Icarus 195 (2008) 871–881www.elsevier.com/locate/icarus

Dust ring formation due to ice sublimation of radially drifting dust particlesunder the Poynting–Robertson effect in debris disks

Hiroshi Kobayashi a,b,∗, Sei-ichiro Watanabe b, Hiroshi Kimura a, Tetsuo Yamamoto a

a Institute of Low Temperature Science, Hokkaido University, Kita-ku Kita 19 Nishi 8, Sapporo 060-0819, Japanb Department of Earth and Planetary Sciences, Graduate School of Environmental Studies, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan

Received 17 October 2007; revised 24 January 2008

Available online 29 February 2008

Abstract

In a disk with a low optical depth, dust particles drift radially inward by the Poynting–Robertson (P-R) drag rather than are blown out by stellarradiation pressure following destructive collisions. We investigate the radial distribution of icy dust composed of pure ice and refractory materialsin dust-debris disks taking into account the P-R drag and ice sublimation. We find that icy dust particles form a dust ring by their pile-ups at theedge of their sublimation zone, where they sublime substantially at the temperature 100–110 K. The distance of the dust ring is 20–35 AU fromthe central star with its luminosity L� � 30L� and 65(L�/100L�)1/2 AU for L� � 30L�, where L� is the solar luminosity. The effective opticaldepth is enhanced by a factor of 2 for L� � 100L� and more than 10 for L� � 100L�. The optical depth of the outer icy dust disk exceeds thatof the inner disk filled with refractory particles, namely, the residue of ice sublimation, which are further subjected to the P-R effect. As a result,an inner hole is formed inside the sublimation zone together with a dust ring along the outer edge of the hole.© 2008 Elsevier Inc. All rights reserved.

Keywords: Debris disks; Ices; Celestial mechanics

1. Introduction

Dust-debris disks around main-sequence stars are inferredby infrared excesses in the spectral energy distribution of thestars and in some cases found by high resolution imagingobservations in a wide range of wavelengths from visual tosubmillimeter. The dust in a debris disk falls into the cen-tral star by the Poynting–Robertson drag (hereafter P-R drag)or is blown out by the radiation pressure following colli-sional breakups. Characteristic timescales for elimination ofdust particles due to these processes are much shorter thanthe ages of the central stars, so that dust particles in dust-debris disks are not primordial but secondary. Namely, the par-ticles are continuously produced through destructive collisionsbetween parent-body planetesimals in the so-called dust pro-duction region. These disks are characterized by deficient gas

* Corresponding author at: Institute of Low Temperature Science, HokkaidoUniversity, Kita-ku Kita 19 Nishi 8, Sapporo 060-0819, Japan. Fax: +81 11 7067142.

E-mail address: hkobayas@lowtem.hokudai.ac.jp (H. Kobayashi).

0019-1035/$ – see front matter © 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.icarus.2008.02.005

and low dust temperatures �100 K in comparison with pro-toplanetary disks (e.g., Kim et al., 2005; Hines et al., 2006;Silverstone et al., 2006). The lack of warm dust in debris disksindicates the presence of inner holes, some of which are directlyresolved by imaging observations. If the effective optical depthis larger than 10−5(rpro/20 AU)−1/2 in the dust production re-gion where rpro is the distance of the dust production regionfrom the central star, the dust becomes small by destructive col-lisions and tiny dust is blown out by radiation pressure, formingan inner hole in the region of r < rpro (Wyatt et al., 1999;Dominik and Decin, 2003; Wyatt, 2005).

The effective optical depth τeff of debris disks decreases withage, because the number of parent planetesimals decreases bydestructive collisions. At τeff � 10−5, the drift timescales ofsmall particles due to the P-R drag are shorter than their col-lision timescales. We focus our attention to such low-optical-depth disks, which are hereafter referred to as drag-dominateddisks. When dust drifts radially inward, the surface density atr < rpro becomes constant with distance from the central star(e.g., Strubbe and Chiang, 2006). In this case, one expectsthat holes are covered with the drifting dust. Even in a drag-

872 H. Kobayashi et al. / Icarus 195 (2008) 871–881

dominated disk, however, another basic mechanism can openan inner hole. This is sublimation of ices. Because the dust pro-duction region is cold, ice is the main composition of dust. Weassume that the composition of parent planetesimals is similarto that of comets (cf. McDonnell et al., 1987); the volume frac-tion of ice is 0.5–0.9 in icy dust (see Section 3). Icy dust driftsinward by the P-R drag and sublimes in the inner region by ahigh temperature. We expect that ice sublimation could produceinner holes as large as those found around main-sequence stars.

Sublimation may form not only an inner hole but also aring at the edge of the ice sublimation zone. In general, theratio β of radiation pressure to stellar gravity increases dur-ing ice sublimation. Subliming dust attempts to drift inwardby the P-R drag and outward by the increase in the β ratio.When the inward drift rate becomes comparable to the out-ward one at a certain location, the dust would pile up at thatplace. This is known for interplanetary dust grains that fall tothe Sun by the P-R drag and sublime. Mukai and Yamamoto(1979) show that spherical dust particles of graphite and ob-sidian pile up at a few solar radii (R�) from the Sun. Thepile-up occurs at the sublimation zone, even for other materialsand shapes of the particles (e.g., Shestakova and Tambovtseva,1995; Kimura et al., 1997). A solar dust ring was observedaround 4R� from the Sun in the period of 1966–1983, althoughit was not detected in the nineties (Kimura and Mann, 1998;Ohgaito et al., 2002). We expect that this pile-up takes place indust-debris disks as well by ice sublimation. Recently, opticallythin debris disks with τeff as low as 10−6 come to be observableat infrared wavelengths (e.g., Chen et al., 2006). For the per-spective on future imaging observations of these disks, it is im-portant to understand the dynamics of dust in drag-dominateddisks.

