SDSS-IV MaNGA: Uncovering the Angular Momentum Content of Central and Satellite Early-type Galaxies

The aim of MaNGA, one of three core SDSS-IV projects, is to obtain integral-field spectroscopy of 10,000 nearby galaxies. Like previous SDSS surveys, MaNGA utilizes the 2.5 m Sloan Foundation Telescope (Gunn et al. 2006) and the BOSS spectrographs (Smee et al. 2013). Unlike previous SDSS surveys, the MaNGA survey groups individual fibers into 17 hexagonal fiber bundles to perform a multiobject IFS survey (Drory et al. 2015). Each fiber has a diameter of 2″, while the bundles range in diameter from 12″ to 32″ with a 56% filling factor. The two dual BOSS spectrographs cover the wide wavelength range of 3600–10300 Å while maintaining a spectral resolution of ${\sigma }_{r}\approx 70$ $\mathrm{km}\,{{\rm{s}}}^{-1}$, appropriate for galaxy studies. Careful spectrophotometry yields a relative calibration accurate to a few percent (Yan et al. 2016b). The survey design is described in Yan et al. (2016a), the observing strategy in Law et al. (2015), and the data reduction pipeline in Law et al. (2016). The MaNGA team has also developed useful tools for data visualization and vetting (Cherinka et al. 2017).

The MaNGA sample is selected from an enhanced version of the NASA-Sloan Atlas (NSA; Blanton M.; http://www.nsatlas.org) with redshifts primarily taken from the SDSS DR7 MAIN galaxy sample (Abazajian et al. 2009) and photometry from reprocessed SDSS imaging. The sample is built in i-band absolute magnitude (M_i) complete shells, with more luminous galaxies observed in more distant M_i shells such that the spatial resolution (in terms of R_e) is roughly constant across the sample (Yan et al. 2016a; Wake et al. 2017). The MaNGA Primary sample, which forms ∼50% of the total sample, is selected such that 80% of the galaxies in each M_i shell can be covered to $1.5\,{R}_{e}$ by the largest MaNGA integral-field unit (IFU). There is also a Secondary sample, accounting for ∼40% of the targets, selected such that 80% of these galaxies are covered to $2.5\,{R}_{e}$. The remainder of the MaNGA main sample is the Color-enhanced supplement, which fills in poorly covered regions of the color–magnitude diagram (e.g., faint red galaxies or the most luminous blue galaxies) and, like the Primary sample, is covered to $1.5\,{R}_{e}$.

Given this sample construction, one must reweight the galaxy distribution according to the volume in each shell to construct volume-limited samples. Whenever possible, we apply these weights before drawing conclusions. In this work we take a conservative approach and consider the Primary and Secondary samples only, excluding the Color-enhanced supplement. We make this choice since the addition of the color term in the Color-enhanced selection potentially increases the uncertainty in volume weights for these galaxies.

The MaNGA survey awarded a small fraction of fiber bundles to “Ancillary” programs that can help boost survey efficiency by serving as filler targets and also enhance the science output of MaNGA with a small time investment. We were awarded an Ancillary program to augment the number of central galaxies in the most massive halos available within the local volume targeted by MaNGA. Here we describe our target selection, along with a brief description of the group catalog that our selection is based on.

While there are several samples of central galaxies in the literature, building an uncontaminated sample of central galaxies spanning a range of halo masses is a nontrivial exercise. Different cluster finders and central galaxy selections, automated and visual, often disagree. We base our target selection on the Yang et al. (2007, hereafter Y07) cluster catalog updated to DR7, created from the SDSS DR7 New York University Value Added Catalog (VAGC; Blanton et al. 2005), a spectroscopic galaxy catalog.

Y07 use an iterative, adaptive group finder to assign galaxies to halos. In short, they first use a friends-of-friends algorithm (e.g., Davis et al. 1985) to identify potential groups. Each group is assigned a characteristic luminosity, defined as the combined r-band luminosity of all group members with ${M}_{r}-5\mathrm{log}(h)\leqslant -19.5$ mag, where M_r is the absolute Galactic-extinction-corrected r-band luminosity, K-corrected to z = 0.1. Roughly speaking, the stellar mass composes 1% of the total mass. This characteristic luminosity is used to assign halo masses to groups and to refine the group identification using an iterative method. Using a mock catalog, Y07 estimate the scatter in the assigned halo masses to be of order ${\sigma }_{Q}\sim 0.35$ dex for groups with ${10}^{13}\,{h}^{-1}\,{M}_{\odot }\lt {M}_{200b}\lt {10}^{14.5}\,{h}^{-1}\,{M}_{\odot }$, where ${\sigma }_{Q}$ is the standard deviation of $Q=[\mathrm{log}({M}_{L})-\mathrm{log}({M}_{200b})]/\sqrt{2}$, with M_L the halo mass inferred from the group luminosity. They report that the overall scatter is dominated by intrinsic scatter between halo mass and M_L, while the details of the group finder (interlopers, incompleteness effects) are a relatively small component. Campbell et al. (2015) demonstrate that in general it is possible to extract meaningful physical correlations from Y07 as a function of color, stellar mass, and halo mass despite misidentifications and errors in halo mass (although see also Teklu et al. 2017).

We adopt the Y07 modelC catalog, which uses the SDSS model magnitudes and includes redshifts from SDSS and the 2dF Galaxy Redshift Survey (Colless et al. 2001) and nearest neighbors from the VAGC. We identify central galaxies as the most luminous (in r band) galaxy, as noted in the Y07 imodelC_1 catalog. We adopt the Y07 group halo mass based on the total luminosity ranking of the groups, which is in the modelC_group file in the Y07 catalog. To build our sample, we avoid edges of the catalog by applying ${grp}\_f\_{edge}\gt 0.6$, as recommended by Y07, and $0.02\lt {z}_{\mathrm{group}}\lt 0.15$, with the lower limit reflecting the Y07 limits and the upper limit set to match the MaNGA sample.

Because massive halos are rare, the default Primary and Secondary MaNGA catalogs do not contain many central galaxies in high-mass halos (Figure 1). To address this lack, in our ancillary program we select central galaxies in halos more massive than ${M}_{200b}={10}^{13.75}\,{h}^{-1}\,{M}_{\odot }$, dividing our sample into four halo mass bins. We construct the ancillary sample such that in each halo bin we add sufficient galaxies to enable stacked stellar population gradient measurements in each halo bin for two bins in stellar velocity dispersion. In practice, we insist on a stacked signal-to-noise ratio (S/N) of 50 at a radius of $1.25\mbox{--}1.75\,{R}_{e}$. In total, the ancillary program aims to observe ∼50 additional central galaxies to add to the Primary and Secondary MaNGA observations, of which seven are included in this paper.

