Clouds of Fluffy Aggregates: How They Form in Exoplanetary Atmospheres and Influence Transmission Spectra
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Clouds of Flu ff y Aggregates: How They Form in Exoplanetary Atmospheres and Influence Transmission Spectra K azumasa O hno , 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, 6-3, Aramaki, Aoba-ku, Sendai, Miyagi, 980-8578, Japan
ABSTRACTTransmission spectrum surveys have suggested the ubiquity of high-altitude clouds in exoplanetary atmo-spheres. Theoretical studies have investigated the formation processes of the high-altitude clouds; however,cloud particles have been commonly approximated as compact spheres, which is not always true for solid min-eral particles that likely constitute exoplanetary clouds. Here, we investigate how the porosity of cloud particlesevolve in exoplanetary atmospheres and influence the cloud vertical profiles. We first construct a porosity evo-lution model that takes into account the fractal aggregation and the compression of cloud particle aggregates.Using a cloud microphysical model coupled with the porosity model, we demonstrate that the particle internaldensity can significantly decrease during the cloud formation. As a result, flu ff y-aggregate clouds ascend toaltitude much higher than that for compact-sphere clouds assumed so far. We also examine how the flu ff y-aggregate clouds a ff ect transmission spectra. We find that the clouds largely obscure the molecular features andproduce a spectral slope originated by the scattering properties of aggregates. Finally, we compare the syntheticspectra with the observations of GJ1214 b and find that its flat spectrum could be explained if the atmosphericmetallicity is su ffi ciently high ( ≥ × solar) and the monomer size is su ffi ciently small ( r mon < µ m). Thehigh-metallicity atmosphere may o ff er the clues to explore the gas accretion processes onto past GJ1214b. Keywords: planets and satellites: atmospheres — planets and satellites: composition — planets and satellites:individual(GJ1214 b) INTRODUCTIONTransmission spectroscopy is a powerful approach to probethe compositions of exoplanetary atmospheres (e.g., Seager& Sasselov 2000; Brown 2001). Recent surveys of transmis-sion spectra have shown that clouds and / or hazes are ubiq-uitous in exoplanetary atmospheres (e.g., Bean et al. 2010;Narita et al. 2013a,b; Kreidberg et al. 2014, 2018; Knutsonet al. 2014a,b; Sing et al. 2016; Crossfield & Kreidberg 2017;Lothringer et al. 2018; Espinoza et al. 2019; Benneke et al.2019). A remarkable feature of the exoplanet clouds / hazes isthat some of them are present at extremely high altitude. Forexample, the Neptune-sized exoplanet GJ436b and super-Earth GJ1214b are suggested to have an opaque cloud / hazeat an altitude as high as ∼ . / hazes are also suggested for many hot Jupiters (e.g.,Sing et al. 2016; Barstow et al. 2017). Understanding howthe high-altitude clouds / hazes form may enable us to inferwhat composition the atmosphere beneath the clouds wouldhave, which in turn might tell us how the planets formed.In hot, close-in transiting planets, clouds made of con-densed minerals potentially form (Morley et al. 2012), andseveral studies have investigated their formation processes using 1D cloud microphysical models (e.g., Helling et al.2008, 2017, 2019; Lee et al. 2015; Ohno & Okuzumi 2018;Powell et al. 2018; Gao & Benneke 2018; Ormel & Min2019) as well as 3D models (e.g., Lee et al. 2016; Lineset al. 2018, 2019; Roman & Rauscher 2019). Neverthe-less, it is still highly uncertain how the high-altitude cloudsare formed. Morley et al. (2013, 2015) and Charnay et al.(2015a,b) showed that a high-altitude cloud producing theflat transmission spectrum of GJ1214 b could be formed ifthe sedimentation velocity of the particles constituting thecloud is su ffi ciently slow. Recently, Ohno & Okuzumi (2018)modeled the formation of clouds in GJ1214 b and GJ436 byexplicitly calculating the size and settling velocity of cloudparticles from the microphysics of particle growth. Theyfound that the cloud particles grow too large to ascend to aheight of 0 .
01 mbar, needed to explain the transmission spec-trum of GJ1214 b. Gao & Benneke (2018) also attempted toreproduce the cloud structure of GJ1214 b using a micro-physical model that fully solves the evolution of size dis-tribution. However, they concluded the high-altitude cloudof GJ1214 b can only be explained when the eddy di ff usioncoe ffi cient in the atmospheres is assumed to be at least anorder of magnitude higher than predicted from the generalcirculation model (GCM) with passive tracers (Charnay et al. a r X i v : . [ a s t r o - ph . E P ] J a n O hno , O kuzumi , & T azaki ff y aggregates with very low internal density (e.g., Do-minik & Tielens 1997; Blum & Wurm 2000; Wada et al.2008). Because the flu ff y aggregate has a sedimentation ve-locity much lower than the compact sphere with same mass,it would easily ascend to the high altitude. Some previ-ous studies pointed out the importance of particle porosityon the vertical structures of mineral clouds (Marley et al.2013; Ohno & Okuzumi 2018). The e ff ect of porosity evolu-tion was recently studied for photochemical haze formation(Adams et al. 2019; Lavvas et al. 2019), while quantitativeinvestigations have not been carried out yet for mineral cloudformation.In this paper, we investigate how the porosity of cloud par-ticles evolve in exoplanetary atmospheres, and how it a ff ectsthe vertical profiles of mineral clouds. Using a cloud micro-physical model coupled with the porosity evolution model,we will demonstrate that cloud particle aggregates (CPAs,hereafter) grow without compression in most cases studiedhere. We will also compute synthetic transmission spectra tostudy how the flu ff y-aggregate clouds influence the observ-able transmission spectra. The organization of this paper is asfollows. We introduce how internal density of an aggregatevaries with microphysical processes and establish a porosityevolution model for CPAs in Section 2. We describe our mi-crophsical model and investigate the vertical structures of theflu ff y-aggregate clouds in Section 3. We present the synthetictransmission spectrum and compare it with the observationsof GJ1214 b in Section 4. We discuss the caveats of this studyand future prospects in Section 5. We summarize this paperin Section 6 MODELING THE FORMATION OF FLUFFYAGGREGATES IN MINERAL CLOUDSFlu ff y aggregates form through the mutual sticking of solidparticles with a low collision energy (e.g., Meakin 1991).The smallest particles constituting an aggregate are called monomers . One of the most important quantities that char- acterize a porous aggregate is the filling factor φ defined by φ = ρ agg ρ mon , (1)where ρ agg is the mean internal density of the aggregate and ρ mon are the bulk density of the individual monomers. Foraggregates made of single-sized monomers, Equation (1) canalso be written as φ = NV mon V agg , (2)where N is the number of the constituent monomers, and V agg and V mon are the volumes of the aggregate and individualmonomers, respectively. Here, the volume of an aggregateis defined as that of a sphere with the same gyration radius.The number of constituent monomers is another importantparameter for aggregate of monodisperse monomers becauseit is directly related to the aggregate mass.The set of N and φ defines the characteristic size, or lengthscale, of a porous aggregate. If we approximate an aggregatewith a sphere of radius r agg , the ratio of volumes V mon to V agg is V agg / V mon = ( r agg / r mon ) , where r mon is the monomer ra-dius. Using this expression with Equation (2), we obtain therelation that determines r agg as a function of N and φ , r agg = (cid:32) N φ (cid:33) / r mon , (3)In atmospheres, the filling factor of an aggregate canchange through various processes, and we introduce themin following subsections.2.1. Evolution of the Filling Factor
We introduce how the filling factor of an aggregate φ evolves via various processes. For convention, we describea filling factor determined by a specific process using a sub-script φ ; for example, φ coll for the collisional compression.2.1.1. Fractal Growth
Aggregates forming through low-energy sticking colli-sions often have an open structure with fractal geometry(e.g., Meakin 1991). A fractal aggregate can be character-ized by the fractal dimension D f defined by N = k (cid:32) r agg r mon (cid:33) D f , (4)where k is a prefactor of order unity, r mon is the radius of in-dividual monomers, and r agg is the characteristic radius ofan aggregate. An aggregate with D f = r agg is proportional to itsmass ( ∝ N ), while an aggregate with D f = ∼ r is proportional toits mass. Experimental and numerical studies show that ag-gregates growing by accreting similar-sized aggregates have louds of F luffy A ggregates D f = . .
2, whereas aggregates growing by accreting in-dividual monomers tend to have D f ≈ D f ≈ . D f = D f ≈ k ≈ φ frac , can be calculated as a function of N .Substituting r agg / r mon = k − / D f N / D f along with D f = k = φ , we obtain φ frac = N − / , (5)which indicates that the filling factor decreases with increas-ing N , i.e., as the aggregate grows. Whenever two aggregatesstick at a low velocity, the newly formed aggregate contains alarge void whose volume is comparable to the volume of thecollided aggregates (see Section 4 of Okuzumi et al. 2009 formore quantitative analysis). This causes the decrease of thefilling factor. 2.1.2. Collisional Compression
The fractal growth described by Equation (5) breaks downif the impact energy is higher than needed for internal restruc-turing of the newly forming aggregate, for which case col-lisional compaction occurs (e.g., Dominik & Tielens 1997;Blum & Wurm 2000; Wada et al. 2007, 2008; Paszun & Do-minik 2009). For a collision between two aggregates withsimilar individual masses ≈ m agg /
2, the collisional energycan approximately be written as E imp ≈ m agg ∆ v , (6)where ∆ v is the collisional velocity. Here, m agg standsfor the mass of the newly forming aggregate, and we haveused that the reduced mass of the collided aggregates is ≈ ( m agg / / = m agg /
4. Restructuring of the new aggregateoccurs if E imp is much higher than the energy E roll neededto roll one monomer over another monomer in contact by90 ◦ against rolling friction (Dominik & Tielens 1997; Blum& Wurm 2000). Following Dominik & Tielens (1995), weevaluate E roll as E roll = π γ r mon ξ crit , (7)where γ is the surface energy of the monomers and ξ crit isthe critical rolling displacement above which inelastic rollingoccurs. A realistic value of ξ crit is somewhat uncertain: the model of Dominik & Tielens (1995) anticipates ξ crit ∼ ∼ ξ crit = N -body dynamical simulations (Wada et al. 2007, 2008;Paszun & Dominik 2009; Suyama et al. 2008, 2012). Ac-cording to Wada et al. (2008), the size of an aggregate after ahigh-energy ( E imp (cid:38) E roll ) collision between two equal-sizedfractal ( D f =
2) aggregates follows r agg r mon = N / (cid:32) E imp . NE roll (cid:33) − / . (8)Using Equation (3), Equation (8) translates into the fillingfactor after a high-energy collision, φ coll = N − / (cid:32) E imp . E roll (cid:33) / . (9)Here, the prefactor N − / corresponds to the filling factorwithout collisional compression (see Equation 5), whereasthe factor ( E imp / . E roll ) / represents compression occur-ring for E imp (cid:38) E roll . Wada et al. (2008) derived Equa-tion (8) for aggregates after a single compressive collision,but Suyama et al. (2008) later confirmed that the expressionapproximately holds for aggregates growing through multi-ple compressive collisions (see their Equation (33)).For particles in atmospheres, the collision velocity inEquation (6) is calculated as the root sum square of thethermal (Brownian) relative velocity and the relative velocity ∆ v t of gravitational settling, i.e., ∆ v = (cid:115) k B T π m agg + ∆ v . (10)Here we write ∆ v t ≈ (cid:15) v (cid:48) t , where v (cid:48) t is the terminal settlingvelocity of individual aggregates before collision and (cid:15) is anumerical factor arising from finite width of actual size dis-tribution of the aggregates. We here adopt (cid:15) = . v (cid:48) t = gr (cid:48) ρ (cid:48) agg η β ( r (cid:48) agg ) + . gr (cid:48) ρ g ρ (cid:48) agg η / − / , (11)where r (cid:48) agg and ρ (cid:48) agg are the characteristic radius and densityof aggregates before collision, respectively, η is the dynamicviscosity of ambient gas, and β is the slip correction factoraccounting for the free-molecular flow regime. In Equation(11), we have approximated the aerodynamic radius of an ag-gregate with its characteristic radius r agg defined by Equation O hno , O kuzumi , & T azaki (3). This approximation is invalid for very flu ff y aggregateswith D f <
2, for which the aerodynamic radius is generallysmaller than the characteristic radius (Okuzumi 2009). Weuse this assumption because we only consider D f ≥ β ( r (cid:48) agg ) = + l g r (cid:48) agg (cid:34) . + . (cid:32) − . r (cid:48) agg l g (cid:33)(cid:35) , (12)where l g is the mean free path of gas molecules. The sec-ond term in the bracket in Equation (11) corrects for highReynolds (turbulent) flow, although it is mostly negligible forslowly settling aggregates considered in this study.2.1.3. Gas-drag Compression
An aggregate moving relative to the surrounding gas canexperience compression when the gas drag force acting on itis strong enough to cause internal restructuring. We employthe model of Kataoka et al. (2013b) to evaluate the filling fac-tor of an aggregate under gas-drag compression (see Kataokaet al. 2013b; Arakawa & Nakamoto 2016 for applications ofthe model to dust evolution in protoplanetary disks). We as-sume that compression occurs when the ram pressure P ram of the gas flow exceeds the static compressional strength P str of the aggregate. The compression thus proceeds until P ram becomes equal to P str . Based on the results of N -body simu-lations, Kataoka et al. (2013b) found that the static compres-sional strength can be written as P str = E roll r φ , (13)where E roll is the rolling energy already introduced in Sec-tion 2.1.2. The ram pressure can be evaluated as the dragforce per cross section of the aggregate. For an aggregatesetting in an atmosphere at a terminal velocity, the drag forceis equal to the gravity m agg g , where g is the gravitational ac-celeration. Thus, P ram is given by P ram ≈ m agg g π r = r agg g ρ mon φ. (14)Solving P str = P ram together with Equation (3) for φ , theequilibrium filling factor under gas-drag compression is ob- The reason can be easily understood for the special case of the freemolecular regime, for which the aerodynamic cross section is approximatelyequal to the projected area (Blum et al. 1996). For D f <
2, the projected areaincreases linearly with mass (e.g., Minato et al. 2006), but the “characteris-tic” cross section π r ∝ N / D f increases faster than mass ( ∝ N ). For D f ≈
2, the characteristic cross section ≈ N π r is only ∼ ff ers from the aerodynamic radius onlyby ∼ D f > tained as φ drag = N / (cid:32) g ρ mon r E roll (cid:33) / . (15)Equation (15) indicates that under gas-drag compression,the filling factor increases with aggregate mass. It is worthnoting that φ drag is independent of the ambient gas densitybecause the gas drag force balances with the gravity, whichdoes not depend on the gas density.2.1.4. A General Formula
For a given number of monomers, equivalent to the ag-gregate mass, one can calculate the equilibrium filling fac-tor from the highest one determined by the fractal growth,gas-drag compression, and collisional compression (Kataokaet al. 2013a), i.e., φ eq = max[ φ frac , φ drag , φ coll ] . (16)2.2. An Example: KCl Cloud Aggregates in GJ1214b
We here illustrate how the filling factor of CPAs in ansuper-Earth atmosphere evolves as they grow. We considerthe cloud of KCl solid particles in the super-Earth GJ1214b.It is assumed that the cloud has its base at P =
