Radial Flow of Dust Particles in Accretion Disks 1

Taku Takeuchi and D. N. C. LinUCO/Lick Observatory, University of California, Santa Cruz, CA95064

ABSTRACT

We study the radial migration of dust particles in accreting protostellar disks analogous tothe primordial solar nebula. Our main objective is to determine the retention efficiency of dustparticles which are the building blocks of the much larger planetesimals. This study takes accountof the two dimensional (radial and normal) structure of the disk gas, including the effects of thevariation in the gas velocity as a function of distance from the midplane. It is shown that thedust component of disks accretes slower than the gas component. At high altitude from the diskmidplane (higher than a few disk scale heights), the gas rotates faster than particles because ofthe inward pressure gradient force, and its drag force causes particles to move outward in theradial direction. Viscous torque induces the gas within a scale height from the disk midplaneto flow outward, carrying small (size . 100 µm at 10 AU) particles with it. Only particles atintermediate altitude or with sufficiently large sizes (& 1 mm at 10 AU) move inward. Whenthe particles’ radial velocities are averaged over the entire vertical direction, particles have a netinward flux. The magnitude of their radial motion depends on the particles’ distance from thecentral star. At large distances, particles migrate inward with a velocity much faster than the gasaccretion velocity. However, their inward velocity is reduced below that of the gas in the innerregions of the disk. The rate of velocity decrease is a function of the particles’ size. While largerparticles retain fast accretion velocity until they approach closer to the star, 10 µm particleshave slower velocity than the gas in the most part of the disk (r . 100 AU). This differentialmigration of particles causes the size fractionation. Dust disks composed mostly of small particles(size . 10 µm) accrete slower than gas disks, resulting in the increase in the dust-gas ratio duringthe gas accretion phase. If the gas disk has a steep radial density gradient or if dust particlessediment effectively to the disk midplane, the net vertically averaged flux of particles can beoutward. In this case, the accretion of the dust component is prevented, leading to the formationof residual dust disks after their gas component is severely depleted.

Subject headings: accretion, accretion disks — planetary systems: formation — solar system: formation

1. Introduction

Planets form in circumstellar disks. In thestandard scenario, formation of earth-size planetsor planetary cores occurs through coagulation ofsmall dust particles (e.g., Weidenschilling & Cuzzi1993). Thus, the total amount, radial density dis-tribution, and size distribution of dust particles inthe disks are the important initial conditions ofplanet formation. In many of the previous studiesof planet formation, the dust-gas ratio of circum-

1To appear in Dec. 20 issue of Astrophys. J.

stellar disks is assumed to be similar to the solarvalue (∼ 10−2), and dust particles are consideredto be well mixed with gas, i.e., the dust-gas ratio isconstant throughout the disk (Hayashi, Nakazawa,& Nakagawa 1985).

However, the motion of dust particles is differ-ent from that of gas. Dust particles have a radialmotion which is induced by the gas drag force.Adachi, Hayashi, & Nakazawa (1976) and Weiden-schilling (1977) studied the motion of particles ingas disks. Due to radially outward pressure gra-dient force, the rotation velocity of the gas is gen-

1

erally slower than particles on nearly Kepleriancircular orbits. Consequently, the gas drag forcetakes angular momentum from particles, resultingin their inward migration (Whipple 1972). At afew AU from the central star, the orbital decaytime is estimated to be ∼ 106 yr for 100 µm par-ticles and 104 yr for 1 cm particles. These timesare much shorter than the life time of gas disks(∼ 107 yr). Thus, dust component may evolvemuch faster than the gas in protostellar disks, inwhich case, the dust-gas ratio would decline. Itappears that the initial conditions of planet for-mation need not to be protostellar disks in whichdust particles are well mixed with the gas and thedust-gas ratio is constant.

In this paper, we start a series of studies todetermine the initial distribution of dust parti-cles in the context of planet formation. Here, westudy the radial migration of dust particles in gasdisks. We focus our attention on particles smallerthan ∼ 1 cm in order to consider evolution of dustdisks before the initiation of planetesimal forma-tion. The migration of these small particles estab-lishes the initial density distribution of the dustcomponent and sets the stage for planetesimal for-mation. In this phase of disk evolution, the gascomponent is considered to be turbulent and is ac-creting onto the star. The discussion in the aboveparagraph is based on the study of particle mi-gration under the assumption that particles havecompletely sedimented on the midplane of laminardisks. However, in turbulent disks, the sedimenta-tion of particles is prevented, because particles arestirred up to high altitude from the midplane byturbulent gas motion. In such disks, particles aredistributed in the vertical direction, and the radialmotion of particles depends on the distance fromthe midplane. Thus, there is a need for study-ing the radial migration of particles which resideabove the midplane.

There are two important factors causing thevertical variation in the migration velocity of par-ticles. The first is the variation of rotation velocityof the gas in the vertical direction. As mentionedabove, the gas rotation differs from the Keplerianrotation because of the gas pressure gradient. Asthe gas density decreases with the distance fromthe midplane, the radial pressure gradient variesand under some circumstances even changes itssign. While at the midplane the pressure gradient

force is outward, it is inward near the disk surfacebecause the disk thickness increases with the ra-dius. The gas drag force on particles also varieswith the height, resulting in a variation of particlemigration velocities. In §3.1, we will see that par-ticles at high altitude, where the pressure gradientforce is inward, flow outward.

The second factor is the variation in the radialgas flow. Small particles (. 100 µm) are well cou-pled to the gas, so that they migrate with the gasflow, as discussed in §3.1 below. That is if the gasis accreted onto the star, the small particles wouldalso be accreted, while if the gas flows outward,particle flow would also be outward. In accretingprotostellar disks, the gas does not always flow ra-dially inward. Radial velocity of the gas varieswith the distance from the midplane. Viscousstress can cause gas outflow near the midplane(Urpin 1984; Kley & Lin 1992). Rozyczka, Bo-denheimer, & Bell (1994) showed that the outflowoccurs if the radial pressure gradient at the mid-plane is steep enough. The density (or pressure)gradient at the midplane is steeper than the av-erage spatial (or surface) density gradient, e.g., ifthe surface density varies as Σg ∝ r−1 and the diskthickness varies as hg ∝ r, the midplane densityhas a steeper variation corresponding to ρg ∝ r−2.Viscous diffusion at the midplane causes the out-flow of the gas to reduce the steep density gra-dient. On the other hand, at high altitude fromthe midplane, the density gradient is shallow andthe viscous torque causes the usual inward flow.Thus, in accretion disks, the outward flow at themidplane is sandwiched by the inflow at the disksurfaces (see §2.2 for details). Because small parti-cles are carried by the gas flow, they are expectedto flow outward near the midplane, and inward athigher altitude.