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We now turn to the properties of the entire sample of galaxies considered here. We are working with the MaNGA Data Reduction Pipeline version 2.0.1 sample (MaNGA Product Launch 5; MPL5; K. Westfall et al. 2017, in preparation). An initial set of central galaxies are selected from the Y07 catalog and are defined as the most luminous galaxies among the group members. Two coauthors performed visual inspection in the r band of all of the Y07 groups with ${M}_{200b}\gt {10}^{14}\,{h}^{-1}\,{M}_{\odot }$, the halo mass range that we focus on for the ancillary sample. In performing these visual checks, we both ensure that the chosen overdensity exists and check the validity of the choice of central galaxy. Based on the visual inspection, we judge that the Y07 algorithm overall selected visually reasonable clusters and central galaxy candidates. We apply a stellar mass cut ${M}_{* }\gt {10}^{10}\,{h}^{-2}\,{M}_{\odot }$ (in practice, 95% of central galaxies have stellar masses ${M}_{* }\gt {10}^{10.5}\,{h}^{-2}\,{M}_{\odot }$) and a halo mass cut ${M}_{200b}\gt {10}^{12.5}\,{h}^{-1}\,{M}_{\odot }$. As shown in Yang et al. (2009), the groups are quite incomplete below this halo mass in the redshift range of interest. As a result, the majority of our central galaxies have stellar masses ${M}_{* }\gt {10}^{10.5}\,{h}^{-2}\,{M}_{\odot }$. Satellite galaxies are those in the Y07 catalog that are not central galaxies. Of course, the satellite galaxies extend to much lower stellar masses. However, our primary goal here is to compare the satellite and central galaxy populations. Furthermore, at low stellar mass, the early-type galaxies in MaNGA are overwhelmingly unresolved (see also Appendix B). Therefore, we apply a stellar mass cut to the satellite galaxies of ${M}_{* }\gt {10}^{10}\,{h}^{-2}\,{M}_{\odot }$.

We adopt measured properties (e.g., stellar mass, galaxy radius, redshift) from the MaNGA source catalog, which in turn is based on version v1_0_1 of the NSA. The galaxy magnitudes are based on elliptical Petrosian apertures, measured as Petrosian magnitudes (Petrosian 1976; Blanton et al. 2001) but using elliptical apertures.²² The stellar masses are derived using the k-correct code (Blanton et al. 2003), which fits spectral energy distributions to the elliptical Petrosian magnitudes to derive the mass-to-light ratio. A Chabrier (2003) IMF is assumed.

Typically ${\lambda }_{R}$ is measured in a fixed aperture (often R_e), so that all galaxies can roughly be on the same footing. There are two complications to this approach for our sample. One is that the galaxies are typically poorly resolved spatially at R_e, which compromises the ${\lambda }_{R}$ measurements (see Appendix B). The other issue is that it is notoriously difficult to measure a uniform effective radius (e.g., Kormendy et al. 2009). In the end, we therefore adopt a measurement of ${\lambda }_{R}$ that is not directly tied to R_e. Nevertheless, we must adopt some measure of size.

There are two sizes tabulated by the NSA that we consider here. The one used by the MaNGA team to define the sample is the elliptical Petrosian radius. The other possibility within the NSA is the R_e provided by a single-Sérsic fit. We adopt the aspect ratio (B/A) and position angle (PA) derived from the parametric Sérsic fit because they are point-spread function (PSF) corrected. Indeed, when we compare B/A derived from the Petrosian and Sérsic fits, we find clear evidence that the PSF correction makes a difference, since the Sérsic B/A is typically 10% smaller than the Petrosian value. To be consistent between the size measurement R_e and the ellipticity and PA measurements, we adopt the circularized R_e from the Sérsic fit. All R_e measurements use the Sérsic fits. This measurement is roughly 30% larger than the elliptical Petrosian measurements used to define the MaNGA targets. Again, we emphasize that this decision does not impact the final ${\lambda }_{R}$ measurements, but the circularized Sérsic-derived R_e is used as a benchmark throughout the paper.

2.3.1. Galaxy Morphologies

Many of the central galaxies are late-type (spiral) galaxies. Late-type galaxies tend to have high ${\lambda }_{R}$ values, and our main goal is to investigate the distribution in ${\lambda }_{R}$ for the early-type galaxies. We have visually classified galaxies into those with or without spiral structure (early- and late-type galaxies). Visual inspection was performed by the first author using the three-color SDSS images. Of the 475 central galaxies, there are 379 early-type central galaxies; of the 241 satellite galaxies, 159 are of early type. Of the 379 central galaxies, there are 217 Primary and 162 Secondary galaxies, while the 241 satellite galaxies comprise 90 Primary and 69 Secondary galaxies (Table 1). There are 15 central and 13 satellite galaxies that we classify as ambiguous, most of which are edge-on galaxies that may be S0 or later spiral types. We exclude these ambiguous cases, although the numbers are too small to impact our conclusions. Throughout the paper we focus on the sample of early-type galaxies unless explicitly noted otherwise.

Table 1.  Final Sample Composition

Sample	Morphology	MaNGA Selection	${N}_{\mathrm{tot}}$	${N}_{\mathrm{final}}$	$\langle {R}_{e}\rangle$
(1)	(2)	(3)	(4)	(5)	(6)
Central	Early	Primary	217	212	8.8
Central	Late	Primary	40	40	9.9
Central	Early	Secondary	162	145	4.7
Central	Late	Secondary	56	55	5.6
Central	Early	Ancillary	7	5	15.9
Satellite	Early	Primary	90	90	5.9
Satellite	Late	Primary	59	59	8.1
Satellite	Early	Secondary	69	69	3.2
Satellite	Late	Secondary	23	23	3.9

Note. Column (1): environment type. Column (2): morphological type. Column (3): MaNGA selection. Column (4): total number in sample. Column (5): number with $\gt 50 \%$ good spaxels and robust ${\lambda }_{R}$ measurements. Column (6): median R_e (circularized Sérsic fit) in arcsec.

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Galaxy classification grows harder at higher redshifts. Bamford et al. (2009) have shown that above $z\approx 0.08$ the fraction of galaxies classified as elliptical rises unphysically as detail is lost in imaging. We could introduce a redshift-dependent fraction of fast-rotating spiral galaxies into our early-type sample if this effect is at play. The morphological bias would cause us to measure a higher early-type fraction at $z\gt 0.08$ relative to the lower-redshift bin, caused entirely by spirals appearing as early-type galaxies. To search for this effect, we take all the galaxies with stellar masses ${10}^{10.75}\lt {M}_{* }/{M}_{\odot }\lt {10}^{11}\,{h}^{-2}$ and examine the early-type fraction with redshifts above and below z = 0.08. There are 69 (63) objects in the low (high) redshift bin. We find consistent early-type fractions of 0.85 ± 0.12 and 0.80 ± 0.13 in the two redshift bins, suggesting that this morphology bias is not impacting our results.

