Dirt-cheap Gas Scaling Relations: Using Dust Absorption, Metallicity, and Galaxy Size to Predict Gas Masses for Large Samples of Galaxies

The formation of stars from the cold gas in the interstellar medium (ISM) is a fundamental process of galaxy formation and evolution. The observed tight relationship between star formation rate (SFR) and stellar mass (M_⋆) in galaxies, known as the star-forming main sequence (Noeske et al. 2007), is interpreted as indicating that galaxies exhibit a self-regulated quasi-equilibrium between external gas accretion, star formation, and gas outflow (Bouché et al. 2010; Dekel et al. 2013; Lilly et al. 2013; Forbes et al. 2014; Peng & Maiolino 2014; Rodríguez-Puebla et al. 2016). However, the mechanisms that regulate the availability of gas for star formation and the efficiency with which it is converted to stars on galactic scales are not well understood. Energy and momentum inputs from massive stars, supernovae, and accreting supermassive black holes may regulate global star formation by driving galactic outflows or heating the ISM (e.g., Dekel et al. 2009; Hopkins et al. 2011, 2014; Ostriker & Shetty 2011; Faucher-Giguère et al. 2013; Harrison et al. 2018).

Although gas observations have strong constraining power for theoretical models of galaxy formation and evolution, directly measuring gas properties for large representative galaxy samples is a difficult and time-consuming endeavor. Neutral atomic hydrogen (H i) can be directly observed via the 21 cm hyperfine line, but molecular hydrogen (H₂) must be inferred indirectly from other tracers, such as the rotational transitions of CO. However, measuring these lines for galaxy sample sizes comparable to those of ultraviolet-optical surveys will remain unattainable for the foreseeable future. Blind H i surveys such as the Arecibo Legacy Fast ALFA survey do provide global (unresolved) measurements of the H i content for ∼31,500 galaxies (Haynes et al. 2018), but are only sensitive to the most gas-rich galaxies located at very modest redshifts (z ≲ 0.06). In contrast, the extended Galaxy Evolution Explorer (GALEX) Arecibo SDSS Survey (xGASS; Catinella et al. 2018) provides more sensitive measurements of atomic gas content for 1179 representative galaxies. A follow-up survey, xCOLD GASS (Saintonge et al. 2017), additionally obtained H₂ measurements for 477 galaxies using the CO(1–0) emission line. Although the GASS samples account for only less than 0.2% of local galaxies, they are large enough to statistically characterize gas scaling relations and their scatter. Saintonge et al. (2017) and Catinella et al. (2018) showed that typical state-of-the-art hydrodynamic and semi-analytic simulations, which currently implement sub-grid ISM models, do not succeed in reproducing the gas scaling relations based on the GASS data. But encouraging progress is being made (e.g., Diemer et al. 2019). In a similar spirit, pushing out to much greater distances, the Plateau de Bure High-z Blue Sequence Survey (Freundlich et al. 2019) legacy program collected CO data of ∼120 galaxies at z = 0.5–3. Detecting H i beyond the local universe remains challenging. Concerted efforts are being made with current interferometers, but the current samples remain frustratingly small (Verheijen et al. 2007; Hess et al. 2019); the most distant H i detection to date is at z = 0.316 (Fernández et al. 2016).

An alternative, albeit indirect, method to probe the gas content of galaxies is to estimate the dust mass from the far-infrared emission and convert it to gas mass using a gas-to-dust ratio. Dust is an important constituent of the ISM, and it has a huge impact on the chemistry and thermodynamics of the gas (Krumholz et al. 2011; Glover & Clark 2012; Gong et al. 2017). Dust shields molecules (e.g., CO) from photodissociation by attenuating the interstellar far-ultraviolet radiation field. The atomic-to-molecular hydrogen transition is governed by the balance between H₂ formation via dust grain catalysis and destruction by ultraviolet photons from the Lyman–Werner band. With the advent of the Herschel Space Observatory and the Atacama Large Millimeter Array, dust continuum emission has been utilized in numerous studies as a surrogate tool to estimate gas masses (e.g., Eales et al. 2012; Genzel et al. 2015; Groves et al. 2015; Scoville et al. 2016; Janowiecki et al. 2018; Shangguan et al. 2018, 2019). The far-infrared emission, however, is not a simple tracer; there are major systematic uncertainties or modeling challenges to exploiting it as a probe of gas content of galaxies (e.g., Berta et al. 2016). And rest-frame far-infrared measurements, while more widely available than H i or CO observations, are still non-trivial to amass for large galaxy samples.

This paper introduces new, efficient, and cost-effective methods to predict gas masses within a factor of ∼2–3 for large samples of galaxies. To that end, we revisit the relationship between the integrated gas mass, SFR, and galaxy morphology of the well-studied local galaxies in the xGASS and xCOLD GASS surveys (Saintonge et al. 2011, 2017; Catinella et al. 2018). We investigate whether parameters such as dust attenuation, galaxy radius, and galaxy morphology are useful for predicting atomic and molecular gas masses. We conclude that the dust extinction A_V of both the continuum and nebular emission significantly correlates with ${M}_{{{\rm{H}}}_{2}}$, after controlling for the effect of SFR. The same does not hold for M_{H i}. We find that R₅₀ also significantly correlates with both ${M}_{{{\rm{H}}}_{2}}$ and M_{H i}.

2.1. Stellar and Dust Properties

We use the publicly available Catalog Archive Server (CAS)⁴ to retrieve measurements of median stellar mass, galaxy size, and optical emission-line fluxes based on the Sloan Digital Sky Survey (SDSS) data release 15 (Aguado et al. 2019). We use the median metallicity estimate given in galSpecExtra table for non-AGN galaxies whose principal optical diagnostics emission-line ratios are measured at >5σ. These data are supplemented with global measurements of V-band dust attenuation (A_V) and SFR from version 2 of the GALEX-SDSS-Wide-field Infrared Survey Explorer Legacy Catalog (GSWLC-2; Salim et al. 2016, 2018), along with improved galaxy morphology measurements, derived using machine learning, from Domínguez Sánchez et al. (2018). Salim et al. (2016, 2018) used spectral energy distribution fitting of ultraviolet, optical, and infrared photometry to calculate the global A_V and SFR. To estimate accurate total infrared luminosities (∼0.1 dex uncertainty), Salim et al. (2018) used luminosity-dependent infrared templates (Chary & Elbaz 2001) and calibrations derived from a subset of galaxies that have far-infrared photometry from the Herschel ATLAS survey. They assumed a Chabrier (2003) stellar initial mass function. By combining accurate visual classifications from the Galaxy Zoo 2 project (Willett et al. 2013) and machine-based classifications from Nair & Abraham (2010), Domínguez Sánchez et al. (2018) provided one of the largest and most reliable (≳97% accuracy) morphological catalogs for the SDSS galaxies. Their classifications apply the convolutional neural networks learning algorithm to reanalyze the SDSS three-color galaxy images to learn the mapping between the images and the measurements in the visual classification catalogs, and also to correct misclassifications in the visual catalogs. The measurements from Domínguez Sánchez et al. (2018) that we use are the T-types and a set of probabilities that quantify whether a galaxy has a disk/feature or a bar, and whether it is an edge-on system or a merger. The T-types range from −3 to 10, whereby 0 corresponds to S0s, <0 corresponds to early-type galaxies, >0 corresponds to spirals (Sa−Sm), and 10 represents irregular galaxies. The disk/feature probability quantifies whether a galaxy is smooth or has a disk or features such as spiral arms.

We calculate the fiber V-band attenuation using the observed Hα/Hβ ratio and the dust attenuation curve, as (Charlot & Fall 2000; Wild et al. 2011)

$$\begin{eqnarray}&&{Q}_{\lambda }=0.6{(\lambda /5500)}^{-1.3}+0.4{(\lambda /5500)}^{-0.7}.\end{eqnarray} \tag{ 1 }$$

Assuming that the intrinsic, dust-free Balmer decrement is Hα/Hβ = 2.86 for inactive galaxies and Hα/Hβ = 3.1 for active galactic nuclei (AGNs; e.g., Ferland & Netzer 1983; Gaskell & Ferland 1984),

$$\begin{eqnarray}&&{A}_{V,\mathrm{fiber}}=\displaystyle \frac{2.5}{({Q}_{4861}-{Q}_{6563})}\times \mathrm{log}\displaystyle \frac{{\rm{H}}\alpha /{\rm{H}}\beta }{3.1\,\mathrm{or}\,2.86},\end{eqnarray} \tag{ 2 }$$

where Q₄₈₆₁ − Q₆₅₆₃ = 0.31. If the observed ratio of an object is below the intrinsic ratio, we set A_V,fiber = 0.

