A Full Accounting of the Visible Mass in SDSS MaNGA Disk Galaxies

SDSS DR7 (Abazajian et al. 2009) observed approximately one quarter of the northern sky in both photometry and spectroscopy. A dedicated 2.5 m telescope at the Apache Point Observatory in New Mexico with a wide-field imager and a pair of double fiber-fed spectrometers was used to conduct the multiband imaging and spectroscopic survey. Photometric data were taken in the five SDSS filters: u, g, r, i, and z (Fukugita et al. 1996; Gunn et al. 1998). Using 320 fibers placed into fiber plug plates with a minimum fiber separation of 55″, follow-up spectroscopic analysis was performed on galaxy targets with Petrosian r-band magnitudes m_r ≤ 17.77 and r-band Petrosian half-light radii μ₅₀ ≤ 24.5 mag arcsec⁻² (Lupton et al. 2001; Strauss et al. 2002). For SDSS DR7, the spectrometers covered a wavelength range of 3800–9200 Å with a resolution of λ/Δλ ∼ 1800 (Smee et al. 2013).

We make use of the photometric data for MaNGA galaxies available from the Korea Institute for Advanced Study Value-Added Galaxy Catalog (KIAS-VAGC; Choi et al. 2010). The KIAS-VAGC is based on SDSS DR7 and the New York University Value-Added Galaxy Catalog (NYU-VAGC; Blanton et al. 2005). The NYU-VAGC contains multiple crossmatched galaxy catalogs, including SDSS and independently reduced data from SDSS. We use the u − r color, Δ(g − i) color gradient, and inverse concentration index from the KIAS-VAGC, which are calculated using the NYU-VAGC data. Colors are calculated using fluxes within the r-band Petrosian radius. Δ(g − i) is the difference in the (g − i) color between the region within 0.5R_Pet and the annulus between 0.5R_Pet and R_Pet, where R_Pet is the i-band Petrosian radius. A galaxy with a negative color gradient is bluer on the outside, whereas a galaxy with a positive color gradient is redder on the outside. The inverse concentration index, c_inv, is measured as c_inv = R_50,i /R_90,i , where R_50,i and R_90,i are the i-band 50% and 90% Petrosian radii, respectively. We use the global emission-line fluxes from the Portsmouth group galaxy properties catalog (Thomas et al. 2013) to calculate the gas-phase metallicity.

H i mass estimates are obtained from the H i–MaNGA DR3 (Stark et al. 2021). H i–MaNGA is a follow-up survey of MaNGA galaxies conducted on the Robert C. Byrd Green Bank Telescope (GBT) in Green Bank, West Virginia. MaNGA galaxies with redshifts z < 0.05 are observed in the GBT L band (1.15–1.73 GHz). The third data release of H i–MaNGA has H i observations of 3358 MaNGA galaxies from GBT and also includes a crossmatch between the Arecibo Legacy Fast ALFA (ALFALFA) survey (Haynes et al. 2018) performed at the Arecibo Observatory in Arecibo, Puerto Rico, and MaNGA DR17 targets. ALFALFA was a blind 21 cm survey with observations of 31,500 H i sources within z < 0.06 using the Arecibo L-band Feed Array (1.355–1.435 GHz). H i–MaNGA DR3 includes 3,274 of these sources, which are crossmatched with MaNGA and have a redshift z < 0.05.

H₂ masses are inferred from measurements of the CO(1-0) line emission from two surveys: the MaNGA Arizona Radio Observatory (ARO) Survey of Targets (MASCOT) first data release (Wylezalek et al. 2022) and the xCOLD GASS survey (Saintonge et al. 2017). The MASCOT survey performs observations of MaNGA galaxies at the ARO using the 12 m ARO antenna with a 3 mm receiver with frequency range 84–116 GHz. MASCOT has observations of the CO(1-0) emission line for 187 galaxies selected from MaNGA DR15 with stellar masses greater than 10^9.5 M_⊙. The xCOLD GASS survey conducted CO(1-0) observations of SDSS galaxies on the IRAM 30-meter telescope in Spain. xCOLD GASS targets were selected from SDSS DR7 with redshifts 0.01 < z < 0.05 and stellar masses greater than 10⁹ M_⊙. xCOLD GASS has observations of 532 SDSS DR7 galaxies. We crossmatch 41 galaxies from xCOLD GASS with MaNGA DR17, two of which also have observations in MASCOT. Excluding CO nondetections, we have a total of 204 galaxies with CO observations in MaNGA DR17 from MASCOT and xCOLD GASS combined.

