ALMA observation of the protoplanetary disk around WW Cha: faint double-peaked ring and asymmetric structure
Kazuhiro D. Kanagawa, Jun Hashimoto, Takayuki Muto, Takashi Tsukagoshi, Sanemichi Z. Takahashi, Yasuhiro Hasegawa, Mihoko Konishi, Hideko Nomura, Hauyu Baobab Liu, Ruobing Dong, Akimasa Kataoka, Munetake Momose, Tomohiro Ono, Michael Sitko, Michihiro Takami, Kengo Tomida
aa r X i v : . [ a s t r o - ph . E P ] J a n D RAFT VERSION J ANUARY
27, 2021Typeset using L A TEX twocolumn style in AASTeX63
ALMA observation of the protoplanetary disk around WW Cha: faint double-peaked ring and asymmetric structure K AZUHIRO
D. K
ANAGAWA ,
1, 2 J UN H ASHIMOTO , T AKAYUKI M UTO , T AKASHI T SUKAGOSHI , S ANEMICHI T AKAHASHI , Y ASUHIRO H ASEGAWA , M IHOKO K ONISHI , H IDEKO N OMURA ,
5, 8 H AUYU B AOBAB L IU , R UOBING D ONG , A KIMASA K ATAOKA , M UNETAKE M OMOSE , T OMOHIRO O NO , M ICHAEL S ITKO ,
11, 12 M ICHIHIRO T AKAMI , AND K ENGO T OMIDA
Research Center for the Early Universe, Graduate School of Science, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan College of Science, Ibaraki University, 2-1-1 Bunkyo, Mito, Ibaraki 310-8512, Japan Astrobiology Center, National Institutes of Natural Sciences, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan Division of Liberal Arts, Kogakuin University, 1-24-2 Nishi-Shinjuku, Shinjuku-ku, Tokyo 163-8677, Japan National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA Faculty of Science and Technology, Oita University, 700 Dannoharu, Oita 870-1192, Japan Department of Earth and Planetary Sciences, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan Institute of Astronomy and Astrophysics, Academia Sinica, 11F of Astronomy-Mathematics Building, AS No.1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan,R.O.C. Department of Physics & Astronomy, University of Victoria, Victoria, BC, V8P 1A1, Canada Department of Physics, University of Cincinnati, Cincinnati, OH 45221, USA Space Science Institute, 475 Walnut Street, Suite 205, Boulder, CO 80301, USA Astronomical Institute, Tohoku University, Sendai 980-8578, Japan (Received January 27, 2021; Revised January 27, 2021; Accepted January 27, 2021)
Submitted to ApJABSTRACTWe present Atacama Large Millimeter/submillimeter Array (ALMA) Band 6 observations of dust continuumemission of the disk around WW Cha. The dust continuum image shows a smooth disk structure with a faint(low-contrast) dust ring, extending from ∼ AU to ∼ AU, not accompanied by any gap. We constructedthe simple model to fit the visibility of the observed data by using MCMC method and found that the bump(we call the ring without the gap the bump) has two peaks at 40 AU and 70 AU. The residual map betweenthe model and observation indicates asymmetric structures at the center and the outer region of the disk. Theseasymmetric structures are also confirmed by model-independent analysis of the imaginary part of the visibility.The asymmetric structure at the outer region is consistent with a spiral observed by SPHERE. To constrainphysical quantities of the disk (dust density and temperature), we carried out radiative transfer simulations. Wefound that the midplane temperature around the outer peak is close to the freezeout temperature of CO on waterice ( ∼ K). The temperature around the inner peak is about K, which is close to the freezeout temperatureof H S and also close to the sintering temperature of several species. We also discuss the size distribution of thedust grains using the spectral index map obtained within the Band 6 data.
Keywords: protoplanetary disks – stars:individual (WW Cha) – stars:pre-main sequence – tech-niques:interferometric INTRODUCTIONPlanets are born in a protoplanetary disk around ayoung star. Recent observations have revealed substruc-
Corresponding author: Kazuhiro D. [email protected] tures such as gaps, rings, and crescents in the protoplane-tary disks (e.g., Fukagawa et al. 2013; Akiyama et al. 2015,2016; ALMA Partnership et al. 2015; Momose et al. 2015;Dong et al. 2018b; Long et al. 2018; van der Marel et al.2019; Soon et al. 2019; Kim et al. 2020). These struc-ture could be formed at an edge of a gap inducedby disk-planet interaction (e.g., Paardekooper & Mellema2004; Muto & Inutsuka 2009; Zhu et al. 2012; Dong et al. K.D. K
ANAGAWA ET AL .2015; Pinilla et al. 2015; Kanagawa et al. 2018). Alter-natively, these could be associated to dust growth re-lated to snowline (e.g., Zhang et al. 2015; Cieza et al. 2017;Mac´ıas et al. 2017; van der Marel et al. 2018; Facchini et al.2020) and the sintering effect (Okuzumi et al. 2016), orsecular gravitational instability (Takahashi & Inutsuka 2014,2016; Tominaga et al. 2018). The ring/gap structures of thedust grains also could be formed by axisymmetric gas per-turbation due to evolution of luminosity of the central star(Vorobyov et al. 2020). In any case, these substructurescould be reflected by planet formation and growth of dustgrains which are building blocks of planets. Direct observa-tions of the disks help to understand how formation of theplanets progresses in the disk.Our target, WW Cha is a young star with a circumstel-lar disk (e.g., Pascucci et al. 2016; Garufi et al. 2020) in theChameleon I star-forming region. The star is located atabout 190 pc (Gaia Collaboration et al. 2018, 2020). Themass of the star is about M ⊙ , the surface temperature is4350 K (Spectral type is K5) (Luhman 2007), and the lu-minosity is L ⊙ (Garufi et al. 2020). The star is veryyoung ( ∼ . Myr) and it could be still embedded intothe molecular cloud core with a high extinction (Ribas et al.2013; Garufi et al. 2020). A high accretion rate onto thestar, − . M ⊙ / yr, is inferred from the photometric and theBalmer continuum observations (Manara et al. 2016). More-over, the binary with the separation of ∼ AU is reportedby VLTI (Anthonioz et al. 2015). The disk of WW Cha maybe a pre-transition disk because strong infrared emission isdetected (Espaillat et al. 2011; Ribas et al. 2013), while therecent modeling using radiative transfer simulations done byvan der Marel et al. (2016) suggested an inner cavity with theradius of ∼ AU (but with a large uncertainty). The diskis very bright in millimeter wavelength (Pascucci et al. 2016)and Lommen et al. (2009) reported the emission at ∼ cm,which indicates the presence of large grains due to growth ofdust grains.In this paper, we report dust continuum observations ofthe disk around WW Cha in ALMA Cycle 5. In Section 2,we describe the setup of the observation and show the ob-servational results. We developed a model for the observedemission by using the Markov Chain Monte Carlo (MCMC)method and found axisymmetric substructures and asymmet-ric structures, which is described in Section 3. Moreover,we carried out radiative transfer simulations to constrain thephysical parameters of the disk which are described in Sec-tion 4. In Section 5, we discuss origins of the substructures,an inner cavity and binary, and the dust growth in the disk.Section 6 contains our conclusion. OBSERVATIONS AND RESULTS The observation was carried out by ALMA in Band 6,which is summarized in Table 1. The data were cali-brated by the Common Astronomy Software Applications(CASA) package (McMullin et al. 2007) version 5.6.1-8, fol-lowing the calibration scripts provided by ALMA. We con-ducted self-calibration of the visibilities. The phases wereself-calibrated once with a fairly long solution intervals(solint=‘inf’) combining all spectral windows (SPWs).We combined the data taken by sparse (C43-8) and com-pact (C43-4) array configurations to recover the missingflux at larger angular scales. By using the CASA tool uvmodelfit , we fitted the data by a Gaussian shape, andthe phase center was corrected to be the center of the Gaus-sian shape by fixvis . The inclination i = 37 . ◦ ± . ◦ and position angle φ = 32 . ◦ ± . ◦ are ob-tained by the Gaussian fit by uvmodelfit . There aretwo SPWs for continuum with . GHz frequency widthwith the central frequency being . GHz (Upper band)and . GHz (Lower band) in both the C43-4 and C43-8data. As shown below, the total flux density of the data in . GHz ( ∼ mJy) is significantly larger than that in . GHz ( ∼ mJy). Hence, the dust continuum im-age of combined data was synthesized by CASA with the tclean task using the mtmfs algorithm (Rau & Cornwell2011) with nterms =2. We obtained a synthesized im-age at . GHz with the beam size of . × . mas( . × . AU) with PA= . ◦ and with the σ RMSnoise level of . mJy/Beam. The imaging parameters aresummarized in Table 1.The synthesized image is shown in Figure 1. The panel(a) shows the synthesized dust continuum image derivedfrom all SPW data, and the brightness temperatures alongthe major and minor axes are shown in the panel (b). Thedust disk is clearly resolved and its size is about . arc-sec, which corresponds to about AU. We see a faintlow-contrast dust ring feature in the radial profile of bright-ness temperature at ∼ . arcsec ( ∼ AU) from the cen-tral star. This ring structure is not accompanied with thegap structure, as different from the rings found in the diskof HL Tau (ALMA Partnership et al. 2015), and DSHARP’ssamples (Andrews et al. 2018). Hence, we call this ringstructure (without the gap) the bump in the following. The to-tal flux density with > σ (0 . mJy ) emission is measuredto be . mJy. We do not see a clear cavity structure inthe image. Therefore, the large cavity such as that predicted The fitted inclination and position angle are slightly different in the C43-8 (sparse configuration) and C43-4 (compact configuration) data. Weadopted the values of the C43-8 data. For the C43-4 data, i = 39 . ◦ and φ = 30 . ◦ with the relatively large reduced χ , 5.78 while, for thefit of the C43-8 data, reduced χ = 1 . . The relatively large χ for theC43-4 data can be due to the asymmetric structure discussed in Section 3.4.Hence, we adopted the values given by the fit of the C43-8 data LMA
OBSERVATION OF THE PROTOPLANETARY DISK AROUND
WW C HA − . − . . . . − . − . − . − . . . . . . D ec . o ff s e t( a r c s ec ) . . . . . . . . . I n t e n s i t y ( m J y / B e a m ) (a) Combined image at . GHz − . − . . . . − . − . − . − . . . . . . D ec . o ff s e t( a r c s ec ) . . . . . . . . . Sp ec tr a li nd e x (c) Image of spectral index − . − . . . . B r i g h t n e ss t e m p e r a t u r e ( K ) T mid along major axis (Eq.1)Major axis Minor axis −
100 0 100offset (AU) (b) Radial slices of Panel (a) − . − . − . − .
