Radiative torque alignment: essential physical processes

Abstract

We study the physical processes that affect the alignment of grains subject to radiative torques (RATs). To describe the action of RATs, we use the analytical model (AMO) of RATs introduced in our previous paper. We focus our discussion on the alignment by anisotropic radiation flux with respect to the magnetic field, which defines the axis of grain Larmor precession. Such an alignment does not invoke paramagnetic dissipation (i.e. the Davis–Greenstein mechanism), but, nevertheless, grains tend to be aligned with long axes perpendicular to the magnetic field. When we account for thermal fluctuations within grains, we show that for grains that are characterized by a triaxial ellipsoid of inertia, the zero-J attractor point obtained in our earlier study develops into a low-J attractor point. The value of angular momentum at the low-J attractor point is of the order of the thermal angular momentum corresponding to the grain temperature. We show that, for situations when the direction of radiative flux is nearly perpendicular to a magnetic field, the alignment of grains with long axes parallel to the magnetic field (i.e. ‘wrong alignment’) reported in our previous paper, disappears in the presence of thermal fluctuations. Thus, all grains are aligned with their long axes perpendicular to the magnetic field. We study the effects of stochastic gaseous bombardment and show that gaseous bombardment can drive grains from low-J to high-J attractor points in cases when high-J attractor points are present. As the alignment of grain axes with respect to angular momentum is higher for higher values of J, counter-intuitively, gaseous bombardment can increase the degree of grain alignment with respect to the magnetic field. We also study the effects of torques induced by H₂ formation and show that they can change the value of angular momentum at high-J attractor points, but marginally affect the value of angular momentum at low-J attractor points. We compare the AMO results with those obtained using the direct numerical calculations of RATs acting upon irregular grains and we validate the use of the AMO for realistic situations of RAT alignment.

polarization, dust, extinction, ISM: magnetic fields

1 INTRODUCTION

Polarization from absorption and emission by aligned grains is widely believed to trace magnetic field topology (see Hough et al. 1989; Hildebrand et al. 2000; Hildebrand 2002; Aiken et al. 2002). The reliability of the interpretation of polarization maps in terms of magnetic fields depends crucially on the understanding of grain alignment theory.

When, nearly 60 yr ago, the polarization of starlight was discovered (Hall 1949; Hiltner 1949), it was immediately explained by absorption by elongated dust grains, which are aligned with respect to the interstellar magnetic field. Since then, the problem of grain alignment has been addressed by many authors (see a recent review by Lazarian 2007, and references therein). As a result, substantial progress has been made towards understanding how these tiny particles can become aligned.

The original theory of grain alignment formulated by Davis & Greenstein (1951) is based on the paramagnetic dissipation of energy for a grain rotating in an external magnetic field. However, the paramagnetic relaxation time-scale for a typical interstellar field strength is long compared to the gas damping time. Moreover, this mechanism has been found to be more efficient for small grains (see Lazarian 1997a; Roberge & Lazarian 1999), contrasting with observation data, which have testified that small grains are not aligned (Kim & Martin 1995; Andersson & Potter 2007).

Study of the paramagnetic alignment mechanism was given new life by Purcell (1979). In his classical paper, Purcell suggested three processes that can spin a grain up to suprathermal rate (i.e. a rate much higher than the thermal value): H₂ formation, photoemission, and the variation of the accommodation coefficient over the grain surface. H₂ formation was identified as the most powerful of these three processes. In terms of alignment, fast rotation is advantageous, as fast rotating grains are immune to randomization by gas bombardment. As a result, paramagnetic dissipation can give rise to good alignment of angular momentum with the magnetic field. An obvious limitation of the Purcell mechanism is that the spin-up process is most efficient, provided that a sufficient fraction of hydrogen is in atomic form. Therefore, this mechanism would fail in dark molecular clouds where most of the hydrogen is in molecular form. In fact, all spin-up processes fail in molecular clouds (see Lazarian, Goodman & Myers 1997; Roberge & Lazarian 1999), while observations have testified that the grains are aligned there (see Ward-Thompson et al. 2000).

Nevertheless, combining the ideas of suprathermal rotation with magnetic inclusions, in the spirit of the classical Jones & Spitzer (1967) paper, researchers hoped to explain the observational data. For instance, a model by Mathis (1986) provided a good fit to the observed Serkowski polarization curve (Serkowski 1973). Problems concerning the Purcell (1979) alignment mechanism became obvious, however, when Lazarian & Draine (1999a) reported the effect of thermal trapping. This effect stems from the coupling of vibrational and rotational degrees of freedom via the internal relaxation (e.g. Barnett relaxation; Purcell 1979). Thermally trapped grains undergo fast flips that average out uncompensated Purcell torques. Later, Lazarian & Draine (1999b) reported on a new effect, which they called nuclear relaxation;1 they showed this to be 10⁶ times stronger than the Barnett relaxation.2

The mechanical alignment mechanism, which is based on the relative motion of gas and dust, was pioneered by Gold (1952a). However, this mechanism, as well as its more sophisticated cousins (Lazarian 1995, 1997a; Lazarian & Efroimsky 1996; Lazarian, Efroimsky & Ozik 1996), requires supersonic motion of gas relative to dust (see Purcell 1969; Lazarian 1994; Roberge, Hanany & Messinger 1995; Lazarian 1997b), and this is only available in special circumstances. In his original paper, Gold (1952b) proposed collisions between clouds as a way to drive supersonic motion. Soon after this, Davis (1955) pointed out that such a process can only align grains in a tiny fraction of the interstellar medium (ISM). Other, more promising means of obtaining supersonic drift are ambipolar diffusion (Roberge & Hanany 1990; Roberge et al. 1995) and magnetohydrodynamics (MHD) turbulence (Lazarian 1994; Lazarian & Yan 2002; Yan & Lazarian 2003; Yan, Lazarian & Draine 2004). Nevertheless, while applicable for particular environments, the mechanical mechanisms are unable to explain the ubiquitous alignment of dust in a diffuse medium and molecular clouds. A more promising mechanism is the mechanical alignment of helical grains, first mentioned in Lazarian (1995) and Lazarian et al. (1997). We briefly described this in Lazarian (2007), Lazarian & Hoang (2007a, hereafter Paper I) and Lazarian & Hoang (2007b). The consequences of this have to be evaluated, but these are beyond the scope of the present paper.

Alignment by radiative torques (RATs) has recently become a favoured mechanism to explain grain alignment. This mechanism was initially proposed by Dolginov & Mytrophanov (1976), but was not sufficiently appreciated at the time of its introduction. Draine & Weingartner,(1996, 1997, hereafter DW96 and DW97) re-invigorated the study of the RAT mechanism by showing that RATs induced by anisotropic radiation upon rather arbitrarily shaped grains can spin-up and directly align them with the magnetic field. These papers refocused attention on the RAT mechanism and made it a promising candidate to change the grain alignment paradigm. Also, Bruce Draine has very kindly modified the publicly available ddscat code (Draine & Flatau 1994) to include RATs. This has enabled other researchers to access a useful tool for studying the RAT alignment.

The RAT mechanism seems to be able to address major puzzles presented by observations. For instance, the observation of polarized emission emanating from starless cores (see Ward-Thompson et al. 2000; Ward-Thompson, Andre & Kirk 2002) initially seemed completely unexplainable.3 Indeed, all mechanisms seem to fail in such cores, which are presumably close to thermodynamic equilibrium (see Lazarian et al. 1997). The RATs seem to be too weak as well (DW96). However, Cho & Lazarian (2005, hereafter CL05) found that the efficiency of RATs increases quickly with grain size (see also Paper I), and thus large grains can still be aligned in dark clouds. They found that grains as large as 0.6 μm can be aligned in dark clouds by radiation attenuated by the column density with A_V≈ 10. This was for much higher extinction than expected in the absence of embedded stars (see DW96), and was claimed to be observed with optical and near-infrared polarimetry (Arce et al. 1998).

The pre-stellar cores studied in Ward-Thompson et al. (2000) correspond to A_V= 50–60; the shielding column, assuming uniformity, is approximately half of these values. However, Crutcher et al. (2004) pointed out that polarization data do not sample the innermost core regions,4 and this provides an explanation for polarization ‘holes’ (see Matthews & Wilson 2000, 2002; Matthews, Wilson & Fiege 2001; Lai et al. 2002). The reported decrease in the percentage polarization with the optical depth also agrees well with the findings in CL05.

The approach of CL05 was further elaborated in the studies by Pelkonen, Juvela & Padoan (2007) and Bethell et al. (2007), in which the synthetic maps obtained via MHD simulations were analysed. In particular, Bethell et al. (2007) obtained the polarization maps for a simulated cloud with peaks of extinction A_V as high as ∼25. The study showed the non-trivial nature of radiative transfer in a fractal turbulent cloud, and confirmed the ability of aligned grains to trace a magnetic field in dark clouds and proved a decrease of percentage polarization at the highest A_V. Note that the studies above were in contrast to the previous studies, which used rather arbitrary criteria (e.g. A_V= 3) for the alignment to shut down, or the even more unrealistic assumption that all grains were perfectly aligned.

Similarly, the change of the degree of polarization with optical depth observed by Whittet et al. (2001) was also explained by RATs in Lazarian (2003, 2007). Detailed modelling of this effect is provided in Whittet et al. (2008) and Hoang & Lazarian (2008a). We believe that similar modelling should help to determine whether grains should have superparamagnetic inclusions. Lazarian & Hoang (2008) show that grains with such inclusions can be perfectly aligned under the action of RATs. We see below that the alignment is substantial, but not perfect, when RATs are not assisted by superparamagnetic relaxation.

The encouraging correspondence of the theoretical expectations5 and the observed polarization arising from aligned grains makes it essential to understand the reason why RATs align grains. In Paper I, we subjected to scrutiny the properties of RATs. Using a simple analytical model (AMO) of a helical grain, we studied the properties of RATs and the alignment driven by such RATs. The analytical results were found to be in good correspondence with numerical calculations for irregular grains obtained with ddscat. Evoking the generic properties of the RAT components, we explained the RAT alignment of grains in both the absence and presence of magnetic fields. Intentionally, for the sake of simplicity, in Paper I we studied a simplified dynamical model to demonstrate the effect of the RAT alignment. This model disregards the wobbling of grain axes with respect to the angular momentum that arises from thermal fluctuations (Lazarian 1994; Lazarian & Roberge 1997), and thermal flipping of grains (Lazarian & Draine 1999a,b). The simplifications allowed us to provide a vivid illustration of the RAT alignment. However, there is still a question concerning the modification of the alignment by thermal fluctuations, as well as the action of additional (e.g. random) torques.

The first study to combine the RAT alignment and the physics of thermal fluctuations and flipping was carried out by Weingartner & Draine (2003, hereafter WD03). They studied the alignment by RATs produced by a monochromatic radiation field and for one particular radiation direction ψ= 70°, taking into account thermal fluctuations and thermal flipping. They observed the appearance of new attractor points at low angular momentum, but the underlying physics of this effect remained unclear.

WD03 do not answer the question of what is happening when the entire spectrum of the interstellar radiation field (ISRF) is accounted for, and when radiation arrives from other directions. Their study does not consider the effect of gaseous bombardment and effects of H₂ formation. We address these and other issues below.

The structure of the paper is as follows. In Section 2, we present current theoretical understandings of the RAT alignment and formulate our theoretical predictions. We briefly describe thermal wobbling and our method of averaging RATs in Section 3. In Section 4 we derive the analytical expressions for RATs for the AMO averaged over thermal fluctuations, and we study the influence of thermal fluctuations to the grain alignment by RATs. In Section 5, we study the alignment for irregular grains with RATs from ddscat. We provide an explanation for the appearance of the low attractor point based on the AMO and we discuss the stability of low- and high-J attractor points in Section 6. We consider fast alignment in the presence of thermal fluctuations in Section 7. In Sections 8 and 9, we show the influences of random collisions and H₂ formation torques in the framework of RAT alignment for both the AMO and irregular grains. An extended discussion is presented in Section 10. Finally, we summarize our results in Section 11.

2 THEORETICAL CONSIDERATIONS

In Paper I, we provided analytical calculations for RATs for a grain model that consists of a perfectly reflecting oblate spheroid with a ‘massless’ mirror attached at its pole (see the upper panel in Fig. 1). This helical grain demonstrates RATs that are similar to those of irregular grains obtained by ddscat (see Paper I).

