Exploring the Structure and Assembly of Galaxies at the Heart of the Coma Cluster

Abstract

We constrain the assembly history of galaxies in the projected central 0.5 Mpc of the Coma cluster by performing structural decomposition on 69 massive $(M_{\star } \geq 10^{9}\ M_{\odot })$ galaxies using high-resolution F814W images from the HST Treasury Survey of Coma. Each galaxy is modeled with up to three Sérsic components having a free Sérsic index n. After excluding the two cDs in the projected central 0.5 Mpc of Coma, 57 % of the galactic stellar mass in the projected central 0.5 Mpc of Coma resides in classical bulges/ellipticals while 43 % resides in cold disk-dominated structures. Most of the stellar mass in Coma may have been assembled through major (and possibly minor) mergers. Hubble types are assigned based on the decompositions, and we find a strong morphology-density relation; the ratio of (E+S0):spirals is (91.0 %):9.0 %. In agreement with earlier work, the size of outer disks in Coma S0s/spirals is smaller compared with lower-density environments captured with the Sloan Digital Sky Survey. Among similar-mass clusters from a hierarchical semi-analytic model, no single cluster can simultaneously match all the global properties of the Coma cluster. The model strongly overpredicts the mass of cold gas and underpredicts the mean fraction of stellar mass locked in hot components over a wide range of galaxy masses. We suggest that these disagreements with the model result from missing cluster physics (e.g., ram-pressure stripping), and certain bulge assembly modes (e.g., mergers of clumps). Overall, our study of Coma underscores that galaxy evolution is not solely a function of stellar mass, but also of environment.

This chapter has been previously published as Weinzirl, T., Jogee, S., Neistein, E., et al.: Mon. Not. R. Astron. Soc. 441, 3083 (2014)

Appendix

3.1.1 Using GALFIT

The proper operation of GALFIT depends on certain critical inputs. We briefly describe below how these important inputs are handled:

Point Spread Function (PSF): Accurate modeling of the PSF is essential in deriving galaxy structural properties. GALFIT convolves the provided PSF with the galaxy model in each iteration before calculating the χ ². Because the PSF varies with position across the ACS Wide Field Camera chips, it is ideal to separately model the PSF for each galaxy position. We use the grid of model ACS PSFs in the F475W and F814W filters from Hoyos et al. (2011). This grid of PSFs was created with TinyTim (Krist 1995) and DrizzlyTim.⁶

For a given set of multidrizzle parameters, DrizzlyTim transforms x − y coordinates in the final science frames back to the system of individually distorted flat-fielded (FLT) images. DrizzlyTim invokes TinyTim to create a PSF with the specified parameters (e.g., position and filter) and then places the PSF at the appropriate position in blank FLT frames. The FLT frames are passed through MULTIDRIZZLE with the same parameters as the science images. Finally, a Charge Diffusion Kernel is applied to the PSFs in the geometrically distorted images. The grid of ACS PSFs from Hoyos et al. (2011) models a PSF for every 150 pixels in the x and y directions. For each galaxy in our sample we select the model PSF closest in proximity to the galaxy.

Sigma Images: A sigma image is the 2D map of the 1σ standard deviations in pixel counts of the input image. GALFIT uses the sigma image as the relative weight of pixels for calculating the goodness of fit. Achieving a reduced χ ² ∼ 1 with a successful model fit requires that the sigma image be correct. A sigma image can either be provided, or GALFIT can be allowed to calculate one based on the properties of the data image (image units of counts or counts/second, effective gain, read noise, number of combined exposures). We choose the latter option and allow GALFIT to calculate the sigma images.

