2. Quantifying the geometry of dust in galaxies

Shorter wavelengths are more greatly impacted by obscuration due to the small characteristic sizes of dust grains. The degree of obscuration a particular wavelength experiences in a given dust cloud is known as the optical depth of the dust. The optical depth can be characterised by the attenuation parameter, $\tau(\lambda)$ , for a simple uniform layer of dust that lies between a source and the observer. This is defined through

(1) \begin{equation} I_\lambda = I^0_\lambda e^{-\tau(\lambda)}\end{equation}

where $I^0_\lambda$ is the intrinsic intensity of the source, and $I_\lambda$ is the intensity observed (Calzetti, Kinney, & Storchi-Bergmann Reference Calzetti, Kinney and Storchi-Bergmann1994). The same dust cloud will have a greater optical depth for shorter wavelengths.

The difference in optical depth for the H $\alpha$ and H $\beta$ emission lines, referred to as the Balmer optical depth, is given by

(2) \begin{equation} \tau^l_B = \tau_\beta - \tau_\alpha = \ln\left(\frac{\text{H}\alpha/\text{H}\beta}{2.86}\right)\end{equation}

(Calzetti, Kinney, & Storchi-Bergmann Reference Calzetti, Kinney and Storchi-Bergmann1994). The equation from Calzetti, Kinney, & Storchi-Bergmann (Reference Calzetti, Kinney and Storchi-Bergmann1994) used the value 2.88. Here we use 2.86 to remain consistent throughout the paper, although all results change only negligibly if the value 2.88 is used. Equation (2) can be rewritten as

(3) \begin{equation} \tau^l_B = 2.303\log\left(\frac{\text{H}\alpha}{\text{H}\beta}\right) - 1.0508.\end{equation}

The geometry of dust with respect to stars is known to have an impact on the observed level of obscuration affecting a galaxy’s emission (e.g., Tuffs et al. Reference Tuffs, Popescu, Völk, Kylafis and Dopita2004; Natale et al. Reference Natale, Popescu, Tuffs, Debattista, Fischera and Grootes2015; Narayanan et al. Reference Narayanan, Conroy, Davé, Johnson and Popping2018; Lin et al. Reference Lin, Hirashita, Camps and Baes2021; Sachdeva & Nath Reference Sachdeva and Nath2022; Witt, Thronson, & Capuano Reference Witt and Thronson1992). In Figure 1, we present illustrations depicting two extreme versions of a dust geometry model. The first such model considered here, known as the foreground screen dust geometry model (Calzetti, Kinney, & Storchi-Bergmann Reference Calzetti, Kinney and Storchi-Bergmann1994), assumes that all the dust lies in a screen between the observer and the stars (Figure 1a-c). Figure 1a shows the case in which there is a low $M_d$ in a foreground screen geometry. In this scenario, there is little obscuration of the light. In Figure 1b, the increase in $M_d$ results in greater obscuration due to the increased optical depth. In Figure 1c the $M_d$ has increased to an extreme limit in which the optical depth is so great that any starlight is completely obscured.

Figure 1. Diagram of different dust geometries. The observer lies to the right of each panel in this figure. Lightly coloured dust screens or clouds indicate low $M_d$ , and darker dust screens or clouds indicate higher $M_d$ . Blue stars (with many points) indicate little or no obscuration, green stars indicate mid to high obscuration, and red stars (with five points) indicate complete obscuration. Each panel in this figure represents a galaxy of the same size, such that an increase in dust mass results in an increase in optical depth.

In the distributed dust geometry model the dust and stars are mixed (Figure 1d-f). Figure 1d illustrates a scenario with low $M_d$ in a distributed geometry, resulting in little obscuration of the light. In Figure 1e the $M_d$ , and therefore optical depth, are increased. The light from deeper within the cloud is more obscured due to travelling through more dust to reach the observer. Conversely, the light from the edges of the cloud closer to the observer are less obscured. In Figure 1f the $M_d$ has been increased to an observational limit in which the light from deeper within the cloud is no longer detected. This light has been completely obscured. The light from the closer edges of the cloud, which travels through less dust, is less obscured, resulting in the detection of some obscured light.

