Understanding Polarized Dust Emission from ρ Ophiuchi A in Light of Grain Alignment and Disruption by Radiative Torques

In this work, we use the archival FIR polarimetric data observed by SOFIA/HAWC+. These data sets are introduced in Santos et al. (2019). The observations were made in 2017 using two bands of HAWC+ instrument, namely C (89 μm) and D (154 μm). The angular resolutions are 78 and 136, respectively. The polarization degree maps in these bands are shown in Figures 1(a) and (b). ⁸ We select the common sky positions in which data is detected in both bands. The local polarization degree varies significantly across the ρ Oph-A cloud, in which the median value is 7.5% in band C and 5.0% in band D as discussed in Santos et al. (2019). Figures 1(a) and (b) show a "tight" spatial correlation in polarization degree in two bands except at the southernmost area (T_d ≃ 25 K), where the data at 89 μm is more polarized than at 154 μm. The reason for this difference is beyond this work's scope because we do not have enough information to investigate it quantitatively. However, a possible explanation could be that there is a warmer outer component (with a local temperature larger than 25 K) and a colder inner component (with a local temperature smaller than 25 K) along the line of sight (LOS) as proposed in Santos et al. (2019). In the warmer component, which favors enhancing the shorter wavelength polarization (band C), grains are well exposed to radiation. Therefore the alignment is efficient, causing a larger value of P_C(%). On the contrary, the colder component along the LOS favors emission at longer wavelengths (band D); however, the grain alignment is less effective due to the shielding. Therefore, the value of P_D(%) becomes smaller. A star symbol indicates the high-mass star Oph S1.

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We adopt the dust temperature (T_d) and the gas column density (N) maps of Santos et al. (2019). These maps were generated by a fit of the modified thermal spectral energy distribution (SED) to each pixel using 70, 100, and 160 μm from Herschel/PACS data (Poglitsch et al. 2010) with the fixed exponential index of the dust opacity 1.6. Figures 1(c) and (d) show the gas density and dust temperature maps in the same regions that HAWC+ detected data. The high-mass star Oph S1 warms up the surrounding environment, causing a large temperature gradient, i.e., from ≃45 K near Oph S1 down to ≃20 K at the edge of the cloud. On the contrary, the gas is densest at the edge of the map and radially diffuses backward to Oph S1.

Figure 2 shows the 2D-histogram dust polarization degree in bands C (left panel) and D (right panel) to dust temperature made of 52 bins. They share the same feature (1) the polarization degree increases as the dust temperature increases up to T_d ≃ T_crit (i.e., positive slope region) and (2) the polarization degree decreases for higher dust temperature (i.e., negative slope region). In the positive slope, the polarization degree in band C is higher than that in band D, while it is lower in the negative slope, which was also shown by the fractional polarization ratio of Figure 6(d) in Santos et al. 2019). In other words, the polarization degree at a shorter wavelength (89 μm) is higher than at the longer wavelength (154 μm) in the denser region (i.e., at the edge of the polarimetric map) while it is the opposite in the less dense region (close central star) in ρ Oph-A. Using the RATs theory, the spherical model of Santos et al. (2019) could explain the increase (decrease) of the P_D/P_C ratio with respect to dust temperature (gas column density) in the dense (N > 10^21.75 cm⁻²) and cold region (T_d ≤ 32–34 K) (see their Figure 6). However, this model could not explain the observational trend in the more diffuse and hotter region. We fitted the data with a piecewise linear function. ⁹ The best fits (i.e., black dashed lines) show that the transition takes place at T_crit ≃ 25 K for band C, and ≃32 K for band D. In the statistical point of view, the reason for low T_crit in band C is that there is an excess of the polarization degree at T_d ≃ 25 K as mentioned in Section 2.1, which makes the piecewise linear fit peak toward this value of T_d. In the framework of our theoretical point of view, however, the transition from positive to negative slope at ≃25 K is unlikely physical because the tensile strength of grains must be extremely small (see Section 4). In addition, the polarization ratio P_D/P_C changes its slope at T_d ≃ 32–34 K as shown in Santos et al. (2019).