Figure 1

Upper panel: geometry for the AMO consisting of a perfectly reflecting spheroid and a weightless mirror. Lower panel: the orientation of a grain, described by three principal axes and ⁠, in the laboratory coordinate system (scattering reference system) and ⁠, is defined by three angles Θ, β and Φ. The direction of the incident photon beam k is along ⁠.

The basic properties of RATs were the subject of a detailed analysis in Paper I. We established that the projection of RATs on to the axis (i.e. Q_e3; see the upper panel in Fig. 1) determines the precession of the grain axis about the radiation direction. This component is also present for non-helical (e.g. simple spheroid) grains, and neither case induces grain alignment. It is the other two RAT components, Q_e1 and Q_e2, that are responsible for grain alignment. In Paper I, we explain why the RAT alignment tends to occur with the long axis of the grain either perpendicular to the magnetic field or perpendicular to the radiation direction (the choice of which, for a given external magnetic field, is determined by the ratio of the rate of radiative precession about the radiation direction and Larmor precession).

In Paper I, we found an important parameter that affects the grain alignment, the ratio Q^max_e1/Q^max_e2, where Q^max_e1 and Q^max_e2 are amplitudes of Q_e1 and Q_e2, respectively. Different grain shapes illuminated by different radiation fields can have different ratios Q^max_e1/Q^max_e2, and thus the resulting alignment is different.

We carried out the study of the RAT alignment in Paper I assuming that the grain always rotates about its principal inertia axis a₁, corresponding to the maximal moment of inertia (hereafter called the maximal inertia axis). However, this assumption is only valid for fast rotating grains. Such grains, are subjected to fast internal relaxation (Purcell 1979; Lazarian & Efroimsky 1999; Lazarian & Draine 1999b) that couples the angular momentum J and a₁. As the grain slows down to thermal angular velocity, the coupling becomes weaker. As a result, a₁ wobbles about J (Lazarian 1994; Lazarian & Roberge 1997). Similar to DW97, in Paper I we disregarded this effect.

Some implications of the thermal wobbling are self-evident. For instance, in Paper I we found that for a narrow range of angles, when the radiation beam is close to being perpendicular to the magnetic field, the alignment becomes ‘wrong’ (i.e. it occurs with the maximal inertia axis perpendicular to the magnetic field).6 Such a ‘wrong’ alignment is in contrast to what is widely observed in the ISM. However, we found that the ‘wrong’ alignment in a diffuse medium corresponds to low angular momentum J. Therefore, we predicted in Paper I that thermal wobbling of grains would destroy the ‘wrong’ alignment.

Qualitatively, in Paper I, we show that, in most cases, a high fraction of grains are aligned at a zero-J attractor point, assuming that a₁ is always parallel to J. In reality, thermal fluctuations induce the wobbling of a₁ with respect to J, resulting in the modification of RATs. This, in turn, changes the alignment of J with respect to B. Hence, we expect grains to be trapped at attractor points with thermal angular momentum J (hereafter called low-J attractor points).

One direct consequence of being trapped at the low-J attractor point is the decrease of the degree of the alignment of grain axes with respect to the magnetic field. The degree of alignment is defined by the Rayleigh reduction factor (Greenberg 1968)

1

where β is the angle between the maximal inertia axis a₁ and the magnetic field B, and 〈…〉 denotes the averaging over an ensemble of grains. Because the process of internal alignment between a₁ and J occurs much faster than the alignment of J with respect to B, R could be approximately7 separated (see Lazarian 1994) as8

2

Here 〈R(θ)〉 and 〈R(ξ)〉 are given by

3

4

where θ is the angle between a₁ and J, and ξ is the angle of J and B.

When thermal fluctuations are absent, we have a perfect alignment of a₁ with J (i.e. 〈R(θ)〉= 1). Therefore, the Rayleigh reduction factor depends only on the degree of alignment of J with B. In the presence of thermal fluctuations, and assuming that the relaxation process obeys the normal distribution, 〈cos²θ〉 is given by

5

Here, T_d is the dust temperature,

is the normalization factor, and E(θ) =J²[1 + (h− 1) sin²θ]/2I₁ is the kinetic energy of the spheroidal grain. Clearly, as the value of angular momentum J becomes comparable with the thermal value, ⁠, where I₁ is the maximal inertia moment of the grain, the aforementioned thermal wobbling should decrease the alignment of a₁ with B.

On the basis of our findings in Paper I, we can qualitatively address some questions posed in WD03. What will be the effect of random collisions on grain alignment? How do suprathermal torques (e.g. torques arising from H₂ formation; Purcell 1979) influence the alignment by RATs?

We expect that grains trapped at low-J attractor points can be significantly affected by random collisions of gas atoms. When the phase trajectory map of grains has attractor points at high J, we expect that random collisions can depopulate grains aligned at the lower attractor point, and stochastically direct grains to the high-J attractor point. As the grains with high J are immune to randomization and internal alignment for high J is close to perfect (Purcell 1979), counter-intuitively, random gaseous bombardment can increase the degree of alignment. However, the removal of grains from the low-J attractor points is expected to make grain alignment time-dependent, although the corresponding time may be long compared with other processes in the system. With regards to H₂ uncompensated torques (Purcell 1979; Lazarian 1995), because they are fixed in the grain body coordinate system, their effect depends on the flipping rate of the grains.

Obviously, at the high attractor point corresponding to J≫J_th, the flipping rate is very low. Therefore, H₂ torques act together with RATs and change the angular momentum depending upon the resurfacing process. In contrast, the grains flip fast in the process of heading to low attractor points. Assuming that the correlation time-scale is shorter than the alignment time, then grains are rapidly driven to low angular momentum at which they flip very fast. Thus, H₂ torques will be averaged out to zero (see equation 57), and their only effect will be an additional randomization at low attractor points.

3 RADIATIVE TORQUES AND THE EFFECT OF THERMAL WOBBLING

In this section, we briefly discuss the general definitions of RAT components, and then present methods to average the torques over torque-free motion and thermal wobbling. We also analyse the role of the third component of RATs, Q_e3.

3.1 Radiative torques: spin-up, alignment and precession

Similar to Paper I, in order to compare our results more easily with those in earlier works, wherever possible, we preserve the notations adopted in DW97.

As in Paper I, we consider only the anisotropic component of the radiation field. The RAT is then defined by

6

where γ is the degree of radiation anisotropy, a_eff is the effective size of the grain (see DW96; Paper I), and and are the mean wavelength and mean energy density of the radiation field, respectively. The RAT efficiency vector Q_Γ is represented in the laboratory system and ⁠:

7

Here Θ, β and Φ are angles describing the orientation of a₁ in the laboratory system (see the lower panel in Fig. 1). It was shown in Paper I that components Q_e1(Θ, β, Φ= 0) and Q_e2(Θ, β, Φ= 0) have universal properties and play a major role in the process of grain alignment, whereas the component Q_e3(Θ, β, Φ= 0) does not affect either spin-up or alignment, provided that is coupled with J.

Here, we only deal with the alignment of angular momentum with respect to the radiation direction or magnetic field (i.e. external alignment). Therefore, it is convenient to consider RATs in the spherical coordinate system J, ξ, φ (see Fig. 2).

Figure 2

Alignment coordinate system in which ψ is the angle between the magnetic field B and the radiation direction k, ξ is the angle between the angular momentum vector J and B, and φ is the Larmor precession angle.

In this coordinate system, the RAT can be written as

8

Here, F, which is the torque component parallel to ⁠, acts to change the orientation of J with respect to B. H, the component parallel to ⁠, is to spin grains up and G induces the precession of J about the magnetic field or radiation. These are given by

9

10

11

where Q_e1(ξ, ψ, φ), Q_e2(ξ, ψ, φ) and Q_e3(ξ, ψ, φ), as functions of ξ, ψ and φ, are components of the RAT efficiency vector in the laboratory coordinate system (see DW97; Paper I). To obtain Q_e1(ξ, ψ, φ), Q_e2(ξ, ψ, φ) and Q_e3(ξ, ψ, φ) from Q_Γ(Θ, β, Φ), which is calculated using the AMO or provided by ddscat for irregular grains, we need to use the relations between ξ, ψ, φ and Θ, β, Φ (see WD03 and Appendix A).

3.2 Torque-free motion

In the absence of external torques, a grain rotates around its principal axes. This motion is called torque-free motion. For a symmetric grain (e.g. the spheroidal body of AMO or brick; Paper I; Spitzer & McGlynn 1979), the torque-free motion consists of the nutation of angular velocity ω around J at a constant angle γ and the rotation around the maximal inertia axis a₁ (see Fig. 3) with a period P_τ.

Figure 3

In the body frame, the motion of a brick consists of the nutation of Ω about and the precession of J around ⁠.

In general, an irregular grain can be characterized by an ellipsoid with moments of inertia I₁, I₂ and I₃ around three principal axes a₁, a₂ and a₃, respectively. For a freely rotating grain, its angular momentum is conserved (i.e. J is fixed in space), while the angular velocity ω nutates around J. We can call the wobbling associated with the irregularity in the grain shape irregular wobbling, to avoid confusion with thermal wobbling (also thermal fluctuations) induced by the Barnett effect (Purcell 1979) and nuclear relaxation (Lazarian & Draine 1999b).

Obviously, the time-scales of the torque-free motion (e.g. rotation period P_τ) are much shorter than other time-scales (e.g. internal relaxations and the gas damping time; see Lazarian 2007). As a result, it is always feasible to average RATs over the torque-free motion. A description of the torque-free motion for an asymmetric top in terms of Euler angles γ, α and ζ (see Fig. A1) can be found in classical textbooks (e.g. Landau & Lifshitz 1976; see also WD03). For an AMO with a spheroidal body, as in Fig. 1, the Euler angle γ is constant, while α and ζ are functions of time (see Spitzer & McGlynn 1979). For irregular grains, we can obtain the Euler angles by numerically solving the equations of motion (see Appendix C).

Figure A1

Orientation of principal axes in the angular coordinate system are described by Euler angles γ, α and ζ.

3.3 Averaging over torque-free motion and role of Q_e3

As mentioned earlier, the rotation period P_τ about the maximal inertia axis and the precession time are both much shorter than the gas damping time. Therefore, we can average RATs over these processes.

In Fig. 4, we show the phase trajectory of the tip of a₁ about J for a spheroid and a triaxial ellipsoid for two initial angles γ₀=π/10 and γ₀=π/4. It is shown that in the case of a spheroid, the trajectory is a circle corresponding to the precession of a₁ about J at a constant angle γ, but the evolution of the tip of vector a₁ produces a torus shape for the irregular grain. As a result, the average over torque-free motion for a spheroid is concerned only with the average over a circle (i.e. over the precession), whereas it is required to average RATs over the torus area (see Fig. 4) for a triaxial ellipsoid.

Figure 4

The evolution of the maximal inertia axis a₁ around the angular momentum for two initial angles γ₀=π/10 (upper panel) and γ₀=π/4 (lower panel) of a₁ and J. Here, ζ is the nutation angle of about J and γ is the azimuthal angle between and J. Filled dots and torus show the evolution of a₁ around J for the spheroid and the irregular grain, respectively.

We briefly describe the averaging algorithm over torque-free motion for a triaxial ellipsoid in Appendix D (see also WD03).

In Paper I, we proved that in the absence of thermal fluctuations, the third component of RATs, Q_e3, does not contribute to the spin-up and alignment process, but it induces the precession of the grain about the radiation direction. To evaluate the role of Q_e3 to the alignment when thermal fluctuations are taken into account, we calculate the average of spin-up and aligning torque components over thermal fluctuations for two cases using the AMO. In the first case, we consider all components, and in the second case, the third component is turned off. As shown in Fig. D1, the resulting torques 〈H〉_φ and 〈H〉_φ are marginally different, and they tend to be the same with the enhancement of the computational precision. Therefore, in this paper, we disregard Q_e3 in the analytical treatment.

Figure D1

Aligning and spinning torques averaged over free-torque motion, 〈F〉_φ and 〈H〉_φ, for the AMO and the AMO in which Q_e3= 0. A slight difference between them is observed.

3.4 Thermal fluctuations and thermal flipping

Thermal fluctuations (i.e. thermal wobbling) and flipping arise from the coupling of rotational and vibrational degrees of freedom induced by internal relaxations. The effect was first discussed in Lazarian (1994). At the time, the strongest internal relaxation was believed to be associated with the Barnett effect (Purcell 1979).

