Thermal emission from the amorphous dust: An alternative possibility of the origin of the anomalous microwave emission
Masashi Nashimoto, Makoto Hattori, Ricardo Génova-Santos, Frédérick Poidevin
aa r X i v : . [ a s t r o - ph . GA ] N ov Thermal emission from the amorphous dust: Analternative possibility of the origin of theanomalous microwave emission
Masashi
NASHIMOTO , ∗ Makoto
HATTORI , Ricardo G ´ ENOVA - SANTOS , Fr ´ed ´erick
POIDEVIN Astronomical Institute, Tohoku University, 6-3, Aramaki-Aza-Aoba, Aoba-ku, Sendai, Miyagi,Japan Graduate Program on Physics for the Universe (GP-PU), Tohoku University, 6-3,Aramaki-Aza-Aoba, Aoba-ku, Sendai, Miyagi, Japan Instituto de Astrofis´ıca de Canarias, E-38200 La Laguna, Tenerife, Canary Islands, Spain Departamento de Astrof´ısica, Universidad de La Laguna (ULL), E-38206 La Laguna,Tenerife, Spain ∗ E-mail: [email protected]
Received ; Accepted
Abstract
Complete studies of the radiative processes of thermal emission from the amorphous dustfrom microwave through far infrared wavebands are presented by taking into account, self-consistently for the first time, the standard two-level systems (TLS) model of amorphous ma-terials. The observed spectral energy distributions (SEDs) for the Perseus molecular cloud(MC) and W43 from microwave through far infrared are fitted with the SEDs calculated withthe TLS model of amorphous silicate. We have found that the model SEDs well reproduce theobserved properties of the anomalous microwave emission (AME). The present result sug-gests an alternative interpretation for the AME being carried by the resonance emission ofthe TLS of amorphous materials without introducing new species. Simultaneous fitting of theintensity and polarization SEDs for the Perseus MC and W43 are also performed. The amor-phous model reproduces the overall observed feature of the intensity and polarization SEDs of he Perseus MC and W43. However, the model’s predicted polarization fraction of the AME isslightly higher than the QUIJOTE upper limits in several frequency bands. A possible improve-ment of our model to resolve this problem is proposed. Our model predicts that interstellar dustis amorphous materials having very different physical characteristics compared with terrestrialamorphous materials. Key words: dust, extinction — infrared: ISM — radiation mechanisms: thermal — radio continuum: ISM— submillimeter: ISM
Studies of the physical processes of thermal emission from Galactic dust have been a long-standing problem and are still of significance. The typical temperature of Galactic interstellardust is about 20 K (Schlegel et al. 1998; Planck Collaboration et al. 2014). Its emissionappears predominantly at long wavelengths from the far infrared through microwave. Sincethe whole sky is covered by the emission from Galactic interstellar dust, thermal emissionfrom the Galactic dust is a serious obstacle for the detection of B-mode polarization signalsfrom cosmic microwave background (CMB) radiation imprinted by primordial gravitationalwaves. The success of CMB B-mode polarization observations relies on how accurately we canremove the Galactic dust signals from observational data. To tackle this difficult task, manyCMB experiments are under way and others are being planned (e.g. ACTPol (Naess et al.2014); BICEP2/3 and the Keck Array (Grayson et al. 2016); CLASS (Essinger-Hileman et al.2014); GroundBIRD (Oguri et al. 2016); LiteBIRD (Matsumura et al. 2014); PIPER (Gandiloet al. 2016); POLARBEAR and the Simons Array (Arnold et al. 2014); QUIJOTE (Rubi˜no-Mart´ın et al. 2012); the Simons Observatory (Ade et al. 2019); SPIDER (Gualtieri et al. 2018);SPTPol (Austermann et al. 2012) ). These surveys provide extremely high precision data onthe microwave sky with wide sky coverage in many different wavebands. It is certain thatsignificant progress in our understanding of interstellar dust will be made with these data.Therefore, theoretical studies of the physical processes of thermal emission from Galactic dustmust be undertaken now to achieve fruitful outcomes from these data.The origin of anomalous microwave emission, which is abbreviated AME found ubiqui-tously in the Galaxy at around 10–30 GHz (e.g. see Dickinson (2013) for a summary of AMEobservations in HII regions) is still under debate. The spatial correlation between AME and2alactic interstellar dust strongly indicates that AME originates from a kind of dust (Davieset al. 2006). The most popular model of the origin of AME is electric dipole emission radiatedby charged rotating dust with a frequency of several tens of GHz, as proposed by Draine &Lazarian (1998); this is referred to as the spinning dust model. A carrier of the spinning dustis supposed to be very small grains producing rotation at ultra high frequencies. The factof the lack of AME in cold dense cores supports the spinning dust origin of AME since thelack of the small grains is expected in dense clouds (Tibbs et al. 2016). Polycyclic aromatichydrocarbon (PAH) has been proposed as one of the plausible candidates for spinning dust(Draine & Lazarian 1998). However, no observational correlation between amount of PAH andthe intensity of AME, as reported by Hensley, Draine, & Meisner (2016), contradicts the PAHpossibility. A new species of very small dust grains named nanosilicates has been introduced asanother possible carrier of the spinning dust (Hoang et al. 2016; Hensley & Draine 2017). Theproblem with this possibility is that up to now, apart from AME, no signature to confirm theexistence of the nanosilicate has been observed. The nanosilicate is only observable as AME.Therefore, it is hard to check whether the carrier of the spinning dust is such a new family ofdust grains or not. Magnetic dust emission has been proposed as another candidate for theAME mechanism (Draine & Lazarian 1999). The spins of electrons inside a magnetic dust grainalign spontaneously to settle down to the minimum energy state. Alignment is disturbed bythermal fluctuation. Owing to the magnetic relaxation, the disturbed state tries to return tothe original minimum energy state. In course of this transition, microwave radiation is emitted.This emission could be the origin of AME if the interstellar dust is magnetic (Draine & Hensley2013). The magnetic dust emission model predicts a positive correlation between the temper-ature and intensity of AME. However, Hensley, Draine, & Meisner (2016) found a negativecorrelation between the AME temperature and intensity that contradicts the predictions of themagnetic dust emission model. A comprehensive review on the state of research of AME isgiven by Dickinson et al. (2018).Crucial clue to distinguishing the emission mechanisms of AME is offered by polariza-tion observations. Draine & Hensley (2016) showed that the quantum effect suppresses thethermalization of the grain rotational kinetic energy of the spinning dust. As a result, thealignment of grains is suppressed and the spinning dust model predicts a very low degree ofpolarization. In the magnetic dust emission model, the high degree of AME polarization isexpected since the main carrier of magnetic dust emission is large grains which are aligned bythe interstellar magnetic field. Although the progress of AME polarization observations havebeen made by several projects (e.g. WMAP and QUIJOTE), there has as yet been no definite3eport of the detection of AME polarization (G´enova-Santos et al. 2015, 2017). The detectionof polarization from AME has been reported for W43, but whether the reported polarization isa residual of the synchrotron emission of Galactic interstellar matters around W43 is still beingdebated. Current observational upper limits somehow rule out the magnetic dust hypothesis,which typically predicts a higher polarization fraction.Almost all types of interstellar dust are supposed to be made of amorphous materials.For example, the broad emission line observed ubiquitously in interstellar space at 9.7 µ mis considered to be a signature that one of the main components of interstellar dust is anamorphous silicate (Kraetschmer & Huffman 1979; Li & Draine 2001). Moreover, laboratorysimulations of cosmic dust analogues suggest that various forms of amorphous carbon grains aremore favorable than graphite grains (Colangeli et al. 1995; Zubko et al. 1996). The observedspectrum of interstellar dust emission in submillimeter wavebands obtained by the Plancksatellite is flatter than the spectrum expected from crystal dust (Planck Collaboration et al.2014). This is further evidence indicating that interstellar dust is composed of amorphousdust. According to recent laboratory measurements of the emissivity of amorphous material,the emissivity of the amorphous material has complex frequency dependences that cannot beapproximated by a single power law at longer than far infrared wavelengths (e.g., Coupeaudet al. 2011). Physical diagnostics of the amorphous material appear in heat capacity and heatconductivity at very low temperature. Zeller & Pohl (1971) found that the heat capacity of theamorphous material below 1 K shows significant deviation from the Debye model and dependslinearly on temperature instead of the cube of the temperature. They also found that heatconductivity below 1 K is in excess of that expected for crystals and depends on the square ofthe temperature. It has also been shown that these characteristics appear universally in anyamorphous material. This universality indicates that above-mentioned diagnostics observed inamorphous materials are governed by universal physics. Anderson, Halperin, & Varma (1972)and Phillips (1972) independently proposed that heat absorption and heat transport by two-level systems from amorphous materials predominate over lattice oscillation below 1 K. Thismodel is referred to as the TLS model. The degree of freedom concerning heat absorptionbecomes one when absorption by the TLS becomes dominant