Thus, if most of dust particles concentrate inthe region where they migrate outward, i.e., nearthe disk surface or around the midplane, the dustcomponent of the disk would accumulate in totalmass and expand in size. In reality, the verticalspatial distribution of particles is determined bythe degree of sedimentation, and their size distri-bution is regulated by the coagulation, condensa-tion, and sublimation processes. In this paper, wecalculate the vertical distribution of particles in or-der to identify the mass flux of the dust particlesas functions of their size and their distance from

2

the midplane and their host stars. The results ofthe present calculation will be used to study thelong term dynamical evolution of the dust compo-nent of the disk. For the present task, we do notconsider the effects of gas depletion, particles’ sizeevolution or their feedback influence on the flowvelocity of the gas. These effects will be consid-ered in a future investigation.

There are several works on the particle mi-gration in turbulent disks. Stepinski & Valageas(1997) derived the vertically averaged velocity ofmigration for two limiting cases. One modelis constructed for small particles which are wellmixed with the gas and the other is for large parti-cles which are totally sedimented to the midplane.The velocity for intermediate sized particles is cal-culated by the interpolation of these limiting cases.Supulver & Lin (2000) investigated the evolutionof particle orbits in turbulent disks numerically.This paper gives analytical expression of the parti-cle velocity for any size of particles, which are esti-mated from interpolation by Stepinski & Valageas(1997). Several studies showed that vortices orturbulent eddies can trap particles within somesize range (Barge & Sommeria 1995; Cuzzi, Dobro-volskis, & Hogan 1996; Tanga et al. 1996; Klahr& Henning 1997; Cuzzi et al. 2001, see howeverHodgson & Brandenburg 1998). Particle trappingby eddies may be important for local collectionof size-sorted particles. In this paper, turbulenteddies are treated just as a source of the particlediffusion and of the gas viscosity. The effect of par-ticle trapping on their coagulation is not discussedhere.

The plan of the paper is as follows. In §2, thevertical variation of gas flow is derived. In §3, wedescribe how the radial velocity of dust particlesvaries in the vertical direction, and then calculatethe net radial migration velocity. In §4, we dis-cuss the steady distribution of particles assumingno particle growth. We show that the size frac-tionation of particles occurs as a result of radialmigration, and that the distribution of particlesdiffers from that of the gas.

2. Viscous Flows of Gas Disks

2.1. Rotation Law of the Gas

Gas disks rotate with a slightly different veloc-ity from the Keplerian velocity. The rotation ve-

locity is determined by the balance between thestellar gravity, the centrifugal force, and the gaspressure gradient. In cylindrical coordinates (r, z),the balance of the forces described by the momen-tum equation in the r-direction is given by

rΩ2g −

GMr

(r2 + z2)3/2− 1

ρg

∂Pg

∂r= 0 , (1)

where Ωg is the angular rotation velocity of thegas, G is the gravitational constant, M is the stel-lar mass, ρg is the gas density, and Pg is the gaspressure. The density distribution in the verticaldirection is determined by the balance between thez-components of the stellar gravity and the pres-sure gradient, which is written as

− GMz

(r2 + z2)3/2− 1

ρg

∂Pg

∂z= 0 . (2)

We assume that the disk is isothermal in the ver-tical direction. Neglecting (z/r)2 and higher orderterms, the vertical density distribution is obtainedas

ρg(r, z) = ρg(r, 0) exp(− z2

2h2g

). (3)

The disk scale height hg is given by

hg(r) =c

ΩK,mid, (4)

where c is the isothermal sound speed, andΩK,mid =

√GM/r3 is the Keplerian angular ve-

locity at the midplane. The radial variations ofphysical quantities are assumed to have power lawforms, such that

ρg(r, z) = ρ0rpAU exp

(− z2

2h2g

),

c2(r) = c20r

qAU ,

hg(r) = h0r(q+3)/2AU , (5)

where the subscript “0” means the values at 1 AU,rAU is the radius in the unit of AU, and power lawindices p and q are usually negative. The surfacedensity is

Σg(r) =∫ +∞

−∞ρgdz =

√2πρ0h0r

ps

AU , (6)

where ps = p + (q + 3)/2. To derive the rotationlaw of the gas, we retain (z/r)2 and lower order

3

terms in equation (1), then obtain

Ωg(r, z) = ΩK,mid

[1 +

12

(hg

r

)2 (p + q +

q

2z2

h2g

)].

(7)The difference of the gas rotation from the Keplerrotation is order of (hg/r)2, and the gas at highaltitude rotates slower than the gas at the mid-plane.

2.2. Radial Velocity of the Gas Flow

In viscous disks, angular momentum is trans-fered from the inner part to the outer part ofthe disk under the action of viscous stress. Theazimuthal component of the momentum equationimplies

2πrρg

(vr,g

∂

∂r+ vz,g

∂

∂z

)(r2Ωg) =

2π

[∂

∂r

(r3ρgν

∂Ωg

∂r

)+

∂

∂z

(r3ρgν

∂Ωg

∂z

)], (8)

where vr,g and vz,g are the r- and z-components ofthe gas velocity, respectively, and ν is the kineticviscosity. The right hand side of equation (8) rep-resents the torque exerted on an annulus with unitwidth in the r- and z-directions. The left hand sideis the variation of the angular momentum of theannulus accompanying with the motion. We as-sume that the angular velocity is always adjustedto the equilibrium (eq. [7]) on a dynamical timescale. Since molecular viscosity is too ineffective,it is customary to assume that angular momentumis primarily transported by the Reynolds stress in-duced by the turbulent motion of the gas. Follow-ing conventional practice, we adopt an ad hoc butsimple to use α prescription for turbulent viscosity(Shakura & Sunyaev 1973), such that

ν = αchg = αc0h0rq+3/2AU . (9)

The mass conservation of the gas is

∂ρg

∂t+

1r

∂

∂r(rρgvr,g) +

∂

∂z(ρgvz,g) = 0 . (10)

In accretion disks considered here, the second termis the same order of the third term. Then, wesee that vz,g ∼ (hg/r)vr,g. Besides, from equa-tion (7) we see ∂(r2Ωg)/∂z ∼ (hg/r)∂(r2Ωg)/∂r.Thus, vz,g∂(r2Ωg)/∂z is factor (hg/r)2 smaller

than vr,g∂(r2Ωg)/∂r and can be neglected in equa-tion (8). Using power law expressions (5) and (9)and retaining terms of order (hg/r)2, the radialvelocity is reduced to

vr,g = −2πα

(h0

AU

)2 (3p + 2q + 6 +

5q + 92

z2

h2g

)

× rq+1/2AU AU yr−1 . (11)

In the following discussions, we adopt these val-ues as being representative of a typical model:M = 1M, ρ0 = 2.83 × 10−10 g cm−3, h0 =3.33 × 10−2 AU, p = −2.25, q = −0.5, andα = 10−3. These values correspond to a min-imum mass solar nebula whose surface densitydistribution has a power law form with indexps = p + (q + 3)/2 = −1.0 and has a gas mass2.5× 10−2M inside 100AU.

