Modeling rotational disruption of grains and microwave emission from spinning dust in AGB envelopes

Radio observations of Asymptotic Giant Branch (AGB) star envelopes frequently show some excess emission at frequencies below 100 GHz which cannot be explained by thermal dust emission (hereafter anomalous microwave emission-AME). Moreover, AGB envelopes are a common place where gas molecules condense to form nanoparticles (e.g., polycyclic aromatic hydrocarbons) and large grains. In this paper, we will study whether electric dipole emission from rapidly spinning nanoparticles can reproduce the AME observed toward AGB stars. To properly model the size distribution of nanoparticles in the AGB envelope, we take into account both the increase of nanoparticles due to rotational disruption of large grains spun-up by radiative torques and the decrease of smallest nanoparticles due to rotational disruption driven by stochastic gas collisions. We then perform detailed modeling of microwave emission from rapidly spinning nanoparticles from both C-rich and O-rich AGB envelopes using the grain size distribution constrained by rotational disruption. We find that spinning dust emission is dominant over thermal dust emission at frequencies below 100 GHz. We attempt to fit the observational data of AME using our spinning dust model and demonstrate that spinning dust can reproduce the observed AME in six AGB stars. Finally, we discuss that microwave emission from spinning dust in AGB envelopes could be observed with high-resolution upcoming radio telescopes such as ngVLA and ALMA Band 1. This would be a major leap for understanding AGB envelopes, formation, evolution, and internal structures of dust. Observations would help to distinguish the carrier of AME from comparing C-rich to O-rich stars, because PAHs are formed in C-rich AGB stars while silicates are formed in O-rich stars.


INTRODUCTION
Late in their evolution, low-and intermediate-mass stars (1-8 M ) reach the Asymptotic Giant Branch (AGB) phase, before they become white dwarfs. During this phase, AGB stars lose most of its material (Lamers & Cassinelli 1999) because radiation pressure accelerates them to speeds above the stars escape velocity. Massloss builds an expanding circumstellar envelope (CSE) around the star, containing dust and gas (e.g., Olofsson et al. 2010;Ramstedt et al. 2011;Cox et al. 2012). CSEs of AGB stars can be considered as the most significant chemical laboratories in the universe because their effec-tive temperatures are usually low (T 2000 K -3500 K), and the mass-loss timescale is long, so that molecules can form in the envelope through chemical and physical processes (e.g., Cernicharo et al. 2000;Tenenbaum et al. 2010).
It is well-known that in the AGB envelopes, simple molecules condense to form complex molecules and then tiny nanoparticles including polycyclic aromatic hydrocarbons (PAHs), and finally to submicron-sized grains before being expelled into the ISM due to radiation pressure. Theoretical studies on the formation of PAHs in the evolved star envelope is well studied (see Cherchneff 2011 and references therein). While observation evidence of dust grains from AGB envelopes is well established thanks to infrared emission from dust grains. However, observational evidence for the existence of PAHs which are demonstrated through mid-IR emission features at 3. 3, 6.2, 7.7, 8.6, 11.3, and 17 µm (Leger & Puget 1984;Allamandola et al. 1985;Smith et al. 2007;Draine & Li 2007) is not yet available (see Tielens 2008 for a review). The underlying reason for that is during the AGB stage, the star temperature is rather low, such that there lacks of UV photons to trigger mid-IR emission of PAHs. One note that prominent PAH features are usually observed at the later stage such as planetary nebula due to higher temperatures of the central star. Therefore, in order to achieve a complete understanding on the formation of dust in AGB star envelopes, it is necessary to seek a new way to observe PAHs/nanoparticles in these environments.
Modern astrophysics establishes that rapidly spinning nanoparticles that have permanent electric dipole moment (e.g., PAHs, nanosilicates, nanoiron particles) can emit electric dipole radiation at microwave frequencies below 100 GHz (Draine & Lazarian 1998;Hoang et al. 2010;Hoang, & Lazarian 2016). The rotational excitation of nanoparticles can be achieved in the absence of UV photons, such that microwave emission from spinning dust can be efficient in the environments without UV photons like AGB envelopes and dense shocked regions . Therefore, microwave emission could be a unique way for us to probe nanoparticles in the envelope of AGB stars. In particular, with upcoming advanced radio telescopes, e.g., ALMA Band 1, the Square Kilometer Array (SKA), and next generation VLA (ngVLA), the potential of observing nanoparticles via spinning dust emission is very promising, which would allow us to have a complete understanding on PAH formation, dust formation, and evolution of solid particles in general.
Interestingly, previous radio observations for AGB envelopes usually show the anomalous microwave emission (AME) at frequencies below 100 GHz that cannot be explained by thermal dust emission, such as from IRC +10216 (Sahai et al. 1989 andMenten et al. 2006), or from VY CMa, IRc +10216, CIT 6 and R Leo (Knapp et al. 1995), or from α Her, IRC +10216, IRC +20370, WX Psc, α Sco, V Hya (Dehaes et al. 2007). To date, the origin of such excess is not well understood. Therefore, the goal of this paper is to perform detailed modeling of spinning dust emission from nanoparticles in AGB stars and explore whether spinning dust mechanism can explain such AME.
To model the emission from spinning dust, we consider the radiation-driven wind model, which starts from dust-formation zone (i.e., ∼ 5-10R * ), where the temperature varying from 1000 K down to 600 K and a total number density varying from 10 10 to 10 8 cm −3 . Furthermore, dust grains are classified by their spectral features and they correspond to a special kind of envelope properties. Silicates, identified at 9.7 and 18 µm are considered as a major feature of oxygen-rich stars (C/0 < 1) (Kwok 2004). In contrast, in carbon-rich stars (C/0 > 1), one expect to observe prominent mid-IR features from PAHs. In this paper, we will model the microwave emission from spinning PAHs for a typical representative of carbon-rich AGB stars (IRC +10216) and from nanosilicates for a typical representative of oxygen-rich AGB stars (IK Tau).
Owing to their large sizes, AGB stars are among the most luminous objects in the sky, with luminosities of order 10 4 L . Thus, the radiation field in their CSEs is remarkable strong. Recently,  showed that a strong radiation field can torque large dust grains up to ∼ 10 9 revolutions per second, at which rate the centrifugal force can disrupt a large grain into smaller ones. In addition,  and  showed that very small grains (referred to nanoparticles whose sizes are lower 10 nm) can be spun up to suprathermal rotation by stochastic torques induced by gas bombardment in a dense and hot medium. As a result, smallest nanoparticles are disrupted into smaller tiny fragments when the centrifugal force excesses the material strength, after which dust particles smaller than 3.5Å are thermally sublimated into the gas phase (Guhathakurta & Draine 1989). These two disruption mechanisms are then taken into account in our model because they can modify the size distribution of dust grains in AGB envelopes.
The structure of this paper is as follows. Section 2 describes the parametric physical model of AGB envelopes. We constrain the upper limit of size of grains by RATD mechanism in Section 3, and the lower limit of size of nanoparticles by mechanical torques in Section 4. In Section 5, we review the spinning dust model, and calculate the SED from spinning dust for two examples of AGB stars. An extended discussion on implications of our results for explaining AME in AGB envelopes in Section 6. A summary of our main results is presented in Section 7.