below 1 K. That is why thetemperature dependence of the heat capacity switches from cubic to linear. The temperaturedependence of the heat conductivity below 1 K is also successfully explained by the TLS model.Agladze et al. (1994) showed that the temperature dependence of the absorption coefficient infar-infrared wavebands measured for amorphous powder are well described by the TLS modelin laboratory experiments. They were the first to propose that the TLS may contribute to the4bserved features of interstellar dust. Meny et al. (2007) performed theoretical calculationsof the frequency dependence of the absorption coefficient based on the TLS model. Paradiset al. (2011) compared their models with the observed spectra of diffuse interstellar dust fromfar infrared through submillimeter wavebands. They showed that the TLS model succeeds inreproducing the observed features, including the inverse correlation of the spectral index withdust temperature.Jones (2009) proposed the idea that the AME might originate from the resonance emis-sion due to radiative transition between the TLS of amorphous dust. The fact that the effect ofthe TLS appears below 1 K, indicates that the energy splitting between the TLS is about 10 − eV, which just coincides with the observed frequency of the AME. Therefore, the resonanceemission from amorphous dust is an attractive possibility for the origin of AME. The negativecorrelation between the AME temperature and intensity is naturally explained by the amor-phous model since the intensity of the resonance emission decreases as the dust temperatureincreases (Meny et al. 2007). They assumed that the peak value of the absorption cross-sectionof the resonance process of the TLS is the geometric cross-section. It is well known, however,that the absorption cross-section of a small particle at microwave wavelengths is much smallerthan the geometric cross-section (e.g., Draine & Lee 1984). It is likely that their model over-estimates the TLS contribution. It is also still unclear what kind of physical characteristics ofthe amorphous dust can be extracted from the observation of AME. Because of the potentialpossibility of the amorphous origin of AME, studies of the thermal emission of amorphous dustrelying on microscopic physical processes based on the TLS model are required.In this paper, the intensity and polarization spectral energy distributions (SEDs) model-ing from far infrared through microwave wavelengths were conducted based on the TLS modelof amorphous dust. By comparing the model with observations, we studied whether the amor-phous dust model is able to explain the diagnostics of the entire frequency range spectrum; e.g.,the emission peak in the far infrared, the spectral index in submillimeter wavebands, the bumpin the emission of the AME, and the low polarization fraction of the AME. We showed whatkind of physical characteristics are required for the amorphous dust in order to explain theobservations. We adopted two archetypical AME objects, the Perseus molecular cloud (MC)and W43 for our comparison with the observations. Both objects have intensive data on theintensity and polarization spectrum over a wide number of frequency bands.In section 2, fundamental quantities of amorphous materials to describe their opticalproperties are summarized. Details of the basics of the standard TLS model are introducedin appendix 1. In section 3, we show how the SEDs of the thermal emission from amorphous5ilicate dust respond to the physical parameters of the TLS model and compare model SEDswith observational data. Then we move on to the polarized emission in section 4. In section5, we examine the properties of the amorphous silicate dust. Limitation of the present modeland possible improvements are discussed in section 6. Our conclusions and a summary arepresented in section 7. Optical properties of an amorphous material are determined by its electric susceptibility (seethe details in sections 3 and 4). In this section, we summarize how the electric susceptibilitiesdue to the TLS and disordered charged distribution (DCD) are related to the micro physics ofeach process, respectively. The details of the standard TLS model are described in appendix1. The basic equations of the TLS model and the DCD model can apply to both amorphoussilicate material and amorphous carbon material since the physical mechanism behind eachmodel does not depend on the material composition. The differences between the amorphoussilicate material and the amorphous carbon material appear in the differences of values ofphysical variables.
The basic idea of the TLS model is that some of the atoms composing an amorphous materialhave two stable positions due to deformation of crystal structures. The mechanical potentialof the atom is described by the double-well potential illustrated in figure 1. The x coordinatemarks the position of the atom. This potential is generally described by a quartic function of x ,which is called the soft-potential model (Karpov et al. 1982). In this paper, in order to describehow the TLS modifies the spectrum of the thermal emission from dust and determine whetherAME can be explained by introducing the TLS model, we adopted the same approximationmade by Anderson, Halperin, & Varma (1972) and Phillips (1972), who first proposed the TLSmodel to describe the physical characteristics of the amorphous materials appearing at very lowtemperature. They expanded the ground state and the first excited state of the Schr¨odingerequation of the atom confined in the double-well potential V by the two ground states whenthe atom is confined in each harmonic potential V and V individually (see figure 1). We referto this model as the standard TLS model.As described in appendix 1, there are three independent processes, that is the resonance6 E d xVV V ∆ V Fig. 1:
A double-well potential, shown by the black solid curve, in which an atom is trapped. Thegray dashed curves denote harmonic potentials V and V . transition, the tunneling relaxation and the hopping relaxation, which contribute to the electricsusceptibilities of the TLS. The complex susceptibilities for the resonance transition χ res0 , thetunneling relaxation χ tun0 and the hopping relaxation χ hop0 are obtained as χ res0 = − i Z ∆ max0 ∆ min0 d ∆ Z √ (∆ max0 ) − ∆ d ∆ f (∆ , ∆) × τ + ¯ h | d | (cid:18) ∆ E (cid:19) tanh (cid:18) E k B T (cid:19) × "
11 + i ( ω − ω ) τ + −
11 + i ( ω + ω ) τ + , (1) χ tun0 = Z ∆ max0 ∆ min0 d ∆ Z √ (∆ max0 ) − ∆ d ∆ f (∆ , ∆) × | d | k B T (cid:18) ∆ E (cid:19) − iωτ tun sech (cid:18) E k B T (cid:19) , (2) χ hop0 = Z ∆ max0 ∆ min0 d ∆ Z √ (∆ max0 ) − ∆ d ∆ f (∆ , ∆) Z ∞ dV g ( V ) × | d | k B T (cid:18) ∆ E (cid:19) − iωτ hop sech (cid:18) E k B T (cid:19) , (3)where the definitions of parameters and distribution functions are as follows: ∆ is the energydifference between the two states located at each minimum of the double-well potential (seefigure 1); ∆ is the parameter that characterizes the degree of the cross correlation betweenthe states located in two minima; f (∆ , ∆) d ∆ d ∆ provides the number density of the atomstrapped in the TLS from ∆ to ∆ + d ∆ and from ∆ to ∆+ d ∆; d is the electric dipole momentfor the state located at the minimum of the potential V ; E ( ≡ (∆ + ∆ ) / ) is the energy We define χ and χ as susceptibilities for the response to a macroscopic internal electric field and to an external electric field, respectively. χ is related toan electric permittivity ε as ε = 1 + 4 πχ . able 1: Typical values of physical variables of an amorphous silicate material
Parameter Meaning Value Unit Refs. ‡ V m /k B mean value of the barrier height distribution 550 K 1 V σ /k B deviation of the barrier height distribution 410 K 1 V min /k B lower cutoff of the barrier height distribution 50 K 1 ρ mass density of a dust particle 3.5 g cm − n atom the number density of atoms composing a dust particle ρ/ . × N A ∗ cm − c t sound velocity for transverse waves 3 × cm s − γ t elastic dipole for transverse waves 1 eV 3 | d | electric dipole moment for the localized state at the bottom of V τ pre-exponential factor for hopping relaxation time (see equation (A52)) 10 − s 1 l c correlation length of propagation of lattice vibration 30 ˚A 1 q electric charge of an atom composing a dust particle e erg cm 2 m mass of an atom composing a dust particle m O † g 2 ω D Debye angular frequency 2 πc t [9 n atom / (8 π )] / s − — C correction factor for the DCD model 4 . × − — — ∗ N A is the Avogadro constant and we consider the composition of amorphous silicate materials as MgFeSiO whose mass number is172.2 g mol − . † m O is the mass of an oxygen atom. ‡ References: 1: B¨osch (1978); 2: Li & Draine (2001); 3: Meny et al. (2007) splitting of the TLS; τ + is phase relaxation time; τ tun is tunneling relaxation time; τ hop is hoppingrelaxation time; ω is defined as E/ ¯ h ; ω is angular frequency of the electromagnetic wave thatstimulates the resonance transition, the tunneling relaxation and the hopping relaxation; V is the height of the potential barrier; g ( V ) is probability density function. Upper and lowercutoff of ∆ , ∆ max0 and ∆ min0 , are introduced to avoid divergence of the probability distributionfunction. The typical values of physical variables of amorphous silicate material are given intable 1. Electric polarization due to the acoustic vibration propagating through the solid also makes asignificant contribution to the absorption coefficient of the amorphous material. To describe theirregular distribution of the lattice in the amorphous material, Schl¨omann (1964) introducedthe DCD model. The electric susceptibility derived from the DCD model, χ DCD0 , is given as χ DCD0 = C q π m Z ω D /c t dk k ( c t k ) − ω + iγω h ( kl c ) , (4)8 ( x ) ≡ − x ) , (5)where l c is the correlation length of propagation of lattice vibration, ω D is the Debye frequency, γ is the damping factor, and C is the correction factor. The correction factor C is introducedfor the imaginary part of the dielectric susceptibility at 300 µ m predicted by DCD model soas to coincide with the imaginary part of the dielectric susceptibility at the same frequencyproposed by Draine & Lee (1984). The uncertainty of the adopted parameters listed in table1 are absorbed by introducing the correction factor. A crystalline material with a regulardistribution is realized in the infinite correlation length limit, i.e., kl c ≫
