THE SAMI GALAXY SURVEY: REVISITING GALAXY CLASSIFICATION THROUGH HIGH-ORDER STELLAR KINEMATICS

The SAMI instrument and Galaxy Survey is described in detail in Croom et al. (2012) and Bryant et al. (2015). Here, we briefly outline the main characteristics of the instrument, sample selection, and global galaxy parameters.

SAMI is a multi-object integral field spectrograph with 13 IFUs deployable over a 1° diameter field of view, mounted at the prime focus of the 3.9 m Anglo Australian Telescope (AAT). Each IFU, or hexabundle, is made out of 61 individual fibers. Even though the hexabundles have a high filling factor of 75%, observations were carried out using a dither pattern to create data cubes with 05 spaxel size (Sharp et al. 2015; Allen et al. 2015). The fibers are 16 in size, and combine into a hexabundle which covers a ∼15'' diameter on the sky. All 819 fibers, including 26 individual sky fibers, are fed into the AAOmega dual-beamed spectrograph (Saunders et al. 2004; Smith et al. 2004; Sharp et al. 2006). For the SAMI Galaxy Survey, the 580V grating is used in the blue arm of the spectrograph, which results in a resolution of R ∼ 1700 with wavelength coverage of 3700–5700 Å. In the red arm, the higher resolution grating 1000R is used, which gives an R ∼ 4500 over the range 6300–7400 Å.

The SAMI Galaxy Survey aims to observe ∼3600 galaxies. The redshift range of the survey, $0.004\lt z\lt 0.095$, was chosen such that Mgb λ5177 and [S ii] $\lambda 6716,6731$ fall within the wavelength range of the blue and red arm, respectively. This limited redshift range results in spatial resolutions of 0.1 kpc per fiber at z = 0.004 to 2.7 kpc at z = 0.095. The survey has four volume-limited galaxy samples derived from cuts in stellar mass in the Galaxy and Mass Assembly (GAMA) G09, G12, and G15 regions (Driver et al. 2011). GAMA is a large campaign that combines large multi-wavelength photometric data with a spectroscopic survey of ∼300,000 galaxies carried out using the AAOmega multi-object spectrograph on the AAT. Furthermore, targets were selected from eight high-density cluster regions sampled within radius R₂₀₀ with the same stellar mass limit as for the GAMA fields (Owers et al. 2017; Bryant et al. 2015). The aim of the SAMI Galaxy Survey selection is to cover a broad range in galaxy stellar mass (${M}_{* }={10}^{8}\mbox{--}{10}^{12}$ M_⊙) and galaxy environment (field, group, and clusters).

For galaxies in the GAMA fields, the aperture matched g and i photometry from the GAMA catalog (Hill et al. 2011; Liske et al. 2015) are used to derive g − i colors, which were measured from reprocessed SDSS Data Release Seven (York et al. 2000; Kelvin et al. 2012). For the cluster environment, we use photometry from the SDSS (York et al. 2000) and VLT Survey Telescope ATLAS imaging data (Owers et al. 2017; Shanks et al. 2013). Stellar masses are derived from the rest-frame i-band absolute magnitude and g − i color by using the color–mass relation following the method of Taylor et al. (2011). For the stellar mass estimates, a Chabrier (2003) stellar initial mass function and exponentially declining star formation histories are assumed. For more details see Bryant et al. (2015).

In Figure 1, we show the g − i color versus stellar mass, color-coded by the number of spaxels (spatial pixels) with stellar continuum signal-to-noise ratio (S/N) $\gt \,20$ Å⁻¹. SAMI galaxies with low stellar mass (${M}_{* }\lt {10}^{10}$ M_⊙) rarely have more than 10 individual un-binned spaxels with relatively high S/N. Galaxies around ${M}_{* }\sim {10}^{10}$M_⊙ have on average 10–40 good quality spaxels, whereas at higher stellar masses (${M}_{* }\gt {10}^{11}$M_⊙) almost all galaxies have more than 40 good quality spaxels.

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Galaxy sizes are derived from GAMA-SDSS (Driver et al. 2011), SDSS (York et al. 2000), and VST (Shanks et al. 2013; Owers et al. 2017) imaging. The Multi-Gaussian Expansion (MGE; Emsellem et al. 1994) technique and the code from Cappellari (2002) is used for measuring effective radii, ellipticity, and positions angles. For more details, we refer to F. D'Eugenio et al. (2017, in preparation). Throughout the paper, ${R}_{{\rm{e}}}$ is defined as the semimajor axis effective radius, and ${R}_{{\rm{e}},{\rm{c}}}$ as the circularized effective radius, where ${R}_{{\rm{e}},{\rm{c}}}$ = ${R}_{{\rm{e}}}$ $\sqrt{q}$, where q is the axis ratio $b/a=1-\epsilon$. The ellipticity used in this paper, ${\epsilon }_{{\rm{e}}}$, is the average ellipticity of the galaxy within one effective radius measured from the best-fitting MGE model, not the global luminosity-weighted ellipticity from the MGE fit.

SAMI is setup to have a resolution of R ∼ 1700 in the blue 3700–5700 Å, and R ∼ 4500 in the red 6300–7400 Å (Croom et al. 2012; Bryant et al. 2015). Given the importance of the instrumental resolution and spectral profile for the stellar kinematic measurements, here we re-derive the full-width-at-half-maximum (FWHM) of the spectral instrumental line-spread-function (LSF) of the extracted spectra, and test whether the instrumental profile is well approximated by a Gaussian function. We use 24 unsaturated, unblended CuAr arc lines in the blue arm, and 12 lines in the red arm, from 16 frames between 2013 March 05 and 2015 August 17, for all 819 fibers.

Two functions are used for fitting the arc lines: a Gaussian and a Gaussian with skewness and kurtosis, as parameterized with Gauss–Hermite polynomials h₃ and h₄, respectively (Gerhard 1993; van der Marel & Franx 1993). There is an excellent agreement between the median resolution from the Gaussian and high-order moment fit. The median value for h₃ and h₄ is −0.01 in the blue arm and 0.00 in the red arm, with a 1σ spread of 0.016. These results indicate that SAMI's instrumental profile is well approximated by a single Gaussian function.

Key resolution quantities for SAMI are given in Table 1. In the blue arm, we find a median resolution of ${\mathrm{FWHM}}_{\mathrm{blue}}=2.65\,\mathring{\rm A}$, and in the red arm of ${\mathrm{FWHM}}_{\mathrm{red}}=1.61\,\mathring{\rm A}$. The fiber-to-fiber FWHM variation is 0.05 Å (1-σ scatter) in the blue, and 0.03 Å in the red. Over a period of two years, we find FWHM variations of 0.04 Å in the blue arm, and 0.03 Å in the red arm. The FWHM decreases with increasing wavelength in the red arm by 0.15 Å, but we do not find a wavelength dependence in the blue.

Table 1.  SAMI Spectral Resolution in Blue and Red

Arm	λ-range (Å)	λ-central (Å)	FWHM (Å)	σ (Å)	${R}_{\lambda \mbox{-} \mathrm{central}}$	${\rm{\Delta }}v$ ( km s−1)	${\rm{\Delta }}\sigma$ ( km s−1)
Blue	3750–5750	4800	${2.65}_{-0.09}^{+0.12}$	1.13	1812	165.5	70.3
Red	6300–7400	6850	${1.61}_{-0.05}^{+0.07}$	0.68	4263	70.3	29.9

Note. Line-spread-function parameters derived from unblended CuAr arc lines. For both the blue and red arm we give the wavelength range (λ-range), the central wavelength (λ-central), the median FWHM of the best-fit Gaussian to the spectral instrumental LSF in Å, the standard deviation of this Gauss in Å, the spectral resolution at λ-central (${R}_{\lambda \mbox{-} \mathrm{central}}$), the velocity resolution (FWHM) in km s⁻¹ (${\rm{\Delta }}v$), and the dispersion resolution ($1\sigma$) in km s⁻¹ (${\rm{\Delta }}\sigma$).

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3.1.1. Combining Blue and Red Arm

3.1.2. Annular Spectra Extraction

We run pPXF in two different modes, producing two final data products. The first data product consists of fits using a Gaussian LOSVD, i.e., fitting only the stellar velocity and stellar velocity dispersion (hereafter ${V}_{{\rm{m}}2}$ and ${\sigma }_{{\rm{m}}2}$). The velocity and dispersion maps from the Gaussian LOSVD fits are used for measuring ${\lambda }_{r}$.

In the second mode, a truncated Gauss–Hermite series (Gerhard 1993; van der Marel & Franx 1993) is used to parameterize the LOSVD. We fit four kinematic moments: ${V}_{{\rm{m}}4}$, ${\sigma }_{{\rm{m}}4}$, h₃, and h₄, where h₃ and h₄ are related to the skewness and kurtosis of the LOSVD.

3.2.1. Additive Polynomials

3.2.2. Noise Estimate

3.2.3. Removing Emission Lines and Outliers

3.2.4. Optimal Template Construction

3.2.5. Full Spaxel Fitting

3.2.6. Quality Cuts

The quality of each stellar kinematic fit depends on a number of factors: most importantly the S/N of the spectra, but also on the age of the stellar population, if the velocity dispersion is close to, or lower than, the instrumental resolution, and in the presence of strong emission lines. If we were to apply a strong cut in mean S/N = 40 Å⁻¹ as in, for example, the ATLAS${}^{3{\rm{D}}}$ survey, a large fraction of the galaxies in the SAMI sample would be excluded from the sample and the spatial coverage for the remaining individual galaxies would decrease as well, or, if we Voronoi-bin the data, the spatial resolution would decrease.

Fogarty et al. (2015) used Monte Carlo (MC) simulations to show that the ${\sigma }_{{\rm{m}}2}$, for SAMI spectra with S/N = 5, can be recovered with an accuracy of ±20 km s⁻¹ at ${\sigma }_{{\rm{m}}2}$ = 50 km s⁻¹. Therefore, instead of setting a strict limit on the minimal S/N, we explore a quality cut based on the velocity dispersion and its uncertainty that keeps the maximum number of spaxels without including unreliable measurements.

Figures 2(a) and (b) show the uncertainty on the stellar velocity and velocity dispersion for all ∼400,000 spaxels with S/N > 3 and ${\sigma }_{{\rm{m}}2}$ > FWHM_instr/2 ∼ 35 km s⁻¹. We exclude all spaxels with ${\sigma }_{{\rm{m}}2}$ < 35 km s⁻¹, where the systematic uncertainties, due to the instrumental spectral resolution, start to dominate over the random uncertainties (Appendix A.5, see also Fogarty et al. 2015). For the SAMI velocity measurements, most spaxels have uncertainties less than 20 km s⁻¹ as seen from the higher density of spaxels in the bottom left corner of Figure 2(a). In order to keep the majority of spaxels without sacrificing the quality of our results, we exclude all spaxels where the maximum velocity uncertainty $\gt 30$ km s⁻¹ (hereafter Q₁), as indicated by the gray dashed line in Figure 2(a).

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For the velocity dispersion, three different quality cuts are tested (Figure 2(b)). ${Q}_{\mathrm{red}}$: a conservative selection in which the uncertainty on the velocity dispersion has to be less than 10% (red line), ${Q}_{\mathrm{green}}$: a less strict quality cut of 25% (green line), and ${Q}_{\mathrm{blue}}$: a relative quality cut where the uncertainty on the velocity dispersion has to be less than 10% plus an additional 25 km s⁻¹ (blue line). We find that ${Q}_{\mathrm{red}}$ and ${Q}_{\mathrm{green}}$ remove a relatively large number of spaxels when the dispersion is low, which is not surprising given that 10% (25%) of 50 km s⁻¹ is only 5 km s⁻¹ (12.5 km s⁻¹). A fractional quality cut based on the velocity dispersion therefore biases our sample toward higher velocity dispersions. The ${Q}_{\mathrm{blue}}$ cut softens the limit on the relative uncertainty for low velocity dispersion. This way, we keep a large fraction of the low velocity dispersion spaxels, while keeping a strict quality cut for the higher velocity dispersions. A total of 347,538 spaxels meet our selection criteria ${Q}_{\mathrm{blue}}$ out of the initial 408,666 SAMI spaxels with S/N > 3 Å⁻¹ and ${\sigma }_{{\rm{m}}2}$ > 35 km s⁻¹.

With ${Q}_{\mathrm{blue}}$, the ratio of the velocity dispersion uncertainty and velocity dispersion is always less than 75% (Figure 2(c)). Furthermore, in Figure 2(d), the median velocity dispersion of the sample with and without the ${Q}_{\mathrm{blue}}$ cut are in good agreement (black versus blue distribution), whereas ${Q}_{\mathrm{red}}$ and ${Q}_{\mathrm{green}}$ bias the sample toward higher velocity dispersions. Therefore, we adopt ${Q}_{\mathrm{blue}}$ as the quality cut for the velocity dispersion measurements, which we hereafter refer to as ${Q}_{2}:{\sigma }_{\mathrm{error}}\lt \sigma \ast 0.1+25$ km s⁻¹.

Finally, in Appendix A.5 we show that a reliable estimate of h₃ and h₄ requires an additional quality cut of S/N > 20 Å⁻¹ and $\sigma \gt 70$ km s⁻¹, which we define as Q₃. This stricter quality cut Q₃ is shown in Figure 2(d) as the purple line. From this figure it is clear that only a relatively small number (∼10%) of spaxels pass Q₃: 35,798 versus 347,538 (Q₂).

