Stellar Population Synthesis with Distinct Kinematics: Multiage Asymmetric Drift in SDSS-IV MaNGA Galaxies

We selected seven galaxies from the MaNGA survey to develop and test the algorithms in this study. The galaxies were chosen to have exceptionally regular gas and stellar kinematics (in terms of azimuthal symmetry, with little evidence for bar distortions), moderate inclinations (20° < i < 50°), and clear evidence for AD between their (single-component) stellar velocities and ionized gas (as reckoned by the MaNGA data analysis pipeline data products, hereafter DAP). This list is by no means exhaustive of good candidates.

These galaxies were also chosen to have stellar masses and colors similar to the MW.¹⁰ Photometric properties of these galaxies, sorted by stellar mass, are summarized in Table 1, where redshift, b/a (minor to major axis ratio), half-light radius, rest-frame g − r and NUV-i colors, SDSS r- and i-band absolute magnitude, stellar mass, and Sérsic index (n_S) are from the NASA-Sloan Atlas (NSA; Blanton et al. 2011).¹¹ Except for n_S, all photometric quantities report their elliptical Petrosian aperture measurements. Magnitudes and colors have AB zero-points; absolute magnitudes and masses are scaled to H₀ = 100 $\mathrm{km}\,{{\rm{s}}}^{-1}$ Mpc⁻¹ throughout. A value of n_S = 6 is a hard upper limit in the Blanton et al. (2011) analysis. The NUV-i color range is blueward of the red sequence (e.g., Figure 2 of Yan et al. 2016a) but in the redder half of the blue cloud. As we will see, the sample has peak rotation speeds between 200 to 300 $\mathrm{km}\,{{\rm{s}}}^{-1}$, with both slow- and fast-rising rotation curves.

Table 1.  Galaxy Sample

MID	Plate-IFU	Redshift	b/a	Re	${\left(g-r\right)}_{0}$	Mr	${\left(\mathrm{NUV}-i\right)}_{0}$	Mi	M*	nS
				(kpc)	(mag)	(mag)	(mag)	(mag)	(1010 M⊙)
1–339041	8138–12704	0.031	0.69	6.7	0.67	−21.73	4.06	−22.09	6.62	4.14
1–209537	8486–12701	0.038	0.63	8.6	0.74	−21.44	4.52	−21.79	5.27	6.00
1–532459	8320–9102	0.052	0.67	7.6	0.60	−21.63	3.07	−21.90	4.77	2.72
1–251279	8332–12705	0.033	0.81	3.6	0.69	−21.16	4.10	−21.52	3.88	3.33
1–542358	8482–3702	0.040	0.93	3.9	0.63	−21.25	3.62	−21.58	3.74	3.19
1–265988	8329–6103	0.031	0.64	4.0	0.62	−20.30	3.68	−20.63	1.65	1.47
1–209199	8485–9102	0.026	0.72	4.1	0.56	−20.33	3.30	−20.68	1.61	2.26

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We refer to these galaxies by their plate-IFU designation throughout the paper. Table 1 provides their corresponding, unique MaNGA ID (MID).

For the spectral analysis throughout this paper, we use the "LOGCUBE" data cube from the data reduction pipeline (DRP) of MaNGA (Law et al. 2016) from an internal release¹² closely related to the versions in Data Release (DR)14 (Abolfathi et al. 2018). We spot-checked our analysis to verify that subsequent subtle changes in the data-processing through DR15 (Aguado et al. 2019) and the next internal data release¹³ do not alter the results in any significant way. Given the significant investment in MCMC analysis, we have retained the results from the older release.

MaNGA data cubes contain the reduced and combined spectra from the dithered IFU observations of a single galaxy. Spaxels have been resampled to a logarithmic bin of 70 $\mathrm{km}\,{{\rm{s}}}^{-1}$ and to a common wavelength grid.¹⁴ Data cubes also contain information on the mean and variance of the spectral line spread function (LSF). We use this LSF to match the resolution of our spectral templates to the data, spaxel by spaxel, before fitting. We measure the S/N of individual spaxels in MaNGA data cubes using the median of the spectra and inverse variance (IVAR in the data cube) within the wavelength range of 360–940 nm. In our analysis we ignore spaxels with S/N per angstrom less than 1.

Throughout this paper we conduct full-spectrum fitting of the observed spectra between 350 and 940 nm. The extensive wavelength coverage yields a wide range of stellar features including the full Balmer series dominated by hot young stars as well as strong stellar features from Ca H and K to the NIR triplet, which are dominated by cool stars; together, these features should contain significant information on the kinematics of the young and old populations.

Full-spectrum fitting uses pPXF "The Full-spectrum fitting is conducted using the pPXF code." This code fits an observed spectrum with a set of spectral templates, convolved with a line-of-sight velocity distribution (LOSVD) in pixel-space. Throughout this study, we fit for the velocity and velocity dispersion moments of the LOSVD of all kinematic components. The code allows multiple kinematic components to be fit simultaneously. In our analysis we have leveraged this to fit the observed galaxy spectrum with a single component for the observed gas emission features and a single or multiple stellar kinematic components.

During our full-spectrum fitting of observed galaxy spectra, we fit for the following gas emission features: Hα, Hβ, Hγ, Hδ, [O ii]λ3727, [O ii]λ3729, [O iii]λ4959, [O iii]λ5007, [O i]λ6300, [O i]λ6364, [N ii]λ6548, [N ii]λ6584, [S ii]λ61716, and [S ii]λ6731. The gas emission lines are modeled as Gaussian features and are treated as a single kinematic component independent of that measured for the stellar component(s). While we fix the relative flux ratio between the [O iii], [O i] and [N ii] doublets based on atomic physics, the fluxes of the gas emission lines are free parameters during the fitting process.

When fitting two stellar components, we take advantage of the pPXF feature to constrain the relative weight of two kinematic components, referred to in the code as fraction but referred to as yfrac in this study. This parameter is a light-weighted quantity defined as the relative weight in the first kinematic component to that in all components:

$\begin{eqnarray}&&{\mathtt{yfrac}}=\displaystyle \frac{\sum {w}_{i}}{\sum {w}_{i}+\sum {w}_{j}}\end{eqnarray} \tag{ 1 }$

where w_i is the weight of the first component while w_j is the weight of second component. In our analysis the first component will always correspond to the young stellar population and hence the use of the subscript "y." We flux normalize the template spectra for our stellar components and hence, if the first (second) component represents the young (old) stellar population, yfrac is a relative indicator of star formation history (SFH). Though by default pPXF limits the user to constrain the relative weight of two kinematic components, in the course of our analysis, we removed this limit to allow for the development the algorithm in Section 6. This allowed us to constrain the relative weight of more than two kinematic components simultaneously during our full-spectrum fitting by providing a vector of relative fractional values (frac_n for the n^th element), i.e.,

$\begin{eqnarray}&&{{\mathtt{frac}}}_{n}=\displaystyle \frac{\sum {w}_{n}}{{\sum }_{k=1}^{c}\sum {w}_{k}},n=[1,2,...,c]\end{eqnarray} \tag{ 2 }$

where c is the number of kinematic components being fit, $\sum {w}_{n}$ is the sum of the weights of the n^th component, and ${\sum }_{k=1}^{c}\sum {w}_{k}$ is the sum of the weights of all components. Throughout our study, we allow the relative weight of the gas component to be free and unconstrained by yfrac.