1. In this limit, theDCD model reduces to the Lorentz model. In the limit of small γ , the following analyticalformulae of χ DCD0 are obtained:Re (cid:16) χ DCD0 (cid:17) = C q π mc " ω D − ω ω D ω + ω )( ω + ω )+ (2 ω + ω ) ω ω + ω ) ln (cid:12)(cid:12)(cid:12)(cid:12) ω D − ωω D + ω (cid:12)(cid:12)(cid:12)(cid:12) − ω ( ω − ω )2( ω + ω ) atan (cid:18) ω D ω c (cid:19) , (6)Im (cid:16) χ DCD0 (cid:17)(cid:12)(cid:12)(cid:12) ω<ω D = C q ω πmc ( − ω/ω c ) ] ) , (7)where ω c ≡ c t /l c . Although Im( χ DCD0 ) becomes zero when ω > ω D , it does not affect our analysissince we are interested in the low frequency range. The absorption cross section of an amorphous dust is obtained by summing up contributionsfrom the TLS and the DCD. The electric susceptibility of the amorphous dust is written as χ = f TLS (cid:16) χ res0 + χ tun0 + χ hop0 (cid:17) + χ DCD0 , (8)where f TLS is the fraction of atoms trapped in the TLS.The absorption cross section is derived under the dipole approximation since the radiusof the dust grain is much smaller than the wavelength, λ , of electromagnetic waves. In thissection, the dust shape is assumed to be spherical. The absorption cross section of the sphericalamorphous dust, C abs ν , is given by the optical theorem as (c.f. Schl¨omann 1964; Bohren &Huffman 1983; Meny et al. 2007), C abs ν = 8 π V λ Im ( χ ) , (9)9here V is the volume of an amorphous dust. An intensity emission spectrum of the thermal emission of amorphous dust is deduced in thissubsection. In this paper, we take into account only an amorphous silicate dust and do notconsider the contribution of carbonaceous dust such as PAHs, graphite, and amorphous carbon(expected effects of including amorphous carbon dust are discussed in section 6). The generalformula for the spectrum of thermal emission from dust grains is given by I dust ν = N dust Z da dnda B ν ( T ) C abs ν ( a, T ) , (10)where N dust is the column density of the dust grains in the line of sight, and dn/da providesthe size distribution of the dust grains and is normalized to be 1 by integrating over the dustsize a . The size distribution function proposed by Draine & Li (2007) is adopted. In this study,we neglect the effect of the time variation in the temperature of small dust grain to the SEDof the thermal emission of amorphous silicate dust. We assume that all dust grains stay at thesame temperature.Figure 2 shows the parameter dependence of thermal emission SEDs from amorphoussilicate dust. The results show that the bump emission appears at around several tens ofGHz. These are caused by the resonance transition of the TLS. Figure 2a shows that the peakfrequency of the bump emission is shifted toward higher frequency as the upper limit of theenergy difference between the TLS, ∆ max0 , increases while R ∆ ( ≡ ∆ min0 / ∆ max0 ) is fixed. Figure2b shows that the bump feature of the resonance emission becomes broader as R ∆ gets smaller,although the response is not prominent. Figure 2c shows that the bump emission due to theresonance process relative to the far-infrared peak becomes higher when the temperature of thedust grain lowers. This is attributed to the fact that, the electric dipole moment caused by theresonance transition rate increases with decreasing temperature because the fraction of atomsin the ground state increases with decreasing temperature (see appendix 1 equation (A46)).Figure 2d shows that the width of the bump emission sensitively responds to the relaxation timescale of the resonance process, τ + . Figures 2a and 2d show that the bump emission becomesprominent when 1 /τ + becomes comparable to, or greater than, ∆ max0 /h . Figure 2e shows thatthe peak intensity of the bump emission caused by the resonance process is about two ordersof magnitude lower than the peak intensity in the far infrared, even when all the atoms aretrapped in the TLS. The relative intensity of the bump emission decreases almost linearly with f TLS . 10 a) ∆ max0 frequency [Hz]10 − − − − − n o r m a r i ze dflu x d e n s i t y × h × h
10 GHz × h
30 GHz × h
100 GHz × h (b) R ∆ frequency [Hz]10 − − − − − n o r m a r i ze dflu x d e n s i t y . . . . . (c) T frequency [Hz]10 − − − − − n o r m a r i ze dflu x d e n s i t y (d) τ + frequency [Hz]10 − − − − − n o r m a r i ze dflu x d e n s i t y × − s3 × − s1 × − s3 × − s1 × − s (e) f TLS frequency [Hz]10 − − − − − n o r m a r i ze dflu x d e n s i t y − − − . Fig. 2:
Parameter dependences of the SEDs of dust thermal emission in the standard TLS model.SEDs are given in arbitrary units normalized to each maximum value. Thick solid curvesin each panel show the SEDs with the same parameter values. In each panel, one of thevariables characterizing the amorphous silicate dust was varied to see how the shape of theSED responds for each variable. The variables are (a) ∆ max0 which are expressed in thecorresponding frequency normalized by the Planck constant h , (b) R ∆ , (c) T , (d) τ + , and (e) f TLS , respectively. Given values for each parameter are shown in legends. frequency [Hz]10 − − − − − n o r m a r i ze dflu x d e n s i t y resonancetunnelinghoppingDCD Fig. 3:
Dust thermal emission SEDs caused by each TLS and DCD process. Parameters have thesame values as the thick solid curves in figure 2. Every curve is normalized to the peak valueof the SED arising from the DCD.
To clarify how the frequency dependence of the thermal emission of amorphous sili-cate dust is defined, figure 3 shows SEDs for each process. The frequency dependence of theabsorption coefficient of the resonance process and the relaxation processes in submillimeterwavebands are described by C res ν ∝ ν and C rel ν ∝ ν , respectively. The frequency dependenceof the resonance process in the long wavelength limit is the same as for crystal. As the wave-length increases starting from the far infrared, the contributions from tunneling and hoppingrelaxation become more significant. As a result, the slope of the absorption coefficient of theamorphous material becomes flatter than that of crystal in the submillimeter wavelength range. Our model SEDs are fitted to the observed spectra from millimeter through far infrared fortwo MCs, Perseus and W43, for which prominent AME is detected. The observed data for thePerseus MC and W43 are taken from table 2 in G´enova-Santos et al. (2015) and table 3 inG´enova-Santos et al. (2017), respectively. Although the temperature fluctuation of the CMBis subtracted from the spectrum of W43, it is not taken into account in the spectrum of thePerseus MC. Therefore, the SED fit with and without the CMB contribution are performed forthe Perseus MC. Planck data at 100 and 217 GHz may still be contaminated by CO residuals(Planck Collaboration et al. 2011). To take this possibility into account, data points at 100and 217 GHz are not included in the fit for the Perseus MC with a CMB contribution. Thesefrequency bands are included in the fit for other cases. The observed spectra of these MCsare shown in figure 4. The contributions of synchrotron emission, free–free emission and dustthermal emission from the Galactic interstellar medium along the line of sight were removed12y subtracting the median value of the intensity surrounding each MC. As for the SED of thefree–free emission originating from each MC, the formulae adopted by Planck Collaborationet al. (2011) are applied in this paper; that is, I ff ν = 2 k B T ff ν c , (11) T ff = T e (1 − e − τ ff ) , (12) τ ff = 3 . × − T − . (cid:18) ν GHz (cid:19) − EMcm − pc ! g ff , (13) g ff = ln " . × − (cid:18) ν GHz (cid:19) − + 1 . (cid:18) T e K (cid:19) , (14)where EM is the emission measure. The electron temperature of each MC is fixed at T e = 8000K for the Perseus MC (Planck Collaboration et al. 2011) and T e = 6038 K for W43 (Alveset al. 2012). Therefore, free parameters to fit the observed SEDs are EM which characterizesthe fraction of the free–free contribution, dust temperature T , dust column density N dust , thefraction of the atoms trapped in the TLS f TLS , the upper and lower bounds of ∆ (that is, ∆ max0 and ∆ min0 ), and the relaxation time scale τ + . In the case of the Perseus MC, the amplitude ofthe temperature fluctuation of the CMB ∆ T CMB is also an additional fit parameter. The SEDof the CMB temperature fluctuation is given as, I CMB ν = B ν ( T CMB ) xe x e x − T CMB T CMB , (15) x = hνk B T CMB , (16)where T CMB = 2 .
725 K (Mather et al. 1999) is the CMB temperature.
We searched the parameters that minimize the chi squared by a brute force. The best-fittingmodel SEDs based on our amorphous model are overlaid on the observed spectra in figure4. As shown in table 2, the best CMB temperature fluctuation takes a negative value. Theabsolute value of the best-fit CMB contribution is shown by dashed-dotted line in figure 4a.The best-fit parameters are summarized in table 2. Assuming an optically thin condition, N dust is interpreted as the optical depth at λ = 250 µ m, τ . Our SED models reproduce the observedSEDs from AME through the far infrared feature very well. In our models, AME originatesmainly from the resonance emission of the TLS of large grains. It should be stressed that theamorphous model is able to explain AME without introducing new species. The reduced chisquared of the best-fit models for the Perseus MC is χ / dof = 2 .
04, where dof = 21 without CMB, χ / dof = 1 .
67, where dof = 18 with CMB, and for W43 χ / dof = 6 .