2.2. Atomic and Molecular Gas Properties

2.3. Statistical Methods

3.1. Predictors of Molecular Gas Mass

Table 1 summarizes the correlation and partial correlation analysis between the molecular or atomic gas and other galaxy properties. The integrated SFR, A_V, galaxy size, and disk/feature probability are among the galaxy properties that exhibit high Kendall's τ correlation coefficients (τ ≥ 0.3) with the gas mass. As expected, SFR shows the strongest correlation with gas content (τ = 0.58 for ${M}_{{{\rm{H}}}_{2}};$ τ = 0.46 for M_{H i}). We find a moderately strong (τ ≈ 0.4) but highly significant (>5σ) correlation between global or fiber A_V and ${M}_{{{\rm{H}}}_{2}}$. A_V correlates better with ${M}_{{{\rm{H}}}_{2}}$ than with M_{H i}, or the combination of the two. The partial correlation analysis indicates that there is a highly significant correlation between A_V and ${M}_{{{\rm{H}}}_{2}}$ (τ ≈ 0.2), after controlling for the effect of the SFR. In contrast, the partial correlation between A_V and M_{H i} is very weak and hardly statistically significant. The stellar r-band half-light radius of the galaxy, R₅₀, also has a partial τ ≈ 0.2 with both ${M}_{{{\rm{H}}}_{2}}$ and M_{H i} at fixed SFR. However, the correlation between R₅₀ and A_V is weak (τ = 0.1), which implies that controlling for the effect of the other hardly changes the partial correlation with the gas mass. After controlling for A_V, M_⋆ shows significant partial correction (τ = 0.24). This could be anticipated from the mass–metallicity relation (Tremonti et al. 2004) and from the dependence of dust-to-gas ratios on metallicity (e.g., Janowiecki et al. 2018). It is not so obvious why R₅₀ is better correlated (τ = 0.32) with ${M}_{{{\rm{H}}}_{2}}$ after fixing A_V. This correlation may reflect the association of R₅₀ with metallicity and its deviation from the mass–metallicity relation (Tremonti et al. 2004), and the processes that set the molecular-to-atomic gas ratio or gas mass to SFR ratio (i.e., the star formation efficiency). The disk/feature probability (P_d/f) shows a partial correlation of τ ≈ 0.3 with ${M}_{{\rm{H}}{\rm{I}}}$ after controlling for SFR or R₅₀. After controlling for SFR, no significant correlation exists between molecular/atomic gas mass and the stellar concentration index, stellar surface mass density, or merger or edge-on probability. The atomic gas partially correlates with T-type (τ = 0.2), given the SFR.

Table 1.  Kendall's τ Correlation and Partial Correlation Coefficients between Gas Mass and Galaxy Properties

Galaxy Property	${\tau }_{{{\rm{H}}}_{2}}$	${\tau }_{{{\rm{H}}}_{2}}|$AV	${\tau }_{{{\rm{H}}}_{2}}|$SFR	${\tau }_{{{\rm{H}}}_{2}}| {R}_{50}$	${\tau }_{{\rm{H}}{\rm{I}}}$	${\tau }_{{\rm{H}}{\rm{I}}}|$SFR	${\tau }_{{\rm{H}}{\rm{I}}}| {R}_{50}$	Significance
Fiber (nebular) dust attenuation AV	0.39	⋯	0.24	0.38	0.16	0.04	0.14	✓—✓ ✓✓ ✓ ✓
Global (stellar) dust attenuation AV	0.41	0.25	0.21	0.41	0.23	0.01	0.21	✓ ✓ ✓ ✓ ✓ ⨯ ✓
Global SFR	0.58	0.54	⋯	0.58	0.46	⋯	0.41	✓ ✓—✓ ✓— ✓
Petrosian radius R50	0.32	0.32	0.19	⋯	0.35	0.24	⋯	✓ ✓ ✓—✓ ✓ —
Petrosian radius R90	0.27	0.29	0.19	0.03	0.29	0.21	0.01	✓ ✓ ✓ ⨯ ✓ ✓ ⨯
Stellar mass M⋆	0.18	0.24	0.19	0.08	0.09	0.10	−0.08	✓ ✓ ✓ ⨯ ✓ ✓ ✓
Disk/feature probability Pd/f	0.30	0.21	0.10	0.19	0.37	0.26	0.28	✓ ✓ ✓ ✓ ✓ ✓ ✓
T-type	0.23	0.11	0.04	0.16	0.30	0.20	0.27	✓ ✓ ⨯ ✓ ✓ ✓ ✓
Bar probability	0.19	0.14	0.03	0.12	0.22	0.12	0.16	✓ ✓ ✓ ⨯ ✓ ✓ ✓
Concentration index R90/R50	−0.20	−0.11	−0.04	−0.17	−0.18	−0.08	−0.18	✓ ✓ ⨯ ✓ ✓ ✓ ✓
Mass density 0.5M⋆/(π R502)	−0.20	−0.14	−0.17	−0.03	−0.10	0.03	−0.03	✓ ✓ ✓ ⨯ ✓ ⨯ ⨯
Edge-on probability	−0.09	−0.16	∼0.00	−0.03	−0.07	−0.02	−0.02	✓ ✓ ⨯ ⨯ ⨯ ⨯ ⨯
Merger probability	0.04	0.08	∼0.01	∼0.01	0.01	0.01	∼0.00	⨯ ⨯ ⨯ ⨯ ⨯ ⨯ ⨯

Note. ${\tau }_{{{\rm{H}}}_{2}}$ and τ_{H i} are Kendall's τ rank-correlation coefficient for the molecular and atomic gas, respectively. ${\tau }_{y}| x\,:\,x\in \{{A}_{V},\mathrm{SFR},{R}_{50}\}\,\&\,y\in \{{{\rm{H}}}_{2},{\rm{H}}\,{\rm{I}}\}$ are the partial Kendall's τ coefficients, after removing the effect of dust attenuation (nebular) or SFR or the half-light radius. The significance of each correlation or partial correlation is indicated in the last column, in the order of the preceding columns: ✓ denotes that the probability of the null hypothesis τ = 0 (no correlation or partial correlation) being correct is p < 0.001; ⨯ denotes that this probability is higher and the null hypothesis cannot be rejected at >3σ. The T-types range from −3 to 10: ≤ 0 corresponds to early-type and S0 galaxies; > 0 corresponds to spiral galaxies.

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Next, we fit the relationship between gas mass and galaxy properties that show strong statistical correlation. Tables 2–4 (the latter two are in Appendices A and B) present the result of fitting a censored quantile regression (q = 0.15, 0.5, and 0.85) or a normal censored linear regression model to the xCOLD GASS molecular mass data (Saintonge et al. 2017). In Table 2, the coefficients of the equations describing the relationship among molecular gas mass, dust attenuation, gas-phase metallicity (Z) and/or galaxy radius are given. We infer Z from the mass–metallicity relation of Tremonti et al. (2004). We present two sets of fits, using either the global (stellar) attenuation or the fiber (nebular) attenuation. Although the scatter is modeled by the q = 0.15 and 0.85 regression fits, we additionally quantify the scatter for each relationship using the root mean square deviation (RMSD) and the median absolute deviation (MAD). For normally distributed residuals, RMSD ≈ 1.48 MAD. The calculations of MAD and RMSD include the detections but not the upper limits, and these measures should be regarded as approximate indicators of the true scatter. Table 3 repeats the analysis for the constant Galactic X_CO. Both cases give consistent fitting formulae.

Table 2.  Regression Model Fits for Molecular Gas, $\mathrm{log}{M}_{{{\rm{H}}}_{2}}=\alpha +{\beta }_{A}{A}_{V}+{\beta }_{Z}\mathrm{log}Z+{\beta }_{R}\mathrm{log}{R}_{50}$

Nebular AV,Fiber					Stellar AV,Global
	15%	Median	85%	Mean	15%	Median	85%	Mean
α	7.70 ± 0.07	8.31 ± 0.04	8.91 ± 0.06	8.34 ± 0.04	7.63 ± 0.22	8.04 ± 0.09	8.71 ± 0.12	8.12 ± 0.05
βA	0.47 ± 0.05	0.47 ± 0.02	0.38 ± 0.05	0.43 ± 0.03	1.52 ± 0.20	1.58 ± 0.13	1.39 ± 0.27	1.49 ± 0.10
Scatter		0.31		0.50		0.35		0.49
α	8.05 ± 0.08	8.27 ± 0.11	8.56 ± 0.07	8.30 ± 0.05	7.96 ± 0.03	8.23 ± 0.12	8.59 ± 0.11	8.21 ± 0.06
βA	0.25 ± 0.10	0.38 ± 0.13	0.38 ± 0.04	0.31 ± 0.05	1.10 ± 0.06	0.91 ± 0.36	0.91 ± 0.30	0.90 ± 0.13
βZ	1.29 ± 0.36	1.44 ± 0.34	1.67 ± 0.19	1.56 ± 0.20	⋯	1.30 ± 0.31	1.65 ± 0.26	1.46 ± 0.20
Scatter		0.22		0.37		0.24		0.35
α	7.41 ± 0.10	7.84 ± 0.09	8.49 ± 0.10	7.95 ± 0.05	7.48 ± 0.09	7.58 ± 0.11	8.16 ± 0.05	7.70 ± 0.06
βA	0.35 ± 0.06	0.42 ± 0.06	0.35 ± 0.04	0.37 ± 0.03	1.24 ± 0.29	1.41 ± 0.13	1.34 ± 0.14	1.34 ± 0.08
βR	1.34 ± 0.28	1.27 ± 0.09	0.89 ± 0.12	1.11 ± 0.09	0.97 ± 0.09	1.35 ± 0.19	1.12 ± 0.10	1.19 ± 0.08
Scatter		0.25		0.41		0.26		0.38
α	7.87 ± 0.07	8.07 ± 0.06	8.56 ± 0.07	8.16 ± 0.06	7.74 ± 0.13	7.98 ± 0.07	8.41 ± 0.18	8.02 ± 0.07
βA	0.24 ± 0.03	0.35 ± 0.06	0.38 ± 0.04	0.31 ± 0.05	1.09 ± 0.16	0.91 ± 0.11	0.98 ± 0.18	0.96 ± 0.12
βZ	0.95 ± 0.35	1.40 ± 0.15	1.67 ± 0.19	1.29 ± 0.20	⋯	0.98 ± 0.23	1.13 ± 0.37	1.02 ± 0.20
βR	0.63 ± 0.12	0.56 ± 0.15	⋯	0.48 ± 0.12	0.63 ± 0.18	0.74 ± 0.11	0.50 ± 0.22	0.62 ± 0.12
Scatter		0.21		0.35		0.25		0.33

Note. The coefficients of the equations describe the relationship among molecular gas mass (${M}_{{{\rm{H}}}_{2}}$), dust attenuation (A_V), metallicity ($\mathrm{log}Z=12+\mathrm{log}[{\rm{O}}/{\rm{H}}]-8.8$), and/or galaxy radius (R₅₀). Fits are given for the attenuation in the fiber (A_V,fiber) and galaxy-wide attenuation (A_V,global). We give two measures of the scatter for the fits: the median absolute deviation (MAD, the left numbers corresponding to the median fits) and the root mean square deviation (RMSD, the right numbers corresponding to the mean fits). $\mathrm{MAD}=\mathrm{median}(| {Y}_{i}-\tilde{Y}|$), where ${Y}_{i}=\mathrm{log}{M}_{{{\rm{H}}}_{2}}$ is the ith observed molecular gas mass and $\tilde{Y}$ is the fitted median relation. Similarly, $\mathrm{RMSD}=\sqrt{\mathrm{mean}({({Y}_{i}-\bar{Y})}^{2})}$, where $\bar{Y}$ is the fitted mean relation. The fits that include $\mathrm{log}Z$ are based solely on strong emission-line star-forming galaxies. They give reasonable prediction when the metallicity is estimated from the stellar mass for all galaxies (see Figure 1).