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Dust grains are heated by the radiation and re-emit in the thermal range. The fractional polarization of the thermal dust emission is the ratio of the polarized intensity (I_pol) to the total emission intensity (I_em), which yields

$$\begin{eqnarray}&&P( \% )=100\times \displaystyle \frac{{I}_{\mathrm{pol}}}{{I}_{\mathrm{em}}}.\end{eqnarray} \tag{ 1 }$$

Assuming a dust environment containing carbonaceous and silicate grains, the total emission intensity is given by

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{I}_{\mathrm{em}}(\lambda )}{{N}_{{\rm{H}}}} & = & \displaystyle \sum _{j=\mathrm{sil},\mathrm{car}}{\displaystyle \int }_{{a}_{\min }}^{{a}_{\max }}{Q}_{\mathrm{ext}}\pi {a}^{2}\\ & & \times \,\displaystyle \int {{dTB}}_{\lambda }({T}_{{\rm{d}}})\displaystyle \frac{{dP}}{{dT}}\displaystyle \frac{1}{{n}_{{\rm{H}}}}\displaystyle \frac{{{dn}}_{j}}{{da}}{da}.\end{array}\end{eqnarray} \tag{ 2 }$$

If silicate and carbon are separated populations, then as paramagnetic grains, silicates can align with the ambient magnetic field, while carbon grains cannot (Hoang & Lazarian 2016). Thus, the polarized intensity resulting from its alignment is given by

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{I}_{\mathrm{pol}}(\lambda )}{{N}_{{\rm{H}}}} & = & {\displaystyle \int }_{{a}_{\min }}^{{a}_{\max }}f(a){Q}_{\mathrm{pol}}\pi {a}^{2}\\ & & \times \,\displaystyle \int {{dTB}}_{\lambda }({T}_{{\rm{d}}})\displaystyle \frac{{dP}}{{dT}}\displaystyle \frac{1}{{n}_{{\rm{H}}}}\displaystyle \frac{{{dn}}_{\mathrm{sil}}}{{da}}{da},\end{array}\end{eqnarray} \tag{ 3 }$$

where B_λ (T_d) is the blackbody radiation at dust temperature T_d, dP/dT is the distribution of dust temperature, f(a) is the alignment function, Q_ext is the extinction coefficient, Q_pol is the polarization coefficient, and dn/da is the grain size distribution. The dust temperature distribution depends on the grain size and radiation strength, which is computed by the DustEM code (Compiègne et al. 2011, see, e.g., Figure 8 in Lee et al. 2020). The extinction and polarization coefficients are computed by the DDSCAT model (Draine & Flatau 1994, 2008; Flatau & Draine 2012) for a prolate spheroidal grain shape with an axial ratio of one-third.

If silicate and carbon grains are mixed together (e.g., Jones et al. 2013), which may exist in dense clouds due to many cycles of photoprocessing, coagulation, shattering, accretion, and erosion, carbon grains could be aligned with the ambient magnetic field and its thermal emission could be polarized. For a simplest case, assuming these grain populations have the same alignment parameters (i.e., a_align, f(a)), the total polarized intensity is

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{I}_{\mathrm{pol}}(\lambda )}{{N}_{{\rm{H}}}} & = & \displaystyle \sum _{j=\mathrm{sil},\mathrm{car}}{\displaystyle \int }_{{a}_{\min }}^{{a}_{\max }}f(a){Q}_{\mathrm{pol}}\pi {a}^{2}\\ & & \times \,\displaystyle \int {{dTB}}_{\lambda }({T}_{{\rm{d}}})\displaystyle \frac{{dP}}{{dT}}\displaystyle \frac{1}{{n}_{{\rm{H}}}}\displaystyle \frac{{{dn}}_{j}}{{da}}{da}.\end{array}\end{eqnarray} \tag{ 4 }$$

Let us consider a radiation field with the energy density of u_rad( erg cm⁻³), the mean wavelength of $\bar{\lambda }$ and an anisotropy degree of γ. Its strength is defined by a dimensionless U = u_rad/u_ISRF, where u_ISRF = 8.64 × 10⁻¹³ erg cm⁻³ is the radiation energy density of the interstellar radiation field (ISRF) in the solar neighborhood (Mathis et al. 1983). This radiation field can spin a dust grain of size a and density ρ up to the rotational rate ¹⁰