The Barnett relaxation can be easily understood. Indeed, a freely rotating paramagnetic grain acquires a magnetic moment that is parallel to angular velocity Ω as a result of the Barnett effect (first discussed in this context in Dolginov & Mytrophanov 1976). The Barnett effect is the phenomenon of transferring the macroscopic angular momentum of a rotating body to electrons through flipping electronic spins. To some degree, the magnetization of the rotating body is analogous with the body at rest, which is magnetized by a rotating external magnetic field. The Barnett equivalent magnetic field is H_Be=Ω/γ where γ=e/2mc is the magnetogyric ratio of electrons (Purcell 1979).

Because Ω may not coincide with the maximal inertia axis a₁, it precesses continuously in the grain coordinate system (see Fig. 3). Therefore, the Barnett equivalent field can be decomposed into the constant component parallel to the precession axis and the rotating component that is perpendicular to it.9 Apparently, the rotating component induces a dissipation of rotational energy. As a result, we have an alignment of angular velocity and maximal inertia axis with angular momentum.

Lazarian & Draine (1999a) found a new effect related to nuclear spins, which they called nuclear relaxation. Similar to the rearranging of electronic spins, nuclear spins also become oriented by angular momentum transferred from the body. Although the nuclear magnetization arising from grain rotation is mostly negligible compared to that arising from electrons (see Purcell 1979), the nuclear relaxation was shown to be much more efficient than Barnett relaxation for 10⁻⁵ < a_eff < 10⁻⁴ cm grains. This can be easily understood. Spin flipping is a mechanical effect that depends on the angular momentum of the species rather than on the magneto-magnetogyric ratio γ. Thus, a rotating grain will have the same portion of nuclear and electron spins flipped. The magnetic field that induces such a flip is different for electrons and nuclei (i.e. it is inversely proportional to γ). The dissipation is proportional to the ‘equivalent’ field squared (i.e. to B²_eq∼ 1/γ²). It does not depend on the value of the nuclear magnetic moment, as the lag between magnetization and Ω increases with the decrease of the magnetic moments. In other words, the coupling between nuclear spins is less efficient than the coupling between electron spins; hence, there is a substantial lag in the nuclear spin alignment when Ω precesses around a₁. All in all, although the magnetic moment arising from nuclear spins of a rotating body is negligible, the corresponding relaxation (i.e. nuclear relaxation) is approximately 10⁶ times higher than the Barnett one.

Lazarian & Hoang (2008) considered grains with superparamagnetic inclusions and showed that the Barnett relaxation is substantially enhanced in this case. They also found that the range of rotational rate for which nuclear relaxation is efficient is also extended to higher frequencies. Interestingly enough, the nuclear relaxation is still likely to dominate for classical aligned insterstellar grains.

Thermal fluctuations within the grain body (see Purcell 1979) coupled via internal relaxations with the macroscopic rotation of the grain can affect the internal alignment, and result in random deviations of the major axis a₁ with respect to J. Following the fluctuation-dissipation theorem (Landau & Lifshitz 1976), the thermal equilibrium distribution of deviations can be established. The average of a torque A over thermal fluctuations for a spheroid with I₁ > I₂=I₃ is defined by (Lazarian & Roberge 1997)

12

Here, h=I₁/I₂, γ is the deviation angle of a₁ and J, and the kinetic energy E of the grain is

13

For irregular grains with I₁ > I₂ > I₃, the corresponding average is given in Appendix C.

Using the RAT expressions obtained for the AMO (see Paper I), we can explicitly integrate equation (12) to obtain RATs induced by thermal fluctuations. However, for irregular grains, we need to numerically average RATs according to equation (D1). The resulting averaged RAT components are inserted into equations (9)–(11) to obtain 〈F(ξ, φ, ψ, J)〉, 〈H(ξ, φ, ψ, J)〉 and 〈G(ξ, φ, ψ, J)〉, which are required to solve the equations of motion.

The internal relaxations can induce the axis a₁ to flip over with respect to J. This phenomenon is called thermal flipping (Lazarian & Draine 1999a). The probability of thermal flipping has been obtained in Lazarian & Draine (1999a):

14

Here, is the thermal angular momentum corresponding to the dust temperature, and t_Bn= (t⁻¹_Bar+t⁻¹_nucl)⁻¹ is the internal relaxation time for the Barnett and nuclear relaxation processes. Because t_nucl is usually much shorter than t_Bar for grains from 10⁻⁵ to 10⁻⁴ cm, the nuclear relaxation dominates the process of thermal flipping for typical aligned interstellar grains (see Kim & Martin 1995).

In our analysis, we assume that thermal fluctuation and flipping time-scales are shorter than the Larmor precession time. Fortunately, for the magnetic field of the ISM and for astronomical silicate material, the Larmor precession time-scale is much longer than that of thermal fluctuations and thermal flipping. Therefore, averaging over the latter motion is appropriate.

For circumstances in which the magnetic field is stronger, the Larmor precession time-scale may be comparable with the thermal flipping time-scale. For instance, Fig. 5 shows that for B= 50 μG and for ordinary paramagnetic grains, t_Lar∼t_tf for J < 10 J_th. For this case, in order to treat properly the grain dynamics, it is necessary to follow both the thermal fluctuation and Larmor precession.

Figure 5

Larmor precession, internal relaxation and thermal flipping time-scales for B= 5 and 50 μG for a grain size of a_eff= 0.2 μm. When the magnetic field increases, the Larmor precession time-scale decreases and becomes comparable to the Barnett relaxation time-scale for small J.

3.5 Equations of motion

The motion of a grain subjected to a net torque is completely determined by three variables: the angle ξ between the angular momentum vector J and the magnetic field direction B, the precession angle φ of J around B and the value of the angular momentum J (see the lower panel of Fig. 1). The equations of motion, if we disregard paramagnetic dissipation for these variables, are

15

16

17

Here, 〈F(ξ, φ, ψ, J)〉, 〈G(ξ, φ, ψ, J)〉 and 〈H(ξ, φ, ψ, J)〉 are defined by equations (9)–(11), which are already averaged over thermal fluctuations (see WD03), t_gas is the gas damping time-scale (see Table 1) and Ω_B is the Larmor precession frequency of the angular momentum about the magnetic field.

Table 1

Physical parameters for diffuse ISM. Here K, g cm⁻³ and a₋₅=a_eff/10⁻⁵ cm. g cm⁻³ where ρ= 3 g cm⁻³ is the mass density of the grain.

Definitions	Values
Gas density	n= 30 cm−3
Gas temperature	Tgas= 100 K
Gas damping time	s
Dust temperature	Td= 20 K
Anisotropy degree	γ= 0.1
Mean wavelength	m
Mean density of ISRF	uISRF= 8.64 × 10−13 erg cm−3
Effective grain size	aeff= 0.2 μm

For the ISM, the Larmor precession time is always shorter than the gas damping; thus, we can average equations (16) and (17) over a precession period. As a result, equations (15)–(17) can be reduced to a set of equations for ξ and J only, in which the spinning and aligning torques 〈F〉 and 〈H〉 are replaced by 〈F〉_φ and 〈H〉_φ, which denote the quantities obtained from averaging corresponding RATs over the Larmor precession angle φ from 0 to 2π.10

A stationary point in a phase trajectory map is determined by ξ_s, J_s, which are solutions of the equations of motion (see DW97)

18

19

The above stationary point is an attractor point if

20

and is a repellor point otherwise (see DW97).

From equations (16) and (17), we obtain ξ_s, J_s that satisfy

21

22

4 RADIATIVE TORQUE ALIGNMENT FOR THE ANALYTICAL MODEL

We first consider the role of thermal fluctuations in grain alignment based on an AMO consisting of a reflecting spheroid (I₂=I₃, hereafter, a spheroidal AMO) and a mirror as in Paper I. Then, in order to see the correspondence of the AMO with irregular grains in terms of dynamics, we replace the spheroid by an ellipsoid with the principal moments of inertia I₁ > I₂ > I₃ (hereafter, an ellipsoidal AMO).

4.1 Radiative torques: general expressions

In terms of RATs, an AMO is formally only applicable for λ≪a_eff as only in this case, is the geometric optics approach, used to derive the analytical formulae in Paper I, appropriate. However, in Paper I we proved that the functional forms of RATs for the AMO and irregular grains for λ≥a_eff are similar. By choosing the appropriate ratio Q^max_e1/Q^max_e2 it is possible to see that the dynamics of the AMO is similar to that of irregular grains, if the torque ratio Q^max_e1/Q^max_e2 is the same. Therefore, the AMO can act as a proxy for actual grains, and the advantage of this is that it allows an analytical insight into grain alignment.

For an AMO in which the mirror is tilted by an angle α with the axis a₂, the RAT components are given by (see Paper I)

23

24

25

Here, Θ is the angle between the axis of major inertia a₁ and the radiation direction k (see lower panel of Fig. 1), l₁ is the distance from the square mirror of side l₂ to the centre of mass, λ is the wavelength, n₁=−sin α and n₂= cos α are components of the normal vector of the mirror in the grain coordinate system (see Fig. 1),11a and b are the minor and major semi-axes of the spheroid, s=a/b < 1, e is the eccentricity of the spheroid, and K(Θ) is the fitting function (see also Paper I). As in Paper I, we treat the AMO with α= 45° as our default model.

RATs at a precession angle Φ (see lower panel of Fig. 1) are given by (see DW97)

26

27

28

4.2 Alignment with respect to k

First, we consider the alignment with respect to the direction of radiation k, which also corresponds to the situation when the direction of light k coincides with that of magnetic field B (i.e. ψ= 0°).

4.2.1 Analytical averaging of radiative torques for one componentQ_e1

In the absence of thermal fluctuations, as discussed in Paper I, the component Q_e3 does not affect RAT alignment, apart from inducing the precession of angular momentum J about k. When thermal fluctuations are accounted for, we see in Appendix D that the thermal average of Q_e3 also plays a minor role in the alignment for the AMO. Thus, the alignment problem only involves Q_e1 and Q_e2 components. To clarify the role of the torque components, we first consider the alignment driven by the component Q_e1 in the presence of thermal fluctuations.

For the default AMO (i.e. α= 45°), the contribution arising from the change in the cross-section is insignificant (see fig. 6 in Paper I). Thus, we can ignore the factor A_⊥= |n₁ cos Θ−n₂ sin Θ cos β| present in equations (23)–(25). As a result, with an accuracy of 5 per cent, equation (23) can be rewritten as

29

where

When there is incomplete alignment of and J, we have

30

where η is the precession angle of a₁ around J, γ is the angle of a₁ and J, and ξ is the angle between J and B. In addition, β is a complicated function of Euler angles.

Substituting equation (30) into equation (29) and averaging the resulting expression over the precession angle η, the second and third terms involving β are averaged to zero. Hence, we obtain

31

Using equations (12) and (13) for expression (31) we obtain

32

where ⁠. The aligning and spin-up torques are respectively given by (see equations 11–13)

33

34

Hence,

35

36

where J′=J/J_th. Equation (35) reveals that 〈F〉_φ= 0 for ξ= 0, π and cos 2ξ=−1/3, regardless of angular momentum J. In addition, zero points cos ξ= 0 and cos 2ξ=−1/3 of 〈H〉_φ do not depend on J either. This indicates that thermal fluctuations within the spheroidal grains do not alter the value of angular momentum (i.e. J= 0) at low-J attractor points produced by RATs when J→ J_th. Therefore, we expect that the resulting alignment is not significantly affected by thermal fluctuations.

The above features of RATs can be seen in Fig. 6, which shows RAT components 〈F〉_φ (solid line) and 〈H〉_φ (dashed line) as functions of ξ at a value of angular momentum J= 0.1 J_th. It follows that there will be four stationary points in the phase map, corresponding to 〈F〉_φ= 0 at ξ= 0, π/3, 2π/3 and π. In addition, the stationary point ξ= 0 is a high-J attractor point as

and 〈H〉_φ(ξ= 0) > 0 (see the upper panel in Fig. 6). The lower panel of Fig. 6 shows 〈F〉_φ and 〈H〉_φ as functions of J/J_th for ξ=π/3. There, it can be seen that for J≫J_th, 〈F〉_φ and 〈H〉_φ are saturated as a result of the perfect coupling of a₁ with J, which makes RATs independent of the angular momentum. As J/J_th decreases, 〈F〉_φ and 〈H〉_φ decrease steeply (see Fig. 6).

