Attenuation Modified by DIG and Dust as Seen in M31

We observed all fields using the Potsdam Multi-aperture Spectrophotometer (Roth et al. 2005) at the 3.5 m telescope at the Calar Alto Observatory on 2011 September 16–24. To split the spatial image, the telescope uses a specialized fiber-bundle, PPaK, which consists of 331 bare science fibers (with an additional 36 sky and 15 calibration fibers) in a hexagonal grid. The grid has a diameter of 75'' (Verheijen et al. 2004) and spatial sampling of 27 per fiber. We used the 4 k × 4 k CCD detector with the V300 grating to achieve a wavelength coverage of 3500–9000 Å (centered at 5400 Å) and resolution of R = 1000.

Each of the five observed fields combines 10 pointings in a mosaic, resulting in an area of 3' × 4' (680 pc × 900 pc) for each field. Resulting mosaics have an effective PPaK resolution of 27. Each pointing was observed with a dither pattern (three dither positions shifted by Δdecl. = +156, Δdecl. = +078, and ΔR.A. = +156, ΔR.A. = −078) with 2 × 600 s exposures in order to fill in gaps between the fibers, thus covering the entire field of view. Dedicated sky observations (with 120 s exposures) were taken in the same manner between each science observation in order to be subtracted later during calibration.

Astrometry for each mosaic position was applied by eye through comparison of compact H ii regions with Hα images from the Local Group Galaxies Survey (Azimlu et al. 2011). Additionally, we compared r- and g-band SDSS images with our data by applying the SDSS response functions for the corresponding bands to our observed spectra. Maximum deviation in astrometry are 2'', with a mean offset of around 1''. Astrometric inaccuracies at this level do not affect our data analysis because we compare the A_V and dust maps at 25'' resolution.

To translate the electron count values into fluxes, we observed the standard stars BD +33d2642 and BD +25d4655 (Oke 1990). The positions of lines and spectra on the detector, the optical path and the transmission are all affected by fiber flexure of the IFU instrument (Sánchez 2006). To correct for these effects, we obtained calibration continuum lamp images (used for positioning of the spectra), He+HgCd arc lamp images (used for wavelengths calibration) and twilight flats (used for accurate flat fielding).

The atmospheric conditions were mostly clear, resulting in approximately uniform imaging of our fields. The observations of some pointings were repeated on September 24th due to bad weather (clouds) or moonlight contamination. Seeing was subfiber (less than 27) for all observations.

2.1.1. Calibration

2.1.2. Sky Subtraction, Relative Flux Calibration, and Data Cubes

2.1.3. Line Maps

The optical galaxy spectrum contains both stellar spectra (with continuum, emission, and absorption lines) and emission lines from the ionized gas. We separate the nebular emission from the stellar spectra using the GANDALF¹¹ software package¹² version 1.5 (Sarzi et al. 2006). GANDALF simultaneously fits both the emission lines and stellar continuum in an iterative approach.

It fits the emission lines using the penalized pixel-fitting method (pPXF; Cappellari & Emsellem 2004). Each prominent emission line is fit by a Gaussian profile with the kinematics tied together (i.e., v and σ) but fluxes left free. To fit the stellar continuum, we use template spectra taken from the Tremonti et al. (2004) library of Bruzual & Charlot (2003) simple stellar population (SSP) templates for a range of stellar ages (100 Myr to 12 Gyr) and metallicities (Z = 0.004 and 0.05). While SSP spectra may not represent the resolved stellar populations in our fields in M31, we chose to use the same templates as K13 for consistency. All templates are convolved to match the spectral resolution of our spectra.

We checked if using other SSP templates or stellar spectra (with higher spectral resolution and different stellar populations) could change the results of our fitting and alter the underlying stellar absorption features. We used the MILES SSP and stellar spectra library templates (Sánchez-Blázquez et al. 2006; Falcón-Barroso et al. 2011) in fitting our spectra and found that the results do not show any significant difference compared to using the Tremonti et al. (2004) library.

In fitting the continuum, we also allow for a multiplicative third-order Legendre polynomial correction. This correction is necessary due to the poor flat-field correction in the blue part of the spectra caused by low CCD sensitivity, and to allow for stellar continuum attenuation by dust intrinsic to the fields of M31.

Foreground extinction from the Milky Way is also accounted for in the spectral fitting. However, this is considered to be uniform across the disk of M31 with an ${A}_{{\rm{V}}}=0.1705$ mag in all fields (based on Schlegel et al. 1998 and Schlafly & Finkbeiner 2011).