100 mbarand T =
700 K, where P is the atmospheric pressure. Thematerial density and surface energy are ρ mon = − and γ = .
11 J m − for KCl crystals (Westwood & Hitch 1963).We note that one cannot calculate the filling factor for colli-sional compression φ coll without a knowledge of filling factorof the aggregates before the collision, as the terminal veloc-ity depends on the aggregate density (see Equation 11). Thus,we first calculate φ eq only from φ frac and φ gas , and then φ coll is calculated with the obtained φ eq .We find that the internal density of CPAs can be lower thanthe material density by several orders of magnitude. The evo-lution pathways of the equilibrium filling factor for r mon = .
01, 0.1, and 1 µ m are shown in Figure 1. Here the equilib-rium filling factor is expressed as a function of the number ofmonomers making up the aggregates, N = m agg / m mon . Onecan see that the aggregates are highly porous, with φ eq (cid:46) . N . For small N , both gas-drag andcollisional compression are negligible and the filling factoris determined by fractal growth. Once an aggregate sizeexceeds a certain value, either collisional or gas-drag com-pression sets in. For all monomer sizes shown in Figure 1( r mon = . µ m), gas-drag compression always domi-nates over collision compression. Collisional compressionis important for larger monomer sizes and occurs only for r mon (cid:38) µ m around at the cloud base. No matter which com-pression mechanism dominates, the filling factor increaseswith N , and hence with aggregates mass. Nevertheless, thefilling factor never exceeds 0.1 as long as the monomer massis in the range 10 (cid:46) N (cid:46) . The results thus demonstrate louds of F luffy A ggregates Number of Monomers E q u ili b r i u m F illi n g F a c t o r F r a c t a l G r o w t h G a s - d r a g C o m p r e s s i o n m m m m m ( r mon = 0.01 m) fracdragcolleq Number of Monomers m m m m ( r mon = 0.1 m) Number of Monomers m m m ( r mon = 1 m) Figure 1.
Equilibrium filling factor of KCl particle aggregates at the base of the KCl cloud in the super-Earth GJ1214b. The left, center, andright panels are for monomer radii r mon = .
01, 0 .
1, and 1 µ m, respectively. The orange, blue, green, and black lines show the filling factorsdetermined by fractal growth ( φ frac ; Equation 5), gas-drag compression ( φ drag ; Equation 15), collisional compression ( φ coll ; Equation 9), and allof them ( φ eq ; Equation 16), respectively. The aggregate radius r agg = .
01, 0 .
1, 1, 10, and 100 µ m are denoted as the triangles. the importance of considering the porosity of mineral cloudaggregates.2.3. Analytic Estimates of Compression Threshold Sizes
To further elaborate how the porosity of CPAs evolve ingeneral cases, we here analytically estimate the thresholdsizes at which the compression sets in.2.3.1.
Gas-drag Compression Threshold
Comparison between Equations (5) and (15) shows that φ drag exceeds φ frac when the number of monomers satisfies N > (cid:32) π γξ crit ρ mon g (cid:33) / r − , (17)where we use Equation (7). Since r agg = N / r mon for D f = D f = r drag = (cid:32) π γξ crit ρ mon g (cid:33) / ≈ µ m (cid:18) g
10 m s − (cid:19) − / (cid:32) ρ mon − (cid:33) − / (cid:18) γ . − (cid:19) / . (18)It is worth noting that r drag is independent of the monomersize and only depends on material properties and planetarygravity. Equation (18) indicates that gas-drag compressionis responsible to aggregates larger than tens micron, whileit will be responsible to micron-sized aggregates on high-gravity objects, such as brown dwarfs.2.3.2. Collisional Compression Threshold
We here estimate the threshold size at which fractal ag-gregates begin to be compressed by high-energy collisions.Since the thermal kinetic energy k B T ∼ − J ( T / E roll ∼ − J ( γ/ . − )( r mon / µ m), one can con-sider that only relative velocity from gravitational settling in-duces collisional compression. For small fractal aggregates,the second term in the bracket in Equation (11) is negligible,and thus we approximately have v (cid:48) t ≈ gr (cid:48) ρ (cid:48) agg η β. (19)For fractal aggregates of D f =
2, we also have ρ (cid:48) agg ≈ ( r mon / r (cid:48) agg ) ρ mon , r (cid:48) agg = − / r agg , and m agg = ( r agg / r mon ) m mon ,where r agg is the radius of the newly formed aggregate. Sub-stituting ∆ v t ≈ (cid:15) v (cid:48) t with these expressions into Equation (6),the collisional energy of a settling-induced collision is givenby E imp ≈ m mon gr ρ mon η β . (20)Collisional compression occurs ( φ coll > φ frac ) when E imp > . E roll (see Equations (5) and (9)). For r agg (cid:29) l g ( β ≈ r coll = ηρ mon g (cid:114) . E roll m mon / ≈ µ m (cid:18) g
10 m s − (cid:19) − / (cid:32) ρ mon − (cid:33) − / (cid:32) r mon µ m (cid:33) − / (21) × (cid:18) γ . − (cid:19) / (cid:18) T (cid:19) / , where we have used the dynamic viscosity for hydrogen-richatmospheres η = . × − Pa s √ T [K] (Woitke & Helling2003). In the opposite limit of r agg (cid:28) l g , for which β ≈ . l g / r (cid:48) agg ≈ . l g / r agg , we obtain the threshold size of r coll = P πρ mon gv th (cid:114) . E roll m mon (22) O hno , O kuzumi , & T azaki ≈ µ m (cid:18) g
10 m s − (cid:19) − (cid:32) ρ mon − (cid:33) − / (cid:32) r mon µ m (cid:33) − × (cid:18) γ . − (cid:19) / (cid:18) v th − (cid:19) − (cid:18) P
100 mbar (cid:19) . Here, we have used l g = η/ ( ρ g v th ) and ρ g = (8 /π ) P / v ,where v th = (cid:112) k B T /π m g is the mean thermal velocity of gasmolecules and m g is the mass of a gas molecule. VERTICAL STRUCTURE OF FLUFFY-AGGREGATECLOUDS3.1.
Model
To demonstrate how the porosity evolution a ff ects thecloud structures, we calculate the vertical transport andgrowth of cloud particles using the double-moment bulkscheme described by Ohno & Okuzumi (2018). The modeladopts a 1D Eulerian framework and calculates the verticaldistributions of the mass density ( ρ c ) and number density ( n c )of the cloud particles. The model assumes that the mass dis-tribution of particles is narrowly peaked at the characteristicmass m agg dominating the total cloud mass. In this context,the mass and number densities are related as ρ c = m agg n c .3.1.1. Prescription of Nucleation and Condensation
Formation of the flu ff y-aggregate cloud will be triggeredby the formation of monomers via nucleation followed bycondensation (Figure 2). The processes will determine thesize of monomers, which predominantly control the poros-ity evolution and thus the particle growth. However, micro-physical processes associated to the monomer formation—especially the nucleation of initial condensates—are highlyuncertain for exoplanetary atmospheres. Although the clas-sical nucleation theory is available, as used in previous stud-ies (e.g., Helling & Fomins 2013; Powell et al. 2018; Gao& Benneke 2018), one should keep in mind that the theorysometimes deviates from the nucleation rate measured by nu-merical and laboratory experiments by several orders of mag-nitudes (e.g., Ford 1997; Tanaka et al. 2011; Lee et al. 2018).In this study, we mimic the monomer formation by set-ting the size of monomers as a free parameter. For the sakeof simplicity, every monomer is assumed to have the samesize. We assume that the nucleation predominantly occurs atthe cloud base, and the formed condensate particles instan-taneously grow until all condensable vapor at the cloud baseis incorporate into the particles. In other words, we calculatethe growth of cloud particles in the region above which themonomer formation is completed (Figure 2).3.1.2. Aggregate Growth and Transport above the Cloud Base
The formed monomers are collided each other and growinto the flu ff y aggregates (Figure 2). The aggregates are thenmixed in the vertical direction by atmospheric circulation, Nucleation & Condensation
Low-speed collision (cid:31)
Setting monomer size
High-speed collisionGas drag
Computation domain
Cloud base
Vertical transport
Figure 2.