With the foreground screen geometry, all light of a given wavelength experiences consistent levels of obscuration, whether little or heavy obscuration, as it must travel through the same amount of dust. With a distributed dust geometry, light of a given wavelength from stars deeper within the cloud will experience more obscuration than light of the same wavelength from stars towards the edges of the cloud as the light from deeper within the cloud must travel through more dust. This means that the relative depth of stars within the dust cloud determines the level of obscuration, with deeper stars experiencing higher attenuation. Additionally, as the $M_d$ increases, a wider range of obscuration levels can be detected due to the varying positions of stars within the dust clouds.

The level of obscuration within galaxies can be quantitatively assessed through the BD, while the optical depth is intrinsically connected to $M_d$ . Given this relationship, combining the BD and $M_d$ offers a promising approach to investigate the intricate geometry of dust in galaxies. We use data from the Galaxy And Mass Assembly (GAMA) survey, described below, following which we introduce two novel parameters that link the BD and $M_d$ . These parameters aim to provide deeper insights into the role of dust geometry in influencing dust properties and SFRs.

2.1 Data

The GAMA survey is a spectroscopic and photometric survey that covers approximately 250 deg $^2$ of the sky over 5 regions using the AAOmega spectrograph of the 3.9m Anglo-Australian Telescope (Driver et al. Reference Driver2011; Liske et al. Reference Liske2015; Driver et al. Reference Driver2016; Baldry et al. Reference Baldry2018; Driver et al. Reference Driver2022).

The GAMA data and derived parameters are organised into Data Management Units (DMUs, Liske et al. Reference Liske2015). We use data and derived parameters from various DMUs, detailed in Table 1. Specifically, emission line flux measurements and uncertainties for H $\alpha$ , H $\beta$ , N[ii], and O[iii], in addition to equivalent width measurements and uncertainties for the H $\alpha$ and H $\beta$ emission lines are obtained from the SpecLineSFR DMU (Gordon et al. Reference Gordon2017). The SpecLineSFR DMU also provides the original survey source and redshift estimates with redshift quality measures, nQ. The MAGPHYS DMU provides dust mass, $M_d$ , estimates and percentile ranges which were used to obtain the uncertainties on $M_d$ . While MAGPHYS is a robust tool for estimating dust masses, it is important to acknowledge that different SED models can yield systematically different $M_d$ estimates. The StellarMasses DMU (Taylor et al. Reference Taylor2011b) provides the stellar mass, $M_*$ , estimates and uncertainties in addition to obscuration-corrected absolute magnitude measurements and uncertainties in the r band. The $M_*$ estimates of Taylor et al. (Reference Taylor2011b) are based on the (Bruzual & Charlot Reference Bruzual and Charlot2003) population synthesis code, which is also used by MAGPHYS (da Cunha, Charlot, & Elbaz Reference da Cunha, Charlot and Elbaz2008; Taylor et al. Reference Taylor2011a). Uncorrected measurements and uncertainties for the r band and FUV band were also used. The FIR flux measurements and uncertainties were taken from the gkvFarIR DMU (Bellstedt et al. Reference Bellstedt2020b). The approximate elliptical semi-major axis and axial ratio values used to calculate the galaxy areas were obtained from the gkvInputCat DMU (Bellstedt et al. Reference Bellstedt2020a).

Table 1. Summary of the data and derived parameters used and their GAMA DMUs.

The sample used here was selected to ensure the data was of high quality. Only objects with redshifts originating from the GAMA, SDSS, and 2dFGRS surveys were used. The sample was limited to objects with redshift quality nQ $\geq$ 3, which indicates a reliable estimate that is suitable for use in scientific analysis (Driver et al. Reference Driver2011). Only galaxies with lines in emission were retained, and the emission line flux measurements (H $\alpha$ , H $\beta$ , N[ii], O[iii]) were constrained to only those with a signal-to-noise ratio (S/N) $\geq$ 5. Similarly, the sample only contains dust mass estimates with S/N $\geq$ 3, and FIR flux measurements with S/N $\geq$ 1. We use the standard optical diagnostic diagram of Baldwin, Phillips, & Terlevich (Reference Baldwin, Phillips and Terlevich1981) (BPT) to classify our galaxies as star forming or hosting active galactic nuclei (AGN). AGN were identified using the Kauffmann et al. (Reference Kauffmann2003) definition, and removed from our sample due to the significant effect AGN can have on the observed emission lines and the inferred properties of dust within galaxies. Table 2 shows how many objects remain after these various criteria are applied. Highly inclined galaxies are not removed from this sample. Less than 5% of the galaxies in this sample are moderately or highly inclined (with an axial ratio < 0.4), and removing these galaxies does not alter our results. Retaining these more highly inclined galaxies maintains the completeness of the sample and demonstrates that our analysis is not sensitive to inclination.