Background Subtraction: While it is possible for GALFIT to freely fit the background sky, this is not recommended (Peng et al. 2002). In a multiple-component fit to a galaxy with at least two components, freely fitting the sky can exaggerate or suppress the wings of the central Sérsic profile and incorrectly measure the bulge half-light radius and Sérsic index. To avoid this, for each galaxy the background sky is measured and held fixed during the fit. The sky background is based on ellipse fitting with the IRAF/ELLIPSE task. Ellipses are fit to the galaxy and the surrounding area, with the ellipses in the surrounding area being fixed to the shape and orientation of the galaxy. The gradient along the semi-major axis is calculated, and the sky is estimated as the mean of elliptical annuli over a span in semi-major axis where the gradient reaches a prescribed small value. In each case, the area fitted by the ellipses exceeds the area subtended by the galaxy. Visual inspection of the ellipse fits shows that the perceived flat gradient corresponds to empty sky and not an extended galaxy outer profile with a very small gradient.

Image Thumbnails and Masks: Thumbnail cutouts of the intermediate-mass galaxies are made to lessen the computational time for fitting. Following Hoyos et al. (2011), square image thumbnails centered on the target galaxy are made using the output from SExtractor. Image size in pixels is determined with

$$\displaystyle{ \mathrm{size} = 4 \times \mathrm{ A\_IMAGE} \times \mathrm{ KRON\_RADIUS}. }$$

(3.8)

The image units are transformed from counts/second to counts by multiplying by the exposure time. Image masks are based on the segmentation images provided from SExtractor. The segmentation images are modified to unmask the background and target galaxy being fitted. Any bright sources that visibly overlap with the target galaxy are also unmasked so that overlapping sources can be fitted simultaneously. Masks for relatively bright sources that do not overlap with the galaxy being fitted are expanded in semi-major axis by a factor of 1.5. We visually check by blinking the data image and modified segmentation image to verify that the unmasked region encompasses all of the target galaxy, including those with large diffuse halos that SExtractor does not capture (Hoyos et al. 2011).

3.1.2 Details of Structural Decomposition

This appendix contains the full details concerning the structural decomposition scheme outlined in Sect. 3.3.2.

3.1.2.1 Single Sérsic Fits

We first fit all galaxies with a single Sérsic profile before attempting the multi-component fits. This step is useful for measuring the total luminosity of a galaxy as well as measuring the centroid (Weinzirl et al. 2009). The Sérsic profile has seven free parameters: centroid, luminosity, half-light radius r _e, Sérsic index n, axis ratio, position angle, and diskiness/boxiness. We fix the diskiness/boxiness so that the fitted structures are perfect ellipses. We estimate the other six parameters based on the parameters in SExtractor and allow them to optimize in the fit. The detailed image preparation and inputs for the proper operation of GALFIT are described in section “Using GALFIT”.

Figure 3.14 compares our results for the single Sérsic fits (with no point source) with those of Hoyos et al. (2011), who also perform single Sérsic fits with GALFIT and GIM2D using Coma ACS Treasury Survey data. Note that the galaxies in our sample requiring one Sérsic profile are distinguished in Fig. 3.14. With the exception of COMAi125935.698p275733.36 (NGC 4874), our results for these sources requiring one Sérsic profile well match those derived by Hoyos et al. (2011). For NGC 4874, we measure the r _e and n of NGC 4874 to be 17.3 kpc and 3.05, respectively, while Hoyos et al. (2011) measure r _e and n to be 3.2 kpc and 1.3.

Fig. 3.14

This figure compares our results for the single Sérsic fits (with no points source) with those obtained by Hoyos et al. (2011) using GALFIT on the same data. The sources in our sample requiring one Sérsic component are labeled separately from sources requiring two or three Sérsic profiles. Our derived magnitudes, r _e, and n for the sources requiring one Sérsic profile agree well with the parameters from Hoyos et al. (2011), with the one exception being cD galaxy NGC 4874 (COMAi125935.698p275733.36) with n ∼ 3. Note that cD galaxy NGC 4889 requires only one Sérsic profile but it is not included here as it is not in the Hoyos et al. (2011) sample. See section “Single Sérsic Fits” for additional details.

For sources requiring more than one Sérsic profile, our single Sérsic magnitudes agree well in general with those of Hoyos et al. (2011), except for one case (COMAi13051.149p28249.90) where Hoyos et al. (2011) underestimate the magnitude by ∼ 5.5 mag. There are also outliers in both r _e and n. In 10 (5) instances (including COMAi13051.149p28249.90), the difference in r _e (n) exceeds a factor of 1.5.