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75 1 . Sp ec tr a li nd e x Major axis − − −
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75 1 . offset (arcsec) Sp ec tr a li nd e x Minor axis − − −
50 0 50 100 150 (d) Radial slice of Panel (c)
Figure 1. panels (a) and (c) show the image and the spectral index maps resulting from the combination of 233.0 GHz and 216.7GHz data.The contours in Panel (c) indicate the intensity levels of . mJy/Beam (3 σ ), . mJy/Beam (10 σ ), . mJy/Beam ( σ ), . mJy/Beam( σ ) and . mJy/Beam ( σ ) in Panel (a). Panels (b) and (d) show brightness temperature and spectral index along the major and minoraxis, respectively. The gray thick lines in Panel (b) denote the midplane temperature given by Equation (1) with L ∗ = 11 L ⊙ . by SED analysis done by van der Marel et al. (2016), is ruledout at least in the millimeter image, while the existence of acavity that is smaller than the beam size is not ruled out.In the panel (c) of Figure 1, we show the spectral indexmap, and panel (d) illustrates the spectral indexes along themajor and minor axes. Within the region of < . arcsec( < AU), the spectral index is ∼ , which is an indica-tive of optically thick dust emission. Around the center ofthe disk, in particular, the spectral index is slightly below ,which may indicate optically thick dust scattering (Liu 2019;Zhu et al. 2019). In the region where the offset is larger than . arcsec, the disk may be optically thin because the spec- tral index is larger than 2, and the spectral index seems toincrease in the outer region, though there is large uncertainlyat > . arcsec.In the panel (b) of Figure 1, we also plot the mid-plane temperature along the major axis, estimated by thesimple expression for a passive heated radiative disk (e.g.,Chiang & Goldreich 1997), T mid = (cid:18) α g L ∗ πR σ SB (cid:19) / , (1)where σ SB is the Stefan-Bolzmann constant, and R is thedistance from the star. The grazing angle α g is set to be . K.D. K
ANAGAWA ET AL . Table 1.
ALMA Band 6 Observations and Imaging Parameters
Observations Sparse configuration Compact configurationObserving date (UT) 2017.Nov.27 2018.Mar.11Configuration C43-8 C43-4Project code 2017.1.00286.STime on source (min) 60.3 29.4Number of antennas 47 42Baseline lengths 92.1 m to 8.5 km 15.1 m to 1.2 kmBaseband Freqs. (GHz) 233.0 (Upper band), 216.7 (Lower band)Channel width (GHz) 1.87Continuum band width (GHz) 4.0Bandpass calibrator J0635 − − − − − − × . × . AU) at PA of . ◦ r.m.s. noise (mJy/beam) 0.029 and the stellar luminosity L ∗ is L ⊙ in the plot. The bright-ness temperature is close to the midplane temperature within . arcsec from the center, which indicate the optically thickemission. The outer region ( R > . arcsec or > AU)can be optically thin, which is consistent with the spectralindex map mentioned above.Using the inclination and position angle, we have depro-jected the face-on equivalent view, to identify substructureson the disk. In Figure 2, we show the intensity profile alongthe major axis and minor axis in the face-on view. In Fig-ure 2, we put gray double-sided arrows to indicate the loca-tion of the faint low-contrast dust bump, which extends from R ≃ . arcsec ( ≃ AU) to R ≃ . arcsec ( ≃ AU).Moreover, one can see this bump has a double-peak feature,which is indicated by the black arrows in the figure: the innerpeak locates at | R | ≃ . arcsec ( AU) and the outer onelocates at | R | ≃ . arcsec ( AU), where R is an offsetfrom the center. Although the double-peak feature is not clearin the image, it is confirmed by the visibility fitting describedin Section 3. Outside of . arcsec, the intensity decreasesquickly, but around | R | ∼ . arcsec ( R ∼ AU), theslope of the intensity becomes moderate.Figure 3 shows the azimuthal distributions of the intensityin the face-on view. There are asymmetric structures in theazimuthal distribution at R = 0 . arcsec ( ∼ AU), and thedeviation from the averaged value of the averaged value atthis radius is at most about 1 mJy/Beam (5% of the averaged − . − . − . − .
25 0 .
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75 1 . F l u x ( m J y / B e a m ) Major axisMinor axis − − −
50 0 50 100 150 offset (AU) − . − . − . − .
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75 1 . offset (arcsec) − F l u x ( m J y / B e a m ) Major axisMinor axis
Figure 2.
Intensity distributions along the major and minor axes ofthe face-on view of Figure 1 (a). We also plot the 1 σ noise level ateach data point. The black arrows indicate the locations of the peakswithin the bump structure, and the gray one indicates location of thebump structure. The dashed horizontal line denotes σ noise level(= . mJy/Beam). The bottom panel is the same as the top panel,but the vertical axis is in logarithmic scale. The gray double-sidedarrows indicate the location of the faint dust bump and the blackarrows denote the locations of the peaks within the bump. F l u x ( m J y / B e a m ) offset=0.05 arcsec0 50 100 150 200 250 300 3506789 offset=0.25 arcsec0 50 100 150 200 250 300 3500 . . . . azimuthal angle (degree) . . . Figure 3.
Azimuthal distributions of observed intensity in the face-on view, at offset = 0 . arcsec ( ≃ AU), . arcsec ( ≃ AU), . arcsec ( ≃ AU), . arcsec ( ≃ AU). The error bars in-dicate σ noise level (0.029 mJy). The horizontal lines are the aver-ages, which indicates . mJy/Beam ( ∼ σ ), . mJy/Beam( ∼ σ ), . mJy/Beam ( ∼ σ ), and . mJy/Beam ( ∼ σ ),from the top panel to bottom panel. LMA
OBSERVATION OF THE PROTOPLANETARY DISK AROUND
WW C HA ρ ( kλ )0100200300400500 F l u x ( m J y ) Upper band(233GHz)Lower band (217GHz)
Figure 4.
Real part of the visibility for the upper and lower banddata. In the plot, we combine the C43-4 and C43-8 data. The insetshows the zoom in of the region with the flux < mJy. value or ∼ σ ). The asymmetry at the innermost radii ofthe disk is also indicated in Figure 2 as the distribution alongthe major and minor axes do not overlap at R < mas or < AU. At R = 0 . arcsec ( ∼ AU), the structure hasthe similar pattern of asymmetricity seen in the distributionat R = 0 . arcsec, and the derivation from the averagedvalue is about . mJy/Beam (it is about 5% of the averagedvalue or ∼ σ ). At R = 0 . arcsec ( ∼ AU), one cansee a significant asymmetric structure. The intensity at < ◦ is larger than averaged value by . mJy/Beam ( ∼ σ )and it is smaller around ◦ by . mJy/Beam ( ∼ σ ).The deviation from the averaged value is about 20% of theaveraged value. At R = 0 . arcsec ( ∼ AU), we may seean asymmetric structure, as the intensity at ∼ ◦ is largerin . mJy/Beam ( ∼ σ ) than the average and it is smallerthan the average at > ◦ in . mJy/Beam ( ∼ σ ). Wediscuss asymmetric structures in Section 3.4 in more detailby directly analyzing the data in visibility domain.In the rest of this section, we show the difference betweenthe upper ( . GHz) and lower ( . GHz) band data.In Figure 4, we compare the real parts of the visibility ofupper and lower band data. The visibility data are depro-jected using the inclination and position angle derived ear-lier. Then, the data with similar uv -distance are binned andaveraged. The bin width are kλ for ρ < kλ while kλ for ρ > kλ , where ρ is the uv -distance of the de-projected visibility data. The error bars in the figure indicatethe standard deviation of the data divided by the square rootof the number of data. The amplitudes of the visibility areclearly different at small uv -distances, namely ρ < kλ ,which corresponds to a spacial scale of ∼ arcsec. More-over, one can find that the visibility of the upper band datais larger than that of the lower band data around the peakaround ρ = 700 kλ . The details of statistics of the data aredescribed in Appendix A. − . − . − . − .
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75 1 . F l u x ( m J y / B e a m ) Upper band (233.0 GHz)Lower band (216.7 GHz) − − −
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75 1 . offset (arcsec) − F l u x ( m J y / B e a m ) Upper band (233.0 GHz)Lower band (216.7 GHz)
Figure 5.
Intensity distributions along the major axis for theupper and lower band data. The error bar denotes σ noise level(= . mJy/Beam) and the dashed horizontal line denotes σ noiselevel ( . mJy/Beam). Figure 5 compares the intensity distributions along the ma-jor axis for the upper and lower band data. The intensity ofthe upper band data is slightly larger than that of the lowerband data. Indeed, for the upper band data, the total fluxdensity with > σ ( . mJy) emission is . mJy andfor the lower band data, it is . mJy. We note that atthe offset ≃ . arcsec, the profile is different in the upperand lower band data, which can be related to the asymmetricstructures discussed in Section 3.4. MCMC MODELING OF DUST CONTINUUMEMISSION3.1.
