Spinning dust emission from circumstellar disks and its role in excess microwave emission
Electric dipole emission from rapidly spinning polycyclic aromatic hydrocarbons (PAHs) is widely believed as an origin of anomalous microwave emission (AME), but recently it encounters a setback due to the non-correlation of AME with PAH abundance seen in a full-sky analysis. Microwave observations for specific regions with well-constrained PAH features would be crucial to test the spinning dust hypothesis. In this paper, we present physical modeling of microwave emission from spinning PAHs from protoplanetary disks (PPDs) around Herbig Ae/Be stars and T-Tauri stars where PAH features are well observed. Guided by the presence of 10 silicate features in some PPDs, we also model microwave emission from spinning nanosilicates. Thermal emission from big dust grains is computed using the Monte Carlo radiative transfer code (radmc-3d; Dullemond et al. 2012). Our numerical results demonstrate that microwave emission from either spinning PAHs or spinning nanosilicates dominates over thermal dust at frequencies GHz, even in the presence of significant grain growth. Finally, we attempt to fit mm-cm observational data with both thermal dust and spinning dust for several disks around Herbig Ae/Be stars that exhibit PAH features and find that spinning dust can successfully reproduce the observed excess microwave emission (EME). Future radio observations with ngVLA, SKA and ALMA Band 1 would be valuable for elucidating the origin of EME and potentially open a new window for probing nanoparticles in circumstellar disks.
Polycyclic aromatic hydrocarbons (PAHs) are an important dust component of the interstellar medium (ISM, see review by Tielens 2008). Following the absorption of ultraviolet (UV) photons, PAH molecules reemit radiation in mid-infrared, producing prominent 3.3, 6.2, 7.7, 8.6, 11.3, and 17 m features (Leger & Puget 1984; Allamandola et al. 1985; Smith et al. 2007; Draine & Li 2007). Rapidly spinning PAHs also emit electric dipole radiation in microwaves via a new mechanism, so-called spinning dust (Draine & Lazarian 1998; Hoang et al. 2010). The latter is the most likely origin of anomalous microwave emission (AME) that contaminates Cosmic Microwave Background (CMB) radiation (Kogut et al. 1996; Leitch et al. 1997; Planck Collaboration et al. 2011; Planck Collaboration et al. 2016).
PAH molecules appear to be a natural carrier of the AME (Draine & Lazarian 1998; Planck Collaboration et al. 2011) because it is an established component of interstellar dust (Draine et al., 2007). Yet such an explanation recently faces a setback due to no correlation between the observed AME and PAH abundance based on a full-sky analysis by Hensley et al. (2016). Due to the spatial variation of PAH properties (e.g., geometry, size, and electric dipole moment), it is rather challenging to achieve a robust constraint for the carrier of AME by means of the full-sky analysis (see Dickinson et al. 2018 for a review). Therefore, observations of AME from specific regions with well-constrained PAH properties are critical to elucidate the exact carrier of AME.
PAH molecules are usually detected in circumstellar disks around Herbig Ae/Be stars and some T-Tauri stars (Habart et al. 2004; Seok & Li 2017). Their presence in PPDs is puzzling because one expects that interstellar PAHs are already depleted in dense cores due to coagulation. Thus, PAH molecules in PPDs may be newly formed particles as a result of dynamical processes, such as desorption of PAHs from the grain surface due to stellar radiation heating, replenishment due to collisions of planetesimals. PAHs can be formed in PPDs near the high temperature and density regions (see Kamp 2011). Interestingly, if PAHs are produced by fragmentation of big carbonaceous grains, then, we expect a population of silicate nanoparticles that can also be produced by fragmentation of big silicate grains. Note that the modeling of very small grains (hereafter VSGs) around Herbig Ae/Be stars is previously studied in Natta et al. (1993). Alternatively, nanoparticles (including PAHs and nanosilicates) may follow a different evolution from classical grains (size of m). Thus, while classical dust grains are depleted in the disk due to coagulation and settling, PAHs/VSGs that are well mixed to the gas can exempt from grain settling and coagulation, and turbulence can be responsible for the mixing (see Dullemond et al. 2007).
Modern understanding of AME indicates that, in addition to spinning PAHs, rapidly spinning silicate nanoparticles can successfully reproduce the observed AME in the diffuse ISM (Hoang et al. 2016; Hensley & Draine 2017). Spinning iron nanoparticles cannot reproduce the entirety of the observed AME (Hoang et al. 2016). Although, the presence of nanosilicates in the ISM remains a hypothesis, in contrast to PAHs, an analysis by Li & Draine (2001) shows that the fraction of total Si abundance (Si/H) contained in ultrasmall grains, denoted by , can reach without violating the observational constraints of the UV starlight extinction and mid-infrared (IR) emission. Hoang et al. 2016 found that their emission and UV absorption do not violate the observational constraints for . As PAHs, we expect nanosilicates are present in PPDs as a component of dust evolution model (see Natta et al. 2007). Indeed, Seok & Li (2017) found strong emission Si-O features by nanosilicates in 40 out of 61 circumstellar disks (cf. Keller et al. 2008).