The pPXF code can also minimize errors caused by imperfect spectral calibration, template mismatch, or scattered light by fitting multiplicative and/or additive Legendre polynomials or trigonometric series. In our analysis we simultaneously use additive and multiplicative polynomials of order eight when fitting MaNGA galaxy spectra for their stellar kinematics with independent stellar population templates.¹⁵ When fitting mock spectra using the same stellar population templates, we do not use any polynomials.

For our analysis, we adopt the SSPs of MIUSCAT (Vazdekis et al. 2012) as our template spectra for the stellar components. These models were developed using empirical stellar spectral libraries, particularly the Medium resolution INT Library of Empirical Spectra (MILES; Falcón-Barroso et al. 2011), Indo-US Library of Coudé Feed Stellar spectra (Valdes et al. 2004), and the CaT Stellar Library (Cenarro et al. 2001). These models were selected primarily due to their significant overlap in rest-frame wavelength coverage with MaNGA. The 346.5–946.9 nm span for MIUSCAT almost exactly matches the 360–1030 nm span for MaNGA at the typical redshift of the MaNGA sample. The spectral resolution (0.251 nm FWHM) is slightly higher at all wavelengths than that of the MaNGA data (see Law et al. 2016; Yan et al. 2016a, respectively, their Figures 18 and 20), which is important for properly convolving the data to the MaNGA instrumental resolution for kinematic analysis, as described in Westfall et al. (2019). MIUSCAT SSPs used in our analysis cover an age range 0.03–14 Gyr in 53 steps, metallicities ([M/H]) −0.66, −0.35, −0.25, 0.06, 0.15, and 0.26 (without α-enhancement), and use the BaSTI isochrones (Pietrinferni et al. 2004) and a Kroupa initial mass function (IMF; Kroupa 2001). The metallicity range of these SSPs encompasses the observed range of metallicities seen in large integral-field surveys sampling all galaxy types over a wide range of luminosity and radius (e.g., Sánchez-Blázquez et al. 2014; González Delgado et al. 2015; Zheng et al. 2017).

For the old stellar population, we use SSPs older than 1.5 Gyr as template spectra, and the remaining SSPs to model the young stellar population. This choice is depicted in Figure 1. Here we conduct a light-weighted full-spectrum fit of each MIUSCAT SSP with a set of empirical stellar spectra representative of a broad range of stellar spectral types (see Appendix A) and sum the weights for the presented bins of stellar types. The plot demonstrates that except for the lowest-metallicity models, the relative weight in hot stars (O-, B-, and A-type) falls below 10% above the age of 1.5 Gyr, while the coolest stars begin to contribute substantially (>30%) after this time, namely the timescale for the formation of the red-giant branch.

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It is also the case that the intermediate stars (F and G) peak around 1 Gyr. While the hottest of these (F0V) will still have significant Balmer absorption and main-sequence (MS) lifetimes of roughly 3 Gyr, for regions with ongoing or recent star formation, the F-star contribution to the Balmer absorption equivalent width will be small compared to contributions from hotter, more luminous MS stars. Nonetheless, based on this Figure, an argument can be made for an intermediate age range between 0.5 and 3 Gyr, particularly for modeling galaxies or regions of galaxies that have not had recent star formation. We take advantage of the multimodal distributions in Figure 1 to constrain the mix of stars used in the stellar library algorithms (Section 6.4). However, while an additional time bin would greatly aid in determining the AVR, for simplicity of our development in this work, we distinguish only between two age bins for measuring differential AD signals.

In the MW solar cylinder from Aumer & Binney (2009) or from MW and M33 star clusters (Beasley et al. 2015), there is observed to be roughly a factor of three change in velocity dispersion between t = 0 and 1.5 Gyr (assuming the birth population has the same dispersion as the molecular gas—something that is not clearly the case for the star-cluster measurements), and about a factor of two increase between 1.5 and 10 Gyr. In M31 (Dorman et al. 2015), the increase in velocity dispersion is roughly the same during these two time periods. Hence our choice of age bins appears a sensible starting point if the three large galaxies in the Local Group are representative of the larger population of intermediate-to-massive spiral disks.

We applied the MCMC algorithm to our sample of seven galaxies on a spaxel-by-spaxel basis. Figure 2 shows the measured, line-of-sight (projected) velocity and velocity dispersion maps for one galaxy for young and old stellar components in the second to fourth columns of the top row. (Maps for the remaining six galaxies in our sample are shown in Appendix D.) There is no smoothing applied to these maps; the color of each data point in the image corresponds to the kinematics of individual spaxels. Blank spaxels in these maps within the MaNGA IFU footprint are either regions masked in the DRP as foreground stars or, in the case of the MCMC only, represent spaxels where the MCMC did not converge within the maximum run-time permitted for jobs on the computer cluster to which we had access.¹⁷ For comparison we also show the derived velocity and velocity dispersion maps for gas and stars for a single stellar component. (In later sections these serve as the initial fitting step in multistep algorithms.) These were measured directly from pPXF. Note the coherency of the velocity fields with the expected spider-diagram of isovels, as well as the smoothness of the kinematics on scales much larger than the beam size (2 arcsec fiber diameter). This smoothness and coherency persists for both single and two-component stellar kinematics. Keeping in mind that the kinematics are derived independently for each spaxel, this coherency indicates that our two-component stellar solutions from the MCMC algorithm are robust and measure astrophysical kinematic signals. While there is much more scatter in the velocity dispersion maps, our approach is to avoid using these and higher kinematic moments.¹⁸ It is reassuring nonetheless to see that on average the younger component has lower dispersions than the older component as anticipated, and the old component has the qualitatively expected radial decline in value. These trends are seen for all galaxies in our sample.

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While the AD signal is readily apparent from Figure 2 to the practiced eye, this is more easily seen by extracting rotation curves using some fiducial geometry. Accordingly, Figure 3 shows the results of our MCMC algorithm in decoupling the deprojected tangential speeds of young and old stellar populations. In most cases and at most radii, we see that both young and old stellar components lag in their tangential speed relative to the ionized gas (i.e., this is AD), with the old-component lag being larger, as one would expect from the presence of AVR. For comparison we show the deprojected tangential speeds for gas and stars for a single stellar component (again, derived directly from pPXF), from which the AD signal also is readily apparent.