Model description
To examine the structure of the disk in detail, we per-formed fitting for dust continuum emission in the visibilitydomain with a simple disk model. As described in the pre-vious section, the disk could have two peaks at ∼ . arc-sec and . arcsec. Moreover, we also include an unre-solved small cavity. By motivated from the observed fea-tures, we adopted a simple power-law intensity profile withexponential cutoff with two Gaussian bumps, two intensityenhanced/depleted regions and an inner cavity, as describedbelow: I ( R ) ∝ " f I (cid:18) RR c (cid:19) − γ exp (cid:18) RR c (cid:19) − ζ + X i =1 H i exp (cid:18) R − R rc , i W i (cid:19) , (2) K.D. K ANAGAWA ET AL .where f I = δ cav < R < R cav R cav < R < R g , in , δ gc , R g , in , < R < R g , out , ) δ gc , R > R g , out , ) (3)The total flux that the intensity given by Equation (2) is inte-grated over the entire disk is normalized to be F tot which isone of the parameters of the model. The intensity distributionof Equation (2) has 16 parameters, namely, two exponents γζ , the depth and radius of the inner cavity δ cav , R cav , charac-teristic radius R c , total intensity of the disk F tot , and param-eters of substructures: R g , in , , R g , out , , δ gc , , δ gc , for twoenhanced/depleted regions and R rc , , W , H , R rc , , W , H for two Gaussian bumps.3.2. Fitting approach
We fit the observation data with the model of Equation (2)in the visibility domain. In this modeling, we focus on asymmetric structure, following Zhang et al. (2015). In thefollowing, ρ indicates the deprojected baseline in the visibil-ity domain. The likelihood function is defined by χ = N X k Re (cid:0) V (cid:1) obs , k − Re (cid:0) V (cid:1) model , k σ obs , k ! , (4)where k indicates the index of the radial bin and N is thetotal number of radial bins. We take an average withinthe bin in radial and azimuthal direction in visibility do-main (the overline indicates the average). The bin size is kλ for ρ > kλ , and kλ for ρ < kλ . Sincethe amplitude of the visibility is comparable with the noiselevel in ρ > kλ , we used the visibility in the range of ρ < kλ in this modeling. The real part of the visibilityis denoted by Re ( V ) and the subscript ’model’ and ’obs’ in-dicate the quantities of model and observation, respectively.The standard deviation of the averaged real part of the visi-bility σ obs , i is calculated by dividing the standard deviationof azimuthal direction in the visibility domain by the squareroot of the number of data within the bin.For the fitting, we utilized the public python code vis_sample (Loomis et al. 2017). We used the MarkovChain Monte Carlo (MCMC) method in the emcee pack-age (Foreman-Mackey et al. 2013). We carried out the fittingwith the MCMC method with χ given by Equations (4). Inthe MCMC fitting, we run 1000 steps with 100 walkers afterthe burnin phase with 1000 steps (2000 steps in total).3.3. fitting result We found that the C43-4 (compact configuration) data isslightly scattered as compared with the C43-8 (sparse con-figuration) data, and the data is slightly statistically different . . . . . . . . offset (arcsec) F l u x ( m J y / B e a m ) Upper band Lower band offset (AU) − − − offset (arcsec) F l u x ( m J y / B e a m ) Figure 6.
The shapes of the best-fit model given by parameterslisted in Table 2, for the upper and lower band data. The inset is thezoom in of the region of offset < . arcsec. around ρ = 200 kλ (see Appendix A). Because of this dif-ference between the C43-4 and C43-8 data, the reduced χ of the best fit model is much deviated from unity, when thedata are combined. If only C43-4 data is used, the resolutionis not enough to identify substructures. Hence, we used onlythe C43-8 data for the MCMC fitting . We performed theMCMC fitting for the upper and lower band data separately.The total flux density of the images synthesized by onlythe C43-8 data is – smaller than that by both C43-8 and C43-4 data due to missing flux. However, since thevisibilities of the C43-8 data are quite similar to these thatcombined by the C43-8 and C43-4 data, as shown in Fig-ure 19), excluding the C43-4 data could not affect the fittingresults.The fitting results are summarized in Table 2. The best fitparameters for the upper and lower band data are slightly dif-ferent, especially on the total flux density, and the parametersrelated the outer structure, namely, ζ , δ g, and the parametersof the outer peak ( R rc , , W , H ), and the depth of the innercavity ( δ cav ).Figure 6 shows the best-fit models for the upper and lowerband data. They look very similar, as both have small cavitywith radius ∼ AU (corresponding to R cav ), a bump-likeexcess emission at R ∼ . – . arcsec with double peaks at . arcsec (corresponding to R g , in , , ≃ AU) and . arc-sec (corresponding to R rc , , ≃ AU). The fitting results ofboth upper and lower bands indicate that the there is a cavitywith the radius of ∼ . arcsec ( ∼ AU). Since the size ofinner cavity is much smaller than the spatial resolution with ρ < kλ , we consider that this structure should be con-firmed in future observations. For the outer region, though When both the C43-4 and C43-8 data are used, the reduced χ is ∼ ,though the best fit parameters are similar to these shown in table 2. Thislarge reduced χ is mainly due to points around ρ = 200 kλ . LMA
OBSERVATION OF THE PROTOPLANETARY DISK AROUND
WW C HA Table 2.
Fitting results for C43-8 data
Upper band (233.0 GHz data) Lower band (216.7 GHz data)Parameters Best fit Range Best fit RangeMin Max Min Max γ . − . . . − . . ζ . − . . . − . . R c (AU) . − . . . − . . R cav (AU) . − . . . − . . R g , in , (AU) . − . . . − . . R g , out , (AU) . − . . . − . . R rc , (AU) . − . . . − . . W (AU) . − . . . − . . ln( H ) − . − . . -1.0 0.5 − . − . . -1.0 0.5 R rc , (AU) . − . . . − . . W (AU) . − . . . − . . ln( H ) − . − . . -3.0 -1.0 − . − . . -3.0 -1.5 ln( δ cav ) − . − . . -3.0 -0.0 − . − . . -3.0 -0.0 ln( δ g , ) 0 . − . . -0.5 1.0 . − . . -0.5 1.0 ln( δ g , ) 0 . − . . -1.0 1.0 . − . . -1.0 0.5 F tot (mJy) . − . . . − . . OTE — Error range of the best parameters are estimated by ± σ . the parameters such as γ , ζ , and R rc , are slightly different,the profiles agree with each other.Figure 7 compares the visibility of the model and observa-tion. As can be seen in the figure, the models well reproducethe observed visibility, and the reduced χ for the model ofthe upper band data is . and that of the lower band data is . , respectively.Figure 8 compares the model and the observation in theimage. The model image is first converted to the ALMAmeasurement set by vis_simple with the observed mea-surement set and we made a mock observational image bythe same procedure of the imaging of the observed data withthe parameters listed in Table 1. In Figure 8, we show theintensity distributions along the major axis in the mock ob-servational image (model) and the observed image. In thebottom panel of the figure, we show the residual between themodel and the observational data. We calculated the residualas the subtraction between the model and the observation inthe visibility domain, by using vis_sample . The visibilityof the residual is converted by tclean task with the imag-ing parameters listed in Table 1. Around the center, one cansee the residual which is larger than σ , though the residualis smaller or comparable with the σ level in other regions.This discrepancy is related to the asymmetry which is indi-cated by the difference between the structures along the ma-jor and minor axes shown in Figure 2. Figure 9 shows the map of the residual between the modeland the observation in the upper and lower band data. Thepattern of the residual in the upper and lower band dataare similar to each other. One can see significant residualsaround the center, which is also shown by Figure 8. More-over, the residual map shows positive and negative structuresat the upper left (R.A. offset ≃ − . arcsec, Dec. offset ≃ . arcsec). The amplitudes of those structures are largerthan σ , which can indicate the real asymmetric structures.3.4. Asymmetric structure
As shown in the previous subsection, the residual map be-tween the model and the observation indicates the asymmet-ric structures at the center and the outer disk. Here we fur-ther investigate this asymmetry of the disk, by using a model-independent analysis.In the visibility domain, the visibility is expressed by V ( ~ρ ) = Z Z I ( ~R ) e − j ~R · ~ρ d ~R, (5)where j = √− is the imaginary unit and ~ρ and ~R indi-cate the position vectors in the deprojected uv plane and theimage, I ( ~R ) is the intensity distribution. When I ( ~R ) is ax-isymmetric, we can express the visibility as V ( ρ ) = 2 π Z I ( R ) J ( Rρ ) RdR, (6) K.D. K
ANAGAWA ET AL . F l u x ( m J y ) Upper band (233.0 GHz) F l u x ( m J y ) Lower band (216.7 GHz) ρ ( kλ ) − R e s i du a l ( m J y ) Upper band Lower band
Figure 7.
Real part of the visibilities for the observation (dot)and best-fit-model (solid)) for the upper and lower band data, in theupper and middle panels, respectively. The inset shows the zoom-upview of the region with < mJy. In the lower panel, we show thedifferent of the visibilities between the observation and the model.The error bars of the observational data and residuals are estimatedby σ deviation of the average. where J ( k ) is 0th-order Bessel function of the first kind.The visibility of the axisymmetric disk has only a real part.The image does not change if the disk is ◦ rotated againstthe disk center. In mathematics, a ◦ rotated image hasthe visibility which is the complex conjugate of that of theoriginal image. Hence, the difference between the originaland ◦ rotated images has only imaginary part, namely,twice the imaginary part of the original image. When thesystem does not have a significant asymmetric structure, thedifference is almost zero because the imaginary part of thevisibility of the original image is very small. On the otherhand, when the disk has asymmetric structures, we couldsee some residual between the original and ◦ rotated im-ages, which corresponds to the imaginary part of the visibil-ity. This approach of investigating asymmetric structures istotally model-independent.We produced the synthesized image using all the data,namely, the C43-4 and C43-8 data in upper and lower bands.Figure 10 shows the azimuthally averaged imaginary part of − . − . − . − .