Figure 1 shows the radial velocity of the gas.The radial velocity is outward (positive) near themidplane, while it is inward (negative) above theheight 0.73hg. In the standard model (q = −0.5),the radial velocity is independent of the distancefrom the star, as seen from equation (11). At themidplane, the radial gradient of the gas densityis so steep (p = −2.25) that the gas receives moretorque from the inner disk than the torque it exertson the outer disk. This radial dependence meansthat viscous stress acts to move the gas outwardin such a steep density gradient. The condition forthe outward gas flow at the midplane is

p +23q < −2 , (12)

which is satisfied in many accretion disk models.The rotation of the gas is slower at higher altitudeas seen in equation (7). Thus, the gas at the mid-plane loses its angular momentum through the ac-tion of viscous stress it has with the gas at higheraltitude. However, in the standard model, thisangular momentum loss is smaller than the an-gular momentum gain through the viscous stressin the r-direction. The outflow zone around themidplane shrinks as the radial density gradient isreduced (p → 0), and when the condition (12) isviolated, gas at all altitudes flows inward. Theradial density gradient is shallower at higher alti-tude, because the disk scale height increases withthe radius. Thus, at high altitude, the gas losesmore angular momentum through exerting torque

4

on the outer disk than it is receiving from the innerdisk. For such gas, the presence of viscous torqueinduces it to flow towards the star.

The net radial velocity of the gas is given byaveraging in the vertical direction:

〈vr,g〉 =1

Σg

∫ +∞

−∞vr,gρgdz

= −6πα

(h0

AU

)2

(ps + q + 2)

× rq+1/2AU AU yr−1 . (13)

The net accretion of the gas toward the star occursif

ps + q = p +32(q + 1) > −2 . (14)

We may consider two models in which a steadystate is achieved. One model is the case in whichthe net radial velocity is zero, i.e., ps+q = −2. Forexample, if ps = −1.5 and q = −0.5, there is nonet accretion of the disk. A more realistic modelis the case in which the mass flux of the gas is con-stant with the radius, i.e., rΣg〈vr,g〉 ∝ rps+q+3/2

is constant. We take this as our standard model,i.e., ps = −1.0 and q = −0.5. This surface den-sity profile is expected to arise after the disk hasundergone a period of initial viscous evolution.

3. Radial Flow of Dust Particles

3.1. Vertical Dependence of Radial Veloc-ity

The azimuthal velocities of dust particles aredifferent from that of the gas. The resulting gasdrag force transfers angular momentum betweenthe particles and the gas and moves particles inthe radial direction. If there is no gas drag force,particles would orbit with the Keplerian angularvelocity, which is approximated as

ΩK(r, z) ≈ ΩK,mid

(1− 3

4z2

r2

). (15)

We express the deviation of the angular velocityof the gas from the Keplerian angular velocity as

Ωg = ΩK(1− η)1/2 . (16)

From equation (1) it is seen thatη = −(rΩ2

Kρg)−1∂Pg/∂r is the ratio of the gaspressure gradient to the stellar gravity in the radial

direction. From equations (7), (15), and (16), η iswritten within the order of (hg/r)2 as,

η = −(

hg

r

)2 (p + q +

q + 32

z2

h2g

). (17)

Note that the pressure gradient force is outward (ηis positive) around the midplane, while it is inward(η is negative) where |z| > √−2(p + q)/(q + 3)hg.In the standard model, η changes sign at z ≈1.5hg. Dust particles near the midplane rotatefaster than the gas, and particles at high altituderotate slower than the gas.

The equations of motion of a particle are

dvr,d

dt=

v2θ,d

r− Ω2

Kr − ΩK,mid

Ts(vr,d − vr,g) , (18)

d

dt(rvθ,d) = −vK,mid

Ts(vθ,d − vθ,g) , (19)

where vr and vθ are the r- and θ- components ofthe velocity, respectively, with the subscripts “g”and “d” distinguishing gas and dust, and vK,mid =rΩK,mid is the Keplerian velocity at the midplane.The gas drag force is expressed through the non-dimensional stopping time Ts (normalized by theKepler time at the midplane). We do not solve theequation of motion in the z-direction. Instead, wesimply assume vz,d = 0. When an equilibrium inthe vertical dust distribution is achieved, the dustsedimentation to the midplane is balanced by thediffusion of particles, as discussed below in §3.2.The vertical velocity of a particle is zero when timeaveraged.

The mean free path of gas molecules is largerthan 1 cm for r & 1 AU in our models (Naka-gawa, Sekiya, & Hayashi 1986). In this paper, weconsider particles smaller than 1 cm and use Ep-stein’s gas drag law. Then, the non-dimensionalstopping time Ts, as given by Takeuchi & Arty-mowicz (2001), is

Ts =ρpsvK,mid

ρgrvT, (20)

where ρp is the particle internal density, s is theparticle radius, and the mean thermal velocity isvT =

√8/πc 2. We take the particle internal den-

sity as ρp = 1.25 g cm−3.

2In Takeuchi & Artymowicz (2001), vT is defined to be 4/3times the mean thermal velocity. This definition causes afactor 4/3 difference between equation (20) and their equa-tion (10).

5

For particles smaller than 1 cm, the non-dimensional stopping time is much smaller thanunity through most of the disk. Figure 2 showsthe radii of particles whose stopping time is unity.At the midplane (see the solid line), the non-dimensional stopping time is smaller than unityfor particles of size s . 1 cm at r . 100 AU. Atz = 2hg (the dashed line), although the gas den-sity is lower than at the midplane and the stoppingtime is longer, particles of size s . 1 mm still havethe non-dimensional stopping time smaller thanunity. These particles are well coupled to the gasand their angular velocity is similar to the gasangular velocity (7). Only particles with s & 1 cmwhich are located at high altitude and r & 100 AUbecome decoupled from the gas.