In this paper, we investigate the effect of ice sublimationon the spatial distribution of dust particles in drag-dominateddust-debris disks. We consider icy dust particles, which arecomposed of pure ice and refractory materials, radially drift-ing from the dust production region outside the ice sublimationzone with its effective optical depth less than 10−5. We willanswer the following questions. At what temperature does icydust sublime and where is its sublimation zone? How does thelocation of the sublimation zone depend on the luminosity ofthe central star? How are dust rings and inner holes formed?To answer these and related questions, we start with reasonableassumptions in drag-dominated dust-debris disks.

In Section 2, we describe equations of motion and sublima-tion rate and give the equation of temporal evolution of orbitalelements due to the P-R drag and sublimation. In Sections 3and 4, we show our results on calculations of the orbital evo-lution for icy dust particles. We determine the distance andtemperature of ice sublimation, the spatial distribution of theicy dust in a disk, and the enhancements of the number densityand the effective optical depth. In Section 5, we discuss the ec-centricities of dust particles drifting from the dust productionregion into the sublimation zone by the P-R drag and the fateof refractory materials in the icy dust after the ice sublimation.We also discuss other mechanisms to open inner holes in dust-debris disks and compare the inner holes from observation with

those due to ice sublimation. Section 6 summarizes our find-ings.

2. Basic equations

2.1. Equation of motion

We consider a dust-debris disk around a main-sequence star.If relative velocities between planetesimals sufficiently exceedtheir surface escape velocities, their collision becomes destruc-tive and produces small fragments (e.g., Kobayashi and Ida,2001). The cause of such collisions is dynamical excitation ofparent-body planetesimals by protoplanets of radii of a few hun-dreds km and an early stellar encounter in the outermost regionwhere a sufficient number density of planetesimals still remains(Kenyon and Bromley, 2004; Kobayashi and Ida, 2001). Col-lisions of fragments produced by mutual collisions of parentplanetesimals supply dust particles in the outer dust produc-tion region (�20 AU) of the disk. We assume that the drifttimescales of dust particles due to the P-R drag are shorter thantheir collision timescales, namely, the disk is drag-dominated atleast in the regions considered. The effective optical depth inthe vertical direction of such a disk is less than 10−5. The op-tical depth in the radial direction is less than unity even in theoutermost regions of the disk, so that the self-shielding effectof the disk is negligible. We also assume that the drift timescaleis shorter than the size decreasing timescale due to UV sputter-ing, namely, icy dust drifts radially inward (see Grigorieva etal., 2007 for the UV sputtering effect).

We consider the gravity, radiation, and wind of a central starand the P-R drag. The motion of a spherical dust particle inorbit around the star is determined by

(1)d2r

dt2= −GM�

r2

[(1 − βtotal)rrr + βPR

(2vr

crrr + vθ

cθθθ

)],

where r is the position vector of the particle with respect tothe central star, G is the gravitational constant, M� is the stel-lar mass, c is the speed of light, and rrr and θθθ are the unit vectorsin the particle’s orbital plane along and perpendicular to r , re-spectively. Further, vr and vθ are the rrr and θθθ components of theparticle’s velocity, βtotal is the ratio of the pressure force dueto the electro-magnetic and particulate radiation to the gravi-tational force, and βPR is the coefficient of the total P-R drag,namely, the sum of drag forces due to stellar radiation and wind.The first term in the brackets represents the sum of the gravita-tional force and the pressure force due to stellar radiation andwind, and the second term represents the total P-R drag. We callthe term involving βtotal in the first term the pressure force andthe second term the P-R drag hereafter.

The βtotal is expressed by βtotal = β + βsw, where β is theratio of the electromagnetic radiation force to the gravitationalforce and βsw is the ratio of the stellar wind pressure to thegravitational force (e.g., Burns et al., 1979). For an icy sphericalparticle with radius s, β and βsw are given by (Gustafson, 1994)

Dust ring formation due to ice sublimation 873

(2)β = 3L�Qpr

16πcGM�ρs,

(3)βsw = 3M�vswQsw

16πGM�ρs,

where L� is the stellar luminosity, ρ = 1 g cm−3 is the ma-terial density of the particle, M� is the stellar mass loss rateby the wind, vsw is the average radial velocity of the stellarwind, and Qsw is the velocity-averaged momentum transfer ef-ficiency from the stellar wind to the dust particle (Minato et al.,2004, 2006). The Planck mean Qpr of the radiation pressure ef-ficiency Qpr(λ, s) is given by

(4)Qpr(T�, s) = 1

σSBT 4�

∞∫0

dλB(λ,T�)Qpr(λ, s),

where B(λ,T�) is the Planck function at a wavelength of λ

with a stellar effective temperature of T� and σSB is the Stefan–Boltzmann constant.

The coefficient of the P-R drag βPR is given by (Minato etal., 2004, 2006),

(5)βPR = β(1 + γ ),

with

(6)γ = M�c2

L�

Qsw

Qpr,

where γ is the ratio of the P-R drag due to the stellar wind tothe P-R drag due to the stellar radiation.

For M� = M� and L� = L�, γ is estimated to be 0.3 andβsw/β is 5 × 10−3, where M� is the solar mass loss rate bythe solar wind. For M�/L� � M�/L�, the P-R effect is deter-mined by stellar wind. An icy dust particle drift radially into thesublimation zone by the P-R drag prior to the mutual collisionof particles and the UV sputtering if the P-R drift timescale isshorter than the collision timescale and the sputtering timescale.Because the P-R drift timescale becomes short with γ , icy par-ticles drift into the sublimation zone without suffering frommutual collisions and UV sputtering for large γ . On the otherhand, the temperature at the sublimation zone varies within 3 Kfor M�/L� = 1–103M�/L�. The distance and the number den-sity at the sublimation zone hardly vary because the temperaturevariation is small. In the following calculations, the stellar windis neglected (M� = 0), because the choice of M� does not sig-nificantly change the distance and the dust number density atthe sublimation zone unless M�/L� � 103M�/L�.