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For Figure 3 we adopt geometric parameters derived from full, two-dimensional kinematic modeling of a monolithic inclined-disk. We use the method described in Westfall et al. (2011) and Andersen & Bershady (2013), applied simultaneously to the DAP gas and stellar kinematics. These parameters are summarized in Table 2. However, these geometric parameters are only used to define radial and azimuthal bins and deproject the kinematics; the individual fits to each spaxel are independent of these geometric parameters. Consequently the rotation curves in Figure 3 make no assumptions about kinematic or rotation-curve models except insofar as they share the geometries given in Table 2. Figure 2 and the associated Figures in Appendix D are completely independent of any assumed geometric or rotation-curve model.

We focus on two fairly extreme cases: 8138–12704 and 8329–6103. Both galaxies have S/N above 10 Å⁻¹ even at large radii, and yfrac values between 0.1 to 0.2 (in the central regions) and 0.4 to 0.55 in the outer regions. The radial trend in yfrac is qualitatively what is to be expected for age gradients in spiral galaxies where the older population is centrally more concentrated.

The former galaxy 8138–12704 was chosen because of its exceptionally large AD signal as measured for a single stellar kinematic component with respect to the ionized gas, yet clear presence of young and old stellar populations. At R = R_e, D_n(4000) = 1.42 while V_gas − V_stars = 41.8 $\mathrm{km}\,{{\rm{s}}}^{-1}$. Of a sample of nearly 500 galaxies in MPL-5 selected to have regular kinematics and moderate inclinations (see Section 2.1), this galaxy is in the upper quartile in terms of deprojected gas rotation speed (V_gas = 304.8 $\mathrm{km}\,{{\rm{s}}}^{-1}$). Since in general the AD signal scales with velocity, it is useful to also look at the fractional difference between tangential speed of the stars and gas relative to the gas (1 − V_stars/V_gas). For 8138–12704, this fraction is 13.7%, and it too is in the upper quartile.

In contrast 8329–6103 has an AD signal that is in the lowest quartile for the sample of galaxies in both an absolute and relative sense: V_gas − V_stars = 11.8 $\mathrm{km}\,{{\rm{s}}}^{-1}$, and 1 − V_stars/V_gas is only 6%. The galaxy has about 25% of the total stellar mass and 41% of the dynamical mass at R = R_e (V_gas = 194.4$\mathrm{km}\,{{\rm{s}}}^{-1}$) compared to 8138–12704, yet has a comparable D_n(4000) of 1.36. Indeed, our analysis shows the two galaxies have yfrac values of 0.42 and 0.53, respectively. Compared to the MW (e.g., Licquia et al. 2016), 8329–6103 is almost a factor of two less luminous.

For the more massive galaxy, 8138–12704, the differential AD signal is readily apparent at all radii sampled. To quantify this excellent kinematic separation of young and old stellar populations, Figure 4 shows the bivariate PPD for velocity and velocity dispersion of two spaxels selected from near the major axis. One spaxel is on the approaching side at 0.3 R_e (3.5 arcsec radius) and the other is on the receding side at 1.1 R_e (12 arcsec radius). In both cases the peaks and 99.7% contours are well separated with the relative velocities and dispersions for the two populations. Moreover, they are differentiated as expected toward higher dispersions and lower speeds for the old population and vice versa for the younger population. While this is consistent with our priors, the presence of these distinct, age-dependent kinematics independently determined for all spaxels using an unconstrained model suggests there is ample information to extract a quantitative measure consistent with our qualitative astrophysical expectations.

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For the less-massive galaxy 8329–6103, Figure 3 shows that the kinematics for the young and old populations are differentiated at smaller radii, as expected, but at larger radii, the differentiation is less evident. This is reflected in Figure 4 as well. One spaxel is on the receding side at 0.6 R_e (4 arcsec radius) and the other is on the approaching side at 1.3 R_e (8.5 arcsec radius). For the inner spaxel, we see the anticipated velocity difference between the young and old populations, although the velocity dispersions are flipped in the opposite sense of what we might physically expect, i.e., the faster-rotating (young component) has a larger dispersion. However, the 67% probability contours are quite broad, and both components are at, or well below, the instrumental resolution. At the larger radius, the young stellar component continues to appear to rotate more quickly, even though the probability contours substantially overlap. The velocity dispersions are well below the instrumental resolution in this limit.

These results are promising, indicating we should be able to derive separate kinematics for most of the MaNGA sample that have regular disk kinematics. Clearly Figure 3 shows some irregularities in the tangential speeds of young and old components at large radii, which correspond to low S/Ns. This behavior reflects an increasing frequency with decreasing signal-to-noise where the measured kinematics of the young stellar populations are dynamically hotter and rotate slower than the old stellar component. In Appendix B, we demonstrate that this behavior is due to degeneracies in disentangling the kinematics of the young and old stellar populations in this regime when using a global minimizer (i.e., MCMC) with a very broad range of template age and metallicity. Our subsequent algorithms that use a local minimizer with carefully constructed initial conditions largely eliminate this problem.

The simplest scheme for measuring the kinematics of two discrete stellar population components consists of giving pPXF the set of SSPs described in Section 2. We again split the assignment of stellar population components at 1.5 Gyr regardless of metallicity. There are no constraints placed on yfrac; pPXF is free to assign weights to the templates as needed to minimize χ². A single full-spectrum fit to the galaxy spectrum is made with these two (young and old) stellar kinematic components and, in the case of real galaxy spectra (as opposed to emission-line free mocks), a kinematically independent gas component. The initial velocities and velocity dispersions for the two stellar components are identical and are set at 0 and 210 $\mathrm{km}\,{{\rm{s}}}^{-1}$, respectively.

This simple algorithm performs well on our mock spectra (metric 1), yielding derived AD signals within 10% of the expected value 67% of the time even for AD signals as low as 5–10 $\mathrm{km}\,{{\rm{s}}}^{-1}$ (see Appendix C.1 and Table C1). While these results look promising, the test is highly idealized, absent of noise, and uses the same templates in the mocks and the fitting templates. Consequently, we turn to apply this simple algorithm to real MaNGA spectra as a more definitive performance test. In applying the simple SSP algorithm to real galaxy data, we fit for the gas kinematic component simultaneously along with the stellar kinematic components and match the LSF of the SSP templates to that of the spaxels.¹⁹

Application of the simple SSP algorithm to 8138–12704 (Section 3) indicates that the algorithm does not appear to reliably converge to the kinematics measured by the MCMC (metric 2). In Figure 2, we observe that for both the MCMC results and the simple SSP algorithm, the younger component appears to have on average faster rotation than its older counterpart, as expected. However, the young velocities for the simple SSP algorithm are systematically lower (in amplitude) than those derived from MCMC, while the older velocities are systematically higher; both young and old velocity fields are less smooth as well. Further, the σ maps are noisy, particularly for the young component, and lack the expected structure of being azimuthally smooth with a radial decline. These qualitative conclusions for 8138–12704 are also found in the remaining six galaxies in our sample (Appendix D). This is quantified in the first three lines of Table 3.

