# Spinning Nanoparticles Impacted by C-shock: Implications for Radio-
millimeter Emission from Star-forming Regions

Ilsang Yoon National Radio Astronomy Observatory, 520 Edgemont Road,
Charlottesville, VA 22903, USA

(Received September 22, 2021; Revised June 29, 2022; Accepted July 25, 2022)

###### Abstract

We investigate the impact of anomalous microwave emission (AME) on the radio-
millimeter spectral energy distribution for three typical interstellar medium
(ISM) conditions surrounding star-forming regions – cold neutral medium, warm
neutral medium, and photodissociation region – by comparing the emissivities
of three major contributors: free-free, thermal dust emission, and AME. In
particular, for spinning nanoparticles (i.e., potential carriers of AME), we
consider a known grain destruction mechanism due to a centrifugal force from
spin-up processes caused by collisions between dust grains and supersonic
neutral streams in a magnetized shock (C-shock). We demonstrate that, if the
ISM in a magnetic field is impacted by a C-shock developed by a supernova
explosion in the early phase of massive star-formation ($\lesssim 10$ Myr),
AME can be significantly or almost entirely suppressed relative to free-free
and thermal dust continuum emission if the grain tensile strength is small
enough. This study may shed light on explaining the rare observations of AME
from extragalactic star-forming regions preferentially observed from massive
star clusters and suggest a scenario of “the rise and fall of AME” in
accordance with the temporal evolution of star-forming regions.

submillimeter: ISM – radio continuum: ISM – galaxies: ISM – ISM:dust –
radiation mechanisms: non-thermal – radiation mechanisms: thermal

††journal: ApJ††software: SpDust (Ali-Haïmoud et al., 2009; Silsbee et al.,
2011), Paris-Durham shock code (Flower & Pineau des Forêts, 2003; Lesaffre et
al., 2013; Godard et al., 2019), numpy (Harris et al., 2020), matplotlib
(Hunter, 2007)

##  1 Introduction

Thermal free-free emission is a good tracer of star formation and widely used
to estimate the star formation rate (SFR) of galaxies using radio continuum
measurements at 10-33 GHz where the free-free emission is dominant (Murphy et
al., 2011). However, anomalous microwave emission (AME) explained by dipole
radiation from spinning nanoparticles (Draine & Lazarian, 1998) is also bright
at this frequency range, and the peak frequency can be as high as 100 GHz
(Ali-Haïmoud et al., 2009), which implies that (1) AME can be a significant
contamination if the SFR is measured by thermal free-free radio continuum
emission from a single frequency (e.g., Murphy et al., 2011) and (2) AME can
affect a measurement of molecular gas mass by a submillimeter (e.g., 850
$\mu$m or 350 GHz) flux density (e.g., Scoville et al., 2016).

Therefore, the observational study of AME in Galactic and extragalactic star-
forming regions is important to understand the impact of AME on the radio-
millimeter continuum spectral energy distribution (SED). The current estimate
of the fraction of AME to the total emission at a frequency near $\approx 30$
GHz in our Galaxy is as large as 50% (Dickinson et al., 2018). However, in
other galaxies, AME does not seem to be strong, as shown by the observations
of two galaxies, NGC 6946 (Murphy et al., 2010; Scaife et al., 2010a; Hensley
et al., 2015) and NGC 4725B (Murphy et al., 2018b) although the integrated
flux over entire M31 galaxy shows significant enhancement of AME (Battistelli
et al., 2019).

The extragalactic AME detection is reported for only two galaxies (NGC 6946
and NGC 4725B) with spatially resolved emission and one galaxy (M31) with
spatially integrated emission. It is not clear whether this discrepancy is due
to a different interstellar medium (ISM) environment associated with the AME
region in our Galaxy and other galaxies or just a bias in the AME estimate in
our Galaxy based on the observation of solar neighbors or a combination of
both (Dickinson et al., 2018). If the observation of AME in our Milky Way is
biased, and AME is indeed weak and rare in star-forming regions in galaxies in
general, it is interesting to understand the possible reasons for the weakness
and rareness of AME in extragalactic star-forming regions.

In general, a star-forming region is complex and consists of a mixture of
different phases of ISM from the fully ionized phase
($\textrm{H}\scriptstyle\mathrm{II}$ region) to the cold phase of gas and dust
shielding strong UV radiation, as illustrated in Figure 1. As illustrated by
Ali-Haïmoud et al. (2009) using models of spinning dust emission, the types of
ISM from which one can expect both thermal free-free emission and significant
AME are cold neutral medium (CNM), warm neutral medium (WNM), and
photodissociation region (PDR) that are usually associated with nearby
$\textrm{H}\scriptstyle\mathrm{II}$ regions (Figure 1).

Although our current understanding of a connection between the physical ISM
conditions of star-forming regions and the AME detection is still incomplete
and evolving, the emerging impression is that the lower-density star-forming
regions with a high interstellar radiation field (ISRF; like WNM) provide the
most promising environment for detecting AME, while a very high ISRF may
destroy the grain population (see Scaife, 2013, for a review).

Recent observations of the resolved star-forming clouds suggest that AME is
correlated with polycyclic aromatic hydrocarbon (PAH) emission associated with
the PDR (Bell et al., 2019; Arce-Tord et al., 2020; Casassus et al., 2021),
which is consistent with the previous observations suggesting a correlation
between AME and PAH tracers (Scaife et al., 2010b; Ysard et al., 2010; Tibbs
et al., 2011; Battistelli et al., 2015).

The AME may also be detected in $\textrm{H}\scriptstyle\mathrm{II}$ regions
(e.g., RCW 175 by Tibbs et al., 2012; Battistelli et al., 2015) if the dust is
present in the interior of the $\textrm{H}\scriptstyle\mathrm{II}$ region
(Paladini et al., 2012; Watson et al., 2008) or in the
$\textrm{H}\scriptstyle\mathrm{II}$ bubble (Anderson et al., 2012; Flagey et
al., 2011). However, in general, it is not likely that AME in the
$\textrm{H}\scriptstyle\mathrm{II}$ regions dominates the free-free emission
because the abundance of dust particles (e.g., PAH) in
$\textrm{H}\scriptstyle\mathrm{II}$ regions is suppressed due to strong UV
radiation (e.g., Peeters et al., 2004; Binder & Povich, 2018; Povich et al.,
2007) and supernova shocks (e.g., Jones et al., 1996), and, even in high-
density $\textrm{H}\scriptstyle\mathrm{II}$ regions with increased dust
abundance (e.g., Draine, 2011a), free-free emission is likely to be dominant
due to the free-free emissivity scaled as $n^{2}_{e}$.

In the current working models of AME (e.g., Draine & Lazarian, 1998; Ali-
Haïmoud et al., 2009), there are several model parameters – distribution of
the grain size, charge, dipole moment, strength of the ambient ISRF, gas
temperature, and hydrogen density (e.g., Ali-Haïmoud et al., 2009; Hensley &
Draine, 2017) – that can change the shape and intensity of the AME emissivity
function such that AME can be suppressed. However, the emissivity of AME from
CNM, WNM, and PDR is still significantly larger than the free-free emissivity
in the frequency of 10-100 GHz for typical parameter values of the ISM and
dust (Ali-Haïmoud et al., 2009). Furthermore, the recently proposed dust
destruction process by radiative torque in a strong radiation field suggests
that it efficiently breaks large grains (Hoang et al., 2019) and increases the
small-grain population, which implies that AME can be even stronger in those
star-forming regions with strong UV radiation from OB star associations (see
Section 6.1).

Another possible (and probably more plausible) way of suppressing AME from
those star-forming regions (CNM, WNM, and PDR) is to reduce the abundance of
small nanoparticles by destroying them, which has not been discussed much.
Recently, Hoang & Tram (2019) suggested a new disruption mechanism of very
small grains (nanoparticles with size $\sim 10^{-9}$m) by increased
centrifugal force due to suprathermal rotation by stochastic mechanical torque
(originally proposed by Gold (1952)) from a magnetized shock (C-shock). The
magnetic field is ubiquitous, and, for the ISM with a lower ionization
fraction, the supersonic drift of neutral particles in a C-type shock (Draine,
2011b) can impact small nanoparticles bound by the magentic field and increase
their angular velocity (Hoang & Tram, 2019). Indeed, relative to other grain
destruction mechanisms (thermal sputtering, nonthermal sputtering, and
grain–grain collision), this rotational disruption appears to be the fastest
mechanism to destroy nanoparticles in C-shocks, as suggested by the comparison
of the time scale for each process (Hoang & Tram, 2019).

In this work, we compute the emissivity of AME by incorporating the disruption
of nanoparticles impacted by C-shocks and investigate the emissivity of the
composite of AME, thermal free-free, and dust continuum emission in the radio-
millimeter wavelength (10–100 GHz) from the ISM associated with star-forming
regions, to characterize how much AME contributes to the radio-millimeter SED.

In Section 2, we introduce a brief overview of the emission mechanism of
spinning nanoparticles and the characteristic properties of the ISM associated
with star-forming regions. In Section 3, the properties of C-shocks and the
rotational dynamics of nanoparticles in the region affected by C-shocks are
discussed. In Section 4, we discuss the suprathermal angular rotation velocity
of spinning nanoparticles and their critical angular rotation speed to resist
centrifugal force, as well as the implications of assumed dust size
distribution impacting the AME emissivity. In Section 5, we show the impact of
C-shocks on the radio-millimeter emissivity by considering grain destruction
mechanisms discussed in Section 4. In Section 6, we propose a possible
scenario of the rise and fall of AME and discuss the relevant issues related
to the current work. Finally, we summarize our work in Section 7.