We assume that motions of both the gasand the particles are close to Keplerian, i.e.,vθ,g ≈ vθ,d ≈ vK,mid, and that d(rvθ,d)/dt ≈vr,dd(rvK,mid)/dr = vr,dvK,mid/2. Then, fromequation (19), we have

vθ,d − vθ,g = −12Tsvr,d . (21)

Using equation (16) and neglecting terms of order(hg/r)4 and higher, equation (18) is reduced to

dvr,d

dt= −η

v2K,mid

r+

2vK,mid

r(vθ,d − vθ,g)

−ΩK,mid

Ts(vr,d − vr,g) . (22)

The left hand side is order of v2r,d/r and is ne-

glected if vr,d c. Substituting equation (21)into equation (22), we find the radial velocity ofthe particle to be

vr,d =T−1

s vr,g − ηvK,mid

Ts + T−1s

. (23)

In the above derivation of the particle’s radial ve-locity, we assume that in the z-direction the par-ticle sedimentation is balanced by the turbulentdiffusion, but that in the r-direction the turbulenteffects are neglected.

For particles well coupled with the gas (Ts 1), the radial velocity reduces to

vr,d = vr,g + vr,drift , (24)

where vr,drift = −ηTsvK,mid is the relative velocity

from the gas. Using equations (5), (17), and (20),

vr,drift = 2π

(h0

AU

)2 (p + q +

q + 32

z2

h2g

)

× rq+1/2AU Ts,mid exp

(z2

2h2g

)AU yr−1 , (25)

where Ts,mid is the non-dimensional stopping timeat the midplane. The radial drift velocity increasesexponentially with the height. Because the parti-cles are strongly coupled with the gas, the gas dragforce suppresses their drift velocity. At the mid-plane where the gas density is highest, the sup-pression of the drift is most effective. The driftvelocity at the midplane is

vr,drift,mid =

√π3

2h0s

AU2

ρp

ρ0(p+q)r−p+q/2−1

AU AU yr−1 ,

(26)which agrees with the derivations of Adachi et al.(1976) and Weidenschilling (1977).

For large particles decoupled from the gas(Ts 1), the radial motion of the gas doesnot affect the particles’ velocity, so that vr,d =−ηT−1

s vK,mid. In this paper, we do not considersuch large particles.

Figure 3 shows the radial velocity vr,d of parti-cles of s = 10 µm, 100 µm, and 1 mm at 10 AUin the standard model disk. Near the midplane,the drift velocity vdrift (relative to the gas veloc-ity shown by the dotted line) of 10 µm particlesis small. The particles move outward almost to-gether with the gas. As the altitude increasesand the gas velocity decreases, the radial veloc-ity of the particles also decreases, then becomesnegative. At z ≈ 1.5hg where η changes its sign,the radial pressure gradient vanishes and the gasrotates with the Keplerian velocity. The parti-cles co-rotate with the gas at that location, andhave the same radial velocity as the gas. Thedrift velocity changes its sign to become positive.At high altitude (z & 2hg) where the gas dragis weak, the particles’ drift velocity begins to in-crease rapidly, and then the radial velocity be-comes positive again at z & 3hg. The behaviorof the radial velocity of 100 µm particles is quali-tatively similar to that of 10 µm particles, i.e., it isoutward at the midplane, inward at intermediatealtitude, and outward again at high altitude. Thecoupling of 100 µm particles to the gas is weaker,

6

so that their radial velocity is much different fromthe gas velocity even at the midplane. However,the drift velocity at the midplane is still slightlysmaller than the gas outflow velocity and the par-ticles move outward. Particles of 1 mm move in-ward even at the midplane, because the inwarddrift velocity of such large particles is larger thanthe outflow velocity of the gas. These large parti-cles also move outward at high altitude where thegas rotates faster than the particles.

3.2. Vertical Distribution of Particles

At the beginning stage of star formation frommolecular cloud cores, dust particles in circum-stellar disks may be well mixed with the gas andthe dust-gas ratio may be uniform throughout thedisk. After the termination of gas infall onto thedisks, the disks become to be hydrostatic and par-ticles begin to sediment toward the midplane be-cause of the z-component of the stellar gravity.The particles reach the terminal velocity, wherethe gravity and the gas drag balance with eachother,

vz,d = −ΩK,midTsz . (27)

The time scale of the sedimentation is tsed ∼z/vz,d ∼ T−1

s Ω−1K . If this time scale is much

smaller than the radial migration time scale, tr ∼r/vr,d ∼ (α + Ts)−1(hg/r)−2Ω−1

K , i.e., if Ts/α (hg/r)2, then the particles sediment before thelarge migration in the r-direction. (At high alti-tudes or around the midplane, particles may driftto the opposite direction to the gas, as seen inFig. 3. Some particles with Ts ∼ α may have theradial migration time scale tr much larger thanthe above estimate. Even for such particles, thecondition Ts/α (hg/r)2 for the fast sedimenta-tion is still valid.) For example, at 10 AU in adisk with α = 10−3 and (hg/r)2 ∼ 10−3, parti-cles larger than 0.2 µm sediment to the midplanewithout large radial movement.

We consider disks at later stages, such as theT Tauri stage. The disks are still turbulent, andthe turbulent motion of the gas stirs dust particlesup to high altitude to prevent dust sedimentation.An equilibrium distribution of particles in the ver-tical direction is achieved by the balance betweenthe sedimentation and the diffusion due to the tur-bulent gas.

The turbulent diffusion is modeled in an anal-

ogy of molecular diffusion, provided that the par-ticles are considered to be the passive tracers offluid, i.e., if the particles have no influence on thegas motion and have the similar velocity of thesurrounding gas [see e.g., chapter 10 in Monin &Yaglom (1971) and section 3.5.1 in Morfill (1985)].The equation of continuity is written as

∂

∂tρd +∇·(ρdvd + j) = 0 , (28)

where ρd is the particle density, and vd is the parti-cle velocity. The diffusive mass flux j is estimatedby

j = −ρgν

Sc∇

(ρd

ρg

), (29)

where the Schmidt number Sc represents thestrength of coupling between the particles andthe gas. For small particles, Sc approaches unityand the particles have the same diffusivity as thegas, while it becomes infinite for large particles.For intermediate particle sizes, Sc can be as smallas 0.1, which means that particle diffusion occurseffectively (see e.g., Fig. 1 in Cuzzi et al. 1993).In our standard model, we use Sc = 1. If Sc isnot unity, the particles are not the passive tracerswhich completely follow the fluid motion. In thiscase, the formulation according to the analogyof molecular diffusion may be inappropriate. Inaddition, if the velocity of sedimentation, vz,d, iscomparable to or larger than the turbulent veloc-ity, the particle could not be the passive tracers.The estimate of the diffusive mass flux by equa-tion (29) should be considered just as a “gradientdiffusion hypothesis.”