Given that the more robust MCMC technique derived a solution for the kinematics different from our simple SSP algorithm, this suggests that the spectral fitting code (pPXF), which uses a Levenberg–Marquardt technique to find the local χ² minima, is not able to converge to the global minima with real data.

To circumvent the pitfalls of a complex χ² terrain, we augment our simple SSP algorithm by adding an additional fitting step. This additional step allows us to improve initial conditions to position pPXF fairly close to where the global minimum is expected.

In a new first step, we conduct a full-spectrum fit with only one stellar kinematic component and one gas emission-line component, using all SSPs (regardless of age) as templates for the single stellar component. This fit is essentially a stellar population synthesis of the spaxel and, accordingly, we use only multiplicative Legendre polynomials during this fit. Expectations based on the MW AVR are that the kinematics of the young stellar population should be close to that of the gas. Simultaneously the kinematics of the old stellar component should be more similar to that from a single stellar component than the gas kinematics. Results from our MCMC analysis support these expectations. Hence, in the second fit, we use the derived gas kinematics from our first fit as the initial conditions for the young component, and the derived stellar kinematics from our first fit as the initial conditions for the old component. During this second fit, we fix the kinematics of the gas to that derived from the first fit and use additive and multiplicative polynomials during the fit. We refer to this algorithm as "2-Step SSP" in Table 3.

As seen in Table 3, this two-step SSP algorithm exhibits some modest quantitative improvement in the derived kinematics compared to the simple SSP algorithm. This demonstrates that the irregular χ² space of the solution affects the reliability of the kinematic decomposition unless reasonable initial conditions can be placed on the kinematics. However, some observable differences persist in the velocities of the young component. The discrepancy suggests a lack of full convergence of the two-step SSP algorithms to the global minima of the χ² space identified by the MCMC analysis. We find that although yfrac determined in the first step of the two-step SSP algorithm correlates well with that determined by the MCMC analysis, the relation has significant scatter; better initial conditions on yfrac could also be beneficial. Previously Katkov & Chilingarian (2012) used an MCMC technique to demonstrate that when conducting a spectral decomposition of co-spatial components, an incorrect estimation of the relative light contribution of the components could bias the results.

To test if constraining the relative weights of the young and old components further improves the results, we provide a data-driven constraint on yfrac by using the template weights from the first full-spectrum fit with a single stellar component. From this weight distribution, yfrac is set in the second fitting step to the sum of the weights associated with the young SSP models (≤1.5 Gyr) over the sum of the weights of all SSPs. This constraint is imposed independently for each spaxel. We then add a third full-spectrum fitting step to our SSP algorithm where we (i) update the initial conditions for the kinematics to that measured in the second step, but we (ii) remove the constraint on yfrac for the fit. The motivation for this step is to assume that the kinematic solution obtained in the second step is close enough to the global minima that the final step can free the fit to converge onto the global minima without constrains put in place on yfrac. We refer to this algorithm as "3-Step SSP" in Table 3. It is immediately evident that across the board, all of the metrics improve significantly, and hence we adopt this as our final SSP algorithm. Figure 2 and Appendix D show the velocity and velocity dispersion fields for MCMC, simple, and three-step SSP algorithms.

The aim of using a multistep approach to nudge pPXF toward a global minimum using iteratively improved initial conditions appears to be on target. In contrast, the initial two-step SSP algorithm had less refined initial conditions on the kinematics, and produced nosier kinematic maps for the two stellar components—compared both to this three-step algorithm and the MCMC results.

Our final SSP algorithm used to measure the two-component AD is a three-step process as follows:

• 1.  
  Use full spectral fitting with pPXF with one stellar and one gas (emission-line) kinematic component. The entire suite of SSP models is used as templates for the single stellar component. Since each SSP is normalized, the resulting spectral fitting weight distribution is light-weighted. The gas emission-line components assume Gaussian line profiles, and the kinematics for all lines are tied to a single value for velocity and dispersion per spaxel. During this fit, the initial conditions for the kinematic for all components are zero velocity with respect to the NSA redshift and velocity dispersion of 210 $\mathrm{km}\,{{\rm{s}}}^{-1}$. All stellar kinematic fitting is done assuming a Gaussian LOSVD and multiplicative Legendre polynomials of order eight are used to match continuum shapes between observed and template spectra. The resulting kinematics and weight distribution of this fit provide informed constraints for the second fitting step.
• 2.  
  Use full spectral fitting with pPXF with two stellar kinematic components (young and old) and one gas (emission-line) kinematic component. The young kinematic component is restricted to use only SSP models with ages ≤1.5 Gyr, while the remainder are used as templates for the old component; yfrac is constrained by the ratio of these weights derived in the first step. The initial kinematics of the young component is set to the derived gas kinematics from the first fit, and the old component is set to those of the single stellar component. The kinematics of the gas component in this second fit is constrained to that derived by the previous fit; however, the fitting code is free to optimize the relative weights of the different emission-line features. Similarly, pPXF is allowed to optimize the relative weights of SSPs within young and old age bins, but the sum of these weights is constrained by yfrac from the first fit.
• 3.  
  Repeat the full spectral fitting in the same setup as the previous step with the initial kinematics of the stellar components updated to those derived in step 2, but with no constraint on yfrac. This full-spectrum fitting provides the final measured kinematics for the two stellar components, and the weights in the two components provides the yfrac in the spaxel.

We use a representative subset of 51 stars from the empirical Indo-US Library of Coudé Feed Stellar Spectra (Valdes et al. 2004). As described in Appendix A, these are well suited for our purposes of analyzing MaNGA spectra. Similar to what was done for the SSP algorithms, we begin by defining the templates for the two age components being fit.

In the context of SSPs (Section 2.5), we define the young component templates to be those containing significant Balmer features. For a stellar library, however, this definition excludes stars cooler than late-F, which we know are present in young stellar populations for any plausible present-day stellar IMF. This is quantified in Figure 7. Since the younger populations have both hot and cool stars, with metal lines contributed to the spectra by the latter, we select three ranges that always include the hottest stars but extended to progressively cooler limits in cases labeled A, B, and C. Our aim is to include the smallest possible range of cool stars to minimize degeneracy with the older stellar populations, and hence simplify the fitting algorithm.

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The choice of stellar templates for the old component is simpler since hot, main-sequence stars are short lived and hence only present in young stellar populations, where they dominate the spectral continuum in the blue and visible portion of the spectrum. We therefore exclude hot stars (hotter than F) from the templates for the old component, effectively ignoring the possibility of extreme (very metal-poor) blue horizontal-branch stars from contributing significantly to the integrated light of a mixed stellar population. In one case (D), we restrict the older population to have only K- and M-type stars to understand if there is any sensitivity to the giant-branch effective temperature.