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Table 3.  Regression Model Fits for Molecular Gas Using Constant X_CO, $\mathrm{log}{M}_{{{\rm{H}}}_{2}}=\alpha +{\beta }_{A}{A}_{V}+{\beta }_{Z}\mathrm{log}Z+{\beta }_{R}\mathrm{log}{R}_{50}$

Nebular AV,fiber					Stellar AV,global
	15%	Median	85%	Mean	15%	Median	85%	Mean
α	7.62 ± 0.06	8.16 ± 0.03	8.99 ± 0.08	8.26 ± 0.05	7.45 ± 0.04	7.92 ± 0.10	8.84 ± 0.08	8.05 ± 0.06
βA	0.54 ± 0.06	0.63 ± 0.08	0.44 ± 0.06	0.53 ± 0.04	1.85 ± 0.10	1.91 ± 0.16	1.33 ± 0.13	1.74 ± 0.12
Scatter		0.38		0.58		0.45		0.56
α	7.89 ± 0.14	8.01 ± 0.11	8.41 ± 0.02	8.11 ± 0.05	7.73 ± 0.25	7.98 ± 0.05	8.32 ± 0.08	7.98 ± 0.06
βA	0.20 ± 0.16	0.45 ± 0.10	0.35 ± 0.05	0.32 ± 0.05	1.05 ± 0.63	1.01 ± 0.08	0.99 ± 0.18	1.00 ± 0.13
βZ	2.46 ± 0.76	2.43 ± 0.42	2.74 ± 0.14	2.61 ± 0.20	1.87 ± 0.47	2.48 ± 0.16	2.74 ± 0.14	2.52 ± 0.20
Scatter		0.23		0.37		0.24		0.34
α	7.15 ± 0.22	7.70 ± 0.07	8.30 ± 0.05	7.76 ± 0.06	7.07 ± 0.16	7.51 ± 0.06	8.01 ± 0.14	7.52 ± 0.06
βA	0.44 ± 0.10	0.50 ± 0.05	0.42 ± 0.07	0.45 ± 0.03	1.80 ± 0.17	1.48 ± 0.12	1.48 ± 0.27	1.52 ± 0.09
βR	1.70 ± 0.55	1.51 ± 0.13	1.27 ± 0.08	1.43 ± 0.10	1.29 ± 0.23	1.57 ± 0.09	1.46 ± 0.18	1.52 ± 0.09
Scatter		0.27		0.45		0.24		0.41
α	7.65 ± 0.16	7.90 ± 0.09	8.25 ± 0.07	7.93 ± 0.06	7.55 ± 0.07	7.80 ± 0.03	8.07 ± 0.13	7.74 ± 0.07
βA	0.26 ± 0.10	0.36 ± 0.10	0.34 ± 0.07	0.32 ± 0.05	1.06 ± 0.10	0.92 ± 0.06	0.93 ± 0.37	1.07 ± 0.11
βZ	2.02 ± 0.19	2.01 ± 0.48	2.23 ± 0.36	2.23 ± 0.20	1.45 ± 0.29	2.14 ± 0.11	2.44 ± 0.40	1.94 ± 0.19
βR	0.73 ± 0.44	0.69 ± 0.19	0.56 ± 0.12	0.63 ± 0.12	0.74 ± 0.29	0.72 ± 0.08	0.73 ± 0.18	0.77 ± 0.11
Scatter		0.21		0.34		0.21		0.30

Note. Similar to Table 2 except here we adopt the constant Galactic CO-to-H2 conversion factor ${\alpha }_{\mathrm{CO}}=4.35\,{M}_{\odot }\ \,{\mathrm{pc}}^{-2}\,{({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1})}^{-1}$.

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As shown in Figures 1 and 2, ${M}_{{{\rm{H}}}_{2}}$ can be indirectly estimated (within a factor of ∼2.5) for a large sample of galaxies using A_V and Z, as follows:

$$\begin{eqnarray}&&\mathrm{log}{M}_{{{\rm{H}}}_{2}}=\alpha +{\beta }_{A}{A}_{V}+{\beta }_{Z}\mathrm{log}Z,\end{eqnarray} \tag{ 3 }$$

where $\mathrm{log}Z=12+\mathrm{log}[{\rm{O}}/{\rm{H}}]-8.8$, (α, β_A, β_Z) = (8.27 ± 0.11, 0.38 ± 0.13, 1.44 ± 0.34) for fiber (nebular) A_V, and (α, β_A, β_Z) = (8.23 ± 0.12, 0.91 ± 0.36, 1.30 ± 0.31) for global (stellar) A_V. Note that the continuum attenuation is ∼2 times lower than the nebular attenuation (Calzetti et al. 1994; Wild et al. 2011). This may explain the lower β_A in the case of fiber A_V. ${M}_{{{\rm{H}}}_{2}}$ can be similarly related to A_V and R₅₀:

$$\begin{eqnarray}&&\mathrm{log}{M}_{{{\rm{H}}}_{2}}=\alpha +{\beta }_{A}{A}_{V}+{\beta }_{R}\mathrm{log}{R}_{50},\end{eqnarray} \tag{ 4 }$$

where (α, β_A, β_R) = (7.84 ± 0.09, 0.42 ± 0.06, 1.27 ± 0.09) for fiber A_V and (α, β_A, β_R) = (7.58 ± 0.11, 1.41 ± 0.13, 1.35 ± 0.19) for global A_V.

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The linear combination of A_V and M_⋆ also has comparable predictive power, but it does not constrain well the scatter from its mean relation. For the mean or median gas relations, however, the coefficients corresponding to M_⋆, Z, and R₅₀ are all significant at α = 0.05. The Akaike information criterion (AIC) and the Bayesian information criterion (BIC), which adjust the likelihood/penalize for the extra parameters, indicate that the normal censored regression model that combines A_V with either Z or M_⋆ or R₅₀ is preferred over A_V by itself. Furthermore, using the inferred mean metallicity from the mass–metallicity relation (e.g., Equation (3) of Tremonti et al. 2004), the gas masses predicted by Equation (3) above agree within a factor of ∼2.5 with observed ${M}_{{{\rm{H}}}_{2}}$ of all detections in xCOLD GASS (MAD = 0.22 and RMSD = 0.36). Thus, this equation may be applied to large samples of galaxies with indirect metallicity estimates. Because the metallicity is measured only for strong emission-line star-forming galaxies, when applying the relation to weak emission-line galaxies, users should verify whether other scaling relations give similar ${M}_{{{\rm{H}}}_{2}}$ estimates. The relation that adds R₅₀ to Equation (3) (see Table 2) or just combines A_V and R₅₀ (Equation (4)) gives a complementary gas mass estimate in such cases.

It is customary to use SFR as a predictor of gas content, especially for high-redshift studies (e.g., Boselli et al. 2014b; Popping et al. 2015; Saintonge et al. 2017; Catinella et al. 2018). As shown in Table 4 and Figure 3 in Appendix B, adding SFR to Equation (4) improves the fit significantly. This table also gives detailed information for various fits when one or more variables are missing or may not be desirable to include. All these combinations can predict gas mass within a factor of ∼3. Users should be careful not to use these relations in a circular manner. For example, if SFR is already used by a scaling relation to get ${M}_{{{\rm{H}}}_{2}}$, such a scaling relation should not be employed to study the correlation between ${M}_{{{\rm{H}}}_{2}}$ and SFR. Users should also take into account the quartile relations in the gas mass estimation, as there are systematic trends in the scatters of the scaling relations.

