A large void exists between thevided
ofts isthe
numerical
simulatmv/ta= πr2 , andthe stopping
time of theaggreFdrag =As
s , Aresultnd they are connected with one connectiongate. While
the pressure has both compressive and tensile comas
follows.
we assume that the pressure is compressive. Thus, we
ct, represented by dashed line in the rightponents,
obtain the pressure as– The compression
The compression of the BCCA cluster oc-・compressive
strength strength can be
mv 1
P = 2 ·
(3)
gas flow requires only rolling ofgas
arge void, whichtπr sEroll 3
onnection.
The typical gas
P drag= law3 is adopted, to obtain ts and v. The gas(d) Self-gravitational compression
drag law is the Epsteinrregime, when
the dust
radiusis less A&A)
than
(Kataokaetal.
2013a,
te the compression strength. In static com-4/9 times the mean free0 path of gas. On the other hand, it is
Stokes regime if
the Reynolds number is less than unity (seee is compressed by external pressure. Eachthe・externalpressure
Eq. (4) in Okuzumi
et al.E
2012,
When the Reynolds
where
the rollingenergyrollforisexample).
similar pressure, P. Using the pressure, thenumber exceeds
unity, the gas drag law changes as a function
radius,and is th
rammonomerpressure
thein gasof Reynolds-thenumber(see Eqs. (8a) of
to (8c)Weidenschilling
uster is approximately
given
by
force
is determinedby thegravitational force
self-gravity
thefillingfactorasrelative=velocity⇢/⇢ of, w
Kataoka et al. 2013b 1977). The -drag0
the gas and dust. The relative velocity is induced by Brownian
Fig. 1. Schematic drawing to illustrate dust growth via fluﬀy aggregates. motion, radial drift, azimuthal drift, and turbulence. We use the08
Akimasa Kataoka (Heidelberg University)a) The dust aggregate hits another aggregate to be stick. This reducesof the whole aggregate, and ⇢ is

3dust-internal
density [g/cmρ [g/cm] 3]

Planetesimal formation via fluffy aggregates
10

0

10

-1

10

-2

10

-3

10
10

Dust grains (0.1 μm)
Compact growth

Radial Drift
Barrier

1μm
1µm

Planetesimals
10 km
10km

fractal growth
100µm
100 μm
100m

100 m

1m
1m

-4

Gas-dragcompression

1cm
1 cm

-5

10

-10

10

-5

Self-gravity
compression

10

0

10

5

10

R=5AU
10

10

15

10

20

m mass[g][g]
solid

Kataoka et al. 2013b, A&A

Akimasa Kataoka (Heidelberg University)9Dependence on the orbital radius(mass-equivalent radius)
1mm
1m
1km
1µm
pressure
ISM as an equilibrium density ρeq . Using Eq. (1), we0
10obtain ρeq asclassical scenario⎛ 3 ⎞1/3
planetesimals⎜⎜⎜ r0 ⎟⎟⎟P⎟⎠ ρ0 .
(2)
10ρ-1eq = ⎜⎝Eroll

A&A 557, L4 (2013)(a) Hit-and-stick(b) Collisional compression(c) Gas compression

gas flow

We consider a source of the pressure to be ram pressure of the-2
10diskgas or self-gravity of the aggregate.
We obtain ram pressure of the disk gas as follows. We conhit-and-stickself-gravity
sider a dust aggregate of mass m and radius r, which is moving-3
10in
the disk gas with velocity v against the gas. The pressure Pgascompression
against the aggregate can be defined5asAU
the gas drag force divided by the geometrical cross section: Pgas ≡ Fdrag /A, where-42
10F
drag = mv/ts , A = πr , and ts is the stopping time of the aggre30 AUand tensile comgate. While the pressure has both compressivegas compression
ponents, we assume that the pressure is compressive. Thus, we-5
10obtain
the pressure as
Pgas =

mv 1 -1010 ·πr2 ts

10-5

100

105

m [g]

1010

(3)
1015

The typical gas drag law is adopted to obtain ts and v. The gas(d) Self-gravitational compression
drag law is the Epstein regime, when the dust radius is less than
4/9 times the mean free path of gas. On the other hand, it is
the Stokes regime if the Reynolds number
is less than1/3
unity (see3
Eq. (4) in Okuzumi et al. 2012, for example).
When the Reynolds
0
number exceeds eq
unity, the gas drag law changes
gas as a function
0of Reynolds number (see Eqs. (8a) to (8c) in Weidenschillingroll1977). The drag force is determined by the relative velocity ofgravitational force
the gas and dust. The relative velocity is induced by Brownian
Fig. 1. Schematic drawing to illustrate dust growth via fluﬀy aggregates. motion, radial drift, azimuthal drift, and turbulence. We use thea) The dust aggregate hits another aggregate to be stick. This reduces
closed formula of the turbulence model (Ormel & Cuzzi 2007)
dust density and occurs in a very early stage of dust growth. b) When theand(Heidelberg
assume the turbulent
parameter αD = 10−3 , except for the
Akimasa
Kataoka
University)
collisional speed is high enough to disrupt the dust aggregates,
they are

⇢

=

✓

r
E

P

◆

1020

⇢ .