As shown in Douglass & Demina (2022), a galaxy’s ratio of total to stellar mass depends on the galaxy’s evolutionary stage. We therefore separate the galaxies into three populations—blue cloud, green valley, and red sequence—in the color–magnitude diagram (CMD) to better understand these relationships. Galaxies in the blue cloud are typically fainter and more blue, while galaxies in the red sequence are brighter and more red. It is believed that galaxies transitioning between the blue cloud and red sequence occupy the green valley (Martin et al. 2007).

We use the same method to classify the galaxies into one of these three populations as used in Douglass & Demina (2022), where the classification is based on the inverse concentration index, c_inv, color, u − r, and color gradient, Δ(g − i). As shown in Figure 1, galaxies that are part of the red sequence are those that generally fall above and to the right of the depicted boundary originally defined by Park & Choi (2005; normal early-type galaxies), while galaxies that are part of the blue cloud are those that generally fall below and to the left of the boundary (late-type galaxies). Galaxies that are part of the green valley are either those above the boundary but with u − r < 2 (blue early-type galaxies) or a high c_inv or those below the boundary with θ < 20°, where

See Douglass & Demina (2022) for a more detailed description of the CMD classification.

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In this study, we analyze the rotation curves of galaxies. Thus, we require our objects to be dominated by rotational motion (described in Section 3 below). Disk galaxies have velocity dispersions that are small compared to their rotational velocities, so the total dynamical mass of a disk galaxy can be calculated assuming that the centripetal acceleration is due to the gravitational force (Sofue 2017). As a result, we expect few galaxies in our sample to be in the red sequence. After visual inspection, we find that the red-sequence galaxies that are in our final sample appear to be either red-disk galaxies with little to no star formation (likely lenticulars) or elliptical galaxies that are still supported by rotation.

We estimate a galaxy’s total dynamical mass using its Hα velocity map obtained from the SDSS MaNGA DAP. We restrict our analysis to spaxels with a data quality bit of 0, as provided by the SDSS MaNGA DAP. Spaxels with a nonzero bit value indicate data that experienced issues in observations or in the data analysis process (Westfall et al. 2019). In addition, we only include spaxels with a signal-to-noise ratio ≥ 5 in the Hα flux to ensure that only spaxels with a significant detection, and therefore a reliable redshift, are considered in the analysis.

We also require all galaxies to have a smooth velocity gradient with a maximum “smoothness score” of 2.0, as described in Douglass et al. (2019). We further restrict the analysis to galaxies with a T-Type > 0 (late-type galaxies), as classified by the MaNGA Morphology Deep Learning DR17 Value Added Catalog (Domínguez Sánchez et al. 2022).

Similar to both Douglass et al. (2019) and Douglass & Demina (2022), the velocity map of each galaxy is fit to the rotation curve parameterization defined in Barrera-Ballesteros et al. (2018):

where V(r) is the rotational velocity at a distance r from the center of the galaxy. The free parameters are , the magnitude of the velocity at which the rotation curve plateaus; R_turn, the radius at which the rotation curve changes from increasing to flat; and α, which describes the sharpness of the curve. The extent of the MaNGA Hα velocity maps and the radii to which we can measure rotational velocities are limited by the visible extent of the galaxy. Rotation curves are only fit out to the maximum radius, , covered by the IFU, the extent of which is shown for an example galaxy in Figure 2.

