An Updated Dust-to-Star Geometry: Dust Attenuation Does Not Depend on Inclination in 1.3 ≤z ≤2.6 Star-forming Galaxies from MOSDEF

For this work, we use galaxy spectra from the MOSFIRE Deep Evolution Field (MOSDEF) Survey (Kriek et al. 2015). MOSDEF obtained moderate-resolution (R = 3000–3650) rest-frame optical spectra of ∼1500 H-band selected galaxies in the CANDELS fields (Grogin et al. 2011; Koekemoer et al. 2011). Observations were taken in the Y, J, H, and K bands using the MOSFIRE spectrograph (McLean et al. 2012) on the Keck I telescope from 2012 December to 2016 May. The sample spans three redshift intervals (1.37 ≤ z ≤ 1.70, 2.09 ≤ z ≤ 2.61, and 2.95 ≤ z ≤ 3.60), targeted such that many prominent rest-frame emission and absorption features fall into the observed atmospheric transmission windows. For complete details of the MOSDEF survey, including sample selection, observation details, data reduction, and initial properties, we refer to Kriek et al. (2015).

We aim to study the effects of galaxy inclination on the dust parameters, so we need robust axis ratio measurements. The structural parameters for all galaxies in MOSDEF, including axis ratio (b/a), were measured by van der Wel et al. (2012) using GALFIT (Peng et al. 2010). Fits were made to HST imaging in three bands (F125W, F140W, and F160W) from CANDELS (Faber 2011).

In addition to spectra from MOSDEF, we have access to deep photometry from the 3D-HST survey (Brammer et al. 2012; Skelton et al. 2014; Momcheva et al. 2016). Stellar masses are determined by fitting the emission-line-corrected multiband photometry (0.3–8.0 μm) using the FAST fitting code (Kriek et al. 2009). We adopt flexible stellar population models (FSPS; Conroy et al. 2009; Conroy & Gunn 2010), MOSFIRE spectroscopic redshifts, a Chabrier stellar initial mass function (IMF; Chabrier 2003), the Calzetti et al. (2000) attenuation curve, delayed exponentially declining star formation histories, and solar metallicity. These best-fit models also yield measurements of the dust attenuation in the V band (A_V) and UV slope (β).

Prominent emission lines (Hα, Hβ, [O III], [N II], and [O II]) from MOSDEF spectra were fit to determine individual galaxy properties such as SFR and metallicity. Most emission lines were measured by fitting Gaussian and linear components (Kriek et al. 2015; Reddy et al. 2015). For the Balmer lines, stellar absorption must be modeled to ensure accurate flux measurements, so the best-fit stellar population models are used to measure the Balmer absorption lines and correct for the Balmer emission. For the full details on the emission-line fits to the individual galaxies, we refer to Reddy et al. (2015).

For our analysis, it is important to have an accurate SFR measurement for all galaxies in our sample. All galaxies in MOSDEF that have a strong $\left(3\sigma \right)$ detection in Hα and Hβ have a measured dust-corrected SFR (Reddy et al. 2015; Shivaei et al. 2015). In summary, the measured Hα flux is dust-corrected using the Balmer decrement (Hα/Hβ) and the Cardelli et al. (1989) extinction curve (which was shown to work well for MOSDEF galaxies by Reddy et al. 2020), and it was then converted into the Hα luminosity using the measured redshift. The SFRs are then determined from the Hα luminosity assuming a Chabrier (2003) IMF and the relation from Hao et al. (2011) for solar metallicity. In this work, we adopt these measured SFRs. However, we only include galaxies with at least a 5σ Hα detection for a more robust analysis and measurement of the emission-line properties after stacking (see Section 2.3).

For the galaxies that have a 5σ Hα measurement but no detected Hβ emission, we can only estimate a lower limit on the Balmer decrement using the previous method, and therefore, a lower limit on the SFR. Instead, for these galaxies that lack significant Hβ detections, we estimate the SFR based on Hα alone, correcting for dust using the A_V from the best-fit FAST models. We use a Calzetti et al. (2000) extinction law to correct from the V band at 5500 Å to the wavelength of Hα at 6565 Å. Finally, using the SFR calculation from Hao et al. (2011) and assuming a Chabrier (2003) IMF, we compute the SFR. We performed this calculation for all galaxies in the sample, including the galaxies that had strong Hβ detections. The SFRs measured from both methods agreed well, with an offset of 0.1 dex and similar scatter (see also Price et al. 2014).