Cartoon illustrating the formation of flu ff y-aggregateclouds. which we approximate as a di ff usion process in the horizontalaveraged sense (Parmentier et al. 2013; Charnay et al. 2015a;Zhang & Showman 2018a,b). The upward transport is lim-ited by the downward settling motion of the particles. Wetreat these processes using 1D vertical transport equationswith a collisional growth term (Ohno & Okuzumi 2018), ∂ n c ∂ t = ∂∂ z (cid:34) n g K z ∂∂ z (cid:32) n c n g (cid:33) + v t n c (cid:35) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ n c ∂ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) coll , (23) ∂ρ c ∂ t = ∂∂ z (cid:34) ρ g K z ∂∂ z (cid:32) ρ c ρ g (cid:33) + v t ρ c (cid:35) , (24)where n g is the atmospheric gas number density and | ∂ n c /∂ t | coll is the decrease of the aggregate number densitydue to collisional growth. We use the eddy di ff usion coe ffi -cient K z for GJ1214b derived by Charnay et al. (2015a): K z = K (cid:18) P (cid:19) − / , (25)where K is the eddy di ff usion coe ffi cient at 1 bar dependingon the atmospheric metallicity, as listed in Table 1.Collisional growth is induced by Brownian motion (coag-ulation hereafter) and di ff erential gravitational settling (coa-lescence hereafter). We write | ∂ n c /∂ t | as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ n c ∂ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) coll = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ n c ∂ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) coag + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ n c ∂ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) coal , (26)where the first and second terms represent the contributionfrom coagulation and coalescence, respectively. One can ap-ply the same formula of collisional growth terms for spheres louds of F luffy A ggregates r agg , the two terms can bewritten as (e.g., Rossow 1978) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ n c ∂ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) coag = min (cid:115) π k B Tm agg r n , k B T β η n (27)and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ n c ∂ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) coal ≈ π r n (cid:15) v t E , (28)where (cid:15) = . E is the collection e ffi ciency originated fromthe fact that an aggregate strongly coupled to the ambient gascannot collide with another aggregate. We use the expression(Guillot et al. 2014) E = max[0 , − . − / ] , (29)where St ≡ ( v t / g ) / ( r agg /(cid:15) v t ) is the Stokes number.3.1.3. Numerical Procedures
We consider that the cloud particles are in solid form andmade of pure KCl, which is a major condensable speciesformed in warm ( T = (cid:46) . (cid:46)
900 K,see e.g., Gao & Benneke 2018) are well below the triple-pointpressure and temperature of KCl (140 bars and 1041 K, Ro-drigues & Silva Fernandes 2007). Thus, the KCl clouds arelikely made of solid particles that could grow into an aggre-gate. We suppose a hypothetical planet that has the PT pro-file and surface gravity ( g = .
93 m s − ) of GJ1214b. The PTprofile is calculated by an analytical model of Guillot (2010)for cloud-free atmospheres as applied in Ohno & Okuzumi(2018), but we additionally include the e ff ect of the convec-tive adjustment by setting the adiabatic lapse late g / c p as anupper limit of a temperature gradient.To obtain the vertical profiles of ρ c and n c , we solve Equa-tions (24) and (23) until the system reaches a steady state.The sizes of aggregates are calculated by using the equi-librium filling factor from Equation (16) at each time step.We note that the collisional compression should occur onlywhen the particle collisions dominate over the vertical trans-port. Otherwise, the compression can occur without colli-sion, which is clearly unrealistic. To take into account it, weswitch o ff the collisional compression if the vertical mixingtimescale τ mix ≡ H / K z is shorter than the collisional growthtimescale | d log n c / dt | − . The upper boundary condition is setto zero-flux, while the flux at the lower boundary is calcu-lated assuming that n c / n g and ρ c /ρ g are constant at the cloudbase. Since we have assumed that all condensable vapor isincorporated in the cloud particles at the cloud base (Section Table 1.
Fiducial Parameters of This Studymetallicity µ g q v , KCl (mol / mol) K (m s − ) ∆ z (km)1 × solar 2.3 1.83 × − . × × solar 2.5 1.80 × − . × × solar 4.3 1.70 × − . × × solar 16.7 1.20 × − . × ρ c ( P b ) = ρ s ( P b ) , (30)where P b is the cloud-base height in pressure and ρ s is thesaturation vapor density of KCl, which is calculated by thesaturation vapor pressure described in Morley et al. (2012).For a given monomer radius, the number density of cloudparticles at the lower boundary is also calculated as n c ( P b ) = ρ s ( P b )4 π r ρ p . (31)The top and bottom of the computation domain are im-posed at 10 − bar and the cloud-base height, which is de-termined by the volume mixing ratio of KCl vapor q v , KCl listed in Table 1 and the saturation vapor pressure. The ver-tically coordinate z is discretized into linearly spaced bins,depending on the atmospheric metallicity (Table 1). Thetime increment is calculated at each time step so that thefractional decrease of n c does not exceed 0 .
5, i.e., ∆ t ≤ . × | ∂ log n c /∂ t | − . 3.2. Results
We here demonstrate how the porosity evolution a ff ectsthe vertical profiles of KCl clouds. Figure 3 shows the ver-tical distribution of the size r agg , cloud mass mixing ratio q c = ρ c /ρ g , and filling factor of aggregates φ eq for variousmonomer sizes and atmospheric metallicities. We also plotthe vertical profiles of compact ( D f =
3) sphere clouds forcomparison. The left panels of Figure 3 show that the cloudparticles produced at the cloud base grow locally until thetimescale of collisional growth becomes comparable to thevertical di ff usion timescale. Well above the cloud base, noappreciable growth occurs because the growth timescale in-creases with height (Ohno & Okuzumi 2018; Powell et al.2018; Gao & Benneke 2018). Notably, the cloud mass mix-ing ratio for submicron monomer cases is high even at a veryhigh altitude of P < − bar as compared to the case ofcompact-sphere clouds. The reason for this will be explainedin a later part of this section.We note that the aggregate sizes in upper atmospheres maydecrease with height in reality, as seen in other studies (Gao& Benneke 2018; Ormel & Min 2019). The trend is not cap-tured in our calculations where the particle sizes are constantat upper atmospheres. This is caused by the fact that our O hno , O kuzumi , & T azaki P r e ss u r e [ b a r ] r drag r mon = 1 mr mon = 0.1 mr mon = 0.01 m Compact 1e-071e-061e-050.00010.0010.01 × s o l a r P r e ss u r e [ b a r ] × s o l a r P r e ss u r e [ b a r ] × s o l a r Aggregate Radius [ m ] P r e ss u r e [ b a r ] Mass Mixing Ratio Volume Filling Factor × s o l a r Figure 3.
Vertical structure of a KCl cloud in GJ1214b from compact and flu ff y aggregate models. The left, center, and right columns showthe radius r agg , mass mixing ratio ρ c /ρ g , volume filling factor of CPAs, respectively. The top, middle, and bottom rows are for atmosphericmetallicities of 1 × , 10 × , 100 × , and 1000 × solar, respectively. The vertical axes are atmospheric pressure for all panels. The light-green, green,and dark-green lines show the profiles for r mon =
1, 0 .
1, and 0 . µ m, respectively. The dotted lines also show the profiles for compact-sphereclouds ( D f =
3) for reference. The black dash-dot lines in the left column denote the compression radius r drag given by Equation (18). louds of F luffy A ggregates ff ect is presumably not crucial forslowly settling CPAs. This is because the e ff ective size be-comes nearly constant in vertical, as seen in our calculations,when the particles have su ffi ciently small sizes and thus smallsettling velocity (see e.g., Figure 4 of Gao et al. 2018).The trend of vertical size distribution is appreciably di ff er-ent between compact-sphere and flu ff y-aggregate cases. Forcompact-sphere case, the particle size well above the cloudtop decreases with decreasing monomer size r mon because ahigher number density at the cloud base (which correspondsto a smaller monomer size at the base; see Equation 31) leadsto a smaller particle size above the base (Gao et al. 2018;Ohno & Okuzumi 2018; Ormel & Min 2019). The trend isoriginated from the fact that, for a given mass mixing ratio,a total amount of condensing materials on each particle de-creases with increasing a number density. The coagulation ise ff ective for a high number density, but halted once the parti-cle size exceeds the threshold above which the number den-sity becomes too low to cause the collisions (see Section 3.2of Ohno & Okuzumi 2018). By contrast, for flu ff y-aggregateclouds, the aggregate radius at high altitude increases withdecreasing monomer radius r mon . As shown below, this isbecause the coagulation timescale is a function of aggre-gate mass and because aggregates made of smaller monomershave to grow to larger in size to obtain a certain mass. Foraggregates larger than the mean free path of themselves, thetimescale of coagulation growth τ coag ≡ | d log n c / dt | − isapproximately given by τ coag ≈ η k B T n c , (32)which follows from Equation (27). Using the relation q c ρ g = m agg n c , we obtain τ coag ≈ η m agg k B T ρ g q c , (33)which indicates that the coagulation timescale is indepen-dent of aggregates properties other than m agg . Since the finalsize is determined by the balance between coagulation andmixing timescales ( τ coag = τ mix ), the final aggregate mass isgiven by m agg ≈ k B T H η K z ρ g ( P ∗ ) q c , (34)where P ∗ is the pressure level where the growth is completed.For D f =
2, the aggregate mass scales as m agg ∝ r r mon ,and hence the final aggregates radius increase with decreas-ing monomer size.The aggregate size slightly increase with increasing on at-mospheric metallicity. In the case of r mon = . µ m, for example, the aggregate radii at high altitude are 2, 3, 5, and5 µ m for the metallicities of 1 × , 10 × , 100 × , and 1000 × solarabundance, respectively. The increase of the aggregate size iscaused by the fact that a higher atmospheric metallicity ( q c atthe cloud base) leads to a higher cloud density that facilitatescoagulation growth. This can also be seen from Equation(34), which shows m agg ∝ q c . However, the aggregate sizealso depends on the mixing timescale H / K z (see Equation34), which decreases with increasing the atmospheric metal-licity in our parameter set. This e ff ect substantially cancelsout the e ff ects of q c , which explains the weak metallicity-dependence of the aggregate size in Figure 3.The key result of this section is that the aggregates neverexperience compression in the cases studied here. The dot-dashed lines in the left panels of Figure 3 show the thresholdsize for the gas-drag compression r drag (Equation 18) abovewhich the aggregates leave fractal growth. Figure 3 showsthat the particle growth is insu ffi cient to reach the thresholdsize for the gas-drag compaction. Although the collisionalcompression can operate on micron-size aggregates in upperatmospheres ( P (cid:46) − bar, see Equation 22), it does nottake place there because the number density is too low tocause the particle collision, i.e., τ coll (cid:29) τ mix . As a result,aggregates are fractal ( D f ≈
2) even at high altitude.The absence of the compression enables us to evaluate thevertical extent of clouds. The cloud particle aggregates canascend to the height of τ mix ∼ τ fall , where τ fall ≡ H / v t is thefalling timescales (e.g., Charnay et al. 2015a). Assuming l g (cid:29) r agg for upper atmospheres, the terminal velocity can beapproximated as v t ≈ ρ mon g ρ g v th r mon , (35)where we use the relation r agg ρ agg = r mon ρ mon for D f = τ mix = τ fall about the pressure, we find the pressurelevel P top to which cloud particles can ascend: P top ≈ ρ mon g H r mon v th K z (36) ∼ .
03 mbar (cid:32) r mon . µ m (cid:33) (cid:32) ρ mon − (cid:33) (cid:18) K z m s − (cid:19) − (cid:18) v th − (cid:19) , where we use v th = (cid:112) (8 /π ) gH . Equation (36) indicates that P top is independent of the size of cloud particle aggregates r agg . This explains why the cloud particle aggregates madeof smaller monomers can ascend higher altitude in Figure 3despite their very large sizes ( (cid:29) µ m). TRANSMISSION SPECTRUM WITHFLUFFY-AGGREGATE CLOUDSThe optical properties of flu ff y aggregates are considerablydi ff erent from those of compact spheres. In addition, flu ff yaggregates are able to ascend to very high altitude as demon-0 O hno , O kuzumi , & T azaki strated in Section 3. In this section, we investigate how thesee ff ects influence the transmission spectra of exoplanets.4.1. Method
We calculate synthetic transmission spectra of GJ1214b, asuper-Earth believed to be covered by clouds (and / or hazes)in very high altitude (e.g., Kreidberg et al. 2014), using thevertical profiles of KCl clouds obtained in Section 3. Wedo this by calculating the wavelength-dependent transit depth D ( λ ) of a planet, which can be expressed as (e.g., Heng &Kitzmann 2017) D ( λ ) = π R + π (cid:82) ∞ R [1 − exp ( − τ s )] rdr π R ∗ , (37)where R is the reference transit radius and τ s is the opti-cal depth for slant viewing geometry, called the slant opticaldepth, and r is the distance from the center of the planet. Wetake R to be the radius at the pressure level of 10 bar follow-ing previous studies (e.g., Kreidberg et al. 2015). The slantoptical depth τ s is calculated by integrating the extinction bygas molecules and cloud particles along the observer’s line ofsight (e.g., Fortney et al. 2003): τ s ( r ) = (cid:90) ∞ r ( α g + α c ) r (cid:48) dr (cid:48) √ r (cid:48) − r , (38)where α g and α c are the extinction e ffi ciencies of gasmolecules and cloud particles, respectively. The stellarradius R ∗ and planet’s semi-major axis a are taken to be R ∗ = . R sun and a = .