Rafikov (2006) carried out a simple modeling of microwave emission by spinning PAHs for the fiducial disks around T-Tauri () and Herbig A/Be stars (). Assuming a one-dimensional disk structure, the author found that spinning dust emission dominates over thermal dust emission for GHz.
Radio observations of circumstellar disks often show excess emission at microwave frequencies, i.e., GHz, above what is extrapolated from thermal dust emission at sub(mm) wavelengths, which we term excess microwave emission (hereafter EME).111EME is different from AME in the sense that the latter is the excess emission left after removing all three known galactic emission components, including thermal emission, free-emission, and synchrotron. For instance, Very Large Array (VLA) observations by Calvet et al. (2002) and Natta et al. (2004) reveal EME at 7 mm (or GHz) from the disk around T-Tauri star, TW Hya, whereas Wilner et al. (2005) found excess emission at 3.5 cm (or 9 GHz). The authors explained EME by thermal emission from very big grains (i.e., cm-sized grains). EME is also detected in circumstellar disks around Herbig Ae/Be stars (Skinner et al. 1993; Dent et al. 2006; Sandell et al. 2011). Although thermal dust from very big grains as well as free-free emission from winds are believed to be responsible for such excess emission, the exact mechanism is still unclear. Very recently, Ubach et al. (2017) found EME in 11 disks around T-Tauri stars and suggested that multiple mechanisms different from thermal dust may be responsible for EME. To better understand the origin of EME, we will explore whether spinning dust could reproduce the observed EME.
With high resolution and low frequencies, next-generation VLA (ngVLA), ALMA Band 1, and SKA would be useful for observing spinning dust emission from circumstellar disks around T-Tauri and Herbig A/Be stars (Di Francesco et al. 2013; Scaife 2013). Radio observations by SKA and ALMA Band 1 would be crucial to study grain growth from mm to cm-sized pebbles as a first step of planet formation (Testi et al., 2015).
To provide a more realistic predictions for future observations, in this paper, we will improve modeling of microwave emission from Rafikov (2006) by (1) treating the realistic geometry (i.e., two-dimensional) of disks, (2) finding the dust grain temperature using the publicly available 3D Monte Carlo radiative transfer code (radmc-3d; Dullemond et al. 2012),222The code and user guide are available at http://www.ita.uni-heidelberg.de/ dullemond/software/radmc-3d/. (3) considering both emission from disk interior and surface layers, and (4) accounting for microwave emission from spinning nanosilicates. We also perform modeling of thermal dust emission with grain growth to 10 cm, in order to quantify the simultaneous effect of grain growth and spinning dust on the spectral energy density (SED).
The structure of this paper is as follows. Section 2 describes the physical model of circumstellar disks. In Section 3, we review the spinning dust model, and calculate the excitation coefficients for the disk conditions, which demonstrate that the collisions dominate both damping and excitation. Section 4 presents dust opacity calculated for the different grain size distribution and thermal emission. Section 5 describes the SED from spinning dust for a wide range of disks. An extended discussion on implications of our results and especially an explanation of EME from circumstellar disks in terms of spinning dust are presented in Section 6. A summary of our main results is presented in Section 7.
2 Circumstellar Disk Model
2.1 Gas density profile
This section briefly describes the disk model adopted for our modeling of spinning dust emission. We adopt a flared, radiative equilibrium disk model from Chiang & Goldreich (1997) (see also Dullemond et al. 2001). The schematic model of a protoplanetary disk is shown in Figure 1. PAHs and nanoparticles (i.e., VSGs) are assumed to be well mixed with the gas, thus present in the entire disk.
The total mass surface density at disk radius is given by
where is the model constant, and is the mass density at . When the total surface density at is given as , then, we have (see Appendix A.2).
Assuming a Gaussian vertical profile, the gas density at radius for the hydrostatic disk model (Lynden-Bell & Pringle 1974) reads
where the pressure height scale is described by
where is the aspect ratio at the reference radius . For AU, is taken to be 0.1 as a fiducial model, which corresponds to at AU. Although the chosen aspect ratio is much lower than predicted by Chiang & Goldreich (1997), it is comparable to observations (Avenhaus et al. 2018).
The typical value is adopted. Other physical parameters, including , are listed in Table 1.
2.2 Gas and Dust temperatures
Following the popular model of protoplanetary disks (Chiang & Goldreich 1997), the surface layer is defined by a path of optical depth . At distance from the star, the surface layer is heated to a high temperature by stellar radiation. Subsequent collisions with gas atoms result in gas heating. Dust grains in these superheated layers reemit radiation in IR that in turn heats gas and dust in the disk interior to a temperature . For an isothermal disk, gas and dust are in thermal equilibrium, so that .
In our paper, instead of using the simplified temperature profile as in Rafikov (2006), we directly compute the dust temperature for the realistic disk model using radmc-3d. The dust opacity is calculated assuming a power grain size, with different values of for silicate grains.
Figure 2 shows the gas density and temperature for a fiducial disk around Herbig Ae/Be stars. The obtained dust temperature depends on because the opacity .
2.3 Gas ionization and charge of PAHs/VSGs
Gas in PPDs can be ionized by X-rays, far-UV photons, cosmic rays (see Perez-Becker & Chiang 2011 and ref therein). Theoretical estimates provide the hydrogen ionization fraction for the surface layer and for the disk interior (Perez-Becker & Chiang, 2011).