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\omega }_{\mathrm{RAT}}}{{\omega }_{{\rm{T}}}} & \simeq & 2.9\times {10}^{2}{\hat{\rho }}^{0.5}\gamma {a}_{-5}^{3.2}U\left(\displaystyle \frac{{10}^{3}{\mathrm{cm}}^{-3}}{{n}_{{\rm{H}}}}\right){\left(\displaystyle \frac{\bar{\lambda }}{0.5\mu {\rm{m}}}\right)}^{-1.7}\\ & & \times \,\left(\displaystyle \frac{20\,{\rm{K}}}{{T}_{\mathrm{gas}}}\right)\left(\displaystyle \frac{1}{1+{F}_{\mathrm{IR}}}\right),\end{array}\end{eqnarray} \tag{ 5 }$$

where a₋₅ = a/(10⁻⁵cm), $\hat{\rho }=\rho /(3{\mathrm{gcm}}^{-3})$, and n_H and T_gas are the gas density and temperature. ${\omega }_{{\rm{T}}}={({k}_{{\rm{B}}}{T}_{\mathrm{gas}}/I)}^{0.5}$ is the thermal angular velocity with I = 8π ρ a⁵/15 the inertia moment of the grain. A rotating grain is damped by gas collisions and IR emission (see Hoang 2019). The dimensionless parameter (F_IR) that describes the ratio of the IR damping to collisional damping ¹¹ is defined as

$$\begin{eqnarray}&&{F}_{\mathrm{IR}}\simeq 0.4\left(\displaystyle \frac{{U}^{2/3}}{{a}_{-5}}\right)\left(\displaystyle \frac{30{\mathrm{cm}}^{-3}}{{n}_{{\rm{H}}}}\right){\left(\displaystyle \frac{100{\rm{K}}}{{T}_{\mathrm{gas}}}\right)}^{1/2}.\end{eqnarray} \tag{ 6 }$$

A grain rotating at rotational velocity ω results in a tensile stress S = ρ ω² a²/4 on the materials making up the grain. Thus, the maximum rotational velocity that a grain can withstand is:

$$\begin{eqnarray}&&{\omega }_{\mathrm{crit}}=\displaystyle \frac{2}{a}{\left(\displaystyle \frac{{S}_{\max }}{\rho }\right)}^{1/2}\simeq \displaystyle \frac{3.6\times {10}^{8}}{{a}_{-5}}{S}_{\max ,7}^{1/2}{\hat{\rho }}^{-1/2},\end{eqnarray} \tag{ 7 }$$

where ${S}_{\max ,7}={S}_{\max }/({10}^{7}\,\mathrm{erg}\,{\mathrm{cm}}^{-3})$.

One can see from Equation (5) that the stronger the radiation field and the larger the grain size, the faster the rotation of the grain. A strong radiation field can thus generate such a fast rotation that the induced stress on large grains can result in a spontaneous disruption. This disruption mechanism is named as RATD and discovered by Hoang et al. (2019). From Equations (5) and (7), we can derive the critical size above which grains are disrupted:

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{a}_{\mathrm{disr}}}{0.1\mu {\rm{m}}}\right)}^{2.7}\simeq 5.1{\gamma }_{0.1}^{-1}{U}^{-1/3}{\bar{\lambda }}_{0.5}^{1.7}{S}_{\max ,7}^{1/2},\end{eqnarray} \tag{ 8 }$$

where ${\bar{\lambda }}_{0.5}=\bar{\lambda }/(0.5\,\mu {\rm{m}})$.

Dust grains are disrupted efficiently (for a greater than a_disr) in stronger radiation fields. The disruption of dust grains by RATD can modify the grain size distribution. Since only the largest grains are affected by the RATD mechanism, RATD determines the upper limit of the size distribution. The disruption is thus expected to enhance more smaller grains, resulting in a steeper grain size distribution than in the standard ISM. In the particular case of ρ Oph-A cloud, Liseau et al. (2015) furthermore showed that the grain size distribution experiences a varying power index across the cloud. In this work, we adopt a power-law grain size distribution assumption for both the original large grains and the smaller grains produced by disruption, with a power-law index β:

$$\begin{eqnarray}&&\displaystyle \frac{1}{{n}_{{\rm{H}}}}\displaystyle \frac{{{dn}}_{\mathrm{sil},\mathrm{car}}}{{da}}={C}_{\mathrm{sil},\mathrm{car}}{a}^{\beta }\ \ \ ({a}_{\min }\leqslant a\leqslant {a}_{\max }),\end{eqnarray} \tag{ 9 }$$

where C_sil and C_car are the normalization constants for silicate and carbonaceous grains, respectively. The smallest grain size is chosen as a_min = 10 Å, while the maximum size is constrained by the RATD mechanism (i.e., a_max = a_disr). The normalization constants are determined through the dust-to-gas mass ratio M_d/g (see Chau Giang & Hoang 2020; Tram et al. 2020) as

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \sum _{j=\mathrm{sil},\mathrm{car}}{C}_{j}{\rho }_{j} & = & \displaystyle \frac{(4+\beta ){M}_{{\rm{d}}/{\rm{g}}}{m}_{\mathrm{gas}}}{\tfrac{4}{3}\pi ({a}_{\max }^{4+\beta }-{a}_{\min }^{4+\beta })}\,\,\,\mathrm{for}\ \beta \ne -4\\ \displaystyle \sum _{j=\mathrm{sil},\mathrm{car}}{C}_{j}{\rho }_{j} & = & \displaystyle \frac{{M}_{{\rm{d}}/{\rm{g}}}{m}_{\mathrm{gas}}}{\tfrac{4}{3}\pi (\mathrm{ln}{a}_{\max }-\mathrm{ln}{a}_{\min })}\,\,\,\mathrm{for}\ \beta =-4,\end{array}\end{eqnarray} \tag{ 10 }$$

where C_sil/C_car is adopted as 1.12 (Draine & Lee 1984), and the dust-to-mass ratio M_d/g is fixed as 0.01 throughout this work. The latter assumption is close to what is derived from X-ray observations ≃0.011–0.0125 (Vuong et al. 2003) or gas tracers ≃0.0114 (Liseau et al. 2015).

An anisotropic radiation field can align dust grains via the RATs mechanism (see Lazarian 2007; Andersson et al. 2015 for reviews). In the unified theory of RATs alignment, grains are first spun-up to suprathermal rotation and then driven to be aligned with the ambient magnetic fields by superparamagnetic relaxation within grains having iron inclusions (Hoang & Lazarian 2016). Therefore, grains are only efficiently aligned when they can rotate suprathermally. This aligned grain size (a_align) is determined by the following condition ω_RAT(a_align) = 3ω_T as in Hoang & Lazarian (2008). From Equation (5), we have:

$$\begin{eqnarray}\begin{array}{rcl}{a}_{\mathrm{align}} & \simeq & 0.024{\hat{\rho }}^{-5/32}{\gamma }^{-5/16}{U}^{-5/16}{\left(\displaystyle \frac{{10}^{3}{\mathrm{cm}}^{-3}}{{n}_{{\rm{H}}}}\right)}^{-5/16}\\ & & \times \,{\left(\displaystyle \frac{\bar{\lambda }}{0.5\mu {\rm{m}}}\right)}^{17/32}{\left(\displaystyle \frac{20{\rm{K}}}{{T}_{\mathrm{gas}}}\right)}^{-5/16}{\left(\displaystyle \frac{1}{1+{F}_{\mathrm{IR}}}\right)}^{-5/16}\,\mu {\rm{m}},\end{array}\end{eqnarray} \tag{ 11 }$$

which implies a_align ∼ 0.02 μm for a dense ISM with γ = 1.0, U = 1, and $\bar{\lambda }=0.3$ μm. In this work, we adopt the alignment function as in Lee et al. (2020):

$$\begin{eqnarray}&&f(a)={f}_{\min }+({f}_{\max }-{f}_{\min })\left\{1-\exp \left[-{\left(\displaystyle \frac{0.5a}{{a}_{\mathrm{align}}}\right)}^{3}\right]\right\}.\end{eqnarray} \tag{ 12 }$$

For those grains with a ≪ a_align, the alignment is minimum as f_min = 10⁻³, while the alignment degree gets maximum f_max for a ≫ a_align. This parametric function agrees with the results obtained from inverse modeling to interstellar polarization data (Draine & Fraisse 2009; Hoang et al. 2014). For the model with only silicate grains aligned, modeling requires f_max = 1, and a mixture model with both carbon and silicate grains aligned requires f_max < 1 (Draine & Fraisse 2009; Guillet et al. 2018).