014 au, which are the values forGJ1214b from the Exoplanet eu catalog .4.1.1. Gas Opacity
To evaluate the gas opacity, we calculate the mixing ra-tio of gas molecules using the open-source ThermochemicalEquilibrium Abundances (TEA) code (Blecic et al. 2016).The TEA calculates the gas mixing ratio in thermochemicalequilibrium for given temperature, pressure, and elementalabundances based on Asplund et al. (2009) using the Gibbsfree-energy minimization method. Following Freedman et al.(2008, 2014), we take into account the molecular absorptionof H , H O, CH , CO, CO , NH , H S, and PH as well asthe Rayleigh scattering of the molecules. We calculate theabsorption and scattering cross sections of the molecules fol-lowing the method of Kawashima & Ikoma (2018) with theline list of HITRAN2016. The Voigt function is calculated bythe polynomial expansion (Kuntz 1997; Ruyten 2004), andthe total internal partition function sums are calculated byTIPS code (Gamache et al. 2017). We refer readers to the rel-evant literature (e.g., Rothman et al. 1998; Sharp & Burrows http: // exoplanet.eu ffi cients (e.g., Tennyson & Yurchenko2018; Gharib-Nezhad & Line 2019) remain for future stud-ies, as our current focus is to study how the flu ff y-aggregateclouds influence the transmission spectra.4.1.2. Aggregates Opacity
The Mie theory (e.g., Bohren & Hu ff man 1983) is usu-ally used for the calculations of the opacity of spherical par-ticles, but is no longer valid for irregular aggregates. TheMie theory coupled with the e ff ective medium theory is oneof the ways to calculate the aggregate opacity (Marley et al.2013). However, this approach also fails to reproduce scatter-ing properties of an aggregate when the relevant wavelengthis much smaller than the aggregate (Tazaki et al. 2016; Tazaki& Tanaka 2018). Aggregates potentially grow to 1–10 µ m insize as shown in Section 3, while current and future obser-vations mainly use shorter wavelengths such as 1 . . µ mfor HST / WFC3, 0 . µ m for JWST / NIRSpec (Batalha et al.2017), and 1 . . µ m for ARIEL (Tinetti et al. 2016).Therefore, the e ff ective medium theory is still not a good ap-proximation especially for upcoming observations.To properly calculate the aggregate opacity, we apply themodified mean field (MMF) theory (Tazaki & Tanaka 2018).The MMF theory is based on the Rayleigh-Gans-Debye(RGD) theory that calculates the interference of single-scattered waves from every monomer by taking the aggregatestructure into account (Tazaki et al. 2016) with modificationsfor multiple scattering within an aggregate using the meanfield assumption (Berry & Percival 1986). The MMF the-ory successfully reproduces the extinction, absorption, andscattering opacities of aggregates calculated by the rigorousT-matrix method in a wide range of wavelength (Tazaki &Tanaka 2018). For calculations, we apply the Gaussian cut-o ff for the two-points correlation function that specifies theaggregate structure (Tazaki et al. 2016).Figure 4 shows the extinction opacity of KCl aggregatesof D f = ff erent aggregate sizes r agg and monomersizes r mon . The refractive index of KCl is taken from Palik(1985) compiled by Kitzmann & Heng (2018). In the exam-ples presented here, the extinction opacity is dominated byscattering in the wavelength range λ ∼ . µ m. At longerwavelengths, absorption dominates over scattering, and theabsorption peak of KCl appears at λ ∼ µ m. It is worthnoting that the absorption feature is visible even if aggregatesize is very large, as seen in the case of r agg = µ m. Thisis because, unless the multiple scattering becomes dominant,the absorption cross section of an aggregate is the sum ofthe absorption of every monomer, and thus the wavelengthdependence is the same as that of an individual monomer(Berry & Percival 1986; Tazaki & Tanaka 2018). louds of F luffy A ggregates Wavelength [ m] E x t i n c t i o n O p a c i t y [ c m g ] = 2 r mon 24 ( r mon = 0.1 m) Aggregate Size Dependency r agg =0.3 mr agg =3 mr agg =30 m Wavelength [ m] ( r agg = 10 m) Monomer Size Dependency r mon =0.01 mr mon =0.1 mr mon =1 m Figure 4.
Extinction opacity of KCl aggregates with D f = ff erent aggregate sizes r agg and monomer sizes r mon , calculated by the MMF theory. The left panel is for aggregates of fixed r mon = . µ m and di ff erent r agg , wheres the right panel is for fixed r agg = µ m and di ff erent r mon . The wavelength corresponding to 2 π r mon and 2 π r agg are denoted as dotted lines and filled circles, respectively. According to the MMF theory, the optical properties ofan aggregate behave di ff erently among three wavelengthregimes λ (cid:28) π r mon , 2 π r mon (cid:28) λ (cid:28) π r agg , and λ (cid:29) π r mon .In the first regime, geometric optics applies to the constituentmonomers, and the scattering cross section is approximatelygiven by σ s ∼ π r , independent of wavelength. In the op-posite limit of λ (cid:29) π r agg , the Rayleigh limit applies to theaggregate, and the scattering cross section obeys the well-known law σ s ∝ λ − . I the left panel of Figure 4, this can beseen in the case of r agg = . µ m, at λ ∼ µ m.The intermediate regime 2 π r mon (cid:28) λ (cid:28) π r agg providesunique opacity properties for aggregates. For this regime, wefind that the scattering opacity scales with wavelength de-pendence as σ s ∝ λ − (see Figure 4). In this intermediateregime, the scattered wave by an aggregate is a superpositionof singly scattered waves from individual monomers, and thescattering cross section of a D f = σ s ∝ r r λ − log (16 π r / b λ ) , (39)where b is a constant order of unity. This explains the scatter-ing slope for r agg = µ m and 30 µ m in the left panel of Fig-ure 4. The unique scattering slope is caused by interferenceamong the scattered waves from individual monomers. Thescattered waves toward large scattering angles ( (cid:38) λ/ π r agg )cancel out because of the presence of waves with oppositephases, leading to the λ − dependence (Kataoka et al. 2014).4.2. Cloud-top Pressure
Before showing the synthetic spectra, we investigate thecloud-top pressure, defined as the pressure level at which thecloud becomes optically thick along the line of sight of anobserver (i.e., τ s = ff y-aggregate clouds as a func-tion of wavelength for di ff erent monomer sizes and the at-mospheric metallicities. In general, the cloud top is locatedat a lower atmosphere for longer wavelength because thescattering opacity decreases with increasing wavelength for λ > π r mon (see Figure 4). We note that the cloud top hardlyexceeds the altitude of τ mix = τ drag (dashed lines in Figure 5)for parameter ranges examined in this study. In near-infraredwavelength, the cloud-top height increases with decreasingmonomer size as long as r mon (cid:38) . µ m, as CPAs consti-tuted by smaller monomers ascend to higher altitude. On theother hand, the cloud-top height also decreases with decreas-ing monomer size for r mon (cid:46) . µ m. This opposite trend iscaused by the monomer size dependence of aggregate scat-tering opacity. Using Equation (39) and an aggregate mass m agg ∝ r r mon , one can see that the scattering mass opacityfollows κ s ≡ σ s m agg ∝ r mon λ − log (16 π r / b λ ) . (40)Thus, the scattering mass opacity decreases with decreas-ing monomer size. On the other hand, in the limit of smallmonomer size (i.e., τ mix (cid:28) τ fall ), the cloud mass mixing ratio q c is vertically constant and independent of monomer size(see Figure 3). Therefore, the scattering e ffi ciency ( α c = ρ g q c κ s ) and thus the cloud-top height decrease with decreas-ing monomer size for very small r mon .We also find that the cloud-top pressure tends to besmaller for higher atmospheric metallicities. This is be-cause the cloud mass mixing ratio increases with increas-2 O hno , O kuzumi , & T azaki C l o u d - T o p P r e ss u r e [ b a r ] mix < fall ( r mon = 0.01 m) mix < fall ( r mon = 0.1 m) mix < fall ( r mon = 1 m) (1×solar) r mon = 1 m r mon = 0.1 m r mon = 0.01 m (10×solar) Wavelength [ m] C l o u d - T o p P r e ss u r e [ b a r ] (100×solar) Wavelength [ m] (1000×solar) Figure 5.
Cloud-top pressure of the flu ff y-aggregate clouds as a function of wavelength. The dark-green, green, and light-green lines are for r mon =
1, 0 .
1, and 0 . µ m, respectively. The dashed lines indicate the pressure level of τ mix = τ fall for each monomer size. Each panel exhibitsthe result for di ff erent atmospheric metallicity. ing the metallicity. Specifically, the cloud-top pressure forthe atmospheric metallicity of 100 and 1000 × solar reach P ∼ − bar at near-infrared wavelength if the monomeris smaller than 1 µ m. It is worth pointing that the flu ff y-aggregate clouds can produce the cloud top at the pressurelevel comparable to that retrieved from the observations ofHST / WFC3 for GJ1214b (Kreidberg et al. 2014), which washardly attained by the compact-sphere clouds in our previousstudy (Ohno & Okuzumi 2018).4.3.
Synthetic Spectra
We begin by studying how the aggregate structure a ff ectstransmission spectra. For later convenience, we introduce ametric characterizing the spectral slope, given by (e.g., Line& Parmentier 2016) S ≡ dD ( λ ) d log λ = π R p H π R ∗ α, (41) where α is the power-law index of the extinction e ffi ciency ofatmosphere, i.e., ( α g + α c ) ∝ λ α . For example, α = − α = H instead of the cloud scale height. Strictly speaking, thethe could scale height is equal to H only when the particlesettling timescale is much longer than the mixing timescale(see e.g., Equation (33) of Ohno & Okuzumi 2018). Thecluod scale height is smaller than H at high altitude wherethe cloud mass mixing ratio decreases with increasing height,implying τ fall (cid:46) τ mix . However, cloud particles at such veryhigh altitude are usually so depleted that their contribution totransmission spectra is small. In fact, as shown in previoussection, the cloud top hardly exceeds the the pressure levelof τ mix = τ fall for the parameter space examined in this study.Therefore, Equation (41) o ff ers a reasonable diagnosis of thespectral slope. louds of F luffy A ggregates Wavelength [ m] T r a n s i t D e p t h [ % ] ( = 4) Compact-Sphere Clouds
Cloud free r mon = 1 m r mon = 0.1 m r mon = 0.01 m Wavelength [ m] ( = 2)Fluffy-Aggregate Clouds
Figure 6.
Synthetic transmission spectra of GJ1214b with a solar-metalicity atmosphere, from compact-sphere and flu ff y-aggregate models(left and right panels, respectively) presented in Section 3. The purple, red, and orange lines are from the models assuming the monomer radiiof r mon = µ m, 0 . µ m, and 0 . µ m, respectively. In the compact-sphere models, the monomer size merely determines the number density ofcloud particles at the cloud base (see Equation 31). For comparison, the spectrum for a cloud-free atmosphere is also shown by the black line.The gray dashed lines denote the spectral slopes corresponding to α ∝ λ − for the left panel, and ∝ λ − for the right panel (see Equation 41). Forclarity, the spectral resolution is binned down to λ/ ∆ λ ≈ / WFC3.
Figure 6 shows the synthetic transmission spectra ofGJ1214b with a solar-metallicity atmosphere and with a KClcloud obtained from compact-sphere and flu ff y-aggregatemodels (see the top rows of Figure 3 for the cloud verticalstructure). We set the reference radius to R = . R Earth sothat the cloud-free solar-composition atmosphere producesthe planet-to-star radius ratio of R p / R ∗ ∼ .
