Condition for the formation of micron-sized dust grains in dense molecular cloud cores
aa r X i v : . [ a s t r o - ph . GA ] J un Mon. Not. R. Astron. Soc. , 1–5 (2012) Printed 14 August 2018 (MN L A TEX style file v2.2)
Condition for the formation of micron-sized dust grains in densemolecular cloud cores
Hiroyuki Hirashita ⋆ and Zhi-Yun Li Institute of Astronomy and Astrophysics, Academia Sinica, P.O. Box 23-141, Taipei 10617, Taiwan Astronomy Department, University of Virginia, Charlottesville, VA 22904, USA
ABSTRACT
We investigate the condition for the formation of micron-sized grains in dense cores of molec-ular clouds. This is motivated by the detection of the mid-infrared emission from deep insidea number of dense cores, the so-called ‘coreshine,’ which is thought to come from scatteringby micron ( µ m)-sized grains. Based on numerical calculations of coagulation starting fromthe typical grain size distribution in the diffuse interstellar medium, we obtain a conservativelower limit to the time t to form µ m-sized grains: t/t ff > /S )( n H / cm − ) − / (where t ff is the free-fall time at hydrogen number density n H in the core, and S the enhancementfactor to the grain-grain collision cross-section to account for non-compact aggregates). Atthe typical core density n H = 10 cm − , it takes at least a few free-fall times to form the µ m-sized grains responsible for coreshine. The implication is that those dense cores observedin coreshine are relatively long-lived entities in molecular clouds, rather than dynamicallytransient objects that last for one free-fall time or less. Key words: dust, extinction — infrared: ISM — ISM: clouds — ISM: evolution — turbu-lence
Dense cores of molecular clouds are the basic units for the for-mation of Sun-like, low-mass stars. A fundamental question aboutthese cores that has not been answered conclusively is: are theylong-lived entities or simply transient objects that disappear in onefree-fall time or less?The core lifetime is important to determine, because it affectsthe rate of star formation as well as the time available for chem-ical reactions, which in turn affect the chemical structure of notonly the cores themselves but also the disks (and perhaps even ob-jects such as comets) that form out of them (Caselli & Ceccarelli2012). It also has implications on how the cores are formed(Ward-Thompson et al. 2007). If the cores are relatively long lived,it would favor those formation scenarios that involve persistent sup-port against gravity from, for example, magnetic fields (Shu et al.1987; Mouschovias & Ciolek 1999), or long mass-accumulationtime (e.g. Gong & Ostriker 2011). If the core lifetime turns outcomparable to the free-fall time or less, then rapid formation andcollapse, through for example turbulent compression, would be pre-ferred (Mac Low & Klessen 2004).One way to constrain the core lifetime is to compare the num-ber of starless cores to that of young stellar objects (YSOs), whoselifetimes can be independently estimated (Ward-Thompson et al.2007; Evans et al. 2009). Ward-Thompson et al. (2007) found that ⋆ E-mail: [email protected] cores of – cm − typically last for ∼ µ m Spitzer
Infrared Array Camera(IRAC) band] from deep inside dense cores of molecular clouds(Pagani et al. 2010; Steinacker et al. 2010). It is found in about halfof the cores where the emission is searched for (Pagani et al. 2010).The emission is thought to come from light scattered by dust grainsup to 1 µ m in size. Such grains are much larger than those in the dif-fuse interstellar medium (e.g. Mathis, Rumpl, & Nordsieck 1977,hereafter MRN). Since it takes time for small MRN-type grainsto grow to µ m-size, the observed coreshine should provide a con-straint on the core lifetime. The goal of our investigation is to quan-tify this constraint. Specifically, we want to answer the question:how long does it take for the grains in a dense core to grow to µ m-size at a given density?Grain growth through coagulation has been studiedfor a long time (e.g. Chokshi, Tielens, & Hollenbach 1993;Dominik & Tielens 1997). Even before the discovery of coreshine,Ormel et al. (2009) was able to demonstrate that coagulation canin principle produce µ m-sized grains in dense cores, provided thatthe grains are coated by ‘sticky’ materials such as water ice andthat the cores are relatively long-lived (see also Ormel et al. 2011).In this paper, we aim to strengthen Ormel et al. (2009)’s results c (cid:13) Hirashita and Li by deriving a robust lower limit to the lifetimes for those coreswith µ m-sized grains inferred from coreshine through a simpleframework that isolates the essential physics of coagulation. Wefind that cores of typical density cm − must last for at leasta few free-fall times in order to produce µ m-sized grains. Ourcoagulation models are explained in Section 2 and the results aredescribed in Section 3. We discuss the robustness and implicationof the results in Section 4, and conclude in Section 5. We consider the time evolution of grain size distribution by co-agulation in a dense core. We adopt the formulation used in ourprevious paper, Hirashita (2012) (see also Hirashita & Yan 2009),with some changes to make it suitable for our purpose. We brieflysummarize the formulation here, and refer to Hirashita (2012) forfurther details.We assume that the grains are spherical with a constant ma-terial density ρ gr . We define the grain size distribution such that n ( a, t ) d a is the number density of grains whose radii are between a and a + d a at time t . For numerical calculation, we consider N = 128 discrete logarithmic bins for the grain radius (or mass),and solve the discretized coagulation equation. In considering thegrain–grain collision rate between two grains with radii a and a ,we estimate the relative velocity by v = p v ( a ) + v ( a ) − v ( a ) v ( a ) µ , (1)where the grain velocity as a function of grain radius, v ( a ) , is givenbelow in equation (3), and µ ≡ cos θ ( θ is an angle between thetwo grain velocities) is randomly chosen between − and 1 in eachtime-step, and the cross-section by σ = Sπ ( a + a ) , (2)where S is the enhanced factor of cross-section, which representsthe increase of cross-section by non-compact aggregates. Note thatwe always define the grain radius a and the grain material density ρ gr for the compact geometry, even if S > , to avoid the extrauncertainty caused by the grain geometry [see also the commentin the item (iii) in Section 2.2]. We adopt the turbulence-drivengrain velocity derived by Ormel et al. (2009), who assume that thedriving scales of turbulence is given by the Jeans length and that thetypical velocity of the largest eddies ( ∼ the Jeans length) is givenby the sound speed (see also Hirashita 2012): v ( a ) = 1 . × (cid:16) T gas
10 K (cid:17) / (cid:18) a . µ m (cid:19) / × (cid:16) n H cm − (cid:17) − / (cid:18) ρ gr . − (cid:19) / cm s − , (3)where T gas is the gas temperature assumed to be 10 K in this pa-per. Thermal velocities are small enough to be neglected. The ro-bustness of our conclusion in terms of the grain velocity is furtherdiscussed in Section 4.1.The form of equation (1) suggests that the motions of dustparticles are random. This treatment is not valid in general, since This treatment is different from Hirashita (2012), who represented thecollisions by µ = − , 0, and 1. Such a discrete treatment of µ cause artifi-cial spikes in the grain size distribution. turbulent motions are correlated. However, we do not include thefull treatment of the probability distribution function of the truerelative particle velocity in turbulence for the following three rea-sons: (i) Our simple formulation is sufficient to give a lowerlimit for the coagulation time-scale (Section 4.1). (ii) The prob-ability distribution function of the true relative particle velocityin turbulence is unknown, and has only recently been investi-gated (Hubbard 2013; Pan & Padoan 2013). (iii) In the environ-ments of interest to this paper, the forcing of turbulent eddiescan be represented by a model where the particle motions ex-perience ‘random kicks’: in this so-called ‘intermediate regime’(Ormel & Cuzzi 2007), the prescription given by equation (1) isapplicable. Indeed, we can confirm that the condition for the inter-mediate regime is satisfied as follows. The intermediate regime isdefined by Re − / < St < , where Re is the Reynolds numberand St is the Stokes number (Ormel & Cuzzi 2007). This condi-tion is translated into a/ µ m ) ( T gas /
10 K) − cm − < n H < . × ( a/ µ m ) ( T gas /
