Alignment of Irregular Grains by Radiative Torques: Efficiency Study

We work in the alignment coordinates (ξ, ψ, ϕ), where ξ is the alignment angle between angular momentum and the magnetic field vectors J and B , ψ is the angle between B and the anisotropic radiation direction k , and ϕ is the Larmor precession angle of J around B . The coordinate system is illustrated in Figure 2. In the following analysis, averaging a quantity q over ϕ is denoted by $\bar{q}$.

Using the equations of motion of RATs where perfect internal alignment is assumed, we have, similarly as in LH07, for the alignment angle ξ and angular momentum $J^{\prime} =J/{I}_{1}{\omega }_{T}$ (Hoang & Lazarian 2016)

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\xi }{{\rm{d}}t^{\prime} }=\displaystyle \frac{M}{J^{\prime} }\bar{F}(\xi ,\psi )-{\delta }_{m}\sin \xi \cos \xi ,\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}J^{\prime} }{{\rm{d}}t^{\prime} }=M\bar{H}(\xi ,\psi )-J^{\prime} (1+{\delta }_{m}{\sin }^{2}\xi ).\end{eqnarray} \tag{ 9 }$$

Above, $t^{\prime} =t/{\tau }_{\mathrm{gas}}$, I₁ is the maximum grain moment of inertia, ${\omega }_{T}={(2{k}_{B}{T}_{\mathrm{gas}}/{I}_{1})}^{1/2}$, $M=\gamma \bar{\lambda }{u}_{\mathrm{rad}}{a}_{\mathrm{eff}}^{2}/2$. Finally, $\bar{F}$ and $\bar{H}$ are the ϕ-averaged aligning and spin-up RAT components.

The stationary points (ξ_s , J_s ) can be found by setting the equations of motion to zero, which leaves us with

$$\begin{eqnarray}&&\displaystyle \frac{\bar{F}({\xi }_{s})(1+{\delta }_{m}{\sin }^{2}{\xi }_{s})}{\bar{H}({\xi }_{s})}={\delta }_{m}\sin {\xi }_{s}\cos {\xi }_{s}.\end{eqnarray} \tag{ 10 }$$

The zeros of Equation (10) now give, as shown in LH07, the stationary points in the (ξ, J) phase space. Specifically, we find universal stationary points $\sin {\xi }_{s}=0$, which have, after ϕ-averaging, $\bar{F}=0$.

Generally, for a stationary point to be an attractor, we have the requirements, as given by Draine & Weingartner (1997; DW97),

$$\begin{eqnarray}&&A+D\lt 0,{BC}-{AD}\lt 0,\end{eqnarray} \tag{ 11 }$$

where

$$\begin{eqnarray}\begin{array}{rcl}A & = & \displaystyle \frac{M{\rm{\nabla }}\bar{F}}{J{{\prime} }_{s}}-{\delta }_{m}\cos 2{\xi }_{s},B=-\displaystyle \frac{M}{J{{\prime} }_{s}^{2}}\bar{F},\\ C & = & M{\rm{\nabla }}\bar{H}-{\delta }_{m}J{{\prime} }_{s}\sin 2{\xi }_{s},\\ D & = & -(1+{\delta }_{m}{\sin }^{2}{\xi }_{s}),\end{array}\end{eqnarray} \tag{ 12 }$$

and ${\rm{\nabla }}\equiv \tfrac{{\rm{d}}}{{\rm{d}}\xi }$. When we consider the universal stationary points, the requirement for high-J attractors simplifies to

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\nabla }}\bar{F}}{\bar{H}}\left|{}_{\sin {\xi }_{s}=0}-{\delta }_{m}\lt 0,\quad \bar{H}\gt 0\right..\end{eqnarray} \tag{ 13 }$$

When the equations of motion are analyzed for grain ensembles as a function of ψ, we can find f_highJ . In the following calculations, we assume that the maximum axes of inertia of grains are aligned with the magnetic field, i.e., the universal stationary points alone are considered. Physically, this corresponds to a situation in which the alignment process has reached a steady state, and all alignable grains are aligned. Thus, the fraction f_highJ is the fraction of grains for which requirement (13) for any of the two universal stationary points is satisfied.

In order to derive predictions for alignment degrees with AMO, a distribution of q_max corresponding to some dust grain population is required. Now, we focus on q_max distributions obtained from the GRE ensemble. In Figure 3 we present distributions of q_max for the three differently sized oblate and prolate GREs, along with lognormal fits for the distributions.