PHYSICAL MODEL OF AGB ENVELOPES
During the AGB phase, a star loses most of its mass through stellar winds. Very close to the central star, where it is too hot for dust grains to form, the ejection mechanism is not well unified yet. The pulsation process is believed to accelerate gas above the escape velocity, sending it radially outward (e.g., Hoefner & Dorfi 1997, Willson 2000; and see Tram et al. 2018 for review). Farther from the star, where the gas temperature decreases to the condensation temperatures, dust grain form. Subsequently, dust grains are accelerated by radiation pressure and transfer their momentum to gas molecules through collisions. Collisions result in a drag force (Gilman 1972) that allows gas beyond the dust-formation distance to overcome the gravitational well of the star. The resulting winds are called dustdriven or radiation-driven winds. Here we briefly describe our parametric physical model of AGB envelopes for a radiation-driven wind, which will be used as the framework for calculating rotational disruption of dust grains and spinning dust emission.

Gas density profile
LetṀ be the rate of mass loss of an AGB star, and we assume a spherical envelope. The gas density at distance r is then given by (1) where v exp is the expansion velocity of the outflow. The typical mass loss rates areṀ ∼ 1-10 × 10 −5 M yr −1 . For cold AGB stars, we take m gas = 2.33 amu because hydrogen in the CSE around these stars is predominantly molecular (Glassgold & Huggins 1983;Tram et al. 2018). For hot AGB stars, on the contrary, the atomic form of hydrogen dominate the CSE, therefore we take m gas = 1.3 amu (90% H and 10% He). The critical temperature to distinguish cold or hot AGB stars depends on the density of stellar photosphere, e.g., T cri * 2500K for stars with density ≥ 10 14 cm −3 (Tram et al. 2018). The expansion velocity is likely constant for a radiation-driven wind model as demonstrated by Tielens (1983) and Krueger et al. (1994).