Of the central, early-type galaxies, seven belong to our ancillary program. Our sample contains 30 central galaxies in halos more massive than ${M}_{200b}\gt {10}^{14}\,{h}^{-1}\,{M}_{\odot }$ and six central galaxies in halos more massive than ${M}_{200b}\gt {10}^{14.5}\,{h}^{-1}\,{M}_{\odot }$. Many central galaxies are known to have a large extended halo, sometimes known as cD galaxies (e.g., Morgan & Lesh 1965; Schombert 1984; Zhao et al. 2015a); as the MaNGA sample grows, it will become possible to examine trends between cD halo and ${\lambda }_{R}$.

We use the kinematic measurements provided by the MaNGA Data Analysis Pipeline (DAP; K. Westfall et al. 2017, in preparation). The individual spaxels are combined using Voronoi binning (Cappellari & Copin 2003) to maintain an S/N of at least 10 per spectral pixel of 70 $\mathrm{km}\,{{\rm{s}}}^{-1}$. The number of combined spectra at $\sim {R}_{e}$ varies considerably from object to object depending on R_e, with a mean value of 12 spectra, a median value of 3 spectra, and a maximum of 200. Spaxels with individual S/N $\lt 1$ are excluded from the binning. In calculating ${\lambda }_{R}$, we use the distance to the center of each spaxel.

The kinematics are measured using the penalized pixel-fitting code pPXF (Cappellari & Emsellem 2004; Cappellari 2017), with emission lines masked. Stellar templates from the MILES library (Sánchez-Blázquez et al. 2006), which cover the spectral range 3525–7500 Å at 2.5 Å (FWHM) spectral resolution (Falcón-Barroso et al. 2011), are convolved with a Gaussian line-of-sight velocity distribution to derive the velocity and velocity dispersion of the stars (values are not corrected for instrumental resolution at this stage). An eighth-order additive polynomial is included to account for flux calibration and stellar population mismatch.

When we extract the σ measurements from the MaNGA catalog, we correct them for instrumental resolution using the measured ${\sigma }_{r}$, which is the unweighted average of the difference in resolution between the templates and the data as measured over all wavelengths and all spectra in the cube (K. Westfall et al. 2017, in preparation). The MILES templates are used at their native resolution of 2.5 Å, which is higher than the MaNGA data over the full spectral range for all cubes (Yan et al. 2016a). With this approach, the pipeline is able to reliably recover intrinsic dispersions of ∼35 $\mathrm{km}\,{{\rm{s}}}^{-1}$ or more. The MaNGA team also tried to employ the wavelength-dependent kernel convolution available within pPXF but could not recover such low velocity dispersions with that functionality enabled. As shown by Penny et al. (2016), even for intrinsic dispersions of 40 $\mathrm{km}\,{{\rm{s}}}^{-1}$, it is possible to recover the dispersion to within 10% for high-S/N spectra, while at our limiting ${\rm{S}}/{\rm{N}}=10$, it is possible to recover the dispersions above the nominal limit of 70 $\mathrm{km}\,{{\rm{s}}}^{-1}$ to within ∼20%. The DAP then supplies a single resolution correction that is the effective difference between the MILES templates and the MaNGA spectra calculated for each cube. The resolution as a function of wavelength is calculated from arcs taken before and after each exposure and then corrected using strong sky lines (Law et al. 2016). Tests indicate that these corrections are better than 5% for dispersions of 70 $\mathrm{km}\,{{\rm{s}}}^{-1}$ or higher (K. Westfall et al. 2017, in preparation). In cases in which σ is below the spectral resolution of the instrument (a condition that does occur in the outer parts of some galaxies), we mask these values. Example maps for interesting subsets of the population are presented in Appendix A.

While in general these kinematic measurements are robust, there are some known systematic failures. There are foreground stars, which have been successfully masked by MaNGA. At lower S/Ns, particularly in the outer regions of the galaxies, the velocity dispersion measurements can peg at the unphysical value of 1000 km s⁻¹. Finally, the superposition of two companion galaxies in the IFU field of view can lead to unphysical velocity and velocity dispersion measurements if the two components are not jointly modeled. The MPL5 catalog has identified these unphysical values; we adopt their “DONOTUSE” flags, as well as flagging all $\sigma \gt 500$ $\mathrm{km}\,{{\rm{s}}}^{-1}$ and $V\gt 400$ $\mathrm{km}\,{{\rm{s}}}^{-1}$ values. We only analyze objects for which at least 50% of the fibers within R_e are not flagged. Visual inspection verifies that we remove most of the clear merger cases through these cuts, and no galaxies are removed by visual inspection. Excluding these problematic cases does not lead to any systematic bias in redshift or stellar mass distributions of the galaxies.

We perform two checks on our measurements. We explore a different analysis of the MPL4 data cubes performed by J. Ge (Zheng et al. 2017). The kinematics are also measured with pPXF, but the binning and masking prescriptions are different. The ${\lambda }_{{R}_{e}}$ measurements agree well in general, with $\langle {\lambda }_{\mathrm{MPL}4}-{\lambda }_{\mathrm{MPL}5}\rangle =0.02\pm 0.09$. This provides confidence that the measurements are robust to detailed choices about masking and binning. We also compare the central σ measurements with the original SDSS measurements that were made on different spectra using a different fitting technique. We find decent agreement, with $\langle ({\sigma }_{\mathrm{MaNGA}}-{\sigma }_{\mathrm{SDSS}})/{\sigma }_{\mathrm{SDSS}}\rangle =-0.05\pm 0.1$.

3.1.1. Radial Coverage

Keeping only galaxies with at least 50% of the fibers unmasked leaves 357 central galaxies. We have verified that no bias in stellar or halo mass is incurred when we remove the galaxies with problematic measurements. Among these 357 central galaxies, 272 have coverage at or beyond R_e, 136 have coverage at or beyond $1.5\,{R}_{e}$, and 73 have coverage at or beyond $2\,{R}_{e}$ (Figure 2; Table 1). The median effective radius of the Primary sample is 52, while the Secondary sample has a slightly smaller median of R_e = 47.

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Turning to the satellites, there are 159 early-type galaxies with $\gt 50 \%$ of their fibers unmasked. Of these, 137 reach R_e and are spatially resolved by that point, 94 reach $1.5\,{R}_{e}$, and 34 galaxies reach $2\,{R}_{e}$. Only 50 of the early-type satellites have ${R}_{e}\gt 4$″ and radial coverage out to $1.5\,{R}_{e}$. In Section 3.2.1 we will return to the issue of spatial resolution and angular momentum measurements.