##  2 AME from Star-forming Regions

In this section, we provide a brief overview of the grain rotational dynamics.
More details are found in Draine & Lazarian (1998) and Ali-Haïmoud et al.
(2009).

###  2.1 Basics of Dipole Radiation from Spinning Nanoparticles

The radiation power at the frequency $\nu=\omega/2\pi$ from spinning particles
with an angular velocity $\boldsymbol{\omega}$ of the electric dipole moment
$\boldsymbol{\mu}$, with a component $\boldsymbol{\mu}_{\perp}$ perpendicular
to $\boldsymbol{\omega}$, is

$P=\frac{2}{3}\frac{\boldsymbol{\mu}^{2}_{\perp}\boldsymbol{\omega}^{4}}{c^{3}}.$
(1)

and the emissivity of the spinning particles per H atom in erg s-1 sr-1 Hz-1
(H atom)-1 is

$\frac{{j_{\tiny\nu,sp}}}{{n_{\tiny\mbox{H}}}}=\frac{1}{4\pi}\int^{a_{\tiny\mbox{max}}}_{a_{\tiny\mbox{min}}}da\frac{1}{{n_{\tiny\mbox{H}}}}\frac{d{n_{\tiny\mbox{gr}}}}{da}4\pi\omega^{2}f_{a}(\omega)2\pi\frac{2}{3}\frac{\mu^{2}_{\perp}\omega^{4}}{c^{3}}$
(2)

where $a_{\tiny\mbox{min}}$ and $a_{\tiny\mbox{max}}$ are the minimum and
maximum size of the dust grain, and
$\frac{1}{{n_{\tiny\mbox{H}}}}\frac{d{n_{\tiny\mbox{gr}}}}{da}$ is the number
of dust grains per unit size per H atom.

The angular velocity distribution function for grain size $a$,
$f_{a}(\omega)$, can be obtained by solving a stationary Fokker–Planck
equation (Ali-Haïmoud et al., 2009),

$\frac{df_{a}(\omega)}{d\omega}+\left[\frac{I\omega}{kT}\frac{F}{G}+\frac{\tau_{\tiny{\mbox{H}}}}{\tau_{\tiny{\mbox{ed}}}}\frac{1}{3G}\frac{I^{2}\omega^{3}}{(kT)^{2}}\right]f_{a}(\omega)=0$
(3)

where $I$ is the moment of inertia of the dust particles,
$\tau_{\tiny{\mbox{H}}}$ is the characteristic rotational damping time-scale
for collisions with neutral H atoms, and $\tau_{\tiny{\mbox{ed}}}$ is the
characteristic damping time scale for electric dipole radiation. Here
$\tau_{\tiny{\mbox{H}}}$ and $\tau_{\tiny{\mbox{ed}}}$ are

$\displaystyle\tau_{\tiny{\mbox{H}}}$ $\displaystyle=$
$\displaystyle\left[{n_{\tiny\mbox{H}}}m_{\tiny\mbox{H}}\left(\frac{2kT}{\pi
m_{\tiny\mbox{H}}}\right)^{1/2}\frac{4\pi
a^{4}_{\tiny\mbox{cx}}}{3I}\right]^{-1}$ (4)
$\displaystyle\tau_{\tiny{\mbox{ed}}}$ $\displaystyle=$
$\displaystyle\frac{I^{2}c^{3}}{2\mu_{\perp}^{2}kT}$ (5)

where $a_{\tiny\mbox{cx}}$ is the “cylinderical excitation-equivalent” radius,
defined as $4\pi
a^{4}_{\tiny\mbox{cx}}\equiv\frac{3}{2}\oint\rho^{2}d\mbox{S}$ (Ali-Haïmoud et
al., 2009).

In Equation 3, $F$ and $G$ are the dimensionless damping and excitation
coefficients of spinning dust particles for various interaction processes
(e.g., collisions with ions and neutrals, plasma drag, infrared radiation,
photoelectric emission, H2 formation) and need to be computed for each
interaction process (e.g., Draine & Lazarian, 1998; Ali-Haïmoud et al., 2009).

Although the small grains may be sheetlike (Draine & Lazarian, 1998; Ali-
Haïmoud et al., 2009), we adopt a spherical geometry for the grains when
computing the damping and excitation coefficients for the impact of C-shocks,
as many previous works do when computing $F$ and $G$ (e.g., Draine & Lazarian,
1998; Ali-Haïmoud et al., 2009; Silsbee et al., 2011; Hoang & Tram, 2019).

For computing $j_{\tiny\nu,sp}$, we use the publicly available code
SpDust(Ali-Haïmoud et al., 2009; Silsbee et al., 2011) that solves Equation 3
numerically to obtain $f(\omega)$ and computes
$\frac{{j_{\tiny\nu,sp}}}{{n_{\tiny\mbox{H}}}}$ for an assumed grain size
distribution $\frac{1}{{n_{\tiny\mbox{H}}}}\frac{d{n_{\tiny\mbox{gr}}}}{da}$,
a dipole moment $\mu$ and ISM environment parameters ($n_{\tiny\mbox{H}}$,
$T$, $x_{\mbox{\tiny H}}$, $x_{\mbox{\tiny M}}$ and $y$ in Table 1).

###  2.2 ISM Associated with Star-forming Regions

The classical multiphase model of the ISM proposed by McKee & Ostriker (1977)
is a simplified but still relevant picture of the ISM in star-forming regions.
In Figure 1, we show an illustrative picture of a star-forming region where
the “magnetized” molecular cloud is associated with an
$\textrm{H}\scriptstyle\mathrm{II}$ region. The molecular cloud consists of a
multiphase ISM (WNM and CNM) and is exposed to UV radiation from the nearby
$\textrm{H}\scriptstyle\mathrm{II}$ region where the ISM is fully ionized at
$T\approx 10^{4}$K by OB stars. A small part of the high-density molecular
cloud forms a PDR where dust is present in abundance as both large grains and
PAHs, as shown by the far-IR/submillimeter emission (from large grains) and
near-IR emission (e.g. 8 $\mu$m from PAHs). A low-velocity shock wave
propagating into the “magnetized” molecular cloud far from the
$\textrm{H}\scriptstyle\mathrm{II}$ region creates a “C-type” shock.

In this study, we assume that a high-velocity strong supernova shock wave
(J-type shock) or strong UV radiation destroys the dust grains (including AME
carriers) in the $\textrm{H}\scriptstyle\mathrm{II}$ region; therefore, AME
does not rise from the $\textrm{H}\scriptstyle\mathrm{II}$ region itself, and
the radio SED from the $\textrm{H}\scriptstyle\mathrm{II}$ region is dominated
by free-free emission. We study the condition for AME formation in the CNM,
WNM, and PDR in molecular clouds and investigate the relative strength of AME
compared with free-free and thermal dust emissions that are also from the
molecular cloud.

The approximate volume filling factor $f_{\tiny V}$ for hot ionized medium
(HIM), warm ionized medium (WIM), WNM and CNM in our Galaxy is 0.5, 0.1, 0.4,
and 0.01, respectively (Draine, 2011b). Although the WNM fills a significant
fraction of the volume of the galactic disk, the effective volume densities
$n_{\tiny\mbox{H}}f_{\tiny V}$ of WNM and CNM are similar (0.2 and 0.3
$\mbox{cm}^{-3}$); therefore, it is likely that the contribution to the
observed AME from CNM and WNM is similarly important for a given observing
beam.

If the OB stars are forming and creating the
$\textrm{H}\scriptstyle\mathrm{II}$ region, the clouds are exposed to strong
UV radiation, and some of them form a PDR. The significance of AME from a PDR
with high density ($n_{\tiny\mbox{H}}\approx 10^{5}{\mbox{cm}^{-3}}$) is
difficult to assess due to its uncertain and probably very small volume
filling factor; however, it could be an important contribution when the next-
generation Very Large Array (ngVLA; Murphy et al., 2018c) starts to resolve
individual star-forming regions in galaxies with high angular resolution
(0.5-50 mas).

Within a few tens of megayears, the supernova explodes and the shock wave
propagates into the clouds. For a low-ionization medium with average or higher
density like CNM, WNM, and PDR, the shock wave is a C-type where the shock
velocities of the ionized and neutral particles are different due to the
existence of a magnetic field, and therefore the neutral particle can have a
supersonic drift velocity relative to the ionized particles (Draine, 2011b).

Figure 1: Illustration of the star-forming region including
$\textrm{H}\scriptstyle\mathrm{II}$ region and associated ‘magnetized’
molecular cloud with the WNM, CNM, and dense PDR, where the potential AME can
be observed before the ‘C-type’ shock wave after the supernova explosion
arrives and impacts the dust particles in the clouds.

In Figure 2, we show a schematic picture of the propagation of a C-shock into
the ISM with dust particles having different sizes. The small orange circles
in the pre-shock region on the left-hand side of the shock layer (hatched red
lines in the middle) represent the nanoparticles as a source of AME, and the
other large dust grains existing in both pre- and post-shock regions are
represented by larger blue circles. The velocity, temperature, and density
profile of the ISM in the shock layer are shown in the zoom-in panel. If a
charged dust grain sticks with the magnetic field when the C-shock propagates
(to the left), the neutral particles with high drift velocity ($V_{n}-V_{i}$)
relative to the ions collide with the dust particles via stochastic
bombardment and introduce additional damping and excitation terms, which
destroys the small dust grains (small orange circles in the pre-shock region).
We will discuss the process in detail in Sections 3 and 4.