The final data products from our fitting for each spaxel in our data cubes are pure stellar continuum spectrum and fractional contribution of various SSP templates; multiplicative polynomial indicative of the intrinsic dust attenuation (and flat-field corrections); gas velocity and gas velocity dispersion; and finally individual line amplitudes and fluxes for the most prominent emission lines. Identified emission lines are Hδ, Hγ, Hβ, [O iii]λ4959 Å, [O iii]λ5007 Å, [N ii]λ6548 Å, Hα, [N ii]λ6583 Å, [S ii]λ6717 Å, and [S ii]λ6731 Å.

Atmospheric optical emission lines (dominant around 5500 Å) can cause problems with the sky subtraction. There is a weak Hg i 4358.34 Å sky line (Osterbrock & Martel 1992; Slanger et al. 2000) near the Hγ line, which can also affect sky subtraction and fitting of the Balmer emission line. This effect is visible in the residuals near Hγ (Figure 2, seen as an absorption feature). Additional contaminants like earthshine and zodiacal light are removed from the spectra during sky subtraction because they exist as faint, extended features on the sky and in the sky spectrum (Reach 1997). Geocoronal emission lines, spatially extended and slowly changing with time, are removed by sky subtraction and do not have a significant effect on our Balmer line fluxes due to their narrowness (Nossal et al. 2001; Haffner et al. 2003; Bishop et al. 2004).

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While the [O ii] λ3727 Å line doublet is detected in many of our spectra, the line is strongly affected by the poor sensitivity and subsequent calibration in the blue part of the spectrum, and hence it was not used in our analysis. Foreground stars (approximately 2–5 spaxels per field) were not removed from our data cubes, but spaxels affected by the stars are masked during our GANDALF spectral analysis.

The median 3σ sensitivities of Hα and Hβ in all fields are $7.6\times {10}^{-18}$ and $4.9\times {10}^{-18}$ erg s⁻¹ cm⁻² arcsec⁻², respectively. However, averaged or median 3σ sensitivities of Balmer lines do not determine whether the data from spaxels will be shown or excluded from the following diagrams and measurements. Data are included only if a line's amplitude is above the 3σ noise on the continuum. GANDALF calculates the noise on the continuum as the standard deviation of the residuals for the entire wavelength range, while uncertainties in the line fits as the amplitude over noise values (AoNs; Sarzi et al. 2006). Figure 2 presents an example of the spectrum fitting for one spaxel in Field 1.

FIR emission is a good tracer of the amount of the dust in the line of sight (Wynn-Williams 1982; Neugebauer et al. 1984). Therefore, we use the FIR data observed by the PACS and SPIRE camera on the ESA Herschel Space Observatory (Griffin et al. 2010; Pilbratt et al. 2010). The dust mass surface density map of M31 presented by Draine et al. (2014), used for the comparison with attenuation maps, was determined by fitting the SED of the near and FIR emission (Groves et al. 2012) with the Draine & Li (2007) dust model. The Draine & Li (2007) model specifies the dust characteristics, like the distribution of grain sizes, the frequency-dependent opacity, the fraction of dust in polycyclic aromatic hydrocarbons (PAHs), and the dust column densities. The models were determined by calculating emission spectra (in near-IR, FIR, and sub-millimeter) and reproducing extinction curves for different abundances of small PAHs and various dust mixtures heated by different starlight intensities.

The resulting dust mass surface density map (Σ_dust) has an effective 249 Gaussian PSF (matched to SPIRE 350 μm resolution). In order to compare our final attenuation maps with the dust mass surface density maps, we convolved our cubes from the effective PPaK resolution of 27 into the SPIRE 249 resolution. To convolve the data we use the kernels and the routine described in Aniano et al. (2011).

We convolve the data cubes by splitting them into images for each wavelength bin, and convolving each image separately. Before the convolution process, we add extrapolated values to the area outside the edges of the image, convolve the image, and then replace those areas with blanks after the process. Then we reassemble the cubes to the SPIRE 350 μm image grid in order to compare the maps. After the convolution, we perform the same fitting routine and spectral analysis on the convolved images as described in Section 2.1.3.

This convolution technique changes the Balmer lines intensities (up to 30%) on the edges of the data cubes, depending on the position of bright H ii regions. It affects both Balmer lines simultaneously, which results in only small changes in A_V. If we compare A_V maps derived from the convolved data cubes and smoothed (but not convolved) cube, changes in A_V can be up to 0.3 mag for the bright regions. The effects of foreground stars are minimized due to the convolution process. Figure 3 shows an example of our un-convolved and convolved maps of Hα emission within Field 3.