Gas Temperatures
In the envelope beginning at the condensation radius (r c ), where the gas expands adiabatically at constant v exp , the radiative heating is dominant because the gas is weakly or not longer shielded by dust so that the temperature drops gradually with the distance, and we approximate its profile as a power law as: (e.g., Mamon et al. 1988;Agúndez & Cernicharo 2006;Decin et al. 2010;Li et al. 2016): where T 0 , r 0 , and α gas are parameters whose values are given in Table 1.

Grain temperature
The grain temperature evolution is determined by the balance between the heating and cooling rates. Grains are heated by collisions with the gas particles and by absorption of stellar or ambient radiation. Grains are cooled by collisional energy transfer and by thermal radiation. The detailed profile of dust temperature in a realistic envelope can be modeled by solving the radiative transfer equation for a given dust opacity (Winters et al. 1994). But we assume here that the stellar radiation dominates. In the simple case when the grain absorption efficiency can be approximated as a powerlaw of waveleght Q abs ∼ λ −s , the dust temperature can be derived as (Habing & Olofsson 2003): where T * and R * are the temperature and radius of star.
Observations suggest s 1.

Physical properties of a C-star and an O-star
We apply the parametric models described above to calculate the physical properties of the freely expanding wind in CSE around the well-known C-rich AGB stars (namely IRC +10216) and O-rich (namely IK Tau) as illustrated in Figure 1. For IRC +10216, the profile of gas temperature is taken from Mamon et al. (1988), who fitted the power-law to the theoretical results of Kwan & Linke (1982). For IK Tau, we similarly fit the powerlaw to the gas temperature profile that is derived from observational 12 CO lines . Other parameters are listed in Table 1 with the references given.

Drift velocity for nanoparticles
Once dust grains have formed, they scatter and absorb stellar photons, leading to a radiative force that pushes them outwards away from the star. The radiation force on a grain of radius a at distance r due to the central star is given by where c is the speed of light, Q rp is the radiation pressure efficiency, andQ rp is the wavelength averaged ra-  diation pressure efficiency weighted by the stellar spectrum, and L is the total stellar luminosity. Let v drift be the drift velocity of grains through the gas. If the drift speed is much larger than the thermal speed of gas particles, the drag force per unit volume of the gas is F drag (a) = πa 2 ρv 2 drift n gas , where a is the radius of a single grain. 1 On the other hand, if the drift speed is lower than the thermal speed of the gas, the drag force F drag (a) = πa 2 ρc s v drift n gas . To combine those limits, we can express the drag force as follows: Because the mean free path of the gas is higher than the typical dust radii, and the velocities of gas and dust are different, the grains are not position coupled to the gas. Despite the fact that the grains collide with only a small References. (1)  fraction of the gas particles, Gilman (1972) indicates that the subsequent collisions among the gas molecules allow the momentum that they receive from the radiation field to be transferred to the gas. Gilman (1972) also demonstrates that the small grains rapidly reach the terminal drift velocity. The grains move at the terminal drift velocity when the radiation force balances the drag force: where c s = (2k B T gas /m gas ) 1/2 is the sound speed in the gas. We define the dimensionless drifting parameter s d = v drift /c s . As a limitation, if the thermal speed is much smaller than speed of outflow and the grain size is much smaller than wavelength,Q rp ∝ a and therefore s d (r) ∼ v drift (r) ∝ a 0.5 , which means the drift between gas and big dust grains is larger than that of small dust grains.
In Figure 2, we show the normalized grain drift velocity v drift /v th calculated for the different grain sizes. The drift velocity s d tends to increase with increasing the distance because of the decrease of the gas temperature. Large grains can be accelerated to supersonic motion, while smallest nanoparticles of size a 1 nm are mostly moving at subsonic speeds.   In the AGB and post-AGB stages, the star is relatively cool, with T ∼ 2000 − 3000 K. Therefore, its ionizing radiation is negligible, and the gas remains mostly neutral. In the stage of PN, the star has contracted to a white dwarf with high temperature, and the envelope is ionized. Nanoparticles in the outer envelope will be affected by interstellar UV photons which also ionize gas atoms. The most abundant species in C-rich and O-rich stars are listed in Table 2 with references.