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Each galaxy’s systemic velocity, kinematic center, inclination angle, and position angle are also free parameters in this fit, resulting in a total of eight free parameters. When determining the best-fit model for each galaxy, we make use of χ² = Σ((data − model)/uncertainty)² and , where we normalize χ² by the difference between the number of unmasked spaxels in the velocity map and the number of free parameters in the fit. We define four best-fit models, as follows:

• Model 1: the model with the smallest χ².
• Model 2: the model with the smallest residual, Σ(data − model)².
• Model 3: to help remove foreground artifacts from the analysis, we define upper- and lower-velocity bounds by binning all unmasked spaxels with a bin width of 10 km s⁻¹. The velocity bounds are defined as the nearest empty bin on either side of the bin with the most spaxels, as shown in the histogram to the bottom left of Figure 3. Spaxels with values outside of this velocity range are masked; see the top center of Figure 3 for an example of the resulting mask. We then select the model with the smallest χ².
• Model 4: to help remove spaxels that are potentially contaminated by emission from active galactic nuclei, which are defined as bins with an unusually high velocity dispersion, we define an upper limit on the velocity dispersion by binning the velocity dispersion of the unmasked spaxels with a bin width of 10 km s⁻¹. The velocity dispersion upper bound is defined as the nearest empty bin to the lowest-velocity-dispersion bin containing spaxels, as shown in the histogram to the bottom right of Figure 3. Spaxels with velocities above this upper limit are masked; see the top right plot of Figure 3 for an example of the resulting mask. We then select the model with the smallest χ².

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Of these four models, we eliminate those with α = 100, where α is the parameter from Equation (2). This eliminates models where the fitting algorithm has failed, as 100 is the upper limit of α while fitting. Of the remaining models, we select the one with the lowest as the best-fit model for each galaxy. An example Hα velocity map and best-fit model map are shown in Figure 4.

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We estimate each galaxy’s stellar mass by fitting a rotation curve due to the stellar component of the galaxy using the stellar mass density maps available through the Pipe3D MaNGA analysis pipeline (Sánchez et al. 2016, 2018). An example stellar mass density map is shown at the top of Figure 5. Using the best-fit model Hα velocity map values for the galaxy’s kinematic center, inclination angle, and position angle described above, we define concentric ellipses that correspond to different orbital radii in the galaxy, with the radius of each ellipse increasing by 2 spaxels. We compute the stellar mass as a discretized function of radius, M_*(r), by summing the stellar mass density per spaxel over all the spaxels within each ellipse.

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We assume that the stellar mass is the primary component of the galaxy’s disk and model the stellar mass as the sum of a central bulge and exponential disk. The rotational velocity due to the bulge and disk is summed in quadrature to get the rotational velocity due to the stellar mass:

where V_*(r) is the rotational velocity due to the stellar mass, V_b (r) is the rotational velocity due to the bulge component, and V_d (r) is the rotational velocity due to the disk component.

The bulge is modeled as an exponential sphere (Sofue 2017) with rotational velocity

where G = 6.67408 × 10⁻¹¹ m³ kg⁻¹ s⁻² is the Newtonian gravitational constant, F(x) = 1 − e^−x (1 + x + 0.5x²), and . The free parameters in this fit are the scale radius of the bulge, R_b , and the central density of the bulge, ρ_b .

The rotational velocity due to the exponential disk (a thin disk without perturbation; Freeman 1970) is

where Σ_d is the central surface mass density of the disk, R_d is the scale radius of the disk, y = r/2R_d , and I_i and K_i are modified Bessel functions (Sofue 2013). The free parameters in this fit are Σ_d and R_d .

We calculate the galaxy’s total dynamical mass within the 90% elliptical Petrosian radius, R₉₀, using the rotational velocity at this radius as determined from the best-fit rotation curve, found as described in Section 3.1. We can calculate the mass of a galaxy within some radius r from the center of the galaxy under the assumption that the galaxy’s rotational motion is dominated by Newtonian orbital mechanics. Assuming axial symmetry, the velocity of a particle at distance r from the center of the galaxy is a function of the mass within that radius, M(r). Assuming that the orbital motion is circular in spiral galaxies, the centripetal acceleration of an orbiting particle is due to the gravitational force:

Here, V(r) is the rotational velocity a distance r from the center of the galaxy. In order to study the same region of each galaxy, we estimate the mass within R₉₀, M(R₉₀) = M_tot, by calculating V(R₉₀) from Equation (2). When , we extrapolate our parameterization of the fitted rotation curve, Equation (2), out to R₉₀. On average, R₉₀ exceeds by about 10%. Figure 6 shows a subset of our rotation curves normalized by , where the curves extrapolated out to R₉₀ for the galaxies with are shown as dashed extensions beyond . The stellar and H i masses are also evaluated within R₉₀. Only global measurements are available for the remaining mass components, but these are expected to be concentrated within the visible disk and thus are also within R₉₀.