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Figure 8.
Intensity distributions along the major axis in the modelsand observations for the upper band data(upper) and the lower banddata (middle). The bottom panel shows the residuals between themodel and the observation, along the major axis. the visibility. The average is calculated in the same way asin Figure 4. The imaginary part of the visibility is muchfainter than the real part. However, one can see the signif-icant signals as large as a few mJy, at ρ . kλ , whichimplies the asymmetric structure with the scale of & . arc-sec ( & AU). Figure 11 shows the synthesized image pro-duced only from the imaginary part of visibility data. Wecan find the asymmetric structure at the center and that at theupper left (R.A offset ≃ − . arcsec ( ≃ − AU) and Dec.offset ≃ . arcsec ( ≃ AU)), which is consistent with thedifference btween the observed image and the axisymmetricmodel shown in Figure 9. In addition, we see structures at thelower right, namely, R.A offset ≃ . arcsec and Dec. off-set ≃ − . arcsec. The Fourier transform of pure imaginaryfunction must be an odd function, which means that thereis a counter part in the opposite location in the synthesizedimage. If the asymmetric structure indicated by the imagi-nary part is real, we should find a signal at the same locationwith the model-subtracted image. Hence, we consider thatthe structure at the lower right could be the counter part ofthe structure at the upper left.LMA OBSERVATION OF THE PROTOPLANETARY DISK AROUND
WW C HA − . − . . . . − . − . − . − . . . . . . D ec . o ff s e t( a r c s ec ) − . − . − . . . . . I n t e n s i t y ( m J y / B e a m ) (a) Residual at . GHz − . − . . . . − . − . − . − . . . . . . D ec . o ff s e t( a r c s ec ) − . − . − . . . . . I n t e n s i t y ( m J y / B e a m ) (b) Residual at . GHz
Figure 9.
Residual between the observational data and model at 233.0 GHz (a), and at 216.7 GHz (b). The contour indicates the levels of ± σ and ± σ . The thick dashed contour lines indicate the observed intensity distribution, which are . mJy/Beam ( σ ), . mJy/Beam ( σ ), . mJy/Beam ( σ ), . mJy/Beam ( σ ), and . mJy/Beam ( σ ) from the outside. ρ ( kλ ) − − − F l u x ( m J y ) Figure 10.
Imaginary part of the visibility combined from all data.
As described in Section 2 and in the result shown above,we determined the phase center by the Gaussian fitting withCASA tool uvmodelfit . However, the above analysis ofthe imaginary part of the visibility depends on the choice ofthe phase center. Figure 12 shows how the images (depro-jected to the disk plane) produced from pure imaginary visi-bility change with the choice of the phase center. In the fig-ure, we shift the center in ± mas ( ± . AU) in horizontaland vertical directions from the phase center of Figure 11.The image is shifted in the visibility domain by the phaseshift defined as exp [2 π ( udx + vdy )] , where u and v are thespatial frequencies and dx and dy are shift values in R.A andDec. directions, respectively. It is reasonable to assume thatthe disk structure is almost axisymmetric, despite the diskhas some asymmetric structures. Under this assumption, the − . − . . . . − . − . . . . D ec . o ff s e t( a r c s ec ) − . − . − . . . . . I n t e n s i t y ( m J y / B e a m ) Figure 11.
Image synthesized from the imaginary part of the visi-bility shown in Figure 10. The contour indicates the levels of ± σ ( ± . mJy/Beam) and ± σ ( ± . mJy/Beam). image deprojected from the imaginary part with the ’correct’phase center has the minimum root mean square of the in-tensity. We calculated the sum of the root mean square valueof the intensity at each pixel within the radius of 0.6 arcsec(114 AU) from the center, which is labeled at the upper leftcorner at each panel (labeled by RMS). The figure with ourfiducial phase center has the minimum value of RMS amongthe listed panels. Hence, the phase center determined by theGaussian fit is consistent with the center which minimalizethe asymmetry. On the other hand, in the panel at the upperleft corner (dx=2 mas and dy=2 mas), the asymmetric struc-ture around the center is almost vanished. This indicates that0 K.D. K ANAGAWA ET AL ..
Image synthesized from the imaginary part of the visi-bility shown in Figure 10. The contour indicates the levels of ± σ ( ± . mJy/Beam) and ± σ ( ± . mJy/Beam). image deprojected from the imaginary part with the ’correct’phase center has the minimum root mean square of the in-tensity. We calculated the sum of the root mean square valueof the intensity at each pixel within the radius of 0.6 arcsec(114 AU) from the center, which is labeled at the upper leftcorner at each panel (labeled by RMS). The figure with ourfiducial phase center has the minimum value of RMS amongthe listed panels. Hence, the phase center determined by theGaussian fit is consistent with the center which minimalizethe asymmetry. On the other hand, in the panel at the upperleft corner (dx=2 mas and dy=2 mas), the asymmetric struc-ture around the center is almost vanished. This indicates that0 K.D. K ANAGAWA ET AL .. − . − . − . − .
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75 1 . R.A. offset − . − . − . − . . . . . . RMS=0.064 dx=-2.0 mas, dy=-2.0 mas − . − . − . . . . . I n t e n s i t y ( m J y ) Figure 12.
Images deprojected only from the imaginary part of the visibility as shown in Figure 11, but with the different phase center. Thecentral figure is the same as that shown in Figure 11 with the same phase center. The horizontal coordinate of phase center is shifted in 2 mas( − mas) from that of the central figure, in left (right) column, and the vertical coordinate of the phase center is shifted in 2 mas ( − mas) inthe top (bottom) row. In the middle column (row), the horizontal (vertical) coordinate of the phase center is the same as that of the center figure. the center of the inner structure is shifted to the center of theouter structure. MODELING BY RADIATIVE TRANSFERSIMULATIONS4.1.
Setup and model description
We now have a model for the intensity distribution of thedisk around WW Cha. In this section, we discuss the physicalcondition (e.g., dust surface density, size distribution of thedust grains, and temperature) of the disk, by using radiative transfer simulations with RADMC-3D (Dullemond et al.2012). We setup a model of an axisymmetric dust surfacedensity distribution which is motivated by the intensity dis-tribution derived in the previous section as Σ( R ) = Σ " f I (cid:18) RR c (cid:19) − s exp (cid:18) RR c (cid:19) − ζ + X i =1 H i exp (cid:18) R − R rc , i W i (cid:19) , (7) ∼ dullemond/software/radmc-3d/index.html LMA
OBSERVATION OF THE PROTOPLANETARY DISK AROUND
WW C HA f I is defined by Equation (3), ζ , R c , H i , W i , R rc , i , andparameters in f I ( δ cav , δ gc , , δ gc , , R cav , R g , in , , R g , out , ) arefixed to the values given in Table 2 (for C43-8 data). For theparameter s that is related to the slope of the dust surfacedensity, we use s = 0 , which is different from the best-fit parameter of γ which is ∼ . . We confirm that thechoice of s hardly affects the estimates of physical param-eters. When we use s = 0 . , the mid-plane temperature isaffected only by a few Kelvin. The parameter Σ controlsthe total mass of the dust M dust and when Σ = 1 . g/cm , M dust = 3 × − M ⊙ .We first vary the stellar luminosity and Σ and check theagreement with observations in order to address the uncer-tainty of the estimate of the disk physical parameters. Wethen vary the dust size in order to address the spectral indexdistribution. Here we present a physical disk and star modelthat reasonably matches observations. Full modeling studiesthat derive the uncertainties of all the parameters are beyondthe scope of this paper.Rest of the setup of the simulation is as follows. In thevertical direction, we adopt a Gaussian shape distribution,namely, ρ ( R, z ) = Σ( R ) √ πh exp (cid:18) − z h (cid:19) , (8)where h is the scale height of the dust layer. The dust scaleheight can be smaller than that of gas structure, due to dustsettling (e.g., Nakagawa et al. 1986). Hence, we considertwo dust components:one is a small dust component with sizedistribution ∝ s − . , where s is the size of the grains, andthe minimum and maximum sizes are . µ m and . mm,respectively. The other is a large dust component with itssize having Gaussian distribution in logarithmic space. Thepeak of the size distribution s d , large is mm with the smallestsize . mm and with the largest size of . mm. We assumethat the scale height of the large grains is 0.1 times the scaleheight of the gas. The mass ratio between the small and largegrains is set to be 1:9. We adopt the same compositions ofthe dust grains to that adopted in Birnstiel et al. (2018) . Theabsorption and scattering coefficients for each component areaveraged by the size distribution.The mass and the surface temperature of the central starare set to . M ⊙ and 4350 K, respectively (Luhman 2007).By the SED modeling, the stellar luminosity is estimated by L ⊙ (Garufi et al. 2020). Since WW Cha is a young new-born star, however, it could be still embedded in the core(Ribas et al. 2013; Garufi et al. 2020). In such a case, theluminosity may be underestimated due to high extinction(Follette et al. 2015). Hence, we also consider the case with L ∗ = 22 L ⊙ . The optical constant file was provided by Dr. Ryo Tazaki. − . − . − . − .
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Figure 13.
