A CATALOG OF BULGE, DISK, AND TOTAL STELLAR MASS ESTIMATES FOR THE SLOAN DIGITAL SKY SURVEY

The basis for our SPS model grid is the flexible stellar population synthesis (FSPS) code of Conroy et al. (2009),⁶ which allows us to generate synthetic SEDs given a range of galaxy properties. All models are constructed using Padova isochrones (Girardi et al. 2000; Marigo & Girardi 2007; Marigo et al. 2008) and the BaSeL3.1 theoretical stellar library (Lejeune et al. 1997, 1998; Westera et al. 2002). We adopt a form for the stellar IMF from Chabrier (2003) with lower and upper integration limits of 0.08 and 120 M_☉. We refer the reader to Conroy et al. (2009) for details of the FSPS model construction, as well as subsequent papers (Conroy et al. 2010; Conroy & Gunn 2010) that describe the calibration of FSPS models with observational data.

We generate a suite of models with smoothly declining star-formation histories characterized by an e-folding timescale, τ, such that the time-dependent star-formation rate ψ_⋆(t) is given by ψ_⋆(t)∝τ⁻¹exp (− t/τ). We follow Taylor et al. (2011) in sampling τ on a semi-uniform grid where log (τ/yr) spans the range 8–9 in steps of 0.2 dex, and from 9.1–10 in steps of 0.1 dex. Stellar metallicities cover the range −1.8 ⩽ log (Z/Z_☉) ⩽ 0.2 in steps of 0.2 dex. We include internal attenuation by dust using the extinction law of Calzetti et al. (2000) with E(B − V) sampled uniformly over the range 0 ⩽ E(B − V) ⩽ 1 (i.e., a range of V-band optical depth of 0 ⩽ τ_V ⩽ 2.855) in steps of 0.05 mag. Finally, population ages are sampled uniformly with 8 ⩽ log (t/yr) ⩽ 10.1 in steps of 0.05 dex. The full SPS grid therefore contains 158,928 individual sets of synthetic ugriz photometry. These properties are summarized in Table 2. We also perform a set of fits where we remove dust as a free parameter and instead fix E(B − V) = 0. We discuss a comparison of dusty and dust-free models in Section 3.5.

There is a well-documented discrepancy between the observed colors of red galaxies and the predicted colors of old, metal-rich populations in SPS models such that synthetic g−r photometry is ∼0.1 mag redder than is observed (e.g., Eisenstein et al. 2001; Maraston et al. 2009; Conroy et al. 2010). This is illustrated in Figure 1, where we plot apparent g−r color as a function of redshift for red galaxies in the SDSS. When compared to the predicted color evolution of a dust-free, metal-rich single stellar population (dashed line in Figure 1), there is a clear tendency for the models to be redder than observations by 0.05 to 0.1 mag.

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Barring changes in our general picture of stellar evolution, rectifying the predicted colors from SPS models with observational data requires including a minority population of hot, blue stars. These can come in a variety of forms, either as a distinct evolutionary phase (e.g., BS and/or BHB stars) or as a more complex stellar population (e.g., an α-rich or metal-poor sub-component). As an example, the solid line in Figure 1 shows the predicted color evolution of a stellar population that includes 5% by mass in metal-poor stars with log (Z/Z_☉) = −1.98. The net effect of including this metal-poor sub-component is relatively minor in terms of the derived mass-to-light ratio (M/L)—masses are on average 0.02 dex lighter relative to a simple single-metallicity population; however, it improves the overall agreement between observed and synthetic ugriz SEDs (see also the discussion by Conroy & Gunn 2010). While arguments following chemical evolution support the incorporation of metal-poor stars into SPS models (e.g., Pagel & Patchett 1975) they are not the only solution to the problem highlighted by Figure 1. Instead it is likely that a combination of variables contribute to the observed offset between SPS models and data, and it is beyond the scope of the present work to investigate this issue in more detail. Nevertheless, given the improved agreement between the observed and synthetic SEDs when incorporating even a small component metal-poor stars we choose to include such a subpopulation in our fiducial SPS model. We stress that the exact properties of the included metal-poor sub-component (in terms of mass fraction and metallicity) are not unique.

Finally, in contrast to the widely used stellar mass catalogs produced by the MPA-JHU group (Kauffmann et al. 2003b; Salim et al. 2007), our SPS grid does not include bursty star-formation histories. With the exception of recent starbursts, the majority of local galaxies' SEDs are well fit by single-component τ models due to the smooth evolution of SED properties with population age. More importantly, properly accounting for bursty star-formation histories is non-trivial. Such galaxies are not easily identified using photometry alone, and the ability to recover reliable M/L estimates depends on the relative frequency with which bursts are incorporated into the underlying SPS library. Gallazzi & Bell (2009) show that including too large a fraction of bursty SPS models can lead to a systematic underestimate of M/L by up to 0.1 dex for galaxies whose star-formation histories are generally smooth, while fitting bursty galaxies with smooth models may overestimate their M/L by up to 0.2 dex. Our decision to exclude bursts is therefore predicated on obtaining robust mass estimates for the majority of (smoothly evolving) galaxies. Conversely, for the minority population of bursty galaxies (<10%; Kauffmann et al. 2003b), we may overestimate stellar masses by a factor of two or more, and systematically underestimate their uncertainties (see also Walcher et al. 2011).

For each photometric object—galaxy, disk, or bulge—our goal is to determine the set of fundamental parameters that most accurately describe its underlying stellar population. It is through this determination that we ultimately derive estimates of stellar mass. We have chosen to approach this task via Bayes' theorem, which allows us to compute the conditional probability for a set of model parameters, θ, given the observational data as

$$\begin{equation} p(\theta |F_\mathrm{obs}) \propto p(F_\mathrm{obs}|\theta) p(\theta). \end{equation} \tag{ 1 }$$

In the above, p(F_obs|θ) is the likelihood of observing a set of fluxes, F_obs, given θ, and p(θ) is the prior probability of observing θ before considering the observations. In general θ specifies a vector, in this case the defining parameters of our SPS grid—i.e., τ, Z, E(B − V), and t. One benefit of adopting a Bayesian approach to SED fitting is that our output, p(θ|F_obs), reflects the full posterior probability density function (PDF), incorporating uncertainties on individual parameters of the SPS grid in our final estimates of stellar mass.