The ionization of PAHs/VSGs can be approximately described by a three-layer vertical structure model (Visser et al. 2007). In the surface layers, PAHs/VSGs are positively charged due to photoelectric emission induced by stellar UV photons. In the intermediate region, PAHs/VSGs are mostly neutral, reflecting the balance of photoelectric emission and electron captures, and PAHs/VSGs are negatively charged in the diskplane due to electron collisions and the lack of UV photons (see Kamp 2011; also Maaskant et al. 2014). Nevertheless, as shown in the next section, the effect of PAH and nanoparticles charge states is not important for grain rotation in very dense environments as PPDs.
3 Spinning dust model
3.1 Electric dipole moment and emission power
The rotational emission mechanism is built upon the assumption that nanoparticles own non-zero electric dipole moments. PAH molecules can acquire intrinsic dipole moments due to polar bonds (see Draine & Lazarian 1998). The attachment of SiO and SiC molecules to the grain surface gives rise to the electric dipole moment for nanosilicates (Hoang et al. 2016).
Let be the total number of atoms in a nanoparticle of effective size that is defined as the radius of an equivalent sphere of the same volume. Assuming PAHs with a typical structure C:H= having mean mass per atom amu, one obtains with , for the mass density (Draine & Lazarian 1998). Assuming nanosilicate with a structure SiOMgFe having amu, one has for (Hoang et al., 2016).
Let be the dipole moment per atom in the grain. Assuming that dipoles have a random orientation distribution, the intrinsic dipole moment of the grain can be estimated using the random walk formula:
for PAHs, and for nanosilicates (Hoang et al., 2016).
The power emitted by a rotating dipole moment at angular velocity is given by the Larmor formula:
where is the angle between ! and ¯. Assuming an uniform distribution of the dipole orientation, , then, is replaced by .
3.2 Rotational damping and excitation coefficients
Rotational damping and excitation for nanoparticles, in general, arise from collisions between the grain and gaseous atoms (neutrals and ions) followed by the evaporation of atoms/molecules from the grain surface, absorption of starlight and IR emission (Draine & Lazarian 1998; Hoang et al. 2010). Moreover, the distant interaction between the grain electric dipole and electric field induced by passing ions results in an additional effect, namely plasma drag. The rotational damping and excitation for these processes are respectively described by the dimensionless damping coefficient and where denotes the neutral-grain collision, ion-grain collision, plasma drag, and IR emission (see Hoang et al. 2010).
We consider the major neutral components in the PPDs, including H, H, He, and ions H and C. The typical values .
Let be the rotational temperature of spinning nanoparticles, so that . Thus, using the rms angular velocity from Draine & Lazarian (1998), we obtain
where and are the characteristic damping times due to gas collisions and electric dipole emission (see Draine & Lazarian 1998). From Figure 1 we see that the majority of the disk has , which results in (see Hoang et al. 2010). Thus, , i.e., the rotational temperature is only determined by and coefficients.
Figure 3 shows the and coefficients for neutral PAHs at three locations in the diskplane at and . The corresponding gas density is and (see Equations 2). The radiation intensity factor is and (see Eq. A1). A typical ionization fraction is chosen. In all three locations, collisional interactions with neutral dominate the damping and excitation. In Figure 4, we show the results for negative charged PAHs. Collisions still dominate the interaction, such that the evaporation is , leading to the detailed balance with .
In the disk interior with anion PAHs, the ionization fraction is too low, i.e, , ion excitation and plasma drag are not important, as shown in Figure 3. In the surface layer with higher ionization fraction (), rotational excitation by neutral-positively charged PAH can be efficient. Nevertheless, the mass of the surface layer is very low compared to the total disk mass, such that the ionization effect has little impact on the net spinning dust emission. As a result, in the following, we can adequately assume for modeling spinning dust throughout the disk.
3.3 Angular momentum distribution function
In high-density conditions where collisional excitations dominate rotation of nanoparticles (e.g., in PPDs), the grain angular momentum can be appropriately described by the Maxwellian distribution:
where is the moment of inertia of the spherical nanoparticle of mass density .
3.4 Size distribution: PAHs and nanosilicates
Following Li & Draine (2001), nanoparticles are assumed to follow a log-normal size distribution:
where corresponds to PAHs and nanosilicate composition, and are the model parameters, and is a constant determined by
where with being the fraction of abundance contained in very small sizes and being the solar abundance of element , and is the grain mass per X atom. In our studies, C for PAHs and Si for nanosilicates.
The peak of the mass distribution occurs at . Three parameters determine the size distribution of nanoparticles, including .
In realistic environments, should depend on the local conditions and is a function of the radial distance . However, due to poorly known nanoparticles in the disk, is kept constant in this paper.
3.5 Spinning dust emissivity and emission spectrum
Let be the emissivity from a spinning nanoparticle of size at location in the disk, where in general is a function of the local conditions. Thus,
where is given by Equation (7).
The rotational emissivity per H nucleon is obtained by integrating over the grain size distribution (see Hoang et al. 2011):
where for spinning PAHs and nanosilicates, respectively.
Thus, the total emission luminosity from the disk is given by
where is given by Equation (2). For a disk at distance from the observer, the spectral flux density of spinning dust .