There are two key differences in our fitting methodology (see section “Using GALFIT”) compared with Hoyos et al. (2011). Most importantly, we entirely unmask the target galaxy and background in the segmentation-based masks so that GALFIT fits to pixels beyond what SExtractor associates with each galaxy. Hoyos et al. (2011) confine a galaxy to a customized mask generated based on the output of SExtractor. This approach misses a finite fraction of the flux in the target galaxy. This may explain why in Fig. 3.14 we measure brighter magnitudes and larger r _e for more extended galaxies, where SExtractor does not detect all of the light in the galaxy. Second, we measure and fix the background sky while Hoyos et al. (2011) keep the sky as a free parameter. Allowing the sky background to freely vary in our fits fails to account for most of the scatter between our results and those of Hoyos et al. (2011). Rather, the disagreement appears to mainly be the result of differences in image masking.

3.1.2.2 Multi-Component Fits

For the Stage 2 Sérsic + Sérsic fits, we model the ‘inner’ and ‘outer’ components (C1 and C2) with Sérsic profiles that can represent physically different components (see Sect. 3.3.2).

Sensible initial guess parameters for Stage 2 are determined from a combination of the data image, Stage 1 model, and Stage 1 residuals. Guesses for the inner Sérsic component (C1) are usually based on the Stage 1 model. The centroid of the Sérsic components (and nuclear point source if present) are fixed to the best-fit centroid from the single Sérsic model. During the fits, we allow all other parameters (luminosity, r _e, n, axis ratio, and position angle) to vary for the inner and outer components without a priori fixing the nature of these components.

With one exception, the χ ² in Stage 2 is always lower compared with χ ² in Stage 1 due to the extra Sérsic component. While the rare increase in χ ² from Stage 1 to Stage 2 is an indication the latter model is not reliable, the almost universal decrease in χ ² is not necessarily a sign that the Stage 2 fit is meaningful because, in principle, such a decrease in χ ² could be driven by the extra free model parameters. We consider a Stage 2 multi-component model to be superior to the Stage 1 fit if (1) χ ² drops, (2) the Stage 2 model parameters are well behaved (i.e., not unphysically large or small), and (3) the Stage 2 residuals are deemed by visual inspection to show a reduction in coherent structure relative to the Stage 1 residuals.

Fig. 3.15

This figure shows examples C1 to C6 where a single Sérsic model (plus point source if needed) does not provide a good fit to coherent galaxy structure that is best modeled with one or more additional Sérsic profiles. Such residual structure includes central compact structures (C2, C3, C4, C5, C6), rings (C3, C4), annuli and extended components (C1, C4), and bars/ovals (C5, C6). These systems are better fitted by models with multiple Sérsic components (see Figs. 3.16 and 3.17). Columns 1 and 3 show the input I-band images. Columns 2 and 4 show the residuals after subtracting the best single Sérsic fit NoteC1 = COMAi125931.893p275140.76,C2 = COMAi125935.286p275149.13,C3 = COMAi13021.673p275354.81,C4 = COMAi13014.746p28228.69,C5 = COMAi13027.966p275721.56, andC6 = COMAi125930.824p275303.05

Fig. 3.16

This figure shows how some of the galaxies (C2 = COMAi125935.286p275149.13 and C4 = COMAi13014.746p28228.69) poorly fitted by a single Sérsic model (plus point source if needed) in Stage 1 can be better fitted by two Sérsic models (plus point source if needed) in Stage 2. Each row shows the data, residual after Stage 1, and the residual after Stage 2. Galaxy C2 is best-fit as having an inner disk (n = 0. 31) and an outer elliptical structure (n = 2. 08). Galaxy C4 is best fit with an inner bulge n = 3. 68 and an outer disk (n = 0. 47)

Fig. 3.17

This figure shows two examples of barred galaxies (C5 = COMAi13027.966p275721.56 and C6 = COMAi125930.824p275303.05) poorly fitted by a Stage 1 single Sérsic model that are better fitted by a Stage 3 triple Sérsic (plus point source if needed) model. Column 1 shows the data images while columns 2 and 3 show the residuals after the Stage 1 and Stage 3 model, respectively