In the case where an object is detected in all bands, the likelihood function for model i is defined as $p(F_\mathrm{obs}|\theta _i) \propto \exp (-\chi _i^2/2)$, where $\chi _i^2$ is given by

$$\begin{equation} \chi _i^2 = \sum _X \left(\frac{F_{\mathrm{obs},X} - A_i F_{\mathrm{mod}_i,X}}{\sigma \left(F_{\mathrm{obs},X}\right)}\right)^2. \end{equation} \tag{ 2 }$$

Here the summation is over the photometric passbands, i.e., X = (u, g, r, i, z). F_{obs, X} and $F_{\mathrm{mod}_i,X}$ are the observed and model fluxes through passband X respectively, and σ(F_{obs, X}) is the uncertainty on F_{obs, X}. The normalization factor A_i accounts for the scaling between the model SED (which is normalized to a time-integrated mass of 1 M_☉) and the observational data, and can be computed analytically by taking the derivative of Equation (2) with respect to A_i.

While galaxies are always detected in all five bands, in some cases bulges and disks may be undetected in one or more photometric passbands, for example when B/T = 0 or 1. In order to incorporate information from these non-detections in our SED fits we modify Equation (2) following Sawicki (2012) such that

$$\begin{eqnarray} \chi _i^2 &= \sum _X \left(\frac{F_{\mathrm{obs},X} - A_i F_{\mathrm{mod}_i,X}}{\sigma \left(F_{\mathrm{obs},X}\right)}\right)^2 \nonumber \\ &\quad {} -2\sum _Y \ln \int _{-\infty }^{F\mathrm{lim,Y}} \exp \left[-\frac{1}{2}\left(\frac{F - A_i F_{\mathrm{mod}_i,Y}}{\sigma \left(F_{\mathrm{obs},Y}\right)}\right)^2\right] {dF}, \nonumber\\ \end{eqnarray} \tag{ 3 }$$

where the first term is now the sum over detected passbands and the second term accounts for the probability of a non-detection given the band-specific flux limit, F_{lim, Y}, and model flux. Note that in the case where an object is detected in all passbands Equation (3) reduces to the standard χ² form of Equation (2). There is no simple analytic solution for A_i when incorporating non-detections into the likelihood function, so we instead solve for the normalization numerically.

3.2.1. Priors

For each set of F_obs and θ_i the most likely stellar mass is given by $A_i M_{\mathrm{mod}_i, \star }$ with probability p(θ_i|F_obs),⁷ where $M_{\mathrm{mod}_i, \star }$ is the normalized stellar mass for model i. For each galaxy or structural component, the stellar mass posterior PDF is obtained through the application of Equation (2) or (3) at each point in our synthetic photometric grid, and is completely specified for a given set of ugriz fluxes and their associated uncertainties.

Uncertainties on the observed fluxes reflect a combination of the statistical uncertainties on the model parameters, output by gim2d, and the rms noise in the sky background, which we estimate using the statistics in each SDSS imaging field. We include an additional uncertainty of 3% in griz and 5% in u to account for both the relative and absolute SDSS photometric calibration. In most cases the photometric calibration and sky background both contribute significantly to the overall uncertainty, with the exception of the u and z bands where the background dominates. In Appendix B we present an analysis of the uncertainties on the S11 photometry based on Monte Carlo simulations. We have verified that uncertainties estimated based on the gim2d and image field statistics (described above) are consistent with those derived from the model galaxies presented in Appendix B; we preferentially use the former as they are independent of the specific form of the galaxy surface-brightness profile (i.e., Sérsic or bulge+disk) used to generate the synthetic galaxies described in Appendix B.

Final estimates of stellar mass are taken as the median of the AM_{mod, ⋆} distribution, weighted by p(θ|F_obs). We account for foreground (Galactic) extinction using the Schlegel et al. (1998) dust maps, and apply this correction directly to the FSPS photometry on a galaxy-by-galaxy basis. We have not corrected for the contribution of emission lines to the observed broadband flux measurements, however, for the majority of galaxies their contribution is negligible. We quote uncertainties as the ±1σ percentiles of the marginalized posterior PDF. Note that we fit the total, bulge, and disk photometry independently, and therefore it is not necessarily true that the total, bulge, and disk masses—M_{B + D}, M_B, and M_D, respectively—are consistent (i.e., we do not explicitly impose that M_B + M_D = M_{B + D}). Nevertheless, in 91% of galaxies M_{B + D} and M_B + M_D differ by less than their 1σ uncertainties. This comparison will be discussed further in Section 3.4.

The median (statistical) uncertainties of the total, bulge and disk stellar masses are ∼0.13, 0.14 and 0.15 dex, respectively. Uncertainties on the total mass derived from bulge+disk or Sérsic photometric models are similar. In Figure 2 we show the distribution for these three components as a function of derived stellar mass to give a sense of their behavior.

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In Figure 3 we show a comparison of masses derived from the bulge+disk and Sérsic photometry. In the left hand panel, we plot the difference in total mass from the bulge+disk and Sérsic fits as a function of stellar mass derived from the Sérsic photometry. At masses below log (M/M_☉) = 10 the two estimates are remarkably similar, with a median offset of 0.02 dex. On the other hand, at log (M/M_☉) > 10 the masses derived from Sérsic photometry are generally larger than those from the bulge+disk fits by up to 0.2 dex. This is consistent with the generally higher luminosities measured using Sérsic profiles relative to other parametric or curve-of-growth approaches (see, e.g., Bernardi et al. 2012, 2013). The simulations discussed in Appendix B suggest that the tendency for Sérsic profiles to recover higher luminosities is actually a combination of two different effects: the known correlation between Sérsic index and stellar mass (e.g., Graham & Guzmán 2003), and the tendency for our bulge+disk models to systematically underestimate the luminosity of galaxies with n > 5 (Appendix B).

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In the right hand panel of Figure 3 we plot the difference between the mass derived from the total bulge+disk photometry, M_{B + D}, and the sum of the independently derived bulge and disk masses, M_B + M_D, as a function of the r-band bulge fraction, (B/T)_r. This provides an indicator of the internal consistency of the bulge+disk stellar masses. At (B/T)_r ≳ 0.4 the sum of the bulge and disk masses typically agree to within 10% of the mass derived from the total galaxy photometry, where M_B + M_D is generally heavier. This systematic offset can be traced back to the larger photometric uncertainties for the individual structural components.

The comparison at low (B/T)_r is somewhat more unsettling: while the median offset stays relatively unchanged, in some cases M_B + M_D exceeds M_{B + D} by more than a factor of 2. The behavior of M_B + M_D at low (B/T)_r is driven by ∼10% of the total sample where the bulge masses are significantly overestimated. Many of these "problem" bulges have extremely red colors (their median g−r color is ∼1.4) and, in some cases, relatively large photometric uncertainties. As a result, the posterior PDFs for these galaxies are dominated by families of heavily extincted models despite our use of a prior on E(B − V) for the bulge and disk fits. The resulting bulge mass fractions for these systems are physically unreasonable given their typical photometric B/T in the i and z bands of ∼25%, and they should be approached with extreme caution. In general, the comparison of the M_{B + D} and M_B + M_D can be used as an indicator of the internal consistency of the bulge and disk photometry and mass estimates, and provides a useful diagnostic for identifying robust subsamples of the data. We will discuss this point further in Section 5.

