The origin of dust polarization in the Orion Bar

We present in Figure 1 the maps of the flux density in the Orion Bar at each of the 4 HAWC+ bandpasses overlaid with the polarization position angles showing the apparent magnetic field lines (rotating the polarization $E$-vectors by 90^∘), adopting a sampling pattern of 1-2 vectors per beam resolution element. The rectangular grey box shows the location of the Orion Bar and denotes the region discussed in detail in the following Sections. The box’s major axis has a position angle with respect to North of 55^∘, and its dimensions are $90^{\prime\prime}\times 340^{\prime\prime}$, centered on the FIR emission. In the four plots of Figure 1, we conserve the original resolution of the observations. However, later on in the analysis, we do regrid and smooth the Band A, C, and D observations to the Band E gridding and resolution³³3Smoothing polarization and their covariance map must account for the fact that the polarization reference direction frame on the celestial sphere varies over a projected map, and thus within the smoothing 2D-kernel. However, this effect is ignored because of the relatively small size of the HAWC+ maps. Therefore, the Stokes maps and their uncertainty are smoothed independently, and we rebuild the polarization quantities from the smoothed and regridded maps., to ensure accurate and quantitative comparisons across different wavelength (using the reproject python package of astropy; see Appendix B). The magnetic field exhibits a complex morphology, with several sub regions having homogeneous polarization patterns, but different from one another. Indeed, different organized components of the magnetic field co-exist, as seen in the South West of the Bar in the Band C, D and E observations, where two patches of polarized dust emission exhibit orthogonal magnetic field orientations. Band A observations, which have the highest spatial resolution, retrieve several structures in Stokes $I$, not observed in the lower resolution Band C, D and E maps. Different wavelengths can preferentially probe different regions if the distribution of dust temperature is not uniform on the line of sight. If those regions have different magnetic field geometries, they will contribute to polarization angles disparities across wavelengths.

To quantitatively estimate whether any regions of the Bar can exhibit dust polarization consistent with $k$-RATs, Figure 2 presents histograms of polarization position angles ($B$-vectors) in five sub-regions, obtained by splitting the Orion Bar along its major axis. The three central regions experience a stronger radiation field given their higher proximity to the Trapezium cluster. Data at the four wavelengths, all with a gridding and resolution corresponding to those of the Band E, are presented. In each histogram, the orientations of the anisotropic radiation field emanating from $\theta^{1}$ Ori C and pointing to the center of the five boxes are indicated by vertical dot-dashed lines. Then, if a given region exhibits $B$-vector angles similar to the radiation field, those vectors would be consistent with the $k$-RAT mechanism. In each box, and at all four observation wavelength, none of the main peak of the different histogram components correspond to the vertical lines. This indicates that the ambient radiation field direction does not dictate the direction of grain alignment in the Bar. However, two small areas in Figure 2 exhibit $B$-vector polarization angles consistent with the radiation field: the center of box # 3 in the Band E data (toward one equivalent beam surface area), and the bottom left of box # 4 in the Band C, D, and E data (toward one equivalent beam surface area at Band C, two at Band E and D). This latter one corresponds to the cold and dense dust component in the South West of the Bar mentioned above, not recovered by the Band A observations (see below).

Toward the northern and middle regions (boxes # 1, 2, 3) of the Bar, the polarization angles across the four wavelengths are consistent within 20^∘ (based on the differences across wavelength of the peaks of the histogram components). However, the southern region (histogram in box # 4) shows clear differences between the Band A observations, and the observations taken at Band C, D, and E, because the band C, D, and E data have picked up the emission from a dense ($N_{\textrm{H}_{2}}\;\geq\;5\,\times\,10^{22}$ cm^-2) and cold ($T_{\textrm{d}}\;\leq\;30$ K) region, that is visible in total intensity (the $N_{\textrm{H}_{2}}$ and $T_{\textrm{d}}$ maps are from Chuss et al. 2019, see their Figure 3., and the Section III.2 of this paper). In contrast, the emission from the Band A observations, sensitive to warmer dust, corresponds to regions closer to the irradiated front of $T_{\textrm{d}}\;\geq\;40$ K. The $B$-vectors in box # 4 of Band A are mainly parallel to the minor axis of the Bar, which is also a pattern of emission seen with the three others wavelengths, alongside this component of colder dust. The polarization angles of box # 5 are consistent across the Band C, D, and E observations within 10^∘. Finally, the polarization angles exhibited by the HAWC+ Band E observations are also consistent with the 850 $\mu$m dust polarization observations obtained with POL-2 on the JCMT (Ward-Thompson et al., 2017; Pattle et al., 2017). The standard deviation of the distribution of polarization angles differences between the HAWC+ Band E and POL-2 data is $\sim$ 20^∘, similar to what is obtained among the HAWC+ bands (see Figure 3).

In order to go further in quantitatively comparing polarization angles across wavelength, Figure 3 presents comparisons (as pair-wise differences) of the polarization angles between the various pairs of observations at different wavelengths, using the OTFMAP mode maps, and the C2N mode data presented in Chuss et al. (2019). The distributions peak around 0^∘ and the standard deviations are within 15-25^∘, while $\sigma_{\phi}\,\leq\,5^{\circ}$ in the regions of Bar where SNR($I$)$\;\geq\;200$, SNR($P$)$\;\geq\;5$. Comparing the difference between the mean and the standard deviation of the polarization angle difference distributions, it is not possible to separate them with statistical significance, even if we increase the SNR criteria on the polarized intensity and Stokes $I$, in which case we also deselect too many data points. The distribution of the polarization difference between Band A and Band C, and between Band A and Band E, show a significant number of points outside the central peak of the distribution, corresponding to typical difference of +30 ^∘ and -30 ^∘. Those data points come from the south-east side of the Bar, where the SNR of Band A detections are the lowest. However, if we weight the polarization difference by their relative uncertainty (i.e., computing $(\phi_{\lambda_{1}}-\phi_{\lambda_{2}})/\sqrt{(\delta\phi_{\lambda_{1}}^{2}+\delta\phi_{\lambda_{2}}^{2})}$), we find that the histograms using the Band A still exhibit larger standard deviation compared to the others. We thus suspect those data points may be artifacts from the data reduction.