The V and σ measurements on the binned data are used to calculate ${\lambda }_{R}$ (Equation (1)) in elliptical apertures, as defined from the single-component Sérsic (1963) fit from the NSA. We adopt the ellipticity $[\epsilon =1-({\rm{B}}/{\rm{A}})]$ and position angle from this fit because the model is corrected for seeing and thus should be the most robust measurement available of these parameters, roughly measured at the effective radius (see also Section 2.3). The R_e value is the circularized half-light radius from the Sérsic fit, and in the following figures and calculations we use the radial coordinate ${R}_{\mathrm{Area}}$, which is calculated as the radius of the circle that would have the equivalent area as the enclosed spaxels (${R}_{\mathrm{Area}}=\sqrt{{R}_{e,a}\times {R}_{e,b}}=\sqrt{{N}_{\mathrm{pixels}}\times {A}_{\mathrm{pixel}}/\pi }$), where ${A}_{\mathrm{pixel}}$ is the area of a pixel (e.g., Cappellari 2013). Examples of ${\lambda }_{R}$ are shown in Figure 3. In internal comparisons between different MaNGA teams, there is good agreement between ${\lambda }_{{R}_{e}}$ values calculated with different prescriptions (M. Graham et al. 2017, in preparation).

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3.2.1. Spatial Resolution Constraints

3.2.2.  ${\lambda }_{\mathrm{out}}$ versus

The distribution of ${\lambda }_{\mathrm{out}}$ as a function of is presented in Figure 4. In addition to the galaxies, we show an analytic model of an edge-on galaxy in which anisotropy is proportional to ellipticity (e.g., Cappellari et al. 2007). The majority of the fast rotators lie above this magenta line as expected from prior work (Emsellem et al. 2011). Galaxies scatter above the magenta line because of inclination and internal variations in ellipticity. The ${\mathrm{ATLAS}}^{3{\rm{D}}}$ survey also defines an empirical division between “slow” and “fast” rotators. We will adopt a similar prescription to separate the two (black line), but in Emsellem et al. (2011) the division between slow and fast rotator is defined at $r={R}_{e}$. An empirical value of ${\lambda }_{{R}_{e}}/\sqrt{\epsilon }=0.31$ best divides the populations. However, we are not adopting R_e as our aperture. We determine the revised value of ${\lambda }_{{R}_{e}}/\sqrt{\epsilon }$ empirically using our data. By comparing ${\lambda }_{\mathrm{out}}$ with ${\lambda }_{{R}_{e}}$, we find that the former is 14% larger than the latter. Therefore, we scale the division between slow and fast rotators to ${\lambda }_{\mathrm{out}}/\sqrt{\epsilon }=0.35$. A number of different divisions into fast and slow rotator have been proposed in the literature (see also Lauer 2012; Cappellari 2016). If we were to adopt the Cappellari (2016) definition instead, $\lt 10 \%$ of galaxies would change designation.

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We draw attention to the lower right region of the figure, galaxies that apparently have relatively high $\epsilon \gt 0.4$ and low ${\lambda }_{\mathrm{out}}\lt 0.2$. Emsellem et al. (2011) do not have galaxies that populate this region (see also Cappellari 2016). Some of these outliers appear round visually, suggesting that they have poorly measured values. A related issue is that we utilize an effective value, but in general these massive galaxies grow more flattened at larger radius (e.g., Huang et al. 2012; Oh et al. 2017), which may contribute to the scatter. However, the majority of these galaxies are quite elongated and display low levels of rotation along their major axis, with a high central dispersion (see example maps in Appendix A). Objects in this region may be galaxies with high angular momentum that masquerade as slow rotators. In particular, galaxies known as “double-σ” galaxies have two well-separated peaks in their σ distributions and have been shown to have counterrotating disks (see Krajnović et al. 2011, and references therein).

We visually examine the low-${\lambda }_{\mathrm{out}}$, high- outliers and show a possible candidate for a double-σ galaxy in Appendix A, although our ability to distinguish such features is limited by spatial resolution (see also a first sample of counterrotating gas disks picked out from MaNGA; Jin et al. 2016). Otherwise, there are no obvious differences between these galaxies and those with low ${\lambda }_{\mathrm{out}}$ but correspondingly low . We conclude that double-σ galaxies are unlikely to dominate this outlier population, and since they constitute such a small fraction of the sample, we do not remove them from consideration in what follows.

3.2.3. Notes on the Ancillary Galaxies

With ${\lambda }_{\mathrm{out}}$ and slow/fast-rotator determinations in hand, we investigate trends in the standard slow-rotator fraction using the Emsellem et al. definition (Figure 5; Table 2). Our stellar mass coverage extends only to ${M}_{* }\gt {10}^{10.5}\,{h}^{-2}\,{M}_{\odot }$ for the central galaxies owing to the halo mass cut, while our satellite sample extends a bit lower and is limited by spatial resolution. Also, this figure includes only the early-type galaxies, but there are likely to be some biases in these by-eye determinations (Bamford et al. 2009). As a sanity check, we recalculate the slow-rotator fraction with spiral galaxies included. The slow-rotator fraction changes by $\sim 50 \%$ at the lowest stellar masses but is unaffected for ${M}_{* }\gt {10}^{11}\,{h}^{-2}\,{M}_{\odot }$. Our conclusions are unchanged if we include the spiral galaxies in the sample.

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Consistent with previous work, we see a steep increase in slow-rotator fraction with stellar mass. At ${M}_{* }\lt 3\times {10}^{10}\,{h}^{-2}\,{M}_{\odot }$, galaxies are overwhelmingly fast rotators, with only a 10%–15% slow-rotator fraction. Galaxies with stellar mass ${M}_{* }\,\gt {10}^{11}\,{h}^{-2}\,{M}_{\odot }$ are mostly slowly rotating, and this mass scale is consistent with the quoted transition mass from Cappellari (2013) once our ${h}^{-2}$ and IMF scales are matched.

Our results are in good agreement with the literature. After correcting the ${\mathrm{ATLAS}}^{3{\rm{D}}}$ masses to be in ${h}^{-2}$ units and shifting them to match our assumed Chabrier IMF from Salpeter (assuming a 0.2 dex decrease in mass; e.g., Conroy 2013), there is good agreement (Emsellem et al. 2011). At the highest masses, our ancillary sample results agree well with the results from both Veale et al. (2017b) and Oliva-Altamirano et al. (2017), as well as a number of studies of individual massive clusters (DÉugenio et al. 2013; Fogarty et al. 2014; Scott et al. 2014). Our results are also in qualitative agreement with the study by Pasquali et al. (2007) based on the mass and luminosity dependence of the disky versus boxy fraction of early-type galaxies, if one associates disky (boxy) galaxies with fast (slow) rotators, as suggested by the seminal work of Bender et al. (1989) and Bender et al. (1989).

M_* clearly correlates strongly with ${\lambda }_{R}$. We now address whether there is an additional dependence on large-scale environment. One approach to evaluating the impact of environment is to compare central and satellite galaxies at fixed stellar mass.

