High circular polarization of near infrared light induced by micron-size dust grains
MMNRAS , 1–6 (2019) Preprint 12 June 2020 Compiled using MNRAS L A TEX style file v3.0
High circular polarization of near infrared light induced bymicron-size dust grains
Hajime Fukushima (cid:63) , Hidenobu Yajima , Masayuki Umemura Center for Computational Sciences, University of Tsukuba, Ten-nodai, 1-1-1 Tsukuba, Ibaraki 305-8577, Japan
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We explore the induction of circular polarization (CP) of near-infrared light in star-forming regions using three-dimensional radiative transfer calculations. The simula-tions trace the change of Stokes parameters at each scattering/absorption process ina dusty gas slab composed of aligned grains. We find that the CP degree enlargessignificantly according as the size of dust grains increases and exceeds ∼ percentfor micron-size grains. Therefore, if micron-size grains are dominant in a dusty gasslab, the high CP observed around luminous young stellar objects can be accountedfor. The distributions of CP show the asymmetric quadrupole patters regardless ofthe grain sizes. Also, we find that the CP degree depends on the relative position ofa dusty gas slab. If a dusty gas slab is located behind a star-forming region, the CPreaches ∼ percent in the case of . µ m dust grains. Hence, we suggest that theobserved variety of CP maps can be explained by different size distributions of dustgrains and the configuration of aligned grains. Key words:
Recent observations show that circular polarization(CP) of near infrared-light exists around young stellar ob-jects (YSOs; e.g., Fukue et al. 2010; Kwon et al. 2013, 2014,2016, 2018). In many cases, the observed CP maps exhibita quadrupole pattern. Although the degree of CP is at alevel of (cid:46) percent in most of the cases, a few YSOs showCP higher than 10 percent. In particular, the star-formingregion NGC6334-V exhibits ∼ percent CP (Kwon et al.2013). However, the origin of such high CP has not revealed.It is argued that interstellar circularly polarized lightcould be responsible for the enantiomeric excesses of bio-logical molecules (Bailey et al. 1998). The homochirality ofbiological molecules on the Earth is a long-standing mystery.In modern astrobiology, seeking for the origin of homochiral-ity in space is a significant theme. It is reported that variousamino acids exist in meteorites, and also, the enantiomericexcesses of L-amino acids are detected in the Murchison andMukundpura meteorites (Engel & Macko 1997; Pizzarello& Yarnes 2018). It has been hitherto proposed that aminoacids were formed in interstellar space and delivered onto theEarth by meteorites (Bonner 1991). As suggested in Baileyet al. (1998), CP of interstellar radiation could break down (cid:63) E-mail:[email protected] the symmetry of chiral amino acids efficiently, resulting inthe enantiomeric excess.Previous works have shown that CP can be producedby the scattering from non-spherical dust grains (Gledhill& McCall 2000; Wolf et al. 2002). Lucas et al. (2005) ar-gued that the high degree of CP than ∼ percent cannotbe produced by single-scattering. Therefore, the dichroic ex-tinction has been frequently considered as the origin of CP(e.g., Kwon et al. 2013). However, their analyses have beenmade for limited physical conditions of dust grains. There isa wide range of physical conditions, e.g., different grain sizes,dust distributions around a YSO, and so on. In particular,dust grains larger than . µ m have not been considered inthe previous works.Recent observations suggest that dust grains growthrough accretion and coagulation in high-density regionsnear massive stars and reach micron size (Pagani et al. 2010;Steinacker et al. 2010). If the dust size is close to the wave-lengths of photons, the change of polarization on scatteringbecomes significantly different from that in cases of smallergrains. However, CP in the environment with such large dustgrains has not been investigated. Therefore, in this Letter , westudy the generation of CP for a wide range of grain sizesby three-dimensional radiative transfer simulations. Espe-cially, we concentrate on the impacts of grain sizes on thedegree and pattern of CP. Also, we explore the dependence © a r X i v : . [ a s t r o - ph . GA ] J un Fukushima et al. (a) Computational setup ~a
Dust grain Y ⇥