10 K) − cm − . Since we are inter-ested in the range of grain radius, . µ m . a . µ m, the inter-mediate regime is applicable to the density range considered in thispaper.We adopt the following coagulation threshold veloc-ity, v ki coag , given by (Chokshi, Tielens, & Hollenbach 1993;Dominik & Tielens 1997; Yan, Lazarian & Draine 2004) v ki coag = 21 . (cid:20) a k + a i ( a k + a i ) (cid:21) / γ / E ⋆ / R / ki ρ / , (4)where γ is the surface energy per unit area, R ki ≡ a k a i / ( a k + a i ) is the reduced radius of the grains, E ⋆ is the the reduced elasticmodulus. This coagulation threshold is valid for collision betweentwo homogeneous spheres and would not be applicable to collisionsbetween aggregates. At low velocities, grains stick with each otherand develop a non-compact or fluffy aggregates. These aggregatesstick with each other at low relative velocities, and start to deformor bounce as the relative velocities increases. Because the defor-mation absorbs the collision energy, the aggregates can stick witheach other at a velocity larger than the above coagulation thresh-old. At very high velocities, cratering and catastrophic destructionwill halt the growth (Paszun & Dominik 2009; Wada et al. 2011;Seizinger & Kley 2013). In this paper, we only limit the applica-tion of this threshold to compact spherical grains [i.e. cases (i) and(ii) in Section 2.2; see Ormel et al. (2009) and references thereinfor a detailed treatment of coagulation of aggregates.] For the initial grain size distribution, we adopt the following power-law distribution, which is typical in the diffuse ISM (MRN): n ( a ) = C a − . ( a min a a max ) , (5)where C is the normalizing constant, with a min = 0 . µ m and a max = 0 . µ m. The normalization factor C is determined ac-cording to the mass density of the grains in the ISM: D µm H n H = Z a max a min π a ρ gr C a − . d a , (6)where n H is the hydrogen number density, m H is the hydrogenatom mass, µ is the atomic weight per hydrogen (assumed to be1.4) and D (0.01; Ormel et al. 2009) is the dust-to-gas mass ratio.We adopt n H = 10 cm − for the typical density of densecores emitting coreshine (Steinacker et al. 2010), but also survey a c (cid:13) , 1–5 ormation of micron-sized grains wide range in n H . We normalize the time to the free-fall time, t ff : t ff = r π Gµm H n H = 1 . × (cid:16) n H cm − (cid:17) − / yr . (7)To isolate the key pieces of physics that determine the rate ofcoagulation, we examine the following three models:(i) Standard silicate model:
We adopt coagulation thresholdgiven by equation (4) with silicate material parameters ( ρ gr =3 . g cm − , γ = 25 erg cm − , and E ⋆ = 2 . × dyncm − ; Chokshi, Tielens, & Hollenbach 1993). We estimate thecross-section by the compact spherical case (i.e. S = 1 in equa-tion 2).(ii) Sticky coagulation model:
We do not apply the coagula-tion threshold; that is, if grains collide with each other, they co-agulate. This is motivated by the fact that grains coated by waterice have a large coagulation threshold velocity (Ormel et al. 2009).We adopt S = 1 .(iii) Maximal coagulation model:
As shown by Ormel et al.(2009), the volume filling factor of the grains after coagulation is ∼ . because of the non-compact structure of aggregates. Thus,we adopt S = 5 [ ∼ (1 / . / ]. Like the sticky coagulationmodel, we do not apply the coagulation threshold. This model pro-vides a conservative estimate for the coagulation time-scale (seeSection 4.1 for discussion). Note that a and ρ gr are defined for thecompact grains. In fact, the grain velocity (equation 3) also hasa dependence on the volume filling factor of aggregates through a and ρ gr in such a way that the non-compact structure enhancesthe gas–grain coupling, leading to a lower velocity. Thus, the max-imal coagulation model overestimates the grain velocity (i.e. thecoagulation rate), which strengthens the case for the model being‘maximal’. We present the evolution of grain size distribution for n H = 10 and cm − at t = 1 t ff , t ff , and t ff . The results are shown inFig. 1 for all three models: (i) the standard silicate model, (ii) thesticky coagulation model, and (iii) the maximal coagulation model.In order to show the grain mass distribution per logarithmic radius,we show a n ( a ) .In the standard silicate model shown in Fig. 1a, the graingrowth stops at a ∼ . µ m because of the coagulation threshold:the grain velocities are too large for coagulation if a & . µ m.Thus, bare silicate cannot grow to µ m sizes, a result found previ-ously by Ormel et al. (2009). We conclude that bare silicate cannotbe the source of coreshine.Indeed, water ice has higher coagulation threshold, so if grainsare coated by water ice, coagulation proceeds further (Ormel et al.2009, 2011). Motivated by this, we examine the sticky coagulationmodel, in which there is no