Figure 5 displays a trend whereby satellite galaxies have a slightly higher slow-rotator fraction than the central galaxies at fixed stellar mass. This behavior is mirrored in Figure 6 (top left; Table 3), where there is a difference in the median ${\lambda }_{\mathrm{out}}/\sqrt{\epsilon }$ and also possibly a larger scatter toward high ${\lambda }_{\mathrm{out}}$ in the central relative to the satellite population. One possible driver of this difference could be spatial resolution. However, because of the design of the MaNGA survey, within the mass range of ${M}_{* }={10}^{10.5}\mbox{--}{10}^{11}\,{h}^{-2}\,{M}_{\odot }$, the central and satellite galaxy samples have similar median apparent sizes of 5″ and comparable median redshifts of $\langle z\rangle =0.065$, suggesting that spatial resolution is not obviously to blame for the systematic difference between the two populations.

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Table 3.  ${\lambda }_{\mathrm{out}}/\sqrt{\epsilon }$ vs. M_*

M*	${\lambda }_{\mathrm{out}}/\sqrt{\epsilon }$	Error
(1)	(2)	(3)
	Central
10.58	0.70	0.40
10.73	0.55	0.37
10.86	0.48	0.30
10.95	0.37	0.26
11.03	0.45	0.27
11.09	0.39	0.31
11.23	0.38	0.33
	Satellite
10.07	0.75	0.36
10.20	0.70	0.41
10.33	0.78	0.18
10.45	0.80	0.27
10.59	0.62	0.36
10.74	0.45	0.32
10.97	0.44	0.30

Note. Column (1): median stellar mass in each bin. Column (2): mean ${\lambda }_{\mathrm{out}}$. Column (3): error in slow-rotator fraction.

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Given that the central sample is much larger than the satellite sample, and given that the detailed mass distributions do not match between the two samples, we perform an additional test to compare the satellite and central galaxies. We focus on the mass range where the two populations overlap: ${M}_{* }\,={10}^{10.5}\mbox{--}{10}^{11}\,{h}^{-2}\,{M}_{\odot }$ (203 central and 54 satellite galaxies). The observed mass distribution of the satellite sample is sharply falling toward the higher-mass end of this bin, while the reverse is true for the central galaxy mass distribution. We thus assign weights to the central galaxies to force the mass distribution between the two populations to match. We then build the weighted distribution in ${\lambda }_{\mathrm{out}}/\sqrt{\epsilon }$ for both the central and satellite galaxies (Figure 7) and compare the two distributions using an Anderson–Darling test (e.g., Babu & Feigelson 2006). There is a probability of $P=33 \%$ that the two samples are drawn from the same underlying distribution. There is no compelling evidence for a difference between the central and satellite galaxies in their distributions of ${\lambda }_{\mathrm{out}}$. However, we caution that some subtle differences may still exist owing to the different radial coverage between the central and satellite samples seen in Figure 2. Furthermore, central galaxy samples are not pure, with the level of contamination depending on halo mass (Skibba et al. 2011). In Section 4.5 we revisit the impact of contamination in the group catalog using the mock catalogs from Campbell et al. (2015).

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We conclude that the central and satellite galaxies have statistically consistent distributions in ${\lambda }_{\mathrm{out}}/\sqrt{\epsilon }$ when their mass distributions are carefully matched. We do not detect any significant difference between the central and satellite galaxies in their slow-rotator fraction.

As an additional probe of the large-scale environment, we attempt to disentangle the stellar from the halo mass dependence (Figure 6, top right panel). We treat the central and satellite populations separately. We divide each into two groups based on their host halo mass and examine the weighted mean ${\lambda }_{\mathrm{out}}/\sqrt{\epsilon }$ of that subpopulation. We divide the central galaxies with a halo mass above and below ${10}^{13.3}\,{h}^{-2}\,{M}_{\odot }$, which is the median halo mass. There are 172 (185) central galaxies in the higher-mass (lower-mass) halo bin. The satellites are divided at their median halo mass of ${10}^{13.8}\,{h}^{-2}\,{M}_{\odot }$. There are 83 (76) satellites in the higher-mass (lower-mass) bin.

We reweight the distributions such that the stellar mass distributions match, over the mass range of overlap (${M}_{* }={10}^{10.8}\mbox{--}{10}^{11.1}\,{h}^{-2}\,{M}_{\odot }$) for the ∼100 galaxies in each mass-limited sample. Although this comparison is limited to a narrow range in stellar mass, the distributions of central and satellite $\lambda /\sqrt{\epsilon }$ are consistent with each other, with an Anderson–Darling test returning a $P=30 \%$ chance that the two samples were drawn from the same distribution. Similar results are found for the satellite galaxies. This group is divided at a higher halo mass of ${M}_{200b}={10}^{13.8}\,{h}^{-1}\,{M}_{\odot }$. In a mass range of ${M}_{* }={10}^{10.0}\mbox{--}{10}^{10.7}\,{h}^{-2}\,{M}_{\odot }$ (∼60 galaxies in each bin) we see that satellites in lower-mass halos tend to have a higher ${\lambda }_{\mathrm{out}}/\sqrt{\epsilon }$. After forcing the mass distributions to match, we find only a marginally significant difference between the high and low halo masses, with an Anderson–Darling test returning a probability $P=4 \%$ that the two samples are drawn from the same distribution. A larger sample is needed to investigate whether there is a real difference in the satellite population as a function of halo mass, but in the central galaxies our finding of no halo mass dependence is consistent with prior work (Brough et al. 2017; Veale et al. 2017a). Finally, we check for a correlation between ${\lambda }_{\mathrm{out}}/\sqrt{\epsilon }$ and the magnitude difference between the central galaxy and the next brightest galaxy and find no correlation.

In a companion paper (Greene et al. 2017), we further compare the fast- and slow-rotator fractions as a function of local overdensity to compare with the recent literature (e.g., Cappellari 2016; Brough et al. 2017; Veale et al. 2017a).

While the majority of galaxies at high stellar mass are slowly rotating, there is a tail of fast-rotating galaxies even in the highest stellar mass bin (see also Jimmy et al. 2013). Maps of massive fast and slow rotators are shown in Appendix A. Nearby small companions add some contamination to this class of objects (at the ∼15% level), but the majority are single objects with real rotation. We now investigate whether properties of the fastest- and slowest-rotating galaxies differ in any other interesting ways.

We select the 32 fast-rotating central galaxies with ${\lambda }_{\mathrm{out}}\gt 0.3$ and ${M}_{* }\gt {10}^{11}\,{h}^{-2}\,{M}_{\odot }$ and compare with the 43 galaxies of the same stellar mass that have ${\lambda }_{\mathrm{out}}\lt 0.1$. The two samples have similar median galaxy sizes of $\langle {R}_{\mathrm{petro}}\rangle =16$ and 15 kpc, respectively, and an Anderson–Darling test shows that they have indistinguishable distributions in size. They are at a similar median redshift of $\langle z\rangle =0.1$ and have similar flattening of $\langle \epsilon \rangle =0.3$ and 0.2, respectively, again with distributions consistent with arising from the same distribution. The galaxies have similar median color, $\langle g-r\rangle =0.94$ mag. The velocity dispersions in the fast rotators are lower (mean of 230 and 260 km s⁻¹, respectively; $P=6\times {10}^{-4}$ that the distributions are the same). This difference is not surprising since ${\lambda }_{\mathrm{out}}$ depends inversely on σ.