Table 1: ISM Environment Parameters from Draine & Lazarian (1998) Parameter | CNM | WNM | PDR
---|---|---|---
$n_{\tiny\mbox{H}}$[$\mbox{cm}^{-3}$] | 30 | 0.4 | $10^{5}$
$T$ [K] | 100 | 6000 | 300
$T_{d}$ [K] | 20 | 20 | 50
$\chi$ | 1 | 1 | 3000
$x_{\mbox{\tiny H}}\equiv n(\mbox{H}^{+})/$$n_{\tiny\mbox{H}}$ | 0.0012 | 0.1 | 0.0001
$x_{\mbox{\tiny M}}\equiv n(\mbox{M}^{+})/$$n_{\tiny\mbox{H}}$ | 0.0003 | 0.0003 | 0.0002
$y\equiv 2n(\mbox{H}_{2})/$$n_{\tiny\mbox{H}}$ | 0 | 0 | 0.5
Figure 2: Schematic view of the dust particles in gas clouds before and after
the impact of the C-shock. The upper panel illustrates the dust particles of
different sizes. The zoomed-in view of the shock layer (hatched by red lines)
in the lower panel illustrates the velocity, temperature, and density profile
of the gas experiencing magnetized shock creating different velocities for
neutrals and ions ($V_{n}$ and $V_{i}$). The small orange circles seen in the
pre-shock region represent the nanoparticles as an AME carrier that are broken
up in the post-shock region by centrifugal force due to the spin-up by the
collision with neutral particles drifting with supersonic drift velocity
($V_{n}-V_{i}$).

##  3 Rotational Dynamics of Spinning Nanoparticles in C-shocks

In the ISM associated with star-forming regions (CNM, WNM, and PDR) from which
we expect to observe both free-free emission and AME, a large abundance of
small nanoparticles with a high spin angular momentum resulting from the
shattering of large grains by supernova winds (Jones et al., 1996) is expected
to increase the strength of AME. However, on the other hand, one can also
expect that the high spin angular velocity of the nanoparticles creates a
strong centrifugal force disrupting the particles themselves. This rotational
disruption can decrease the abundance of the smallest nanoparticles, which can
decrease the resulting AME.

Recently, Hoang & Tram (2019) introduced a model of spinning dust in C-shocks,
accounting for this destruction effect. We will incorporate this process in
the SpDust code (Ali-Haïmoud et al., 2009; Silsbee et al., 2011) and compute
the emissivity of spinning nanoparticles in the presence of C-shocks.

###  3.1 Structure of C-shocks in CNM, WNM, and PDR

Shocks in molecular clouds where H2 rovibrational cooling is efficient are
common and expected to often be C-type (Draine, 2011b). The energy dissipation
in C-shocks with two fluids (ions and neutrals) is continuous rather than
impulsive, and efficient radiative cooling keeps the gas cool (Draine, 2011b).
We use the Paris–Durham shock code (Flower & Pineau des Forêts, 2003; Lesaffre
et al., 2013; Godard et al., 2019) to compute the velocity profiles of neutral
and ion particles in C-shocks for the ISM with typical environmental
parameters (Table 1) for CNM, WNM, and PDR (Draine & Lazarian, 1998).

The simulated shock is static. We assume that ISM parameters like temperature
and density are continuous and not drastically different in the pre- and post-
shock region because of the nature of the continuity of C-shocks. For the pre-
shock parameters, we use the default values in the code, except for hydrogen
density, gas temperature, and dust temperature ($n_{\tiny\mbox{H}}$, $T$, and
$T_{d}$) which are chosen for the CNM, WNM, and PDR from Table 1. For the
initial shock velocity, we note that the velocity of C-type shocks is low
(5-25km/s) in general, and the high-velocity shocks are J-type (Godard et al.,
2019). Therefore, we choose 20km/s for the initial shock velocity. For the
magnetic field strength, the Paris–Durham shock code uses the following
parameterization (Godard et al., 2019):

$B=b\sqrt{\frac{{n_{\tiny\mbox{H}}}}{1cm^{-3}}}~{}\mu\mbox{G}$ (6)

where $b=1$ by default in the shock code.

Figure 3: Upper panels: shocked-frame velocity profiles of neutrals (blue
dashed line) and ions (red dotted-dashed line) with a thermal velocity profile
(solid black line) Lower panels: drift velocity of neutrals with respect to
the thermal velocity, $s_{d}=\frac{v_{n}-v_{i}}{v_{th}}$. The initial shock
velocity (20 km/s) is the same for all cases.

In the upper panels of the plots in Figure 3, we show the velocity profiles of
neutrals ($v_{n}$, blue dashed line) and ions ($v_{i}$, red dotted-dashed
line) in the shock frame for C-shocks in the magnetized ISM, for the physical
conditions of CNM, WNM, and PDR in Table 1 and a 20 km/s shock velocity. The
solid black line is the thermal velocity profile ($v_{th}$). In the lower
panels of the plots in Figure 3, we show the profile of the ratio between
drift velocity ($v_{n}-v_{i}$) and thermal velocity ($v_{th}$):

$s_{d}=\frac{v_{n}-v_{i}}{v_{th}}$ (7)

Although the detailed shapes of the profile vary depending on the pre-shock
density, temperature, and strength of irradiated radiation (e.g., Godard et
al., 2019), the neutrals and ions have different velocity profiles for all
three ISM conditions (CNM, WNM, and PDR) because ions are decelerated by the
magnetic field, and the drift velocity becomes supersonic ($s_{d}\gg 1$) at
the shock layer (Figure 3). Note that the computation of the velocity profiles
stops when the neutrals and ions are about to recouple; however a full profile
is not necessary to confirm that the supersonic drift is developed in the
C-shock.

The Mach number of the neutral drift velocity, $s_{d}$ depends on the shock
velocity and sound speed, and the C-shock develops supersonic drift for the
typical values of shock velocity (a few tens of kilometers per second) found
in the ISM velocity–density plane (e.g., Draine & McKee, 1993). The C-shock
with supersonic drift ($s_{d}\gg 1$) has an impact on the calculation of the
damping and excitation coefficients ($F$ and $G$) described in the following
section (Section 3.2).

###  3.2 Rotational Damping and Excitation in C-shocks

Various damping and excitation processes in the rotational dynamics of grain
particles are captured by the dimensionless damping and excitation
coefficients $F$ and $G$ in Equation 3. In this work, we consider additional
damping and excitation processes due to the presence of supersonic neutral
drift relative to charged grains (Hoang & Tram, 2019).

As discussed in Section 2.1, we assume a spherical geometry when computing
damping and excitation coefficients. For the damping and excitation
coefficients for the supersonic ($s_{d}\gg 1$) and transonic cases ($s_{d}\sim
1$), we follow Roberge et al. (1995) and adopt their results, as Hoang & Tram
(2019) does.

Let $\hat{\boldsymbol{x}}\hat{\boldsymbol{y}}\hat{\boldsymbol{z}}$ be the
reference frame fixed to the gas, such that the $\hat{\boldsymbol{z}}$-axis is
directed along the magnetic field, and the drift velocity $v_{d}$ lies in the
$\hat{\boldsymbol{y}}\hat{\boldsymbol{z}}$-plane with an angle $\alpha$ with
$\hat{\boldsymbol{z}}$. We consider a perpendicular shock
($\alpha=90^{\circ}$).

From Roberge et al. (1995), the dimensionless damping coefficients
$\langle\Delta j_{i}\rangle$ are

$\langle\Delta
j_{i}\rangle=\begin{cases}-\left[\delta+M_{0}(s_{d})\right]j_{i},&i=x,y,\\\
-M_{0}(s_{d})j_{i},&i=z.\end{cases}$ (8)

In this equation,

$M_{0}(s_{d})=\frac{\sqrt{\pi}}{4s_{d}}\left[2(1+s^{2}_{d})\mbox{erf}(s_{d})-P\left(\frac{3}{2},s^{2}_{d}\right)\right]$
(9)

where $P$ is an incomplete gamma function, and $\delta$ is a magnetic damping
parameter measuring the relative efficiency of magnetic versus gas damping, in
the sense that a large $\delta$ value would correspond to efficient magnetic
alignment (Roberge et al., 1995), and is written as (Roberge et al., 1993)

$\delta=\frac{3KVB^{2}}{4\sqrt{\pi}{n_{\tiny\mbox{H}}}m_{\tiny\mbox{H}}v_{th}b^{4}\Gamma_{\|}}.$
(10)

where $K=10^{-13}\left(\frac{T_{d}}{18\mbox{\tiny K}}\right)^{-1}$ (Draine,
2011b), $V=\frac{4\pi}{3}a_{\mbox{\tiny cx}}^{3}$,
$v_{th}=\sqrt{\frac{2kT}{m_{\tiny\mbox{H}}}}$, $b\approx a_{\mbox{\tiny cx}}$,
and $\Gamma_{\|}=1.0$ for spherical grains (Roberge et al., 1993). For the
magnetic field strength ($B$), we use Equation 6. Although $\delta$ is not
well constrained due to the highly uncertain material parameter $K$(Roberge et
al., 1993), we find that $\delta\ll M_{0}(s_{d})$ for the ISM condition that
we are considering (i.e., high temperature in the C-shock region).

