Recent progress in theory and observational study of dust grain alignment and rotational disruption in star-forming regions

Let us define an irregular grain of size $a$ and mass density $\rho$ (Figure 1). In the fixed grain’s frame, this irregular shape is described by there principal axes $\hat{a}_{1},\hat{a}_{2},\hat{a}_{3}$ with the inertia moments of $I_{1}>I_{2}\geq I_{3}$, respectively. For the sake of simplicity, we assume an oblate spheroidal shape (i.e., $I_{1}>I_{2}=I_{3}$) with the length of the semi-minor axis of $c$ and the lengths of the semi-major axes of $a$. The corresponding inertial moments are $I_{\|}\equiv I_{1}=\frac{8\pi}{15}\rho a^{4}c=\frac{8\pi}{15}\rho sa^{5}$ and $I_{\perp}\equiv I_{2}=I_{3}=\frac{4\pi}{15}\rho a^{2}c\left(a^{2}+c^{2}\right)=\frac{4\pi}{15}\rho sa^{5}\left(1+s^{2}\right)$, where $s=c/a$ is the axial ratio of oblate grains. The volume of this grain is $V=4/3\pi sa^{3}$.

For an isolated grain, the angular momentum $\boldsymbol{J}$ is conserved, and the grain axis of maximum inertia in general makes an angle with $\boldsymbol{J}$. Figure 1(a) describes the torque-free motion of the oblate grain, which includes the precession of angular velocity of $\boldsymbol{\Omega}$ or momentum $\boldsymbol{J}$ with a frequency of $\omega$ along the grain shortest axis. In the presence of an external magnetic field, radiation field or a gas flow, the angular momentum $\boldsymbol{J}$ precesses around the magnetic field, radiation field or incident gas flow, Figure 1(b) (Lazarian and Hoang, 2007a, b).

Magnetic properties of dust are essential for their interaction with the ambient magnetic field and grain alignment. We consider grains having embedded iron atoms (such as silicate grains). The presence of iron atoms with unpaired electrons makes interstellar dust a natural paramagnetic material. When iron atoms are diffusely distributed, the grain is called paramagnetic (PM). When iron atoms are distributed as iron clusters, dust grains become superparamagnetic material (SPM, see Hoang and Lazarian 2016a).

The magnetic susceptibility of a paramagnetic grain at rest is given by

$$\chi_{\rm PM}(0)=0.03n_{23}f_{p}\hat{p}^{2}\left(\frac{20\,\rm K}{T_{\rm d}}\right)~{}{,}$$

(1)

with $\hat{p}=p/5.5$ ($p\simeq 5.5$ for silicate grains; see Hoang and Lazarian 2016a, and references therein). This equation shows the dependence on $T_{\rm d}$. The magnetic susceptibility reduces for higher $T_{\rm d}$ because of the halt in magnetization due to the thermal fluctuation of free-electron spins.

For a superparamagnetic grain, we get

$$\chi_{\rm SPM}(0)=0.026N_{\rm cl}\phi_{\rm sp}\hat{p}^{2}\left(\frac{20\,\rm K}{T_{\rm d}}\right)~{}{.}$$

(2)

where $N_{\rm cl}$ is the number of Fe atoms per cluster, $\phi_{sp}$ is the volume filling factor of iron cluster in grain. $N_{\rm cl}$ could vary from 20 to $10^{5}$ (Jones and Spitzer 1967). One can see that iron inclusion increases significantly the magnetic property of grains.

For a grain rotating at a frequency $\omega$, the magnetic susceptibility is a complex number with the imaginary part that describes the absorption properties of magnetic energy given by

$$\chi_{2}(\omega)=\frac{\chi(0)\tau_{\rm el}\omega}{[1+(\omega\tau_{\rm el}/2)^{2}]^{2}}~{}{,}$$

(3)

where $\tau_{\rm el}=2.9\times 10^{-12}/f_{p}n_{23}$ is the spin-spin relaxation time with $n_{23}=n/10^{23}\,\rm cm^{-3}$ the atomic density of material, $f_{p}$ the fraction of Fe atoms in the grain (e.g., $f_{p}=1/7$ for MgFeSiO₄ grains, see Hoang et al. 2014).

Let us define a quantity $K(\omega)=\chi_{2}(\omega)/\omega$, which yields

$$K_{\rm PM}(\omega)\simeq 8.7\times 10^{-14}\hat{p}\left(\frac{20\,\rm K}{T_{\rm d}}\right)\left(\frac{1}{[1+(\omega\tau_{\rm el}/2)^{2}]^{2}}\right)~{}{,}$$

(4)

and

$$K_{\rm SPM}(\omega)\simeq 2.6\times 10^{-11}N_{\rm cl}\phi_{\rm sp}\hat{p}^{2}\left(\frac{20\,\rm K}{T_{\rm d}}\right)k_{\rm SPM}(\omega)~{}{,}$$

(5)

with

$$k_{\rm SPM}(\omega)=\exp\left(\frac{{\rm 0.011\,K}\times N_{\rm cl}}{T_{\rm d}}\right)\left[1+\left(\frac{\omega\tau_{\rm sp}}{2}\right)^{2}\right]^{-2},$$

(6)

where $\tau_{\rm sp}$ is the timescale of thermally activated remagnetization given by

$$\tau_{\rm sp}\simeq 10^{-9}\exp\left(\frac{0.011\,{\rm K}\times N_{\rm cl}}{T_{\rm d}}\right)~{}{\rm s.}$$

(7)

The supermagnetic grain has enormously increased magnetic susceptibility (Jones and Spitzer 1967), and the coupling with the magnetic field is stronger than in an ordinary paramagnetic grain. Note that a superparamagnetic grain behaves like a paramagnetic grain in the absence of an external magnetic field.