4 Thermal dust emission
4.1 Dust opacity
Let be the absorption efficiency for a grain of radius at frequency . The density of dust grains is given by the grain size distribution . The dust opacity, defined as the total absorption cross-section per unit of dust mass, is given by
where are the lower and upper cutoffs of the size distribution of big grains. Here is chosen, and is varied to account for grain growth.
We compute the absorption cross-section for spherical grains using the Mie theory coded from Bohren & Huffman (1983), assuming the optical constant of amorphous silicate (MgFeSiO) 333http://www.astro.uni-jena.de/Laboratory/OCDB/data/silicate/amorph/pyrmg70.lnk. The opacity is then calculated by Equation (13), assuming a power law of the grain size distribution .
Figure 5 shows the dust opacity for the different values of , assuming the typical value and a more shallow distribution of . The grain growth from 1 cm to 10 cm can increase the opacity at GHz by a factor of 2. The distribution of results in the increase of the opacity at GHz. Note that the opacity at GHz does not increase monotonically with .
4.2 Thermal dust emission
In the case of an isothermal disk along the vertical direction, the spectral flux density of thermal emission from the disk in the optically thin regime can be calculated by (see Rafikov 2006)
where is given by Equation (13).
In the present paper, we directly compute using radmc-3d for the different grain size distributions and . This allows us to relate the SED of thermal dust to the effect of grain growth.
5 Spinning dust emission spectrum from circumstellar disks
5.1 Numerical method and Model Setup
Our modeling strategy is depicted in Figure 6. We adopt the fiducial model of a circumstellar disk around a Herbig Ae/Be star and T-Tauri star, with physical parameters listed in Table 1. For a set of the disk parameters, , we create a disk physical model as described in Section 2 to generate the gas density profile . We then use radmc-3d to calculate for the constructed disk. We consider the different grain size distributions, which have opacity given by Figure 5. Viscous heating and internal heating are not considered. For MC simulations, we use the default value of photon packages. The grid resolutions are , in which spans to , from to , and from to .
At each location with given local physical parameters (), we can calculate the damping and excitation coefficients and to obtain using Equation (6). This process can be simplified by the fact that, in the dense conditions, . The spinning dust emissivity is then calculated using Equation (11). Finally, the energy flux density of spinning dust is calculated by integrating over the symmetric disk as given by Equation (12).
5.2 Microwave emission from spinning PAHs
We first consider the emission from spinning PAHs. The PAH size distribution is varied from to . Here we fix the C abundance contained in PAHs, , to be similar to the diffuse ISM, of (see Draine & Li 2007).444The effect of varying is analogous to spinning nanosilicates, which will be quantified in the next section. The lower and upper cutoff of the PAH size distribution Å and Å.
Figure 7 shows the spectral flux density of spinning PAH emission from both the disk interior and surface layer for a Herbig Ae/Be (upper panels) and T-Tauri (lower panels) disks. Models with smaller PAHs tend to have stronger emission and higher peak frequency.
Figure 8 shows the ratio of spectral flux density from spinning PAHs to thermal dust. Emission from spinning PAHs dominates the thermal dust for frequency GHz for Herbig Ae/Be disk, and GHz for T-Tauri disks.
5.3 Microwave emission from spinning nanosilicates
Nanosilicates are expected to have a larger lower cutoff due to more efficient sublimation, as Hensley & Draine (2017), thus, we adopt Å. We now vary the Si abundance contained in nanoparticles from to , while the size distribution parameters are fixed with Å and . Indeed, the variation of () should produce the similar behavior as in spinning PAH emission because the physics is the same.
In Figure 9, we plot five spectra of the spinning dust from the disk for the different values of . Emission from spinning nanosilicates is as strong as spinning PAHs for , as expected, although the maximum emissivity occurs at a lower frequency because of smaller . Spinning dust flux is increased with increasing . The total emission from spinning PAHs and nanosilicates is much greater than the thermal dust emission at GHz.
Figure 10 shows the ratio of spinning dust to thermal dust flux densities. The spinning dust dominates over the thermal dust for GHz, even with only 1 percent of Si abundance contained in nanoparticles.
5.4 Effect of grain growth on mm-cm thermal emission
To quantify the effect of grain growth on mm-cm thermal emission, in Figure 11, we show the thermal emission for the different spanning 0.1 mm to 10 cm. The variation of thermal emission is noticeable for GHz, but the increase in thermal emission from mm to cm is negligible at GHz.
Figure 12 shows the ratio for the different values of . The increase from mm to cm increases the thermal dust emission significantly, resulting in the reduction of by an order of magnitude. The more shallow size distribution helps to enhance thermal dust emission. However, spinning dust is still dominant at frequencies below 60 GHz. The dashed lines show that even only of Si contained in nanosilicates can still produce substantial microwave emission compared to thermal dust with grain growth at GHz.
6.1 PAHs/Nanoparticles traced by mid-IR emission and implication for spinning dust
PAH molecules are widely detected in circumstellar disks around Herbig Ae/Be stars and some T-Tauri stars (Habart et al. 2004; Seok & Li 2017). The presence of nanosilicates is also demonstrated by m emission features present in many PPDs (Seok & Li 2017).