Radiation field from an AGB star
In this paper, we are considering the radiative-driven wind, which should start from the dust condensation zone. Normally, this zone is located quite far from the central star (e.g., 5R * for the case of IRC +10216). Thus we can neglect the angle-dependence of radiation field to dust grain particles. Therefore, the radiation energy density from an AGB star at a distance r is: where L 4 = L * /10 4 L , and τ is the total optical depth. The averaged interstellar radiation field (ISRF) from other Milky Way stars has energy density u ISRF = 8.64 × 10 −13 erg cm −3 (Mathis et al. 1983). The relative strength of the radiation field from the AGB star is then For IRC +10216 at 5R * , U 2 × 10 9 , whereas at 10 4 R * is U 3 × 10 2 (neglecting extinction).

Rotational disruption by radiative torques
As revealed in Hoang et al. (2019), a strong radiation field can spin large grains up to a maximum angular velocity: where I is the grain inertial moment, Γ RAT is the radiative torque (Draine & Weingartner 1996;Lazarian & Hoang 2007;Hoang & Lazarian 2008;Herranen et al. 2019), and τ damp is the total damping timescale.
The averaged radiative torque applied on a grain of irregular shape with effective size a is (see  and reference theirin): where γ is the degree of anisotropy of the radiation field (0 ≤ γ ≤ 1),λ is the average wavelength of the radiation field (i.e.,λ = 2.42µm for IRC +10216 and λ = 2.53µm for IK Tau). Above, the radiative torque efficiency averaged over the radiation spectrum is approx-imatelyQ Γ 2(λ/a) −2.7 for a ≤λ/1.8, andQ Γ ∼ 0.4 for a >λ/1.8. However, we adopt the maximum of grain size to be 0.25µm as deduced from observations (Mathis et al. 1977). Therefore, we just consider a ≤λ/1.8 in this work.
The total rotational damping of dust grains consists of collisional damping due to collisions with gas species (Jones & Spitzer 1967) and the IR damping due to IR emission, and the characteristic damping time is described by (see : where the characteristic timescale of collisional damping is with n H = n(H) + 2n(H 2 ) being the proton number density, and the dimensionless IR damping coefficient

IRC +10216
Species abundances Note Species abundances Note Glassgold & Huggins (1983), Tram et al. (2018); (2) Agúndez & Cernicharo (2006); (3) Millar et al. (2000); (4) Li et al. (2016) where a −5 = a/(10 −5 cm), andρ = ρ/(3 g cm −3 ). Here we have assumed that dust grains are in the thermal equilibrium established by radiative heating of starlight and radiative cooling by IR emission. Plugging Γ RAT and τ damp into Equation (9), yields the maximum rotation rate spun-up by RATs: with U 6 = U/10 6 , n 5 = n H /10 5 cm −3 , T 2 = T gas /100K, andλ 2.42 =λ/2.42µm. Equation (14) reveals that the rotational rate induced by RATs depends on the term U/n H T 1/2 gas and F IR . Thus, grains at 154R * in CSE of IRC +10216, where U = 10 6 , can be spun-up to ω RAT ∼ 10 9 rad s −1 . However, when a grain of size of a is rotating at an angular velocity ω, it develops a centrifugal stress due to the centrifugal force which scales as S = ρa 2 ω 2 /4 . Then, if the rotation rate increases to a critical limit such that the stress induced by centrifugal force exceeds the tensile strength of the material, grains are disrupted instantaneously. The critical angular velocity for the disruption is given by: where S max is the maximum tensile strength of dust material and S max,10 = S max /(10 10 erg cm −3 ). 2 The exact value of S max depends on the dust grain composition and structure. Compact grains can have higher S max than porous grains. Ideal material without impurity, such as diamond, can have S max ≥ 10 11 erg cm −3 . Burke & Silk (1974) suggested that S max ∼ 10 9 − 10 10 erg cm −3 for polycrystalline bulk solids (see also Draine & Salpeter 1979). Composite grains as suggested by Mathis & Whiffen (1989) would have much lower strength. In Draine & Salpeter (1979), the value S max = 5 × 10 9 erg cm −3 is taken for small graphite grains. In the following, nanoparticles with S max 10 10 erg cm −3 are referred to as strong materials, and those with S max < 10 10 erg cm −3 are called weak materials.
For each location in the AGB envelope with given n H , T gas , and U , one can obtain the critical grain size of rotational disruption by radiative torques by setting ω RAT ≡ ω cri . Figure 3 shows the disruption size of dust grains by radiative torques with respect to distance from C-rich star IRC +10216 and O-rich star IK Tau, assuming the different tensile strengths. Very close to the star, rotational damping by gas collisions is very efficient because n H 1 and T gas 1 and dominates over IR damping (i.e., F IR 1), resulting in a very short damping timescale (τ damp ∼ τ gas 1 yr). As a result, grains lose their angular momentum rapidly and cannot be spun up to critical rotation rates, and large grains can survive rotational disruption. Farther away from the star, the rotational damping rate is decreased due to the rapid decrease of n H and T gas with distance r, large grains can be spun-up to extremely fast rotation by radiative torques and are disrupted into small sub-fragments. The disruption size decreases with increasing distance from the star. Finally, when grains are far enough from the star, the collisional damping is subdominant compared to IR damping, i.e., F IR 1 and U 1 and the disruption size of grains can be analytically estimated as (Hoang et