Table 4 summarizes the four distinct combinations of stellar templates we explore for the young and old stellar components. These template combinations place no restrictions on luminosity class.

As a starting point for the development of our algorithm, we test the following simple algorithms to identify pitfalls to be ameliorated by more complex techniques. These pitfalls highlight subtle but important aspects of fitting multiple kinematic components where velocities are offset by values comparable to the spectral resolution. There are no constraints on yfrac in any of these algorithms.

• 1.  
  Full-spectrum fitting (full): pPXF fits two kinematic components to the provided (mock or real) spectrum over the full wavelength range using the above mentioned template sets.
• 2.  
  Feature fitting (feature): based on the knowledge that we expect to see hot stars only within young stellar populations, the kinematics of the young component are fit with pPXF using only wavelength regions of ± 1500 $\mathrm{km}\,{{\rm{s}}}^{-1}$ about Balmer lines up to n = 18 in the spectra while masking out the remainder, as shown in Figure 8. Because of the strong contamination from Ca H, we exclude H; and because of the G band (CN), we only use ± 500 $\mathrm{km}\,{{\rm{s}}}^{-1}$ for Hγ. Given the ± 1500 $\mathrm{km}\,{{\rm{s}}}^{-1}$ velocity width, the region containing Hη (n = 9) and above form a continuous band from 385.36 to 367.21 nm. During this fit, two kinematic components are still applied in pPXF despite the dominance of Balmer features in these regions for young populations, and there remains an imprint of the kinematics of the older population in these regions, both in the Balmer lines as well as the metal lines within these windows. This fit is then followed by a fit using pPXF that excludes the Balmer regions. During this second fit, the velocity moments of the young component are kept fixed to the values derived from the first step.

Based on our tests using mock spectra (Appendix C.2 and Table C2), we find that the performance of these two algorithms is comparable. This confirms expectations about where most of spectral information is stored for discriminating between young and old stellar populations. For simplicity and consistency with the SSP algorithms, we adopt the "full" algorithm.

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Table C2 also reveals important performance differences between templates. Velocity systematics for the young component are minimized for template set C, while velocity systematics for the old component are minimized for template set B. Perhaps this is unsurprising given Figures 1 and 7. However it is concerning that neither template set optimizes both young and old-component systematics. A further concern arises with template sets C and D. In these cases there are many occurrences where yfrac > 0.85, i.e., the best-fitting solution yields too large a young component contribution. This is due to the degeneracy in the contribution of cool stars in the young and old components. The pPXF likelihood minimization falls into a local minima where the stellar kinematics of the mock spectra is almost entirely modeled by the young component. An example of this is given in Appendix C.2, Figure C2. We conclude that template set B is the best compromise. This leaves us with performance results, based on mocks, that are somewhat unsatisfactory.

Following the success of adding additional steps to improve the kinematic initial conditions with our SSP-based algorithms, we further developed the "full" algorithm using the template set B by adding a step to provide initial conditions in an identical fashion as our two-step SSP algorithm. Foreshadowing further development, we refer to this algorithm as "2-Step 1-Bin SL," where the "1-Bin" designation refers to the grouping of the templates as a single unit for each of the stellar components.

An evaluation of the performance of this algorithm using mocks (Appendix C.2, Table C3) indicates no improvement, but we suspect this is due to systematics between the mocks generated by SSPs and the fits using the stellar library. We turn instead, and henceforth, to a comparison with the MCMC for real data. Velocity and velocity dispersion maps (Figure 2 and Appendix D) show marked qualitative improvement in the smoothness of the kinematic maps between our one-step and two-step stellar library algorithms. Table 5 quantifies this contrast, most pronounced for the young component velocities. There also are performance improvements for the old-component velocities.

Table 5.  SL Algorithm Performance Metrics Referenced to MCMC Results

Algorithm	S/N	${\rm{\Delta }}{V}_{y}$		${\rm{\Delta }}{V}_{o}$		$\delta {V}_{\mathrm{Algo}}/\delta {V}_{\mathrm{mock}}$		${\rm{\Delta }}{\mathtt{yfrac}}$
	Range	Med	${\sigma }_{\mathrm{MAD}}$	Med	σMAD	Med	σMAD	median	σMAD
		($\mathrm{km}\,{{\rm{s}}}^{-1}$)		($\mathrm{km}\,{{\rm{s}}}^{-1}$)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
Simple SL	[10,35]	13.76	51.44	4.02	14.21	0.90	1.18	0.09	0.12
Simple SL	[35,50]	21.50	43.47	8.21	10.88	0.88	1.38	0.13	0.07
Simple SL	>50	69.56	87.29	5.23	8.82	1.05	2.36	0.05	0.05
2-Step 1-Bin SL	[10,35]	1.70	29.94	3.70	15.27	0.73	0.81	0.05	0.12
2-Step 1-Bin SL	[35,50]	1.50	17.25	6.13	7.86	1.00	0.40	0.10	0.07
2-Step 1-Bin SL	>50	0.90	24.86	5.46	6.69	1.09	0.74	0.04	0.06
2-Step 3-Bin SL	[10,35]	−2.02	26.27	6.17	18.38	0.62	0.87	0.01	0.08
2-Step 3-Bin SL	[35,50]	0.66	11.50	5.44	6.54	0.98	0.35	0.05	0.05
2-Step 3-Bin SL	>50	1.45	26.45	7.56	5.54	1.16	0.70	0.03	0.04

Note. SL algorithms (column 1) are defined in Section 6.2 and 6.4. Columns (2) through (10) are defined identically to those in Table 3 using the same spaxels.

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However, further inspection of the kinematic maps reveals that the velocities and velocity dispersions are quite noisy for the young component of galaxies with lower mass (<2 × 10¹⁰ M_⊙) or bluer color (${\left(\mathrm{NUV}-i\right)}_{0}\lt 3$). (These properties are correlated in general, and particularly so for our small sample.) It is also the case that for ∼33% of the spaxels with 10 < S/N < 35, both the simple and two-step one-bin algorithms find very high values of yfrac compared to the MCMC results. This is indicative of an incorrect allocation of cool stars to the young kinematic component in the stellar library algorithms, and suggests that further algorithm improvement is desirable.

Since problems with our stellar library algorithms arise due to incorrect allocation of cool stars to the young kinematic component, we need an objective mechanism to appropriately assign weights for these stars in young and old components. The above stellar library algorithms attempt to steer the allocation of weights via a prescriptive template set (B). Additional constraints on the relative weights of the young and old component (yfrac) fail to improve algorithm performance with nearly all our quantitative metrics. Further improvement requires a better guid.