2.2. Sublimation rate

As most dust particles in debris disks are colder than 100 K,their major component is water ice. When dust particles spiraltoward the central star by the P-R drag, sublimation of ice willreduce their size. Since βtotal in Eq. (1) changes with dust size s,sublimation affects the orbital evolution of dust particles. Thesize decreasing rate due to sublimation is given by (e.g., Mukaiand Yamamoto, 1979)

(7)ds

dt= −

√μmu

2πkT

Pv

ρ,

where mu is the atomic mass unit, μ = 18 is the mean molecularweight of water molecules, and k the Boltzmann constant. Thevapor pressure Pv of ice strongly depends on the temperature T

of dust particles, and is given by (Washburn, 1928)

logPv = −2445.5646

T+ 8.2312 logT − 0.01677006T

(8)+ 1.20514 × 10−5T 2 − 3.63227,

where Pv is in dyn cm−2 and T in K. As shown in the latersection, sublimation is effective at T = 100–110 K. AlthoughEq. (8) is fitted for T � 170 K, Pv is in good agreement withexperimental data within a factor of 1.5 at T = 100–110 K (seeMurphy and Koop, 2005 for a review).

The temperature T of a dust particle is determined by thebalance equation of energy between the absorption of incidentstellar radiation, particle’s thermal emission, and the latent heatof sublimation:

(9)L�

16πr2Q� − QdσSBT 4 − ρ

∣∣∣∣ds

dt

∣∣∣∣� = 0,

where � is the latent heat of sublimation of ice per unit mass,and Q� is the Planck mean absorption efficiency with T� andQd is the efficiency with T given by

(10)Q�(T�, s) = 1

σSBT 4�

∞∫0

dλB(λ,T�)Qabs(λ, s),

(11)Qd(T , s) = 1

σSBT 4

∞∫0

dλB(λ,T )Qabs(λ, s),

where Qabs(λ, s) is the absorption efficiency of a spherical par-ticle with radius s at a wavelength of λ.

From Eqs. (1) and (7), we can obtain the orbital evolution ofsublimating particles.

2.3. Changes in semimajor axis, eccentricity, and size

The osculating elements of semimajor axis a and eccentric-ity e for the orbit of a dust particle, which are defined in termsof the position r and velocity v of the dust particle at an instant,are given by

(12)a =[

2

r− v2

(1 − βtotal)GM�

]−1

,

(13)e =√

1 − r2v2θ

(1 − βtotal)GM�

[2

r− v2

(1 − βtotal)GM�

].

The secular changes in semimajor axis a, eccentricity e, andsize s of a dust particle averaged over a Keplerian period aregiven by (Kobayashi et al., submitted for publication)⟨

da

dt

⟩= α

(1 + e2

1 − e2ψ + 2e

1 − e2φ

)βtotal

1 − βtotal

a

s

(14)− βPRGM� 2 + 3e2

2 3/2,

c a(1 − e )

874 H. Kobayashi et al. / Icarus 195 (2008) 871–881

⟨de

dt

⟩= α(eψ + φ)

βtotal

1 − βtotal

1

s

(15)− 5βPRGM�

2c

e

a2(1 − e2)1/2,

(16)

⟨ds

dt

⟩= −ψ

with

(17)ψ = 1

2π

2π∫0

df

√μmu

2πkT

Pv

ρ

(1 − e2)3/2

(1 + e cosf )2,

(18)φ = 1

2π

2π∫0

df

√μmu

2πkT

Pv

ρcosf

(1 − e2)3/2

(1 + e cosf )2,

where α ≡ −d lnβtotal/d ln s, f is the true anomaly, and 〈A〉denotes a quantity A averaged over the Keplerian period TK,

(19)〈A〉 = 1

TK

TK∫0

dt A = 1

2π

2π∫0

df A(1 − e2)3/2

(1 + e cosf )2.

In Eqs. (14) and (15), the first terms on the r.h.s. indicate thechanges due to sublimation, and the second terms the changesdue to the P-R drag. The first terms on the r.h.s. of Eqs. (14)and (15) are always positive and the second terms are alwaysnegative. We integrate Eqs. (14)–(16) with Eqs. (17) and (18)to reveal the orbital evolution of dust particles.

3. Orbital and size evolution

For an illustrative purpose, we first consider an ideal casethat dust particles can be regarded as perfect absorbers. In thiscase, we can choose Qph = Q� = Qd = 1 and βtotal = β =βPR ∝ s−1.

Fig. 1 shows the evolution of semimajor axis a, eccentric-ity e, and radius s of a perfectly absorbing particle for initialparameters of a0 = 10 AU, e0 = 10−4, and s0 = 3 µm. The stel-lar parameters are set to be L� = L� = 3.85 × 1033 erg s−1 andM� = M� = 1.99×1033 g. We confirm that the results obtainedfrom the direct integration of Eqs. (1) and (7) well coincidewith those from the integration of the orbital-averaged equa-tions of Eqs. (14)–(16).1 This is because the orbital evolutiontime (∼105 yr) is much longer than the local Keplerian period.

