Tracing Magnetic Fields with Aligned Grains

Tracing Magnetic Fields with Aligned Grains

A. Lazarian University of Wisconsin-Madison, Astronomy Department, 475 N. Charter St., Madison, WI 53706, e-mail: lazarian@astro.wisc.edu
Abstract

Magnetic fields play a crucial role in various astrophysical processes, including star formation, accretion of matter, transport processes (e.g., transport of heat), and cosmic rays. One of the easiest ways to determine the magnetic field direction is via polarization of radiation resulting from extinction on or/and emission by aligned dust grains. Reliability of interpretation of the polarization maps in terms of magnetic fields depends on how well we understand the grain-alignment theory. Explaining what makes grains aligned has been one of the big issues of the modern astronomy. Numerous exciting physical effects have been discovered in the course of research undertaken in this field. As both the theory and observations matured, it became clear that the grain-alignment phenomenon is inherent not only in diffuse interstellar medium or molecular clouds but also is a generic property of the dust in circumstellar regions, interplanetary space and cometary comae. Currently the grain-alignment theory is a predictive one, and its results nicely match observations. Among its predictions is a subtle phenomenon of radiative torques. This phenomenon, after having stayed in oblivion for many years after its discovery, is currently viewed as the most powerful means of alignment. In this article, I shall review the basic physical processes involved in grain alignment, and the currently known mechanisms of alignment. I shall also discuss possible niches for different alignment mechanisms. I shall dwell on the importance of the concept of grain helicity for understanding of many properties of grain alignment, and shall demonstrate that rather arbitrarily shaped grains exhibit helicity when they interact with gaseous and radiative flows.

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1 Introduction

Magnetic fields are of an utmost importance for most astrophysical systems. Conducting matter is entrained on magnetic field lines, and magnetic pressure and tension are very important for its dynamics. For instance, galactic magnetic fields play key role in many processes, including star formation, mediating shocks, influencing heat and mass transport, modifying turbulence etc. Aligned dust grains trace the magnetic field and provide a unique source of information about magnetic field structure and topology. The new instruments, Sharc II (Novak et al. 2004), Scuba II (Bastien, Jenness & Molnar 2005), and an intended polarimeter for SOFIA open new horizons for studies of astrophysical magnetic fields via polarimetry.

The enigma that has surrounded grain alignment since its discovery in 1949 (Hall 1949; Hiltner 1949) makes one wonder how reliable is polarimetry as a way of magnetic field studies. In fact, for many years grain alignment theory used to have a very limited predictive power and was an issue of hot debates. Works by great minds like Lyman Spitzer and Edward Purcell moved the field forward, but the solution looked illusive. In fact, the reader can see from the review, that a number of key physical processes have been discovered only recently. Fig. 1a demonstrates the complexity of grain motion as we understand it now.

The weakness of the theory caused a somewhat cynical approach to it among some of the polarimetry practitioners who preferred to be guided in their work by the following rule of thumb: All grains are always aligned and the alignment happens with the longer grain axes perpendicular to magnetic field. This simple recipe was shattered, however, by observational data which indicated that
I. Grains of sizes smaller than a critical size are either not aligned or marginally aligned (Mathis 1986, Kim & Martin 1995).
II. Carbonaceous grains are not aligned, while silicate grains are aligned (see Mathis 1986).
III. A substantial part of small grains grains deep within molecular clouds are not aligned (Goodman et al. 1995, Lazarian, Goodman & Myers 1997, Cho & Lazarian 2005 and references therein).
VI. Grains might be aligned with longer axes parallel to magnetic fields (Rao et al 1998).

These facts were eloquent enough to persuade even the most sceptical types that the interpretation of interstellar polarimetric data does require an adequate theory. A further boost of the interest to grain alignment came from the search of Cosmic Microwave Background (CMB) polarization (see Lazarian & Finkbeiner 2003, for a review). Aligned dust in this case acts as a source of a ubiquitous foreground that is necessary to remove from the data. It is clear that understanding of grain alignment is the key element for such a removal.

While alignment of interstellar dust is a generally accepted fact, the alignment of dust in conditions other than interstellar has not been fully appreciated. A common explanation of light polarization from comets or circumstellar regions is based on light scattering by randomly oriented particles (see Bastien 1988 for a review). The low efficiency and slow rates of alignment were sometimes quoted to justify such an approach. However, it has been proved recently that grain alignment is an efficient and rapid process. Therefore, we do expect to have circumstellar, interplanetary, and cometary dust aligned. Particular interesting in this respect are T-Tauri accretion disks (see Cho & Lazarian 2006). Tracing magnetic fields in these environments with aligned grains opens new exciting avenues for polarimetry.

Taken a broader view, grain alignment is a part of a wider range of alignment astrophysical processes that can provide the information about magnetic fields. Molecules aligned in their excited states can trace magnetic field (Goldreich & Kylafis 1982), the effect that was first successfully used in Girart, Crutcher & Rao (1999) to map magnetic field in molecular clouds. Atoms and ions with fine and hyperfine structure can be aligned by radiation in their ground or metastable states. The magnetic field then mixes up the states due to the Larmor precession, which allows studies of interstellar and circumstellar magnetic fields via absorption and emission lines (Yan & Lazarian 2006ab)111Those studies potentially can restore 3D direction of magnetic fields, compared to the plane-of-sky projection of magnetic field available via dust polarimetry.. Making use of several alignment processes is another avenue for observational studies of astrophysical magnetic fields (see Lazarian & Yan 2005).

Getting back to dust, one should mention that in the past the linear starlight polarimetry was used. These days, far infrared polarimetry of dust emission has become a major source of magnetic field structure data in molecular clouds (see Hildebrand 2000). It is likely that the circular polarimetry may become an important means of probing magnetic fields in circumstellar regions and comets.

In this review I shall show that the modern grain alignment theory allows us for the first time ever make quantitative predictions of the polarization degree from various astrophysical objects. A substantial part of the review is devoted to the physics of grain alignment, which is deep and exciting. Enough to say, its progress resulted in a discovery of a number of new solid state physics effects. The rich physics of grain alignment presents a problem, however, for its presentation. Therefore I shall describe first the genesis of ideas that form the basis of the present-day grain alignment theory. The references to the original papers should help the interested reader to get the in-depth coverage of the topic. Earlier reviews on the subject include Lazarian (2003), Roberge (2004), Lazarian & Cho (2005) and Vaillancourt (2006). Progress in testing theory is addressed in Hildebrand et al. (2000), while the exciting aspects of grain dynamics are covered in Lazarian & Yan (2004).

Below, in §2 I shall show how the properties of polarized radiation are related to the statistics of aligned grains. In §3 I shall discuss the essential elements of grain dynamics. In §4 I shall analyze the main alignment mechanisms. In §5 I shall compare the mechanisms and discuss new processes related to subsonic mechanical alignment of irregular grains. In §6, I shall discuss the observational data that put the grain-alignment theory to test. An outlook on the prospects of the polarimetric studies of magnetic fields will be presented in §7.

2 Aligned Grains & Polarized Radiation

A practical interest in aligned grains arises from the fact that their alignment results in polarization of the extinct starlight as well as in polarization in grain emission. Below we discuss why this happens.

2.1 Linear Polarized Starlight from Aligned Grains

For an ensemble of aligned grains the degrees of extinction in the directions perpendicular and parallel to the direction of alignment are different222According to Hildebrand & Dragovan (1995), the best fit of the grain properties corresponds to oblate grains with the ratio of axis about 2/3.. Therefore initially unpolarized starlight acquires polarization while passing through a volume with aligned grains (see Fig. 2a). If the extinction in the direction of alignment is and in the perpendicular direction is one can write the polarization, , by selective extinction of grains as

(1)

where the latter approximation is valid for . To relate the difference of extinctions to the properties of aligned grains one can take into account the fact that the extinction is proportional to the product of the grain density and their cross sections. If a cloud is composed of identical aligned grains and are proportional to the number of grains along the light path times the corresponding cross sections, which are, respectively, and .

In reality one has to consider additional complications (like, say, incomplete grain alignment) and variations in the direction of the alignment axis relative to the line of sight. (In most cases the alignment axis coincides with the direction of magnetic field.) To obtain an adequate description, one can (see Roberge & Lazarian 1999) consider an electromagnetic wave propagating along the line of sight (the axis, as on Fig. 1b). The transfer equations for the Stokes parameters depend on the cross sections and for linearly polarized waves with the electric vector, , along the and axes perpendicular to (Lee & Draine 1985).

Figure 1: (a)Left panel– Alignment of grains implies several alignment processes acting simultaneously and covering various timescales. Internal alignment was introduced by Purcell (1979) and was assumed to be a slow process. Lazarian & Draine (1999a) showed that the internal alignment is times faster if nuclear spins are accounted for. The time scale of and alignment is given for diffuse interstellar medium. It is faster in circumstellar regions and for cometary dust. (b) Right panel– Geometry of observations (after Roberge & Lazarian 1999).

To calculate and , one transforms the components of to the principal axes of the grain, and takes the appropriately-weighted sum of the cross sections and for polarized along the grain axes (Fig 1b illustrates the geometry of observations). When the transformation is carried out and the resulting expressions are averaged over the precession angles, one finds (see transformations in Lee & Draine 1985 for spheroidal grains, and in Efroimsky 2002a for the general case) that the mean cross sections are

(2)
(3)

being the angle between the polarization axis and the plane, and being the effective cross section for randomly-oriented grains. To characterize the alignment, we used in eq. (3) the Rayleigh reduction factor (Greenberg 1968) defined as

(4)

where angular brackets denote ensemble averaging, , is the angle between the axis of the largest moment of inertia (henceforth the axis of maximal inertia) and the magnetic field , while is the angle between the angular momentum and . To characterize the alignment with respect to the magnetic field, the measures and are employed. Unfortunately, these statistics are not independent and therefore is not equal to (see Lazarian 1998, Roberge & Lazarian 1999). This considerably complicates the description of the alignment process.

2.2 Polarized Emission from Aligned Grains

Aligned grains emit polarized radiation (see Fig. 2b). The difference in and for aligned dust results in the emission polarization:

(5)

where both optical depths are were assumed to be small. Taking into account that both and are functions of the wavelength and combining eqs.(1) and (6), one obtains for

(6)

which establishes the relation between the polarizations in emission and absorption. The minus sign in eq (6) reflects the fact that emission and absorption polarizations are orthogonal. This relation enables one to predict the far infrared polarization of emitted light if the starlight polarization is measured. This opens interesting prospects of predicting the foreground polarization arising from emitting dust using the starlight polarization measurements (Cho & Lazarian 2002, 2003). As depends on , also depends on the Rayleigh reduction factor.

