Paramagnetic alignment of small grains: a novel method for measuring interstellar magnetic fields
aa r X i v : . [ a s t r o - ph . GA ] M a y Draft version June 30, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
PARAMAGNETIC ALIGNMENT OF SMALL GRAINS: A NOVEL METHOD FOR MEASURINGINTERSTELLAR MAGNETIC FIELDS
Thiem Hoang , A. Lazarian , and P. G. Martin Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, ON M5S 3H8, Canada Institut f¨ur Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik, Ruhr-Universit¨at Bochum, 44780 Bochum, Germany Department of Astronomy, University of Wisconsin, Madison, WI 53706, USA
Draft version June 30, 2018
ABSTRACTWe present a novel method to measure the strength of interstellar magnetic fields based on ultra-violet (UV) polarization of starlight, which is in part produced by weakly aligned, small interstellargrains. We begin with calculating degrees of alignment of small (size a ∼ . µ m) and very small( a ∼ . µ m) grains in the interstellar magnetic field due to the Davis-Greenstein paramagnetic re-laxation and resonance paramagnetic relaxation. We compute the degrees of paramagnetic alignmentwith the ambient magnetic field B using Langevin equations. In this paper, we take into accountvarious processes essential for the dynamics of small grains, including infrared (IR) emission, elec-tric dipole emission, plasma drag and collisions with neutral and ionized species. We find that thealignment of small grains is necessary to reproduce the observed polarization in the UV, although thepolarization arising from these small grains is negligible at the optical and IR wavelengths. Basedon fitting theoretical models to observed extinction and polarization curves, we find that the best-fitmodel requires a higher degree of alignment of small grains for the case with the peak wavelength ofpolarization λ max < . µ m, which exhibits an excess UV polarization relative to the Serkowski law,compared to the typical case λ max = 0 . µ m. We interpret the correlation between the systematicincrease of the UV polarization relative to maximum polarization (i.e. of p (6 µ m − ) /p max ) with λ − by appealing to the higher degree of alignment of small grains. We identify paramagnetic relaxation asthe cause of the alignment of small grains and utilize the dependence of the degree of alignment on themagnetic field strength B to suggest a new way to measure B using the observable parameters λ max and p (6 µ m − ) /p max . Applying our new technique to the available observational data, we estimatethe upper limit of interstellar magnetic field B ∼ − µ G for the typical sightline ( λ max = 0 . µ m),which is consistent with the strength obtained from other available techniques. For the sightlines withlower λ max , the magnetic field strengths tend to be higher, assuming that the interstellar radiationfield is similar along these sightlines. Subject headings: dust, extinction–ISM: magnetic fields–polarization INTRODUCTION
The polarization of starlight discovered more than ahalf century ago (Hall 1949; Hiltner 1949) revealed thatinterstellar dust grains must be nonspherical and alignedwith respect to interstellar magnetic field. Since then, anumber of alignment mechanisms have been proposed toexplain why dust grains become aligned in the magneticfield, which include paramagnetic relaxation, mechanicaltorques, and radiative torques (see Lazarian 2007 for areview).In the present study we revisit the consequences ofparamagnetic alignment, which was one of the first align-ment mechanisms proposed to explain the polarizationof starlight by Davis & Greenstein (1951). The mecha-nism relies on paramagnetic relaxation within rotatinggrains to align them with the magnetic field. Quantita-tive studies for the paramagnetic alignment mechanism(hereafter Davis-Greenstein (D-G) mechanism) were onlyconducted about two decades later. For instance,Jones & Spitzer (1967) quantified the efficiency of theD-G mechanism using Fokker-Planck (FP) equations,while Purcell (1969) and Purcell & Spitzer (1971) dealtwith the problem by means of the Monte Carlo method.Their works showed that the D-G mechanism is ineffi-cient for aligning grains in the typical interstellar mag- netic field. Later, Purcell (1979) suggested that the jointaction of pinwheel torques and paramagnetic relaxationwould result in efficient alignment of suprathermally ro-tating grains (see also Spitzer & McGlynn 1979). Lazarian (1997) investigated analytically the D-Galignment for thermally rotating grains (grains not sub-ject to pinwheel torques) while accounting for the Bar-nett relaxation effect (Barnett 1915a) and internal ther-mal fluctuations (Lazarian 1994; Lazarian & Roberge1997). Roberge & Lazarian (1999) (hereafter RL99)quantified the efficiency of the D-G mechanism for ther-mally rotating grains by numerically solving Langevinequations that describe the temporal evolution of grainangular momentum. These studies assumed a constantmagnetic susceptibility K ( ω ) and considered the rota-tional damping and excitation of grains by gas atombombardment. Also, the studies were performed forthe ∼ . µ m interstellar grains that rotate slowly ( ω < s − ) not subject to spin-up systematic torques. Theauthors concluded that the D-G mechanism was ineffi-cient to account for the dust polarization observed inmolecular clouds, where the temperatures of dust and The improved theory of paramagnetic alignment of suprather-mally rotating grains is presented in Lazarian & Draine (1997).
Hoang, Lazarian, & Martingas are expected to be comparable.The first attempt to infer grain alignment from obser-vations was performed by Kim & Martin (1995). The au-thors employed maximum entropy method to fit theoret-ical polarization curves to observational data and foundthat interstellar silicate grains of size a ≥ . µ m (here-after typical interstellar grains) are efficiently alignedwhile smaller grains are very weakly aligned. Theyfound that there exists some residual alignment for0 . µ m − . µ m grains (hereafter small grains, see alsoMartin 2007). Recently, in the interest of polarized sub-millimeter emission motivated by the Planck mission,Draine & Fraisse (2009) derived the alignment functionfor interstellar grains by fitting simultaneously to the ob-served extinction and polarization curves for the typicaldiffuse ISM with R V = 3 . λ max = 0 . µ m.They came to the same conclusion as Kim & Martin(1995) that the typical interstellar grains are efficientlyaligned. In addition, they found that the degree of align-ment of small grains is f ∼ .
01 for the model with onlysilicate grains aligned. Due to its minor contributionto the polarization of starlight and to submillimeter/IRpolarized emission, the problem of alignment of smallgrains was mostly forgotten. Through this study we ar-gue that an in-depth understanding on the alignment ofsmall grains can provide us new insight into the ISM.Clayton et al. (1992) and Clayton et al. (1995) re-ported an excess polarization in the UV from theSerkowski law (Serkowski et al. 1975) extrapolation for anumber of stars with λ max ≤ . µ m. It is worth notingthat the optical and IR polarization is mostly producedby typical interstellar grains aligned. The fact that theSerkowski law could fit well to the observational datafrom the optical to IR wavelength but failed for the UV(Clayton et al. 1995) reveals that the excess UV polar-ization should originate from some aligned grains that donot contribute to the optical and IR polarization. Thiscould be a potential evidence for the alignment of smallgrains.Clayton et al. (1995) found a tight correlation be-tween the excess UV polarization, characterized by p (6 µ m − ) /p max , and λ − for a number of stars. Inparticular, they showed no difference in the UV ex-tinction for these stars, indicating that the propertiesof dust along these sightlines are not distinct fromthe general ISM. Using updated observational data,Martin et al. (1999) have confirmed the correlation be-tween p (6 µ m − ) /p max and λ − and showed that the UVpolarization can be described by a modified-Serkowskirelation. A systematic change in the size distribution ofaligned grains was suggested as a potential cause of therelationship (see Martin et al. 1999). Lazarian (2003)and Lazarian (2007) discussed the paramagnetic align-ment of small grains and pointed out that it can provideupper limits on the interstellar magnetic field as strongmagnetic fields in the diffuse ISM can overproduce thepolarization from small grains distorting Serkowski re-lations. Lazarian (2007) argued that the paramagneticalignment of small grains can be used for magnetic fieldstudies. In the absence of quantitative studies, however,the issue of the mechanisms responsible for the alignmentof small grains remains open.
Modern understanding on grain alignment has estab-lished a leading alignment mechanism based on radia-tive torques (RATs) induced by anisotropic radiationacting on realistic irregular grains. The mechanismwas proposed by Dolginov & Mitrofanov (1976) and nu-merically studied by Draine & Weingartner (1996) andDraine & Weingartner (1997). The analytical model ofRAT alignment was introduced in Lazarian & Hoang(2007). This model explained many puzzling featuresof the RAT alignment and provided the basis for quan-titative predictions of the RAT alignment efficiencies.The theory was further elaborated in Hoang & Lazarian(2008) and Hoang & Lazarian (2009a), which improvesits predictive abilities. Observational evidence for theRAT alignment was reported by a number of pa-pers (Andersson & Potter 2007; Whittet et al. 2008;Andersson et al. 2011).While the RAT alignment proves to be a robust align-ment mechanism for large grains in various environmentconditions, it appears to be inefficient for small grainsdue to the RAT magnitude decreasing with the decreas-ing grain size as ( λ/a ) − α with α = 3 − a ≪ λ (Lazarian & Hoang 2007). The spin-up mechanisms pro-posed by Purcell (1979) are also inefficient because thefast flipping of small grains tends to cancel out system-atic torques fixed in the grain body (Lazarian & Draine1999b; Hoang & Lazarian 2009b). Therefore, a promis-ing mechanism responsible for the weak alignment ofsmall grains is the paramagnetic relaxation. As the de-gree of alignment by the paramagnetic mechanism de-pends strongly on the magnetic field strength, the grainalignment of small grains can open a new way to measurethe ISM magnetic field based on the UV polarization.While the idea of using the UV polarization to measuremagnetic fields was mentioned in Lazarian (2007), nodetailed study of the process has been carried out yet. Quantitative study of paramagnetic alignment of smallgrains is the main goal of the present study.
We should mention that special attention was paidto the alignment of very small ( a ∼ − λ = 2175˚A. Such a UV bump is widelybelieved to originate from the electronic transition π − π ∗ in sp -bonded carbon sheets of small graphitegrains (Stecher & Donn 1965; Draine 1989) or PAHs(Li & Draine 2001; Weingartner & Draine 2001). In themodels by Draine and his co-workers, PAHs are sug-gested to be the dominant carrier of the 2175˚A feature. Among about 30 stars for which the UV polariza-tion data are observationally available to date, most ofthem do not show the polarization feature (bump) at λ = 2175˚A as seen in the extinction curves, except fortwo stars HD 197770 and HD 147933-4. This indicatesthat small carbonaceous grains may be only very weaklyaligned.Taking advantage of the special UV polarizationbumps at 2175˚A seen in HD 197770 and HD 147933-4, Hoang et al. (2013) carried out the fitting to the ob-served data and inferred the alignment function for theentire range of grain size distribution. We found that thealignment of ultrasmall carbonaceous grains with the effi-ciency of ∼ .
5% is required to reproduce the 2175˚A po-larization bump of HD 197770.
The question now is thatwhether ultrasmall grains are aligned by the same mech-anism as small grains.
The goal of the present study is (1) to calculate thedegree of alignment for small grains by the paramag-netic relaxation (e.g., D-G paramagnetic relaxation andresonance paramagnetic relaxation), taking into accountvarious processes of rotational damping and excitation;(2) to derive the degree of alignment of small grains thatreproduces observed polarization curves of the different λ max ; and (3) to employ the inferred degree of align-ment combined with the theoretical predictions to esti-mate the strength of interstellar magnetic fields. Thepaper is structured as follows.In §
2, we describe the basic assumptions and principaldynamical timescales involved in the alignment problem. Li & Greenberg (2003) listed many other candidates that havebeen proposed as carriers of the 2175˚A feature, including amor-phous carbon, graphitized (dehydrogenated) hydrogenated amor-phous carbon (Hecht 1986), nano-sized hydrogenated amorphouscarbon (Schnaiter et al. 1996), quenched carbonaceous compos-ite (Sakata et al. 1995), coals (Papoular et al. 1995), and OH ionin low-coordination sites on or within silicate grains (Duley et al.1989). In § § §
5, we describe a numerical method to com-pute the degree of grain alignment using the Langevinequations and present the obtained results for both sili-cate and carbonaceous grains. In § § § §
9, respectively. ASSUMPTIONS AND DYNAMICAL TIMESCALES
Grain geometry
We consider oblate spheroidal grains with the momentsof inertia I > I = I along the grain’s principal axesdenoted by ˆ a , ˆ a and ˆ a . Let I k = I and I ⊥ = I = I .They take the following forms: I k = 25 M a = 8 π ρa a , (1)(2) I ⊥ = 4 π ρa a (cid:0) a + a (cid:1) , (3)where a and a = a are the lengths of semimajor andsemiminor axes of the oblate spheroid with axial ratio r = a /a >
1, and ρ is the grain material density. Afrequently used parameter in the following, h = I k /I ⊥ ,is equal to h = 2 a a + a = 21 + s , (4)where s = 1 /r = a /a < a is defined as the radius of a sphere ofequivalent volume, which is given by a = (cid:18) π (4 π/ a a (cid:19) / = a s / . (5) Barnett relaxation
Barnett (1915b) first pointed out that a rotatingparamagnetic body can get magnetized with the mag-netic moment along the grain angular velocity. Later,Dolginov & Mytrophanov (1976) introduced the magne-tization via the Barnett effect for dust grains and con-sidered its consequence on grain alignment.The instantaneous magnetic moment due to the Bar-nett effect is equal to µ Bar = χ (0) ω γ g V = − χ (0) ~ Vg e µ B ω , (6)where V is the grain volume, γ g = − g e µ B / ~ ≈ − e/ ( m e c )is the gyromagnetic ratio of an electron, g e ≈ g − factor, µ B = e ~ / m e c ≈ . × − ergs G − is the This is an inverse of the Einstein-de Haas effect that was usedto measure the spin of the electron.
