Nonsticky Ice at the Origin of the Uniformly Polarized Submillimeter Emission from the HL Tau Disk
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Nonsticky Ice at the Origin of the Uniformly Polarized Submillimeter Emission from the HL Tau Disk S atoshi O kuzumi and R yo T azaki Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan Astronomical Institute, Tohoku University, Sendai 980-8578, Japan
ABSTRACTRecent (sub)millimeter polarimetric observations toward the young star HL Tau have successfully detectedpolarization emission from its circumstellar disk. The polarization pattern observed at 0.87 mm is uniform andparallel to the disk’s minor axis, consistent with the self-scattering of thermal emission by dust particles whosemaximum radius is ≈ µ m. However, this maximum size is considerably smaller than anticipated from dustevolution models that assume a high sticking e ffi ciency for icy particles. Here, we propose that the unexpectedlysmall particle size can be explained if CO ice covers the particles in the outer region of the HL Tau disk. CO ice is one of the most major interstellar ices, and laboratory experiments show that it is poorly sticky. Basedon dust evolution models accounting for CO ice mantles as well as aggregate sintering and post-processingradiative transfer, we simulate the polarimetric observation of HL Tau at 0.87 mm. We find that the models withCO ice mantles better match the observation. These models also predict that only particles lying between theH O and CO snow lines can grow to millimeter to centimeter sizes, and that their rapid inward drift results ina local dust gap similar to the 10 au gap of the HL Tau disk. We also suggest that the millimeter spectral indexfor the outer part of the HL Tau disk is largely controlled by the optical thickness of this region and does notnecessarily indicate dust growth to millimeter sizes. Keywords: dust, extinction — planets and satellites: formation — polarization — protoplanetary disks — stars:individual (HL Tau) — submillimeter: planetary systems INTRODUCTIONDust particles in protoplanetary disks are the ultimatebuilding blocks of planets and also the primary opacitysources in the disks. Understanding how dust grows in thedisks is essential to understanding how planet formation be-gins.The disk around the young star HL Tau is an interestingobject for studying how dust particles in young disks growand evolve. The HL Tau disk is well known for its spectac-ular multiple dust rings revealed by ALMA (ALMA Partner-ship et al. 2015), but is also known for its detectable polar-ized emission at millimeter and submillimeter wavelengths(Stephens et al. 2014; Kataoka et al. 2017; Stephens et al.2017). The morphology of the (sub)millimeter polarizationpatterns of this system is particularly interesting becauseit is highly wavelength dependent. The polarization mapat 0.87 mm shows a uniformly polarized pattern in whichthe polarization vectors are parallel to the disk’s minor axis
Corresponding author: Satoshi [email protected] (Stephens et al. 2017). In contrast, the map at 3.1 mm showsa completely di ff erent polarization pattern with azimuthallyaligned polarization vectors (Kataoka et al. 2017). The po-larization pattern at 1.3 mm is a mix of the unidirectionaland azimuthal patterns (Stephens et al. 2017), indicating thatthe uniformly polarized emission component diminishes asthe wavelength increases from 0.87 mm. There is growingevidence that many other disks also produce uniformly po-larized emission at (sub)millimeter wavelengths (Lee et al.2018; Cox et al. 2018; Sadavoy et al. 2018; Hull et al. 2018;Harris et al. 2018; Bacciotti et al. 2018; Dent et al. 2019;Takahashi et al. 2019).A likely mechanism for producing such uniformly polar-ized emission is self-scattering of thermal radiation by dustparticles (Kataoka et al. 2015; Yang et al. 2016, see also Sen-gupta & Krishan 2001; Marley & Sengupta 2011 for a similarpolarization mechanism for brown dwarfs and self-luminousplanets). This mechanism well explains the wavelength-dependent nature of the uniform polarization for the HL Taudisk because dust particles can only e ffi ciently produce po-larized scattered light at a wavelength similar to their ownsize (Kataoka et al. 2015). Kataoka et al. (2016, 2017) andStephens et al. (2017) conclude that the uniformly polarized a r X i v : . [ a s t r o - ph . E P ] M a y O kuzumi et al .emission seen in the 0.87 and 1.3 mm images of the HLTau disk are most likely produced by dust particles of radii ∼ µ m. Kataoka et al. (2016, 2017) also conclude thatmillimeter- to centimeter-sized particles should be much lessabundant than 100 µ m-sized particles, at least in the regionwhere the polarized emission is observed, because such largeparticles would produce unpolarized thermal emission withlittle polarized scattered light at these wavelengths (Kataokaet al. 2016, 2017). The uniform (sub)millimeter polarizationseen in other disks also gives a similar constraint on the max-imum particle size (see the references listed in the previousparagraph).The above interpretation for the uniform (sub)millimeterpolarization raises the question why the 100 µ m-sized par-ticles are so prevalent. Theoretically, it has long been be-lieved that icy grains in the cold outer part of the disks arehighly sticky thanks to strong hydrogen bonding betweenH O molecules (e.g., Chokshi et al. 1993; Dominik & Tie-lens 1997; Wada et al. 2009, 2013). Dust growth models as-suming a high sticking e ffi ciency for icy particles generallypredict that the particles should grow to 1 mm or even larger(e.g., Brauer et al. 2008; Birnstiel et al. 2010; Okuzumi et al.2012; Dra¸ ˙zkowska & Dullemond 2014; Banzatti et al. 2015),thus unable to explain the observed (sub)millimeter polariza-tion of the HL Tau disk with the dust self-scattering mecha-nism.To summarize, the uniformly polarized (sub)millimeteremission observed toward the HL Tau disk and many otherprotoplanetary disks imply that the growth of icy particles inthe outer regions of the disks may not be as e ffi cient as previ-ously anticipated. It could merely indicate that the previousdust growth models underestimated the stickiness of waterice. In fact, some recent experiments (Gundlach et al. 2018;Musiolik & Wurm 2019) question a high adhesion energy forwater ice at temperatures below 150–200 K. However, thesenew experiments are apparently inconsistent with earlier ex-periments (Gundlach & Blum 2015) that confirmed e ffi cientsticking of H O-ice grains at temperatures down to 100 K.Alternatively, it is possible that 100- µ m-sized aggregatesmade of µ m-sized grains are considerably less sticky thanthe grains themselves. For instance, experiments show thatmacroscopic aggregates made of silica grains do not stick butbounce o ff at moderate collision velocities (e.g. G¨uttler et al.2010). In principle, aggregates of water ice grains could alsoexperience bouncing at similar velocities if they are compact(Wada et al. 2011). If this is the case, bouncing could limitthe growth of icy particles more severely than fragmentation(e.g., Zsom et al. 2010, 2011).In this study, we explore another possibility that the growthof of 100 µ m-sized particles in the outer part of the disks issuppressed by a nosticky solid material—CO ice. Modelsof interstellar and circumstellar ices suggest that dust grains in dense and cold environments are covered by “apolar” icesof CO and CO (e.g., Boogert et al. 2015). Recent laboratoryexperiments by Musiolik et al. (2016a,b) show that CO iceparticles are considerably less sticky than H O particles ofthe same size, presumably because of the absence of hydro-gen bonding. Musiolik et al. (2016a) and Pinilla et al. (2017)point out that the CO ice mantles can suppress dust growthin the outer part of protoplanetary disks. Here, we incorpo-rate the low stickiness of CO ice mantles into our previousdust evolution model for the HL Tau disk (Okuzumi et al.2016; henceforth O16), and demonstrate that this e ff ect canindeed cause the uniform submillimeter polarimetric patternseen in the HL Tau disk.The structure of this paper is as follows. Section 2 de-scribes our dust evolution model and radiative transfer ap-proach used to synthesize disk polarimetric images. Sec-tion 3 presents main results from the dust evolution calcula-tions and compares them with the submillimeter polarimetricobservation of the HL Tau disk. Section 4 presents some im-plications for the ring–gap substructure of the HL Tau diskand also discusses model and parameter dependences. Sec-tion 5 presents a summary. MODELTo produce synthetic polarimetric images of the HL Taudisk, we perform global dust evolution simulations and post-processing radiative transfer calculations. In this section, wedescribe the assumptions made in the calculations.2.1.