Rotational dynamics of nanoparticles
In dense regions, such as the CSE of AGB stars, Jones & Spitzer (1967) showed that collisions with gas atoms and molecules rotate dust grains. In addition, nanoparticles are directly bombarded by molecules and atoms (including elements heavier than H), and they experience long-distance interactions with passing ions (Draine & Lazarian 1998;Hoang et al. 2010;. Following Jones & Spitzer (1967) and , one can define the dimensionless damping and excitation coefficients for interaction processes with respect to the damping and excitation coefficients of purely hydrogen neutral-grain collisions as: where ∆J r and (∆J ) 2 are the mean decrease of grain angular momentum along the r-axis and the mean increase of rotational energy per unit time by colliding with pure hydrogen, respectively, and ω T is the thermal angular momentum of grains at temperature T gas . The dimensionless damping (F ) and excitation (G) coefficients are defined respectively as: where suffixes sd, IR, n, p, and i stand for gas-grain drift, Infrared, neutral, plasma, and ion, respectively. F sd and G sd are computed as in Roberge et al. (1995), whereas other coefficients are computed as in Draine & Lazarian (1998). Note that F j = G j = 1 for grain collisions with purely atomic hydrogen gas. Figure 4 shows the obtained values of F and G for various processes for PAH and silicates grains. As shown, IR emission is dominant for rotational damping of the smaller grains (< 0.005 µm) and gas-grain drift is dominant for larger grains. Let T rot be the rotational temperature of spinning nanoparticles, so that I ω 2 = 3k B T rot . Thus, using the rms angular velocity from Draine & Lazarian (1998), we obtain where τ H and τ ed are the characteristic damping times due to gas collisions and electric dipole emission (see Draine & Lazarian 1998). As τ ed /τ H ∼ (a/3.5Å) 7 (n H /10 4 cm −3 ) (see Hoang et al. 2010), this ratio is much bigger than 1 as long as n H > 10 5 cm −3 . Therefore, T rot /T gas ∼ G/F , i.e., the rotational temperature is only determined by the rotational damping (F ) and excitation (G) coefficients.
Accordingly, the rotation rate at the rotational temperature T rot is given by  To calculate the smallest size a min that nanoparticles can withstand the rotational disruption, we compute ω 2 using the rotational temperature T rot as given by Equation (21) at each radius for a grid of grain sizes from 0.35 − 10 nm and compare it with ω cri . Figure 5 shows the obtained minimum size a min as a function of radius in CSE for different values of S max and two AGB models. Strong nanoparticles can survive (green line), while weak nanoparticles can be destroyed (blue and orange lines). The disruption size drops gradually with distance from central star due to the rapid decrease of gas temperature with distance.

SPINNING DUST EMISSION
The emission from spinning dust is detailed in . In this section, we briefly recall the principles of this model.

Size distribution of nanoparticles modified by rotational disruption
In the case without rotational disruption, we assume that dust grains in AGB envelopes consist of two pop- ulations, very small grains (nanoparticles) and larger grains, which we refer to as the original dust populations. In the presence of rotational disruption, smallest nanoparticles are removed by mechanical torques, whereas large grains are disrupted into smaller fragments which comprise nanoparticles, corresponding to the modified dust population. The size distribution of these modified dust populations is unknown, and below we adopt a simplified strategy to model their size distributions.
Since disruption by mechanical torques occurs at the lower end of the size distribution and disruption by radiative torques occur at the high end, we can account for these disruption effects separately. Indeed, the sizes of original nanoparticles can be assumed to follow a lognormal size distribution (Li & Draine 2001): with a 0 = 3 − 6Å, σ = 0.3 − 0.6 the model parame-ters (see Tabel 1 in Li & Draine 2001), and B constant determined by a 0 , σ and Y X (fraction of total silicate (X = sil) or carbon (X = C) abundance contained in grains (see Eq 51 in . The effect of rotational disruption by mechanical torques is then to increase the lower cutoff of the log-normal size distribution from a min = 3.5Å to a cri (see Figure 5).
To model the effect of rotational disruption by RATs, we assume that both original large grains and nanoparticles produced by disruption follow a power-law distribution with slope η: where A is a normalization constant determined by the dust-to-gas mass ratio (M d/g ) as: with a max the upper constraint of grain size (Fig. 3), and a min the lower cutoff of grain sizes (Fig. 5).
For example, using a typical value of M d/g = 0.01, η = −3.5, a min = 50Å, and a max = 0.25µm we estimate A 10 −25.16 cm −2.5 for carbonaceous grains (as known as the MRN distribution, Mathis et al. 1977). With these parameters, the contribution of the powerlaw distribution is negligible compared to the log-normal size distribution (see e.g., Draine & Lazarian 1998).
In the presence of RATD, large grains are disrupted into smaller ones, so we can extend the power-law to a min = 3.5Å. Hence, the size distribution (Eq. 24) is modified such that the slope η becomes steeper, and the contribution of the power-law becomes more important. Therefore, the net size distribution of nanoparticles is given by Since the slope η is unknown, in the following, we will consider several values for η. In this work we adopt the standard dust-to-gas mass ratio M d/g = 0.01, even though this ratio is likely to vary in practice (e.