For this work, we need a complete sample of star-forming galaxies with measured masses, SFRs, and axis ratios. To carry out measurements of the emission features, we require galaxies in our sample to have a detected Hα line and a covered Hβ line. Consequently, we restrict our sample to 1.37 ≤ z ≤ 2.61 so that the Hα and Hβ emission features are targeted for all galaxies. This reduces the original ∼1500 galaxies to 1061.

We then apply the following selection criteria to limit the sample to star-forming galaxies with accurate redshifts, SFRs, and little contamination from other sources. First, we remove any galaxies that do not have a high-quality redshift measurement from MOSDEF because we need to be able to accurately measure the location of the emission features (114 galaxies). Some galaxies do not have coverage of the Hα (43) or Hβ (82) emission lines, which are essential for our measurement of the Balmer decrements. Similarly, galaxies that do not have at least a signal-to-noise ratio of 5 for their Hα measurement are discarded (267). In addition, any galaxies classified as active galactic nuclei (AGN) in MOSDEF are removed (Coil et al. 2015; Azadi et al. 2017), ensuring that any emission lines are almost entirely associated with star formation (144). Next, we remove galaxies with axis ratio measurements that were flagged as suspicious, bad, or nonexistent by GALFIT and for which the differences between the F125W and F160W axis ratios exceed 0.08 (15). We restrict our sample to galaxies with $9\leqslant {\mathrm{log}}_{10}({M}_{* }/{M}_{\odot })\leqslant 11$ because the sample is only complete down to a stellar mass of 10⁹ M_⊙ (25). Finally, any galaxy in which Hα or Hβ is contaminated by a sky line is removed, where sky lines are identified by pixels for which the measured error is at least 10 times larger than the median measured error of the spectrum (63). After this process, 308 galaxies remained in our sample. Their masses, SFRs, and Balmer decrements or lower limits are displayed in Figure 1. All remaining galaxies fall in the star-forming region of the UVJ diagram (see Section 4; e.g., Wuyts et al. 2007; Williams et al. 2009).

Zoom In Zoom Out Reset image size

We note that these selection criteria may introduce some biases into our sample. In particular, we caution that the sample may be missing the most heavily obscured systems because a signal-to-noise ratio of 5 is required for their Hα measurement.

To determine the effects of inclination on dust attenuation, we require accurate axis ratio measurements for all galaxies. As mentioned in Section 2.1, axis ratios $\left(b/a\right)$ were measured in F125W, F140W, and F160W by van der Wel et al. (2012) using GALFIT (Peng et al. 2010). Due to their higher coverage and signal-to-noise ratio, we limit our analysis to the F125W and F160W axis ratio measurements. We checked the consistency of the F125W and F160W axis ratios, and for the vast majority of galaxies, they are consistent to better than 0.1 (see Figure 2). Therefore, either the F125W or the F160W axis ratio should be a robust way to split the galaxies into groups, as described in Section 3.1. Since the galaxies come from two distinct redshift samples, 1.37 ≤ z ≤ 1.70 and 2.09 ≤ z ≤ 2.61, using only one filter would measure axis ratios at a wide range of rest-frame wavelengths. To better compare the galaxies, we use the F125W axis ratios for the lower-redshift sample (resulting in a median rest wavelength of 4900 Å) and the F160W axis ratios for the higher-redshift sample (resulting in a median rest wavelength of 4700 Å).

Zoom In Zoom Out Reset image size

For our work, we examine galaxy parameters as a function of galaxy inclination. We use the axis ratio as a proxy of galaxy inclination, which assumes that all galaxies are disks. Therefore, we must ensure that the galaxies in our sample are consistent with being disky. Given the axis-ratio distribution of the galaxies, we can infer the properties of the population. As described in Lambas et al. (1992), a randomly oriented distribution of disk galaxies is expected to have a relatively flat distribution of axis ratios, while a randomly oriented distribution of ellipticals will have an increasing distribution that peaks at roughly (b/a) = 0.8. Figure 3 shows the distribution of axis ratios for the galaxies in our sample, which is consistent with the expected distribution for disk galaxies and inconsistent with the expected distribution for elliptical galaxies. Finally, Price et al. (2016, 2020) showed that the star-forming galaxies in MOSDEF at z ≈ 1.5 and z ≈ 2.2 have an average v/σ of ≈3.5 and ≈2.0, respectively, consistent with being rotating disks. Thus, we can assume in this work that galaxies are disky, so the axis ratio is a good indicator of the galaxy inclination.