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Due to reddening by dust (correlated with the extinction) the Balmer line ratios (known as Balmer decrements) are altered from their intrinsic ratios. The total V-band extinction (A_V) is related to nebular reddening ${E}_{{\rm{B}}-{\rm{V}}}$ by

$$\begin{eqnarray}&&{A}_{{\rm{V}}}\equiv {R}_{{\rm{V}}}{E}_{{\rm{B}}-{\rm{V}}},\end{eqnarray} \tag{ 1 }$$

where ${R}_{{\rm{V}}}$ is the selective extinction. ${R}_{{\rm{V}}}$ depends on the physical characteristics of the extinguishing dust grains. In the diffuse ISM of the Milky Way, ${R}_{{\rm{V}}}$ has an average value of 3.1, which is the value typically assumed for massive star-forming galaxies (Schultz & Wiemer 1975; Cardelli et al. 1989; Calzetti et al. 2000). We assume the same ${R}_{{\rm{V}}}$ value for M31 because our fields are at similar galactic radii as the Sun (at 0.3–0.6 R₂₅; Bigiel & Blitz 2012) and with similar metallicities (Zurita & Bresolin 2012; Draine et al. 2014). The reddening between two lines (F₁ and F₂) is calculated as (Calzetti et al. 1994)

$$\begin{eqnarray}&&{E}_{{\rm{B}}-{\rm{V}}}=\displaystyle \frac{2.5}{{k}_{2}-{k}_{1}}{\mathrm{log}}_{10}\left(\displaystyle \frac{{F}_{1}/{F}_{2}}{{R}_{\mathrm{int}}}\right),\end{eqnarray} \tag{ 2 }$$

where k is the extinction as a function of wavelength for the corresponding lines. Here we use the extinction curve from Cardelli et al. (1989). By using the Calzetti et al. (2000) attenuation curve instead, the resulting inferred A_V would decrease by 9%. That systematic shift should be kept in mind when comparing our results with those found by K13, where they used the Calzetti et al. (2000) curve. The difference between those two curves is because the Cardelli et al. (1989) curve solely accounts for foreground extinction, while the Calzetti et al. (2000) curve takes into account geometrical effects on attenuation.

We assume an intrinsic flux ratio of ${R}_{\mathrm{int}}={\rm{H}}\alpha /{\rm{H}}\beta =2.86$, corresponding to an ionized gas temperature of $T\approx {10}^{4}$ (assuming case B recombination; Miller 1974; Osterbrock 1974; Osterbrock & Martel 1992, 2006).

We show attenuation (A_V) maps and the dust mass surface density maps (Σ_dust) in Figures 4 and 5. For all our fields, contours show the Hα intensities, tracing the position of the H ii regions. All maps are at the same scales, which offers a direct comparison between the fields. A_V spans values between 1.5 and 4.5 mag in our fields. Similar results are observed by Sanders & Caldwell (2012), where observed H ii regions in M31 show values between 1 and 5 mag. The fact that there are no M31 data points with attenuation lower than ${A}_{{\rm{V}}}=1$ is an effect of targeting dense spiral arm regions in M31, which biases us to regions of high Σ_dust. If our fields were larger and included less dusty regions, we would expect our maps to have more data points with ${A}_{{\rm{V}}}\lt 1$.

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In general, we find that the H ii regions are situated near or in the regions of higher dust mass surface densities. This is expected if the dust traces the regions of high gas density where new stars are formed. Contrary to other fields, in Field 1, the H ii regions are situated in less attenuated and less dusty areas. We explain this as an effect of stellar feedback from more evolved H ii regions that have already destroyed the dusty birth cloud.

We exclude from the following analysis and map regions where the Balmer lines do not exceed a threshold of AoN ≳ 3 (corresponding to S/N ≳ 3). Due to the low flux and hence S/N of the emission in Field 5, there are few spaxels that exceed this threshold. This causes some statistical difficulties in the analysis of this field. There is also a possibility that a young H ii region is buried in the dense cloud, seen in the center of Field 5 (Figure 5).

Our calculated attenuation depends on the physical condition of the ionized gas (which can cause different intrinsic line ratios) and on the dust composition (resulting in different values of ${R}_{{\rm{V}}}$), both of which affect the extinction curve. Therefore, the attenuation values of some data points can be different due to intrinsically different physical conditions in those regions. However, these effects are presumed to be small relative to the real variation in A_V due to the dust distribution.

The main goal of this paper is to compare the dust in M31 as determined via two independent methods; the attenuation derived from the Balmer decrement, and the dust mass surface density derived from IR photometry. This follows on from the work of K13 who found a relation between these measures within eight nearby galaxies, but with large scatter between galaxies.

The dust attenuation and dust mass surface density are connected via the distribution of dust. We consider here two simplistic models of the dust distribution, derived from Caplan & Deharveng (1986) and Calzetti et al. (1994). The Calzetti models predict the correlation between attenuation and the dust mass surface density for five different spatial distributions of the dust. The two major models that are used for this work are the "foreground screen" model and the "mixed" model (in Calzetti et al. 1994, these are referred to as the "Uniform dust screen" and the "Internal dust" models). The remaining models are variations on those two extremes.