41 where dof = 23. For W43,13 a) Perseus MC flu x d e n s i t y [ J y ] frequency [Hz]1 . . d a t a / m o d e l (b) W43 flu x d e n s i t y [ J y ] frequency [Hz]1 . . d a t a / m o d e l Fig. 4:
Spectra of two archetypal dust-rich objects accompanying prominent AME, (a) the PerseusMC and (b) W43, are fitted by our amorphous model. Dashed curves are the best fit ther-mal emission model from amorphous silicate dust, and dotted lines are the best fit free–freeemission contribution. In both data, contributions from the Galactic interstellar medium aresubtracted. Although temperature fluctuation of the CMB is subtracted from the spectrumof W43, it is not taken into account in the spectrum of the Perseus MC. Therefore, theCMB contribution is taken into account in the SED fit for the Perseus MC. As shown intable 2, the best CMB temperature fluctuation takes negative value. The absolute value ofthe best-fit CMB contribution is shown by dashed dotted line in (a). The solid curves showthe total SEDs of the best-fit models. The bottom panel inserted in each figure shows thedata-to-model ratio. the overall observed feature is also well reproduced by our model although the quality of thefit is not so good. The bottom panels of figures 4a and 4b show that our models underestimatethe observed intensities in the frequency range from 100 GHz through 500 GHz.
The observations of polarization emission is one of the crucial keys to discriminating amongthe models of the origin of AME. Dust thermal emission is supposed to be polarized becausethe shapes of the dust grains are non-spherical and align with the magnetic field. Hereafter,dust shape is represented by an ellipsoid for simplicity. In this section, the theoretical model ofpolarized emission from amorphous silicate dust based on the standard TLS model is establishedand the model predictions are compared with the observed results obtained for the Perseus MCand W43. 14 able 2:
Best-fit parameters for the Perseus MC and W43
Perseus MC W43without with CMB T (K) 16.78 ± ± ± τ ( × − ) 4.08 ± ± ± f TLS ± ± ± max0 /h (GHz) 15.2 ± ± ± R ∆ ± ± ± τ + ( × − s) 2.24 ± . +0 . − . ± − pc) 26.9 ± ± ± T CMB ( µ K) — − . +6 . − . —dof 21 18 23 χ / dof 2.04 1.67 6.41 The errors are at 1 σ errors. The shape of an ellipsoid is characterized by the radii of three axes, a radius of semi-major axis a x , semi-middle axis a y and semi-minor axis a z , that is a x ≥ a y ≥ a z . We take the semi-majoraxis along the x -axis, the semi-middle axis along the y -axis, and the semi-minor axis along the z -axis. The absorption cross sections of an ellipsoidal particle for radiation linearly polarizedalong each axis are given by following equation: C abs ν,i = 8 π V λ Im (cid:16) χ i (cid:17) , (17)where i = x, y, z and χ i are the complex susceptibilities responding to an external electric fieldparallel to each axis. As shown in appendix 2, χ i are given by, χ i = χ π ( L i − / χ , (18)where L i are geometric factors defined as (Bohren & Huffman 1983), L i ≡ V π Z ∞ dq ( q + a i ) q ( q + a x )( q + a y )( q + a z ) . (19)Figure 5 shows the frequency dependences of C abs ν,x , C abs ν,y , and C abs ν,z for amorphous silicate dust.The adopted values of the geometrical factors are L x = 1 / L y = 1 /
3, and L z = 1 /
2. Ingeneral, C abs ν,x takes the largest value and C abs ν,y takes the median value of the three. This canbe understood by a change of sign of the term 4 π ( L i − / χ appearing in the denominatorof equation (18). For the semi-major axis, this term takes a negative sign. On the other hand,15 − − C a b s ν , i / π V [ c m − ] xyz . . C a b s ν , i / C a b s ν , x frequency [Hz]0 . . . χ ′ , χ ′′ χ ′ χ ′′ Fig. 5:
The frequency dependence of absorption cross sections for an electric field parallel to eachaxis of an ellipsoid normalized by 8 π V (top panel), of relative absorption cross sectionsfor electric field parallel to semi-middle and semi-minor axis to that of the semi-major axis(middle panel), and of the real and imaginary parts of the electric susceptibility for a sphericaldust (bottom panel). A ratio of geometric factors is fixed to L x : L y : L z = 1 : 2 : 3. Otherparameters are set to the best-fit values for W43 listed in table 2. this term is zero for the semi-middle axis and is positive for the semi-minor axis. Therefore, thedenominator of equation (18) takes the smallest value for the semi-major axis and the largestvalue for the semi-minor axis. The above-mentioned order of the amplitude of the absorptioncross section is a consequence of this result. However, the order of the amplitude becomesreversed at around the resonance peak. This is evident in the middle panel of figure 5. Figure5 shows that the resonant peak frequency for the semi-middle axis coincides with that of theimaginary part of the electric susceptibility for the spherical particle. This is the expected resultsince L y = 1 /
3. The resonant peak for the semi-major axis appears at a slightly lower frequencythan the peak frequency of the imaginary part of the electric susceptibility for the sphericalparticle. This reflects the fact that L x is smaller than 1 /
3. The resonant peak frequency of theabsorption cross section for the semi-minor axis is shifted to a higher frequency since L z > / h C abs ν i and h C pol ν i are deduced by averaging over thedirection of the semi-major axis around the magnetic field, as in Draine & Hensley (2017): h C abs ν i = h C abs ν,x i + h C abs ν,y i + 2 h C abs ν,z i , (20) h C pol ν i = h C abs ν,x i + h C abs ν,y i − h C abs ν,z i , (21)where h C abs ν,x i , h C abs ν,y i and h C abs ν,z i are the shape-averaged absorption cross sections for the linearlypolarized radiation in the direction of each axis as defined in appendix 3. Except around theresonance peak frequency, h C pol ν i takes positive value since h C abs ν,x i > h C abs ν,y i > h C abs ν,z i . Therefore,the predicted direction of the polarization emission is perpendicular to the magnetic field.The degree of polarization Π ν is obtained by taking the ratio of |h C pol ν i| to |h C abs ν i| .Figure 6 shows the frequency dependence of Π ν of the thermal emission from amorphous silicatedust. In the frequency range, except in the waveband around the resonance peak, Π ν is nearlyconstant. Since h C pol ν i takes a positive value, the direction of polarization is perpendicularto the magnetic field, as expected. Since the imaginary part of the susceptibility, χ ′′ , is muchsmaller than the real part, χ ′ , in these frequency ranges, the ensemble averages of the absorptionand polarization cross sections are expressed as h C abs ν i = χ ′′ f ( χ ′ ) and h C pol ν i = χ ′′ g ( χ ′ ) in thefirst-order of the imaginary part. Therefore, Π ν is independent of χ ′′ and depends only on χ ′ .Since the frequency dependence of χ ′ is very small, Π ν of the high and low frequency ranges,except around the resonance peak, become almost constant against frequency change. In thehigh frequency range, the DCD contribution of χ ′ is dominant. On the other hand, in the lowfrequency part, the contribution of the resonance process to χ ′ is dominant. This results inthe discrepancy of Π ν found in figure 6 between the high and low frequency region across theresonance peak.At around the resonance peak, the degree of polarization shows a prominent behaviorfor some sets of parameters. Figures 6a and 6d show that the degree of polarization decreasesabruptly and takes the local minima around the peak frequency of the resonance process when∆ max0 > h/τ + . This is because the amplitude of the polarization cross sections for all threeaxes of the ellipsoid get closer near the resonance peak, as shown in figure 5. As a result,the polarization cross section defined by equation (21) approaches zero. In extreme cases, the17 a) ∆ max0 frequency [Hz]10 − − − p o l a r i ze d f r a c t i o n × h × h
10 GHz × h
30 GHz × h
100 GHz × h (b) R ∆ frequency [Hz]10 − − − p o l a r i ze d f r a c t i o n . . . . . (c) T frequency [Hz]10 − − − p o l a r i ze d f r a c t i o n (d) τ + frequency [Hz]10 − − − p o l a r i ze d f r a c t i o n × − s3 × − s1 × − s3 × − s1 × − s (e) L min frequency [Hz]10 − − − p o l a r i ze d f r a c t i o n . . . . Fig. 6:
Parameter dependences of the degree of polarization of the thermal emission from amorphoussilicate dust as a function of frequency. For reference, the polarization degree with the sameparameter set is shown by a solid line in each panel. The fraction of the atoms trapped in theTLS is set to be 0.01. In each panel, one of the variables characterizing the amorphous silicatedust is varied with respect to the reference model to see how the shape of the polarizationdegree responds to each variable. In the case of ∆ max0 /h = 100 GHz shown in (a), h C pol ν i definedby equation (21) takes negative values between 100 GHz and 500 GHz. The polarization crosssections take positive values for all other cases shown in these figures. The variables are (a)∆ max0 , which are expressed in the corresponding frequency normalized by h , (b) R ∆ , (c) T ,(d) τ + , and (e) L min , respectively. The given values for each parameter are shown in thelegends. a) Perseus MC I ν , P ν [ J y ] I ν P ν frequency [Hz]10 − − Π ν (b) W43 I ν , P ν [ J y ] I ν P ν frequency [Hz]10 − − Π ν Fig. 7:
Intensity and polarized SEDs, and the polarization fraction for (a) the Perseus MC and (b)W43. The error bars are 1 σ but the upper limits are at the 95% confidence level. polarization cross section changes its sign. This can be seen in the case of ∆ max0 /h = 100 GHzin figure 6a. In this case, the order of the amplitude of the absorption cross section reverses: h C abs ν,x i < h C abs ν,y i < h C abs ν,z i . As a result, h C pol ν i becomes negative. This means that the directionof the polarization changes and becomes parallel to the magnetic field near the resonance peakfrequency. Figure 6c shows that the polarization degree takes a minimum value at the resonancepeak when the temperature of the dust is as low as 10 K. This is because the relative intensityof the resonance peak to far infrared emission increases when the dust temperature decreases,as shown in figure 2c. There is no definite report on the detection of the polarization from AME. The upper limits onΠ ν for the Perseus MC and W43 are given by G´enova-Santos et al. (2015) and G´enova-Santoset al. (2017), respectively. In this subsection, we attempt to fit intensity and polarization datasimultaneously with our model. In figure 7, the polarized SEDs for the Perseus MC and W43 are shown. The data for polarizedAME are taken from table 4 in G´enova-Santos et al. (2015) and table 8 in G´enova-Santoset al. (2017). We exclude the DRAO 1.4 GHz data point because G´enova-Santos et al. (2017)suspected that Faraday rotation affects the data point. The polarization flux at 143, 217,and 353 GHz are extracted from the Planck second data release (Planck Collaboration et al.19 able 3: best-fit parameters for the Perseus MC and W43 with polarization