Zoom In Zoom Out Reset image size

We also note that at this epoch of peak star formation, galaxy disks are thick (Elmegreen & Elmegreen 2006) and turbulent (Genzel et al. 2008; Simons et al. 2017), unlike the more rotationally supported disks that are observed locally. We further explore the implications of disk morphology as we discuss our results (Section 5.1).

Since we require accurate emission-line measurements to determine Balmer decrements, we divide the galaxies into eight groups with similar properties and stack their spectra. Many galaxy properties are known to vary strongly with mass and SFR, such as metallicity, morphology, and kinematics. Therefore, we ensure that galaxies of similar masses and SFRs are stacked together. For this sample, we divide the SFR-M_* plane into four bins, as shown in Figure 1. In particular, the galaxies are divided by mass into $9\leqslant {\mathrm{log}}_{10}({M}_{* }/{M}_{\odot })\lt 10$ (which we now refer to as "low-mass") and $10\leqslant {\mathrm{log}}_{10}({M}_{* }/{M}_{\odot })\lt 11$ (referred to as "high-mass"). Then, to complete dividing the SFR-M_* plane, we use the star-forming main sequence (SFMS) rather than the SFR alone. The SFMS is a tight relation between the SFR and mass of galaxies that varies with redshift. The position of a galaxy compared to the SFMS is an indicator of how quickly it is forming stars relative to galaxies of similar mass and redshift. We linearly scale the SFMS fit from Whitaker et al. (2014) to our data, which results in

$\begin{eqnarray}&&{\mathrm{log}}_{10}{\rm{SFR}}\left(\displaystyle \frac{{M}_{\odot }}{{\rm{yr}}}\right)=a+b{\mathrm{log}}_{10}\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right)-c{\mathrm{log}}_{10}{\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right)}^{2},\end{eqnarray} \tag{ 1 }$

where a = − 24.0415, b = 4.1693, and c = 0.1638. To ensure that there was no difference in the SFMS between the edge-on and face-on groups of galaxies, we fit the SFMS again to each of the groups separately, and found that the separate fits have very similar values to those of the total sample. Additionally, we recover the known redshift dependence of the SFMS by fitting the low-redshift and high-redshift samples separately. We find similar results whether we use a redshift-dependent SFMS or an SFMS that is constant with redshift. Therefore, for simplicity, we use the constant SFMS. We divide the sample into two groups of SFR using Equation (1), with one group above and one group below the SFMS.

Finally, because our motivating question is how galaxy inclination affects dust properties, we further divide each of the four bins in the SFR-M_* plane into two equal-sized groups in axis ratio, using the median axis ratio of the sample, b/a = 0.55, as a divider. Given that the galaxies are consistent with being disks (see Section 2.5), these two samples roughly correspond to edge-on (which refers to the groups with b/a < 0.55, the left panel of Figure 1) and face-on galaxies (referring to b/a ≥ 0.55, the right panel of Figure 1). The edge-on groups have a median b/a ≈ 0.4, while the face-on groups have a median b/a ≈ 0.7.

In summary, we have eight groups, where each of the four bins in the SFR-M_* plane are divided into edge-on and face-on axis ratio groups. We note that our results are unaffected by the choice of SFRs for this division. For example, we reach similar conclusions when using spectral energy distribution SFRs rather than Hα SFR. This result is not surprising, as several works have shown that the SFRs agree well (e.g., Steidel et al. 2014; Shivaei et al. 2016; Theios et al. 2019; Price et al. 2020).

In order to substantially increase the signal-to-noise ratio of the spectra to measure an accurate Balmer decrement for a complete sample of galaxies, we employ spectral stacking within each of the eight groups.

First, we move the spectra of all galaxies to the rest frame and correct for the luminosity distance. As all MOSDEF spectra were flux calibrated, the resulting spectra present the luminosity density. Therefore, stacking in this manner effectively weights the spectra by SFR, as a galaxy with higher SFR will have more flux in its Balmer lines and thus will contribute more to the Balmer decrement of the stack. In essence, this weights the spectra such that each star-forming region within these galaxies is counted roughly equally. Therefore, studying these stacks gives insight into how dust affects the star-forming regions at high redshift.

To compute a stacked spectrum, we first take the rest-frame spectrum of each galaxy in a group and create a mask to identify sky lines and regions with poor atmospheric transmission. We mask pixels where the measured error is at least 10 times larger than the median measured error of the spectrum. Next, all galaxies must be on the same wavelength axis to perform the stacking, so we linearly interpolate the spectra to 0.5 Å per pixel, which is roughly the dispersion of the spectra in the rest frame. Finally, before stacking, we extend the sky line mask to encompass one additional pixel to either side of the sky lines, as these pixels are far more uncertain due to the interpolation.