Hoang, Lazarian, & MartinBohr magneton. In the above equation, χ (0) is the zero-frequency paramagnetic susceptibility (i.e., at ω = 0),which reads χ (0) = 4 . × − f p (cid:18) T d
15 K (cid:19) − , (7)where T d is the grain temperature, and f p is the frac-tion of paramagnetic atoms (i.e., atoms with partially-filled shells) in the grain (see Draine 1996 and referencestherein). An extended discussion on the magnetic prop-erties of interstellar dust is presented in § ω coupledto µ Bar around the grain symmetry axis ˆ a producesa rotating magnetization component within the grainbody coordinates. As a result, the grain rotational en-ergy is gradually dissipated until ω becomes aligned withˆ a – an effect which Purcell termed ”Barnett relaxation”.Lazarian & Draine (1999a) (henceforth LD99a) revisitedthe problem by taking into account both spin-lattice andspin-spin relaxation (see Morrish 1980). Another inter-nal relaxation process discussed in Purcell (1979) is re-lated to imperfect elasticity of the grain material, whichwas expected to be important for grains of suprathermalrotation only (see e.g., Lazarian & Roberge 1997).Following LD99a, the Barnett relaxation time is de-fined as: τ Bar = γ g I k V K ( ω ) h ( h − J , (8)where K ( ω ) is related to the imaginary part of the mag-netic susceptibility χ ” as follows: K ( ω ) = χ ′′ e ( ω ) ω = χ (0) τ el [1 + ( ω τ el / ] (9) ≈ . × − s[1 + ( ω τ el / ] (10)where ω = ( h − J cos θ/I k is the precession frequencyof ω around ˆ a , and τ el is the relaxation time of electronicspins.For oblate spheroidal grains, we obtain τ Bar ≈ .
33 ˆ ρ a − ˆ s − / (cid:18) s . (cid:19) (cid:18) J d J (cid:19) × ˆ K − h ω τ el / i yr , (11)where a − = a/ − cm, ˆ s = s/ .
5, ˆ ρ = ρ/ − , τ el ∼ τ ∼ . × − f − p s with assumption of f p = 0 . K = χ (0) τ el / . × − s,and J d = p I k k B T d / ( h −
1) is the dust thermal angularmomentum. Although Purcell (1979) considered the grains havingboth electronic and nuclear spins, his study missed the ef-fect of internal relaxation related to nuclear spins. LD99a Due to a typo, the term ( ~ /g n µ N ) in Eq. 7 of LD99a shouldbe replaced by ( ~ /g e µ B ) ≡ /γ g which is applied for electronspins. The relaxation of electronic spins results from the spin-latticeand spin-spin relaxation, with time scales τ ≫ τ , so here weadopted τ el ∼ τ (Draine 1996). found that for astrophysical grains of realistic composi-tion nuclear spins induce a new type of relaxation, whichwas termed ”nuclear relaxation” by LD99a. This relax-ation process was shown to be dominant for large grainsbut it is negligible for small grains considered in this pa-per.Internal relaxation involves the transfer of grain ro-tational energy to vibrational modes. Naturally, if thegrain has nonzero vibrational energy, energy can alsobe transferred from the vibrational modes to grain ro-tational energy (Jones & Spitzer 1967). For an isolatedgrain, a small amount of energy gained from the vibra-tional modes can induce fluctuations of the rotationalenergy E rot when the grain angular momentum J is con-served (Lazarian 1994). Over time, the fluctuations in E rot establish a local thermal equilibrium (LTE).Using the rotational energy of oblate spheroid E rot = J (cid:2) h −
1) sin θ (cid:3) / I k , the fluctuations of the rota-tional energy can be described by the Boltzmann distri-bution (Lazarian & Roberge 1997): f LTE ( J, θ ) = A exp (cid:18) − J I k k B T d (cid:2) h −
1) sin θ (cid:3)(cid:19) , (12)where A is a normalization constant such that R π f LTE ( J, θ ) sin θdθ = 1.
Larmor precession of J around B A rotating paramagnetic grain can acquire a mag-netic moment due to the Barnett effect (Eq. 6) andthe Rowland effect if the grain is electrically charged(Martin 1971). The former is shown to be much strongerthan that arising from the rotation of its charged body(Dolginov & Mitrofanov 1976).The interaction of the grain’s magnetic moment dueto the Barnett effect with an external static mag-netic field B , governed by the torque [ µ Bar × B ] = −| µ Bar | B sin β ˆ φ ≡ I k ω sin βdφ/dt ˆ φ , causes the rapidprecession of J k ω around B . The period of such a Lar-mor precession denoted by τ B , is given by τ B = 2 πdφ/dt = 2 πI k ω | µ Bar | B = 2 πI k gµ B χ V ~ B ≈ . a − ˆ s − / ˆ ρ ˆ χ − ˆ B − yr , (13)where ˆ B = B/ µ G, ˆ χ = χ (0) / − . Measures of Alignment and Rayleigh ReductionFactor
Let G X be the degree of alignment of the axis of majorinertia ˆ a of the grain with its angular momentum J (i.e.,internal alignment) and G J be the degree of alignmentof J with the ambient magnetic field B (i.e., externalalignment, see Figure 14). They are respectively givenby G X = 12 (cid:0) θ − (cid:1) , (14) G J = 12 (cid:0) β − (cid:1) . (15)Since we are interested in the mean alignment of anensemble of grains with different orientations, the degreesof internal alignment and external alignment of grainsmall grain alignment and UV polarization 5are usually given by their ensemble averages, i.e., Q X = h G X i and Q J = h G J i .The net degree of alignment of the grain axis of majorinertia with the magnetic field, namely Rayleigh reduc-tion factor, is defined as R = h G X (cos θ ) G J (cos β ) i . (16)In the regime of efficient Barnett relaxation, the fastvariable θ can be separated from the slow variables J and β (Roberge 1997). Therefore, the internal alignment canbe described by the mean degree of alignment q X ( J ) = Z G X f LTE ( J, θ ) sin θdθ (17)and the Raleigh reduction factor becomes R = Z G J (cos β ) q X ( J ) f ( J x , J y , J z ) d J, (18)where the distribution of grain angular momentum f ( J )is used. ROTATIONAL DAMPING AND EXCITATIONPROCESSES
For typical and big interstellar grains, theoretical cal-culations show that the rotational damping by randomcollisions of the grain with gas atoms and molecules isdominant. For small grains under interest, in additionto the gas collisions, the damping is caused by variousprocesses, e.g., IR emission (Purcell 1969), interactionswith passing ions, electric dipole emission.Draine & Lazarian (1998) investigated in detail rota-tional damping and excitation processes for VSGs, in-cluding PAHs. They derived diffusion coefficients forplanar PAHs rotating around its symmetry axis. HDL10improved DL98 results and calculated the diffusion coef-ficients for planar PAHs with its rotation axis disalignedwith grain angular momentum. Here we deal with thealignment of small grains and VSGs of oblate spheroidalshape.
Rotational damping and excitation coefficients
We follow the definitions of rotational damping F andexcitation coefficients G from Draine & Lazarian (1998).The dimensionless damping coefficient for the j process, F j , is defined as the ratio of the damping rate inducedby that process to that induced by the collisions of gasspecies, τ − , assuming that the gas consists of purelyatomic hydrogen: F j = (cid:18) − dωωdt (cid:19) j (cid:18) τ − (cid:19) (19)and the excitation coefficient is defined as G j = (cid:18) Idω dt (cid:19) j (cid:18) τ H k B T gas (cid:19) , (20)where j =n, i, p and IR denote the grain collisions withneutral and ion, plasma-grain interactions, and IR emis-sion, (cid:0) Idω / dt (cid:1) j is the rate of increase of kinetic energyfor rotation along the axis that has moment of inertia I due to the excitation process j , T gas is the gas tempera-ture. For an uncharged grain in a gas of purely atomichydrogen, F n = 1 and G n = 1. To calculate the damping and excitation coefficientsfor wobbling grains, we follow the same approach as inHDL10, where the parallel components F j, k and G j, k ,and perpendicular components F j, ⊥ and G j, ⊥ with re-spect to ˆ a are computed using the general definitions(Equations 19 and 20). The only modification is the mo-ments of inertia I k and I ⊥ , which are given by Equations(1) and (3) for oblate spheroid instead of those for disk-like grains in HDL10.For example, the characteristic damping times of anoblate spheroidal grain with s = a /a < a are respectively given by τ H , k = 3 I k √ πn H m H v th a Γ k , (21) τ H , ⊥ = 3 I ⊥ √ πn H m H v th a Γ ⊥ , (22)where τ H , k ≡ τ H , z , τ H , ⊥ ≡ τ H , y = τ H , x with z the grainsymmetry axis, and x and y being the axes perpendicularto the symmetry axis (see Lazarian 1997). In the aboveequation, n H is the gas density, m H is the hydrogen mass, v th is the thermal velocity of hydrogen, and the geomet-rical factors Γ k and Γ ⊥ were derived in Roberge et al.(1993) and given in Appendix A.For the typical parameters of the ISM, Equations (21)and (22) become τ H , k ≈ . × ˆ ρ (cid:16) s . (cid:17) / a − × (cid:16) n H
30 cm − (cid:17) − (cid:18) T gas
100 K (cid:19) − / Γ − k yr (23)and τ H , ⊥ ≈ . × ˆ ρ (cid:16) s . (cid:17) / (cid:18) s . (cid:19) a − × (cid:16) n H
30 cm − (cid:17) − (cid:18) T gas
100 K (cid:19) − / Γ − ⊥ yr (24)Likewise, the characteristic damping times due to theelectric dipole emission from HDL10 can be rewritten as τ ed , k = 3 I k c k B T gas µ ⊥ , (25) τ ed , ⊥ = 3 I ⊥ c k B T gas (cid:16) µ ⊥ / µ k (cid:17) , (26)where µ k and µ ⊥ are the components of the electricdipole moment µ parallel and perpendicular to the grainsymmetry axis. Here we assume an isotropic distributionof µ , which corresponds to µ k = µ ⊥ / µ / µ is given by Equation (11) in Draine & Lazarian (1998). Relative importance of the different interactionprocesses
Depending on environment conditions, the dampingand excitation process by gas-dust interactions (i.e., col-lisions and plasma drag) or IR emission dominates. Forsmall grains, in the hot diffuse ISM, including warm neu-tral medium (WNM), warm ionized medium (WIM), or Hoang, Lazarian, & Martinin reflection nebula with strong radiation, the dampingby IR emission is the most important process. In the coldneutral medium (CNM) and molecular clouds where gasdensity is higher and starlight photons are shielded, thedamping by gas-dust interactions dominate. For ultra-small grains (e.g., PAHs), electric dipole emission inducesthe most significant damping (see Draine & Lazarian1998 for detailed discussion).Table 1 presents physical parameters for idealized envi-ronments where χ = u rad /u ISRF is the ratio of radiationenergy density u rad to the mean radiation density forthe diffuse interstellar medium u ISRF (see Mathis et al.1983), n (H ) , n (H + ) , n (M + ) are the molecular hydrogendensity, ion hydrogen density and ionized metal density,respectively. TABLE 1Idealized Environments For Interstellar Matter
Parameters
CNM WNM WIM n H (cm − ) 30 0.4 0.1 T gas (K) 100 6000 8000 χ x H = n (H + ) / n H x M = n (M + ) / n H y = 2 n (H ) /n H
0. 0. 0. PARAMAGNETIC ALIGNMENT MECHANISM FORSMALL GRAINS
Davis-Greenstein Paramagnetic Relaxation
A classical mechanism of grain alignment basedon paramagnetic relaxation was proposed byDavis & Greenstein (1951). The underlying idea ofthe mechanism is that, a paramagnetic grain gets mag-netized with an instantaneous magnetization M parallelto the induced magnetic field. If the grain angularmomentum makes an angle β with B , then B can bedecomposed into the parallel B k and perpendicular B ⊥ to J . Since the paramagnetic material gets magnetizedinstantaneously in response to the induced magneticfield, the magnetization component M k parallel to J remains constant during the grain rotation, whilethe perpendicular component M ⊥ , fixed to the labsystem, is rotating with respect to the grain body. Asa result, the rotating magnetization experiences energydissipation, which results in the gradual alignment of J with B .Due to the magnetic dissipation, the angle between J and B decreases as I k ω dβdt = − K ( ω ) V B ω sin β cos β, (27)where K ( ω ) = χ ′′ ( ω ) /ω with χ ”( ω ) being the imaginarypart of complex magnetic susceptibility of the grain ma-terial at the rotation frequency ω . In deriving the aboveequation, ω and ˆ a are assumed to be aligned with J dueto fast internal relaxation. Equation (27 can be rewritten as dβdt = − sin β cos βτ DG , (28)where τ DG = I k K ( ω ) V B (29)is the characteristic timescale of paramagnetic alignment.For the normal paramagnetic material, τ DG can bewritten as τ DG = 2 ρa K ( ω ) B ≈ . × ˆ ρ ˆ s − / a − (cid:18) B µ G (cid:19) − (cid:18) . × − s K ( ω ) (cid:19) yr . (30)Jones & Spitzer (1967) employed the Fokker-Planckequations to compute the degree of alignment of angularmomentum Q J in the magnetic field subject to the gasatom bombardment. Their obtained value Q J is equalto Q J = q ( x ) , (31)where x = (cid:18) T av T gas − (cid:19) = (cid:18) δ δ × T d − T gas T gas (cid:19) (32)with δ = τ gas /τ DG and τ gas = τ H , k . Here T av is regardedas the rotational temperature, and q ( x ) takes the follow-ing form: q ( x ) = −
13 + 1 x "(cid:18) xx (cid:19) / arcsinh √ x − , (33)for x >
0. For x <
0, the term x − / arcsinh √ x is re-placed by ( − x ) − / arcsin √− x , hence q ( x ) = −
13 + 1 x "(cid:18) x − x (cid:19) / arcsin √− x − . (34)The degree of the internal alignment for the case T d = T gas (i.e., the distribution of angular momen-tum is Maxwellian) is equal to (Jones & Spitzer 1967;Lazarian & Roberge 1997) Q X, Mw = 32 (1 − h − ) (cid:20) − √ h − − h − ) (cid:21) − . (35)Using Monte Carlo simulations, Purcell & Spitzer(1971) showed that Equation (31) provides a good agree-ment with their numerical calculations. The paramag-netic alignment of oblate grains was studied analyticallyin Lazarian (1997) and numerically in RL99, accountingfor the Barnett relaxation. Magnetic properties of interstellar dust
Following Draine & Lazarian (1999), the criticallydamped susceptibility is given by χ ”( ω ) = ωK ( ω ) with K ( ω ) = χ (0) τ [1 + ( ωτ / ] , (36)mall grain alignment and UV polarization 7where χ (0) is the magnetic susceptibility at the zero ro-tation frequency. Using the Curie’s law for paramagneticmaterial, we have χ (0) = n p µ k B T d , (37)where the effective magnetic moment µ reads µ ≡ p µ = g e µ B [ J ( J + 1)] = γ g (cid:2) ~ J ( J + 1) (cid:3) , (38)with J being the angular momentum quantum numberof electrons in the outer partially filled shell and p ≈ . In Equation (36), τ is the spin-spin relaxation time,which is equal to the precession time of the grainmagnetic moment µ around the magnetic field H i =3 . n p µ B : τ = ~ g e pµ B H i ≈ ~ . n p g e pµ B ≈ . × − (cid:18) . f p (cid:19) (cid:18) cm − n tot (cid:19) s , (39)where n p = f p n tot is the number density of paramagneticatoms and n tot ≈ cm − is the total atomic numberdensity within the grain (Draine 1996). Amorphous silicate grains usually contain Si, Mg, Fe,and O atoms. Assuming the silicate material with struc-ture MgFeSiO containing Fe ( S / ), the fraction ofparamagnetic atoms is f p = 1 / ≈ .