The most important process relies on the Barnett relaxation (Purcell 1979). A rotating (super-)paramagnetic grain will be magnetized due to the Barnett effect, which results in the dissipation of rotational energy due to the rotating component perpendicular to $\boldsymbol{a_{1}}$. The dissipation will eventually bring $\boldsymbol{\Omega}$ and $\boldsymbol{J}$ aligned with $\boldsymbol{a_{1}}$, which is the minimum energy state. The stage of $\boldsymbol{J}\perp\boldsymbol{a_{1}}$ is called ”right internal alignment”.

The relaxation times by the Barnett effect for a paramagnetic (so-called Barnett relaxation) and a supermagnetic grain (so-called super-Barnett relaxation) are given by

$$\tau_{\rm BR,PM}=\frac{\gamma_{e}^{2}I_{\|}^{3}}{VK_{\rm PM}(\omega)h^{2}(h-1)J^{2}}\simeq 0.5\hat{\rho}^{2}a^{7}_{-5}f(\hat{s})\left(\frac{J_{\rm d}}{J}\right)^{2}\left[1+\left(\frac{\omega\tau_{\rm el}}{2}\right)^{2}\right]^{2}~{}{\rm yr,}$$

(15)

$$\tau_{\rm BR,SPM}=\frac{\gamma_{e}^{2}I_{\|}^{3}}{VK_{\rm SPM}(\omega)h^{2}(h-1)J^{2}}\approx 0.16\hat{\rho}^{2}f(\hat{s})a_{-5}^{7}\left(\frac{1}{N_{\rm cl}\phi_{\rm sp,-2}\hat{p}^{2}}\right)\left(\frac{J_{\rm d}}{J}\right)^{2}\left(\frac{T_{d,1}}{k_{\rm SPM}(\omega)}\right)~{}\rm yr,\\$$

(16)

where $V=4\pi/3a^{3}$ is the grain volume, $\gamma_{e}=-g_{e}\mu_{B}/\hbar$ is gyromagnetic ratio of an electron ($g_{e}\simeq 2$ and $\mu_{B}\simeq 9.26\times 10^{-21}\,\rm erg\,G^{-1}$), $\hat{s}=s/0.5,f(\hat{s})=\hat{s}[(1+\hat{s}^{2})/2]^{2}$, and $J_{\rm d}=\sqrt{I_{\|}k_{\rm B}T_{\rm d}/(h-1)}$ is the dust thermal angular momentum (see also Hoang and Lazarian 2014) with $h=I_{\|}/I_{\perp}=2/(1+s^{2})$ and $\phi_{\rm sp,-2}=\phi_{\rm sp}/0.01$.

However, gas collisions can strongly randomize the internal alignment if the Barnett relaxation is slower than the grain randomization by gas collisions. Therefore, grains have efficient internal alignment when the timescale for the Barnett relaxation is shorter than that for gas rotational damping. Thus, the maximum size that a paramagnetic and superparamagnetic grain has the internal alignment with $\boldsymbol{J}$ along the short axis $\boldsymbol{a_{1}}$ can be determined by the balance between these two timescales, which yields (Hoang et al. (2022a))

$$a^{\rm PM}_{\rm max,aJ}\simeq 0.39\times h^{1/3}St^{1/3}\left(\frac{\hat{p}^{2}}{\hat{\rho}n_{3}T^{1/2}_{1}}\right)^{1/6}\times\left(\frac{1}{1+(\omega\tau_{\rm tel}/2)^{2}}\right)^{1/3}\left(\frac{(h-1)T_{\rm gas}}{T_{\rm d}}\right)^{1/6}~{}{\rm\mu m},$$

(17)

and

$$a^{\rm SPM}_{\rm max,aJ}\simeq 2.18h^{1/3}St^{1/3}\left(\frac{N_{\rm cl,4}\phi_{\rm sp,-2}\hat{p}^{2}}{\hat{\rho}n_{3}T_{1}^{1/2}\Gamma_{\|}}\right)^{1/6}\times\left(\frac{1}{k_{\rm SPM}(\omega)}\right)^{1/6}\\ \times\left(\frac{(h-1)T_{\rm gas}}{T_{\rm d}}\right)^{1/6}~{}{\rm\mu m},$$

with $N_{\rm cl,4}=N_{\rm cl}/10^{4}$ (Hoang et al., 2022a). One can see that $a_{\rm max,aJ}$ increases with grain rotation as $St^{1/3}$ and decreases with gas density as $\sim n^{-1/6}_{\rm H}$. With iron inclusion, superparamagnetic grains could achieve efficient internal alignment for larger grain sizes.