As discussed in Section 3, the parameters of the model include the gas properties of gas number density (n_H) and gas temperature (T_gas); the dust properties of size (a), shape, internal structure (i.e., tensile strength S_max), and size distribution power index (β); and the ambient properties of radiation field strength U (which is in fact equivalent to the dust temperature T_d), mean wavelength ($\bar{\lambda }$), and an anisotropy degree (γ) of the radiation field. Figure 1(c) shows that the gas is denser at the edge of the polarimetric map area and more diffuse close to the Oph S1 star. We derive the relation between the gas number density and the dust temperature by assuming that the dust temperature linearly decreases from 45 K down to 20 K at the edge of the polarimetric map area, and the gas number density is calculated from a spherical model as in Section 3.5 of Santos et al. (2019). This relation is shown in Figure 4. Throughout this work, we fix the value of the gas temperature as T_gas = 20 K, which is fairly common for dense molecular clouds.

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In a dense molecular cloud, large grains are expected to be present thanks to the coagulation process. We set the initial maximum value of grain size as 1 μm, then the RATD mechanism constrains the actual maximum value. The smallest value for the grain sizes is kept fixed at 10 Å. The internal structure of grains is determined via their tensile strength (e.g., large composite grains have ${S}_{\max }\simeq {10}^{7}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$, stronger grains have a higher value of S_max), which is a free parameter. The grain size distribution could change across the ρ Oph-A cloud (Liseau et al. 2015), thus we vary the power index β as another free parameter. In our model, the local value of the radiation strength is determined by the dust temperature as shown in Figure 1(d) via the relation ${T}_{{\rm{d}}}=16.4{a}_{-5}^{1/15}{U}^{1/6}{\rm{K}}$ (Draine 2011). The mean wavelength is $\bar{\lambda }\simeq 0.3\,\mu {\rm{m}}$ corresponding to a B-like star with T_* ≃ 1.5 × 10⁴ K. The anisotropy degree is γ = 1 for the unidirectional radiation field from a nearby star.

Here, we show the numerical results of the multiwavelength polarization degree of thermal dust emission using RATs theory in two cases: without disruption (namely classical RATs) and with disruption for comparison.

Figure 5 shows the polarization spectra obtained with grain alignment by RATs only (without RATD), computed for several values of the dust temperature with different grain size distributions, i.e., β = −3.5 (left panel) and β = −4.0 (right panel). One can see that (1) the polarization degree proportionally increases as the dust temperature increases, and (2) the polarization degree is lower for lower values of β for the same T_d. The first effect is due to the fact that a higher dust temperature (equivalent to higher radiation strength) causes larger torques acting on dust grains, which decreases the alignment size a_align and then increases the polarization degree of dust emission. Moreover, for a lower β, the dust mass contained in large grains is smaller, decreasing the polarization degree of the thermal dust emission that is dominantly produced by aligned, large grains. This explains the second effect.

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Figure 6 shows the polarization spectra obtained with both grain alignment and disruption by RATs (with RATD), assuming different values of tensile strength, i.e., ${S}_{\max }\,={10}^{7}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$ (left panel) and ${S}_{\max }={10}^{9}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$ (right panel). In the left panel, the low-T_d curves are the same as those in Figure 5 (blue, orange, green, and dashed–dotted red lines). However, differing from Figure 5, the polarization degree decreases as dust temperature increases beyond a critical value (i.e., ≃34 K, the dotted violet and dashed brown lines). Higher S_max leads the disruption to occur at a higher critical dust temperature (i.e., ≃38 K, the dashed brown line). The reason is that dust grains, exposed to strong radiation (indicated by where the dust temperature is high, see Figure 1(d)), can be rotated extremely fast due to strong radiative torques while damping is inefficient (because of a low gas density, Figure 1(c)), resulting in RATD as described in Section 3.2. For T_d lower than the critical temperature, on the contrary, the radiative torques are weaker, and the damping process is more substantial (because gas is denser) so that the RATD cannot occur and thus the results are the same as those for the classical RATs calculations. The disruption leads to a drop in the polarization degree. The critical temperature above which RATD occurs and the level of the decline depend on the internal structure of the grains controlled by S_max. The composite grains (${S}_{\max }={10}^{7}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$) are more easily disrupted, resulting in a significant decrease of the polarization degree (Figure 6, left panel), than for the compact grains (${S}_{\max }={10}^{9}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$) (Figure 6, right panel).