With all galaxies on the same wavelength axis and bad pixels masked out, we perform the stacking: at each pixel, we set the value of the stack to the median of all of the nonmasked points contributing to it. Due to the different redshifts of the galaxies, this procedure results in all of the galaxies contributing to the stacks in the regions of 4800–5100 Å and 6400–6700 Å rest frame, but as low as 10% contributing to the regions between these bands, such as 5500 Å. For the purposes of this work, we are interested the emission-line regions, so the larger uncertainties in the regions far from the lines are not relevant. Uncertainties on the spectrum are computed by adding the weighted errors of each contributing point in quadrature.

One concern of this technique is how well the emission lines from galaxies with different velocity dispersions will combine. To assess this potential issue, we generated synthetic Hα and Hβ lines with a realistic range of Balmer decrements (3–6), velocity dispersions (70–150 km s⁻¹), and redshifts (1.6 ≤ z ≤ 2.6). Using the same stacking technique that we applied to the data, we stacked these synthetic lines and attempted to recover their mean Balmer decrements, changing several parameters to assess the recovery accuracy. First, increasing the resolution of the spectral interpolation from 0.5 to 0.001 Å had no effect on the recovered Balmer decrements, so the effects of the Nyquist bias seem negligible. Then, we tested smoothing the synthetic emission lines to the same velocity dispersion and observed no effect on the recovered Balmer decrements. In all cases, stacking using the median was better at recovering the sample median Balmer decrement, whereas stacking using the mean typically underestimated the median Balmer decrement by roughly 2%. As a result of these tests, we proceeded with the spectral stacking by taking the median of all pixels that contribute to a given point. However, we also repeated the analysis in this paper with mean stacking, and found similar results that are consistent within the uncertainties.

We also stack the FAST stellar continuum models using the same method that is used for the spectra. We interpolate the models to the same wavelength axis as the spectra. Then, at each point, we take the median of all of the scaled FAST models that contribute to it. These continuum models include Balmer absorption and are therefore necessary for deriving accurate measurements of the Balmer-line fluxes in the stacks.

Since the FAST continuum models were generated from photometry, they need to be scaled to match the spectra. Assuming that the continuum has the correct shape, we compute two scale factors: one for the emission lines near Hα (using 6300 Å<λ < 6900 Å), and one for the lines near Hβ (using 4600 Å<λ < 5200 Å). When scaling the continuum, we mask out a broad region over the emission lines. Then, in each group, we subtract the scaled continuum model from the spectrum, leaving just the emission lines. We present a subset of the continuum-subtracted Hα and Hβ regions for the stacked spectra in Figure 4.

Zoom In Zoom Out Reset image size

For each of the stacks, we fit Hβ 4863 Å, [O III] 4960 Å and 5008 Å, Hα 6565 Å, and [N II] 6550 Å and 6585 Å. To first order, the lines are Gaussian, so all six emission lines are fit simultaneously with a Gaussian located at the expected center of each line at a fixed galaxy rotation velocity. We fit for redshift (which should be nearly zero because these are rest frame), velocity dispersion, and the amplitude of each line. With these fluxes, we compute the Balmer decrements and metallicities for each group.

To obtain the uncertainties on all of the emission-line measurements and resulting properties (i.e., Balmer decrement and metallicity), we use bootstrapping. Within each group, we randomly draw n galaxies with replacement, where n is the size of the group. Then we stack their spectra as described in Section 3.2 and repeat this process 100 times. Each of the 100 bootstrapped spectra in each group was fit and integrated with the same method as above, and then a lower and upper 1σuncertainty was computed as the 16th and 84th percentile of the resulting distribution, respectively.

Since the gas-phase metallicity is an important parameter for understanding dust in galaxies, we also measure the metallicities for the eight groups. Not all galaxies in the stacks cover the [O II] 3726–3729 Å doublet, so we are limited in options for our metallicity calibration. Fortunately, we have all the required lines for O3N2, which is one of the more robust calibrators because it is computed from multiple line ratios and independent of dust effects (e.g., Liu et al. 2008; Steidel et al. 2014; Sanders et al. 2015). Metallicities for the stacks are computed from this O3N2 line ratio, defined as

$\begin{eqnarray}&&{\rm{O}}3{\rm{N}}2={\mathrm{log}}_{10}\left(\displaystyle \frac{[{\rm{O}}\,{\rm\small{III}}]\ 5008\,\mathring{\rm A} \ /{\rm{H}}\beta }{[{\rm{N}}\,{\rm\small{II}}]\ 6585\,\mathring{\rm A} \ /\ {\rm{H}}\alpha }\right).\end{eqnarray} \tag{ 2 }$