From Fig. 1, we can define three stages of the orbital andsize evolution of a dust particle. In the first stage (Stage I:t < 1.2 × 105 yr), the temperature of the particle is low becauseof large distance from the central star. In consequence, the sub-limation rate is small (s ∼ 0) and both ψ and φ in Eqs. (14)and (15) are negligible. In this stage, a and e are determined bythe P-R drag, under which a dust particle spirals toward the cen-tral star and its temperature gradually increases. In the secondstage (Stage II: t = 1.2–1.7 × 105 yr) when the dust reaches a

1 Here we use the α-Hermite method (Kokubo and Makino, 2004) for theformer and the fourth-order Runge–Kutta method (Press et al., 1992) for thelatter.

Fig. 1. Eccentricity e and size s as a function of semimajor axis a for their initialvalues of a0 = 10 AU, e0 = 1 × 10−4, and s0 = 3 µm. Note that the results onthe numerical integration of Eqs. (1) and (7) (white dashed lines) agree withthose of Eqs. (14)–(16) (black solid lines).

certain temperature, the size s begins to decrease rapidly bysublimation. This leads to a slow increase in a and a rapidincrease in e, although the e value is still small. As the dustparticle gets small by sublimation, the relative importance ofthe stellar gravity becomes weaker compared with the radia-tion pressure, namely, dβ/dt > 0. This increases the semimajoraxis a and the eccentricity e. The increase in β prevents the in-ward drift due to the P-R drag, yielding a pile-up of dust gainsat the sublimation zone. In Stage II, the orbit is almost circu-lar because of low e and the orbital radius is almost constant,resulting in almost a constant sublimation rate. This constantsublimation rate allows the dust particle to stay in the subli-mation zone for a long period. In the third stage (Stage III:t > 1.7 × 105 yr), the orbit becomes eccentric and the dust par-ticle begins to drift outward with a high speed because of theincrease in β due to sublimation.

We show the dependence of the orbital evolution on initialparameters e0 (Fig. 2) and s0 (Fig. 3); the eccentricity is de-termined only by the P-R drag in Stage I, so that the results oflarger a0 is same as those of smaller e0. Fig. 2 indicates that dustparticles with different e0 stay at the sublimation zone withouta substantial change of a in Stage II so far as e � 0.1, althoughthe duration of their staying at the sublimation zone becomesshort with increasing e0. A similar situation occurs if startingfrom different s0, as is seen from Fig. 3 that dust particles with

Dust ring formation due to ice sublimation 875

Fig. 2. Evolution of the eccentricities and the sizes of perfectly absorbingparticles as a function of semimajor axis, for a0 = 10 AU, s0 = 3 µm, ande0 = 10−4 (solid lines), 10−3 (dotted lines), 10−2 (short dashed lines), and10−1 (long dashed lines). The vertical lines in the bottom panel indicate thedistance where the dust temperatures equal 100 and 110 K.

different s0 stay at similar distances from the central star in thesame sublimation zone, so far as e � 0.1. In summary, dust par-ticles with different e0 and s0 pile up at the sublimation zone,forming a dust ring. The distance of dust ring is 6–7 AU for theperfectly absorbing dust with a temperature of 100–110 K.

Next we consider icy particles as a more realistic dust model.The icy particles are assumed to consist of organic matter em-bedded in a matrix of pure ice. The average dielectric functionof the icy particle is estimated by the Maxwell–Garnett mixingrule (e.g., Bohren and Huffman, 1983):

(20)εav = εm

[2εm + εi − 2Γ (εm − εi)

2εm + εi + Γ (εm − εi)

],

where εm and εi are the dielectric functions of the ice matrixand the organic inclusions, respectively, and Γ is the volumefraction of the inclusions. From Giotto’s in situ measurementsof Comet 1P/Halley, an estimate of Γ is 0.1–0.5 (McDonnell etal., 1987). We set Γ = 0.1; the size of the sublimation zoneincreases with increasing Γ within a factor of 1.6 so far as0.1 < Γ < 0.5. The temperature at the sublimation zone hardlydepends on Γ . We will discuss that in Section 4. We use therefractive indices of water ice from Warren (1984) and thoseof organic refractory material from Li and Greenberg (1997).Once the refractive indices of the icy particle are determined,we calculate Qpr and Qabs using Mie theory (Bohren and Huff-

Fig. 3. Same as in Fig. 2 but for e0 = 10−4 and s0 = 1.1 µm (dashed lines),3 µm (solid lines), and 10 µm (dotted lines).

man, 1983). The results are used to obtain Qpr, Q�, and Qd

as a function of temperature and size from Eqs. (4), (10),and (11). In Fig. 4, we show Qpr, Q�, and Qd for T� = 5800 Kand T = 110 K. The efficiencies decrease with decreasing sizewhen s � 1 µm for Q� and Qpr and s � 10 µm for Qd. The ef-ficiencies are almost constant with size if the size is larger thanthe wavelength at which the Planck function reaches its maxi-mum. Because of Q� > Qd for icy particles smaller than 10 µm,their temperatures are always higher than the temperature forperfectly absorbing dust.

Fig. 5 shows the orbital and size evolution of icy dust. Thedistance of the outer edge of the sublimation zone depends ons0 because the temperature of the dust depends on the size aswell as the distance from the central star; the smaller its size,the higher its temperature as is seen from Fig. 5. The evolutionconsists of three stages similar to perfectly absorbing dust. InStage I, a dust particle drifts radially inward by the P-R drag.In Stage II, its size decreases rapidly while its samimajor axisincreases slowly. The distance of the transition from Stage I toII is given roughly by tsub tPR, where tsub is the sublimationtimescale and tPR is the P-R drift timescale. In Stage III, thedust particles are blown out when the eccentricities exceed 0.1as is in the case of perfectly absorbing dust.

