SPARC: MASS MODELS FOR 175 DISK GALAXIES WITH SPITZER PHOTOMETRY AND ACCURATE ROTATION CURVES

The overall properties of SPARC galaxies are shown with histograms in Figure 1. SPARC spans a broad range in morphologies (S0 to Im/BCD), luminosities (∼10⁷ to ∼10¹² L_⊙), effective radii (∼0.3 to ∼15 kpc), effective surface brightnesses (∼5 to ∼5000 L_⊙ pc⁻²), rotation velocities (∼20 to ∼300 km s⁻²), and gas content (0.01 ≲ M_{H i} /L_[3.6] ≲ 10). Galaxies with 10¹⁰ < L_[3.6]/L_⊙ < 10^10.5 are slightly underrepresented (in a volume-weighted sense): apparently H i   studies have tended to focus on bright spirals and dwarf galaxies. As expected, SPARC does not contain galaxies with L_[3.6] > 10¹² L_⊙, which typically are gas-poor ellipticals in clusters, nor does it contain galaxies with L_[3.6] < 10⁷ L_⊙, for which our knowledge is restricted to gas-poor spheroidals around the Milky Way and M31.

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Most galaxies in SPARC have late Hubble types (9–10), corresponding to Sm and Im galaxies. This happens because the Hubble classification is intrinsically biased toward HSB features and assigns most LSB galaxies within these two latest stages (McGaugh et al. 1995b). Currently, the SPARC data set contains only three lenticulars because early-type galaxies generally lack a high-density H i   disk and have been historically neglected by interferometric H i   studies. Recent surveys, however, show that several early-type galaxies in the group and field environments possess an outer low-density H i   disk (Serra et al. 2012), which can be used for kinematic investigations (den Heijer et al. 2015).

The ratio R_{H i}/R_eff varies from ∼1 to ∼14 (panel 8 of Figure 1). Clearly, the H i   rotation curve probes the gravitational potential out to much larger radii than existing IFU surveys like CALIFA (García-Lorenzo et al. 2015) or MANGA (Bundy et al. 2015). Historically, the relative extent of H i to stellar disks has been measured using either R_{H i}/R₂₅, where R₂₅ is the B-band isophotal radius at 25 mag arcsec⁻² (Broeils & Rhee 1997; Noordermeer et al. 2005), or R_{H i}/R_⋆, where R_⋆ = 3.2R_d (Swaters et al. 2009; Lelli et al. 2014). The latter definition allows for comparing stellar disks with different central surface brightnesses: a pure exponential disk with Σ_d = 21.5 B mag arcsec⁻² (Freeman 1970) has R_⋆ = R₂₅. We find with a median value of 1.7. This is in line with previous estimates of for different galaxy types: 1.7 ± 0.7 (Noordermeer et al. 2005) for early-type galaxies (S0 to Sab), 1.7 ± 0.5 (Broeils & Rhee 1997) for spiral galaxies (mostly Sb to Scd), 1.8 ± 0.8 (Swaters et al. 2009) for late-type dwarfs (Sd to Im), and 1.7 ± 0.5 (Lelli et al. 2014) for starburst dwarfs (Im and BCD). Martinsson et al. (2016), however, find R_{H i} /R₂₅ = 1.35 ± 0.22 for 28 late-type spirals. We note that SPARC galaxies are representative of the disk population in the field, nearby groups, and diffuse clusters like Ursa Major (Tully & Verheijen 1997). Galaxies in dense clusters like Virgo may show smaller values of R_{H i}/R_⋆ due to gas stripping (e.g., Chung et al. 2009).

The galaxies in SPARC can be classified in three main groups depending on the distance estimate.

I. Accurate Distances: 50 objects have distances from the tip of the red giant branch (45), Cepheids (3), or supernovae (2). These distances have errors ranging from ∼5% to ∼10% and are generally on the same zero-point scale (see discussion in Tully et al. 2013). The vast majority of these distances are drawn from the Extragalactic Distance Database (Jacobs et al. 2009; Tully et al. 2013).