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We chose a lognormal distribution to provide the fits for the adequate quality of the fit and because the generation of GRE shapes involve lognormal statistics. It should be noted that no claims on the connection between the shape of the q_max distribution and the underlying GRE statistics are made, and such an analysis is well beyond our current scope.

The lognormal distribution, which can be generated using logmean μ and logdeviation σ as parameters, provides adequate fits for all the separate-size-shape grain populations. Furthermore, for the total ensemble of 2400 grains, the lognormal distribution provides a good fit with σ = 0.37, μ = 1.17 (see Figure 10, right panel).

The mean value of the subensemble q_max distribution increases with the grain size in the ISRF. For extrapolations, it is important to identify the functional form of the average q_max with respect to the ratio λ/a_eff of the wavelength and grain size. In Figure 4 we present the mean values for the oblate and prolate subensembles.

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We identify a significant peak in ${\bar{q}}_{\max }$ (bar indicating an ensemble mean) at λ/a_eff = 2 for both the prolate and oblate subensembles. In terms of grain sizes, there thus will be some grain size for any given incident energy density distribution for which a maximum ${\bar{q}}_{\max }$ is obtained. Now, the ISRF according to Mathis et al. (1983) peaks at 700 nm, and the UV peak at 120 nm is left outside the wavelength range considered.

When we extrapolate a similar functional form of ${\bar{q}}_{\max }$ for any grain size, the peak location will correspond to a grain size of 0.35 μm. For the ISRF, the distribution of q_max for grains with a_eff = 0.35 μm would therefore likely tend most toward high values.

Additionally, we note that ${\bar{q}}_{\max }\approx 1$ is a local minimum between λ/a_eff = 4 and 5 with a low standard deviation, especially in the oblate case. At this size range, we therefore expect the ensemble behavior to be one of the most representative behaviors of the difference between irregular grains and AMO, which predicts no high-J attractors without magnetic inclusions at ${q}_{\max }=1$.

The distributions of q_max have mean values close to ${q}_{\max }=3$. Considering AMO, using Figure 1, we predict that if most grains have q_max values close to 2 or higher, the fraction f_highJ will quickly decrease to zero at around ψ = 30°. Next, we test how the GRE ensemble results compare to the AMO prediction.

We compute the high-J alignment fractions for the GRE subensembles using the procedure described in Section 4.1. For simplicity, magnetic inclusions are not considered, i.e., δ_m = 0.

First, in the left panel of Figure 5, the f_highJ of the subensembles are compared with the prediction of AMO with a lognormal (σ = 0.37, μ = 1.17) q_max distribution. As δ_m = 0, the f_highJ of AMO falls to zero just after ψ has reached 30°, as is predictable from Figure 1 for a distribution of q_max that does not contain grains with ${q}_{\max }\lt 0.5$.

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The GRE ensembles exhibit high-J attractors more universally than predicted by AMO. Two important observations can immediately be identified from the data: First, f_highJ both increases and becomes more uniform with respect to ψ along with increasing grain size. Second, for all subensembles, the minimum value of f_highJ is centered around ψ = 40°, near the angular range for which AMO predicts no high-J attractors without paramagnetic inclusions, and the smaller the grain, the wider the minimum. Most grain types also exhibit a local maximum around ψ = 70°–80°.

The difference between f_highJ of the oblate and prolate grains is slightly in favor of oblate grains. The difference between base shape types is mostly smaller than 10 %, with the exception of the 0.2 μm cases in the approximate range ψ ∈ (60°, 70°).

Second, in the right panel of Figure 5, a comparison of rearranged subensembles is performed. The rearrangement is done so that grains of a certain size with q_max values lower or higher than the mean q_max value of all grains at this size are separated into different subensembles.

According to Figure 1, grains with large q_max are expected to have higher f_highJ at small ψ and lower f_highJ large ψ when compared to the counterpart with low q_max values. However, we see that this is not the case for irregular grains. Instead, grains with larger q_max have almost universally at least as large f_highJ as the counterpart with smaller q_max, the most notable exception being the 0.75 μm case around ψ = 40°.

Third, in order to identify the contribution of RATs by different wavelengths on f_highJ , Figures 6 and 7 show f_highJ as a function of λ/a_eff and ψ.

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Comparing the columns of Figures 6 and 7 to the functional form of ${\bar{q}}_{\max }$ confirms that a high value of q_max is required to produce high-J attractors at small ψ. Furthermore, as ${\bar{q}}_{\max }$ of both the oblate and prolate cases are highly similar, so are Figures 6 and 7.