Fig. 3.18

This figure shows the decomposition of moderately inclined, barred galaxy COMAi125950.105p275529.44, in which we measure the highest outer disk Sérsic index n = 1. 66. Thus, this galaxy sets the empirically determined upper limit on disk Sérsic index, $n_{\mathrm{disk\_max}} = 1.66$. Column 1 shows the data images while columns 2 and 3 show the residuals for the Stage 2 and Stage 3 model, respectively. The bar signature is clearly present in the residuals

Figure 3.15 provides examples where a single Sérsic model fails to model the entire galaxy well and leaves behind coherent structure in the residuals. Such coherent structure is indicative of additional components such as compact central structures, rings, annuli and extended components, and bars/ovals. We illustrate in Figs. 3.16 and 3.17 how some of these examples are best fitted by models with multiple Sérsic components.

If a galaxy does not require a Stage 2 model, or if the Stage 2 model fails to meet the above criteria, then the galaxy is described by a single Sérsic profile + point source, if present. Six galaxies are best represented by Stage 1. Two (COMAi13017.683p275718.93 and COMAi13018.093p275723.59) cannot be fit with multiple Sérsic models because they are interacting. In the third case, (COMAi125931.103p275718.12), the χ ² increases from Stage 1 to Stage 2. The final three cases (NGC 4874, NGC 4889, and COMAi125909.468p28227.35) show evidence of a core (see section “Identifying Core Ellipticals”).

Galaxies for which the Stage 2 model is deemed an improvement are interpreted as follows. Since the outer component C2 could represent a disk, we must specify criteria for identifying an outer disk. The outer component C2 is a disk if it satisfies at least one of the following. (1) The galaxy is highly inclined such that C2 has a low axis ratio b∕a ≤ 0. 25 that is below the axis ratios found for ellipticals. (2) The galaxy is moderately inclined and C2 shows disk signatures (e.g., bars, rings, or spiral arms) in the data images and/or Stage 2 residuals. (3) For moderately inclined galaxies without disk features that do not satisfy (1) or (2), we require Sérsic n be less than the threshold value $n_{\mathrm{disk\_max}}$.

Theoretical considerations show that pure disks have n = 1, suggesting the threshold should be n ∼ 1. However, real galaxy disks are not fitted perfectly by Sérsic profiles. We determine the value empirically from the maximum disk Sérsic index in galaxies satisfying (1) and (2). Highly inclined disks show a range in Sérsic index of 0.48–0.86. Moderately inclined galaxies identified as having spiral arms but no bar have outer disks with Sérsic index 0.63–1.20. Note that some of the highly inclined galaxies could be barred, and this may account for the small difference in average Sérsic index between the highly inclined and moderately inclined barred galaxies.

In order to accurately model the outer disk of moderately inclined barred galaxies, a triple Sérsic profile (see below) is required. After taking this extra step, the outer disk Sérsic index among moderately inclined barred galaxies is 0.25–1.66. The maximum Sérsic index among outer disks in galaxies satisfying requirement (1) and (2) is 1.66, and we therefore set $n_{\mathrm{disk\_max}}$ to this value. Thus, outer disks span the range 0.25–1.66 in Sérsic index and have a median n of 0.84. Figure 3.18 shows the galaxy (COMAi125950.105p275529.44) on which we base our measurement of $n_{\mathrm{disk\_max}}$. Section “Systematics of $n_{\mathrm{disk\_max}}$” discusses the uncertainties in the adopted value of $n_{\mathrm{disk\_max}}$.

Galaxies that satisfy any of requirements (1), (2), or (3) are deemed to have an outer disk. Galaxies without an outer disk are considered photometric ellipticals.

We test all galaxies having an outer disk for the presence of a large-scale bar/oval in Stage 3 by fitting a triple Sérsic profile + point source, if present. Bars/ovals are modeled with an elongated, low Sérsic index (n ∼ 0.5) profile (Peng et al. 2002; Weinzirl et al. 2009). In the text, we do not distinguish between bars and ovals, and we use “bar” to describe both.