Table 2. The number of objects remaining in the sample after each selection criteria was applied to the data.

2.2 BD, $\boldsymbol{M}_{\boldsymbol{d}}$ $\&$ dust geometry

To quantify the BD we use the H $\alpha$ and H $\beta$ emission line flux measurements. These were first corrected for stellar absorption following Hopkins et al. (Reference Hopkins2003), as

(4) \begin{equation} F_{\text{H}\alpha} = \frac{(\text{H}\alpha \text{EW} + \text{EW}_c)}{\text{H}\alpha \text{EW}} f_{\text{H}\alpha}\end{equation}

(5) \begin{equation} F_{\text{H}\beta} = \frac{(\text{H}\beta \text{EW} + \text{EW}_c)}{\text{H}\beta \text{EW}} f_{\text{H}\beta}\end{equation}

where $f_{\text{H}\alpha}$ is the H $\alpha$ emission line flux, $f_{\text{H}\beta}$ is the H $\beta$ emission line flux, H $\alpha$ EW is the equivalent width of the H $\alpha$ emission line, H $\beta$ EW is the equivalent width of the H $\beta$ emission line, and we adopt the same stellar absorption equivalent width correction for both H $\alpha$ and H $\beta$ of EW $_c$ = 2.5 following Gunawardhana et al. (Reference Gunawardhana2013). Finally, the BD values are calculated as

(6) \begin{equation} \text{BD} = \frac{F_{\text{H}\alpha}}{F_{\text{H}\beta}}.\end{equation}

The BD values calculated from each of GAMA, SDSS & 2dFGRS were compared to ensure that all three were providing consistent BD values within acceptable ranges. Although 2dFGRS spectra are not flux calibrated, the BD values are reliable, as they span the same range as those seen with the flux calibrated GAMA and SDSS spectra. If we omit them from our analysis our results remain unchanged apart from having slightly fewer galaxies represented.

Figure 2 shows the relationship between the BD and both $M_d$ (Figure 2a) and dust surface density ( $\Sigma_{M_d}$ ) (Figure 2b). There is an upper envelope for the observed BD values evident, which increases with increasing $M_d$ and $\Sigma_{M_d}$ . The dotted lines shown tracing these envelopes are empirical characterisations. For this dataset, the two different envelope lines are given by:

(7) \begin{equation} \log\!(\text{BD}_{\text{Env}}) = 0.185 \log(M_d) - 0.481,\end{equation}

and

(8) \begin{equation} \log\!(\text{BD}_{\text{Env}}) = 0.459 \log(\Sigma_{M_d}) - 1.801.\end{equation}

Figure 2. BD as a function of (a) $M_d$ , and (b) dust surface density, coloured by $M_*$ . The black dotted line represents the observed upper envelope of the data. The black dashed line represents the BD Case B value of 2.86. This Case B value of 2.86 is the BD value which corresponds to no obscuration (Osterbrock Reference Osterbrock1989). The correlation coefficient for panel (a) is 0.022 and the correlation coefficient for panel (b) is 0.013.

It is important to note that if different samples are being used, while qualitatively similar, such envelopes may differ quantitatively, especially if $M_d$ is estimated with different population synthesis tools.

The envelope is not a consequence of the S/N limits placed on the dataset, as an envelope of the same shape is present even when no S/N limits are imposed. Investigating the H $\beta$ line flux as a function of $M_d$ does reveal that at the lowest values of $M_d$ there is a tendency for the faintest H $\beta$ fluxes to be absent. This, however, does not explain the existence of the envelope at the high $M_d$ end, nor its relatively linear shape over the full range of $M_d$ . As a result, we are confident that the envelope seen here is not an observational bias, nor is it a result of our sample selection limits.