We convert the measured O3N2 values into metallicity from the relation described in Bian et al. (2018), derived from observations of galaxies at low redshift that are analogs of high-redshift galaxies. This relation is

$\begin{eqnarray}&&12+{\mathrm{log}}_{10}\left({\rm{O}}/{\rm{H}}\right)=8.97\mbox{--}0.39\times {\rm{O}}3{\rm{N}}2.\end{eqnarray} \tag{ 3 }$

We also measure the metallicities of the 100 bootstrapped spectra in each group, fitting their emission lines and using the same calculation. Uncertainties on the metallicities are then derived from the 16th and 84th percentile of the resulting measured metallicity distribution.

The optical depth of dust depends on the opacity and density of the obscuring material, as well as on the path length of the light through the galaxy. Thus, we may expect that dust attenuation would correlate with the axis ratio, as the light from more edge-on galaxies may encounter more obscuring material. However, we find that the Balmer decrement does not depend on the inclination, implying that light from the youngest stars in observed edge-on, and face-on galaxies encounter the same amount of obscuration.

Interestingly, similar to the Balmer decrement, A_V and β do not change with inclination either. A_V and β measure the attenuation toward all stars, not just the young stellar population. Typically, the older stellar population is only attenuated by the diffuse ISM, as older stars are no longer in star-forming regions. Thus, we would expect to see a clear dependence of A_V and β on the axis ratio. Our finding that there is no trend of the Balmer decrement, A_V, or β with axis ratio seems to contradict any model in which ISM dust plays a large role in attenuation. For example, the two-component model mentioned in the introduction does not appear to hold, as it predicts a substantial ISM dust component, which would lead to inclination-dependent attenuation. However, the galaxies in our sample have relatively high sSFRs, even below the star-forming main sequence ($-9.0\lt {\mathrm{log}}_{10}\left(\mathrm{sSFR}\ {\mathrm{yr}}^{-1}\right)\lt -8.1$). Therefore, a substantial fraction of the stellar continuum light from these galaxies may be emitted from young stars in star-forming regions rather than from the older stellar populations, and ISM dust might not significantly attenuate the stellar light.

For a dust model to be consistent with our results, the model must be able to explain the following findings:

• 1.  
  Dust attenuation for both the Balmer lines and stellar continuum does not depend strongly on inclination (Figures 5, 6).
• 2.  
  The total amount of attenuation in the Balmer lines and in the continuum increases with galaxy mass (Figures 5, 6).
• 3.  
  The Balmer lines are increasingly more attenuated than the continuum as galaxy mass and SFR increase (Figure 7).

Based on these constraints, we propose a dust model in which attenuation occurs in three main components: large star-forming clumps, typical star-forming regions, and a very small amount from the ISM (see also Reddy et al. 2020). This model is similar to the two-component model that was found to be valid at low redshift, with the addition of large, dusty, star-forming clumps.

Large star-forming clumps are indeed observed in galaxies at z ∼ 2 in Hα and in rest-frame UV maps. These large, 1 kpc scale structures can contribute up to 20% of the star formation in the galaxy (Schreiber et al. 2011; Wuyts et al. 2012). Moreover, as galaxy mass increases, the clumps tend to be larger (Swinbank et al. 2012) and more common (Tadaki et al. 2014). As they are detected through Hα, these clumps are a strong source of Balmer emission.

Our dust model meets all of the constraints imposed by the data. First, to meet the constraint of independence from inclination effects, the ISM is not a large contributor to attenuation in our model. The observed galaxies have high sSFRs, which causes a significant fraction of the stellar light to be coming form star-forming regions rather than from the older population. Then, because most attenuation occurs in (roughly spherically symmetric) star-forming regions and clumps, changes in attenuation with inclination would not be observed (constraint 1). Second, clumps are larger and more common as galaxy mass increases. Since these clumps emit a fraction of the blue stellar light, as well as a large amount of Balmer emission, the average path length of the light increases with mass for both the continuum and Balmer lines (constraint 2). Third, Balmer attenuation is dominated by clumps, while stellar attenuation primarily probes the less obscured, typical, star-forming regions. Thus, the model predicts that the Balmer attenuation will exceed the continuum attenuation because clumps are more dusty. Then, as galaxy mass and SFR increase, the clump size and prevalence increases, and so the model predicts an increasing difference between the Balmer and stellar attenuation (constraint 3). Our model meets all constraints of the data, and is presented pictorially in Figure 9.