II. Ursa Major: 28 objects lie in the Ursa Major cluster (Verheijen & Sancisi 2001), which has an average distance of 18 ± 0.9 Mpc (Sorce et al. 2013b). This explains the peak at log(D/Mpc) ≃ 1.25 in panel 10 of Figure 1. The Ursa Major cluster has an estimated depth of ∼2.3 Mpc (Verheijen 2001); hence, we adopt an error of Mpc for individual galaxies. Two galaxies from Verheijen (2001) are considered as background/foreground objects: NGC 3992, having D ≃ 24 Mpc from a Type Ia supernova (Parodi et al. 2000), and UGC 6446, lying near the cluster boundary in both space and velocity (D ≃ 12 Mpc from the Hubble flow).

III. Hubble Flow: 97 objects have Hubble-flow distances corrected for Virgocentric infall (taken from NED). We assume H₀ = 73 km s⁻¹ Mpc⁻¹, giving distances on a similar zero-point scale as those in groups I and II (e.g., Tully et al. 2013). Hubble-flow distances are very uncertain for nearby galaxies, where peculiar velocities may strongly contribute to the systemic velocities, but become more accurate for distant objects. We adopt the following error scheme: 30% for D ≤ 20 Mpc, 25% for 20 Mpc < D ≤ 40 Mpc, 20% for 40 Mpc < D ≤ 60 Mpc, 15% for 60 Mpc < D ≤ 80 Mpc, and 10% for D > 80 Mpc. This scheme considers that peculiar velocities can be as high as ∼500 km s⁻¹ and H₀ has an uncertainty of ∼7%. Lelli et al. (2016) show that the intrinsic scatter in the BTFR for galaxies in groups I+II is similar to that for the entire sample, indicating that the assumed errors on Hubble-flow distances are realistic.

For the 175 galaxies in SPARC, we derived homogeneous surface photometry at 3.6 μm following the procedures of Schombert & McGaugh (2014b). The frames were flat-fielded and calibrated using the standard Spitzer pipeline. Subsequently, they were interactively cleaned by masking bright contaminating sources, like foreground stars and background galaxies. This is a crucial step because [3.6] images contain nearly 10 times more point sources than similar exposures in optical bands. Masked pixels were subsequently replaced with the local mean isophotal value for aperture luminosity determination.

The cleaned frames were analyzed using the Archangel software (Schombert 2011). The absolute sky value and its error were estimated using multiple sky boxes, i.e., regions in the frame that are free of background/foreground objects. The standard Spitzer pipeline provides a formal zero-point error smaller than 2%, but a more realistic photometric error should also consider the sky brightness error, which dominates all other sources of uncertainty in the surface photometry and aperture magnitudes (Schombert & McGaugh 2014b). The mean error on sky brightness is ∼3%.

To derive azimuthally averaged surface brighntess profiles, we fitted ellipses to the frames using a standard Fourier-series least-square algorithm. During the ellipse fitting, pixels with values above 3σ of the mean isophote were automatically masked and replaced with the mean value at that radius. These pixels typically correspond to unresolved background/foreground objects, but we visually inspected each frame to ensure that real asymmetric features in the galaxy (like prominent spiral arms or H ii regions) were not removed. Most low-mass and LSB galaxies have irregular morphologies; hence, ellipses only provide a zeroth-order apporximation of their shape. The errors in the surface brightness profiles, however, are dominated by the uncertainties in the sky value rather than variations around each ellipse. For thin, egde-on galaxies, ellipse fitting does not provide satisfactory results in the central regions; hence, we derived surface brightness profiles considering a narrow circular section around the semimajor axes and average both sides.

To estimate aperture magnitudes, we built curves of growth by summing both raw and replaced pixels within the best-fit ellipse at a given radius (subpixels are used at the ellipse edges). The amount of light from replaced pixels varies from 2% to 15%. Summing pixels generally leads to unstable curves of growth at large radii, where the galaxy surface brightness is comparable to the contaminating halos of imperfectly masked sources. Hence, we replaced the aperture values with the mean isophotal values at large radii (beyond the radius containing ∼90% of the raw luminosity). This provides curves of growth that approach a clear asymptotic value at large radii. We calculated total magnitudes using asymptotic fits to the curves of growth. Corresponding errors were estimated by recalculating the total magnitudes using a 1σ variation in the sky value. Total luminosities assume a solar absolute magnitude of 3.24 at 3.6 μm (Oh et al. 2008). We also calculated the effective radius (R_eff), i.e., the radius encompassing half of the total luminosity, and the effective surface brightness (Σ_eff), i.e., the average surface brightness within R_eff.