In the case of Figure 2a this envelope traces the BD values resulting from a foreground screen geometry as the optical depth of the screen increases. We focus here on BD and $M_d$ for the purpose of this illustrative discussion of dust geometry. The low optical depth foreground screen geometry lies at the low BD, low $M_d$ end of the envelope. The high optical depth foreground screen geometry lies at the high BD, high $M_d$ end of the envelope. The models from Figure 1 are positioned in Figure 3, to capture their representative locations in the diagram, and to emphasise how each model would be reflected in the quantitative measurements.

Figure 3. Schematic diagram conceptualising where the different dust geometries shown in Figure 1 are expected to fall in the BD vs $M_d$ diagram. The data from Figure 2a is represented here as a grey data density plot.

In the case of low dust content, the foreground screen geometry and distributed geometry both fall in the low BD region, as neither provides enough attenuation to produce high BD measurements. This indicates that with less dust, the differences in geometries become more redundant in terms of their impact on the BD. However, even in low dust content scenarios, certain configurations of distributed geometry, such as those depicted in Figure 1f, can result in variations in the BD values.

There is a point at which the dust becomes so optically thick that the Balmer lines are too attenuated to escape. In the case of the foreground screen this results in no measurement of the Balmer lines at all, and corresponds to the high BD, high $M_d$ region that is lacking any data, as seen in the upper right of figures 2a and 3.

For the case of the distributed dust geometry, the emission lines from stars buried within the dust cloud escape less than emission at the outer boundary. This results in the emission lines originating in the edges of the cloud experiencing less attenuation and being observed, allowing for measurements that lie in the low BD, high $M_d$ region. Therefore, the data points along the Case B line, with a low BD, correspond to a maximal distributed dust geometry. A mixture of these two extreme dust geometries allows for the spread of values observed between the envelope line and the Case B line.

It is well established that $M_d$ in galaxies is correlated with the $M_*$ . This is less the case for $\Sigma_{M_d}$ , and these relationships are shown together in Figure 4. Panel 4a shows the direct relationship between stellar mass and $M_d$ , while panel 4c shows the ratio of $M_d$ to stellar mass as a function of stellar mass. Panel 4b displays the relationship between $\Sigma_{M_d}$ and stellar mass. There is no panel that depicts the relationship between $\Sigma_{M_d}$ and $\Sigma_{M_*}$ as that would in effect be the same as panel 4a. However, panel 4d does show the ratio of $\Sigma_{M_d}$ to $\Sigma_{M_*}$ as a function of $\Sigma_{M_*}$ . These relationships are explored further by Cortese et al. (Reference Cortese2012), Clemens et al. (Reference Clemens2013), Calura et al. (Reference Calura2017), De Vis et al. (Reference De Vis2017), Orellana et al. (Reference Orellana2017), Casasola et al. (Reference Casasola2020); Casasola et al. (Reference Casasola2022). The trends in Figure 4 are colour coded by the SFR estimated from the H $\alpha$ luminosity, which we calculate as follows. The obscuration corrected $L_{\text{H}\alpha}$ comes from

(9) \begin{align} L_{\text{H}\alpha} &= (\text{EW}_{\text{H}\alpha} + \text{EW}_c) 10^{-0.4(M_r-34.10)}\nonumber \\ &\qquad \times\frac{3\times10^{18}}{[6564.61(1+z)]^2}\left(\frac{\text{BD}}{2.86}\right)^{\!\!2.36}\end{align}

where $M_r$ is the obscuration corrected r band absolute magnitude, and z is the redshift (Gunawardhana et al. Reference Gunawardhana2011, Reference Gunawardhana2013). If the observed BD is less than 2.86, then it is set to 2.86 here, equivalent to having no obscuration correction term. Equation (9) also includes terms that apply aperture and stellar absorption corrections, which are required for the H $\alpha$ . The SFR $_{\text{H}\alpha}$ is then calculated as

(10) \begin{equation} \text{SFR}_{\text{H}\alpha} = \frac{L_{\text{H}\alpha}}{1.27\times 10^{34}}\end{equation}

(Kennicutt Reference Kennicutt1998; Gunawardhana et al. Reference Gunawardhana2011).

Figure 4. (a) $M_d$ as a function of $M_*$ , (b) $\Sigma_{M_d}$ as a function of $M_*$ , (c) $M_d/M_*$ as a function of $M_*$ , and (d) $\Sigma_{M_d}/\Sigma_{M_*}$ as a function of $\Sigma_{M_*}$ , with all panels coloured by H $\alpha$ SFR. The correlation coefficients are 0.66 for panel (a), 0.28 for panel (b), −0.032 for panel (c), and −0.47 for panel (d).