Noisy or spurious photometric data can lead to unpredictable results when fitting broadband SEDs, and the combination of significant photometric uncertainties and degeneracy between dust extinction and stellar age can result in anomalously high stellar mass estimates (e.g., Pforr et al. 2012). In the particular case of extremely red objects, such as those discussed in the previous section, masses can be significantly overestimated when using the full SPS grid due to the abundance of extincted models. Therefore, we have produced an additional set of bulge+disk and Sérsic stellar mass estimates in which E(B − V) has been set to zero. These "dust-free" models are useful in instances where very uncertain and/or bad photometry result in spurious estimates of the stellar mass, particularly for bulges and disks.

In Figure 4 we show a comparison of the dusty and dust-free models for each of the photometric components. In each case we plot the mass derived from the dust-free model minus the mass derived based on the full SPS grid. In terms of the total photometry, fixing E(B − V) = 0 results in a relatively minor systematic shift in the stellar masses of ∼0.04 dex when using either the bulge+disk or Sérsic photometry. In both instances there is a tail of galaxies whose masses are up to 0.2 dex higher in the dust-free fits. This tail is populated primarily by relatively red star-forming galaxies—that is, they populate the red side of the blue cloud in the color versus stellar mass plane. In general terms, removing dust as a free parameter makes blue galaxies heavier by ∼0.1 dex, while red galaxies get lighter by ∼0.03 dex.

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Turning to the individual sub-components, we find no evidence for a systematic offset in bulge stellar mass when dust is excluded for most galaxies; this confirms our expectation that spheroids contain relatively little dust. However, there is a tail of the bulge population whose masses are significantly higher when dust is included as a free parameter. These are the particularly red, low B/T systems highlighted in the comparison of M_{B + D} and M_B + M_D discussed in Section 3.4. In Figure 5 we show the comparison between M_{B + D} and M_B + M_D as a function of (B/T)_r, analogous to the right-hand panel of Figure 3, for the dust-free masses. Contrary to the comparison in Figure 3, the dust-free fits remain relatively well behaved even at low B/T, and likely provide a more robust estimate of the bulge mass in these systems. The behavior of disks in the dusty versus dust-free fits is similar to that of the total photometry: the reddest disks get slightly lighter in the dust-free fits, while star-forming disks get slightly heavier.

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The MPA-JHU group provide a catalog of stellar masses which have been widely used in analyses of the SDSS⁸ (Kauffmann et al. 2003b; Brinchmann et al. 2004; Salim et al. 2007). These data therefore provide a well-tested reference point for the mass estimates presented here. It is important to note that there are two effects at play in these comparisons: the first reflects differences between the SDSS- and gim2d-derived photometry, which are discussed extensively in S11, while the second relates to differences expected as a result of variations in the underlying SPS models. Masses derived in the MPA-JHU catalogs for DR7 are based on SED fits to the SDSS model magnitudes, which are estimated from parametric fits to the observed surface-brightness profile. In each case, galaxies are fitted separately with exponential and de Vaucouleurs profiles, and the relative likelihood of these fits is used to determine which profile provides the most appropriate description of the data. The final model magnitudes are derived by integrating the best-fit (de Vaucouleurs or exponential) profile out to large radii. The underlying SPS models are computed using the galaxev code (Bruzual & Charlot 2003) and a Monte Carlo approach to construct a large suite of templates with varying star-formation histories, formation times, and metallicities. As discussed in Section 3.1, one of the most significant structural differences between the MPA-JHU SPS library and our own is their inclusion of bursts at late times, which results in relatively small differences in the derived stellar mass for most galaxies. It is beyond the scope of the present work to address the effects of this difference in detail, and we refer the reader to Gallazzi & Bell (2009) and Walcher et al. (2011) for a more extensive discussion of this issue.

In the left panels of Figure 6 we plot the median difference in total stellar mass (top panel) and M/L (bottom panel) between the MPA-JHU mass estimates and our own as a function of stellar mass from the bulge+disk profile fits. We find that masses computed using gim2d photometry and FSPS are ∼0.02 dex heavier on average than the MPA-JHU mass estimates, with slightly larger deviations for low-mass galaxies. Comparing the top and bottom panels, we see that the offset in total mass generally exceeds that of M/L, which is a consequence of the higher average flux derived by S11 relative to the SDSS model magnitudes. At low masses, offsets in total mass are almost entirely accounted for by variation in M/L, suggesting that the underlying models may play a more important role in star-forming galaxies. As discussed in Section 3.1, application of smoothly evolving models—such as those used here—to bursty systems can lead to a systematic overestimate of their M/L. The increased M/L relative to the MPA-JHU catalog may therefore reflect a combination of differences in the two models grids and the increasingly stochastic nature of star formation in low-mass galaxies; however, we see no evidence that this results in a significant offset in the derived M/L on average. The comparison discussed above is qualitatively similar if we use masses determined from the Sérsic photometry, which are on average ∼0.08 dex heavier than the MPA-JHU masses. At high mass the Sérsic-profile based masses deviate more significantly from the MPA-JHU masses, and are up to 0.2 dex heavier at log (M/M_☉) = 12. This same behavior is shown in Figure 3 in comparison to the bulge+disk masses.

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The discussion above is focused primarily on galaxy properties (mass or M/L) as a function of their stellar mass; however, we have largely neglected details of the photometric measurements that can significantly affect the recovery of galaxies' stellar mass. For example, both S11 and Taylor et al. (2011) have shown that the SDSS model magnitudes have a tendency to bifurcate for galaxies with intermediate structural properties (i.e., 0.25 < B/T < 0.6 or 1.5 < n < 3), resulting in a spurious bimodality in the derived photometry for such systems. This structural dependence can manifest itself as a systematic shift in the derived stellar masses of objects that host both bulges and disks (e.g., Taylor et al. 2011). In the right panels of Figure 6 we investigate this structural dependence explicitly in our data by comparing stellar mass and M/L as a function of fracDeV, the fraction of light contained within the de Vaucouleurs component fit by the SDSS photo pipeline. In both the top and bottom panels, the median offsets for galaxies with model magnitudes derived from the de Vaucouleurs (exponential) fits are shown by the solid (dot–dashed) lines. Here we can see that galaxies with intermediate fracDeV can have their masses over- or underestimated by ∼0.2 dex, but that there is good agreement between M/L estimates for any given fracDeV. The disagreement between the masses presented here and those in the MPA-JHU catalog at intermediate fracDeV is primarily due to fitting a single, fixed profile (de Vaucouleurs or exponential) as opposed to something that can better accommodate complex structure; any differences between the SED fitting procedures appear to be small once an appropriate model is fit to the galaxy light profile. Looking at it another way, there is relatively good agreement between the model and gim2d colors, but the total fluxes can vary significantly depending on galaxies' underlying structure.