The "foreground screen" model describes a situation where all the dust sits in a smooth, uniform screen between the emitter and the observer (the dust is not mixed with the sources of radiation). Assuming the dust model from Draine & Li (2007) and the dust-to-gas ratio (DGR) from Draine et al. (2014), the attenuation is linearly related to Σ_dust via

$$\begin{eqnarray}&&{A}_{{\rm{V}}}^{\mathrm{screen}}=0.74\cdot \displaystyle \frac{{{\rm{\Sigma }}}_{\mathrm{dust}}}{{10}^{5}{M}_{\odot }\,{\mathrm{kpc}}^{-2}}\,(\mathrm{mag}).\end{eqnarray} \tag{ 3 }$$

Given our assumptions, this equation provides a theoretical upper limit on the attenuation possible at a given dust mass surface density.

The "mixed" model assumes that the dust is uniformly distributed with the sources of radiation. Therefore, the attenuation in the mixed model is much lower than in the foreground screen model. The resulting mixed model attenuation, assuming isotropic scattering and the same connection with dust mass as in Equation (3), is (based on Calzetti et al. 1994)

$$\begin{eqnarray}&&{A}_{{\rm{V}}}^{\mathrm{mixed}}=-2.5{\mathrm{log}}_{10}({\gamma }_{{\rm{V}}})\,(\mathrm{mag}),\end{eqnarray} \tag{ 4 }$$

where

$$\begin{eqnarray}&&{\gamma }_{{\rm{V}}}=\displaystyle \frac{1-{e}^{0.57{A}_{{\rm{V}}}^{\mathrm{screen}}}}{0.57{A}_{{\rm{V}}}^{\mathrm{screen}}}.\end{eqnarray} \tag{ 5 }$$

γ functionally limits the value of the optical depth in the optically thick medium (Calzetti et al. 1994). The screen model correlates linearly with the dust column, while the mixed model saturates and for high dust mass surface densities yields a moderate expected attenuation (Figure 6).

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Figure 6 shows a comparison of the dust mass surface densities and the attenuation for M31 and for the nearby galaxies observed by K13, together with the two Calzetti models. While the data from K13 span the area between the foreground screen and mixed models, the more resolved M31 data follows a foreground screen model more closely. No clear trends are seen within each Field of M31, but when the fields are considered together the data fits the foreground screen model very well. We find the correlation between A_V and Σ_dust to be well fit by scaling down the foreground screen model by a factor of 3.8 for the nearby galaxies (K13) and 1.35 for M31.

We see some regions in M31 that have larger attenuation than is expected from our foreground screen model (Equation (3)). Approximately 25% of the data points are above the foreground screen model, or 3% of the data if we take into account their corresponding 1σ errors. These high A_V regions can be explained by (1) "clumpiness" in the dust below our resolution that is affecting our measured Σ_dust, (2) an underestimation of the dust mass surface density, (3) poor calibration of the optical spectra, (4) values of ${A}_{{\rm{V}}}/{{\rm{\Sigma }}}_{\mathrm{dust}}$ different from what we have assumed, and (5) variation in the extinction curve ${\rm{k}}(\lambda )$ that affect ${R}_{{\rm{V}}}$ and thus A_V.

If an area with low Σ_dust has locally a high density clump of dust covering the H ii region, averaging the dust surface density due to the low spatial resolution of our observations could misleadingly show a low Σ_dust and a high A_V. However, when comparing the value of A_V around H ii regions at PPaK 27 resolution (∼10 pc), we find that A_V does not change rapidly with different aperture sizes.

Previous work has tested the accuracy of estimating dust masses using the Draine & Li (2007) model (Alton et al. 2004; Dasyra et al. 2005; Aniano et al. 2012; Galametz et al. 2012; Dalcanton et al. 2015; Planck Collaboration et al. 2016). Most recently, Dalcanton et al. (2015) and Planck Collaboration et al. (2016) measured dust column density within the Milky Way and M31 by measuring the extinction of the light from background sources. Both studies concluded that the Draine & Li (2007) dust model may overestimate the mass of the dust by a factor of ∼2.5. Comparisons with independent FIR observations have shown that this offset is not due to uncertainties in the Herschel photometry (Verstappen et al. 2013; Planck Collaboration et al. 2016).

Planck Collaboration et al. (2016) suggest that this offset is dependent upon the heating radiation field intensity. In the Appendix, we explore the effects of renormalizing Σ_dust in M31 and the K13 (Appendix A; Figure 12). Based on this, we would expect that such a normalization would only further the disagreement between the K13 galaxies and M31. However, because this renormalization is not well calibrated in the high interstellar radiation field regime (corresponding to most of the K13 regions), we do not include the renormalization in the following analysis and figures.