The $M_d$ and stellar mass are correlated, although with a scatter of about 1 dex around the broad trend. This is reflected in the relatively flat relationship seen in Figure 4c, albeit with the scatter emphasised in this representation. It is apparent that much of this scatter is related to the SFR. As the SFR increases with stellar mass, galaxies of a given dust mass with high SFRs have larger stellar masses than those with low SFRs. Figure 4d shows the ratio of surface densities for dust and stellar mass, which is actually identical to $M_d/M_*$ as the surface area term cancels. Looking at this parameter as a function of $\Sigma_{M_*}$ , however, highlights that galaxies with the largest stellar mass surface density favour a lower proportion of dust mass, although the trend is mostly driven by the relatively small number of galaxies with $\Sigma_{M_*} \gtrsim 10^9\,{\rm M}_{\odot}$ /kpc $^2$ . This may suggest that the most compact galaxies may have proportionally less dust than more typical star forming galaxies. The interplay between stellar mass, $M_d$ , and SFR implicitly includes the contribution of BD, as it is incorporated in the H $\alpha$ SFR estimate. In order to tease out these related parameters further, it is helpful to explore new ways of quantifying the links between BD, $M_d$ , and $\Sigma_{M_d}$ . We start with an investigation of SFR estimators that are sensitive in different degrees to the presence of obscuring dust.

3. The role of BD in understanding SFR tracers

Koyama et al. (Reference Koyama2015) studied the relationship between H $\alpha$ attenuation and the ratio of SFR $_{\text{H}\alpha}$ to SFR $_{\text{FUV}}$ . This ratio can highlight the effect of optical depth due to the different effect at the different wavelengths. When the dust is more optically thick, proportionally less of the FUV emission will be detected, resulting in higher values of SFR $_{\text{H}\alpha}$ /SFR $_{\text{FUV}}$ . Koyama et al. (Reference Koyama2015) found a positive correlation between the H $\alpha$ attenuation and SFR $_{\text{H}\alpha}$ /SFR $_{\text{FUV}}$ , but noted that there is substantial scatter surrounding this relationship. Due to the scatter, they determined that dust attenuation levels could be roughly estimated using SFR $_{\text{H}\alpha}$ /SFR $_{\text{FUV}}$ , but that it could not be used as a more precise method of deriving the dust attenuation.

We reproduced their figure comparing H $\alpha$ attenuation and SFR $_{\text{H}\alpha}$ /SFR $_{\text{FUV}}$ , but instead used the BD rather than the H $\alpha$ attenuation, as a starting point to explore the relationships surrounding dust geometry and optical depth (Figure 5a). This figure shows the relationship when no obscuration corrections are applied to the SFR estimates. For SFR $_{\text{H}\alpha,Obs}$ , this simply corresponds to omitting the BD term from Equation (9). For SFR $_{\text{FUV},Obs}$ we use

(11) \begin{equation} L_{\text{FUV,Obs}} = 4\pi\text{D}_L^2 \times 10^{-0.4(m_{AB,\text{FUV}} + 56.1)}\end{equation}

where D $_L$ is the luminosity distance, $m_{AB,\text{FUV}}$ is the FUV band apparent AB magnitude. This is converted to SFR through

(12) \begin{equation} \text{SFR}_{\text{FUV,Obs}} = \frac{L_{\text{FUV},\text{Obs}}}{7.14\times 10^{20}}\end{equation}

(Kennicutt Reference Kennicutt1998; Hopkins et al. Reference Hopkins2003).

Figure 5. (a) BD as a function of SFR $_{\text{H}\alpha, \text{Obs}}$ /SFR $_{\text{FUV, Obs}}$ , and (b) BD as a function of SFR $_{\text{H}\alpha}$ /SFR $_{\text{FUV}}$ , both coloured by $M_*$ . The correlation coefficient for panel (a) is 0.21 and the correlation coefficient for panel (b) is −0.399.