1. The magne-tization is induced by electrons in the outer partiallyfilled shell of Fe ion having the structure S / . Using γ g (Fe) = − g e µ B / ~ = − . × s − G − , one can esti-mate the static magnetic susceptibility for silicate grainsas follows: χ sil (0) ≈ . × − (cid:18) f p . (cid:19) (cid:16) p . (cid:17) (cid:16) n tot cm − (cid:17) (cid:18) T d
15 K (cid:19) − . (40)Plugging in Equation (40) into (36), one obtain K sil ( ω ) ≈ . × − (cid:18) T d
15 K (cid:19) − ωτ / ] s . (41)From Equation (39) and (41) one can see that fortypical interstellar grains ( a > . µ m) rotating with ω ∼ ω th = (cid:0) k B T gas /I k (cid:1) / = 1 . × a − / − ˆ T gas s − ,the term ωτ ≪
1. Thus, it is disregarded in earlierstudies on paramagnetic alignment of interstellar grains(e.g., Lazarian 1997; Roberge & Lazarian 1999). On theother hand, small grains ( a ≤ . µ m) are expectedto spin rapidly with ω > s − . Thus, the term ωτ becomes important, and the paramagnetic relaxation issuppressed due to the decrease of K ( ω ). For VSGs thatrotate extremely fast of ω > s − , K ( ω ) is substan-tially reduced. Thus, VSGs cannot be aligned by theclassical D-G paramagnetic relaxation.For ultrasmall carbonaceous grains or PAHs , the mag-netization arises from the presence of free radicals, Draine (1996) presented the total magnetic moment as µ = pµ B with p = 5 .
9. One can see that for Fe (6 S / ) ion with S = 5 / , L = 0 and J = 5 / g e ≈
2, one obtain g e J ( J + 1) ≡ p = 5 . paramagnetic carbon rings, and captured ions (seeLazarian & Draine 2000 and references therein). Follow-ing Lazarian & Draine (2000), we take f p = 0 .
01 corre-sponding to n p = 10 cm − for the typical atom numberdensity n tot = 10 cm − .For graphite grains , known as diamagnetic material,the magnetization originates from the attachment of Hatoms to the grain through hydrogenation. Since an Helectron is already used to make a covalent bond witha C atom, the magnetization is only produced by theH nucleus (proton). The gyromagnetic ratio for the Hnucleon is γ g (H) = g n µ N / ~ ≈ . × s − G − where g n ≈ .
59 and µ N = e ~ / m p c ≈ . × − ergs G − ,which is three orders of magnitude smaller than that ofa Fe atom present in silicate grains. Plugging in J = 1 / γ g (H) into Equation (37) we obtain χ gra (0) ≈ . × − (cid:18) f p . (cid:19) (cid:16) n tot cm − (cid:17) (cid:18) T d
15 K (cid:19) − , (42)where f p is the fraction of H atoms. If f p (H) is too small( ≪ . C that has f p ( C) ≈ .
01 (see also LD99a).The function K ( ω ) for graphite grains is given byEquation (36) but the spin-spin relaxation time τ nowis replaced by the nuclear relaxation time τ n with τ − n = τ − ne + τ − nn . Following LD99a, τ ne and τ nn are given by τ ne = ~ g e . n e g n µ ≈ × − (cid:18) . g n (cid:19) (cid:18) cm − n e (cid:19) s , (43) τ nn = ~ . g n n n µ ≈ . τ ne (cid:18) n e n n (cid:19) . (44)Plugging in the above equation into Equation (36), oneobtain K gra ( ω ) ≈ . × − (cid:18) T d
15 K (cid:19) − ωτ n / ] s , (45)for n e = n n = f p n tot = 10 cm − , assuming f p = f p (H) = 0 . Since τ n ≫ τ , one can see that K gra ( ω ) ≪ K sil ( ω ).Indeed, for a grain of a = 10 − cm rotating at thethermal velocity ω th ∼ s − , Equation (45) yields K gra ( ω th ) ≈ − s, compared to K sil ( ω th ) ≈ − sfor silicate grains. Thus, the paramagnetic alignment ofgraphite grains is rather inefficient. Resonance Paramagnetic Relaxation
The traditional treatment of the paramagnetic mag-netization by the Barnett effect within a rotating bodyrelies on the following assumption: the magnetizationwithin a rotating body in a static magnetic field is equiv-alent to the magnetization of a body at rest in a rotatingambient magnetic field. This assumption was adoptedin Davis & Greenstein (1951). LD00 realized that theabove treatment of paramagnetic relaxation is not exactbecause it neglects the splitting of rotational energy lev-els. They pointed out that the Barnett effect can help the Jones & Spitzer (1967) suggested that due to nuclear paramag-netism, the lower bound for interstellar grains K ( ω ) ∼ − /T d sregardless of their composition. Hoang, Lazarian, & Martinparamagnetic dissipation to occur resonantly at a maxi-mum rate thanks to the splitting of energy levels. Such anew effect, termed by LD00 the resonance relaxation, canoccur whenever the grain rotates in the ambient magneticfield.Assuming the critically damped balance(Draine & Lazarian 1999), LD00 found that K ( ω ) = χ (0) τ γ g e τ τ H sin θ , (46)where γ = e/ m e c (e.g., γ = γ e /g e ), τ is the spin-latticerelaxation time. Their estimate yields γ g e τ τ H sin θ ≈ (cid:16) τ s (cid:17) (cid:18) τ × − s (cid:19) (cid:18) H µ G (cid:19) (cid:18) sin θ / (cid:19) . (47)Following LD00, the spin-lattice relaxation time ofdust grains at a temperature T d , τ ( T d ) is given by τ ( T d ) τ , ∞ (77 K) ≈ (cid:18)
77 K T d (cid:19) m +1 (cid:18) T d T l (cid:19) m exp (cid:18) T d T l (cid:19) m ! ζ ( m ) , (48)where ζ ( m ) is the Riemann zeta function for m = 6 or m = 8, and T l is the lowest grain vibrational tempera-ture, which is equal to T l = ~ ω min k B ≈ (cid:18) − cm a (cid:19) K , (49)and τ , ∞ (77 K) ≈ − for the spin-lattice relaxation.The uncertainty of the resonance relaxation arises fromuncertainties of the microphysics of spin-lattice relax-ation within VSGs. For such grains, LD00 used plausi-ble arguments, but the laboratory testing would be mostuseful.The timescale of magnetic alignment due to the D-Gand resonance paramagnetic relaxation is equal to τ m = min ( τ DG , τ res ) , (50)where τ DG and τ res are obtained by plugging in K ( ω )from Equations (36) and (46) into Equation (30), respec-tively. NUMERICAL CALCULATIONS OF DEGREE OFPARAMAGNETIC ALIGNMENT
Numerical Method
RL99 have statistically calculated the efficiency of D-Galignment mechanism for dust grains using the Langevinequations, which was first suggested for studies of graindynamics in Roberge et al. (1993). RL99 also took intoaccount the Barnett effect and internal thermal fluctua-tions. Here, we study the paramagnetic alignment usingthe same approach as in RL99 but account for a vari-ety of damping and excitation processes that are impor-tant for small grains, including the dust-gas collisions, IRemission, plasma drag, and electric dipole emission (seeHDL10, HLD11).Following HDL10, to study the alignment of the grainangular momentum J with the ambient magnetic field B , we solve Langevin equations for the evolution of J in time in an inertial coordinate system using the Euler-Maruyama algorithm. The inertial coordinate system is denoted by ˆ e ˆ e ˆ e where ˆ e -axis is chosen to be parallelto B . The Langevin equations (LEs) read dJ i = A i dt + p B ii dq i for i = x, y, z, (51)where dq i are the random variables generated from a nor-mal distribution with zero mean and variance h dq i i = dt , A i = h ∆ J i / ∆ t i and B ii = h (∆ J i ) / ∆ t i are drifting(damping) and diffusion coefficients defined in the in-ertial coordinate system.The drifting and diffusion coefficients in the referencesystem fixed to the grain body, A bi and B bij , are relatedto the damping and excitation coefficients as follows: A bi = − J bi τ gas , i = − J bi τ H , i F tot , i , (52) B bzz = B k = 2 I k k B T gas τ H , k G tot , k , (53) B bxx = B byy = B ⊥ = 2 I ⊥ k B T gas τ H , ⊥ G tot , ⊥ , (54)where F tot , i and G tot , ii for i = x, y, z (or ⊥ , k ) are thetotal damping and excitation coefficients from variousprocesses which are defined by Equations (19) and (20),and τ gas , i = F tot , i /τ H , i . Using the transformation of dif-fusion coefficients from the body system ˆ a ˆ a ˆ a to theinertial system ˆ e ˆ e ˆ e (see Appendix C), we obtain thedrifting and diffusion coefficients A i and B ii in the iner-tial system.To account for the magnetic alignment, we need to adda damping term − J x,y /τ m to the drifting coefficient A x,y and an excitation term B m ,xx = B m ,yy to the diffusioncoefficient B xx and B yy (see Appendix B).In dimensionless units, J ′ ≡ J/I k ω T, k with ω T, k ≡ (cid:0) k B T gas /I k (cid:1) / being the thermal angular velocity ofthe grain along the grain symmetry axis, and t ′ ≡ t/τ H , k ,Equation (51) becomes dJ ′ i = A ′ i dt ′ + p B ′ ii dq ′ i for i = x, y, z, (55)where h dq ′ i i = dt ′ and A ′ i = − J ′ i τ ′ gas , eff − J ′ i τ ′ ed , eff − J ′ i τ ′ m (1 − δ zi ) , (56) B ′ ii = B ii I k k B T gas τ H , k + T d T gas δ m (1 − δ zi ) , (57)where δ m = τ H , k /τ m , δ zi = 1 for i = z and δ zi = 0 for i = z , τ ′ gas , eff = τ gas , eff τ H , k , τ ′ ed , eff = τ ed , eff τ H , k , (58)where τ gas , eff and τ ed , eff are the effective damping timesdue to dust-gas interactions and electric dipole emissionthat result from transforming damping coefficients A i from the body system to the inertial system (see Eq.E4 in HDL10).Equations (55) together with (56) and (57) aresolved by the iterative method for N step with thetime step dt ′ . As in HDL10, we choose dt ′ =0 . /F tot , k , /G tot , k , τ ed , k /τ H , k ] and N step = 10 forall calculations. If the returning timestep dt ′ > − ,mall grain alignment and UV polarization 9then we take dt ′ = 10 − . At each time step, the an-gular momentum J and the angle β between J and B obtained from the Langevin equations are employed tocompute the degrees of grain alignment.Indeed, at each time step, the components of angu-lar momentum J x , J y and J z are computed. Then, wecalculate J and the angle β between J and B such ascos β = J z /J . We can calculate R as follows: R ≡ N step − X l =0 G X (cos θ ) G J (cos β ) N step . (59)Above, the angle θ is kept unchanged during the timeinterval of dt ′ , which is invalid when the fast internalrelaxation is assumed. Therefore, G X would be replacedby q X ( J ).In practical, the actual value R and its approximation Q X Q J have some correlation, which can be described by R = h G X G J i = Q X Q J (1 + f corr ) , (60)where f corr is a correlation factor (see RL99). The case f corr = 0 corresponds to no correlation, i.e., θ and β arecompletely independent. Davis-Greenstein Alignment of Thermally RotatingGrains
We first study the paramagnetic alignment of grainssubject to a single rotational damping and excitation pro-cess by gas bombardment as in RL99. In this case, grainsare expected to be rotating at thermal velocity.