The one independent difference between the two samples appears to be in their emission-line properties. Focusing on the Hα equivalent width within 3″, and including only systems where the emission line within that radius is measured with $\gt 3\sigma$ significance, there is a measurable difference in the equivalent width distribution (P = 0.003 of belonging to the same distribution), with 47% of the fast rotators having Hα EW $\gt 1$Å, while only 16% of the slow rotators have Hα EW $\gt 1$Å (Figure 8). This difference in emission-line properties suggests that the fast rotators do typically have higher gas content, as we might expect for galaxies with a disk component, perhaps associated with the event that increased their spin (see Section 4.6 below). Unfortunately, the S/Ns of the other strong lines are not high enough to examine the sources of photoionization (e.g., star formation, active nuclei) in these objects without more work. We perform a similar test with the satellite galaxies, but to construct a decent sample, we must shift the mass limit down to log(${M}_{* })=10.5\,{h}^{-2}\,{M}_{\odot }$. We see similar trends, although the difference in emission properties is less significant for the satellites.

In the future, it will be interesting to investigate additional galaxy properties, such as luminosity-weighted mean age, which seems to correlate with the outer ellipticity of halos (Oh et al. 2017), disky or boxy isophotes (e.g., Bender et al. 1989), stellar population gradients (e.g., Greene et al. 2015; Goddard et al. 2017), or even cD envelope fraction, as Zhao et al. (2015b) have argued that the cD envelope is an alternate tracer of merger history.

Group finders cannot perfectly identify dark matter halos. For our purposes, there are two main concerns. There is scatter in the assignment of halo mass to groups, leading to uncertainties in the halo masses. Also, there are errors in the identification of central galaxies (e.g., Skibba et al. 2011; Lin et al. 2016; Lange et al. 2018). To explore how these two errors impact our results, we employ a mock catalog built by Campbell et al. (2015). Campbell et al. use N-body simulations to create a “true” galaxy catalog matched to the SDSS with perfect knowledge of the groups, halo masses, and central galaxies. They then run the Y07 algorithm on their mock galaxy catalog to create a mock Y07 group catalog. This catalog contains the same scatter in halo mass and in central designation as Y07, being constructed in the same manner.

We assign every mock galaxy a value of ${\lambda }_{\mathrm{out}}/\sqrt{\epsilon }$ based on its stellar mass, with a linear dependence designed to roughly reproduce our measurements. We model the ${\lambda }_{\mathrm{out}}/\sqrt{\epsilon }$ distribution as a sum of two Gaussians. The primary Gaussian (containing 80% of the galaxies) has a narrow width of 0.2 and a central value that varies with stellar mass as ${\lambda }_{\mathrm{out}}$/$\sqrt{\epsilon }=-0.7\mathrm{log}{M}_{* }-0.14$. The secondary Gaussian has no mass dependence and a large scatter and is added in an ad hoc way to match the scatter that we see at all masses. The center of the distribution is fixed at ${\lambda }_{\mathrm{out}}/\sqrt{\epsilon }=0.8$ with a dispersion of 0.5, and this component composes 20% of the total. In the fiducial assignment, satellite and central galaxies are treated the same way at fixed stellar mass.

First, we use these catalogs to test our sensitivity to differences between central and satellite populations. We create a suite of simulated satellite galaxies in which ${\lambda }_{\mathrm{out}}/\sqrt{\epsilon }$ is boosted relative to the default values for central galaxies at a given stellar mass by $\delta {\lambda }_{\mathrm{out}}/\sqrt{\epsilon }=0.1,0.2,0.3,0.35$ on average, with a scatter of 0.05. These differences are introduced in the true catalog, and then we ask whether we can recover this difference in the Y07 mock catalog. We then create 100 data sets with statistics matched to our true sample by selecting a mock galaxy with stellar mass within 0.05 dex and halo mass within 0.1 dex of each sample galaxy, for both central and satellite galaxies. We then calculate the dependence of ${\lambda }_{\mathrm{out}}/\sqrt{\epsilon }$ and slow-rotator fraction on M_* as with the real data (adopting the appropriate MaNGA weight for each mock galaxy). We find that the difference between satellite and central becomes measurable when the offset is $\delta {\lambda }_{\mathrm{out}}/\sqrt{\epsilon }=0.2$ and significant when the offset is $\delta {\lambda }_{\mathrm{out}}/\sqrt{\epsilon }=0.35$ (Figure 9). As is apparent from the figure, there is a bias introduced by the mixing between satellite and central galaxies.

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Second, we test whether we could uncover a secondary trend in ${\lambda }_{\mathrm{out}}/\sqrt{\epsilon }$ with M_200b at fixed stellar mass. We scatter the default values in the catalog by an amount that depends on halo mass. Specifically, we perturb the values according to $\delta {\lambda }_{\mathrm{out}}/\sqrt{\epsilon }$ = m (log M_200b - log ${M}_{200{\rm{b}},\mathrm{median}}$), for slope m, with $m=0.1,0.2,0.3$ and a scatter in m of 0.05. In Figure 9 we see that there is minimal bias introduced by scatter in halo masses, and the two halo mass bins become measurably different for a slope of $\gt 0.3$, corresponding to changes in ${\lambda }_{\mathrm{out}}/\sqrt{\epsilon }\sim 0.2$ at ${M}^{* }\approx {10}^{11}$ ${M}_{\odot }$.

We see that the errors in the satellite/central designation suppress the input difference between central and satellite. The amount of suppression ($\lambda /\sqrt{\epsilon }\sim 0.2$) is of the same order as the current uncertainty in the mean value. Thus, simply increasing the total number of objects will not help uncover subtle differences between central and satellite galaxies; rather, we are limited by the systematic errors in the designation of central and satellite galaxies by group finders. While it is possible to uncover trends in halo mass for central galaxies (test 2 above), to distinguish differences in satellite versus central galaxies, the impact of group-finding methods will have to be forward-modeled via mock catalogs.

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Several papers have used simulations to investigate the primary mechanisms that impact ${\lambda }_{R}$ and galaxy flattening () in galaxies. We compare our results with their predictions here.