Figure 4 presents the variation of Stokes $I$, $P$, $\phi$, and $\mathcal{P_{\rm{frac}}}$ for the four wavelengths OTFMAP observations as a function of the position along the minor axis of the Bar (i.e., the minor axis of the gray rectangle used as a reference so far that is shown in Figure 1), which is directed toward the irradiation front. We highlight that the data points of Figure 4 are SNR-selected independently across bands. The goal is to determine how important the effects of the environmental conditions are for the emission. Indeed, our region of interest, defined by the rectangle highlighted in Figure 1, shows a clear dust temperature gradient (see Chuss et al. 2019). The peak in Stokes $I$ and $P$ for Band A is clearly closer by 15^′′ to the irradiation front compared to the Stokes $I$ and $P$ peaks of Band C, D, and E observations, which is likely a dust temperature effect. As suggested above, Band A is more sensitive to the hot dust layer located at the irradiated edge of the Bar, compared to the dust emission emanating from the colder region not directly exposed to the irradiation of the Trapezium cluster.

The $B$-vector map exhibits several components, which explains why the variations of the mean polarization angle across the Bar in Figure 4 shows significant discrepancies between the bands. The Band A observations at Band E resolution do not have a large number of independent polarization detections. This explains the marked rotations at the beginning and end of the profile of $\phi$ as a function of position across the Bar for this band, which are not statistically significant. Band C data, having a much higher signal to noise than the other bands, trace more reliably the underlying polarized dust emission emanating from the more tenuous outer part of the Bar, and/or from the background OMC1 cloud. Band D and E polarization angle profiles also show significant differences (up to 80^∘ between the mean polarization angles) compared to Band A and C for positions $\leq$ 20^′′ from the cold side of the PDR. Again, lack of uniform detections toward these faint regions explains the retrieved discrepancies. Finally, the polarization fraction profiles follow opposite trends compared to the total intensity profiles. Effects due to both dust temperature and gas density must play a role in the resulting $\mathcal{P_{\rm{frac}}}$ profiles. We will investigate these in Section III.2.

In this data analysis, no variations of the polarization position angles as a function of wavelength are detected with statistical significance. In all our observations, the polarization angles analyzed alongside the radiation field direction along the Bar show no evidence for $k$-RAT aligned grains. However, the analysis of the dust polarization observational data alone is limited by the complexity of the apparent magnetic field lines structure, the observational wavelengths probing different environmental conditions across the Orion Bar, and the observation data quality.

The effects of the radiation field and the gas density govern the radiative torques and the efficiency of de-alignment by gaseous collisions, respectively. This in turn regulates the polarization level of the dust emission. We use the work of Chuss et al. (2019) (see also Arab et al. 2012), who gathered HAWC+, Herschel PACS & SPIRE (André et al., 2010; Abergel, 2010; André, 2011; Bendo et al., 2013), JCMT/SCUBA-2 (Mairs et al., 2016), GBT and VLA (Dicker et al., 2009) observations of the Orion nebula and fitted modified blackbody spectra for each pixel. These fits yielded column density, dust temperature and emissivity maps with 18.7^′′ angular resolution and a 3.7^′′ square pixel size. To ensure statistical independency, we regrid these maps to a Nyquist sampled map of 4 pixels per beam area, resulting in a pixel size of 8.3^′′. We then smooth and regrid the maps of the four bands of HAWC+ observations to the resolution and pixel sampling of the dust temperature and gas column density maps, i.e., to 18.7^′′ angular resolution with 8.3^′′ square pixel size.

We compare our polarization results with three physical quantities describing the local environmental conditions: the line of sight dust temperature & column density, and the derived gas column density between the Bar and the Trapezium cluster $N_{\textrm{H}_{2},}\;{}_{\textrm{from}}\;{}_{\,\theta^{1}\,\textrm{Ori}\,\textrm{C}}$. That is, the column density “seen” towards the Trapezium by each location we map in the Orion Bar. The dust temperature will serve as a proxy for the radiation field intensity, a major parameter that govern the efficiency of radiative torques, and we use $N_{\textrm{H}_{2}}\times\sqrt{T_{\textrm{d}}}$ as a proxy for the efficiency of de-alignment by gaseous collisions because the gaseous collisional rate is proportional to this quantity (Draine & Weingartner, 1996).

In order to compute $N_{\textrm{H}_{2},}\;{}_{\textrm{from}}\;{}_{\,\theta^{1}\,\textrm{Ori}\,\textrm{C}}$, we first estimate the gas column density of the Bar itself $N_{\textrm{H}_{2}}$, by subtracting 2$\times$10²¹ cm^-2 from the derived column densities, corresponding to the background cloud OMC1 based on the SED fitting of the maps. We then estimate the gas volume density in the Bar $n_{\rm{H}}$ by dividing $N_{\textrm{H}_{2}}$ by 0.28 pc, the size of the Bar along the LOS derived by Salgado et al. (2016), who used an estimation of the absorbing area at the surface of the PDR. We finally sum the gas density values along the lines separating each pixel in the Bar from the Trapezium cluster to derive $N_{\textrm{H}_{2},}\;{}_{\textrm{from}}\;{}_{\,\theta^{1}\,\textrm{Ori}\,\textrm{C}}$.