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Figure 7 shows the polarization spectrum for the case of mixed silicate and carbon grains in which both grain populations are aligned by RATs. Similar to Figure 6, the disruption occurs for T_d > 34 K. In this case, the spectrum shows an increase and then a plateau feature, which differs from Figure 6. The reason is that the polarization degree is the ratio of the polarized intensity (I_pol) to the total intensity (I_em) (Equation (1)). Since the T_d of silicate grains is lower than that of carbon grains, their spectrum slopes differ from each other. When only silicate grains are aligned the different spectrum slopes of I_pol to I_em result in a slope in polarization spectrum (see, e.g., Figure 6). When both silicate and carbon grains are aligned, I_pol and I_em differ by a factor of a degree of grain alignment, which results in a flat spectrum. The degree of grain alignment is defined by f_max (Equation (12)). For a combination of carbon and silicate grains, the imperfectly aligned grains (f_max < 1) can reproduce observations (see, e.g., Draine & Fraisse 2009; Guillet et al. 2018). Grains with higher values of f_max (right panel) produce more polarized thermal emission than those with lower values of f_max (left panel).

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Figure 8 shows the polarization degree at 89 μm (left panel) and 154 μm (right panel) with respect to dust temperature. In the case without RATD (dotted lines), the polarization degree first increases rapidly with increasing dust temperature and then slowly changes (as shown in Figure 5). Accounting for RATD, the polarization degree first increases with T_d and then rapidly declines once the dust temperature exceeds a critical value, which depends on the grains' tensile strength as shown in Figure 6. The critical dust temperature is lower for weaker grains (i.e., lower value of S_max) because of an effective disruption, which leads to a deeper decrease of the polarization degree in comparison to stronger grains.

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Above, we assume that all grains have the same tensile strength (S_max). However, the tensile strength of composite grains scales with the radius of monomoers as ${a}_{p}^{-2}$ (see Hoang 2019 for a detailed demonstration), which implies that large grains (comprising many monomers) are more breakable than smaller grains presumably having a compact structure. As an example, we set ${S}_{\max }={10}^{7}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$ for all grains with size a ≥ 0.1 μm, while we set ${S}_{\max }={10}^{9}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$ for smaller grains. The results are shown in Figure 9, black solid lines. The trend of the polarization degree also shows an increase and decrease feature to dust temperature. However, for T_d > T_crit (i.e., denoted by the position B), its amplitude is higher than in the case of fixed ${S}_{\max }$. The reason is as follows. When the RATD does not occur, the polarization is higher for higher dust temperature/stronger radiation. When the dust temperature is just enough for RATD to occur, the disruption mostly effects the largest grains (i.e., low ${S}_{\max }={10}^{7}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$ in this example), so that the curve follows the case of fixed ${S}_{\max }={10}^{7}\,\mathrm{erg}\,{\mathrm{cm}}^{3}$ (e.g., BB1 slope). As the dust temperature increases, the RATD can affect smaller grains (i.e., higher ${S}_{\max }$). Because the decline of polarization is smaller for higher ${S}_{\max }$, there is a short increasing interval in the polarization (see the B1, B2 segment). Finally, once RATD only effects "strong" grains, the trend of the polarization follows the fixed ${S}_{\max }={10}^{9}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$ case, as shown in Figure 8.

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Figure 10 shows the polarization degree of thermal dust as a function of the grain temperature, assuming that both carbon and silicate grains are aligned. The results generally show that the polarization degree drops at the same critical dust temperature as Figure 8 in which only silicate grains are aligned. However, the mixed grain model results in higher polarization, as well as in less decline than in the case of single grains due to the contribution of aligned carbon grains. With the T_d − n_H relation, we varied the value of n_H by 10% but we do not see a significant change (i.e., the correlation coefficient is ≃0.99). However, another relation of n_H to T_d could have a more significant effect.