Perseus MC W43 T (K) 16.67 ± ± τ ( × − ) 4.25 ± ± f TLS ± ± max0 /h (GHz) 14.8 ± ± R ∆ ± ± τ + ( × − s) 2 . +0 . − . ± L min ± ± − pc) 28.2 ± ± χ / dof 3.59 7.65 The errors are 1 σ . L min among the fitparameters. The free–free emission is assumed to be unpolarized. The observed SEDs of the intensity and polarization flux are fitted simultaneously. The brute-force fitting method adopted in subsection 3.3.2 is used. The best-fit parameters are summarizedin table 3. The model predictions of the polarized SED with these best-fit parameters areoverlaid on the observed SED in figure 7.It shows that our model is able to reproduce the overall features of both the intensityand polarization SEDs simultaneously. In the best-fit model for the Perseus MC, there is avalley in the frequency dependence of the polarization fraction, and the polarization fractionreaches its minimum value at 20 GHz. The polarization fraction increases abruptly towardlower frequencies and approaches the asymptotic value. The asymptotic polarization fraction isfactor 5 larger than the polarization fraction in submillimeter wavebands. The model predictionis marginally consistent with the QUIJOTE 2 σ upper limits but is slightly higher than the20UIJOTE upper limits in several frequency bands. In the best-fit model for W43, there is adip in the polarization fraction defined by the ratio of equation (21) to equation (20) from 10to 50 GHz. In this case, h C pol ν i changes sign from 13 to 30 GHz. Therefore, a 90 degree flip inthe polarization direction in this frequency range is predicted. The polarization fraction below10 GHz is factor ten larger than the polarization fraction in submillimeter wavebands. Themodel prediction is marginally consistent with the QUIJOTE 2 σ upper limits but is slightlyhigher than the QUIJOTE upper limits in several frequency bands. To reproduce the relative intensity of AME to the far infrared peak intensity, our model requiresvery different physical characteristics for amorphous silicate dust in comparison with amorphoussilicate materials found in the laboratory. In the laboratory, the fraction of atoms trapped inthe TLS is reckoned to be of the order of 10 − . This comes from reproducing the experimentalfact that the diagnostics dominated by the TLS in the heat capacity only appears below 1 K(Phillips 1987). On the other hand, for amorphous silicate dust, the required fraction of atomstrapped in the TLS is a few percent in order to reproduce the observed ratio of the AMEpeak intensity to the far infrared peak intensity with dust temperature of about 20 K. Figure 8shows the frequency dependence of the absorption efficiency Q abs ν , which is the absorption crosssection normalized by the geometrical cross section, of spherical amorphous silicate dust forvarious TLS fractions. It shows that the peak value of the absorption efficiency of the resonanceprocess of the TLS with f TLS = 1 is factor 5 larger than the absorption efficiency at 2 THz wherethe far-infrared peak appears. As shown in equation (10), the thermal emission spectrum isthe product of the absorption cross section and the Planck function B ν ( T ). The ratio of thevalue of B (20 K) at the peak frequency of AME to B (20 K) at the frequency of thefar-infrared peak is about 0.0025. Therefore, f TLS ∼ − is required to reproduce the observedratio of the peak intensity of AME to the far-infrared peak intensity of 10 − . Figure 8 alsoshows that the peak value of the absorption cross section of the resonance process of the TLS istwo orders of magnitude less than the geometrical cross section, even in the case where f TLS = 1.It shows that this is two orders magnitude less than the absorption cross section adopted byJones (2009). Therefore, his predicted SED due to the resonance process of the TLS was twoorders of magnitude overestimated.The allowed ranges of ∆ are narrowly limited to reproduce the bump structure in theSED. Because of these results, the temperature dependence of the heat capacity of amorphous21 frequency [Hz]10 − − − − − Q a b s ν f TLS = 1 f TLS = 0 . f TLS = 10 − f TLS = 10 − f TLS = 10 − Fig. 8:
Frequency dependences of absorption efficiencies for spherical amorphous silicate dust. Thegiven values of f TLS are shown in legends. For the other parameters, the best-fit parametervalues for the Perseus MC listed in table 2 are adopted. The thin solid line is the absorptionefficiency provided by the Draine & Lee (1984) model. silicate dust has peculiar characteristics, as shown in figure 9. The heat capacity of the TLSwith energy difference, E , is described by the Schottky heat capacity as follows: C V = E k B T sech (cid:18) E k B T (cid:19) . (22)Since the energy difference, E , has a distribution in the amorphous silicate dust, the heat ca-pacity of the amorphous silicate dust, C TLS V , is obtained by integrating over it. The contributionof the TLS to the heat capacity of the amorphous silicate dust is then calculated as C TLS V = P Z ∆ max0 ∆ min0 d ∆ ∆ Z √ (∆ max0 ) − ∆ d ∆ C V = P Z ∆ max0 ∆ min0 dE Z τ max τ min dτ C V τ q − τ min /τ = P ∆ min0 Z /R ∆ dx C V asinh (cid:16) √ x − (cid:17) , (23)where τ is the relaxation time caused by the tunneling effect in equation (A51) and x ≡ E/ ∆ max0 .There are two distinctive diagnostics compared with amorphous silicate materials in the labo-ratory. The heat capacity has a bump at an extremely low temperature and is not proportionalto the temperature. This is because the allowed ranges of ∆ are narrowly restricted. Ourmodel predicts that the amorphous silicate dust is composed of amorphous silicate materials,which have very distinctive characteristics compared with amorphous silicate materials foundin the laboratory.Speck et al. (2011) proposed the possible forms of amorphous silicate dust in space.If a few percent of atoms are trapped in the double-well potential caused by deformation ofthe crystal structure, our results are applicable to any forms of amorphous silicate dust. The22 − T [K]10 − − − − − − C V / N k B Fig. 9:
Temperature dependence of the heat capacities of the amorphous silicate dust in each MCpredicted by our model. The contributions from the TLS are shown by a thick dotted curvefor the Perseus MC and a thin dotted curve for W43. The contribution from the Debye modelshown by the dashed curve is the same for both MCs. The thick and thin solid curves are thetotal heat capacities for the Perseus MC and W43, respectively. classic 10 µ m amorphous silicate feature observed in the interstellar medium (Knacke et al.1969; Hackwell et al. 1970) is not affected by this. Although our amorphous models reproduce the observed intensity SEDs for the Perseus MCand W43, the fits were not satisfactory. Our models underestimate the observed intensities inthe frequency range from 100 GHz through 500 GHz. The model prediction of the polarizationfraction of AME is slightly higher than the QUIJOTE upper limits in several frequency bands.The model prediction of the polarization fraction below 10 GHz is too high compared withthat for submillimeter frequencies. Possible improvements to our model for each unsatisfactorypoint will now be discussed.The TLS model describes the very low temperature limit of the soft-potential model.By fully taking into account the soft-potential model, the model SED above 100 GHz couldbe improved. The TLS describes the physical behavior of amorphous materials below 1 K.There are still deviations in the heat capacity from the Debye model in amorphous materialsabove 1 K, and the deviation at temperatures above 1 K is different from that below 1 K. Theplateau in the heat conductivity at T ∼ − − − χ ′ − − Π ν χ ′′ /χ ′ = 0 . χ ′′ /χ ′ = 0 . χ ′′ /χ ′ = 1 χ ′′ /χ ′ = 10 Fig. 10:
The polarization fraction as a function of the real part of the electric susceptibility ( χ ′ ) forvarious ratios of the imaginary part to the real part ( χ ′′ /χ ′ ). The dotted, dashed, solid, anddashed–dotted curves correspond to χ ′′ /χ ′ = 0.01, 0.1, 1, and 10, respectively. The lowercut-off of L x , L min , is set at 0. TLS model is incorporated in the soft-potential model as its very low temperature limit. Sincethe typical temperature of interstellar dust is about 20 K, the physical processes beyond thestandard TLS model may have a significant effect on the SED of the thermal emission fromamorphous dust above 100 GHz.One of the possibilities for reducing the model prediction of the polarization fractionin AME frequency bands is to replace the current DCD model by some other model thatprovides a higher value of the real part of the electric susceptibility χ than that of the currentDCD model. Figure 10 shows how the polarization fraction depends on the real part of theelectric susceptibility ( χ ′ ). The plots calculated for various ratios of the imaginary part to thereal part of the electric susceptibility ( χ ′′ /χ ′ ) are shown. The polarization fraction increasesmonotonically with increasing χ ′ . As the ratio of the imaginary part to the real part decreases,the polarization fraction converges to the asymptotic value for each value of χ ′ . This is becausethe polarization fraction depends only on the real part of the electric susceptibility when theimaginary part is much smaller than the real part, as shown in subsection 4.1. When the ratiois less than 0.1, the polarization fraction is proportional to χ ′ below χ ′ < .