An astrophysically motivated mechanism to properly weight stellar contributions to young and old stellar populations is to assign priors based on what we would expect from stellar evolution. A limiting case is just to use SSPs, but since the entire exercise here is to not restrict ourselves to the specific weights produced by the mapping of theoretical isochrones to a template library, we aim to impose a less restrictive scheme that still is informed by the relative weights of different stellar types as we would expect from stellar evolution. As we show, it is possible to do this by coarsely categorizing stellar types in effective temperature while leaving pPXF the freedom to optimize template weights within these coarse (broad) categories.

Figure 7 shows the derived weights for our stellar library from fitting SSPs with pPXF. These results broadly match our expectations for stellar evolution and the impact of metallicity on, e.g., the temperature of the giant branch. The youngest populations, dominated by the hottest stars, indeed have most of their weight in the B and A templates, but do contain significant weight in cooler stars. As the populations age (and the main sequence burns down while the asymptotic and then red-giant branches are populated), the relative weight of the cooler stars gradually increases and the weight of the hotter stars decreases. The oldest SSPs are entirely dominated by the coldest stars, but the specific age where this occurs depends on metallicity because of the temperature of the stars on the horizontal and giant branches. Hence by constraining the relative weights of the templates used in the kinematic fitting such that they are at least consistent with that expected from stellar evolution at some granularity, we could disentangle the relative weight of the cool stars in the two kinematic components in our fit. What particular granularity should be adopted is one question we answer.

To constrain the relative weights of the stellar templates in the two kinematic components, we use pPXF to do a full-spectrum fit of the target spectrum with SSPs for a single stellar kinematic component (plus a second component for gas, if the target is a real galaxy spectrum; see Section 2). The resulting SSP weight distribution is translated into weights of our stellar templates for a young and old population via a reference look-up table containing the stellar weights derived from fitting our Indo-US stellar subset to each SSP (Figure 7). The weights are divided between young and old populations by summing over SSP weights younger or older than 1.5 Gyr.

While this scheme provides reasonable estimates for individual stellar template weights, the detailed result can be strongly effected by noise in the galaxy spectra. Further, fixing the relative weights is almost equivalent to using the SSP spectra as templates themselves, which defeats the purpose of this algorithm. To relax these constraints, we divided the stellar template into temperature bins. The purpose of this division is to constrain the relative weights only between but not within these bins for young and old populations during the full-spectrum fitting of the galaxy spectrum. The detailed distribution of weights for the individual stellar templates within each bin is a free parameter, tuned (by pPXF) to optimize the likelihood of the solution.

We divide templates into three temperature bins (each contains stars with a range of luminosity classes and metallicity) motivated by the fact that the young and old kinematic components both have cooler stellar types (F–M) as common templates, but the mix of intermediate-temperature stars (F–G) changes strongly with population age (see Figure 1), while only the young component has the hottest stellar types (O–A). The three bins are: hot (which contains templates of stellar types O, B, and A with T_eff roughly above 9000° K), intermediate (F and G with T_eff roughly between 5000° and 9000° K), and cold (K and M roughly below 5000° K). Since the unrealistic distribution of weights for the cool stellar types is causing the catastrophic failure in yfrac when deriving the kinematics of the young and old kinematic components for our simple and two-step one-bin algorithms, this three-bin division fixes the ratio of the contributions of hot, intermediate and cool stars in each kinematic component to reasonable values consistent with both the observed spectra and rough expectations from stellar evolution. Consequently we can dispense with the restrictions of the template sets A–D from Section 6.1. We refer to this as our two-step three-bin stellar library algorithm.

To implement constraints on the relative weights of the bins, we utilize features in pPXF that (i) allow the relative weight to be fixed between multiple kinematic components, and (ii) grant the user the ability to tie together the kinematics of different components. Effectively we fit a set of template bins with tied kinematics for each stellar component. By fixing the relative weights of the bins of the young and old components, we can avoid unrealistic template weights while giving freedom for template optimization. Nominally pPXF limits users to fixing the relative weight between only two kinematic components. For this study, we modified the pPXF code to disable this limit, enabling us to split the templates for each component into as many bins as needed.

We find that the addition of these multiple constraints on yfrac yield significant qualitative improvements in the smoothness of the velocity and velocity dispersion maps for the lowest-mass and bluest galaxies in Appendix D. As seen in these Figures, this stellar library algorithm provides comparable qualitative performance to the SSP algorithm, and based on this, we adopt this three-bin approach. Further comparison with the SSP algorithm is presented in Section 7.

The final algorithm (SL hereafter) can be summarized as a two-step process. The first step is identical to what is described in Section 4.3; it is a full-spectrum fit for a single stellar kinematic component using SSPs. This step is used to constrain the relative weights of empirical stellar spectra via a decomposition of the SSPs into relative weights for the empirical stellar spectra. Henceforth SSPs are not used. A second and final, two-stellar-component fit (also full-spectrum) uses the empirical stellar spectra with weights broadly constrained in three effective-temperature classes (hot, intermediate, and cool). The second step uses a modified version of pPXF with fixed fractional contributions to the total fit for two stellar kinematic components, each with three different subpopulations. By definition, the three subpopulations for each kinematic component share the same kinematics. In detail:

• 1.  
  The relative weights for the hot, intermediate, and cool stellar bins for each (young and old) kinematic component are derived from the decomposition of the SSPs into individual stellar templates, as illustrated in Figure 1. The stellar template weights for the SSPs younger or older than 1.5 Gyr, respectively, are assigned to the young and old stellar components via the look-up table generated from the decomposition of each SSP (Figure 7).
• 2.  
  The initial kinematics of the young component is set to the derived gas kinematics from the initial fit, and those of the old component are set to those of the single stellar component.
• 3.  
  The kinematics of the gas component in this second fit is constrained to that derived in the first fit. The fitting code is free to optimize the relative fluxes of the different emission lines.
• 4.  
  The fitting code is allowed to optimize the relative weights of the stellar templates within each temperature bin for each age component (young and old), but the relative sum of these weights is constrained by the six yfrac values defined above. This flexibility minimizes the effect of template mismatch.

The velocity and velocity dispersion maps in Figure 2 and Appendix D show that our different algorithms display qualitatively similar kinematics. We focus further analysis on kinematics from spaxels within a ±30° wedge of each galaxy's major axis, defined using the geometry in Table 2. Tangential velocities for each galaxy and tracer are corrected for their own systemic velocity determined by minimizing the difference in the approaching and receding components, excluding data at radii less than 2 arcsec. We find that these systemic velocity values are consistent (at the ∼1 $\mathrm{km}\,{{\rm{s}}}^{-1}$ level, on average) with values estimated from full two-dimensional kinematic modeling of a monolithic inclined-disk using the method described in Westfall et al. (2011) and Andersen & Bershady (2013). There are variations between systemic velocities between tracers that are of the order of 3 to 5 $\mathrm{km}\,{{\rm{s}}}^{-1}$. These differences are consistent with random errors, and they are negligible in our overall error budget.