Atomic nuclei within the grain can also have nuclear paramagnetism due to unpaired protons and nucleons. The attachment of H atoms to the grain or water ice mantle induce nuclear magnetism due to proton spin (Purcell 1979). Nuclear paramagnetism can also induce internal relaxation as Barnett effect for electron spins (Lazarian and Draine 1999). The nuclear relaxation (NR) time is given by

	$\displaystyle\tau_{\rm NR}$	$\displaystyle=$	$\displaystyle\frac{\gamma_{n}^{2}I_{\|}^{3}}{VK_{n}(\omega)h^{2}(h-1)J^{2}},$		(18)
		$\displaystyle\simeq$	$\displaystyle 125\hat{\rho}^{2}a_{-5}^{7}f(\hat{s})\left(\frac{n_{e}}{n_{n}}\right)\left(\frac{J_{d}}{J}\right)^{2}\left(\frac{g_{n}}{3.1}\right)^{2}\left(\frac{2.79\mu_{N}}{\mu_{n}}\right)^{2}\times\left[1+\left(\frac{\omega\tau_{n}}{2}\right)^{2}\right]^{2}{\rm s},$

where $\gamma_{n}$ is nuclear gyromagnetic ratio, $n_{e}$ is the number density of unpaired electrons in the dust, $n_{n}$ is the number density of nuclei that has the magnetic moment $\mu_{n}$, and $\mu_{N}=e\hbar/2m_{p}c=5.05\times 10^{-24}$ erg G^-1 with $m_{p}$ the proton mass is the nuclear magneton, $\tau_{n}\sim 10^{-5}(3.1/g_{n})^{2}(10^{22}{\rm cm}^{-3}/n_{e})$s is the nuclear spin-spin relaxation time, and $K_{n}=\chi_{n}/\omega$ with $\chi_{n}$ the nuclear magnetic susceptibility (see Hoang 2022).

The maximum size for internal alignment induced by nuclear relaxation is given by Hoang (2022),

$\displaystyle a_{\rm max,aJ}({\rm NR})<3.03h^{1/3}St^{1/3}\left(\frac{(n_{e}/n_{n})(g_{n}/3.1)^{2}}{n_{3}T_{1}^{1/2}\Gamma_{\|}}\right)^{1/6}\left(\frac{1}{1+(\omega\tau_{\rm n}/2)^{2}}\right)^{1/3}\times\left(\frac{(h-1)T_{\rm gas}}{T_{\rm d}}\right)^{1/6}~{}{\rm\mu m.}$

(19)

which increases with the suprathermal rotation as $St^{1/3}$ for $\omega<2/\tau_{n}$, but decreases as $St^{-1/3}$ for $\omega>2/\tau_{n}$ due to suppression of nuclear magnetism at fast rotation. For the typical density of $n=10^{3}\,\rm cm^{-3}$, large grains of size upto $a\sim 3\,\mu$m can have efficient internal alignment by nuclear relaxation.

Atoms and molecules in a rotating grain experience a centrifugal force that stress them out from the grain central of mass. The chemical attractive forces, on the other hand, tend to tight them together. These effects cause the grain deformation. If grains are made up inelastic materials, the deformation results in the dissipation of rotational energy, which eventually brings the angular momentum $\boldsymbol{J}$ to align with the grain’s shortest axis $\boldsymbol{a_{1}}$ (see Purcell 1979; Lazarian and Efroimsky 1999). This internal alignment mechanism is named as inelastic relaxation.

The characteristic time of inelastic relaxation for an oblate spheroidal grain can be estimated as (see Hoang et al. 2022a)

$$\tau_{\rm iER}=\frac{\mu Q}{\rho a^{2}\Omega_{0}^{3}}g(s)=\frac{\rho^{1/2}a^{11/2}\mu Q}{(kT_{\rm gas})^{3/2}St^{3}}g^{\prime}(s)=0.034\hat{\rho}^{1/2}a_{-5}^{11/2}\frac{\mu_{8}Q_{3}}{T_{1}^{3/2}}\frac{g^{\prime}(s)}{St^{3}}~{}{\rm yr},$$

(20)

where $Q$ is the quality factor of grain material (Q=100 for a silicate rocks, Efroimsky and Lazarian 2000), $\mu$ is the modulus of rigidity ($\mu\sim 10^{7}\,\rm erg\,cm^{-3}$ for cometary, Knapmeyer et al. 2018). $\Omega_{0}=St\Omega_{T}$, $\mu_{8}=\mu/(10^{8}\,\rm erg\,cm^{--3})$, $Q_{3}=Q/10^{3}$, and $g^{\prime}=2.2s^{3/2}g(s)$ with $g(s)$ the geometrical factor as

$$g(s)=\frac{2^{3/2}7}{8}\frac{(1+s^{2})^{4}}{s^{4}+1/(1+\sigma)},$$

(21)

which corresponds to $g(s)=7.0$ and $4.6$ for $s=1/2$ and $1/3$, respectively.