3.2.7. High-order Moments

We parameterize the skewness and kurtosis of the LOSVD, i.e., deviations from Gaussian line profiles, by using Gauss–Hermite polynomials, with h₃ being related to the skewness and h₄ related to the kurtosis (Gerhard 1993; van der Marel & Franx 1993). Gauss–Hermite polynomials are used as opposed to using, for example, a decomposition into a double Gaussian for two reasons. First, the uncertainties in the six parameters in a two-Gaussian decomposition are highly correlated, whereas Gauss–Hermite polynomials are orthogonal, reducing the degeneracy in the LOSVD fit (van der Marel & Franx 1993). Second, a two-Gaussian decomposition a priori assumes that two kinematic distinct components are present, which is not always the case.

pPXF was designed to employ a maximum penalizing likelihood, i.e., forcing a solution to a Gaussian LOSVD, if the high-order moments are unconstrained by the data (Cappellari & Emsellem 2004). Following the code documentation, we derive an optimal penalizing bias value for SAMI spectra (see also Cappellari et al. 2011a), as the automatic penalizing bias in pPXF is often too strong. We define the ideal bias as one that reduces the scatter in the velocity dispersion, h₃ and h₄, without creating a systematic offset in the velocity and velocity dispersion. By running a large ensemble of MC simulations, a simple analytic expression was obtained for the ideal penalizing bias for SAMI spectra as a function of S/N (see Appendix A.5):

$$\begin{eqnarray}&&\mathrm{Bias}=0.0136+0.0023{({\rm{S}}/{\rm{N}})-0.000009({\rm{S}}/{\rm{N}})}^{2}.\end{eqnarray} \tag{ 1 }$$

For every spaxel, this optimal bias setting is then fed into pPXF.

If the LOSVD is a Gaussian, the m = 2 and m = 4 parameters are the same. In the case of a non-Gaussian LOSVD, due to the Gauss–Hermite polynomial parameterization of the skewness and kurtosis of the LOSVD fit, the velocity ${V}_{{\rm{m}}4}$ and velocity dispersion ${\sigma }_{{\rm{m}}4}$ deviate from the velocity ${V}_{{\rm{m}}2}$ and velocity dispersion ${\sigma }_{{\rm{m}}2}$ of a pure Gaussian LOSVD. Furthermore, the best-estimates of the true LOSVD moments can be calculated by (Equation (18), van der Marel & Franx 1993)

$$\begin{eqnarray}&&\tilde{V}\approx {V}_{{\rm{m}}4}+\sqrt{3}\,{\sigma }_{{\rm{m}}4}\,{h}_{3}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\tilde{\sigma }\approx {\sigma }_{{\rm{m}}4}(1+\sqrt{6}\,{h}_{4}).\end{eqnarray} \tag{ 3 }$$

In Figure 3, we show the difference between ${V}_{{\rm{m}}2}$ and ${V}_{{\rm{m}}4}$, ${\sigma }_{{\rm{m}}2}$ and ${\sigma }_{{\rm{m}}4}$ versus h₃ and h₄, and compare these values to the best-estimates of the true moments $\tilde{V}$ and $\tilde{\sigma }$. Only spaxels that meet selection criteria Q₃ are shown. As expected, there is a strong correlation between ${V}_{{\rm{m}}2}$–${V}_{{\rm{m}}4}$ and h₃: if the LOSVD is highly skewed, the velocity difference between the second and fourth order moments fits will be larger (h₃ = 0.1, Δ(${V}_{{\rm{m}}2}$–${V}_{{\rm{m}}4}$) ≃ 10 km s⁻¹). For the velocity dispersion we find a strong correlation between ${\sigma }_{{\rm{m}}2}$–${\sigma }_{{\rm{m}}4}$ and h₄. If the line has strong kurtosis, then the velocity dispersion difference between the second and fourth order moments fits will be larger (h₄ = 0.1, Δ(${\sigma }_{{\rm{m}}2}$–${\sigma }_{{\rm{m}}4}$) ≃ 15 km s⁻¹).

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The dashed lines in Figure 3 show the median values of the data. There is no systematic offset from zero in the relation between ${V}_{{\rm{m}}2}$–${V}_{{\rm{m}}4}$ and h₃. For ${\sigma }_{{\rm{m}}2}$–${\sigma }_{{\rm{m}}4}$ and h₄, however, we find a systematic offset toward lower values of ${\sigma }_{{\rm{m}}4}$, and higher values of h₄. Moreover, there is more scatter in the ${\sigma }_{{\rm{m}}2}$–${\sigma }_{{\rm{m}}4}$ and h₄ relation toward the positive h₄ side. We investigated whether the positive h₄ values are the result of instrumental resolution, template mismatch, or different seeing conditions. However, no correlations of h₄ with ${\sigma }_{\mathrm{obs}}$, S/N, FWHM of the point-spread function (PSF), host galaxy's stellar mass, or color were detected, which excludes the aforementioned possible issues.

Figures 3(c) and (d) show the best-estimates of the true LOSVD moments $\tilde{V}$ and $\tilde{\sigma }$ versus the best-fitted Gaussian moments. We find that ${V}_{{\rm{m}}2}$ agrees well with $\tilde{V}$, albeit with a small offset of 10 km s⁻¹ for $| {V}_{{\rm{m}}2}|$ > 100 km s⁻¹. For ${\sigma }_{{\rm{m}}2}$ versus $\tilde{\sigma }$, there is an offset in the one-to-one relation, such that the best-estimate of the true σ is larger than ${\sigma }_{{\rm{m}}2}$. At ${\sigma }_{{\rm{m}}2}$ = 200 km s⁻¹ there is an offset of ∼10 km s⁻¹ that goes up to ∼25 km s⁻¹ at ${\sigma }_{{\rm{m}}2}$ = 300 km s⁻¹). The offset can be largely explained by the overall positive h₄ that we measure.

In Cappellari et al. (2006) a similar test was performed, and they find that their results are consistent, within the errors, if Equation (3) is used. Veale et al. (2017) also find positive h₄ values for 41 massive early-type galaxies in the MASSIVE survey, similar to the results presented here. This suggests that some physical effect could be responsible for our positive h₄ values, rather than residual template mismatch. However, given the current unknown nature of the positive h₄ values in our data we do not explore this matter further. Furthermore, we caveat that ${\sigma }_{\mathrm{obs}}$ is highly sensitive to the wings of the LOSVD, which are generally poorly constrained by the data. For this reason, it is unclear whether Equation (3) can provide a better estimate of the true velocity dispersion rather than ${\sigma }_{\mathrm{obs}}$ alone. Detailed N-body simulations, with realistic LOSVDs, and accounting for template mismatch, are needed to demonstrate this.

We visually checked all 1380 unique SAMI kinematic maps and flagged 75 galaxies with irregular kinematic maps due to mergers or nearby objects that influence the stellar kinematics of the main object. These objects are excluded from further analysis in this paper.

For each galaxy, we calculate the maximum aperture out to which there are reliable data. ${R}_{\sigma }^{\max }$ and ${R}_{{\rm{h}}3}^{\max }$ are defined as the semimajor axis of an ellipse where at least 75% of the spaxels meet our velocity dispersion (Q₂) or h₃ quality criteria (Q₃) respectively. The axis ratio and position angle of the ellipse are obtained from the 2D MGE fits to the imaging data. A total of 270 galaxies have ${R}_{{\rm{e}}}$ or ${R}_{\sigma }^{\max }$ less than the half width at half maximum (HWHM) of the PSF (${\mathrm{HWHM}}_{\mathrm{PSF}}$), and are therefore excluded from the sample. This brings the number of galaxies with usable stellar velocity and stellar velocity dispersion maps to 1035.

Figure 4 shows the ratio of the maximum aperture radius and the effective semimajor radius for galaxies in the SAMI Galaxy Survey. The left panel shows the results for ${R}_{\sigma }^{\max }$, the right panel for ${R}_{{\rm{h}}3}^{\max }$. In red we show the cumulative fraction of galaxies with ${R}_{\sigma }^{\max }$/${R}_{{\rm{e}}}$ > value (scale on right-axis). Out of the 1035 galaxies with usable V and σ data, 76% (N = 784) have ${R}_{\sigma }^{\max }$> ${R}_{{\rm{e}}}$, and 24% (N = 247) have ${R}_{\sigma }^{\max }$> 2${R}_{{\rm{e}}}$. With the stricter quality cut Q₃ for the high-order moment fits, a total of 479 SAMI galaxies have ${R}_{{\rm{e}}}$ and ${R}_{{\rm{h}}3}^{\max }$ > ${\mathrm{HWHM}}_{\mathrm{PSF}}$. Of these 479 galaxies, 69% (N = 332) have ${R}_{{\rm{h}}3}^{\max }$> ${R}_{{\rm{e}}/2}$, and 13% (N = 63) have ${R}_{{\rm{h}}3}^{\max }$ > ${R}_{{\rm{e}}}$.

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In order to measure reliable high-order signatures of galaxies within the SAMI Galaxy Survey, we apply another quality cut to our sample. We only consider galaxies that have enough high-quality spaxels (Q₃) to fill an area greater than the maximum seeing aperture. The maximum allowed seeing in our data is 3'' (FWHM), which corresponds to an aperture containing 30 spaxels. In our sample, we find that 321 galaxies have a minimum of 30 spaxels that meet Q₃ and the other quality flags as mentioned before. Finally, given the low number (six) of low-mass galaxies with 30 high-quality spaxels or more (see also Figure 1), the sample is further restricted to galaxies with stellar mass ${M}_{* }\gt {10}^{10}$M_⊙. Our final sample contains a total of 315 galaxies that meet all these selection criteria.

Our sample has no selection on morphology, age, or galaxy type. However, due to the quality cuts in S/N and requiring that ${\sigma }_{\mathrm{obs}}\gt 70$ km s⁻¹, our sample might be biased toward early-type galaxies. We therefore perform a basic visual classification on our sample using the available GAMA-SDSS, SDSS, and VST imaging. Within our sample of 315 galaxies, 82% are early-type and 18% are late-type galaxies. Note that with the relatively poor imaging-quality, and the fact that visual-classifications can vary from observer to observer, this number if a rough approximation only.

Following Emsellem et al. (2007, 2011), we use the spin parameter approximation ${\lambda }_{R}$ to investigate separating fast-rotating galaxies from slow-rotating galaxies. For each galaxy, ${\lambda }_{R}$ is derived from the following definition (Emsellem et al. 2007):

$$\begin{eqnarray}&&{\lambda }_{R}=\displaystyle \frac{\langle R| V| \rangle }{\langle R\sqrt{{V}^{2}+{\sigma }^{2}}\rangle }=\displaystyle \frac{{\displaystyle \sum }_{i=0}^{{N}_{\mathrm{spx}}}{F}_{i}{R}_{i}| {V}_{i}| }{{\displaystyle \sum }_{i=0}^{{N}_{\mathrm{spx}}}{F}_{i}{R}_{i}\sqrt{{V}_{i}^{2}+{\sigma }_{i}^{2}}},\end{eqnarray} \tag{ 4 }$$

where the subscript i refers to the spaxel position within the ellipse, V_i is the stellar velocity in km s⁻¹, ${\sigma }_{i}$ the velocity dispersion in km s⁻¹, R_i the radius in arcseconds, and F_i the flux in units of erg cm² s⁻¹ Å⁻¹ of the ith spaxel. We sum over all spaxels N_spx that meet the quality cut for the second-moment fits as described in Section 3.2.6 within an ellipse with semimajor axis ${R}_{{\rm{e}}}$ and axis ratio b/a. Note that R_i is the semimajor axis of the ellipse on which spaxel i lies, not the circular projected radius to the center as is used by e.g., ATLAS${}^{3{\rm{D}}}$ (Emsellem et al. 2007). A different approach of using the intrinsic radius R_i (semimajor axis) over the projected radius (circular) is used here, as the intrinsic radius follows the light profile of the galaxy more accurately. Our current method assigns the same weight R_i for all spaxels on the same isophote, and will thus be less dependent on inclination; a spaxel on the minor axis will be weighted the same whether the galaxies is observed face-on or edge-on. However, by using the intrinsic radius rather than the projected, ${\lambda }_{R}$ is expected to be lower, as more weight will be given to spaxels on the minor-axis, which typically have low velocity values. We quantify the effect by measuring ${\lambda }_{{R}_{{\rm{e}}}}$ using both methods. For round objects ($\epsilon \lt 0.4$), the effect is small, i.e., we find a median ${\lambda }_{{R}_{\mathrm{proj}}}\,-{\lambda }_{{R}_{\mathrm{intr}}}=0.01$. The effect becomes more pronounced for flattened objects ($\epsilon \gt 0.4$), for which we find a median ${\lambda }_{{R}_{\mathrm{proj}}}-{\lambda }_{{R}_{\mathrm{intr}}}=0.04$, with a maximum difference of 0.09. ${\lambda }_{R}$ within one effective radius is only considered reliable and used in our analysis when the fill factor of good spaxels (Q₃; Section 3.2.6) within ${R}_{{\rm{e}}}$ is greater than 75%.¹⁹ Out of our 315 galaxies, 269 (85%) have ${\lambda }_{R}$ measurements out to one ${R}_{{\rm{e}}}$. For more details on ${\lambda }_{R}$ from SAMI data, see also Cortese et al. (2016). All derived ${\lambda }_{{R}_{{\rm{e}}}}$ values are given in Table 2.