The dimensionless angular momentum $j_{i}(i=x,y,z)$ is
$j_{i}\equiv\frac{J_{i}}{\sqrt{IkT}}$ and $j_{i}\approx 1$ for a spherical
grain rotating with the equipartition energy (Roberge et al., 1995). The
dimensionless damping coefficient $F_{sd}$ for spherical grains due to
supersonic neutral drift in the C-shock becomes

$F_{sd}=\frac{1}{3}\displaystyle\sum_{i=x,y,z}\langle\Delta j_{i}\rangle$ (11)

Also from Roberge et al. (1995), the dimensionless excitation coefficients
$\langle(\Delta j_{i})^{2}\rangle$ are

$\langle(\Delta
j_{x})^{2}\rangle=D_{T}+\frac{T_{d}}{T}\left[2\delta+M_{0}(s_{d})\right]\\\ $
(12) $\langle(\Delta
j_{y})^{2}\rangle=D_{P}\sin^{2}\alpha+D_{T}\cos^{2}\alpha+\frac{T_{d}}{T}\left[2\delta+M_{0}(s_{d})\right]\\\
$ (13) $\langle(\Delta
j_{z})^{2}\rangle=D_{P}\cos^{2}\alpha+D_{T}\sin^{2}\alpha+\frac{T_{d}}{T}M_{0}(s_{d})\\\
$ (14)

The quantities

$D_{T}(s_{d})=\frac{3}{4}\left[\left(1+2s_{d}^{2}\right)M_{0}(s_{d})+\left(1-2s_{d}^{2}\right)M_{2}(s_{d})\right]$
(15)

and

$D_{P}(s_{d})=\frac{3}{2}\left[M_{0}(s_{d})-M_{2}(s_{d})\right]$ (16)

where

$\displaystyle
M_{2}(s_{d})=\frac{\sqrt{\pi}}{4}s_{d}\mbox{erf}(s_{d})-\frac{3\sqrt{\pi}}{16}s_{d}^{-3}P\left(5/2,s_{d}^{2}\right)$
(17) $\displaystyle+\frac{\sqrt{\pi}}{4}s_{d}^{-3}P\left(3/2,s_{d}^{2}\right)$

are dimensionless, monotonically increasing functions of $s_{d}$ that have the
limiting values $D_{T}(0)=D_{P}(0)=1$ and satisfy the inequality
$D_{T}(s_{d})>D_{P}(s_{d})$ for $s_{d}>0$ (Roberge et al., 1995). The
dimensionless excitation coefficient $G_{sd}$ for spherical grains due to
supersonic neutral drift in C-shocks becomes

$G_{sd}=\frac{1}{3}\displaystyle\sum_{i=x,y,z}\langle\left(\Delta
j_{i}\right)^{2}\rangle$ (18)

.

##  4 Rotational Disruption of Spinning Nanoparticles

###  4.1 Critical Angular Velocity for Disruption

The tensile stress of a spherical grain with mass $m$, radius $r$ and angular
velocity $\omega$ is $S=\frac{1}{4}\rho\omega^{2}a^{2}$, which is converted to
the relation between the critical angular velocity for grain disruption and
the maximum grain tensile strength, $S_{\mbox{\tiny max}}$ (Hoang, 2020),

$\frac{\omega_{cri}}{2\pi}\simeq 5.72\times
10^{10}a^{-1}_{-7}S^{1/2}_{\mbox{\tiny
max,9}}\hat{\rho}^{-1/2}~{}~{}[\mbox{Hz}]$ (19)

where $a_{-7}=a/(10^{-7}\mbox{cm})$, $S_{\mbox{\tiny max,9}}=S_{\mbox{\tiny
max}}/(10^{9}~{}\mbox{erg cm}^{-3})$ and $\hat{\rho}=\rho/(3~{}\mbox{g
cm}^{-3})$.

The exact value of the maximum tensile strength $S_{\mbox{\tiny max}}$ depends
on the composition and structure of the dust grain and is largely unknown;
compact grains can have higher tensile strength than porous/composite grains
(e.g., Hoang & Tram, 2019). For example, a polycrystalline bulk solid has
$S_{\mbox{\tiny max}}\sim 10^{9}-10^{10}$erg cm-3 while ideal material, like
diamond, can have $S_{\mbox{\tiny max}}\gtrsim 10^{11}$erg cm-3(Hoang, 2020).
Since the smallest grains may be sheetlike and have approximately 100 carbon
atoms corresponding to the size of large PAH (Draine & Lazarian, 1998; Ali-
Haïmoud et al., 2009), it is a reasonable assumption that the tensile strength
of the nanoparticles considered in this study is not very large because they
are not compact. Therefore, in this study, we consider $S_{\mbox{\tiny max}}$
to be varied for a range, $10^{8}$–$10^{10}$erg cm-3. However, we emphasize
that the result of the grain destruction model in this study depends on the
poorly understood maximum tensile strength parameter.

###  4.2 Spin Angular Velocity of Nanoparticles

If one assumes Maxwellian distribution for $f_{a}(\omega)$ with gas
temperature $T$, one can solve Equation 3 to obtain $\langle\omega^{2}\rangle$
(Draine & Lazarian, 1998):

$\langle\omega^{2}\rangle=\frac{2}{1+\left[1+(G/F^{2})(20\tau_{\tiny{\mbox{H}}}/\tau_{\tiny{\mbox{ed}}})\right]^{1/2}}\left(\frac{G}{F}\right)\left(\frac{3kT}{I}\right)$
(20)

This analytic expression lets us compute the angular velocity
$\omega_{rot}\equiv\sqrt{\langle\omega^{2}\rangle}$ of grain particles with a
rotational temperature $T_{rot}$ for a given grain size $a$ (Hoang & Tram,
2019),

$\displaystyle\frac{\omega_{rot}}{2\pi}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\left(\frac{3kT_{rot}}{I}\right)^{1/2}$ (21)
$\displaystyle\simeq$ $\displaystyle 1.4\times
10^{10}a^{-5/2}_{-7}\left(\frac{T_{rot}}{10^{3}\mbox{K}}\right)^{1/2}\hat{\rho}^{-1/2}~{}[\mbox{Hz}]$

which will be compared to the critical angular velocity $\omega_{cri}$ in
Equation 19 for centrifugal disruption of grain particles. The rotational
temperature $T_{rot}$ is related to the gas temperature $T$ by

$\frac{T_{rot}}{T}=\frac{2}{1+\left[1+(G/F^{2})(20\tau_{\tiny{\mbox{H}}}/\tau_{\tiny{\mbox{ed}}})\right]^{1/2}}\left(\frac{G}{F}\right)$
(22)

and if the nanoparticles are in suprathermal rotation, $T_{rot}>T$.

Figure 4: Grain volume per H atom per logarithmic interval in dust size,
$(4\pi a^{3}/3)d{n_{\tiny\mbox{gr}}}/d\mbox{\small ln}a$. Blue dotted-dashed,
green solid, and red dashed lines are the models with parameters in lines 1,
4, and 7 of Table 1 in Weingartner & Draine (2001), labeled as “case1”,
“case2”, and “case3”, respectively, for subsequent analysis. The gray vertical
line indicates the minimum dust size, $a_{\mbox{\tiny min}}=3.5$Å, used in
SpDust.

###  4.3 Grain size distribution

The emissivity of AME is determined by the ensemble of electric dipole
emission from nanoparticles with a distribution of their size $a$
($\frac{1}{{n_{\tiny\mbox{H}}}}\frac{d{n_{\tiny\mbox{gr}}}}{da}$ in Equation
2), and most of the AME emissivity is from the smallest particle
($a_{-7}\lesssim 0.5$), as shown in Ali-Haïmoud et al. (2009). Therefore, the
shape of the grain size distribution, especially for small grains, is a very
important component of the AME models (Hensley & Draine, 2017); however, is
not well understood for different ISM environments. Therefore, for
carbonaceous grains, we assume that the grain size distribution follows the
commonly used form of the composite of log-normal and power-law distribution
by Weingartner & Draine (2001) for grain radii $a_{\mbox{\tiny min}}=3.5$Å
$<a<a_{\mbox{\tiny max}}=100$Å as adopted in the SpDust code (Ali-Haïmoud et
al., 2009),

$\displaystyle\frac{1}{{n_{\tiny\mbox{H}}}}\frac{d{n_{\tiny\mbox{gr}}}}{da}=D(a)+\frac{C}{a}\left(\frac{a}{a_{t}}\right)^{\alpha}F(a;\beta;a_{t})$
(23) $\displaystyle\times\begin{cases}1&a_{\mbox{\tiny min}}<a<a_{t},\\\
e^{-\left[(a-a_{t})/a_{c}\right]^{3}}&a>a_{t}\end{cases}$

where

$F(a;\beta;a_{t})=\begin{cases}1+\beta\frac{a}{a_{t}}&\beta\geq 0,\\\
\left(1-\beta\frac{a}{a_{t}}\right)^{-1}&\beta<0\end{cases}$ (24)

and the log-normal distribution $D(a)$ is

$D(a)=\displaystyle\sum^{2}_{i=1}\frac{B_{i}}{a}\mbox{exp}\left\\{-\frac{1}{2}\left[\frac{\mbox{ln}(a/a_{0,i})}{\sigma}\right]^{2}\right\\}.$
(25)

with the normalization $B_{i}$ defined to place a total number $b_{C,i}$ of
carbon atoms per H atom in the $i$th log-normal distribution. Here
$b_{C,1}=0.75b_{C}$ and $b_{C,2}=0.25b_{C}$, $b_{C}$ being the total carbon
abundance per H atom, $a_{0,1}=3.5$Å, $a_{0,2}=30$Å, and $\sigma=0.4$. This
size distribution has a total of six parameters ($b_{C},C,a_{t},a_{c},\alpha$,
and $\beta$).