In a steady state of axisymmetrical disks, ∂/∂tand ∂/∂θ in equation (28) is zero. In addition, forparticles satisfying the condition Ts/α (hg/r)2

(i.e., for particles sedimenting fast without largeradial migration), we see that ∂(ρdvz,d)/∂z ∂(ρdvr,d)/∂r. Therefore, the mass flux in the z-direction must be zero, i.e.,

ρdvz,d − ρgν

Sc∂

∂z

(ρd

ρg

)= 0 . (30)

This equation is the same as the one derived byDubrulle, Morfill, & Sterzik (1995). Solving equa-tion (30) with equations (3), (4), (9), (20), and

7

(27) gives the particle density

ρd(r, z) = ρd(r, 0) exp[− z2

2h2g

− ScTs,mid

α

(exp

z2

2h2g

− 1)]

.

(31)The first term in the exponential comes from thegas distribution and the second term representsthe sedimentation. In the limit of tight coupling ofthe particles and the gas (Ts,mid = 0), the particledistribution is the same as that of the gas. Thesurface mass density of dust particles is

Σd(r) =∫ +∞

−∞ρd(r, z)dz . (32)

Figure 4a shows variation of the dust-gas ratioin the vertical direction at 10 AU in the standardmodel disk. We see the sedimentation of particles.3 Large particles (& 1 mm) sediment around themidplane, while small particles (. 10 µm) spreadover a few scale height of the gas disk. These smallparticles are stirred up to high altitude by the tur-bulence of the gas. The gas drag force becomesweaker at higher altitude where the gas densitydecreases, so the gas cannot sustain the dust parti-cles there. Thus, the particle density drops rapidlyat some altitude and the dust disk obtains a rela-tively sharp surface (e.g., at z ∼ 2.5hg for 10 µmparticles). The sedimentation of particles is moreeffective in the outer part of the disk, farther awayfrom the star (see Fig. 4b). For example, 100 µmparticles at 1 AU spread over the gas disk, whilethe particles at 100 AU concentrate around themidplane. At the outer part of the disk, the gasdensity is lower, so that the gas drag is weakerand turbulence cannot loft the particles to highaltitudes.

Dust particles are well mixed with the gas if

ScTs,mid

α

(exp

z2

2h2g

− 1) 1 , (33)

i.e., below the height

z

hg=

[2 log

(α

ScTs,mid+ 1

)]1/2

. (34)

3We obtained slightly thinner distribution of particles thanthat by Dubrulle et al. (1995) which is shown in their Fig-ure 3. They assumed that the stopping time of particles,which is inversely proportional to the gas density, to be con-stant and used the value at the midplane. This assumptioncauses the dust disk to puff out.

In the standard model, particles smaller than

s 20 r−1AU µm (35)

uniformly distribute with regard to the gas underthe height z = 3hg.

3.3. Net Radial Velocity

As discussed in §3.1, the radial velocity of par-ticles varies with the altitude. In this subsection,we discuss the net radial velocity averaged in thevertical direction.

3.3.1. Particles Well Mixed with the Gas

At the beginning stage of formation of circum-stellar disks, small particles would be distributeduniformly in the disk gas. Before the particles haveundergone enough sedimentation, their net radialvelocity averaged in the vertical direction is

〈vr,d〉 =1

Σg

∫ +∞

−∞ρgvr,ddz . (36)

Figure 5 shows the net radial velocities of smallparticles. The velocity of particles approachesthe gas velocity as the distance from the starbecomes smaller. Near the star, the gas den-sity is high enough to induce almost all particlesto move together with the gas. As the distancefrom the star increases, the gas density decreasesand particles at high altitude begin to drift out-ward from the gas. At sufficiently large distancefrom the star, the net radial velocity can be pos-itive, i.e., particles move outward. However, be-fore they move over a large distance in the ra-dial direction, they also sediment to the midplane.The time scale of the outward radial motion istout ∼ r/〈vr,d〉 ∼ 1/(ηTsΩK) (for outflowing parti-cles, |vr,g| is smaller than |vr,drift| and is neglectedin the estimate of tout), while the sedimentationtime scale is tsed ∼ z/vz,d ∼ 1/(TsΩK). Becauseη 1, tout tsed. The outward motion of thoseparticles which are well mixed with the gas mayproceed shortly after the formation of circumstel-lar disks, but it does not significantly modify theradial distribution of dust particles. The evolutionof radial distribution occurs primarily through themotion of sedimented particles.

8

3.3.2. Removal of Outflow at High Altitude bythe Sedimentation

The net radial velocity of sedimented particlesis

〈vr,d〉 =1

Σd

∫ +∞

−∞ρdvr,ddz . (37)

The function ρdvr,d/Σd, which is proportional tothe mass flux, is shown in Figure 6. Because of thesedimentation to the midplane, which is efficientfor larger particles, the mass flux is dominatedby the particles near the midplane. For example,the mass flux of 10 µm particles is mainly car-ried by particles moving outward around the mid-plane (|z| . 0.7hg) and by those moving inwardat intermediate altitude (0.7hg . |z| . 2.9hg).Although particles at high altitude (|z| & 2.9hg)move outward, their contribution to the mass fluxis negligibly small. Very large particles, for exam-ple 1 mm particles, move inward even at the mid-plane. Thus, the sedimentation causes the vastmajority of particles to move inward.

Sedimentation effectively removes outflowingparticles at high altitude. This flow pattern is seenas follows. The particles flowing outward have adrift velocity larger than the gas inflow velocity,i.e., |vr,drift| > |vr,g|. From equations (11) and(25), it is seen that α−1Ts,mid exp(z2/2h2

g) & 1 forsuch particles. However, from equation (31), wealso see that the density of these particles is assmall as ρd(z)/ρd(0) . e−1. Therefore, very fewparticles remain in the outflow region at high al-titude.