The initial guesses for the three-component models come from the best Stage 2 model combined with visual inspection. The Sérsic index for the bar is initially guessed to be 0.5, and the shape and position angle of the bar are visually estimated using the data image or the residuals of the Stage 2 fit. When selecting between the Stage 2 and Stage 3 fits, we applied the same constraints described above for the behavior of χ ². An additional complication is that in galaxies with unbarred outer disks, GALFIT may fit a ‘bar’ to any existing spiral arms, rings, or clumpy disk structure. Stage 3 fits in these cases could be discarded by noting the resulting discrepancies in appearance between the galaxy images and the Stage 3 model images. Figure 3.17 shows examples of two disk galaxies where adding the third Sérsic component removes the bar signature from the residuals.

3.1.2.3 Nuclear Point Sources

Nuclear point sources are found in galaxies of all Hubble types. The frequency of nuclear point sources is very sample dependent and is particularly sensitive to range of galaxy luminosity. HST studies of early-type galaxies (e.g., Ravindranath et al. 2001; Côté et al. 2006) have measured nucleation rates of 50 % or more. Ravindranath et al. (2001) find about half of early-type (E, S0, S0/a) galaxies have nuclear point sources. Côté et al. (2006) show that the frequency of nucleation in ACS images of the Virgo cluster is at least 66 % in galaxies with M _B  ≤ −15. Graham and Guzmán (2003) discuss 13/15 examples of dwarf ellipticals in the Coma cluster showing evidence for nucleation. Balcells et al. (2007a) measure a frequency of 58 % for S0 to Sbc galaxies. Böker et al. (2002) measure the frequency of point sources to be 75 % in spirals with Hubble types Scd to Sm.

Although nuclear point sources account for a small percentage ( < 1 %) of a galaxy’s light, it is important to include them during multi-component structural decomposition. Neglecting nuclear point sources can have a significant effect on derived parameters of bulges (Balcells et al. 2003; Weinzirl et al. 2009). We assess the presence of nuclear point sources with visual inspection. If a compact light source is visible by eye in the residuals of the single Sérsic fit, the galaxy is flagged as having a potential point source. With this procedure, 49/69 galaxies in sample S2 have a potential nuclear point source.

Galaxies having a potential nuclear point source are fitted with an extra nuclear point source component in the best-fit single or multi-component model. GALFIT models the point source with the user-input PSF. More than half (38/69, 55. 1 ± 6. 0 %) of objects in sample S2 have a nuclear point source in the final, best-fit structural decomposition. Figure 3.19 shows examples of residual galaxy images with point sources.

Fig. 3.19

This figure compares residuals after fitting a single Sérsic model (top row) versus the best fit double Sérsic + nuclear point source model (bottom row) for an elliptical (COMAi13030.954p28630.22), S0 (COMAi13021.673p275354.81), and spiral (COMAi13041.193p28242.34). The nuclear point source is visible in the residuals in the top row

Figure 3.20 shows the derived point source luminosities correlate with total galaxy magnitude such that more luminous point sources are found in brighter galaxies. Similar results been found in earlier work (e.g., Graham and Guzmán 2003; Balcells et al. 2007a).

Fig. 3.20

This panel shows the relation between total galaxy luminosity and point source luminosity for objects having a nuclear point source in the final, best structural decomposition

3.1.2.4 cD Galaxies

cD galaxies are defined by having extra light on cluster-sized ( ∼ 1 Mpc) scales with respect to the outward extrapolation of the Sérsic profile fit to the inner ( ∼ 100 kpc) portion of the galaxy. Such galaxies are luminous and are found in regions of high galaxy number density (Binney and Merrifield 1998). Of the three cD galaxies in Coma, two (NGC 4874 and NGC 4889) lie in the projected central 0.5 Mpc and are therefore in our sample. The third cD (NGC 4839) lies is in the outer southwest region of Coma and is not part of this study.