Following Hoang et al. (2022a), the maximum grain size for efficient inelastic relaxation is defined by $\tau_{\rm iER}=\tau_{\rm gas}$ and given by

$$a_{\rm max,aJ}({\rm iER})\simeq 1.55\left(\frac{\mu_{8}Q_{3}}{\sqrt{\hat{\rho}}}\right)^{-2/9}\left(\frac{T_{1}}{n_{3}}\right)^{2/9}\times\left(\frac{s}{g^{\prime}(s)\Gamma_{\|}}\right)^{2/9}St^{2/3}~{}{\rm\mu m,}$$

(22)

which implies that large grains of $a\sim 1.55\,\rm\mu m$ can be efficiently aligned via inelastic relaxation even at thermal rotation of $St=1$, and large grains can be aligned for $St>1$.

Figure 2 shows the comparison of grains having internal alignment by Barnett relaxation for superparamagnetic grains (color lines) and inelastic relaxation (black lines) as functions of the gas density. Grains are assumed to be spun-up by RATs. First, the Barnett relaxation is not effective in very dense regions (e.g., $n_{\rm H}>10^{9}\,\rm cm^{-3}$) because of the gas rotational damping and lower $St_{\rm RAT}$. For a given gas density, one can see that Barnett relaxation is more efficient, but the inelastic relaxation is able to align larger grains (panel (a)). The reason is that the maximum grain size for inelastic relaxation scales as $\sim St^{2/3}$, while it scales as $\sim St^{1/3}$ for Barnett relaxation. For stronger radiation fields, the inelastic relaxation is efficient in denser gas (panel (b)), which is due to the fact that inelastic timescale reduces as $\sim St^{-1/3}$.

For a typical density of the ISM ($n_{\rm H}\simeq 10^{3}\,\rm cm^{-3}$), one can see that the Barnett relaxation for PM/SPM grains are much more effective than the inelastic relaxation. However, unlike the PM/SPM, diamagnetic grains (e.g., carbonaceous dust) does not experience the Barnett relaxation effect, but they could still have efficient internal alignment for size $a<a_{\rm max,aJ}(\rm iER)$ due to inelastic relaxation, because this relaxation is independent from the grain magnetic properties.

Larmor precession: The interaction between the grain magnetic moment due to the Barnett effect with the magnetic field induces the Larmor precession of $\boldsymbol{J}$ around $\boldsymbol{B}$, resulting the first coupling of the grain with the magnetic field (see e.g., Hoang 2022).

The rate of the Larmor precession for (super)paramagnetic grains are respectively given by

$$\tau_{B,\rm PM}=\frac{2\pi}{|d\phi/dt|}=\frac{2\pi I_{\|}\Omega}{|\mu_{\rm Bar}|B}=\frac{2\pi|\gamma_{e}|I_{\|}}{\chi_{\rm PM}(0)VB}\simeq 3.5\times 10^{-4}\frac{\hat{\rho}T_{d,1}a^{2}_{-5}}{n_{23}f_{p}\hat{p}^{2}}\left(\frac{B}{20\,\rm\mu G}\right)^{-1}~{}{\rm yr,}$$

(23)

and

$$\tau_{B,\rm SPM}=\frac{2\pi|\gamma_{e}|I_{\|}}{\chi_{\rm SPM}(0)VB}\simeq 4.05\times 10^{-2}\frac{\hat{\rho}T_{d,1}a_{-5}^{2}}{N_{\rm cl}\phi_{sp,-2}\hat{p}^{2}}\left(\frac{B}{20\,\rm\mu G}\right)^{-1}\rm yr.~{}~{}~{}~{}$$

(24)

When the Larmor precession is faster than the gas randomization, the magnetic field becomes an axis of grain alignment, which is known as magnetic alignment. The maximum size for the fast Larmor precession around the B-field is defined by $\tau_{B}=\tau_{\rm gas}$, which is

$$a^{\rm Lar,PM}_{\rm max,JB}\simeq 1.2\times 10^{6}\hat{s}n^{-1}_{3}T^{-1/2}_{1}T^{-1}_{d,1}\left(\frac{B}{20\,\rm\mu G}\right)\left(\frac{n_{23}f_{p}\hat{p}^{2}}{\Gamma_{\|}}\right)~{}{\rm\mu m,}$$

(25)

and

$$a^{\rm Lar,SPM}_{\rm max,JB}\simeq 1.02\times 10^{9}\hat{s}n^{-1}_{3}T_{1}^{-1/2}T^{-1}_{d,1}\left(\frac{B}{20\,\rm\mu G}\right)\left(\frac{N_{\rm cl,4}\phi_{\rm sp,-2}\hat{p}^{2}}{\Gamma_{\|}}\right)~{}{\rm\mu m.}~{}~{}~{}$$

(26)

for PM and SPM grains, respectively.