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Table 4.  Regression Model Fits for Molecular Gas, $\mathrm{log}{M}_{{{\rm{H}}}_{2}}=\alpha +{\beta }_{A}{A}_{V}+{\beta }_{R}\mathrm{log}{R}_{50}+{\beta }_{M}\mathrm{log}{M}_{\star }+{\beta }_{S}\mathrm{logSFR}$

Without SFR					With SFR
	15%	Median	85%	Mean	15%	Median	85%	Mean
α	9.30 ± 0.91	6.62 ± 0.38	6.46 ± 0.36	6.91 ± 0.43	8.44 ± 0.08	6.67 ± 0.21	5.92 ± 0.37	6.34 ± 0.28
${\beta }_{A,\mathrm{fiber}}$	0.44 ± 0.07	0.39 ± 0.03	0.29 ± 0.03	0.35 ± 0.03	0.13 ± 0.07	0.16 ± 0.02	0.15 ± 0.02	0.15 ± 0.02
βR	1.61 ± 0.25	1.13 ± 0.13	0.62 ± 0.06	0.96 ± 0.11	0.34 ± 0.11	0.30 ± 0.05	0.22 ± 0.09	0.29 ± 0.07
βM	−0.22 ± 0.10	0.13 ± 0.04	0.22 ± 0.04	0.11 ± 0.05	⋯	0.20 ± 0.02	0.29 ± 0.04	0.23 ± 0.03
βS	⋯	⋯	⋯	⋯	0.70 ± 0.09	0.59 ± 0.03	0.53 ± 0.05	0.52 ± 0.02
Scatter		0.23		0.40		0.12		0.24
α	7.41 ± 0.10	7.84 ± 0.09	8.49 ± 0.10	7.95 ± 0.05	8.44 ± 0.08	8.53 ± 0.22	8.69 ± 0.03	8.49 ± 0.04
βA,fiber	0.35 ± 0.06	0.42 ± 0.06	0.35 ± 0.04	0.37 ± 0.03	0.13 ± 0.07	0.22 ± 0.02	0.20 ± 0.04	0.20 ± 0.02
βR	1.34 ± 0.28	1.27 ± 0.09	0.89 ± 0.12	1.11 ± 0.09	0.34 ± 0.11	0.49 ± 0.09	0.73 ± 0.06	0.59 ± 0.07
βS	⋯	⋯	⋯	⋯	0.70 ± 0.09	0.61 ± 0.06	0.56 ± 0.05	0.54 ± 0.03
Scatter		0.25		0.41		0.14		0.27
α	7.70 ± 0.07	5.21 ± 0.38	4.44 ± 0.45	5.10 ± 0.41	8.57 ± 0.05	6.05 ± 0.41	5.58 ± 0.33	5.84 ± 0.25
βA,fiber	0.47 ± 0.05	0.38 ± 0.05	0.27 ± 0.05	0.35 ± 0.03	0.14 ± 0.06	0.14 ± 0.03	0.14 ± 0.03	0.14 ± 0.02
βM	⋯	0.32 ± 0.04	0.44 ± 0.04	0.33 ± 0.04	⋯	0.27 ± 0.04	0.34 ± 0.03	0.29 ± 0.02
βS	⋯	⋯	⋯	⋯	0.75 ± 0.06	0.64 ± 0.05	0.55 ± 0.04	0.55 ± 0.02
Scatter		0.25		0.44		0.12		0.25
α	7.70 ± 0.07	8.31 ± 0.04	8.91 ± 0.06	8.34 ± 0.04	8.57 ± 0.05	8.72 ± 0.06	9.04 ± 0.06	8.76 ± 0.03
βA,fiber	0.47 ± 0.05	0.47 ± 0.02	0.38 ± 0.05	0.43 ± 0.03	0.14 ± 0.06	0.23 ± 0.05	0.19 ± 0.03	0.20 ± 0.02
βS	⋯	⋯	⋯	⋯	0.75 ± 0.06	0.73 ± 0.03	0.69 ± 0.08	0.63 ± 0.03
Scatter		0.31		0.50		0.16		0.30
α	⋯	5.20 ± 0.85	3.16 ± 0.53	4.67 ± 0.50	7.47 ± 0.19	6.14 ± 0.16	5.74 ± 0.22	5.83 ± 0.26
βM	⋯	0.34 ± 0.09	0.60 ± 0.05	0.39 ± 0.05	0.12 ± 0.02	0.28 ± 0.02	0.34 ± 0.02	0.31 ± 0.02
βS	⋯	⋯	⋯	⋯	0.83 ± 0.01	0.78 ± 0.04	0.66 ± 0.04	0.66 ± 0.02
Scatter		0.34		0.61		0.14		0.30
α	⋯	7.83 ± 0.15	8.70 ± 0.05	8.02 ± 0.05	8.54 ± 0.04	8.77 ± 0.05	8.91 ± 0.08	8.70 ± 0.04
βR	⋯	1.84 ± 0.31	1.25 ± 0.07	1.48 ± 0.11	0.35 ± 0.08	0.46 ± 0.10	0.70 ± 0.14	0.59 ± 0.07
βS	⋯	⋯	⋯	⋯	0.79 ± 0.06	0.81 ± 0.03	0.69 ± 0.07	0.68 ± 0.03
Scatter		0.36		0.55		0.17		0.31
α	7.48 ± 0.09	7.58 ± 0.11	6.76 ± 0.65	7.04 ± 0.40	8.33 ± 0.15	6.56 ± 0.37	6.16 ± 0.47	6.32 ± 0.27
βA,global	1.24 ± 0.29	1.41 ± 0.13	1.28 ± 0.16	1.32 ± 0.08	0.39 ± 0.12	0.41 ± 0.09	0.53 ± 0.09	0.50 ± 0.06
βR	0.97 ± 0.09	1.35 ± 0.19	0.84 ± 0.16	1.08 ± 0.10	0.49 ± 0.26	0.30 ± 0.10	0.31 ± 0.10	0.37 ± 0.07
βM	⋯	⋯	0.15 ± 0.07	0.07 ± 0.04	⋯	0.21 ± 0.04	0.26 ± 0.05	0.22 ± 0.03
βS	⋯	⋯	⋯		0.59 ± 0.16	0.61 ± 0.03	0.47 ± 0.05	0.48 ± 0.03
Scatter		0.26		0.38		0.13		0.24
α	7.48 ± 0.09	7.58 ± 0.11	8.16 ± 0.05	7.70 ± 0.06	8.33 ± 0.15	8.51 ± 0.06	8.60 ± 0.13	8.40 ± 0.05
βA,global	1.24 ± 0.29	1.41 ± 0.13	1.34 ± 0.14	1.34 ± 0.08	0.39 ± 0.12	0.44 ± 0.08	0.58 ± 0.20	0.57 ± 0.07
βR	0.97 ± 0.09	1.35 ± 0.19	1.12 ± 0.10	1.19 ± 0.08	0.49 ± 0.26	0.60 ± 0.06	0.83 ± 0.13	0.69 ± 0.07
βS	⋯	⋯	⋯		0.59 ± 0.16	0.61 ± 0.06	0.47 ± 0.06	0.49 ± 0.03
Scatter		0.26		0.38		0.15		0.27
α	7.70 ± 0.07	5.37 ± 0.32	4.30 ± 0.20	4.96 ± 0.38	8.57 ± 0.08	6.40 ± 0.19	5.27 ± 0.15	5.72 ± 0.25
βA,global	0.47 ± 0.05	1.49 ± 0.14	1.17 ± 0.15	1.38 ± 0.09	0.31 ± 0.15	0.39 ± 0.05	0.50 ± 0.09	0.43 ± 0.06
βM	⋯	0.27 ± 0.04	0.43 ± 0.02	0.32 ± 0.04	⋯	0.24 ± 0.02	0.36 ± 0.02	0.30 ± 0.02
βS	⋯	⋯	⋯	⋯	0.75 ± 0.10	0.65 ± 0.03	0.52 ± 0.03	0.53 ± 0.03
Scatter		0.32		0.43		0.14		0.25
α	7.70 ± 0.07	8.31 ± 0.04	8.91 ± 0.06	8.34 ± 0.04	8.57 ± 0.08	8.78 ± 0.09	9.05 ± 0.07	8.77 ± 0.04
βA,global	0.47 ± 0.05	0.47 ± 0.02	0.38 ± 0.05	0.43 ± 0.03	0.31 ± 0.15	0.41 ± 0.11	0.52 ± 0.11	0.46 ± 0.08
βS	⋯	⋯	⋯	⋯	0.75 ± 0.10	0.73 ± 0.04	0.71 ± 0.06	0.63 ± 0.03
Scatter		0.31		0.50		0.18		0.31

Note. The coefficients of the equations describe the relationship among molecular gas mass (${M}_{{{\rm{H}}}_{2}}$), SFR, stellar mass (${M}_{\star }$), dust attenuation (A_V), and/or galaxy radius (R₅₀). Dust absorption is estimated from Balmer decrement in the fiber (A_V,fiber) or by galaxy-wide SED fitting (A_V,global). The scatter for the fits are quantified by the median absolute deviation (MAD, the left numbers corresponding to the median fits) and the root mean square deviation (RMSD, the right numbers corresponding to the mean fits).

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In summary, dust attenuation is significantly correlated with the molecular gas mass. A linear combination of A_V and metallicity or A_V and R₅₀ can indirectly estimate molecular gas masses within ∼2.5–3 times the observed masses. If SFR is further used, ${M}_{{{\rm{H}}}_{2}}$ can be predicted to within a factor ≲2. In this case, A_V and R₅₀ are the two best secondary parameters that improve the primary correlation between ${M}_{{{\rm{H}}}_{2}}$ and SFR.

3.2. Predictors of Atomic Gas Mass

For xGASS data (Catinella et al. 2018), the atomic gas mass can be predicted within a factor of ∼2.5 from the SFR and half-light radius (Equation (5)). Table 5 in Appendix B presents the result of fitting censored regression models to the xGASS data, describing the relationship among the atomic gas mass, SFR, the half-light radius, T-type, disk/feature probability, and/or dust attenuation. We present two sets of fits, by analyzing the central and satellite galaxies together, or by restricting the sample to central galaxies only. The classification of satellites and centrals is based on the galaxy group catalog of Yang et al. (2007). The median fit for the whole sample is

$$\begin{eqnarray}&&\mathrm{log}{M}_{{\rm{H}}{\rm{I}}}=\alpha +{\beta }_{S}\mathrm{logSFR}+{\beta }_{R}\mathrm{log}{R}_{50},\end{eqnarray} \tag{ 5 }$$

where (α, β_S, β_R) = (9.07 ± 0.04, 0.47 ± 0.02, 1.08 ± 0.11). The regression coefficients for central galaxies alone are consistent with those for the total sample. In comparison, the corresponding median relation for the molecular gas yields (α, β_S, β_R) = (8.77 ± 0.05, 0.81 ± 0.03, 0.46 ± 0.11). The molecular gas has weaker dependence on radius. Hence, the atomic-to-molecular ratio depends on R₅₀. Table 5 also gives three other scaling relations that can predict gas mass with accuracy of a factor ∼3 without using SFR but combining R₅₀ with either T-type or disk/feature probability or dust attenuation.