It should be noted that the detailed features of the evolutionof icy particles in Stage II are different from that of the per-fectly absorbing dust. For particles of the initial sizes s0 = 3and 10 µm in Stage II, a decrease in size increases the dust

876 H. Kobayashi et al. / Icarus 195 (2008) 871–881

Fig. 4. Qd averaged with T = 110 K (solid line), Q� with T� = 5800 K (dottedlines), and Qpr with T� (dashed lines) as a function of dust size in µm.

Fig. 5. Evolution of the eccentricities and the sizes of icy particles as a functionof semimajor axis for a0 = 26 AU, e0 = 10−4, and s0 = 10 µm (dotted lines),3 µm (solid lines), and 1.1 µm (dashed lines). The thin solid lines in the bottompanel indicate the distance where the dust temperatures equal 100 and 110 K.

temperature rapidly because of a strong size dependence of thedust temperature. The increase in temperature leads to a furtherdecrease in size, and eventually the sublimation rate becomeshigher than that for the perfectly absorbing dust. This impliesthat the outward drift rate for the icy particle is much fasterthan that for the perfectly absorbing dust. This indicates thattsub is much shorter than tPR. In consequence, icy dust particles

Fig. 6. Schematic illustration to derive the number density ns(q, s) in the q–s

plane. Two adjacent evolution lines (thick lines) do not intercept in the q–s

plane. We consider one point (q, s) is on one evolution line and the other point(q + δq, s + δs) in the vicinity of (q, s) is on the other evolution line. Thenumber of particles passing through a line connecting the two points (q, s) and(q + δq, s + δs) per unit time given by ns(q, s)|sδq − qδs| is kept constantalong the “tube” defined by the evolution lines.

of large initial sizes do not form a ring. For s0 = 1.1 µm, onthe other hand, the outward drift rate due to sublimation at thedistance given by tsub tPR is larger than that for larger initialsize s0 with smaller β . The particle radially drifts outward be-fore active sublimation takes place at temperatures higher than110 K. Namely, its temperature in Stage II is kept low com-pared to that of larger s0. The outward drift rate in Stage IIis smaller than that for large s0 because of the low tempera-ture. In consequence, dust particles of s0 = 1.1 µm pile up inthe sublimation zone. The numerical simulations and an analyt-ical estimate show that dust particles pile up if s0 � 2 µm forL� = L�. The dependence of the accumulation on the luminos-ity of the central star will be discussed in the next section (seeFig. 9).

4. Distribution of dust in disks

We investigate the distribution of dust in disks using thenumerical integration of Eqs. (14)–(16). We consider a steadystate distribution of dust particles. The equation of continuitygives the number density ns(q, s) in the q–s plane in the steadystate because evolution lines in the q–s plane with different s0

values do not intersect in this plane for any given set of e0

and q0. We consider two points (q, s) and (q + δq, s + δs),each of which is on two different adjacent evolution lines (seeFig. 6). The number of particles passing through a line connect-ing the two points (q, s) and (q + δq, s + δs) per unit time isgiven by N = ns(q, s)|qδs − sδq|. In the steady state, N is keptconstant along the “tube” defined by the two different evolutionlines. Calculating q and s from Eqs. (14)–(16) along the evolu-tion lines starting from initial sets of e0, q0, s0, and ns(q0, s0),we obtain the number density ns(q, s) in the q–s plane at anypoint reached by the evolution lines. Integrating the numberdensity ns(q, s) over q and s, we have the surface number den-

Dust ring formation due to ice sublimation 877

Fig. 7. The surface number density Ns (top panel) and the effective opticaldepth (bottom panel) for the icy dust as a function of the distance from thecentral star with the luminosity of L�. We set e0 = 1 × 10−4, q0 = 26 AU andthe power-law index p = 3 (short dashed lines), p = 3.5 (solid line), and p = 4(long dashed lines), where the size distribution ns(q0, s0) ∝ s

−p0 .

sity Ns(r)2 and the effective optical depth τeff(r) for given e0,

q0, s0, and ns(q0, s0),

(21)Ns(r) =∫ ∫

dq dsfd(r;q, e)

2πrns(q, s),

(22)τeff(r) =∫ ∫

dq ds πs2 fd(r;q, e)

2πrns(q, s),

where r is the distance from the central star. Here, fd(r;q, e)dr

is a probability that a dust grain having q and e is in the region[r, r +dr], where e(q, s) is calculated with the use of Eqs. (14)–(16) on the evolution lines for given q0 and e0. We assume auniform distribution of the true anomaly f . For small e, onemay expect such a distribution of the true anomalies in the re-gion [r, r + dr] in practice. A pile-up occurs only for smalle in Stage II, whereas it is negligible in Stage III where e islarge, and one might take into account the distribution of thetrue anomalies. For a uniform distribution of the true anomalyf in the region [r, r + dr], we have fd(r;q, e) = (1 − e)/2eq

2 The surface number density Ns is the number density integrated verticallyin a disk. The vertically averaged number density is given by the surface numberdensity divided by the thickness of the disk, which is proportional to r becausethe radiation pressure and the P-R drag do not change orbital planes of dustparticles.

Fig. 8. The number-density enhancement factor fNs and the optical-depth en-hancement factor fτeff as a function of initial orbital eccentricity e0 of the dustgrains for q0 = 26 AU.

at q < r < q(1 + e)/(1 − e) and fd(r;q, e) = 0 at r < q orr > q(1 + e)/(1 − e).