Spinning dust emissivity
At any location in the CSE, the emissivity j a ν (µ, T rot ) from a spinning nanoparticle of size a is: where P (ω, µ) is the power emitted by a rotating dipole moment µ at angular velocity ω given by the Larmor formula, with θ the angle between spin vector ω and moment vector µ. Assuming a uniform distribution of θ, then sin 2 θ = 2/3. The dipole moment µ 2 86.5(β/0.4D) 2 a 3 −7 D 2 for PAHs, and µ 2 66.8(β/0.4D) 2 a 3 −7 D 2 for nanosilicates, in which β is the dipole per atom . For PAHs, β 0.4D, while nanosilicates are expected to have a large dipole moment depending on selection of molecules (see Table 1 in . Therefore, we set β as a free parameter when calculating spinning emission for nanosilicates grains. The grain angular velocity f MW can be appropriately described by a Maxwellian distribution in high-density conditions where collisional excitations dominate rotation of nanoparticles: The rotational emissivity per H nucleon is obtained by integrating over the grain size distribution (see Hoang et al. 2011): where dn/da is the size-distribution for PAHs and nanosilicates (see Section 5.1).

Emission spectrum of spinning dust
Assuming CSEs are spherically symmetric, which is justified in the outer region (e.g., see Figure 1 in Sahai & Chronopoulos (2010)), the spectral flux density of spinning dust is: where D is the distance from the AGB star to the observer (see Table 1). Figure 6 shows the emission spectrum of spinning PAHs in a C-rich star IRC +10216 (top panel) and nanosilicates in O-rich star IK Tau (bottom panel) for different values of material strengths. For strong grains (e.g., S max = 10 10 erg cm −3 ) for which rotational disruption does not occur, the spinning dust emissivity is strong and can peak at high frequencies (see dashed lines). For weaker grains with rotational disruption, both the peak emissivity and peak frequency are reduced significantly because the smallest (and fastest-spinning) nanoparticles are destroyed into molecule clusters (see dashed-dotted and solid lines).
The spectrum of spinning dust emission depends on both the dipole moment and the size distribution. The top panel of Figure 7 shows the increase of the emission flux from spinning nanosilicate grains with increasing the dipole moment because the power emitted by a rotating dipole moment is proportional to β 2 . The peak frequency is insensitive to the β value because in the dense envelopes, electric dipole damping is subdominant compared to gas damping. The bottom panel of Figure 7 shows the dependence of spectrum on the slope of size-distribution, η. One can see that the peak emission flux tends to increase with increasing the magnitude of slope, but the peak frequency does not change. Note that for the dust in the standard ISM, η = −3.5 (Mathis et al. 1977). For circumstellar dust, Dominik et al. (1989) estimated this index to be ∼ −5. The steeper size distribution might be explained by the enhancement of small particles owing to the disruption of large grains due to RATD mechanism. Therefore the constant A of the grain-size distribution increases for steeper slope (see Eq. 25) and results in an increment of flux.
Above, we calculated the emission spectrum of spin- ning dust by using the same value of S max for both large grains and small grains. In reality, the structure of small grains, however, would be more compact and that should cause a higher value of S max than for the large ones. Figure 8 shows the effect of the size-dependent tensile strength on the emission spectrum of spinning grains, in which we assume S u max = 10 7 erg cm −3 for the upper constraint on grains sizes by RATD, and S l max = 10 9 erg cm −3 for the lower constraint on those by mechanical torques. In this case, the emission flux has higher amplitude (solid line) than the cases of constant S max = 10 9 erg cm −3 . The underlying reason is owing to the enhancement of nanoparticles because weaker large grains (i.e., low value of S max ) are more easy to be disrupted by RATD. Moreover, comparing to the case of the same S l max (dashed dotted line), one sees that the peak frequency does not change because the disruption effect of smallest nanoparticles by mechanical torques are the same in both cases. So, the peak frequency is We have studied the rotational disruption of large grains spun-up by radiative torques using the theory developed by . As shown in Figure  3, owing to their extremely fast rotation, large grains are being disrupted into smaller fragments, including nanoparticles, which places a constraint on the upper limit of the grain size distribution in CSEs around AGB stars. The efficiency of rotational disruption is different for different locations along the radial trajectory of stellar winds (see Sec. 3). In addition, weak grains (i.e., tensile strength S max 10 9 erg cm −3 ) are quite easily disrupted, while the disruption of stronger grains (i.e., tensile strength S max 10 10 erg cm −3 ) is less efficient.