Gas Disk Model
Following O16, we use simple prescriptions for the gassurface density and temperature profiles of the HL Tau disk.The gas surface density is given by Σ g = (2 − γ ) M disk π r c (cid:32) rr c (cid:33) − γ exp − (cid:32) rr c (cid:33) − γ , (1)where r is the distance from the central star, r c and M disk arethe characteristic radius and total mass of the gas disk, re-spectively, and γ is a dimensionless number characterizingthe radial slope of Σ g . Equation (1) is motivated by the sim-ilarity solution for evolving viscous accretion disks (Lynden-Bell & Pringle 1974; Hartmann et al. 1998), although we donot follow the evolution of the gas disk for simplicity. Thevalues of r c and M disk are taken to be 150 au and 0 . M (cid:12) , re-spectively.The actual profile of Σ g in the HL Tau disk may not besmooth. Yen et al. (2016) suggest that the distribution ofHCO + in the HL Tau disk has radial gaps, indicating that theprofile of Σ g (which is dominated by H ) might also havegaps. Hu et al. (2019) show that the combined e ff ects of non-ideal magnetohydrodynamics and dust-dependent ionizationchemistry can produce gas gaps at dust gaps. Such compli-cations are not considered in this study.The value of γ is taken to be either 1 or 0. The modelwith γ = α parameter (Hart-mann et al. 1998). The model with γ = r (cid:28) r c . Such a flat profile can besee in some recent simulations of dust accretion includingmagnetically driven disk winds (Suzuki et al. 2016). In fact,HL Tau is accompanied by strong jets and winds (Klaassenet al. 2016), and the winds may significantly contribute tothe evolution of the HL Tau disk (Hasegawa et al. 2017). Wenote, however, that the global picture of wind-driven accre-tion is still uncertain and likely depends on the assumed ra-dial distribution of magnetic field strength (Bai 2016). In anycase, the gas distribution of the HL Tau disk is essentiallyunknown, and therefore it is important to assume di ff erentsurface density profiles.The temperature profile is given by T = (cid:32) r (cid:33) − . = (cid:32) r
60 au (cid:33) − . K . (2)O16 derived this profile based on the assumption that thebright central part and rings of the HL Tau disk are opti-cally thick at the wavelength of 0.87 mm. Similar midplanetemperature profiles were also derived by previous radiativetransfer models for the HL Tau disk (Men’shchikov et al.1999; Kwon et al. 2015).The disk is assumed to be vertically isothermal, and thevertical distribution of the gas is given by a Gaussian with thescale height H g = c s / Ω , where c s and Ω are the sound speedand local Keplerian frequency, respectively. The Keplerianfrequency depends on the stellar mass, which we take to be1 . M (cid:12) following recent mass estimates (Pinte et al. 2016;Yen et al. 2017).Figure 1 shows the radial distribution of Σ g for γ = Q ≡ c s Ω / ( π G Σ g ) is greater than unity atall r . 2.2. Dust Model
We use a modified version of the dust evolution modelO16, which was originally developed to explain the multipledust ring structure of the HL Tau disk. The model tracks thegrowth, fragmentation, and radial inward drift of dust aggre-gates at di ff erent distances from the central star. The single-size approach (Sato et al. 2016) is adopted, in which aggre-gates at each radial distance are characterized by a single ra-dius a ∗ . E ff ectively, this approach tracks the evolution of thelargest, mass-dominating aggregates, and the characteristicsize a ∗ corresponds to the size of the largest aggregates. Ag-gregates smaller than the largest are neglected in the calcu- γ = γ = (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) (cid:3) (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) (cid:4) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13)(cid:14) (cid:15)(cid:6)(cid:7)(cid:12) (cid:1) [ (cid:7)(cid:16) ] (cid:1) (cid:2)(cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:2)(cid:8)(cid:9) (cid:10) (cid:9) (cid:11) (cid:3) (cid:12) (cid:13) (cid:14) Σ (cid:1) [ (cid:15) (cid:8) (cid:16) - (cid:1) ] Q = Q = Figure 1.
Gas surface density profiles for γ = Q ≡ c s Ω / ( π G Σ g ) = Q (cid:38) lations of dust evolution, but are taken into account in radia-tive transfer calculations (see Section 2.3). Each aggregateis composed of icy dust grains whose composition changeswith temperature and hence with radial distance. The graincomposition controls the stickiness of aggregates, thus cen-tral to this study. Aggregate sintering (Sirono 2011; O16) isalso included to reproduce multiple dust rings as seen in thehigh-resolution image of the HL Tau disk (ALMA Partner-ship et al. 2015).In the following subsections, we describe the main aspectsof our dust model. We refer interested readers to Sections 3and 4 of O16 for a more detailed description of the model.2.2.1. Grain and Aggregate Structure
Dust particles are assumed to be initially in the form ofmicron-sized spherical grains. Each grain consists of a sili-cate core and ice mantles. The ice mantles are mainly com-posed of H O, CO , and CO with a molar ratio of 1:0.1:0.1.We also consider less abundant CH and C H ice but only inmodeling aggregate sintering (see below). The initial solid-to-gas mass ratio of the disk is set to 0.01, and the positionsof the snow lines for all ices are computed by comparing theirequilibrium vapor pressures and partial pressures in the ini-tial state. The equilibrium vapor pressures depend on thesublimation energies, which are typically uncertain to 10%.Following O16, we take the sublimation energies of H O andC H to be 10% lower than the fiducial values to tune the lo-cations of their snow lines within the uncertainty of the sub-limation energies (see Section 3.2 of O16 for details). We donot track the evolution of the snow lines for simplicity.The core–mantle grains stick together to form aggregates.To distinguish between initial core–mantle grains and theiraggregates, we refer to the former as monomers as wellas grains. The internal density ρ int of the aggregates is O kuzumi et al .taken to be 0 .