Para- and superpara-magnetic relaxation: The angle between $\boldsymbol{J}$ and the magnetic field can be reduced by paramagnetic relaxation, resulting in the magnetic alignment of $\boldsymbol{J}$ with $\boldsymbol{B}$. This mechanism is known as the DG relaxation (Davis and Greenstein 1951). As numerically shown in Hoang and Lazarian (2016a, b), magnetic relaxation alone is not enough to produce efficient grain alignment when grains have thermal rotation (i.e., $St\leq 1$) due to strong internal thermal fluctuations and gas randomization. Yet, the joint action of suprathermal rotation by RATs and magnetic relaxation however can make grains to achieve high degree of alignment.

The characteristic times of the DG relaxation for the PM and SPM grains are given by (see e.g., Hoang and Lazarian 2016a)

$$\tau_{\rm mag,PM}=\frac{I_{\|}}{VK_{\rm PM}(\Omega)B^{2}}=\frac{2\rho a^{2}}{5K_{\rm PM}(\Omega)B^{2}}\simeq 8.6\times 10^{4}\frac{\hat{\rho}a^{2}_{-5}T_{d,1}}{\hat{p}}\left(\frac{B}{20\,\rm\mu G}\right)^{-2}\left(1+(\omega\tau_{\rm el}/2)^{2}\right)^{2}~{}{\rm yr,}~{}~{}~{}$$

(27)

and

$$\tau_{\rm mag,SPM}=\frac{2\rho a^{2}}{5K_{\rm SPM}(\Omega)B^{2}}\simeq 3.75\times 10^{2}\frac{\hat{\rho}a_{-5}^{2}}{N_{\rm cl}\phi_{\rm sp}\hat{p}^{2}}\frac{T_{d,1}}{k_{\rm SPM}(\Omega)}\left(\frac{B}{20\,\rm\mu G}\right)^{-2}~{}{\rm yr.}~{}~{}~{}$$

(28)

The respective maximum of size that grain alignment can still be affected by the DG magnetic relaxation is defined by $\tau_{\rm mag}=\tau_{\rm gas}$ and is equal to (Hoang et al., 2022a)

$$a^{DG,\rm PM}_{\rm max,JB}\simeq 0.0097\times n^{-1}_{3}T^{-1/2}_{1}T^{-1}_{d,1}\left(\frac{B}{20\,\rm\mu G}\right)^{2}\left(1+(\omega\tau_{el}/2)^{2}\right)^{-2}~{}{\rm\mu m,}$$

(29)

and

$$a_{\rm max,JB}^{DG,\rm SPM}\approx 224\times n^{-1}_{3}T^{-1/2}_{1}T^{-1}_{d,1}\left(\frac{B}{20\,\rm\mu G}\right)^{2}N_{\rm cl,4}\phi_{\rm sp,-2}\hat{p}^{2}k_{\rm SPM}(\omega)~{}{\rm\mu m.}$$

(30)

One can see that the gas density and B-field strength determine which grains are aligned with B-field as $a^{DG}_{\rm max,JB}\sim n^{-1}_{\rm H}B^{2}$. The DG efficiency drops dramatically with increasing the gas density. Paramagnetic relaxation is negligible for large grains, but superparamagnetic relaxation can be efficient even for large grains up to $\sim 20\,\rm\mu m$ for MCs with $n_{\rm H}\sim 10^{3}\,\rm cm^{-3}$.

Here we describe the basic elements of the RAT alignment paradigm, including radiative precession, grain alignemnt at low-$J$ and high-$J$ attractors, and the minimum size for RAT alignment.

Radiative precession: In the presence of an anisotropic radiation field, irregular grains experience radiative precession due to RATs, with the grain angular momentum precessing around the radiation direction ($\boldsymbol{k}$). For a grain with angular momentum $J$, the radiative precession time is given by (Lazarian and Hoang 2007a; Hoang and Lazarian 2014),

$$\tau_{k}=\frac{2\pi}{|d\phi/dt|}\approx\frac{2\pi J}{\gamma u_{\rm rad}\lambda a_{\rm eff}^{2}Q_{e3}}\simeq 56.8\hat{\rho}^{1/2}T_{1}^{1/2}\hat{s}^{-1/6}a_{-5}^{1/2}St\left(\frac{1.2\,\rm\mu m}{\gamma\bar{\lambda}\hat{Q}_{e3}U}\right)\rm yr,~{}~{}~{}$$

(31)

where $\hat{Q}_{e3}=Q_{e3}/10^{-2}$ with $Q_{e3}$ the third component of RATs that induces the grain precession around $\boldsymbol{k}$ and the normalization is done using the typical value of $Q_{e3}$ (see Lazarian and Hoang 2007a).

In the absence of magnetic field, radiative precession can be much faster than gas damping time for a thermal rotation of $St\sim 1$ (Eq. 9). In this case, the anisotropic radiation direction is the axis of grain alignment, which is known as k-RAT alignment.