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While the majority of the stellar mass is encompassed by R₉₀, gas and dark matter profiles are known to extend much farther than that (e.g., Ostriker et al. 1974; Begeman 1989; Kamphuis & Briggs 1992; Pohlen et al. 2010). Extrapolating the rotation curves to higher radii would significantly increase the uncertainty on the rotational velocity, so we focus our study on the mass content within the visible extent of the galaxy.

To calculate the total dynamical mass within R₉₀, we require:

• 1.  
  α ≤ 99;
• 2.  
  Velocity maps with less than 95% of their spaxels masked;
• 3.  
  10 km s⁻¹ < V(R₉₀) < 1000 km s⁻¹; and
• 4.  
  , where is the uncertainty in the best-fit value of .

The first, third, and fourth conditions eliminate unsuccessful fits that result in nonphysical models. The first condition eliminates fits where α approaches the maximum allowed value, which indicates an unsuccessful fit. The third condition eliminates galaxies where the inclination angle in the fit is incorrect and approaches the boundary values for the parameter, resulting in a very high or very low V(R₉₀). The fourth condition removes models with large uncertainties in , also indicative of an unsuccessful fit. The second condition removes models for velocity maps where too many spaxels are masked and they therefore have too few data points to result in a trustworthy model.

To estimate the total stellar mass of each galaxy, M_*, within R₉₀, we use the parameters from the best-fit disk and bulge rotation curve, as described in Section 3.2. The total mass of the bulge and disk at some radius r is

where the mass of the bulge component is

and the mass of the disk component is

So that we study the stellar mass within the same region of each galaxy as the total mass, we evaluate Equation (7) at R₉₀. We apply a stellar mass cut and remove galaxies with M_*(R₉₀) < 10⁹ M_⊙ from our analysis so that we can perform the H i mass scaling described below (Section 4.3). After applying the quality cuts described in Section 4.1 and this stellar mass cut, our final sample consists of 5503 galaxies with best-fit rotation curves.

We use the H i mass from the H i–MaNGA DR3 survey to quantify the neutral atomic gas content within each galaxy. H i mass estimates are available for 2588 galaxies in our sample.

We estimate the H i mass within R₉₀ from the total H i mass following the procedure in Wang et al. (2020) for galaxies with M_* > 10⁹ M_⊙. Using the total H i mass, we calculate R_{H I}, the radius where the H i density is 1 M_⊙ pc^–2 (Wang et al. 2016):

Here, M_HI,tot is the total H i mass obtained from H i–MaNGA and R_{H I} is in units of kiloparsecs. We assume that within R_{H I}, the H i density follows the median profile from Wang et al. (2016), and outside of R_{H I}, it follows an exponential profile with scale radius 0.2R_{H I}:

Σ_{H I} is the H i surface density at radius r. We consider 1.5R_{H I} to be the edge of the H i disk, following Wang et al. (2020). The H i mass outside some radius r can then be calculated by integrating over the H i surface density from r to 1.5R_{H I}:

If R₉₀ is greater than 1.5R_{H I}, then we define the H i mass within R₉₀, M_{H I}, as the total H i mass, M_HI,tot. If R₉₀ is less then 1.5R_{H I}, then we calculate the H i mass between R₉₀ and 1.5R_{H I} using Equation (13) and subtract this value from the total H i mass to define M_{H I} for the galaxy.

Molecular hydrogen, H₂, is a low-mass, symmetric molecule without a dipole moment, and therefore it does not produce a significant amount of radiation, making it notoriously difficult to detect. Hence, to evaluate the molecular hydrogen content in a galaxy, it is customary to parameterize it with respect to some other observable. The most commonly used method is to use another molecular gas, particularly CO. We obtain mass estimates of H₂ parameterized by the CO(1-0) line emission from the MASCOT and xCOLD GASS surveys for 107 galaxies that also have total mass, stellar mass, and H i mass estimates (as described above).