Definitive proof that NGC 4874 and NGC 4889 are cDs is the detection of intra-cluster light in Coma (Kormendy and Bahcall 1974; Melnick et al. 1977; Thuan and Kormendy 1977; Bernstein et al. 1995; Adami et al. 2005; Arnaboldi 2011).

The single Sérsic indices reported in section “Details of Structural Decomposition” and Table 3.3 for the these cD galaxies are n ∼ 3–4.4 because the decompositions also include the central core. The central core is a clear deviation from the inward extrapolation of the Sérsic profile that characterizes the outer galaxy structure. For this reason, masking the core regions (i.e., the central ∼ 2^{′ ′}) is more physically motivated and would yield higher single Sérsic indices n $\gtrsim$ 8. This is demonstrated in section “Identifying Core Ellipticals” and Table 3.2. We note that both approaches (masking or not masking the core during the 2D decomposition) lead us to the same conclusion that all of the cD light is associated with structures of n ≫ n _{disk_max} (in section “Identifying Core Ellipticals”). Note in Table 3.5 we list the cD galaxies the structure parameters from the 2D decomposition where the core is masked.

The high n ≫ n _{disk_max} values in the cD galaxies are due to the extended wings in the Sérsic profile resulting from the extended low surface brightness envelope of the cD. This extended envelope is likely made up of intra-cluster light and the cumulative debris from galaxies, consistent with the view that cD galaxies arise from repeated bouts of galactic cannibalism and tidal stripping of satellite galaxies in a cluster (Ostriker and Tremaine 1975; Aragon-Salamanca et al. 1998; De Lucia and Blaizot 2007).

3.1.2.5 Cosmic Variance

The Coma ACS data only cover 19.7 % of the projected central 0.5 Mpc radius of Coma. The relative fractional numbers of E+S0:spiral, or specifically the ratio of E/S0s, we derive from this data may not be representative of the full region in the projected central 0.5 Mpc radius of Coma due to the incomplete sampling and cosmic variance. In order to assess the effect of incomplete sampling and cosmic variance on our results, we perform the following test.

First, we define the region covered by ACS in the projected central 0.5 Mpc radius of Coma as R1, and the full area in the projected central 0.5 Mpc radius of Coma as R2. We use the Hubble morphological types (MT) from the GOLD Mine database⁷ (Gavazzi et al. 2003) to compute the fraction of E+S0:spiral galaxies in region R1 and R2 with $M_{\star } \geq 4.4 \times 10^{9}\ M_{\odot }$, the mass limit of the Coma GOLD Mine sample. The MT reported by GOLD Mine are sourced from the literature. If we take the visual MT from GOLD Mine at face value then we draw the following conclusions:

• 1. The effect cosmic variance causes the ratio of E/S0 within the GOLD Mine MT to vary by a factor of 1.11 between region R1 and R2 for $M_{\star } \geq 4.4 \times 10^{9}\ M_{\odot }$.

• 2. The partial ACS coverage of the projected central 0.5 Mpc and associated cosmic variance thus causes our study based on region R1 to

  • a. overestimate the ratio of S0/E in the ACS sample for $M_{\star } \geq 4.4 \times 10^{9}\ M_{\odot }$ by a factor of 1.4.

  • b. overestimate the fraction f _cold of dynamically cold stellar mass (43 %) by a factor of 1.27 (Sect. 3.5.6) for $M_{\star } \geq 10^{9}\ M_{\odot }$. We note that the over-estimation of f _cold is not by the same factor as in 2(a) because S0s have a significant fraction of their mass in dynamically hot bulges.

• 3. Currently, our conclusion in Sect. 3.5.6, based on region R1 is that the hierarchical models are over-predicting the empirical fraction f _cold. It is clear from 2(b), that correcting for partial ACS coverage and cosmic variance would only strengthen this conclusion further.

3.1.2.6 Galaxy Color Gradients

In Sect. 3.4.2 we suggest that galaxy color gradients should not bias our conclusions concerning the distribution of dynamically hot and cold stellar mass. Here, we explicitly test this idea.