Schematic view of our radiative transfer calculation:(a) the spatial positions of the radiation source, the dusty gasslab, and the observer; and (b) definitions of angles related tosingle scattering. The vector (cid:174) a and l represent respectively thedirection of the dust minor axis and the distance between thestar and the dusty gas slab. In the fiducial model, we set the starin the center of the slab as l = . In the panel (b), Θ and ( θ, φ ) represent the opening angle between the incident light and theminor axis of the dust grains and the scattered direction (see,Draine & Flatau 2013). of CP on the spatial distributions of aligned grains arounda star-forming region. We develop a numerical code for calculating radiationtransfer of the Stokes parameters based on the Monte Carlotechnique. Here, we consider the near-infrared wavelengthof λ = . µ m , in which the high degree of CP is detectedin observations (Kwon et al. 2013). We set a circular slabof dusty gas with a radius of and a thickness of . near a massive star. In this Letter , we study the generationmechanism of high CP degree with this idealized setup. Notethat, in reality, the spatial distribution of dusty gas arounda star can be complicated with clumps, holes, and shell-likestructure.In a fiducial model, the star is located in the centerof the slab. To study the impact of the relative positionbetween the star and the dusty slab, we consider the casesin which the dusty slab is placed at . behind or in frontof the star (Fig. 1-a). The number density of dust grainsis set as the scattering optical depth along the horizontaldirection is unity at λ = . µ m . If the optical depth islarger than unity, the number of multiple scattering photonsincreases. In this case, due to mismatch of the sign of CP,the CP degree is likely to be decreased. Here, photon packetsare emitted from the star and assumed to be unpolarizedinitially. For the radiative transfer calculations, the dustyslab is divided by × × cells.In the transport of a photon packet, the path lengthfrom a scattering position to next scattering one is evaluatedby the optical depth that is stochastically determined by τ = − ln ( R ) , where R is the random number in the range of − . We pursue photon packets, which can satisfactorilyreduce the shot noise from the random number.Each photon packet has information of the Stokes vector I = ( I , Q , U , V ) T . The transformation of the Stokes vector inone scattering event is determined by the M¨uller matrix as(e.g., Bohren & Huffman 1983), (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) IQUV (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) sc = k r (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) S S S S S S S S S S S S S S S S (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) IQUV (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) i , (1)where k and r are the photon wavenumber and the distancefrom the scattering point. Using Equation (1), the differen-tial cross section for scattering is estimated as dC sca d Ω = k ( S I + S Q + S U + S V ) I . (2)We obtain the cross section by integrating Equation (2) withrespect to solid angle Ω . The direction of scattering light isstochastically chosen based on the phase function obtainedfrom Equation (2). Note that the Stokes parameters ( Q and U ) depend on the coordinate. In Equation (1), the directionof the linear polarization of ( Q , U ) = ( , ) corresponds to theplane which contains incident and scattering directions. Wealso include the absorption of photons estimated as (e.g.,manual for RADMC-3D, Dullemond et al. 2012), dds (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) I h I v UV (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) = − (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) α h α v α uv
00 0 0 α uv (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) I h I v UV (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) , (3)where I h = ( I + Q )/ and I v = ( I − Q )/ . In Equation (3), α h and α v are absorption coefficients for the horizontal andvertical polarization light for the scattering plane, and α uv = ( α h + α v )/ .We evaluate the M¨uller matrix using the DDSCAT code(Draine & Flatau 1994), where dust grains are represented asan array of electric dipoles. Given that a dust grain is muchsmaller than the wavelength of incident light ( π a d (cid:28) λ ), theelectromagnetic field in the dust grain can be understood asthe behavior of a couple of dipoles along intersecting axes atright angles on a plane perpendicular to the incident light,i.e., Rayleigh approximation. On the other hand, accordingas the grain size increases, the interaction among multipledipoles leads to the complicated phase distributions of cir-cular polarization. In this case, the polarization of scatteredlight is given by superposition of electromagnetic waves fromall dipoles. We discrete the opening angle Θ with 91 bin,and the scattered angles ( θ, φ ) with 1024 bin (see Fig. 1) byHEALPix (G´orski et al. 2005).In this study, we assume dust grains of oblate spheroidswith the axial ratio of 2:1. The oblate dust grains are pre-ferred rather than prolate ones, and this axis ratio is similarto that was suggested for the aligned interstellar dust grains(Hildebrand & Dragovan 1995; Whitney & Wolff 2002). Thedust grains are placed as the minor axes of grains are alignedwith the radial direction of the dusty slab. In diffuse in-terstellar medium, dust grains larger than a d ∼ . µ m are aligned (Kim & Martin 1995; Andersson et al. 2015).The CP degree is mainly generated by the larger grains( a d (cid:61) . µ m ) in this study, and thus we adopt this simpleassumption. We use the dielectric function of astronomical MNRAS000