Recently, modeling works have been done to constrain the physical properties of PAHs. Li & Lunine (2003) inferred the PAH size distribution () by fitting the mid-IR spectrum for the disk around HD 141569A. Seok & Li (2017) derived the PAH size distribution and the total mass of PAHs in about 60 disks around Herbig Ae/Be and T-Tauri stars. The authors found that small PAHs, characterized by , are ubiquitous in PPDs.
If the size distribution of nanoparticles from the shielded region is not different from the surface layer,555Apparently, the PAH parameters describe PAH molecules from the surface layer directly illuminated by UV radiation. Nevertheless, the vertical mixing is efficient due to turbulence (Siebenmorgen & Krügel 2010; Siebenmorgen & Heymann 2012), leading to the frequent circulation of PAHs and nanoparticles between the surface layer and disk interior. as constrained by mid-IR features, then many disks that have small PAHs inferred in Seok & Li (2017) would provide strong spinning dust emission, provided that C abundance in PAHs (see Figure 8). These disks appear to be the most favorable targets for future observations of spinning dust.
6.2 Comparison to previous works
Rafikov (2006) carried out a one-dimensional (1D) modeling of microwave emission from spinning PAHs for the disk interior for fiducial disks around Herbig Ae/Be, T-Tauri, and brown dwarf stars. Rafikov (2006) assumed the thermal rotation (i.e., ) and adopted the standard size distribution of PAHs from the diffuse ISM, with . The gas and dust temperature is assumed to follow an analytical formula as a function of the radial distance. Thermal dust is modeled by a power law with a constant spectral slope , although the slope at GHz is not a simple function of the maximum grain size (see Figure 5).
In this paper, we have performed self-consistent, two-dimensional (2D) modeling of spinning dust emission (including radial and vertical structures), which is combined with Monte Carlo radiative transfer modeling of thermal dust emission using radmc-3d. In this way, we naturally account for spinning dust emission from both the surface layer and disk interior. We considered a variety of PAH size distributions () that captures the inferred distribution from mid-IR emission (see the preceding section). Moreover, we took into account the emission from rapidly spinning silicate nanoparticles (Hoang et al. 2016; Hensley & Draine 2017). We found that microwave emission from nanosilicates could significantly increase the SED at GHz, making the detection more easy than spinning PAHs alone. Previous studies by Hoang et al. (2016) show that Si abundance can reach without violating observational constraints in UV extinction, AME polarization, and IR emission. We find that even can produce the AME by a factor of 10 larger than the thermal dust for GHz.
In particular, the flux density of thermal dust emission from a circumstellar disk is calculated by radmc-3d for the different dust size distributions in the presence of grain growth with spanning from 0.1 mm to 10 cm. The simultaneous modeling of spinning dust and thermal dust with grain growth allows us to quantify the respective contribution to microwave emission by these two mechanisms as a function of the nanoparticle size distribution and maximum values .
6.3 Can spinning dust explain excess microwave emission from circumstellar disks?
6.3.1 Excess microwave emission (EME) from disks
EME is often found in radio observations from circumstellar disks around Herbig Ae/Be stars (Skinner et al. 1993; Meeus et al. 2001; Dent et al. 2006; Sandell et al. 2011; van der Plas et al. 2016), as well as T-Tauri stars (Calvet et al. 2002; Natta et al. 2004; Wilner et al. 2005; Ubach et al. 2012). The popular explanations for such EME include thermal dust emission from cm-sized grains and free-free emission from winds/ jets (see e.g., Ubach et al. 2012). Recently, Ubach et al. (2017) observed the emission excess from 11 disks around T-Tauri stars and suggested that multiple mechanisms should be responsible for EME.
6.3.2 Spinning dust as an origin of EME
Investigating observational data collected from the literature presented in Sandell et al. 2011, one can see that the Herbig Ae/Be disks with prominent EME include R Mon, HD 35187, HD 163296, HD 169142. Interestingly, the three latter disks also exhibit prominent PAH emission (see Seok & Li 2017), while weak PAH emission is observed in the R Mon disk (Verhoeff et al., 2012). The HD 35187 and HD 163296 disks also exhibit strong 9.7 m silicate emission. Thus, we expect some contribution of spinning dust to the observed EME.
To explore whether spinning dust can explain EME from circumstellar disks, we first fit the observational data with a two-component model, including thermal dust and spinning dust. The total flux density is described by:
where and are the frequency and the flux density at the peak of the spinning dust spectrum (see Draine & Hensley 2012), is the thermal emission flux density measured at 100 GHz, and with the spectral slope of the dust opacity. The model parameters include , and . 666Due to the limited observational data at GHz, we fit with a parametric model instead of performing physical modeling of spinning dust because the physical model depends on 10 parameters (e.g., dipole moment, size distribution, and gas density and temperature). A three-component fitting, including thermal dust, spinning dust, and free-free emission is also not feasible due to the same reason.
The goodness of fit to the observed flux, , is measured by , as defined by
where is the data uncertainty at the data point , which is fixed to be of . We infer the best-fit model parameters by minimizing using the Levenberg-Marquart method from a publicly available package lmfit (Newville et al., 2014). We fit to the mm-cm (i.e., GHz) data obtained from Sandell et al. (2011) and Meeus et al. (2001).