Figure 5 shows the two-dimensional histogram of the gas column density after the correction for the background OMC1 material as a function of dust temperature toward the Orion Bar. The gas column density is inversely proportional to the dust temperature for the complete range of $T_{\textrm{d}}$ values probed here. We note that Chuss et al. (2019) performed Markov Chain Monte Carlo tests to evaluate the systematic covariance between the SED fitting parameters (i.e., $T_{\textrm{d}}$, $N_{\textrm{H}_{2}}$, and $\beta$ the dust emissivity spectral index, see their Section 3.1.5). While there is some covariance between $T_{\textrm{d}}$ and $N_{\textrm{H}_{2}}$, the width of the likelihood functions are such that we can consider that the trends of Figure 5 have a physical origin. The RAT alignment theory predicts that the minimum size of aligned dust grains decreases with increasing radiation field, and with decreasing volume density. We note this point because Tram et al. (2021c) flagged values below a specific dust temperature, below which gas column density was positively proportional to the dust temperature. In our case, we expect the polarization degree to increase (decrease) as a function of $T_{\textrm{d}}$ ($N_{\textrm{H}_{2}}\times\sqrt{T_{\textrm{d}}}$), for the full range of our $T_{\textrm{d}}$ ($N_{\textrm{H}_{2}}\times\sqrt{T_{\textrm{d}}}$) values. A departure from this trend can be explained by an evolution of dust properties as a function of the environmental conditions. In this context, Tram et al. (2021a, b, c) proposed that the decrease of $\mathcal{P_{\rm{frac}}}$ as a function of $T_{\textrm{d}}$ observed at high dust temperature ($T_{\textrm{d}}\,\gtrsim\,40-50$ K) with HAWC+ toward star forming clouds/dense cores can be caused by RATD. In addition, we note that $N_{\textrm{H}_{2}}\times\sqrt{T_{\textrm{d}}}$ and $N_{\textrm{H}_{2},}\;{}_{\textrm{from}}\;{}_{\,\theta^{1}\,\textrm{Ori}\,\textrm{C}}$ are used here to quantify the effect of two different physical quantity on the efficiency of radiative torques. While we use $N_{\textrm{H}_{2}}\times\sqrt{T_{\textrm{d}}}$ as a proxy for the efficiency of collisional gaseous de-alignment, we use $N_{\textrm{H}_{2},}\;{}_{\textrm{from}}\;{}_{\,\theta^{1}\,\textrm{Ori}\,\textrm{C}}$ as a proxy for the reddening of the radiation field emanated from the Trapezium cluster. Because the radiative torque efficiency strongly decreases for grains smaller than the wavelength of impinging photons, the grain alignment efficiency induced by RATs should be affected by the reddening (Lazarian & Hoang, 2007).

The disorganized component of the magnetic field can be a source of depolarization due to cancellation of the polarization signals on the line of sight, as shown by Jones et al. (1992). It can also cause a decrease in the observed polarization when the typical scale of the POS magnetic field fluctuations are smaller or comparable to the spatial resolution. Therefore, alongside the polarization fraction $\mathcal{P_{\rm{frac}}}$, we also consider the dispersion of polarization angles in the POS $\mathcal{S}$, which allows us to quantify the level of disorganization of the apparent magnetic field lines. Loss of grain alignment also directly impacts the fractional polarization. To disentangle between these effects, one needs to analyze $\mathcal{S}$ and $\mathcal{P_{\rm{frac}}}$ together, using the disorganized component of the magnetic field provided by $\mathcal{S}$ to determine whether changes in $\mathcal{P_{\rm{frac}}}$ can be attributed to changes in the grain alignment efficiency. Several studies have revealed that $\mathcal{P_{\rm{frac}}}$ and $\mathcal{S}$ are correlated, showing the role of depolarization due to magnetic field disorganization (Fissel et al., 2016; Planck Collaboration et al., 2020; Chuss et al., 2019; Le Gouellec et al., 2020). We use the nearest neighbors approach to derive $\mathcal{S}$ (Le Gouellec et al., 2020), given the pixel sampling of our maps. To obtain $\mathcal{S}$, we debias the maps of measured dispersion $\mathcal{S}_{m}$ by subtracting its uncertainty: $\mathcal{S}\,=\,\sqrt{\mathcal{S}_{\textrm{m}}^{2}-\sigma_{\mathcal{S}}^{2}}$, following the relation of Alina et al. (2016). We then use $\mathcal{S}\times\mathcal{P_{\rm{frac}}}$ as a proxy for the polarization degree corrected for the depolarization effects induced by the disorganized component of the magnetic field. We use SNR($\mathcal{P_{\rm{frac}}}$)$\,\geq\,$5 and SNR($I$)$\,\geq\,$200 for our data selection criteria.

Figure 6 presents the evolution of $\mathcal{P_{\rm{frac}}}$ (right-hand axes), $\mathcal{S}$, and $\mathcal{S}\times\mathcal{P_{\rm{frac}}}$ (left-hand axes) as functions of $T_{\textrm{d}}$, $N_{\textrm{H}_{2}}\times\sqrt{T_{\textrm{d}}}$, and $N_{\textrm{H}_{2},}\;{}_{\textrm{from}}\;{}_{\,\theta^{1}\,\textrm{Ori}\,\textrm{C}}$, for the four wavelength observations (see Section V.3 for the polarization fraction spectra specifically). With this approach, we want to determine whether the evolution of polarization degree (tentatively corrected from depolarization caused by the disorganized component of the magnetic field) as a function of environmental conditions suggests varying alignment conditions. Given the limited sensitivity of the Band A observations, and the large dynamic range between the Band A initial resolution and the angular scale at which we smooth the maps to, i.e., 18.7^′′, the corresponding distribution shown in Figure 6 are not conclusive and must be interpreted with caution. However, Band C, D, and E, present clearer systematic trends.

To estimate which grain alignment mechanism is dominating, we derive the different timescales that describe the efficiency of the different phenomena involved in the alignment of dust grains by radiative torques. RATs can be presented as a balance between gaseous de-alignment induced by collisions of gas particles onto dust grains, the precession speed of the grain’s magnetic moment induced by grain rotation (for paramagnetic grains) around the magnetic field, and the efficiency of the radiative torques applied by the anisotropic radiation field impinging onto grains. We neglect the infrared emission damping and plasma drag effects (Draine & Lazarian, 1998). The efficiency of these three processes can be compared to one another from comparing the relevant characteristic timescales, i.e., the collisional gaseous damping timescale $\tau_{\rm{gas}}$, the Larmor precession timescale $\tau_{\rm{Larmor}}$, and the radiative precession time-scale $\tau_{\rm{rad}}$ (Lazarian & Hoang, 2007; Tazaki et al., 2017). Ultimately, the minimum size of aligned grains $a_{\textrm{align}}$ denotes the grain size over which dust grains can be considered aligned (Hoang & Lazarian, 2008).