2. Therefore, byreplacing the current DCD model by some other model that provides a higher value of χ ′ , theellipticity required to reproduce the observed polarization fraction in submillimeter wavebandsis expected to be smaller than the current model; in other words, L min takes a value closer to1/3. As the result, a reduction is expected in our model predictions of the polarization fractiondue to the resonance process of the TLS. Figure 11 compares the frequency dependence ofthe real and imaginary parts of the electric susceptibility predicted by our DCD model withthat of Draine & Lee (1984) model. The imaginary parts of both models are identical in24 frequency [Hz]10 − − − − − χ ′ , χ ′′ Fig. 11:
Comparison of frequency dependence of the electric susceptibility of our DCD model (solidcurves) and Draine & Lee (1984) model (dashed curves). Thick curves are real parts andthin curves are imaginary parts. submillimeter wavebands. Therefore, the intensity SEDs in submillimeter wavebands wouldnot be changed by replacing our DCD model by the Draine & Lee (1984) model. The realpart is an order of magnitude larger than the imaginary part in submillimeter wavebands inboth models. This shows that the real part of the Draine & Lee (1984) model is an order ofmagnitude larger than that of our DCD model. Therefore, replacing the current DCD modelby the Draine & Lee (1984) model is one possible solution to reducing the model prediction ofthe polarization fraction with a small change in the model prediction of the intensity SED. Adetailed quantitative study of this possibility is beyond the scope of the present paper and willbe carried out in a forthcoming paper.We have to mention that in the range of frequencies where AME is detected the interpre-tation of the nature of the polarization signals is quite complex. The total polarized emissioncould be increased or decreased because of a polarized synchrotron residual component. Inaddition to this, the band pass and the beam of the telescope could mitigate the total level ofpolarization of AME, particularly if this happens in the frequency range where the polarizationof AME is expected to change sign.Although we have neglected the contribution of the carbonaceous dust, it is known thatamorphous carbon dust is closer to the realistic form of carbonaceous dust in the interstellarmedium (Zubko et al. 1996) and almost half of the mass of interstellar dust is shared bycarbonaceous dust (Hirashita & Yan 2009). Since the physical processes of the TLS are universalamong the amorphous materials and independent from elemental compositions, intensity andpolarization SEDs of thermal emission from amorphous dust derived in this paper are applicableto the amorphous carbon dust. However, the physical parameters which described the TLS ofthe amorphous carbon dust could be different from those of the amorphous silicate dust. In25ddition, free electrons might contribute to the electric susceptibility of the amorphous carbondust. Further, amorphous carbon dust might have isotropic structure (Draine & Lee 1984).It results in the anisotropic dielectric function tensor. Since the main scope of this paper isproviding the framework to evaluate the intensity and polarization SEDs of thermal emissionfrom amorphous dust by self-consistently taking into account the TLS model and demonstratingthat this model is promising, the studies of the effect of the amorphous carbon dust on SEDsare beyond the scope of this paper and are shown in the forthcoming paper.Although we assumed that all dust grains stayed at the same temperature, significanttime variation of the temperature of the small dust grains are expected according to the stochas-ticity of the heating process (e.g. Draine & Li 2001). In a significant fraction of time, smalldust grains stay much lower temperature than that of the large dust grain which is definedby thermal equilibrium. As shown in figure 2c, the relative intensity of the emission from theresonance process to the contribution from the lattice vibration becomes higher as the dusttemperature becomes lower. Therefore, quantitative studies of the stochastic heating and sizedistribution of the dust grains are important.As shown in figure 4, there are significant differences in the shapes of the spectra of thethermal emission from amorphous dust between the Perseus and W43 MCs. The variation of thespectra shape originates from the fact that the physical parameters describing the amorphousdust, such as ∆ max0 , R ∆ , τ + , and f TLS , take different values for each MC (see table 2). Possibleorigins of the variations are now summarized.1. The elemental composition of amorphous silicate dust is different for the Perseus MC andW43. The shape of SEDs is sensitive to the values of ∆ max0 and R ∆ . The peak frequencyand width of the AME spectra due to the resonance process in the standard TLS modelare defined by ∆ max0 and R ∆ , respectively. As presented in equation (A7), ∆ is directlyrelated to the tunneling parameter λ . The value of the tunneling parameter depends onthe masses of the atoms constituting amorphous silicate dust, the width and height of thepotential barrier of the double-well potential in which an atom is trapped. It is natural thatthe typical values of these parameters change if the elementary composition of amorphoussilicate dust is different. In this paper, we have assumed the number ratio of Fe to Mg inthe amorphous silicate dust to be 1 to 1 (see table 1), however, that value might be differentin each MC. Although we have not considered contributions from amorphous carbon dustin SEDs, there are several arguments that indicate their existence (e.g., Compi`egne et al.2011; Jones et al. 2017). It may also affect the spectral shape of AME. Since the elemental26omposition of the amorphous carbon dust is expected to be insensitive to the environmentalmetal abundance, the variations of the ∆ max0 and R ∆ of the amorphous carbon dust amongthe MCs could not be attributed to the variation of the metal abundance of the environment.2. Differences of cooling processes which solidify gas and form an amorphous dust in each MCmay result in a variation of the bonding structure of the atoms in a dust and in a variationin amorphous nature of a dust particle. Amorphous materials are generated by rapid coolingfrom the liquid phase to the solid phase in laboratory. In interstellar space, the solidificationmay happen from the gas phase without passing through the liquid phase in an extremelylow pressure environment. This could be one of the sources for which f TLS takes an extremelyhigh value compared with terrestrial amorphous materials.3. Since the shape of the AME spectrum depends sensitively on the shape of the spectrum offree–free emission in the microwave region, which depends sensitively on the temperature ofthe ionized gas, it is possible that component separation between the free–free emission andAME is not sufficient. If the magnitude or shape of the free–free emission SED changes, thebest-fit values of these parameters also change easily.The fact of the lack of AME in cold dense cores (Tibbs et al. 2016) could be explained by a kindof variation in the amorphous nature of the dust due to a difference in environment conditions.
Complete studies of the radiative processes of thermal emission from amorphous dust from themillimeter through far infrared wavebands were presented by, for the first time, self-consistentlytaking into account the standard TLS model. How the intensity and the polarization SEDsrespond in physical parameters characterizing the standard TLS model was shown. The amor-phous model could reproduce very well the observed SEDs from AME up to the far-infraredfeature. In our models, AME is originated mainly from the resonance emission of the TLS oflarge grains. The amorphous model is able to explain AME without introducing new species.Simultaneous fitting of the polarization and intensity SED for the Perseus MC and W43 werealso performed. Since there is no definite detection of polarization emission from AME, theadopted polarization intensities in the AME frequency range were upper limits. The polar-ization intensities measured by Planck at 143, 217, and 353 GHz were also included. Theamorphous model could reproduce the overall observed feature of the intensity and polariza-tion SEDs of the Perseus MC and W43. However, the model prediction of the polarizationfraction of AME was slightly higher than the QUIJOTE upper limits in several frequency27ands. Possible improvements to our model to resolve this problem were proposed in the pre-vious section. Our model predicts that amorphous silicate dust have very different physicalcharacteristics compared with amorphous silicate materials found in the laboratory. We haveshown that thermal emission from amorphous dust is an attractive alternative possibility asthe origin of AME.
Acknowledgments
We thank Tetsuo Yamamoto for helpful discussions throughout the course of this work. We thank the referee, Itsuki Sakon,for constructive comments. We thank T.J. Mahoney for revising the English of the draft. MN acknowledges support from theGraduate Program on Physics for the Universe (GP-PU), Tohoku University. FP acknowledges the European Commission and theMINECO. This work is partially supported by MEXT KAKENHI Grant Number 18H05539 and MEXT KAKENHI Grant Number18H01250. This project has been partially funded by the European Union’s Horizon 2020 research and innovation programme undergrant agreement number 687312 (RADIOFOREGROUNDS), and by the SPACE IR MISSIONS II project under grants agreementsESP2015-65597-C4-4-R and ESP2017-86852-C4-2-R, respectively. MH would like to express his sincere condolences to Prof. TsaiAn-Pang who was the world authority on quasicrystal and passed away in May 2019 at age of 60. In the course of this study, asthe resident who lived in the same apartment by chance, MH received fruitful comments on the study and great support in theprivate life.