Table 5.  Regression Model Fits for Atomic Gas, $\mathrm{log}{M}_{{\rm{H}}{\rm{I}}}=\alpha +{\beta }_{A}{A}_{V}+{\beta }_{R}\mathrm{log}R+{\beta }_{M}\mathrm{log}{M}_{\star }+{\beta }_{S}\mathrm{logSFR}+{\beta }_{T}{\rm{T}}\,\ \mathrm{types}+{\beta }_{D}\mathrm{Prob}.\ \ \mathrm{disk}$

All Galaxies					Centrals Only
	15%	Median	85%	Mean	15%	Median	85%	Mean
α	8.09 ± 0.17	8.58 ± 0.11	9.02 ± 0.05	8.54 ± 0.07	8.07 ± 0.35	8.52 ± 0.20	9.02 ± 0.06	8.50 ± 0.08
${\beta }_{{R}_{90}}$	0.69 ± 0.20	0.98 ± 0.10	0.93 ± 0.05	0.98 ± 0.07	0.75 ± 0.24	1.05 ± 0.20	0.96 ± 0.06	1.04 ± 0.09
βT	0.16 ± 0.01	0.06 ± 0.01	0.03 ± 0.01	0.05 ± 0.01	0.16 ± 0.06	0.06 ± 0.01	0.02 ± 0.01	0.05 ± 0.01
βS	0.63 ± 0.05	0.44 ± 0.05	0.42 ± 0.02	0.40 ± 0.03	0.63 ± 0.12	0.46 ± 0.09	0.43 ± 0.07	0.42 ± 0.03
Scatter		0.22		0.39		0.22		0.39
α	8.40 ± 0.14	8.69 ± 0.12	9.08 ± 0.09	8.69 ± 0.07	8.18 ± 0.19	8.55 ± 0.07	9.01 ± 0.09	8.60 ± 0.08
${\beta }_{{R}_{90}}$	0.72 ± 0.13	1.01 ± 0.13	0.94 ± 0.09	0.96 ± 0.08	1.02 ± 0.27	1.21 ± 0.07	1.05 ± 0.08	1.09 ± 0.09
βS	0.77 ± 0.10	0.49 ± 0.06	0.42 ± 0.09	0.48 ± 0.03	0.75 ± 0.14	0.49 ± 0.06	0.41 ± 0.06	0.48 ± 0.03
Scatter		0.23		0.41		0.23		0.41
α	8.59 ± 0.04	9.07 ± 0.04	9.45 ± 0.04	9.04 ± 0.04	8.57 ± 0.05	9.02 ± 0.05	9.44 ± 0.05	9.01 ± 0.05
${\beta }_{{R}_{50}}$	1.05 ± 0.11	1.08 ± 0.11	0.98 ± 0.07	1.03 ± 0.08	1.16 ± 0.07	1.20 ± 0.11	1.03 ± 0.10	1.12 ± 0.09
βS	0.69 ± 0.05	0.47 ± 0.02	0.43 ± 0.05	0.44 ± 0.03	0.69 ± 0.07	0.47 ± 0.04	0.42 ± 0.10	0.45 ± 0.03
Scatter		0.23		0.41		0.24		0.41
α	12.32 ± 0.77	11.16 ± 0.71	9.18 ± 0.16	10.30 ± 0.43	11.34 ± 0.60	10.22 ± 0.48	9.16 ± 0.05	9.82 ± 0.50
βA,global	0.97 ± 0.08	0.73 ± 0.23	0.33 ± 0.14	0.64 ± 0.08	0.63 ± 0.13	0.54 ± 0.11	0.29 ± 0.13	0.55 ± 0.10
${\beta }_{{R}_{50}}$	2.42 ± 0.13	1.99 ± 0.20	1.09 ± 0.22	1.68 ± 0.11	2.47 ± 0.12	1.97 ± 0.13	1.18 ± 0.08	1.77 ± 0.12
${\beta }_{M}$	−0.50 ± 0.09	−0.30 ± 0.08	⋯	−0.20 ± 0.05	−0.38 ± 0.07	−0.20 ± 0.05	⋯	−0.15 ± 0.05
Scatter		0.31		0.49		0.29		0.48
α	7.85 ± 0.08	8.41 ± 0.07	9.14 ± 0.07	8.49 ± 0.04	7.52 ± 0.20	8.42 ± 0.17	9.13 ± 0.05	8.47 ± 0.04
${\beta }_{{R}_{50}}$	1.75 ± 0.43	1.06 ± 0.11	1.06 ± 0.23	1.06 ± 0.09	2.65 ± 0.44	1.32 ± 0.22	1.20 ± 0.10	1.28 ± 0.11
βD	⋯	0.71 ± 0.05	0.26 ± 0.09	0.55 ± 0.05	⋯	0.52 ± 0.23	0.19 ± 0.06	0.45 ± 0.06
Scatter		0.29		0.47		0.28		0.47
α	7.48 ± 0.14	8.54 ± 0.06	9.18 ± 0.08	8.56 ± 0.04	7.44 ± 0.10	8.47 ± 0.08	9.13 ± 0.03	8.52 ± 0.04
${\beta }_{{R}_{50}}$	1.94 ± 0.27	1.28 ± 0.19	1.17 ± 0.13	1.29 ± 0.09	2.08 ± 0.10	1.56 ± 0.17	1.30 ± 0.08	1.45 ± 0.10
βT	0.16 ± 0.03	0.10 ± 0.01	0.03 ± 0.01	0.08 ± 0.01	0.15 ± 0.02	0.07 ± 0.02	0.03 ± 0.01	0.06 ± 0.01
Scatter		0.30		0.47		0.28		0.47
α	8.98 ± 0.06	9.55 ± 0.04	9.96 ± 0.03	9.50 ± 0.02	9.00 ± 0.14	9.55 ± 0.04	9.99 ± 0.03	9.51 ± 0.03
βS	0.91 ± 0.08	0.66 ± 0.05	0.56 ± 0.03	0.60 ± 0.03	0.88 ± 0.17	0.67 ± 0.06	0.62 ± 0.05	0.65 ± 0.04
Scatter		0.30		0.49		0.29		0.48

Note. Subsets of the general equation are fitted. The notation is similar to Table 4. Morphology as quantified by T-types or disk probability is useful, in combination with radius, to predict atomic gas mass, when SFR is not used.

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For completeness, we give the relations for the total gas mass (${M}_{{\rm{H}}{\rm{I}}}+{M}_{{{\rm{H}}}_{2}}$), $\mathrm{log}{M}_{\mathrm{gas}}=\alpha +\beta \mathrm{log}{R}_{50}+\gamma \mathrm{logSFR}$, using galaxies contained in both the xGASS and xCOLD GASS surveys. The median relation yields α = 9.28 ± 0.04, β = 0.87 ± 0.11, and γ = 0.70 ± 0.04; for q = 0.15, α = 8.83 ± 0.05, β = 1.05 ± 0.07, and γ = 0.88 ± 0.02; and for q = 0.85, α = 9.55 ± 0.05, β = 0.85 ± 0.11, and γ = 0.59 ± 0.07.

Dust is often used to estimate indirectly the gas content in galaxies (e.g., Devereux & Young 1990). We find that dust attenuation, A_V, is significantly correlated with the molecular gas mass, ${M}_{{{\rm{H}}}_{2}}$. A linear combination of A_V and metallicity or A_V and R₅₀ can indirectly estimate ${M}_{{{\rm{H}}}_{2}}$ within ∼2.5–3 times the observed masses. However, the correlation between A_V and M_{H i} is weak. A combination of R₅₀ and SFR can give an estimate of M_{H i} accurate to within a factor of ∼3. Next, we briefly discuss previous studies that aid the interpretation of our results. We surmise that the correlation of ${M}_{{{\rm{H}}}_{2}}$ with metallicity and R₅₀ likely comes from physical processes that determine the gas-to-dust ratio and molecular gas fraction.