coagulation threshold. (The coagulationthreshold of water ice is separately discussed in Section 4.1 to min-imize the uncertainty in the material properties adopted.) Fig. 1bshows that grains grow beyond 0.1 µ m. For n H = 10 cm − , µ m-sized grains form at t ff , while for the standard density n H =10 cm − , the typical grain radius does not reach 1 µ m even at t ff .In reality, aggregates are thought to form as a result of co-agulation (Ossenkopf 1993). Thus, the cross-section is effectivelyincreased compared with the spherical and compact case. Fig. 1c shows that the maximal coagulation model in which the cross-section is elevated by a factor of 5 (i.e. S = 5 ) successfully pro-duces µ m-sized grains within t ff even for n H = 10 cm − . Itremains difficult, however, to produce µ m-sized grains in t ff for n H = 10 cm − and in t ff for n H = 10 cm − . µ m-sized grains As mentioned in Introduction, the aim of this paper is to deter-mine the condition for the formation of µ m-sized grains thoughtto be responsible for the observed coreshine (Pagani et al. 2010;Steinacker et al. 2010). According to Steinacker et al. (2010), scat-tering dominates over absorption by an order of magnitude at λ = 3 . µ m if a & µ m. Since the peak of the grain size dis-tribution in a n ( a ) is well defined (see Fig. 1), we simply findthe condition for the radius at the peak, a peak , reaches or exceeds1 µ m. We also examine a more conservative criterion by using a peak = 0 . µ m instead of 1 µ m, motivated in part by the fact that a ∼ . µ m is the grain radius at which scattering is comparableto absorption at λ = 3 . µ m (Steinacker et al. 2010).We will concentrate on the maximal coagulation model withan enhancement factor for cross-section S = 5 ; the result from thesticky coagulation model with S = 1 can be obtained through asimple scaling. In Fig. 2, we show a grid of models with differentcore densities and times (in units of the free-fall time at the coredensity). The solid line marks roughly the critical time t grow at agiven density n H above which µ m-sized grains are produced. It isgiven by t grow t ff = A (cid:16) S (cid:17) (cid:16) n H cm − (cid:17) − / , (8)where A = 5 . and 3.0, respectively, if we adopt a peak = 1 µ mand . µ m for the criterion of micron-sized grain formation. Thecondition for forming µ m-sized grains is therefore t > t grow . Thesame condition applies to the sticky coagulation model (with S =1 ) as well, since coagulation time is inversely proportional to thecross-section for grain-grain collision.Equation (8) can be understood in the following way.Since coagulation is a collisional process, t grow should begiven by the collision time-scale, t coll = ( vSπa n dust ) − =4 aρ gr / (3 D µm H n H vS ) , where v and n dust are the velocityand the number density of grains, respectively (Ormel et al.2009). The growth time-scale in terms of grain radius is t grow ≃ t coll (note that t coll is the time-scale of grainvolume being doubled by coagulation). Then, t coll /t ff isevaluated by using equations (3) and (7) as t grow /t ff ≃ . a/ µ m ) / ( n H / cm − ) − / ( S/ − ( T gas /
10 K) − / · ( ρ gr / . − ) / ; that is, A = 7 . (5.2) for a = 1 µ m( . µ m), in a fair agreement with the above numerical estimate.Thus, t grow can be understood in terms of collision time-scale,which strengthens our numerical results.Note that, to form µ m-sized grains in one free-fall time, thedensity n H must be of order cm − or higher, even in the max-imal coagulation model. In the sticky coagulation model, the re-quired density would be higher still. Such densities are much higherthan the typical core value (of order cm − ). At cm − ,Fig. 2 and equation (8) indicate that, under reasonable conditions,it takes at least several free-fall times for the grains to grow to µ m-size (see Section 4.2 for more discussion). The implication is thatthose dense cores detected in coreshine should be rather long-livedentities rather than transient objects that disappear in one free-fall c (cid:13) , 1–5 Hirashita and Li
Figure 1.
Evolution of grain size distribution. The solid, dashed, and dot-dashed, lines show the grain size distributions at t = 1 t ff , t ff , and t ff , respectively,for (a) the standard silicate model, (b) the sticky coagulation model, and (c) the maximal coagulation model. The dotted line presents the initial condition. Theupper and lower panels show the cases with n H = 10 cm − and cm − , respectively. Figure 2.