00 0 0 α uv (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) I h I v UV (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) , (3)where I h = ( I + Q )/ and I v = ( I − Q )/ . In Equation (3), α h and α v are absorption coefficients for the horizontal andvertical polarization light for the scattering plane, and α uv = ( α h + α v )/ .We evaluate the M¨uller matrix using the DDSCAT code(Draine & Flatau 1994), where dust grains are represented asan array of electric dipoles. Given that a dust grain is muchsmaller than the wavelength of incident light ( π a d (cid:28) λ ), theelectromagnetic field in the dust grain can be understood asthe behavior of a couple of dipoles along intersecting axes atright angles on a plane perpendicular to the incident light,i.e., Rayleigh approximation. On the other hand, accordingas the grain size increases, the interaction among multipledipoles leads to the complicated phase distributions of cir-cular polarization. In this case, the polarization of scatteredlight is given by superposition of electromagnetic waves fromall dipoles. We discrete the opening angle Θ with 91 bin,and the scattered angles ( θ, φ ) with 1024 bin (see Fig. 1) byHEALPix (G´orski et al. 2005).In this study, we assume dust grains of oblate spheroidswith the axial ratio of 2:1. The oblate dust grains are pre-ferred rather than prolate ones, and this axis ratio is similarto that was suggested for the aligned interstellar dust grains(Hildebrand & Dragovan 1995; Whitney & Wolff 2002). Thedust grains are placed as the minor axes of grains are alignedwith the radial direction of the dusty slab. In diffuse in-terstellar medium, dust grains larger than a d ∼ . µ m are aligned (Kim & Martin 1995; Andersson et al. 2015).The CP degree is mainly generated by the larger grains( a d (cid:61) . µ m ) in this study, and thus we adopt this simpleassumption. We use the dielectric function of astronomical MNRAS000 , 1–6 (2019) ircular polarization of near infrared light silicate ( m = . + . i , Draine 2003b) to calculate theM¨uller matrix.The size distributions n d of dust grains in the diffuseinterstellar medium is often modeled as the MRN mixture( n d ∝ a − . ) in the range . µ m (cid:53) a d (cid:53) . µ m , where a d is the grain size (Mathis et al. 1977). Around YSOs, onthe other hand, the size distributions can differ from theMRN model because the grain size increases via metal ac-cretion and coagulation (e.g., Ormel et al. 2009; Hirashita &Li 2013). The mid-infrared scattered light from dense coresin star-forming regions suggests that there are micron-sizeddust grains (Pagani et al. 2010; Steinacker et al. 2010). How-ever, the grain size distributions around YSOs are still un-clear. Here, we employ three models with a single size of . , . or . µ m as well as the MRN mixture case in the sizerange of − µ m (cid:53) a d (cid:53) µ m . In this work, the grain sizeis defined as a d = ( V / π ) / , where V is the volume of theoblate dust. The dust grains are assumed to be composed ofdielectric materials. We note that the albedos for a d = . , . and . µ m are . , . and . .Finally, we construct the observational images on thescreens in arbitrary directions using the Stokes vectors atscattering points. For the purpose, we solve the radiativetransfer to the observer as (e.g., Yajima et al. 2012) I ob = I sc exp [−( τ ab + τ sc )] , (4)where I ob and I sc are the Stokes vectors at the observerand at a scattering point, τ ab and τ sc are the optical depthsof absorption and scattering from the scattering point tothe observer. By integrating the Stokes vectors calculatedin Equation (4), the polarization map is built up. In thiswork, the propagation angle of photon packets to the exter-nal screen is discretized by HEALPix (G´orski et al. 2005)with 1024 angle bins. The resultant CP maps for different grain size modelsare shown in Figure 2. In this work, we present the face-on views of the dusty gas slab. The top row shows theratio of the emergent intensity I to the intrinsic intensity I ∗ emitted from the star, the middle does linear polarization,and the bottom does circular polarization. According as thegrain size becomes small, the scattering albedo decreases,and the absorption optical depth increases. Therefore, inthe case of a d = . µ m , the emergent intensity is as low as ∼ − I ∗ at the edge of