If the sedimentation is so effective that mostparticles concentrate around the midplane (|z| .hg), e.g., for particles larger than 1 mm at 10 AU(see Fig. 4a), the inward drift velocity at the mid-plane is larger than the gas outflow velocity. Thus,such particles flow inward. Again from equation(31), we see that Ts,mid/α & 1 for such particles.Then, it is seen from equations (11) and (25) that|vr,drift| & |vr,g| at the midplane. This inequalityimplies that when particles grow up sufficientlylarge sizes to sediment toward and to concentratearound the midplane, they become decoupled fromthe outflow motion of the gas. If the sedimenta-tion is so efficient that all particles concentrate in athin layer at the midplane (Ts,mid/α 1), the ra-dial motion of the particles would not affected bythe radial motion of the gas. In this limit, the par-

ticles’ motion is similar to that deduced by Adachiet al. (1976) and Weidenschilling (1977).

3.3.3. Net Velocity of Sedimented Particles

Figure 7a shows the net radial velocities of sed-imented particles. These net radial velocities arenegative for particles of all sizes. Particles rapidlymove inward when they are at large distances fromthe star. In the outer regions of the disk, par-ticles sediment and concentrate at the midplane,and their net radial velocity, which is much fasterthan the gas outflow velocity vr,g, is approximatedby the drift velocity at the midplane (eq. [26]).Their inward velocity decreases as they approachthe star, because the suppression of inward veloc-ity by the gas drag becomes stronger. The timescale of their orbital decay is r/〈vr,d〉 ∝ r2+p−q/2.When they approach the location where the gasdensity is dense enough to make Ts,mid/α . 1,the inward drift velocity becomes smaller than thegas outflow velocity at the midplane. The parti-cles begin to move with the gas flow. At the sametime, particles are spreaded over more than thedisk scale height. The particles’ net radial veloc-ity approaches the net gas velocity (the dashed linein Fig. 7a). However, because the particle distri-bution concentrates slightly more to the midplanethan the gas distribution, the number of particlesriding on the outflowing gas around the midplaneis larger than in the case where the particle dis-tribution is same as the gas. The enhancement ofoutflowing particles retards their net inflow veloc-ity to values below that of the gas. At the innerpart of the disk, the net inward velocity of particlesis always slower than the gas velocity. Near thestar, the particles mix more with the gas and theirmotion converges with that of the gas. The par-ticles’ inward velocity has a minimum magnitude,〈vr,d〉 = 5.2× 10−6 AU/yr, which is about half ofthe gas velocity. Particles smaller than 10 µm havea slower inward velocity than the gas throughoutnearly the entire disk (r . 100 AU), while parti-cles larger than 1 mm have slower velocity only inthe innermost part of the disk (r . 1 AU). Thedifference in the inward velocity induces the sizefractionation of particles, as discussed in §4.1 be-low. The dust-gas ratio may increase during theviscous evolution of disks, because the accretionvelocity of small particles is slower than that ofthe gas.

9

Small particles which satisfy the condition (35)are mixed well with the gas. The net radial ve-locities of such small particles may be calculatedby equation (36), even after larger particles haveundergone the sedimentation. It is seen from com-parison between Figures 5 and 7a that, e.g., for10 µm particles, the radial velocities are about thesame in both figures for r . 1 AU, while they de-viate significantly between these figures for r & 10AU. As shown in Figure 5, the radial velocity of10 µm particles at r & 10 AU is outward whenthe particles are mixed with the gas, i.e., in thevery first stage of disk formation. However, af-ter the particle sedimentation, the radial velocitybecomes inward even at r & 10 AU, as shown inFigure 7a. It is concluded that the particle sedi-mentation suppresses the outward flow of particlesin the standard model.

3.3.4. Various Models

In Figure 7b, we show the case with ps = −0.5where the surface density profile of the gas is flat-ter than in the standard model. The velocity pro-files are qualitatively similar to those in the stan-dard model. Particles of 10 µm move slower thanthe gas throughout nearly the entire disk, while1 mm particles rapidly migrate inward.

In Figure 7c, the case with a steeper surfacedensity gradient is shown. In this model, we adoptps = −1.3 which satisfies the condition (14) for in-ward gas accretion flow. In this case, we see thatthe net radial velocity of particles becomes posi-tive at some locations. In these regions, the accre-tion of particles is prevented. The particles flowinginward from large distances terminate their migra-tion at the location where the net radial velocitybecomes zero, and then accumulate at that par-ticular orbital radius. Particles located just insidethis critical radius flow outward and accumulatethere also. Particles at the innermost part of thedisk flow inward onto the star. Size fractiona-tion occurs because the location of the accumu-lation depends on the particle size. The accumu-lation continues as particles drift radially inwardfrom large radii until particles in the outer diskare significantly depleted or the number densityof accumulated particles becomes so high that theremoval of particles through coagulation or col-lisional destruction becomes efficient. Althoughaccretion of dust particles terminates at some lo-

cations, the gas disk continues to accrete onto thestar. The dust-gas ratio increases as the accretionof the gas proceeds.

3.3.5. Self-similarity of the Velocity Profile

The functions of net radial velocities shown inFigure 7 for various sizes have self-similar forms.From equations (11), (24), (25), (31), (32), and(37), the net radial velocity can be written as

〈vr,d〉 = 2πα

(h0

AU

)2

rq+1/2AU F

(p, q, Sc;

Ts,mid

α

)AU yr−1 ,

(38)where F is a function of p, q, Sc, and Ts,mid/α,

F

(p, q, Sc;

Ts,mid

α

)=

∫ +∞−∞

−3p− 2q − 6− (5q + 9)z′2

+[p + q + (q + 3)z′2

] Ts,midα exp z′2

× exp

[−z′2 − ScTs,mid

α

(exp z′2 − 1

)]dz′

×∫ +∞−∞ exp

[−z′2 − ScTs,mid

α

(exp z′2 − 1

)]dz′

−1

,(39)

and z′ = z/√

2hg. If we vary the properties of par-ticles, keeping Ts,mid/α to be constant, the func-tion F yields the same value. For example, whenwe vary the particle size s → s′, then transformthe orbital radius as r → r′ = (s′/s)1/[p+(q+3)/2]r,we have the same value of Ts,mid/α and the samefunctional form of F . In our models, q = −1/2, so〈vr,d〉 depends on r only through the function F .In this case, the profiles of the net radial velocitiesfor different sizes are obtained by the transforma-tion in the r-direction from one profile. In Figure8, the function F is plotted for various values ofthe Schmidt number Sc.