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Since the T_crit ≃ 30–34 K critical dust temperature above which the polarized thermal dust emission drops for ${S}_{\max }={10}^{6}\mbox{--}{10}^{7}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$ (see Figures 8, 10) is consistent with observations (see Figure 2), Figure 11 shows only the numerical results for ${S}_{\max }={10}^{7}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$ with a variation of the silicate grain size distribution power index β overlaid over the observational data. For illustration, we also show the results from the RATs model without the disruption effect (dashed line). Since RATs theory implies that stronger radiation strength causes more torques acting on grains, which results in higher polarization, then this classical RATs model can only lead to an increase of the dust polarization degree with respect to dust temperature and fails to explain its decrease beyond T_crit dust temperature (as discussed in Section 4.2). When the rotational disruption mechanism is incorporated into RATs (solid lines), the model can reproduce the increase and decrease features from observations. For T_d < T_crit, the disruption does not proceed; hence, the model is exactly as classical RATs, which accounts for the increase of the polarization degree. For T_d ≥ T_crit, on the contrary, the disruption occurs so that large grains are disrupted into many smaller fragments. The enhancement of smaller grains causes a decrease in the polarization degree at these FIR wavelengths.

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Furthermore, different solid lines correspond to different values of the grain size distribution power index β. In the case of silicate grains alone, a simple χ² calculation as in Table 1 shows that it gets minimum at β ≃ −4.0 to observational data at 89 μm, while it is slightly lower as β ≃ −4.1 to 154 μm data. Hence, the slope of the size distribution is steeper than the MRN size distribution for the standard interstellar medium (Mathis et al. 1977), which is evidence of enhancement of small grains by RATD. Note that 154 μm probes different (more embedded) layers of the ρ Oph-A than the 89 μm observations do, thereby the size distribution could be slightly different. Polarimetric data at longer wavelengths (e.g., 850 μm JCMT/SCUBA-2 observations, see Kwon et al. 2018), which trace bigger grain size, are desired to get a more comprehensive picture.

Figure 12 shows the comparison for a mixture of carbon and silicate grains with respect to observations. As shown in Section 4.2, both grain size distribution (β) and degree of alignment (${f}_{\max }$) control the amplitude of the polarization degree, but they do not affect the spectrum trend. We found that the same range of β as in Figure 11 also nicely fits the observational trend with ${f}_{\max }\simeq 0.35$. In this case, the χ² calculation in Table 1 indicates that β ≃ −4.0 and β ≃ −4.1 also result in minimum χ² to observed data at 89 μm and 154 μm, respectively.

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Our model's primary and most sensitive input parameters are the local gas column density and the local dust temperature. The first controls the damping process of the rotating grains, while the second defines the angular rotational rate of grains. The value for the gas column density is derived from a spherical model, whereas the value for the dust temperature is adopted from observations. Therefore, our results contain uncertainties, and we would like to address here the main limitations of our model. First, the adopted value of dust temperature is, in fact, the projection on the plane of the sky, the actual value could be higher than these. Second, the dust temperature and gas density maps are derived from only three FIR bands of Herschel/PACS (60, 100, and 160 μm). The derivation could be more accurate if the (sub)millimeter and radio bands are taken into account as they were in Chuss et al. (2019). However, we expect that accounting for local variations of dust temperature and gas number density could explain the observational scatter, but should not change the trend or our conclusions.

Because our main input parameters are the local values, our prescription will be easy to incorporate into more elaborate models that have better physical treatments for the gas and dust properties, such as 3D radiative dust modeling codes (e.g., Dullemond et al. 2012; Liseau et al. 2015).

Finally, we note that the magnetic field geometry is assumed to not vary along the line of sight toward ρ Oph-A in the modeling. The effect of turbulent magnetic field would reduce the polarization degree predicted, but the trend P(%) versus T_d is not affected. Nevertheless, the inferred magnetic field direction shown in Figure 2 in Santos et al. (2019) indicates the coherent magnetic stream lines in ρ Oph-A. The turbulence, therefore, may occur at a very small scale.