Alignment with constant K ( ω ) As in RL99, we assume the magnetic susceptibility K ( ω ) to be constant by disregarding the term containing ω in Equation (41). This assumption is valid for typicalinterstellar grains that rotate thermally at ω ≪ τ − .We consider two cases of low ( T d = 4 K) and normal( T d = 20 K) grain temperature and a variety of the mag-netic field strength B for the CNM (see Table 1 for morephysical parameters). Oblate spheroidal grains with ax-ial ratio r = 2 and r = 1 . f p = 0 . Q J as a func-tion of δ m = τ gas /τ DG . Q J appears to increase with theincreasing δ m as expected. The analytical results fromJones & Spitzer (1967) for spherical grains are similar toour numerical results in the case T d = 4 K. As T d in-creases to T d = 20 K, our numerical result is a factor of1 . χ (0) decreases when the thermal fluc-tuations within the grain (i.e., T d ) increase. Effect of fast rotation
When the grain rotation frequency becomes compara-ble to 2 τ − , K ( ω ) decreases sharply according to Equa-tion (41), resulting in the decrease of the paramagneticalignment rate. For ultrasmall grains of 4˚A, the damping time by electric dipoleemission dominates with τ H /τ ed ∼ (see e.g., HDL10). At thissize, the resonance paramagnetic alignment occurs over τ D − G, res ∼ τ H . Therefore, the chosen timestep dt ′ remains valid for solving theLangevin equations. C N M Q J LEJS671.3xLE -0.5 0.0 0.5 1.0alog ( δ DG ) LEJS671.3xLE T d =4KT d =20K Fig. 1.—
Degree of grain alignment by the paramagnetic (D-G) relaxation for thermally rotating grains and constant K ( ω ) inthe CNM as a function of δ m = τ gas /τ DG for T d = 4 K (upperpanel) and T d = 20 K (lower panel). The solid lines show ournumerical results obtained by solving LEs, and dotted lines showthe analytical results predicted for T d → K from Jones & Spitzer(1967). To clearly see the effect of fast rotation on the degreeof paramagnetic alignment, we repeat calculations in theprevious subsection using K ( ω ) from Equation (41). Theobtained degrees of alignment Q J and R are shown inFigure 2 for the CNM. As shown, both Q J and R in-crease when a decreases from a = 0 . µ m to a ∼ . µ mduring which the grain still rotates slowly and the para-magnetic relaxation rate increases. Below a ∼ . µ m, Q J and R fall sharply as a result of the suppression ofparamagnetic relaxation when the grain spins sufficientlyfast, producing a peak alignment at this grain size. Resonance Paramagnetic Alignment ofSubthermally Rotating Grains
Below, we investigate the paramagnetic alignment bytaking into account additional damping and excitationprocesses due to the collisions with ions, electric dipoleemission, IR emission, and plasma drag. Due to theseinteraction processes, grains are expected to be rotatingsubthermally (i.e., ω < ω th , see HDL10). We first con-sider the alignment by the D-G relaxation and then byboth D-G relaxation and resonance relaxation. Silicate Grains
Figure 3 shows Q J and R due to the D-G relaxation forsilicate grains of axial ratio r = 2. Q J and R are shownin the upper and lower panels respectively. Similar toFigure 2, one can see the sharp decline of Q J and R at a ∼ × − µ m as a result of the suppression of para-magnetic relaxation due to fast rotation. In particular,one can see a substantial decrease of grain alignment of ∼ . µ m grains compared to Figure 2. This is a directconsequence the additional damping processes included,which make grains to rotate subthermally and hence de-crease the D-G alignment.Figure 4 shows Q J and R as functions of grain size a when the resonance relaxation is included for silicate0 Hoang, Lazarian, & Martin C N M Q J B=10 µ GB=5 µ GB=1 µ G10 -3 -2 -1 a (μm)0.020.040.06 R Fig. 2.—
Degrees of grain alignment by the D-G relaxation forthermally rotating grains with K ( ω ) changing with ω . Upper andlower panels show Q J and R as functions of a . Constant graintemperature T d = 20 K and three magnetic field strengths for theCNM are assumed. Results for silicate grains with axial ratio r = 2are shown. CNM0.000.0 Q J -3 -2 -1 a (μm)0.010.020.030.040.05 R B=20µGB=10µGB=5µGB=1µG
Fig. 3.—
Similar to Figure 2 but including various interactionprocesses (e.g., IR emission, plasma drag, and dipole emission).Upper and lower panels show Q J and R as functions of grain size a . grains with axial ratio r = 2 (upper panel) and r = 1 . a ∼ − µ m. Q J is similar for two cases of grain shapewhile R is smaller for the less elongated shape (lowerpanel) due to lower internal alignment. Some fluctua-tions in Q J and R can be seen for a > . µ m whenthey are as small as their numerical errors. (cid:0)✁(cid:0)(cid:0)(cid:0)✁(cid:0)✂(cid:0)✁✄(cid:0)(cid:0)✁✄✂(cid:0)✁☎(cid:0)(cid:0)✁☎✂✆✝✞ ✟ ✠✡☛ ☞✌✍✎ ✏✑ ✒☎(cid:0)✓ ✂✔✕ ✒ ✖✄(cid:0)✔✕✄✂✔✕✄(cid:0)✗✘ ✄(cid:0)✗✙ ✄(cid:0)✚✛✜ ✢✣✤✥(cid:0)✁(cid:0)(cid:0)(cid:0)✁(cid:0)✂(cid:0)✁✄(cid:0)(cid:0)✁✄✂(cid:0)✁☎(cid:0)(cid:0)✁☎✂✆✝✞ ✟ ✠✡☛ ✦ ✧ ★✦ ✧ ✩✪✫ Fig. 4.—
Degrees of grain alignment by both the D-G relaxationand resonance relaxation for silicate grains with axial ratio r = 2(upper) and r = 1 . T d = 20 K isconsidered. Resonance relaxation induces the peaks of alignmentaround 10 − µ m. µ G)10 − − − R , Q J , Q X a = 0.01 µ mCNM, T d =20K Q J Q X R Fig. 5.—
Dependence of Q X , Q J and R as functions of the mag-netic field strength for the 0 . µ m silicate grains of T d = 20 K. From the figure it can also be seen that the small ( ∼ . µ m) grains are aligned much less efficient than theultrasmall ( ∼ − µ m) grains if they have the same tem-perature T d . In fact, the temperature of ultrasmall grainsis expected to be transient with temperature spikes dueto UV heating, which decreases their alignment signifi-cantly. The temperature of the a > . µ m grains in theISM is estimated at T d ≈
20 K, thus from Figure 4, wecan see that the paramagnetic alignment is rather smallwith
R < .
05 for B ≤ µ G.Figure 5 shows the increase of Q X , Q J and R with B for the 0 . µ m silicate grains of T d = 20 K in the CNM.mall grain alignment and UV polarization 11 R , Q J RQ J G = B10 µ G15 µ G CNM, PAHs20K60K − − a (μm)0.000.010.020.030.040.050.060.07 R , Q J RQ J Fig. 6.— Q J and R of very small carbonaceous grains with axialratio r = 2 predicted for different B . Grain temperature T d = 20 K(upper) and 60 K (lower) are considered. As shown, Q J and R increase rapidly with the increasing B whereas Q X declines slowly with B . Carbonaceous Grains
Figure 6 shows Q J and R computed for very small car-bonaceous grains (i.e. PAHs) with the axial ratio r = 2.Two grain temperature T d = 20 K and 60 K are consid-ered. As shown, Q J and R vary with the grain size a inthe same trend as silicate grains, although the degreesof alignment of PAHs are slightly lower than those pre-dicted for silicate grains of the same T d (see upper panel)due to the lower value of f p . The degrees of alignmentare subtantially decreased when the temperature is in-creased from 20 K to 60 K.One interesting feature for the higher T d case is that Q J starts to rise at a ≈ B xx, m . Forlarge grains, the excitation by other processes dominate,but when a ∼ Paramagnetic alignment in the WIM
Figure 7 shows the degrees of alignment of grains theWIM. It can be seen that the efficiency of paramagneticalignment in the WIM is higher than in the CNM. More-over, in contrast with the increase of Q J with the decreas-ing a from 0 . − . µ m in the CNM, Q J decreases or R , Q J RQ J WIM, r = 25µG = B10µG15µG10 − − 2 −1 a (μm)0.000.050.100.150.20 R , Q J RQ J T d =60K Fig. 7.—
Similar to Figure 4 but for the WIM. The efficiency ofmagnetic alignment in the WIM is higher than in CNM due to itslower gas damping rate. almost is flat for a in this range in the WIM. This is dueto the fact that, as a decreases, the ratio τ gas /τ DG doesnot increase as in the CNM because in the WIM the dom-inant contribution to the rotational damping arises fromIR emission, which has the timescale increasing with thedecreasing grain size a (see HDL10). OBSERVATIONAL CONSTRAINTS FOR ALIGNMENT OFSMALL GRAINS
In this section, we are going to derive the grain sizedistribution and degree of grain alignment as a functionof grain size (i.e., alignment function) that fit simultane-ously to observed extinction and polarization curves. Letus first start with a summary on observational results forthe starlight polarization.
Observed Polarization Curves of Starlight
Observational data in Serkowski et al. (1975) showthat the polarization of starlight can be described wellby an empirical law, usually referred to as the Serkowskilaw: p ( λ ) = p max exp (cid:20) − K ln (cid:18) λ max λ (cid:19)(cid:21) , (61)where K is a parameter, which depends on λ max (Wilking et al. 1980). Whittet et al. (1992) derived therelationship K = c λ max + c with c = 1 . ± .
09 and c = 0 . ± .
05 for most of sightlines.The observational data in Serkowski et al. (1975) alsoshow that the maximum polarization of starlight is con-strained by an upper limit p max < ∼ E B − V , (62)2 Hoang, Lazarian, & Martinwhich corresponds to p max < ∼ A V (63)for the typical diffuse ISM with R V = 3 .
1. For the gen-eral case, one expect that p max /A ( λ max ) < ∼ − .For some sightlines with low λ max (e.g., λ max < . µ m), there exist an excess UV polarization from theSerkowski law (Clayton et al. 1992; Clayton et al. 1995).The UV polarization for such sightlines can be describedby a modified-Serkowski relation (Martin et al. 1999): p UV = p max exp (cid:20) − K UV ln (cid:18) λ max λ (cid:19)(cid:21) , (64)where K UV = (2 . ± . λ max + ( − . ± . p max /A ( λ max ) from the up-per limit can arise from fluctuations of the magnetic fielddirection from the perpendicular direction, the variationof the degree of grain alignment along the sightline, andthe variation of grain properties (composition, shape).For instance, in molecular clouds, the decline of polariza-tion efficiency p max /A ( λ max ) can be explained by the de-cline of the degree of grain alignment by radiative torqueswhen going deeper into the cloud (Cho & Lazarian 2005;Whittet et al. 2008) or by the effect of magnetic turbu-lence (Jones et al. 1992). The question is what is the im-print of the variation of the strength of magnetic fieldson the polarization curves, provided that small grains areweakly aligned by paramagnetic relaxation?