Figure 2: (a)Left panel– Polarization of starlight passing through a cloud of aligned dust grains. The direction of polarization () is parallel to the plane of the sky direction of magnetic field. (b) Right panel– Polarization of radiation from a optically thin cloud of aligned dust grains. The direction of polarization () is perpendicular to the plane of the sky direction of magnetic field.

2.3 Circular Polarization from Aligned Grains

One way of obtaining circular polarization is to have a magnetic field that varies along the line of sight (Martin 1972). Passing through one cloud with aligned dust the light becomes partially linearly polarized. On passing the second cloud with dust gets aligned in a different direction. Hence, the light gets circular polarization. Literature study shows that this effect is well remembered (see Menard et al 1988), while another process entailing circular polarization is frequently forgotten. We mean the process of single scattering of light on aligned particles. An electromagnetic wave interacting with a single grain coherently excites dipoles parallel and perpendicular to the grain’s long axis. In the presence of adsorption, these dipoles get phase shift, thus giving rise to circular polarization. This polarization can be observed from an ensemble of grains if these are aligned. The intensity of circularly polarized component of radiation emerging via scattering of radiation with wavenumber on small () spheroidal particles is (Schmidt 1972)

(7)

where and are the unit vectors in the directions of incident and scattered radiation, is the direction along the aligned axes of spheroids; and are the particle polarizabilities along and perpendicular to it.

The intensity of the circularly polarized radiation scattered in the volume at from the star at a distance from the observer is (Dolginov & Mytrophanov 1978)

(8)

where is the stellar luminosity, is the number of dust grains per a unit volume, and is the cross section for producing circular polarization, which for small grains is . According to Dolginov & Mytrophanov (1978) circular polarization arising from single scattering on aligned grains can be as high as several percent for metallic or graphite particles, which is much more than one may expect from varying magnetic field direction along the line of sight (Martin 1971). In the latter case, the linear polarization produced by one layer of aligned grains passes through another layer where alignment direction is different. If on passing through a single layer, the linear polarization degree is , then passing through two layers produces circular polarization that does not exceed .

3 Grain Dynamics: Never Ending Story

Grain dynamics is really rich, as it involves an abundant variety of effects. We provide a brief over-review of this exciting field of research.

3.1 Wobbling Grains

To produce the observed starlight polarization, grains must be aligned, with their long axes perpendicular to magnetic field. According to eq. (4) this involves alignment not only of the grains’ angular momenta with respect to the external magnetic field , but also the alignment of the grains’ long axes with respect to . Jones & Spitzer (1967) assumed a Maxwellian distribution of the angular momentum, distribution that favored the preferential alignment of with the axis of a maximal moment of inertia (henceforth, maximal inertia, using Purcell’s terminology). As we already mentioned in §3.2, Purcell (1979, henceforth P79) later considered grains rotating much faster than the thermal velocities and showed that the internal dissipation of energy in a grain will make grains spin about the axis of maximal inertia.

Indeed, it is intuitively clear that a tumbling and precessing grain should, due to the internal dissipation, tend to get into the state of minimal energy, i.e., to spin about the axis of maximal inertia. P79 discussed two possible causes of internal dissipation – one due to the well known inelastic relaxation (see also Lazarian & Efroimsky 1999), another due to the mechanism that he discovered and termed “Barnett relaxation”.

We would remind to the reader that the Barnett effect is inverse to the Einstein-de Haas effect. The essence of the Einstein-de Haas effect is that a paramagnetic body acquires rotation during remagnetizations, when the flipping electrons transfer to the lattice their spin angular momentum. The essence of the Barnett effect is that the rotating body shares its angular momentum with the electron subsystem, thus causing magnetization. The magnetization is directed along the grain’s angular velocity, and the value of the Barnett-induced magnetic moment is  erg gauss (where )333Therefore the Larmor precession has a period  s (where ), and the magnetic field defines the axis of alignment (see also §5.4).

Into the grain-alignment theory, the Barnett effect was introduced by Dolginov & Mytrophanov (1976), who noticed that the magnetization of rotating grains due to this effect far exceeds the one arising from their typical charge. By itself, this was a big advance in understanding the grain dynamics. Moreover, it induced Purcell to think about the relaxation that this magnetization could cause.

The Barnett relaxation process may be easily understood. We know that a freely rotating grain preserves the direction of , while the (body-axes-related) angular velocity precesses about (see Fig. 3a). The “Barnett equivalent magnetic field”, i.e. the equivalent external magnetic field that would cause the same magnetization of the grain material, is  G. Due to the precession of the angular velocity, the co-directed “Barnett equivalent magnetic field” precesses in the grain axes. This causes remagnetization accompanied by the inevitable dissipation.

Figure 3: (a) Left panel– Grain dynamics as seen in the grain frame of reference. The Barnett magnetization is directed along , and it causes a gradual grain remagnetization as the precesses in the grain axes. (b) Right panel– Time scale for the internal alignment due to nuclear and Barnett relaxation processes. . Also shown the “crossover time” , where the torques are due to the H formation, with a density of active sites cm. From Lazarian & Draine (1999a).

The Barnett relaxation takes place over the time scale of  sec, which is very short compared to the time over which randomization through gas-grain collisions takes place. As a result, models of interstellar-dust polarization developed since 1979 have often assumed that the Barnett dissipation aligns with the major axis of inertia. However, Lazarian (1994, henceforth L94) showed that this approximation is invalid if the grains rotate with thermal kinetic energies: thermal fluctuations in the Barnett magnetization will excite rotation about all 3 of the body axes, preventing perfect alignment unless either the rotation velocity is suprathermal () or the grain’s material temperature is zero.

Following Lazarian & Roberge (1997, henceforth LR97), consider an oblate grain (see Fig. 3a) with an angular momentum . Its energy can be written as

(9)

where is the ratio of the maximal to minimal moments of grain inertia. Internal forces cannot change the angular momentum, but it is evident from eq.(9) that the energy can be decreased by aligning the axis of maximal inertia along , i.e. by decreasing . However, whatever the efficiency of internal relaxation, in the presence of thermal fluctuations the grain energy as a function of should have a Boltzmann distribution, i.e. , where is the grain temperature, rather than the -function distribution assumed in the literature thitherto. The quantitative analysis offered in LR97 allowed many further theoretical advances.

As the numbers of parallel and antiparallel spins become different, the body develops magnetization, even if the unpaired spins are nuclear spins. The relation between and the strength of the “Barnett-equivalent” magnetic field (P79) that would cause the same nuclear magnetization (in a non-rotating body) is given by

(10)

where is the so-called nuclear -factor (see Morrish 1980), and is the nuclear magneton, which is equal to the Bohr magneton multiplied by the electron to proton mass ratio, .

The nuclear magnetization was mentioned in P79 as an subdominant effect that can induce Larmor precession. The same paper discussed the Barnett relaxation, but did not address a possible effect of the nuclear spins on the internal relaxation. Presumably, this was due to the fact that the nuclear moments induce the magnetization of grains that is smaller that the magnetization by electrons.

The nuclear relaxation was considered by Lazarian & Draine (1999a, further on LD99a). Surprisingly and rather counter-intuitively, the effect happened to be very strong. Indeed, a striking feature of eq. (10) is that the Barnett-equivalent magnetic field is inversely proportional to the species’ magnetic moment. As grain tumbles, the magnetization changes in the grain’s body coordinates, and this causes paramagnetic relaxation. This relaxation is proportional to (where is the imaginary part of the nuclear contribution to the susceptibility) and is approximately times faster for nuclear moments than for their electron counterparts (see Fig. 3b).

In terms of parameters involved, our arguments may be summarized as follows. The Barnett equivalent field is , while the paramagnetic relaxation is proportional (for sufficiently slow rotation) to , which means that the relaxation rate is proportional to . As , the heavier the species to align along the higher the relaxation rate.

Curiously enough, while the Barnett effect is reduced for nuclear spins by a factor of , the relaxation increases by a factor of . Therefore it would be incorrect to identify this relaxation as a modification of the Barnett relaxation for nuclear spins. This is a separate relaxation process. In terms of its domain of applicability it is limited by the spin-spin relaxation rate. Indeed, the nuclear spins precess in the field of their neighbors, which is approximately (van Vleck 1937), where is the magnetic moment of the nuclei, is the density of the nuclei. For hydrogen nuclei , for (see Robinson 1991). The rate of precession in such a field is , where is the nuclear -factor, which is, for instance, for . According to LD99a the interaction of nuclei in the interstellar grains with electrons induce a nuclei-electron relaxation rate which is comparable with and the actual spin-spin relaxation rate is the sum of the two. If grain rotational frequency exceeds the rate of spin-spin relaxation, the internal nuclear dissipation rate gets suppressed by a factor (Draine & Lazarian 1998b). This explains why for small fast rotating grains the Barnett relaxation may be more efficient than the nuclear one (see Fig. 3).

However, the nuclear relaxation dominates the Barnett one for grains larger than  cm, the range that includes most of the aligned interstellar grains. In general, for several relaxation processes acting simultaneously, the overall internal relaxation rate is .

3.2 Grains that are Swiftly Rotating, Flipping, and Thermally Trapped

All the studies undertaken prior to 1979, with a notable exception of Dolginov & Mytrophanov (1976) that we shall discuss separately, assumed the Brownian grain rotation with the effective temperature equal to the mean of the grain and gas temperatures (see Jones & Spitzer 1967). The greater complexity of grain rotation was appreciated only later. Purcell (1975; 1979) realized that grains may rotate at a much faster rate resulting from systematic torques. P79 identified three separate systematic torque mechanisms: inelastic scattering of impinging atoms when gas and grain temperatures differ, photoelectric emission, and H formation on grain surfaces (see Fig. 4a). Below we shall refer to the latter as ”Purcell’s torques”. These were shown to dominate the other two for typical conditions in the diffuse ISM (P79). The existence of systematic H torques is expected due to the random distribution over the grain surface of catalytic sites of H formation, since each active site acts as a minute thruster emitting newly-formed H molecules. The arguments of P79 in favor of suprathermal rotation were so clear and compelling that other researchers were immediately convinced that the interstellar granules in diffuse clouds must rotate suprathermally.

Figure 4: (a) Left panel– A grain acted upon by Purcell’s torques before and after a flipover event. As the grain flips, the direction of torques alters. The H formation sites act as thrusters. (b) Right panel– A regular crossover event as described by Spitzer & McGlynn (1979). The systematic torques nullify the amplitude of the component parallel to the axis of maximal inertia, while preserving the other component, . If is small then the grain is susceptible to randomization during crossovers. The direction of is preserved in the absence of random bombardment.