In Figure 13, we show our two-component fits to the observational data for four disks around Herbig Ae/Be stars. For R Mon and HD 163296 disks, the required spinning dust flux is mJy, which can easily be reproduced with the low PAH/Si abundance of (see Figure 9). To reproduce the data for HD 163296, it requires spinning dust to peak at GHz, while HD 169142 requires GHz. Incidentally, mid-IR modeling by Seok & Li (2017) find that the HD 169142 disk contains small PAHs (i.e., ), while the HD 163296 disk contains larger PAHs (i.e., ). Such small/large PAHs are predicted to strongly emit microwave emission with a high/low peak frequency (see Figure 7), consistent with the peak frequencies inferred from the model fitting.
Our best-fit thermal dust yields , implying the presence of cm-sized grains in these disks (see Draine 2006). The study of pebbles and planetesimals in PPDs using ALMA Band 1-3 and SKA (see e.g., Testi et al. 2015) would suffer contamination from spinning dust at GHz. Therefore, to achieve a realistic measurement of the thermal dust spectral slope and realistic understanding of planet formation, spinning dust needs to be carefully modeled and separated from the observational data.
6.3.3 On the importance of free-free emission
At microwave frequencies, free-free emission from stellar winds or ionized jets is expected to be important in circumstellar disks. Its emission flux can be described by a power law, , where is the spectral slope. For optically thin region, , but becomes positive and can reach for optically thick regions (Reynolds 1986). With this wide range of values, free-free emission is a leading mechanism to explain the EME (cf., see Ubach et al. 2017). Here, we have also attempted to fit the EME with a model consisting of free-free emission and thermal dust emission. As expected, free-free emission can provide an equally good fit to the observational data as spinning dust. Specifically, the best-fit spectral index is for R Mon, for HD 164192, -0.02 for HD 163296, and -0.1 for HD 35187.
Finally, we should stress that, except R Mon and HD 163296, two other disks (HD 35187 and HD 169142) have insufficient data points below 100 GHz to allow a robust constraint on the actual role of spinning dust for EME. Future multi-frequency observations between 1-60 GHz by SKA, ngVLA, and ALMA Band 1 and 2 (Fuller et al., 2016) would be valuable to differentiate spinning dust and free-free emission as an origin of EME in circumstellar disks. Moreover, polarization observations would be particularly useful because free-free emission is unpolarized. It also can constrain the carriers of AME because the polarization of spinning nanosilicate emission is expected to be higher than spinning PAHs (Hoang & Lazarian, 2016).
6.4 Towards probing nanoparticles in circumstellar disks via spinning dust
PAHs and nanoparticles are expected to play an important role in gas heating, chemistry and dynamics of disks because they contribute the largest surface area for charge carrier and astrochemical activities (see Akimkin et al. 2013). Indeed, nanoparticles characterize the ionization level of the disk interior, which affects the magnetohydrodynamic instability activity and the dead zones (Fleming & Stone 2003). The probe of PAHs through mid-IR emission is limited mostly to the surface region where PAHs are directly exposed to the stellar UV radiation. Therefore, the detection of spinning dust emission in PPDs is not only a smoking-gun for the PAHs as a carrier of AME, but it also provides a new diagnostic for nanoparticles in the entire volume of PPDs.
The non-detection of AME from the disk with prominent PAH features but no silicate emission features would provide a convincing test for spinning PAHs as a carrier of the AME. Similarly, the detection/non-detection of AME from the disks with silicate features would provide a valuable test for the spinning nanosilicates as a carrier of AME.
It is worth to mention that (sub)mm-wavelength observations usually reveal central cavities and gaps in transitional disks (e.g., HD 169142 Fedele et al. 2017). This indicates that significant grain growth has occurred so that its thermal emission is substantially reduced in (sub)mm wavelengths. If the assumption of PAHs/VSGs mixed to the gas is valid, then, we expect to detect spinning dust emission by these nanoparticles from cavities and gaps. Therefore, transitional disks appear to be excellent target to study spinning dust with future high-resolution experiments like ALMA Band 1, ngVLA, SKA. Interestingly, a marginal detection of 33 GHz signal from the intracavity in MWC 758 is recently reported by Casassus et al. (2018), which is suggested to be spinning dust emission.
We studied microwave emission from rapidly spinning nanoparticles from circumstellar disks around Herbig Ae/Be stars and applied to explain the observed excess microwave emission. The principal results are summarized as follows:
We performed a physical, two-dimensional modeling of microwave emission from both rapidly spinning PAHs and spinning nanosilicates in circumstellar disks that include both for the disk interior and surface layers. The dust temperature is numerically computed using the Monte Carlo radiative transfer code (radmc-3d).
We found that microwave emission from either spinning PAHs or spinning nanosilicates can dominate over thermal dust at frequencies GHz in circumstellar disks. The presence of both spinning nanosilicates and PAHs can significantly increase the spectral flux density at GHz. Our obtained results imply that the possibility to detect spinning dust emission in PPDs is much higher than previously thought.