10

3
dust-internal
density [g/cmρ [g/cm] 3]

What is the observed "mm-sized grains"?
1001mm compact grains

10

-1

10

-2

10

-3

10

-4

10

-5

Planetesimals

1μm
1µm

10 km
10km

100µm
100 μm
100m

100 m

1m
1m
1cm
1 cm

10

-10

10

-5

R=5AU
10

0

10

5

10

10

10

15

10

20

m mass[g][g]
solid
Akimasa Kataoka (Heidelberg University)

11

Opacity evolution(a)af=1µmaf=10µm
af=100µm
af=1mmaf=1cm

100

10-2
10-3
10-4

(e) af = 1 mm
10-4

10-2

100

102

5

104

10

a [cm]

0

f=10-1
f=10-2
f=10
f=10-3
-4
f=10

104
3

10

fluﬀy
growthradius a,filling factor f

κabs [cm2/g]

f

10-1

102
101
100
10-1
10-2
10-3
10-4 -4
10

10-3

10-2

10-1

100

101

λ [cm]
Akimasa Kataoka (Heidelberg University)

12How can we understand porosity?

A&A proofs: manuscri

Stokes number(a) x<1

3⌦ m
St =4⇢g vth ⇡a2(Epstein regime)(b) x>1, optically thin

Opacity⇡a22
abs =
Qabs = f (m/⇡a )m(c) x>1, optically thick(mass)
a3 f⇠ 2 ⇠ af
(area)aaf = const

mass-to-area ratio = const(Kataoka et al. 2014)

Fig. 6. The three limiting regime. (a) When x ⌧ 1, the opacity goes into
the Rayleigh regimes. (b) When x 1 and is optically thin (kx ⌧ 3/8),
the opacity goes into the optically thin geometric regime. (c) When x
1 and is optically thick (kx3/8), the opacity goes into the opticallythick geometric regime.

scribed as
Qabs ' Qabs,3 ⌘Akimasa Kataoka (Heidelberg University)

Z ⇡/2
0

(1

R(✓i )) sin 2✓i d✓i ,

(15)

13

What are “mm-sized grains’’?
af=0.1µm

-1

10

f

10-2
10

af=1m

0

3

ρ [g/cm ]10

af=1mm

1µm

-3

10

-4

10

-5

10km

100µm
100m

1m

1cm

10

-10

10

-5

R=5AU
0

5

10
10
10
solid
m mass[g] [g]Akimasa Kataoka (Heidelberg University)

10

10

15

10

20

14

Akimasa Kataoka (Heidelberg University)

15

Dust density

filling factor

Optical depth at 1 micron

Optical depth at 1 mm

Planetesimals

Fluffy aggregateslack of micron-sized grains→needs fragments

Akimasa Kataoka (Heidelberg University)

mm-wave opacity

17

Conclusions•

We investigate the the static compressive strength of highly
porous aggregates (f<0.1)••

we reveal the overall porosity evolution
the path avoids the three barriers: radial drift, fragmentation,and bouncing barriers(Kataoka et al. 2013a, A&A, 554, A4, Kataoka et al. 2013b, A&A, 557, L4)•

Opacities are characterized by af(Kataoka et al., 2014, A&A, 568, A42)••

Planetesimals are formed inside ~ 10 AU
Fluffiness does not help for the radial drift barrier in outer disk.

Akimasa Kataoka (Heidelberg University)

18

How to observe fluffy aggregates?
a×f(e) af == 1 1mmmm

Opacity(b)index β

5

10

0

f=10-1
f=10-2
f=10-3
f=10-4
f=10

104

2

10

1

10

3.5
3

1mm compact grain

2.5βκabs [cm2/g]

103

0

10

2

10-1

1.5

10-2
10-3
10-4
10-4

f=1
f=10-1-2
f=10

4

fluﬀy aggregates

1
0.5

10-3

10-2

10-1

100

101

0
10-4

10-3

λ [cm]

a: radius
f: filling factor

10-2

10-1

100

101

102

103

(af)max [cm]β>1.7 only when grains are compact

Akimasa Kataoka (Heidelberg University)19

tocalculatedimensionle-3
10
Understanding
opacity
A&A proofs: manuscript no. opacity_publish
Qabs and Qsca . Then, we
opacities
as wi
Figure 7 shows
the absorption definedmass opacities calculated(a) x<1