Several studies (e.g., Khochfar & Burkert 2005; Naab et al. 2006; Kang et al. 2007) have argued that the dichotomy of early-type galaxies can be explained in a scenario whereby boxy, slowly rotating ellipticals have their origin in a merger that is both major (i.e., mass ratio of progenitors close to unity) and dry (i.e., progenitors have small gas mass fractions). In particular, Kang et al. (2007) conclude that the observed stellar mass dependence of the boxy fraction requires that slow rotators result from mergers with a progenitor mass ratio <2 and with a combined cold gas mass fraction <0.1. Lagos et al. (2017) also find that major merging is a primary driver of angular momentum evolution. Interestingly, Choi & Yi (2017) find that the cluster galaxies in their simulations with no major merging are the ones with the most rapid decrease in ${\lambda }_{R}$, but they are not certain what physical process drives this decline.

Penoyre et al. (2017) examine the distribution of ${\lambda }_{R}$ with stellar mass using the Illustris simulation (Vogelsberger et al. 2014). They report that major mergers and gas accretion have the strongest impact on ${\lambda }_{R}$, with the former typically spinning down and the latter spinning up galaxies, although they do not yet consider black hole feedback or large-scale intrinsic alignments of galaxies as possible factors (see Martizzi et al. 2014, for the possible importance of black hole feedback). They find that lower-mass galaxies can be spun up by accretion of gas, while at higher mass the accretion rates are not high enough to change ${\lambda }_{R}$. They suggest that high-mass galaxies above ${M}_{* }\approx {10}^{11}\,{M}_{\odot }$ have uniformly low ${\lambda }_{R}$ because at later times mergers and accretion are unable to significantly change their angular momentum content. In contrast, lower-mass galaxies can be spun up at late times by accretion and star formation. Penoyre et al. (2017) also find that faster-rotating galaxies are more gas-rich (and more metal enhanced), and they do not find a discernible difference between satellite and central galaxies at fixed mass.

Naab et al. (2014) use cosmological simulations of 44 massive galaxies to examine the relationship between merger history and angular momentum content. In addition to ${\lambda }_{R}$, they compare their simulations with observations of the flattening of the merged remnant () and higher-order moments of the line-of-sight velocity distribution parameterized with Gauss–Hermite polynomials; h₃ is the asymmetry parameter (e.g., van der Marel & Franx 1993). They identify three pathways that form fast rotators: galaxies with little merging but some gas (e.g., a faded disk galaxy), late gas-rich merging that spins up the remnant (these have anticorrelated h₃ and $V/\sigma$), and late dissipationless merging that spins up the remnant (no anticorrelation between h₃ and $V/\sigma ;$ Naab & Burkert 2001; Naab et al. 2006). Van de Sande et al. (2017) have identified SAMI galaxies without an h₃-$V/\sigma$ anticorrelation that may indeed be the remnants of late dissipationless merging. There are also three main pathways to make slow rotators in their simulations: galaxies that form early and only experience minor merging since, galaxies with a gas-rich major merger that spins down the remnant, and galaxies with a gas-poor major merger that spins down the remnant. Apparently these last two events produce highly flattened configurations and may correspond to our small tail of high-${\lambda }_{\mathrm{out}}$, high- galaxies.

Voronoi binning is done to keep the S/N above a baseline level of 10. However, some of the outermost bins have lower S/N in practice. They also can cover large regions of the galaxy. It is natural to ask whether the ${\lambda }_{R}$ measurements are sensitive either to degraded S/N in the outermost parts of the galaxy or to irregular bin shapes. We briefly address each concern here.

First, we ask whether including noisy measurements in the outer bins has a strong impact on the value of ${\lambda }_{\mathrm{out}}$ that we infer. We perform a simple test. We simply perturb each velocity measurement with a random Gaussian variate using twice the reported error in the measurement and then recompute the perturbed ${{\lambda }_{\mathrm{out}}}_{,p}$. The distribution in $\delta {\lambda }_{\mathrm{out}}={\lambda }_{\mathrm{out}}-{{\lambda }_{\mathrm{out}}}_{,p}$ is strongly peaked at zero, with a negative tail. That is, adding noise tends to increase ${\lambda }_{\mathrm{out}}$ slightly, with a median $\delta {\lambda }_{\mathrm{out}}=-0.001$, with 30% of objects having $\delta {\lambda }_{\mathrm{out}}\lt -0.05$ and 5% of objects having $\delta {\lambda }_{\mathrm{out}}\lt -0.1$. We do the same test with σ rather than V. This change has even less effect, presumably because σ is an even quantity, and so the average σ is the same everywhere at a given radius, unlike V. The distribution in $\delta {\lambda }_{\mathrm{out}}$ is peaked at zero, with an FWHM of 0.01. Only 8% of systems have $| \delta {\lambda }_{\mathrm{out}}| \gt 0.02$, and only one object has $| \delta {\lambda }_{\mathrm{out}}| \gt 0.04$. Finally, we repeat the same test but introduce scatter in both V and σ at the same time. In this case the systematic bias is slightly larger, with a median $\delta {\lambda }_{\mathrm{out}}=-0.02\pm 0.01$ (Figure 13). Only 6% of systems have $| \delta {\lambda }_{\mathrm{out}}| \gt 0.04$ and 1% of systems have $| \delta {\lambda }_{\mathrm{out}}| \gt 0.05$.

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The punchline is that scatter in the outer measurements does not influence ${\lambda }_{\mathrm{out}}$ strongly. While we experimented with excluding measurements based on large uncertainties, this test suggests that it is worth keeping all the measurements, even those that are noisy. If we exclude lower-S/N measurements, we simply restrict the radial range of the data unnecessarily.

Second, we perform the following test to see how the large and irregular bin sizes impact ${\lambda }_{\mathrm{out}}$. We run our same ${\lambda }_{R}$ measuring apparatus on the unbinned spaxels as we do on the Voronoi-binned data. Since the unbinned data can get very noisy in the outer parts, we restrict attention in this test to ${\lambda }_{{R}_{e}}$. Taking only galaxies that are binned (with at least five or more spaxels per bin at R_e), we again find that the majority of objects have $\delta {\lambda }_{{R}_{e}}=0$, and again we find a bias toward larger ${\lambda }_{{R}_{e}}$ in the binned versus unbinned data. Again, the bias is quite small, with only 11% of galaxies having $| \delta {\lambda }_{{R}_{e}}| \gt 0.01$ and only 5% having $| \delta {\lambda }_{{R}_{e}}| \gt 0.02$.

Because of the design of the MaNGA survey, most galaxies are moderately resolved, with typical angular radii (in our early-type, central galaxy sample) of 5″. We cannot afford to exclude galaxies that are only marginally resolved at R_e (Figure 2). In this appendix, we explore creative ways to use the available information to derive robust ${\lambda }_{R}$ measurements that can be compared across the entire sample.