Figure 5a shows a positive correlation between the BD and SFR $_{\text{H}\alpha, \text{Obs}}$ /SFR $_{\text{FUV, Obs}}$ . This is expected as both are tracers of the obscuration present, consistent with the results of Koyama et al. (Reference Koyama2015). To implement an obscuration correction for SFR $_{\text{FUV}}$ , we follow Calzetti et al. (Reference Calzetti, Armus, Bohlin, Kinney, Koornneef and Storchi-Bergmann2000), using

(13) \begin{equation} L_{\text{FUV}}(\lambda) = L_{\text{FUV,Obs}}(\lambda) 10^{0.4 E(B-V) k(\lambda)}\end{equation}

(Hopkins et al. Reference Hopkins, Connolly, Haarsma and Cram2001) where $k(\lambda)$ is the reddening curve (Calzetti et al. Reference Calzetti, Armus, Bohlin, Kinney, Koornneef and Storchi-Bergmann2000). For FUV the wavelength is $\lambda = 1\,500$ Å $ = 0.15\,\mu$ m and k(FUV) = 10.33. $E(B-V)$ is given by

(14) \begin{equation} E(B-V) = 0.44\frac{\log(\text{BD})}{0.4(k(\text{H}\beta)-k(\text{H}\alpha))}\end{equation}

(Calzetti Reference Calzetti2001a; Hopkins et al. Reference Hopkins, Connolly, Haarsma and Cram2001), with k(H $\alpha$ ) = 2.38 and k(H $\beta$ ) = 3.65. The obscuration corrected SFR $_{\text{FUV}}$ values are again calculated with the same SFR calibration factor

(15) \begin{equation} \text{SFR}_{\text{FUV}} = \frac{L_{\text{FUV}}}{7.14\times 10^{20}}\end{equation}

(Kennicutt Reference Kennicutt1998; Hopkins et al. Reference Hopkins2003).

With obscuration corrections in place, the correlation between BD and SFR $_{\text{H}\alpha}$ /SFR $_{\text{FUV}}$ is no longer present (Figure 5b). This difference when the obscuration corrections are applied is a direct consequence of the degree of obscuration correction, and that the FUV measurements require a larger correction for a given value of BD compared to the H $\alpha$ measurements. This is reflected in the apparent lower diagonal envelope to the data distribution, where higher values of BD lead to proportionally higher SFR $_\textrm{FUV}$ compared to SFR $_{\text{H}\alpha}$ . This causes the SFR ratio to decrease as BD increases, changing the positive correlation with the uncorrected ratio to a more or less vertical, uncorrelated, distribution. There is also a trend for galaxies of higher $M_*$ to show higher BD, seen in the vertical colour gradient. This is expected from the fact that higher $M_*$ implies higher $M_d$ , and high values of BD are only seen in high $M_d$ galaxies.

Instead of just investigating the H $\alpha$ and UV luminosities and their ratios, we can introduce other wavelength measurements as well. In particular the FIR emission is a well known tracer of the total dust-reradiated emission. We now use it as well in exploring this approach. Here we refer to the ratio of the FIR luminosity to the BD corrected H $\alpha$ luminosity as the ‘H $\alpha$ deficit.” We choose this term since, if some degree of optically thick dust is present, the H $\alpha$ luminosity will be reduced in comparison to the FIR luminosity. In the absence of optically thick dust affecting the H $\alpha$ , the two should be proportional, both tracing the underlying SFR.

The FIR luminosity is calculated simply with

(16) \begin{equation} L_{\text{FIR}} = 4\pi\text{D}_L^2 \times f_{\text{FIR}}\end{equation}

where $f_{\text{FIR}}$ is the Herschel-ATLAS $100\,\mu$ m flux, and D $_L$ is the luminosity distance. These results are qualitatively unchanged if we use the $160\,\mu$ m flux instead, or a linear combination of both. We choose to present the results here in terms of a single FIR band for simplicity. At this stage, we are ready to return to quantifying links between the BD, $M_d$ , and $\Sigma_{M_d}$ .

4. A new approach

To link the BD with the $M_d$ and $\Sigma_{M_d}$ , we introduce two new parameters. The first of these is a new parameter aimed at quantifying the mixing of the foreground screen and distributed dust geometries. This parameter, $F_\textrm{dust}$ , is calculated as

(17) \begin{equation} F_\textrm{dust} = \frac{\log\text{BD}-\log(2.86)}{\log\text{BD}_{\text{Env}}-\log(2.86)}\end{equation}

where BD $_{\text{Env}}$ is the BD value at the envelope line from Figure 2a (Equation 7).