For a subset of ten galaxies spanning types G3 to G5 and matching the morphology distribution of the mass-selected sample (E+S0:spiral = 2+7:1) in Table 3.6, we re-evaluated the fractional mass in hot and cold components based on combining structural decompositions of both the F814W and F475W images. The new F475W-band decompositions were performed identically to the existing F814W decompositions, except that the position angle and axis ratio of the galaxy structures were fixed to their values from the F814W-band decompositions. Stellar masses of the structural components were calculated according to Into and Portinari (2013) after converting the F475W-F814W color and the F814W luminosity into a B − I color and I-band luminosity, respectively, using the procedure in Sect. 3.2.2.

In the new F475W decompositions for this subset of galaxies, the half-light radii and Sérsic n are similar to the corresponding values in the F814W band. The average offset is 5.4 % with a standard deviation of 5.6 %. Furthermore, the fractional hot stellar mass inferred from a constant global F814W M∕L ratio is 53.4 %. After calculating the stellar mass of each galaxy component from the B − I color, the fractional hot stellar mass is found to be 50.5 %. Thus, M∕L gradients within a galaxy do not appear to have a significant effect on the fractional masses measured in cold versus hot components.

3.1.3 Identifying Core Ellipticals

While elliptical galaxies are remarkably well-fit by Sérsic profiles over large dynamic ranges, giant elliptical galaxies contain cores, or “missing light” at small radii that constitute a downward deviation from the inward extrapolation of the outer Sérsic profile (Graham et al. 2003; Trujillo et al. 2004; Kormendy et al. 2009). Such cores are hypothesized to form from scouring induced by binary black holes during dry, dissipationless mergers.

Because cores, which have traditionally been identified with 1D radial light profiles, are not an obvious feature of the galaxy’s 2D light distribution, global Sérsic fits will encompass any existing core. This is potentially problematic for at least two reasons. Including the core in the Sérsic fit will lower the global Sérsic index. This is of concern in this paper where the Sérsic index plays a key role in interpreting galaxy structure (Sect. 3.3.1). Secondly, fitting the core region may produce features in the residuals that prompt addition of extra nuclear components that have no physical justification.

We systematically search for cores in all sample galaxies. For this task, we use 1D light profiles generated from ellipse fitting of deconvolved images. The ACS images were deconvolved using a simulated PSF (section “Using GALFIT” for details) and 40 iterations of Lucy-Richardson deconvolution with the IRAF task LUCY Lucy 1974; Richardson 1972. Our approach uses the criteria for identifying core galaxies from Trujillo et al. (2004) by fitting Sérsic and core-Sérsic profiles (Graham et al. 2003) to the 1D light profiles.

For simplicity, we use the version of the core-Sérsic profile that assumes an infinitely sharp transition between the outer Sérsic and inner power-law regions, namely

$$\displaystyle{ \mathrm{I}(\mathrm{r}) =\mathrm{ I}_{\mathrm{b}}[(\mathrm{r}_{\mathrm{b}}/\mathrm{r})^{\gamma }\mathrm{u}(\mathrm{r}_{\mathrm{b}} -\mathrm{ r}) +\mathrm{ e}^{\mathrm{b}(\mathrm{r}_{\mathrm{b}}/\mathrm{r}_{\mathrm{e}})^{1/\mathrm{n}} }\mathrm{e}^{-\mathrm{b}(\mathrm{r}/\mathrm{r}_{\mathrm{e}})^{1/\mathrm{n}} }\mathrm{u}(\mathrm{r} -\mathrm{ r}_{\mathrm{b}})]. }$$

(3.9)

Here, r _b denotes the division between the outer Sérsic and inner power-law profiles, I _b is the intensity at this radius, γ is the inner power-law slope, and u(x − a) is the Heaviside step function. Parameters n and r _e refer to the shape and half-light radius of the outer Sérsic profile. Additionally, b is a constant that depends on several free parameters (r _b , γ, r _e, and n).