Removal of nanoparticles due to rotational disruption by mechanical torques
Due to stochastic collisions with neutral and ionized gas, plasma drag, and infrared emission, nanoparticles tend to rotate thermally/subthermally ( Draine & Lazarian 1998;Hoang et al. 2010). Nevertheless, due to their small sizes (inertia moment), nanoparticles can rotate extremely fast, at rates more than 10 10 rad s −1 (see Eq. 22). Subject to a supersonic gas flow induced by shocks  or radiation pressure as shown in this paper, nanoparticles can be spun-up to suprathermal rotation, resulting in the disruption of smallest nanoparticles into molecule clusters because the centrifugal stress exceeds the maximum tensile strength of nanoparticles. This mechanism places a contraint on the lower limit of the grain- size distribution. As shown in Figure 5, the disruption of nanoparticles is strongest near the central star and rapidly decreases outward. Moreover, nanoparticles of strong materials are hardly disrupted by mechanical torques in the AGB envelopes.
6.3. Can spinning dust explain excess microwave emission from AGB envelopes?
The early detection of cm-wave observations toward to the AGB stars, i.e., at 15 GHz (or 2 cm) and 20 GHz (or 1.5 cm) from IRC+10216 (Sahai et al. 1989), and at 8.4 GHz (or 3.57 cm) from 4 AGB stars over 21 samples (Knapp et al. 1995) cannot be explained by thermal dust emission. Recently, Dehaes et al. (2007) presented the SED observations from a large sample of O-rich and C-rich AGB stars envelopes and showed emission excess at cm wavelengths for many stars, including some post-AGB and supergiants with circumstellar shells. The authors divided them into two groups. Group I could be fitted successfully with thermal dust. Group II, on the other hand, is being fitted very well at optical and IR bands, but not being reproduced at cm-mm bands, including: IRC +10216, α Her, IRC +20370, WX Psc, α SCo, and V Hya (with central peak at 100 GHz); and AFGL 1922, IRAS 15194-5115 (with central peak at 300 GHz).
As shown in Section 5, the emission of spinning dust is strong and dominant over the thermal emission at the frequency 100 GHz. Let us now use our model calculated for IRC +10216 and IK Tau to fit with the observational data of two C-rich and O-rich AGB stars, respectively. To fit the data, we vary three parameters S max , β and η while fixing other physical parameters until we obtain the best-fit models. Figure 9 shows our best-fit models to observational data for three Crich (upper panel) and three O-rich (lower panel) stars with given the corresponding set of dim parameters in the caption. For thermal dust emission, we adopt the best-fit models from Dehaes et al. (2007), which were modelled by DUSTY (Ivezic et al. 1999). Apparently, thermal dust and spinning dust are able to reproduce the mm-cm emission for both C-rich star (top panel, Figure 9) and O-rich stars (bottom panel, Figure 9).
We note that some AGB stars such as AFGL 1922, IRAS 15194-5115 exhibit submm emission excess (i.e., at higher frequencies of 300 GHz). Such submm excess cannot be reproduced by our standard spinning dust model presented here because spinning dust is known to be efficient at microwave frequencies. Very recently,  and  show that spinning dust can be efficient at ν 300 GHz in magnetized shocks where nanoparticles can be spunup to suprathermal rotation by supersonic neutral drift, provided that nanoparticles are strong enough to withstand disruption by centrifugal force. Indeed, termination shocks (or reversed shocks) could occur in CSEs when the stellar wind interact with the surrounding ISM because the terminal velocity of the wind is supersonic (about ∼ 10 − 30km s −1 ). Therefore, we propose that observing both the gas and the dust emission in the AGB envelopes should be crucial in order to have better interpretation to the observable submm emission excess because we could then have a better idea of where the termination shocks occur so that we could take the spinning dust emission from this shock into account.
It is worth to mention that Sahai et al. (2011) suggested that the submm emission excess by means of thermal emission from cold, very large grains (above 1 mm) in post AGB or pre-PN. However, how dust can grow to such big sizes in AGB envelopes is difficult to reconcile, because theoretical calculations show that dust released by AGB outflows is small grains (see, e.g., Jura 1994).