Low$-J$ and high$-J$ Attractor Alignment by RATs: In addition to radiative precession, RATs can align grains at high$-J$ attractors and low$-J$ attractors. In general, a fraction of grains ($f_{\rm high-J}$) can be aligned at a high$-J$ attractor due to the aligning and spin-up effects of RATs (Lazarian and Hoang 2007a; Hoang and Lazarian 2008). Grains aligned at the high$-J$ attractor have the suprathermal rotation with $St_{\rm high-J}>1$. On the other hand, due to the aligning and spin-down effects of RATs, the $1-f_{\rm high-J}$ fraction of grains, are aligned at the low$-J$ attractor with $St_{\rm low-J}\sim 1$ (Hoang and Lazarian, 2014). The RAT alignment process can occur on a timescale smaller than the gas damping time, which is called fast alignment (Lazarian and Hoang, 2007a, 2021) with a strong radiation field. Grains can be stably aligned at low$-J$ attractors when the grain randomization by gas collisions is not considered. Otherwise, the gas randomization could randomize grains at low$-J$ attractors and eventually transport them to the high$-J$ attractor after a timescale greater than the gas damping time, which is usually called slow alignment (Hoang and Lazarian, 2008; Lazarian and Hoang, 2021). The fast RAT alignment occurs in the local environment of intense transient radiation sources such as supernova (Giang et al., 2020), kilonova, and gamma-ray bursts (Hoang et al., 2020), whereas the slow RAT alignment occurs in most astrophysical environments, from the ISM, MCs, to SFRs. A detailed discussion of fast and slow alignment by RATs is discussed in detail in Lazarian and Hoang (2021).

The exact value of $f_{\rm high-J}$ depends on the grain properties (shape and size) and magnetic susceptibility. For grains with ordinary paramagnetic material (e.g., silicate), Herranen et al. (2021) found that $f_{\rm high-J}$ can be about $10-70\%$ based on calculations of RATs for an ensemble of Gaussian random shapes. The presence of iron inclusions embedded in the grains increases grain magnetic suceptibility and superparamagnetic relaxation, which can produce in the universal high$-J$ attractors(i.e., $f_{\rm high-J}\sim 100\%$) (Hoang and Lazarian, 2016a; Lazarian and Hoang, 2021).

Minimum size of RAT alignment: Due to gas collisions, grains can be efficiently aligned only when they rotate suprathermally at a high$-J$ attractor. Hoang and Lazarian (2008) and Hoang and Lazarian (2014) demonstrated that grain alignment is stablized under the gas randomization when $\Omega_{\rm RAT}\equiv 3\times\Omega_{\rm T}$ (or $St_{\rm RAT}=3$). This criteria yields the minimum size for grain alignment by RATs as

$$a_{\rm align}\simeq 0.055\hat{\rho}^{-1/7}\left(\frac{n_{3}T_{1}}{\gamma_{-1}U}\right)^{2/7}\times\left(\frac{\bar{\lambda}}{1.2\,\rm\mu m}\right)^{4/7}\left(\frac{1}{1+F_{\rm IR}}\right)^{-2/7}~{}\rm{\mu m},$$

(32)

where $\gamma_{-1}=\gamma/10^{-1}$. One can see that for a typical MC with $n_{\rm H}=10^{3}\,\rm cm^{-3}$, grains larger than $a_{\rm align}=0.055\,\mu$m can be aligned by RATs. However, the radiation strength is not an observable quantity, thus we could use the dust temperature $T_{\rm d}$ as a proxy for this strength. If we adopt $U\simeq(T_{\rm d}/16.4\,\rm K)^{6}(a/10^{-5}\,\rm cm)^{6/15}$ (Draine 2011), $a_{\rm align}\sim n^{2/7}_{\rm H}T^{-12/7}_{\rm dust}$ that illustrates that higher $T_{\rm d}$ is able to bring smaller grains to align. Vice-versa, only larger grains are able to align in higher $n_{\rm H}$ (denser gas). Note that the power-index in Equation 32 slightly differs from Hoang et al. (2021b) and Tram et al. (2021b). The reason is due to the slightly difference in the scaling of the average RAT efficiency, but this difference results in a negligible discrepancy of $a_{\rm align}$ (see Figure 1 in Hoang et al. 2021b).

k-RAT vs. B-RAT Alignment: Using Equation (31) one calculates the radiative precession (k-RAT) timescale for grains aligned at low$-J$ attractors with $St=1$ as

$\displaystyle\tau_{k}^{\rm low-J}=56.4\hat{\rho}^{1/2}T_{1}^{1/2}\hat{s}^{-1/6}a_{-5}^{1/2}\left(\frac{1.2\mu m}{\gamma\bar{\lambda}\hat{Q}_{e3}U}\right){\rm yr},~{}~{}~{}~{}~{}$

(33)

and for grains aligned at high$-J$ attractors with $St=St_{\rm RAT}$ from Equation (14), one has

$\displaystyle\tau_{k}^{\rm high-J}$

$\displaystyle=$

$\displaystyle 1.17\times 10^{6}\hat{s}^{-1/3}\left(\frac{\hat{\rho}a_{-5}}{\hat{Q}_{e3}}\right)\left(\frac{1}{n_{3}T_{1}^{1/2}}\right)\left(\frac{1}{1+F_{\rm IR}}\right){\rm yr}.~{}~{}~{}~{}~{}$