-4

10

2

3

10

10

4

λ [µm](b) x>1, optically thin

10

10

5

both
the Mie calculation and the analyticopacity
formula. The absorp
absorption/scatteringtion mass opacity of Figs. 7 (a) and (b) are the same becaus
2 factor is di↵erent ( f = 1 ana f = 0.1 µm but only the fillingf = 0.01), as shown in the previous sections (see Fig. 3). Fig
ure 7 (a) showsabs
the case of a f = 0.1 µm
absand f = 1 (compact
The whole wavelengths in this panel satisfy x < 1, and thu
Qabs ' Qabs,1 . The analytic formula greatly reproduce the M
calculations. Figure 7 (b) shows the case of a f = 0.1 µm an
f = 0.01 (flu↵y). In this case,2 a = 10 µm, and thus x =at = 2⇡a ' 63 µm. We use Qabs = Qabs,1 for x > 1 an
Qabs = Qabs,2 forthem at x = 1. This als
scax < 1 and connect sca
reproduces the Mie calculation.
Figures 7 (c) and (d) show(mthe: graincase ofa f = 1 mm, but thmass)filling factor is 1 and 0.01, respectively. The absorption ma
opacity of Figs. 7 (c) and (d) are almost the same except for th→Qabs,structure.
Qsca is The
calculated
Miecorresponds
theoryinterference
interferencewith
structure
where x > 1 and kx < 3/8, the optically-thin geometric optic
regime.
We notethat the di↵erence between the Mie calculatio
Usefulparameters
and the analytic formulae is the interference structure. In Fi
7 (c), which is the compact case, the real part of the refractiv
・size
parameterindex
is greaterthan unity. Thus, Qabs has an enhancement b
cause of the interference. In Fig. 7 (d), which is the flu↵y cason the other hand, the real part of the refractive index is almo
unity
and thus nodepth
enhancement appears and smoothly connec
・opticalto x < 1 region at the longer wavelengths. From the analytic fomula, we conclude that the interference structure only appears
the compact
casesthebecausen is still larger equations
than unity when x >-> derive
approximatedand kx < 3/8.

index of the mixture of silicate, organics,



⇡a
=
Q ,
m

⇡a
 =
Q .
m
We note that the mass oobtain the mass opacitie
mass opacities by the dual abundance based on Pollack et al.
2⇡a(c) x>1, optically thickx⌘
ed in several papers (e.g., D’Alessio
2005; Isella et al. 2009; Ricci et al. 3. Results
kx). Some recent studies use the dieleceous material (Zubko et al. 1996) in- 3.1. Absorption mass o
Kataoka(Heidelberg
20
three limiting
When x ⌧ 1,Akimasa
the opacity
goes into
etFig.al.6. The1994;
Liregime.
&(a)Greenberg
1997).
InUniversity)

Qabs ' 4x Im

=

2 µm,
2 +100f scattering mass opacity over absorption mass opacity sca /abs in the casesµm,
m2 +of
2 a f =(n0.1
k2 +12)
(2

(c)
x>1,
optically
thick
1
2
3
ses with the same a f , but the filling factor ranges are in f = 1, 10 , 10 , 10 , and 10 4 .

Understanding opacity
absorption(see Eq. :(5.11)
in BH83). The imaginary par
A&A proofs: manuscript no. opacity_publish