The ratio of dust mass to total gas mass ($\xi =\mu {M}_{{\rm{H}}}/{M}_{\mathrm{dust}}$, where ${M}_{{\rm{H}}}={M}_{{\rm{H}}{\rm{I}}}+{M}_{{{\rm{H}}}_{2}}$ and the correction factor for helium μ ≈ 1.4) varies approximately linearly with metallicity ($\mathrm{log}\xi \propto \beta \mathrm{log}Z\propto \beta \mathrm{log}[{\rm{O}}/{\rm{H}}]$) such that higher metallicity galaxies have lower ξ (Issa et al. 1990; Lisenfeld & Ferrara 1998; Draine et al. 2007; Galametz et al. 2011; Leroy et al. 2011; Brinchmann et al. 2013; Berta et al. 2016; De Vis et al. 2019). The amount of dust along a line of sight can be estimated using dust attenuation A_V, while the amount of gas is probed by the total hydrogen column density N_H. A_V/N_H can then be used to infer the dust-to-gas ratio. It is known that A_V/N_H positively correlates with metallicity in nearby galaxies (e.g., Kahre et al. 2018). Therefore, it is natural to anticipate that the molecular gas mass depends on both A_V and metallicity. If ${f}_{{{\rm{H}}}_{2}}\equiv {M}_{{{\rm{H}}}_{2}}/{M}_{{\rm{H}}}$ and ${R}_{{{\rm{H}}}_{2}}\equiv {M}_{{{\rm{H}}}_{2}}/{M}_{{\rm{H}}{\rm{I}}}$, then the molecular mass can be expressed as

$$\begin{eqnarray}\begin{array}{rcl}{M}_{{{\rm{H}}}_{2}} & = & {M}_{{\rm{H}}}\times {M}_{{{\rm{H}}}_{2}}/{M}_{{\rm{H}}}\\ & = & \xi {\mu }^{-1}{M}_{\mathrm{dust}}{f}_{{{\rm{H}}}_{2}}=\xi {\mu }^{-1}{M}_{\mathrm{dust}}{(1+{R}_{{{\rm{H}}}_{2}}^{-1})}^{-1}.\end{array}\end{eqnarray} \tag{ 6 }$$

Because previous observations indicate $\xi \propto {Z}^{{\beta }_{\xi }}$ and ${R}_{{{\rm{H}}}_{2}}\,\propto {Z}^{{\beta }_{{R}_{{\rm{H}}2}}}$, ${M}_{{{\rm{H}}}_{2}}$ depends on both dust mass and metallicity. Depending on the metallicity calibration, X_CO, and sample selection, ${\beta }_{\xi }\approx -0.7$ to −2.5 (Issa et al. 1990; Muñoz-Mateos et al. 2009; Leroy et al. 2011; Rémy-Ruyer et al. 2014; Janowiecki et al. 2018; De Vis et al. 2019) and ${\beta }_{{R}_{{\rm{H}}2}}\,\approx 2.7-3.4$ (Boselli et al. 2014b). Furthermore, we expect that ${f}_{{{\rm{H}}}_{2}}$ and ${R}_{{{\rm{H}}}_{2}}$ also depend on R₅₀. We showed in Section 3.2 that ${M}_{{{\rm{H}}}_{2}}$ and M_{H i} scale differently with R₅₀. Based on the empirical correlation between mid-plane gas pressure of galactic disks and ${R}_{{{\rm{H}}}_{2}}$ (e.g., Blitz & Rosolowsky 2006), Obreschkow & Rawlings (2009) presented a phenomenological model that predicts that ${R}_{{{\rm{H}}}_{2}}\propto {M}_{{\rm{H}}}^{-0.24}\propto {R}_{50}^{-0.39}$. If ${f}_{{{\rm{H}}}_{2}}\propto {R}_{50}^{{\beta }_{R}^{{\prime} }}{Z}^{{\beta }_{Z}^{{\prime} }}$, then Equation (6) implies that $\mathrm{log}{M}_{{{\rm{H}}}_{2}}\,\propto \mathrm{log}{M}_{\mathrm{dust}}+{\beta }_{R}^{{\prime} }\mathrm{log}{R}_{50}\,+{\beta }_{Z}\mathrm{log}Z$, where ${\beta }_{Z}={\beta }_{\xi }+{\beta }_{Z}^{{\prime} }$. For galaxies in the Herschel Reference Survey (HRS), Janowiecki et al. (2018) found β_ξ = −0.72 ± 0.19 and β_Z = 0.67 ± 0.26 (without including the R₅₀ term; see equations in their Section 2.2). In other words, they found a negative correlation between metallicity and gas-to-dust ratio defined using the total gas phase (ξ) or H i (ξ^HI = M_{H i}/M_dust), but a positive correlation using H₂ (${\xi }^{{{\rm{H}}}_{2}}={M}_{{{\rm{H}}}_{2}}/{M}_{\mathrm{dust}}$). Their ${\xi }^{{\rm{H}}{\rm{I}}}\mbox{--}Z$ relationship has the highest correlation coefficient ($| r| =0.50\pm 0.08$), while their relationship with ξ has the smallest scatter (σ = 0.23). The correlation of metallicity with ${\xi }^{{{\rm{H}}}_{2}}$ is weak (r = 0.29 ± 0.09; see their Figure 1 for all three correlations). Likewise, Bertemes et al. (2018) found β_Z = 0.12 using β_ξ = −0.85 (Leroy et al. 2011). Our molecular gas scaling relations (e.g., Equation (3)) also imply a positive ${\xi }^{{{\rm{H}}}_{2}}\mbox{--}Z$ correlation. When A_V is estimated from Hα/Hβ in the fiber and ${A}_{V}\ne 0$, we find a median relation $\mathrm{log}{M}_{{{\rm{H}}}_{2}}\,\propto \mathrm{log}{A}_{V}+{\beta }_{Z}\mathrm{log}Z$ with β_Z = 1.11 ± 0.26. In Equation (3), we used the term ${\beta }_{A}{A}_{V}$ in lieu of log A_V and found β_Z = 1.44 ± 0.34. But M_dust likely depends not only on A_V (dust column or surface density) but also on galaxy radius.

The radial profile of dust attenuation or dust surface mass density and the dependence of gas-to-dust ratio on galactocentric distance have been well-studied for local galaxies (e.g., Issa et al. 1990; Boissier et al. 2004, 2007; Muñoz-Mateos et al. 2009; Pappalardo et al. 2012; Sandstrom et al. 2013; Smith et al. 2016; Giannetti et al. 2017; Casasola et al. 2017; Chiang et al. 2018; Vílchez et al. 2019). Accordingly, in most galaxies the attenuation decreases with the galactocentric distance. The quantity A_V/N_H (or ξ⁻¹) also decreases with radius: it is higher in the central, metal-rich regions than in the outer parts of galaxies (Issa et al. 1990; Kahre et al. 2018). By combining the signal of 110 spiral galaxies in HRS, Smith et al. (2016) detected dust emission out to about twice the optical radius. They found that the radial distribution of dust is consistent with an exponential, ${{\rm{\Sigma }}}_{d}(r)={{\rm{\Sigma }}}_{0}{10}^{{\alpha }_{r}r}$, with a gradient of α_r = −1.7 dex/r₂₅. Here r is the galactocentric radius and r₂₅ is the optical radius at the 25 mag arcsec⁻² isophote. Moreover, Σ_d declines with radius at a similar rate to the stellar mass surface density but more slowly than the surface density of molecular gas or star formation rate (Schruba et al. 2011; Bigiel & Blitz 2012; Smith et al. 2016, see Table 1 of the latter work). Schruba et al. (2011) showed that the CO radial profiles have exponential distributions with a gradient of α_r = −2.2 dex/r₂₅. Similarly, using galaxies in the H i Nearby Galaxy Survey, Bigiel & Blitz (2012) showed that the total H i and H₂ gas profiles of these galaxies follow exponential distributions beyond the inner 20% of the optical radius with a gradient of α_r = −0.7 dex/r₂₅. Because mass is the integral of the surface density profile, if the shape of the profile is "universal," the gas and dust masses of (unresolved) galaxies should vary in a predictable way with the optical radius, consistent with what we found. The origin of the universal gas profile may be related to the halo angular momentum or the radial distribution of the accretion rate (e.g., Mo et al. 1998; Kravtsov 2013; Wang et al. 2014).

For the profile of Smith et al. (2016), ${M}_{\mathrm{dust}}=2\pi {\int }_{0}^{2r}{{\rm{\Sigma }}}_{d}(r){rdr}\approx 0.4{{\rm{\Sigma }}}_{d}(0){r}_{25}^{2}$. Then, for ${f}_{{{\rm{H}}}_{2}}\,\propto {R}_{50}^{{\beta }_{R}^{{\prime} }}{Z}^{{\beta }_{Z}^{{\prime} }}$ and $\xi \propto {Z}^{{\beta }_{\xi }}$, Equation (6) leads to ${M}_{{{\rm{H}}}_{2}}\propto {\beta }_{A}{A}_{V}+{\beta }_{R}\mathrm{log}{R}_{50}\,+{\beta }_{Z}\mathrm{log}Z$. Now, β_R is at least ${\beta }_{R}^{{\prime} }+2$ and ${\beta }_{Z}={\beta }_{\xi }+{\beta }_{Z}^{{\prime} }$. The conversion of Σ_d to A_V is not straightforward. It depends on radiative transfer effects of the relative geometric configuration of dust and ionized gas (Calzetti et al. 1994). Kreckel et al. (2013), however, showed that empirical fits to A_V and Σ_d data of local galaxies, with Σ_d expressed in both linear and logarithmic form, can predict Σ_d given A_V to within a factor of ∼3. Because in our analysis (Table 2) β_R = 0.56 ± 0.15 and β_Z = 1.40 ± 0.15, Equation (6) implies ${f}_{{{\rm{H}}}_{2}}\propto {R}_{50}^{{\beta }_{R}^{{\prime} }}{Z}^{{\beta }_{Z}^{{\prime} }}$ if the dust profile of our sample is an exponential, whose scale length is a fixed fraction of R₅₀. Note that combining equations in Janowiecki et al. (2018) for the ${\xi }^{{{\rm{H}}}_{2}}-Z$ and $\xi \mbox{--}Z$ relations gives ${\xi }^{{{\rm{H}}}_{2}}/\xi ={f}_{{{\rm{H}}}_{2}}\propto {Z}^{1.39\pm 0.45}$.