Zoom In Zoom Out Reset image size

A different explanation for our results might be that dusty star formation in these galaxies is concentrated toward the center, perhaps in a bulge-like structure. There are some indications for this in high-redshift galaxies, where ALMA observations detected high central infrared SFRs in high-mass galaxies at z = 2.5 (e.g., Barro et al. 2016). If the dominant source of the Balmer emission and attenuation is centrally located and occurs in a structure that is roughy spherical, then the encountered dust column density might not change substantially with inclination (constraint 1). In this central star formation model, larger galaxies have larger regions of central star formation, and thus a longer path length through attenuating material to escape (constraint 2). Finally, to explain the higher differential attenuation between the Balmer lines and continuum emission, the youngest stars must be formed closer to the center, while other star-forming regions might be located farther out. As the galaxy mass increases, the fraction of star formation from the dusty central starburst increases, and thus the nebular attenuation increases faster than the stellar attenuation (constraint 3).

Finally, it is expected that scattering of thermal emission in the disk of edge-on galaxies would lead to radiation loss in directions perpendicular to the disk (Misiriotis et al. 2001). We note that for our sample, we do not observe any difference in MIPS 24 micron emission between edge-on and face-on galaxies with similar properties. Although the MIPS 24 micron band does not probe thermal emission well because it is in the regime of polycyclic aromatic hydrocarbons, this result points to the fact that the energy balance is similar for both orientations. The reason that we do not see any difference may again be the high sSFR of our sample, which might cause most dust obscuration to occur in spherically symmetric regions. If indeed dust spread throughout the whole disk does not play a large role in scattering thermal emission, there would not be an expected trend with the axis ratio.

In the future, our proposed geometry can be tested with dust measurements of individual galaxies across the full range of axis ratios and SFRs, as well as in further investigations of the distribution of star formation and clumps in high-redshift galaxies.

Our data support a geometry in which attenuating dust is mostly concentrated around star-forming regions and clumps, explaining the observations that the Balmer decrement does not vary with axis ratio. The ISM does not seem to play a large role in attenuation, especially due to the relatively high sSFR of the galaxies. Our findings fit the framework suggested by Reddy et al. (2020), in which young star-forming regions may be dustier and contribute strongly to nebular attenuation. Our observations also partially agree with the patchy dust model proposed by Reddy et al. (2015), according to which the youngest stars become increasingly more obscured as the SFR increases. However, this model is not fully consistent with our observations because it also predicts that the stellar attenuation is dominated by the more diffuse ISM dust, and thus we would expect to see a trend between the axis ratio and A_V or β. Another patchy dust model is presented in Fetherolf et al. (2023), in which the dust transitions from smooth to patchy as galaxy mass increases, and dusty clumps are predicted at high masses. This model is consistent with our results as long as the ISM dust does not cause much of the observed stellar attenuation.

The lack of any Balmer decrement correlation with axis ratio is consistent with Yip et al. (2010), Battisti et al. (2017), and Yuan et al. (2021), who all found analogous results for samples at similar stellar masses at low redshift, and consequently lower SFR. Yip et al. (2010) and Yuan et al. (2021) observed that the stellar continuum attenuation is higher in low-redshift edge-on galaxies than in face-on galaxies. Nonetheless, they concluded that dust attenuation primarily takes place in H II regions and that attenuation in the ISM of the thick disk is insignificant, similar to our results. Our results also contradict Patel et al. (2012) at lower redshift $\left(0.6\lt z\lt 0.9\right)$, who found that galaxies with higher V − J values have lower axis ratios. Similar results were found by Zuckerman et al. (2021), who showed that the wide range of dust attenuation values measured for star-forming galaxies at a given redshift and stellar mass is almost entirely due to the effect of inclination. This result is mostly valid for z < 1.5, but it is less apparent at higher redshifts. These inconsistencies may be partly explained by the lower sSFR at later times, as less stellar light would originate from star-forming regions. Furthermore, there may be more ISM dust at lower redshifts. Indeed, at z ≈ 1.4 and at lower sSFRs than in this work, Price et al. (2014) demonstrated a relation between the sSFR and the difference between nebular and stellar attenuation for star-forming galaxies in the 3D-HST survey.