By simultaneous modeling of spinning dust and thermal dust emission for a physical disk model, we showed that the thermal dust is still much lower than spinning dust at GHz, even the maximum grain size is increased 10 cm. The presence of spinning dust emission would complicate the probe of grain growth and formation of planetesimals using radio observations.
Our two-component (thermal dust and spinning dust) model fitting to the mm-cm observational data of several Herbig Ae/Be disks (R Mon, HD 163296, HD 35187, and HD 169142) reveal that spinning dust can reproduce excess microwave emission from the disks. Future multi-frequency observations by ALMA, ngVLA, and SKA would be valuable for elucidating the origin of EME as well as AME. Polarization observations would help to distinguish the carriers (PAHs or nanosilicates) of AME. Detection of spinning dust emission in circumstellar disks would open a powerful way to probe nanoparticles and understand its role on disk astrochemistry.
Appendix A Review of Circumstellar Disk Physics
a.1 Stellar radiation
The surface layer has energy density given by
where with is the typical energy density of the diffuse interstellar radiation from Mathis et al. (1983).
a.2 Disk mass and PAH mass
The total gas and dust mass of a disk is estimated as
where has been used. For a fiducial disk of , and , we get , assuming . The dust disk mass is .
The total mass of X nanoparticles (PAHs or nanosil) from both disk and surface layer is evaluated as
where is the average atomic mass of PAHs, is the abundance of C in nanoparticles. For graphene of purely carbon, .
a.3 Thermal dust emission
In addition to spinning emission, the grains thermally heated (by starlight, etc.) in the disk emit thermal emission. The luminosity of emission from the entire disk is equal to
where is the absorption coefficient, is the optical depth along z-direction, and measures the optical depth from to the infinity (Chiang et al. 2001). For an isothermal disk, this integral yields
where at the far-side surface layer.
- Akimkin et al. (2013) Akimkin, V., Zhukovska, S., Wiebe, D., et al. 2013, ApJ, 766, 8
- Allamandola et al. (1985) Allamandola, L. J., Tielens, A. G. G. M., & Barker, J. R. 1985, ApJ, 290, L25
- Avenhaus et al. (2018) Avenhaus, H., Quanz, S. P., Garufi, A., et al. 2018, arXiv:1803.10882
- Bohren & Huffman (1983) Bohren, C. F., & Huffman, D. R. 1983, Absorption and scattering of light by small particles (New York: Wiley)
- Calvet et al. (2002) Calvet, N., D’Alessio, P., Hartmann, L., et al. 2002, ApJ, 568, 1008
- Casassus et al. (2018) Casassus, S., Marino, S., Lyra, W., et al. 2018, arXiv:1805.03023
- Chiang & Goldreich (1997) Chiang, E. I., & Goldreich, P. 1997, ApJ, 490, 368
- Chiang et al. (2001) Chiang, E. I., Joung, M. K., Creech-Eakman, M. J., et al. 2001, ApJ, 547, 1077
- Dent et al. (2006) Dent, W. R. F., Torrelles, J. M., Osorio, M., Calvet, N., & Anglada, G. 2006, Monthly Notices of the Royal Astronomical Society, 365, 1283
- Di Francesco et al. (2013) Di Francesco, J., Johnstone, D., Matthews, B. C., & et al. 2013, arXiv:1310.1604, 74
- Dickinson et al. (2018) Dickinson, C., Ali-Haïmoud, Y., Barr, A., et al. 2018, New Astronomy Reviews, 80, 1
- Draine (2006) Draine, B. T. 2006, ApJ, 636, 1114
- Draine & Hensley (2012) Draine, B. T., & Hensley, B. 2012, ApJ, 757, 103
- Draine & Lazarian (1998) Draine, B. T., & Lazarian, A. 1998, ApJ, 508, 157
- Draine & Li (2007) Draine, B. T., & Li, A. 2007, ApJ, 657, 810
- Draine et al. (2007) Draine, B. T., Dale, D. A., Bendo, G., et al. 2007, ApJ, 663, 866
- Dullemond et al. (2001) Dullemond, C. P., Dominik, C., & Natta, A. 2001, ApJ, 560, 957
- Dullemond et al. (2007) Dullemond, C. P., Henning, T., Visser, R., et al. 2007, A&A, 473, 457
- Dullemond et al. (2012) Dullemond, C. P., Juhasz, A., Pohl, A., et al. 2012, RADMC-3D: A multi-purpose radiative transfer tool, Astrophysics Source Code Library
- Fedele et al. (2017) Fedele, D., Carney, M., Hogerheijde, M. R., et al. 2017, A&A, 600, A72
- Fleming & Stone (2003) Fleming, T., & Stone, J. M. 2003, ApJ, 585, 908