Our molecular gas scaling relation likely reflects physics that governs the amount and spatial distribution of dust ($\xi \propto {Z}^{{\beta }_{\xi }},{\beta }_{\xi }\lt 0$) and molecular hydrogen (${f}_{{{\rm{H}}}_{2}}\,\propto {R}_{50}^{{\beta }_{R}^{{\prime} }}{Z}^{{\beta }_{Z}^{{\prime} }},{\beta }_{Z}^{{\prime} }\gt 0$ and ${\beta }_{R}^{{\prime} }\lt 0$) in the ISM. We believe that it can usefully constrain theoretical models of galaxy formation and evolution. The observed ξ–Z relation has been useful to constrain dust evolution models (e.g., Lisenfeld & Ferrara 1998; Draine et al. 2007; Mattsson & Andersen 2012; Bekki 2013; Rémy-Ruyer et al. 2014; Zhukovska 2014; Aoyama et al. 2017; Galliano et al. 2018; De Vis et al. 2019; Hou et al. 2019; Li et al. 2019). The models range from analytical models to cosmological hydrodynamic galaxy simulations. They attempt to characterize the complex dust physics by including its production by stellar evolution, growth by accretion of metal in the ISM, destruction by supernova shocks and by thermal sputtering in hot gas, and its dependence on star formation history and gas inflows and outflows. The models are able to reproduce reasonably well the observed ξ–Z relation. They indicate that dust growth in the ISM is crucial, and dust destruction by supernova may greatly influence the chemical evolution of galaxies. Dust-related physical processes, however, have only been self-consistently included in a few simulations. Studying the time evolution of dust and ${f}_{{{\rm{H}}}_{2}}$ in galaxies using simulations with a self-consistent model for the formation and evolution of dust and H₂, Bekki (2013) found that ${f}_{{{\rm{H}}}_{2}}$ rapidly increases in the first several Gyr of a star-forming galaxy because of more efficient dust production; it is higher in the inner regions of galaxies than in their outer parts. Regions with low ξ (high dust-to-gas ratio) likely have higher ${f}_{{{\rm{H}}}_{2}}$. The author also broadly reproduced (with a systematic offset) the observed ${M}_{{{\rm{H}}}_{2}}\propto {M}_{\mathrm{dust}}^{0.77}$ relation, which is based on dust emission of only 35 galaxies in the Virgo cluster (Corbelli et al. 2012). Bekki (2015) suggested that the evolution of dust distributions driven by stellar radiation pressure is important for the evolution of SFR, ${f}_{{{\rm{H}}}_{2}}$, and Z in galaxies.

We analyze the atomic and molecular gas masses of local representative galaxies (${M}_{\star }={10}^{9}-{10}^{11.5}$ M_⊙ and z = 0.01–0.05) from the xGASS and xCOLD GASS surveys (Saintonge et al. 2017; Catinella et al. 2018) in conjunction with a number of galaxy properties derived from SDSS. The central result of this work is that we have discovered remarkable new empirical relations that allow us to predict gas masses easily from galaxy survey data. Using different statistical approaches (partial correlation and censored quantile regression analyses), we provide useful and convenient formulae in Tables 2, 4, and 5 that summarize the median, mean, and 0.15/0.85 quantile multivariate relationships between gas mass and other galaxy properties that correlate best with it.

Our main conclusions are as follows:

• 1.  
  The dust attenuation A_V, of both the stellar continuum and gas emission, is significantly correlated with the molecular gas mass (Kendall's rank coefficient τ = 0.4). After controlling for the effect of the SFR, the dust attenuation is still correlated with the molecular gas. Without using SFR, ${M}_{{{\rm{H}}}_{2}}$ can be indirectly estimated (within a factor of ∼2.5) using A_V and gas-phase metallicity (Z) or A_V and R₅₀ (Figures 1 and 2 and Tables 1 and 2). In contrast, the correlation between dust attenuation and the atomic gas mass is weak and may be explained by other covariates such as SFR. This result is theoretically expected since in poorly dust-shielded regions molecular hydrogen is dissociated by the far-ultraviolet photons and the atomic gas dominates.
• 2.  
  The galaxy half-light radius R₅₀ is significantly correlated with both molecular and atomic gas (Kendall's τ = 0.3). The correlation is still significant after controlling for the effect of SFR or A_V. In practice, R₅₀ can be used with A_V or SFR to predict the atomic and molecular gas mass within a factor of ∼2–3 for large samples of galaxies and AGNs (detailed equations are given in Tables 2, 4, and 5).
• 3.  
  Depending on how it is quantified, galaxy morphology may be correlated with the atomic gas. In particular, the disk/feature probability and the T-type show significant correlation with the atomic gas after accounting for the effect of SFR or R₅₀. Combining these morphology indicators with R₅₀ (and/or A_V) may be useful in predicting atomic gas masses (Table 5), especially for future AGN studies, as they are easier to measure than the SFR. On the other hand, the merger probability and the edge-on probability have little or no significance in explaining the variability in the gas masses of the xGASS sample. The correlations between the molecular gas and all morphology indicators are insignificant, or weak after accounting for the effects of SFR or R₅₀.

As this work was nearing completion, we became aware of a similar effort by Concas & Popesso (2019), who proposed an empirical scaling relation between Balmer decrement and molecular gas mass. They reported a significant correlation (with Spearman rank coefficient of 0.7 at >3σ) between the Balmer decrement and ${M}_{{{\rm{H}}}_{2}}$ in a subsample of 222 star-forming galaxies, also from the xCOLD GASS survey. Similar to what we found in Table 1 using the edge-on probability, they showed that star-forming galaxies with high disk inclination angles tend to exhibit high Hα/Hβ ratios, for the same ${M}_{{{\rm{H}}}_{2}}$, compared to their less inclined counterparts. After correcting for the inclination effect, they did not find residual dependence on galaxy size or star formation rate. This is contrary to what we present in Table 1 for the total xCOLD GASS sample. We show that ${M}_{{{\rm{H}}}_{2}}$ is primarily correlated with SFR and secondarily with A_V. Our analysis presented in Appendix C shows that the partial correlation of the inclination angle, i, with ${M}_{{{\rm{H}}}_{2}}$ at fixed ${\rm{H}}\alpha /{\rm{H}}\beta$ is much weaker than the corresponding partial correlation of SFR. Likewise, fitting together ${M}_{{{\rm{H}}}_{2}}$, SFR, i, and $\mathrm{logH}\alpha /{\rm{H}}\beta$ indicates that the inclination angle does not bring additional information, once SFR and $\mathrm{logH}\alpha /{\rm{H}}\beta$ are used. Using information criteria AIC/BIC and the rms of the residuals, we conclude that the combination of A_V and Z or A_V and R₅₀ is a better predictor of molecular gas than the combination of A_V and i. Unlike the combination of Hα/Hβ and i proposed by Concas & Popesso (2019), our gas scaling relations work well for all galaxies, star-forming or not, regardless of whether A_V is measured from Hα/Hβ or stellar continuum absorption.

We note that our atomic and molecular gas scaling relations predict gas masses consistent with independent gas data of galaxies in the Herschel Reference Survey compiled by Boselli et al. (2014a; see Appendix D).

We thank the anonymous referee for helpful comments and suggestions. This work was supported by the National Science Foundation of China (11721303) and the National Key R&D Program of China (2016YFA0400702).

Table 3 repeats Table 2 for the constant Galactic CO conversion factor ${\alpha }_{\mathrm{CO}}=4.35\,{M}_{\odot }\ \,{\mathrm{pc}}^{-2}\,{({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1})}^{-1}$ (e.g., Bolatto et al. 2013). The results in the two tables are not significantly different. In other words, there is a significant relationship among molecular gas, metallicity, radius, and dust absorption, $\mathrm{log}{M}_{{{\rm{H}}}_{2}}\propto {\beta }_{A}{A}_{V}+{\beta }_{R}\mathrm{log}{R}_{50}+{\beta }_{Z}\mathrm{log}Z$, whether the metallicity-dependent or the constant α_CO is used. Although the difference is not statistically significant, β_Z tend to be higher in the latter case. By definition, the absolute amount of molecular gas depends on α_CO. Thus, the intercepts of the scaling relations change with the definition. They are lower for the constant α_CO, as expected.

Figure 2 repeats Figure 1 using galaxy-wide stellar A_V instead of nebular A_V. Table 4 extends the analysis in the main section by including SFR and stellar mass as predictors of molecular gas mass, in addition to A_V or/and R₅₀. Figures 3–5 show how the fits change when R₅₀ and/or A_V are added to the primary correlation between ${M}_{{{\rm{H}}}_{2}}$ and SFR. Figure 6 additionally visualizes the dependence of ${M}_{{{\rm{H}}}_{2}}$ on A_V, R₅₀, or metallicity. Figure 7 demonstrates that combining M_⋆ with SFR fails to predict median gas mass for gas-poor galaxies (for xCOLD GASS). When galaxies are on the star formation main sequence, high SFR and high M_⋆ give high gas mass, but when they are off the sequence the relation predicts higher gas mass for high-mass galaxies and overpredicts the gas upper limits. As Table 4 shows, the mass dependence for gas-poor galaxies is weaker (the coefficients change by a factor of 2 and give large changes when multiplied by M⋆). The true scatter from the median gas relation predicted by the combination of SFR and M_⋆ is much higher if the upper limits are observed. A priori it is hard to tell which galaxies will be off the median relation. So using this median relation in practice will give less accurate estimates for gas-poor massive galaxies. If SFR is not used, the 0.15 quartile relations (for gas-poor galaxies) seem not to depend on M⋆. Table 5 presents details of the atomic gas scaling relations, including ones that combine galaxy size with T-type morphology or disk/feature probability.