There are usually multiple ways to approach any problem, and stellar population synthesis is no exception. While we have chosen to use FSPS as the fiducial model in our mass estimates, there are many other SPS models based on a variety of stellar evolutionary tracks and spectral libraries that could have been used to the same end (e.g., Fioc & Rocca-Volmerange 1997; Leitherer et al. 1999; Bruzual & Charlot 2003; Maraston 2005; Vazdekis et al. 2012). It is therefore crucial to understand the level at which these different models produce consistent predictions, and the extent to which (and where) they disagree.

In Figure 7 we show the variation in ugriz M/Ls relative to FSPS as a function of population age for two different SPS models: Bruzual & Charlot (2003, top panel, hereafter BC03) and Maraston (2005, bottom panel, hereafter M05). In all cases we show the results for a dust-free single stellar population with solar metallicity. M/L values from the BC03 models are computed using a Chabrier (2003) IMF, and so are directly comparable to the FSPS models. M05 models are based on a Kroupa (2001) IMF, which we convert to Chabrier (2003) using an offset of 0.05 dex (Section 4.2; Bernardi et al. 2010). The BC03 models are based on an earlier version of the Padova isochrones than FSPS (Girardi et al. 1996), and the luminosity at all phases of stellar evolution is computed via an isochrone synthesis technique. M05 models are based on isochrones and evolutionary tracks from Cassisi et al. (1997), and use a similar approach to BC03 and FSPS to compute the luminosity of stars up to the main sequence turnoff. For evolved stars (post-main sequence), M05 compute luminosities via fuel consumption theory rather than isochrone synthesis. The most significant difference between the M05 and BC03 or FSPS models is that the luminosity of thermal-pulsing AGB (tp-AGB) stars derived from fuel consumption theory is higher relative to other calculations.

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On average the M/Ls predicted by FSPS are higher by 0.04–0.1 dex (10%–25%)—that is, the models have a higher stellar mass per unit luminosity—relative to either the BC03 or M05 models. This systematic offset notwithstanding, the FSPS and BC03 models agree reasonably well at all wavelengths with a typical spread of 0.04–0.06 dex that increases slightly toward old ages.

The comparison with M05 models is less favorable; however, if we restrict the comparison to relatively old ages (≳ 2 Gyr) then the scatter is generally less than 0.1 dex. At ages less than 1 Gyr there is significant spread in the relative M/L depending on wavelength: there is relatively good agreement between the u- and g-band M/Ls, while the i- and z-band M/Ls are up to 0.2 dex lower in M05 relative to FSPS. This decrease in M/L can be traced to the increased luminosity of tp-AGB stars in the M05 models relative either FSPS or BC03, which contribute significantly in the near-IR. We will discuss the influence of tp-AGB stars more thoroughly in Section 4.3, and here simply note that they represent a significant uncertainty in the determination of stellar masses. In both comparisons, the feature at ∼1 Gyr is due to differences in the onset of core helium burning in the stellar evolutionary tracks used by the different models.

The above comparison points to an inherent (and unavoidable!) uncertainty in any stellar mass estimates based on broadband photometry of 0.1–0.2 dex, depending on the properties of the underlying stellar population. This is in relatively good agreement with the results of previous studies which have addressed the differences between various SPS models (e.g., Bell & de Jong 2001; Maraston 2005; Marchesini et al. 2009; Ilbert et al. 2010; Swindle et al. 2011).

Among the assumptions that go into the construction of our SPS grid, changes in the stellar IMF and extinction law can lead to significant systematic shifts in galaxies' derived stellar mass, and are therefore worth a moment of discussion.

In addition to the relatively large uncertainty on the slope of the IMF locally (Kroupa 2001), over the past several years there has been mounting evidence for systematic variation of the stellar IMF as a function of galaxy properties (e.g., Cenarro et al. 2003; Hoversten & Glazebrook 2008; van Dokkum & Conroy 2010; Gunawardhana et al. 2011; Dutton et al. 2012). While the exact form of this variation is still poorly understood, it seems that some variability in either or both the high- and low-mass IMF slope should be considered when discussing the overall influence of IMF uncertainties on our final mass catalog.

Changes in the IMF slope at M_☉ ≲ 1 can be approximated by a simple offset in M/L (or stellar mass) for all galaxies, as low-mass stars contribute relatively little to galaxies' overall luminosity. Of the IMFs most frequently adopted in the literature, those of Salpeter (1955), Kroupa (2001), and Chabrier (2003) have similar logarithmic slopes at high mass, and can therefore be transformed in this fashion. Roughly speaking, masses derived using a Salpeter (1955) or Kroupa (2001) IMF are 0.25 and 0.05 dex heavier than those estimated assuming a Chabrier (2003) IMF. On the other hand, changing the slope of the IMF at M_☉ ≳ 1 has a significant impact on the rate at which the luminosity of a stellar population evolves (Tinsley 1980). Therefore, unlike changing the IMF for low-mass stars, changing the slope of the IMF at high-mass results in changes in M/L which depend on the (luminosity-weighted) age of the stellar population. To place this in the context of our mass estimates, adopting a power-law IMF slope of x = 1.0 for M_☉ > 1 (in units where a Chabrier 2003 IMF has x = 1.3) leads to an increase in the M/L of red galaxies of ∼0.1 dex, but relatively little change in the M/L of blue galaxies. Similarly, adopting x = 1.6 leads to no change in M/L for red galaxies, but a 0.05 dex increase for blue galaxies.

We showed in Section 3.5 that removing E(B − V) as a free parameter can lead to significant changes in the derived mass, particularly when models are poorly constrained by the photometric data. Even when considering dust properties as a variable, the particular choice of extinction law can have a significant effect on the derived masses. In constructing our fiducial grid we have used the Calzetti et al. (2000) extinction curve, however empirical attenuation laws based on the Milky Way (MW; e.g., Cardelli et al. 1989) and Small Magellanic Cloud (SMC; e.g., Pei 1992), as well as single- or multi-component power-law curves (e.g., Charlot & Fall 2000) are also common in the literature. Changing among the Calzetti et al. (2000), MW, and SMC extinction curves leads to variations in total mass of 0.1–0.2 dex depending on the wavelengths used in the SED fitting procedure (e.g., Marchesini et al. 2009; Ilbert et al. 2010; Swindle et al. 2011; Pforr et al. 2012); differences are typically largest when incorporating data which extend into the far-UV. The dust model of Charlot & Fall (2000) includes additional attenuation for young objects to allow for the effects of stars embedded in H ii regions, and can lead to even more significant variation for particular subsamples of (primarily young) galaxies (e.g., Ilbert et al. 2010; Swindle et al. 2011). At optical wavelengths, however, variations in the extinction law lead to uncertainties of ∼0.15 dex.