Brightness temperature distributions along the majoraxis of the disk, for the observation (black dot), the results of thesimulations with L ∗ = 11 L ⊙ , Σ = 3 g/cm (red dashed) and with L ∗ = 22 L ⊙ , Σ = 1 . g/cm (cyan solid). The radial coordinate extends from . AU to
AU,which is divided into 256 meshes with logarithmic spacing.The azimuthal angle θ and polar angle φ are divided into 256meshes in < θ < π and in < φ < π (the midplane islocated at φ = π/ ), respectively. We adopted × pho-tons for thermal Monte Carlo radiative transfer and imaging,and for the SED, photons are adopted.We first carried out radiative transfer simulations withthe disk scale height calculated by the empirical for-mula given by Dong et al. (2018a), namely, T =220( L ∗ / L ⊙ ) / ( R/ − . (it is quite similar to Equa-tion 1). After the first run, we calculate the temperature onthe midplane at each radius and we preformed simulationagain with the scale height given by the midplane temper-ature. We repeated the above cycles until the temperaturedistribution is converged. In our case, the temperature distri-bution is converged after 2–3 iterations.For the imaging, we converted the output of RADMC-3Dto the ALMA measurement set with observed measurementset by use of vis_sample . Then, we deprojected the imagefrom the model measurement set with the imaging parameterlisted in Table 1. 4.2. Results
Intensity and spectral energy distribution
Figure 13 compares the brightness temperatures given bythe observations and simulations. With L ∗ = 11 L ⊙ , we need Σ = 3 g/cm which corresponds to the total dust mass of × − M ⊙ . In this case, the disk is highly gravitationallyunstable in most part if the gas-to-dust mass ratio is 100, asshown below. If the stellar luminosity is larger by a factor oftwo ( L ∗ = 22 L ⊙ ), we found that Σ = 1 . g/cm is enoughto reproduce the observation.Figure 14 illustrates the SED at . µ m – 2 cm, givenby the previous observations (Lommen et al. 2007, 2009;2 K.D. K ANAGAWA ET AL ..
Figure 13 compares the brightness temperatures given bythe observations and simulations. With L ∗ = 11 L ⊙ , we need Σ = 3 g/cm which corresponds to the total dust mass of × − M ⊙ . In this case, the disk is highly gravitationallyunstable in most part if the gas-to-dust mass ratio is 100, asshown below. If the stellar luminosity is larger by a factor oftwo ( L ∗ = 22 L ⊙ ), we found that Σ = 1 . g/cm is enoughto reproduce the observation.Figure 14 illustrates the SED at . µ m – 2 cm, givenby the previous observations (Lommen et al. 2007, 2009;2 K.D. K ANAGAWA ET AL .. λ ( µ m) − − − − − ν F ν ( e r g/ c m / s ) L ∗ = 11 L ⊙ , Σ = 3 g/cm L ∗ = 22 L ⊙ , Σ = 1 . Our observationObservations
Figure 14.
Spectral energy distribution given by observations(dots) and the simulations with L ∗ = 11 L ⊙ , Σ = 3 g/cm (reddashed) and with L ∗ = 22 L ⊙ , Σ = 1 . g/cm (cyan solid).The crosses indicate the total fluxes given by our observation at233 GHz (494 mJy) and 216.7 GHz (418 mJy). The observa-tional data with λ < mm are extracted from VizieR database(https://vizier.u-strasbg.fr/) and the references are in the main text. Gutermuth et al. 2009; Ishihara et al. 2010; Cutri et al. 2014;Pascucci et al. 2016; Ribas et al. 2017), including our result,and the simulations. Both simulations with L ∗ = 11 L ⊙ and L ⊙ can reproduce the ALMA band 6 flux ( ∼ GHz).For fluxes at the longer wavelengths, namely mm < λ < cm, simulations agree with the observation. The fluxes ofthe simulations at λ > cm is smaller than the observedflux, though it could be due to the contribution from the free-free emission from the star (Rodmann et al. 2006). In thecase with L ∗ = 11 L ⊙ , the flux at far infrared wavelengthsis about a factor of two smaller than the observed values. Inthis case, we need some contribution from the envelope toaccount for infrared flux. On the other hand, the case with L ∗ = 22 L ⊙ can reproduce the fluxes from the infrared to theradio, by only the emission from the star.4.2.2. Dust density and temperature
The stellar luminosity affects estimate on the mass ofthe gas disk, but it does not significantly affect the mid-plane temperature. In the upper panel of Figure 15, when Σ = 1 . g/cm , we show the distribution of the dust surfacedensity and Toomre’s Q-value (Toomre 1964), assuming thegas-to-dust ratio of 100. The lower panel of Figure 15 showsthe midplane temperatures given by simulations. The mid-plane temperature with L ∗ = 22 L ⊙ is just about 1.16 timeshigher than that with L ∗ = 11 L ⊙ , roughly corresponding to L / ∗ dependence as expected from Equation (1).We find that the disk is expected to have relatively low val-ues of Toomre’s Q-value. As shown in the upper panel ofFigure 15, in the case of Σ = 1 . g/cm , the Q-value issmaller than or close to unity at 20 AU < R <
100 AU.In particular, the dust bump ( AU – AU) is gravita- . . . . . . Σ ( g/ c m ) T oo m r e ’ s Q Σ dust Toomre’s Q . . . . . . offset (arcsec) R (AU) T m i d ( K ) L ∗ = 11 L ⊙ , Σ = 3 g/cm L ∗ = 22 L ⊙ , Σ = 1 . CO on water iceCO sintering . . . . . . Figure 15. ( Upper ) Dust surface density and Toormre’s Q-value(with gas-to-dust ratio being 100) in the case of Σ = 1 . g/cm (in the case of Σ = 3 g/cm , Σ simply becomes two times largerand Q-value decreases with the increase of Σ .). The horizontal thinline indicates Q = 1 . ( Lower ) Midplane temperatures given bythe radiative transfer simulations. The two horizontal lines indicate K and K from the top. tionally unstable because Q . and the dust bump mayfragment. In the case of Σ = 3 g/cm , the Q-value are de-creased by a factor of two. In this case, the disk is still nearlygravitationally unstable and the fragmentation is expected at R > AU and turbulence due to gravitational instability isalso expected. However, no fragment-like structure is seen inthe dust-continuum image. This may indicate that the gas-to-dust ratio is smaller than 100 in the region with > AU. Itis discussed in Section 5.1.We note that the temperatures at the peak locations are ∼ – K, which is close to the freezeout temperature of theCO on water ice. This may be related to the origin of thebump structure, as discussed in Section 5.1.4.2.3.
Spectral index
Using the images given by the simulations at upper andlower bands, we produced the map of the spectral index bythe use of tclean task with nterm =2. Although it is cal-culated from the very narrow frequency range, we may con-strain the dust size distribution from it. The spectral index de-pends on the peak size of the large grain s d , large , rather thanstellar luminosity and the dust surface density. Hence, wecarried out additional simulations with s d , large = 0 . mm.In Figure 16, we show the spectral index map in the caseof s d , large = 1 . mm and . mm, when L ∗ = 22 L ⊙ and Σ = 1 . g/cm . In the case of s d , large = 1 . mm, the spec-LMA OBSERVATION OF THE PROTOPLANETARY DISK AROUND
WW C HA − . − . . . . − . − . . . . D ec . o ff s e t( a r c s ec ) . . . . . . . . . Sp ec tr a li nd e x (a) s large = 1 . mm − . − . . . . − . − . . . . D ec . o ff s e t( a r c s ec ) . . . . . . . . . Sp ec tr a li nd e x (b) s large = 0 . mm Figure 16.
Spectral index maps given by radiative transfer simulations, with s d , large = 1 mm (left panel) and s d , large = 0 . mm (right panel).The contours indicates the same flux levels shown in the panel (b) of Figure 1. − . − . − . − .
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Figure 17.
Distributions of the spectral index along the major axisin the case of s d , large = 1 . mm (red) and . mm (cyan). In theupper panel, L ∗ = 22 L ⊙ and Σ = 1 . g/cm , while in the lowerpanel, L ∗ = 11 L ⊙ and Σ = 3 . g/cm . The dots indicates theobserved one shown in the panel (d) of Figure 1. tral index increases in the outer region, which agrees withthe observations (panel (b) of Figure 1), but it is larger than2 around the center. In the case of s d , large = 0 . mm, onthe other hand, the spectral index is below 2 around the cen-ter, whereas it is much smaller than the observed index at theouter region.Figure 17 compares the distributions of the spectral indexalong the major axis between the simulations and the obser-vation. As can be seen in the figure, the distributions are sim-ilar between the case with L ∗ = 22 L ⊙ and Σ = 1 . g/cm and the case with L ∗ = 11 L ⊙ and Σ = 3 . g/cm . Inboth the cases, for the outer region of offset > . arcsec,the observational feature that the spectral index increases in the outer region is consistent with the radiative transfer cal-culations when s d , large = 1 . mm. For the inner region of < . arcsec, the spectral index below 2 is consistent withthe case of s d , large = 0 . mm.The spectral index also depends on the size distribution ofthe large grains, though the intensity does not significantlydepends on that. In Appendix C, we demonstrate radia-tive transfer simulations with the large grains which have apower-law size distribution like that of the small grains. Withthe power-law distribution, the spectral index does not be-come below 2, even if the maximum size of the dust grainsis 0.5 mm. Hence, the log-normal size distribution of largegrains may be preferred for the inner region. On the otherhand, the power-law distribution may be preferred for theouter region as can be seen in Figure 23.We should note that the spectral index discussed in this pa-per is provided from the very narrow range of the observedfrequency within band 6. The spatial variation of spectral in-dex should be investigated by future observations at multiplewavelengths. In Appendix D, we show a few examples of thespectral index based on ALMA bands, which are calculatedfrom the results of radiative transfer simulations. The spec-tral index can be different by the choice of the bands whichare taken to calculate the spectra index. We may be able toconstrain the dust size distribution from the spectral indexesin multiple bands. DISCUSSION5.1.