Figure 6

Spheroidal AMO: aligning 〈F〉_φ and spinning 〈H〉_φ torques averaged over thermal fluctuations for the case ψ= 0° and Q_e2= 0. Upper panel: 〈F〉_φ and 〈H〉_φ as functions of ξ for J= 0.1 J_th. Lower panel: 〈F〉_φ and 〈H〉_φ for ξ=π/3 decrease rapidly with J/J_th decreasing and become saturated as J/J_th≫ 1.

4.2.2 Averaged radiative torques for both componentsQ_e1andQ_e2

The analytical averaging over thermal fluctuations for Q_e2 is more complicated because of its dependence upon Φ, which is a function of Euler angles α, γ and ξ, φ, ψ (see equation 27 and Appendix A). Therefore, we use numerical averaging for RATs, rather than deriving analytical expressions for them. The resulting torques 〈F〉_φ and 〈H〉_φ are used to solve the equations of motion (16) and (17).

Fig. 7 shows spin-up and alignment torques for J= 10⁻³J_th, 0.9J_th and 10J_th corresponding to cases in which thermal fluctuations are dominant, important and negligible, respectively. It can be seen that for J≫J_th, 〈F〉_φ= 0 for cos ξ=∓1. It turns out that for J≫J_th, the AMO creates only two stationary points corresponding to perfect alignment of J with B. When J= 10⁻³J_th, there are new stationary points at cos ξ=±0.9 corresponding to 〈F〉_φ= 0. This means that thermal fluctuations can produce new stationary points. At the same time, the magnitude of spinning torque 〈H〉_φ decreases as J decreases (i.e. when thermal fluctuations increase; see the lower panel of Fig. 7). As a result, the alignment of grains is expected to be similar to the case without thermal fluctuations.

Figure 7

Spheroidal AMO: aligning (upper panel) and spinning (lower panel) torques averaged over thermal fluctuations for different J/J_th. The upper panel shows that as J= 10⁻³J_th, one new zero-point of 〈F〉_φ appears at cos ξ=−0.9. Both 〈F〉_φ and 〈H〉_φ exhibit a rapid decrease with J/J_th decreasing.

4.2.3 Trajectory maps

Let us consider the dynamics of the spheroidal AMO driven by RATs averaged over thermal fluctuations, and test the predictions using the analytical results above.

We solve the equations of motion for grains having the same initial angular momentum and uniform orientation distribution with respect to the interstellar magnetic field. Parameters necessary for calculations for the ISM conditions are given in Table 1. Phase trajectory maps are plotted in coordinates (cos ξ, J/I₁ω_T) where ξ is the angle between J and B, and ⁠. The upper and lower parts of these correspond to a₁ initially parallel and antiparallel to J, respectively. For a grain size a_eff= 0.2 μm and shape 1 (see Fig. 8), the ISRF can produce high stationary points with J_high∼ 200 I₁ω_T for ψ= 0°. To represent high and low attractor points together, we consider the ISRF with u_rad=u_ISRF/10 where u_ISRF= 8.64 × 10⁻¹³ erg cm⁻³ is the energy density of the ISRF. Thus, the phase trajectory maps are shown with J_max= 20 I₁ω_T in the present paper.

Figure 8

Irregular grains of shape 1 and shape 3 (see DW97) are taken to study RAT alignment by ddscat.

Further in the paper, a stationary point on the phase trajectory maps marked by a circle is an attractor point, which is the point to which adjacent trajectories tend to converge, and a stationary point marked by a cross denotes a repellor point (i.e. the trajectories are repulsed while approaching it; see also Paper I).12 In general, a phase trajectory map may have low-J and high-J attractor points or only low-J attractor points (see more in Paper I). To see clearly the modification induced by thermal fluctuations on grain dynamics, we frequently show side by side the trajectory maps for the case without thermal fluctuations as in Paper I, and with thermal fluctuations. Phase trajectory maps for the spheroidal AMO are shown in Figs 9, 10 and 11, where the upper and lower panels correspond to the cases without and with thermal fluctuations, respectively.

Figure 9

Spheroidal AMO: phase trajectory maps for the alignment by Q_e1 only. The upper panel shows the phase map in the absence of thermal fluctuations and the lower panel shows the phase map when thermal fluctuations are taken into account. The panels show that the position of the zero-J attractor point at cos ξ=−0.6 is not changed even in the presence of thermal fluctuations.

Figure 10

Spheroidal AMO: similar to Fig. 9, but grains are aligned by all torque components. The figures show the shift of the zero-J attractor point from cos ξ=−1 (upper panel) to cos ξ=−0.9 (lower panel), but its angular momentum remains zero even when thermal fluctuations are taken into account.

Figure 11

Spheroidal AMO: similar to Fig. 10, but for ψ= 70°. The figures show that the zero-J attractor point C at cos ξ=−0.9 (upper panel) is unchanged in the presence of thermal wobbling (lower panel).

When the grain alignment is only driven by Q_e1 and ψ= 0° (Fig. 9), we see that each phase map has two high-J attractor points A and B. In the absence of thermal fluctuations, the torque 〈H〉_φ decelerates grains to the attractor point C with J= 0 (upper panel). When thermal fluctuations are present, the attractor point C is not affected. This result is consistent with our analytical predictions in Section 4.2.1

When the components Q_e1 and Q_e2 act together, Fig. 10 shows that the angular momentum of low attractor points remains the same (i.e. J remains equal zero). However, its position is slightly shifted to cos ξ=±0.9.

4.3 Alignment with respect to B

Below we consider grain dynamics when the magnetic field plays the role of the alignment axis. As an example, the radiation direction ψ= 70° is adopted.

Fig. 11 shows that thermal fluctuations do not increase the value of angular momentum at the attractor point C for the case ψ= 70°. For other angles ψ, we also found that the zero-J attractor point C is unchanged in the presence of thermal fluctuations.

It is easy to see that with the assumption of a₁ ∥ J, the irregular shape of the ellipsoid of inertia (i.e. I₁≠I₂≠I₃), is not important for the grain dynamics. However, the irregularity in the grain shape becomes important in the case of a wobbling grain because the averaging of RATs over torque-free motion (see Fig. 4) depends on its ellipsoid of inertia. To address such effects, the spheroidal AMO can be modified. For instance, if the mirror is not weightless, the free motion of the spheroidal AMO will be of a triaxial ellipsoid of inertia, rather than a spheroid. Further, in Section 4.4, we replace the spheroid of the AMO by a triaxial ellipsoid.

4.4 Alignment for an ellipsoidal AMO

In this section, we study the alignment of an ellipsoidal AMO in which the spheroid body (see Fig. 1) is replaced by an ellipsoid with moments of inertia I₁ : I₂ : I₃= 1.745 : 1.62 : 0.876, which is similar to shape 1 (see Fig. 8).

We recall that stationary points, which determine the alignment of grains, correspond to 〈F〉_φ= 0 (see equation 22). Fig. 12 shows the torque components for ψ= 70°. It can be seen that for J≫J_th (i.e. thermal fluctuations are negligible), 〈F〉_φ= 0 for cos ξ=∓1. This indicates that the ellipsoidal AMO produce two stationary points corresponding to perfect alignment of J with B. As J→ 0.1 J_th, there appears a new stationary point at cos ξ=−0.8. This means that thermal fluctuations produce a new stationary point at cos ξ∼− 0.8. However, this new stationary point is a repellor point as

(see Fig. 12). In addition, the lower panel shows the change in sign of the spinning torque from negative to positive as J∼ 0.1J_th for cos ξ∼−1. Thus, the stationary point cos ξ=−1 is an attractor because

(see Fig. 12). Therefore, thermal fluctuations within the ellipsoidal AMO modify the properties of RATs as J→ 0, and split a zero-J attractor point in the absence of thermal wobbling into two low-J attractor points. The corresponding trajectory maps for this case are shown in the upper panel of Fig. 13.

Figure 12

Ellipsoidal AMO: aligning and spinning torques averaged over thermal fluctuations as a function of ξ for several angular momenta. The upper panel shows that as J= 1 J_th, one new zero-point of 〈F〉_φ appears at cos ξ=−0.8. The change in sign of 〈H〉_φ in the vicinity of cos ξ=−1 for J= 0.1J_th is shown in the lower panel.

Figure 13

Ellipsoidal AMO. Upper panel: phase map for the default AMO, for ψ= 70°, with two low attractor points, C' and D', and two repellor points, A and B. Lower panel: similar to upper panel, but the torque ratio Q^max_e1/Q^max_e2 is changed from the default value ∼1.2 to 0.78, and the two repellor points A and B become high-J attractor points, A and B, with J= 18 J_th.

Now we rescale the amplitude of Q_e1, Q_e2 so that Q^max_e1/Q^max_e2= 0.78. For this AMO, Paper I shows that the grains are aligned with two high-J attractor points and a zero-J attractor point. However, in the presence of thermal fluctuations and for the ellipsoidal AMO with I₂≠I₃, the zero-J attractor point becomes the attractor point with the value of angular momentum of the order of the thermal value, as seen in the lower panel of Fig. 13.

5 RADIATIVE TORQUE ALIGNMENT FOR IRREGULAR GRAINS

5.1 Properties of averaged radiative torques

We use the publicly available ddscat code (Draine & Flatau 1994) to calculate RATs for irregular grains (i.e. shapes 1 and 3; see Fig. 8). The optical constant function for astronomical silicate is adopted (Draine & Lee 1984). We computed RAT efficiency Q_λ(Θ, β, Φ) in the laboratory coordinate system (see Fig. 1), over 32 directions of Θ in the range [0, π] and 33 values of β between 0 and 2π for Φ= 0 (angle produced by a₁ and the plane where are three unit vectors of the laboratory coordinate system; see Fig. 1). The RAT efficiency Q_λ(Θ, β, Φ) for an arbitrary angle Φ is easily calculated using equations (26)–(28). In our paper, we calculate RATs for irregular grains of size a_eff= 0.2 μm induced by monochromatic radiation field of λ= 1.2 μm and the spectrum of the ISRF (see Mathis, Mezger & Panagia 1983).

Fig. 14 shows 〈F〉_φ and 〈H〉_φ (i.e. aligning and spinning torques) obtained by averaging the corresponding expressions (see equations 9 and 11) over thermal fluctuations for different angular momenta, and for the monochromatic radiation at angle ψ= 70° toward B.

Figure 14

〈F〉_φ and 〈H〉_φ for the monochromatic radiation field at ψ= 70° as functions of the angle ξ between J and B for three values of J.

From Fig. 14, we can see that for J/J_th≫ 1, 〈F〉_φ= 0 at cos ξ=±1, −0.65, corresponding to three stationary points. However, when J/J_th decreases (i.e. thermal fluctuations become stronger), the intermediate stationary point shifts to the left, and disappears as J/J_th→ 0.

Fig. 15 is similar to Fig. 14, but presents torques for the entire spectrum of the ISRF and ψ= 0°. It is also seen that the intermediate stationary point (i.e. point with 〈F〉_φ= 0) shifts to the left as J/J_th decreases. It disappears when J/J_th= 0. Furthermore, following both Figs. 14 and 15 (lower panels), we can see that the spinning torque 〈H〉_φ decreases with J decreasing. This is because for low angular momentum, thermal fluctuations become stronger, so the grain axis a₁ fluctuates with a wider amplitude around J. As a results, RATs decrease (see analytical results for the spheroidal AMO in the lower panel of Fig. 6).

Figure 15

〈F〉_φ and 〈H〉_φ for the ISRF at ψ= 0° as functions of the angle ξ between J and B for three values of J.

Now let us consider the property of the intermediate zero point of the aligning torque 〈F〉_φ shown in the upper panel of Fig. 15. We can check that this stationary point corresponding to J/J_th∼ 2.5 is an attractor point. According to equation (20), the criteria in order for one stationary point to be an attractor point is

With the stationary point cos ξ_s=−0.85, we have 〈H〉_φ > 0 (see the lower panel), and

as a result of the decrease of 〈F〉_φ with ξ in the vicinity of ξ_s (see the upper panel of Fig. 15). Thus,

which satisfies the criteria for an attractor point.

5.2 Phase trajectory maps for shape 1

Let us consider first the trajectory map for grains of size a_eff= 0.2 μm driven by RATs produced by a monochromatic radiation of λ= 1.2 μm, in the direction ψ= 70°, which is similar to the setting in WD03.