Theoretical Considerations for Alignment Function
Recent advances in grain alignment theory allow usto predict the alignment of a variety of interstellar dustpopulation, ranging from ultrasmall grains of a fewAngstroms to micron-sized grains. As shown in Section5, ultrasmall and small grains can be aligned weakly byresonance paramagnetic and D-G paramagnetic relax-ation while large grains are believed to be aligned ef-ficiently by RATs. The grain size at which the RATalignment starts to dominate is given by a ali , which isusually referred to as the critical size of aligned grains(see e.g., Hoang & Lazarian 2014).For the diffuse interstellar radiation field (ISRF,see Mathis et al. 1983), the value a ali is determinedby the maximum angular momentum induced byRATs, which is equal to (see Hoang & Lazarian 2008;Hoang & Lazarian 2014): J RATmax J th = (cid:18)Z Γ λ dλ (cid:19) τ drag J th , (65) ≈ γ rad ˆ ρ / a / − (cid:18)
30 cm − n H (cid:19) (cid:18)
100 K T gas (cid:19) × (cid:18) ¯ λ . µ m (cid:19) (cid:18) u rad u ISRF (cid:19) (cid:18) Q Γ − (cid:19) (cid:18)
11 + F IR (cid:19) , (66)where τ drag = τ gas / (1 + F IR ) with F IR being the dampingcoefficient due to IR emission, ˆ γ rad = γ rad / . γ rad the anisotropy degree of radiation field, and¯ λ = R λu λ dλu rad , (67) Q Γ = R Q Γ λu λ dλλu rad , (68) are the wavelength and RAT efficiency averaged over theentire radiation field spectrum, respectively. For grainsof a ≪ λ in the ISM, Q Γ is approximately equal to Q Γ ≈ (cid:18) λa (cid:19) − . ≈ . × − (cid:18) λ . µ m (cid:19) − . a . − . (69)For the ISRF of λ = 1 . µ m, the above equationsyield a critical size (i.e., size for which J RATmax = 3 J th )of aligned grains a ali ≈ . µ m. As shown previously(e.g., Cho & Lazarian 2005; Hoang & Lazarian 2009b),the value a ali becomes larger for grains located deeperin molecular clouds (i.e., larger A V ). Thus, grains largerthan a ali are aligned efficiently by RATs while smallergrains ( a < a ali ) should be aligned weakly by the para-magnetic relaxation.The degree of alignment R of the a > a ali grainstends to increase with increasing a due to the increase of J RATmax (i.e., less affected by randomization by gas bom-bardment). For small grains ( a < a ali ) that are beingaligned by the paramagnetic relaxation, our computedresults show that R decreases with the decreasing a (seeFigures 4). The alignment of ultrasmall silicate grains( a < × − µ m) is peaky, but their contribution tothe UV polarization is negligibly small. As a result, thealignment function of silicate grains that are importantfor producing the polarization curves is expected to in-crease with the increasing a . Observationally Inferred Grain Size Distributionsand Alignment Functions
To explore the variation of the alignment function f ( a ) with λ max , we will find the best-fit models by fit-ting our theoretical models p mod and A mod (EquationsD7 and D8) to the observed polarization curves with λ max = 0 . µ m , . µ m and 0 . µ m. The observedpolarization curves are calculated using Equations (61)(for optical and IR wavelengths) and (64) (for UV wave-lengths), taking the mean values of K and K UV . Theobserved extinction curves are calculated using the ex-tinction law (Cardelli et al. 1989; O’Donnell 1994) for R V = 3 .
1. The search for best-fit models is performedby minimizing an objective function χ (see Appendix Efor detail). We consider N λ = 100 bins of the wavelengthfrom λ = 0 . − . µ m and N a = 100 bins of grain sizefrom a = 3 . µ m. We aim to perform the fit-ting for the case of maximum polarization efficiency, i.e., p max /A ( λ max ) = 3%mag − .We adopt a mixture dust model consisting of amor-phous silicate grains, graphite grains and PAHs (seeWeingartner & Draine 2001 ; Draine & Li 2007). Sinceobservational evidence for alignment of graphite is stillmissing, we conservatively assume that only silicategrains are aligned while carbonaceous grains are ran-domly oriented. Oblate spheroidal grains with axial ratio r = 2 as in Kim & Martin (1995) and r = 1 . n ( a ) that best reproduces the observationaldata for the typical ISM, which corresponds to model 3in Draine & Fraisse (2009). By doing so, we assume thatdust properties are similar throughout the ISM and thedifference in the polarization of starlight is mainly due tomall grain alignment and UV polarization 13the efficiency of grain alignment, which depends on en-vironment conditions along the sightlines, e.g., radiationfield, magnetic fields and gas density. We take the align-ment function for the ISM from Draine & Fraisse (2009)as an initial alignment function.One particular constraint for the alignment func-tion is that, for the maximum polarization efficiency p max /A ( λ max ) = 3%mag − , we expect that the condi-tions for alignment are optimal, which corresponds tothe case in which the alignment of big grains can beperfect, and the magnetic field is regular and perpen-dicular to the sightline. Thus, we set f ( a = a max ) =1. For a given sightline with lower p max /A ( λ max ), theconstraint f ( a = a max ) should be adjusted such that f ( a = a max ) = (1 / p max /A ( λ max ). As discussed in Sec-tion 6.2, we expect the monotonic increase of f ( a ) versus a , thus a constraint for this is introduced. Other con-straints include the non-smoothness of dn/da and f ( a )(see Draine & Allaf-Akbari 2006).The nonlinear least square fitting is carried out usingthe Monte Carlo direct search method. Basically, foreach size bin, we generate N rand random samples in therange [ − ζ, ζ ] from a uniform distribution for f ( a ) and n ( a ), α f and α n , respectively. The new values of f and n are given by ˜ f = ( α f + 1) f ( a ) and ˜ n = ( α n + 1) n ( a ).Then we calculate p mod and A mod for the new values ˜ f and ˜ n using Equations (D7) and (D8). The values of χ obtained from Equation (E1) are used to find the mini-mum χ . The range [ − ζ, ζ ] of the uniform distributionis adjusted after each iteration step. Initially ζ = 0 . χ is small) ζ is decreased to ζ = 0 . χ after one step: ǫ = ( χ ( n, f ) − χ (˜ n, ˜ f )) /χ ( n, f ). If ǫ ≤ ǫ with ǫ suffi-ciently small, then the convergence is said to be achieved(see also Hoang et al. 2013). With the value ǫ = 10 − adopted, the convergence is slow for some sightlines, thenwe stop the iteration process after 60 steps.Figure 8 (upper panel) shows the extinction cross sec-tion σ ext as a function of λ − for our best-fit models andthe observed extinction curve with R V = 3 .
1, assum-ing oblate spheroidal grains with axial ratio r = 2. Thelower panel shows σ pol for our best-fit models and the ob-served polarization curves of different λ max . As shown,our models provide an excellent fit to the observationaldata in all cases of λ max .Figure 9 (upper panel) shows the mass distributions ∝ a dn/da that reproduce the best-fit models in Figure8. From the figure, one can see that our best-fit massdistributions of silicate grains have three peaks at a ≈ . µ m , .
07 and 0 . µ m. The mass of small grains inthe range a = 0 . − . µ m is higher for lower λ max .Figure 9 (lower panel) shows the alignment functions f ( a ) for our best-fit models. One can see that the a > . µ m grains are efficiently aligned with f ( a ) > . f ( a ) drops rapidly for a < . µ m. Interest-ingly, a prominent transition from efficient alignment toweak alignment occurs at a ∼ . − . µ m for all threecases of λ max , suggesting that this can be indicative of λ − ( µ m − )10 − − − σ e x t ( c m H − ) r = 2 obs. λ max = 0.55 µ m λ max = 0.53 µ m λ max = 0.51 µ m λ − ( µ m − )10 − − σ po l ( c m H − ) r = 2 λ max = 0.55 µ m λ max = 0.53 µ m λ max = 0.51 µ m Fig. 8.—
Upper panel: observed extinction curve (symbols) of R V = 3 . λ max and oblate grains with axial ratio r = 2are considered. the change in the alignment mechanism (e.g., from RATalignment to paramagnetic alignment). Moreover, thealignment degree of typical interstellar ( a > . µ m)grains tends to shift to the range of smaller a as λ max decreases. In particular, as λ max decreases, the degreeof alignment of small grains a ∼ . − . µ m mustincrease considerably in order to reproduce the observedpolarization curves (see Figure 9, lower panel).Similar to Figures 8 and 9, Figures 10 and 11 show ourbest-fit models to the observed data for the case withaxial ratio r = 1 .
5. As shown, our models also pro-vide good fit to the observational data. The alignmentfunctions (see Figure 11, lower) exhibit the same fea-tures (e.g., transition from efficient to weak alignment)as those in the case r = 2. However, to reproduce theobserved data, small grains with r = 1 . r = 2 by afactor of ∼ . MEASURING MAGNETIC FIELDS USING THE UVPOLARIZATION
In this section, we employ the degrees of alignment(from theoretical calculations and best-fit models) andsize distributions obtained in the previous sections topredict theoretical polarization curves (see Section D fortheory) for the diffuse ISM of the different magnetic fieldstrengths.
Theoretical Polarization Curves
Since the fitting is performed for the case of maximumpolarization efficiency p max /A ( λ max ) for which the mag-4 Hoang, Lazarian, & Martin − − − a( µ m)10 − − − − ( π / ) a dn / da ( c m H − ) carbon s ili c a t e r = 2 λ max = 0.55 µ m λ max = 0.53 µ m λ max = 0.51 µ m − − − a( µ m)10 − − − f ( a ) r = 2 λ max = 0.55 µ m λ max = 0.53 µ m λ max = 0.51 µ m Fig. 9.—
Grain size distributions (upper panel) and alignmentfunctions (lower panel) of our best-fit models for oblate grains withaxial ratio r = 2. The dotted vertical line marks a = 0 . µ m.The size distribution appears quite similar for the different λ max ,whereas the alignment of small grains ( a ∼ . − . µ m) increaseswith the decreasing λ max . netic field should lie in the sky plane, the inferred align-ment function is then equal to the Raleigh reduction fac-tor, i.e., f ( a ) = R ( a ).In the previous section, we found that the best-fitmodel requires the increased alignment of small grainsas λ max decreases. Such increased alignment of smallgrains in general can arise from (i) the increase of mag-netic fields as calculated in Section 5.3 and (ii) the in-crease of RAT alignment due to enhanced radiation fieldby some hot stars in the vicinity of the sightline. In thelatter case, the excess thermal emission is expected sincedust is warmer due to higher radiation field. Below weconsider the first situation and leave the second one forthe discussion section.To explore the effect of paramagnetic alignment ofsmall grains on polarization curves, we distinguish thealignment of the typical interstellar grains with a ≥ a ali and that of smaller grains with a < a ali , which areexpected to be induced by RATs and the paramag-netic relaxation, respectively. Moreover, there is alwayssome intermediate range from the paramagnetic align-ment to RAT alignment. Thus we assume that grainswith a ≤ a mag (i.e. a mag < a ali ) are solely aligned byparamagnetic relaxation and take the degree of align-ment computed in Section 5 for the CNM of differentmagnetic field strengths. The degree of alignment ofgrains with a > a mag is taken from the best-fit alignmentfunctions. The precise value of a mag is uncertain, and wetake a mag ≈ . µ m, which is equal to the grain size at λ − ( µ m − )10 − − − σ e x t ( c m H − ) r = 1.5 obs. λ max = 0.55 µ m λ max = 0.53 µ m λ max = 0.51 µ m λ − ( µ m − )10 − − σ po l ( c m H − ) r = 1.5 λ max = 0.55 µ m λ max = 0.53 µ m λ max = 0.51 µ m Fig. 10.—
Similar to Figure 8 but for oblate grains of axial ratio r = 1 . − − − a( µ m)10 − − − − ( π / ) a dn / da ( c m H − ) carbon s ili c a t e r = 1.5 λ max = 0.55 µ m λ max = 0.53 µ m λ max = 0.51 µ m − − − a( µ m)10 − − − f ( a ) r = 1.5 λ max = 0.55 µ m λ max = 0.53 µ m λ max = 0.51 µ m Fig. 11.—
Similar to Figure 9 but for oblate grains with axialratio r = 1 . mall grain alignment and UV polarization 15which J RATmax /J th = 1 for the diffuse ISM, i.e, when theRAT alignment is negligible. Moreover, since large grainsare likely in thermal equilibrium with the ISRF whileVSGs are expected to undergo thermal spikes due tothe absorption of UV photons (Guhathakurta & Draine1989), we assume T d = 18 K for the a > T d = 60 K for very small ( a < σ pol produced by aligned silicategrains for the different values B for three selected λ max .Upper panels show results for the case axial ratio r = 2and lower panels show results for r = 1 .