To understand how the resolution impacts our measurements, we define a subset of 50 galaxies that have sizes ${R}_{e}\gt 5\buildrel{\prime\prime}\over{.} 5$ (more than four beams across R_e, for a median seeing FWHM of 25) and radial coverage to at least $1.5\,{R}_{e}$. While some previous work (e.g., van de Sande et al. 2017) has performed simulations starting with the ${\mathrm{ATLAS}}^{3{\rm{D}}}$ sample, which has very high sampling and spatial resolution, working directly with the most spatially extended MaNGA cubes provides two key advantages. First, because of the different-sized MaNGA bundles, we are able to find galaxies that both are well resolved and have coverage beyond $1.5\,{R}_{e}$. Very few ${\mathrm{ATLAS}}^{3{\rm{D}}}$ cubes extend to such radii, but this is the regime where the MaNGA data are best. Second, because of the small volume of ${\mathrm{ATLAS}}^{3{\rm{D}}}$, the sample is heavily weighted toward low-mass galaxies. Our well-resolved sample spans the same range in stellar mass as our parent sample (Figure 14).

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Taking this sample, we degrade the spatial resolution such that the final $\mathrm{FWHM}={R}_{e}/3.6$ (which is typical for the sample as a whole). In degrading the data, we assume a Gaussian PSF and that the line-of-sight velocity distribution (LOSVD) for each fiber is a Gaussian, simply adopting the measured velocity and velocity dispersion as measured by pPXF for each fiber. A new LOSVD is constructed at each position by weighting each spectrum with the new PSF and combining the weighted spectra, while the new velocity and velocity dispersion are derived from a fit to the new LOSVD. We then recompute λ(R) and compare the values of ${\lambda }_{{{R}}_{e}}$ between the input and smoothed cube.

In general, ${\lambda }_{{{\rm{R}}}_{e}}$ is sensitive to the resolution, and the situation is most severe for systems with ${\lambda }_{{R}_{e}}\lt 0.2$. For galaxies with input ${\lambda }_{{R}_{e}}$ between 0 and 0.2, ${\lambda }_{{R}_{e}}$ values from the simulations are typically lower by 45% compared to the true values (assuming the typical MaNGA galaxy ratio of seeing to R_e of ∼2.5). Galaxies with ${\lambda }_{{R}_{e}}\gt 0.2$ are impacted at the 15% level on average (Figure 15, left panel). If we measure λ at $1.5\,{R}_{e}$ instead, the bias decreases. Galaxies with intrinsic ${\lambda }_{{R}_{e}}\lt 0.2$ decline by only 35%, while galaxies with higher ${\lambda }_{{R}_{e}}$ are biased low by 10% (with an absolute offset of −0.02 in ${\lambda }_{R}$). Using λ at $1.5\,{R}_{e}$, however, excludes 5% of the galaxies whose rotation curves do not quite reach this radius. Thus, we introduce a final measurement, the mean λ as measured in the outermost 10% of the ${\lambda }_{R}$ curve, ${\lambda }_{\mathrm{out}}$. The quantity ${\lambda }_{\mathrm{out}}$ matches λ($1.5\,{R}_{e}$) with no bias and a scatter of 20%. To maximize the sample size, we compare objects using ${\lambda }_{\mathrm{out}}$, but we repeat all of our analysis for λ at $1.5\,{R}_{e}$, and none of our conclusions change.

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We apply one more correction to mitigate the final 10% bias. For obvious reasons, within the central beam (∼2 arcsec) all the rotation is measured as dispersion, naturally lowering λ everywhere. If we excise this inner region, then both ${\lambda }_{\mathrm{out}}$ and λ at $1.5\,{R}_{e}$ increase by $\sim 10 \%$. We demonstrate this in two ways. First, we take our smoothed models, remove the central spaxels that are within the smoothed FWHM, and remeasure λ($1.5\,{R}_{e}$) (Figure 16). We find that λ($1.5\,{R}_{e}$) increases on average such that ${\rm{\Delta }}\lambda =({\lambda }_{1.5{R}_{e},m}-{\lambda }_{1.5{R}_{e}}$)/${\lambda }_{1.5{R}_{e}}=0.04$. There is quite a bit of scatter introduced in these models because of their relatively limited radial coverage. As a second test, we go back to the full sample and experiment with removing the central 2″, in Figure 17. Interestingly, the slowest rotators have their ${\lambda }_{\mathrm{out}}$ values drop slightly when the center is excluded (by $\sim 0.01\mbox{--}0.02$, which is in the noise of our ${\lambda }_{\mathrm{out}}$ measurements), but at ${\lambda }_{\mathrm{out}}\gt 0.2$, ${\lambda }_{\mathrm{out}}$ systematically increases by $\sim 10 \%$. Thus, we are able to fully recover the input λ($1.5\,{R}_{e}$) values in our simulations, on average. This corrected ${\lambda }_{\mathrm{out}}$ is the value utilized here.

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We can use these simulations to calculate an error bar on the slow-rotator fraction, by asking how many galaxies might change designation based on residual bias in our ${\lambda }_{\mathrm{out}}$ values. For ${\lambda }_{\mathrm{out}}\gt 0.2$, there is effectively no bias. At lower ${\lambda }_{\mathrm{out}}\lt 0.2$, we tend to systematically underestimate ${\lambda }_{\mathrm{out}}$ by 30% (Figure 17). Here we estimate the resulting uncertainty in slow-rotator fraction that ensues from this uncertainty, using two methods.

First, we take all the objects in the sample with a measured ${\lambda }_{\mathrm{out}}\lt 0.14$ (corresponding to an intrinsic ${\lambda }_{\mathrm{out}}\lt 0.2$). If we assume that these all have an intrinsic ${\lambda }_{\mathrm{out}}$ that is 30% higher, we can ask what fraction of these would be considered fast rotators if we did not have a biased ${\lambda }_{\mathrm{out}}$ measurement for them. Of the 72 sources with ${\lambda }_{\mathrm{out}}\lt 0.14$, 16 are misidentified as slow rotators, constituting a 10% error in the slow-rotator fraction. The mass distribution of the low-${\lambda }_{\mathrm{out}}$ sources is comparable to that of the full sample, with a median $\langle \mathrm{log}{M}_{* }\rangle =10.9$ ${M}_{\odot }$. Thus, we feel confident that the residual errors incurred from our lower-than-optimal spatial resolution do not impact the main conclusions of this paper.

Second, we perform an actual Monte Carlo experiment. In three bins of ${\lambda }_{\mathrm{out}}(\lt 0.1$, $0.1\mbox{--}0.2$, $\gt 0.2$), we measure the mean offset and scatter from Figure 17. In each Monte Carlo run, we perturb $\lambda /\sqrt{\epsilon }$ based on this mean offset and scatter. Based on these perturbed values, we calculate a new slow-rotator fraction. We then calculate the mean and scatter in the slow-rotator fraction from these runs at each mass. In Figure 18, we show the perturbed values in the dotted lines; there is little difference between the dotted lines and our calculated values, shown with solid lines.

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