Appendix 1 Standard TLS model
The basics of the standard TLS model are as follows. The ground states confined in eachharmonic potential are denoted by ϕ and ϕ , respectively. They are defined by the Schr¨odingerequations as H i | ϕ i i = − ¯ h m d dx + V i ! | ϕ i i = ǫ i | ϕ i i , (A1)where i runs from 1 to 2, ǫ i is the energy of the ground state, and m is the mass of an atom.The wave functions h x | ϕ i and h x | ϕ i are localized around the bottom of the potentials V and V , respectively. In other words, there is almost no overlap between h x | ϕ i and h x | ϕ i ; thus h ϕ | ϕ i vanishes in practice; i.e., h ϕ | ϕ i = Z dx ′ h ϕ | x ′ ih x ′ | ϕ i ≃ . (A2)The Hamiltonian of the standard TLS model is described by H = H − V + V = H − V + V. (A3)The ground state, ψ , and the first excited state, ψ , confined in the double-well potentialsatisfy the following Schr¨odinger equations: 28 k | ψ k i = H | ψ k i , (A4)where E and E are the energy eigenvalues of the ground state and the first excited state,respectively. Under the standard TLS approximation, these states are represented in the form | ψ i i = P j =1 , c ij | ϕ j i . Using equation (A3), equation (A4) is expressed in matrix form: E i c i c i = H c i c i , (A5) H = ǫ + h ϕ | ( V − V ) | ϕ i h ϕ | H | ϕ ih ϕ | H | ϕ i ǫ + h ϕ | ( V − V ) | ϕ i . (A6)Since each ϕ i is located at the bottom of V i , h ϕ i | ( V − V i ) | ϕ i i ≪ ǫ i , and the diagonal elementsare approximated by ǫ i . Two variables, ∆ ≡ ǫ − ǫ and ∆ ≡ h ϕ | H | ϕ i = 2 h ϕ | H | ϕ i , areintroduced to characterize the TLS. ∆ is the energy difference between the two states locatedat each minimum of the double-well potential and characterizes the degree of asymmetry of thepotential. ∆ is the parameter that characterizes the degree of the cross correlation betweenthe states located in two minima and can be approximated by∆ = ¯ h Ω e − λ , (A7)where ¯ h Ω is the order of ǫ and ǫ . Note that λ is used for the tunneling parameter in thissection. By shifting the meaningless zero level of the energy eigenvalues, H is rewritten withthese two parameters as H = 12 − ∆ ∆ ∆ ∆ . (A8)The energy eigenvalues E i are obtained by deducing the eigenvalues of the matrix H , writtenin equation (A8) as, E = − E , (A9) E = E , (A10) E ≡ q ∆ + ∆ , (A11)where E is the energy splitting of the TLS. By normalizing the states ψ and ψ as h ψ | ψ i = h ψ | ψ i = 1, the expansion coefficients are expressed by using a single parameter as c = cos θ , c = − sin θ , c = sin θ , and c = cos θ where cos 2 θ ≡ ∆ /E , sin 2 θ ≡ ∆ /E . Therefore, | ψ i and | ψ i are represented by | ψ i = cos θ | ϕ i − sin θ | ϕ i , (A12)29 ψ i = sin θ | ϕ i + cos θ | ϕ i . (A13)Suppose the two energy eigenstates set up a complete system. Then an arbitrary state of theTLS | ψ i can be described by | ψ i = a ( t ) | ψ i + a ( t ) | ψ i , (A14)where a and a are time-dependent complex numbers and satisfy the normalization condition( | a | + | a | = 1). Furthermore, the identity operator can be defined as ˆ σ I ≡ | ψ ih ψ | + | ψ ih ψ | .The following operators are useful for seeing the physical behavior of the TLS, such asˆ σ + ≡ | ψ ih ψ | , (A15)ˆ σ − ≡ | ψ ih ψ | , (A16)ˆ σ w ≡ | ψ ih ψ | − | ψ ih ψ | . (A17)In order to clarify the physical meanings of these operators, let them act on | ψ i . We thenobtain ˆ σ + | ψ i = a | ψ i , (A18)ˆ σ − | ψ i = a | ψ i , (A19)ˆ σ w | ψ i = a | ψ i − a | ψ i . (A20)The expectations of each operator, u ± and w , can be calculated as, u + = h ψ | ˆ σ + | ψ i = a a ∗ , (A21) u − = h ψ | ˆ σ − | ψ i = a ∗ a , (A22) w = h ψ | ˆ σ w | ψ i = a a ∗ − a a ∗ . (A23)Therefore, ˆ σ ± are something like ladder operators and ˆ σ w measures a difference of the probabil-ities of finding an atom in each state. The operator ˆ σ + represents the excitation of the groundstate ψ to the excited state ψ . The operator ˆ σ − represents the downward transition from theexcited state to the ground state.The interaction Hamiltonian between the TLS and an electromagnetic field, H ′ , can bewritten, H ′ = − q r · E local , (A24)where q is the charge of an atom trapped in the double-well potential, r is its position vector,and E local is the local electric field at the position of the atom. The magnetic effect is negligiblesince the velocity of the atom is much smaller than the speed of light. Evolution of the atomicstate ψ caused by the perturbation is described by the following Schr¨odinger equation,30 ¯ h ∂ | ψ i ∂t = ( H + H ′ ) | ψ i . (A25)The electric dipole moment arising from the TLS, d TLS , is given by, d TLS = h ψ | q r | ψ i≃ [ − ( a a ∗ − a a ∗ ) cos 2 θ − ( a a ∗ + a ∗ a ) sin 2 θ ] d , (A26) d ≡ −h ϕ | q r | ϕ i = h ϕ | q r | ϕ i . (A27)The cross terms, h ϕ | q r | ϕ i and h ϕ | q r | ϕ i , are neglected in comparison with the diagonalterms. We can choose the origin of the coordinate to realize equation (A27) without losinggenerality. d is the electric dipole moment for the state located at the minimum of thepotential V . The time evolution equations for a and a are led by equation (A25), da dt = ia (cid:18) + ω cos 2 θ (cid:19) − ia Ω sin 2 θ, (A28) da dt = ia (cid:18) − ω cos 2 θ (cid:19) − ia Ω sin 2 θ, (A29)where ¯ hω ≡ E and ¯ h Ω ≡ d · E local . Defining a Bloch vector R ≡ ( u, v, w ) ≡ ( a a ∗ + a ∗ a , − i ( a a ∗ − a ∗ a ) , a a ∗ − a a ∗ ), each component of R satisfies the following equations: dudt = − v ( ω − cos 2 θ ) , (A30) dvdt = u ( ω − cos 2 θ ) + 2 w Ω sin 2 θ, (A31) dwdt = − v Ω sin 2 θ. (A32)Equations (A30)–(A32) are called Bloch equations . Equations for u ± are led by equations(A30) and (A31) as, du ± dt = ± iu ± ( ω − cos 2 θ ) ± iw Ω sin 2 θ. (A33)Due to the spontaneous transition, w is relaxed to the instantaneous thermal equilibrium state¯ w ( t ). The energy levels of the TLS are shifted as E → E + ( d cos 2 θ ) · E local and E → E − ( d cos 2 θ ) · E local owing to the interaction of the atom with the local electric field. As a result,the population of the thermal equilibrium states changes to ¯ w ( t ). This is the instantaneousthermal equilibrium state of the population. By introducing the relaxation time τ w , equation(A32) is modified as, dwdt = − v Ω sin 2 θ − w − ¯ w ( t ) τ w . (A34) u + represents the excitation of the atom in the TLS by the absorption of the electromagnetic By introducing a vector Ω ≡ ( − sin 2 θ, , ω − cos2 θ ) , equations (A30)–(A32) are rewritten as d R /dt = Ω × R . u − represents the transition from the excited state to the ground state stimulated by theelectromagnetic wave. The relaxation of the states to the original states is taken into accountin the evolution equations of u ± by introducing the phase relaxation time τ + such that du ± dt = ± iu ± ( ω − cos 2 θ ) ± iw Ω sin 2 θ − u ± τ + . (A35)When the local electric field carried by the electromagnetic wave is weak enough, the Blochequations (A35) and (A34) can be treated perturbatively. The zeroth order solutions are givenby u (0) ± = 12 sech (cid:18) E k B T (cid:19) e ± i ( ω t + δ ) , (A36) w (0) = tanh (cid:18) E k B T (cid:19) , (A37)where k B is the Boltzmann constant. Since the zeroth order states are the thermal equilibriumstates, the population of each level is given by | a (0)1 | = 1 / { exp[ − E/ ( k B T )] + 1 } and | a (0)2 | =1 / { exp[ E/ ( k B T )] + 1 } , respectively. Since the phase coefficients exp( ± iδ ) take random valuesthrough the whole amorphous material, we may assume u (0) ± = 0.In the first order of perturbations, the Bloch equations are written du (1) ± dt = ± iω u (1) ± ± iw (0) Ω sin 2 θ − u (1) ± τ + , (A38) dw (1) dt = − w (1) − ¯ w (1) τ w . (A39)¯ w (1) is estimated by expanding ¯ w with E ,¯ w (1) = ∂ ¯ w∂E [ − d cos 2 θ ) · E local ]= − d · E local cos 2 θk B T sech (cid:18) E k B T (cid:19) . (A40)By decomposing the incident electric field into the Fourier spectrum, E local ( t ) = R dω ˆ E local ( ω ) e − iωt , the first order solutions of the Bloch equations are:ˆ u (1) ± = ± i τ + ¯ h d · ˆ E local sin 2 θ i ( ω ∓ ω ) τ + tanh (cid:18) E k B T (cid:19) , (A41)ˆ w (1) = − d · ˆ E local cos 2 θk B T − iωτ w sech (cid:18) E k B T (cid:19) . (A42)To obtain the absorption coefficient of the amorphous material against electromagneticwaves, the electric susceptibilities based on the standard TLS model are deduced. Before goinginto detail on each physical process, we have to model the distributions of ∆ and ∆ . Anderson,Halperin, & Varma (1972) and Phillips (1972) proposed that the probability of finding λ and∆ at some value is uniform since the possible range of these variables is narrowly limited. The32istribution function of ∆ and ∆ , f (∆ , ∆) is then given by: f (∆ , ∆) d ∆ d ∆ = P dλd ∆ = P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂λ∂ ∆ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ∆ d ∆ ,f (∆ , ∆) = P ∆ , (A43)where f (∆ , ∆) d ∆ d ∆ provides the number density of the atoms trapped in the TLS from ∆ to ∆ + d ∆ and from ∆ to ∆ + d ∆, and P is the constant providing the number density ofatoms trapped in the TLS, n TLS , which is deduced by integrating the distribution function over d ∆ and d ∆ as, n TLS = P Z ∆ max0 ∆ min0 d ∆ ∆ Z √ (∆ max0 ) − ∆ d ∆= P ∆ max0 ln q − R + 1 R ∆ − q − R , (A44) R ∆ ≡ ∆ min0 ∆ max0 . (A45)∆ max0 and ∆ min0 are introduced to avoid divergence of the probability distribution function.