The collisional gaseous damping time of the grain is estimated by:

where $v_{\textrm{th}}\,=\,\sqrt{2k_{\textrm{B}}T_{\textrm{gas}}/\mu m_{\textrm{H}}}$ is the thermal velocity of gas atoms of mass $m_{\rm H}$, $n_{\rm H}$ is the gas density, $T_{\textrm{gas}}$ is the gas temperature (we will use $T_{\textrm{d}}$ as a proxy for $T_{\textrm{gas}}$, which is not necessarily true in PDRs; see Koumpia et al. 2015), $I_{1}$ is the principal grain moment of inertia (which scales as $\rho a^{5}$, where $\rho$ the dust grain density), $a$ is the grain effective size, and $\mu$ is the mean molecular weight per hydrogen molecule.

The Larmor precession timescale, describing the precession of the magnetized grain’s angular momentum around an external magnetic field $B$ resulting from the interaction of the grain magnetic moment with $B$ is given by:

where $\hat{B}\,=\,B/5\,\mu$G is the magnetic field, $\hat{\rho}\,=\,\rho/3\,$g cm^-3, $\hat{s}\,=\,s/0.5$ the grains’ aspect ratio, and $\hat{\chi}\,=\,\chi(0)/10^{-4}$ the grains’ paramagnetic zero-frequency susceptibility. In our calculations we fix the magnetic field strength to $B\;=\;200\;\mu$G, given the results of Chuss et al. (2019); Guerra et al. (2021). We adopt $\rho\,=\,3$ g cm^-3 and s = 0.5 (Hildebrand & Dragovan, 1995). The zero-frequency susceptibility of a dust grain depends on its paramagneticity. For a super-paramagnetic grain, we have from Morrish (2001):

where $\phi_{\rm{sp}}$ is the fraction of atoms that are super-paramagnetic, and $N_{\rm{cl}}$ is the number of atoms per cluster. The GEMS measurements derived values $\phi_{\rm{sp}}$ = 0.03 (Bradley, 1994; Martin, 1995; Goodman & Whittet, 1995), and $N_{\rm{cl}}$ is expected to be $N_{\rm{cl}}$ = 10³-10⁵ (Kneller & Luborsky, 1963; Jones & Spitzer, 1967). Lastly, we consider an ordinary paramagnetic grain, with:

where $f_{p}$ is the fraction of atoms in the grain that are paramagnetic, evaluated at 10% (Tazaki et al., 2017). Results obtained from modeling and observations of dust polarization of the diffuse ISM and dense cores tend to favor the scenario where the efficiency of RATs can only be reproduced if grains’ magnetic relaxation is sufficiently fast, i.e., for super paramagnetic grains (Hoang & Lazarian, 2016; Reissl et al., 2020; Le Gouellec et al., 2020). We also note that H₂ formation occurring at the grains’ surface provide additional torque that also increase the grain rotational velocity, eventually bringing the grain to suprathermal rotation (Purcell, 1979; Hoang et al., 2015), especially toward PDRs (Le Bourlot et al., 2012; Andersson et al., 2013; Soam et al., 2021). These both effects of grains’ super-paramagneticity and H₂ formation supplemental torque increase the fraction of grains with high angular momentum, i.e., the fraction of grains at the the so-called high-$J$ attractor point (Lazarian & Hoang, 2007; Hoang & Lazarian, 2009b, 2016). While grains at high-$J$ can be considered perfectly aligned, grains with a low angular momentum (i.e., at the the so-called low-$J$ attractor point) would be poorly aligned and produced, exhibiting a Rayleigh reduction factor of $\sim$ 0.1 (Hoang & Lazarian, 2016). The relative fraction of grains at high-$J$, i.e., $f_{\textrm{high-J}}$, would increase with increasing grains’ magnetic susceptibility and magnetic field strength, and decreasing grain size, gas density and temperature (Chau Giang et al., 2022).

Finally, the radiative precession time-scale $\tau_{\rm{rad}}$ is given by:

where $a_{-5}\,=\,a/10^{-5}$ cm, $\hat{T}_{\textrm{d}}\,=\,T_{\textrm{d}}/100$ K, $u_{\rm{rad}}$ is the radiation field intensity, $\bar{\gamma}$ is the anisotropy of the mean radiation field (equation from Tazaki et al. 2017), $\bar{\mathcal{Q}}_{\Gamma}(\bar{\lambda})$ is the RAT efficiency (see the relation 10 in Hoang et al. 2021), and $\bar{\lambda}\,=\,\frac{\int^{\infty}_{0}\lambda u_{\rm{rad}}d\lambda}{u_{\rm{rad}}}$ is the mean wavelength of the radiation field. We estimate the radiation field using the relation from Draine (2011):

To estimate $\bar{\lambda}$, i.e., the reddening of the radiation field, we use the relation from Hoang et al. (2021), who, using an analytical model, derives the mean wavelength $\bar{\lambda}$ as a function of the blackbody temperature of a star and the gas column density integrated between the star and the region where $\bar{\lambda}$ is estimated. We use T${}_{\textrm{eff}}\,=\,$39 000 K (corresponding to $\theta^{1}$ Ori C; Simón-D´ıaz et al. 2006), and the $N_{\textrm{H}_{2},}\;{}_{\textrm{from}}\;{}_{\,\theta^{1}\,\textrm{Ori}\,\textrm{C}}$ value derived above with the SED fitting performed in Chuss et al. (2019) to estimate the reddening of the radiation field $\bar{\lambda}$ in every pixel of the Bar.