Table 2.  Compilation of All Measured Quantities

CATID	${z}_{\mathrm{spec}}$	$\mathrm{log}{M}_{* }$/M⊙	g − i	${R}_{{\rm{e}}}$ (kpc)	${\epsilon }_{{\rm{e}}}$	${R}_{\sigma }^{\max }$/${R}_{{\rm{e}}}$	${R}_{{\rm{h}}3}^{\max }$/${R}_{{\rm{e}}}$	${\lambda }_{{R}_{{\rm{e}}}}$	$\overline{{k}_{5}/{k}_{1}}$
15165	0.0775	11.15	1.31	5.854	0.072	1.562	0.464	0.429 ± 0.008	0.0281 ± 0.0090
15481	0.0541	11.08	1.27	4.877	0.014	1.357	0.557	0.057 ± 0.006	0.2123 ± 0.0685
22582	0.0778	11.06	1.28	4.189	0.142	2.040	0.781	0.141 ± 0.007	0.0299 ± 0.0150
22595	0.0790	11.12	1.36	4.479	0.308	1.424	0.599	0.388 ± 0.006	0.0237 ± 0.0055
22887	0.0363	10.47	1.09	3.146	0.440	1.395	0.384	0.521 ± 0.006	0.0334 ± 0.0038

Note. This Table will be published in its entirety in the electronic edition of ApJ. A portion is shown here for guidance regarding its form and content.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Figure 5 shows ${\lambda }_{{R}_{{\rm{e}}}}$ versus ellipticity ${\epsilon }_{{\rm{e}}}$. For each galaxy, we show the velocity map to highlight the rotational properties. To avoid overlap between the galaxy velocity maps, the data are first put on a regular grid with spacing 0.02 in ${\lambda }_{{R}_{{\rm{e}}}}$ and ${\epsilon }_{{\rm{e}}}$. We position every galaxy to a closest grid point, or its neighbor, if its closest grid point is already filled by another galaxy. The size of the grid and velocity maps are chosen such that no galaxy is offset by more than one grid point from its original position. The stellar mass Tully–Fisher (Dutton et al. 2011) relation is used for the velocity scale: for a galaxy with stellar mass ${M}_{* }\gt {10}^{10}$M_⊙ the velocity scale of the velocity map ranges from −95 < V [ km s⁻¹] < 95, whereas a galaxy with stellar mass ${M}_{* }\gt {10}^{11}$M_⊙ is assigned a velocity scale from $-169\lt V\,$[ km s⁻¹]$\,\lt \,169$. The kinematic position angle ${{PA}}_{\mathrm{kin}}$ is used to align the major axis of all galaxies to 45°.

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In the SAURON and ATLAS${}^{3{\rm{D}}}$ survey, fast and slow rotators are separated as based on their position in ${\lambda }_{{R}_{{\rm{e}}}}$–${\epsilon }_{{\rm{e}}}$ space. In the SAURON survey, galaxies above and below ${\lambda }_{{R}_{{\rm{e}}}}$ = 0.1 were defined as fast and slow rotator respectively (Emsellem et al. 2007), whereas in the ATLAS${}^{3{\rm{D}}}$ survey slow rotators are defined to have ${\lambda }_{{R}_{{\rm{e}}}}$ $\lt \,0.31\sqrt{{\epsilon }_{{\rm{e}}}}$, and fast rotators are selected by ${\lambda }_{{R}_{{\rm{e}}}}$ $\geqslant \,0.31\sqrt{{\epsilon }_{{\rm{e}}}}$ (Emsellem et al. 2011). We show the SAURON and ATLAS${}^{3{\rm{D}}}$ relation in Figure 5 as the dashed and dotted gray line. Recent results from the SAMI Pilot Survey (Fogarty et al. 2014) and the CALIFA survey (Sánchez et al. 2012) motivated Cappellari (2016) to propose a refinement of the fast–slow rotators division, presented here as the solid line.

For SAMI galaxies, the majority of the galaxies with clear rotation reside above the ATLAS${}^{3{\rm{D}}}$ and Cappellari (2016) relations. In the bottom left region, however, where ${\epsilon }_{{\rm{e}}}\lesssim 0.15$ and ${\lambda }_{{R}_{{\rm{e}}}}$ ≲ 0.15, we find a number of galaxies with no clear sign of rotation that would be fast rotators according to the ATLAS${}^{3{\rm{D}}}$ relation. In addition, there are several galaxies with regular velocity fields that are below the ATLAS${}^{3{\rm{D}}}$ and Cappellari (2016) relations. The SAURON relation of ${\lambda }_{{R}_{{\rm{e}}}}$ = 0.1 appears to be most effective in separating galaxies with and without regular velocity fields. We will return to this issue in Section 6.

We use kinemetry(Krajnović et al. 2006, 2008) to estimate the kinematic asymmetry of the galaxies in our sample. Our aim is to separate galaxies with regular rotation from galaxies with non-regular rotation following the method by Krajnović et al. (2006, 2011). In kinemetry, the assumption is that the velocity field of a galaxy can be described with a simple cosine law along ellipses: $V(\theta )={V}_{\mathrm{rot}}\cos \theta$, with ${V}_{\mathrm{rot}}$ the amplitude of the rotation and θ is the azimuthal angle. Kinematic deviations from the cosine law can be modeled by using Fourier harmonics. The first order decomposition k₁ is equivalent to the rotational velocity, whereas the high-order terms (k₃, k₅) describe the kinematic anomalies. The kinematic asymmetry can be quantified by using the amplitudes of the Fourier harmonics. Following Krajnović et al. (2011) the kinematic asymmetry is defined as the mean (dimensionless) ratio ${k}_{5}/{k}_{1}$.

Our method for measuring the kinematic asymmetries is as follows. For each galaxy in our sample, we first mask all spaxels that do not pass the velocity quality cut Q₁ (see Section 3.2.6). For determining the amplitude of the Fourier harmonics the kinemetry routine (Krajnović et al. 2006) is used. In the fit, the position angle is a free parameter, whereas the ellipticity is restricted to vary between ±0.1 of the photometric ellipticity. This approach was chosen as opposed to leaving both parameters completely free for kinemetry to determine, because ellipticity is not well constrained from the velocity field alone. An average separation of 1.75 spaxels between the semimajor axis of the kinemetry ellipses is used, because of the covariance of the spaxels in the SAMI data.

For each ellipse, the kinemetry routine determines a best-fitting amplitude for k₁, k₃, and k₅. The kinemetry routine is also used to determine the mean surface brightness in each ellipse from the SAMI flux images, with the same input radii, ellipticity, and ${{PA}}_{\mathrm{kin}}$. For each galaxy we then determine the luminosity-weighted average ratio $\overline{{k}_{5}/{k}_{1}}$ within one effective radius. The uncertainty on $\overline{{k}_{5}/{k}_{1}}$ is estimated from MC simulations. The radial ${k}_{5}/{k}_{1}$ values are perturbed randomly within their measurement uncertainty range, and the mean $\overline{{k}_{5}/{k}_{1}}$ value is re-derived. The process is repeated 1000 times, and the uncertainty on $\overline{{k}_{5}/{k}_{1}}$ is then estimated from the standard deviation of the distribution of simulated $\overline{{k}_{5}/{k}_{1}}$ values. The derived $\overline{{k}_{5}/{k}_{1}}$ values are given in Table 2.

We compare our kinematic asymmetry values to those of the ATLAS${}^{3{\rm{D}}}$ survey (Krajnović et al. 2011) in Figure 6(a). The distribution for SAMI galaxies is shown in blue and for ATLAS${}^{3{\rm{D}}}$ in red. There is an excellent agreement between the results from the two surveys, but in the SAMI sample there are slightly more galaxies with low $\overline{{k}_{5}/{k}_{1}}$ values. We note that our sample contains both early-type and late-type galaxies, whereas the ATLAS${}^{3{\rm{D}}}$ sample only consisted of early-type galaxies. For ATLAS${}^{3{\rm{D}}}$ galaxies, a limit of $\overline{{k}_{5}/{k}_{1}}\lt 4$% was chosen for the velocity map to be well-described by the cosine law. Galaxies below that limit were named regular rotators and galaxies above the limit were called non-regular rotators. Here, the same limit of $\overline{{k}_{5}/{k}_{1}}\leqslant 0.04$ is adopted for regular rotators, but we use $\overline{{k}_{5}/{k}_{1}}\gt 0.08$ for non-regular rotators. We define the class between $0.04\lt \overline{{k}_{5}/{k}_{1}}\leqslant 0.08$ as quasi-regular rotators, because the distribution of $\overline{{k}_{5}/{k}_{1}}$ does not show a sharp transition between regular and non-regular rotators.

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Within one effective radius, 71% of galaxies are classified as regular rotators ($\overline{{k}_{5}/{k}_{1}}\leqslant 4$%) and 29% are classified as quasi-regular or non-regular rotators ($\overline{{k}_{5}/{k}_{1}}\gt 0.04$). We also perform a visual classification of the velocity maps into regular versus non-regular rotation. We find that 76% of galaxies have regular velocity fields, and 24% have non-regular velocity fields within one effective radius. This ratio agrees well with the automated kinemetry classification. For the 23 galaxies that were visually "mis-classified" as regular rotators, we find a median $\overline{{k}_{5}/{k}_{1}}$ = 0.049, with a scatter of 0.013, close to the regular/non-regular selection criteria.

Note that Krajnović et al. (2011) used their kinemetry results to come up with a more elaborate classification scheme, by including the kinematic position angle as a function of radius (${{\rm{\Gamma }}}_{\mathrm{kin}}$) and visual classification of the stellar velocity and dispersion fields. They split non-regular rotators into four subgroups (low-velocity, counter rotating cores, kinematically decoupled cores, and galaxies with two-sigma peaks), and regular rotators are split into two groups (regular morphology and bar/ring galaxies). We do not extend our kinemetry analysis beyond the use of $\overline{{k}_{5}/{k}_{1}}$, as this is beyond the scope of the paper.

Figure 6(b) compares the $\overline{{k}_{5}/{k}_{1}}$ values to a different definition of the kinematic asymmetry by Bloom et al. (2017): $({k}_{3}+{k}_{5})/2{k}_{1}$ (see also Shapiro et al. 2008). The second definition uses both k₃ and k₅ and is slightly more robust when the S/N is low. We find the mean $({k}_{3}+{k}_{5})/2{k}_{1}$ values to be slightly higher when compared to the $\overline{{k}_{5}/{k}_{1}}$ values, with a $1-\sigma$ scatter of 0.009. If we adopt the same selection criteria for both $\overline{{k}_{5}/{k}_{1}}$ and the mean $({k}_{3}+{k}_{5})/2{k}_{1}$, more galaxies (119 versus 82, respectively) would be classified as quasi-regular or non-regular if the mean $({k}_{3}+{k}_{5})/2{k}_{1}$ definition were used.

The color (g − i) versus stellar mass relation is shown in Figure 6(c). The data are color-coded by kinematic asymmetry $\overline{{k}_{5}/{k}_{1}}$. Galaxies with regular rotation fields are shown in blue, and non-regular rotators are shown as red, with quasi-regular rotators in between. Most galaxies with high $\overline{{k}_{5}/{k}_{1}}$ values are on the massive end of the red sequence above ${\mathrm{log}}_{10}{M}_{* }/$M_⊙ $\gt \,10.7$. There are also a few non-regular rotating galaxies below $\mathrm{log}{M}_{* }/$M_⊙$\,\lt \,10.5$.

We re-investigate the relation between ${\lambda }_{{R}_{{\rm{e}}}}$ and ellipticity in the left panel of Figure 7, but this time color code the data by kinematic asymmetry. Our observational data are first compared to simple galaxy models with different intrinsic ellipticities and viewing angles as presented in Cappellari et al. (2007) and Emsellem et al. (2011). These tracks are derived from models based on $V/\sigma$ and make use of the tight relation between $V/\sigma$ and ${\lambda }_{R}$, with a conversion factor κ (e.g., Equation (B1) from Emsellem et al. 2011). We remeasure κ because our definition of ${\lambda }_{R}$ (Equation (4)) is slightly different from Emsellem et al. (2011). We find a lower value for κ than Emsellem et al. (2011): 0.94 versus 1.1 respectively. For the models shown here we use $\kappa =0.94.$ Using the SAURON sample, Cappellari et al. (2007) showed that regular rotating galaxies appear to be bounded by the anisotropy parameter ${\beta }_{z}=0.70\times {\epsilon }_{\mathrm{intr}}$, where ${\beta }_{z}=1-{({\sigma }_{z}/{\sigma }_{R})}^{2}$. This relation is illustrated by the solid magenta line in Figure 7 for an asymmetric galaxy viewed edge-on (see e.g., Emsellem et al. 2011; D'Eugenio et al. 2013). The same model observed under different viewing angles, from edge-on (magenta line) to face-on (toward origin), is shown by the dotted gray lines. Furthermore, we show models with different intrinsic ellipticities (${\epsilon }_{\mathrm{intr}}=0.85-0.35$) as gray dashed lines.

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We find that the majority (92%) of galaxies with regular velocity fields ($\overline{{k}_{5}/{k}_{1}}\leqslant 0.04$) are consistent with being rotating axisymmetric systems with a range in intrinsic ellipticities ${\epsilon }_{\mathrm{intr}}=0.85-0.35$. This confirms previous results from Emsellem et al. (2011). However, an observational bias may be present in ${\lambda }_{{R}_{{\rm{e}}}}$ as our results differ from Emsellem et al. (2011) in two ways: (1) there is lack of galaxies with ${\lambda }_{{R}_{{\rm{e}}}}\lt 0.05$, and (2) there is dearth of flat objects with ${\epsilon }_{{\rm{e}}}\gt 0.4$ and ${\lambda }_{{R}_{{\rm{e}}}}\gt 0.6$. The first could be explained due to noise, which increases ${\lambda }_{{R}_{{\rm{e}}}}$ (e.g., Emsellem et al. 2007), whereas the second might be due to the effect of seeing, which decreases ${\lambda }_{{R}_{{\rm{e}}}}$and because we use elliptical apertures rather circular (as adopted by Emsellem et al. 2011), which lower ${\lambda }_{{R}_{{\rm{e}}}}$ on average by 0.05 when ${\epsilon }_{{\rm{e}}}\gt 0.4$.

In Appendix A.2 we investigate the effect of seeing and measurement on ${\lambda }_{{R}_{{\rm{e}}}}$ and $\overline{{k}_{5}/{k}_{1}}$ as measured by SAMI. Kinematic maps from the ATLAS${}^{3{\rm{D}}}$ survey are used as an input, which are rebinned to match the SAMI spatial resolution. The reconstructed LOSVD is then smeared by a Gaussian PSF with varying FWHM, mimicking the different seeing conditions for the SAMI Galaxy Survey. For the typical seeing of ${\mathrm{FWHM}}_{\mathrm{PSF}}=2\buildrel{\prime\prime}\over{.} 0$, we find that the increase due to measurement errors and the decrease due to seeing cancel out for galaxies with ${\lambda }_{{R}_{{\rm{e}}}}\lt 0.2$. For galaxies with ${\lambda }_{{R}_{{\rm{e}}}}\gt 0.2$, seeing is the dominant effect and causes a decrease in ${\lambda }_{{R}_{{\rm{e}}}}$ varying between −0.02 and −0.09 with a median of −0.05.