The upper panel in Fig. 16 shows the trajectory map for the case without thermal fluctuations with two high-J attractor points (A, B) and one zero-J attractor point C. When thermal fluctuations are accounted for, the lower panel reveals the split of the attractor point C to C and D as seen in WD03.

Figure 16

Phase trajectory maps for λ= 1.2 μm and a= 0.2 μm, and ψ= 70° corresponding to no thermal fluctuations (upper panel) and with thermal fluctuations (lower panel). For the case in which J is fixed, when grains are driven to the crossover J= 0, a₁ flips over. So grains frequently change their states corresponding to switching between the upper and lower frames in the map.

The lower panel in Fig. 16 also shows that 20 per cent of grains are aligned at two high-J attractor points, A and B, and 80 per cent grains are driven by RATs to the low-J attractor points, C and D. There, grains flip to the opposite flipping state (i.e. from upper to lower frame) and back. However, immediately after entering the opposite flipping state, grains are decelerated by RATs to that point again and they flip back. So grains flip back and forth between C and D. In fact, C and D are indistinguishable because grains flip very fast between them, and the probability of finding grains on each point is ∼0.5. In this sense, points C and D are also crossover points.13

Figs 17 and 18 show the phase maps of grain alignment for the full spectrum of the ISRF, and for two directions of radiation ψ= 0° and 90°. For the direction ψ= 0°, the map in the lower panel consists of three attractor points, A', B' and C, in which point C is the old attractor point at high angular momentum, and A′ and B′ are new low-J attractor points originating from A and B at J= 0 for the case of no thermal fluctuations (see the upper panel). In other words, averaging over thermal fluctuations of grains for the ISRF and ψ= 0° also changes the zero-J attractor point to the new thermal J attractor point.

Figure 17

Trajectory maps for the entire spectrum of the ISRF and irregular grain of size a_eff= 0.2 μm when thermal fluctuations are absent (upper panel) and when thermal fluctuations and thermal flipping are included (lower panel), corresponding to the alignment with respect to k or ψ= 0° (equivalent to the grain alignment in the absence of the magnetic field).

Figure 18

Similar to Fig. 17, but for ψ= 90°. Upper panel: a map with two high-J attractor points, A and B, two zero-J attractor points, C and E, and a ‘wrong’ attractor point D at cos ξ= 0. Lower panel: with thermal fluctuations, and the ‘wrong’ attractor point D disappears.

In Paper I, we found that at ψ close to 90° for both the AMO and irregular grains, the grain alignment tends to take place with a₁⊥B, which is in contrast to the classical Davis & Greenstein (1951) expectations. We called this ‘wrong’ alignment. An attractor point D at the ‘wrong’ alignment angle (i.e. ξ= 90°) is shown in Fig. 18 (upper panel) for the direction ψ= 90°, together with four other attractor points A, B, C and E, in the absence of thermal wobbling (see also Paper I). However, in the presence of thermal wobbling, the ‘wrong’ attractor point D no longer exists. Instead, we obtain four new attractor points, A', B', C' and D', corresponding to J∼J_th and cos ξ=±1 (see the lower panel in Fig. 18). This means that the ‘wrong’ alignment is indeed eliminated by averaging induced by thermal wobbling (see also Section 2).

5.3 Dynamics of shape 3

For the sake of completeness, let us study the effect of thermal fluctuations on RAT alignment for another irregular grain (shape 3 in DW97). As an example, we consider a particular grain size a_eff= 0.2 μm and one light direction ψ= 70°.

Fig. 19 shows the trajectory maps obtained for this grain corresponding to the case without (upper panel) and with thermal fluctuations (lower panel). It is shown that for ψ= 70°, this grain produces a phase map with two high-J attractor points, A and B, and two low-J attractor points, C' and D', in the presence of thermal fluctuations (see lower panel of Fig. 19). The lifting of the low-J attractor point C (see the upper panel of Fig. 19) from J= 0 to J= 2J_th (C' and D') when thermal fluctuations are taken into account, is also seen for this shape. However, grains can rapidly flip back and forth between C' and D'.

Figure 19

Phase maps for the grain shape 3 and the ISRF with two high-J attractor points corresponding to the cases without thermal fluctuations (upper panel) and with thermal fluctuations (lower panel). Thermal fluctuations shift the zero-J attractor point C to C' and D' with J∼ 2 J_th= 2I₁ω_T.

6 LOW-J ATTRACTOR POINTS

Our study indicates that a high fraction of grains are aligned at low-J attractor points. Therefore, the origin and stability of this low-J attractor point are important for grain alignment. We address these questions below.

6.1 Origin

The results above show that thermal fluctuations produce a new low-J attractor point from the zero-J attractor point, for both the ellipsoidal AMO and irregular grains, but this effect has not been seen for the spheroidal AMO. Now let us explore why this occurs by using the AMOs; however, the argument can be applicable to irregular grains.

First, let us study RATs averaged over the torque-free motion for the spheroidal and ellipsoidal AMOs. Following equation (D1) (see Appendix D), the average over thermal fluctuations can be rewritten as

37

where A is the torque component that arises from the average over torque-free motion and q= 2I₁E/J² with E is the total kinetic energy. It is convenient to define

38

which represents the torque resulting only from the average over torque-free motion as a function of J and the density of states in phase space s (see Appendix D).

In Fig. 20 we show torque components defined by equation (38) for spheroidal and ellipsoidal AMOs with J= 10J_th and J= 0.5J_th, corresponding to cases in which the role of thermal fluctuations is marginal and important, respectively.

Figure 20

Spin-up and alignment torques as functions of s(γ) for two values of angular momentum J= 20 J_th and J=J_th corresponding to an angle ξ= 160° between angular momentum and the magnetic field, and the precession angle φ= 180°. A fast drop of F and H for small J is observed, which gives rise to the fact that the sign of the averaged value over s is opposite to its sign at s= 0.

It can be clearly seen that for J= 10 J_th (i.e. J≫J_th), the obtained torques are nearly similar for irregular and axisymmetric grains. Yet the torques drop very rapidly to zero as s (note that s= sin γ for the spheroid) increases (see dashed lines in Fig. 20). The former stems from the fact that for the suprathermal rotation, a good coupling between the maximal inertia axis and the angular velocity is achieved. Therefore, there is no difference between the torque-free motion of irregular and axisymmetric grains.

However, the averaged torques become very different as J decreases. In fact, Fig. 20 (see the curves with J= 0.5 J_th) reveals that for axisymmetric grains, RATs decrease regularly with respect to s. Whereas, RATs for irregular grains drop rapidly, change the sign and rise again with increasing s.

The effect of such a variation on the average of RATs over thermal fluctuations (see equation 37) is evident. In Fig. 21, we show the corresponding torque components 〈F〉_φ and 〈H〉_φ obtained by averaging F and H using equation (37), as functions of J for several angles ξ between J and k for ψ= 0°. It can be seen that for J≫J_th, the averaged torques of irregular grains are similar to those of axisymmetric grains, and they become saturated in both cases. However, their averaged torques become very different when J decreases. For instance, the averaged torques for the former case change their sign at J∼J_th, while the torques for the latter case do not. We note that for the angle cos ξ < 0, the torque 〈H〉_φ is negative (i.e. torques tend to drive grains to the zero-J attractor point) for J≫J_th, equivalent to the absence of thermal fluctuations. As a result, the change of sign of 〈H〉_φ for J∼J_th in the presence of thermal fluctuations reveals that grains can be spun up again for J∼J_th. To some value of J, 〈H〉_φ continues to reverse its sign, and grains are decelerated. In other words, the irregular grains can be aligned at low attractor points with angular momentum J∼J_th.

Figure 21

Aligning 〈F〉_φ and spinning 〈H〉_φ torque components averaged over the Larmor precession angle φ and thermal fluctuations (i.e. over s using equation 37), normalized over their amplitude values for three angles ξ, as functions of angular momentum for the spheroid and ellipsoid AMOs. Plots show the change of sign of both torques when J²/J²_th decreases (i.e. thermal fluctuations increase) for the ellipsoid AMO, but not seen for spheroid. For J≫J_th, the torques are constant.

Consider now a stationary point ξ_s. As discussed previously, the stationary point ξ_s is either an attractor or a repellor point depending on the sign of the spin-up torque and the derivative of the alignment torque at that point; it is an attractor point if

Therefore, the change in sign found above for 〈F〉_φ and 〈H〉_φ can give rise to the thermal attractor points observed in maps for the ellipsoidal AMO and irregular grains.

In summary, the analytical analysis for the AMO results indicates that as thermal fluctuations become more important (i.e. J small), RATs as a function of ξ can have different forms (sign and magnitude) that increase the angular momentum of the low attractor points from J= 0 to J≈J_th. We also see a radical difference between torques averaged over the free precession of spheroidal grains and the free wobbling of ellipsoidal grains.

6.2 Stability of low- and high-J attractor points

From Section 6.1, we see that the effect of thermal fluctuations on RATs is to produce attractor points at low J. In addition, there are also attractor points at high J to which thermal fluctuations have marginal influence. It can be shown that most of grains in the ensemble tend to be at low-J attractor points. However, having low J, they may be significantly influenced by randomization processes (e.g. collisions by gaseous atoms).

For high-J attractor points present in the trajectory map (see Figs 9, 13 and 16), the angular momentum is determined by the radiation energy density u_rad (see equation 22), that is, J/I₁ω_T∼u_rad〈H〉_φ. As a result, depending on the distance to a given radiation source, the angular momentum of the high-J attractor points changes.

However, for low-J attractor points, because the angular momentum depends on thermal fluctuations (see Hoang & Lazarian 2008a), it does not change when radiation intensity varies. So grains of the same size, near and far from the radiation pumping source, have low-J attractor points of similar angular momentum. Therefore, because of thermal wobbling (see Lazarian, 1994), even near the radiation source, the alignment degree is not high for the case of trajectory maps without a high-J attractor point. Consequently, even in regions close to the strong radiation source (e.g. star-forming regions) most grains may rotate slowly.

Furthermore, the angular momentum of low attractor points depends on the grain size a_eff because internal relaxations are a function of a_eff (scaled as a⁷_eff; see also Table 1). So we expect that for large grains that have weak internal fluctuations, the value of angular momentum at low-J attractor points is very close to zero as a result of the deceleration action of RATs. In contrast, small grains having strong internal relaxations can have low-J attractor points of higher angular momentum.

7 FAST ALIGNMENT

In Paper I we showed that the grains can be aligned over a time-scale much smaller than the gas damping time. Now let us consider this problem when thermal wobbling and flipping are important.

Fig. 22 shows the trajectory map constructed for an ellipsoidal AMO with Q^max_e1/Q^max_e2= 0.78 and the light direction ψ= 70°. The arrow represents a time interval Δt= 10 t_phot, where t_phot defines the time-scale over which RATs decelerate grains from J=J_th to J= 0 in the absence of gas damping. It can be seen that the angular momentum of the low attractor point is the same as the case in which the gas damping is included (cf. lower panel of Fig. 13). We observe that a significant number of grains are aligned on low-J attractor points A, B, C and D over t_ali∼ 40 t_phot (see Fig. 22).

Figure 22

Similar to Fig. 13 (lower panel) but the gas damping is neglected. The arrow represents a time interval 10t_phot.

As discussed in Paper I, this type of fast alignment can occur in a diffuse medium with high radiation intensity, such as in the vicinity of a supernova or the wake of a comet. This stems from the fact that strong radiation can drive grains faster to the low-J attractor points where the thermal fluctuations act to maintain the stability of grain alignment.

8 INFLUENCE OF RANDOM BOMBARDMENT BY ATOMIC GAS

In Paper I we showed that, in most cases, RATs align grains with respect to magnetic fields while decreasing the angular momentum of the grains to J= 0. The results above indicate that thermal fluctuations can change a zero-J attractor point to a J∼J_th attractor point. In addition, atomic collisions can affect the alignment established by RATs in the presence of thermal fluctuations. In this section, we first briefly discuss the method of studying gas bombardment based on the Langevin equation. Then, we show the influence of this process on RAT alignment.