Origin of substructures
Ring
ANAGAWA ET AL .We found the bump with double peaks at 40 – 80 AU fromthe central star by the model fitting in the visibility domain.One of the peaks is at ∼ AU and the other is at ∼ AU.The bump structure can be formed by dust trapping due tothe pressure bump (e.g., Pinilla et al. 2012; Dullemond et al.2018). However, the bump that is formed by the above mech-anisms likely has a single peak, while the bump of our modelhas double peaks. If there are two pressure bumps, it mightexplain a shape with double peaks. Another possible loca-tion where a dust bump is likely to form is the outer edge ofthe planet-induced gap (e.g., Paardekooper & Mellema 2004;Muto & Inutsuka 2009; Zhu et al. 2012; Pinilla et al. 2015;Dong et al. 2015; Kanagawa et al. 2018). However, we didnot detect any gap structure interior to the bump structure.Hence, it might not be the structure induced by the dust trapof the pressure bump and planetary gap.One plausible scenario of producing a bump with doublepeaks uses the snowline. The bump can be formed due tovolatile freeze-out altering the coagulation and fragmenta-tion of dust grains (e.g., Zhang et al. 2015; Okuzumi et al.2016). As shown in Figure 15, the temperature around theouter peak is about K, which is close to the freezeouttemperature of CO at water ice Huang et al. (2018). Aroundthe inner peak, the temperature is about K. Although it isnot close to the condensation temperatures of major volatiles(e.g., CO, CO ), it is close to the condensation temperatureof H S (Zhang et al. 2015). Moreover, in T ∼ K, thesintering can occur for some species such as H S and C H (Okuzumi et al. 2016). Hence, this bump with double peaksmay be formed by the snowline and the sintering effect.5.1.2. Asymmetric structure
As discussed in Section 3.4, the asymmetric structure issuggested at the outer region of the disk, the positive (bright)structure at R.A. offset ≃ − . arcsec and the negative (dark)structure at R.A. offset ≃ − . arcsec. Recently Garufi et al.(2020) have observed a bright spiral in the disk of WW Chaby SPHERE observation, which wraps around from east tonorth in clockwise. In Figure 18, we compare the infraredimage given by SPHERE and the image synthesized fromthe imaginary part of the visibility (the same as that shownin Figure 11). The spiral feature in the SPHERE image hasbright and faint parts. The location of the positive asymmet-ric feature in sub-mm observations coincides with the brightNIR spiral while that of the negative asymmetric feature co-incides with the location of faint spiral region of the NIRobservation.In tandem with the radiative transfer modeling, we suggestthat the spiral features are formed by gravitational instability.When the value of Toomre’s Q is smaller than ∼ , the ef-fects of self-gravity become prominent spiral features appear(e.g., Lodato & Rice 2004). As shown in Figure 15, hence, − . − . . . . R.A. offset (arcsec) − . − . . . . D ec . o ff s e t( a r c s ec ) I n t e n s i t y ( J y ) Figure 18.
Comparison between the SPHERE image (Garufi et al.2020) (color) and the residual map for the upper band data shown inFigure 9 (contour). The contours show ± σ and ± σ levels. Thesolid and dashed contours indicate positive and negative excesses,respectively. the disk can be gravitational unstable at R = 40 – AU,when Σ = 1 . g/cm , if the gas-to-dust ratio is 100. Whenthe disk was highly gravitationally unstable, we could ob-serve clear spiral waves or fragmented structures. However,we cannot see clear significant spirals or fragments on thedisk, except the relatively weak structure at the upper left.Hence, the gas-to-dust ratio may be smaller than 100, espe-cially within the bump region. The faint positive structureat the upper left might be explained by the gravitational in-stability if the disk is marginally stable with a smaller gas-to-dust ratio. Alternatively, we cannot see significant spiralpattern because the disk is optically thick. In this case, thespiral might be observed at longer wavelengths.Another possible mechanism for making asymmetry isdust concentration in the vortex. Several protoplanetary diskshave been observed to have vortex structures, for instance,HD 142527 (Fukagawa et al. 2013; Soon et al. 2019), Sz 91Tsukagoshi et al. (2014); Canovas et al. (2016), Oph IRS 48(van der Marel et al. 2015), MWC 758 (Dong et al. 2018b),and etc. One possible mechanism to form such a vortex istrapping dust grains into vortex formed by Rossby wave in-stability (e.g., Lovelace et al. 1999; Li et al. 2000; Lin 2014;Fu et al. 2014; Ono et al. 2016). However, as different fromthe above disks, the asymmetric structure found in the diskof WW Cha is so faint that it is not visible in the raw image(Figure 1). It is visible only in the image with subtraction(the right panel of Figure 9 and Figure 11). The region of R > . arcsec can be optically thin and the intensity of theasymmetric structure ( ∼ σ ) is just 5 % of that of the sym-metric structure ( ∼ σ at R ≃ . arcsec, as can be seenin Figure 1). Such a faint structure might be difficult to beformed by the dust trap into the vortex.LMA OBSERVATION OF THE PROTOPLANETARY DISK AROUND
WW C HA A V ∼ . mag (Ribas et al.2013), compared to e.g., – mag of HL Tau system(ALMA Partnership et al. 2015)). Since the star has a highaccretion rate, the asymmetry may be associated to the accre-tion variability or a jet. So far, we do not see a significant jet-like structures both in the SPHERE and our dust continuumimages so it is difficult to address this further with currentobservations. Further observations (e.g., H α observations tosee accretion variability or jet structure) would be required.5.2. Inner cavity and binary
Although a large cavity is ruled out by the observed imageshown in Figure 1, our model allows the existence of a smallcavity with the radius of about AU. However, further obser-vations are required to confirm the small cavity, because it ismuch smaller than the angular resolution with the baselinesof < kλ . Moreover, we should note that there is a rela-tively large uncertainties on the MCMC fitting of the cavitysize ( R cav ) and depth ( δ cav ) as seen in Table 2. It indicatesthat our fitting cannot rule out the solution with no cavity.If the disk has the small cavity, it can be formed by binaryinteraction (e.g. Artymowicz & Lubow 1994; Dunhill et al.2015; Miranda et al. 2017; Thun et al. 2017; Price et al.2018). The binary separation a bin may be estimated from thecavity radius R as R = 2 . a bin (Artymowicz & Lubow1994), and equivalently a bin ≃ . AU. This separationis roughly consistent with the binary motion observed byAnthonioz et al. (2015).As shown in Section 3.4, the center of the inner structurecan be different in ∼ mas from the center of the outer struc-ture. When the eccentricity of the binary motion is relativelylarge, the shape of the cavity induced by the binary interac-tion is also eccentric (e.g. Thun et al. 2017). On the otherhand, the shape of the outer disk can keep a symmetric shapebecause of the small binary separation. Hence, the binaryeccentricity may be relatively large, if the star is binary.5.3. Dust size distribution
Finally, we discuss the size distribution of the dust grains inthe disk from the spectral index map, though it is calculatedfrom the narrow range of the observed frequency. As canbe seen in the panels (b) and (d) of Figure 1, the spectralindex below 2 around the center can be induced by the opticalthick scattering emission (Zhu et al. 2019; Liu 2019), and the model with s d , large = 0 . mm can reproduce such a spectralindex. In the outer region of > . arcsec from the center( R > AU), the spectral index increases in the outer region,which is consistent with the model with s d , large = 1 . mm.This feature implies that the size of the large dust is largerin the outer disk. Here, we discuss how such a distributioncan be realized. When the Stokes number of the dust grainsis much smaller than unity, the radial drift velocity can bewritten by (e.g., Nakagawa et al. 1986) v R , dust ≃ − StηV R , (9)where V R denotes the Keplerian rotation velocity, St is theStokes number of the dust grains given by πρ d s d / (2Σ gas ) ( ρ d is the internal density of the dust), and η = d ln P/d ln R is ∼ − in conventional disk models. Our model indicates Σ dust ∼ . g/cm (Figure 15), and hence, the Stokes num-ber of 1 mm-sized dust is about . when the gas-to-dustratio is 100 and ρ d = 3 . The radial drift timescale of thegrains, τ drift = − R/v R , dust , can be estimated by τ drift ≃ . Myr (cid:18) Σ dust g/cm (cid:19) (cid:16) ǫ (cid:17) (cid:18) ρ d g/cm (cid:19) − × (cid:16) s dust mm (cid:17) − (cid:16) η − (cid:17) − (cid:18) R AU (cid:19) / , (10)where ǫ is the gas-to-dust ratio and s dust is the size of the dustgrain, respectively. WW Cha is young and the age is about . Myr, which is shorter than the drift timescale of mm-sized dust and longer than the growth timescale of the dust, τ growth ≃ ǫ/ Ω K = 0 . Myr (50 AU /R ) / with ǫ = 100 (Brauer et al. 2008). Hence, the 1 mm-sized dust grains ob-served in the outer region is consistent with the dust drift.In the inner region, the drift timescale of the dust grains be-comes shorter. Considering the star is as young as < Myr,the drift timescale can be comparable with the stellar age at