An additional effect that can play a role in our derived attenuation is the "mid-plane" effect, because we expect half of the dust to be situated behind the ionizing sources. That can partly explain the offset of M31 data from the "foreground screen" model line. This effect should be strongest in the case where the scale height of the ionized gas is much smaller than the scale height of the dust. However, in the case of similar scale heights, this effect should play a much smaller role.

Given our confidence in our spectrophotometric calibration, we explain higher attenuation values of some regions as a result of possible variations in the extinction law tied to variations in the dust properties, which we expect to be related to variations in our assumed DGR and ${R}_{{\rm{V}}}$.

While K13 probed spatial scales between ∼0.3 and ∼2 kpc, the proximity of M31 means that the SPIRE 350 μm physical resolution is ∼100 pc. The fields in M31 are located within the most dense and dusty spiral arms and cover only a small fraction of the galaxy. Therefore, it is possible that our M31 results are biased by dusty, star-forming regions, where a foreground screen model more closely represents the dust distribution. The larger physical sizes of the regions sampled in K13 can probe both regions where a foreground screen model is more representative and regions where a mixed model more closely matches reality. This yields a result where together all regions show a mixture between the two models. K13 searched for a correlation between A_V–Σ_dust slope and different spatial scales, but the changes in slope were not significant.

To directly compare our result with K13 (whose best spatial resolution is ≈300 pc), we smoothed our data (Figure 7) to 65'' resolution (corresponding to spatial scales of ≈260 pc) and find no difference in the slope of the correlation. Moreover, we also integrate all spectra from each field into a single data point and re-extract the line fluxes to determine the average attenuation for each field. Resulting integrated field data points span spatial scales of around 0.6 kpc × 0.9 kpc. We then compare this value to the average dust mass surface density for the field. Integrated field data points are shown as circles in Figure 7, and it is clear that even at these scales we see the same relation between attenuation and dust mass surface density. Some fields are not presented, due to the low number and brightness of H ii regions. Because the Hβ line is faint, this results in low AoN values for those fields.

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The regions in K13 target wide areas (covering both spiral arms and inter-arm regions) and can consist of regions with or without low-brightness H ii regions. It is possible that the dust probed in some of those regions is heated by an old stellar population and not solely by H ii regions. However, because the Draine & Li (2007) model explicitly takes into account starlight heating, the modeled dust mass should not suffer from any bias in either K13 or our results on any of the scales considered. Using the close alignment of disk galaxies in front of early-types, Holwerda & Keel (2013) found that inter-arm regions contain less dust relative to spiral arms. While K13 included regions with a broad range of dust mass surface densities, the regions with weak Hα flux (where dust may be predominantly heated by the old stellar population) are generally removed by the S/N criteria in their work. Therefore, we conclude that the difference between our results and theirs is not related to the treatment of the inter-arm regions. However, the integrated fields of M31, which have a similar spatial resolution to K13 ones, have SFR surface densities (${{\rm{\Sigma }}}_{\mathrm{SFR}}$) that are a factor of 10 lower. This makes a direct comparison between fields at fixed ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ impossible. The mean ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ in M31 is around ∼0.01 M_⊙ yr⁻¹ kpc⁻², while for the K13 galaxies ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ spans a 0.03–10 M_⊙ yr⁻¹ kpc⁻² range (Appendix B; Figure 13).

Another possibility to explain the difference in the A_V–Σ_dust relation between M31 and the K13 galaxies is that, in the K13 galaxies, some fraction of the ionized gas resides outside the dust disk. This emission would not be extinguished, therefore, lowering the observed A_V. In this section, we propose the DIG as a candidate for this gas outside the dust disk. We also explain why the previously used Calzetti et al. (1994) model of dust/gas distribution (combining only H ii regions, stars and dust) requires an additional diffuse and spatially extended component.

4.2.1. Additional Flux from Non-attenuated Gas

The Calzetti et al. (1994) model for the dust/gas distribution combines only dust and stars (that ionize the gas within the H ii regions). That model neglects the possibility of additional flux from ionized gas spatially even more extended than the dust. In the scenario where additional gas resides outside the dust disk, the observed attenuation would be lower than in the case of the Calzetti et al. (1994) model, even for the same observed amount of dust. If we assume that the M31 regions have all of the ionized gas embedded in the dust disk, we explore how flux from a non-attenuated ionized gas, which may lie outside the dust disk, would effect the A_V–Σ_dust relation.

In Figure 8, we investigate how large the contribution from a non-attenuated ionized gas component above the dust disk would need to be in order for the M31 data to follow the trend of K13. Hα flux from that non-attenuated ionized gas, ${F}^{\mathrm{non} \mbox{-} \mathrm{ext}}$, is presented as a percentage of the total observed flux, F^tot. For M31, we assume that no ionized gas is outside the dust disk. The left panel in Figure 8 shows the impact on our integrated fields in M31 when we add flux from such an additional ionized gas outside the dust disk. Similarly, the right panel shows the impact of that additional ionized gas on the foreground screen model. Even a small amount of flux from the added ionized gas lowers the attenuation enough that we observe values similar to those found by K13.