P79 considered changes of the grain surface properties and noted that those should stochastically change the direction (in body-coordinates) of the systematic torques. Spitzer & McGlynn (1979, henceforth SM79) developed a theory of such crossovers. During a crossover, the grain slows down, flips, and thereafter is accelerated again (see Fig. 4b).

From the viewpoint of the grain-alignment theory, the important question is whether or not a grain gets randomized during a crossover. If the value of the angular momentum is small during the crossover, the grains are susceptible to randomization arising from atomic bombardment. The original calculations in SM79 obtained only marginal correlation between the values of the angular momentum before and after a crossover, but their analysis disregarded thermal fluctuations within the grain material. Indeed, if the alignment of with the axis of maximal inertia is perfect, all the time through the crossover the absolute value of passes through zero during the crossover. Therefore gas collisions and recoils from nascent molecules would completely randomize the final direction of during the crossover. Thermal fluctuations partially decouple from the axis of maximal inertia (see §3.1). As a result, the minimal value of during a crossover is equal to the component of perpendicular to the axis of maximal inertia. This value for moderately oblate grains is approximately , and the randomization during a crossover decreases (Lazarian & Draine 1997, henceforth LD97). LD97 obtained a high degree of correlation between the angular-momentum directions before and after the crossover for grains larger than the critical radius cm. This is the radius for which the time for internal dissipation of the rotational kinetic energy is equal to the duration of a crossover.

As nuclear relaxation is faster than the Barnett one for grains larger than  cm (see Fig. 3), the actual grain critical size gets larger than  cm. In view of this, the results of LD97 study are related only to very large grains, e.g. grains inside molecular clouds or accretion disks.

Figure 5: (a) Left panel– Grain trajectory on the plane, where and are components of perpendicular or parallel to the grain’s principal axis of largest moment of inertia. The solid trajectory shows a “thermal flip”, while the broken line shows the “regular” crossover which would occur in the absence of a thermal flip. (b) Right panel– Top: Thermal flipping to damping ratio as a function of for grains of given size [broken lines, labeled by ] and for grains with a given value of systematic torques [solid lines, labeled by ]. Dot shows for flipping-assisted crossover of grain with . Bottom: Thermal trapping for grains of given size [broken lines, labeled by ], and given value of torques [solid lines, labeled by ]. From Lazarian & Draine (1999b).

What would happen for grains that are smaller than ? The SM79 theory prescribed that the granules should follow the phase-space trajectory along which is approximately constant while the component of parallel to the axis of maximal inertia changes sign. Later, though, Lazarian & Draine (1999b, henceforth LD99b) demonstrated that in reality the grains undergo flipovers (see Fig. 5a) during which the absolute value does not change. If these flipovers repeat, the grains get “thermally trapped” (LD99b and Fig. 5b). This process can be understood in the following way. For sufficiently small , the rate of flipping becomes large. Purcell’s torques change sign as grain flips, and they cannot efficiently spin the grain up. As a result, a substantial part of grains smaller than cannot rotate at high rates predicted by P79, even in spite of the presence of systematic torques that are fixed in the body axes (LD99a). A more elaborate study of the phenomenon in Roberge & Ford (preprint; see also Roberge 2004) supports this conclusion.

While the thermal trapping limits the range of grain sizes which can be spun up by Purcell’s torques, a natural question arises: do the astrophysical grains rotate suprathermally?

Earlier than Purcell, Dolginov (1972) and Dolginov & Mytrophanov (1976) identified radiative torques as the way of spinning up a subset of the interstellar grains. Unlike Harwit (1971), who addressed the issue of interaction of symmetric, e.g. spheroidal, grains with a radiative flow, Dolginov and Mytrophanov considered “twisted grains” that can be characterized by some helicity. They noticed that “helical” grains would scatter differently the left- and right-polarized light, for which reason an ordinary unpolarized light would spin them up. The subset of the “helical” grains was not properly identified, and the later researchers could assume that it is limited to special shapes/materials. One way or another, this ground-breaking work did not make much impact to the field until Draine & Weingartner (1996, henceforth DW96) numerically showed that grains of rather arbitrary irregular shapes get spun up efficiently.

DW96 and Draine & Weingartner (1997, henceforth DW97) demonstrated that radiative torques can be separated into isotropic and anisotropic parts. While the isotropic torques that are fixed in body coordinates are averaged out similarly to the Purcell torques, the anisotropic torques do not change sign when the grain flips. If those spin-up grains are fast enough to avoid constant flipping, the Purcell torques can also act on a grain in a regular way. Do all grains get spun up efficiently by anisotropic radiative torques? While DW97 provide arguments in favor of the positive answer, it should be mentioned that they treated crossovers in a crude way, i.e., as singularities at which the grain does not flip, while the direction of changes to the opposite one. This is different from the crossover prescriptions in SM79 and Lazarian & Draine (1997). On the contrary, the study in Weingartner & Draine (2003, henceforth WD03), that accounted for thermal wobbling of grains (LR97, LD99b), indicated that only a fraction of grains rotates suprathermally when acted upon by anisotropic radiative torques. Lazarian & Hoang (2006, henceforth LH07) showed that the same effect is also present when thermal wobbling is absent, but a more rigorous treatment of crossovers is applied. In fact, LH06 showed that at and no gaseous bombardment most grains undergo multiple crossovers and get settled with . For finite , the same subset of grains settles with in accordance with the findings in WD03. The effective temperature of grain rotation increases to approximately when gaseous bombardment is present (Hoang & Lazarian 2007).

This presents an unexpected twist in the theory of radiative torques. Interestingly enough, for most grains their alignment by radiative torques is a way to minimize their rotational velocity. Therefore most grains in the diffuse interstellar gas, contrary to the common belief, do not rotate suprathermally. In addition, essentially none of the small grains (i.e. ones with  cm), rotate suprathermally as the radiative torques are too weak to spun up the grains of size much less than the wavelength444In the vicinity of stars with UV excess smaller grains can be spun up as well.. On the contrary, grains deep within starless molecular clouds were usually assumed to rotate thermally. However, Cho & Lazarian (2005) showed that the radiative torques efficiency increases with the grain size. Therefore some fraction of large grains will rotate suprathermally even in dark cores of molecular clouds. As we explain further, rapid rotation is not a necessary requirement for the efficient alignment, if radiative torques are concerned.

3.3 Grains Zooming in Space

Grains can stream through ambient gas. One of the processes to induce such streaming was suggested by Gold (1952) who considered penetration of grains from one cloud to another as the clouds collide. Later, though, Davis (1955) showed that the applicability realm of the process is quite limited.

A more standard way of driving grain-gas motion is by radiation pressure (see Purcell 1969). Grains are exposed to various forces in anisotropic radiation fields. Apart from radiation pressure, grains are subjected to forces due to the asymmetric photon-stimulated ejection of electrons. A detailed discussion can be found in Weingartner & Draine (2001). They demonstrated that the emission caused force is comparable to the one arising from the usual radiation pressure, provided that the grain potential is low and the radiation spectrum is hard. Another photon-stimulated ejection process showing up in the picture is photodesorption of atoms absorbed on grain surface. The force due to photodesorption of atoms is comparable to the radiation and photoelectric ones (Draine 2003). However, none of these forces is expected to induce a supersonic grain drift under the typical interstellar conditions.

A residual imbalance arises from the difference of the number of catalytic active sites for H formation on upper and lower grain surfaces (P79). The nascent H molecules leave the active sites with kinetic energy , and the grain experiences a push in the opposite direction. The uncompensated force is parallel to the spin direction as the other components of force are averaged out due to the grain’s fast rotation. Applying the best-guess values555The number of H formation sites is highly uncertain. It may also depend on the interplay of the processes of photodesorption and poisoning (Lazarian 1995b; 1995c). adopted in LD97, Lazarian & Yan (2002) got the “optimistic” velocity cmcm/s for the Cold Neutral Medium (CNM) and cmcm/s for the Warm Neutral Medium (WNM), provided that grains do not flip (see §3.2). In dark clouds, a similar effect arising from variations of the accommodation coefficient can induce translational motion of grains.

Turbulence is another driver for grain drift with respect to gas. It is generally accepted that the interstellar medium is turbulent (see Elmegreen & Scalo 2002). Turbulence has been invoked by a number of authors (see Weidenschilling & Ruzmaikina 1994, Lazarian & Yan 2002 and references therein) to induce grain motion relative to the gas. In hydrodynamic turbulence, the grain motions are caused by the frictional interaction with the gas. At large scales, grains are coupled with the ambient gas, and the fluctuating gas motions mostly cause an overall advection of the grains with the gas (Draine 1985). At small scales, grains are decoupled. The largest velocity difference occurs on the largest scale at which the grains are still decoupled. Thus the characteristic velocity of a grain with respect to the gas corresponds to the velocity dispersion of the turbulence on the scales corresponding to eddies with turnover time equal to (Draine & Salpeter 1979). Using the Kolmogorov scaling relation , Draine (1985) obtained the largest velocity dispersion in hydrodynamic turbulence , where is the eddy turnover time at the injection scale.

A complication, though, comes from the fact that most astrophysical fluids are magnetized. Therefore magnetohydrodynamic (MHD) turbulence should be used to characterize interstellar turbulence. This was attempted first in L94. A more quantitative approach was adopted in Lazarian & Yan (2002) and Yan & Lazarian (2003, henceforth YL03). There, in accordance with the simulations in Cho & Lazarian (2002), the MHD turbulence was decomposed into an Alfven, slow and fast modes. The particular scalings of the modes were applied, i.e., Goldreich & Sridhar (1995) scaling for Alfven and slow modes, and acoustic turbulence scaling for fast modes. Moreover, in YL03 we considered a gyro-resonance between the fluctuating magnetic field and charged grains, and thus identified a new mechanism of grain acceleration.

Specifically, the resonance condition that the Doppler-shifted frequency of the wave in the grain’s guiding center rest frame is a multiple of the particle gyrofrequency : , (). Basically, there are two main types of resonant interactions: gyroresonance acceleration and the transit one. The transit acceleration () requires longitudinal motions that are present only for compressible modes. As the dispersion relation for fast waves is , it is clear that it is applicable only to the super-Alfvenic (for a low medium, i.e. with magnetic pressure higher than the thermal one, as ) or supersonic (for a high medium) grains. For low speed grains that we deal with, gyroresonance is the dominant MHD interaction. The calculation by YL03 showed that grains gain the maximum velocities perpendicular to the magnetic field, so the averaged decreases. This is understandable since the electric field accelerating the grain is perpendicular to the magnetic field.