- Fuller et al. (2016) Fuller, G. A., Avison, A., Beltran, M., & et al. 2016, arXiv:1602.02414
- Habart et al. (2004) Habart, E., Natta, A., & Krügel, E. 2004, A&A, 427, 179
- Hensley & Draine (2017) Hensley, B. S., & Draine, B. T. 2017, ApJ, 836, 179
- Hensley et al. (2016) Hensley, B. S., Draine, B. T., & Meisner, A. M. 2016, ApJ, 827, 45
- Hoang et al. (2010) Hoang, T., Draine, B. T., & Lazarian, A. 2010, ApJ, 715, 1462
- Hoang & Lazarian (2016) Hoang, T., & Lazarian, A. 2016, ApJ, 821, 91
- Hoang et al. (2011) Hoang, T., Lazarian, A., & Draine, B. T. 2011, ApJ, 741, 87
- Hoang et al. (2016) Hoang, T., Vinh, N. A., & Quynh Lan, N. 2016, ApJ, 824, 18
- Kamp (2011) Kamp, I. 2011, EAS Publications Series, 46, 271
- Keller et al. (2008) Keller, L. D., Sloan, G. C., Forrest, W. J., et al. 2008, ApJ, 684, 411
- Kogut et al. (1996) Kogut, A., Banday, A. J., Bennett, C. L., et al. 1996, ApJL, 464, L5
- Leger & Puget (1984) Leger, A., & Puget, J.-L. 1984, A&A, 137, L5
- Leitch et al. (1997) Leitch, E. M., Readhead, A. C. S., Pearson, T. J., & Myers, S. T. 1997, ApJL, 486, L23
- Li & Draine (2001) Li, A., & Draine, B. T. 2001, ApJ, 550, L213
- Li & Lunine (2003) Li, A., & Lunine, J. I. 2003, The Astrophysical Journal, 594, 987
- Lynden-Bell & Pringle (1974) Lynden-Bell, D., & Pringle, J. E. 1974, MNRAS, 168, 603
- Maaskant et al. (2014) Maaskant, K. M., Min, M., Waters, L. B. F. M., & Tielens, A. G. G. M. 2014, A&A, 563, A78
- Mathis et al. (1983) Mathis, J. S., Mezger, P. G., & Panagia, N. 1983, A&A, 128, 212
- Meeus et al. (2001) Meeus, G., Waters, L. B. F. M., Bouwman, J., et al. 2001, A&A, 365, 476
- Natta et al. (1993) Natta, A., Prusti, T., & Krügel, E. 1993, A&A, 275, 527
- Natta et al. (2007) Natta, A., Testi, L., Calvet, N., et al. 2007, Protostars and Planets V, 767
- Natta et al. (2004) Natta, A., Testi, L., Neri, R., Shepherd, D. S., & Wilner, D. J. 2004, A&A, 416, 179
- Newville et al. (2014) Newville, M., Stensitzki, T., Allen, D. B., & Ingargiola, A. 2014, LMFIT: Non-Linear Least-Square Minimization and Curve-Fitting for PythonÂ¶
- Perez-Becker & Chiang (2011) Perez-Becker, D., & Chiang, E. 2011, ApJ, 735, 8
- Planck Collaboration et al. (2016) Planck Collaboration, Adam, R., Ade, P. A. R., Aghanim, N., & et al. 2016, Astronomy and Astrophysics, 594, A10
- Planck Collaboration et al. (2011) Planck Collaboration, Ade, P. A. R., Alves, M. I. R., & et al. 2011, A&A, 536, A20
- Rafikov (2006) Rafikov, R. R. 2006, ApJ, 646, 288
- Reynolds (1986) Reynolds, S. P. 1986, ApJ, 304, 713
- Sandell et al. (2011) Sandell, G., Weintraub, D. A., & Hamidouche, M. 2011, The Astrophysical Journal, 727, 26
- Scaife (2013) Scaife, A. M. M. 2013, Advances in Astronomy, 2013, 1
- Seok & Li (2017) Seok, J. Y., & Li, A. 2017, ApJ, 835, 291
- Siebenmorgen & Heymann (2012) Siebenmorgen, R., & Heymann, F. 2012, Astronomy and Astrophysics, 543, 25
- Siebenmorgen & Krügel (2010) Siebenmorgen, R., & Krügel, E. 2010, Astronomy and Astrophysics, 511, 6
- Skinner et al. (1993) Skinner, S. L., Brown, A., & Stewart, R. T. 1993, ApJS, 87, 217
- Smith et al. (2007) Smith, J.-D. T., Draine, B. T., Dale, D. A., & et al. 2007, ApJ, 656, 770
- Testi et al. (2015) Testi, L., Perez, L., Jimenez-Serra, I., et al. 2015, in Proceedings of Advancing Astrophysics with the Square Kilometre Array (AASKA14). 9 -13 June, 117
- Tielens (2008) Tielens, A. G. G. M. 2008, ARA& A, 46, 289
- Ubach et al. (2012) Ubach, C., Maddison, S. T., Wright, C. M., et al. 2012, MNRAS, 425, 3137
- Ubach et al. (2017) Ubach, C., Maddison, S. T., Wright, C. M., et al. 2017, MNRAS, 466, 4083
- van der Plas et al. (2016) van der Plas, G., Wright, C. M., Menard, F., et al. 2016, A&A, 597, A32
- Verhoeff et al. (2012) Verhoeff, A. P., Waters, L. B. F. M., van den Ancker, M. E., et al. 2012, A&A, 538, A101
- Visser et al. (2007) Visser, R., Geers, V. C., Dullemond, C. P., et al. 2007, Astronomy and Astrophysics, 466, 229
- Wilner et al. (2005) Wilner, D. J., D’Alessio, P., Calvet, N., Claussen, M. J., & Hartmann, L. 2005, ApJ, 626, L109