Fig. 7 shows the surface number density Ns/Ns0 and the ef-fective optical depth τeff/τeff0 in an icy dust disk, where Ns0and τeff0 are, respectively, Ns and τeff of inward drifting dust atr = q0. We set initial orbital elements of icy dust as q0 = 26 AUand e0 = 10−4 and the stellar parameters (L�, M�, and T�) asthose of the Sun. We assume a power-low size distribution fordust particles drifting inward by the P-R drag. In general, thesize distribution of dust particles in a dust production region isnot a simple power law but is modulated by a wavy pattern ofshort “wavelength” (Krivov et al., 2006). However, for dust par-ticles with which we are concerned, the P-R drift timescale isshorter than the timescale of collisions in the dust productionregion. Therefore, the size distribution of the dust particles isdetermined by that of collisional fragments modulated by theP-R drag. We express this size distribution as ns(q0, s0) ∝ s

−p

0with the initial size s0 ranging from smin to smax. We take thesmallest and largest sizes smin and smax to be the sizes corre-sponding to β = 0.5 and 0.1, respectively.

Fig. 7 shows that the distributions of both Ns and τeff havepeaks at r 22 AU. The smallest dust particles contribute mostto the peaks of Ns and τeff. Their temperature is 100–110 Kat the peaks. The Ns and τeff decrease sharply inside 22 AUbecause of their sublimation. The basic features of Ns and τeffdo not strongly depend on p so far as p > 3.

Fig. 8 shows the number-density enhancement factor fNs

and the optical-depth enhancement factor fτeff , as a function ofinitial eccentricity e0 of dust particles, where fNs = Ns/Ns0 andfτeff = τeff/τeff0 at the peak. The enhancement factors becomesmall with increasing e0 (e.g., Mukai and Yamamoto, 1979;Mann et al., 2000). This is explained as follows. The eccen-tricities of dust particles increase at the edge of the sublimationzone, where they pile up, and eventually they drift radially out-ward with a high speed for e � 0.1 (see Fig. 2). Dust particleswith small initial eccentricities stay at the sublimation zone fora long period, resulting in high enhancement factors.

The top panel in Fig. 9 shows the enhancement factors fNs ofthe surface number density Ns and fτ of the effective optical

eff

878 H. Kobayashi et al. / Icarus 195 (2008) 871–881

Fig. 9. The number-density enhancement factor fNs and optical-depth enhance-ment factor fτeff (top panel) and the sublimation distance asub (bottom panel)as a function of luminosity of the central star. Top panel: The circles and the tri-angles indicate fNs and fτeff , respectively, calculated numerically for the fourstars listed in Table 1. The solid lines indicate Eqs. (29) and (30) for the dirty-icedust. Bottom panel: The circles indicate the distance at the peaks of the surfacenumber density and the effective optical depth calculated numerically for thefour stars. The solid line indicates the distance of the dust having the size sminestimated from Eq. (23). The dotted lines in the panels show fNs , fτeff , andasub for the perfectly absorbing dust for references.

depth τeff for the icy dust as a function of stellar luminosity. Thebottom panel is the distance asub of the peaks of Ns and τeff. Thedotted lines indicate the enhancement factors fNs and fτeff (thetop panel) and the sublimation distance asub (the bottom panel)for the perfectly absorbing dust for a reference.

The position of the peaks of Ns and τeff is estimated fromthe condition that

(23)tPR = tsub

for s = ssub with ssub being the dust size that contributes most tothe enhancement of Ns and τeff at the peaks. The size ssub nearlyequals smin, for which β = 0.5, in practice. Here tPR = a/|aPR|is the P-R drift timescale and tsub = s/|s| is the sublimationtimescale with aPR being the P-R drift rate given by the secondterm on the r.h.s. of Eq. (14). The solid line on the bottom panelindicates the distance asub at the peak position determined fromEq. (23) by using the relation of L� ∝ M3.5

� (Allen, 1976). Notethat the distance asub coincides with the edge of the sublima-tion zone obtained from the numerical calculations for the starslisted in Table 1 (Fig. 7). The distance of the peaks of Ns and

Table 1Stellar parameters

L�/L� M�/M� T� [K]

Sun 1 1 5780β Pic 8.7a 1.75a 8200a

Vega 35b 2.3b 9988b

λ Casc 250d 4.0d 13,290d

a Crifo et al. (1997).b Peterson et al. (2006).c Binary star.d Chen et al. (2006).

τeff is approximated by asub 65(L�/100L�)1/2 AU for L� �30L�, whereas asub = 22(L�/L�)1/7 AU for L� � 30L�, as isseen from Fig. 9. For L� � 30L�, asub increases with Γ . If Γ

ranging from 0.1 to 0.5, asub = 22–35 AU for L� = L�. On theother hand, asub hardly changes with Γ for L� � 30L�. Thetemperature Tsub of the dust of the size ssub is 100–110 K atthe sublimation distance of r = asub for the luminosities and Γ

considered.The enhancement factors are approximately given by (Koba-

yashi et al., submitted for publication)

fNs p − 1

p − 1 + α(smin)

α(ssub)βPR(smin)βtotal(ssub)

[1 − βtotal(ssub)]βPR(ssub)

(24)× g(Tsub, ssub)h1

(smax

smin

)+ 1,

fτeff p − 3

p − 1 + α(smin)

(ssub

smin

)2α(ssub)βPR(smin)βtotal(ssub)

[1 − βtotal(ssub)]βPR(ssub)

(25)× g(Tsub, ssub)h2

(smax

smin

)+ 1

to the accuracy of O(e) unless p = 1, 1 − α(smin), and 3. Herethe function g(Tsub, ssub) depends on the temperature Tsub at thesublimation zone and the optical properties of dust particles.

(26)g(T , s) =2

4+qT

d lnPvd lnT

− 2

1 + α − αPR + αβtotal

1−βtotal− d lnα

d ln s− q�−qd

4+qT

d lnPvd lnT

with αPR = −d lnβPR/d ln s, and

h1

(smax

smin

)=

[1 −

(smax

smin

)−p+1−α(smin)]

(27)×[

1 −(

smax

smin

)−p+1]−1

,

h2

(smax

smin

)=

[1 −

(smax

smin

)−p+1−α(smin)]

(28)×[

1 −(

smax

smin

)−p+3]−1

.