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If 30%–60% of the observed total Hα flux arises from gas that sits outside the dust disk, we can recover the relation observed by K13 within nearby galaxies. This provides a better qualitative fit than simply scaling the foreground screen model by a factor of 3.8, as suggested by K13 (Figure 8, right panel). When the contribution from the additional ionized gas component to the emission is large, our model drops even below the "mixed" model curve. This is because, unlike the Calzetti model, we are including a component with no attenuation at all in the model. At extreme contributions, we would only see the non-attenuated component and measure little to no total attenuation.

It is not expected that the ionzied gas directly associated with H ii regions will span large spatial scales because typical H ii region sizes are less than 100 pc (Azimlu et al. 2011). However, a good candidate for the extended ionized gas component we propose here is the DIG.

4.2.2. DIG as an Additional Component

4.2.3. Effects of Different Dust-H ii-DIG Distributions on A_V

In the case where the DIG resides outside the dust disk and does affect our observations, we expect areas with higher [S ii]/Hα and the same Σ_dust to have a lower ${A}_{{\rm{V}}}$. There would be no trend between A_V and [S ii]/Hα (at the same Σ_dust) if both components lie within the dust disk.

In Figure 8, our M31 data are color-coded according to their [S ii]/Hα ratio (with 0.35 as a threshold). Data points with lower ratios indicate areas with H ii regions, while those with higher ratios mark DIG-dominated areas. Our M31 fields do not show a clear trend between the [S ii]/Hα ratio and ${A}_{{\rm{V}}}$, with Field 4 even exhibiting lower line ratios for regions lying further away from the foreground screen model. Due to possible changes of the intrinsic Balmer line ratio in DIG-dominated spaxels, A_V values of those spaxels could drop down to 0.3–0.5 mag from those currently shown in Figure 8. K13 showed that there is a trend for a higher [S ii]/Hα ratio at lower attenuation in all their galaxies, which they attribute to the fact that H ii regions are located within dusty birth clouds (Figure 4 in K13). Because the two lines are very close in wavelength, extinction alone cannot account for the change in the [S ii]/ Hα line ratio (at A_V = 3 mag the change is only 0.02 in the ratio).

Figure 9 shows [S ii]/Hα as a function of A_V for regions within nearby galaxies (left panel) and M31 (right panel), color-coded by the dust column density. In both panels, the contours indicate the distribution of the data for nearby galaxies at 27 resolution. In the left panel of Figure 9, the regions within nearby galaxies are convolved to 18'' resolution (spanning physical scales between ≈300 and ≈2000 pc). NGC 2146 (X symbols) is an outlier in this diagram as it has a high inclination, high dust column galaxy, with clear shock driven outflows (Kreckel et al. 2014). In the right panel of Figure 9, our M31 data are shown at 249 resolution (≈100 pc) and the integrated fields.

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In general, the change in spatial scales affects our line ratios by diluting the flux from compact, resolved objects like H ii regions. H ii regions within M31 (with [S ii]/Hα < 0.3 at 10 pc scales) exhibit higher [S ii]/Hα ratios when convolved to 100 pc scales. The same diluting effect causes a decrease in the [S ii]/Hα ratio for regions dominated by extended DIG emission.

Some areas (20% of all regions and almost half of Fields 2, 4, and 5) are consistent with shock excitation ([S ii]/Hα > 0.5, Kewley et al. 2006); however, they also show large uncertainties (due to low S/Ns, as they have low surface brightness).

The left panel in Figure 9 does not show a clear trend for lower attenuation at higher [S ii]/Hα ratios. However, some of the individual galaxies in the K13 sample (like NGC 3627, NGC 6946, and NGC 3077) do show a weak trend for increased [S ii]/Hα toward lower A_V. Shown in the right panel of Figure 9, the M31 data actually suggests an increase of the [S ii]/Hα ratio with A_V. However, there is a large scatter, and this trend is predominantly driven by variation between the fields, suggesting some dependence on location and physical characteristics of ISM (see Section 4.4).

We could miss certain spaxels with both higher [S ii]/Hα ratio and A_V due to the faintness of the lines (caused by both low surface brightness and high attenuation). Such spaxels could possibly change the trends seen in Figure 9.

There is a possibility that the gas-phase metallicity has an effect on the global trend in line ratios. Like most galaxies, there exists a radial metallicity gradient in M31 meaning the metallicity increases from Fields 1 to 5 (Table 3 in Kapala et al. 2015; Zurita & Bresolin 2012; Draine et al. 2014). This higher metallicity is associated with the higher DGR and the higher [S ii]/Ha ratio. However, this ratio is affected not only by metallicity, but also by ionization parameter and temperature of the gas, and this degeneracy causes issues in the analysis of the ratio and A_V.