In Figure 4, we plot the grain volume per H atom per logarithmic interval in
dust size, $(4\pi a^{3}/3)d{n_{\tiny\mbox{gr}}}/d\mbox{\small ln}a$ for three
different models in Weingartner & Draine (2001) that matches the extinction
curve of the diffuse ISM. Blue dotted-dashed, green solid, and red dashed
lines are the models with parameters in lines 1, 4, and 7 of Table 1 in
Weingartner & Draine (2001), and the gray vertical line indicates the minimum
dust size, $a_{\mbox{\tiny min}}=3.5$Å, used in SpDust. In Figure 4, we label
each model as “case1”, “case2”, and “case3”, respectively.

The exact shape of the size distribution of the dust grains varies depending
on the ISM condition and is not well known for the ISM in star-forming regions
exposed to extreme physical conditions (strong radiation and shock). In this
study, we use the three labeled models in Figure 4 to investigate a range of
the dust size distribution from the case without additional enhancement of
small grains to the case with significant enhancement of small grains as
represented by the log-normal component, which leads to stronger AME. In
principle, strong UV radiation from a massive star-forming region can exert
radiation torque on dust grains, and the fraction of small grains can increase
by disruption of large grains (Hoang et al., 2019), which then may increase
the AME in an early stage of massive star formation before the C-shock impacts
the ISM and destroys the small nanoparticles. However, the grain size
distribution is poorly constrained, and extreme ISRF may be required to
destroy large grains (see the discussion in Section 6.1).

##  5 Radio-Millimeter Emission from Star-forming Regions

Recent observations of AME from star-forming regions challenge the widely
accepted notion about the radio-millimeter SED; the radio-millimeter SED from
star-forming regions in the frequency range $\approx 10$–$100$ GHz is
dominated by free-free emission and thermal dust emission. Since the observed
SED is based on flux measurements with a finite beam, unknown “environmental”
factors, such as the geometry and volume filling factor of the ISM that are
not directly related to the physics of the emission, affect the SED.
Therefore, we only focus on the emissivity of each radiation process in order
to isolate the environmental factors and characterize the significance of AME
relative to the free-free and thermal dust emission.

Originally, the emissivity $j_{\nu}$ is defined as energy emitted at a
frequency $\nu$ per unit volume, time, and solid angle (erg s-1sr-1cm-3 Hz-1).
However, in this work, we use the emissivity per H atom
$\frac{j_{\nu}}{{n_{\tiny\mbox{H}}}}$ in erg s-1 sr-1 Hz-1 (H atom)-1 (see
Equation 2) following previous works (Draine & Lazarian, 1998; Ali-Haïmoud et
al., 2009). The emissivities of free-free and thermal dust emission are
computed as a function of frequency $\nu$ for CNM, WNM, and PDR ISM conditions
(Table 1) and the emissivity of AME (Equation 2) before and after the impact
of the C-shock will be computed by following the process in Section 3 and
compared with the emissivity of free-free and thermal dust emission.

###  5.1 Calculating the Emissivity of Free-Free, Thermal Dust Emission and
AME

####  5.1.1 Free-Free emission

The emissivity per H atom of free-free emission with electrons of temperature
$T$ is (Rybicki & Lightman, 1986)

$\frac{{j_{\tiny\nu,ff}}}{{n_{\tiny\mbox{H}}}}=\frac{1}{4\pi}2^{5}\pi\left(\frac{e^{6}}{3m_{e}c^{3}}\right)\left(\frac{2\pi}{3km_{e}}\right)^{1/2}g_{ff}\frac{{n_{\tiny\mbox{H}}}}{\sqrt{T}}e^{-\frac{h\nu}{kT}}$
(26)

where $g_{ff}$ is the Gaunt factor for the free-free transition. The free-free
emissivity is implemented in SpDust(Ali-Haïmoud et al., 2009; Silsbee et al.,
2011) using $g_{ff}$ tabulated from Sutherland (1998).

####  5.1.2 Thermal dust emission

The emissivity per H atom of thermal dust emission with dust temperature
$T_{d}$ is (Rybicki & Lightman, 1986)

$\displaystyle\frac{{j_{\tiny\nu,bb}}}{{n_{\tiny\mbox{H}}}}=\frac{\kappa_{0}}{{n_{\tiny\mbox{H}}}}\left(\frac{\nu}{\nu_{0}}\right)^{\beta}\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{\frac{h\nu}{kT_{d}}}-1}$
(27)

where $\kappa_{0}$ is the dust volume absorption coefficient (i.e., cross
section per unit volume), and $\beta$ is the emissivity spectral index.
Instead of $\kappa_{0}$, often the more frequently used coefficient is the
mass absorption coefficient $\kappa_{0}^{\prime}$ (cm2/g) and
$\kappa_{0}=\kappa_{0}^{\prime}\rho_{d}$ for dust mass density $\rho_{d}$
(total dust mass/ISM volume). If gas and dust are well mixed in the same
volume, $\rho_{d}$ can be inferred by the gas mass density ($\rho_{\mbox{\tiny
gas}}={n_{\tiny\mbox{H}}}m_{\mbox{\tiny H}}$) and dust-to-gas mass ratio
$M_{d}/M_{g}=0.0083\rho_{\mbox{\tiny ref}}/(3~{}\mbox{g cm}^{-3})$ (Draine,
2011b),

$\displaystyle\rho_{d}={n_{\tiny\mbox{H}}}m_{\mbox{\tiny H}}\times
0.0083\left(\frac{\rho_{\mbox{\tiny ref}}}{3~{}\mbox{g/cm}^{-3}}\right)$ (28)

where the reference density $\rho_{\mbox{\tiny ref}}$ is the solid density of
a grain particle and different for different types of dust particles with an
intermediate value of $3~{}\mbox{g/cm}^{-3}$ (Draine, 2011b). Both
$\kappa_{0}^{\prime}$ and $\beta$ vary depending on the type of dust and
wavelength. In this work, we adopt $\kappa_{0}^{\prime}=1.8$ (cm2/g) at
$\nu_{0}=599.98$ GHz from Clark et al. (2016) and $\beta=2.0$ (e.g., Schnee et
al., 2010). Then $j_{\tiny\nu,bd}$ can be written as

$\displaystyle\frac{{j_{\tiny\nu,bb}}}{{n_{\tiny\mbox{H}}}}=0.015\left(\frac{\rho_{d}}{3\mbox{g/cm}^{3}}\right)\frac{{n_{\tiny\mbox{H}}}m_{\mbox{\tiny
H}}}{{n_{\tiny\mbox{H}}}}\left(\frac{\nu}{\nu_{0}}\right)^{2}\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{\frac{h\nu}{kT_{d}}}-1}$
(29)

We assume that even though the small nanoparticles (i.e., the source of AME)
are destroyed by shocks, the bulk of the grain volume is still dominated by
large grain particles (Figure 4), and the thermal dust emission is not much
affected by the disruption of the small nanoparticles. Therefore, we use the
same $j_{\tiny\nu,bd}$ before and after the shock. We implemented the
emissivity of thermal dust emission in the SpDust code.

####  5.1.3 AME

The emissivity per H atom of AME is shown in Equation 2. We modify the SpDust
code and add the damping and excitation coefficients $F_{sd}$ and $G_{sd}$ due
to the interaction with supersonic neutral drift in C-shocks (Section 3) to
the other damping and excitation coefficients, $F_{i}$ and $G_{i}$ when
computing the total damping and excitation coefficients:

$\displaystyle F=\sum_{i}F_{i}+F_{sd}$ (30) $\displaystyle
G=\sum_{i}G_{i}+G_{sd}.$

Since grains might not spin around the axis of their greatest inertia, Silsbee
et al. (2011) introduced the correction terms to the damping coefficient by
incoming particles. We also introduce the correction terms to the damping
coefficient, $F_{sd}$ in order to account for the randomized orientation of
grains relative to their angular momentum vector. The impact of C-shocks is
the collision between neutrals (H, H2 and He) and charged grains, and we use
the correction terms (Equation (140) in Silsbee et al. (2011)) for the charged
grain with neutral impactors.

####  5.1.4 Total Emissivity in Radio-millimeter Wavelength

Using SpDust, we compute the total emissivity per H atom for the emission in
the radio-millimeter wavelength, including free-free, thermal dust, and AME
emissivity, using Equation 26, 29, and 2:

$\frac{j_{\nu}}{{n_{\tiny\mbox{H}}}}=\left(\frac{{j_{\tiny\nu,ff}}}{{n_{\tiny\mbox{H}}}}+\frac{{j_{\tiny\nu,bb}}}{{n_{\tiny\mbox{H}}}}+\frac{{j_{\tiny\nu,sp}}}{{n_{\tiny\mbox{H}}}}\right)$
(31)

The original input parameters to run SpDust are the ISM environment parameters
(Table 1). Two additional parameters that we add to these standard SpDust
parameters are the dust emissivity spectral index ($\beta$ in Equation 27) and
the minimum size of nanoparticles ($a_{\mbox{\tiny min}}$ in Equation 23).

The total emissivity of the radio-millimeter emission is computed for three
different ISM environments, CNM, WNM, and PDR, before and after the C-shock
impacts the ISM.