Next, Figures 6 and 7 can be compared against Figure 1 with the help of ${\bar{q}}_{\max }$ data from Figure 4. At λ/a_eff ≈ 5, where ${\bar{q}}_{\max }$ is at a minimum, the contour structure of Figure 1 predicts that the cutoff angle ψ for which high-J begins to occur should also be at a minimum. This slight drift is visible in both Figures 6 and 7. Additionally, in the middle of the cross-shaped contour structure, no high-J attractors should exist without magnetic inclusions. For AMO, the high-J poor region occurs at ${q}_{\max }\approx 1.2$, while for irregular grains, the region of the least high-J attractors occurs at λ/a_eff ≈ 3, at ${q}_{\max }\approx 2$, or at 3 (for small prolates).

As a last observation, the 0.75 μm cases clearly show the discrepancy between f_highJ of AMO and the GRE ensemble for ψ < 40°. As shown in Figure 4, only a small fraction of the grains have ${q}_{\max }\lt 1$, but more high-J attractor points exist at values of ψ higher than 40°.

It is important to note that because q_max is not a free parameter for irregular grains and because the irregular grain radiative torques can have a highly variable functional form for two grains with similar q_max, a figure such as Figure 1 is difficult if not impossible to produce for the GRE ensemble. Thus, a comparison of the radiative torques of AMO and of the GRE ensemble is crucial in order to explain the discrepancy between f_highJ of AMO and the GRE ensemble of Figure 5.

Potentially, the main explanation for the differences between obtained GRE ensemble results and the predictions derived from AMO is the different functional forms of the RATs themselves. In this section, we compare the RAT components in both the scattering frame and in the alignment frame. In both cases, the mean square differences $\left\langle {{\rm{\Delta }}}^{2}\right\rangle q$ of the GRE and AMO quantities q,

$$\begin{eqnarray}&&\left\langle {{\rm{\Delta }}}^{2}\right\rangle q=\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}{({q}_{\mathrm{GRE}}-{q}_{\mathrm{AMO}})}^{2},\end{eqnarray} \tag{ 14 }$$

are determined.

First, we compare in Figure 8 the shapes of the scattering frame RAT components Γ₁ and Γ₂. In the procedure, the AMO RAT is produced according to the q_max of each grain in the GRE ensemble and by normalizing the torques with the maximum values of Γ₁. The mean square difference is then calculated over the angle Θ, and the result is presented as a function of λ/a_eff.

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According to Figure 8, the AMO is generally more likely to produce similar RATs to GRE grains the smaller the grains are. The result is consistent with the results of Figures 6 and 7, which show that smaller grains exhibit f_highJ values closer to those predicted by the AMO.

Additionally, the difference $\left\langle {{\rm{\Delta }}}^{2}\right\rangle {{\rm{\Gamma }}}_{2}$ reaches a peak value around λ/a_eff = 2. The difference for Γ₂ remains large for larger grains, and for Γ₁, it increases scrictly as the grains grow larger. For large grains, the predictions of the AMO can therefore deemed to be relatively inaccurate.

Next, alignment frame RATs are compared in order to potentially identify critical differences between the AMO prediction of Figure 1 and f_highJ of the GRE subensemble. The differences are determined over the whole GRE ensemble as a function of the angle ψ. Additionally, the sign difference percentages of ∇F between the AMO and the GRE subensembles are determined. The results are collected in Figure 9.

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Irregular grain RATs, particularly ∇F and its sign, differ from those of AMO at high values of ψ. The spin-up component H appears to approach that of the AMO as ψ approaches 90°.

Differences of ∇F are locally very small at values around ψ = 40°, where f_highJ reaches the minimum value with both the AMO and the GRE subensembles (Figure 5, left panel).

The sign difference of the AMO and the GRE subensembles reaches up to 50% at ψ = 90°. However, f_highJ tends to decrease at ψ > 80° for all grains to as low as 0.2, implying that the attractors in this region are low high-J ones.

To summarize the section, the RAT differences show an expected size dependence that is evident from the f_highJ results. Particularly the differences in the scattering frame component Γ₂, which is expected to be important near ψ = 90° (where the radiation direction is perpendicular to J ), and the differences of the ξ-derivative of aligning the RAT component F are likely to be related to high f_highJ values that are not predicted by the AMO at large ψ.

According to Figure 1, high-J attractors are universal with moderate magnetic inclusions, expect in the limit of large q_max and ψ. We test the existence of high-J attractors when magnetic inclusions are added for the total GRE ensemble in Figure 10.