Implications for future observations
Observations would help to distinguish the carrier of AME by comparing C-rich vs. O-rich stars, because PAHs are formed in C-rich AGB stars while silicates are formed in O-rich stars. The interferometers like ALMA and the VLA are capable of mapping the gas distribution around AGB stars and polarization observations to constrain the dust grain properties in the shell (Khouri et al. 2018;Brunner et al. 2019). Such investigations are few in number, but they could help understand dust nucleation and growth in wind-driven AGB stars. Figure 9 in this paper presents the dust models fitted to available low-resolution radio observations of AGB stars. High resolution dust continuum observations from ALMA in bands 3, 4, and 5 at ∼ 84, 125 and 163 GHz, respectively, can improve the constraints on fitted model parameter much better than done for available low-resolution observations. Additionally, carbon rich stars such as IRC +10216 can be observed for different transitions of CO molecules (e.g., Cernicharo et al. 2015). The observations covering the ring structure will map the CO gas emission and hence the CSE of the star. Simultaneously, dust continuum observations at comparable resolution will be useful to map the dust shell and hence spatially correlating it with the gas emissions. Multiband observations of AGB stars from VLA with frequencies from 1-40 GHz may help in understanding the dust distribution by investigating the fluxes obtained in these frequencies and fitting SEDs. VLA plays an important role in covering the regimes of frequencies less than 40 GHz (see Figure 9) at much higher resolution than that of available data in literature. Therefore, a combination of VLA and ALMA dust continuum observations in some of the bright AGB shells will help to pin down the dust characteristics using our dust models.
VLA and ALMA also provide unique opportunities to map high resolution dust continuum and line polarization. We can investigate whether wind-swept shell region of these ABG stars is polarized. The polarization measurements in the envelope of AGB stars will help in investigation the dust-grain properties such as size and alignment efficiency. 6.5. Toward constraining internal structure of dust grains with microwave emission By modeling rotational disruption of grains and resulting microwave emission from spinning dust, we find that microwave emission has strong correlation with the tensile strength of grain materials. Both the peak flux and peak frequency tend to increase with increasing the tensile strength (see Figure 7). This can be a powerful constraint on the internal structure of newly formed dust grains in AGB envelopes, which is still a mystery in dust astrophysics. Recently, Hoang (2019) suggested that the upper cutoff of the grain size distribution in the ISM can be constrained by rotational disruption.

SUMMARY
We have studied rotational disruption of dust grains by radiative and mechanical torques in the AGB envelopes, performed detailed modeling of microwave emission from rapidly spinning nanoparticles, and applied the models to explain the observed excess microwave emission. The principal results are summarized as follows: 1 We model the rotational disruption of large grains by centrifugal stress induced by radiative torques from central star. We find that large grains (e.g., a > 0.1 µm) made of weak materials (tensile strength S max 10 9 erg cm −3 can be disrupted into nanoparticles within a radius of 10 16 cm from the star.
2 We also study the disruption of nanoparticles by centrifugal stress due to stochastic collisions of grains with supersonic gas flow driven by radiation pressure and find that smallest nanoparticles of weak materials located close to the star can be destroyed.
3 We model microwave emission from spinning PAHs and silicate nanoparticles in C-rich and Orich envelopes. We find that due to the radial dependence of the gas temperature, the spinning dust emits over a wide range of microwave frequencies.
4 We found that microwave emission from either spinning PAHs or spinning nanosilicates can dominate over thermal dust at frequencies ν < 100 GHz in the AGB envelopes.
5 By fitting the spinning dust to observed data from mm-cm wavelengths, we find that AME observed in AGB envelopes can be successfully reproduced by microwave emission from carbonaceous or silicate nanoparticles.
6 Thanks to the correlation of spinning dust flux with the grain tensile strength, we suggest that internal structure of newly formed dust in AGB envelopes can be probe with microwave emission observations.