Finally, we note that line-of-sight integration of the background OMC1 cloud must affect the results from the SED fitting that we use, especially for the determination of the dust temperature. In addition, the coarse resolution of this SED modeling, i.e., 18.7^′′, precludes us from resolving the hotter layer of dust. This is more efficiently traced by the mid-infrared photometry study performed with FORCAST in Salgado et al. 2016, where a cold and a warm modified blackbody component were used. Therefore, the grain alignment timescale analysis does not probe physical scales corresponding to this hot dust layer where $T_{\textrm{d}}$ $\,\geq\,$100 K. However, the Orion Bar is resolved in its minor axis such that we can establish a clear gradient in dust temperature with a range of 35$-$75 K and in projected column density with a range of 10${}^{21}-2\,\times\,10^{22}$ cm^-2.

Equation 5 is only valid for grains aligned at low-$J$. High-$J$ aligned grains are most likely be aligned with the magnetic field, because the spin-up effect of RATs dominate over the radiative precession (Hoang et al., 2022). We will thus compare the radiative precession timescale with the Larmor precession timescale of ordinary paramagnetic grains.

Figure 7 presents the grain alignment conditions of the Orion Bar, showing the different timescales discussed above. The Bar is divided into 3$\times$5 sub regions, i.e., 5 along its major axis, and 3 along its minor axis (i.e., each of the five boxes displayed in Figure 2 are simply separated in three sub-boxes along the minor axis of the Bar). Dust grains larger than the typical size at which $\tau_{\rm{Larmor}}\;>\;\tau_{\rm{rad}}$ can be considered as aligned at low-$J$ with the radiation field (Lazarian & Hoang, 2007). This transitional grain size is referred to as $a_{\textrm{trans}}$ hereafter. Toward the region probed by the box #2, corresponding to the highest irradiation conditions that we probe, $a_{\textrm{trans}}$ ranges from $\sim$ 0.1 to 0.5 $\mu$m. In the southern section of the region in box #5, corresponding to the coldest and densest conditions that we probe, $a_{\textrm{trans}}$ would be $\sim$ 1 $\mu$m.

The evolution of the transitional $a_{\textrm{trans}}$ values as a function of the proximity to the irradiated front is shown within the 5 boxes distributed along the major axis of the Bar in Figure 8. The southern boxes #4 and 5 exhibit on average higher values of $a_{\textrm{trans}}$ and $a_{\textrm{align}}$ compared to boxes #1$-$3. These latter ones must correspond to most of the radiation absorbing area at the surface of the PDR.

The constraints on the grain alignment timescales obtained above determine which grains are potentially aligned with the radiation field, depending on its strength. However, aligned dust grains subject to high irradiation can also trigger the so-called RAdiative Torque Disruption (RATD; Hoang et al. 2019) mechanism, which causes in the disruption of grains into fragments, occurring when the grain rotational energy induced by radiative torques exceed the grain cohesion forces. Comparing the angular velocity at which dust grains are rotationally disrupted with the rotation speed induced by RATs, allows us to constrain which grain sizes are affected by the RATD mechanism. Grains subject to RATD would thus be those with high angular momentum, i.e., at high-$J$. Given the two regimes of radiative torque efficiency, i.e., when $a\;>\;\bar{\lambda}$ or $a\;<\;\bar{\lambda}$, it is possible to define the grain size interval $[a_{\rm{disr}}\,;\,a_{\rm{disr,max}}]$, inside which the aligned dust grains are rotationally disrupted. From Hoang et al. (2021), we have :

where $S_{\rm{max}}$ is the tensile strength of dust grains, $F_{\rm{IR}}$ is the rotational damping coefficient due to the emission of infrared photons emitted by the grain which reduces the grain’s angular momentum (see Draine & Lazarian 1998; we adopt the relation of Hoang et al. 2021). The tensile strength of interstellar dust is uncertain, because it depends on both the grain structure, i.e., compact versus composite, and the grain composition. Dust grain evolution in the ISM is ruled by fragmentation and coagulation processes, as well as the formation of ice mantles. Dust grains with a core-mantle structure (Desert et al., 1990; Jones et al., 1990), and dust composite models composed of silicate and carbon grain aggregates (Mathis & Whiffen, 1989; Zubko et al., 2004; Köhler et al., 2015; Draine & Hensley, 2021), have been proposed. We explore the range $10^{5}-10^{9}\,\rm{erg}\,\rm{cm}^{-3}$ for $S_{\rm{max}}$ in order to consider both the case of large aggregates, and the case of compact silicate core grains (Hoang, 2019). Figures 7 and 8 show $[a_{\rm{disr}}$ and $a_{\rm{disr,max}}]$ for the two extreme values of $S_{\rm{max}}$ that we consider here.

We find that for the most irradiated region of the Bar (box #1, 2, and 3 columns in Figure 7), the RATD mechanism can disrupt aggregates of tensile strength $S_{\rm{max}}\;=\;10^{5}\,\rm{erg}\,\rm{cm}^{-3}$ and size $a\;\gtrsim$ 0.01 $\mu$m toward the northern side exposed to the irradiation (top-row panels), and grains $\gtrsim$ 0.1 $\mu$m toward the southern, colder regions (bottom-row panels). Compact grains of tensile strength $S_{\rm{max}}\;=\;10^{9}\,\rm{erg}\,\rm{cm}^{-3}$ appear to be affected by RATD only toward the northern irradiated side of the Bar, where grains $\gtrsim$ 0.1 $\mu$m can be rotationally disrupted. Setting aside the considerations about $f_{\textrm{high-J}}$, that does not depend only on the grain’s characteristics, Figure 8 clearly shows that in box #1, 2, and 3, grains potentially aligned with the radiation field, i.e., larger than $a_{\textrm{trans}}$, are generally disrupted by RATD if they are aggregates. Compact grains of high tensile strength are also affected by RATD close to the irradiation front. Large and compact grains can in theory be aligned via $k$-RAT in this region if $a\;\geq\;a_{\textrm{trans}}$ and $a\;\geq\;a_{\rm{disr,max}}$, i.e., $a\;\gtrsim\;1-10\;\mu$m. However, such large grains, which are expected to be formed by collisions, are likely to be aggregates-type, and likely to be rotationally destroyed if they are efficiently aligned. In addition, it is unlikely that such large grains dominate the polarized dust emission in the FIR.