115 (i.e., D ∼ . ff man 1983). For comparison, wealso plot the transmission spectrum for the cloud-free atmo-sphere, which exhibits molecular absorption signatures ofmainly H O molecules and the spectral slope in λ (cid:46) . µ mcaused by the Rayleigh scattering of H molecules. In theleft panel of Figure 6, the compact-sphere clouds producea floor of the transit depth at λ (cid:46) µ m. In the compact-sphere model, a cloud deck that is gray in visible is producedno matter how small the monomers at the cloud base are,because they always grow to (cid:38) µ m in size through coag-ulation as shown in Section 3 (see also Ohno & Okuzumi2018).The transmission spectrum for flu ff y-aggregate clouds ex-hibit a considerably di ff erent shape from that for compact-sphere clouds. Since the flu ff y-aggregate cloud is lofted tomuch higher altitude, the absorption features in the spectraare largely obscured as compared to the cases of the compact-sphere clouds except for the case of r mon = . µ m in whichthe e ff ect that decreases the cloud extinction e ffi ciency isimportant (Section 4.2). Furthermore, the flu ff y-aggregateclouds produce a spectral slope at λ (cid:46) µ m, in particular when the monomers are small. The spectrum for r mon = µ mis nearly identical between the flu ff y-aggregate and compact-sphere models because the monomers satisfy λ > π r mon atnear-infrared wavelengths. The spectral slope for r mon = . . µ m is well characterized by S ( α = − π r mon < λ < π r agg (see Section 4.1.2).Since the spectral slope with S ( α = −
2) originates fromthe scattering property of aggregates, it could potentially beused as an observational signature for CPAs when the at-mospheric scale height H is well constrained. We find thatthe slope with S ( α = −
2) also emerges for many other ma-terials that may build up mineral clouds on exoplanets (Ap-pendix A). However, caution should be taken regarding thisinterpretation because S ( α = −
2) may also be caused by thecombination of small and large compact spheres.Although the flu ff y aggregates can largely obscure themolecular features in visible to near-infrared, they are op-tically too thin to hide the features at longer wavelengths( λ (cid:38) µ m), as can be seen in Figure 7. This implies thatfuture transmission spectroscopy at λ (cid:38) µ m with JWSTand ARIEL could detect molecular features in super-Earthsthat look cloudy in visible and near-infrared.The transmission spectrum from the flu ff y-aggregatemodel also substantially depends on the atmospheric metal-licity. Figure 7 shows the transmission spectra from theflu ff y-aggregate model for various atmospheric metallicities,where R = . R Earth is assumed for every case. One cansee that the higher the atmospheric metallicity is, the flatterthe spectral slope is. This is because the gradient of spec-4 O hno , O kuzumi , & T azaki T r a n s i t D e p t h [ % ] (1×solar) Cloud free r mon = 1 m r mon = 0.1 m r mon = 0.01 m (10×solar) Wavelength [ m] T r a n s i t D e p t h [ % ] (100×solar) Wavelength [ m] (1000×solar)
Figure 7.
Synthetic transmission spectra of GJ1214b with a cloud of flu ff y KCl aggregates for various atmospheric metallicities. tral slope is proportional to the pressure scale height H (seeEquation 41), which decreases with increasing the atmo-spheric metallicity. The e ff ect is notable for (cid:38) × solarmetallicity, and the spectral slope is almost flat for (cid:38) × solar metallicity.4.4. Comparison with Observations of GJ1214b
Here, we compare our synthetic transmission spectra withthe observed transmission spectrum of GJ1214b. We calcu-late the cloud profiles as well as the synthetic spectra for theatmospheric metallicities of 1–1000 × solar abundances andmonomer sizes of 0 . µ m. We also vary the reference ra-dius R from 2 to 3 R Earth to be consistent with the observedplanet radius. The relative goodness-of-fit for each model isquantified by the reduced chi-square χ . The model free-dom is the number of data points minus three, the numberof the fitting parameters (atmospheric metallicity, monomersize, and reference radius). For instance, Morley et al. (2015)assumed that an acceptable model for GJ1214b produces χ < .
14 if only data points from the HST / WFC3 observa-tions (Kreidberg et al. 2014) are used.The left panel of Figure 8 shows the best-fit transmis-sion spectra for the metallicity of 1, 10, 100, and 1000 × solar abundance, compared with the observational datafor GJ1214b from HST / WFC3 (Kreidberg et al. 2014) andSpitzer / IRAC (Gillon et al. 2014). All available observa-tional data are also denoted as gray dots in the left panelof Figure 8. The right panel shows the best-fit spectra onlyfor HST / WFC3. For all data points (left panel), the small-est reduced chi-square for the atmospheric metallicities of1, 10, 100, and 1000 × solar abundance are χ = . .
04, 3 .
28, and 2 .
41, respectively. However, these χ val-ues are significantly a ff ected by the large scatter in the dataat visible wavelengths. If we only focus on the data pointof HST / WFC3 (Kreidberg et al. 2014) and Spitzer / IRAC(Gillon et al. 2014), which are less scattered than the visibledata, we obtain the reduced chi-squared of χ = . .
13, 6 .
62, and 2 .
03 for the metallicities of 1, 10, 100, and louds of F luffy A ggregates (cid:1) (cid:1)(cid:2)(cid:3) (cid:1) (cid:1)(cid:2)(cid:4) (cid:1) (cid:1)(cid:2)(cid:5) (cid:1) (cid:1)(cid:2)(cid:6) (cid:1) (cid:1) (cid:1) (cid:3) (cid:1) (cid:4) (cid:1) (cid:5) (cid:1) (cid:6) (cid:1) (cid:2) (cid:3)(cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:8) (cid:9)(cid:10) (cid:7) (cid:11) (cid:1) (cid:12) (cid:13) (cid:14) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:4)(cid:6)(cid:7)(cid:8)(cid:9) (cid:1) (cid:10)(cid:11)(cid:12)(cid:13) (cid:1)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:1)(cid:12)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:1)(cid:12)(cid:12)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:1)(cid:12)(cid:12)(cid:12)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11) (cid:13)(cid:1)(cid:6)(cid:12)(cid:13)(cid:1)(cid:12)(cid:12)(cid:13)(cid:6)(cid:12) (cid:1) (cid:12) (cid:1) (cid:6)(cid:12) (cid:1) (cid:1)(cid:12)(cid:12) (cid:1) (cid:1)(cid:6)(cid:12) (cid:1) (cid:3)(cid:12)(cid:12) (cid:1) (cid:1)(cid:2)(cid:1) (cid:1) (cid:1)(cid:2)(cid:3) (cid:1) (cid:1)(cid:2)(cid:4) (cid:1) (cid:1)(cid:2)(cid:5) (cid:1) (cid:1)(cid:2)(cid:6) (cid:1) (cid:1)(cid:2)(cid:14) (cid:1) (cid:1)(cid:2)(cid:15) (cid:15) (cid:9) (cid:16) (cid:3) (cid:7) (cid:6) (cid:17) (cid:9) (cid:1) (cid:1) (cid:2) (cid:3)(cid:4) (cid:5) (cid:6) (cid:7) (cid:1) (cid:8) (cid:9)(cid:10) (cid:7) (cid:11) (cid:1) (cid:12) (cid:10)(cid:10) (cid:18) (cid:14) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:4)(cid:6)(cid:7)(cid:8)(cid:9) (cid:1) (cid:10)(cid:11)(cid:12)(cid:13) (cid:1)(cid:12)(cid:12)(cid:12)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:16) (cid:1) (cid:17)(cid:9)(cid:8)(cid:18)(cid:19)(cid:13)(cid:20)(cid:11)(cid:21)(cid:21)(cid:22)(cid:11)(cid:21)(cid:23)(cid:19)(cid:24)(cid:21)(cid:11)(cid:25) (cid:1) (cid:21)(cid:26) (cid:1) (cid:10)(cid:9)(cid:2) (cid:1) (cid:27)(cid:3)(cid:12)(cid:1)(cid:5)(cid:28) Figure 8.
Synthetic transmission spectra of GJ1214 b (colored lines) compared with the observational spectrum to date (black and gray points).The left panel shows all observed transit depth ranging from 0 . µ m and the best-fit spectra for comparisons with data of HST / WFC3(Kreidberg et al. 2014) and Spitzer / IRAC (Gillon et al. 2014). The right panel shows the comparisons with the data points of only HST / WFC3.The horizontal axis are wavelength, and the vertical axises are transit depth. The blue, green, orange, and red lines show the spectra with themetallicity of 1 × solar and with r mon = . µ m (reduced chi-square is χ = . × solar and r mon = . µ m ( χ = . × solarand r mon = . µ m ( χ = . × solar and r mon = . µ m ( χ = . × solar metallicity. The spectral resolution is binned down to λ/ ∆ λ ≈
100 for clarity.The gray dots exhibit currently available observational data (Bean et al. 2011; Croll et al. 2011; D´esert et al. 2011; de Mooij et al. 2012; Bertaet al. 2012; Murgas et al. 2012; Col´on & Gaidos 2013; Narita et al. 2013a,b; Fraine et al. 2013; Rackham et al. 2017). Specifically, the blackdots indicate the data from the observations by the HST / WFC3 (Kreidberg et al. 2014) and the Spitzer / IRAC (Gillon et al. 2014). × solar abundance, respectively. For a comparison withthe HST data only (right panel), the reduced chi-squared val-ues are χ = . .
50, 6 .
78, and 1 .
16 for atmosphericmetallicities of 1, 10, 100, and 1000 × solar abundance, re-spectively. Overall, a higher atmospheric metallicity leads toa smaller reduced chi-squared value. We also find that thepresence of the flu ff y-aggregate cloud appreciably improvesthe goodness-of-fit of the model as compared to the cloud-free case. For the comparison with the HST data as anexample, the cloud-free atmosphere with 1000 × solar metal-licity yields χ = .
44 (the black line in the right panelof Figure 8), whereas the model with the flu ff y-aggregateclouds yields χ = .
16. The reduced chi-square χ foreach parameter set is summarized in Figure 9.Our results show that the model with a higher atmosphericmetallicity yields a better match to the observational data.The high-metallicity atmospheres supply su ffi cient KCl con-densates, and the produced clouds can obscure the molecularfeatures if monomer size is su ffi ciently small, namely (cid:46) µ m.Indeed, the molecular absorption at around λ = . µ m, no-ticeable in cloud-free atmospheres even with 1000 × solarmetallicity, is significantly weakened by the cloud opacity(right panel of Figure 8). The spectral slope is also closerto the observed flat spectrum because of the relatively smallscale height. Notably, the model with 1000 × solar metallic-ity yields χ ν = .