Thus, seeing and the use of elliptical apertures for deriving ${\lambda }_{R}$ are the likely causes for the dearth of flat objects with ${\lambda }_{{R}_{{\rm{e}}}}\gt 0.6$. However, we do not believe the lack of galaxies with ${\lambda }_{{R}_{{\rm{e}}}}\lt 0.05$ as compared to Emsellem et al. (2011) to be fully caused by measurement noise. Instead, we note that the galaxies with ${\lambda }_{{R}_{{\rm{e}}}}\lt 0.05$ in Emsellem et al. (2011) are only measured out to 0.25–0.6 ${R}_{{\rm{e}}}$. If these galaxies had been observed out to one ${R}_{{\rm{e}}}$ the minimum ${\lambda }_{{R}_{{\rm{e}}}}$ values in Emsellem et al. (2011) would have been higher. This is also clear from Figure 7(b), which shows that there would be significantly more galaxies with ${\lambda }_{R}\lt 0.05$ if the aperture would be only go out to ${R}_{{\rm{e}}/2}$.

From the kinemetry and spin parameter results combined, we find that galaxies with ${\lambda }_{{R}_{{\rm{e}}}}$ $\gt 0.2$ are predominantly regular rotators, whereas galaxies below ${\lambda }_{{R}_{{\rm{e}}}}=0.1$ are almost all non-regular rotators. There is no evidence for a strong dichotomy between regular and non-regular galaxies, but a transition zone at $0.1\lt$ ${\lambda }_{{R}_{{\rm{e}}}}$ $\lt 0.2$, where galaxies go from slow and non-regular rotation to fast and regular. A clear dichotomy is also missing in Figure 7(b), where we show the radial ${\lambda }_{R}$ profiles color-coded by $\overline{{k}_{5}/{k}_{1}}$. Non-regular rotating galaxies ($\overline{{k}_{5}/{k}_{1}}\gt 0.08;$ red) show a slow linear increase in ${\lambda }_{R}$, whereas for regular rotators ($\overline{{k}_{5}/{k}_{1}}\leqslant 0.04;$ blue color) we find a steep ${\lambda }_{R}$ relation with a turnover around $1\lt R/{R}_{{\rm{e}}}\lt 2$. Quasi-regular rotators ($0.04\lt \overline{{k}_{5}/{k}_{1}}\lt 0.08;$ beige) show a variety of ${\lambda }_{R}$ profiles, but most reside within the transition zone between non-regular and regular rotators. Figure 7(b) suggests that our results from Figure 7(a) do not depend on the choice of aperture for ${\lambda }_{R}$. We would find the same results if ${\lambda }_{{R}_{{\rm{e}}/2}}$ or ${\lambda }_{2{R}_{{\rm{e}}}}$ instead of ${\lambda }_{{R}_{{\rm{e}}}}$ are used.

In Section 4.2, we divided galaxies into three groups: regular rotators ($\overline{{k}_{5}/{k}_{1}}\leqslant 0.04$), quasi-regular rotators ($0.04\lt \overline{{k}_{5}/{k}_{1}}\,\leqslant 0.08$), and non-regular rotating galaxies ($\overline{{k}_{5}/{k}_{1}}\gt 0.08$). In Figures 8–9, we show the high-order kinematic signatures of these three groups. The density of spaxels that meet the strict selection criteria Q₃ are indicated by the contours drawn at 68% and 95%. The data in the top row are color-coded by the mean azimuthal deviation $\cos \theta$ from the galaxy's minor axis (top row), such that spaxels along the major axis are shown in blue and spaxels along the minor axis are shown in red. In the bottom row, we color code the data by the mean distance from the center in units of ${R}_{{\rm{e}}}$.

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Different high-order stellar kinematic signatures are clearly visible for regular and non-regular rotating galaxies (Figure 8). Regular rotators ($\overline{{k}_{5}/{k}_{1}}\leqslant 0.04$) show a strong anti-correlation between h₃ and $V/\sigma$, indicative of a stellar disk within these galaxies. We find that the strongest h₃ signal originates from spaxel along the major axis at large radii. There is a weak anti-correlation, close to being vertical, for quasi-regular rotating galaxies ($0.04\lt \overline{{k}_{5}/{k}_{1}}\leqslant 0.08$). We still detect a correlation between the azimuthal angle and h₃, which shows that quasi-regular galaxies still have rotation with a possible small disk. Non-regular rotators ($\overline{{k}_{5}/{k}_{1}}\gt 0.08$) show a steep vertical relation in h₃ versus $V/\sigma$ with no relation between $\cos \theta$ and h₃. As a function of radius, we find that spaxels at larger radii have a stronger h₃ signal.

Regular rotators show a distinct, heart-shaped pattern in h₄ versus $V/\sigma$ (Figure 9). The highest h₄ values originate from spaxels at large radii, but not from a specific azimuthal direction, while the lowest h₄ spaxels tend to lie in the center along the minor axis. Quasi-regular and non-regular galaxies show no relation in h₄ with $V/\sigma$.

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The high-order kinematic signatures that we find with SAMI are similar to the results from Krajnović et al. (2008, 2011). Note, however, that they used their kinemetry results to come up with a more elaborate classification scheme, which is beyond the scope of this paper.

In this section, we explore classifying individual galaxies from their high-order signatures alone. We do this by representing the h₃ versus $V/\sigma$ distribution with a 2D elliptical Gaussian, with dispersion ${\sigma }_{x}$, (${\sigma }_{\mathrm{major}}$, along the major axis of the ellipse), ${\sigma }_{y}$ (${\sigma }_{\mathrm{minor}}$, along the minor axis of the ellipse), and angle ϕ, centered on the origin:

$$\begin{eqnarray}&&f(x,y)=\displaystyle \frac{1}{2\pi {\sigma }_{x}{\sigma }_{y}}\exp [-({{ax}}^{2}-2{bxy}+{{cy}}^{2})],\end{eqnarray} \tag{ 5 }$$

with

$$\begin{eqnarray}&&a=\displaystyle \frac{{\cos }^{2}\phi }{2{\sigma }_{x}^{2}}+\displaystyle \frac{{\sin }^{2}\phi }{2{\sigma }_{y}^{2}}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&b=-\displaystyle \frac{\sin 2\phi }{4{\sigma }_{x}^{2}}+\displaystyle \frac{\sin 2\phi }{4{\sigma }_{y}^{2}}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&c=\displaystyle \frac{{\sin }^{2}\phi }{2{\sigma }_{x}^{2}}+\displaystyle \frac{{\cos }^{2}\phi }{2{\sigma }_{y}^{2}}.\end{eqnarray} \tag{ 8 }$$

A maximum log-likelihood approach is used to determine how well our Gaussian model approximates the n number of data-points. The log-likelihood is defined as

$$\begin{eqnarray}&&\mathrm{ln}{ \mathcal L }=\displaystyle \prod _{i=1}^{n}f({x}_{i},{y}_{i}),\end{eqnarray} \tag{ 9 }$$

Here, $f({x}_{i},{y}_{i})$ is the probability function for a given data-point i at x_i and y_i. The log-likelihood is then calculated from the product of all probability functions. For our 2D Gaussian this becomes

$$\begin{eqnarray}&&\mathrm{ln}{ \mathcal L }=\displaystyle \sum _{i=1}^{n}-\mathrm{ln}(2\pi {\sigma }_{x}{\sigma }_{y})-{{ax}}_{i}^{2}+2{{bx}}_{i}{y}_{i}-{{cy}}_{i}^{2}\end{eqnarray} \tag{ 10 }$$

with a, b, and c defined in Equations (6)–(8). We calculate the log-likelihood for a large range of values for ${\sigma }_{x}$, ${\sigma }_{y}$, and ϕ, and then derive for which values the maximum log-likelihood is reached. Each galaxy is assigned the corresponding value of ${\sigma }_{x}$, ${\sigma }_{y}$, and ϕ. Hereafter we will refer to ${\sigma }_{x}$ and ${\sigma }_{y}$ as ${\sigma }_{\mathrm{major}}$ and ${\sigma }_{\mathrm{minor}}$ for clarity. In order to get a model independent measurement of the anti-correlation strength, we also calculate the Pearson correlation coefficient for each galaxy. The derived quantities are given in Table 3.

Table 3.  Log-likehood Estimates and High-order Classification

CATID	${\sigma }_{\mathrm{major}}$	${\sigma }_{\mathrm{minor}}$	ϕ	Corr. Coeff.	Class
15165	0.30	0.032	−8.90	−0.82	3
15481	0.04	0.023	−8.70	−0.16	1
22582	0.15	0.026	−8.30	−0.62	2
22595	0.42	0.023	−8.20	−0.93	3
22887	0.44	0.056	−2.50	−0.33	5

Note. This Table will be published in its entirety in the electronic edition of ApJ. A portion is shown here for guidance regarding its form and content.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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We show the distribution of the four parameters that quantify each galaxy's h₃–$V/\sigma$ relation in Figure 10. While each parameter reveals a new insight on the different families of high-order kinematic signatures, no strong groups are immediately apparent in each of the six panels, only small over-densities. Inspired by Milone et al. (2015), we adopt a method based on the finite mixture models by McLachlan & Peel (2000). We use the Mcluster CRAN package (Fraley & Raftery 2002; Fraley et al. 2012) in the statistical software system R, designed for model-based clustering and classification. The package performs a maximum likelihood fit assuming different groups, where the significance of each group is determined from the Bayesian information criterion (BIC) given the log-likelihood, the dimension of the data, and number of mixture components in the model.

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We first run a Gaussian finite mixture model with ellipsoidal, varying volume, shape, and orientation (VVV). Five classes are identified with a best-fit BIC = 1378. Using the bootstrap method, i.e., random sampling with replacement, we repeat the fit 10,000 times to estimate the uncertainty on the number of classes. The distribution of the recovered optimal number of classes is shown in the top-right panel of Figure 10. We find a clear peak at N = 5 which confirms the initial classification fit. Next, we fix the number of classes to five and repeat the fit 10,000 times using bootstrapping to identify how often a galaxy is classified into one of the five groups. Figure 10 shows the results of the bootstrap analysis, where each galaxy is assigned the color of the class it most often resides in.

There is no clear distinct separation of the classes in any of the parameters, instead every distribution shows a gradual transition from one class to another. However, out of all four parameters, ϕ and ${\sigma }_{\mathrm{major}}$ are the cleanest set to separate the five classes that we found by using the Gaussian finite mixture models. Given that the five classes are most easily separated in ϕ and ${\sigma }_{\mathrm{major}}$, we propose a simplified classification based on ϕ and ${\sigma }_{\mathrm{major}}$ alone, as indicated by the blue dashed lines in the bottom left panel of Figure 10. The selection criteria are given in Table 4. From Class 1 to 3, the mean value in ${\sigma }_{\mathrm{major}}$ increases, which coincides with a steep vertical h₃ versus $V/\sigma$ relation in Class 1 to a strong anti-correlation in Class 3, respectively. Class 3–5 are selected by similar ${\sigma }_{\mathrm{major}}$, but are separated by their angle ϕ. From Class 3 to 5, the angle ϕ goes from a negative angle to zero angle, i.e., the h₃ versus $V/\sigma$ relation goes from steep to horizontal.

Table 4.  Stellar Kinematic Classes and their Mean Properties

Class	Range ${\sigma }_{\mathrm{major}}$	Range ϕ	${N}_{\mathrm{gal}}$	$\langle {\mathrm{log}}_{10}{M}_{* }/{M}_{\odot }\rangle$	$\langle g-i\rangle$	$\langle {R}_{{\rm{e}}}\rangle$ (kpc)	$\langle {\epsilon }_{{\rm{e}}}\rangle$	$\langle {\lambda }_{{\rm{e}}}\rangle$	$\langle \overline{{k}_{5}/{k}_{1}}\rangle$	$\langle {\mathrm{FWHM}}_{\mathrm{PSF}}\rangle$	$\langle {R}_{(\max ,{\rm{h}}3)}/{R}_{{\rm{e}}}\rangle$
All			315	10.74	1.18	3.52	0.27	0.31	0.08	2.10	0.71
1	$0\lt {\sigma }_{\mathrm{major}}\leqslant 0.125$	$-45^\circ \lt \phi \lt 0^\circ \ \ \ \ \$	52	11.03	1.23	5.57	0.15	0.08	0.41	2.14	0.60
2	$0.125\lt {\sigma }_{\mathrm{major}}\leqslant 0.3\,\ \ \$	$-45^\circ \lt \phi \lt 0^\circ \ \ \ \ \$	97	10.68	1.19	3.34	0.20	0.23	0.04	2.13	0.73
3	${\sigma }_{\mathrm{major}}\gt 0.3\ \ \$	$\phi \leqslant -6^\circ \ \ \$	64	10.67	1.18	2.96	0.28	0.36	0.03	2.07	0.81
4	${\sigma }_{\mathrm{major}}\gt 0.3\ \ \$	$-6^\circ \lt \phi \leqslant -3\buildrel{\circ}\over{.} 5$	57	10.70	1.14	3.11	0.35	0.42	0.02	2.07	0.71
5	${\sigma }_{\mathrm{major}}\gt 0.3\ \ \$	$\phi \gt -3\buildrel{\circ}\over{.} 5$	45	10.69	1.17	2.86	0.46	0.44	0.02	2.07	0.66

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Figure 11 shows examples of individual galaxies in Class 1–5, which we will use to describe the five classes in more detail. Class 1 shows an h₃ versus $V/\sigma$ relation that is steep to vertical with little spread in ${\sigma }_{\mathrm{major}}$ direction. The galaxy has a high average velocity dispersion, and the velocity field shows no sign of rotation, except in the core, where there is evidence for a kinematically decoupled core. The h₃ map shows no strong directional anti-alignment, except in the core, where there is an anti-alignment with the kinematic decoupled core. From a visual inspection of the broad-band color images and kinematic maps, we find that Class 1 galaxies are related to non-rotating or slow-rotating elliptical galaxies.