To estimate the disk scale length (R_d) and central surface brightness (μ_d), we fitted exponential functions to the outer parts of the surface brightness profiles. We stress that the results depend on the fitted radial range, which may be ambiguous for galaxies with outer down-bending (type II) or up-bending (type III) profiles (see Erwin et al. 2008). For most low-mass and LSB galaxies, exponential functions provide a good description of the surface brightness profiles over a broad radial range, but several galaxies show light enhancements/depressions in the inner parts (similar to optical bands; e.g., Swaters & Balcells 2002; Schombert et al. 2011). As expected, the vast majority of spiral galaxies show deviations from exponential profiles in the inner parts due to bulges, bars, and lenses. All surface brightness profiles are available for download at astroweb.cwru.edu/SPARC.

The majority (∼75%) of SPARC rotation curves are the result of Ph.D. theses from the University of Groningen (Begeman 1987; Broeils 1992; de Blok 1997; Verheijen 1997; Swaters 1999; Noordermeer 2006; Lelli 2013). They were derived using similar techniques and software, forming a relatively homogeneous data set. These 175 rotation curves are drawn from the following publications (in parentheses we provide the corresponding number of objects excluding duplicates): Swaters et al. (2009; 32), Sanders & Verheijen (1998; 30), Sanders (1996; 17), Noordermeer et al. (2007; 12), de Blok et al. (2001; 10), Begeman et al. (1991; 9), Lelli et al. (2012a, 2012b, 2014; 8), de Blok & Bosma (2002; 7), de Blok et al. (1996; 7), Kuzio de Naray et al. (2008; 6), Spekkens & Giovanelli (2006; 5), Gentile et al. (2004, 2007; 4), van Zee et al. (1997; 4), Côté et al. (2000; 3), Fraternali et al. (2002, 2011; 3), Hallenbeck et al. (2014; 2), Trachternach et al. (2009; 2), Barbieri et al. (2005; 1), Battaglia et al. (2006; 1), Begum et al. (2003; 1), Begum & Chengalur (2004; 1), Blais-Ouellette et al. (2004; 1), Boomsma et al. (2008; 1), Chemin et al. (2006; 1), Elson et al. (2010; 1), Kepley et al. (2007; 1), Richards et al. (2015; 1), Simon et al. (2003; 1), Verdes-Montenegro et al. (1997; 1), Verheijen & de Blok (1999; 1), and Walsh et al. (1997; 1). For duplicate galaxies, we select the most reliable rotation curve. Table 1 provides the reference for each galaxy.

We do not consider rotation curves from THINGS (de Blok et al. 2008) and LITTLE-THINGS (Oh et al. 2015) for the sake of homogeneity. These rotation curves are characterized by many small-scale bumps and wiggles that are not observed in other rotation curve samples. Most likely these small-scale features do not trace the smooth, axisymmetric gravitational potential, but may be due to noncircular components like streaming motions along spiral arms (e.g., Khoperskov & Bertin 2015). SPARC rotation curves are generally smooth but can show large-scale features with a direct correspondence in the surface brightness profile, in agreement with Renzo’s rule (Sancisi 2004): “For any feature in the luminosity profile there is a corresponding feature in the rotation curve and vice versa.”

3.2.1. Beam-smearing Effects and Hybrid Rotation Curves

3.2.2. Error Budget and Rotation Curve Quality

The errors on the rotation velocities are estimated as in Swaters et al. (2009):

where V_app and V_rec are the rotation velocities obtained from fitting separately the approaching and receding sides of the disk, respectively, and is the formal error from fitting the entire disk. We consider only velocity points that are derived from both sides of the disk. Equation (1) considers both local deviations from circular motions and global kinematic asymmetries between the two sides of the disk. It does not include, however, systematic errors due to the assumed inclination. If the H i   disk is not strongly warped, a change in i would change the normalization of the rotation curve without altering its shape. Inclination corrections go as sin(i); thus, they are small for edge-on galaxies but become large for face-on ones.