###  5.2 AME before the Impact of the C-shock

Figure 5: Before the impact of the C-shock: Angular rotation velocity of the
dust particle (blue line) emitting AME and the critical angular rotation
velocity (red lines) for a range of the maximum tensile strength
($10^{8}-10^{10}$ erg cm-3) of the spinning dust particles Figure 6: Before
the impact of the C-shock: radio-millimeter emissivity including free-free
emission, thermal dust emission, and AME for typical ISM conditions for CNM,
WNM, and PDR before the impact of the C-shock

Using SpDust, we compute the AME emissivity $j_{\tiny\nu,sp}$ (Equation 2)
without including damping and excitation coefficients due to the supersonic
neutral drift ($F_{sd}$ and $G_{sd}$ in Equation 30). In Figure 5, we show the
angular velocity $\omega_{rot}$ (connected blue dots) in Equation 21 as a
function of grain size $a$ based on the computed damping and excitation
coefficients for CNM, WNM, and PDR, together with the critical angular
velocity $\omega_{cri}$ (red lines) in Equation 19 for a range of maximum
tensile strength, $S_{\mbox{\tiny max}}=10^{8}$ – $10^{10}$ [erg cm-3]. As the
grain size becomes smaller, both $\omega_{rot}$ and $\omega_{cri}$ increase.
For a grain whose size is smaller than a certain length, where
$\omega_{rot}=\omega_{cri}$, one can expect that $\omega_{rot}$ will be larger
than $\omega_{cri}$ and grains smaller than that size will be broken apart due
to centrifugal force, which, however, is not likely to happen in the cases for
the given range of $S_{\mbox{\tiny max}}$ (Figure 5), where the impact of the
C-shock is not being considered.

Therefore, the integration over grain size in Equation 2 is performed over the
full range (3.5-100Å). The resulting emissivities, including AME (dotted-
dashed, solid and dashed blue line for case1, 2, and 3, respectively), free-
free (red dashed line), and thermal dust emission (green dashed line) for CNM,
WNM, and PDR, are shown in Figure 6. In addition to the fact that the strength
of free-free emissivity increases as we go from CNM to PDR, the AME emissivity
is significantly larger than the free-free emissivity for CNM, WNM, and PDR.
In particular, for PDR, the peak frequency even moves to the higher frequency
($>100$ GHz) and impacts the emissivity curve at millimeter wavelength that is
usually dominated by the emissivity of thermal dust emission if there is no
AME.

###  5.3 AME after the Impact of the C-shock

We compute the AME emissivity $j_{\tiny\nu,sp}$ (Equation 2) including damping
and excitation coefficients due to the supersonic neutral drift ($F_{sd}$ and
$G_{sd}$ in Equation 30) by following the formalism described in Section 3.
The $s_{d}$ can be varied depending on the exact shock condition in the ISM
and we use $s_{d}=10$ from Figure 3.

In Figure 7, we show the angular velocity $\omega_{rot}$ (connected blue dots)
as a function of grain size $a$ based on the computed damping and excitation
coefficients, including $F_{sd}$ and $G_{sd}$, for CNM, WNM, and PDR, together
with the critical angular velocity $\omega_{cri}$ (red lines). Due to the
excitation by the interaction with supersonic neutral drift, $\omega_{rot}$
increases significantly, and the critical grain size below which the
destruction process happens for the grain with the maximum tensile strength
$S_{\mbox{\tiny max}}=10^{8}$ [erg cm-3], becomes larger ($a>1$ nm or $10$Å).
For each ISM condition in Figure 7, we find the critical grain size where
$\omega_{rot}$ and $\omega_{cri}$ for $S_{\mbox{\tiny max}}=10^{8}$ [erg cm-3]
become equal and use this grain size as $a_{\tiny\mbox{min}}$ for integrating
Equation 2, which reduces the AME emissivity.

The resulting emissivities, including AME (dotted-dashed, solid, and dashed
blue lines for case1, 2, and 3, respectively) impacted by the C-shock, free-
free (red dashed line), and thermal dust emission (green dashed line) for CNM,
WNM, and PDR, are shown in Figure 8. Compared to Figure 6, we find that the
AME emissivity is significantly (for CNM) and almost entirely (for WNM and
PDR) suppressed because the small grains are destroyed by centrifugal force
from increased spin angular velocity.

Figure 7: After the impact of the C-shock: angular rotation velocity of dust
particles (blue line) emitting AME and critical angular rotation velocity (red
lines) for a range of the maximum tensile strength ($10^{8}$– $10^{10}$ erg
cm-3) of the spinning dust particles Figure 8: After the impact of the
C-shock: radio-millimeter emissivity including free-free emission, thermal
dust emission, and AME for typical ISM conditions for CNM, WNM, and PDR after
the impact of the C-shock. Note that AME emissivity in the WNM condition is
entirely suppressed and negligible compared to free-free and thermal dust
continuum emissivity.

##  6 Discussion

In Section 5, we demonstrate that the impact of C-shocks can suppress the AME
emissivity for typical conditions of the ISM (CNM, WNM, and PDR). Although
progress has been made (for a review, see Dickinson et al., 2018), the ISM
condition of star-forming regions for strong AME is still not well known; the
number of objects that are not detected as AME still have a physical condition
similar to those that are detected (Scaife, 2013). In this section, we propose
a hypothesis of the rise and fall of the AME in star-forming regions according
to their evolutionary stage (Seections 6.1 and 6.2), consider the ‘observed’
SED predicted from the volume-integrated composite emissivity for
monotonically or mildly varying ISM physical parameters (Section 6.3), and
discuss other rotational disruption processes (Section 6.4), dust
fragmentation (Section 6.5), and the difficulty and prospect of the
extragalactic AME detection in the era of high-resolution radio observation
(Section 6.6).

###  6.1 The Rise of AME in the Early Stage of Star Formation

A typical lifetime of OB stars is 1-10 Myr. In the early stage of star-
formation ($\lesssim 10$ Myr), the strong radiation from young stars exerts a
radiative torque on the dust particles and increases their angular velocity
(Draine & Weingartner, 1997; Lazarian & Hoang, 2007a). The radiative torque
breaks large grains more efficiently, such that grains with a size larger than
$a_{\mbox{\tiny disr}}$ are disrupted by the centrifugal force due to the
increased angular velocity (Hoang et al., 2019),

$\left(\frac{a_{\mbox{\tiny disr}}}{0.1\mu m}\right)^{2.7}\approx
0.046\gamma^{-1}\bar{\lambda}_{0.5}^{-1.7}\left(\frac{U_{6}}{1.2}\right)^{-1/3}S_{\mbox{\tiny
max,9}}^{1/2}$ (32)

where $\gamma$ is the anisotropy parameter of the radiation field
$0<\gamma<1$, and $\bar{\lambda}_{0.5}=(\bar{\lambda}/0.5\mu m)$ for the mean
wavelength of the radiation field $\bar{\lambda}$. Here $U_{6}=U/10^{6}$ where
$U$ is the radiation energy density normalized by the ISRF energy density at
the solar neighborhood ($u_{\mbox{\tiny ISRF}}=5.29\times 10^{-14}$ erg cm-3
Draine (2011b)). This radiative torque disruption is more efficient than other
disruption mechanisms (thermal sublimation, thermal sputtering, grain
shattering) in destroying large ($a\gtrsim 0.1\mu m$) grains (Hoang et al.,
2019).

Since the radiation field energy density $u_{\mbox{\tiny rad}}$ is $F/c$ where
$F$ is flux density $L/(4\pi R^{2})$ at a distance $R$ from the source with
bolometric luminosity $L$, we can write $U$ as follows:

$U=\frac{u_{\mbox{\tiny rad}}}{u_{\mbox{\tiny ISRF}}}\approx
2100\left(\frac{L}{10^{7}L_{\odot}}\right)\left(\frac{R}{10\mbox{pc}}\right)^{-2}$
(33)

If a massive star-forming region contains 100 O stars of bolometric luminosity
$\sim 10^{5}L_{\odot}$ with anisotropic illumination ($\gamma=0.1$) of
radiation and $\bar{\lambda}=0.5\mu m$, $a_{\mbox{\tiny disr}}\approx 0.14\mu
m$ at 10 pc away and $a_{\mbox{\tiny disr}}\approx 0.25\mu m$ at 100 pc away
from the center of the star-forming region for the grain particle with the
maximum tensile strength of $S_{\mbox{\tiny max}}=10^{9}$erg cm-3. So, in
typical extragalactic star-forming regions with massive star formation, grains
with sizes larger than 0.2$\mu m$ (or $2\times 10^{-5}$cm) will be disrupted
by the radiation torque, while small nanoparticles ($a\sim 10^{-7}$cm) are
still resilient against the centrifugal force.

Detailed dust grain modeling by Silsbee & Draine (2016) suggests that ‘fluffy’
grains exposed to solar radiation can be disrupted by increased spin. However,
in order to make the radiation pressure an efficient mechanism for breaking
large grains, it may require an extreme radiation field (Draine 2021 private
communication), which is much larger than the value from Equation 33.

The disrupted large grains are likely to be added as small grains and change
the shape of the dust size distribution by steepening the power-law
distribution in Equation 23. This implies that the AME might be stronger in
young (10 Myr or less) and massive ($\sim 100M_{\odot}$) star-forming regions
before a supernova explosion drives a shock wave into the ISM, which is
consistent with the interpretation of the extragalactic AME observation by
Murphy et al. (2018b), where the best AME model for the source suggests that
AME is from a nascent star-forming region with very young ($<3$ Myr) massive
stars.

A compilation of the observations of AME from Galactic star-forming regions
associated with $\textrm{H}\scriptstyle\mathrm{II}$ region is reported in
Dickinson (2013). It is difficult to compare the reported values of the
significance of AME detection for individual regions because the observations
are made with different angular scales and the exact definitions of the
significance of AME detection are different. However, amongst the observed
star-forming regions, those that show a significant flux enhancement over the
thermal free-free emission in $\sim 30$ GHz frequency are powered by OB stars
(Finkbeiner et al., 2004; Dickinson et al., 2009; Planck Collaboration et al.,
2011; Shuping et al., 2012; Tibbs et al., 2012). On the other hand, the AME is
not detected from a star-forming region that is relatively older and
previously impacted by shocks, like the Orion Nebula (Planck Collaboration et
al., 2011), and from the supernova remnant like 3C 396 (Cruciani et al.,
2016). These observations suggest that another important environmental issue
for understanding the observed AME might be whether or not the ISM is impacted
by a shock, which is correlated with the age of the star-forming region.