16 for the comparisons with Kreidberg et al.(2014), which is comparable to the χ obtained by Gao &Benneke (2018) who assumed the eddy di ff usion coe ffi cient much larger than that predicted by 3D GCM (Charnay et al.2015a). Our results suggest that it would be able to explainthe observed spectra of GJ1214b in the range of K z predictedby the GCM, if the mineral cloud consist of flu ff y aggre-gates. We emphasize that as the metallicity is increased,the resulting synthetic spectrum better matches the transitdepth at mid-infrared wavelengths, especially at 4 . µ m, ob-served by the Spitzer / IRAC (see the left panel of Figure 8).This is thanks to the absorption of CO whose abundance in-creases with increasing the atmospheric metallicity (Moseset al. 2013).There are two reasons why the low-metallicity models (1and 10 × solar) fail to match the observations. The first is aninsu ffi cient cloud abundance: the mixing ratio of KCl in thelow-metallicity atmosphere is too low to produce su ffi cientlyopaque clouds. The second reason is that, more importantly,the spectral slope caused by the aggregate opacity is toosteep to match the flat spectrum observed by the HST / WFC3(Kreidberg et al. 2014) because of the large scale height (seeSection 4.3). Therefore, our flu ff y-aggregate cloud modelstill requires a small atmospheric scale height to explain theflat spectrum of GJ1214b, which is achieved by the high-metallicity atmosphere.Although the flu ff y-aggregate clouds potentially explainthe featureless spectrum of GJ1214b, we note that the CPAsneed to be constituted by monomers with r (cid:46) µ m (see Fig-ure 9). The monomer size is presumably controlled by theformation of condensation nuclei, namely the nucleation, and6 O hno , O kuzumi , & T azaki (cid:1)(cid:2)(cid:2) (cid:1) (cid:3)(cid:4)(cid:5)(cid:4) (cid:1) (cid:1)(cid:2) (cid:1)(cid:2)(cid:2) (cid:1)(cid:2)(cid:2)(cid:2) (cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14) (cid:1) (cid:15)(cid:11)(cid:5)(cid:4)(cid:2)(cid:2)(cid:13)(cid:14)(cid:13)(cid:5)(cid:16) (cid:1) (cid:17)(cid:18)(cid:8)(cid:7)(cid:2)(cid:4)(cid:12)(cid:19) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:10)(cid:1)(cid:2)(cid:11)(cid:12)(cid:2)(cid:11)(cid:1)(cid:2)(cid:11)(cid:2)(cid:12)(cid:2)(cid:11)(cid:2)(cid:1) (cid:1) (cid:2)(cid:3)(cid:2) (cid:4) (cid:5) (cid:6) (cid:1) (cid:7) (cid:8)(cid:9) (cid:10) (cid:11) (cid:12) (cid:1) (cid:13) (cid:14) (cid:4) (cid:15) (cid:20)(cid:21)(cid:22) (cid:1) (cid:23) (cid:1) (cid:21)(cid:9)(cid:13)(cid:5)(cid:24)(cid:11)(cid:12) (cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:5)(cid:7)(cid:16)(cid:8)(cid:19)(cid:20)(cid:11)(cid:7)(cid:21)(cid:22)(cid:13) (cid:1) (cid:6) (cid:1) (cid:1)(cid:23)(cid:20)(cid:23)(cid:12)(cid:1) (cid:1)(cid:2) (cid:1)(cid:2)(cid:2) (cid:1)(cid:2)(cid:2)(cid:2) (cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14) (cid:1) (cid:15)(cid:11)(cid:5)(cid:4)(cid:2)(cid:2)(cid:13)(cid:14)(cid:13)(cid:5)(cid:16) (cid:1) (cid:17)(cid:18)(cid:8)(cid:7)(cid:2)(cid:4)(cid:12)(cid:19) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:10)(cid:1)(cid:2)(cid:11)(cid:12)(cid:2)(cid:11)(cid:1)(cid:2)(cid:11)(cid:2)(cid:12)(cid:2)(cid:11)(cid:2)(cid:1) (cid:20)(cid:21)(cid:22)(cid:25)(cid:26)(cid:27)(cid:28)(cid:29) (cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:5)(cid:7)(cid:16)(cid:8)(cid:19)(cid:12)(cid:11)(cid:7)(cid:21)(cid:22)(cid:13) (cid:1) (cid:6) (cid:1) (cid:1)(cid:23)(cid:20)(cid:23)(cid:12)(cid:1) (cid:1)(cid:2) (cid:1)(cid:2)(cid:2) (cid:1)(cid:2)(cid:2)(cid:2) (cid:1)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14) (cid:1) (cid:15)(cid:11)(cid:5)(cid:4)(cid:2)(cid:2)(cid:13)(cid:14)(cid:13)(cid:5)(cid:16) (cid:1) (cid:17)(cid:18)(cid:8)(cid:7)(cid:2)(cid:4)(cid:12)(cid:19) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:10)(cid:1)(cid:2)(cid:11)(cid:12)(cid:2)(cid:11)(cid:1)(cid:2)(cid:11)(cid:2)(cid:12)(cid:2)(cid:11)(cid:2)(cid:1) (cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:1) (cid:1)(cid:2)(cid:2) (cid:30)(cid:11)(cid:31) (cid:14)(cid:11)(cid:31) (cid:1) (cid:28)(cid:10)(cid:13)(cid:25)(cid:21)! (cid:4)(cid:12)(cid:11) Figure 9.
Reduced chi-squared values for the synthetic transmission spectra of GJ1214b as a function of the monomer radius and atmosphericmetallicity. The left panel shows the chi-squred values obtained by fitting models to all observational data. The middle panel shows the resultsfrom the analysis that only uses the data of HST / WFC3 (Kreidberg et al. 2014) and Spitzer / IRAC (Gillon et al. 2014). The right panel is fromthe analysis that only uses the data of HST / WFC3. subsequent condensation growth that keeps a spherical shape(Lavvas et al. 2011). If one adopts the classical nucleationtheory, the homogeneous nucleation followed by condensa-tion yields KCl particles with the e ff ective size of ∼ µ m(Gao & Benneke 2018). This is substantially larger thanthe required monomer size. This could suggest that classi-cal nucleation theory underestimates the nucleation rate ofKCl, because a larger number of condensation nuclei gener-ally leads to a smaller monomer size (Gao et al. 2018; Ohno& Okuzumi 2018). In fact, Lee et al. (2018) reports that clas-sical nucleation theory underestimates the nucleation rate ofTiO . Alternatively, a number of stable, small nuclei could beproduced by the heterogeneous nucleation of ZnS onto KCl(Gao & Benneke 2018), although its nucleation rate dependson the desorption energy of ZnS that is currently unknown.Laboratory studies of nucleation and condensation would beimportant to predict the monomer size in exoplanetary atmo-spheres, which in turn test the scenario of the flu ff y-aggregatecloud for GJ1214b. DISCUSSION5.1.
Model Caveats
In this study, we have adopted simplified porosity and mi-crophysical model. The models are useful to clarify the ef-fects of the porosity evolution, but involves some caveats be-cause of its simplicity. In what follows, we state the caveatsof our model and discuss their possible impacts on the results. 5.1.1.
Validity of D f = for Other Size Distributions The most strong assumption of our porosity model maybe the fractal dimension of 2 for the fractal growth (Sec-tion 2.1.1). We have adopted this assumption since our cloudmicrophysical model assumes the narrowly peaked size dis-tribution, for which the equal-sized collision is a dominantgrowth process. However, the cloud particles could have dif-ferent shape of size distributions (Powell et al. 2018; Gao &Benneke 2018), and the monomer-aggregate collision mightbe dominant. In that case, CPAs grow into more sphericalshapes (e.g., D f ≈ D f = D f = m t ( m ≤ m t ), defined as(Okuzumi 2009) C m t ( m ) = mK ( m t , m ) f ( m ) (cid:82) m t m (cid:48) K ( m t , m (cid:48) ) f ( m (cid:48) ) dm (cid:48) , (42)where K ( m , m (cid:48) ) is the collision kernel between particles withmasses m and m (cid:48) , and f ( m ) dm is the number density of par-ticles with masses between m and m + dm . Equation (42)measures the contributions of aggregates with masses of m on the growth of aggregate with mass of m t . We assume thatthe cloud particles obey the Hansen size distribution (Hansen1971), described as f ( r ) ≡ dn ( r ) dr ∝ r (1 − b ) / b exp (cid:18) − rab (cid:19) , (43) louds of F luffy A ggregates Mass Ratio m / m t m C m t ( m ) b = 0.1 b = 0.5 b = 1.010 r / a N o r m a li z e d d n / d L n ( r ) Figure 10.
Normalized mass-weighted collision rate between par-ticles with masses m t and m . The black, gray, and silver lines showthe collision rate for the Hansen size distributions with b = .
1, 0 . .
0, respectively. The corresponding size distributions normal-ized by a f ( a ) are also shown in the inner panel. where a is the mean e ff ective radius and b is the e ff ectivevariance. The shape of the size distribution is controlled bythe e ff ective variance b ; for example, b < . b > . b = .
1, 0 .
5, and 1 .
0. We assume a = µ mand m t calculated from the mass-weighted particle size: m t = πρ p (cid:82) ∞ rm f ( r ) dr (cid:82) ∞ m f ( r ) dr = πρ p a (1 + b ) . (44)We use the collision kernel described in Chapter 15 of Ja-cobson (2005) assuming a constant particle density. Figure10 demonstrates that the growth is largely contributed by thecollisions of particles with masses of m / m t ∼ . . D f ∼ . . D f = D f = Limitation of the Compression Model
Here we state several limitations of a compression modeladopted in Section 2.1.3. First, the relation between massand size of the collisionally compressed aggregates (Equa-tion 8) was derived for collisions between two equal-massaggregates with D f ≈ ff erent-mass collisions, the degree of compression is evaluated from the comparison of the impact energy with work done by dy-namic compression strength (Suyama et al. 2012). we alsonote that the head-on-collision is assumed here, but o ff setcollisions could induce the elongation of aggregates, furtherhinders the compression (Paszun & Dominik 2009). Sec-ond, the static compression strength used for the gas-dragcompression (Equation 13) was derived for an aggregatewhose internal structure is characterized by D f ≈ ff erent D f wasrecently proposed by Arakawa et al. (2019) from a semi-analytical argument. Although the verification with numer-ical experiments remain to be carried out, their formula ispotentially applicable to our compression model. Further nu-merical experiments will be helpful to extend the compres-sion model to more universal cases.5.1.3. Simplified Nucleation and Condensation
In this study, we have assumed that saturated vapor is in-stantaneously incorporated into the condensation nuclei atthe cloud base. This assumption would be reasonable sincethe condensation timescale is much shorter than the verticalmixing timescale near at the cloud base (Ohno & Okuzumi2018; Powell et al. 2018; Gao & Benneke 2018). Additionalcondensation could transform the CPAs to sphere-like parti-cles if the surface growth rate via condensation dominatesover the coagulation rate (Lavvas et al. 2011). However,the e ff ect is presumably insignificant for KCl clouds becauseother condensing species, such as Na S and MnS, have thecloud bases at deeper atmospheres and are likely depleted atthe KCl cloud formation region (e.g., Mbarek & Kempton2016). ZnS is an only species whose cloud base is placednear at the KCl cloud base (e.g., Morley et al. 2012). But, weexpect that CPAs are still present as aggregates even if ZnScondensation takes place. This is because the abundance ofZnS is 2–3 times lower than KCl and likely insu ffi cient to fillall pores.We have also assumed that the nucleation followed by con-densation, namely the monomer formation, occurs right atthe cloud base. This would be true if the condensation nu-clei are supplied from deep atmospheres, as argued in Leeet al. (2018). On the other hand, the monomer formationcould occur above the cloud base in the context of homoge-neous nucleation that needs significant supersaturation to setin (e.g., Helling & Fomins 2013). The monomers formed inupper atmospheres might increase the D f of CPAs throughdi ff erent-size collisions. We expect that the resulting D f isstill close to 2, as discussed in Section 5.1.1, though a micro-physical model solving size distributions will be needed toverify it.5.2. Comparison with Other Porosity Models
Some previous studies of haze microphysics adopted aporosity model di ff erent from ours (Wolf & Toon 2010;8 O hno , O kuzumi , & T azaki Number of Monomers F r a c t a l D i m e n s i o n D f r mon = 3 nmr mon = 10 nmr mon = 30 nmAdams et al. (2019) Figure 11.
Comparison of our porosity model with that used inAdams et al. (2019). The vertical and horizontal axes show the frac-tal dimension D f and number of monomers N mon . Di ff erent coloredlines exhibit the evolution track of D f for di ff erent monomer size,and the gray line shows the track assumed in Adams et al. (2019).We assume P = .
01 mbar to evaluate the collision velocity.