the slab. For a d (cid:61) . µ m , the albedois close to unity, and diffuse photons can propagate even atthe edge. In the case of the MRN model, small grains withlow scattering albedos absorb photons efficiently, resulting inthe suppression of photon propagation over a long distance.The distributions and directions of linear polarizationare shown in the middle row in Figure 2. In the cases with a d = . and . µ m , the maximum linear polarization degreereaches ∼ percent and the polarization vectors exhibit theconcentric pattern. In both cases, the photon scattering is The details of the dependence on viewing angles and dust prop-erties will be presented in our forthcoming paper (Fukushima etal. in prep.). understood by the Rayleigh approximation ( π a d (cid:28) λ ), inwhich the scattered light is linearly polarized along the ver-tical plane to incident and outgoing vectors. On the otherhand, as the grain size is close to a d = . µ m , the linear po-larization pattern becomes different from the Rayleigh limit.If photons propagate along the face-on or edge-on directionto . µ m grains and they are scattered to the observer,the oscillation of electric dipole moment along the parallelplane to the incident and outgoing vectors becomes domi-nant. Thus, the polarization vectors point to the radial direc-tions. If the grain size is much larger than the wavelength ofphotons, the direction of linear polarization is aligned withthe major axis of the oblate dust grain regardless of theincident angle.The bottom row of Figure 2 shows the CP maps,i.e., | V |/ I . The angular distributions show the symmetricquadrupole patterns. These patterns nicely match recent ob-servations (e.g., Kwon et al. 2014). Compared to the smalldust grains ( . and . µ m ), the sign of CP is reversed inthe cases of a d = . µ m and the MRN mixture. In all cases,the absolute value of CP becomes maximum in the diago-nal direction. In Figure 3, we show the degrees of CP along ◦ direction of the slab as a function of distance from thestar. Here, ◦ is defined as the direction of the minor axisof dust grains. The CP degree increases as the grain sizebecomes larger, and it reaches percent for a d = . µ m .The CPs for the models of single size grains are roughly con-stant. On the other hand, in the case of the MRN mixture, | V |/ I decreases as the radial distance increases. Because ofthe contribution of small dust grains, the absorption opticaldepth is larger than the single size models. After the absorp-tion of photons, penetrating photons are linearly polarized.The direction of the linear polarization is perpendicular tothat of photons after scattering. According to the M¨ullermatrix, the sign of S is inverse to S . Therefore, if photonpackets after absorption and scattering do the final interac-tion with dust, and they are collected in the same pixel onthe imaging screen, the CP becomes weak due to the offset.This is why the CP decreases at the large radius in the caseof the MRN.In the MRN mixture, the CP degree cannot becomehigher than percent, although micron-sized grains are con-tained. As shown in the single size models, the CP from thescattering on large dust grains with a d (cid:38) . µ m offsets thatby small grains, resulting in the low level of CP. If the typicaldust size is as large as ∼ µ m through accretion and coagu-lation, the CP degree becomes higher as a result of reducedoffset. Recent observations indicate that the CPs in star-forming regions are mostly at a level of less than ∼ percent,while some observations show the CP degree as high as ∼ percent (Kwon et al. 2014). We suggest that the diversity ofobserved CP maps can originate from different size distri-butions of dust grains. Actually, the observations of . µ m emission from dense cores in the star-forming clouds sup-port µ m dust grains (Steinacker et al. 2010; Pagani et al.2010), and metal accretion or coagulation is thought to in-duce such grain growth (e.g., Ormel et al. 2009; Hirashita& Li 2013). Also, the positive correlation of the CP degreewith the luminosity of YSOs is shown in Kwon et al. (2014).The high-density regions are preferred for the formation sitesof massive stars (e.g., McKee & Tan 2003; Fukushima et al.2018), and such environments may enhance the dust growth. MNRAS , 1–6 (2019)
Fukushima et al. V / I
The distributions of intensity (top row), linear polarization (middle) and circular polarization (bottom) in the face-on viewto the dusty slab. The four columns of panels represent the results for (a) . µ m , (b) . µ m , (c) . µ m , and (d) the MRN mixture.In the top panels, we plot the intensity ratio of the scattered light to the stellar direct light ( I ∗ ). In the middle panels, the directions ofpolarization are also shown by bars. − . − . . . . r [pc]10 − − − | V | / I . µ m0 . µ m1 . µ m MRN
Figure 3.