R < AU. This may explain the reason why the size of thedust grains is smaller than in the outer region.The size of the grains can be determined by turbulentfragmentation (Birnstiel et al. 2010). In this case, the α -parameter relevant to that the fragmentation threshold sizeequals to ∼ mm is α = 1 . × − (cid:18) Σ dust g/cm (cid:19) (cid:16) ǫ (cid:17) × (cid:18) ρ d g/cm (cid:19) − (cid:16) s dust mm (cid:17) − (cid:16) v f m/s (cid:17) (cid:18) T K (cid:19) − , (11)where v f is the fragmentation threshold velocity. Hence,considering the mid-plane temperature shown in Figure 15,we can estimate α ≃ − . This relatively large α is consis-tent with the high accretion rate onto the star (Manara et al.6 K.D. K ANAGAWA ET AL .2016), whereas it is larger than a value of α estimated bythe recent observations of the disks around the aged stars,namely, HD 163296 (Flaherty et al. 2015), and TW Hya(Teague et al. 2016; Flaherty et al. 2018), namely, α . − .Such a high viscosity may be due to turbulence induced bygravitational instability (e.g., Boley et al. 2006).As mentioned in Section 4.2.3, we note that the above dis-cussion is based on the spectral index provided only from thevery narrow range within band 6. The spectral index shouldbe investigated by future multi-wavelength observations. CONCLUSIONWe presented the dust continuum emission of the proto-planetary disk around WW Cha, observed by ALMA Band 6.Our conclusions are summarized as follows:1. The dust continuum image clearly shows no large cav-ity, and a faint dust bump (Figure 1). We also found theasymmetric structure at the center (Figure 2). More-over, since the visibility is clearly different in the upper( . GHz) and lower bands ( . GHz) (Figure 4),we can obtain the spectral index map. The spectral in-dex around the center is below 2, and it becomes largerin the outer region.2. We constructed a model to fit the observation in visibil-ity domain by MCMC fitting (Section 3). Our modelhas a bump extending from ∼ AU from the cen-tral star to ∼ AU, with two local peaks located at ∼ AU and ∼ —AU. As a result of radiative trans-fer simulations (Figure 15), the midplane temperaturearound the outer peak is close to the freezeout temper-ature of CO on water ice ( ∼ K). The midplane tem-perature around the inner peak is about K, which isclose to the condensation temperature of H S and it isalso close to the temperature that the sintering can becaused for several species. Hence, this bump may beinduced by the snowline and the sintering effect.3. The residual map between the observed data and bestfit model indicates asymmetric structures at the cen-ter and the upper left of the disk. We also confirmedthose asymmetric structures by a model-independentmethod, which is imaging of the imaginary part of thevisibility of the observed data (Section 3.4). These structure could be robust, though the amplitude of theasymmetric structures are faint ( ∼ σ level ) as com-pared with the symmetric structure.4. The spectral index map given by the observation maybe consistent with the result of radiative transfer sim-ulations with the relatively large dust grains (1 mm)in the outer region, whereas the result of the simula-tion with the smaller dust grains (0.5 mm) can be suit-able for the region close to the center (Figures 16 and17). This implies that the size of the dust grains islarger than that in the inner region. As discussed inSection 5.3, such a distribution is consistent with theradial drift and collisional growth of the dust grains,because of the massive disk around a young star.We thank Ryo Tazaki for providing the optical constantdata to calculate opacity of dust grains. KDK was also sup-ported by JSPS Core-to-Core Program “International Net-work of Planetary Sciences” and JSPS KAKENHI. Thiswork is in part supported by JSPS KAKENHI grant Nos.18H05441 and 17H01103. Y.H. is supported by the JetPropulsion Laboratory, California Institute of Technology,under a contract with the National Aeronautics and Space.H.B.L. is supported by the Ministry of Science and Tech-nology (MoST) of Taiwan (grant Nos. 108-2112-M-001-002-MY3). This paper makes use of the following ALMAdata: ADS/JAO.ALMA Software: A. STATISTICS OF VISIBILITY DATAIn this appendix, we show the statistics of the visibility data. In the upper panel of Figure 19, we shows the real parts of thevisibility for the C43-4 (compact configuration) and C43-8 (sparse configuration) data in the upper and lower bands, separately.LMA
OBSERVATION OF THE PROTOPLANETARY DISK AROUND
WW C HA ρ ( kλ ) F l u x ( m J y ) Upper band
C43-8C43-4 ρ ( kλ ) F l u x ( m J y ) Lower band
C43-8C43-4 ρ ( kλ ) S t a nd a r dd e v i a t i o n ( m J y ) C43-8 C43-4 ρ ( kλ ) S t a nd a r dd e v i a t i o n ( m J y ) C43-8 C43-4
Figure 19. ( Upper ) Real parts of the visibility of the C43-4 (compact array configuration) and C43-8 (sparse array configuration) data in theupper and lower bands. The visibilities are averaged within the bin which the radial width is the same as that described in Section 2 and theazimuthal width is . π . The thin solid lines indicates the visibilities shown in Figure 4. ( Lower ) Standard deviation of the data within the bin.
The visibility in the figure is averaged the bin of the same uv -distance width as Figure 4 but with the azimuthal width of . π ,instead of whole π in Figure 4. Hence, the figure enable us to see the scatter of data in the azimuthal direction. For reference, weplot the visibility shown in Figure 4. In the lower panel of Figure 19, we shows the standard deviation of the data within the bin.The visibilities of the C43-4 and C43-8 data are similar to each other. The standard deviations are similar in all the data, namely σ ≃ mJy. However, around ρ = 200 kλ , there are some points with larger standard deviations in the C43-4 data (especially atthe upper band data). Because of this data scatter, the value of χ increases around ρ = 200 kλ when the visibility combined bythe short and long baseline data for the MCMC fitting.Figure 20 is the same as Figure 19 but for the imaginary part of the visibility. The standard deviation of the imaginary part islarger than that of the real part in the shorter baseline, namely ρ . kλ , whereas it is comparable with that of the real parts atlong baseline. Hence, the imaginary part of the visibility shown in Figure 10 has a relatively large error at short baseline. B. POSTERIORS OF MCMC FITTINGThe posterior of the MCMC fitting for the upper hand data is shown in Figure 21 and that for the lower band data is shown inFigure 22. C. SPECTRAL INDEX MAP WITH A POWER-LAW SIZE DISTRIBUTION FOR LARGE DUST GRAINSIn Section 4, we carried out radiative transfer simulations with large grains which has a Gaussian size distribution around thespecific radius of the dust grains. Here, we show the results of radiative transfer simulations with large grains which has a power-law size distribution like that of the small grains, and investigate the dependence of the dust size distribution on the spectral indexmap.As in the cases described in Section 4, we consider two-kind of dust grains: one represents small grains, other is large grains.The size distribution of the small dust is the same as that described in Section 4. For the large grains, the number density of thedust grains is proportional to s p , and we adopted p = − . and − . cases. The minimum size of the large grains is . mm, andwe consider three maximum sizes of the large grains, namely, s max = L ⊙ and Σ = 1 . g/cm , and other parameters are the same these described in Section 4.8 K.D. K ANAGAWA ET AL ..
The visibility in the figure is averaged the bin of the same uv -distance width as Figure 4 but with the azimuthal width of . π ,instead of whole π in Figure 4. Hence, the figure enable us to see the scatter of data in the azimuthal direction. For reference, weplot the visibility shown in Figure 4. In the lower panel of Figure 19, we shows the standard deviation of the data within the bin.The visibilities of the C43-4 and C43-8 data are similar to each other. The standard deviations are similar in all the data, namely σ ≃ mJy. However, around ρ = 200 kλ , there are some points with larger standard deviations in the C43-4 data (especially atthe upper band data). Because of this data scatter, the value of χ increases around ρ = 200 kλ when the visibility combined bythe short and long baseline data for the MCMC fitting.Figure 20 is the same as Figure 19 but for the imaginary part of the visibility. The standard deviation of the imaginary part islarger than that of the real part in the shorter baseline, namely ρ . kλ , whereas it is comparable with that of the real parts atlong baseline. Hence, the imaginary part of the visibility shown in Figure 10 has a relatively large error at short baseline. B. POSTERIORS OF MCMC FITTINGThe posterior of the MCMC fitting for the upper hand data is shown in Figure 21 and that for the lower band data is shown inFigure 22. C. SPECTRAL INDEX MAP WITH A POWER-LAW SIZE DISTRIBUTION FOR LARGE DUST GRAINSIn Section 4, we carried out radiative transfer simulations with large grains which has a Gaussian size distribution around thespecific radius of the dust grains. Here, we show the results of radiative transfer simulations with large grains which has a power-law size distribution like that of the small grains, and investigate the dependence of the dust size distribution on the spectral indexmap.As in the cases described in Section 4, we consider two-kind of dust grains: one represents small grains, other is large grains.The size distribution of the small dust is the same as that described in Section 4. For the large grains, the number density of thedust grains is proportional to s p , and we adopted p = − . and − . cases. The minimum size of the large grains is . mm, andwe consider three maximum sizes of the large grains, namely, s max = L ⊙ and Σ = 1 . g/cm , and other parameters are the same these described in Section 4.8 K.D. K ANAGAWA ET AL .. ρ ( kλ ) − − F l u x ( m J y ) Upper band
C43-8 C43-4 ρ ( kλ ) − − F l u x ( m J y ) Lower band
C43-8 C43-4 ρ ( kλ ) S t a nd a r dd e v i a t i o n ( m J y ) C43-8 C43-4 ρ ( kλ ) S t a nd a r dd e v i a t i o n ( m J y ) C43-8 C43-4
Figure 20.
The same as Figure 19, but for the imaginary parts.