We now turn to a consideration of systematic effects as a result of uncertain phases of stellar evolution, focusing primarily on the contributions of blue horizontal branch (BHB), blue straggler (BS), and tp-AGB stars.

In the context of FSPS, variations in horizontal branch (HB) morphology are parameterized by the fraction of HB stars bluer than the red clump. These stars are assumed to uniformly populate the HB between the temperature of the red clump and 10,000 K. We adopt two values for the fraction of blue HB stars, f_BHB, of 0% and 20%, which encompass the abundance of very blue stars necessary to explain the UV upturn in elliptical galaxies (Dorman et al. 1995). We note that our fiducial model includes a 5% contribution by mass in metal-poor stars ([Z/H] ≈ −2), and therefore a minimum 5% contribution of BHB stars might be more appropriate; however, in order to facilitate comparison with previous SPS models which do not consider contributions from BHB stars (e.g., Bruzual & Charlot 2003) we have elected to adopt a minimum model with f_BHB = 0. BS stars are defined in terms of the specific frequency relative to HB stars, S_BS, and populate a parameter space between 0.5 and 2.5 mag brighter than the zero-age main sequence. We consider models in which S_BS is either 0 or 2. Both BHB and BS stars are assumed to contribute only in populations older than 5 Gyr.

Luminous tp-AGB stars contribute significantly to flux in the near-infrared, particularly in populations with ages of ∼1–2 Gyr, and are therefore an important consideration for the estimation of stellar mass in such galaxies. Currently, the overall contribution of tp-AGB stars is extremely uncertain. On the one hand, the stellar populations of intermediate-age star clusters seem to require a significant contribution from tp-AGB stars to describe their observed photometric and spectral properties (e.g., Maraston 2005; Lyubenova et al. 2010, 2012). However, the extent to which such stars contribute to the SEDs of intermediate-age galaxies is less clear (e.g., Kriek et al. 2010; Conroy & Gunn 2010; Zibetti et al. 2013). FSPS allows for shifts in the characteristic temperature T and luminosity L of tp-AGB stars, and we adopt a relatively broad range of Δlog L = ±0.4 dex and Δlog T = ±0.2 dex, reflecting the significant uncertainty of this evolutionary phase. The fiducial FSPS models already incorporate an empirical adjustment to the tp-AGB temperature and luminosity, described by Conroy & Gunn (2010). The values of Δlog L and Δlog T used here characterize shifts in tp-AGB luminosity and temperature relative to the already-calibrated FSPS models.

In Figure 8 we plot the offset in r-band M/L relative to our fiducial model for various contributions of BHB, BS and tp-AGB stars as indicated in each panel. Dotted line represent no shift in M/L, and are shown for reference only. Figure 8 suggests that varying stellar evolutionary phases result in relatively small systematic deviations in the derived M/L, typically less than ∼0.05 dex, in good agreement with the findings of Conroy et al. (2009). Both BHB and BS stars appear to play only a minor part in the overall systematic uncertainty on the stellar masses, though the inclusion of 20% BHB stars can result in a shift of the derived M/L of 0.02–0.03 dex. Uncertainties in the tp-AGB phase introduce a much larger spread in the mass estimates, particularly given increases in the tp-AGB luminosity (bottom right panel of Figure 8). This is consistent with the SPS model comparison discussed in Section 4.1. It is important to note, however, that the relative improvement in the quality of the model fits (as judged by their χ²) is minimal for the adopted variations in both Δlog L and Δlog T. As discussed by Conroy et al. (2009), the primary effect of treating Δlog L and Δlog T as free parameters is to increase the uncertainty in M/L, but not necessarily change its value systematically. In this sense, the offsets shown in Figure 8 represent upper limits on the expected variation of mass given the adopted parameter range.

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4.3.1. Color Dependence

In addition to the globally averaged uncertainties in M/L discussed above, it is equally important to understand how these uncertainties vary as a function of galaxy properties.

In Figure 9 we show the change in stellar M/L as a function of galaxies' g−r color for the same models considered in Figure 8. In each panel, solid lines show the median offset in M/L, while dashed lines show the ±1σ percentile confidence intervals. Shading shows the underlying galaxy distribution.

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The systematic offsets shown in Figure 8 are echoed in Figure 9, but it highlights the dependence of these systematic effects on the underlying stellar population. The effects of both the BHB and BS contributions are relatively minor, at most ∼0.05 dex. While there is no obvious systematic trend for BS stars, including 20% BHB stars can lead to an increase in the recovered stellar mass by 0.02–0.03 dex for red galaxies relative to blue. Recall that BHB (and BS) stars are only implemented in populations older than 5 Gyr, so that this color dependence reflects the correlation between observed color and mean (luminosity-weighted) age of the stellar population. In part, the lack of sensitivity to BS and BHB stars in our fits can be attributed to the large observational uncertainties on u-band flux estimates relative to other bandpasses.

As discussed in previous sections, uncertainties in the properties of tp-AGB stars are significant, and these uncertainties manifest themselves most clearly in young stellar populations. The relative sensitivity of a given galaxy's recovered stellar mass to the temperature and luminosity of tp-AGB stars therefore hinges on the average age of its stellar population, and is emphasized by the strong dependence of recovered mass on galaxy color in Figure 9. For reference, the division between red sequence and blue cloud galaxies falls at g − r ∼ 0.4–0.5. For blue galaxies, variations in Δlog L and Δlog T can lead to systematic shifts in r-band M/L of up to 0.1 dex for most galaxies, and up to 0.2–0.3 dex in extreme cases. Conversely, older (redder) galaxies are relatively insensitive to the properties of tp-AGB stars, as RGB stars dominate the flux output of post main sequence stars in populations more than 2–3 Gyr old. Recent observational evidence suggests that the lifetimes of low-mass AGB stars may be significantly overestimated (Girardi et al. 2010; Melbourne et al. 2012), and therefore the total flux from tp-AGB stars may be too high by a factors of a few in the models of Marigo & Girardi (2007). These findings are supported by the comparison of post-starburst galaxy spectra with synthetic infrared SPS SEDs, which point to a relatively small contribution from tp-AGB stars even in young galaxy populations (e.g., Kriek et al. 2010; Zibetti et al. 2013). As mentioned previously, FSPS already incorporates an empirical calibration of Δlog L and Δlog T relative to the fiducial tp-AGB evolutionary tracks of Marigo & Girardi (2007), described by Equations (2) and (3) of Conroy & Gunn (2010). These corrections range from Δlog L = −1.0 to −0.2 and Δlog T = +0.1 to −0.1 depending on the age and metallicity of the stellar population. Girardi et al. (2010) show that the evolutionary tracks of Marigo & Girardi (2007) may overestimate the frequency of AGB stars by factors of 3–7, naively suggesting that at Δlog L of −0.3 to −0.8 may be necessary to match the observed data, within the range of the FSPS calibration. Kriek et al. (2010) find that a similar offset (Δlog L ≈ −0.6) is required to fit their sample of post-starburst galaxies in the NEWFIRM medium-band survey. Therefore, while adopting Δlog L = 0.4 leads to a significant differential offset between young and old galaxies of 0.1 dex, empirical evidence suggests that typical tp-AGB luminosities are unlikely to be underestimated (and may, in fact, be overestimated).