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For Class 2 the h₃ versus $V/\sigma$ relation is steep, with relatively little spread in the ${\sigma }_{\mathrm{major}}$ direction, but with more spread than Class 1 by definition. Class 2 objects sometimes show a weak vertical boxy signature in h₃ versus $V/\sigma$. From a visual inspection, we find that this class could be further separated into galaxies with boxy-round or weak anti-correlated signatures, but the spatial sampling of the data in combination with the fitting method do not allow for this. The velocity maps show rotation, but the rotation is not as strong as compared to Class 3–5.

Classes 3 and 4 have a strong anti-correlation between h₃ and $V/\sigma$ which is also clearly evident from the anti-alignment of the velocity field and the h₃ map. From the strength of their velocity fields relative to their velocity dispersions, these galaxies would be classified as fast rotators. By definition, the anti-correlation of Class 4 is less steep as compared to Class 3, but Class 4 also has a larger perpendicular spread in the anti-correlation.

For Class 5 galaxies the relation between h₃ and $V/\sigma$ is mostly horizontal to slightly inclined, and sometimes show signs of a combined weak anti-correlation and correlation with h₃. From the kinematic maps there is evidence for strong rotational support, as is also evident by the large range in $V/\sigma$. Furthermore, in the kinematic maps we find that the 2D h₃ signal shows no directional anti-alignment with the 2D velocity field. All Class 5 galaxies would be classified as fast-rotating galaxies based on their positions in the ${\lambda }_{{R}_{{\rm{e}}}}$–${\epsilon }_{{\rm{e}}}$ diagram.

We show the stacked high-order kinematic signatures of the five classes in Figures 12 and 13. The contours indicate 68% and 95% of the data, where we applied a boxcar smoothing filter with a width of two. In the top row, we color-coded by the mean azimuthal deviation $\cos \theta$ from the galaxy's minor axis, and by the mean distance from the center in units of ${R}_{{\rm{e}}}$ in the bottom row. We find five specific h₃ versus $V/\sigma$ signatures, but with a gradual transition from one class into another. The gradual transition was already indicated by Figure 10, in which we found that our five classes highlight well-defined regions in the 4D parameter space but without strong overdensities. Class 1–4 show similar h₃ versus $V/\sigma$ relations as compared to the individual examples in Figure 11, whereas Class 5 now shows a weak anti-correlation that was absent in the example galaxy. Class 3 and 4 show the strongest anti-correlation between h₃ and $V/\sigma$, indicative of a rotating stellar disk within these galaxies.

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For galaxies in Class 2–5, the strongest h₃ signal originates from spaxel along the major axis at large radii. However, for galaxies in Class 2–4, spaxels that are located along the minor axis (red) show a vertical h₃ versus $V/\sigma$ relation, whereas minor-axis spaxels in Class 5 galaxies show a slight positive h₃ versus $V/\sigma$ relation. We detect no relation between $\cos \theta$ and h₃ for Class 1 galaxies, but spaxels at larger radii do show a stronger h₃ signal.

We show h₄ versus $V/\sigma$. in Figure 13. For Class 1 galaxies, the range of $V/\sigma$ is very narrow as compared to the range of h₄, with no trend, similar to the h₃ signatures. Class 2 shows a broader spread in $V/\sigma$ as compared to Class 1. For Class 3 and 4 galaxies, we find the heart-shape that was also clear for regular rotators in Figure 9, whereas Class 5 galaxies show a rounder distribution in h₄ and $V/\sigma$. For Class 1 and 2 there is no correlation with radius and h₄ strength, whereas for Class 3–5 the lowest h₄ values are found in the center. For Class 3 and 4, the highest h₄ values originate from spaxels at large radii, but not from a specific azimuthal direction.

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The main conclusion from Figures 12 and 13 is that for all five classes we find well-defined signatures in h₃ versus $V/\sigma$ with a gradual transition from one class into another. In the next section, we investigate how the kinematic signatures relate to integrated global galaxy properties such as stellar mass, color, and ${\lambda }_{R}$.

In the previous section, galaxies were separated into five classes based on their high-order stellar kinematic signatures alone. Here, we will look at the integrated galaxy properties of these classes, and investigate where they lie in known relations between color versus stellar mass, effective radius versus stellar mass, and proxy for the spin parameter versus ellipticity.

5.3.1. Global Properties

Our full sample contains 315 galaxies, with 52 galaxies in Class 1, 97 in Class 2, 64 in Class 3, 57 in Class 4, and 45 in Class 5 (see also Table 4). In Figure 14(a), we show the g − i color versus stellar mass for all galaxies for which a high-order stellar kinematic class could be determined. Class 1 galaxies (red circles) are mostly found on the red sequence and dominate at the high-mass end (${M}_{* }\gt {10}^{11}\,{M}_{\odot }$). There are few galaxies at relatively low stellar masses and only two galaxies with blue g − i colors ($\lt 1.0$). Galaxies in Class 2 (orange hexagons) have lower mean stellar masses than Class 1, but the bulk resides on the red sequence. Galaxies from Class 3 (beige diamonds), Class 4 (light-blue squares), and Class 5 (blue pluses) have a large range in both color and stellar mass.

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We show the mass–size relation in Figure 14(b). Unsurprisingly, Class 1 galaxies are among the largest galaxies in our sample, whereas Class 3–5 are distributed evenly on the mass–size plane. In Figures 14(c) and (d), we show the spin parameter approximation (${\lambda }_{{R}_{{\rm{e}}}}$) versus ellipticity (${\epsilon }_{{\rm{e}}}$). Note that ${\lambda }_{R}$ within an effective radius could be determined for 269 out of the 315 galaxies for which we derived a high-order stellar kinematic class. The data for all individual galaxies are shown in Figure 14(c), while kernel density estimates are used in Figure 14(d); the contours show 68% out of the total probability. For the kernel density estimates we use a Gaussian kernel with a bandwidth of 0.076.

Galaxies in Class 1 populate the region below the magenta line that indicates an edge-on view of axisymmetric model galaxies with ${\beta }_{z}=0.70\times {\epsilon }_{\mathrm{intr}}$. From panels (a) and (b) we already learned that Class 1 galaxies are among the most massive, large, red galaxies in our selected sample, so it comes as no surprise that these galaxies will have complex dynamical structure, and are also classified as slow rotators by the ${\lambda }_{{R}_{{\rm{e}}}}$–${\epsilon }_{{\rm{e}}}$ criterion.

Class 2 galaxies have slightly higher ${\lambda }_{{R}_{{\rm{e}}}}$ and ellipticity values than Class 1. Most Class 2 galaxies reside close to the fast–slow separation criteria of Emsellem et al. (2011) and Cappellari (2016). A closer inspection of Class 2 galaxies that sit above ${\lambda }_{{R}_{{\rm{e}}}}$ $\gt \,0.35$ reveals that significant number of these outliers have bars (6/10). Class 3 and 4 are true fast rotators as indicated by their high ${\lambda }_{{R}_{{\rm{e}}}}$ values, but Class 4 galaxies have on average higher ${\lambda }_{{R}_{{\rm{e}}}}$ values than Class 3 galaxies (${\lambda }_{{R}_{{\rm{e}}}}$ = 0.42, versus ${\lambda }_{{R}_{{\rm{e}}}}$ = 0.36, respectively). For the 36 galaxies in Class 5 with ${\lambda }_{{R}_{{\rm{e}}}}$ measurements, we find on average high ellipticity and high ${\lambda }_{{R}_{{\rm{e}}}}$; Class 5 galaxies populate the extreme regions. The nine galaxies without ${\lambda }_{{R}_{{\rm{e}}}}$ measurements also have high average ellipticity.

In Figure 14(d) we find that the five classes occupy distinct regions in the ${\lambda }_{{R}_{{\rm{e}}}}$–${\epsilon }_{{\rm{e}}}$ diagram, but there is significant overlap between the contours. The key result, however, is that galaxies with similar ${\lambda }_{{R}_{{\rm{e}}}}\mbox{--}{\epsilon }_{{\rm{e}}}$ values can show distinctly different ${h}_{3}\mbox{--}V/\sigma$ signatures. Thus it is important to realize that the overlapping regions observed here separate more clearly in a higher dimensional space.

5.3.2. Class 5 Morphologies

In the previous section, Class 5 galaxies were found to be fast-rotating without an h₃–$V/\sigma$ anti-correlation. Given the interesting properties of this class, here we will compare the morphologies and stellar kinematic maps of Class 5 to Class 2–4 galaxies. For comparison, galaxies are selected that occupy the same region in the ${\lambda }_{{R}_{{\rm{e}}}}$–${\epsilon }_{{\rm{e}}}$ diagram, i.e., for each Class 5 galaxy we select its nearest neighbor from any other class.

Figure 15 shows color images and stellar kinematic maps of Class 5 galaxies on the left, and for their nearest ${\lambda }_{{R}_{{\rm{e}}}}$–${\epsilon }_{{\rm{e}}}$ neighbors on the right. We find morphologies ranging from spirals to fully edge-on disks, but there are no morphological differences between Class 5 and the selected Class 2, 3, and 4 galaxies. For example, galaxies 9403800814 and 9239900390 are similar in morphology (sixth row); both galaxies are edge-on disks with a central bulge.

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All galaxies shown in Figure 15 are strongly rotating, and some show a dispersion dominated bulge. The h₃ maps look different for Class 5 galaxies as compared to Class 2–4 galaxies. For Class 5 galaxies, we find no anti-alignment of the h₃ signal with the velocity field, whereas similar ${\lambda }_{{R}_{{\rm{e}}}}$–${\epsilon }_{{\rm{e}}}$ galaxies do show this strong anti-alignment. This is also visible from the h₃ versus $V/\sigma$ panel, where Class 2–4 galaxies show a strong anti-correlation and Class 5 galaxies do not.

Our h₃ spatial detection limit (i.e., our minimum requirement of 30 good spaxels) is also not the cause for the discrepancy between the classes. For example, galaxy 586330 (Class 3, second row) and galaxy 382158 (Class 4, third row) have 30–35 spaxels for which h₃ could be reliably measured, yet the h₃–$V/\sigma$ anti-correlation is clearly visible. All Class 5 galaxies are also well above the spatial detection limit. However, if Class 5 on average has lower ${R}_{{\rm{h}}3}^{\max }$/${R}_{{\rm{e}}}$ as compared with Class 3–4, then we could be tracing different regions of the galaxies, i.e., bulge versus disk. Class 5 galaxies have lower median ${R}_{{\rm{h}}3}^{\max }$/${R}_{{\rm{e}}}$ (0.66) as compared to Class 3 and 4 (0.73; 0.81; 0.71, respectively; see also Table 4). From Figure 4(b), however, we find that ${R}_{{\rm{h}}3}^{\max }$/${R}_{{\rm{e}}}$ ranges from 0.1 to 1.5, so the difference between the median ${R}_{{\rm{h}}3}^{\max }$/${R}_{{\rm{e}}}$ of the classes is small. Furthermore, Class 5 galaxies have higher ellipticity as compared to Class 3 and 4. For edge-on galaxies ${R}_{{\rm{h}}3}^{\max }$ could be smaller due to observational effects. If we compare the median ${R}_{{\rm{h}}3}^{\max }$/${R}_{{\rm{e}}}$ for galaxies with ${\epsilon }_{{\rm{e}}}\gt 0.5$ (0.59) to galaxies with ${\epsilon }_{{\rm{e}}}\lt 0.5$ (0.70), a similar trend is detected. This means that any galaxy with high ellipticity would have slightly lower ${R}_{{\rm{h}}3}^{\max }$, irrespective of its class. We conclude therefore that the spatial detection limit is not the cause for the different identified classes.

Finally, we also look into the effects due to seeing, which could be affecting our classification. Because all classes have similar median seeing (see Table 4), this is not likely to impact our results. Moreover, the size of the PSF is indicated by the circle on the bottom right in the h₃ map (Figure 15), and is always smaller than the h₃ detection map.

With the introduction of the SAURON instrument (Bacon et al. 2001) and its survey (de Zeeuw et al. 2002), a visual inspection of the stellar kinematics maps of 66 galaxies by Emsellem et al. (2004) led to a simple classification of early-type galaxies into two groups. The first group were galaxies with rotating disks seen at different inclinations, whereas the galaxies in the other group were inconsistent with having simple disks. This confirmed earlier results obtained with long-slit spectrographs which revealed that luminous ellipticals are found to rotate slowly (Bertola & Capaccioli 1975; Illingworth 1977; Binney 1978; Bertola et al. 1989), whereas intrinsically faint ellipticals rotate as rapidly as disk bulges (Davies et al. 1983). A more quantitative classification of fast and slow rotators was later proposed that used an approximation for the spin parameter ${\lambda }_{R}$ (Cappellari et al. 2007; Emsellem et al. 2007). Early-type galaxies were separated into slow and fast rotators, depending on whether their ${\lambda }_{R}$ within an effective radius was below or above 0.1, respectively.

Subsequently, the ATLAS${}^{3{\rm{D}}}$ survey expanded the sample to 260 galaxies and the classification was further refined as ${\lambda }_{{R}_{{\rm{e}}}}$ = 0.31 $\sqrt{{\epsilon }_{{\rm{e}}}}$ (Emsellem et al. 2011). Galaxies above this limit were classified as fast rotators, galaxies below were defined as slow rotators. This refined classification was motivated by a different classification based on the kinematic asymmetry of the velocity field (Krajnović et al. 2011). Their results showed that galaxies with regular rotation fields can also be classified as fast rotators, whereas non-regular rotators either had (i) no rotation at all, (ii) irregular rotation, (iii) signs of kinematically decoupled cores, or (iv) two counter-rotating disks.