We assign a quality flag (Q) to each rotation curve using the following scheme: Q = 1 for galaxies with high-quality H i   data or hybrid Hα/H i rotation curves (99 objects); Q = 2 for galaxies with minor asymmetries and/or H i   data of lower quality (64 objects); Q = 3 for galaxies with major asymmetries, strong noncircular motions, and/or offsets between H i   and stellar distributions (12 objects). Galaxies with Q = 3 are not suited for detailed dynamical studies: we build mass models for completeness but do not consider them in our analysis. Similarly, we provide mass models for face-on galaxies with i < 30° but exclude them in our analysis. This does not introduce any selection bias since galaxy disks are randomly oriented on the sky. Our final science sample is made of 153 galaxies.

We derive the contributions of gas disk (V_gas), stellar disk (V_disk), and bulge (V_bul) to the observed rotation curves. The total baryonic contribution is then given by

where ϒ_disk and ϒ_bul are the stellar mass-to-light ratios of disk and bulge components, respectively. Note that absolute values are needed because V_gas can sometimes be negative in the innermost regions: this occurs when the gas distribution has a significant central depression and the material in the outer regions exerts a stronger gravitational force than that in the inner parts. In this paper, we investigate the ratio V_bar/V_obs for different assumptions of ϒ_⋆ (Section 5.3). Rotation curve fits with DM halos are presented in Katz et al. (2016) and will be exploited in future publications.

The gas contribution is calculated using the formula from Casertano (1983), which solves the Poisson equation for a disk with finite thickness and arbitrary density distribution. We assume a thin disk with a total mass of 1.33 M_{H i}, where the factor 1.33 considers the contribution of helium. V_gas is either computed using H i   surface density profiles or taken from published mass models. For D512-2, D564-8, and D631-7, H i   surface density profiles are not available; hence, we compute V_gas assuming an exponential profile with a scale length of 2R_d. This is a reasonable approximation for such late-type galaxies (e.g., Swaters et al. 2002). We neglect the contribution of molecular gas because CO data are not available for most SPARC galaxies. Fortunately, molecules constitute a minor dynamical component and are generally subdominant with respect to stars and atomic gas (Simon et al. 2003; Frank et al. 2016). If molecules are distributed in a similar way to the stars, their contribution is implicitly included in the assumed value of ϒ_⋆.

The stellar contribution is calculated using the observed [3.6] surface brightness profile and extrapolating the exponential fit to , unless the profile shows a clear truncation. For galaxies with T < 4 (Hubble types earlier than Sbc), we perform bulge–disk decompositions in a nonparametric way. First, we identify a fiducial radius R_bul within which the bulge light dominates. We then subtract the stellar disk from the profile by extrapolating an exponential fit at R < R_bul: this fit describes the inner disk structure (like lenses, rings) and does not necessarily correspond to the outer disk fit. Finally, the residual luminosity profile is used to compute V_bul and linearly extrapolated at R > R_bul to avoid unphysical truncations. We assume that the bulge is spherical and the disk has an exponential vertical distribution with scale height z_d = 0.196R_d^0.633 (Bershady et al. 2010b). The uncertainties in V_bul and V_disk are dominated by ϒ_⋆ rather than geometry. For example, for oblate bulges the peak in V_bul would increase by less than 20% (Noordermeer 2008). For galaxies with T ≥ 4, we assume that all stars lie in a disk and interpret possible light concentrations as disky pseudobulges (Kormendy & Kennicutt 2004). For only five Sbc-to-Scd galaxies (IC 4202, NGC 5005, NGC 5033, NGC 6946, UGC 2885) do bulge–disk decompositions improve the results. Conversely, we do not decompose NGC 3769 despite it being an Sb.

In Table 2, we provide the values of V_bul and V_disk for ϒ_⋆ = 1 M_⊙/L_⊙. This is just a convenient value but often exceeds a maximum-disk fit. V_bul and V_star can be trivially rescaled to any arbitrary ϒ_⋆.

Figure 2 illustrates the covariance between L_[3.6], R_eff, and Σ_eff (top panels) as well as L_[3.6], R_d, and Σ_d (bottom panels). As expected, the total luminosity broadly correlates with both radius (R_eff and R_d) and surface brightness (Σ_eff and Σ_d). In the top left panel, the solid line shows the K-band mass–radius relation from the GAMA survey (Lange et al. 2015) for morphologically identified late-type galaxies. For comparison, the stellar masses from Lange et al. (2015) are converted into [3.6] luminosities adopting ϒ_⋆ = 0.5 M_⊙/L_⊙ (Schombert & McGaugh 2014a). The dashed lines show a 0.4 dex deviation from the GAMA relation, corresponding to ∼2 times the observed rms scatter (σ_obs). We find overall agreement with Lange et al. (2015), indicating that SPARC is not missing any type of galaxy disk in terms of luminosity, size, or surface brightness.