We note that with its 18.7 ^′′ (i.e., 0.035 pc at 390 pc) angular resolution, our grain alignment timescale study does not resolve the hot dust layer close to the dissociation front, where the temperature can reach 400-700 K (Goicoechea et al., 2011, 2017; Parikka et al., 2017). In addition, observations and models have suggested that the Orion Bar PDR actually consists of high density clumps $n_{\textrm{H}}\,=\,10^{6}-10^{7}\;\textrm{cm}^{-3}$ embedded in a lower density medium of $n_{\textrm{H}}\,\approx\,5\,\times\,10^{4}\;\textrm{cm}^{-3}$ mainly responsible for the extended PDR emission (Lis & Schilke 2003; Andree-Labsch et al. 2017; Habart et al. 2022 and references therein). These substructures could have been induced by UV radiation driven compression (Gorti & Hollenbach, 2002; Tremblin et al., 2012), advecting the molecular gas through the atomic gas (Goicoechea et al., 2016). Based on the results of Sections IV.1 and IV.2, it is possible to predict if those specific environmental conditions, toward these denser and warmer substructures, could trigger $k$-RAT. Fixing all other parameters, we have : $a_{\textrm{disr,min}}\,\propto\,u_{\textrm{rad}}^{-1/2}n_{\textrm{H}}^{1/2}\bar{\lambda}$ and $a_{\textrm{trans}}\,\propto\,$ $u_{\textrm{rad}}^{-0.6}\bar{\lambda}^{-0.65}$. From Hoang et al. (2021), $\bar{\lambda}\,\propto\,A_{V,\star}^{\alpha}$, with $\alpha\,\approx\,0.6$, where $A_{V,\star}$ is the visual extinction measured from the illuminating star to a given location in the cloud. Therefore, an increase in density of two orders of magnitude, with a stronger or equal apparent radiation field, could sufficiently decrease the ratio $a_{\textrm{trans}}/a_{\textrm{disr,min}}$ such that grains with radii of $0.1-1\,\mu$m can potentially subject to $k$-RATs, depending on their compactness. In parallel, the fraction $f_{\textrm{high-J}}$ would also decrease with increasing dust temperature and density, enabling a larger fraction of $k$-RAT aligned grains to contribute to the polarization.

The results from the HAWC+ polarimetric observations presented in the histograms of Figure 2 does not show that grains are aligned with the radiation field emanating from the Trapezium cluster. The low-$J$ aligned grains cannot dominate the FIR dust polarized emission because the polarization fraction would be much lower (i.e., $\leq\,1\,\%$) than what is observed. In addition, if large low-$J$ aligned grains were to dominate the polarization, they would generate a rotation of the polarization angle with increasing wavelength, where those large grains contribute more to the total emission, which is not observed. This latter effect would also induce a clear decrease of polarization fraction with wavelength, which is not the case in the Bar (see Section V.3 and Figure 11). Finally, given the transition sizes of $\sim\,0.1\,-\,0.5\,\mu$m probed in Section IV.1 and the maximum size of the dust size distribution ($\leq\,0.5\,\mu$m; see Schirmer et al. 2022, who proposed that fragmentation of large grains due to collisions caused by radiative pressure is an efficient mechanism in the Bar), the low-$J$ $k$-RAT aligned grains are not expected to dominate the FIR emission. Therefore, while there may be a population of $k$-RAT aligned grains, there is no evidence that they are the dominant cause of the FIR dust polarization observations. FIR polarization thus probe grains efficiently aligned at high-$J$ with the magnetic field. We also note that mechanically aligned grains can produce the same polarization pattern that $k$-RAT aligned grains. Because such pattern are not favored in our analysis, we do not consider this mechanism further in our discussion. However, Appendix C present an analytic exploration of this grain alignment mechanism, where we find that it should not dominate the origin of the polarized dust emission.

We find that a population of high-$J$ aligned grains should contribute significantly to the dust polarization because the $k$-RAT non-detection implies that the population of low-$J$ aligned grains is negligible. However, the analysis of Figure 6 showed that the level of grain alignment efficiency toward the irradiated side (i.e., $\mathcal{P_{\rm{frac}}}$$\,\sim\,2-10$ %) of the Bar suggests that a significant fraction of grains is still aligned. We now discuss the efficiency of RATD for those efficiently aligned grains. RATD is a mechanism that is relatively hard to precisely constrain observationally from emission data, as the abundance and size of the largest aligned grains is difficult to measure. Indeed, as shown in Section IV.2, RATD can be efficient at fragmenting all the large grains of $0.1-10\,\mu$m in size if they are aggregates of low tensile strength. However, we also note that RATD cannot be so efficient that it would deplete all grains $\geq\,0.1\,\mu$m in size, because we expect the grain alignment efficiency to be limited for small grains (Lazarian & Hoang, 2007; Andersson et al., 2015), i.e., $\leq\,0.01\,\mu$m in size. Besides, the polarized dust we observe with HAWC+ does not indicate a scenario where only small poorly aligned grains are present. Therefore, one needs to determine whether the efficiency of this mechanism is limited, and/or to characterize the structure of grains in such environment.