We require a core galaxy to meet the following criteria: (1) the core-Sérsic model provide a better fit than the Sérsic profile; (2) the cores are well-resolved so that the break radius r _b is greater than the second innermost data point in the profile; (3) the inner power-law slope γ is less than the logarithmic slope of the Sérsic profile (1∕n) in the core region.

Three sample galaxies meet the above criteria for having a core. Two of these are the central cD galaxies NGC 4874 and NGC 4889. Table 3.2 summarizes the r _b and γ measured from the core-Sérsic fit.

We further explore the best way to handle these cored galaxies in the 2D luminosity decompositions. Two natural approaches are to fit the whole galaxy, including the core, or to mask the galaxy over r ≤ r _b . Masking is more physically motivated because the central core is a clear deviation from the inward extrapolation of the Sérsic profile that characterizes the outer galaxy structure. We try both approaches and summarize the results in Table 3.2. Applying a mask versus no mask has a nominal effect on COMAi125909.468p28227.35, but there is a significant increase in the r _e and n of the cD galaxies when their larger core regions are masked.

Performing the 2D fit with the core masked is more physically motivated, and we consider these models to represent the best fits for the cD galaxies. It is worth noting, however, that our result from section “cD Galaxies” that 100 % of the mass in the cDs is associated with structures of $n \gg n_{\mathrm{disk\_max}}$ remains unchanged irrespective of which approach (mask or no mask) we take.

3.1.4 Systematics of n _{disk_max}

Our effort in this paper to make a census (Sect. 3.4.2) of dynamically cold versus dynamically hot stellar mass depends fundamentally on the upper limit, $n_{\mathrm{disk\_max}}$ (Sect. 3.3.1), measured for the Sérsic index of a disk. In our approach, all structures with Sérsic index $n \leq n_{\mathrm{disk\_max}}$ are considered disk dominated, while all other structures with higher Sérsic index are considered classical components built in mergers.

The value of $n \leq n_{\mathrm{disk\_max}}$ is set by the moderately inclined barred galaxy (COMAi125950.105p275529.44) having the highest outer disk Sérsic index. The accuracy of $n_{\mathrm{disk\_max}}$ depends on how representative the sample is as well as the robustness of the multi-component structural decompositions. Figure 3.18 shows for this galaxy the data image and residuals of the multi-component decompositions. While this galaxy was identified as an ambiguous E/S0 galaxy in Fig. 2 of Marinova et al. (2012), the barred nature of this galaxy seems clear based on the image residuals produced by our improved method (Sects. 3.3.2 and 3.3.3) of structural decomposition.

The value of $n_{\mathrm{disk\_max}}$ is subject to sky subtraction errors because it is measured from the outermost Sérsic profile of disk galaxies, and this is likely the dominant systematic effect on $n_{\mathrm{disk\_max}}$. As described in section “Using GALFIT”, we measure the background sky value with a robust method and hold the sky fixed at this value during the fit. To test the importance of the sky subtraction, we refitted COMAi125950.105p275529.44 while adjusting the mean sky background by ± 1σ. This produces a range in outer disk n of n ∼ 1.57-1.77, which spans ∼ 0.1 above and below the adopted n _{disk_max} value of 1.66. Based on the narrow error bars for n _{disk_max}, we do not expect the uncertainty to have a significant impact on our conclusions.

For completeness, we explore for an alternate value of n _{disk_max} the relative stellar mass fractions that would be interpreted as belonging to cold versus hot stellar components. The value n _{disk_max} = 2 is in line with estimates of the Sérsic index of small-scale disks (e.g., Fisher and Drory 2008; Weinzirl et al. 2009) yet is still above the anticipated range in n _{disk_max} due to sky subtraction errors in this study. With this higher n _{disk_max}, we would find that ∼ 51 % stellar mass is in disk-dominated components while ∼ 49 % is still in classical bulges/ellipticals assembled in major and minor mergers. These values are somewhat different from the corresponding values (43 % in disk-dominated structures versus 57 % in non-disks) derived in Sect. 3.4.2 excluding the two cD galaxies. Choosing a higher n _{disk_max} would increase the importance of disk-building processes relative to processes that build classical bulges/ellipticals.