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In Figure 2, one may discern two separate sequences in the luminosity–size planes, corresponding to early-type HSB spirals and late-type LSB galaxies. This separation may be either real or due to the underrepresentation of galaxies with L_[3.6] ≃ 10¹⁰ L_⊙ in SPARC. Interestingly, a similar separation has been found by Schombert (2006) and may indicate the existence of a surface brightness bimodality for stellar disks (Tully & Verheijen 1997; McDonald et al. 2009a, 2009b; Sorce et al. 2013a).

Figure 3 (top left) shows the M_{H i} –R_{H i} relation (Broeils & Rhee 1997; Verheijen 2001). This correlation is extremely tight. We fit a line using LTS_LINEFIT (Cappellari et al. 2013), which considers errors in both variables and allows for intrinsic scatter (σ_int). LTS_LINEFIT assumes that the errors on the two variables are independent, but M_{H i} and R_{H i} are affected by galaxy distance as D² and D, respectively. Hence, we introduce distance uncertainties only in M_{H i} with a linear dependence. We also consider typical errors of 10% on M_{H i} and R_{H i} due to flux calibration. We find

with σ_obs = 0.13 dex and σ_int = 0.06 ± 0.01 dex. This is one of the tightest scaling relations of galaxy disks. For comparison, the BTFR has σ_obs ≃ 0.18 dex and σ_int ≲ 0.11 dex (Lelli et al. 2016).

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A tight M_{H i} –R_{H i} relation with a slope of ∼2 implies that the mean surface density of H i   disks is constant from galaxy to galaxy (Broeils & Rhee 1997). For comparison, the dotted line in Figure 3 shows the expected relation for H i   disks with a mean surface density of 3.5 M_⊙ pc⁻². We find a slope of 1.87, which is similar to previous estimates: 1.96 (Broeils & Rhee 1997), 1.88 (Verheijen 2001), 1.86 (Swaters et al. 2002), 1.80 (Noordermeer et al. 2005), and 1.72 (Martinsson et al. 2016). These small differences likely arise from different sample sizes and ranges in galaxy properties: in this respect SPARC is the largest sample and spans the broadest dynamic range. The small tilt from the “expected” value of 2 implies that the H i   disks of low-mass galaxies are denser than those of high-mass ones (by a factor of ∼2; see also Noordermeer et al. 2005; Martinsson et al. 2016).

Figure 3 (top right) shows the relation between M_{H i} and L_[3.6]. A linear fit returns

with σ_int = 0.35 ± 0.02 dex. A slope of ∼0.6 implies that the ratio of gas mass (M_gas = 1.33 M_{H i} ) to stellar mass (M_⋆ = ϒ_⋆ L_[3.6]) is not constant in galaxies, but becomes systematically higher for low-luminosity galaxies. For comparison, the dotted line shows the equality M_gas = M_⋆ assuming ϒ_⋆ = 0.5 M_⊙/L_⊙. Interestingly, this line broadly demarcates the transition from spirals (T = 1–7) to irregulars (T = 8–11). Gas fractions are explored in detail in Section 5.2.

Finally, the bottom panels of Figure 3 show the relation between H i   radius and stellar radius. The latter is estimated as R_eff (left) and R_d (right). Assuming a 10% error on these radii, a linear fit returns

with σ_int = 0.22 ± 0.01 dex, and

with σ_int = 0.20 ± 0.01 dex. The slope and normalization of these relations imply that H i   disks are generally more extended than stellar disks, as we discussed in Section 2.1.

The major uncertainty in the mass modeling of galaxies is the value of ϒ_⋆. To address this issue, the DiskMass Survey (DMS; Bershady et al. 2010a) measured the vertical velocity dispersion of disk stars in 30 face-on spirals and employed the theoretical relation between vertical velocity dispersion, disk scale height, and dynamical surface density (valid for a self-graviting disk). Since z_d cannot be directly measured in face-on galaxies, the DMS statistically estimated the disk scale height using the empirical correlation between R_d and z_d, calibrated using NIR photometry of edge-on galaxies (Kregel et al. 2002).