8.1 Method

The effects of collisions in the framework of paramagnetic alignment have been studied by many authors (Jones & Spitzer 1967; Purcell 1975; Purcell & Spitzer 1971; Lazarian & Roberge 1997) using the Fokker–Planck equations. However, the Langevin approach was used to study this problem numerically in Roberge, Degraff & Flatherty (1993) and Roberge & Lazarian (1999). The afore-cited papers used the equivalence of the Fokker–Planck and the Langevin equations to simulate the evolution of angular momentum of grains in a gas coordinate system. They derived explicit diffusion coefficients for random torques acting on spheroidal grains. According to the above works, an increment of angular momentum resulting from gas–grain collisions within an infinitesimal time interval dt is (see Roberge et al. 1993)

39

Here, dω_j are Wiener coefficients, and A_i and B_ij are diffusion coefficients, given by

40

41

where B^T is the transposal matrix of B. Similar to Roberge et al. (1993), diffusion coefficients are first calculated in the grain body system, and then transformed to the laboratory system. The diffusion coefficients are averaged over the precession of around J and over the Larmor precession angle of J about B, given by

42

where t_gas is the gaseous damping time. For a spheroidal grain, the gaseous damping time is

43

where I^b_zz is the moment of inertia of the grain along zz axis (i.e. along the maximal inertia axis ⁠), is the thermal velocity of atom, and b_m is the semi-axis of the grain. Γ_∥ and Γ_⊥ are factors characterizing the geometry of grain given by

44

45

g(e_m) is related to the eccentricity of the grain through the expression:

46

where ⁠. Diffusion coefficients are diagonal and given by the following expressions in the laboratory coordinate system in which the z-axis is along the magnetic field (Roberge et al. 1993)

47

48

49

Note that the above diffusion coefficients are derived by assuming perfect internal alignment of a₁ with J and for spheroidal grains. However, for the sake of simplicity, we can adopt these diffusion coefficients for studying the influence of gas bombardment on the alignment of irregular grains.

8.2 Results

First, we solve the Langevin equation (i.e. equation 39) for grains subjected to random torques as a result of gas collisions, assuming that the diffusion coefficients remain the same for irregular grains. We use the initial condition is generated from a uniform distribution in the range 0 to π and φ is a free parameter. Wiener coefficients dω in equation (39) are generated from a Gaussian distribution function at each time-step. Then the resulting solutions J_x, J_y and J_z are taken as input parameters for solving the equations of motion of grains driven by RATs in the spherical coordinate system described by J, ξ, φ (see equations 15–17) to obtain new values of J and ξ.14 This process is performed over N= 10⁶ time-steps Δt= 10⁻⁴t_gas. Other physical parameters for the ISM are taken from Table 1. The alignment angle ξ is used in averaging over the total time to obtain the degree of external alignment, and J is used to calculate the internal alignment, assuming that thermal fluctuations follow a Gaussian distribution.

For the AMO, we consider our default model of α= 45° (see Paper I). For ddscat, we study the entire spectrum of the ISM for grain shapes 1 and 3 with size a_eff= 0.2 μm.

8.2.1 Influence of randomization to the phase trajectory map in the presence of a high-J attractor point

We can see that collisions have an interesting effect on grain dynamics when the high attractor point is present. Random collisions are very efficient for low J. So, the motion of grains is disturbed by random collisions when they approach the low-J attractor point. After a time interval, grains enter the region of cos ξ > 0 (i.e. positive spinning RAT) for which RATs can spin up to the high-J attractor point. Therefore, for this case, random collisions increase the degree of alignment. The percentage of grains present in the vicinity of the high-J attractor points as a function of time is shown in the upper panel of Fig. 23. The respective degree of alignment is shown in the lower panel.

Figure 23

For the ISRF, shape 3, and ψ= 70°. The upper panel shows the variation of the per cent of grains present in the vicinity of the high-J attractor points A and B in time, and the lower panel shows the Rayleigh reduction factor for this case.

From the upper panel of Fig. 23, we find that, during the time interval t=t_gas, only about 10 per cent of grains are present at the high-J point. Then it increases with time and attains the saturated value of 75 per cent after t= 40t_gas.

Following the lower panel of Fig. 23, we see the rise of the degree of alignment with time. It has a significant value after t= 10t_gas, and the alignment is nearly perfect with R= 0.8 after t= 80t_gas.

Because of the stochastic properties of gas bombardment, the degree of alignment depends on the angular momentum of the high attractor point J_high(ψ). A detailed study of the degree of alignment with J_high(ψ) in Hoang & Lazarian (in preparation) shows that the alignment at high attractor points is nearly perfect if J_high(ψ) ≥ 3J_th,gas where J_th,gas is the angular momentum corresponding to the temperature of the ambient gas.

8.2.2 Influence of randomization to the phase trajectory map in the absence of a high-J attractor point

The degree of alignment for the spheroidal AMO and irregular grains in the absence of high-J attractor points is shown in Fig. 24. For these cases, random collisions act to decrease the alignment. This arises from the fact that random collisions remove grains from the low-J attractor points. We have found that if the angular momentum of the high-J stationary point, J_high(ψ), is larger than J_th,gas, RATs move grains to the vicinity of the high-J stationary points, and decelerate them again. It is seen that although R is decreased by gas bombardment, it is still not negligible (e.g. about 0.1 and 0.2 for AMO and irregular grains; see Fig. 24).

Figure 24

The dynamics of external and internal degrees of alignment and the Rayleigh reduction factor in time corresponding to RATs from the AMO (upper panel) and ddscat (lower panel) when the phase trajectory maps do not have high-J attractor points.

For J_high(ψ) < J_th,gas, grains are in a fully thermal regime. Therefore, the phase map of grains is mostly random. From the lower panel of Fig. 24, it follows that the degree of alignment is marginal. However, a more elaborate treatment in Hoang & Lazarian (2008a) does not use equation (2) and demonstrates a higher degree of alignment.

9 INFLUENCE OF H₂ FORMATION TORQUES ON GRAIN ALIGNMENT

When a hydrogen atom sticks to a grain, it starts its diffusion over the grain surface. In general, the grain surface is never uniform, and there are always certain special catalytic sites where hydrogen atoms can be trapped (see Purcell 1979; Lazarian 1995). A wandering H atom on the grain surface may encounter another atom dwelling at the catalytic site and a reaction takes place, producing a H₂ molecule. The ejected H₂ molecule acts as a miniature rocket thruster. Averaging over the grain surface (e.g. a brick surface), H₂ rockets produce a net angular torque that is parallel to the maximal inertia axis (Purcell 1979). However, resurfacing or poisoning by accretion of heavy elements can destroy the catalytic sites and create new ones. As a result, H₂ torques change both magnitude and direction over a definite time-scale t_L, the so-called resurfacing time.

9.1 Method

Purcell (1979) used the Monte Carlo method to simulate the variation of the torque as a result of grain resurfacing. Roberge & Ford (2000) used the equivalence of the Langevin and the Fokker–Plank equations to simulate the H₂ torque. They model L_z as a Gaussian process that is averaged to zero in time, and the correlation function that is exponential:

50

51

In equation (51), 〈(ΔL_z)²〉 is the magnitude of the H₂ torque given by

52

where f is the fraction of H atoms absorbed by the grain and converted to H₂, n(H) is the H density, E is the kinetic energy of each departing H₂ and l is the side of the individual catalytic site (Purcell 1979; Lazarian & Roberge 1997).

In an interval of time dt, the torque along the axis a₁, L^b_z, can be simulated by a Gaussian process with the correlation time-scale t_L, given by the Langevin equation (Roberge & Lazarian 1999)

53

where d w is a random variable, independent of time, sampled from a Gaussian distribution function, and L^b_z(t) is the instant torque at time t. This equation allows us to follow the evolution of the H₂ torque in time.

When the internal relaxations (Purcell 1979; Lazarian & Draine 1999b) are taken into account, the angle γ between the maximal inertia axis a₁ and the angular momentum fluctuates in time. However, as the fluctuation time-scale is much shorter than the alignment time-scale, we do not follow the evolution of this angle. Instead, we average over this. Following Spitzer & McGlynn (1979), the mean H₂ torque for a spheroid is given by

54

where denotes the average of cos γ over thermal fluctuations as defined by equation (12), which defines the value of the projection of L^b_z on to the angular momentum axis ⁠. For irregular grains, in equation (54) is replaced by the thermal average, ⁠, of C, which is given by

55

and

56

where k² and F are given in Appendix C (see WD03).

Because of the grain flipping, equation (54) becomes

57

where f₊ and f₋ are probabilities of finding the grain in positive and negative flipping states, respectively.

The equations of motion (i.e. equations 16 and 17) of grains by RATs and the H₂ torque are

58

59

where we have averaged RAT components over the Larmor precession angle φ.

9.2 Results

We solve the equations of motion for grains subjected to RATs and H₂ torques using the Runge–Kutta method with a finite time-step, Δt= 10⁻⁴t_gas and N= 10⁵ time-steps. We use the initial condition J= 20 J_th, and ξ is generated from a uniform angle distribution in the range from 0 to π. At each time-step, we first solve the Langevin equation (equation 53) for components of H₂ torques. Then, by substituting the resulting H₂ torques into equations (58) and (59), we solve the obtained equations for J and ξ. The resulting solutions J and ξ are used to construct trajectory maps and to calculate the degree of alignment.

9.2.1 Trajectory maps

Fig. 25 shows the map when the variation of H₂ torques is accounted for (lower panel), compared with the map driven by RATs (upper panel) produced by a radiation field of λ= 1.2 μm for a grain with size a_eff= 0.2 μm. We assume that the resurfacing process on the grain surface occurs rapidly, so we model the variation of H₂ torques with the correlation time-scale t_L= 0.1t_gas. In the absence of H₂ formation torques, RATs drive grains to attractor points. A significant fraction of grains are driven to low attractor points marked by C and D, at J/I₁ω_T∼ 1, and some grains are aligned at attractor points A and B corresponding to high angular momentum (see the lower panel of Fig. 25).

Figure 25

Upper panel: map trajectory of grain a_eff= 0.2 μm and λ= 1.2 μm by RATs. Lower panel: when the H₂ torque varies as a result of resurfacing of active sites on the grain surface. The results show that, during the spin-down, grains are driven by RATs to thermal angular momentum. However, because grains flip very fast, H₂ torques become inefficient to spin up grains at low-J attractor points.

Consider now a case when the H₂ torque amplitude is larger than that of RATs (e.g. consider the radiation direction ψ= 70°). It is obvious that RATs still dominate the Purcell torque for this case, and drive a significant number of grains to the low attractor points C and D. In fact, at the initial phase of J≫J_th, grains are spun up by the Purcell torque; however, during the spin-down, RATs lead grains to low-J attractor points. Because of grains having low angular momentum at the low-J attractor points, grains flip fast and the Purcell torques are averaged to zero (see the lower panel of Fig. 25) similar to what is described in Lazarian & Draine (1999a). Therefore, the trajectory map with two attractor points C and D of low angular momentum is determined by RATs as in the case without H₂ torques (see the upper panel of Fig. 25). However, for the attractor points A and B with high angular momenta, thermal flipping does not occur because of J≫J_th. In other words, during the spin-down period, grains are driven by H₂ torques to reach the low angular velocity regime. As J decreases, RATs become dominant and drive grains to attractor points C and D, as shown in the lower panel of Fig. 25. However, as J increases, H₂ formation torques become dominant and do not allow grains to have attractor points. Therefore, attractor points A and B become unstable, and grains are spun up and down by H₂ torques and RATs in the range between X and Y (see the lower panel of Fig. 25), corresponding to J/J_th= 5–40. The range of such a variation depends on the relative amplitude of H₂.

When the correlation time-scale t_L= 10t_gas, our results show that trajectories of grains ending at high-J attractor points C and D are marginally affected by H₂ torques, because RATs eventually drive grains to these attractor points at which the fast thermal flipping destroys completely the influence of H₂ torques. In contrast, grains that are driven to X and Y experience the same effect as in the case t_L= 0.1t_gas.

For t_L→∞, H₂ torques act to spin grains up. Therefore, the low-J attractor points become high-J attractor points, provided that H₂ torques dominate RATs.

For larger grains, thermal flipping is less important (see Lazarian & Draine 1999a), so H₂ torques are important even during the slow rotation period, influencing significantly the radiative alignment. We provide the treatment of this case elsewhere.

9.2.2 Dynamics of degree of grain alignment

Fig. 26 shows the variation of the degree of alignment 〈R〉 in time for the cases in which only RATs are considered (upper panel) and RATs plus H₂ torques with the correlation time-scale t_L= 0.1t_gas taken into account (lower panel).