In this paper, we confirm the results from the SAURON and ATLAS${}^{3{\rm{D}}}$ survey: the majority of early-type galaxies agree with being a family of oblate rotating systems viewed at random orientation (Figure 7). The other group of early-type galaxies show complex dynamical structures, with irregular velocity fields, 2-sigma peaks, or kinematic misalignment, indicating that they are triaxial systems.

Cappellari (2016) interprets these results as a kinematic dichotomy: slow and fast rotators are distinct classes that can be separated by a selection in the ${\lambda }_{{R}_{{\rm{e}}}}$–${\epsilon }_{{\rm{e}}}$ diagram. Further evidence for a dichotomy is derived from Jeans anisotropic modeling, where the distribution of κ, the ratio of ${V}_{\mathrm{obs}}/V({\sigma }_{\phi }={\sigma }_{R})$ of the observed velocities and a model with oblate velocity ellipsoid, shows two clear distributions (Cappellari 2016).

However, both the ATLAS${}^{3{\rm{D}}}$ and the Cappellari (2016) fast/slow separation in the ${\lambda }_{{R}_{{\rm{e}}}}$–${\epsilon }_{{\rm{e}}}$ space are based upon the regular versus non-regular classification by Krajnović et al. (2011). From Figure 6 we find that there is no sharp transition in the $\overline{{k}_{5}/{k}_{1}}$ distribution from regular to non-regular rotation. Additionally, the regular/non-regular selection may depend on data quality: for galaxies in the SAURON survey, regular rotating galaxies were selected as $\overline{{k}_{5}/{k}_{1}}\lt 0.02$ (Krajnović et al. 2008), whereas in the ATLAS${}^{3{\rm{D}}}$ survey a limit of $\overline{{k}_{5}/{k}_{1}}\lt 0.04$ was chosen due to lower S/N and higher average $\overline{{k}_{5}/{k}_{1}}$(Krajnović et al. 2008). Furthermore, both measurement uncertainties and seeing also impact the SAMI kinemetry measurements (see e.g., Appendices A.1 and A.2). Measurement uncertainties increase the $\overline{{k}_{5}/{k}_{1}}$ parameter, whereas seeing brings it down. With typical 20 seeing, both effects on average cancel out, but with added scatter that could be large enough to wash out a sharp transition in the $\overline{{k}_{5}/{k}_{1}}$ distribution.

From Figure 7 we find that galaxies with $0.1\,\lt$ ${\lambda }_{{R}_{{\rm{e}}}}$ $\lt \,0.2$ have a relatively large range in $\overline{{k}_{5}/{k}_{1}}$, with no clear separation. Instead we find a transition zone where galaxies go from having regular and fast rotation to non-regular and slow rotation. For SAMI galaxies, the ${\lambda }_{{R}_{{\rm{e}}}}$–${\epsilon }_{{\rm{e}}}$ classification is furthermore sensitive to the data quality (see e.g., Appendices A.1 and A.2). We find outliers with ${\lambda }_{{R}_{{\rm{e}}}}$ $\sim 0.05\mbox{--}0.1$ in repeat observations when the difference in seeing is large (i.e., from $1\buildrel{\prime\prime}\over{.} 6$ to $2\buildrel{\prime\prime}\over{.} 8$). This is confirmed by SAMI "re-observed" simulated ATLAS${}^{3{\rm{D}}}$ galaxies, from which we find that both seeing and measurement uncertainties impact the ${\lambda }_{{R}_{{\rm{e}}}}$ measurements. While the effect of seeing is strongest for galaxies with ${\lambda }_{{R}_{{\rm{e}}}}\gt 0.25$, a sharp transition zone between ${\lambda }_{{R}_{{\rm{e}}}}=0.1\mbox{--}0.2$ could disappear because of this.

Thus, from the directly observable properties ${\lambda }_{{R}_{{\rm{e}}}}$ and $\overline{{k}_{5}/{k}_{1}}$ we find that a dichotomy is unlikely to be detected due to measurement uncertainties and inclination effects. The application of Jeans anisotropic modeling to recover intrinsic properties, however, indicates a dichotomy in the internal velocity moments (Cappellari 2016), and does not suffer from these problems. While the division into fast and slow rotators is useful for many studies, e.g., the kinematic-morphology density relation (Cappellari et al. 2011b; Fogarty et al. 2014, S. Brough et al. 2017, in preparation; J. van de Sande et al. 2017, in preparation), we therefore caution against using only the ${\lambda }_{{R}_{{\rm{e}}}}$–${\epsilon }_{{\rm{e}}}$ diagram for kinematically classifying galaxies. The addition of kinemetry, Jeans anisotropic modeling, and/or high-order stellar kinematics should be used to fully understand the stellar kinematic properties of galaxies.

In this paper, we find that galaxies in similar regions of the ${\lambda }_{{R}_{{\rm{e}}}}$–${\epsilon }_{{\rm{e}}}$ diagram can show distinctly different relations between h₃ and $V/\sigma$. While we show that there are five kinematic classes that best describe the different high-order kinematic signatures, we emphasize that the 4D space for parameterizing these signatures shows a continuum of properties with five attractors. There are, however, observational and physical motivations for the five classes. Orientation, inclination, flattening, rotation versus pressure support, and the presence of a bar, all impact the high-order stellar kinematic features. Indications for these effects in different kinematic classes are presented in Sections 6.2 and 6.3. Higher spatial and spectral resolution data and/or more detailed simulation are needed, however, to provide a clean separation between all these effects.

One of the goals of this paper is to compare the high-order stellar kinematic signatures in the SAMI Galaxy Survey to those as predicted by Naab et al. (2014). They use cosmological hydrodynamical zoom-in simulation to link the assembly history of galaxies to their present day shapes and kinematics. They analyze 44 central galaxies and divide these galaxies into six different model classes. Class A and B are fast rotators (${\lambda }_{{R}_{{\rm{e}}}}$ $\sim 0.3\mbox{--}0.6$) that experienced gas-rich mergers, and show a strong anti-correlation between h₃ and $V/\sigma$. Class C, E, and F are slow-rotating galaxies that either had late gas-rich mergers or gas-poor minor and/or major mergers. These model classes have a range in ellipticity of ${\epsilon }_{{\rm{e}}}\sim 0.3\mbox{--}0.5$ with low spin parameter ${\lambda }_{{R}_{{\rm{e}}}}$ $\sim \,0.05\mbox{--}0.2$, and show a vertical relation between h₃ and $V/\sigma$.

Class D consists of galaxies with a late gas-poor merger that either leads to a significant spin-up of the stellar merger remnant, or leaves the rotating properties of galaxy pre-merger intact. These galaxies have low to intermediate spin parameters (${\lambda }_{{R}_{{\rm{e}}}}$ $\sim 0.1\mbox{--}0.3$); roughly half would be classified as fast rotators as based on the slow-fast selection criteria from Emsellem et al. (2011). Most interestingly, despite the relative fast rotation, Class D galaxies do not show a strong anti-correlation between h₃ and $V/\sigma$. There are also no other signs of embedded disk-like components. We note that this signature has been found before in fast-rotating merger remnants from binary merger simulations (Naab & Burkert 2001; Naab et al. 2006a, 2006b; Jesseit et al. 2007). Due to the expected absence of gas in the late major mergers, there is no dissipative component during the merger. Therefore, no significant disk is able to regrow and the galaxy cannot support stars on tube orbits with high-angular momentum (Barnes & Hernquist 1996; Bendo & Barnes 2000; Naab & Burkert 2001; Naab et al. 2006a, 2014; Jesseit et al. 2007; Hoffman et al. 2009, 2010).

The stellar orbits of the model classes are further investigated in Röttgers et al. (2014). They show that the h₃–$V/\sigma$ anti-correlation in rotating galaxies (model Class A and B) originate from a high-fraction of stars on z-tube orbits. Slow rotators that experienced a recent merger (model Class C, E, and F) are dominated by stars on box and x-tube orbits in the center with an increasing contribution of z-tube orbits beyond one effective radius. For model Class D galaxies they show that the majority of the stars are on box orbits. While stars on prograde z-tubes are present in Class D galaxies, their contribution is too low to generate an LOSVD with a steep leading wing (h₃).

In Figure 16 we compare the high-order kinematic signatures our five classes to the six model classes from Naab et al. (2014). The model classes are qualitatively selected to best-match the h₃–$V/\sigma$ relations of our observed class. In the top row of Figure 16 we find that Class 1 and 2 have very similar high-order stellar kinematic signatures as model Classes C, E, and F, but the h₃ amplitudes in model Class E and F are more extreme than in the observed data. Class 3 and 4 with the strong anti-correlations are comparable to model Class A and B, and both Class 5 and model Class D galaxies show a weak or no h₃–$V/\sigma$ anti-correlation. In the bottom row of Figure 16 we show h₄ versus $V/\sigma$. We lowered the observed values h₄ values by 0.05 in order to compare the h₄ shape to those of the models, as we found earlier that the median h₄ in the SAMI data is 0.05 (Figure 3(b)), whereas the models have mean h₄ = 0.0. We find a good agreement in h₄ versus $V/\sigma$ between all observed classes and models. The models do show more scatter in h₄ as compared to the observational data, in particular for Class F and D. Interestingly, the heart-shaped h₄–$V/\sigma$ signature of Class 3 and 4 is also seen in model Class A and B, which is opposite to the h₄–$V/\sigma$ relations from the 3:1 merger simulations by Naab et al. (2006a).

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When we compare the observed and model classes with regards to their positions in the ${\lambda }_{{R}_{{\rm{e}}}}$–${\epsilon }_{{\rm{e}}}$ diagram, there are some differences however. Observed slow rotators (Class 1) have lower ellipticity than slow rotators from the simulations (model Class C, E, and F), although they agree well in stellar mass. Class 5 galaxies have higher average spin parameter and slightly higher ellipticity (${\epsilon }_{{\rm{e}}}\sim 0.46$) than model D, which have on average elipticities of $\epsilon \sim 0.35$. Moreover, two of the five model Class D galaxies have ${\lambda }_{{R}_{{\rm{e}}}}$ $\,\lt \,0.2$, whereas most Class 5 galaxies are well above ${\lambda }_{{R}_{{\rm{e}}}}$ $\,\gt \,0.2$. Thus, while the high-order stellar kinematic signatures are similar, discrepancies in ${\lambda }_{{R}_{{\rm{e}}}}$ and ${\epsilon }_{{\rm{e}}}$ between observed Class 5 and model Class D galaxies shows that the two classes likely have a different formation history.

Given the lack of overlap between Class 5 and Class D galaxies in the ${\lambda }_{{R}_{{\rm{e}}}}$–${\epsilon }_{{\rm{e}}}$ diagram, we then wonder if our classification missed other galaxies that do not show an h₃–$V/\sigma$ anti-correlation. First, the h₃–$V/\sigma$ relations are investigated for all galaxies with similar ${\lambda }_{{R}_{{\rm{e}}}}$ and ${\epsilon }_{{\rm{e}}}$ values as Class D galaxies. We select ellipticities $0.3\lt {\epsilon }_{{\rm{e}}}\lt 0.5$ and spin parameter approximation $0.1\lt$ ${\lambda }_{{R}_{{\rm{e}}}}$ $\lt 0.4$. Within this region, we find galaxies from Class 2–4, but all Class 3 and 4 galaxies show a strong anti-correlation in h₃–$V/\sigma$. Out of 16 Class 2 galaxies in this selected region, five show a possible lack of h₃–$V/\sigma$ relation. All galaxies show little spread in $V/\sigma$ and a square distribution in h₃–$V/\sigma$. For two galaxies we are only tracing the inner part of the bulge, but the three other candidates could be similar to model Class D, although these galaxies suffer from relatively poor spatial h₃ sampling. From the available imaging two out of five galaxies show a clear bulge and disk, which leaves three possible matches with model Class D.

We extend our search to all Class 2 galaxies, without the restriction on ${\lambda }_{{R}_{{\rm{e}}}}$ and ${\epsilon }_{{\rm{e}}}$. Out of our 85 Class 2 galaxies, for eight additional galaxies we find boxy-round h₃–$V/\sigma$ profiles. From a visual inspection of the imaging data, five galaxies show clear signs of an inner bar, while the other three galaxies are round (${\epsilon }_{{\rm{e}}}\lt 0.15$) with low spin parameter (${\lambda }_{{R}_{{\rm{e}}}}$ $\lt 0.2$).

In conclusion, the absence of model Class D galaxies in the SAMI Galaxy Survey data suggests that most fast-rotating galaxies are formed through gas-rich mergers. From the Naab et al. (2014) simulations, we would have expected approximately 10% (5 out of 44) of the SAMI galaxies to be rotating without showing an h₃–$V/\sigma$ anti-correlation. There are several limitations in the simulations, however, that could skew these predictions. First, only a small number (44) of galaxies were analyzed, but this number could be increased by repeating this analysis using the Illustris (Genel et al. 2014; Vogelsberger et al. 2014) and EAGLE (Crain et al. 2015; Schaye et al. 2015) simulations. Second, the model from Naab et al. (2014) disfavors the formation of disks, which could indicate that these scenarios are missing in the cosmological simulation. For example, Naab et al. (2006a) present isolated binary mergers where a remnant stellar disk was observed without a correlation in h₃–$V/\sigma$. Third, Sharma et al. (2012) show that the orientation of the instrinsic spin of merging halos strongly impacts the orientation and amplitude of the angular momentum in the merger remnant. Therefore, it is vital for binary merger simulations to probe a large ensemble of realistic spin orientations, as observed in large-scale structures. Otherwise, certain merger scenarios might be missed. Fourth, the feedback prescription in Naab et al. (2014) could be improved upon such that the model would reproduce reasonable galaxy population properties, as the current model favors the formation of early-type galaxies. With the latest cosmological simulations such as Illustris and EAGLE, many of these issues can be resolved, which could lead to a better understanding of linking the high-order kinematic signatures in galaxies to their assembly histories.