64 g cm − assuming a mean material density of1 .
28 g cm − and an aggregate porosity of 50%. In protoplan-etary disks, aggregates could have much lower porosities, inparticular when they grow through mutual collisions with-out fragmentation (Suyama et al. 2008; Okuzumi et al. 2012;Kataoka et al. 2013). Some issues with our assumption onthe aggregate porosity are discussed in Section 4.3.The stickiness of grains and their aggregates depends onthe materials coating the grain surfaces (Dominik & Tielens1997). The structure of the ice mantles is therefore of partic-ular importance in our model. According to the experimentsby Musiolik et al. (2016b), pure CO ice is poorly sticky,whereas a homogeneous mixture of H O and CO ices witha ratio of 1:0.1 would be almost as sticky as pure H O ice(see their Figure 6). The original model of O16 did not ac-count for the low stickiness of CO ice, and therefore e ff ec-tively assumed a homogeneous H O–CO ice mantle. In thisstudy, we also consider a two-layer mantle model in whichthe silicate cores are covered with an H O-dominated lowermantle and a CO -dominated upper mantle, similar to mod-els for interstellar ice grains (see, e.g., Figure 10 of Boogertet al. 2015). We refer to the former and latter models as themodels with and without CO ice mantles, respectively. TheH O and CO ice mantles are assumed to sublimate instanta-neously at the corresponding snow lines.2.2.2. Fragmentation Threshold
The grain aggregates are assumed to fragment rather thanstick if they collide at velocities greater than a threshold v frag .Bouncing collisions, often observed in laboratory collisionexperiments for rocky aggregates (e.g., G¨uttler et al. 2010),are not considered in this study to focus on the role of CO mantles. Our future modeling will take in account bouncingcollisions.The value of v frag is primarily determined by the materialthat covers the grains. For aggregates of H O-mantled grains,we follow O16 in adopting the following relation between v frag and a mon , v frag = (cid:32) a mon . µ m (cid:33) − / m s − = . (cid:32) a mon . µ m (cid:33) − / m s − . (3)This scaling is based on collision simulations for equal-sizedH O ice aggregates (Wada et al. 2009). Equation (3) is con-sistent with the experiments by Gundlach & Blum (2015),which suggest v frag ∼
10 m s − for aggregates of 1 . µ m-sized H O-ice grains at T ≈ ∼ -ice aggregates, so we adopt v frag = (cid:32) a mon . µ m (cid:33) − / m s − = . (cid:32) a mon . µ m (cid:33) − / m s − . (4)Experiments by Poppe et al. (2000) show a capture velocityof ∼ − for silica grains of 0 . µ m radius (1 . µ m di-ameter), consistent with Equation (4) ( v frag ≈ . − for a mon = . µ m). For aggregates of CO -mantled grains, ex-periments by Musiolik et al. (2016a,b) suggest that the frag-mentation threshold is close to that for silicate grain aggre-gates, and therefore we use Equation (4).Our model also accounts for a decrease in v frag due tosintering (Sirono 1999; Sirono & Ueno 2017). We assumethat volatiles included in monomer grains are able to desorbfrom or di ff use over the grain surface when the temperatureis close to their sublimation points. Sintering refers to thephenomenon in which the mobile volatile molecules recon-dense around the contact points of the monomers and therebyfuse them together (see, e.g., Poppe 2003; Blackford 2007).This phenomenon makes the aggregates harder but less liableto stick upon high-speed collisions (Sirono 1999; Sirono &Ueno 2017). Sintering only requires a small amount of ma-terials (see Figure 2 of Sirono & Ueno 2017) and can in prin-ciple occur near the snow lines of various volatiles (Sirono2011). As shown by O16, the local reduction of the frag-mentation threshold in the sintering zones can lead to theformation of the multiple dust rings observed in the HL Taudisk. Inclusion of this e ff ect thus allows us to model the sub-millimeter polarization of this disk consistently with its dustring structure. We emphasize, however, that the goal of thisstudy is to explain the submillimeter polarization pattern, notto reproduce the multiple ring structure in full detail. Wealso note that sintering may not be the common origin of thedust rings observed in many protoplanetary disks (Long et al.2018; Huang et al. 2018; van der Marel et al. 2019). In anycase, our CO mantle model for submillimeter polarizationis in principle compatible with other ring formation mecha-nisms, such as planet–disk interaction (e.g., Dong et al. 2015;Dipierro et al. 2015), gas–dust instabilities (e.g., Takahashi &Inutsuka 2014), and disk dynamics (e.g., Flock et al. 2015).Our treatment of sintering is similar to that by O16, butsome simplifications are applied given its relatively minorimportance in this study. We consider sintering by H O, CO ,C H , CH , and CO. Unlike O16, we neglect NH and H Sbecause their sintering zones are close to the snow line ofof CO (in particular, the NH sintering zone overlaps withthe CO snow line). Neglecting these species allows us tohighlight the e ff ects of CO sublimation on dust growth nearthe CO snow line. The sintering zones are defined by thelocations where the timescale of sintering is shorter than themean collision interval. In these zones, we use the analyticprescription presented in Section 4.4 of O16 to reduce the (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) - (cid:1) (cid:1)(cid:2) (cid:3) (cid:1)(cid:2) (cid:1) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13)(cid:14) (cid:15)(cid:6)(cid:7)(cid:12) (cid:1) [ (cid:7)(cid:16) ] (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:8) (cid:3) (cid:8) (cid:9) (cid:10)(cid:7)(cid:11)(cid:12) (cid:2) (cid:6)(cid:13) (cid:12)(cid:10) (cid:14) (cid:15) (cid:1) (cid:1) (cid:2) (cid:3) (cid:4) [ (cid:5) (cid:13) - (cid:5) ] γ = (cid:2)(cid:17) (cid:2) (cid:4)(cid:5)(cid:6) = (cid:1)(cid:18)(cid:19) μ (cid:14) H O C O C H CH C O w / o CO mantlew / CO mantle Figure 2.