Figure 23 shows the spectral index along the major axis for the cases with p = − . and p = − . . In all the cases, the spectralindex is about (but slightly larger than 2), and it increases in outer region. D. SPECTRAL INDEX MAP FOR ALMA BANDSFor future observations, in this appendix, we present a few examples of the spectral index map given by radiative transfersimulations. As in the previous section, L ⊙ = 22 L ⊙ and Σ = 1 . g/cm , and other parameters are the same these describedin Section 4. As different from these shown in Section 4, the spectral index presented in this section is not calculated throughthe tclean task. The spectral index is calculated by just the differences of the fluxes convoluted with a Gaussian filter with . arcsec standard deviation. We calculated the spectral index as the difference of the fluxes among ALMA band 3, band 6,band 7, and band 9.Figure 24 shows the spectral index along the major axis, when the large grains has a Gaussian (log-normal) distribution (for thedetail, see Section 4). The spectral index depends on the choice of the bands used. The spectral index calculated from bands withlonger wavelengths becomes larger , as the opacity is smaller and it is optically thin. For instance, the spectral index calculatedfrom the band 3 and band 6 fluxes are larger than the spectral index calculated from other pairs of bands. The spectral indexcalculated from the band 7 and band 9 is below 2 around the center, regardless of s d , large . It is worth pointing out that the spectralindex calculated from band 6 and band 7 significantly different from that calculated from the upper and lower bands of band 6,in the cases with . mm. Figure 25 shows the same as that shown in Figure 25, but for the cases that the large dust grains hasa power-law distribution of ∝ s − . . The distributions of the spectral index calculated by band 6 and 7 is similar to that in thecase with the Gaussian distribution shown in Figure 24, which is below 2 around the center. On the other hand, the spectral indexcalculated from the band 3 and 6 is larger than that in the case with the power-law distributions, as compared with that in thecases with the Gaussian distribution. We may be able to constrain the dust size distribution from the difference of the spectralindexes calculated from the different pair of the bands. REFERENCES Akiyama, E., Muto, T., Kusakabe, N., et al. 2015, ApJL, 802, L17,doi: 10.1088/2041-8205/802/2/L17 Akiyama, E., Hashimoto, J., Liu, H. B., et al. 2016, AJ, 152, 222,doi: 10.3847/1538-3881/152/6/222
LMA
OBSERVATION OF THE PROTOPLANETARY DISK AROUND
WW C HA γ = 0.349 +0.030−0.025 . . . . . γ c γ c = 1.946 +0.079−0.061 R c R c = 51.160 +2.249−1.777 . . . . R c a v R cav = 1.296 +0.348−0.490 R g , i n , R g,in,1 = 35.908 +0.216−0.211 R g , o u t , R g,out,1 = 81.994 +3.488−2.973 R r c , R rc,1 = 67.046 +0.216−0.240 W W = 10.835 +0.613−0.553 − . − . − . . . l n ( H ) ln(H ) = −0.627 +0.026−0.019 R r c , R rc,2 = 122.223 +5.861−6.281 W W = 56.530 +3.597−3.527 − . − . − . − . − . l n ( H ) ln(H ) = −1.708 +0.051−0.031 − . − . − . − . . l n ( δ h o l e ) ln(δ hole ) = −1.055 +0.498−0.763 − . . . . . l n ( δ g , ) ln(δ g,1 ) = 0.194 +0.009−0.010 − . − . . . . l n ( δ g , ) ln(δ g,2 ) = 0.305 +0.037−0.045 . . . . . γ F t o t . . . . . γ c
40 44 48 52 56 R c . . . . R cav
32 36 40 44 R g,in,1
60 75 90 105 R g,out,1
60 64 68 72 76 R rc,1 W − . − . − . . . ln(H )
105 120 135 150 R rc,2
15 30 45 60 75 W − . − . − . − . − . ln(H ) − . − . − . − . . ln(δ hole ) − . . . . . ln(δ g,1 ) − . − . . . . ln(δ g,2 )
472 480 488 496 504 F tot F tot = 493.995 +0.527−0.553 Figure 21.
Posterior of the MCMC fitting for the upper band data.ALMA Partnership, Brogan, C. L., P´erez, L. M., et al. 2015, ApJL,808, L3, doi: 10.1088/2041-8205/808/1/L3Andrews, S. M., Huang, J., P´erez, L. M., et al. 2018, ApJ, 869,L41, doi: 10.3847/2041-8213/aaf741Anthonioz, F., M´enard, F., Pinte, C., et al. 2015, A&A, 574, A41,doi: 10.1051/0004-6361/201424520Artymowicz, P., & Lubow, S. H. 1994, ApJ, 421, 651,doi: 10.1086/173679 Birnstiel, T., Dullemond, C. P., & Brauer, F. 2010, A&A, 513, A79,doi: 10.1051/0004-6361/200913731Birnstiel, T., Dullemond, C. P., Zhu, Z., et al. 2018, ApJL, 869,L45, doi: 10.3847/2041-8213/aaf743Boley, A. C., Mej´ıa, A. C., Durisen, R. H., et al. 2006, ApJ, 651,517, doi: 10.1086/507478Brauer, F., Dullemond, C. P., & Henning, T. 2008, A&A, 480, 859,doi: 10.1051/0004-6361:20077759
ANAGAWA ET AL ..
ANAGAWA ET AL .. γ = 0.280 +0.023−0.021 . . . . . γ c γ c = 1.489 +0.062−0.041 R c R c = 45.881 +1.710−1.499 . . . . R c a v R cav = 0.986 +0.353−0.450 R g , i n , R g,in,1 = 35.630 +0.245−0.314 R g , o u t , R g,out,1 = 71.894 +1.804−1.949 R r c , R rc,1 = 68.310 +0.775−0.742 W W = 10.990 +1.308−0.621 − . − . − . . . l n ( H ) ln(H ) = −0.826 +0.036−0.024 R r c , R rc,2 = 146.630 +4.100−6.825 W W = 42.890 +5.221−3.547 − . − . − . − . − . l n ( H ) ln(H ) = −2.036 +0.060−0.050 − . − . − . − . . l n ( δ h o l e ) ln(δ hole ) = −1.517 +0.781−0.845 − . . . . . l n ( δ g , ) ln(δ g,1 ) = 0.199 +0.008−0.010 − . − . − . . . l n ( δ g , ) ln(δ g,2 ) = 0.091 +0.055−0.063 . . . . . γ F t o t . . . . . γ c
30 40 50 60 70 R c . . . . R cav
32 36 40 44 R g,in,1
60 75 90 105 R g,out,1
60 64 68 72 76 R rc,1 W − . − . − . . . ln(H )
120 140 160 180 R rc,2
15 30 45 60 75 W − . − . − . − . − . ln(H ) − . − . − . − . . ln(δ hole ) − . . . . . ln(δ g,1 ) − . − . − . . . ln(δ g,2 )
392 400 408 416 424 F tot F tot = 418.431 +0.558−0.593 Figure 22.
Posterior of the MCMC fitting for the lower band data.Canovas, H., Caceres, C., Schreiber, M. R., et al. 2016, MNRAS,458, L29, doi: 10.1093/mnrasl/slw006Chiang, E. I., & Goldreich, P. 1997, ApJ, 490, 368,doi: 10.1086/304869Cieza, L. A., Casassus, S., P´erez, S., et al. 2017, ApJL, 851, L23,doi: 10.3847/2041-8213/aa9b7bCutri, R. M., et al., dummy, & dummy. 2014, VizieR Online DataCatalog, II/328 Dong, R., Najita, J. R., & Brittain, S. 2018a, ApJ, 862, 103,doi: 10.3847/1538-4357/aaccfcDong, R., Zhu, Z., & Whitney, B. 2015, ApJ, 809, 93,doi: 10.1088/0004-637X/809/1/93Dong, R., Liu, S.-y., Eisner, J., et al. 2018b, ApJ, 860, 124,doi: 10.3847/1538-4357/aac6cbDullemond, C. P., & Dominik, C. 2005, A&A, 434, 971,doi: 10.1051/0004-6361:20042080
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OBSERVATION OF THE PROTOPLANETARY DISK AROUND
WW C HA − . − . − . − .
25 0 .
00 0 .
25 0 .
50 0 .
75 1 . offset (arcsec) . . . . . . . Sp ec tr a li nd e x s max =1.0 mm, p=-3.5 s max =0.5 mm, p=-3.5 s max =1.0 mm, p=-2.5 s max =0.5 mm, p=-2.5Obs Figure 23.
The same as Figure 17, but for the power-law sized large grains, and in the case of p = − . (solid lines) and the case with p = − . (dashed lines). − . − . . . . . . . . . Sp ec tr a li nd e x Gaussian dist. with s large =1.0 mm b3-6b6-7 b7-9b6 up-low − . − . . . . offset (arcsec) . . . . . Sp ec tr a li nd e x Gaussian dist. with s large =0.5 mm b3-6b6-7 b7-9b6 up-low Figure 24.
The spectral index distributions along the major axis in the cases that the large dust grains have a Gaussian size distribution with s d , large = 1 mm (top), . mm (bottom). The dashed, dotted, and solid lines indicate the spectral indices calculated from the fluxes at band 3(3.1 mm) and at band 6 (1.3 mm), the fluxes at band 6 and at band 7 (0.87 mm), and the fluxes at band 7 and band 9 (0.45 mm), respectively.The thin solid line indicates the spectral index given by the upper and lower bands of band 6 as the same as presented in the main text, forreference. ANAGAWA ET AL ..
The spectral index distributions along the major axis in the cases that the large dust grains have a Gaussian size distribution with s d , large = 1 mm (top), . mm (bottom). The dashed, dotted, and solid lines indicate the spectral indices calculated from the fluxes at band 3(3.1 mm) and at band 6 (1.3 mm), the fluxes at band 6 and at band 7 (0.87 mm), and the fluxes at band 7 and band 9 (0.45 mm), respectively.The thin solid line indicates the spectral index given by the upper and lower bands of band 6 as the same as presented in the main text, forreference. ANAGAWA ET AL .. − . − . . . . Sp ec tr a li nd e x Power law dist. with s max =1.0 mm b3-6b6-7 b7-9b6 up-low − . − . . . . offset (arcsec) Sp ec tr a li nd e x Power law dist. with s max =0.5 mm b3-6b6-7 b7-9b6 up-low Figure 25.
The same as Figure 24, but the large grains have a power-law size distribution.
LMA
OBSERVATION OF THE PROTOPLANETARY DISK AROUND
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