Equations (24) and (25) hold for e0 � 3 × 10−4 that is shownin Fig. 8. The solid lines on the top panel in Fig. 9 indicateEqs. (24) and (25) for the icy dust, where we used L� ∝ M3.5

�

and T� ∝ L0.12� (Allen, 1976). The enhancement factors calcu-

lated from Eqs. (24) and (25) are in fairly good agreement with

Dust ring formation due to ice sublimation 879

those of numerical calculations shown in Fig. 7 for the starslisted in Table 1.

As seen in Fig. 9, the enhancement factors are large for highluminosity stars. That is explained as follow. The enhancementfactors are inversely proportional to the radial drift velocity ofthe dust particle of size ssub smin at the sublimation zone. Forlow luminosity stars, small dust survives against radiation pres-sure. Because the smallest size smin is much smaller than thepeak wavelength (∼30 µm) of thermal emission from dust at thesublimation zone, the dust temperature significantly increasesduring sublimation at a fixed distance in the sublimation zone.The increase of the dust temperature leads to acceleration ofthe radial drift velocity. The high radial drift velocity disturbsdust pile-ups, resulting in low enhancement factors. For highluminosity stars, only large dust survives against high radia-tion pressure. If the smallest dust smin is larger than the peakwavelength (∼30 µm) of thermal emission, the dust tempera-ture slightly changes during sublimation at a fixed distance inthe sublimation zone. In this case, the radial drift velocity iskept low, resulting in high enhancement factors.

Equations (24) and (25) give another interpretation of thebehavior of curves of enhancement factors versus stellar lumi-nosity. For L� � 100L�, ssub is smaller than 30 µm and qd =∂ ln Qd/∂ ln s is close to unity because of the Rayleigh scatter-ing regime (see Fig. 4), resulting in g(Tsub, ssub) 2/(qd − q�).Equations (24) and (25) reduce to

fNs p − 1

p − 1 + α(smin)

βtotal(ssub)

1 − βtotal(ssub)

(29)× 2α(ssub)

qd(Tsub, ssub) − q�(Tsub, ssub)+ 1,

fτeff p − 3

p − 1 + α(smin)

βtotal(ssub)

1 − βtotal(ssub)

(30)× 2α(ssub)

qd(Tsub, ssub) − q�(Tsub, ssub)+ 1

for p > 3, because smax � smin and ssub smin. The slope ofa dust emissivity versus size (qd = ∂ ln Qd/∂ ln s) decreaseswith size, because the size is smaller than the peak wave-length of the thermal emission. The slope of Q� versus s (q� =d ln Q�/d ln s) is much smaller than unity because the size islarger than the peak wavelength of incident stellar radiation (seeFig. 4). The terms of 1/(qd − q�) ∼ 1/qd increase with stellarluminosity in Eqs. (29) and (30), resulting in increases of theenhancement factors as seen in Fig. 9. For L� � 0.4L�, β doesnot exceed 0.5 and its maximum decreases with decreasing L�.Therefore, the enhancement factors become smaller for smallerluminosity, because they are proportional to βtotal/(1 − βtotal)

in Eqs. (29) and (30). For L� � 100L�, on the other hand, ssubbecomes large (ssub � 30 µm) since the blown-out size of thedust becomes large at large luminosities. This implies that theoptical properties of the dust are governed by the geometricaloptics, thus qd = q� 0. The enhancement factors are approxi-mated to be

(31)fNs p − 1

2p

[1

2

(d lnPv

d lnT

)− 2

]+ 1,

Tsub

(32)fτeff p − 3

2p

[1

2

(d lnPv

d lnT

)Tsub

− 2

]+ 1

for p > 3, because of α 1 and β(ssub) β(smin) = 0.5. Equa-tions (31) and (32) indicate the dotted lines on the top panel inFig. 9.

5. Discussion

The enhancement factors depend on the eccentricities ofdrifting dust particles, which are produced through collisionsbetween planetesimals. If the dust particles are formed througha collision of large parent bodies in circular orbits of radius rproin the dust production region, the semimajor axis ad and eccen-tricity ed of the produced dust are given by

(33)ad = 1 − βtotal

1 − 2βtotalrpro,

(34)ed = βtotal

1 − βtotal,

if the ejection velocities of the dust are much smaller than theorbital velocities of the parent bodies as is usually the case.Although the eccentricities ed are of the order of βtotal at theejection, their eccentricities may decrease by the P-R drag dur-ing their radial drifts. If dust particles with β = 0.49 producedby a collision at rpro = 50 AU, they have high eccentricityed 0.96 and large semimajor axis ad = 1275 AU. We esti-mate the eccentricities to be e ∼ 0.1 in the sublimation zoneat asub = 20 AU. The pile-ups hardly occurs for e ∼ 0.1 (seeFig. 8). However, if debris disks still have gas component, ec-centricities of dust particles drifting into the sublimation zonemay become smaller. Therefore, we would have uncertainty inthe distribution of eccentricities of dust particles. Taking intoaccount this uncertainty, our estimate gives possible maxima ofenhancement factors.