Adams et al. 2019). The porosity model adopted in thehaze models assumes that the fractal dimension approaches D f ≈ . D f = . ff erent from our model.Figure 11 shows the fractal dimension as a function of thenumber of monomers in Adams et al. (2019) and our model,where we calculate D f from Equations (3) and (4): D f = (cid:32) − log φ eq log N mon (cid:33) − . (45)For comparisons, we use the surface energy of tholine γ = . − (Yu et al. 2017) and material density ρ p = − . In the model of Adams et al. (2019), the fractaldimension increases to ≈ . N mon (cid:38) , while our modelpredicts that the compression sets in N mon > –10 , de-pending on the monomer size. Thus, the aggregate hazes inprevious studies were assumed to be compressed much easierthan our prediction. This is presumably a reason why aggre-gate hazes in Adams et al. (2019) tend to produce flat spectrarather than those with spectral slopes.The easily compressed aggregates in previous studies werespeculated from the laboratory study of soot formation in aflame. In the experiments, it was observed that the soot-aggregates are restructured by the Coulomb interaction be-tween oppositely charged parts (Onischuk et al. 2003). How-ever, one should take a caution about the compression due tothe Coulomb interaction because the charge states of aerosols in exoplanetary atmospheres are poorly known. Investigat-ing the aerosol charge processes (e.g., Helling et al. 2011a,b)might help to evaluate if the restructuring due to Coulombinteraction is possible.5.3. Implications for Spectral Slopes of Hot-Jupiters
The presence of mineral clouds has also been suggested fora number of hot-Jupiters (e.g., Sing et al. 2016; Barstow et al.2017). A recent retrieval study by Pinhas et al. (2019) sug-gested that the hot-Jupiters whose transmission spectra wereprovided by Sing et al. (2016) typically exhibit transmissionspectral slopes of α (cid:46) −
5. This is considerably steeper thanthe slope originated from the aggregate scattering opacity( α = −
2) and even steeper than the Rayleigh slope ( α = − ff y-aggregate cloud formation, such asNUV absorbers like SH (Evans et al. 2018). Alternatively,the slope potentially implies physical processes that halt theaggregation, leading to a tiny particle size. Electrostatic re-pulsion (e.g., Okuzumi 2009) may be promising because theionization of alkali metals, likely produces charged cloudparticles, takes place at hot-Jupiters (e.g., Batygin & Steven-son 2010). We will examine this possibility in future studies.5.4. Implications for High Metallicity Atmospheres onPlanetary Formation
The high-metallicity atmosphere is of interest from the per-spective of planetary formation theory. Our results suggestthat, if the flat spectrum of GJ1214b is caused by the con-densation clouds, high-metallicity atmosphere ( (cid:38) × soar)is plausible to explain the observations, as suggested by otherstudies (Morley et al. 2015; Gao & Benneke 2018). This isin contrast to some other super-Earths or exo-Neptunes thatlikely retain metal-poor ( < × soar) atmospheres, such asGJ3470b (Benneke et al. 2019) and HAT-P-26b (Wakefordet al. 2017; MacDonald & Madhusudhan 2019). On the otherhand, metal-rich ( > × soar) atmospheres have also beensuggested for some exo-Neptunes, such as GJ436b (Morleyet al. 2017) and HAT-P-11b (Fraine et al. 2014). The diver-sity of the atmospheric metallicity potentially suggests dif-ferent formation processes of these planets. For example,planets with a low-metallicity atmosphere may have formedfrom large building blocks, such as protoplanets, that lessa ff ect atmospheric composition (Fortney et al. 2013). A louds of F luffy A ggregates / or pebbles (Fortneyet al. 2013; Lambrechts et al. 2014; Venturini et al. 2016;Venturini & Helled 2017), occurred during the formation ofthe planet.The presence of high-metallicity atmospheres poses an-other interesting question associated to the past formationprocess: how did the super-Earths avoid to be gas giants?It has been suggested that the high atmospheric metallicityleads to the runnaway gas accretion even for planets withEarth-masses (Hori & Ikoma 2011; Venturini et al. 2015).Thus, the gas accretion must be inhibited in order to forma super-Earth rather than a gas giant. One of the scenario isthat they were formed in the late stage of protoplanetary diskswhere the disk gasses were almost dissipated (e.g., Ikoma& Hori 2012; Lee et al. 2014; Lee & Chiang 2016). Alter-natively, the high-metallicity atmospheres may suggest thepresence of mechanisms regulating the gas accretion, such asthe gap formation and weak viscous accretion of disc gasses(Tanigawa & Ikoma 2007; Tanigawa & Tanaka 2016; Ogi-hara & Hori 2018). Rapid recycling of the atmospheric gasembedded in protoplanetary disc, which is observed in recenthydrodynamical simulations (e.g., Ormel et al. 2015; Lam-brechts & Lega 2017; Kurokawa & Tanigawa 2018; Kuwa-hara et al. 2019), also delays the runnaway gas accretion,but it might be di ffi cult to produce the high-metallicity at-mosphere unless the disc gas is highly enriched in heavy el-ements. The evolution of atmospheric composition after thedisk dissipation, like that suggested for solar-system terres-trial planets (Sakuraba et al. 2019), might increase the atmo-spheric metallicity even if the planet originally possessed alow-metallicity atmosphere. Further studies linking the for-mation processes to the atmospheric metallicity would bewarranted to explore the past formation processes of super-Earths with high-metallicity atmospheres. SUMMARYWe have investigated how the porosity of cloud particle ag-gregates (CPAs) evolve in exoplanetary atmospheres. Basedon the results of numerical experiments investigating the ag-gregate restructuring, we have constructed a porosity evolu-tion model that takes into account the fractal growth, colli-sional compression, and the compression caused by gas drag.Using a cloud microphysical model coupled with the porositymodel, we have examined how the porosity evolution influ-ences the cloud vertical distributions and observed transmis-sion spectra of GJ1214b. Our findings are summarized asfollows.(1) The internal density of CPAs can be much lower thanthe material density by 1–3 orders of magnitudes (Section2), depending on the size of monomers. The gas-drag com- pression sets in once the CPA becomes larger than ≈ µ m(Equation 18). The collisional compression is less importantthan the gas-drag compression in most cases studied here.(2) The compression of CPAs hardly occurs during the KClcloud formation since the particle growth is not su ffi cient toinduce the compression (Section 3). Thus, the porosity evo-lution in general results in the cloud vertical extent muchlarger than that of the compact-sphere clouds. Without thecompression, the flu ff y-aggregate clouds can ascend to theheight where the monomer can ascend to (Equation 36).(3) The flu ff y-aggregate clouds largely obscure the absorp-tion signatures of gas molecules in transmission spectra be-cause of the large vertical extent if the aggregates are consti-tuted by submicron monomers (Section 4.3). Although thespectra in visible to near-infrared tend to be featureless, theflu ff y-aggregate clouds become optically thin at longer wave-length ( (cid:38) µ m). Future observations probing mid-infraredwavelength, such as JWST and ARIEL, may be able to detectmolecular signatures even if the spectrum looks featureless invisible to near-infrared wavelength.(4) CPAs also produce the spectral slope originated by thescattering properties of aggregates (Section 4.1.2). The slopereflects the wavelength dependence of the aggregate scatter-ing opacity, α c ∝ λ − (Section 4.3). This could be potentiallyused as an observable signature of CPAs if the atmosphericscale height is well constrained.(5) Comparing our synthetic spectra with the observationsof GJ1214b, we find that the models of the high-metallicityatmospheres ( ≥ × solar) well matches the observations ifthe CPAs are constituted by submicron monomers (Section4.4). This is due to the fact that the spectral slope producedby CPAs mismatches the observed flat spectrum as long asthe atmospheric scale height is large. The predicted high-metallicity atmosphere potentially suggests the presence ofmechanisms regulating the gas accretion onto past GJ1214b.We note that our results do not rule out other scenarios ex-plaining the flat spectrum of GJ1214b, such as photochem-ical hazes (Morley et al. 2015; Kawashima & Ikoma 2018,2019; Kawashima et al. 2019; Adams et al. 2019; Lavvaset al. 2019). The spectrum with hazes could also match theobservations if the haze production rate is su ffi ciently high(Lavvas et al. 2019). On the other hand, the hazes tend toproduce the spectral slope caused by the haze opacity in theRayleigh regime (Kawashima & Ikoma 2018, 2019; Lavvaset al. 2019). Therefore, from the same reason what we dis-cussed for flu ff y-aggregate clouds, the high-metallicity atmo-sphere may be still needed to explain the flat spectrum withphotochemical hazes. However, if hazes grow into moder-ately compressed aggregates ( D f ≈ . hno , O kuzumi , & T azaki Wavelength [ m] E x t i n c t i o n O p a c i t y [ c m g ] ( r mon = 0.01 m)( r agg = 10 m) ZnSNa SMnSCrMgSiO Mg SiO FeAl O TiO C Figure 12.
Extinction mass opacity of aggregates with D f = of hazes will be helpful to constrain the atmospheric metal-licity of GJ1214b.The compact-sphere cloud is still not ruled out (Gao &Benneke 2018). K z for settling aerosols is still uncertain,and there is an order of magnitude uncertainty among di ff er-ent model predictions (e.g., Komacek et al. 2019). AlthoughGJ1214b is a close-in planet (semi-major axis is 0 .
014 AU),it might retain a non-zero eccentricity (Charbonneau et al.2009; Carter et al. 2011) that yields distinct atmospheric cir-culations (e.g., Kataria et al. 2013; Lewis et al. 2017; Ohno& Zhang 2019) and possibly K z . If it is possible to distin-guish the flu ff y-aggregate and compact-sphere clouds from observations, it might help to understand the aerosol trans-port processes in exoplanetary atmospheres.We thank Yasunori Hori for motivating this study and YuiKawashima for helpful comments on the modeling of trans-mission spectra. We also thank Ryan Macdonald, GrahamLee, and Xi Zhang for insightful comments. We are gratefulto the reviewer, Peter Gao, for useful comments that greatlyimproved the paper. This work was supported by JSPS KAK-ENHI Grant Numbers JP18J14557 and JP18H05438. Software:
TEA (Blecic et al. 2016), TIPS (Gamacheet al. 2017), Matplotlib (Hunter 2007)APPENDIX A. AGGREGATE OPACITY FOR OTHER MINERAL CLOUDSIn this appendix, we show the opacity of an aggregate made of various materials that potentially build up exoplanetary mineralclouds. We have selected a variety of condensable materials (ZnS, Na S, MnS, Cr, MgSiO , Mg SiO , Fe, Al O ) listed inMorley et al. (2012) and some nucleating species (TiO , C, see e.g., Helling et al. 2017, 2019). The refractive indices of thematerials are taken from Kitzmann & Heng (2018). Figure 12 summarizes the calculated extinction opacities for r mon = . µ mand r agg = µ m. Some materials exhibit characteristic absorption features at λ > µ m; for example, λ ≈ µ m for Na S, λ ≈ µ m for MnS, and λ ≈ µ m for MgSiO . Absorption also dominates over extinction at λ < . µ m for all materials. Onthe other hand, the extinction opacity at λ = . µ m is mostly dominated by scattering. Therefore, many minerals other thanKCl also produce an aggregate scattering slope of ∝ λ − (Section 4.1.2) at visible to near-infrared wavelengths. The exceptionwe found is graphite, C, whose opacity is dominated by absorption even at near-infrared wavelengths.REFERENCES Adams, D., Gao, P., de Pater, I., & Morley, C. V. 2019, ApJ, 874,61Arakawa, S., & Nakamoto, T. 2016, ApJL, 832, L19Arakawa, S., Takemoto, M., & Nakamoto, T. 2019, arXiv e-prints,arXiv:1908.03125 Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009,ARA&A, 47, 481Barstow, J. K., Aigrain, S., Irwin, P. G. J., & Sing, D. K. 2017,ApJ, 834, 50Batalha, N. E., Mandell, A., Pontoppidan, K., et al. 2017, PASP,129, 064501 louds of F luffy A ggregates Batygin, K., & Stevenson, D. J. 2010, ApJL, 714, L238Bean, J. L., Miller-Ricci Kempton, E., & Homeier, D. 2010,Nature, 468, 669Bean, J. L., D´esert, J.-M., Kabath, P., et al. 2011, ApJ, 743, 92Benneke, B., Knutson, H. A., Lothringer, J., et al. 2019, NatureAstronomy, 361Berry, M. V., & Percival, I. C. 1986, Optica Acta, 33, 577Berta, Z. K., Charbonneau, D., D´esert, J.-M., et al. 2012, ApJ, 747,35Blecic, J., Harrington, J., & Bowman, M. O. 2016, ApJS, 225, 4Blum, J., & Wurm, G. 2000, Icarus, 143, 138Blum, J., Wurm, G., Kempf, S., & Henning, T. 1996, Icarus, 124,441Bohren, C. F., & Hu ff man, D. R. 1983, Absorption and scatteringof light by small particlesBrown, T. M. 2001, ApJ, 553, 1006Carter, J. A., Winn, J. N., Holman, M. J., et al. 2011, ApJ, 730, 82Charbonneau, D., Berta, Z. K., Irwin, J., et al. 2009, Nature, 462,891Charnay, B., Meadows, V., & Leconte, J. 2015a, ApJ, 813, 15Charnay, B., Meadows, V., Misra, A., Leconte, J., & Arney, G.2015b, ApJL, 813, L1Col´on, K. D., & Gaidos, E. 2013, ApJ, 776, 49Croll, B., Albert, L., Jayawardhana, R., et al. 2011, ApJ, 736, 78Crossfield, I. J. M., & Kreidberg, L. 2017, AJ, 154, 261de Mooij, E. J. W., Brogi, M., de Kok, R. J., et al. 2012, A&A, 538,A46D´esert, J.-M., Bean, J., Miller-Ricci Kempton, E., et al. 2011, ApJ,731, L40Dominik, C., & Tielens, A. G. G. M. 1995, PhilosophicalMagazine, Part A, 72, 783—. 1997, ApJ, 480, 647Espinoza, N., Rackham, B. V., Jord´an, A., et al. 2019, MNRAS,482, 2065Evans, T. M., Sing, D. K., Goyal, J. M., et al. 2018, AJ, 156, 283Ford, I. J. 1997, PhRvE, 56, 5615Fortney, J. J., Mordasini, C., Nettelmann, N., et al. 2013, ApJ, 775,80Fortney, J. J., Sudarsky, D., Hubeny, I., et al. 2003, ApJ, 589, 615Fraine, J., Deming, D., Benneke, B., et al. 2014, Nature, 513, 526Fraine, J. D., Deming, D., Gillon, M., et al. 2013, ApJ, 765, 127Freedman, R. S., Lustig-Yaeger, J., Fortney, J. J., et al. 2014, ApJS,214, 25Freedman, R. S., Marley, M. S., & Lodders, K. 2008, ApJS, 174,504Gamache, R. R., Roller, C., Lopes, E., et al. 2017, Journal ofQuantitative Spectroscopy and Radiative Transfer, 203, 70Gao, P., & Benneke, B. 2018, ApJ, 863, 165Gao, P., Marley, M. S., & Ackerman, A. S. 2018, ApJ, 855, 86 Gao, P., Marley, M. S., Zahnle, K., Robinson, T. D., & Lewis,N. K. 2017a, AJ, 153, 139Gao, P., Fan, S., Wong, M. L., et al. 2017b, Icarus, 287, 116Gharib-Nezhad, E., & Line, M. R. 2019, ApJ, 872, 27Gillon, M., Demory, B.-O., Madhusudhan, N., et al. 2014, A&A,563, A21Guillot, T. 2010, A&A, 520, A27Guillot, T., Ida, S., & Ormel, C. W. 2014, A&A, 572, A72Hansen, J. E. 1971, Journal of Atmospheric Sciences, 28, 1400He, C., H¨orst, S. M., Lewis, N. K., et al. 2018a, arXiv e-prints,arXiv:1812.06957—. 2018b, AJ, 156, 38Heim, L.-O., Blum, J., Preuss, M., & Butt, H.-J. 1999, PhysicalReview Letters, 83, 3328Helling, C., & Fomins, A. 2013, Philosophical Transactions of theRoyal Society of London Series A, 371, 20110581Helling, C., Jardine, M., & Mokler, F. 2011a, ApJ, 737, 38Helling, C., Jardine, M., Witte, S., & Diver, D. A. 2011b, ApJ, 727,4Helling, C., Tootill, D., Woitke, P., & Lee, G. 2017, A&A, 603,A123Helling, C., Woitke, P., & Thi, W.-F. 2008, A&A, 485, 547Helling, C., Iro, N., Corrales, L., et al. 2019, arXiv e-prints,arXiv:1906.08127Heng, K., & Kitzmann, D. 2017, MNRAS, 470, 2972Hori, Y., & Ikoma, M. 2011, MNRAS, 416, 1419H¨orst, S. M., He, C., Lewis, N. K., et al. 2018, Nature Astronomy,2, 303Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90Ikoma, M., & Hori, Y. 2012, ApJ, 753, 66Jacobson, M. Z. 2005, Fundamentals of Atmospheric Modeling,828Kataoka, A., Okuzumi, S., Tanaka, H., & Nomura, H. 2014, A&A,568, A42Kataoka, A., Tanaka, H., Okuzumi, S., & Wada, K. 2013a, A&A,557, L4—. 2013b, A&A, 554, A4Kataria, T., Showman, A. P., Lewis, N. K., et al. 2013, ApJ, 767, 76Kawashima, Y., Hu, R., & Ikoma, M. 2019, ApJ, 876, L5Kawashima, Y., & Ikoma, M. 2018, ApJ, 853, 7—. 2019, ApJ, 877, 109Kitzmann, D., & Heng, K. 2018, MNRAS, 475, 94Knutson, H. A., Benneke, B., Deming, D., & Homeier, D. 2014a,Nature, 505, 66Knutson, H. A., Dragomir, D., Kreidberg, L., et al. 2014b, ApJ,794, 155Komacek, T. D., Showman, A. P., & Parmentier, V. 2019, arXive-prints, arXiv:1904.09676Kreidberg, L., Line, M. R., Thorngren, D., Morley, C. V., &Stevenson, K. B. 2018, ApJL, 858, L6 hno , O kuzumi , & T azaki Kreidberg, L., Bean, J. L., D´esert, J.-M., et al. 2014, Nature, 505,69Kreidberg, L., Line, M. R., Bean, J. L., et al. 2015, ApJ, 814, 66Kuntz, M. 1997, JQSRT, 57, 819Kurokawa, H., & Tanigawa, T. 2018, MNRAS, 479, 635Kuwahara, A., Kurokawa, H., & Ida, S. 2019, A&A, 623, A179Lambrechts, M., Johansen, A., & Morbidelli, A. 2014, A&A, 572,A35Lambrechts, M., & Lega, E. 2017, A&A, 606, A146Lavvas, P., Koskinen, T., Steinrueck, M., Garc´ıa Mu˜noz, A., &Showman, A. P. 2019, arXiv e-prints, arXiv:1905.02976Lavvas, P., Sander, M., Kraft, M., & Imanaka, H. 2011, ApJ, 728,80Lavvas, P., Yelle, R. V., & Gri ffi th, C. A. 2010, Icarus, 210, 832Lee, E. J., & Chiang, E. 2016, ApJ, 817, 90Lee, E. J., Chiang, E., & Ormel, C. W. 2014, ApJ, 797, 95Lee, G., Dobbs-Dixon, I., Helling, C., Bognar, K., & Woitke, P.2016, A&A, 594, A48Lee, G., Helling, C., Giles, H., & Bromley, S. T. 2015, A&A, 575,A11Lee, G. K. H., Blecic, J., & Helling, C. 2018, A&A, 614, A126Lewis, N. K., Parmentier, V., Kataria, T., et al. 2017, arXive-prints, arXiv:1706.00466Line, M. R., & Parmentier, V. 2016, ApJ, 820, 78Lines, S., Mayne, N. J., Manners, J., et al. 2019, MNRAS, 488,1332Lines, S., Mayne, N. J., Boutle, I. A., et al. 2018, A&A, 615, A97Lothringer, J. D., Benneke, B., Crossfield, I. J. M., et al. 2018, AJ,155, 66MacDonald, R. J., & Madhusudhan, N. 2019, MNRAS, 486, 1292Malik, M., Kitzmann, D., Mendonc¸a, J. M., et al. 2019, AJ, 157,170Marley, M. S., Ackerman, A. S., Cuzzi, J. N., & Kitzmann, D.2013, Clouds and Hazes in Exoplanet Atmospheres, ed. S. J.Mackwell, A. A. Simon-Miller, J. W. Harder, & M. A. Bullock,367–391Mbarek, R., & Kempton, E. M. R. 2016, ApJ, 827, 121Meakin, P. 1991, Reviews of Geophysics, 29, 317Minato, T., K¨ohler, M., Kimura, H., Mann, I., & Yamamoto, T.2006, A&A, 452, 701Morley, C. V., Fortney, J. J., Kempton, E. M.-R., et al. 2013, ApJ,775, 33Morley, C. V., Fortney, J. J., Marley, M. S., et al. 2012, ApJ, 756,172—. 2015, ApJ, 815, 110Morley, C. V., Knutson, H., Line, M., et al. 2017, AJ, 153, 86Moses, J. I., Line, M. R., Visscher, C., et al. 2013, ApJ, 777, 34Murgas, F., Pall´e, E., Cabrera-Lavers, A., et al. 2012, A&A, 544,A41Narita, N., Nagayama, T., Suenaga, T., et al. 2013a, PASJ, 65, 27 Narita, N., Fukui, A., Ikoma, M., et al. 2013b, ApJ, 773, 144Ogihara, M., & Hori, Y. 2018, ApJ, 867, 127Ohno, K., & Okuzumi, S. 2017, ApJ, 835, 261—. 2018, ApJ, 859, 34Ohno, K., & Zhang, X. 2019, ApJ, 874, 1Okuzumi, S. 2009, ApJ, 698, 1122Okuzumi, S., Tanaka, H., & Sakagami, M.-a. 2009, ApJ, 707, 1247Onischuk, A., di Stasio, S., Karasev, V., et al. 2003, Journal ofAerosol Science, 34, 383Ormel, C. W., & Min, M. 2019, A&A, 622, A121Ormel, C. W., Shi, J.-M., & Kuiper, R. 2015, MNRAS, 447, 3512Palik, E. D. 1985, Handbook of optical constants of solidsParmentier, V., Showman, A. P., & Lian, Y. 2013, A&A, 558, A91Paszun, D., & Dominik, C. 2006, Icarus, 182, 274—. 2009, A&A, 507, 1023Pinhas, A., & Madhusudhan, N. 2017, MNRAS, 471, 4355Pinhas, A., Madhusudhan, N., Gandhi, S., & MacDonald, R. 2019,MNRAS, 482, 1485Powell, D., Zhang, X., Gao, P., & Parmentier, V. 2018, ApJ, 860,18Rackham, B., Espinoza, N., Apai, D., et al. 2017, ApJ, 834, 151Rodrigues, P. C. R., & Silva Fernandes, F. M. S. 2007, The Journalof Chemical Physics, 126, 024503Roman, M., & Rauscher, E. 2019, ApJ, 872, 1Rossow, W. B. 1978, Icarus, 36, 1Rothman, L. S., Rinsland, C. P., Goldman, A., et al. 1998, JQSRT,60, 665Ruyten, W. 2004, JQSRT, 86, 231Sakuraba, H., Kurokawa, H., & Genda, H. 2019, Icarus, 317, 48Sato, T., Okuzumi, S., & Ida, S. 2016, A&A, 589, A15Seager, S., & Sasselov, D. D. 2000, ApJ, 537, 916Seinfeld, J., & Pandis, S. 2012, Atmospheric Chemistry andPhysics: From Air Pollution to Climate Change (Wiley)Sharp, C. M., & Burrows, A. 2007, ApJS, 168, 140Sing, D. K., Fortney, J. J., Nikolov, N., et al. 2016, Nature, 529, 59Suyama, T., Wada, K., & Tanaka, H. 2008, ApJ, 684, 1310Suyama, T., Wada, K., Tanaka, H., & Okuzumi, S. 2012, ApJ, 753,115Tanaka, K. K., Tanaka, H., Yamamoto, T., & Kawamura, K. 2011,JChPh, 134, 204313Tanigawa, T., & Ikoma, M. 2007, ApJ, 667, 557Tanigawa, T., & Tanaka, H. 2016, ApJ, 823, 48Tazaki, R., & Tanaka, H. 2018, ApJ, 860, 79Tazaki, R., Tanaka, H., Okuzumi, S., Kataoka, A., & Nomura, H.2016, ApJ, 823, 70Tennyson, J., & Yurchenko, S. 2018, Atoms, 6, 26Tinetti, G., Drossart, P., Eccleston, P., et al. 2016, in Proc. SPIE,Vol. 9904, Space Telescopes and Instrumentation 2016: Optical,Infrared, and Millimeter Wave, 99041XVenturini, J., Alibert, Y., & Benz, W. 2016, A&A, 596, A90 louds of F luffy A ggregates23