Figure 26

The variation of degree of alignment 〈R〉, of J with B in time for the monochromatic light illuminating the grain in two directions ψ= 30° and 70°. Upper panel: only RATs align grains and they have stable orientation. Lower panel: the same as the lower panel, but H₂ torques with a correlation time-scale t_L= 0.1t_gas are included.

It is seen from Fig. 26 that the degree of alignment 〈R〉 is the same for the case ψ= 70° because of the dominance of RATs in driving grains to attractor points. At ψ= 30°, the alignment degree exhibits small variations. In addition, for both directions, 〈R〉 does not increase smoothly to a stable value as in the case of alignment by only RATs, as grain dynamics is affected by the stochastic variation of H₂ torques. When grains are driven to the attractor points with thermal angular momentum A and B, the fast thermal flipping of grains wipes out the effect of H₂ torques, which are fixed in the grain body coordinate system. Therefore, the grain alignment is mostly determined by RATs.

9.3 Alignment by radiative torques, H₂ formation torques and gas bombardment

In most cases, RATs play the key role in aligning grains with a magnetic field, while H₂ is only important to spin up grains. Moreover, for grains smaller than ∼10⁻⁴ cm, thermal flipping is very fast, and thus the influence of H₂ torques is rather marginal. In contrast, for large grains ≥10⁻⁴ cm, H₂ torques could be significant for the grain alignment. However, random collisions affect the alignment of grains mostly when grains rotate at thermal velocity. The upper panel in Fig. 27 shows both internal and external degrees of alignment as well as the Rayleigh reduction factor for the alignment induced by RATs and H₂ torques. It is seen that R∼ 0.25 for this case. However, when gas bombardment is taken into account, we find that R increases to R= 0.75 (see the lower panel of Fig. 27). In fact, this increase is not unexpected, as for this case grains are driven by gas bombardment to a high attractor point, which increases the degree of alignment, as discussed in Section 8. A more detailed discussion of the effects of H₂ torques is provided in Hoang & Lazarian (2008b).

Figure 27

The Rayleigh reduction factor for grain a_eff= 0.2 μm aligned by RATs, H₂ formation torques (upper panel) and by RATs, H₂ and gas bombardment (lower panel).

10 DISCUSSION

10.1 Central role of the analytical model

Our understanding of the RAT alignment was improved considerably when a simple model of a helical grain (i.e. the AMO) was introduced in Paper I. With the assumption of perfect internal alignment, the AMO reproduced the alignment effects similar to irregular grains studied by ddscat (see Paper I). For instance, the AMO shows that, for a given shape, size and radiation field, some grains can be spun up to suprathermal rotation and aligned at high-J attractor points, while most grains are driven to low-J attractor points. Moreover, the AMO demonstrated that whether RATs can spin-up grains to high-J attractor points depends on the ratio of the torque components Q_e1 and Q_e2.

The correspondence of the AMO to ddscat calculations of RATs for irregular grains in Paper I encouraged us to use the AMO for the case when thermal wobbling is taken into account. In our study, we used the AMO assuming that its inertial properties are defined by a triaxial ellipsoid rather than a spheroid. This was very easy to accomplish, as, within the AMO, torques acting upon either a spheroidal or an ellipsoidal body induce only precession, while the actual important torques (i.e. Q_e1 and Q_e2) arise from the mirror. We found a number of differences in the AMO dynamics for the two cases. We treat the AMO with an ellipsoid of inertia as our generic case.

10.2 Treatment of free wobbling

In Paper I we showed that Q_e3 in the AMO can be negligible when considering the alignment and spin-up torques. We have found again that the component Q_e3 for the AMO also plays a marginal role in alignment, as well as spinning up grains if we follow the algorithm in WD03 and average RATs over a sufficient time-steps (e.g. about N= 10³; see Section 3.3). Using our new algorithm of averaging in which we solve the differential equation for the rotation angle ζ instead of assuming that this angle ranges from 0 to 2π, we obtained similar results when the averaging is performed over a much longer time (e.g. P= 10³P_τ). This contributes mainly to the precession about the radiation beam axis k (see Fig. 1). As a result, in most practical situations, it is sufficient to deal with only two RAT components, Q_e1 and Q_e2.

We studied the alignment for two cases: (i) when the thermal relaxation time-scale is much shorter than the crossover time (i.e. averaging RATs over thermal fluctuations); (ii) when the thermal relaxation time-scale is much longer than the crossover time. For the former case, grains are directly aligned at low-J attractor points. In contrast, the grains undergo multiple crossovers in the second case.

The AMO enables us to average RATs analytically over thermal fluctuations assuming a Gaussian distribution. The obtained expressions allowed us to gain important insights into the role of thermal wobbling.

10.3 Direction of grain alignment

It has been shown in Paper I that the alignment by RATs occurs mainly with the longer axis perpendicular to the magnetic field. However, there was still a narrow range of radiation directions for which grains were aligned with longer axes parallel to magnetic field (i.e. ‘wrong’ alignment). We have shown that the ‘wrong’ alignment can no longer exist because of the fast wobbling of the grains. Indeed, in Paper I we found that ‘wrong’ alignment takes place for a low-J attractor point for a narrow range of angles around the angle between the light direction and magnetic field ψ=π/2. This range is narrower than the range of grain wobbling at the low-J attractor point. The disappearance of the ‘wrong’ alignment was, in fact, predicted in Paper I. A direct implication of this effect is that it allows a more reliable interpretation of the observation data: grains are aligned with long axes perpendicular to magnetic field when they are aligned by RATs.

10.4 Degree of grain alignment and critical size

The degree of external alignment of angular momentum with the magnetic field can achieve unity. However, because of the weak coupling of the maximal inertia grain axis with the angular momentum at low-J attractor points, the Rayleigh reduction factor R is lower than unity. A new effect that we discuss in this paper is the transition of grains from low-J to high-J attractor points, in situations when such attractor points coexist. This increases the degree of alignment. Interestingly enough, the transition that makes grains better aligned is induced by random bombardment (see Section 9).

The degree of alignment depends on the angular momentum J_high of the high attractor points. The value of J_high is a function of RATs, which increases with grain size. Because of the gas bombardment, only grains aligned with J_high > J_th can maintain the stable alignment with the magnetic field. The minimal size corresponding to J_high,min is called the critical size. For the irregular shape 1 and the ISRF, by studying the alignment for grains with size spanning from 0.025 to 0.2 μm, we found that the critical size of aligned grains is about 0.05 μm, assuming that J_high,min= 3J_th. Apparently, the critical size depends on the radiation field. Therefore, for an attenuated field (e.g. dark molecular clouds), the critical size is shifted to a larger value (i.e. only large grains can be aligned by RATs).

10.5 Role of paramagnetic dissipation

Paramagnetic dissipation does not play any role in the RAT alignment. Although the predictions of RAT alignment coincides with the predictions by the Davis–Greenstein mechanism (i.e. the alignment with long axes perpendicular to the magnetic field), RAT alignment occurs over a shorter time-scale compared to the paramagnetic relaxation timet_DG, provided that grains are paramagnetic.

As a result, unless grains are superparamagnetic (see Jones & Spitzer 1967; Mathis 1986), paramagnetic relaxation is irrelevant. A common fallacy entrenched in the literature is that RAT alignment is a sort of paramagnetic alignment of suprathermally rotating grains (i.e. a type of alignment of fast rotation suggested by Purcell 1979, but with fast alignment produced by RATs). This way of thinking about RAT alignment is erroneous, as, unlike the Purcell (1979) torques, RATs induce their own alignment, which is faster than the paramagnetic alignment.

Grains with superparamagnetic inclusions are different. A study by Lazarian & Hoang (2008) shows that, in typical interstellar magnetic fields of 10⁻⁶ G in situations when without paramagnetic dissipation grains have only low-J attractor points, the superparamagnetic grains always have high-J attractor points. In other words, superparamagnetic dissipation, unlike ordinary paramagnetic dissipation, can convert the high-J repellor point into an attractor point. Our study of the effect of gaseous bombardment shows that if the high-J attractor point exists, then grains eventually accumulate there. As a result, a very non-trivial effect takes place: for a given illumination, paramagnetic grains may rotate subthermally, while superparamagnetic grains rotate suprathermally.

10.6 Effects of radiation field intensity

When grains are aligned at high-J attractor points, the variations of radiation intensity induce the variation in the J value. However, a significant fraction of grains align at low-J attractor points, at which J does not vary with radiation intensity. Therefore, after a certain threshold at which grains are aligned suprathermally (i.e. with ω≫ω_thermal) at the high-J attractor point, the increase of radiation intensity does not increase the degree of alignment for grains aligned at such points.

When grains are aligned at low-J attractor points, a weaker alignment is expected, because of the poor coupling of the maximal inertia axis with angular momentum for low J (see Lazarian 1994; Lazarian & Roberge 1997), irrespective of the intensity of the radiation field. The increase of the radiation intensity does not affect the position of lower-J attractor points.

10.7 Role of gas bombardment and H₂ torques

This is the first study to take into account the effects of gaseous bombardment and H₂ formation within the RAT alignment mechanism. Random gas bombardment itself has always been considered as the cause of grain randomization. However, in the framework of the RAT alignment, its effect can be different.

As discussed in the paper, RATs align grains with respect to the magnetic field while driving grains to high-J and low-J attractor points. Naturally, the latter case is more sensitive to gas bombardment. This random process can increase the alignment degree by driving grains from the low-J attractor points to high-J attractor points. Such a process gives rise to the time-dependent alignment. The time over which the alignment can become stable is about 10²t_gas. For high-J attractor points, the alignment is marginally affected by the bombardment.

H₂ torques were believed to play a major role in grain alignment. However, we found in the present study that H₂ torques are indeed important for high attractor points for which they can spin grains up. For low attractor points with low angular momentum, the fast flipping of the grain axis as a result of thermal fluctuations averages out any torques that are fixed in the grain body (e.g. H₂ torques). However, when combined with RATs, Hoang & Lazarian (2008a) have shown that H₂ torques with a resurfacing time longer than the alignment time can produce new high-J attractor points.

10.8 Time-scales of alignment

If paramagnetic dissipation is neglected, we can guess that the only characteristic time that can determine RAT alignment could be t_gas (see DW97). As pointed out in Paper I, a strong radiation field can provide grain alignment over a time-scale shorter than the gas damping time (i.e. fast alignment). The characteristic time of this alignment can be ∼30t_phot, where t_phot is the time over which the amplitude value of RATs can deposit a grain with the angular momentum of the order of its initial angular momentum. The large coefficient in front of t_phot reflects the relative inefficiency of RATs in the vicinity of low-J attractor points. The supernova shell or star-forming regions are favourable mediums for such a fast alignment.

Our present study shows that another time-scale is appropriate, when the phase trajectory map contains both low-J and high-J attractor points. This time is about ∼30t_gas and it corresponds to the time-scale over which the gaseous bombardment transfers grains from low-J to high-J attractor points.

10.9 Grain dynamics at

In Paper I, with the assumption of simplified dynamics (i.e. assuming that J is always aligned with the maximal inertia axis a₁), we showed that the AMO induces a good alignment with respect to the radiation direction or magnetic field. The assumption of J and coupling is correct only for J≫J_th. In this paper, we use the analytical formulae for RATs and obtain the analytical expressions for torque components when thermal fluctuations are accounted for; therefore, the angle between J and fluctuates. Results show that torque components decrease substantially as thermal fluctuations become stronger (see Fig. 6), but the dynamics of grains does not change radically. This interesting finding testifies that thermal fluctuations as a result of internal relaxations do not play a key role in creating low-J attractor points as was believed earlier, but they ‘lift up’ the earlier existing attractor points.

In order to see whether the irregularity of grain shape is important, we modify the AMO slightly by replacing the spheroidal body by the ellipsoidal body. As discussed earlier, the ellipsoidal body does not affect Q_e1 and Q_e2 for the AMO, but changes the dynamics of fluctuations and therefore the averaging. As a result, the torques resulting from averaging over thermal wobbling are different, which induces the difference in the dynamics for irregular grains. Thus, the irregularity of shape and the presence of thermal fluctuations act together when grain angular momentum is comparable with J_th.

10.10 Helicity of grains

Helicity of grains as a factor of RAT processes was first mentioned in Dolginov & Mytrophanov (1976); however, they did not identify what can make grains helical. Moreover, it was claimed that some regions of space can have non-helical grains, which implied that helicity of grains is something that requires rather special conditions to emerge.15 Later, the RAT action was demonstrated for arbitrarily shaped grains in DW96, but this and subsequent studies (i.e. DW97; WD03) did not use the concept of helicity.