When δ_m is increased to 10, this increases f_highJ to 0.9 when ψ < 70°, i.e., high-J attractors become universal at ψ < 70°. Above ψ = 70°, f_highJ decreases to 0.5, as seen in the left panel of Figure 10.

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Figure 1 predicts that at ψ = 90°, for grains with ${q}_{\max }\gt 2.5$, high-J attractors do not exist without extreme magnetic inclusions, if at all. According to the right panel of Figure 10, the mean value of ${q}_{\max }=2.762$. Thus, approximately half of the grains have ${q}_{\max }\gt 2.5$, which makes the result in the left panel of Figure 10 consistent with Figure 1 and also leads to the conclusion that in the limit of large q_max and ψ, high-J attractors do not exist.

The total degree of alignment is approximately given by Equation (2), and it depends on the degree of internally aligned grains with either low-J or high-J attractors. We recall that for grains with a low-J attractor only, the degree of internal alignment was chosen at Q_X,lowJ = 0.25. This results in the total degrees of alignment presented in Figure 11.

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The resulting degrees of alignment, where prolate and oblate subensembles are combined, resemble the f_highJ results from Figure 5, as expected. The choice of Q_X,lowJ guarantees that AMO at ψ > 30° gives the lowest possible degree R = 0.25. However, especially the large 0.75 μm grains clearly have a higher degree of alignment.

In the approximative scheme, where the total degrees of alignment are derived from Equation (2), f_highJ provides a straightforward means to solve for R with any assumption on Q_X,lowJ/highJ values.

LH07 identified the grain helicity as the driver of the RAT alignment and presented a toy model of a helical grain that allowed a simple analytical description, i.e., the model of the AMO. A significant progress in understanding the properties of grain alignment was achieved by the AMO. Nevertheless, one would not expect the toy model to perfectly represent the RAT alignment of an ensemble of irregular grains. In fact, deviations of the RAT functional form from AMO predictions were reported in LH07. ⁹ However, with just a few shapes studied there, it was not possible to evaluate the consequences of these deviations on the degree of alignment achievable by an ensemble of irregular grains. The present paper provides the statistical data needed by using an extensive number of irregular grains subjected to the interstellar radiation field.

The polarization predicted by the AMO was insufficient to explain the maximum values of polarization observed in the ISM, at least in the dust model where only silicate grains are aligned. ¹⁰ This made the AMO model with enhanced magnetic dissipation, introduced in Lazarian & Hoang (2008), relevant. The detailed predictions of the AMO were presented in Hoang & Lazarian (2016, HL16).

An important point of the present study is that our results show a higher alignment efficiency than for the AMO, opening the possibility that the observed grain alignment can be accounted for by the RAT alignment of ordinary paramagnetic grains. However, magnetic inclusions cannot be immediately disregarded based on our results, where R ∼ 0.4–0.5 for grain sizes <0.3 μm. Recently, an analysis of spectral polarization by Draine & Hensley (2021) required R ∼ 0.7 to explain the 10 μm polarization feature, assuming a reasonable axis ratio of spheroidal grains (between 1/3 and 3, favoring more modest ratios). The axis ratios of the undeformed spheroids in this study were 0.5 and 2. However, the higher alignment degrees of irregular grains, compared to AMO, allow the possibility that in some cases, the RAT alignment degrees of irregular grains may be sufficient to explain some observations even if grains do not have magnetic inclusions.

HL16 presented a contour map for the critical magnetic relaxation parameter, δ_m,cri for AMO (adapted in Figure 1), required to produce high-J attractor points. There, it was found that for δ_m > 10, high-J attractors are universal, and for lower values, there are regions of interest both in terms of q_max and alignment angle ψ.

As a rule of thumb, the q_max values near unity are most problematic for the existence of high-J attractors. HL16 found that when ψ = 45° is approached from smaller angles, δ_m,cri is smallest for grains with q_max smaller than unity. The same is true for q_max larger than unity when it is approached from the large angle side. This implies that f_highJ values should drop for numerical ensembles, given that the distribution of the q_max factor contains values near unity, which was systematically observed in Section 3. The exact value for the δ_m,cri and the critical value for the q_max likely differ from those obtained from the AMO, but the trends suggested by the AMO nevertheless appear.