The results of the theory were applied to various idealized phases of the interstellar medium in Yan, Lazarian & Draine (2004). In Fig. 6, we show the velocity of grain as a function of the grain size in CNM.

Figure 6: (a) Left panel.– Relative velocities as functions of grain radius for silicate grains in the Cold Neutral Medium. The dotted line represents the gyroresonance with fast modes. The dash-dot line refers to the gyroresonance with Alfvén modes. The cutoff is due to viscous damping. From Yan & Lazarian (2003). (b) Right panel.– Grain velocities in CNM gained from gyroresonance for different magnetic field strengths. From Yan, Lazarian & Draine (2004).

The acceleration by gyroresonance in both MC and DC are not so efficient as in other media. This happens in MC and DC because the time for the gyroresonant acceleration, , are much shorter that in the WNM. In MC and DC, due to high density, the drag time is less than the gyro-period for grains larger than  cm.

For molecular clouds Roberge & Hanany (1990) and Roberge, Hanany & Messinger (1995) considered ambipolar diffusion666A similar process was considered by Roberge & Desch (1990) for molecular accretion disks.. They demonstrated that this diffusion entails supersonic relative drift. The action of the mechanism is expected to be localized, however.

To finish our brief discussion of grain motion in magnetized medium consider magnetized shocks. The basic idea is that the weakly charged grains are like ions with high mass to charge ratio (Epstein 1980). Thus they can easily diffuse farther back upstream of the shock and be accelerated more efficiently to suprathermal energies. Nevertheless, the shock acceleration is inefficient for low speed grains. The reason for this is that the efficiency of the shock acceleration depends on the scattering rate, which is determined by the stochastic interaction with the turbulence. For low speed particles, the scattering rate is lower than the rate of momentum diffusion. In this case, the stochastic acceleration by turbulence happens faster than dust acceleration by shocks (YL03).

4 Grain Alignment Theory: Major Mechanisms

4.1 Tough Problem

We have seen in the previous sections that both linear and circular polarizations depend on the degree of grain alignment given by the Rayleigh reduction factor (see Eq. (4)). Therefore it is the goal of the grain alignment theory to determine this factor. Table 1 shows that the wide range of different time scales involved makes the brute force numerical approach doomed.

A number of different mechanisms that produce grain alignment has been developed by now. Dealing with a particular situation one has to identify the dominant alignment process. Therefore it is essential to understand different mechanisms.

The history of grain alignment is really exciting. A real constellation of illustrious scholars, e.g. L. Spitzer and E. Purcell contributed to the field. Our earlier discussion of the complex dynamics of a grain explains why the grain alignment theory still requires theoretical efforts. Note, that most of the effects we discussed in the previous section were discovered in the process of work on grain alignment.

A drama of ideas in historic perspective is presented in Lazarian (2003). It was shown there that the work on grain alignment can be subdivided into a number stages, such that at the end of each the researchers believed that the theory was adequate. However, higher quality observational data made it clear that more work was required.

Notations:
: minor axis                                                                 : major axis
                                                            : ratio of axes
: ratio of moment inertia                                    : normalized grain density
: normalized gas temperature                          : normalized dust temperature
: rotation temperature
: normalized gas density                              : normalized magnetic field
: real part of magnetic susceptibility            
: imaginary part of magnetic susceptibility by electron and nuclear spin
: grain magnetic moment                                              : magnetogyric ratio for electron
: magnetogyric ratio nuclei
: normalized magnetic moment of nucleus           ergs G
: grain angular momentum at        : grain angular momentum at
: total nuclear relaxation time             can also include inelastic relaxation
: energy density of radiation field           : wavelength of radiation field
: third component of radiative torques       : electric field
: electric dipole moment                          : electric constant
: normalized voltage                                       : angular velocity
: magnitude of torque                                              fraction of absorbed atoms
: kinetic energy of H                                      cm: density of recombination sites
                                                  : nuclear spin-spin relaxation rate
: electron spin-spin relaxation rate                                ; : Bohr magneton
                                                     

Table 1: Time-scales relevant for grain alignment

4.2 Paramagnetic Alignment

The Davis-Greenstein (1951) mechanism (henceforth D-G mechanism) is based on the paramagnetic dissipation that is experienced by a rotating grain. Paramagnetic materials contain unpaired electrons that get oriented by the interstellar magnetic field . The orientation of spins causes grain magnetization and the latter varies as the vector of magnetization rotates in the grain body coordinates. This causes paramagnetic loses at the expense of the grain rotation energy. Be mindful, that if the grain rotational velocity is parallel to , the grain magnetization does not change with time and therefore no dissipation takes place. Thus the paramagnetic dissipation acts to decrease the component of perpendicular to and one may expect that eventually grains will tend to rotate with provided that the time of relaxation is much shorter than the time of randomization through chaotic gaseous bombardment, . In practice, the last condition is difficult to satisfy. It is clear from Table 1 that for cm grains in the diffuse interstellar medium, is of the order of s , while is s if magnetic field is G and temperature and density of gas are K and cm, respectively.

The first detailed analytical treatment of the problem of D-G alignment was given by Jones & Spitzer (1967) who described the alignment of using the Fokker-Planck equation. This approach allowed them to account for magnetization fluctuations within the grain material, and thus provided a more accurate picture of the alignment. The first numerical treatment of D-G alignment was presented by Purcell (1969). By that time, it became clear that the original D-G mechanism is too weak to explain the observed grain alignment. However, Jones & Spitzer (1967) noticed that if interstellar grains contain superparamagnetic, ferro- or ferrimagnetic inclusions777The evidence for such inclusions was found much later through the study of interstellar dust particles captured in the atmosphere (Bradley 1994)., the may be reduced by orders of magnitude. Since of atoms in interstellar dust are iron, the formation of magnetic clusters in grains was not far fetched (see Martin 1995). However, detailed calculations in Roberge & Lazarian (1999) showed that the degree of alignment achievable cannot account for the observed polarization coming from molecular clouds if grains rotate thermally. This is the consequence of the thermal suppression of paramagnetic alignment first discussed by Jones & Spitzer (1967). These internal magnetic fluctuations randomize grain orientation with respect to the magnetic field if the grain body temperature is close to the rotational one.

P79 pointed out that fast rotating grains are immune to both gaseous and internal magnetic randomization. Thermal trapping that we discussed in §3.2 limits the range of grain sizes for which Purcell’s torques can be efficient (LD99ab). For grains that are less than the critical size, which can be  cm and larger, rotation is essentially thermal (see section 3.2). The alignment of such grains is expected to be in accordance with the DG mechanism predictions (see Lazarian 1997, Roberge & Lazarian 1999), and seem to be sufficient to explain the residual alignment of small grains that is seen in the Kim & Martin (1995) inversion (see §6.5).

Lazarian & Draine (2000) predicted that PAH-type particles can be aligned paramagnetically due to the relaxation that is faster than the DG predictions. In fact, they showed that the DG alignment is not applicable to very swiftly rotating particles, for which the Barnett magnetic field gets comparable to magnetic fields induced by uncompensated spins in the paramagnetic material. For such grains, this relaxation is more efficient than the one considered by Davis & Greenstein (1951). This effect, that is termed “resonance relaxation” in Lazarian & Draine (2000), allows the alignment of PAHs. These tiny “spinning” grains are responsible for the anomalous foreground microwave emission (Draine & Lazarian 1998, see also Lazarian & Finkbeiner 2003 for a review).

4.3 Mechanical Alignment

The Gold (1951) mechanism is a process of mechanical alignment of grains. Consider a needle-like grain interacting with a stream of atoms. Assuming that collisions are inelastic, it is easy to see that every bombarding atom deposits with the grain an angular momentum , which is directed perpendicular to both the needle axis and the velocity of atoms . It is obvious that the resulting grain angular momenta will be in the plane perpendicular to the direction of the stream. It is also easy to see that this type of alignment will be efficient only if the flow is supersonic888Otherwise grains see atoms coming not from one direction, but from a wide cone of directions (see Lazarian 1997a) and the efficiency of alignment decreases..

Suprathermal rotation introduced in Purcell (1979) persuaded researchers that mechanical alignment is marginal. Indeed, it seems natural to accept that fast rotation makes it difficult for gaseous bombardment to align grains. However, the actual story is more interesting. First of all, it was proven that mechanical alignment of suprathermally rotating grains is possible (Lazarian 1995). Two mechanisms that were termed “crossover” and “cross section” alignment were introduced there. The mechanisms were further elaborated and quantified in Lazarian & Efroimsky (1996), Lazarian, Ozik & Efroimsky (1996), Efroimsky (2002b). Second, as we discussed in §3.3, the supersonic velocities are available over substantial regions of interstellar medium, both due to MHD turbulence and ambipolar diffusion.

In fact, the discovery of thermal trapping (§3.2) made the original Gold (1951) mechanism more relevant. Therefore when grains are not thermally trapped and rotate suprathermally the crossover and cross section alignments should take place, while for thermally trapped grains the original Gold mechanism remains in force. The quantitative numerical study of the Gold alignment in Roberge et al. (1995) was done under the assumption of the perfect coupling of with the axis of maximal inertia (cf §3.1). This study shows a good correspondence with an analytical formulae for the alignment of vector in L94 when the gas-grain velocities are transsonic. An analytical study in Lazarian (1997) accounts for the incomplete internal alignment in a more sophisticated way, compared to L94, and predicts the Rayleigh reduction factors of and more for grains interacting with the Alfven waves. A detailed numerical study would be in order to test the predictions.

4.4 Radiative Torque Alignment

Anisotropic starlight radiation can both spin the grains and align them. This was first realized by Dolginov & Mytrophanov (1976), that radiative torques are bound to induce alignment. In their paper they considered a tilted oblate grain with the helicity axes coinciding with the axis of maximal inertia, as well as a tilted prolate grain for which the two axes were perpendicular. They concluded, that subjected to a radiation flux, the tilted oblate grain will be aligned with longer axes perpendicular to magnetic field, while the tilted prolate grain will be aligned with the longer axes parallel to magnetic field. At that time the internal relaxation was not yet a part of accepted grain dynamics. The problem was revisited by Lazarian (1995), who took into account the internal relaxation and concluded that both prolate and oblate grains will be aligned with longer axes perpendicular to the magnetic field. However, Lazarian (1995) did not produce quantitative calculations and underestimated the relative importance of radiative torque alignment compared to other mechanisms.