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The aim here is to assess the effects of galaxy inclination on molecular gas mass as proposed by Concas & Popesso (2019). To that end, we take inclination angle (i) measurements from the bulge-disk decomposition catalog of Simard et al. (2011), in which the bulge Sérsic index was assumed to be n_b = 4. For fair comparison, we also use the best SFR estimates from the xCOLD GASS catalog in this section. Here, we focus on star-forming galaxies, as inclination is proposed to be a useful predictor only for these galaxies. Similar to what we showed in Table 1 using the edge-on probability, the partial Kendall ${\tau }_{{{\rm{H}}}_{2}}$ for the correlation of i with ${M}_{{{\rm{H}}}_{2}}$ at fixed $\mathrm{logH}\alpha /{\rm{H}}\beta$ is −0.15 in star-forming galaxies, while the corresponding partial ${\tau }_{{{\rm{H}}}_{2}}$ of SFR is 0.64. Although the partial correlation of i after taking into account the effect of the Balmer decrement is significant (at α = 0.001), the strength of its correlation is much smaller than that of SFR. Likewise, given the effect of SFR, the partial ${\tau }_{{{\rm{H}}}_{2}}$ of i is −0.1 (p value 0.005). In contrast, given i, the partial ${\tau }_{{{\rm{H}}}_{2}}$ is 0.7 for SFR. Furthermore, fitting together ${M}_{{{\rm{H}}}_{2}}$, SFR, i, and $\mathrm{logH}\alpha /{\rm{H}}\beta$ with the multiple regression models used in the main text indicates that the coefficient for i is not significant (we fail to reject the null hypothesis that its coefficient is zero at α = 0.05). In other words, once SFR and $\mathrm{logH}\alpha /{\rm{H}}\beta$ are used, i does not carry additional information. Using AIC and BIC also indicates that the normal censored regression model in which ${M}_{{{\rm{H}}}_{2}}$ is fitted with SFR and $\mathrm{logH}\alpha /{\rm{H}}\beta$ describes the data better than the one in which $\mathrm{log}{M}_{{{\rm{H}}}_{2}}$ is fitted with i and $\mathrm{logH}\alpha /{\rm{H}}\beta$.

Because it has weak but significant partial correlation, it may be useful to include i when SFR measurements are not available. Following Concas & Popesso (2019), fitting the relation $\mathrm{log}{M}_{{{\rm{H}}}_{2}}=a+b(\mathrm{logH}\alpha /{\rm{H}}\beta -0.62)+c(i-56^\circ )$ for star-forming galaxies gives a = 8.88 ± 0.03, b = 5.70 ± 0.16 and c = −0.007 ± 0.003 and MAD = 0.22 dex for the median relation. The mean relation, estimated using the normal censored regression model, has coefficients consistent with those of the median relation and RMSD = 0.4 dex. Our median relation is not inconsistent with the relation of Concas & Popesso (2019). They found a = 8.93 ± 0.03, b = 5.86 ± 0.35 and c = −0.010 ± 0.002. Their fitting does not include star-forming galaxies with ${M}_{{{\rm{H}}}_{2}}$ upper limits. Similarly, fitting only the detections with a simulation-extrapolation (SIMEX) weighted least-squares model (which includes measurement errors), we get a = 8.93 ± 0.03, b = 5.09 ± 0.48 and c = −0.008 ± 0.002 with RMSD of 0.39. In contrast, the same statistical model gives RMSD of 0.18 when fitting SFR instead of i. Namely, $\mathrm{log}{M}_{{{\rm{H}}}_{2}}=d\,\,+e(\mathrm{logH}\alpha /{\rm{H}}\beta -0.62)+f\mathrm{logSFR}$, where d = 8.95 ± 0.14, e = 1.92 ± 0.30, and f = 0.80 ± 0.05. This gas scaling of star-forming galaxies with SFR and Hα/Hβ is similar to what we found for the total sample in Section 3. To convert e to the coefficient of A_V,fiber, we divide it by 8.06 (from Equation (2)).

Having established that we get a consistent result to that of Concas & Popesso (2019) for star-forming galaxies, we fit the relation among $\mathrm{log}{M}_{{{\rm{H}}}_{2}}$, i and A_V estimated from the Balmer decrement or stellar continuum. In other words, we add i to the analysis presented in Table 4 and present them in Table 6. After A_V is combined with R₅₀ and M_⋆, the q = 0.15 gas relations do not depend on i. Similarly, when A_V, metallicity (Z) and R₅₀ are used i does not improve the gas mass prediction.

In conclusion, given other variables, the effect of inclination is marginal. The combination of A_V and Z or A_V and R₅₀ is a better predictor of molecular gas than the combination of A_V and i. The latter combination, however, does marginally improve the fit of A_V by itself, more significantly in star-forming or gas-rich galaxies.

The Herschel Reference Survey (HRS; Boselli et al. 2010) is a volume-limited (D = 15–25 Mpc) and K-band-selected sample of 322 galaxies. This sample has observations at 250, 350, and 500 μm, and is a benchmark for studies of dust in the nearby universe. The sample spans a wide range of stellar mass (${M}_{\star }={10}^{9}\mbox{--}{10}^{11}\,{M}_{\odot }$), morphological types, and environments (from the field to the center of the Virgo Cluster).

We use the publicly available Herschel Database in Marseille¹¹ to retrieve measurements of gas masses, SFR, and Hα/Hβ of the HRS galaxies. Boselli et al. (2014a) obtained new CO(1–0) observations for 59 objects, which when combined with literature data produced a molecular gas catalog for 225 galaxies. Among these, only 158 have SFR measurements, and only 115 have both SFR and Hα/Hβ measurements derived from long-slit spectroscopic observations. As shown in Figure 8, the massive quiescent HRS galaxies lack SFRs, and the xCOLD GASS sample we analyze in Section 3 is more representative in terms of coverage of the the M_⋆ versus SFR plane. The SFRs of HRS were derived from a variety of available data (Boselli et al. 2015). We divide the SFRs by 1.58 to convert them to the Chabrier (2003) initial mass function. The aperture-corrected CO fluxes (with 3D exponential disk model) are converted to molecular gas masses using constant (Milky Way) and variable (H-band luminosity-dependent) X_CO conversion factors. Boselli et al. (2014a) also compiled H i data for 315 HRS galaxies from the literature. For 52 H i non-detections, 5σ upper limits were estimated assuming a velocity width of 300 km s⁻¹. Likewise, for the 82 H₂ non-detections, 5σ upper limits were estimated by assuming a CO velocity width equal to the H i width, when detected, or to 300 km s⁻¹ otherwise. Stellar masses and r-band half-light radii were taken from Cortese et al. (2012). The stellar masses were derived from i-band luminosities using a relation between g − i color and stellar mass-to-light ratio, assuming a Chabrier initial mass function.

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Boselli et al. (2014b) used the HRS sample and scaling variables M_⋆, stellar surface density, specific star formation rate SSFR ≡ SFR/M_⋆, and metallicity to extend the gas scaling relations for massive galaxies in COLD GASS (which is the precursor to xCOLD GASS, Saintonge et al. 2011). They found a significant correlation between ${M}_{{{\rm{H}}}_{2}}/{M}_{\star }$ and SSFR and between ${M}_{{{\rm{H}}}_{2}}/{M}_{{\rm{H}}{\rm{I}}}$ and metallicity. Here we also do a regression analysis of the HRS sample using M_⋆ and SFR as independent variables and ${M}_{{{\rm{H}}}_{2}}$ as dependent variable. As shown in Table 7, once the SFR dependence is taken into an account, the M_⋆ dependence (for the mean relation) is much weaker for this sample compared to xCOLD GASS (Table 4). Furthermore, for late-type galaxies, Boselli et al. (2014b) found $\mathrm{log}{M}_{{{\rm{H}}}_{2}}/{M}_{\star }\propto a\mathrm{logSSFR}$, where a = 0.94 for a constant X_CO and a = 1.01 for a luminosity-dependent X_CO (see their Table 3). As $\mathrm{log}{M}_{{{\rm{H}}}_{2}}\propto (1-a)\mathrm{log}{M}_{\star }+a\mathrm{logSFR}$, their fit can be interpreted as indicating that the mass dependence is very weak. Thus, for the HRS sample the ${M}_{{{\rm{H}}}_{2}}/{M}_{\star }$ versus SSFR relation is similar to the ${M}_{{{\rm{H}}}_{2}}$ versus SFR relation. Boselli et al.'s method of fitting—using M_⋆ on both sides—may induce correlations in the residuals, in addition to forcing SFR and M_⋆ to share correlated coefficients for the mean gas mass relation. Because the method is implicit, it does not allow physical insights to be gleaned from the coefficients of SFR and M_⋆ separately. In any event, our fits for the xCOLD GASS/xGASS data are not inconsistent with the HRS sample (Figures 9 and 10), despite the difference in the two samples, the X_CO used, and how SFR and A_V are measured.

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Similar to Boselli et al. (2014b), Saintonge et al. (2017) found that ${M}_{{{\rm{H}}}_{2}}/{M}_{\star }$ shows the tightest correlation with SSFR for xCOLD GASS data. Unlike previous work that use these data, we include the upper limits in our analysis and give simple and convenient formulae summarizing the data. We also present scaling relations, for the first time, that do not use SFR. Despite the emphasis of previous scaling relations on M_⋆, Tables 1 and 4 show that the improvement brought by a linear term of M_⋆ is small if half-light radius is used. Using Random Forest, we confirm that A_V and R₅₀ have better predictive power than M_⋆. The scaling relations in the literature using the equivalent (rearranged) relation may give the impression that they have lower dispersions because they did not include upper limits in their analyses. In addition, the ${M}_{{{\rm{H}}}_{2}}/{M}_{\star }$-SSFR relation in Boselli et al. (2014b) looks tighter because the galaxies in their sample with molecular gas measurements are preferentially star-forming, unlike the galaxies in xCOLD GASS, which contain many quiescent galaxies. Nearly all quiescent galaxies in xCOLD GASS are upper limits, and these upper limits have less constraining power when M_⋆ is combined with SFR (e.g., Figure 7). Note also that xCOLD GASS/xGASS gives 3σ upper limits, while HRS uses 5σ upper limits, which further reinforces the impression that the scatter is small because the upper limits overlap with the detections. Lastly, adding M_⋆, in almost all cases, does not statistically significantly improve the prediction of atomic gas masses.