Fragmentation threshold v frag as a function of orbital ra-dius r for models with and without CO ice mantles (black and graycurves, respectively). The vertical dotted lines indicate the locationsof the snow lines of five volatile species, with the shaded regionsmarking the sintering zones (O16). The plot assumes γ = a mon = . µ m, although the results are very insensitive to γ . value of v frag by up to 60% (Sirono & Ueno 2017). Note thatsintering is assumed to further reduce the sticking e ffi ciencyof CO -mantled grain aggregates . For simplicity, the meancollision interval is approximated by 100 Ω − (Takeuchi &Lin 2005; Brauer et al. 2008). This approximation hardlya ff ects the locations of the sintering zones because the sinter-ing timescale is a very strong function of T (see Figure 4 ofO16).The fragmentation thresholds as well as the widths of thesintering zones depend on the uncertain monomer size a mon .Our sintering model only requires a mon (cid:46) µ m for the sin-tering zones to have a non-zero width (O16). Near-infraredpolarimetry of HL Tau suggests that the maximum grain sizesin the surrounding envelope is more or less 1 µ m (Lucas et al.2004; Murakawa et al. 2008), so it may be reasonable to as-sume a mon ∼ µ m. In this study, we adopt a mon = . µ m forsimulations, but also present an analytic estimate of how thesimulation results depend on v frag .In Figure 2, we plot the fragmentation threshold v frag for a mon = . µ m as a function of r for two models with andwithout CO ice mantles. In the former model, the frag-mentation threshold outside the CO snow line falls in therange 0.2–0 . − . Figure 2 also shows the locations of the This assumption is based on the finding by Sirono & Ueno (2017) thatthe sticking threshold for sintered aggregates is determined by the strengthof non-sintered contacts, and hence by the material that dominates themonomer surface (CO ice in this context). When two sintered aggregatescollide, they temporary stick by forming new, non-sintered contact points.However, the aggregates immediately separate if the tensile forces actingbetween the aggregates are large enough to break the non-sintered contacts.The role of sintering here is to hinder the monomers from dissipating kineticenergy through rolling friction. snow lines and sintering zones. This locations and widths ofthe sintering zones are almost the same as those in our previ-ous model La0-tuned (see Figure 18 of O16), which adoptedthe same sublimation energies and a similar monomer size.Because we neglect NH and H S, our sintering zone at r ∼
15 au is narrower than in the previous model by a factorof ∼ Dust Evolution
We calculate the time evolution of a ∗ and the dust surfacedensity Σ d as a function of r by solving a set of di ff erentialequations with advection and collision terms (Equations (7)and (8) of O16). The radial advection is due to the drift mo-tion of dust in the sub-Keplerian rotating gas disk (Whipple1972; Adachi et al. 1976; Weidenschilling 1977). Turbulentdi ff usion of dust in the radial direction is neglected assumingthat turbulence in the HL Tau disk is weak. This assumptionmay be justified by the observational evidence for a signifi-cant level of dust settling in this disk (Pinte et al. 2016). Thecomputational domain is taken to be 1 au (cid:54) r (cid:54) Ω . We use the expres-sion (e.g., Birnstiel et al. 2010)St = πρ int a ∗ Σ g (5)that applies to aggregates smaller than the gas mean free path.In our simulations, all aggregates fulfill this criterion.The collision velocity ∆ v of aggregates is induced by theirBrownian motion, radial and azimuthal drift (Whipple 1972;Adachi et al. 1976; Weidenschilling 1977), and gas turbu-lence. We assume that collisions with similar-sized aggre-gates determine the evolution of the largest aggregates, andtake the ratio of the Stokes numbers of two colliding aggre-gates to be 0.5 (Sato et al. 2016). Analytic expressions byOrmel & Cuzzi (2007) are used to calculate the turbulence-induced collision velocity ∆ v t .The dust scale height H d is determined by the balance be-tween dust settling and turbulent di ff usion (Dubrulle et al.1995; Youdin & Lithwick 2007). As long as St (cid:28) H d canbe written as H d = (cid:32) + St α Dz (cid:33) − / H g , (6)where α Dz is the vertical di ff usion coe ffi cient for dust nor-malized by c s / Ω .The strength of turbulence is parametrized by a dimension-less number α turb , which is defined so that the velocity dis-persion of the turbulent gas is given by v g , turb = √ α turb c s .The vertical di ff usion coe ffi cient for dust is assumed to be α Dz = . α turb based on the results of MHD simulations fordisk turbulence (Okuzumi & Hirose 2011). We assume a low O kuzumi et al .level of turbulence with α turb = × − to allow dust settling(see Section 3.1). In our models, the turbulence parameter α turb only a ff ects the collision velocity and vertical di ff usionof dust; the gas surface density and temperature are givenindependently of α turb by Equations (1) and (2).2.3. Radiative Transfer
The simulation results are converted into polarimetric im-ages using the 3D Monte Carlo radiative transfer code radmc -3 d (Dullemond et al. 2012). The dust opacity is calcu-lated using the method detailed in Section 4.5 of O16. Inshort, we assume that aggregates at each r obeys a power-lawsize distribution whose total surface density and maximumcut-o ff size are given by Σ d ( r ) and a ∗ ( r ), respectively. Theoptical properties of the aggregates are computed using theMie theory combined with the Maxwell–Garnett mixing rule(Bohren & Hu ff man 1983). We use the same e ff ective refrac-tive index for the ice–dust mixture as in O16. For non-fractalaggregates considered in this study, the absorption and scat-tering opacities obtained from the e ff ective medium approachagree with those from the more rigorous T-matrix method toan accuracy of (cid:46)