The condition for the formation of µ m-sized grains. The successand failure of the formation of a > µ m grains in the maximal coagulationmodel are shown by ‘o’ and ‘x’, respectively. The solid and dashed linesshow the boundary of those two cases in the maximal coagulation modeland the sticky coagulation model, respectively, if we adopt a peak = 1 µ mfor the criterion for coreshine. The dot-dashed line marks the boundary forthe maximal coagulation model for a more conservative criterion: a peak =0 . µ m. time; the latter objects would simply not have enough time to formthe µ m-sized grains responsible for coreshine. µ m-sized grain formation time One may argue that coagulation would be faster if the grains wereto collide at higher speeds than adopted in our model. However,it will be difficult for this to happen because of the existence of acoagulation threshold. As mentioned earlier, bare silicate grains al- ready acquire velocities larger than the threshold at a rather smallsize a ∼ . µ m; they do not grow beyond . µ m under reason-able conditions. To grow to larger sizes, the grains must be ‘moresticky’ than silicate, as is the case when the grains are coated withwater ice (Ormel et al. 2009). For such coated grains, we can esti-mate the coagulation threshold for equal-sized grains from equation(4) using ρ gr = 3 . g cm − , γ = 370 erg cm − , E ⋆ = 3 . × dyn cm − . The result is v coag = 9 . × ( a/ µ m ) − / cms − . For the micron-sized grains that we aim to form, this thresh-old is already smaller than the typical grain velocity v ∼ . × ( a/ µ m ) / that was used in our model. In other words, ourmodel is already generous with the grain-grain collision speed.(Collisions at the relatively high speed that we adopted may leadto the compaction of aggregates, which should reduce the enhance-ment factor S for grain-grain collision cross-section and hence therate of grain growth.). Increasing the collision speed further shouldnot lead to faster growth to µ m-size. For this reason, we believethat the critical time t grow for the formation of µ m-sized grainsestimated in equation (8) is a robust lower limit. Equation (8) indicates that it takes more than ∼ free-fall times toform µ m-sized grains at the typical core density n H = 10 cm − if the enhancement factor for cross-section is S = 5 . If the en-hancement factor is larger, the coagulation would be faster. In par-ticular, if S = 25 , the formation of micron-sized grains may occurin a single, rather than 5, free-fall time. However, S = 25 requiresthe grain volume filling factor to be − / ∼ per cent, which isextreme. For example, to form such a grain of a = 1 µ m with com-pact spherical grains with a = 0 . µ m, one need to connect 1,000grains linearly , which is unlikely. We doubt that there is muchroom to increase S well beyond , which corresponds aggregates ofrather low volume filling factor ( ∼ . ) already. If the cross-sectionenhancement factor S is not much larger than 5, it would take sev-eral free-fall times (or more) to form µ m-sized grains at typical coredensities. The long formation time would indicate that those densecores with observed coreshine are relatively long-lived entities,rather than transient objects that form and disappear in one free- c (cid:13) , 1–5 ormation of micron-sized grains fall time. This estimate of core lifetime based on grain growth isconsistent with that inferred from the number of starless cores (rel-ative to YSOs) (Ward-Thompson et al. 2007). It is also consistentwith the observational results that only a small fraction of densecores show any detectable sign of gravitational collapse and thateven those collapsing cores tend to have infall speeds less than halfthe sound speed (Di Francesco et al. 2007). Such slowly-evolving,relatively long-lived cores can form, for example, as a result of am-bipolar diffusion in magnetically supported clouds (Shu et al. 1987;Mouschovias & Ciolek 1999), even in the presence of a strong, su-personic turbulence (Nakamura & Li 2005). They are less compat-ible with transient cores that are formed rapidly through fast com-pression by supersonic turbulence without any magnetic cushion(Mac Low & Klessen 2004), unless the core material is slowly ac-cumulated in the post-shock region over several free-fall times (e.g.Gong & Ostriker 2011). Large grains ( a & . µ m), once they are injected into the dif-fuse ISM, are rapidly shattered into smaller grains (Hirashita & Yan2009; Asano et al. 2013). Thus, there should be a continuous sup-plying mechanism of large grains (Hirashita & Nozawa 2013). Ifdense molecular cores has lifetimes long enough to produce µ m-sized grains, they can be an important source of large grains. In-cluding the supply of large grains from dense cores will be an in-teresting topic in modeling the evolution of dust in galaxies. Motivated by recent coreshine observations, we have examined thecondition for the formation of µ m-sized grains by coagulation indense molecular cloud cores. We obtained a simple, conservativelower limit to the core lifetime t for the formation of 0.5 µ m-sizedgrains: t/t ff > /S )( n H / cm − ) − / , where t ff is the free-fall time at the core density n H and S the enhancement factor forgrain-grain collision that accounts for aggregates. The formationtime for 1 µ m-sized grains is roughly a factor of 2 longer. Since S isunlikely much larger than 5, we conclude that dense cores of typicaldensity n H = 10 cm − must last for at least several free-fall timesin order to produce the µ m-sized grains thought to be responsiblefor the observed coreshine. Such cores are therefore relatively long-lived entities in molecular clouds, rather than dynamically transientobjects. ACKNOWLEDGMENTS
We are grateful to C. W. Ormel for comments that greatly im-proved the presentation of the paper. This research is supportedthrough NSC grant 99-2112-M-001-006-MY3 and NASA grantNNX10AH30G.
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