$F_\textrm{dust}$ quantifies where a BD value lies vertically in relation to the Case B and envelope lines. In doing so, it quantifies the proportion that each geometry contributes in that galaxy, with higher values of $F_\textrm{dust}$ indicating a more foreground screen geometry and lower values of $F_\textrm{dust}$ indicating a more distributed geometry. A value of $F_\textrm{dust}=1$ may be referred to as a ‘maximal foreground screen’ geometry, and $F_\textrm{dust}=0$ as a ‘maximal distributed dust’ geometry. In our sample, for the small number of galaxies with BD lying above the envelope line or below the Case B line, we set $F_\textrm{dust}$ to be 1 or 0 respectively.

It is important to acknowledge that the terms ‘foreground screen’ and ‘distributed dust’ are used as convenient descriptors of dust distribution regimes and should not be interpreted literally. The primary distinction is that the ‘distributed dust’ regime can result in significant portions of the SFR and associated Balmer lines being entirely obscured by dust, while other regions experience only slight extinction. As a result, the BD can be small despite substantial loss of H $\alpha$ emission. In contrast, the ‘foreground screen’ regime implies a moderate and more uniform obscuration of all emission regions, leading to a larger BD while still allowing a considerable amount of H $\alpha$ emission to be observed.

Since Figure 2b shows a similar structure with an envelope line, a version of $F_\textrm{dust}$ may also be calculated from this $\Sigma_{M_d}$ version of the figure. The version of $F_\textrm{dust}$ calculated from the $\Sigma_{M_d}$ figure (Figure 2b) will be referred to as $\Sigma_{F_\textrm{dust}}$ and uses BD $_{\text{Env}}$ from Equation (8).

Figure 6 compares the $F_\textrm{dust}$ and $\Sigma_{F_\textrm{dust}}$ values. The data are centred quite evenly around the 1:1 line, although there is a slight tendency towards somewhat higher $F_\textrm{dust}$ values compared to $\Sigma_{F_\textrm{ dust}}$ . The mostly even distribution about the 1:1 line indicates that $F_\textrm{dust}$ and $\Sigma_{F_\textrm{ dust}}$ are sufficiently similar that either may be used in exploring dust geometry. However, it is worth noting that the interpretation of the envelope line in these diagrams cannot imply the same dust geometry in both cases. We argue that $F_\textrm{dust}$ is the choice that better matches a model where the envelope line represents a foreground screen dust geometry.

Figure 6. $\Sigma_{F_\textrm{dust}}$ as a function of $F_\textrm{dust}$ , coloured by $M_*$ . The dotted line is a 1:1 line. The correlation coefficient is 0.94.

Independent of any envelope line, the BD and $M_d$ may still be combined in such a way that more information about the geometry and optical depth of the dust may be inferred than if they are used independently. We introduce a second new parameter, $H_\textrm{dust}$ , that links the BD and $M_d$ in a different way, in order to quantify the dust geometry. $H_\textrm{dust}$ is defined in terms of the BD as

(18) \begin{equation} H_\textrm{dust} = 10^{1.0508}\frac{M_d}{\text{BD}^{2.303}}.\end{equation}

Equation (18) was derived using

(19) \begin{equation} H_\textrm{dust} = \frac{M_d}{10^{\tau^l_B}}\end{equation}

which is equivalent in the case of a foreground screen geometry. Therefore, $H_\textrm{dust}$ can be interpreted as a normalised dust mass, providing a way to quantify dust geometry beyond using BD and $M_d$ independently. As with $F_\textrm{dust}$ , an alternate version of $H_\textrm{dust}$ can be calculated in which $\Sigma_{M_d}$ is used in place of $M_d$ . This dust surface density version will be referred to as $\Sigma_{H_\textrm{dust}}$ .

Figure 7 compares $H_\textrm{dust}$ and $\Sigma_{H_\textrm{dust}}$ . Although the $H_\textrm{dust}$ values are consistently higher than the $\Sigma_{H_\textrm{dust}}$ values, the two parameters are still correlated. For the purposes of this investigation $H_\textrm{dust}$ is the more intuitive quantity. $H_\textrm{dust}$ is a global parameter as it uses the total dust content of the galaxy, whereas $\Sigma_{H_\textrm{dust}}$ , using a surface density, is related to the spatial distribution of the dust. In this analysis we are interested in exploring parameters like star formation and dust geometry as global galaxy properties. Were another study to be conducted focusing on surface densities and the spatial distribution of galaxy properties, then perhaps $\Sigma_{H_\textrm{dust}}$ would be the more useful parameter.