40% unless the aggregates are much largerthan the wavelength (see Section 4.1.2 of Tazaki & Tanaka2018). The inclination angle of the HL Tau disk is assumedto be 46 ◦ . photon packets.We calculate the thermal emission and scattering by dustparticles at the wavelength of 0.87 mm under the prescribedtemperature profile given by Equation (2). Because the actualdisk temperature profile should depend on the dust distribu-tion in the disk, our current modeling is not self-consistent.Our calculations do not treat radiative transfer of starlight atvisible wavelengths, and therefore cannot be used to derivethe disk temperature profile. In the particular case of the HLTau system, modeling the temperature structure with radia-tive transfer calculations is di ffi cult, if not impossible, be-cause the reflected light from the surrounding envelope cancontribute to disk heating (Men’shchikov et al. 1999; Kikuchiet al. 2002). RESULTSWe carried out four simulation runs with and without CO ice mantles, and with γ = t ∼ t ∼ . ice mantles, respectively, where t is the time af-ter the start of dust evolution. The CO mantles slow down dust depletion because they induce collisional fragmentation:smaller aggregates drift more slowly (Birnstiel et al. 2009).In the following, we select the snapshots at t = . . ice mantles, respectively.3.1. Radial Profiles of a ∗ , Σ d , and H d The top row of Figure 3 shows the radial distribution of theaggregate size a ∗ in the quasi-steady state from all simulationruns. We find that the models with CO ice mantles success-fully produce a ∗ ∼ µ m outside the CO snow line. Theaggregate size is particularly close to 100 µ m in the sinteringzones (see Figure 2) owing to the combined e ff ects of CO mantles and sintering. Without CO ice mantles, the ag-gregates grow beyond 1 mm except interior to the H O snowline and at the outer edge of the disk.The results presented here can be understood with simpleanalytic arguments (Birnstiel et al. 2009, 2012; O16). In thesimulations with CO ice mantles, the maximum aggregatesize is predominantly determined by collisional fragmenta-tion. For intermediate-sized aggregates, the collision velocityis mainly induced by turbulence, and can be approximatelywritten as ∆ v ≈ √ . α turb St (Ormel & Cuzzi 2007). Sincefragmentation dominates over sticking for ∆ v (cid:62) v frag , aggre-gates can only grow to St ≈ St frag , whereSt frag = v . α turb c s ≈ × − (cid:32) v frag . − (cid:33) (cid:32) α turb × − (cid:33) − (cid:32) T
30 K (cid:33) − . (7)Using Equation (5), the above expression can be translatedinto the maximum aggregate size in the fragmentation-limitgrowth, a frag ≈ v Σ g α turb c s ρ int ≈ (cid:32) v frag . − (cid:33) (cid:32) α turb × − (cid:33) − (cid:32) ρ int . − (cid:33) − × (cid:32) Σ g
10 g cm − (cid:33)(cid:32) T
30 K (cid:33) − µ m . (8)Expressions similar to Equation (7) and (8) can also be foundin the literature (e.g., Birnstiel et al. 2009, 2012; Pinilla et al.2017). Equation (8) implies that the ten-fold decrease in v frag due to the presence of CO ice mantles leads to a 100-folddecrease in a frag outside the CO snow line. In the first and Here, the factor √ . √ γ = / o CO mantle γ = mantle (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) - (cid:2) (cid:1)(cid:2) - (cid:1) (cid:1)(cid:2) (cid:3) (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) (cid:4) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13)(cid:14) (cid:15)(cid:6)(cid:7)(cid:12) (cid:1) [ (cid:7)(cid:16) ] (cid:1) (cid:2)(cid:2) (cid:3) (cid:4) (cid:2) (cid:5) (cid:6) (cid:4) (cid:7) (cid:5) (cid:8) (cid:9) (cid:10) (cid:11) (cid:1) * [ (cid:12)(cid:12) ] (cid:1) (cid:1) (cid:2) (cid:3) (cid:2) (cid:1) (cid:1) (cid:1)(cid:2)(cid:3)(cid:4) (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) - (cid:5) (cid:1)(cid:2) - (cid:4) (cid:1)(cid:2) - (cid:2) (cid:1)(cid:2) - (cid:1) (cid:1)(cid:2) (cid:3) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13)(cid:14) (cid:15)(cid:6)(cid:7)(cid:12) (cid:1) [ (cid:7)(cid:16) ] (cid:13) (cid:6) (cid:14) (cid:15)(cid:4)(cid:11) (cid:16) (cid:10) (cid:12) (cid:17) (cid:4) (cid:3) (cid:13) (cid:6) St frag (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) - (cid:2) (cid:1)(cid:2) - (cid:1) (cid:1)(cid:2) (cid:3) (cid:1)(cid:2) (cid:1) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13)(cid:14) (cid:15)(cid:6)(cid:7)(cid:12) (cid:1) [ (cid:7)(cid:16) ] (cid:18) (cid:10) (cid:11) (cid:6) (cid:13) (cid:10) (cid:3) (cid:19) (cid:5)(cid:20)(cid:4) (cid:18) (cid:4) (cid:21) (cid:11) (cid:9) (cid:6) (cid:22) Σ (cid:1) [ (cid:2) (cid:20) (cid:12) - (cid:1) ] . Σ g (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) - (cid:2) (cid:1)(cid:2) - (cid:1) (cid:1)(cid:2) (cid:3) (cid:1)(cid:2) (cid:1) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13)(cid:14) (cid:15)(cid:6)(cid:7)(cid:12) (cid:12) [ (cid:7)(cid:16) ] (cid:18) (cid:10) (cid:11) (cid:6) (cid:13) (cid:20)(cid:5) (cid:23) (cid:4) (cid:24) (cid:4) (cid:9) (cid:2)(cid:25) (cid:6) (cid:2) (cid:1) [ (cid:5) (cid:10) ] H d = H g γ = / o CO mantle γ = mantle (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) - (cid:2) (cid:1)(cid:2) - (cid:1) (cid:1)(cid:2) (cid:3) (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) (cid:4) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13)(cid:14) (cid:15)(cid:6)(cid:7)(cid:12) (cid:1) [ (cid:7)(cid:16) ] (cid:1) (cid:2)(cid:2) (cid:3) (cid:4) (cid:2) (cid:5) (cid:6) (cid:4) (cid:7) (cid:5) (cid:8) (cid:9) (cid:10) (cid:11) (cid:1) * [ (cid:12)(cid:12) ] a frag (cid:1) (cid:1) (cid:2) (cid:3) (cid:2) (cid:1) (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) - (cid:5) (cid:1)(cid:2) - (cid:4) (cid:1)(cid:2) - (cid:2) (cid:1)(cid:2) - (cid:1) (cid:1)(cid:2) (cid:3) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13)(cid:14) (cid:15)(cid:6)(cid:7)(cid:12) (cid:1) [ (cid:7)(cid:16) ] (cid:13) (cid:6) (cid:14) (cid:15)(cid:4)(cid:11) (cid:16) (cid:10) (cid:12) (cid:17) (cid:4) (cid:3) (cid:13) (cid:6) (cid:1)(cid:2) (cid:1)(cid:2)(cid:3)(cid:4) (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) - (cid:2) (cid:1)(cid:2) - (cid:1) (cid:1)(cid:2) (cid:3) (cid:1)(cid:2) (cid:1) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13)(cid:14) (cid:15)(cid:6)(cid:7)(cid:12) (cid:1) [ (cid:7)(cid:16) ] (cid:18) (cid:10) (cid:11) (cid:6) (cid:13) (cid:10) (cid:3) (cid:19) (cid:5)(cid:20)(cid:4) (cid:18) (cid:4) (cid:21) (cid:11) (cid:9) (cid:6) (cid:22) Σ (cid:1) [ (cid:2) (cid:20) (cid:12) - (cid:1) ] Σ g (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) - (cid:2) (cid:1)(cid:2) - (cid:1) (cid:1)(cid:2) (cid:3) (cid:1)(cid:2) (cid:1) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13)(cid:14) (cid:15)(cid:6)(cid:7)(cid:12) (cid:12) [ (cid:7)(cid:16) ] (cid:18) (cid:10) (cid:11) (cid:6) (cid:13) (cid:20)(cid:5) (cid:23) (cid:4) (cid:24) (cid:4) (cid:9) (cid:2)(cid:25) (cid:6) (cid:2) (cid:1) [ (cid:5) (cid:10) ] H d = H g Figure 3.