We have CO observations for only a small fraction of our galaxies, so we use CO observations of SDSS DR7 galaxies to derive a parameterization of the H₂ mass as a function of galaxy luminosity in the r band, M_r . A galaxy’s H₂ mass has been shown to be strongly correlated with its star formation rate (e.g., Robertson & Kravtsov 2008). We choose to parameterize the H₂ mass with luminosity, since this quantity is related to the star formation rate (e.g., Hirashita et al. 2003) and luminosity is directly measured, whereas star formation rate is a derived quantity. As shown in Figure 7, we use χ² minimization to find the coefficients that describe the linear relationship between and M_r :

where is the mass of molecular hydrogen. The values of a and b are listed in Table 1 and depend on the color–magnitude classification. We use this parameterization to estimate when CO observations are not available for galaxies in our sample.

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We assume that molecular hydrogen is concentrated within the optical disk of galaxies, so we use the global H₂ mass of each galaxy in this analysis as the mass of H₂ within R₉₀.

In this study, we define the total gas mass, M_gas, as the sum of the H i mass, H₂ mass, and helium mass:

We approximate the helium mass, M_He, by assuming a mass fraction of 25%:

This is the amount of helium measured in the intergalactic medium and agrees well with the prediction from big bang nucleosynthesis (Cooke & Fumagalli 2018). In this equation, M_{H I} is the dominant component and is scaled to R₉₀, and is assumed to be contained within R₉₀. Hence, the estimate for M_He, and as a result the estimated total gas mass, can also be considered to be within R₉₀.

The heavy metals and dust mass, M_dust, is approximated from a galaxy’s gas-phase metallicity. We compute the gas-phase metallicity, , following the R-calibration method described in Pilyugin & Grebel (2016), using the flux of the [O ii] λ λ 3727,3729 doublet and the [N ii]λ6548, [N ii]λ6584, [O iii]λ4959, and [O iii]λ5007 emission lines. The fluxes are extinction-corrected using the Balmer decrement, assuming a flux ratio Hα/Hβ = 2.86 (Osterbrock & Ferland 2006). We compute the metallicity as

where

are ratios of the specified emission-line fluxes. The values of the coefficients in Equation (17) depend on the value of logN₂ and are listed in Table 2.

We assume a constant dust-to-metals ratio corresponding to the metallicity calibration, M_dZ /M_dust = 0.206, for galaxies with a gas-phase metallicity greater than 8.2 (De Vis et al. 2019). M_dZ is the dust mass of each galaxy. The total mass of heavy metals and dust is then

where f_Z is the mass fraction of metals,

and M_g is the gas mass of the galaxy as defined in De Vis et al. (2019):

where

We define the total visible mass of a galaxy, M_vis, as the sum of the stellar mass, M_*, the gas mass, M_gas (Equation (15)), and the heavy metals and dust mass, M_dust (Equation (21)):

A summary of the relative contributions of each individual mass component to the total visible mass for SDSS DR7 galaxies within R₉₀ as a function of the r-band luminosity, M_r , is shown in Figure 8. For galaxies with M_r > −17, gas is the dominant component of the visible mass, whereas for galaxies with M_r < −18, the stellar mass dominates the visible mass. Heavy metals and dust contribute on the order of 1%, regardless of magnitude.

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As shown in Figure 11, we find that once we account for all of the visible mass components of a galaxy, the ratio of M_vis/M_tot shows an upward trend with galaxy luminosity. This is in agreement with the previous work by Torres-Flores et al. (2011), who consider the relationship between M_vis, defined as stellar mass and H i mass, and total mass. Torres-Flores et al. (2011) find a correlation between the mass ratio and evolutionary stage, in that late-type low-mass spirals are dominated by dark matter in comparison to early-type high-mass spirals.

M_vis/M_tot is a version of the SHMR, typically described as the ratio of stellar mass to halo mass. Models predict an SHMR that deviates from a flat distribution (e.g., Behroozi et al. 2019), with lower values for the faintest and brightest galaxies. These galaxies are thought to be dominated by dark matter. We find that the faint end of blue-cloud galaxies shows this expected decrease in M_vis/M_tot, suggesting an additional abundance of dark matter within the visible extent of these galaxies that is not present in brighter galaxies.