In Section 4 we found that the Balmer decrement is independent of the SFR at fixed stellar mass for the low-mass galaxies and shows only a minor dependence on SFR for the more massive galaxies. This result may seem puzzling because galaxies with higher SFRs at fixed mass have higher gas fractions (e.g., Kennicutt 1998), and thus may be expected to have more dust. The amount of dust in a galaxy is connected to the amount of gas through the gas-phase metallicity. Thus, in order to explain our results, the next natural step is to investigate the metallicities of our galaxies.

In Figure 10 we show that the median mass of each group and the measured metallicity (from stacked spectra) recovers the well-known mass–metallicity relation (Tremonti et al. 2004): higher-mass galaxies have higher gravitational potentials, so they retain more metals. Figure 10 also shows that we recover the fundamental metallicity relation (FMR; Mannucci et al. 2010): a higher SFR leads to lower metallicity at fixed mass. This anticorrelation between the SFR and metallicity is thought to arise because high SFR tends to be indicative of a high inflow of metal-poor gas from the intergalactic medium (IGM), and vice versa. We refer to Sanders et al. (2015, 2018, 2020, 2021) for detailed studies of the FMR in the MOSDEF survey.

Zoom In Zoom Out Reset image size

Since the Balmer decrement increases with mass, we might expect from the mass–metallicity relation that the Balmer decrement would also increase with metallicity. While this trend holds at a large-scale level, our data reveal that the Balmer attenuation does not strongly depend on the metallicity at fixed mass. The left panel of Figure 11 shows that one value of the Balmer attenuation (calculated directly from the Balmer decrement, as in Price et al. 2014) corresponds to a wide range of metallicities for low-mass galaxies, spanning 0.2 dex. At high mass, there seems to be a trend in the opposite direction than expected, where higher metallicities exhibit lower Balmer decrements. Similarly, we show that there is no correlation between the Balmer attenuation and SFR for the low-mass galaxies (Figure 11, right panel).

Zoom In Zoom Out Reset image size

Our results of a roughly constant Balmer attenuation at constant mass imply that these galaxies, despite the range in SFRs and metallicities, must have approximately the same dust column density. Fundamentally, this arises from the following argument. Assuming a fixed metal yield, the amount of metals in a galaxy only depends on the total mass that has been formed. Since dust is formed from metals, we expect that the dust column density (and thus the Balmer decrement) similarly only depends on the galaxy mass. On the other hand, because the metallicity is defined with respect to the hydrogen content, the metallicity depends on both the dust column density and the gas column density. For example, adding more low-metallicity gas (i.e., infalling from the IGM) will not change the dust column density, but it will lower the metallicity. Therefore, we find a range of metallicities at fixed Balmer decrement due to changes in the gas fraction.

Another way to understand the result that the Balmer decrement only depends on stellar mass is to use the observed relations to examine the interplay between the SFR and metallicity. To compare these properties, we consider both the FMR and the Kennicutt–Schmidt (K-S) relation (Kennicutt 1998). These relations together indeed provide a qualitative explanation for how a wide range of SFRs and metallicities can have the same Balmer attenuation. The K-S relation states that at fixed mass, galaxies with a higher SFR will have a higher hydrogen gas content, fueled by infalling IGM gas. This IGM gas is metal poor, resulting in a decrease in the metallicity as the SFR increases—this relation is the FMR. By definition, a low metallicity implies a low dust-to-gas ratio. Therefore, galaxies with higher SFRs have more gas, but a lower dust-to-gas ratio, and thus can have the same dust column density as galaxies with low SFRs and consequently high dust-to-gas ratios. The relation between metallicity, SFR, and the Balmer decrement is shown in the diagram of Figure 12.

Zoom In Zoom Out Reset image size

To assess whether these different relations indeed lead to a range of SFRs and metallicities at fixed Balmer decrement, we now explore this model quantitatively. As supported by our results, we assume a dust geometry that does not depend on inclination (e.g., Section 5.1). We assume that this attenuating dust has a dust mass absorption coefficient, κ_λ , which is spatially independent and that the dust is spread evenly throughout the attenuating region (either typical star-forming regions or dusty star-forming clumps). From these two assumptions, Shapley et al. (2022) derived that the dust attenuation is proportional to the dust-to-gas ratio and gas surface density,

$\begin{eqnarray}&&{A}_{\lambda }\approx 1.086\times {\kappa }_{\lambda }\times \left(\displaystyle \frac{{M}_{\mathrm{dust}}}{{M}_{\mathrm{gas}}}\right)\times {{\rm{\Sigma }}}_{\mathrm{gas}}.\end{eqnarray} \tag{ 4 }$

We can replace $\left({M}_{\mathrm{dust}}/{M}_{\mathrm{gas}}\right)$ using the power-law relation from De Vis et al. (2019), which is