Besides the difference in trends, thresholds, and biases on A_V, M31 shows higher attenuation at fixed dust mass surface density, and a higher [S ii]/Hα ratio at fixed A_V (especially at higher A_V) compared to other nearby galaxies (as seen in Figures 6 and 9). More complicated models are required to better understand the affects of the different relative scale heights of dust, H ii regions, and DIG on the line ratios and on A_V.

In this sub-section, we investigate two simple models that examine how the vertical distribution (scale heights) and intensity of the H ii and DIG phases compared to the dust layer can change the observed values and relations between Σ_dust, A_V, and the [S ii]/Hα ratio.

The first simple model (Figure 10) represents a face-on galactic disk composed of only a thin layer of H ii regions (with [S ii]/Hα = 0.15), a uniform dust screen, and a DIG component in front (with [S ii]/Hα = 0.75), all along one line of sight. We took those specific S ii/Hα ratios for simplicity, but in a real ISM those values could change. In this model, one resolution element includes emission from an H ii region, that suffers an attenuation ${A}_{{\rm{V}}}^{\mathrm{real}}$ from the dust screen. The emission from the DIG is assumed to be unattenuated. The variables that are changed in the model are (1) the intrinsic attenuation ${A}_{{\rm{V}}}^{\mathrm{real}}$ (directly proportional to Σ_dust) and (2) the ratio of intrinsic intensities from the DIG and the H ii region (parameter X = I^DIG/I^{H ii}). We emphasize that ${I}^{{\rm{H}}{\rm{II}}}$ is not attenuated and cannot be associated with the value ${F}^{\mathrm{ext}}$ described in the previous section. By varying these parameters, we can then explore the changes in the observed [S ii]/Hα ratio (hereafter labeled S2) and ${A}_{{\rm{V}}}^{\mathrm{obs}}$. This toy model is insensitive to spatial scales and to the amount of flux because all variables are relative to each other.

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In the left panel of Figure 10, we show the correlation between S2, the intrinsic and observed attenuation, and the ratio of intrinsic intensities from the DIG compared to H ii regions (parameter X). Our model indicates that different combinations of emission from DIG and H ii regions can generate the same observed S2 and A_V values for a range of dust masses (i.e., $\sim {A}_{{\rm{V}}}^{\mathrm{real}}$). This is because of the large impact the DIG emission has on the S2 and A_V values when the flux from the H ii region is highly attenuated. Combining that effect with the dependence of S2 on the dust and the DIG, our model resembles the behavior observed for nearby galaxies (Figure 9). However, this model cannot explain the data for M31, where we observe high values of [S ii]/Hα and A_V, nor explain the offset in attenuation between regions at fixed dust mass surface density in M31 and the other nearby galaxies.

The second model (Figure 11) has a more realistic ISM distribution where the layers of DIG and diffuse dust lie on both sides of the H ii region and are mixed, as observed by Howk (2012). The left panel shows [S ii]/Hα versus ${A}_{{\rm{V}}}^{\mathrm{obs}}$, while the right panel shows ${A}_{{\rm{V}}}^{\mathrm{obs}}$ versus ${{\rm{\Sigma }}}_{\mathrm{dust}}^{\mathrm{tot}}$. In this model, we assume that a certain amount of diffuse dust is mixed with the DIG. The mass of mixed dust is given by the ratio of mixed to total dust mass (parameter $Y={{\rm{\Sigma }}}_{\mathrm{dust}}^{\mathrm{DIG}}$/${{\rm{\Sigma }}}_{\mathrm{dust}}^{\mathrm{tot}}$). ${{\rm{\Sigma }}}_{\mathrm{dust}}^{\mathrm{tot}}$ is equivalent to the total observed dust and it is a sum of the dust layer around the H ii region, which acts as a screen (labeled as Dust^FSC in Figure 11) and the dust layer mixed with DIG (labeled as Dust^MM in Figure 11).

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In the second model, shown in Figure 11, a case of Y = 0 represents the case where the dust is not mixed with DIG and there is no Dust^MM. That is the case similar to the first model shown in Figure 10. When we redistribute the total dust so that some part of it is mixed with the DIG (thus rising value Y), the dust dims the DIG emission and increases the measured attenuation. We present an extra scenario in Figure 11, where there is only DIG and dust and the scale height of the DIG is smaller than the scale height of the dust layer. That scenario shows higher ${A}_{{\rm{V}}}^{\mathrm{obs}}$ and higher S ii/Hα ratios compared to previous cases. With that scenario, we can reproduce the regions with higher attenuation and higher [S ii]/Hα ratio noticed in Figures 8 and 9.