The DMS concluded that stellar disks are strongly submaximal, contributing less than 30% of the dynamical mass within 2.2 R_d and having ϒ_⋆ = 0.31 M_⊙/L_⊙ in the K band (Martinsson et al. 2013). After correcting for the DM contribution, the DMS gives ϒ_⋆ = 0.24 M_⊙/L_⊙ in the K band (Swaters et al. 2014). Considering that L_K ≃ 1.3 L_[3.6] (Schombert & McGaugh 2014a), these values correspond to ϒ_⋆ ≃ 0.2 M_⊙/L_⊙ at [3.6]. Angus et al. (2016) reanalyzed the DMS data, pointing out that the DM contribution to the vertical force becomes significant for such submaximal disks: when DM halos are included in a self-consistent way, the mean value of ϒ_⋆ further decreases to ∼0.18 M_⊙/L_⊙ in the K band (∼0.14 M_⊙/L_⊙ at [3.6]). These results are in tension with SPS models (McGaugh & Schombert 2014; Meidt et al. 2014; Schombert & McGaugh 2014a), which find mean values between ∼0.6 and ∼0.8 M_⊙/L_⊙ in the K band (∼0.4 to ∼0.6 M_⊙/L_⊙ at [3.6]) depending on the model and IMF.

Interestingly, both the DMS and SPS models suggest that ϒ_⋆ is nearly constant in the NIR (within ∼0.1 dex) among galaxies of different masses and morphologies. The discrepancy is only in the overall normalization of ϒ_⋆. A nearly constant ϒ_⋆ at [3.6] is also suggested by the BTFR (McGaugh & Schombert 2015), which is remarkably tight for a fixed ϒ_⋆ (Lelli et al. 2016). Thus, we assume that ϒ_⋆ is constant among different galaxies and explore different normalizations. For 32 galaxies with significant bulges, however, we adopt ϒ_bul = 1.4 ϒ_disk as suggested by SPS models (Schombert & McGaugh 2014a). Hereafter, the different normalizations of ϒ_⋆ always refer to the stellar disk, i.e., ϒ_⋆ = ϒ_disk.

We define the gas fraction as

where L_disk and L_bul are estimated using the nonparametric decompositions in Section 3.3. Figure 4 plots f_gas versus L_[3.6] assuming ϒ_⋆ = 0.5 M_⊙/L_⊙ (left) and ϒ_⋆ = 0.2 M_⊙/L_⊙ (right). For any realistic value of ϒ_⋆, the gas fraction anticorrelates with L_[3.6], but the overall shapes of these relations significantly change with ϒ_⋆.

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A value of ϒ_⋆ ≃ 0.5 M_⊙/L_⊙ minimizes the scatter around the BTFR (Lelli et al. 2016) and is found using self-consistent SPS models with standard IMFs (McGaugh & Schombert 2014; Schombert & McGaugh 2014a). This normalization leads to the usual picture for the composition of galaxy disks:

1. 1.  
   The relation is roughly linear. Dwarf galaxies are generally gas dominated but show a broad range in f_gas from ∼0.5 to ∼0.9.
2. 2.  
   The transition between spirals (T = 1 to 7) and irregulars (T = 8 to 11) roughly occurs at f_gas ≃ 0.5. This is in line with density wave theory (Lin & Shu 1964), since well-developed spiral patterns cannot be supported in gas-dominated disks.

A value of ϒ_⋆ = 0.2 M_⊙/L_⊙ at [3.6] is comparable to the DMS normalization. This low value of ϒ_⋆ leads to some strange results:

1. 1.  
   The relation shows a flattening at L_[3.6] ≲ 3 × 10⁹ L_⊙ (vertical dashed line). Consequently, the vast majority of dwarf irregulars are heavily gas dominated with f_gas ≃ 0.8–1.0. Dwarf irregulars are generally thought to be gas dominated (e.g., McGaugh 2012), but this seems a very extreme situation.
2. 2.  
   Several bright spirals (10¹⁰ < L_[3.6]/L_⊙ < 10¹¹) have f_gas ≳ 0.5. This also includes four early-type spirals (Sa to Sb), corresponding to T = 1–3. It is unclear how these bright galaxies could sustain stellar bars and grand-design spiral patterns in such gas-dominated disks (see, e.g., Athanassoula 1984).