5. Filled cir-cles show the observed polarization curves that are de-termined by λ max (see Section 6).From the figure, we can see that the polarization at λ − < µ m − remains similar when changing B , indi-cating that the polarization at these wavelengths is de-termined by the alignment of typical interstellar grains( a > . µ m). On the other hand, the polarization inthe UV ( λ − > µ m − ) increases with the increasingmagnetic field, which demonstrates that the alignment ofthe a < . µ m grains by the paramagnetic relaxationplays an important role for the UV polarization. The ris-ing feature of σ pol computed at λ − > . µ m − for somelarge B arises from the fact that the best-fit alignmentfunctions of small grains fall more rapidly with a thancomputed theoretically assuming a constant T d .For the case of r = 2, the theoretical curve with B = 10 µ G (dashed line, also indicated by the arrow) ap-pears to be in good agreement with the observed curveof λ max = 0 . µ m (panel (a)). The corresponding valuesare B ∼ − µ G for the cases with λ max = 0 . µ mand 0 . µ m (panels (b) and (c)). For the smaller ax-ial ratio r = 1 .
5, higher magnetic fields are required toreproduce the observed polarization curves in UV. Forinstance, B ∼ µ G for λ max = 0 . µ m (panel (d)) and B ∼ − µ G for two other cases ((e) and (f)).
Inferred Magnetic field Strengths
As shown in the preceding subsection, higher magneticfields are required to reproduce the observed UV polar-ization with lower λ max . To see clearly the dependenceof the UV polarization on B and λ max , we estimate theratio p (6 µ m − ) /p max for the different B and λ max .Figure 13 shows p (6 µ m − ) /p max as a function of λ − predicted for the different values of B . Square and cir-cle symbols show p (6 µ m − ) /p max calculated using Equa-tions (61)(Serkowski law) and (64) (modified Serkowskilaw) with the mean values of K and K UV . The mag-netic field of the ISM seems to be well constrained in therange B ∼ − µ G for λ max = 0 . − . µ m (upperpanel) if the grain axial ratio r = 2 is assumed. Forless elongated spheroids of r = 1 .
5, the range of mag-netic field is B ∼ − µ G for the same range of λ max (lower panel). Specifically, for λ max = 0 . µ m the mag-netic field strength is estimated at B ∼ µ G for axialratio r = 2, assuming the grain temperature T d = 18 K.Estimated magnetic fields for r = 1 . λ − > µ m − ,the estimated magnetic field tends to be lower becausethe slope of σ pol computed is shallower than the observedone (see arrows in Figure 12). DISCUSSION
Comparison to previous studies on paramagneticalignment
The paramagnetic relaxation was introduced byDavis & Greenstein (1951) to explain the alignment ofinterstellar grains with the Galactic magnetic field. Thefirst quantitative study of grain alignment by param-agnetic relaxation (i.e., Davis-Greenstein (D-G) align-ment) was carried out by Jones & Spitzer (1967) us-ing the Fokker-Planck (FP) equations. Purcell (1969)and Purcell & Spitzer (1971) studied the D-G alignmentby means of the Monte Carlo method and showed thatthis mechanism is inefficient in aligning the typical in-terstellar grains. Latter, Purcell (1979) suggested thatthe joint action of spin-up systematic (pinwheel) torquesthat can drive grains to suprathermal rotation and theparamagnetic relaxation could result in efficient align-ment of suprathermally rotating grains. However, the ef-ficiency of pinwheel torques is believed to be significantlysuppressed due to the rapid thermal flipping of smallgrains, and small grains are expected to be thermallytrapped (Lazarian & Draine 1999b; Hoang & Lazarian2009b). Therefore, we disregarded minor effects of pin-wheel torques on the alignment of small grains.For thermally rotating grains, Lazarian (1997) calcu-lated the paramagnetic alignment using an analyticalmethod based on the FP equations. RL99 have com-puted the efficiency of the D-G mechanism for thesegrains using the Langevin equations. Both papers tookinto account the Barnett relaxation effect and internalthermal fluctuations (an inverse process associated withthe Barnett relaxation). Nevertheless, these aforemen-tioned studies assumed a constant magnetic susceptibil-ity K ( ω ) and considered the rotational damping and ex-citation due to dust-gas collisions only. Such assump-tions are obviously valid for large ( > . µ m) grains thatrotate slowly in the absence of pinwheel torques. Theiressential conclusion is that the D-G mechanism is in-efficient in aligning the interstellar grains and failed toaccount for the observed polarization in the molecularclouds where the dust and gas are likely in thermal equi-librium.This paper investigates the paramagnetic alignmentfor a wide range of grains, from a few Angstroms to0 . µ m, using the Langevin equations (RL99; HDL10).This grain population is expected to rotate subthermallybut rapidly with ω > s − . We take into account thevarious damping and excitation processes that are essen-tial for the rotational dynamics of small grains, includ-ing gas-dust collisions, plasma drag, IR emission, andelectric dipole damping (see Draine & Lazarian 1998;Hoang et al. 2010). For small grains, we found that theefficiency of paramagnetic alignment is indeed rather lowdue to subthermal rotation; the degree of paramagneticalignment R increases with the magnetic field strength B , but R < .
05 for
B < µ G for the typical ISM con-ditions.Lazarian & Draine (2000) (LD00) have identified anew physical process, namely, resonance paramagneticrelaxation, which is shown to enhance the alignment ofultrasmall grains. The efficiency of resonance param-agnetic alignment was estimated at a level of 10% forthe 10˚A grains in LD00 where the idealized model ofspinning dust emission from Draine & Lazarian (1998)6 Hoang, Lazarian, & Martin (cid:0)✁✂✄☎(cid:0)✁✂✄✆✝✞✟✠✡☛☞✌ ✍✎✏ ✑ ✒✁✓✔ ✕ ✖(cid:0)✗✓✔(cid:0)✁✓✘✙✓✔✚✛✜✢✣ ✤✥✦✧ ★✩✪✙✙✫✬ ✒✁✓✔ ✕ ✖(cid:0)✗✓✔(cid:0)✁✓✔✗✓✔✚✛✜✢✣✤✥✦✧ ✕✁✢✗✭✓✮ ✒✁✫✔ ✕ ✖(cid:0)✗✓✔(cid:0)✁✓✘✙✫✔✚✛✜✢✣✤✯✰✱ ✕✁✢✗(cid:0)✓✮✲ ✳ ✴✩✪✵ ✶ ✶✩✷✸✹✺✓✮✸✹✻(cid:0)✁✸✄☎(cid:0)✁✂✄✆✝✞✟✠✡☛☞✼ ✍✎✽ ✾ ✭✁✫✔ ✕ ✖✒✗✓✔✒✁✓✔(cid:0)✗✫✔✚✛✜✢✣ ✷✥✦✧ ✕✁✢✗✗✫✬ (cid:0) (cid:0)✁✷✂✿✺✓✮✸✹❀✭✁✫✔ ✕ ✖✒✗✓✔✒✁✓✔(cid:0)✗✓✔✚✛✜✢✣✤✯✰✱ ✕✁✢✗✭✓✮ (cid:0) (cid:0)✁✤✂✹✺✓✬✂✹❀✭✁✓✔ ✕ ✖✒✗✓✔✒✁✓✔(cid:0)✗✓✔✚✛✜✢✣ ✤✯✰✱ ✕✁✢✗(cid:0)✓✮✲ ✳ ❁❂❃❄❅❆ ❄❇❆ ❄❈❆❄❉❆ ❄❊❆ ❄❋❆
Fig. 12.—
Polarization cross-section arising from aligned silicate grains predicted for various values of the magnetic field B versus theobserved polarizations (filled circles) with different λ max . Upper and lower panels show results for oblate spheroids with axial ratio r = 2and r = 1 .
5, respectively. The arrows indicate the theoretical curves that are close to the observational data. mall grain alignment and UV polarization 17 (cid:0)✁✂✄☎✆✝✞✟✠✝✡✟✠☛☞✌✍✎✏✑✍✎✒✍✍✎✒✑✍✎✓✍✍✎✓✑✍✎✑✍✔✕✖✗✘✙✚ ✛✜✢ ✣✤✥ ✍✎✑✑ ✍✎✑✒ ✍✎✑✦✧★✩✪✫✬✭✮✯✰✱✲✳✴✵✲✶✭✳✷✶✸✰✷ ✯✰✱✲✳✴✵✲✶✦✎✹✍ ✦✎✹✏ ✦✎✹✓ ✦✎✹✺ ✦✎✹✹ ✦✎✻✍ ✦✎✻✏ ✦✎✻✓ ✦✎✻✺✧✼✽★✩✪✫✾✭✼✿✮✍✎✏✑✍✎✒✍✍✎✒✑✍✎✓✍✍✎✓✑✔✕✖❀✘❁✚ ❂❃✔ ✣✤✥ ✯✰✱✲✳✴✵✲✶✭✳✷✶✸✰✷ ✯✰✱✲✳✴✵✲✶ ❄❅❆❇❈❉❊❋❈❉❊❇❈❉●❍■❏●❑■❏▲ ▼ ◆▲ ▼ ❖P◗
Fig. 13.—
Variation of p (6 µ m − ) /p max versus λ − predictedfor different B for spheroids with axial ratio r = 2 (upper) and r = 1 . p (6 µ m − ) /p max truncated by the predictions from the Serkowski and modified-Serkowski laws. (DL98) was adopted. The present work used an im-proved model of spinning dust emission from HDL10which accounts for the grain wobbling and quantified theefficiency of grain alignment by resonance paramagneticrelaxation. Our results in general confirmed the predic-tions by Lazarian & Draine (2000). The only differenceis that our results predict a lower grain size (about 10˚A)of the peak alignment than earlier predicted by LD00.This difference arises from the improved model of spin-ning dust that predicts lower rms grain angular momen-tum than the DL98 model. Excess UV polarization and Alignment of SmallGrains
The excess of continuum polarization in the UVwith respect to the Serkowski law, usually character-ized by p (6 µ m − ) /p max , was observationally reportedin Clayton et al. (1992) and Clayton et al. (1995) (seealso Martin et al. 1999). However, it is still unclear whysuch an excess UV polarization only exists for λ max < . µ m. To resolve this question, we first need to under-stand which grain population is responsible for the UVpolarization.
The original Serkowski law fits well to the observedpolarization at IR and optical wavelengths. At thesewavelengths, we showed that the polarization is mostlyproduced by typical silicate grains ( a > . µ m) aligned in the magnetic field (see Section 7). However, the po-larization in the UV arising from these relatively largegrains is insufficient to reproduce the observed polar-ization; the contribution of weakly aligned small silicategrains ( a < . µ m) allows us to successfully reproducethe UV polarization.If the excess UV polarization is indeed produced bysmall aligned grains, then why the alignment of thisgrain population increases, as it is required by higher p (6 µ m − ) /p max , in the cases λ max < . µ m?Compared to the typical polarization curve of the ISMwith λ max = 0 . µ m, we found that, for the cases with λ max < . µ m, the alignment function of grains tendsto shift to the smaller grain size, corresponding to the de-crease of critical size of aligned grains a ali . Thus, thereexists some additional alignment of intermediate size grains ( a = 0 . − . µ m) by the same alignment mecha-nism as typical interstellar grains (most likely driven byRATs), which gives rise to shift the polarization curveto the shorter λ . At the same time, we found that thealignment of small grains a < . µ m must be enhancedto reproduce the excess UV polarization. Thus, thereseems to exist some correlation between the alignmentof typical interstellar grains, which is most likely drivenby RATs, and the alignment of small grains. Below, wediscuss some possible reasons why this could happen.If the enhanced alignment of small grains is inducedby increased RATs due to nearby hot stars, then sucha correlation is obvious. However, some stars that havethe excess UV polarization do not exhibit excess thermalemission at 60 µ m (see Clayton et al. 1995). Interestingenough, the HD197770 star possesses an excess emissionat 60 µ m, but has actually a lower excess UV polarization(see Gaustad & van Buren 1993; Clayton et al. 1995).This indicates that dust along these sightlines with theexcess UV polarization is actually not hotter than thedust along the stars without the excess. In addition, theamount of dust near the stars may be rather small com-pared to the total dust mass along the entire sightline, assuggested in (Clayton et al. 1995). Furthermore, if theenhanced alignment of small grains is caused by RATs,then the sharp transition in the alignment function at a ∼ . µ m for the best-fit models is unexpected because f ( a ) should decrease monotonically from a = 0 . µ m to a ∼ . − . µ m as seen in the alignment functionobtained for HD 197770. Therefore, the enhanced align-ment of small grains by increased RATs may not be adominant reason for the excess UV polarization.If the enhanced alignment of small grains is induced byan increased magnetic field strength, then the correlationcan be due to the following reasons.First, the RAT alignment tends to increase with in-creasing magnetic field strength as paramagnetic align-ment. Indeed, in the RAT alignment paradigm, wefind that the increase of the paramagnetic relaxationcan result in the increase of the fraction of grainsaligned with high- J attractor points, which increasesthe degree of RAT alignment (Lazarian & Hoang 2007;Hoang & Lazarian 2008; Lazarian & Hoang 2008).Second, the grain randomization due to the elec-tric field acting on the electric dipole moment ofgrains that are accelerated by interstellar turbulence(Lazarian & Yan 2002; Yan & Lazarian 2003; Yan et al.2004; Yan 2009; Hoang et al. 2012) is found to decrease8 Hoang, Lazarian, & Martin(i.e., the degree of RAT alignment is increased) whenthe magnetic field is increased. The effect of such arandomization is described in Weingartner (2006) and(Jordan & Weingartner 2009). For a weak magneticfield, the randomization is thought to be more impor-tant because the rate of Larmor precession is lower thanthe rate of dipole fluctuations. As the magnetic fieldincreases, the RAT alignment is expected to increase be-cause the Larmor precession frequency becomes larger,reducing the randomization effect by dipole fluctuations.