Figure 7. $\Sigma_{H_\textrm{dust}}$ as a function of $H_\textrm{dust}$ , coloured by $M_*$ . The dotted line is a 1:1 line. The correlation coefficient is 0.77.

Seeing as the dust mass and dust surface density definitions of F and H are correlated with one another, either may be used and yield similar results. To avoid repetition and eliminate redundancy, this analysis continues with only one set. As discussed above, $F_\textrm{dust}$ and $H_\textrm{dust}$ provide the more intuitive choice for this investigation. Thus, the remainder of this analysis focuses on $F_\textrm{dust}$ and $H_\textrm{dust}$ .

Figure 8 presents the relationship between $F_\textrm{dust}$ and $H_\textrm{dust}$ , showing a negative correlation, as expected from the way the two parameters are defined. There is a strong $M_*$ dependence visible, in the sense that the inverse correlation between $F_\textrm{dust}$ and $H_\textrm{dust}$ moves to higher values of $H_\textrm{dust}$ as $M_*$ increases. Galaxies with higher $F_\textrm{dust}$ tend to have lower values of $H_\textrm{dust}$ . High $F_\textrm{dust}$ and low $H_\textrm{dust}$ correspond to a more foreground screen dust geometry with more optically thin dust. For galaxies with lower values of $F_\textrm{dust}$ , the spread of $H_\textrm{dust}$ values increases. This indicates that as the geometry moves towards a greater proportion of distributed dust, there is greater variation in the optical depth observed. A numerical quantification of the anticorrelation between $F_\textrm{dust}$ and $H_\textrm{dust}$ is not especially illuminating, as this will be dependent on the specific choice of envelope line, which in turn will be dependent on the specific sample being used, and the determination of dust masses.

Figure 8. $F_\textrm{dust}$ as a function of $H_\textrm{dust}$ , coloured by $M_*$ . The correlation coefficient is −0.56.

We can use another approach to consider the qualitative geometry of the dust as well. If we assume the simplest scenario, that every galaxy has dust arranged in a foreground screen, that has an implication for how we can interpret $\Sigma_{M_d}$ . This can be thought of as representing the thickness of the screen, if we consider the dust mass, $M_d$ , as representative of the volume over which it is spread. In this scenario, we would expect that the BD would be correlated with $\Sigma_{M_d}$ , since a thicker screen would produce a larger BD. We explore this explicitly in Figure 9, which shows $\Sigma_{M_d}$ as a function of BD, coloured by $H_\textrm{dust}$ . We can see here that there is no correlation between $\Sigma_{M_d}$ and BD, with the possible exception of galaxies with the lowest $H_\textrm{dust}$ . Galaxies with the highest $H_\textrm{dust}$ show the largest $\Sigma_{M_d}$ values, but are restricted to a narrow range of the lowest BD. If the foreground screen interpretation is correct we would expect the thickest screen (largest $\Sigma_{M_d}$ ) to have the highest BD, but this is not the case. Accordingly, this implies that these galaxies do not favour a foreground screen geometry, and our interpretation of them as having dust that is mixed and distributed throughout the galaxy is more likely. This permits low BD values to arise from the edges of the dust distribution, where low levels of obscuration can occur. The model proposed above (Figure 3) is supported by this result. Subsequently, we continue with our interpretation that galaxies with low BD, high $M_d$ , and thus high $H_\textrm{dust}$ , have a distributed dust geometry. This in turn is associated with dust that is more optically thick to H $\alpha$ emission.

Figure 9. $\Sigma_{M_d}$ as a function of BD, coloured by $H_\textrm{dust}$ . The dashed line represents the Case B value at BD = 2.86. The correlation coefficient is 0.013.

We now have two parameters which each provide a new way to quantify the optical depth and geometry of a galaxy’s dust, with $F_\textrm{ dust}$ being more directly related to the geometry and $H_\textrm{dust}$ being more directly related to the optical depth. The following analysis explores how these parameters are related to other observational quantities which are themselves expected to strongly trace the degree of optical depth of the dust.