index is much smaller than the real part n (Figureabsorption mass opacities
calculated with(a)7 shows
x1≪ 1theexp(→Rayleighregime(a) x<1
↵⇠)
where
↵
4⇡k/
, and
⇠ is the pSo,
can approximate
both the MieA.1).
calculation
andwe
the analytic
formula.=Theabsorp-Q
abs astion mass
opacityofFigs. 7 (a) and
(b)
are<
the1,
same
becauseif n-1≪1,
k≪1
thematerial.
If↵⇠theincident
light is
a
f
=
0.1
µm
but
only
the
fillingfactorisdi↵erent
(
f
=
1
and
adius a is much smaller than the wavelength,
24nkx is3).opticallyf = 0.01), asthe
shownin the previous
sections (seeitFig.
Figmaterial
because
thin
on
Q
'
Q
⌘
.
(a) shows abs
the case of aabs,1
f = 0.1 µm and
f = 1 2(compact).
into Reyleigh regime. This corresponds tourex7 =2+(n2)1, and thusof the sphere.
length⇠=2a,thediameter
The whole wavelengths in this panel satisfy x <is case, Qabs can be written as
Qabs if' f≪1,
Qabs,1 . n-1∝f,
The analytic
formulagreatly reproduce the Mie
k∝f,
thus
↵⇠
<
1
corresponds
to kx
< 1.
calculations. This
Figure 7equation(b) shows the explainscase of a f = 0.1µmfactand that the abs
the!
f = 0.01 (flu↵y).
In this
case,
a =
10
µm,
and 1
thus (orx = 1equivalently x
2
In
the
limit
of
a
Q
/
af
for
f
⌧
abs,1
1(b) x>1, optically
24nkx
ratio
ot
at = 2⇡a ity' 63isµm.characterized
We use Qabs = Qabs,1by
for mass-to-areax > 1 and
thin
Fig.
6.
The
three
limiting
regime.
(a)
When
x
⌧
1,
=
.
(11)
obtainQabs = Qabs,2thin,
for x <we
1 andconnect them at x = 1. This also
2+2
2
2
2
2
wavelengths,
n is (b)almostunity
whilekoptica/ f(n k + 2) + (2nk)
reproduces
theRayleighMie calculation.
theregimes.
When
x
1
and
is
(b) x≫7 (c)1&kx
2Figuresand(d) ≪1
showthatthe caseofis
a f proportional= 1 mm, but theweobtainkx
tom/a⇣⌘ , whtheisopacity
goes
into 8kx
thethingeometricregfilling factor
1 and 0.01,
respectively.
Theopticallyabsorption
mass
3
2
3/2
n BH83). The imaginary part k of the refractive
Q
Q⌘
nexcept(nthe the1)opacity, goratio.
Since
isdividedbymass-to-a
opacity of1Figs.
7 (c)
and
(d)abs,2are 
almost
the Q
same
for
and
is '
optically
thick
(kx
3/8),
abs
abs
abs
3nproperties.
structure. The interference structure
corresponds tomaller than the real part n (see Fig. 1 andinterference
Fig. thickdependentof
dust
geometric
regime.
where x > 1 and kx < 3/8, the optically-thin geometric optics
Qabs,2/ afbetween
for thef⌧1
approximate Qabs as
regime. We notethat the di↵erence
Mie calculation(see
Eq. is(7.2)
in BH83).
and the analytic
formulaethe interference
structure. In Fig.
7 (c), which is the compact case, the real part of the refractive
24nkx
We
note
that
ifannenhancement= regime
1, which
is usually
scribedasx≫1(c) x>1,
opticallythick
4.2.2.
1
thinindex
is greaterthanunity.
Thus,
Qandbeabs has optically
(c)
x≫
1
&
kx
→
geometric.
(12)of the interference.
Z whichcause
Fig. 7 (d),is theyields
flu↵y case,2
2of flu↵yInmedium,⇡/2Qabsn + 2)
on the other hand, the real part of the refractive index is almostWhenthe dustaR(✓
isconnects
much larger thanQabs,3⌘
unity andQ
thus
enhancement
appearsradiusand (1smoothlyabsno'i )) sin 2✓i d✓i ,to x < 1 region
at thelonger wavelengths.
thegeometricanalytic
forplains the fact that the absorption mass opac8kx
0 Fromtheopacitygoes
into
optics regi
mula, we conclude
thattheinterference
structure
only
appearsinQ= Qnabs,1=⇡Q1than=.
absbecause
abs,2
ed by mass-to-area ratio or a f . At the longer
the compact
cases
is
still
larger
unity
when
x
>
1
the
opticalproperties
canbeunderstoodf⌧where the reflectanceforR(✓)
is31written
as by trand kx3 < 3/8.
s almost unity while k / f . Using f / m/a, the material. The fraction of2 energy that tran
1 cos ✓t is mcoscharacterized✓icos ✓i bym co
2 x ⌧ 1, the opacity goes into
6. The three limiting regime.
(a) WhenThisequationalso
theisFig.
proportional
to
m/a
,
which
is
mass-to-area
Scattering
R(✓i )mass= opacity+
isis optically
proportionalaf
the Rayleigh regimes. (b)1.
Whenκxabs
1 andthin (kx ⌧ 3/8), 4.3.tocosthe✓analytic+mcos✓ofi the xab- /cosa.✓i We+ malscoopacitygoes into theby
optically
thin geometric regime.
(c) Whenxabs isor a off 2obtaining
because/formulaf andtks Qthedividedmass-to-arearatio,
inIn thesame mannerabs
1 and is optically thick (kx3/8), thederive
opacity goes into
the optically
2. we
the
piecewise
formulaofbetween
abs.
sorption masslytic
opacity,
we also derivetheanalyticopacity
formulaof the thick and th
formulaoptically
thick geometric regime.
t properties.
scatteringand
mass opacity. In addition, by using the analysis, we
explainwhy changed
the mass
opacitycannotischaracterized
unity. Thus, we 21
defi
Akimasa Kataoka(Heidelberg
University)
abs besin when✓ can and Qρ [g/cm ]disk gasself gravity
Brownian motion
Radial drift motion
Azimuthal motion
Turbulent motion
dust growth without static compression

0

10
10

-1

10

-2

10

-3

10

-4

10

-5

St=1

10

-10

10

-5

10

0

m [g]

10

5

10

10

10

15