In order to construct mass-complete samples from the flux-limited SDSS we require some knowledge of galaxies underlying SEDs. Since our estimates of stellar mass rely on determining the most likely SED given a set of observed ugriz photometry, we can use the best-fit model (or the likelihood distribution over a set of models) to estimate the redshift range over which a given galaxy satisfies the flux limits of our sample (i.e., 14 ⩽ m_r ⩽ 17.77).

In Figure 10, we show the distribution of z_max—the maximum redshift at which a galaxy still satisfies our sample flux limit—as a function of stellar mass. In the top panel, we have coded tiles according to galaxies' g−r color using the scale shown in the upper left. This shows clearly that blue galaxies are observable over a larger volume than red galaxies at fixed stellar mass—or, alternatively, blue galaxies are sampled to lower stellar masses than red in a fixed volume—as one might expect based on SED shape alone. After some experimentation we found that the redshift-dependent stellar mass limit is well described by a relation of the form

$$\begin{equation} \log (M_\mathrm{lim}/M_\odot) = \alpha + \beta \log z + \gamma \log (1+z). \end{equation} \tag{ 4 }$$

By fitting the stellar mass–z_max relation for galaxies with different rest-frame colors, we can provide a more quantitative description of the sampling effects discussed above. The dotted, dot–dashed and dashed lines in Figure 10 show the resulting functional fits for 3 different color cuts at the green valley, red sequence, and 3σ above the red sequence, which have been determined based on the color–stellar mass distribution in the bottom panel of Figure 10. The relevant parameters for these fits (as well as several others) are given in Table 7, and can be used to select mass-complete samples as required.

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Up to this point we have proceeded without considering the physical reality of structural components recovered by our bulge+disk decompositions. While by construction the recovered components represent the best numerical match to the data, for most scientific investigations we require that the recovered components and their properties be astrophysically meaningful as well. In practice, it is useful to consider a given fit's reliability as the combination of two separate questions: (1) to what extent does a given galaxy image support modeling by a more complex photometric model, e.g., two components versus one? and (2) when can we treat multiple fitted components as physically meaningful structures?

In S11, the distinction between single- and multi-component decompositions was made using an F-test to compare the χ² values of their three fitted models (single Sérsic, de Vaucouleurs bulge+disk, and Sérsic bulge+disk). They define a parameter, P_pS, as the probability that a multi-component model is not required relative to a single-component (i.e., Sérsic profile) fit. To put it another way, P_pS gives the probability that a Sérsic model is adequate to describe the data, and can therefore be used to distinguish genuine structural components (see also Meert et al. 2013). For this reason, we include the P_pS values computed by S11 in our bulge+disk mass catalogs. As shown by S11, the SDSS data are generally only sufficient to distinguish multiple components in galaxies clearly hosting both bulges and disks (0.2 ⩽ B/T ⩽ 0.45), and adopting even modest cuts on the F-test probability, e.g., P_pS ⩽ 0.32, seems to limit the incidence of spurious bulges and disk based on, for example, the distribution of disk axial ratios. As discussed by Meert et al. (2013), the F-test can be used to select relatively pure samples of two-component galaxies, but these samples are not necessarily complete. Some caution must therefore be used in blindly applying P_pS to select bulge and disk subsamples.

Alternatively, one can attempt to classify the fitted profiles in terms of their properties, singling out galaxies where two distinct components appear to be physically motivated and discarding cases where multiple components are either unjustified by the data, as above, or nonsensical. Both Allen et al. (2006) and Lackner & Gunn (2012, hereafter L12) have employed such an approach in their analyses of galaxy structure, and we follow them in filtering the S11 data. In their work, Allen et al. (2006) identify eight profile types that encompass the range of decompositions observed in their sample of ∼10,000 galaxies. These can be broadly understood in terms of the physical systems they describe: two profile types describe single-component models, either pure bulge or disk, while the remaining six describe variations of two-component bulge+disk systems. Allen et al. (2006) describe the application of a logical filter to classify decompositions as reliable, in which case parameters of the bulge+disk fits are retained, or unreliable, in which case structural parameters of single-component fits are adopted. In classifying the S11 data we adopt a simplified framework in which we divide the galaxy population into only four classes based on their best-fitting gim2d profiles. This scheme is summarized below.

• (i)  
  Type 1. A de Vaucouleurs component dominates the surface-brightness profile at all radii, or no exponential component is fit (i.e., B/T = 1). We also include here galaxies for which the best-fit disk scale length is less than 0.2 pixels—i.e., the disk is collapsed to a point source—regardless of B/T (∼0.5% of fits). In terms of the single-component Sérsic profiles fit by S11, the distribution of n for these profiles peaks at n ∼ 4–5. These are single-component elliptical galaxies, and are related to profile types 5 and 8 from Allen et al. (2006).
• (ii)  
  Type 2. The exponential component dominates the surface-brightness profile at all radii, or no de Vaucouleurs component is fit (i.e., B/T = 0). As for type 1 profiles, we include here any fits where the half-light radius of the best-fit de Vaucouleurs profile is less than 0.2 pixels (∼2.6% of fits). These galaxies have low n in the single Sérsic fits, and are either pure disk systems or disks hosting a weak central pseudo-bulge. These galaxies are related to profile types 2, 5, 7, and 8 from Allen et al. (2006).
• (iii)  
  Type 3. The de Vaucouleurs component dominates the surface-brightness profile in the central regions, while the exponential component dominates at large radii. The de Vaucouleurs and exponential profiles cross only once at an r-band surface brightness, μ_r, less than 26 mag arcsec⁻². For the most part these are genuine bulge+disk systems, however in some cases these are single-component (elliptical) galaxies in which a disk has been included to account for deviations from a pure de Vaucouleurs profile, e.g., tidal features, isophotal twists, or n ≳ 4. Such spurious fits stand out in the distribution of axis ratios as an excess of face-on disks (e.g., Figure 13 of S11; Allen et al. 2006; Benson et al. 2007; L12). These are classified as type 1 profiles by Allen et al. (2006).
• (iv)  
  Type 4. These galaxies include everything that cannot be classified as types 1, 2, or 3. These include galaxies where the bulge and disk surface-brightness profiles cross twice at μ_r < 26 mag arcsec⁻², or where the disk and bulge profiles are inverted—i.e., the disk profile dominates in the central regions and the bulge dominates at large radii. These encompass profile types 3, 4 and 6 from Allen et al. (2006).