It then remains unclear, however, why Class 5 galaxies show weak or no h₃–$V/\sigma$ anti-correlation. When comparing the morphologies, we find that many Class 5 galaxies have clear signs of disks, which are absent in the model Class D galaxies from Naab et al. (2014). When we look at the residuals from the photometric Sérsic profile fits (Kelvin et al. 2012), some galaxies show strong tidal features or large residuals in their center. A recent merger could have destroyed the inner parts of the disks or have altered the orbits of the central stars in such a way that a strong h₃ signal is not detected.

Bars in edge-on spirals could also have a strong impact on the h₃ signatures (Chung & Bureau 2004; Bureau & Athanassoula 2005). Depending on the orientation and bar strength of a galaxy, Bureau & Athanassoula (2005) show that the h₃ radial profile can show a strong correlation with radius rather than the more common disk-like anti-correlation to no correlation at all. Furthermore, their results show that the radial h₄ signal appears to be V-shaped, similar to the heart-shaped patterns we find for fast rotators in Class 3 and 4. Seidel et al. (2015) also study the influence of bars on the kinematics of nearby galaxies. They show interesting relations between h₃ and $V/\sigma$ at different bar radii. For some of their galaxies, however, the h₃ and h₄ results are hard to interpret because σ falls well below the instrumental resolution and the h₃ and h₄ drop to zero. Nonetheless, for galaxy NGC 4643 for example, they find a strong h₃–$V/\sigma$ anti-correlation at 0.1${R}_{\mathrm{bar}}$, whereas the anti-correlation disappears at 0.5${R}_{\mathrm{bar}}$ and 1${R}_{\mathrm{bar}}$.

A complex inner h₃–$V/\sigma$ structure is also seen in the edge-on S0 galaxy NGC 3115 (Guérou et al. 2016). While the outer regions show a strong anti-correlation, the inner regions reveal a zig-zag pattern in h₃–$V/\sigma$. Furthermore, a thin, fast-rotating stellar disc is embedded in the fast-rotating spheroid which leads to another h₃–$V/\sigma$ inner anti-correlation. Should a similar galaxy be present in our sample, it seems unlikely that the strong anti-correlation from the outer disk would be observed: both the S/N and the velocity dispersion would not meet our selection criterion Q₃. However, the inner zig-zag structure in h₃–$V/\sigma$ could be observed. When we search for this pattern in Class 5 galaxies, we indeed detect three galaxies that might show similar inner high-order signatures. In Figure 15 (left panel, rows 3–5), galaxies 9016800039, 935880004037, and 9403800637 all show an h₃–$V/\sigma$ pattern similar to the edge-on S0 NGC 3115, which has tentative evidence for an inner bar (Guérou et al. 2016). Thus, it is clear that bars can have a strong impact on h₃. While the evidence is far from conclusive for Class 5 galaxies, the connection to edge-on bars and different orientations and inclinations is worth pursuing further with hydrodynamical simulations.

We mention two other studies that also compare their 2D high-order stellar kinematic results to the simulation from Naab et al. (2014). Spiniello et al. (2015) study the fast-rotating galaxy NGC 4697 using eight VLT-VIMOS pointings, and measure the stellar kinematics out to 0.7${R}_{{\rm{e}}}$. They find a strong anti-correlation between h₃ and $V/\sigma$ for this system. From a combined stellar kinematic and stellar population analysis, their findings suggest that this system assembled its mass through gas-rich minor mergers. With relatively limited data, Forbes et al. (2016) find a large range in high-order kinematic signatures but identify one galaxy (NGC 4649²⁰ ) that does not show a strong h₃–$V/\sigma$ anti-correlation. However, visual classification of the h₃–$V/\sigma$ signatures is open to different interpretations, particularly when the spatial sampling and data quality are relatively low.

Due to the optimal tiling of the SAMI fields and plate configuration, 24 galaxies were observed twice. These sources are ideal for estimating uncertainties due to weather conditions, seeing, and the use of different hexabundles. We pre-select on quality for comparing between the two observations, i.e., galaxies from the first observation (obs₁) were observed under better seeing conditions as compared to the second observation (obs₂).

In Figure 19 we compare the aperture velocity dispersion, ${\lambda }_{{R}_{{\rm{e}}}}$, and kinematic asymmetry measurements for obs₁ and obs₁. Note that not all galaxies have full ${R}_{{\rm{e}}}$ coverage. Galaxies that are in our final high-quality sample are shown as circles, all other galaxies as squares. There is a good agreement between the stellar velocity dispersion measurements of obs₁ and obs₂. For ${\lambda }_{{R}_{{\rm{e}}}}$, however, all galaxies have smaller measurement uncertainties than expected from the offset of the one-to-one relation. Moreover, for three, the repeat observations with worse seeing show significantly lower values of ${\lambda }_{{R}_{{\rm{e}}}}$ (0.05–0.1). For the $\overline{{k}_{5}/{k}_{1}}$, measurements, there is significant scatter and an offset from the one-to-one relation, but the measurement uncertainty for most outliers is also large. Nonetheless, more galaxies are classified as non-regular or quasi-regular rotators when the seeing is worse. Note, however, that only three galaxies with repeat observations passed quality cut Q₃.

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In Figure 20, we show the relation between h₃ and $V/\sigma$ for galaxies from obs₁ (red) and obs₂ (blue). The first repeat observation (galaxy 41144, left panel), shows a similar anti-correlation. With better seeing we find a larger ${\sigma }_{\mathrm{major}}$, but the kinematic classification is the same. We find a good agreement between the measurements from the second repeat observation (galaxy 56140), but the anti-correlation is slightly steeper under worse seeing conditions ($\phi =-8.5$ versus $\phi =9.4$). Note, that for this galaxy, we find a significant difference in ${\lambda }_{{R}_{{\rm{e}}}}$ between the two observations: 0.5 versus 0.41. For the third repeat observation (galaxy 9008500239, right panel) we find similar values of ${\sigma }_{\mathrm{major}}$ and ${\sigma }_{\mathrm{minor}}$ but a difference in angle ϕ that is large enough for the galaxy to change classification.

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Given the relatively large systematic uncertainty in the ${\lambda }_{{R}_{{\rm{e}}}}$ measurements for the repeat observations, here we further investigate this by looking at the growth curves of the spin parameter proxy. Figure 21 shows all repeat observations for which ${\lambda }_{R}$ could be measured as a function of radius. Note that there are more galaxies with ${\lambda }_{R}$ measurements compared to Figure 19, because not all galaxies have measurements out to 1${R}_{{\rm{e}}}$.

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For about half the galaxies, there is a relatively large difference in the growth curves between the repeat observations. In particular, galaxies 15165, 227607, 106016, 106042, and 91959 have been observed under very different seeing conditions (16 versus 28), and are affected the most. In worse seeing conditions, the radial ${\lambda }_{R}$ profiles are significantly lower by 0.05–0.1. Note, that all of these galaxies are from the same aperture plate, hence were observed at the same time. Not all galaxies, however, are affected by the seeing to the same extent. For example, galaxies 56064, 32362, and 41144 have a seeing difference of 053 (198 versus 251), but show nearly identical ${\lambda }_{R}$ profiles.

In conclusion, from the repeat observations we find that different atmospheric conditions can impact the ${\lambda }_{R}$ measurements on the order of 0.05–0.1. Furthermore, these results show the importance of repeat observations for large integral field surveys such as SAMI and MaNGA.

From repeat observations we found that different atmospheric conditions can impact our ${\lambda }_{R}$ and $\overline{{k}_{5}/{k}_{1}}$ measurements. Here, we use existing ATLAS${}^{3{\rm{D}}}$ kinematic measurements to study the effect of seeing and measurement uncertainty on SAMI observations. ATLAS${}^{3{\rm{D}}}$ measurements are used for this purpose, due to the data's higher spatial sampling as compared to SAMI, and because many of the results presented in this paper are compared to key results from the SAURON and ATLAS${}^{3{\rm{D}}}$ surveys.

First, we use the publicly available ATLAS${}^{3{\rm{D}}}$ stellar kinematic data products²¹ and tabulated data from Cappellari et al. (2011a); Emsellem et al. (2011), and Krajnović et al. (2011). We select galaxies that have full coverage out to at least one effective radius; however, only binned data derived from four or less original spaxels are used in order to avoid step functions in the velocity and dispersion maps. The following 23 galaxies meet these selection criteria: NGC 0680, NGC 1121, NGC 2577, NGC 2592, NGC 2594, NGC 2695, NGC 2699, NGC 2824, NGC 2852, NGC 3458, NGC 3610, NGC 3648, NGC 3757, NGC 3838, NGC 4262, NGC 4283, NGC 4342, NGC 4660, NGC 5103, NGC 5507, NGC 5845, NGC 6149, UGC 09519. All selected galaxies are regular rotators (Krajnović et al. 2011), with a broad range in ${\lambda }_{R}$ (0.05–0.6) and ellipticity (0.05–0.6) (Emsellem et al. 2011).

The flux, velocity, and velocity dispersion data are interpolated onto a regular grid. Outside the maximum measurement radius, we extrapolate the data to avoid step functions because later on the LOSVD is smoothed to mimic seeing. We rebin the data in order to obtain a similar angular size distribution as SAMI galaxies, i.e., to have effective radii between 2'' and 6'' in a 25'' × 25'' size box with 05 pixel size. Next, we create a 3D cube, where for each (x, y) pixel we construct a Gaussian LOSVD in the z coordinate using the flux, velocity and velocity dispersion.

To mimic seeing, we convolve all x, y "images" in the LOSVD cube along the z direction with a 2D Gaussian function with an FWHM ranging from 01, 05, 10, ..., 30. Note that it is not correct to simply smooth the flux, velocity, and velocity dispersion maps independently, as all three moments are correlated components in an observed LOSVD. For fast-rotating galaxies this is particularly important, as a steep gradient in flux and velocity is present in the center. When convolved with the seeing, the gradient in flux and velocity go down, whereas the velocity dispersion increases.

A.2.1. Simulated Uncertainty Estimates on ${\lambda }_{{R}_{{\rm{e}}}}$

For each simulated galaxy, we measure ${\lambda }_{{R}_{{\rm{e}}}}$ as described in Section 4.1. Figures 22(a) and (b) shows the results for ${\lambda }_{{R}_{{\rm{e}}}}$ under different simulated seeing conditions. We define Δ${\lambda }_{{R}_{{\rm{e}}}}$ as ${\lambda }_{{R}_{{\rm{e}}}}$ measured under different seeing conditions normalized by ${\lambda }_{{R}_{{\rm{e}}}}$ measured when the seeing is 01. At this point we do not include measurement errors, in order to separate the effect of seeing and measurement errors on ${\lambda }_{{R}_{{\rm{e}}}}$. Different colors show different realizations of the seeing, from 01 in blue to 30 in red. We note that typical seeing for the SAMI Galaxy Survey is 20, indicated by the beige data.

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In Figure 22(a) we show Δ${\lambda }_{{R}_{{\rm{e}}}}$ versus ${\lambda }_{{R}_{{\rm{e}}}}$. We find that ${\lambda }_{{R}_{{\rm{e}}}}$ decreases with increasing ${\mathrm{FWHM}}_{\mathrm{PSF}}$. Furthermore, galaxies with higher ${\lambda }_{{R}_{{\rm{e}}}}$ are more strongly impacted by seeing than galaxies with low ${\lambda }_{{R}_{{\rm{e}}}}$. For ${\lambda }_{{R}_{{\rm{e}}}}\lt 0.2$ we find that ${\rm{\Delta }}{\lambda }_{{R}_{{\rm{e}}}}$ is small (−0.025) even when the seeing reaches 30. When ${\lambda }_{{R}_{{\rm{e}}}}\gt 0.2$, the impact of the seeing is stronger, with a median Δ${\lambda }_{{R}_{{\rm{e}}}}$ = 0.08 when ${\mathrm{FWHM}}_{\mathrm{PSF}}=3\buildrel{\prime\prime}\over{.} 0$. As expected, galaxies with small angular sizes are on average more strongly impacted by seeing than bigger galaxies (Figure 22(b)).

Next, we look at the impact of measurement errors on the ${\lambda }_{{R}_{{\rm{e}}}}$ measurements. For every galaxy, we add normal random noise to the flux, velocity, and velocity dispersion as typically measured for galaxies in the SAMI Galaxy Survey (see Section 3.2.6). The noise is weighted by the S/N such that spaxels in the center will have lower velocity and velocity dispersion errors as compared to spaxels in a galaxy's outskirts. For every galaxy, we repeat the process of adding random noise a 1000 times and remeasure ${\lambda }_{{R}_{{\rm{e}}}}$. Figure 22(c) shows the median of the distributions with an ${\mathrm{FWHM}}_{\mathrm{PSF}}$ of 01 (blue) and 20 (beige), normalized by the ${\lambda }_{{R}_{{\rm{e}}}}$ 01 results without measurement errors from Figures 22(a)–(b). The errors bars show the 16th (lower) and the 84th (higher) percentile of the distribution.

When measurement uncertainties are included, with minimal seeing (${\mathrm{FWHM}}_{\mathrm{PSF}}$  = 01), we find a slight increase in ${\lambda }_{{R}_{{\rm{e}}}}$ by ∼0.02–0.03 for galaxies with ${\lambda }_{{R}_{{\rm{e}}}}\lt 0.2$. With 20 seeing, we find that the increase due to measurement errors and the decrease due to seeing cancel out for these two galaxies. For galaxies with ${\lambda }_{{R}_{{\rm{e}}}}\gt 0.2$, seeing is the dominant effect and causes a median decrease in ${\lambda }_{{R}_{{\rm{e}}}}$ of 0.05.