(34)

In the presence of a magnetic field, the grain experiences simultaneous Larmor precession and radiative precession. If the Larmor precession is faster than the radiative precession, i.e., $\tau_{B}<\tau_{k}$, the axis of grain alignment by RATs changes from with $\boldsymbol{J}$ along $\boldsymbol{k}$ to the magnetic field, $\boldsymbol{B}$, which is known as B-RAT. The minimum grain size that grain alignment still occurs via k-RAT, which is also the maximum size grains can be aligned via B-RAT, is defined by $\tau_{k}=\tau_{B}$ and is derived in Hoang et al. (2022a). For grains aligned at a low$-J$ attractor, one has

$$a^{\rm low-J,PM}_{\rm min,Jk}\equiv a_{\rm max,JB}^{\rm low-J,PM}\simeq 2.96\hat{\rho}^{-1/2}\hat{s}^{-1/9}\left(\frac{\bar{\lambda}}{1.2\,\rm\mu m}\right)^{-2/3}\left(\frac{n_{23}f_{\rm p}\hat{p}^{2}T^{1/2}_{1}}{T_{d,1}\gamma U_{3}\hat{Q}_{e3}}\right)^{2/3}\left(\frac{B}{20\,\rm\mu G}\right)^{2/3}~{}{\rm\mu m,}$$

(35)

and

	$\displaystyle a^{\rm low-J,SPM}_{\rm min,Jk}\equiv a_{\rm max,JB}^{\rm low-J,SPM}$	$\displaystyle=$	$\displaystyle 57.8\hat{\rho}^{-1/2}\hat{s}^{-1/9}\left(\frac{\bar{\lambda}}{1.2\,\rm\mu m}\right)^{-2/3}\left(\frac{\hat{p}^{2}T^{1/2}_{1}}{T_{d,1}\gamma_{\rm rad}U_{3}\hat{Q}_{e3}}\right)^{2/3}$
		$\displaystyle\times$	$\displaystyle\left(\frac{B}{20\,\rm\mu G}\right)^{2/3}\left(N_{\rm cl,4}\phi_{sp,-2}\right)^{2/3}~{}{\rm\mu m.}~{}~{}~{}~{}~{}$

for PM and SPM grains, respectively. The above equations imply that the minimum size for k-RAT when grains aligned at low$-J$ attractors is smaller for higher radiation strength as $U^{-2/3}$, but larger for stronger magnetic field as $B^{2/3}$. It is harder to bring SPM grains to align with the radiation direction because of their larger magnetic susceptibility and then faster Larmor precession.

Similarly, for grains aligned at a high$-J$ attractor (Hoang et al., 2022a)

$$a^{\rm high-J,PM}_{\rm min,Jk}\equiv a_{\rm max,JB}^{\rm high-J,PM}\simeq 3.3\times 10^{8}\hat{s}^{-1/3}\left(\frac{n_{23}f_{\rm p}\hat{p}^{2}}{n_{3}T_{d,1}T^{1/2}_{1}\hat{Q}_{e3}}\right)\left(\frac{B}{20\,\rm\mu G}\right)\left(\frac{1}{1+F_{\rm IR}}\right)~{}{\rm\mu m,}$$

(37)

and

$$a^{\rm high-J,SPM}_{\rm min,Jk}\equiv a_{\rm max,JB}^{\rm high-J,SPM}\simeq 2.8\times 10^{10}\hat{s}^{-1/3}\left(\frac{\hat{p}^{2}}{n_{3}T_{d,1}T_{1}^{1/2}\hat{Q}_{e3}}\right)\left(\frac{B}{20\,\rm\mu G}\right)\left(\frac{N_{\rm cl,4}\phi_{sp,-2}}{1+F_{\rm IR}}\right)~{}{\rm\mu m.}~{}~{}~{}$$

(38)

One realizes that the minimum size for k-RAT alignment is very large for typical molecular clouds, especially when grains are aligned at high$-J$ attractors. Therefore, in these conditions, grains are aligned with $\boldsymbol{J}$ along the magnetic field via B-RAT. Yet, for very strong radiation fields of strength of $U\gg 1$, small grains at low$-J$ attractors can have k-RAT alignment.

A dust grain rotating with an angular velocity $\Omega$ develops a tensile stress $S=\rho\Omega^{2}a^{2}/4$ on grain material. This stress points outward from the grain center of mass and tends to tear the grain apart. When the tensile stress exceeds the maximum tensile strength of the grain material, $S_{\rm max}$, the grain is instantaneously disrupted in fragments. The maximum angular velocity beyond which the grain is disrupted is defined by $S=S_{\rm max}$ and reads

$$\Omega_{\rm disr}=\frac{2}{a}\left(\frac{S_{\rm max}}{\rho}\right)^{1/2}\simeq\frac{3.65\times 10^{8}}{a_{-5}}S^{1/2}_{\rm max,7}\hat{\rho}^{-1/2}{~{}\rm rad/s},$$

(39)

where $S_{\rm max,7}=S_{\rm max}/(10^{7}\rm erg\,cm^{-3})$. The exact value of $S_{\rm max}$ depends on the grain internal structure (see Hoang 2020, and references therein). For example, composite grains have $S_{\rm max}\sim 10^{7}\,\rm erg\,cm^{-3}$, whereas compact grains have a higher value.