Examples of the first three profile types (i.e., elliptical, disk, and bulge+disk) are shown in Figure 11. In our sample, 12.4% of galaxies are classified as type 1, 11.7% as type 2, 71.7% as type 3, and 4.2% as type 4 profiles. In Figure 12 we show the distribution of Sérsic indices for profile types 1, 2, and 3. The classification of type 1 and 2 profiles as ellipticals and disks respectively is supported by the distribution of n for these galaxies. While the majority of galaxies are classified as type 3 (bulge+disk) profiles, relatively few of these seem to require a multi-component model to describe their surface-brightness profiles; only 37% of type 3 profiles satisfy the cut of P_pS ⩽ 0.32 discussed above. The broad range of galaxies classified as type 3 is also evident in their Sérsic index distribution. In most instances, it is therefore useful (or required!) to combine the profile type with cuts on the F-test probability when selecting bulge and/or disk subsamples.

As a test of our classifications, we can compare the profile types determined here with those assigned by L12 in their analysis of SDSS galaxies. Briefly, L12 fit a sample of 71,825 galaxies with five different image models: two multi-component (bulge+disk) and three single-component profiles. In order to determine the most appropriate model for each galaxy, they use a combination of physically driven quality assessment and statistically motivated goodness of fit. Their final classification is in to one of five categories: de Vaucouleurs bulge plus disk, pseudo-bulge (n = 1) plus disk, pure de Vaucouleurs profile, pure exponential profile, or Sérsic profile. In total there are 66,693 galaxies in common between our sample and that of L12. It is worth noting that the classifications of L12 incorporate additional information about the likelihood of a bulge+disk fit relative to other fitted profiles, similar to the P_pS parameter described above. We have chosen to leave these as two separate diagnostics in order to allow greater flexibility in how samples are selected from within our catalog. Nevertheless, a comparison of the raw profile classifications, independent of the F-test probabilities, is still useful in understanding the behavior of these different diagnostics.

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Of type 1 galaxies, 44% are classified as ellipticals by L12, with an additional 21% and 28% classified as de Vaucouleurs bulge+disk and single Sérsic profiles, respectively. Despite the relatively poor agreement between the raw classifications for type 1 profiles, the distribution of Sérsic indices determined by L12 for these galaxies is clearly peaked around n ≈ 4 (median value of 3.9), supporting our classification of these systems as ellipticals. The agreement between our pure disk classifications is significantly better, with 66% of type 2 galaxies classified as pure exponential disks by L12 and a further 32% classified as single Sérsic profiles. As for type 1 profiles, the distribution of Sérsic indices for type 2 galaxies seems to support their classification as disks (median n ≈ 0.84). In total, 24% of galaxies are classified as being best fitted by either a pure de Vaucouleurs or pure exponential disk model.

There is less clear agreement between our type 3 (bulge+disk) galaxies and the classifications of L12; only 36% of type 3 galaxies are classified as hosting two components by L12, with the majority of remaining galaxies classified as unknown (i.e, fitted with Sérsic profiles). As discussed above, our raw classifications make no account for the degree to which a two component is required by the data, which the classifications of L12 do. If we adopt a cut of P_pS ⩽ 0.32—i.e., require that a two-component models is preferred at the ∼1σ level—then 51% of the remaining type 3 galaxies are classified as bulge+disk systems by L12 (38% de Vaucouleurs bulge+disk and 13% pseudo-bulge+disk). As might be expected, increasing the required significance of the two-component fit improves agreement between the two classification schemes, but at the expense of the total number of bulge+disk galaxies. Overall, the comparison with L12 supports running theme of this section: identifying genuine bulge+disk systems requires a combination of quality assessment metrics which incorporate information about both the physical and numerical likelihood of a given decomposition.

Although we have taken steps to limit the number of galaxies with suspect mass estimates, the final catalogs will inevitably contain values that, for one reason or another, are unreliable. As discussed in Section 3.4, one way to identify such spurious values is to identify data which are internally inconsistent, e.g., when M_B + M_D significantly exceeds M_{B + D}. In Tables 4 and 6 we include an additional parameter, Δ_{B + D}, which gives the offset between M_B + M_D and M_{B + D} in units of the standard error. Galaxies for which the bulge and disk masses exceed the total mass by more than 1σ should be approached with extreme caution, and we recommend in these cases using the dust-free masses as more reliable estimate for the "true" stellar mass.

We generate a large number of smooth galaxy image models—either a single-component Sérsic profile or a two-component de Vaucouleurs bulge plus disk profile—with a range of structural parameters described in Table 8. Synthetic galaxy images are generated using the gsimul task included as part of gim2d. In each case, parameters are randomly drawn with uniform probability from their allowed ranges, and the resulting image model is injected into a randomly selected SDSS-corrected frame using the sextractor segmentation image as a guide. We require that the model center falls on a sky pixel, however place no such constraint on the simulated galaxies' isophotal area (see Appendix B.3 for a discussion of crowding effects).

The methodology described by S11 relies on simultaneously fitting the g- and r-band images, and our Monte Carlo procedure therefore requires that we generate coupled pairs of g- and r-band models. In the case of single-component galaxies generation of these coupled pairs is relatively straight forward: given a randomly generated r-band magnitude, we generate g-band magnitudes such that 0.3 ⩽ g − r ⩽ 1.1. Other parameters—i.e., size, Sérsic index, ellipticity and position angle—are drawn uniformly from their respective ranges given in Table 8. In the case of bulge+disk models the procedure is somewhat more complicated. For each (randomly generated) r-band magnitude and (B/T)_r, we use empirical relationships between (B/T)_r, (B/T)_g, and g−r color (given in Table 9) to generate the corresponding g-band magnitude and (B/T)_g. The typical scatter about the empirical (B/T)_r versus color and (B/T)_r versus (B/T)_g relationships are 0.16 mag and 0.08, respectively, and are incorporated in our generation of (B/T)_g and color from (B/T)_r. Bulge and disk size, ellipticity/inclination, and position angle are generated uniformly over the ranges specified in Table 8. For simplicity we require that the bulge and disk position angles are the same.