Snapshots of the characteristic size a ∗ (top row), Stokes number St (second row), surface density Σ d (third row), and scale height H d (bottom row) of dust aggregates as a function of radial distance r for four models with γ = mantle (thick and thin lines, respectively). The gray and orange vertical lines mark the locations of the H O and CO snow lines. The dashed lines in the panels for a ∗ and St indicate the growth limit set by fragmentation (Equations (8) and (7), respectively).The dashed lines in the panels for Σ d and H d mark Σ d = . Σ g and H d = H g , respectively. The purple bars in the bottom panels mark the upperlimit of H d at r ≈
100 au inferred from the geometric thickness of the dust rings in the HL Tau disk (Pinte et al. 2016). O kuzumi et al .second rows of Figure 3, we compare a and St from the sim-ulations with a frag and St frag . For the models with CO icemantles, we find that a frag well explains the aggregate sizeoutside the CO snow line. This confirms our expectationthat fragmentation induced by CO mantle controls the ag-gregate size in the outer part of the disk. Note that St fallsbelow St frag when dust growth is limited by radial drift. Thiscan be seen in the models without CO ice mantles.Since α frag ∝ Σ g , the choice of γ a ff ects the radial slope of a ∗ as can be seen in the top panels of Figure 3. In the caseof γ = a ∗ increases toward the disk center, and thereforeconsiderably deviates from 100 µ m at r (cid:46)
50 au even withCO ice mantles. In contrast, the gas disk model with γ = a ∗ and thus ensures thatCO -mantled grain aggregates have a size of ∼ µ m allthe way down to the CO snow line. As we see in Section 3.2,this di ff erence a ff ects the polarized emission from the inner ∼
50 au region of the disk.The third row of Figure 3 shows the radial profiles of thedust surface density Σ d . In a quasi-steady flow of radiallydrifting aggregates, Σ d is inversely proportional to their driftspeed, and hence to St when St < Σ g in the sintering zones, as already pointedout by O16. The models with CO ice mantles predict a deepgap in Σ d between the H O and CO snow lines. In thesemodels, the inward mass flux of solids across the CO snowline is low because the aggregates outside the snow line aresmall. This causes a large deficit of solids between the H Oand CO snow lines, where the aggregates grow large anddrift rapidly thanks to sticky H O mantles. This gap forma-tion mechanism was also pointed out by Pinilla et al. (2017).The bottom panels of Figure 3 show the dust scale height H d as well as the gas scale height H g ≈ r /
100 au) . au.The well-separated dust rings of the HL Tau disk suggest thatthe dust rings are vertically thin, with H d ≈ ∼ H g /
7) at r ≈
100 au (Pinte et al. 2016). With the choice of α turb = × − , all our models reproduce dust settling at the observedlevel. 3.2. Synthetic Polarimetric Images
Figure 4 presents the synthetic polarimetric images of theHL Tau disk at wavelength λ = Strictly speaking, St frag given by Equation (7) slightly overestimates themaximum Stokes number in the sintering zones. In these regions, St is toosmall for the approximate expression ∆ v ≈ √ . α turb St for the turbulence-induced collision velocity to be valid. volved the “raw” images from radiation transfer calculationswith an elliptic Gaussian kernel of size 0 . (cid:48)(cid:48) × . (cid:48)(cid:48)
35, whichis equal to the beam size for the corresponding ALMA ob-servation (see Figure 5 for a raw image). At this wavelength,the observed polarimetric map shows a uniform polarizationpattern with PI ∼ − and P ∼ PI and P are the linear polarized intensity and the degree of lin-ear polarization, respectively. We find that the models withCO ice mantles successfully reproduce a similar unidirec-tional polarization pattern. The mechanism responsible forthis polarization is the self-scattering by ∼ µ m-sized ag-gregates. In contrast, the models without CO ice mantlespredict much weaker polarized emission. This is not surpris-ing since the aggregate size in those models is in much excessof 100 µ m over almost the entire part of the disk.Of the two models with CO ice mantles, the one with γ = r (cid:46)
100 au. In this region, the polariza-tion degree of the observed emission is spatially uniform towithin a factor of ≈
2. The model with γ = ∼ µ m all the way down to the CO snow line.In contrast, the model with γ = µ m, and con-sequently underestimates the polarization degree there. Wenote, however, the latter model could also provide an equallgood match to the observation if v frag , α turb , and ρ int are al-lowed to vary with r . For instance, according to Equation (8),a model with Σ g ∝ r − and α turb ∝ r − would yield the sameradial dependence for a frag as the model with radially con-stant Σ g and α turb .Given their relatively low angular resolutions ( ≈
60 au),the previous polarimetric observations of the HL Tau diskwere not able to resolve any substructure on 10 au scales,where multiple dust rings are visible (ALMA Partnershipet al. 2015). However, for future high-resolution polarimetricobservations, it is important to predict how the polarizationpattern of the disk would look like on the substructure scale.We show in the first row of Figure 5 the raw maps of I , PI ,and P for the model with CO ice mantles and γ = . (cid:48)(cid:48) × . (cid:48)(cid:48)
35, which areessentially the same as the lower right panel of Figure 3. Thebright rings visible in the total intensity map before smooth-ing (the upper left panel of Figure 5) correspond to the pile-ups of dust in the sintering zones. The map for PI (the uppercenter panel of Figure 5) indicates that these bright rings arealso responsible for the dominant fraction of the polarizedemission. The emission from the dust rings has high PI andhigh P , because the aggregate size in the sintering zones isparticularly close to 100 µ m. However, this correlation be- Figure 4.
Left column: ALMA polarimetric image of the HL Tau disk at λ = .
87 mm (Stephens et al. 2017). The color scale is the polarizedintensity, and the red line segments show the direction and degree of polarization. Contours are the total intensity, with the contour intervalstaken to be the same as in Figure 1 of Stephens et al. (2017). Center and right columns: synthetic polarimetric images at λ = .
87 mm fromfour dust evolution models. The model images are after convolution with an elliptic Gaussian kernel of 0 . (cid:48)(cid:48) × . (cid:48)(cid:48)
35, the beam size for thecorresponding observation. tween P and I is model-dependent; if a ∗ in the model weresmaller by a factor of 3 at all r , the degree of polarizationwould be lower in brighter regions than in darker regions.In any case, future polarimetric observations of the HL Taudisk with higher angular resolutions may test these predic-tions. More detailed comparisons between our models andthe long-baseline observations of HL Tau are presented inSection 4.1.We also performed radiative transfer calculations at longerwavelengths and confirmed that polarization due to dust self-scattering diminishes at λ (cid:29) ice man-tles. This is in qualitative agreement with the fact that the uni-directional polarized emission is not observed at λ = . DISCUSSION4.1.
Implications for the Ring–Gap Substructure andMillimeter Spectral Slope of the HL Tau Disk
So far we have focused on the polarized emission of theHL Tau disk on a 100 au scale. It is also interesting to seewhat our model implies for the ring–gap intensity profilesobserved in the earlier ALMA observations (ALMA Partner-ship et al. 2015). In the center and right panels of Figure 6,we plot the radial profiles of the Planck brightness temper-atures T B at λ = kuzumi et al . Figure 5.
Model polarimetric image at λ = ice mantles and γ =
0. The beam shape is taken to be the same as that for the ALMA polarimetric image by Stephens et al.(2017). The color scales for the left, center, and right panels show the total intensity I , polarized intensity PI , and the degree of polarization P ,while the vectors in the center and right panels indicate the direction and degree of polarization. man-tles and γ =
0, where the aggregates in the optically thicksintering zones have a relatively low albedo of ∼ .