Measuring Magnetic Fields using the UVPolarization
Magnetic fields are no doubt important for numer-ous astrophysical processes, including star formation,transport and acceleration of cosmic rays, and accretiondisks. Dust polarimetry proves being a useful techniqueto trace the magnetic field direction in molecular clouds,and when combined with the Chandrasekhar-Fermi (CF)technique (Chandrasekhar & Fermi 1953) one can mea-sure the magnetic field strength.While the variation of the local magnetic field direc-tion along a sightline is usually referred to explain whythe observed p max /A ( λ max ) is lower than its upper limit p max /A ( λ max ) = 3%mag − , the effect of the magneticfield strength on the polarization curve has not been ex-plored yet. The present study showed that the magneticfield strength can have important imprints on the ob-served polarization curves, particularly, it results in theexcess UV polarization for cases λ max < . µ m. Us-ing this subtle effect, we can estimate the strength ofinterstellar magnetic fields.Assuming the average ISRF and grain axial ratio r = 2, we find that, for the typical diffuse ISM with λ max = 0 . µ m, the magnetic field strength is esti-mated at B ∼ µ G. This magnetic strength appearsto be consistent with the Zeeman measurements (seeCrutcher 2012 for a recent review). For the sightlinewith λ max = 0 . µ m and λ max = 0 . µ m, the estimatedmagnetic fields are B ∼ µ G and B = 16 µ G (see Fig-ure 13, upper). Therefore, the magnetic field tends toincrease with the decreasing λ max . When the grain ax-ial ratio r = 1 . p max /A ( λ max )for different sightlines (see Planck Collaboration et al.2014). For some sightline having p max /A ( λ max ) < − but the same λ max and p (6 µ m − ) /p max as ourselected sightlines (i.e., p max /A ( λ max ) = 3%mag − ), themagnetic field strength would be similar to that withthe maximum p max /A ( λ max ) if we assume the increaseof p max /A ( λ max ) is due to the fluctuation of B and thatthe biggest grains can still be perfectly aligned. The We disagree with the conclusions of these studies, but acceptthe existence and potential importance of the randomization. reason for that is that the strength of B depends onthe Rayleigh reduction factor R , which is the same intwo sightlines while the effective degree of alignment f changes as f = R cos ξ . If both the fluctuations of B andunfavorable conditions of grain alignment responsible forlower p max /A ( λ max ), then the magnetic fields should belower than the magnetic fields estimated for the idealizedsightlines.One of the important implications of this study is thatit provides us a novel way to measure the strength of themagnetic field vector using three observational polariza-tion parameters p max , λ max and p (UV). This techniqueis more useful for the sight lines with low λ max becausethe UV polarization is not too low compared to the p max .The presented method allows us to obtain a constrainton the strength of the total magnetic field, which is moreadvantageous than other methods that return the pro-jected magnetic field only. It is also worth to mentionthat the usage of λ max as an input parameter for mea-suring B is cautious because of its complicated depen-dence on other parameters, including R V and A V (seeAndersson & Potter 2007).The present method for measuring magnetic fieldsmakes use of the polarization data in UV wavelengthrange from the Wisconsin Ultraviolet Photo-PolarimeterExperiment (WUPPE), which is below the atmosphericcut-off ( ∼ . µ m). Therefore, to apply this methodfor beyond WUPPE data set, new space/rocket missionswould be needed. Dependence of Inferred Magnetic fields on physicalparameters
There exists a number of parameters that appear toaffect the inferred magnetic field strength using the UVpolarization.First, grain geometry (i.e., asphericity) can affect theinferred magnetic fields. Our study considered two casesof oblate spheroidal grains with axial ratio r = 2 and r = 1 .
5. The latter grain shape has lower polarizationcross-section C pol , and the degree of alignment requiredto reproduce the observational data is higher, resultingin the stronger inferred magnetic fields.Second, the grain temperature of small grains may alsoplay an important role on the estimated magnetic field.Because the temperature of small dust grains determinesthe level of thermal fluctuations of grains axes with itsangular momentum, which constrains the degree of in-ternal alignment, our estimated magnetic field strengthsbased on the UV polarization should vary with the dusttemperature chosen. Nevertheless, the temperature ofsmall ∼ . µ m grains is expected to be nearly stablein thermal equilibrium (see Draine 2003), so we expectthe effect of grain temperature fluctuations plays a minorrole for constraints of B field.Third, the alignment of small grains is completelyattributed to the paramagnetic alignment. Indeed,the alignment may be enhanced due to the additionaleffect of pinwheel torques (e.g., H formation, seeAndersson et al. 2013).Fourth, our finding that the magnetic field tends to in-crease with the decreasing λ max is based on the assump-tion that the average ISRF (e.g., a ali ) is similar along thethree sightlines. This assumption is valid for most of thesightlines with the excess UV polarization but do not ex-mall grain alignment and UV polarization 19hibit excess thermal emission. For some sightlines withboth the excess UV polarization and thermal emission,the magnetic field required to reproduce the observedpolarization may not need to be increased.Finally, when the strength of magnetic field is known,we can constrain the grain physical properties, such asgrain geometry, using the UV polarization. Earlier stud-ies (Kim & Martin 1995; Draine & Allaf-Akbari 2006;Draine & Fraisse 2009) and our present work show that awide range of axial ratio of oblate spheroid can reproducethe observed extinction and polarization curves. How-ever, grains with a small/large axial ratio (i.e. less/moreelongated) will require a higher/lower degree of align-ment of small grains, which corresponds to higher/lowermagnetic fields, to reproduce the observed polarization.Thus, it is potential to constrain the grain geometrywhen the magnetic fields are known. Resonance Paramagnetic Alignment of ultrasmallGrains and Polarization of Spinning Dust Emission
Hoang et al. (2013) showed that the 2175˚A polariza-tion bump of HD 197770 can be reproduced successfullyby a model of aligned silicate plus weakly aligned PAHs.The alignment function for their best-fit model has peakof R cos ξ ≈ .
004 at ∼ p max /A V ( λ max ) < p max /A V ( λ max ) of this star, one obtain R ≈ . T d = 60 K, we found the peak alignment R ∼ .
006 for B ∼ µ G (see Figure 6), which is equal to thealignment degree for the best-fit model in Hoang et al.(2013).The question is why only HD 197770 posses the2175˚A polarization bump but other stars with the similar λ max do not?It is noted that the possibility to observe the2175˚A polarization bump depends on both the align-ment of PAHs and small silicate grains because the lat-ter is responsible for the UV continuum polarization at λ − > µ m − . If the alignment of small silicates is in-efficient, then the bump can be detected due to highcontrast. If the alignment of small silicate grains is con-siderable, the UV polarization produced by such grainstends to smooth out the bumpy polarization by PAHs,which makes the detection of 2175˚A bump more difficult.One interesting point in the polarization curve of HD197770 is that its excess UV polarization is much lowerthan other stars with the same λ max = 0 . µ m (seeClayton et al. 1995). On the other hand, the HD 197770has an excess emission at 60 µ m, indicating that the ra-diation field is higher than the averaged ISRF and thedust is hotter than the typical ISM. Since hotter dusttends to reduce the alignment of small grains, the UVcontinuum polarization is reduced as well, favoring thedetection of the 2175˚A polarization bump.A related issue is the alignment of carbonaceous grainsand its consequence. PAHs are thought to have attach-ment of aliphatic structures to its surface, producinglarge carbonaceous grains (Kwok et al. 2011). However,the idea that PAHs can be weakly aligned by resonance relaxation seems not to contradict with the unpolarized3 . µ m aliphatic features (Chiar et al. 2006). Indeed, ifthere is attachment of aliphatic structures to a PAH, thenet size of aliphatic-PAH grain will increase, which makesthe grain to rotate slower, assuming the same gas tem-perature. As a result, the alignment of the aliphatic-PAHgrain by resonance relaxation would become negligible.The alignment of large carbonaceous grains by radiativetorque may also be inefficient as discussed in a recentreview by Lazarian et al. (2014). Relating the UV polarization of starlight tospinning dust polarization
Based on the UV polarization of starlight, one can inferthe degree of alignment of small grains. Since the align-ment of small grains and ultrasmall grains is most likelyinduced by the same paramagnetic mechanism, we canderive the alignment of ultrasmall grains. Then, the po-larization of spinning dust can be constrained using theinferred degree of alignment of VSGs (see Hoang et al.2013). SUMMARY
We calculated the degree of grain alignment by theDavis-Greenstein relaxation and resonance paramagneticrelaxation for subthermally rotating grains, and sug-gested a new way to constrain magnetic field strengthusing UV polarimetry. Our principal results can be sum-marized as follows.1. The degrees of grain alignment by paramagnetic re-laxation (classical Davis-Greenstein and resonanceone) were calculated for both small grains ( a ∼ . µ m) and ultrasmall grains ( a ∼ . µ m). Wefound that the alignment of small grains is domi-nated by the D-G relaxation while the alignmentof ultrasmall grains is dominated by the resonancerelaxation. The degree of alignment for normalparamagnetic material in the typical ISM is ratherlow, e.g. a few percent. For the same tempera-ture, ultrasmall grains appear to be more efficientlyaligned than small grains, with the peak alignmentaround 10˚A due to the resonance relaxation. Whenaccounting for the fact that the temperature of ul-trasmall grains is higher with strong fluctuations,the degree of alignment of ultrasmall grains is re-duced.2. We derived the alignment functions that reproducethe observed polarization curves of the differentpeak wavelengths λ max . We identified that the op-tical and IR polarization characterized by λ max ismostly produced by RAT-aligned grains with sizeslarger than ∼ . µ m, while the UV polarization isproduced by both the a > . µ m grains and the a < . µ m grains. The sightlines with lower λ max require the higher degrees of alignment of smallgrains to reproduce the observational data.3. We showed that the excess UV continuum polariza-tion relative to the Serkowski law for the sightlineswith low λ max ( λ max < . µ m) can be reproducedby the enhanced paramagnetic alignment of smallsilicate grains, which higher efficiency arises fromthe increased magnetic field strength.0 Hoang, Lazarian, & Martin4. We suggested a novel method to measure thestrength of magnetic fields based on UV and op-tical polarization observations. Applying our tech-nique for three sightlines with maximum polariza-tion efficiency, we estimated the upper limit ofmagnetic field B ∼ µ G for the typical diffuseISM of λ max = 0 . µ m and larger magnetic fieldsfor the sightlines with λ max ≤ . µ m, assumingoblate spheroid with axial ratio r = 2 for interstel-lar grains and average ISRF. Higher magnetic fieldsare estimated if the oblate spheroid with axial ratio r = 1 . B ∼ µ G.We thank the anonymous referee for valuable com-ments and suggestions that improved our paper. T.H.is supported by Alexander von Humboldt Fellowship atthe Ruhr-Universit¨at Bochum. P.G.M. acknowledges thesupport from the Natural Sciences and Engineering Re-search Council of Canada (NSERC). A.L. acknowledgesthe financial support of NASA grant NNX11AD32G andthe Center for Magnetic Self-Organization.