A.2.2. Simulated Uncertainty Estimates on Kinemetry

The selected ATLAS${}^{3{\rm{D}}}$ galaxies for studying the impact of seeing and measurement errors on ${\lambda }_{{R}_{{\rm{e}}}}$ and $\overline{{k}_{5}/{k}_{1}}$ are all regular rotators with $\overline{{k}_{5}/{k}_{1}}\lt 0.04$ (Krajnović et al. 2011), whereas the full observed range in $\overline{{k}_{5}/{k}_{1}}$ is 0.0–0.25 (see Figure 6). If we were to use these galaxies for testing the impact of seeing on kinemetry measurements the results would be trivial: regular velocity fields smoothed by the seeing will become more regular. Instead, it would be more interesting to test which kinematic features or disturbances disappear due to the effect of seeing.

Therefore, we add an artificial kinematically decoupled core to every velocity field in order to create a range in observable $\overline{{k}_{5}/{k}_{1}}$ parameters. We mimic a kinematically decoupled core by adding to the velocity field: two 2D Gaussians with opposite sign, at a random orientation, and peak velocity strength of 75% of ${V}_{\mathrm{maximum}}$. The positive and negative peak of the decoupled core are placed at 3/4 of the semiminor axis to ensure that the decoupled core is always observable.

The results without measurements uncertainties are shown in Figures 23(a) and (b). We find that Δ$\overline{{k}_{5}/{k}_{1}}$ decreases when the seeing ${\mathrm{FWHM}}_{\mathrm{PSF}}$ increases. The effect of seeing is also stronger for higher $\overline{{k}_{5}/{k}_{1}}$. We find that the median $\overline{{k}_{5}/{k}_{1}}$ decreases by 0.01 when the seeing ${\mathrm{FWHM}}_{\mathrm{PSF}}$ = 10, to Δ$\overline{{k}_{5}/{k}_{1}}$ = −0.03 when the seeing is 30. Opposite to ${\lambda }_{{R}_{{\rm{e}}}}$ we don't find a strong correlation between the impact of seeing on $\overline{{k}_{5}/{k}_{1}}$ and the angular size of the galaxy. This could be partially due to the fact that we artificially inserted kinematically decoupled cores, which might not be a fully realistic description of observed kinematic disturbances.

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Next, we include noise in the same way as we did for the ${\lambda }_{{R}_{{\rm{e}}}}$ simulations. Figure 23(c) shows the impact of both measurement uncertainties and seeing on $\overline{{k}_{5}/{k}_{1}}$ versus the maximum of the velocity field. With minimal seeing (${\mathrm{FWHM}}_{\mathrm{PSF}}$ = 01), we find that $\overline{{k}_{5}/{k}_{1}}$ increases. In particular for galaxies with little to mild rotation, noise on the velocity measurement is misinterpreted as a kinematic disturbance by kinemetry. With 20 seeing, typical for the SAMI Galaxy Survey, we find that the decrease in $\overline{{k}_{5}/{k}_{1}}$ caused by seeing is counteracted by the increase due to measurement uncertainties. The median decrease in $\overline{{k}_{5}/{k}_{1}}$ is 0.01 when the maximum $V\gt 50$ km s⁻¹, but for the two galaxies with little rotation, the measurement errors are dominant over seeing effects (Δ$\overline{{k}_{5}/{k}_{1}}$ = 0.10–0.16).

Template mismatch can significantly impact the measured stellar kinematics (van de Sande et al. 2013), and the high-order moments (Gerhard 1993). In this section, we investigate the impact of the optimal template on the measured SAMI stellar kinematics. Two effects are analyzed: the spatial region in which the optimal template has been determined, and the choice of template library. Figure 24 shows the stellar kinematic maps for galaxy 504713, where the optimal template was derived using two different methods. For the first method, the central 2'' spectrum is used for deriving a single optimal template, and every spaxel in the galaxy is then fit with this optimal template. For the second method, we use binned annuli spectra for constructing an optimal template. The second method is the SAMI default, and is described in more detail in Section 3.2.4. Note that we do not derive optimal templates from individual spaxels, which generally do not meet our S/N requirement of 25 that is needed to derive a reliable optimal template.

The first method shows larger values of h₄ and σ in the disk of the galaxy as compared to method number two. In the center of the galaxy, the templates gives reasonable results, but in the outskirts h₄ reaches relatively high values (∼0.2), which is often associated with template mismatch (Gerhard 1993).

On the bottom right of Figure 24, the reconstructed color image from the SAMI spectra is displayed. Galaxy 504713 shows a clear red bulge and a blue disk. The template from method one was derived from the central bulge, whereas method two has optimal templates that vary as a function of radius. It is therefore not too surprising that the template from method one does not provide an adequate match to the stellar population in the disk.

We conclude that using annular bins for constructing optimal templates provides better results as compared to using centrally derived optimal template.

Next, we test the impact of using different stellar libraries for measuring stellar kinematics. The default MILES stellar library is compared to the Flexible Stellar Population Synthesis models (FSPS; v2.5) of Conroy et al. (2009) and Conroy & Gunn (2010), and to the models by (v8.0; Vazdekis et al. 2010, 2015). One advantage of using SPS models over a stellar library with single stars is reducing the degrees of freedom in deriving an optimal template, which is particularly useful when the S/N is low. Because SPS models are already pre-made optimal templates with only age and metallicity as a degree of freedom, using SPS can reduce the uncertainty on the stellar kinematic parameters (e.g., van de Sande et al. 2013). Another advantage of using SPS models is an increased fitting speed. Stellar libraries typically contain a thousand stars, whereas SPS models are distilled to a few hundred templates. A disadvantage of SPS models is the lack of exotic templates. If the stellar populations in a galaxy are highly mixed due to multiple epochs of star formation and merger events, a combination of SPS templates may no longer describe the integrated light adequately.

For both SPS models, we picked the version that use the stellar MILES library as their input. All galaxies from the test sample are fit with the three models, and the optimal templates are constructed from the binned annular spectra. In general, no systematic offsets in the kinematic maps are found (${V}_{{\rm{m}}4}$, ${\sigma }_{{\rm{m}}4}$, h₃, h₄) when we use different libraries. The scatter is consistent with the random uncertainties that are expected from the MC simulations.

There is one exception: galaxy 230776, a massive, red (${\mathrm{log}}_{10}\,{M}_{* }/$ M_⊙ = 11.6), slow-rotating elliptical galaxy with a counter-rotating core. For the measured velocity there is a good agreement between different models, but for the velocity dispersion we find a systematic difference of  30 km s⁻¹. This difference is much larger than expected, given the high S/N of the galaxy. Also, there is a systematically higher value of h₄ (0.05) when we use the FSPS and Vazdekis models as compared to when using the MILES models. Figure 25 shows the central 2'' spectrum of galaxy 230776, with the best-fitting templates when using the MILES stellar library (red), the FSPS models (blue), and the Vazdekis models (green). Both the FSPS and Vazdekis models show an obvious residual, whereas the best-fit template with the MILES library stars shows very little residual.

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SAMI galaxies are selected to have a large range in mass and star formation activity, for which we expect mixed stellar populations. In light of the results shown here, we therefore decided to use the MILES stellar library for deriving optimal templates as opposed to using SPS models.

Small errors in the flux calibration can create possible mismatches between the stellar continuum emission from the observed galaxy spectrum and the stellar template, that could impact the estimated kinematic parameters. Here, we test which additive order Legendre polynomial is needed for the SAMI data in order to account for flux calibration errors. Multiplicative polynomials are not used here, because these could change the depth of the line as a function of wavelength, which would impact the measurement of kurtosis (h₄) of the absorption lines. For the ATLAS${}^{3{\rm{D}}}$ data, a fourth order additive polynomial was used over a wavelength range of 113 Å. With the SAMI wavelength range of ∼3650 Å, this would imply a 32nd order polynomial. Every added polynomial order makes the fit more time consuming, which is why we aim to use the lowest order possible while still correcting for possible flux calibration errors.

Using the test sample, for each galaxy the central circular 2'' spectrum is fit with a range in additive polynomials of 1–20. We then attempt to find the additive Legendre polynomial degree that gives consistent and stable results, as based on the residuals, velocity, velocity dispersion, and reduced ${\chi }^{2}$. Figure 26 shows the observed flux of galaxy 47500 in the central circular 2'' in black, the optimal template in gray, and the best-fit template with a 12th order polynomial in red. Below the galaxy spectrum, the additive polynomials are shown as derived by pPXF for the optimal fit. Different colors indicate different polynomial degrees, and below the polynomials we show the fit residual (galaxy—best-fit template). A clear non-constant residual, that varies with wavelength, appears when a second order polynomial is used. This non-constant residual disappears when a 12th order polynomial or higher is used.

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In Figure 27, we show the stellar velocity, velocity dispersion, and reduced ${\chi }^{2}$ versus the additive order of the polynomials, where the data are from the stellar kinematic fits on all galaxies in the test sample. In each panel, and for each galaxy individually, the data are normalized to the mean of the five fits with the highest order polynomials. For ${V}_{{\rm{m}}2}$ and ${\sigma }_{{\rm{m}}2}$, the scatter decreases as a function of polynomial order. The least amount of scatter is reached when we use a polynomial with order $\gt 12$. The reduced ${\chi }^{2}$ also decreases as a function of polynomial order up to order 12, after which the ${\chi }^{2}$ no longer varies. We conclude that an additive Legendre polynomial with order 12 suffices to account for possible mismatches between the stellar continuum emission from the observed galaxy spectrum and the template.

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Fitting high-order moments in spectra can be problematic when the S/N is low or if the velocity dispersion is close to, or lower than, the instrumental resolution (see e.g., Cappellari et al. 2011a). pPXF was designed to employ a maximum penalized likelihood, i.e., forcing a solution to a Gaussian if the high-order moments are unconstrained by the data (Cappellari & Emsellem 2004). Whereas pPXF can derive an automatic penalizing bias value based on the ${\chi }^{2}$, but this is often too strict. A penalizing bias value that is too high can systematically offset the recovered velocity and velocity dispersion so should be avoided if possible. Therefore, we have derived the optimal penalizing bias value as a function of S/N for SAMI spectra from MC simulations.

From the MILES-based optimal template for galaxy 215292, we create a representative template galaxy spectrum. This spectrum is rebinned onto a logarithmic wavelength scale with constant velocity spacing, using the code log_rebin provided with the pPXF package. A large ensemble of mock galaxy spectra is created by convolving the template spectrum with a LOSVD. For the LOSVD, we use h₃ = 0.1, h₄ = 0.1, and 2500 random velocity dispersions between 0 and 200 km s⁻¹, with random velocities between −50 and 50 km s⁻¹. Random Gaussian noise is added to the spectra to obtain a full sample of mock galaxy spectra with S/N of 5–100 Å⁻¹.

pPXF is used to measure the LOSVD from the simulated galaxy spectra. The program is limited to only one template, because we intend to study the impact of the penalizing bias and S/N, not template mismatch. A fourth order additive Legendre polynomial removes possible continuum mismatching between the templates and the galaxy.

In Figure 28, we demonstrate the impact of the penalizing bias value on the recovered LOSVD for an S/N = 20 Å⁻¹, in Figure 29 the impact of the S/N on the recovered LOSVD with the optimal penalizing bias for SAMI applied is shown. The solid colored lines are the median of MC simulations, whereas the dotted lines show the 16th and the 84th percentile ($1\sigma$). In the two left panels, we present the differences between the measured values and the input values of the velocity dispersion ${\sigma }_{{in}}$. In the panels on the right, the recovered values of h₃ and h₄ are shown than the input values of 0.1 (dashed line). Note that the curves in the following figures have been smoothed with a box-car filter (10 km s⁻¹), in order to wash out numerical noise due to a limiting number of MC simulations.

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Figures 28 (a)–(d) shows the stellar kinematic parameters and recovered LOSVD for six different penalizing bias values. In Figure 28(d), we see that when the penalizing bias is higher, the fit is more penalized toward ${h}_{3}=0$ and ${h}_{4}=0$. A penalizing bias value that is too high will also systematically offset the recovered velocity and velocity dispersion (Figure 28(b) around $\sigma =50$ km s⁻¹). An optimal penalizing bias value, however, will reduce the scatter in the velocity dispersion, h₃ and h₄, without creating a systematic offset in the velocity and velocity dispersion. For an S/N = 20 Å⁻¹, we find that the optimal bias value is between 0.04 and 0.06.

For 13 different S/N values between 0 and 100 Å⁻¹, we estimated the optimal penalizing bias value (Figures 29(a)–(d)). The different colored lines show the recovered LOSVD for six S/N values and their optimal bias-value. For spectra with S/N < 20 Å⁻¹ the S/N is too low for an accurate measurement or the high-order moments h₃ and h₄. With an S/N = 20 Å⁻¹, at $\sigma =200$ km s⁻¹ the typical uncertainties are ${v}_{\mathrm{err}}=8$ km s⁻¹, ${\sigma }_{\mathrm{err}}=12.5$ km s⁻¹, ${h}_{3,\mathrm{error}}=0.04$, ${h}_{4,\mathrm{error}}=0.05$. Below $\sigma =70$ km s⁻¹ the systematic offset in h₃ and h₄ becomes too large ($\gt 0.02$) for reliable estimates.

We use the results from the MC simulation to derive a relation between the optimal penalizing bias value and the S/N. Figure 30 shows the optimal bias values as a function of S/N, together with a second order polynomial fit. From the best-fit relation, we obtain a simple analytic expression for the ideal penalizing bias as a function of S/N:

$$\begin{eqnarray}&&\mathrm{Bias}=0.0136+0.0023{({\rm{S}}/{\rm{N}})-0.000009({\rm{S}}/{\rm{N}})}^{2}.\end{eqnarray} \tag{ 11 }$$

Note that our penalizing bias values are significantly lower as compared to the ATLAS${}^{3{\rm{D}}}$ values Cappellari et al. (2011a). This can be explained, however, by the larger wavelength range and higher instrumental resolution of the SAMI instrument.

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