3.3.6. Various Schmidt Numbers

In the standard model, the net radial veloc-ity of particles is inward everywhere in the disk.This flow pattern is partially due to the removalof particles, through sedimentation, from high al-titudes where they would flow outward. For par-ticles which are sufficiently large to concentratenear the midplane, the magnitude of their inwarddrift velocity is larger than that of the outflow-ing gas. Thus, the strength of the sedimentationis always moderate enough to remove outflowing

10

particles. The sedimentation is controlled by cou-pling of particles to the turbulent motion of thegas. If coupling is relatively strong (Sc is smallerthan unity), the sedimentation would be less effec-tive and particles would mix more completely tothe gas, while if it is relatively weak (Sc is larger),particles would concentrate more to the midplane.Usually Sc is assumed to be unity, but the actualvalue may be different from unity. Some exper-imental data are plotted in Figure 1 in Cuzzi etal. (1993) which show Sc as small as 0.1 whenthe Stokes number St is about 0.01 (the Stokesnumber is considered to be the same order of thenon-dimensional stopping time Ts in our models).For large particles (St ∼ Ts & 1), Sc increases withthe particle size.

Figure 7d shows the net radial velocities in thecase where the sedimentation is efficient (Sc = 10).The net radial velocity becomes outward at itspeak. Thus, an accumulation of particles and anincrease in the dust-gas ratio would occur. Be-cause of the strong sedimentation, the dust disk ismuch thinner than in the standard model (Sc = 1)and concentrates more to the midplane where thegas flows outward. In the outer part of the disk(for example r & 10 AU for 100 µm particles), theinward drift velocity of particles (relative to thegas) at the midplane is faster than the outward gasvelocity. As the particles drift in, the inward veloc-ity decreases, and at some distances from the star(r ∼ 10 AU for 100 µm particles), the inward driftvelocity at the midplane becomes smaller than theoutward gas velocity, and the particles around themidplane flow outward. At such distances, thedust disk is still concentrated around the mid-plane, and the net radial velocity is dominatedby the particles around the midplane and is out-ward. As the distance from the star decreases, theparticles mix more with the gas and the concen-tration to the midplane becomes weaker. The netradial velocity approaches the gas velocity, whichis inward, at the innermost part of the disk.

If the Schmidt number Sc is less than unity, thesedimentation of particles is less efficient. There-fore, the particles are always well mixed with thegas at where the inward drift velocity of parti-cles is smaller than the gas outflow velocity (i.e.,at where the particle outflow at the midplane oc-curs). Thus, the net radial velocity of particles isinward everywhere.

Figure 8 shows the function F for various valuesof the Schmidt number, Sc = 0.1, 1, and 10. Thenet radial velocity has the same sign as F (see eq.[38]). It is seen that the outflow of dust particlesoccurs if the sedimentation is efficient (Sc & 10).

4. Discussion and Summary

4.1. Steady Density Distribution of DustParticles

Because the inward velocity of particles de-pends on their size, particles of different sizes ac-cumulate at different locations. In this subsection,we discuss the density distribution of dust parti-cles as a consequence of their radial flow and showhow their size fractionation may occur. We adoptthe models in which the net velocity of particlesis always inward (Sc = 1 and ps ≥ −1.0). We as-sume that the distribution of particles approachesa steady state after a brief stage of initial evolu-tion. When a steady state is achieved, the massflux of dust particles becomes constant in the ra-dial direction. We calculate the mass flux of par-ticles as

dMd

ds= 2πr〈vr,d〉dΣd

ds, (40)

where Md and Σd are the mass flux and the sur-face density, respectively, of particles smaller thansize s. For simplicity, we neglect three physicalprocesses. First, the evolution of the particle sizethrough the coagulation, collisional destruction,condensation, and sublimation is neglected, i.e.,we assume the mass flux dMd/ds for each sizerange is constant in the radial direction. Second,in turbulent disks, the particles’ mass flux comesnot only from the mean flow with an average veloc-ity 〈vr,d〉 but also from the turbulent diffusion ofparticles, which appears as j in equation (28) forexample. We neglect the mass flux from the tur-bulent diffusion. Third, we assume the structureof the gas component of the disk is entirely deter-mined by the gas itself. We neglect the feedbackdrag induced by the particles on the gas. In thelimit that the spatial density of the dust compo-nent becomes comparable to that of the gas nearthe midplane, this effect would speed up the az-imuthal velocity of the gas to the Keplerian valueand quench the radial migration of the dust (Cuzziet al. 1993). Such a dust concentration requiressubstantial sedimentation of relatively large parti-

11

cles. Contributions from all of these factors will beinvestigated in future grain-evolution calculations.Here, we adopt the simplest assumptions to focuson showing the size fractionation of particles. Themass flux of particles in all size range is

Md,all =∫ smax

smin

dMd

dsds , (41)

where smin and smax are the minimum and max-imum sizes of particles, respectively. The surfacedensity in the steady state is calculated from equa-tion (40) with given Md. Figure 9a shows the sur-face density of the dust component composed ofsingle size particles. Particles of different sizeshave different density profiles. If the dust com-ponent is composed of relatively large particles, itwould be concentrate at the inner region of thedisk. Thus, as particles grow in size, their sur-face density distribution becomes more centrallyconcentrated. Figure 9b shows the surface den-sity, assuming the size distribution of particles isa power law with index −3.5, smin = 0.1 µm, andsmax = 10 cm. The surface density distributionof the dust particles is different from that of thegas. The power law index of the dust distribu-tion Σd is about −1.5 for a gas disk with indexps = −1.0 (−1.2 for a gas disk with ps = −0.5).The implied power law index, −1.5, is similar tothe value anticipated from the present mass distri-bution of planets in the solar system (Hayashi etal. 1985). However, note that the density distri-butions in Figure 9 are derived assuming no sizeevolution of particles, and no turbulent diffusionin the radial direction. The growth of particlesduring the radial flow adds a source term in theequation of continuity. The evolution of the dustdensity profile should be investigated further bytaking particle growth and turbulent diffusion intoaccount.