So far we have ignored the number density of refractoryparticles after ice sublimation, although the refractory particlesembedded in an ice matrix of icy dust do not disappear duringice sublimation. If βtotal of the refractory particles are smallerthan 0.5, they would still drift inward by the P-R drag after icesublimation. We denote the size of the refractory particles by srand their volume fraction by Γ . We estimate the effective opti-cal depth of the refractory particles revolving in nearly circularorbits (e 0) after ice sublimation. The number of refractoryparticles released from an icy dust particle by sublimation ofthe ice matrix is Γ s3/s3

r . The drift rate of the refractory par-ticles is approximately proportional to 1/sr, while that of theicy dust particle is approximately proportional to 1/s. Hence,the effective optical depth of the refractory particles in a steadystate is expressed by

(35)τr smax∫

smin

ds πs2r Γ

s3

s3rnsurf(r, s)

sr

s Γ τd,

where

(36)τd =smax∫

ds πs2nsurf(r, s)

smin

880 H. Kobayashi et al. / Icarus 195 (2008) 871–881

equals the effective optical depth τeff of the icy dust parti-cles in nearly circular orbits outside the sublimation zone andnsurf(r, s)ds is the surface number density of icy particles inthe size range [s, s + ds]. Interestingly, τr does not dependon sr but Γ . The optical depth τr decreases by a factor of Γ

after ice sublimation. Actually, the τr value is smaller if wetake the blow-out of the refractory particles into account. Ifβtotal(sr) > 0.5 for all refractory particles, the disk is empty in-side the sublimation region (τr = 0). Regardless of βtotal values,the effective optical depth τr of the refractory dust is smallerthan that of icy dust. As a result the ice sublimation forms aninner hole inside the sublimation zone.

Dust-debris disks around main sequence stars showing an in-ner edge of the disk in their observed images are so far not in aP-R drag dominated regime. An inner hole in a drag-dominateddisk cannot be imaged directly with currently available observa-tional facilities because of its low optical depth, but an infraredexcess in the spectral energy distribution (SED) would provideevidence for an inner hole. Some of the dust temperatures de-rived from the fitting of the SEDs of low optical-depth diskswith τeff ∼ 10−6–10−3 from Chen et al. (2006) are consistentwith the temperature for subliming icy dust determined sharplyas 100–110 K from our calculations. These observed disks mayhave inner holes that are caused by ice sublimation.

Holes with the temperature different from 100–110 K wouldnot be caused by ice sublimation, because ice sublimation de-termines the temperature of the outer edge of the inner holessharply. If the observationally derived temperature is far fromthe temperature of subliming dust in a drag-dominated disk,the presence of a hole would indicate the existence of plan-ets, which can form a hole through gravitational scattering andresonance captures (e.g., Moro-Martín et al., 2005). When thetemperature at the outer edge of the hole is lower than the tem-perature of ice sublimation, planets eliminating dust would belocated outside the ice sublimation zone and the dust productionregion would be farther out. When the temperatures are higherthan 110 K, on the other hand, planets and the dust productionregion should exist inside the ice sublimation zone. Dust in thisregion consists mainly of particles more refractory than ice suchas silicate and organic materials. Consequently, the determina-tion of dust temperature at the outer edge of an inner hole willgive clues to the formation mechanism of the hole.

High resolution imaging of the morphology of dust-debrisdisks with low optical depth τeff � 10−5 will clarify the ori-gin of inner holes. If planets form a hole in a disk, the holeis not axisymmetric and the disk has a clumpy structure be-cause of dust captures in the mean motion resonances of theplanets (e.g., Quillen and Thorndike, 2002). The inner edge ofthe dust production region would be axisymmetric if the largeplanet does not exist close to the dust production region. If theremoval of dust by radiation pressure following highly frequentcollisions between dust particles in a disk is the cause, an in-ner hole will form with its outer edge corresponding to theinner edge of the dust production region (Wyatt et al., 1999;Dominik and Decin, 2003; Wyatt, 2005). If an inner hole iscaused by ice sublimation, the hole is axisymmetric and rings(bumps) is formed along the edge of the hole through pile-up of

dust at temperatures around 100–110 K. The progress of obser-vations of dust-debris disks would clarify the origin of holes ofdebris disks.

6. Summary

1. Icy dust radially drifting by the P-R effect in dust-debrisdisks around Vega-like stars substantially sublimes at tem-perature of 100–110 K, where the drift timescale due tothe P-R drag is comparable to the size decreasing timescaledue to sublimation. The sublimation distance from thecentral star is located at 20–35 AU for L� � 30L� and65(L�/100L�)1/2 AU for L� � 30L� (see Fig. 9), consid-ering icy dust that consists of ice and organics. Namely, thesublimation distance varies with the luminosity of the cen-tral star, whereas the dust temperature at the sublimationzone is irrelevant to the luminosity.

2. Icy dust piles up at the sublimation zone, resulting in an en-hancement of the dust density. The enhancement factor ofthe number density fNs is 3 and that of the effective opticaldepth fτeff is 2 for the solar luminosity. The enhancementfactors increase with the luminosity of the central star. Forthe luminosity larger than 100L�, fNs and fτeff are largerthan 20 and 10, respectively.

3. Ice sublimation forms an inner hole at the sublimationzone. After ice sublimation, a disk composed of refractoryparticles is sustained by their further inward drift due to theP-R drag. However, the refractory-dust disk is fainter thanthe icy disk, because of the small volume fraction of the re-fractory particles in icy dust and possible blown-out of therefractory particles.

Acknowledgments

We are grateful to Ingrid Mann and Alexander V. Krivov forinvaluable comments and for a careful reading of this manu-script. We also thank an anonymous reviewer for his/her con-structive suggestions. This research was supported by MEXTJapan, Grant-in-Aid for Scientific Research on Priority Areas,“Development of Extra-solar Planetary Science” (16077203)and “Astrophysical Observations of New Phases of the Inter-stellar Gas at Sub-mm and THz Regions” (18026001), and byJSPS, Grant-in-Aid for Young Scientists (B) (18740103) andGrant-in-Aid (C) (19540239).

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