The absolute value of CP degree along the ◦ di-rection in the observation maps. Blue, red, green and black linescorrespond to the cases at a d = . , . , . µ m and the MRNmixture. The solid and dashed lines represent V / I > or V / I < cases, respectively. Note that, however, the growth rate and size distributionsof dust grains are still under debate, because they are sensi-tive to physical conditions in molecular clouds that are notunderstood well.The distributions and degrees of CP are closely linkedwith the M¨uller matrix. In particular, S component pro-duces the CP from non-polarized light, and mainly deter-mines the pattern of CP as shown in Figure 2 (e.g., Gledhill& McCall 2000). In Figure 4, we show the distributions of ✓
The distributions of S / S in the scattering plane of ( φ, θ ) for the cases with a d = . , . and . µ m .MNRAS000
The distributions of S / S in the scattering plane of ( φ, θ ) for the cases with a d = . , . and . µ m .MNRAS000 , 1–6 (2019) ircular polarization of near infrared light − . − . . . . r [pc]0 . . . . . . | V | / I behindcenterforward Figure 5.
Same as Figure 3, but for the cases where the slabcenter is at l = . (dot), . (solid) and − . (dot-dashed,respectively) at a d = . µ m . S / S against scattering angles for the cases that the open-ing is Θ = ◦ . In this study, we assume oblate dust grains,and thus the phase function of each component of the M¨ullermatrix becomes antisymmetric to φ with respect to the cen-ter φ = ◦ . In the case of a d = . µ m , the S componentis estimated based on the Rayleigh approximation as (e.g.,Gledhill & McCall 2000; Tazaki et al. 2017) S ∝ (cid:16) α α ∗ − α ∗ α (cid:17) sin θ sin φ sin 2 Θ . (5)Therefore, S / S is symmetric to θ with respect to thecenter θ = ◦ . In this case, the CP becomes maximum ifthe scattering angle is ( θ, φ ) = ( ◦ , ◦ ) . In addition, S becomes maximum for Θ = ◦ or ◦ . This reflects theangular distributions of CP as in Figure 2.As the grain size increases, the distributions of S be-come different from equation (5). With a d = . µ m , thepeak value of S / S is around unity, and much larger thanthat for . µ m grains, and the distributions in the phasespace are obviously different. In particular, the sign is in-verse around ( θ, φ ) = ( ◦ , ◦ ) . These differences result inthe different CP distributions between the cases for smalland large dust grains as shown in Figure 2. For the MRNmodel, the pattern of the sign is similar to that for . µ m dust grains. In the Rayleigh approximation, the scatteringcoefficient is proportional to the grain size with the depen-dence of a . The abundance of dust grains is given as ∝ a − . in the MRN mixture. By multiplying the scattering coeffi-cient of the geometrical cross section ( π a ) and the MRNsize distribution ( ∝ a − . ), the scattering optical depth pera specific grain size bin is estimated as d τ / da d ∝ a . . There-fore, in the MRN model, the contribution of the larger dustgrains ( ∼ µ m ) is dominant for the scattering process.The distributions of dusty gas around a massive starhave not been understood well because of the complicatedstellar feedback (e.g., Zinnecker & Yorke 2007; Fukushimaet al. 2018). Therefore, we here investigate the impact ofthe configuration of dusty slabs on the CP map. In thephase distributions of S / S for a d = . µ m , the maxi-mum value appears at ◦ (cid:46) θ (cid:46) ◦ , corresponding to the back-scattering. It means that the CP degree increasesif dust grains are located behind the radiation source. Fig-ure 5 shows the CP degrees in the cases that the dusty slabis shifted from the star to forward or backward by 0.1 pc.As expected from Figure 4, the CP degree becomes largerif the slab is behind the radiation source. For that case,the maximum of the CP degree reaches 60 percent. On theother hand, it is lower if the slab is in front of the radiationsource, and it becomes ∼ percent. In the forward