∆ max and ∆ min0 are related to each other through the maximum energy splitting of the TLS as E = ∆ + (∆ max0 ) = ∆ + (∆ min0 ) (see equation (A11)). We treat ∆ max as a dependentvariable of ∆ min0 . For simplicity, we set ∆ min to zero.The expectations u ± represent the transition between the TLS due to absorption andemission of the electromagnetic wave. These processes refer to the resonance transition. Wederive the electric susceptibility due to the resonance transition. The electric dipole momentcaused by the resonance transition, d res , stimulated by an electromagnetic wave of angularfrequency ω can be written as d res ≃ − (cid:16) ˆ u (1)+ + ˆ u (1) − (cid:17) d sin 2 θ = − i τ + ¯ h d · ˆ E local (cid:18) ∆ E (cid:19) tanh (cid:18) E k B T (cid:19) × "
11 + i ( ω − ω ) τ + −
11 + i ( ω + ω ) τ + d . (A46)The electric polarization P res is calculated by averaging the electric dipole moment over thesolid: P res = 1 V X i d i res = Z d ∆ Z d ∆ f (∆ , ∆) d res , (A47)where V is the volume of an amorphous material. In generally, the electric polarization P ofan isotropic and spherical particle is related to the external electric field E ext ,33 = χ E ext , (A48)where χ is the electric susceptibility for the response to an external electric field. In sphericaldielectric material, a local electric field equals an externally applied field (see appendix 2).Thus, χ res0 is given by, χ res0 = P res · ˆ E ext | ˆ E ext | = P res · ˆ E local | ˆ E local | . (A49)By assuming that the directions of d relative to the local electric field ˆ E local are randomlydistributed, the average of ( d · ˆ E local ) becomes | d | | ˆ E local | /
3. Then, the electric susceptibilitydescribed by equation (1) is obtained.The expectation w relaxes to the instantaneous thermal equilibrium value. We describehow the relaxation process contributes to the electric susceptibility. The electric dipole momentdue to the relaxation process is written as, d rel ≃ − ˆ w (1) d cos 2 θ = d · ˆ E local k B T (cid:18) ∆ E (cid:19) − iωτ w sech (cid:18) E k B T (cid:19) d . (A50)There are two main relaxation processes. One is quantum tunneling in which an atom passesthrough the potential barrier by the quantum effect. The other is hopping where an atomclimbs over the barrier by gaining enough energy due to thermal fluctuation.The tunneling relaxation time τ tun was deduced by Phillips (1972) as τ − = γ c + 2 γ c ! ω ∆ πρ ¯ h coth (cid:18) E k B T (cid:19) , (A51)where γ t(l) and c t(l) are the elastic dipole and sound velocity for the transverse (longitudinal)waves, respectively. ρ is the mass density. Typical values of physical variables of amorphoussilicate material found in laboratory experiments are listed in table 1 (where c − ≪ c − ). Then,the complex susceptibility for the tunneling relaxation χ tun0 is obtained as equation (2).The hopping relaxation time τ hop is given by Arrhenius equation as, τ − = 1 τ exp (cid:18) − V k B T (cid:19) , (A52)where the values of τ are defined by the physical characteristics of each amorphous material.The relaxation time scale of the hopping is sensitive to the height of the potential barrier V ,which must vary in value across a single dust grain. The probability density function g ( V ) for V is introduced. B¨osch (1978) proposed the Gaussian distribution function of g ( V ) as34 ( V ) = C V exp (cid:20) − (cid:16) V − V m V σ (cid:17) (cid:21) , V > V min ;0 , V < V min ; (A53) C V = 2 V σ √ π (cid:20) Erf (cid:18) V m − V min V σ (cid:19) + 1 (cid:21) − , (A54)Erf( x ) ≡ Z x dt e − t . (A55)By taking into account the distribution of V , the complex susceptibilities for the hoppingrelaxation χ hop0 is obtained as equation (3). Appendix 2 Extension of the Clausius-Mossotti relation for an ellipsoidal particle
Consider a homogeneous ellipsoid located in a uniform electric field E i ext , whose direction isparallel to the i th semi-axis of the ellipsoidal particle. An electric polarization P i arising from E i ext aligns in the same direction. P i is given as (e.g., Bohren & Huffman 1983), P i = 14 π ε −
11 + L i ( ε − E i ext ≡ χ i E i ext , (A56)where L i is a geometrical factor defined by equation (A65), ε is the electric permittivity ofthe particle, and χ i is the electric susceptibility along the i th semi-axis of the ellipsoid for theexternal field.The local electric field E local , based on Lorentz’s approach, is the sum of the externalfield and the electric field generated by the electric polarization (see Kittel 2004), E local = E ext + E + E + E , (A57)where E is the depolarization field generated from a surface charge density, E is the electricfield produced by a surface electric charge density on a virtual spherical cavity, and E isthe electric field created by dipole moments inside the cavity. E is related to the electricpolarization according to E i = − πL i P i . (A58) E is expressed by the electric polarization as, E i = 43 π P i . (A59)In amorphous material, it is expected that the position of each atom is completely random,and that the electric fields from dipole moments cancel each other out; therefore, E = 0. Usingequations (A56)–(A59), E local may be calculated:35 i local = 13 ε + 21 + L i ( ε − E i ext . (A60)In a spherical particle, the local field is equal to the external field because L i = 1 / P i = X j N j α j E i local , (A61)where α j is the polarizability of each atom j and N j is the concentration. From equations(A56), (A60) and (A61), we may obtain the following relation: X j N j α j = 34 π ε − ε + 2 . (A62)Equation (A62) is the Clausius-Mossotti relation, which is satisfied regardless of the shape ofthe particle. In other words, the electric permittivity is a physical parameter independent of theparticle shape. This equation relates microscopic physical parameters to macroscopic physicalparameters. From equation (A60), we can see that the local field equals the external field fora spherical particle. Therefore, P j N j α j is equal to χ which is the electric susceptibility of aspherical particle for the external electric field. From equation (A62) we get χ = 34 π ε − ε + 2 . (A63)We can derive the relation between χ i and χ from equations (A56) and (A63), χ i = χ π ( L i − / χ . (A64)This equation shows that χ i = χ when L i = 1 /
3, as expected.
Appendix 3 General optical properties of ellipsoidal particle
The shape of an ellipsoidal particle is characterized by geometric factors (Bohren & Huffman1983): L i ≡ V π Z ∞ dq ( q + a i ) q ( q + a x )( q + a y )( q + a z ) , (A65)where i = x, y and z . The volume of the ellipsoid is given by V = 4 πa x a y a z /
3. The geometricalfactors satisfy the following inequality: L x ≤ L y ≤ L z . In addition, since these variables satisfythe identity of L x + L y + L z = 1, one of the three is not an independent variable. We treat L x and L y as independent variables. The continuous distributions of ellipsoids (CDE: Bohren &Huffman 1983) with a lower cut-off of L x at L min is adopted as the shape parameter distribution.This distribution is referred to the externally restricted CDE (ERCDE: Zubko et al. 1996). A36phere is reproduced by setting L x = L y = L z = 1 / C abs ν,i = 8 π V λ Im (cid:16) χ i (cid:17) , (A66)where χ i is the complex susceptibility responding to an external electric field parallel to eachaxis (equation (A64)).The electric susceptibilities averaged over the shape distribution described by theERCDE are deduced by Draine & Hensley (2017) as, h χ x i = 32 πA ( ε − (cid:20) − A ( ε −
1) + 3 X ln (cid:18) XY (cid:19)(cid:21) , (A67) h χ y i = 3 πA ( ε − (cid:20) X ln (cid:18) ZX (cid:19) + Y ln (cid:18) YZ (cid:19)(cid:21) , (A68) h χ z i = 32 πA ( ε − (cid:20) X ln (cid:18) XZ (cid:19) + W ln (cid:18) WZ (cid:19)(cid:21) , (A69)where A ≡ − L min , X ≡ ε − / Y ≡ L min ( ε − Z ≡ − L min )( ε − / W ≡ − L min )( ε − ε is the electric permittivity of the amorphous dust. Theshape-averaged absorption cross sections for the electric field in the direction of each axis, h C abs ν,i i , are derived by substituting equations (A67)–(A69) for equation (A66). References
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