In our wavelength-dependent study, we find that as grains become smaller, q_max generally tends to decrease gradually, although the special significant drop around λ/a_eff = 5 introduces another general rule to the shape of ${q}_{\max }$ as a function of λ/a_eff. Thus, the decrease in f_highJ as a function of ψ (Figures 6 and 7, 0.1 μm and 0.2 μm cases) should slightly drift toward lower values of ψ in this size range. The effect is produced by our study, but the drift is very subtle and agrees well with the results predicted using the AMO.

In the leftmost columns, the 0.1 μm and 0.2 μm cases in Figures 6 and 7, the values of f_highJ become more uniform as ψ > 50°. The size region is probed in more detail for 0.75 μm grains (N = 400). If q_max and f_highJ are connected, the spread of q_max values as grains grow results in f_highJ values that can be explained by the AMO prediction from HL16. HL16 states that for q_max values greater than unity at 0° < ψ < 45°, grains with large q_max should be well aligned, and again, this is similar for ψ > 45° and small q_max. Here, the idealized AMO predicts similar results, but for grains with large q_max, the upper limit of the range of well-aligned particles is reduced to about ψ = 35°. As the spread of the q_max distribution is not extreme for large grains, the more uniform f_highJ must also be attributed to the RAT shape differences between the AMO and irregular grains.

Using numerical evidence of ensembles of irregular grains, it was found that AMO can successfully reproduce the basic features of the RAT alignment observed with the large sample of GRE grains. In particular, this explains why the alignment takes place with long axes perpendicular to magnetic field, it explains that the alignment can occur at low-J and high-J attractor points, and it explains the alignment properties that are achieved at these points. It also explains the decreasing maximum rotational rate with increasing ψ, which becomes rapid as ψ exceeds 60°. For irregular grains, this decrease is observed around ψ ∼ 45°, beyond which f_highJ fluctuates. The fluctuations in the grain-ensemble-averaged f_highJ can likely be attributed to the irregular shape itself, due to which ϕ-averaged RATs at certain angles provide attractor points.

The importance of q_max was further solidified by the evidence because it correlates with the likelihood of existence of high-J attractor points. The numerical results also imply that while the exact point of critical values for the parameters may differ for different types of grains due to the complicated RAT shapes of irregular grains, the general behavior is also correctly produced by the AMO.

At the same time, our results show the limitations of the AMO in terms of a qualitative description of the properties of some grains in the ensemble of shapes we studied. We will address elsewhere to what extent this is limited to the shapes of higher elongation and whether the correspondence can be obtained with a modified AMO, e.g., by an AMO with a mirror turned at a different angle, as suggested as a modification in LH07. The advantage of the AMO is its extreme simplicity, and having a set of different modifications might not be convenient. The development that we find very positive is that the RATs acting on the studied ensemble of grains can provide a higher alignment degree than the classical model of the AMO for large angles between the radiation direction and the magnetic field (i.e., ψ > 45°, see Figure 11). This gives us hope that the observed interstellar alignment can be explained without evoking an enhanced magnetic susceptibility of grains.

As we mentioned above, the most important question addressed in this paper is whether the enhanced magnetic dissipation is necessary to explain high degrees of alignment. The LH07 theory was formulated for no additional relaxation, while it was shown in Lazarian & Hoang (2008) that higher degrees of alignment can be achieved if the high-J stationary points are stabilized in the presence of a strong magnetic response of the grains. The arguments in favor of the magnetic grains are presented in Hoang & Lazarian (2016), while Lazarian & Hoang (2019) discussed the observational procedures for quantifying the grain magnetic properties.

Our present work opens up the discussion about the composition of interstellar grains. An important question is whether magnetic inclusions (Lazarian & Hoang 2008; Hoang & Lazarian 2016) are necessary to explain the observed interstellar alignment. The increased grain-alignment efficiency reported in this study facilitates explaining interstellar grain alignment. The task becomes even more simple if a significant part of interstellar carbon is locked in the composite grains consisting of carbonaceous and silicate fragments. We note, however, that our ability to explain the interstellar polarization with paramagnetic grains does not actually mean that the grains do not contain magnetic inclusions.

For instance, most of the interstellar iron is locked in grain material, but we do not know of which chemical substance iron is a part and what the magnetic properties of the substance are. Thus, it is important to test the composition of grains by measuring the polarization and its changes in special settings, for instance, near stars of transient radiation sources, e.g., novae. Furthermore, existing studies have constrained the possibility of magnetic inclusions by comparing predictions (Draine & Hensley 2013) with observations (Planck Collaboration et al. 2020). The corresponding tests of the magnetic properties of grains based on the grain-alignment theory are discussed in Lazarian & Hoang (2019), while in Lazarian & Hoang (2021), the tests related to the alignment-dependent grain disruption are proposed.