4. Thesolid and dotted lines in Figure 6 show the profiles beforeand after Gaussian smoothing at the spatial resolutions of thecorresponding ALMA observations. The local bumps visi-ble in the model T B profiles correspond to the dust rings inthe sintering zones. For comparison, we also show in theleft panel Figure 6 the radial brightness temperature profilesof the HL Tau disk derived from the ALMA high-resolutionimages (ALMA Partnership et al. 2015). O16 produces theseprofiles by azimuthally averaging the observed maps (seeSection 2.1 of O16 for more details).The most prominent feature in the intensity profiles causedby CO ice mantles is a deep gap at r ≈
10 au. Interest-ingly, the location of this deep gap coincides with that ofHL Tau’s innermost gap (ALMA Partnership et al. 2015). Inthese models, the deep gap is caused by the rapid growth anddepletion of H O-mantled grain aggregates between the H Oand CO snow lines as described in Section 3.1. The mod-els with CO ice mantles generally predict that the 10 au gap is the deepest among the multiple dust rings, in qualitativeagreement with the ALMA images of the HL Tau disk. How-ever, these models overestimate the depth of the 10 au gap,implying that further tuning of model parameters is neededto achieve a more quantitative match. Sintering also pro-duces a 10 au gap O16 as seen in the models without CO ice mantles. However, this sintering-induced is substantiallyshallower than the 10 au gap in the models with CO ice man-tles.Our results also o ff er an important interpretation for themillimeter spectral slope of the HL Tau disk. We define thespectral index at λ = α . – . ≡ ln( I / I ) / d ln( ν /ν ), where I , and ν , are the intensities and frequencies at λ = α . – . from theobservations and models are shown in Figure 6 (for compar-ison, the observed profile is also overplotted in the panels forthe model profiles). The profile from the observations shows α . – . ∼ r (cid:46)
50 au and α . – . ∼ ∼ T B = T (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1) (cid:1) [ (cid:1) ] ALMA Observations (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:3)(cid:4)(cid:5)(cid:3)(cid:2)(cid:5)(cid:3)(cid:4)(cid:6)(cid:3)(cid:2)(cid:6)(cid:3)(cid:4) (cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14) (cid:15)(cid:16)(cid:17)(cid:18) (cid:19)(cid:10)(cid:11)(cid:16) (cid:1) [ (cid:11)(cid:20) ] α (cid:1) (cid:2) (cid:3)(cid:4)(cid:5)(cid:6) (cid:2) (cid:7) (cid:8)(cid:8) (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) w / o CO mantle, γ = (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:3)(cid:4)(cid:5)(cid:3)(cid:2)(cid:5)(cid:3)(cid:4)(cid:6)(cid:3)(cid:2)(cid:6)(cid:3)(cid:4) (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) w / o CO mantle, γ = (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:3)(cid:4)(cid:5)(cid:3)(cid:2)(cid:5)(cid:3)(cid:4)(cid:6)(cid:3)(cid:2)(cid:6)(cid:3)(cid:4)(cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1) (cid:1) [ (cid:1) ] w / CO mantle, γ = (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:3)(cid:4)(cid:5)(cid:3)(cid:2)(cid:5)(cid:3)(cid:4)(cid:6)(cid:3)(cid:2)(cid:6)(cid:3)(cid:4) (cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14) (cid:15)(cid:16)(cid:17)(cid:18) (cid:19)(cid:10)(cid:11)(cid:16) (cid:1) [ (cid:11)(cid:20) ] α (cid:1) (cid:2) (cid:3)(cid:4)(cid:5)(cid:6) (cid:2) (cid:7) (cid:8)(cid:8) (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) w / CO mantle, γ = (cid:1)(cid:2) (cid:1) (cid:1)(cid:2) (cid:2) (cid:1)(cid:3)(cid:4)(cid:5)(cid:3)(cid:2)(cid:5)(cid:3)(cid:4)(cid:6)(cid:3)(cid:2)(cid:6)(cid:3)(cid:4) (cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14) (cid:15)(cid:16)(cid:17)(cid:18) (cid:19)(cid:10)(cid:11)(cid:16) (cid:1) [ (cid:11)(cid:20) ] Figure 6.
Left column: azimuthally averaged millimeter emission profiles from the long-baseline observations of the HL Tau disk (ALMAPartnership et al. 2015; see O16 for azimuthal averaging). The upper panel shows the Planck brightness temperatures T B at λ = λ = α . – . . The dashed line shows the power-law disk temperatureprofile assumed in this study. Center and right columns: profiles of T B and α . – . from four models. The dotted lines show the raw profilesdirectly obtained from simulation results, while the solid lines show the profiles obtained by Gaussian smoothing the model intensities at thespatial resolutions of the observations. The gray dot-dashed line shows the spectral index profile from the observations for comparison. millimeter sizes, or to a moderately large optically thickness(e.g., Ricci et al. 2010). The model with CO ice mantlesand γ = a ∗ ∼ µ m at r (cid:38)
50 au but still successfully repro-duces α . – . in this outer part (optically thin emissionfrom 100 µ m particles would have α . – . ∼ . O and CO snow lines. Our interpretation is consis-tent with the optical depth estimates by Carrasco-Gonz´alez et al. (2016), who suggest the HL Tau disk is marginally op-tically thick (with optical depth ∼ . λ (cid:46) .
87 and 1.3mm (see their Figure 3).4.2.