$\begin{eqnarray}&&{\mathrm{log}}_{10}\left(\displaystyle \frac{{M}_{\mathrm{dust}}}{{M}_{\mathrm{gas}}}\right)=a\times [12+{\mathrm{log}}_{10}\left({\rm{O}}/{\rm{H}}\right)]+b.\end{eqnarray} \tag{ 5 }$

Empirically, they find a ≈ 2.15 for the O3N2 metallicity calibration, and we can fold b into our constants. We also note that there does not appear to be significant evolution in the dust-to-gas ratio from z = 0 to z ≈ 2, so this relation is applicable (Shapley et al. 2020; Popping et al. 2023). We can use the K-S relation (Kennicutt 1998) to replace Σ_gas from Equation (4), given by

$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{gas}}\propto {\left({{\rm{\Sigma }}}_{\mathrm{SFR}}\right)}^{\left(\tfrac{1}{n}\right)}.\end{eqnarray} \tag{ 6 }$

We now replace Σ_SFR with its definition, focusing specifically on the attenuating region, which is

$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{SFR}}=\displaystyle \frac{\mathrm{SFR}}{2\pi {R}_{\mathrm{sf}}^{2}}.\end{eqnarray} \tag{ 7 }$

Here, R_sf is the effective size of all of the star-forming regions and clumps. Since nearly all of star formation in these galaxies takes place in star-forming regions and clumps, the numerator of the K-S relation is the global SFR of the galaxy. Substituting everything into Equation (4), we arrive at

$\begin{eqnarray}&&{A}_{\lambda }\propto {10}^{a\times [12+{\mathrm{log}}_{10}\left({\rm{O}}/{\rm{H}}\right)]}\times {\left(\displaystyle \frac{\mathrm{SFR}}{{R}_{\mathrm{sf}}^{2}}\right)}^{\left(\tfrac{1}{n}\right)}.\end{eqnarray} \tag{ 8 }$

Therefore, we find that the Balmer attenuation, A_Balmer, should be higher with increasing metallicity (higher dust-to-gas ratio), increasing SFR (more gas in the galaxy), and decreasing R_sf (more gas per unit area). Finally, this gives a prediction for the relation between metallicity and SFR that can control the Balmer attenuation. At fixed mass, if the metallicity decreases linearly with ${\mathrm{log}}_{10}\left(\mathrm{SFR}\right)$, we would find a constant Balmer attenuation.

Coincidentally, the FMR does indeed show that the metallicity decreases roughly linearly with ${\mathrm{log}}_{10}\left(\mathrm{SFR}\right)$. To illustrate the relation between the FMR and our dust model, we plot our model, along with our measured SFRs, metallicities, and the FMR, in Figure 13. We fix a constant of proportionality and solve for the best-fitting R_sf using least squares for the low- and high-mass galaxies separately. The FMR is from Sanders et al. (2021) for MOSDEF z ≈ 2.3 and z ≈ 3.3 star-forming galaxies. This relation is

$\begin{eqnarray}&&12+{\mathrm{log}}_{10}\left({\rm{O}}/{\rm{H}}\right)=8.80+0.188y-0.220{y}^{2}-0.0531{y}^{3},\end{eqnarray} \tag{ 9 }$

where y = μ_0.60 − 10 and ${\mu }_{0.60}=\mathrm{log}\left({M}_{* }/{M}_{\odot }\right)-0.6\times \mathrm{log}\left(\mathrm{SFR}/{M}_{\odot }\ \ {\mathrm{yr}}^{-1}\right)$.

Zoom In Zoom Out Reset image size

Figure 13 shows that the groups divide clearly by mass, each falling on their own roughly linear relation between the metallicity and SFR, as dictated by the FMR from Sanders et al. (2021). We also see the lines of constant Balmer attenuation as predicted by our model in Equation (8). Most notably, we find that the slope of the FMR at fixed mass (≈ − 0.25) is very similar to the slope of a constant Balmer attenuation according to the model (−0.33). Therefore, at fixed mass, any SFR has a corresponding metallicity from the FMR that results in a constant Balmer decrement. This correlation implies that the galaxy mass is the only driver of the Balmer attenuation, which is highlighted in the diagram in Figure 12.

Finally, we note that for either of the proposed dust geometries from Section 5.1, this dust model can still hold. Although the dust may be distributed among star-forming regions and clumps, the attenuating dust is roughly spherically symmetric. The assumptions for this model require only a uniform dust distribution and a spatially independent κ_λ , which are both plausible.