In the present paper we considered the B-alignment, i.e., the alignment with respect to the magnetic field. As was demonstrated in LH07, the alignment can also occur with respect to the direction of radiation, provided that the precession induced by the Γ₃ component of the radiative torque is faster than the grain precession about the ambient magnetic field. The process of k-alignment can be analyzed with our present approach by assuming ψ = 0.

The RAT alignment process is complex, with only a small fraction of grains initially reaching the high-J attractor points. In the presence of time-dependent radiation sources, LH07 demonstrated that transient alignment takes place. During the process, most grains are aligned in the low-J attractor point on the timescale of the order of grain precession induced by the Γ₃ RAT component. However, on the time of several τ_ran, the grains reach stationary alignment at high-J attractor points if such points are present for the given combination of q_max and ψ.

For our calculation in the paper we assumed that the grains are subject to the sufficiently strong radiation field that causes grains at high-J attractor points to rotate with velocities significantly faster than the thermal velocity. These conditions are well satisfied in the diffuse ISM for ∼0.1 μm grains. For grains shielded from the radiation in molecular clouds or accretion disks, as well as for grains with sizes appreciably different from the typical wavelength of the impinging radiation, the rotational rates of grains at high-J points can be reduced to become comparable to the thermal rotation speed. Such grains can be randomized by gaseous collisions and other randomizing factors. The RAT alignment of these grains in these conditions is decreased for ψ larger than 80 degrees. According to Figure 8 in Lazarian & Hoang (2019), the RAT strength decreases for this range.

The current study is for the interstellar radiation field. The ISM is the area in which the grain alignment has been studied most frequently (see Andersson et al. 2015). However, the RAT alignment of dust is a ubiquitous process that can take place in different astrophysical environments and in planet atmospheres (see Lazarian 2007). Therefore detailed studies of the peculiarities of grain alignment for different spectra of radiation-illuminating ensembles of irregular grains of different compositions are required. This work can therefore be viewed as the first step in this direction.

The properties of RATs we explored in this paper are applicable for grains made of different materials. The change in optical constants is not expected to radically change the grain alignment. On the basis of the obtained results, we can therefore consider the alignment of not only silicate, but also of carbonaceous grains. The RAT alignment of carbonaceous grains was considered in Lazarian (2020), and it was shown that carbonaceous grains can exhibit a different mode of RAT alignment, i.e., the alignment with long grain axes parallel to the magnetic field. The anomalous mode arises due to charged carbonaceous grain precession along the electric field. The latter arises due to the grain motion with respect to magnetic field.

The difference between the alignment of carbonaceous and silicate grains considered in this study arises from the difference of the magnetic moment of the grains. The latter is significantly smaller for carbonaceous grains. Our qualitative conclusions on the modification of grain alignment compared to the AMO prediction remain true, but in terms of observed polarization, the observed degrees can be lower due to grain rotation about the magnetic field.

If carbonaceous grains reach the state of high-J rotation, their magnetic moment increases, and it was shown in Lazarian (2020) that such grains can align with long grain axes perpendicular to the magnetic field, i.e., they align in a similar way to silicate grains. Our present study shows that the percentage of grains in the high-J rotation state for irregular grains is higher than in the AMO prediction. Therefore we may expect more carbonaceous grains to become aligned in the regular fashion, i.e., with long grain axes perpendicular to the magnetic field.

The alignment of grains with long axes parallel to the magnetic field can, as explained in Lazarian (2020), result in a more complicated pattern of polarization that is more difficult to interpret in terms of the underlying magnetic field. At the same time, if actual interstellar grains are composite, i.e., contain both carbonaceous and silicate fragments, their alignment is similar to that of silicate grains.

The ensemble of irregular grain shapes used for our study and comparison to AMO is generated using a Gaussian random algorithm, which is not subject to any astrophysical constraints. Therefore a considerable fraction of generated grains has extreme shapes characterized by a high ratio of RAT components, q^max < 0.5 or ${q}^{\max }\gt 5$, which induces a higher degree of alignment for ψ > 45° than predicted by the AMO. However, whether such highly irregular shapes can exist in the ISM remains unclear. Highly irregular shapes are likely to have a low tensile strength, so that they are more easily disrupted by by RATs (Hoang et al. 2019; Hoang 2019, 2020). More observational studies on grain shapes are required to test the RAT alignment and disruption theory as well as the relation of the alignment and the disruption (see Lazarian & Hoang 2021).