Subject to a strong radiation field, dust grains will be spontaneously fragmented into many smaller species when the grain angular velocity spun-up by RATs, $\Omega_{\rm RAT}$, exceeds the critical value $\Omega_{\rm disr}$. This is called the radiative torque disruption (RAT-D) mechanism, which was first introduced in Hoang et al. (2019). Due to the dependence of $\Omega_{\rm RAT}$ on the grain size and local conditions, there is a range of grain sizes that can be disrupted, as illustrated by the shaded areas in Figure 3. The minimum size of grains that can be disrupted by RAT-D is

	$\displaystyle a_{\rm disr}$	$\displaystyle=$	$\displaystyle\left(\frac{0.8n_{\rm H}\sqrt{2\pi m_{\rm H}kT_{\rm gas}}}{\gamma u_{\rm rad}\bar{\lambda}^{-2}}\right)^{1/2}\left(\frac{S_{\rm max}}{\rho}\right)^{1/4}(1+F_{\rm IR})^{1/2}$		(40)
		$\displaystyle\simeq$	$\displaystyle 1.7\left(\frac{\gamma_{-1}U}{n_{3}T_{1}^{1/2}}\right)^{-1/2}\left(\frac{\bar{\lambda}}{1.2\,\rm\mu m}\right)\hat{\rho}^{-1/4}S_{\rm max,7}^{1/4}(1+F_{\rm IR})^{1/2}\,\rm\mu m,$

and the maximum size of grains that can still be disrupted by RAT-D is

	$\displaystyle a_{\rm disr,max}$	$\displaystyle=$	$\displaystyle\frac{\gamma u_{\rm rad}\bar{\lambda}}{16n_{\rm H}\sqrt{2\pi m_{\rm H}kT_{\rm gas}}}\left(\frac{S_{\rm max}}{\rho}\right)^{-1/2}(1+F_{\rm IR})^{-1}$		(41)
		$\displaystyle\simeq$	$\displaystyle 0.03\left(\frac{\gamma_{-1}U}{n_{3}T_{1}^{1/2}}\right)\left(\frac{\bar{\lambda}}{1.2\,\rm\mu m}\right)\hat{\rho}^{1/2}S_{\rm max,7}^{-1/2}(1+F_{\rm IR})^{-1}\,\rm\mu m,$

which depend on the local gas properties, radiation field, and the grain tensile strength (Hoang and Tram, 2020; Hoang et al., 2021b).

As shown, the RAT-D efficiency strongly depends on the local conditions, i.e., more efficient for higher radiation strength (U) or $T_{\rm d}$, but less efficient for higher $n_{\rm H}$.

Detection of interstellar complex organic molecules (COMs) are extensively reported toward low-/high-mass protostars and protoplanetary disks (see e.g., van Dishoeck 2014, and references therein). As a popular scenario, COMs are believed to first form on the icy mantle of dust grains (e.g., Garrod et al. 2008; Jiménez-Serra et al. 2016), and subsequently desorbed to the gas phase during the warm and hot phase, where icy grain mantles can be heated to high temperatures $T_{\rm d}\sim 100–300\,\rm K$ (see e.g., Blake et al. 1987; Brown et al. 1988; Bisschop et al. 2007). However, icy mantles might not be survived under an intense radiation field before the critical temperature for thermal sublimation is reached (Hoang and Tram 2020). The centrifugal force can break the icy mantel from the bare core, then the evaporation and sublimation processes are enhanced owing to smaller size of the icy fragments, enabling COMs to release (see Figure 4(a)). This non-thermal process is called the rotational desorption, and found to be effective at lower dust temperatures than the classical thermal sublimation, which is able to explain the detection of COMs in cold regions (see Hoang and Tram 2020; Tram et al. 2021c for further details).

Figure 4(b) shows an evidence for rotation desorption of COMs from Orion BN/KL region. The timescales of rotational desorption (color lines) and thermal sublimation (dashed dotted black lines) for three COMs with very high binding energy, including ethyl formate ($\rm C_{2}H_{5}OCHO$), methoxymethanol ($\rm CH_{3}OCH_{2}OH$) and ethylene glycol ($\rm OHCH_{2}CH_{2}OH$) that are reported in Tercero et al. (2018), are shown for comparison. One can see that for given low dust temperatures constrained by SOFIA/FORCAST observations ($T_{\rm d}\simeq 80-100\,\rm K$; De Buizer et al. 2012), rotational desorption is more effective than thermal sublimation in desorbing COMs from the grain surface. With regard to the UV photo-desorption (dashed horizontal lines), the rotational desorption could be as efficient as (or more effective than) the UV photo-desorption regarding to the photo-desorption yield uncertainty. However, UV photons are expected to be quickly attenuated by dust absorption, so that the volume in which photo-desorption is important is less extended than the rotational desorption which can work with optical-IR photons.