To investigate the hypothesis of high RATD efficiency precluding the survival of $\gtrsim\;0.1\;\mu$m size grains, we investigate the impact of such a depletion of large silicate grains on the SED of the Orion Bar PDR. To do this, we use the photometry files used in Chuss et al. (2019) for their SED fitting, as well as the SOFIA FORCAST MIR observations (19.4, 31.7, and 37.1 $\mu$m) from Salgado et al. (2016), that we smooth and regrid to 18.7 ^′′ resolution and 3.7 ^′′ pixel size with a flux conserving algorithm. In parallel, we use DustEM (Compiègne et al., 2011) to simulate the effects of both the irradiation level $G_{0}$ and the diminution of the maximum size of silicates on the SED of a standard ISM dust grain population, using compositions of Compiègne et al. (2011). Results are shown in Figure 9. For each level of $G_{0}$ that we implement, we vary the maximum size of silicates $a_{\textrm{Sil-max}}$ among the following values of 2, 0.5, 0.05, and 0.005 $\mu$m. The impact of removing large silicates on the SED is to increase the emission in the MIR, thus reducing the SED slope between the MIR and FIR (the MIR-to-FIR slope we are referring to here is the slope of the flux versus wavelength, calculated with the FORCAST data between 19.4 and 37.1 $\mu$m)⁴⁴4For reference, we also did this analysis using the THEMIS dust model (Jones et al., 2013, 2017), and also noticed a evolution of the MIR-to-FIR slope when decreasing the maximum size of the silicates grains. However, this evolution is different that of using the Compiègne et al. (2011) model. Precise dust modeling would be required, but this goes beyond the scope of this paper.. Indeed, as the mass of dust is kept constant, cutting out the distribution toward the largest grains redistribute the mass toward small grains, whose emission is thus increased. We also note that the spectral signature of the $\sim$ 15 $\mu$m band can also help to identify a scenario of a higher abundance of small silicates.

Also shown in Figure 9 are two sets of flux measurements, taken from the irradiated side of the PDR ($T_{\textrm{d}}$ $\,\sim\,$65 K; star symbols) and on the cold embedded ($T_{\textrm{d}}$ $\,\sim\,$40 K; plus symbols) side of the PDR ⁵⁵5We do not expect to fit the SED with the DustEM models. We only illustrate the impacts of the depletion of large silicates on the SED.. These data points correspond to relatively low $G_{0}$, $\sim\,10-100$ in Habing units, compared to the known FUV radiation field incident on the Orion Bar PDR, i.e., $G_{0}$ $\sim\,1-4\,\times\,10^{4}$ (Marconi et al., 1998), because of the low angular resolution of our photometric maps. The two SEDs shown in Figure 9 exhibit a shallower MIR-to-FIR slope than the model with no large silicate depletion, i.e., with $a_{\textrm{Sil-max}}\,=\,2\,\mu$m.

In order to investigate whether the evolution in the MIR-to-FIR slope in the observational data can be caused by a depletion of silicates, we show in Figure 10 the slope versus the minimum size of rotationally disrupted grains $a_{\textrm{disr,min}}$ derived in Section IV.2 for $S_{\rm{max}}\;=\;10^{5}\,\rm{erg}\,\rm{cm}^{-3}$. We find a clear bimodal change in the MIR-FIR slope as a function of $a_{\textrm{disr,min}}$, where it initially increases with $a_{\textrm{disr,min}}$, up to $a_{\textrm{disr,min}}\approx\,$1$\times$10${}^{-2}\mu$m, after which it decreases. The different slopes are mirrored in the dust temperature, such that positive slopes correspond to larger dust temperatures ($T_{\textrm{d}}$ > 50 K) with negative slope corresponding to cooler dust temperatures. This decrease of the MIR-to-FIR slope slope, for $a_{\textrm{disr,min}}<\,$1$\times$10${}^{-2}\mu$m, for decreasing (increasing) $a_{\textrm{disr,min}}$ ($T_{\textrm{d}}$) is consistent with RATD affecting the size distribution of the silicate grains by increasing the abundance of small silicates, assuming that the MIR-to-FIR slope is sensitive to such effects. For larger $a_{\textrm{disr,min}}$, the decrease of the MIR slope for increasing (decreasing) $a_{\textrm{disr,min}}$ ($T_{\textrm{d}}$) is likely due to a dust temperature effect.

The interpretation that RATD is responsible for the decrease of the MIR slope for decreasing $a_{\textrm{disr,min}}$ at $T_{\textrm{d}}$ > 40 K, is, however, subject to several caveats. For example, in a PDR the NIR to FIR flux ratio is smaller than in the diffuse ISM because of the intense dust evolution processes at the PDR irradiation front (Goicoechea & Le Bourlot, 2007). While nano-grains are efficiently photo-destroyed, large grains fragment due to collisions caused by radiative pressure, in addition to the RATD effect, which in turn can re-form nano-grains via sticking collisions at higher extinction (Schirmer et al., 2020, 2022). This also can have an impact on the MIR slope of the SED⁶⁶6Using the PDR-constrained THEMIS parameters derived in Schirmer et al. (2022), we find that, however, this only affects marginally the MIR-to-FIR slope.. In addition, the characteristic timescale of the photo-fragmentation of large grains, i.e., the collision timescale $\tau_{\rm{coll}}$, is of the same order of magnitude of the rotational disruption timescale $\tau_{\rm{disr}}$. Indeed, Schirmer et al. (2022) derived $\tau_{\rm{coll}}\,\approx\,5\,\times\,10^{1}-10^{2}$ years for the Orion Bar, while we find $\tau_{\rm{disr}}\,\approx\,10^{0}-10^{2}$ years (using Hoang et al. 2019; Lazarian & Hoang 2021), for grains of 0.1$-$1 $\mu$m. Therefore, RATD is not the only process responsible for the fragmentation of the large grains in the Orion Bar. These two timescales are grain size dependent, and the effects of RATD is more important for large grains (Hoang, 2019), while the photo-fragmentation of grains is more efficient for small grains. Therefore, the idea of using the MIR-to-FIR slope to probe the depletion of the largest aligned grains caused by RATD remains to be explored.

Finally, the effect of RATD is also hard to calibrate because of the uncertain degree of mixing of carbon and silicate grains in aggregates. Indeed, as carbon grains are diamagnetic and unlikely to align (Andersson et al., 2022), they are not subject to RATD. Because we have assumed a maximum efficiency of RATs when calculating the RATD parameters, this effect is not included here. The effects of “wrong” alignment of grains with respect to the magnetic field for large grains in the absence of internal alignment is also not included in this study (Hoang & Lazarian, 2009a). An even more important factor affecting the efficiency of RATD is the size of iron clusters embedded in dust, which significantly changes the magnetic susceptibility of dust grains, and in turn their inclination to experience fast alignment and potentially disruption (Chau Giang et al., 2022).