###  6.2 The Fall of AME in the Later Stage of Star Formation

As we show in Section 5, the C-shock in the magnetized medium can suppress the
AME by disrupting small nanoparticles ($a\sim 10^{-7}$ cm). This implies that
if a supernova explodes within 10-100 Myrs when massive stars become old
enough to die, the star-forming region will be impacted by the C-shock and may
not be the source of strong AME anymore. Here we argue that this scenario is
plausible by comparing several important timescales: shock propagation time,
neutral flow time, rotational disruption time, and dust reformation time.

Shock propagation time is the time for the shock front with speed $v_{s}$ to
propagate into the ISM to a distance $l$ and can be written as

$\tau_{\mbox{\tiny prop}}=4.8\times
10^{5}\left(\frac{l}{10\mbox{pc}}\right)\left(\frac{20\mbox{km/s}}{v_{s}}\right)\mbox{yr}.$
(34)

Neutral flow time is the time for neutrals with drift velocity $v_{\mbox{\tiny
drift}}$ to pass the length scale of shock $l_{s}$ and can be written as
(Hoang & Tram, 2019)

$\tau_{\mbox{\tiny
flow}}=30\left(\frac{l_{s}}{10^{15}\mbox{cm}}\right)\left(\frac{10\mbox{km/s}}{v_{\mbox{\tiny
drift}}}\right)\mbox{yr}.$ (35)

Rotational disruption time is the time required to spin-up nanoparticles with
maximum tensile strength $S_{\mbox{\tiny max}}$ to $\omega_{cri}$ and can be
written as (Hoang & Tram, 2019)

$\tau_{\mbox{\tiny disr}}\simeq
0.005a_{-7}^{4}\left(\frac{{n_{\tiny\mbox{H}}}}{10^{5}\mbox{cm}^{-3}}\right)^{-1}S_{\mbox{\tiny
max,9}}\left(\frac{v_{\mbox{\tiny
drift}}}{10\mbox{km/s}}\right)^{-3}\mbox{yr}.$ (36)

Dust reformation time is the time required to replenish the disrupted
nanoparticles, which is most likely done by the fresh dust formation process
via stellar winds from the low- and intermediate-mass stars evolved from the
main sequence because dust growth via coagulation is a very slow process with
gigayears of timescale (Asano et al., 2013). In particular, the thermally
pulsating asymptotic giant branch (AGB) phase of intermediate-mass stars is
likely to be the most promising site for dust formation (Morgan & Edmunds,
2003). Given that the main-sequence lifetime of intermediate-mass ($M\sim
3-5M_{\odot}$) stars is $\gtrsim 100$ Myr (Schaller et al., 1992), the dust
reformation time via stellar wind from evolved stars is at least larger than
100 Myr, $\tau_{\mbox{\tiny reform}}\gtrsim 100~{}\mbox{Myr}$, and this will
probably be the fastest way of replenishing the nanoparticles.

To understand the effect of gas bombardment on disruption of small
nanoparticles in C-shocks, we need to compare $\tau_{\mbox{\tiny disr}}$ with
$\tau_{\mbox{\tiny flow}}$:

$\displaystyle\frac{\tau_{\mbox{\tiny flow}}}{\tau_{\mbox{\tiny disr}}}\simeq
6000a^{-4}_{-7}\left(\frac{l_{s}}{10^{15}\mbox{cm}}\right)\left(\frac{{n_{\tiny\mbox{H}}}}{10^{5}\mbox{cm}^{-3}}\right)S_{\mbox{\tiny
max,9}}^{-1}$ (37) $\displaystyle\times\left(\frac{v_{\mbox{\tiny
drift}}}{10\mbox{km/s}}\right)^{2}.$

In the shock layer ($l_{s}\approx 10^{15}$cm), the $\frac{\tau_{\mbox{\tiny
flow}}}{\tau_{\mbox{\tiny disr}}}$ for CNM and PDR is significantly larger
than 1 for typical values of nanoparticle size $a\approx 10^{-7}$cm, drift
velocity $v_{\mbox{\tiny drift}}\approx 10$km/s and maximum tensile strength
$S_{\mbox{\tiny max}}\approx 10^{9}$ erg cm-3. Although, for WNM,
$\frac{\tau_{\mbox{\tiny flow}}}{\tau_{\mbox{\tiny disr}}}$ is less than 1
($\frac{\tau_{\mbox{\tiny flow}}}{\tau_{\mbox{\tiny disr}}}\sim 0.024$), for
the same values of $a$, $v_{\mbox{\tiny drift}}$ and $S_{\mbox{\tiny max}}$,
$\tau_{\mbox{\tiny flow}}$ and $\tau_{\mbox{\tiny disr}}$ become easily
comparable as the particle size becomes smaller ($a_{-7}<1$) and
$v_{\mbox{\tiny drift}}$ becomes larger (due to a high-velocity shock).
Therefore, a $\tau_{\mbox{\tiny flow}}$ much larger than $\tau_{\mbox{\tiny
disr}}$ implies that the grain disruption process is almost instantaneous
after the shock sweeps the ISM.

Even though the nanoparticles are quickly disrupted in the shock layer
($\frac{\tau_{\mbox{\tiny flow}}}{\tau_{\mbox{\tiny disr}}}\gg 1$), AME will
not be suppressed if the small nanoparticles, after the shock propagation, are
reformed quickly as a result of the dust formation process via stellar wind
from AGB stars. However, the shock propagation time scale, $\tau_{\mbox{\tiny
prop}}$ is much shorter than the dust reformation time $\tau_{\mbox{\tiny
reform}}\gtrsim 100~{}\mbox{Myr}$. It implies that the shock propagates almost
instantaneously through the ISM and disrupts the nanoparticles without quick
replenishment of dust grains by the AGB star.

Dust grains are also produced in the supernova explosion (for a review, see
Sarangi et al., 2018). But in such high-temperature gas ($T>10^{6}$K), a
0.1$\mu$m size grain survives for $<0.1$Myr (Draine, 2011b), and the
theoretical modeling works of dust formation and evolution in the supernova
explosion suggest that the dust produced in the supernova explosion is mostly
large grains ($>1\mu$m; Brooker et al., 2022; Sarangi et al., 2018). Also,
Gall et al. (2014) reported that only grain size distributions with grain
radii larger than 0.25$\mu$m with a lower limit of 0.7$\mu$m can reproduce the
observed supernova extinction curves. The large dust grains formed in the
high-temperature supernova explosion do not survive long and are not likely to
be the source of AME.

###  6.3 Observed Radio-millimeter SED

What we compute is the emissivity, not the observed flux density, as we note
in Section 5. However, we can infer the significance of the observed AME
relative to the free-free and thermal dust emission by assuming a simple
profile of the physical parameters.

In a simple spherical geometry, the observed SED of the star-forming region at
distance $D$ is determined by the measured flux density $F_{\nu}$ at $\nu$.

$F_{\nu}=\frac{1}{4\pi D^{2}}\int^{R_{f}}_{R_{i}}dr4\pi
r^{2}{n_{\tiny\mbox{H}}}\left(\frac{4\pi j_{\nu}}{{n_{\tiny\mbox{H}}}}\right)$
(38)

where

$\frac{4\pi
j_{\nu}}{{n_{\tiny\mbox{H}}}}=\left(\frac{4\pi{j_{\tiny\nu,ff}}}{{n_{\tiny\mbox{H}}}}+\frac{4\pi{j_{\tiny\nu,bb}}}{{n_{\tiny\mbox{H}}}}+\frac{4\pi{j_{\tiny\nu,sp}}}{{n_{\tiny\mbox{H}}}}\right)$
(39)

and the volume integration is performed from the inner radius $R_{i}$ to the
outer radius $R_{f}$ of the star-forming region.

From Equation 38 and the emissivity equation for each emission (Equation 26,
29 and 2), we can infer that the observed flux density of free-free emission
depends on the radial distribution of $n_{\tiny\mbox{H}}$ and $T$, the flux
density of thermal dust emission depends on the radial distribution of
$n_{\tiny\mbox{H}}$ and the flux density of AME depends on the radial
distribution of $n_{\tiny\mbox{H}}$, $T$, $\chi$, $x_{\mbox{\tiny H}}$.

In general, for each ISM, the CNM, WNM, and PDR around the star-forming
regions, ${n_{\tiny\mbox{H}}}(r)$, $T(r)$, $\chi(r)$, and $x_{\mbox{\tiny
H}}(r)$, are monotonically decreasing and not likely to vary much. Therefore,
the ‘relative’ strength of the emissivity for free-free and thermal dust
emission and AME will not change much with $r$. For each ISM condition (CNM,
WNM, and PDR), we indeed made ${n_{\tiny\mbox{H}}}(r)$, $T(r)$, $\chi(r)$, and
$x_{\mbox{\tiny H}}(r)$ an order of magnitude lower than the typical values
and computed the total emissivity. We find that although the peak of AME
emissivity changes, the relative strength of each emission does not change
much by the variation of these parameters.