APPENDIX
A. COLLISIONAL DAMPING TIMES
The process of gas-grain collisions consists of the sticking collisions followed by the evaporation of molecules from thegrain surface. In the grain frame of reference, the mean torque arising from the sticking collisions on an axisymmetricgrain rotating around its symmetry axis ˆ a tends to zero when averaged over grain revolving surface. On the otherhand, the evaporation induces a non-zero mean torque, which is parallel to the rotation axis (see Roberge et al. 1993).The damping times for the rotation parallel and perpendicular to the grain symmetry axis ˆ a were derived in Lazarian(1997). Basically, the collisional damping time for the rotation along an axis is given by h ∆ J bi i ∆ t = − J bi τ H , i for i = x, y, z, (A1)where the superscript b indicates the grain body system ˆ a ˆ a ˆ a , x, y, z denote the components of J i along ˆ a ˆ a ˆ a , τ H , x = τ H , y ≡ τ H , ⊥ , and τ H , z = τ H , k . τ H , k and τ H , ⊥ are given by Equations (21) and (22).Usually, we represent grain angular momentum J in units of the thermal angular momentum and the gaseousdamping time. For oblate spheroid, the thermal angular momentum is given by J th = q I k k B T gas = r πρa s k B T gas ≈ . × − a / − ˆ s / ˆ ρ / ˆ T / g cm rad s − , (A2)where ˆ s = s/ . s = a /a and ˆ T gas = T gas /
100 K.The thermal angular velocity is equal to ω th = (cid:18) k B T gas I k (cid:19) / ≈ . × ˆ s / a − / − ˆ T / ˆ ρ − / s − . (A3)The geometrical factors in Equations (21) and (22) are given byΓ k = 316 (cid:2) − e ) g ( e ) − e − (1 − (1 − e ) ) g ( e ) (cid:3) , (A4)Γ ⊥ = 332 (cid:2) − e + (1 − e ) g ( e ) + (1 − e )(1 + e − [1 − (1 − e ) ) g ( e )]) (cid:3) , (A5)where e = √ − s and g e = 12 e ln (cid:18) e − e (cid:19) . (A6) B. DIFFUSION COEFFICIENTS FOR MAGNETIC ALIGNMENT
Davis & Greenstein (1951) derived the mean torque for rotational damping by paramagnetic relaxation. In dimen-sionless units of τ gas , the drifting components in the inertial coordinate system are given by A m , x = − Z ( θ ) δ m J x , A m , x = − Z ( θ ) δ m J y , A m , z = 0 , (B1)where δ m = τ gas /τ m with τ m being the magnetic alignment timescale due to paramagnetic and resonance paramagneticrelaxation given by Equation (50), and Z ( θ ) = 1 + ( h −
1) sin θ, (B2)mall grain alignment and UV polarization 21 a a a zx yJ θϕ ψ (a) N Fig. 14.—
Coordinate systems used for calculations. (a): Orientation of grain principal axes in the coordinate system ˆ x ˆ y ˆ z with ˆ z parallelto the grain angular momentum J . (b): orientation of J in the inertial coordinate system ˆ e ˆ e ˆ e with ˆ e parallel to the magnetic field B . is a correction term for the spheroidal grain shape from its sphere.In addition to the rotational damping, the paramagnetic relaxation also induces rotational excitation, which is adirect result from the principle of detailed balance, i.e., the probability current at each point in phase space tendsto vanish in thermal dynamic equilibrium (see Jones & Spitzer 1967; RL99). Thus, one can obtain the excitationcoefficient as follows: A m , x f ( J ) − ∂∂J x ( B m , xx f ( J )) = 0 , (B3)where f = C exp (cid:16) ZJ T d /T gas (cid:17) (see also Jones & Spitzer 1967).Following RL99, one obtain, B m , xx = T d T gas δ m , B m , yy = B m , xx , B m , zz = 0 . (B4) C. TRANSFORMATION OF COORDINATE SYSTEMS
Damping coefficient A i = h ∆ J i / ∆ t i and diffusion coefficients B ij = h ∆ J i ∆ J j / ∆ t i are usually derived in the bodycoordinate system, while we are interested in the evolution of grain angular momentum in the inertial coordinate system.Let us define an inertial coordinate system ˆ e ˆ e ˆ e in which the direction J is described by the angle β between J withˆ e k B , and the azimuthal angle η (see Figure 14(b)). To obtain these coefficients in the lab coordinate system, we firsttransform the body system ˆ a i to the external system ˆ x ˆ y ˆ z (see 14(a)). Then, we perform the transformation from ˆ x ˆ y ˆ z system to the inertial system ˆ e ˆ e ˆ e .In the body system, the damping coefficients are given by A bi = h ∆ J bi ∆ t i = − J i τ gas ,i − J i τ ed ,i (cid:18) I i k B T gas (cid:19) , (C1)where τ gas ,i = F tot ,i /τ H , k and i = x, y, z with z k ˆ a .The diffusion coefficients in the grain body system, B bij = h ∆ J bi ∆ J bj / ∆ t i with B bij = 0 for i = j are related to theexcitation coefficients as follows: B bzz = B k = 2 I k k B T gas τ H , k G tot , k , and B bxx = B byy = B ⊥ = 2 I ⊥ k B T gas τ H , ⊥ G tot , ⊥ . (C2)The diffusion coefficients in the inertial system ˆ e i have components B , B and B , which are denoted by B zz , B xx , B yy for consistency. Using the method in Lazarian (1997) to perform the transformations from the body sys-tem to inertial system, after averaging over the fast precession of the grain symmetry axis around angular momentum,we obtain B zz = B k (cid:18)
12 sin θ sin β + cos θ cos β (cid:19) + B ⊥ (cid:18)
12 [1 + cos θ ] sin β + sin θ cos β (cid:19) , (C3) B xx = B k (cid:18)
12 sin θ [cos η + sin η cos β ] + cos θ sin η sin β (cid:19) + B ⊥ (cid:18)
12 [1 + cos θ ][cos η + sin η cos β ] + sin θ sin η sin β (cid:19) , (C4) There is a typo in Eq. (3-21) of RL99 for which the correctform should not have the last term of ( T d /T gas ) δ m . Our expres- sions differ from those of RL99 by a factor 2 because we adoptedthe normalized units J th = (cid:0) I k k B T gas (cid:1) / . B yy = B k (cid:18)
12 sin θ [sin η + cos η cos β ] + cos θ sin η sin β (cid:19) + B ⊥ (cid:18)
12 [1 + cos θ ][sin η + cos η cos β ] + sin θ sin η sin β (cid:19) , (C5)where β is the angle between J and ˆ e , and η is the azimuthal angle of J in the inertial system ˆ e i .In the presence of fast internal fluctuations, we need to average the damping and diffusion coefficients over θ . Therefore, the terms containing θ in above equations are replaced by the averaged values, i.e., h cos θ i = R π cos θf LTE ( J, θ ) sin θdθ , h sin θ i = R π sin θf LTE ( J, θ ) sin θdθ .In the presence of ambient magnetic field, the grain angular momentum precesses around B on a timescale τ Lar (Equation 13), which is short compared to the dynamical timescales due to gas bombardment, electric dipole emission,and IR emission. Therefore, one can average the damping and diffusion coefficients over the uniform distribution ofthe precession angle η . Thus, sin η and cos η are replaced by their averaged values equal to 1 /
2. In this case, ourdiffusion coefficients (Eqs C3-C5) become similar to those in (Lazarian 1997).
D. EXTINCTION AND POLARIZATION
D1. Dust Extinction and Polarization
To find the extinction and polarization of background starlight by interstellar grains, let us define an observer’scoordinate system in which the sightline is directed along the Z − axis, and the X − and Y − axes constitute the skyplane. The polarization of starlight arising from the dichroic extinction by aligned grains in a cell of dZ is computedas dp ( λ ) = dτ X − dτ Y Z a max a min
12 ( C X − C Y ) ( dn/da ) dadZ, (D1)where dn/da is the grain size distribution function with the lower and upper cutoff a min and a max , C X and C Y arethe grain cross-section along the X − and Y − axes, respectively.For the case of perfect internal alignment (i.e., grain symmetry axis ˆ a perfectly aligned with its angular momentum),by transforming the grain’s reference system to the observer’s reference system and taking corresponding weights, weobtain C X = C ⊥ − C pol β, (D2) C Y = C ⊥ − C pol β cos ξ + sin β sin ξ ) , (D3)where ξ is the angle between the magnetic field assumed to be in the Y Z plane and the sky plane, β is the anglebetween the grain angular momentum and the magnetic field, and C pol = C k − C ⊥ is the polarization cross-section foroblate spheroidal grains. By convention, C k and C ⊥ are the extinction cross-section for the electric field of incidentradiation parallel and perpendicular to the grain symmetry axis, respectively.The polarization efficiency then becomes C X − C Y = C pol (cid:0) β − (cid:1) ξ. (D4)Taking the average of C X − C Y over the distribution of the alignment angle β , it yields C X − C Y = C pol h Q J i cos ξ, (D5)where Q J = h G J i is the ensemble average of G J = (cid:0) β − (cid:1) / C X − C Y = C pol h Q J Q X i cos ξ ≡ C pol R cos ξ, (D6)where R = h Q J Q X i is the Rayleigh reduction factor (see also RL99).Let f = R cos ξ be the effective degree of grain alignment. Thus, for the case of perpendicular magnetic field, i.e., B lies on the sky plane f = R , Equation (D6) simply becomes C X − C Y = C pol f .Plugging in Equation (D6) into this above equation, we obtain p ( λ ) = Z dZ X j =carb , sil Z a max a min C j pol f j ( a )( dn j /da ) da, (D7)where f j ( a ) denotes the alignment function of grain specie j of size a .The extinction in units of magnitude is defined by A ( λ ) = 2 . (cid:18) F obs λ F ⋆λ (cid:19) , mall grain alignment and UV polarization 23= 1 . τ λ = 1 . Z dZ X j =carb , sil Z a max a min C j ext ( dn j /da ) da, (D8)where F ⋆λ is the intrinsic flux from the star, F obs λ = F ⋆λ e − τ λ is the observed flux, and τ λ is the optical depth.Frequently, it is more convenient to represent the polarization (extinction) through the polarization (extinction)cross-section. Hence, the above equations can be rewritten as p ( λ ) = σ pol ( λ ) × N H , (D9) A ( λ ) = σ ext ( λ ) × N H , (D10)where N H ( cm − ) is the column density and σ ext and σ pol in units of cm H − are the dust extinction cross-sectionand dust polarization cross-section, respectively.We take C ext ( a, λ ) and C pol ( a, λ ) computed for silicate and carbonaceous grains in Hoang et al. (2013). E. NONLINEAR LEAST CHI-SQUARE FITTING
Following Kim & Martin (1995), we find the grain size distribution and alignment function by minimizing an objectivefunction χ , which is constructed as follows: χ = χ + χ + χ , (E1)where χ = w ext N λ − X i =0 [ A mod ( λ i ) − A obs ( λ i )] , (E2) χ = w pol N λ − X i =0 [ p mod ( λ i ) − p obs ( λ i )] , (E3)with w ext and w pol being the fitting weights for the extinction and polarization, respectively. Here, the summationis performed over N λ wavelength bins. For this study, we adopt N a = 100 size bins from a = 3 . µ m and N λ = 100 from λ = 0 . µ m to 2 . µ m. The last term χ = P Ψ contains the constraints of the fitting model, whichare similar to Equations (A5)-(A9) in Draine & Allaf-Akbari (2006). Below we provide them here for consistency.Ψ N λ + j +2 = α ( N a − / min "(cid:18) d ln fdu (cid:19) j +1 / , , (E4)Ψ N λ + N a +2 = α max [ f ( a N a ) − , , (E5)Ψ N λ + N a +1+ j = α ( N a − / (cid:18) d y sil du (cid:19) , a = a j , j = 2 , ..., N a − , (E6)Ψ N λ +2 N a − j = α ( N a − / (cid:18) d y carb du (cid:19) , a = a j , j = 2 , ..., N a − , (E7)Ψ N λ +3 N a − j = α ( N a − / (cid:18) d ln fdu (cid:19) , a = a j , j = 2 , ..., N a − , (E8)(E9)where du = ln a j +1 − ln a j , ( df /du ) j +1 / = (cid:0) f j +1 / − f j (cid:1) / ∆ u and (cid:0) d f /du (cid:1) j = ( f j +1 + f j − − f j ) / (∆ u ) , and α − α are weights, which are quite arbitrary.The objective functions for extinction and polarization are different from those of Draine & Allaf-Akbari (2006) ina sense that our objective functions are constructed from the difference between the model and observation.We find the minimum χ using the Monte-Carlo direct search method in which the fitting process is iterated untilthe convergence criterion is achieved. REFERENCESAndersson, B.-G., Pintado, O., Potter, S. B., Straiˇzys, V., &Charcos-Llorens, M. 2011, A&A, 534, 19Andersson, B.-G., & Potter, S. B. 2007, ApJ, 665, 369Andersson, B.-G., Piirola, V., De Buizer, J., et al. 2013, ApJ, 775,84Barnett, S. J. 1915a, Science, 42, 163Barnett, S. J. 1915b, Physical Review, 6, 239 Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345,245Chandrasekhar, S., & Fermi, E. 1953, ApJ, 118, 113Chiar, J. E., Adamson, A. J., Whittet, D. C. B., et al. 2006, ApJ,651, 268Cho, J., & Lazarian, A. 2005, ApJ, 631, 3614 Hoang, Lazarian, & Martin