Figure 7: (a) Left panel.– A model of a “helical” grain, that consists of a spheroidal grain with an inclined mirror attached to it, reproduces well the radiative torques (from LH06). (b) Right panel.– The “scattering coordinate system” which illustrates the definition of torque components: is directed along the maximal inertia axis of the grain; is the direction of radiation. The projections of normalized radiative torques , and are calculated in this reference frame for .

It happened that the Dolginov & Mytrophanov (1976) study came before its time. The researchers themselves did not have reliable tools to study the dynamics of irregular grains and the impact of their work was initially low. Curiously enough, Purcell studied the aforementioned work, appreciated the Barnett magnetization described there, but did not recognized the importance to the radiative torques. In fact, he had means to calculate them numerically using the Discreet Dipole Approximation (DDA) code available to him.

The explosion of interest to the radiative torques we owe to Bruce Draine, who realized that the torques can be treated with the DDA code by Draine & Flatau (1994) and modified the code correspondingly. The magnitude of torques were found to be substantial and present for grains of all irregular shapes studied in Draine 1996, DW96 and DW97. After that it became impossible to ignore the radiative torque alignment. More recently, radiative torques have been studied in laboratory conditions (Abbas et al. 2004).

Potentially, the isotropic radiative torques could ensure suprtathermal rotation and provide the alignment in the spirit of P79 mechanism. Indeed, radiative torques are related to the volume of the grain. Therefore a deposition of a monolayer of atoms over the grain surface, i.e. resurfacing, that can reverse the direction of Purcell’s torques, does not affect the radiative torques. Long-lived suprathermal torques may ensure efficient paramagnetic alignment. In this way the idea of radiative torques is presented in a number of research papers. This way of thinking about radiative torque alignment is erroneous, however.

In fact, isotropic torques are fixed in grain coordinates and in all respect are similar to the Purcell’s torques. Therefore, typical interstellar grains driven only by isotropic radiative torques cannot rotate suprathermally due to the thermal trapping effect that we discussed in §3.2.

Moreover, in most cases the radiation field that we deal with has an appreciable anisotropic component. This component induces torques that can align grains. DW97 study confirmed that the torques tend to align grains with long axes perpendicular to magnetic field.

Objectively, the DW96 and DW97 papers signified a qualitative change in the landscape of grain alignment theory. These papers claimed that radiative torques alignment may be the dominant alignment mechanism in the diffuse interstellar medium. However, questions about the nature of the alignment mechanism, the particular choice of grains studied, as well as the efficiency of radiative torques in different environments remained. In addition, the DW97 treatment ignored the physics of crossovers (see §3.2). In view of that, I recall my conversations with Lyman Spitzer, who was excited about the efficiency of radiative torque, but complained that he was lacking a clear physical picture of the alignment mechanism.

Figure 8: Examples of irregular shapes studied in LH07.
Figure 9: (a) Left panel.– Two components of the radiative torques are shown for our analytical model (solid lines) in Fig. 7a and for an irregular grain in Fig. 8 (dashed lines). (b) Right panel.– Radiative torques for different grain shapes. From Lazarian & Hoang (2006).

To address this concern LH07 proposed a simple model that reproduces well the essential basic properties of radiative torques. The model consists of an oblate grain with a mirror attached to its side (see Fig. 7a). This model allows an analytical treatment and provides an physical insight why irregular grains get aligned. In fact, it shows that for a range of angles between the radiation and the magnetic field the alignment gets “wrong”, i.e. with the long axes parallel to magnetic field. However, this range is rather narrow (limited to radiation direction nearly perpendicular to magnetic field) and tends to disappear in the presence of internal wobbling (see §3.1).

In LH07 we concluded that the alignment of grains with longer axes perpendicular to magnetic field lines is a generic property of radiative torques that stems from the basic symmetry properties of the radiative torque components. Our work showed that the entire description of alignment may be obtained with the two components of the radiative torques and as they are defined in the caption of Fig. 7. The third component is responsible for grain precession only. The functional dependences of the torque components that are experienced by our model grain are similar to those experienced by irregular grains shown in Fig. 8. It is really remarkable that our model and grains of very different shapes have very similar functional dependences of their torque components (see Fig. 9)! Note that the particular set of grains is “left-handed”. For “right handed” grains both and change simultaneously in a well defined manner. For our grain model to become “right handed” the mirror should be turned by 90 degrees.

The phase trajectories in Fig. 10 show that only a small fraction of grains get to attractor points with high angular momentum. It is most probable for a arbitrary chosen grain to end up at the attractor point that, in the absence of grain thermal wobbling and gaseous bombardment, corresponds to . Within this model it is only natural to get grain aligned with when thermal wobbling is included, as this is observed in WD03 (see §3.2).

Figure 10: (a) Left panel.– Phase trajectory map obtained for the model grain given shown in Fig. 7. (b) Right panel.– The same for an irregular grain in Fig. 8 (shape 1). From Lazarian & Hoang (2006).

What does make grains helical? Both rotation about a well defined axis and grain irregularity do this. For instance, if we attach the weight-less mirror to a sphere rather than an oblate body, this would average out the radiative torques as the mirror will be constantly changing its orientation in respect to the rotational axis.

4.5 Sub-sonic Mechanical Alignment as Next Challenger

As we mentioned earlier, the requirement of the supersonic drift limits the applicability of mechanical alignment. Such drift is, however, not necessary for helical grains. The model grain in Fig. 7a is helical not only in respect to radiation, but also to mechanical flows (see also LH07). In fact, the functional dependence of the torques that we obtain for our model grain does not depend on whether photons or atoms are reflected from the mirror. Therefore we may predict that, if atoms bounce from the grain surface elastically, the helical grains999The mechanical alignment of helical grains was briefly discussed in Lazarian (1995) and Lazarian, Goodman & Myers (1997), but was not elaborated there. will align with long grain axes perpendicular to the flow in the absence of magnetic field. If the dynamically important magnetic field is present, the alignment is expected with long axes perpendicular to . If atoms stick to the grain surface and then are ejected from the place of their impact, this changes the values of torques by a factor of order unity.

It is easy to understand why supersonic drift is not required for helical grains. For such grains the momentum deposited by regular torques scales in proportion to the number of collisions, while the randomization adds up only as a random walk. In fact, the difference between the mechanical alignment of spheroidal and helical grains is similar to the difference between the Harwit (1971) alignment by stochastic absorption of photons and the radiative torque alignment. While the Harwit alignment requires very special conditions to overpower randomization, the radiative torques acting on a helical grain easily beat randomization.

Similarly, as in the case of the radiative torques, it is possible to disregard the Harwit process, it may be possible to disregard the Gold-type processes (see Table 2 and §4.3) for irregular grains. As the grain helicity does not change sign during grain flipping, the thermal trapping effects described in LD99a are absent for the mechanical spin-up of helical grains.

The properties of helical grains require detailed studies. For instance, in the presence of Purcell’s thrusters and no flipping (see Fig. 4a), the helical grain may induce its own translational motion as it rotates.

What would it take to make a grain helical for mechanical interactions? This is a question similar to one that worried researchers with the radiative torques before Bruce Draine made his simulations. We do not have the simulations of mechanical torques on irregular grains, but in analogy with the radiative torques, I would claim that such torques should be generic for an irregular grain, provided that there is a correlation of the place where an atom hits the grain and where it evaporates from the grain. It is intuitively clear that the effects of helicity should be more important for larger grains. As the relative gas-grain drift induced by gyroresonance (Yan & Lazarian 2003) is faster for larger grains this can be used as another argument for relatively better alignment of large helical grains.

As the physics of helical grain alignment and those previously known mechanical alignment mechanisms is different, we can talk of a completely new process of alignment that can be tentatively termed “sub-sonic mechanical alignment” to stress its independence of supersonic drift. The traditional supersonic mechanical alignment mechanisms we discussed in §4.3 tend to minimize grain cross section. This means, for instance, that for grains streaming along magnetic fields, the stochastic torques tends to align grains with longer axes parallel to magnetic field. On the contrary, our study in LH07 showed that the mechanical torques on helical grains tend to align grains in the same way as the radiative torques do, i.e., the helical grains will tend to be aligned perpendicular to magnetic field irrespectively of the direction of the drift. Further work should show in what situations the “sub-sonic mechanical alignment” can reveal magnetic fields when radiative torques fail to do this.

All in all, our considerations above suggest that the helicity is an intrinsic property of rotating irregular grains and therefore the mechanical alignment of helical grains should overwhelm any mechanical alignment process discussed in §4.3 when the two mechanism tend to align grains in opposite directions. This raises questions of whether we can ever expect to have alignment with grain long axes parallel to magnetic field (cf. Rao et al. 1998, Cortes et al. 2006). Can the alignment of helical grains fail? This can happen, for instance, in the absence of correlation of the impact and evaporation sites of impinging atoms. This issue can be clarified by laboratory studies.

5 Dominant Mechanism: Progress and Problems

5.1 Niches for Mechanisms and Quantitative Theory

It is clear that the major alignment mechanisms discussed in §4 have their own niches. For instance, Davis-Greenstein mechanism should be important for small paramagnetic grains as the ratio of the paramagnetic alignment rate to the gaseous randomization rate scales inversely proportional to grain size (see Lazarian & Draine 2000). At the same time, the most promising mechanism, the radiative torque one, is not efficient for sufficiently small grains (i.e. ). We summarize the current situation with the known alignment mechanisms by Table 2. Conservatively, we did not include in the table the mechanical alignment of helical grains, an interesting mechanism that have not been properly studied yet.

If grains are superparamagnetic (Jones & Spitzer 1967, Mathis 1986), they can be aligned, provided that their rotational temperature is larger than the grain temperature. As the rate of paramagnetic relaxation for “super” grains is larger than the rate of collisional damping, it is this faster rate that should strongly affect the phase trajectory of grains subjected to radiative torques.

We showed that the gyroresonance acceleration of grains discussed in §3.3 allowed an efficient acceleration of grains to supersonic velocities. Note, that the processes enabling a supersonic drift have been the stumbling block for the mechanism. In this sense the Gold-type mechanisms for thermally rotating grains and crossover and cross section mechanisms for suprathermally rotating grains might look currently promising. However, the competition with the mechanical alignment of helical grains and the radiative torque alignment limits the range of circumstances where the process dominates.