The Bar is not optically thin along the direction of the radiation field, and the UV photons are quickly attenuated. This affects the radiation field spectrum responsible for the heating of the dust that produce most of the IR emission across the Bar. Therefore, a radiative transfer modeling of the Orion Bar can be useful to produce a spatially resolved SED model across the PDR. Synthetic observations of such polarization radiative transfer modeling could thus evaluate the potential contribution of $k$-RAT aligned grains to the polarized dust emission retrieved by HAWC+, and the RATD-survival of large silicates grains in supra-thermal rotation. While the radiative transfer code POLARIS (Reissl et al., 2016) has been recently updated to include the calculation of the fractions of high-$J$ aligned grains as a function of grain size and environmental conditions (Chau Giang et al., 2022), the $k$-RAT and RATD mechanisms are not implemented yet. Such modeling would need to be run on a radiation-magneto-hydrodynamic simulation code that includes a proper gas cooling scheme, in order to reliably reproduce the magnetic field morphology, and the sub density structures, lying below the spatial resolution of our SOFIA observations. The magnetic field morphology that affects the polarized dust emission via depolarization effects may be hard to reproduce, and can make the comparisons between such models and dust polarization observations challenging. In theory, if deeper and higher angular resolution MIR to FIR dust polarization observations are made to resolve the typical dust evolution spatial scale, one can constrain the dust evolution scenario via a joint modeling of the grain alignment mechanisms and SED of the (polarized) dust emission.

Additionally, the polarization fraction spectrum can be a powerful tool to constrain grain alignment theories or dust characteristics. Indeed, it is known to vary with the local physical conditions (i.e., dust temperature, grain alignment efficiency, dust grain characteristics such as their composition), that affects the polarized dust emission differently across wavelength (Hildebrand et al., 1999; Vaillancourt et al., 2008; Vaillancourt & Matthews, 2012; Fanciullo et al., 2022). With optical thickness decreasing with wavelength, the polarization fraction is expected to increase with wavelength toward the dense interior of cores (Hildebrand et al., 1999). However, for less dense regions, the polarization fraction is expected to decrease with wavelength, as warm dust traced at short wavelength is more easily aligned due to the higher efficiency of radiative torques. The available polarization spectrum models (Draine & Fraisse, 2009; Guillet et al., 2018; Hensley & Draine, 2022) predict the 50$-$200 $\mu$m spectra to be $\sim\,$flat toward irradiated regions, i.e., regions close to the irradiation front in our case. Figure 11 presents the normalized polarization fraction spectra of the Orion Bar, for four ranges of $T_{\textrm{d}}$, $N_{\textrm{H}_{2}}\times\sqrt{T_{\textrm{d}}}$, and $N_{\textrm{H}_{2},}\;{}_{\textrm{from}}\;{}_{\,\theta^{1}\,\textrm{Ori}\,\textrm{C}}$, and averaging all data points (selecting the pixels meting the SNR($I$)$\;\geq\;200$ & SNR($P$)$\;\geq\;5$ & $\mathcal{P_{\rm{frac}}}\;\leq\;30\,\%$ conditions at all wavelengths). We note here again that the radiation field interacting with the grains is a function of both the input radiation field and the density of the cloud, which make the $T_{\textrm{d}}$ and $N_{\textrm{H}_{2},}\;{}_{\textrm{from}}\;{}_{\,\theta^{1}\,\textrm{Ori}\,\textrm{C}}$ quantities probe the same area of the Bar (i.e., the high $T_{\textrm{d}}$ locations correspond to low $N_{\textrm{H}_{2},}\;{}_{\textrm{from}}\;{}_{\,\theta^{1}\,\textrm{Ori}\,\textrm{C}}$, and inversely; see Figure 5). Data from the SOFIA HAWC+ OTFMAP mode observations are used alongside the JCMT POL-2 observations presented in Ward-Thompson et al. 2017; Pattle et al. 2017. The data have been smoothed to the lowest resolution of the observations we consider, i.e., the 214 $\mu$m data, and regridded to Nyquist-sample the beam. Michail et al. (2021) measured a falling 50$-$220 $\mu$m polarization fraction spectrum toward OMC-1, that they attributed to variation of dust grain population and grain alignment efficiency in the LOS. Given the uncertainties in the spectra of Figure 11, we find that the spectra are on average flat. However, there is a slight decreasing tendency—the spectra have either a minimum at 154 $\mu$m or at 850 $\mu$m, and appear tentatively falling between 53 and 154 $\mu$m (this latter trend has also been observed in starburst galaxies, see Lopez-Rodriguez et al. 2022b). We also notice that the apparent dip at 154 $\mu$m is more prominent at high temperature and low reddening (i.e., the low density region toward the irradiated side of the Bar) within the covered range of physical conditions. However, the error bar on the the mean polarization of the Band A data is large, and as noted in Appendix A, the polarization fractions values of the Band A data may suffer from systematics. Future FIR observatories with multi-wavelength polarization capabilities will be able to put stronger constraints on the dust grain properties and grain alignment mechanisms at play in PDRs.

Finally, optical and NIR spectro-polarimetry observations of background stars represent a strong tool to constrain the size distribution of foreground aligned grains. The shape of the polarization fraction versus wavelength curve, i.e., the Serkowski curve, is sensitive to the total-to-selective extinction $R_{\rm{V}}$, which is a good tracer of dust grain evolution (Serkowski et al., 1975; Whittet & van Breda, 1978; Andersson & Potter, 2007; Fanciullo et al., 2017; Giang et al., 2020; Vaillancourt et al., 2020). Such method can precisely constrain the maximum size of aligned grains, but it would require a good number of polarization detections of background stars as a function of the depth in the Bar, along the direction of the radiation field, to constrain the dust evolution throughout the PDR. Such work shall also be enabled by future FIR polarization capabilities (see Appendix D).