Our two algorithms (SSP, SL) both use pPXF to minimize χ². Because pPXF does not map the shape of the minima in χ² space, these algorithms do not provide a direct estimate of the measurement uncertainty. In contrast, the MCMC analysis we conducted for our sample galaxies does provide this information. From the MCMC, we have marginalized the PPD for the kinematics (velocities, velocity dispersions) and relative light contribution of the young and old stellar populations of the galaxies. As mentioned in Appendix B, the shapes of the PPDs for individual spaxels can sometimes be multimodal, particularly at low S/N. We quantify the PPD width using a standard deviation (rather than, e.g., the median absolute deviation) specifically to include the impact of multi-modality.

We find that the uncertainty in the measured velocities of the two components correlates with the S/N of the spectra and yfrac. Figure 9 illustrates these trends by plotting the median width of the PPDs (${\tilde{w}}_{\mathrm{ppd}}$) as a function of yfrac and S/N. The middle panel shows that for the lowest values of yfrac, the uncertainty in the measured velocity of the young stellar component is the lowest for a given S/N. This is expected given that lower value of yfrac corresponds to a lower contribution of the young stellar component to the observed galaxy spectrum. Likewise, the right panel illustrates that at a given S/N, the reliability of the measured velocity of the old stellar component is higher when yfrac is lower, and the relative contribution of the old stellar component is higher. Both panels illustrate the expected correlation between ${\tilde{w}}_{\mathrm{ppd}}$ and S/N. Since the systematic differences (seen in the next section) and the scatter between methods are smaller than the random errors exhibited in Figure 9, these values are well suited for characterizing the random errors in the application of either the SSP or SL algorithms.

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For reference, at S/N (Å⁻¹) = 10 and yfrac = 0.4, the random errors in the young and old velocities are roughly 30 $\mathrm{km}\,{{\rm{s}}}^{-1}$, while at S/N (Å⁻¹) = 30, the errors are between 5 and 10 $\mathrm{km}\,{{\rm{s}}}^{-1}$ for young and old components, respectively. Online tables of quantities plotted in the middle and left panels of Figure 9 are available.

Figure 10 compares the median difference (spaxel by spaxel) between the derived velocities from our two algorithms binned in S/N and yfrac. This shows that the systematic differences are small in absolute value, but nonetheless nonzero. The SL algorithm's (Section 6.5) measurement of the velocity of the young component is consistent with that of the SSP algorithm (Section 4.3) across S/N and yfrac to within ± 8 $\mathrm{km}\,{{\rm{s}}}^{-1}$, with an overall median difference of −5 $\mathrm{km}\,{{\rm{s}}}^{-1}$ such that the SL algorithm tends to find the young component rotating faster. For the old component, the median difference is 7 $\mathrm{km}\,{{\rm{s}}}^{-1}$ such that the SL algorithm tends to find the old component rotating slower. These systematic differences may reflect template mismatch or systematics in the final yfrac values of the two algorithms. We will further explore the cause of these systematics in future work, but it suffices to conclude here that the systematics are small and have little correlation with S/N or yfrac. As shown in the bottom panels of Figure 10, random errors almost always dominate over systematic differences between algorithms at the spaxel level.

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As a summary measurement of AD for two independent stellar components differentiated by age, we take the mean velocity from our two algorithms (SSP and SL), spaxel by spaxel, as our measure. For each spaxel, we take ${\tilde{w}}_{\mathrm{ppd}}$ as the random error, and half the difference between the velocities from the two algorithms as the systematic error. When averaging the kinematics derived from, e.g., a radial bin of spaxels, we assume the random errors diminish with the usual quadrature averaging, while the systematic errors do not. Systematic errors in the binned data are taken as the median of the systematic error for individual spaxels in the bin.

Figure 11 shows the implementation of this summary scheme to a measure of AD for young and old components relative to the ionized gas velocity for the seven galaxies in our sample. Line ratios (e.g., [O iii]λ5007/Hβ versus [N ii]/Hα) are consistent with H ii–like regions for most galaxies at radii outside several arcsec; hence, ionized gas velocities should be close to the circular speed of the potential. Results from Section 7.2 indicate that young component velocities are unreliable for yfrac < 0.15, and both young- and old-component velocities are unreliable when individual spaxels have S/N < 10. These radial points have been indicated with open symbols. Referring to Figure 3, we see that most of the observed radial range for the galaxies in our sample is above these limits.

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This final figure shows that we can indeed measure a distinct AD for two stellar components for this sample of galaxies. Systematic errors dominate random errors in radial bins, but in most cases and at most radii, the errors are substantially smaller than the difference between the ADs of young and old components. While the difference between ionized gas and the young stellar component is small, it is nonzero (6 ± 2 $\mathrm{km}\,{{\rm{s}}}^{-1}$ in the mean, with a median value of 8 $\mathrm{km}\,{{\rm{s}}}^{-1}$). In all cases the AD signals for young and old components straddle what is measured for a single stellar component, as one would expect. The AD signal for the young component is generally small, with a value of the order of 20 $\mathrm{km}\,{{\rm{s}}}^{-1}$ or less, also as one might expect. The net effect of isolating the two population components by age is to elevate the AD signal for the old component relative to the single-component AD signal.

The results from Figure 11, combined with population age information will be exploited in future work to measure the AVR in MaNGA galaxies. However, it is already interesting to remark on the large range of AD signals observed in our sample of seven galaxies. There is also some indication of a trend with mass: the differential asymmetry drift signal for older populations is largest in the more massive galaxies, but this trend is far less regular (noting the small and large AD signals for 8320–9102 and 8482–3702, respectively) compared to the standard mass–rotation-speed scaling seen in Figure 3. While the addition of age estimates may transform the AD signal into a smoother trend in AVR with mass, the rather uniform g − r colors of our sample suggest that this expectation may be illusory. Given that there are at least three likely astrophysical paths to generating an AVR (see Section 1), perhaps it is unsurprising to see a wide range in AD over such a modest range of stellar mass. Application of our methods to larger samples will provide statistics on the distribution of AD as a function of galaxy mass.

It is worth noting the limitations of the present methods and scope of analysis. We have selected galaxies with regular kinematics and with modest values of yfrac. One may well expect that interpretation of AD measurements will be problematic in cases where the kinematics are less regular or there are significant bar or oval distortions. It is imperative, therefore, to have an assessment of galaxy kinematic regularity, and to limit analysis to galaxies or radial zones within galaxies where the gas is on near-circular orbits and low-order distortions (e.g., m = 1, 2 modes) are weak or absent for both stars and gas.

We have also seen that our algorithms are limited in precision once the continuum S/N falls far below ∼20 Å⁻¹. However, there is nothing preventing the methods described here to be applied in a more sophisticated analysis that fits multiple spaxels simultaneously in a parameterized model to increase the effective S/N. Even a relatively simple tilted-ring approach would likely improve upon the S/N limitations of our current spaxel-by-spaxel scheme, e.g., at larger radii.