The radiative torque alignment looks the most promising at the moment. As we have discussed in §4.3 it allows predictions that correspond well to observational data. Nevertheless, both the observational testing of the theory and the improvement of the radiative torque “cookbook” are essential. Some of the required improvements are obvious. For instance, the position of the low- attractor points (see Fig. 10), at which most of the aligned grains reside, show variations with the grain shape. Therefore to predict the expected alignment measure, i.e., (see eq. 4), more precisely, one may need to consider a variety of grain shapes. The calculation of radiative torques for a given radiation spectrum, a given distribution of grain sizes and a variety of shapes is a challenging computational task. Fortunately, LH07 showed that, with satisfactory accuracy the radiative torques demonstrate self-similarity, i.e. can be presented as a function of only (see also Cho & Lazarian 2006).

The quantitative theory for different mechanisms is at different stages of development. For some mechanisms, e.g. for the Davis-Greenstein alignment, the theory is detailed and well-developed for spheroidal grains (see Roberge & Lazarian 1999 and references therein). There are reasons to believe that these results should be applicable also to realisticly irregular grains. However, at the moment the mechanism does not looks promising for alignment of grains larger than  cm. The paramagnetic alignment theory for suprathermally rotating grains had been developed before the discovery of thermal flipping and thermal trapping effects (Lazarian & Draine 1999ab). Therefore the model of alignment in Lazarian & Draine (1997) is applicable to grains larger than a critical size which is approximately  cm. The relative role of the Purcell suprathermal torques and the radiative torques requires further studies for various astrophysical environments. Cho & Lazarian (2005) claimed that the radiative torques are dominant for the molecular cloud interiors where large grains are present. Similarly, the domain of applicability of the suprathermal mechanical alignment (Lazarian 1995, Lazarian & Efroimsky 1996, Lazarian et al. 1996, Efroimsky 2002a) is also limited by the grains larger than .

The radiative torque alignment mechanism has undergone dramatic changes in the last 10 years. From the mostly forgotten one it has risen to the dominant one. The alignment has been studied for grains assuming perfect alignment (DW97, LH07), as well as taking into account thermal fluctuations (WD03, Hoang & Lazarian 2007). Moreover, the process of alignment is not any more a result of numerical experimentation. A simple analytical model in LH07 does reproduce the essential features of the alignment. However, a more rigorous studies of the effects of the incomplete internal alignment on radiative torques are necessary. An approach based on the elimination of the fast variable presented in Roberge (1997) seems promising if we want to get precise measures for the grain alignment (see eq. 4)).

Obtaining alignment measures when several alignment processes act simultaneously is another challenge for quantitative studies. It has been addressed in Roberge et al. (1995) numerically and in Lazarian (1997) analytically for the situation when the mechanical and paramagnetic alignment mechanisms act simultaneously. In reality, a number of possible combinations is higher and the interaction of different mechanisms may be very non-linear. For instance, radiative torques can prevent some grains from thermal flipping thus changing the conditions for other mechanisms to act. The studies in WD03, LH07 and Hoang & Lazarian (2007) show that the fraction of such suprathermal grains is not large, however.

5.2 New Situations, New Challenges

As the grain alignment theory matures, it starts to deal with a wider variety of astrophysical situation, rather than just interstellar grains. This opens new opportunities for astrophysical magnetic fields studies, but also poses new challenges.

Consider, for instance, the alignment of grains in accretion disks. The grains their may be very large, up to “pebble” size. As grains get larger the physics of their alignment changes101010The observations of very large aligned grains is a separate issue that we do not dwell upon. If aligned grains are much larger than the wavelength of observations, they do not produce polarized signal. This means that to study the alignment of large grains one should increase the wavelength of observations. The magnetic field mapping with aligned large grains may require taking into account polarized synchrotron foreground.. For instance, for grains larger than cm the mechanical alignment arising from the difference in the positions of the center of pressure and the center of gravity, the so-called “weathercock mechanism” (Lazarian 1994b), gets important. In addition, for larger grains, the internal alignment through nuclear relaxation gets subdominant compared to inelastic relaxation (P79, Lazarian & Efroimsky 1999). Eventually, all internal alignment mechanisms get inefficient. This is a regime that earlier researchers, who were unaware of internal relaxation processes, dealt with (see Dolginov & Mytrophanov 1976).

Interestingly enough, some earlier abandoned mechanisms may get important in new situations. Take, for instance, the “iron fillings” mechanism, that considers alignment of iron needles along magnetic fields. This mechanisms proposed by Spitzer & Tukey (1951) at the very beginning of the grain alignment studies, may still be important if grains are sufficiently large and magnetic fields are strong.

Environments for alignment may be quite exotic. For instance, it is a good bet to disregards electric fields in interstellar gas. However, it may not always be true. According to private communication from Jim Hough electric fields could align dust grains in the Earth atmosphere. Serezhkin (2000) estimated electric fields that may be present in comet comas. This opens a completely new avenue for research. Indeed, first of all, electric fields can serve as the “axis of alignment” provided that grains have dipole moments111111Even in the absence of electric field grain dipole moments can affect grain dynamics (see Draine & Lazarian 1998b, Yan et al. 2004, Weingartner 2006). (see a discussion of the latter point in Draine & Lazarian 1998). Thus, the radiative torque, subsonic and supersonic mechanical alignment processes can happen in respect to the electric field. Then, an analog of the “iron fillings” alignment is possible, especially, if grains have properties of electrets (Hilczer & Malecki 1986). Moreover, an electric analog of paramagnetic relaxation is possible as grains rotate in electric field. Some materials, e.g. segnetoelectrics (see Mantese and Alphay 2005), are particularly dissipative and can act the same way as superparamagnetic inclusions act to enhance the efficiency of the D-G relaxation.

The issue of the direction of alignment requires care when the parameters of the environment changes. For instance, it was discussed in Lazarian (2003) that the alignment in typical interstellar medium conditions would happen in respect to magnetic field, irrespectively of the mechanism of alignment. This is the consequence of the fast Larmor precession. Even if magnetic field changes its direction over the time scales longer compared to the Larmor period, the angle between and local is preserved as the consequence of the preservation of the adiabatic invariant. Note, that depending on the mechanism, the grains may align with their longer axes either perpendicular or parallel to magnetic field, however.

Other situations when magnetic field is not the axis of alignment are also possible. Consider, for instance, radiative torques. Whether the radiation direction or magnetic field is the axis of alignment depends on the precession rate around these axes (see Table 1). For instance, in the vicinity of stars the grains can to get aligned in respect to the radiation flux, however. For a star the radius at which the light acts as the axis of alignment changes from of AU for magnetic field of  G to AU for the field of  G (LH07). Light flashes from supernovae explosions may impose the direction of the photon flux as the alignment axis over larger scales. At the same time, one can check that for typical diffuse ISM (see Table 1) is , i.e. the Larmor precession is much faster than precession induced by radiation. Therefore magnetic field stays the alignment axis as it was assumed in the earlier work.

Similarly, gas streaming can induce its own alignment direction. Dolginov & Mytrophanov (1976) assumed that whether magnetic field or a gaseous flow defines the axis of alignment depended on the ratio of Larmor precession time to that of mechanical alignment. LH07, however, concluded that the precession time of a grain in a gaseous flow (an analog of in Table 1) should be taken instead. The latter time is orders of magnitude less than the time assumed in Dolginov & Mytrophanov (1976). As the result, high density molecular outflows can overwhelm the magnetic field and impose its direction as the direction of alignment, provided that the directions of the outflow and magnetic field do not coincide (LH07). Interestingly enough, the mechanical flows can define the axis of alignment even for subsonic flow velocities, i.e. at those velocities for which the process considered by Dolginov & Mytrophanov (1976) is not efficient.

Other processes may also be important in more restricted situations. Consider, for instance, the grain spun-up by cosmic rays (Sorrell 1995ab). The calculations by Lazarian & Roberge (1997b) show, that for the cosmic-ray-induced torques to be important, the enhancement of the low energy cosmic ray flux over its typical interstellar value by a factor of more than is necessary. Therefore this process could only be important over localized regions near cosmic ray sources.

5.3 Avenues for Theory Advancement

It is easy to notice that both studies of irregular grains and subtle physical effects have provided the major boost for the grain alignment theory. Indeed, the theory started with the favorite with physicists “spherical cow” model, which literally corresponded to the assumption of spherical grains in D-G model. Later, the studies of the alignment of oblate and prolate grains have been undertaken. However, completely new effects were revealed when irregular grains were considered. Indeed, the helicity, which is the key ingredient for both the operation of the radiative torques (see §4.4) and the subsonic mechanical alignment (see §4.5), is zero for spheroidal grains.

Similarly, an adequate treatment is necessary for grain properties. Originally grains were considered as solid absolutely rigid passive bricks without internal structure. It is only later, that effects of elasticity as well as magneto-mechanical effects were considered. The back-reaction of thermal fluctuations on grain dynamics through these effects changed drasticly our understanding of both grain dynamics and alignment. Improvements in this direction can be obtained by accounting for the triaxial ellipsoids of inertia corresponding to irregular grains. Some work in this direction has been already done for the inelastic relaxation (see Efroimsky 2000).

We believe that more effects will be considered as grain alignment theory matures and is being applied to new astrophysical environments. For instance, we have discussed above, that potentially grain surface physics may be essential for the mechanical alignment of helical grains. Plasma-grain interactions seem to be another promising direction, which has been marginally developed so far (see Draine & Lazarian 1998b, Yan et al. 2004, Shukla & Stenflo 2005).

6 Polarimetry and Grain alignment

6.1 Grain Alignment in Molecular Clouds

Polarization arising from aligned grains provides a unique source of information about magnetic fields in molecular clouds. For many years this has been the most important practical motivation for developing the grain alignment theory.

The data on grain alignment in molecular clouds looked at some point very confusing. On one hand, optical and near-infrared polarimetry of background stars did not show an increase of polarization degree with the optical depth starting with a threshold of the order of a few (Goodman et al. 1995, Arce et al. 1998). This increase would be expected if absorbing grains were aligned by magnetic field within molecular clouds. On the other hand, far-infrared measurements (see Hildebrand 2000, henceforth H00) showed strong polarization that was consistent with emission from aligned grains. A quite general explanation to those facts was given in Lazarian, Goodman & Myers (1997, henceforth LGM97), where it was argued that all the suspected alignment mechanisms are based on non-equilibrium processes that require free energy to operate. Within the bulk of molecular clouds the conditions are close to equilibrium, e.g. the temperature difference of dust and gas drops, the content of atomic hydrogen is substantially reduced, and the starlight is substantially attenuated. As the result the major mechanisms fail in the bulk part of molecular clouds apart from regions close to the newly formed stars as well as the cloud exteriors that can be revealed by far-infrared polarimetry.