4.2. Evolution of the Dust-gas Ratio

The accretion velocity of dust particles is dif-ferent from that of the gas. This difference causesevolution of the dust-gas ratio. In the standardmodel, 10 µm particles between 10 and 100 AUhave an inflow velocity which is about half of thegas velocity. Thus, if the mass of the dust diskis dominated by 10 µm particles (for example, ifthe particles have size distribution n ∝ s−3.5 withmaximum size 10 µm), the dust-gas ratio would

increase through the gas accretion. For example,if the initial mass of the gas disk is 0.11M andit reduces to 0.01M after viscous accretion, thedust-gas ratio would increase to be 6 times of itsinitial value. The increase in the dust-gas ratiospeeds up the formation of planetesimals throughmutual collisions. Their enhanced abundance mayalso cause the dust component to become unstableand promote the planetesimal formation throughthe gravitational instability (Goldreich & Ward1973; Sekiya 1998).

If the particle growth proceeds to make 1 mmto 1 cm particles during the gas accretion phase,such large particles migrate inward rapidly andaccumulate in the inner part of the disk (see Fig9a). This size fractionation causes an increase inthe dust-gas ratio at the inner disk, while this pro-cess may decrease the dust-gas ratio at the outerdisk, resulting in a dust disk concentrated to theinner part of the gas disk.

4.3. Summary

The radial migration of dust particles in accre-tion disks is studied. Our results are as follows.

1. Dust particles move radially both inward andoutward by the gas drag force. Particles at highaltitude (|z| & 2hg) move outward because theyrotate slower than the gas whose pressure gradientforce is inward. Small particles (s . 100 µm at10 AU) near the midplane (|z| . hg) are advectedby the gas outflow. On the other hand, particlesat intermediate altitude and large particles (s &1 mm at 10 AU) move inward.

2. The net radial velocity, averaged in the verti-cal direction, is usually inward, provided that theradial gradient of the gas surface density is not toosteep (ps & −1.3). Sedimentation removes out-flowing particles from high altitudes. Small par-ticles, which can be advected by the outflowinggas around the midplane, do not concentrate atthe midplane. In the inner part of the gas disk(r . 100 AU for 10 µm particles), the inflow ve-locity of particles is smaller than the gas accretionvelocity, resulting in an increase in the dust-gasratio.

3. The particle sedimentation would be efficientif dust-gas coupling is relatively weak (Sc > 1.0).If the sedimentation is so efficient (Sc & 10), thenumber of outflowing particles around the mid-

12

plane would be large, and the direction of the netradial velocity of particles would change to out-ward at some distances from the star. Accumula-tion of particles at such locations serves to increasethe local dust-gas ratio.

4. The inflow velocity of particles depends onthe particle size. Therefore, the inflow causes thesize fractionation of particles. Larger particles ac-cumulate at distances closer to the star.

We are grateful to the anonymous referee whogave us important criticism, especially on the im-proper treatment of particle diffusion in the orig-inal version of the manuscript. His/her sugges-tions also considerably improved the paper. Wethank Greg Laughlin for careful reading of themanuscript and Jeff Cuzzi for discussions on theturbulent motion of dust particles. This work wassupported in part by an NSF grant AST 99 87417and in part by a special NASA astrophysical the-ory program that supports a joint Center for StarFormation Studies at UC Berkeley, NASA-AmesResearch Center, and UC Santa Cruz.

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AAS LATEX macros v5.0.

14

Fig. 1.— Radial velocity of gas vr,g. The distancefrom the midplane z is normalized by the disk scaleheight hg. In the standard model (q = −0.5), theradial velocity is independent of the distance fromthe star. The dotted line shows the zero velocityfor reference.

Fig. 2.— The particle radius whose non-dimensional stopping time Ts is unity in the stan-dard disk (ps = −1.0 and q = −0.5). The solid linecorresponds to particles at the midplane, whilethe dashed line corresponds to particles at altitudez = 2hg. Particles smaller than the lines have Ts

smaller than unity and couple well with the gas,while particles larger than the lines have Ts > 1.

Fig. 3.— Radial velocity vr,d of dust particles ofs = 10 µm (solid line), 100 µm (dashed line), and1 mm (dot-dashed line) at 10 AU. The distancefrom the midplane z is normalized by the disk scaleheight hg = 0.59 AU. The dotted line shows theradial velocity of the gas vr,g.

15

Fig. 4.— Dust-gas ratio, ρd/ρg, against the dis-tance from the midplane (a) for various sizes at 10AU and (b) at various locations for 100 µm. Thedistance from the midplane z is normalized by thedisk scale height hg. The averaged dust-gas ratiois assumed to be 0.01. (a) Three solid lines showthe dust-gas ratio of particles of 1 mm, 100 µm,and 10 µm from the left line. (b) Three solid linesshow the dust-gas ratio at 100 AU, 10 AU, and 1AU from the left line.

Fig. 5.— Net radial velocity of particles wellmixed with the gas, 〈vr,d〉. The three solid linescorrespond to s = 10 µm, 1 µm, and 0.1 µm par-ticles from the upper line. The dashed line showsthe gas accretion velocity.

Fig. 6.— Normalized mass flux ρdvr,d/Σd at 10AU. The solid, dashed, and dot-dashed lines arefor s = 10 µm, 100 µm, and 1 mm particles, re-spectively. For 1 mm particles (dot-dashed line),1/10 of the value is plotted (or refer the ordinateat the right hand side). The distance from themidplane z is normalized by the disk scale heighthg = 0.59 AU. The dotted line shows the zero ve-locity for reference.

16

Fig. 7.— Net radial velocity 〈vr,d〉 of sedimentedparticles. Three solid lines correspond to s = 1mm, 100 µm, and 10 µm particles from the leftline. The accretion velocity of the gas is shownby the dashed line, and the dotted line shows thezero velocity for reference. (a) the standard model(ps = −1.0, q = −0.5, and Sc = 1.0) (b) ps =−0.5. (c) ps = −1.3. (d) Sc = 10.

17

Fig. 8.— Function F for various values of theSchmidt number, Sc = 10, 1, and 0.1 from theupper line. The dotted line shows the zero forreference.

Fig. 9.— Surface density of dust disks, Σd. (a)The disk is composed of single sized particles ofs = 1 mm, 100 µm, and 10 µm from the left line.The mass flux Md is assumed to be 10−10M yr−1.(The mass flux of the gas is 10−8M yr−1.) Thegas disk is the standard model (ps = −1.0). (b)The disk is composed of particles with a powerlaw size distribution. The indices of the surfacedensity profiles of the gas disks are ps = −1.0and −0.5 from the upper line. The particle num-ber flux is assumed to be proportional to s−3.5.The minimum and maximum sizes of particles are0.1 µm and 10 cm, respectively. The total massflux is Md,all = 10−10M yr−1.

18