Simulations can only be used to understand the reliability of the S11 flux measurements inasmuch as they reflect the properties of real galaxies, and we therefore impose two additional criteria on our simulations. First, the ability of gim2d to accurately recover flux depends sensitively on surface brightness, so that flux estimates of low surface-brightness galaxies or structural components are significantly uncertain. We therefore require that simulations have total half-light radii less than 15 pixels in order to limit any bias incurred as a result of considering large, low surface-brightness galaxies (for reference, ∼91% of observed galaxies satisfy our adopted size limit). Second, in observational samples there is significant covariance between bulge and disk size; however, in generating our models we have explicitly decoupled the properties of bulges and disks. Although we do not want to impose a priori a strong relationship between bulge and disk sizes which may bias our results, we do impose a cut such that r_e ⩽ 1.67r_d. We generate a total of 46,000 simulations matching these criteria: 23,000 each with either single- or two-component light profiles.

Finally, it is worth stressing that these simulations are intended to sample the parameter space that galaxies may occupy (in terms of the parameters listed in Table 8) without any a priori assumptions about how real galaxies actually populate this space.

B.2.1. The Reliability of Total Flux Measurements

We first consider the recovery of galaxies' total flux, which is summarized in Figures 14 and 15 for bulge+disk and single-component input models, respectively. For each simulated image we consider the result of fitting with either a two- or single-component photometric model, indicated as either "Bulge + Disk" or "Sérsic." In each Figure, left panels show the systematic uncertainty in the recovered magnitudes (m_recovered − m_input) for a range of apparent magnitudes and structural parameters (either B/T or n), while the right panels show the statistical uncertainty (i.e., scatter) in offset values. Open and filled tiles indicate negative and positive values, and the size of each tile indicates the magnitude of the offset following the scale on the right. For example, a large filled square represents a significant underestimate of the true flux.

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Focusing first on two-component galaxies (Figure 14), both photometric models produce reasonable estimates of galaxies' total flux for galaxies with m_r ≲ 16 regardless of bulge fraction, which typical systematic offsets <0.1 mag. We stress that these error estimates incorporate both the formal uncertainties output by gim2d as well as uncertainties in the sky background. Pushing toward fainter magnitudes, there is a tendency for Sérsic profile fits to systematically overestimate the true flux by up to ∼0.2 mag, particularly in galaxies with intermediate B/T; this overestimate is also reflected in the increased scatter in recovered flux (bottom right panel of Figure 14. This primarily reflects the difficulty in constraining the wings of the Sérsic profile when the absolute sky level is uncertain, particularly at low surface brightness. In contrast, bulge+disk fits are extremely well behaved over the range of simulated properties, suggesting that they are able to reliably recover the total flux of "classical" bulge+disk galaxies.

Turning to single-component galaxies, the upper left panel in Figure 15 highlights a particular shortcoming of the bulge+disk modeling adopted here: the combination of a de Vaucouleurs bulge exponential disk profile is unable to accurately recover flux in galaxies with n ≳ 5. In this case, the systematic underestimation of flux at high n is governed by an inability of the bulge+disk model to simultaneously reproduce both the bright inner and extended outer profiles of the simulated galaxies, and at least in part motivates the more careful consideration of "genuine" bulges and disks using the metrics discussed in Section 5.2. By comparison, the fluxes recovered by Sérsic profile fits appear to be extremely robust when applied to either simulated sample: systematic uncertainties are less than 0.15 mag across a broad range of input galaxy parameters, and ∼0.02 mag for the sample as a whole (i.e., irrespective of m_r, B/T, or n).

In cases where the only requirement is for robust and reliable total mass measurements, we suggest that masses based on Sérsic profile fits are to be preferred over bulge+disk models.

B.2.2. Recovering Bulge and Disk Flux

As described in Appendix B.1, our image models are generated from a combination of bulge and disk components. Therefore, in addition to quantifying the reliability of our overall flux measurements, as above, we can also study separately the recovery of bulge and disk fluxes, which are shown in Figure 16. The format of this figure is the same as for Figure 14, except that the size of each tile now reflects the systematic or statistical uncertainty in the recovered disk or bulge (top and bottom panels, respectively) separately—note also the increased scale on the righthand side of Figure 16. These figures serve to make the point that the ability to reliably recover the flux of an individual structural sub-component is strongly dependent on the galaxies' overall structure, particularly at faint galaxy magnitudes; the recovered bulge (disk) fluxes in galaxies that are disk (bulge) dominated is uncertain by more than 1 mag, while the statistical uncertainty is ∼0.5 mag for galaxies with intermediate (B/T)_r. While the precision of bulge and disk fluxes suffer relative to the total flux measurements, they remain relatively accurate in the sense that the systematic uncertainties are generally 0.3–0.4 mag or less, with the exception of disks in bulge-dominated galaxies, whose flux can be significantly overestimated. For galaxies which host both bulge and disk components—e.g., 0.2 ⩽ (B/T)_r ⩽ 0.8—the typical systematic uncertainty is of order 0.3 mag or less, while statistical uncertainties range from 0.3 to 0.5 mag depending on bulge fraction.

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As discussed in the Introduction, one of the principal motivations for the re-derivation of photometric quantities by S11 is the relatively poor performance of SDSS DR7 photometry in crowded environments. Given this, it is worthwhile to assess the performance of sextractor and gim2d for galaxies with near neighbors.

As described previously, our simulated galaxy images are injected into the SDSS-corrected frames requiring only that the model center falls on a "sky" pixel (as classified by sextractor); therefore, our simulations include instances where the model galaxy's isophotal area overlaps significantly with another photometric object in the image frame. In order to assess the influence of these near neighbors on the recovered flux, we first find for each simulated image the closest (observed) photometric object with an extinction-corrected r-band Petrosian magnitude no more than 2.5 magnitudes fainter than the input model galaxy (i.e., we require that m_r ⩽ m_r, input + 2.5).

As shown in Figures 14 and 15, the accuracy of gim2d's flux measurement varies with both apparent magnitude and structure, and we need to account for these variations in order to assess any residual dependence on near neighbors. Therefore, for each galaxy we compute the flux offset relative to isolated models—i.e., those with no near neighbors—in a moving window of Δm_r = 0.25 mag and Δ(B/T)_r = 0.2 or Δn = 2 (depending on the underlying model) centered on the galaxy. This "residual" offset can then be used to assess the effects of crowding independent of the global trends shown in Figures 14 and 15. In Figure 17 we plot the residual offset measured in this way as a function of projected separation in pixels for both one- and two-component simulated galaxies. The gim2d fits are extremely well behaved at separations greater than 10 pixels (∼4''), and any residual dependence of the recovered flux on nearby neighbors is typically <0.02 mag, i.e., less than the typical photometric uncertainty quoted in Appendix B.2. Even in the worse case scenario of fitting a (simulated) single-component model with a bulge+disk profile at small separations, the ∼0.04 mag offset corresponds to <0.02 dex in terms of the derived mass. While we show results only for the r band, we have verified that any trends with separation are mirrored by the g-band models, and show no evidence for a dependence of galaxy color on the presence of near neighbors.

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