This implies that the observed radio-millimeter SED resulting from the volume
integration of the emissivity reflects the relative strength of the emissivity
for each emission, and the relative strength of each emission in the radio-
millimeter SED is not much altered by the spatial variation of the ISM
physical parameters. This justifies our argument; for the smooth and monotonic
variation of ISM parameters, the shape of the observed radio-millimeter SED, a
result of volume integration of the free-free, thermal dust, and AME
emissivity, is determined by the relative contribution of each emissivity to
the total emissivity.

###  6.4 Regular Mechanical Torque by Subsonic Drift

In this study, we assume spherical dust affected by a stochastic mechanical
torque from supersonic drift of dust relative to the neutral atoms (Hoang &
Tram, 2019). However, realistic nanoparticles are expected to be irregular and
have helicity, which increases the spin angular momentum by the interaction
with subsonic astrophysical flow (Lazarian & Hoang, 2007b; Hoang et al.,
2018). This mechanical torque (called MET) was originally proposed for grain
alignment (Lazarian & Hoang, 2007b) and known to be much more efficient for
spinning up nanoparticles than stochastic mechanical torque (Lazarian & Hoang,
2021). Our understanding of the impact of MET is primarily based on
qualitative guidance from a simple model (Lazarian & Hoang, 2007b); however,
more complicated models and analyses (e.g., Das & Weingartner, 2016; Hoang et
al., 2018; Reissl et al., 2022) support the idea that irregular grains exhibit
helicity while interacting with gaseous flows, and the corresponding regular
torques dominate the stochastic mechanical torques (Lazarian & Hoang, 2021).
The MET is difficult to calculate and depends strongly on grain shape (Hoang &
Lazarian, 2018), and there is no available tool for numerical studies to
characterize its performance in astrophysical situations (Lazarian & Hoang,
2021). Although it is hard to make a quantitative argument on the impact of
MET on dust disruption, the efficiency of rotational disruption by the
centrifugal force due to increased spin from mechanical torque (either the
regular or the stochastic mechanical torque, or both) may be perhaps more
efficient in real astronomical situations (Hoang & Tram, 2019).

###  6.5 Grain Fragmentation

In this study, we assume that small nanoparticles with higher angular velocity
than the critical angular velocity for centrifugal disruption are
instantaneously and completely destroyed. However, because the rotational
disruption process cannot destroy grains to atoms, one can expect a continuous
hierarchical disruption where large grains become small grains, and as a
result, the grain size distribution might change (e.g., Guillet et al., 2011).
Our study does not model the fragmentation of grains. If the grains are
disrupted to smaller grains and those small grains survive for long enough
time, one can expect increased AME. However, we argue that smaller grains
fragmented from large grains with an angular momentum larger than the critical
value have higher angular velocity and will be disrupted quickly.

Let us suppose that one large spherical grain with angular velocity $\omega$
and moment of inertia $I=\frac{2}{5}MR^{2}$ for mass $M$ and radius $R$ has
angular momentum $J=I\omega$ and fragments into two equal-sized spherical
grains. Then, the two small grains will have the same angular momentum,
$J^{\prime}=\frac{2}{5}\left(\frac{M}{2}\right){R^{\prime}}^{2}{\omega^{\prime}}$.
If one assumes that the angular momentum is conserved ($J=2J^{\prime}$),
$\omega^{\prime}=\omega\left(\frac{R}{R^{\prime}}\right)^{2}$. In this simple
fragmentation scenario, $\omega$ is inversely proportional to the square of
the dust size ($\omega\sim\frac{1}{a^{2}}$), which increases faster than the
critical angular velocity $\omega_{cri}$ in Equation 19 as the dust size
decreases. Although this simple fragmentation scenario does not represent
reality, it is likely that small grains fragmented from large grains have a
higher angular velocity than $\omega_{cri}$ and therefore are disrupted
easily.

###  6.6 AME from Extragalactic Star-forming Region in the Era of ngVLA

Detection of AME from extragalactic star-forming regions is rare (reported for
only two galaxies). Assuming that there is no difference in the ISM condition,
on average, in our Milky Way and other galaxies, the following observational
selection effects affecting the detection of
$\textrm{H}\scriptstyle\mathrm{II}$ regions and AME may explain the rarity of
extragalactic AME.

(i) The AME detection experiment for other galaxies starts from the existing
radio map at the frequency where thermal free-free emission is dominant (e.g.,
Murphy et al., 2018a). For the radio sources detected in the map, a follow-up
multifrequency observation is made if the radio spectral SED of the source is
unusually shallow or suggests an increased flux at higher frequency. It means
that detecting the star-forming region in radio should not be biased to an
extreme system. However, detecting thermal free-free emission from
extragalactic $\textrm{H}\scriptstyle\mathrm{II}$ regions has systematic
selection effects: beam dilution, confusion with disk emission, and confusion
with a nonthermal discrete source (Israel, 1980). These systematic effects
make the detection of $\textrm{H}\scriptstyle\mathrm{II}$ regions difficult
because the contrast between the individual
$\textrm{H}\scriptstyle\mathrm{II}$ region and the underlying extended disk
emission of the galaxy becomes small. As a result, the detected extragalactic
$\textrm{H}\scriptstyle\mathrm{II}$ regions with radio free-free emission may
be biased towards giant $\textrm{H}\scriptstyle\mathrm{II}$ regions formed by
massive stars, as illustrated by Israel et al. (1975), showing that at better
resolution, the detected $\textrm{H}\scriptstyle\mathrm{II}$ regions at larger
beams break up into groups or chains of smaller clumps. Therefore, the number
of detected extragalactic $\textrm{H}\scriptstyle\mathrm{II}$ regions with
free-free emission would be less than that of our Milky Way, where we see many
individual (small and large) $\textrm{H}\scriptstyle\mathrm{II}$ regions,
which limits our ability to find AME candidates.

(ii) The observations suggest that AME is likely to be a local phenomenon. For
example, recent studies of the spatially resolved AME region show that the
31GHz emission (as a proxy for AME) is related to the local PAH column density
(Arce-Tord et al., 2020), and the “excess” emission at 15 GHz (interpreted as
AME) larger than the “expected” radio synchrotron and free-free emission
varies locally (Battistelli et al., 2015). If the AME is governed by local ISM
conditions on small scales, one expects to see a smaller number of AME sources
in other galaxies than our Milky Way because the same radio beam encompasses a
much larger area ($\gtrsim 100$pc) in nearby galaxies, and the emission in the
beam including $\textrm{H}\scriptstyle\mathrm{II}$ regions is dominated by
free-free emission from the $\textrm{H}\scriptstyle\mathrm{II}$ regions, not
by AME from the surrounding molecular clouds. In addition to the dust
destruction mechanism proposed in this study, the systematic effect in radio
observations makes the detection of extragalactic AME difficult. Therefore, if
detected, the extragalactic AME is likely to be from a large molecular cloud
around a young massive star cluster before a supernova explodes ($<10$Myr),
which also explains why extragalactic AME is not detected very often compared
to our Milky Way, where AME is detected more frequently.

When the ngVLA operates with high angular resolution and sensitivity (Selina
et al., 2018), we will be able to locate individually resolved extragalactic
star-forming regions (at a similar scale as our Milky Way) and find AME
candidates. We can follow up on them with multifrequency observations to
confirm the SED shape of AME, which will enable us to test the AME formation
hypothesis based on large samples.

##  7 Summary

We consider the impact of C-shocks on the rotational grain destruction process
for small nanoparticles emitting AME (Hoang & Tram, 2019) and compute the
emissivity of AME, free-free, and thermal dust continuum emission in the
radio-millimeter wavelength for the typical CNM, WNM, and PDR ISM conditions
surrounding star-forming regions. Our model of the rotational destruction
process for AME suppression is based on a spherical dust shape and a specific
range of the maximum tensile strength to withstand centrifugal stress, which
is very poorly understood. With these caveats in mind, our study can be
summarized as follows.

  1. $\bullet$

The magnetized shock (C-shock) for the typical conditions of CNM, WNM, and PDR
can create a supersonic neutral drift that collides with grains and increases
their spin angular momentum.

  2. $\bullet$

If the spin angular velocity is larger than the critical angular velocity that
is resilient to the centrifugal force for given maximum tensile strength, the
AME emitting grains break up, and the emissivity of AME is suppressed relative
to the others: emissivity of free-free and thermal dust continuum emission.

  3. $\bullet$

If a C-shock destroys the small nanoparticles in the ISM surrounding the star-
forming region, the AME that might be prominent in the early stage of star-
formation ($\lesssim 10$Myr) becomes significantly suppressed and might not be
detectable after $\sim 10$ Myr, when the shock from the supernova explosion
develops and impacts the surrounding ISM in a massive star-forming region,
which might explain the rare observations of extragalactic AME.

  4. $\bullet$

This study suggests that the presence of AME can be indicative of an early
stage of star formation, and the strength of AME depends on the local
conditions of the ISM, which implies that spatially resolved high-resolution
radio observations are required to understand the detailed physics of AME and
its connection to the ISM.

I.Y. is grateful to the referee who provided valuable comments that greatly
improved the current work. I.Y. thanks Bruce Draine for his valuable comment
on the $\textrm{H}\scriptstyle\mathrm{II}$ region and the grain destruction
process and Antoine Gusdorf and Sylvie Cabrit for answering questions
regarding the shock code. I.Y. also thanks Eric Murphy and Sean Linden for
useful conversations regarding the observations of extragalactic AME. I.Y.
acknowledges kind financial support from the National Radio Astronomy
Observatory for the publication of this work. The National Radio Astronomy
Observatory is a facility of the National Science Foundation operated under
cooperative agreement by Associated Universities, Inc.

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