The alternative explanations look less appealing. For instance, Wiebe & Watson (2001) noted that small scale turbulence in molecular clouds can reduce considerably the polarization degree even if grain alignment stays efficient. This, however, is inconsistent with the results of the far-infrared polarimetry that revealed quite regular pattern of magnetic field in molecular clouds (see H00).

An extremely important study of alignment efficiency has been undertaken by Hildebrand and his coworkers (Hildebrand et al. 1999, Hildebrand 2000, 2002). They pointed out that for a uniform sample of aligned grains, made of dielectric material consistent with the rest of observational data, polarization degree, , should stay constant for within the far-infrared range. The data at 60 m, 100 m from Stockes on the Kuiper Airborne Observatory, 350 m from Hertz on Caltech Submillimeter Observatory, and 850 m from SCUBA on the JCMT revealed a very different picture. This was explained (see Hildebrand 2002) as the evidence for the existence of the populations of dust grains with different temperature and different degree of alignment. The data is consistent with cold (T=10 K) and hot (T=40 K) dust being aligned, while warm (T=20 K) grains being randomly oriented (H00). If cold grains are identified with the outer regions of molecular clouds, hot grains with regions near the stars and warm with the grains in the bulk of molecular clouds the picture gets similar to that in LGM97.

Figure 11: (a) Left Panel: Aligned grain size vs. visual extinction . For the threshold suprathermal angular velocity 5 times larger than the thermal angular velocity was chosen. It is clear that increase of grain size can compensate for the extinction of light in cloud cores. Solid line: ; Dotted line: in the cloud. (from Cho & Lazarian 2005) (b) Right Panel: The 850 m emission map of the model cloud. Superimposed are the projected polarization vectors (from Bethell et al. 2006).

However, the data obtained for pre-stellar cores in Ward-Thompson et al. (2000) seem to be at odds with the LGM97 predictions. Indeed, the properties of these cores summarized in Ward-Thompson et al. (2002) and Crutcher et al. (2004) fit into the category of zones that, according to LGM97, should not contain aligned grains.

What could be wrong with the LGM97 arguments? The latter paper treats grains of  cm size. The grains in prestellar cores can be substantially larger. Grain alignment efficiency depends on grain size. Therefore the estimates in LGM97 had to be reevaluated.

Cho & Lazarian (2005, henceforth CL05) revealed a steep dependence of radiative torque efficiency on grain size. While an earlier study by Draine & Weingartner (1996) was limited by grains with size  cm, CL05 studied grains up to  cm size subjected to the attenuated radiative field calculated in accordance with the prescriptions in Mathis, Mezger & Panagia (1983). Fig. 11a shows that large grains can be efficiently span up by radiative torques even at the extinction of of 10 and higher. A numerical treatment of the radiative transfer was used in the papers that followed, e.g. Pelkonen et al. (2006), Bethell et al. (2006) (see Fig. 11b and 12).

Figure 12: (a) Left Panel.– The polarization degree for 850 m emission from a cloud as a function of normalized emission intensity for the actual calculated degrees of alignment (real alignment) and assuming that all grains are perfectly aligned. (b) Right Panel.– The polarization spectra of a model core and a “starless” molecular cloud. The projected Hale polarimeter wave-band coverage is also shown. From Bethell et al. (2006).

Fig. 12a illustrates that a naive assumption of the perfect alignment results in a substantial overestimation of the polarization degree. While the polarization spectra in Fig. 12b is obtained for a starless core/cloud, a more non-trivial behavior is expected for a cloud with active star formation. This calls for multi-frequency observations (see also H00).

We note, that in CL05 and the subsequent papers the efficiencies of radiative torques in terms of alignment were parameterized in terms of maximal rotational velocities achievable by the torques. As we discussed in §4.4, most of the interstellar grains do rotate thermally in the presence of radiative torques. Nevertheless, the above parameterization does characterize the relative role of the randomizing atomic collisions and aligning effects of the radiative torques. Our tests that include simulated gaseous bombardment in Hoang & Lazarian (2007) show that grains are being aligned by radiative torques when .

6.2 Testing Alignment at the Diffuse/Dense Cloud Interface

The grain alignment theory can be directly tested at the cloud interface. Mathis (1986) explained the dependence of the polarization degree versus wavelength , namely the Serkowski law (Serkowski 1973) (see also Fig. 13a)

(11)

(where corresponds to the peak percentage polarization and is a free parameter), assuming that it is only the grains larger than the critical size that are aligned. Those grains were identified in Mathis (1986) as having superparamagnetic inclusions and therefore subjected to more efficient paramagnetic dissipation.

The ratio of the total to selective extinction reflects the mean size of grains present in the studied volume. It spans from in diffuse ISM to in dark clouds (see Whittet 1992 and references therein) as the mean size of grain increases due to coagulation or/and mantle growth. The earlier studies were consistent with the assumption that the growth of was accompanied by the corresponding growth of (see Whittet & van Brenda 1978). The standard interpretation for this fact was that as grains get bigger, the larger is the critical size starting with which grains get aligned. This interpretation was in good agreement with Mathis’ (1986) hypothesis of larger grains having superparamagnetic inclusions. However, a more recent study by Whittet et al. (2001) showed that grains at the interface of the Taurus dark cloud do not exhibit the correlation of and . This surprising behavior was interpreted in Whittet et al (2001) as the result of size-dependent variations in grain alignment with small grains losing their alignment first as deeper layers of the cloud are sampled. Whittet et al (2001) did not specify the alignment mechanism, but their results posed big problems to the superparamagnetic mechanism (see §4.2). Indeed, the data is suggestive that and therefore the mean grain size may not grow with extinction while the critical size for grain alignment grows.

Figure 13: (a) Left Panel: Serkowski curves and fits by radiative torque models. (b) Right Panel: as function of from our calculations with radiative torques (solid line) and the observation data by Whittet et al. (2001). The interface is simulated as a homogeneous slab. The MRN distribution of dust with was used. From Hoang & Lazarian (in preparation).

Lazarian (2003) noticed that the Whittet data agrees well with the expectations of the radiative torque mechanism. We present in Fig. 13b our recent fit for the data using the radiative torques that arise from the attenuated interstellar radiation field.

6.3 Alignment in Magnetized Disks around stars

Magnetic field plays important roles in the evolution of protostellar disks. Magnetic pressure can provide extra support to the disks and magnetic field can promote removal of angular momentum from disks (see Velikov 1959; Chandrasekhar 1961; Balbus & Hawley 1991). However, there are many uncertainties in the structure and the effects of the magnetic field in protostellar disks. Quantitative studies of magnetic fields in the disks are essential therefore.

Consider T Tauri stars first. Tamura et al. (1999) detected polarized emission from T Tauri stars, which are low mass protostars. Aitken et al. (2002) studied polarization that can arise from magnetized accretion disks. They considered a single grain component consisting of the 0.6 silicate and used an ad hoc assumption that all grains at all optical depths are aligned with .

Cho & Lazarian (2006) used a more sophisticated model for grain alignment. They calculated the radiative torques acting on grains, assuming the model of the disk in Fig. 14a. The results of their calculations are shown in Fig. 14b. It is clear that with multiwavelengths observations it should be possible to separate the contributions arising from the disk surface and interior.

Figure 14: (a) Left panel.– A schematic view of the disk model. The surface layer is hotter and heated by the star light. The disk interior is heated by re-processed light from the surface layers. We assume that the disk height, , is 4 time the disk scale height, . (b) Right panel.– Spectral energy distribution. The vertical axis (i.e. ) is in arbitrary unit. Results are for oblate spheroid grains with axis ratio of 1.5:1. From Cho & Lazarian 2006.

This is the first attempt to simulate polarization from a disk on the basis of grain alignment theory. More attempts should follow. In fact, it has been known for decades that various stars, both young and evolved, exhibit linear polarization (see a list of polarization maps in Bastien & Menard 1988). While multiple scattering has been usually quoted as the cause of the polarization, recent observations indicate the existence of aligned dust around eta Carinae (Aitken et al. 1995) and evolved stars (Kahane et al. 1997). This suggests that for other stars the dust should be also aligned (Chrisostomou et al 2000). In fact, some of the arguments that were used against aligned grains are, in fact, favor them. For instance, Bastien & Menard (1988) point out that if polarization measurements of young stellar object were interpreted in terms of grain alignment with longer grain axes perpendicular to magnetic field, the magnetic field of accretion disks were in the disk plane. This is exactly what the present day models of accretion disks envisage.

Interestingly enough, alignment of dust in environments different of diffuse ISM and molecular clouds was professed by a number of pioneers of the grain alignment research. For instance, Greenberg (1970) claimed that interplanetary dust should be mechanically aligned. Dolginov & Mytrophanov (1976) conjectured that comet dust and dust in circumstellar regions was aligned. However, both the problems in understanding of grain alignment and the inadequacy of polarimetric data did not allow those views to become prevalent (although see Wolstencroft 1985, Briggs & Aitken 1986 where alignment was supported). I feel that now we have a much better case to include alignment while dealing with polarization from dust in various environments. Quantitative modeling should both test grain alignment theory and environments under study.

6.4 Grain Alignment in Comets

The “anomalies” of polarization from comets121212When light is scattered by the randomly oriented particles with sizes much less than the wavelength, the scattered light is polarized perpendicular to the scattering plane, which is the plane passing through the Sun, the comet and the observer. Linear polarization from comets has been long known to exhibit polarization that is not perpendicular to the scattering plane. (see Martel 1960, Beskrovnaja et al 1987, Ganesh et al 1998) as well as circular polarization from comets ( Metz & Haefner 1987, Dollfus & Suchail 1987, Morozhenko et al 1987) are indicative of grain alignment.

However, conclusive arguments in favor of grain alignment were produced for the Levi (1990 20) comet through direct measurements of starlight polarization, as the starlight was passing through comet coma (Rosenbush et al 1994). The data conclusively proved the existence of aligned grains in comets.

Note, that the issue of circular polarization was controversial for a while. When both left and right handed polarization is present in different parts of coma the average over entire coma may get the polarization degree close to zero. This probably explains why earlier researchers were unsuccessful attempting to measure circular polarization while using large apertures. Recent measurements by Rosenbush et al. (1999), Manset et al. (2000) of circular polarization from Hale-Bopp Comet support the notion that circular polarization is a rule rather than an exception.

Figure 15: Zones of grain alignment in respect to magnetic field and in respect to radiation/electric field for a comet at 1AU from the Sun. Radiative torque alignment. From Hoang & Lazarian (2007).