Comparing the K13 and M31 data seen in Figures 6 and 9 with the models in Figures 10 and 11, we notice complex behaviors and the strong effect the DIG vertical distribution has on the observed ${A}_{{\rm{V}}}^{\mathrm{obs}}$ and S ii/Hα ratios.

In the nearby galaxies studied by K13, the DIG components could be more extended and not so well mixed with the dust. We draw this conclusion by comparing the observed ratios in Figure 9 with the simple models on Figures 10 and 11. The likely difference between those galaxies and M31 is the intensity and relative vertical distribution of dust and DIG.

Unlike K13, the higher spatial resolution data in M31 enables us to clearly distinguish between compact H ii regions and those regions with dust and DIG only. Our results indicate that the DIG component in M31 could be smaller in intensity and scale height, and be well mixed with the dust layers in the disk. An additional layer of diffuse dust, extending to even larger scale heights than the DIG, would increase the attenuation even more.

The high inclination of M31 likely plays an additional role in increasing the attenuation. Nevertheless, K13 did not report any significant correlation or change in their relation with inclination. For example, the highly inclined galaxy NGC 3627 (similar to M31) shows lower A_V than M31, while NGC 4321 has higher A_V values despite its lower inclination (30°) and similar Σ_dust as NGC 3627. NGC 2146 and NGC 7331 are both highly inclined (60° and 70° as M31) but have slightly lower A_V value than those seen in M31.

Given these findings, the question that arises is, why in M31 do we observe a different mixture and scale heights between all three components (DIG, H ii regions, and the dust) compared to other galaxies? To explain this, we consider the effects of probing different locations within the galactic disk.

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We note that the locations and radii of the fields observed in M31 and in K13 are different. In M31, we observe five small fields within the spiral arms of M31, where the most central field is still ∼6 kpc from the galaxy center (equal to 0.28 R₂₅; Table 1, de Vaucouleurs et al. 1991; Zurita & Bresolin 2012). The regions observed by K13 are more central (<0.2 R₂₅, except for one galaxy with <0.4 R₂₅), and composed of bulges, spiral arms, and inter-arm regions. The global SFR and ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ for the nearby galaxies studied by K13 are much higher than that of M31 (SFR ∼ 1 M_⊙ yr⁻¹; Williams 2003) and the observed fields (${{\rm{\Sigma }}}_{\mathrm{SFR}}\sim 0.01$ M_⊙ yr⁻¹ kpc⁻²; see Figure 13 in Appendix B).

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If we assume that the dust follows the gas, as seen in Hughes et al. (2014) and Holwerda et al. (2012) with some dependence on dust temperature, then we can argue that the scale height of the dust is equal to the gas scale height and follows it. In several observations, the scale heights of the gas have been found to increase with galactic radius (Sancisi & Allen 1979; Braun 1991; Rupen 1991; Scoville et al. 1993; Olling 1996; García-Burillo et al. 1999; Wong & Blitz 2002; Matthews & Wood 2003; O'Brien et al. 2010; Velusamy & Langer 2014; Yim et al. 2014; Zschaechner & Rand 2015). The increase of the gas scale height can be understood via the gravitational and hydrodynamical equilibrium of the gas, and its dependence on gas vertical velocity dispersion, stellar volume density, and gas surface density (Koyama & Ostriker 2009; Pety et al. 2013). If the dust and molecular gas are correlated, then the dust scale height is rising with galactic radius.

On the other hand, the DIG scale height and intensity are correlated with the number and brightness of H ii regions and star-formation activity (Dettmar 1990; Rand 1996; Wang et al. 1999; Collins et al. 2000; Oey et al. 2007). Numerous, brighter star-forming regions that are more energetically active (Tyler et al. 2004; Yasui et al. 2015) can affect the production, destruction, and distribution of dust and also the ionization of the DIG. This can result in a more prominent and extended DIG component. Moreover, Dettmar (1990) noticed a decrease of the Hα scale height with galactic radius (with the highest scale height seen in the starbust regions in the center). We argue that if the scale height of the DIG is proportional to the star-formation intensity, then the decrease of star formation with radius would indicate a decrease of the scale height of the DIG component. Regan et al. (2006) and Bigiel et al. (2010) found the FUV and 8 μm emission (both correlated to star formation) are decreasing with galactic radius and have their highest values in the centers of galaxies. Furthermore, García-Burillo et al. (1999) noticed the existence of Hα chimneys spanning the extra-planar area from the galactic center with scale height larger than that of the CO.

Since the dust scale height is likely to increase with galactic radius while the DIG scale height should decrease with galactic radius, differences in the relative DIG/dust geometry and scale heights of galaxy centers (as in K13 galaxies) and galactic outskirts (as in M31) may explain the differences in our derived attenuation.