This suggests that ϒ_⋆ ≃ 0.5 M_⊙/L_⊙ gives a more realistic galaxy population than ϒ_⋆ ≃ 0.2 M_⊙/L_⊙.

Historically, the degree of disk maximality is estimated by measuring the ratio V_⋆/V_obs at 2.2R_d (Sackett 1997). This choice is inherited from early works on late-type spiral galaxies (e.g., van Albada et al. 1985) and is motivated by two facts: (i) in spiral galaxies V_gas is typically much smaller than V_⋆, and (ii) for a pure exponential disk V_⋆ peaks at 2.2R_d. When comparing galaxies of different masses and morphologies, however, this choice becomes inadequate because (i) in dwarf galaxies V_gas can be larger than V_⋆, and (ii) the vast majority of spiral galaxies show deviations from exponential profiles in the inner parts due to bars, lenses, and bulges. Hence, we consider the ratio V_bar/V_obs (instead of V_⋆/V_obs) and explore several characteristic radii.

5.3.1. The Choice of the Characteristic Radius

Figure 5 shows mass models for two galaxies with diametrically opposed properties: a bulge-dominated spiral galaxy (NGC 7814) and a gas-dominated dwarf galaxy (UGCA 442). Top panels show [3.6] surface brightness profiles and exponential fits to the outer disk; bottom panels show rotation curve decompositions adopting ϒ_disk = 0.5 M_⊙/L_⊙ and ϒ_bul = 0.7 M_⊙/L_⊙. The arrows illustrate three different characteristic radii: (i) the commonly used radius at 2.2R_d (R_2.2; gray), (ii) the effective radius encompassing half of the total [3.6] luminosity (R_eff; red), and (iii) the radius where V_bar is maximum (R_bar; blue). The last radius was denoted as R_p by McGaugh (2005a).

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NGC 7814 (left) is an early-type spiral (Sab). The surface brightness profile shows a central light enhancement due to the bulge. The rotation curve displays a very steep rise in the inner parts, a decline of ∼50 km s⁻¹, and a flat outer portion. These are typical features for early-type spirals (Noordermeer et al. 2007). For ϒ_⋆ = 0.5 M_⊙/L_⊙ baryons explain the inner parts of the rotation curve, while DM is needed at large radii. Note that R_eff occurs along the declining part of the rotation curve, while R_2.2 occurs along the flat part: R_bar is the only radius that captures the concept of baryonic maximality.

UGCA 442 (right) is a late-type dwarf (Sm). The surface brightness profile is well described by an exponential law, but there is a central flattening due to a cored light distribution. The rotation curve shows a slow rise in the central regions and a flat part beyond the stellar component. For ϒ_⋆ = 0.5 M_⊙/L_⊙ baryons cannot fully explain the inner rise of the rotation curve and DM is needed down to small radii. This is a typical behavior for low-mass and LSB galaxies: maximum-disk decompositions require high values of ϒ_⋆ that are ruled out by standard SPS models (de Blok & McGaugh 1997; Swaters et al. 2011). Note that the shape of V_bar is mostly driven by the gas contribution and R_bar occurs well beyond R_eff or R_2.2 due to the gas dominance.

These two examples illustrate that both R_2.2 and R_eff can be misleading in quantifying the baryonic maximality because they can be far from the radius where V_bar peaks. In the following, we consider R_bar as our fiducial characteristic radius, but we also show V_bar/V_obs at R_2.2 (hereafter V_2.2/V_obs) to compare with previous works.

In the previous examples, the value of R_bar does not depend on the assumed ϒ_⋆ because the shape of V_bar is dominated by either V_⋆ or V_gas. In galaxies where V_⋆ ≃ V_gas, however, the value of R_bar may depend on the assumed ϒ_⋆ because the overall shape of V_bar is affected by the relative contributions of V_⋆ and V_gas: these cases represent a minority among SPARC galaxies.

Finally, we note that V_bar depends on galaxy distance as other than on the assumed ϒ_⋆. Uncertainties in galaxy distances, therefore, may affect the value of V_bar/V_obs. Moreover, the normalization of V_obs depends on the assumed inclination as , but this is a minor source of uncertainty for galaxies with i ≳ 30°.