scat-tering, the component of S / S becomes smaller, becausethere are photons with different signs of the CP. Therefore,we realize that the CP degrees change by a factor of − depending on the configuration between a star and a dustyslab. Recent theoretical works suggest that the stellar feed-back works anisotropically (e.g., Geen et al. 2017; Kim et al.2018). Hence, if the dusty gas along a line of sight is attenu-ated due to the feedback, we observe mostly back-scatteredphotons, which produce higher CP as shown in our simula-tions. We have explored the generation of CP at the wave-length of . µ m in star-forming regions using three-dimensional radiative transfer simulations. Assuming adusty gas slab composed of aligned oblate grains, we havesuccessfully reproduced the quadrupole patterns of CP,which is similar to the observed CP maps (Fukue et al.2010). We have found that the distributions and degreesof CP sensitively depend on the size of dust grains. The CPdegree shows | V |/ I ∼ percent for a grain size of a d = . µ m and | V |/ I ∼ percent for a d = . µ m . This can explain theobserved variety of CP maps (Kwon et al. 2014). In previ-ous works, the dichroic extinction model was suggested toproduce the observed high degree of CP (Kwon et al. 2013).In that model, multiple scattering processes on several dustscreens/clouds are requisite. In this Letter , we have demon-strated that a single scattering can produce the high degreeof CP, if micron-size grains are dominant in a dusty gas wall.Recent observations show that the CP positively corre-lates with the luminosity of the central YSOs (Kwon et al.2014). High-mass YSOs are likely to form in massive high-pressure cores in molecular clouds (e.g., McKee & Tan 2003),where the growth rate of dust size can be enhanced (Spitzer1978; Asano et al. 2013). Therefore, we suggest that theobserved correlation may be linked to the grain size distri-butions around the massive YSOs, although the growth rateof dust grains is still under the debate quantitatively.In this study, we have also found that the CP degreeis subject to the configuration between a star and a dustyslab. If a dusty slab is located behind the star, the CP degreebecomes higher by a factor of − compared to a slab in frontof a star. Although we assumed a simple slab model in thisstudy, the actual configuration is related to the evolution ofthe star-forming clouds. The directions of the minor axis ofdust grains depend on the streaming of gas, magnetic, andradiation field (Andersson et al. 2015). Thus, to model theCP more precisely, we need to conduct magneto-radiationhydrodynamics simulations in star-forming clouds.In this work, we have considered the contribution of thesilicate dust grains alone, but there are graphite ones in the MNRAS , 1–6 (2019)
Fukushima et al. interstellar medium. The dielectric functions at the infraredwavelength are different between silicate and graphite grains(Draine 2003a), and thus the CP degree can change with thecomposition of dust grains. Besides, the bare silicate may notgrow up to the micron size solely by coagulation (Hirashita& Li 2013). The micron-size dust grains must contain bufferssuch as water ice. We will investigate the dependence of theCP degrees on the composition of dust grains in the futureworks.
ACKNOWLEDGEMENTS
The authors would like to thank Ryo Tazaki for fruit-ful discussions. This research has been supported by Mul-tidisciplinary Cooperative Research Program at the Centerfor Computational Sciences, University of Tsukuba, MEXTas Exploratory Challenge on Post-K Computer (Elucida-tion of the Birth of Exoplanets [Second Earth] and theEnvironmental Variations of Planets in the Solar System),hp190185, and partially by Grants-in-Aid for Scientific Re-search (JP19H00697, JP17H04827 and JP18H04570) fromthe Japan Society for the Promotion of Science (JSPS).
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