A more recent paper by Rosenbush et al. (2006) reports circular polarization from a comet C/1999 S4(LINEAR). The data indicates that the polarization arises from aligned grains. The mechanism of alignment requires further studies, however. If magnetic fields do not penetrate into coma, the alignment happens in respect to direction of radiation (see §5.2) no circular polarization is possible (see Eq. (8)). Outflow velocities are not vividly supersonic to allow efficient Gold alignment. What is the mechanism that produces the circular polarization? Several explanations are possible on the basis of our earlier discussion. First of all, the alignment observed may be the sub-sonic mechanical alignment of irregular grains (see §4.5). Second, the alignment may be due to radiative torques, but the outflow could alter the direction of the axis of alignment. The structure of the “precessing” radiative torque component is such that the precession rate goes to zero as the grain gets aligned in respect to the radiation. Therefore it is easy to perturb the alignment axis for radiative torques. Third, as we discussed in §5.3 electric field could cause grain precession and even grain alignment. The choice between these possibilities should be made on the basis of comparing the results of modeling with observations. We illustrate the model of alignment in a comet in Fig. 15.

6.5 Alignment of Small Grains

For particles much less than the wavelength the efficiency of radiative torques drops as (see L95). Within circumstellar regions, where UV flux is enhanced smaller grains can be aligned by radiative torques. This could present a possible solution for the reported anomalies of polarization in the 2175  Å   extinction feature (see Anderson et al 1996) which have been interpreted as evidence of graphite grain alignment (Wolff et al 1997). If this alignment happens in the vicinity of particular stars with enhanced UV flux and having graphite grains in their circumstellar regions, this may explain why no similar effect is observed along other lines of sight.

The maximum entropy inversion technique in Kim & Martin (1995) indicates that grains larger than a particular critical size are aligned. This is consistent with our earlier discussion of radiative torques and the Serkowski law (see §6.2). However, an interesting feature of the inversion is that it is suggestive of smaller grains being partially aligned. Initially, this effect was attributed to the problems with the assumed dielectric constants employed in the inversion, but a further analysis that we undertook with Peter Martin indicated that the alignment of small grains is real. Indeed, paramagnetic (DG) alignment must act on the small grains131313To avoid a confusion we should specify that we are talking about grains of  cm. For those grains the results of DG relaxation coincide with those through resonance relation in Lazarian & Draine (2000). It is for grains of the size less than  cm that the resonance relaxation is dominant.. An important feature of this weak alignment is that it is proportional to the energy density of magnetic field. This opens a way for a new type of magnetic field diagnostics. As very small grains may emit polarized radiation as they rotate (see §4.2) both UV and microwave polarimetry may be used to estimate the intensities of magnetic field.

7 Concluding remarks

7.1 Present situation

Historically the goal of the grain alignment theory was to account for puzzling polarimetric observations. The situation has changed, however, as grain alignment became a predictive theory. This calls for more quantitative modeling and for more further polarimetry data acquisition, to test the models.

It was not possible in the present review to discuss all the interesting cases where grain alignment may be important. Theoretical considerations suggest that grain alignment should take place within various astrophysical environments. Polarized radiation from neighboring galaxies (Jones 2000), galactic nuclei (see Tadhunter et al 2000), AGNs, Seyfet galaxies (see Lumsden et al 2001) can be partially due to aligned particles. Revealing this contribution would allow us to study magnetic fields in those and other interesting settings.

Polarization from aligned grains can benchmark other techniques for magnetic field studies. For instance, anisotropies of the magnetohydrodynamic (MHD) turbulence reveal the local direction of the magnetic fields; those can be measured with observations of the Doppler-shifted spectral lines (Lazarian et al. 2001, Esquivel & Lazarian 2005 and references therein). Polarimetry of aligned grains provides a way of testing the accuracy of this new technique. Similarly, aligned grains can remove the 90 degrees uncertainty arising in the magnetic field studies based upon the Goldreich-Kylafis effect or alignment of atoms/ion with fine or hyperfine structure141414For the Goldreich-Kylafis (1982) effect this uncertainty is intrinsic, while for the technique proposed in Yan & Lazarian (2006, 2007) the uncertainty can be removed by using several aligned species., as proposed by Yan & Lazarian (2006, 2007).

Recently, promising attempts have been made to test the predictions of the grain alignment theory (see Hildebrand 2003, Andersson & Potter 2005, 2006, 2007), or to use the grain alignment theory to explain observations (see Frisch 2006, Cortes & Crutcher 2006). It is significant that the numerical simulations that include theory-motivated prescriptions for grain alignment (see §6) allow easy comparisons with observations. If combined with new polarimetric instruments, that have been built recently or are to be built in the near future, this ensures progress in reliable tracing of magnetic fields using aligned grains.

7.2 Important questions

In regards to practical studies of magnetic fields a few questions will be in order.

What is the advantage of the far-infrared polarimetry for studies of magnetic fields in molecular clouds compared to the optical and near-infrared observations? An immediate answer would be that the far infrared polarimetry reveals aligned grains near newly born stars, unaccessible to optical or near-infrared photons. An additional advantage of the far infrared spectropolarimetry stems from the fact that it allows us to separate contributions from different parts of the cloud (see Hildebrand 2000). This will enable us to carry out tomography of the magnetic field structure.

What is the future of the optical and near-infrared polarimetry? It would be wrong to think that with the advent of the far-infrared polarimetry there is a bleak future for extinction polarimetry at shorter wavelengths. In fact, its potential for studies of magnetic fields in the Galaxy is enormous (see Fosalba et al. 2002, Cho & Lazarian 2002a). The possibility of using stars at different distances from the observer allows to get an insight into the 3D distribution of magnetic fields. In general, however, it is extremely advantageous to combine polarimetric measurements in optical/near-infrared and far-infrared wavelengths. For instance, it may be pretty challenging to trace the connection of Giant Molecular Clouds (GMCs) with the ambient interstellar medium using just far-infrared measurement. However, if extinction polarimetry of the nearby stars is included, the task gets feasible (see Poidevin & Bastien 2006). Similarly, testing modern concepts of MHD turbulence (see Goldreich & Shridhar 1995) and turbulent cloud support (see McKee 1999) would require data from both diffuse and dense media.

Is it possible to study magnetic fields using radiation scattered by aligned grains? The studies of molecular cloud column densities with the near infrared scattered light were presented in Padoan et al (2006) and Juvela et al. (2006). Those have shown that large scale mapping of scattered intensity is possible up to even for clouds illuminated by the average interstellar radiation field. The polarization of scattered light should be affected by grain alignment. This opens interesting prospects of detailed mapping of magnetic fields at sub-arcsecond resolution, which for the closest star forming regions corresponds to the scale of  AU. This can bring to a new level both the studies of magnetic fields in star forming regions and observational studies of magnetic turbulence.

What is the advantage of doing polarimetry for different wavelengths? The list of advantages is rather long. It is clear that aligned grains can be successfully used as pick up devices for various physical and chemical processes, provided that we understand the causes of alignment. Differences in alignment of grains of different chemical composition (see Smith et al. 2000) provides a unique source of the valuable information. Comets present another case in support of simultaneous multifrequency studies. There the properties of dust evolve in a poorly understood fashion and this makes an interpretation of optical polarimetry rather difficult. Degrees and directions of dust alignment, that can be obtained that can be obtained via far infrared polarimetry, can be used to get a self-consistent picture of the dust evolution and grain alignment.

Do we need the grain alignment theory to deal with polarized CMB foregrounds? Polarized emission spectra arising from aligned dust may be very complex if grains of different temperatures exhibit different degrees of alignment. In this situation, the use of the naive power-law templates may result in huge errors unless we understand grain alignment properly. Needless to say, a well developed grain alignment theory is required to predict the spectra of polarized emission from PAHs in the range of 10-100 GHz.

What is the optical depth at which aligned grains fail to trace magnetic fields? The answer depends on the grain size and the grain environments. If we consider a starless cloud/core illuminated by the interstellar radiation field, for grains of  cm the radiative alignment fails at (see Arce et al. 1998). However, larger grains in cloud cores can be aligned at as was shown by Cho & Lazarian (2005, 2006), which is a great news for polarimetric studies of star formation. In the vicinity of stars and in the presence of grain-gas drift smaller grains can also be aligned.

What is the niche for the magnetic field studies with aligned grains? If we try to answer this question briefly, we can point out that the aligned grains trace magnetic fields in molecular clouds and cold diffuse gas, where so far they have little competition from other techniques. Both observations and theory show that grain alignment is a robust process that can operate in the presence of very weak magnetic fields. I would like to stress the synergy of the starlight/dust emission polarimetry and other techniques of magnetic field studies. Indeed, the different techniques provide us with the data on magnetic fields in different environments, e.g. different phases of the interstellar medium. We can obtain an adequate picture of magnetized astrophysical settings by combining the techniques, e.g. dust polarimetry, synchrotron polarimetry, polarimetry of aligned atoms/ions and molecules.

7.3 Brief Summary

The principal points discussed above are as follows:

  • Grain alignment results in linear and circular polarization. The degree of polarization depends on the degree of grain alignment, the latter being the subject of the grain alignment theory.

  • Substantial advances in understanding grain dynamics, subtle magneto-mechanical effects, as well as the back-reaction of thermal fluctuations on grain rotation have paved the way for the advances in understanding of grain alignment.

  • The grain helicity has been established as an essential property of irregular grains rotating about their axis of the maximal inertia. This allowed for a better physical understanding of the radiative torque’s role, and allowed to introduce new alignment mechanisms, e.g. the sub-sonic mechanical alignment.

  • The grain alignment theory has, at last, reached its mature state when predictions are possible. In most cases grain alignment takes place with respect to magnetic field, thereby revealing the magnetic field direction, even if the alignment mechanism is not magnetic.

  • The radiative torque alignment, after having been ignored for many years, has become the most promising mechanism which predictions agree well with interstellar observations. To create alignment, this mechanism does not rely on paramagnetic relaxation.

  • It is clear that the importance of grain alignment is not limited to interstellar medium and molecular clouds. Polarimetry can be used to study magnetic fields in accretion disks, AGN, circumstellar regions, comets etc.

  • As astrophysical environments exhibit a wide variety of conditions, various alignment mechanisms have their own niches. The importance of studying the alternative mechanisms increases as attempts are made to trace magnetic fields with aligned grains in the environments other than the interstellar one.

Acknowledgments. I thank Bruce Draine, Michael Efroimsky, Roger Hildebrand, and Giles Novak for illuminating discussions. Help by Hoang Thiem was extremely valuable. I acknowledge the support by the NSF grant AST-0507164, as well as by the NSF Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas.

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