5.3.2. The Effect of the Stellar Mass-to-light Ratio

Figures 6 and 7 show V_bar/V_obs measured at R_2.2 and R_bar, respectively. We illustrate the covariance with L_[3.6] (top) and Σ_eff (bottom). We consider two different normalizations of ϒ_⋆ as described in Section 5.1. First, we note that the degree of maximality of dwarf galaxies does not strongly depend on ϒ_⋆ because these objects are gas dominated: the values of V_bar/V_obs stay in the range of 0.3–0.6 (apart from a few outliers) using both R_2.2 and R_bar. Conversely, the degree of maximality of spirals depends strongly on ϒ_⋆.

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For ϒ_⋆ = 0.5 M_⊙/L_⊙, high-luminosity HSB galaxies are nearly maximal, whereas low-luminosity LSB galaxies are submaximal. By definition, the value of V_bar/V_obs is higher at R_bar than R_2.2, especially in galaxies with bulges. Some objects have V_bar/V_obs ≳ 1, but this is not surprising. We expect some intrinsic scatter around the mean value of ϒ_⋆ (of ∼0.1 dex; see Meidt et al. 2014; Lelli et al. 2016); hence, a few objects may have V_bar/V_obs > 1 for a fixed ϒ_⋆. We also note that several galaxies with V_bar/V_obs > 1 have uncertain distances and are consistent with V_bar/V_obs ≲ 1 within the errors. The degree of baryonic maximality correlates better with surface brightness than luminosity, in line with previous studies (e.g., McGaugh 2005, 2016).

For ϒ_⋆ = 0.2 M_⊙/L_⊙, spiral galaxies are submaximal as found by the DMS (Martinsson et al. 2013). For this low value of ϒ_⋆, most spirals are nearly as submaximal as dwarf galaxies (which are not included in the DMS). This would imply that there is no strong dynamical difference among objects spanning ∼5 dex in luminosity and ∼4 dex in surface brightness. One does expect a difference over this range because HSB and LSB galaxies lie on the same BTFR. At a given baryonic mass, V_obs is nearly the same but V_bar goes as ; hence, it is higher in HSB galaxies (with small R_bar) than in LSB galaxies (with large R_bar). Therefore, V_bar/V_obs should correlate with surface brightness, but this trend is barely perceptible for ϒ_⋆ = 0.2 M_⊙/L_⊙. If R_bar is adopted, V_bar/V_obs shows a parabolic trend with Σ_eff (Figure 7, bottom right). The LSB end is driven by gas-dominated dwarf galaxies, while the HSB end is driven by bulge-dominated spiral galaxies: galaxies with intermediate surface brightnesses are generally disk dominated and would be less maximal than gas-dominated LSB galaxies for ϒ_⋆ = 0.2 M_⊙/L_⊙. This seems strange.

It is important to derive an upper limit for the mean value of ϒ_⋆ since this can be used to constrain different IMFs and SPS models. In Figure 8 we show V_bar/V_obs at R_bar for ϒ_⋆ = 0.8 M_⊙/L_⊙. For this high normalization, a large fraction of high-mass HSB galaxies have V_bar/V_obs > 1. It is conceivable that ϒ_⋆ may scatter up to 0.8 M_⊙/L_⊙ for some individual galaxies, but this cannot be the correct mean ϒ_⋆. In the extreme scenario where the “true” mean ϒ_⋆ gives V_bar/V_obs = 1 in bright galaxies, the scatter in ϒ_⋆ should drive ∼50% of galaxies to values of V_bar/V_obs > 1. If we consider galaxies with L_[3.6] > 2 × 10¹⁰ L_⊙, a value of ϒ_⋆ = 0.8 M_⊙/L_⊙ implies over-maximal disks in ∼67% of the cases, which is statistically unacceptable. For truly maximal disks with V_bar/V_obs = 1, we find that the upper limit on the mean value of ϒ_⋆ is ∼0.7 M_⊙/L_⊙. If we adopt the customary definition of maximum disk as V_bar/V_obs = 0.85 (Sackett 1997), then the mean maximum-disk value of ϒ_⋆ is ∼0.5 M_⊙/L_⊙, which is similar to the predictions of SPS models with standard IMFs (e.g., McGaugh & Schombert 2014; Meidt et al. 2014; Schombert & McGaugh 2014a).

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