Parameter Dependence
We have demonstrated that the low stickiness of CO -mantled grain aggregates reasonably explains the abundantpresence of 100 µ m-sized particles in disks. However, it isclear from Equation (8) that the maximum particle size de-pends not only on the sticking threshold v frag but also on otherparameters such as turbulence strength and aggregate poros-2 O kuzumi et al .ity. In this subsection, we show that a low fragmentationthreshold is still the most likely explanation for 100 µ m-sizedparticles in the particular case of the HL Tau disk.A key constraint comes from the evidence for dust set-tling in the HL Tau disk. The distinct morphology of theobserved dust rings suggests H d (cid:46) r ≈
100 au inthis disk (Pinte et al. 2016), which translates into H d (cid:46) H g / α turb . From Equation (6)with α Dz ≈ . α turb , one has H d / H g ≈ (0 . α turb / St) / for H d (cid:28) H d . When dust growth is limited by turbulence-induced fragmentation, St is given by Equation (7), and weobtain α turb ≈ H d H g v frag c s ≈ × − (cid:32) H d H g (cid:33)(cid:32) v frag . − (cid:33)(cid:32) T
30 K (cid:33) − / . (9)Equation (9) can in turn be used to eliminate α turb from themaximum aggregate size a frag given by Equation (8). Theresult is a frag ≈ (cid:32) H d H g (cid:33) − v frag Σ g c s ρ int ≈ (cid:32) H d H g (cid:33) − (cid:32) v frag . − (cid:33)(cid:32) ρ int . − (cid:33) − × (cid:32) Σ g
10 g cm − (cid:33)(cid:32) T
30 K (cid:33) − / µ m . (10)The weak temperature dependence of the right-hand side ofEquation (10) is negligible as long as we consider r ∼ H d (cid:46) H g / a frag ∼ µ mmust satisfy the condition (cid:32) v frag − (cid:33)(cid:32) ρ int − (cid:33) − (cid:32) Σ g
10 g cm − (cid:33) (cid:46) . . (11)Any realistic dust aggregate has ρ int (cid:46) − , andthe high millimeter flux from the HL Tau disk points to Σ g ∼
10 g cm − at r ∼ (cid:46) . − canexplain H d (cid:46) H g / a frag ∼ µ m simultaneously.The requirement v frag (cid:46) . − translates intomonomers sizes of a mon (cid:38) µ m for CO -mantled grain ag-gregates and a mon (cid:38) µ m for H O ice aggregates. The for-mer is more consistent with the maximum grain size ∼ µ mfor the HL Tau envelope inferred from near-infrared po-larimetry (Lucas et al. 2004; Murakawa et al. 2008), suggest-ing that CO -mantles o ff er a more reasonable explanation forthe origin of 100 µ m-sized aggregates. 4.3. Are Aggregates Compact or Flu ff y? We have assumed that the aggregates in the HL Tau diskhave a moderately high filling factor of 50%. If the aggre-gates are flu ff y, i.e., ρ int (cid:28) − , Equation (11) demandsan even lower value of v frag . The question is then whethersuch a low fragmentation threshold is realistic for flu ff y ag-gregates. Although aggregate collision simulations do showa trend of decreasing v frag with increasing aggregate poros-ity (Wada et al. 2009; Gunkelmann et al. 2016), it is unclearat the present whether this trend can compensate for the de-crease of ρ int in Equation (11).A more fundamental question about flu ff y aggregates iswhether they can produce strongly polarized scattered light.The e ff ective medium approach adopted in this study is in-applicable to flu ff y aggregates, in particular to fractal aggre-gates of fractal dimension (cid:46) From mechanical point of view, it is also unclear how icyaggregates in the disk could become compact. Okuzumi et al.(2012) and Kataoka et al. (2013) showed that neither aero-dynamical nor collisional compression is e ffi cient enough tocompress icy aggregates in protoplanetary disks to a fillingfactor of (cid:38) .
1. However, these previous studies assumedthat the ice aggregates are sticky and grow without fragmen-tation. It is yet to be explored how the porosity of aggre-gates evolves when they experience highly destructive col-lisions and subsequently reaccrete a large amount of smallfragments. Poorly sticky aggregates may also lose a highporosity through bouncing collisions (Weidling et al. 2009;Zsom et al. 2011). CONCLUSIONSWe have demonstrated that aggregates with a low stick-ing e ffi ciency can reasonably produce the uniformly polar-ized emission seen in the HL Tau disk. In our scenario, thegrowth of the icy aggregates is primarily limited by the man-tles of nonsticky CO ice, and to a less extent by aggregatesintering. Dust aggregates still can grow beyond 1 mm be-tween the H O and CO snow lines, where CO ice mantlesare absent and the sticky H O mantle facilitates dust growth.The e ffi cient growth and subsequent rapid infall of the dustin this region might explain the 10 au dust gap in the HL Taudisk. The models with CO ice mantles also suggest that thelow spectral index of the HL Tau disk at 0.87–1.3 mm pri-marily reflects the optical thickness at this wavelength, not Our preliminary calculations using the modified mean field theoryseem to show that polarized intensity due to self-scattering diminishes as theaggregate porosity increases (Tazaki et al., in prep.). ice (Musiolik et al.2016a,b; Pinilla et al. 2017). The CO -induced fragmen-tation barrier may also solve the long-standing problem ofdust retention in protoplanetary disks over several millionyears, which requires substantial particle fragmentation toslow down their radial infall (Dullemond & Dominik 2005;Birnstiel et al. 2009). Future applications of our model toother disks with a similar (sub)millimeter polarization pat-tern will enable us to better understand the role of CO iceon dust evolution in protoplanetary disks.Although we have assumed that H O ice is stickier thanCO ice, recent laboratory experiments suggest that H O icemight also be poorly sticky at low temperatures (Gundlachet al. 2018; Musiolik & Wurm 2019). The two scenarioswould equally well reproduce the uniformly polarized sub-millimeter emission from many protoplanetary disk, but thelatter scenario would predict no deep dust gap between theH O and CO snow lines. Therefore, the two scenarios couldbe tested by observing the dust continuum emission from thevicinity of these snow lines in di ff erent protoplanetary disks.Bouncing collisions, which we neglected in this study,could also cause a low sticking thresholds for icy aggregatesand should be taken into account in future work. Compactionof aggregates through bouncing collisions (Weidling et al.2009; Zsom et al. 2011) could also justify our assumptionthat icy aggregates have a relatively high filling factor.It should be noted that whether CO ice suppresses dustgrowth depends on how CO ice is distributed inside grains.As already mentioned in Section 2.2.1, the sticking e ffi -ciency of mixtures CO and H O ices would be similar tothat of pure H O ice if CO / H O ∼ . ice does not pro-duce ∼ µ m grains in every protoplanetary disk. Such avariety can also be expected from recent ALMA observationsof disk substructures. While not all observed disks possesssubstructures clearly associated with snow lines (Long et al. 2018; Huang et al. 2018; van der Marel et al. 2019), somesystems (HD 135544 B, HD 169142, and HD 97048) do ap-pear to have a deficit of dust interior to the CO snow line(see Figure 9 of van der Marel et al. 2019).We also note here that sintering may not lead to the forma-tion of dust rings in all protoplanetary disks. Sirono (2011)suggests that icy aggregates can be sintered over a wide re-gion of a protoplanetary disk if they are temporally trans-ported to the disk surface and get heated there. If this is thecase, sintering would not lead to local concentration of dustin the radial direction. This might be another reason whynot all spatially resolved protoplanetary disks seem to havedust rings associated with snow lines. The HL Tau disk isperhaps an ideal environment for sintering-induced ring for-mation because vertical mixing of dust is ine ff ective in thisdisk (Pinte et al. 2016).Finally, we note that the models presented in this paper donot address the origin of the polarized emission of the HL Taudisk at 3.1 mm. At this wavelength, the polarimetric imageexhibit no unidirectional polarization pattern, only showingazimuthal polarization (Kataoka et al. 2017). Mechanismsthat can produce azimuthally polarized emission, such as ra-diative and aerodynamic grain alignment (Tazaki et al. 2017;Kataoka et al. 2019), must also be taken into account to fullyunderstand the polarized emission from the HL Tau disk, andperhaps from other protoplanetary disks as well.We are grateful to Ian Stephens and Akimasa Kataoka forkindly providing us with the FITS image of the HL Taudisk (Stephens et al. 2017). 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