Grain helicity, as an essential requirement for alignment was identified in Paper I, where, by comparing the results of numerical calculations for irregular grains with the AMO, it was established that grains can be classified as either of left or right helicity. Moreover, it was shown that irregular grains demonstrate helicity and, therefore, alignment, not only subject to radiation, but also to gaseous flows (see more in Lazarian & Hoang 2007b).

In Paper I, an irregular grain rotating about its axis corresponding to the maximal inertia axis was identified as a helical grain. Our present study demonstrates the advantage of good internal relaxation of the maximal inertia axis and angular momentum. Indeed, we show that torques acting on a grain decrease when the grain starts wobbling.

10.11 Irregularity of shape and inertia

Irregular grains, in general, have an irregular shape and have to be characterized by a triaxial ellipsoid of inertia, rather than a spheroid. The irregularity of grain shape gives rise to grain helicity, while the irregularity in terms of the moment of inertia modifies grain dynamics. One of the interesting effects that we have observed for grains, which can be characterized by a spheroid of inertia, is a transfer of grains from a low-J attractor point (actually, zero-J attractor point) to a high-J attractor point in the absence of gaseous bombardment. This effect disappears when thermal fluctuations are taken into account.

10.12 Comparison with earlier studies

Harwit (1970) discussed the irregular torques arising from emission and absorption photons randomly depositing angular momentum to grains. The marginal importance of the mechanism for the alignment was shown, however, in Purcell & Spitzer (1971). Dolginov (1972) proposed the first model of RATs that act on chiral (e.g. quartz) grains. Later, Dolginov & Mytrophanov (1976) considered a more generic case of RATs (i.e. RATs arising from ‘twisted’, irregular grains). The idea of RAT alignment was mostly ignored (see Lazarian 1995, as an exception) until numerical calculations of RATs became available in DW96 and DW97. From the earlier studies, the most relevant to this work is WD03, where the effects of thermal fluctuations within the RAT alignment were empirically studied.

This work extends our study in Paper I, which demonstrated the ability of the AMO to represent helical grains. The AMO allowed us to address the generic features of RAT alignment. We note that our procedures of torque averaging and our results for angular momentum for low-J attractor points differ from those in WD03. Our interpretation of the origin of those points is also different.

Extending the AMO, we have proven that thermal trapping of grains at the low-J attractor points reported in WD03 is generic; that is, it does not depend on the particular direction between the beam and the magnetic field, or on the particular choice of the irregular grain or on the particular wavelength/radiation spectrum chosen. Moreover, we have shown that we expect the new low-J attractor points to appear close to cos ξ= 1, which makes grains aligned with B at these new attractor points. All the results we have obtained with the AMO were also tested with two irregular grain shapes. In addition, we took into account gaseous bombardment and H₂ formation.

10.13 Accomplishments and limitations of the present study

The study above has clarified a number of outstanding issues in RAT alignment theory. It has relied on the guidance of the analytical studies of RATs and crossovers in Paper I and has taken into account the proper averaging of RATs arising from thermal wobbling. It has confirmed the prediction in Paper I of the gradual change of the position of lower attractor points on the trajectory maps of irregular grains from zero to the thermal value of angular momentum. In addition, our study has accounted for the effects of gaseous bombardment and H₂ formation on RATs. This has resulted in the discovery of a new important effect, that is, the transfer of grains from lower attractor points to high attractor points, when the latter points exist. This is the single most important finding of our paper. It is also a practically important effect, as the internal alignment is perfect for grains at high attractor points and reduced for grains at lower attractor points. All in all, the present paper substantially extends our analysis in Paper I.

The limitations of the present study stem from the fact that, while addressing the issue of how the degree of alignment changes in different circumstances, it does not calculate the degree of alignment exactly. This is done in Hoang & Lazarian (in preparation).

10.14 Towards modelling of grain alignment

Present-day modelling of grain alignment in both molecular clouds (CL05; Pelkonen et al. 2007; Bethell et al. 2007) and accretion discs (Cho & Lazarian 2007) uses heuristic recipes for determining whether grains are aligned. In these studies, the amplitude values of the RATs are calculated and used to calculate the maximal angular velocities achievable for a given damping of grain rotation. Such velocities parametrize the efficiencies of grain alignment in the chosen environments. In this paper, we have confirmed the utility of such a parametrization and improved the criterion for the alignment to be efficient. However, we still have to obtain better measures of the expected alignment.

11 SUMMARY

In the present paper, we have studied the role of thermal fluctuations, thermal flipping, efficiency of H₂ torques and influence of random collisions on the RAT grain alignment for both the AMO and irregular grains. Our main results are summarized as follows.

• We have studied analytically the alignment by RATs in the presence of thermal fluctuations, for the spheroidal AMO, and found no increase in the value of angular momentum of low-J attractor points.

• We have also used an AMO with inertia defined by a triaxial ellipsoid and averaged numerically RATs over thermal fluctuation. For this ellipsoidal AMO, we have found the increase of angular momentum of low-J attractor points from J= 0 to J∼J_th. We have also found a similar effect for irregular grains in which RATs are calculated using ddscat and for the entire spectrum of the ISRF.

• We have proved that ‘wrong’ alignment corresponding to low angular momentum for a narrow range of radiation direction reported in Paper I is eliminated when thermal fluctuations and thermal flipping are considered.

• We have found that random collisions of atomic gas increase the degree of alignment when grains are aligned by RATs with both low-J and high-J attractor points by driving grains from low-J to high-J attractor points. When the angular momentum of the high-J attractor point J_high(ψ) is greater than 3 J_th,gas, a significant degree of alignment can be achieved.

• We have studied the influence of H₂ formation torques with a short resurfacing time, in the framework of the RAT alignment, showing that they can, in particular circumstances, enhance the degree of alignment with respect to the magnetic field.

1

The Barnett relaxation arises from the unpaired electron spins in the grain. The nuclear relaxation results from unpaired nuclear spins. The mechanisms are different and should not be mixed up.

2

A recent study by Lazarian & Hoang (2008) proves that in the presence of superparamagnetic inclusions both Barnett and nuclear relaxation are stronger.

3

These findings also contrast with observational claims based on visible and near-infrared radiation (Goodman et al. 1995). The difference in results was explained in Lazarian (2003).

4

Peaks A_V of 150 were claimed for the clouds in Pagani et al. (2004). According to Crutcher (private communication), these peaks are likely not to produce polarized dust emission.

5

Although being a step forward compared to the earlier naıve predictions of polarization, which were mostly detached from the grain alignment theory, CL05 and the follow-up studies (e.g. Pelkonen et al. 2007; Bethell et al. 2007; Cho & Lazarian 2007) are also not exact, as they are based on the alignment efficiencies inferred from idealized numerical studies, rather than on the exact RAT alignment theory.

6

‘Wrong’ alignment without specifying the conditions for it was also reported in DW97.

7

In Hoang & Lazarian (2008a), we perform exact calculations and show that this approximation underestimates the actual degree of alignment. However, it is still useful in the present paper for our simplified qualitative discussion of alignment.

8

This is an approximation, which as we show elsewhere underestimates the actual degree of alignment.

9

A more accurate description of the process that accounts for the finite spin-lattice relaxation is provided in Lazarian & Draine (1999b).

10

For the sake of simplicity, hereafter, we denote 〈F〉=〈F(ξ, φ, ψ, J)〉, 〈H〉=〈H(ξ, φ, ψ, J)〉 and 〈F〉_φ=〈F(ξ, φ, ψ, J)〉_φ, 〈H〉_φ=〈H(ξ, φ, ψ, J)〉_φ.

11

Note that RATs in equations (23)–(25) have opposite signs compared with those in Paper I, because in Paper I we have defined n₁= sin α (i.e. we incorporated the minus sign of RATs into n₁).

12

In some trajectory maps, we do not label all existing repellor points.

13

These crossover points are different from those described in DW97, in which the grains were spinning up right after crossover. This stems from inaccurate treatment of crossovers in DW97; see more detail in Paper I.

14

Here we average over the Larmor precession angle φ.

15

An additional confusion in the aforementioned study stemmed from the claim that the prolate grains should align with long axes perpendicular to the alignment of the long axes of oblate grains. However, it was shown in Lazarian (1995) that both types of grains align the same way, if internal relaxation is taken into account.

We thank Bruce Draine for clarifying for us some points in the ddscat code. TH acknowledges Joseph Weingartner for clarifications provided at the initial stage of this work. We thank Wayne Roberge and E. Ford for sharing with us their results from Roberge & Ford (2000). We also acknowledge helpful comments by the anonymous referees. We acknowledge the support from the National Science Foundation (NSF) Centre for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas and NSF grant AST 0507164.

Appendices

APPENDIX A: RELATION OF ANGLES Θ, β, Φ AND ξ, φ, ψ

(A1)

(A2)

(A3)

where and are unit vectors in the grain reference system, and and are unit vectors in the laboratory system. Dot products in the above equations can be obtained from the following expressions:

(A4)

(A5)

(A6)

Here,

(A7)

(A8)

(A9)

(A10)

(A11)

(A12)

where and are unit vectors of the magnetic field coordinate system in which ⁠. Here α, γ and ζ are Euler angles shown in Fig. A1.

APPENDIX B: RADIATIVE TORQUES

Similar to Paper I, in order to compare our results more easily with those in earlier works, wherever it is possible, we preserve the notations adopted in DW97. The mean RAT efficiency over wavelengths, is defined as

(B1)

where u_λ is the energy density (see Mathis et al. 1983), and Q_λ is the RAT efficiency corresponding to wavelength λ. The RAT from the anisotropic component of radiation is defined by

(B2)

Here, γ is the anisotropy degree of radiation, a_eff is the effective size of the grain (see DW96; Paper I), and and are the mean wavelength and mean energy density of the radiation field, which are respectively given by

(B3)

(B4)

APPENDIX C: AVERAGING OVER TORQUE-FREE MOTION

Consider an ellipsoid with three principal axes a₁, a₂ and a₃ and moment of inertia I₁ > I₂ > I₃. The dynamics of such an ellipsoid is clearly presented in Landau & Lifshitz (1976). Let us define a dimensionless quantity

(C1)

where

is the ratio of total kinetic energy to the rotational energy along the maximal inertia axis.

For q < I₁/I₂, k² < 1, the solution of Euler equations is

(C2)

(C3)

(C4)

where cn, sn and dn are hyperbolic trigonometric functions, and τ is given by

(C5)

The rotation period around the maximal inertia axis is

(C6)

where F is the elliptic integral defined by

(C7)

For q > I₁/I₂, angular velocities are given by

(C8)

(C9)

(C10)

where

(C11)

The rotation period for this case is given by

(C12)

In the abovC16xe equations, ± stand for the positive and negative flipping states.

The Euler angles α, γ, ζ (see Fig. A1) can be deduced from angular velocity as follows:

(C13)

(C14)

(C15)

WD01 averaged RATs over torque-free motion as follows:

(C16)

We have implemented an algorithm to average RATs over torque-free motion in which the angle ζ is obtained by solving an Euler differential equation (see Landau & Lifshitz 1976). We found that RATs obtained by our method are in good agreement with what is obtained using equation (C16) when averaged over a time-scale greater than 10³P_τ. Therefore, to save computing time, equation (C16) is used to average RATs over thermal fluctuations in our paper.

APPENDIX D: AVERAGING OVER THERMAL FLUCTUATIONS

The average of a quantity A over thermal fluctuations is defined by

(D1)

The number of state s in equation (D1) is given by

(D2)

with α₁ given by

(D3)

for q > I₁/I₂, and α₁=π/2 for q≤I₁/I₂ (see WD03).

We compare the results obtained by averaging RATs from the AMO (see equations 23–25) with the body being a triaxial ellipsoid over thermal fluctuations with results for Q_e3= 0 in Fig. D1. It can be seen that the torque components 〈F〉_φ and 〈H〉_φ for the former case are slightly different from those in the latter case. When the averaging of RATs is performed with sufficient accuracy (i.e. with a sufficiently high time-step), we expect the contribution of Q_e3 to the spinning and aligning torques to be negligible. Therefore, in our study for the AMO, Q_e3 is disregarded for both the alignment and spin-up effect.