Nonetheless, the kinematics of the young component are not well measured for yfrac < 0.2. This means that the AVRs within the centers of galaxies with prominent, old bulge or pseudo-bulge populations, while typically observed at sufficient S/N, are not easily accessible given our current methods. While we have not probed yfrac > 0.8 with our modest sample, we can expect that at least in the outskirts of later-type galaxies, this condition will exist. We can also expect that measuring the kinematics for the old component will be difficult in this regime. Hence there is likely to be a sweet spot for our methods at an intermediate radius, where both young and old disk components are comparably present. Such a sweet spot is likely well matched to a comparison with the MW solar neighborhood.

Finally, while we cannot offer a definitive recommendation between our SSP and SL algorithms, our preference is for the SL algorithm. Despite its greater complexity, the SL algorithm in principle should suffer less from template-mismatch systematics. Possibly supporting this statement is the observation that the SL algorithms' velocity dispersion maps for the young stellar component look smoother and more realistic (in amplitude) than for the SSP algorithm (Figure 2 and Appendix D).

7.5.1. Gas Kinematics

"Realistic" mock spectra were used to test the abilities of the simple (one-step) SSP algorithm to disentangle the velocities of young and old stellar populations. We define metrics similar to those given in Tables 3 and 5, presented here in Table C1. These metrics reference the mock spectra model values rather than the MCMC derived values from real data. We also tabulate failure fractions defined as follows: "Cat" represents instances where either of the measured velocities is more than 100 $\mathrm{km}\,{{\rm{s}}}^{-1}$ from the model velocity. "Flip" represents instances where V_y − V_o < 0. "High" represents instances where yfrac > 0.85. High yfrac values are indicative of an incorrect allocation of cool stars to the young kinematic component.

Table C1.  Simple SSP Algorithm Performance Metrics Based on Realistic Mock Spectra

Algorithm	δVmock	ΔVy		ΔVo		$\delta {V}_{\mathrm{Algo}}/\delta {V}_{\mathrm{mock}}$		${\rm{\Delta }}{\mathtt{yfrac}}$		Failure Fraction
	Range	Med	${\sigma }_{\mathrm{MAD}}$	Med	${\sigma }_{\mathrm{MAD}}$	Med	${\sigma }_{\mathrm{MAD}}$	Median	σMAD	Cat	Flip	High
	($\mathrm{km}\,{{\rm{s}}}^{-1}$)	($\mathrm{km}\,{{\rm{s}}}^{-1}$)		($\mathrm{km}\,{{\rm{s}}}^{-1}$)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)
Simple SSP	[10,35]	0.12	0.20	0.08	0.13	1.00	0.02	−0.03	0.01	0.00	0.00	0.02
Simple SSP	[35,50]	0.22	0.16	0.16	0.24	1.00	0.01	−0.03	0.01	0.00	0.01	0.06
Simple SSP	>50	0.23	0.15	0.15	0.28	1.00	0.01	−0.03	0.01	0.00	0.02	0.03

Note. The Simple SSP algorithm, Column (1), is described in Section 4.1. The three bins in $\delta V\equiv {V}_{y}-{V}_{o}$ for the mocks (δV_mock) given in Column (2) contain 678, 263, and 996 mocks, respectively, from low to high δV_mock. Columns (3) to (10) are defined identically to the corresponding columns in Table 5 except that they reference the "realistic" mock model values instead of the MCMC results. Columns (11) to (13) are defined in the text.

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Figure C1 presents the performance of the metrics; δV_Algorithm/δV_mock where δV ≡ V_y − V_o, the "mock" values are the input velocities used to generate the models, and the "algorithm" values are the recovered values. The histograms are divided into four groups based on the strength of the signal in the mock spectra: δV > 40 $\mathrm{km}\,{{\rm{s}}}^{-1}$, 25 < δV < 40 $\mathrm{km}\,{{\rm{s}}}^{-1}$, 10 < δV < 25 $\mathrm{km}\,{{\rm{s}}}^{-1}$, and 5 < δV < 10 $\mathrm{km}\,{{\rm{s}}}^{-1}$. For this metric, a value of 1 suggests a successful disentanglement of the velocities of the two components, while other values can give clues to the cause of failure in the measurement. For example, a value of −1 indicates instances when the derived velocities of the two components are flipped (the velocity of the young is measured by the old component and vice versa). Values between −1 and 1 are likely caused by an overestimation of the velocity dispersion of one component. Very large positive or negative values suggest the observed spectrum has been reproduced by only one component and the other has been used only to minimize the χ² of the solution.

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For all mock δV groups, the distribution of the performance metric peaks at 1, as it should, which is consistent with the result seen in Section 3 where we demonstrate that MaNGA/MaNGA-like spectra of galaxies can contain sufficient information on the kinematics of co-spatial stellar populations for disentanglement. The distribution width increases at smaller model velocity differences, δV, because the uncertainty in the measured velocity difference is approximately constant in kilometers per second. Even for AD signals as low as 5–10 $\mathrm{km}\,{{\rm{s}}}^{-1}$, 67% of the derived AD signals are within 10% of the expected value. Performance evaluation of this algorithm against real data is presented in Section 4.1.

Table C2 presents metrics using our "simple" mocks for all simple stellar library algorithms described and discussed in Section 6.2. The adopted best simple algorithm corresponds to "full" with template set B.

Table C2.  Simple Stellar Library Algorithm Performance Metrics Based on Simple Mock Spectra

Algorithm	TPL	δVmock	ΔVy		${\rm{\Delta }}{V}_{o}$		$\delta {V}_{\mathrm{Algo}}/\delta {V}_{\mathrm{mock}}$		${\rm{\Delta }}{\mathtt{yfrac}}$		Failure Fraction
	set	Range	Med	${\sigma }_{\mathrm{MAD}}$	Med	${\sigma }_{\mathrm{MAD}}$	Med	σMAD	Median	σMAD	Cat	Flip	High
		($\mathrm{km}\,{{\rm{s}}}^{-1}$)	($\mathrm{km}\,{{\rm{s}}}^{-1}$)		($\mathrm{km}\,{{\rm{s}}}^{-1}$)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)
Feature	A	[10, 35]	26.8	14.8	−6.8	5.5	−0.92	1.07	0.24	0.13	0.09	0.82	0.00
Feature	A	[35, 50]	21.5	13.4	−19.9	11.2	−0.05	0.58	0.20	0.12	0.10	0.52	0.00
Feature	A	>50	20.8	10.5	−44.1	24.4	0.10	0.42	0.21	0.13	0.09	0.43	0.00
Feature	B	[10, 35]	18.9	10.4	−7.3	7.0	−0.78	1.06	0.10	0.11	0.02	0.74	0.00
Feature	B	[35, 50]	14.4	6.0	−8.6	11.1	0.47	0.40	0.01	0.08	0.02	0.35	0.00
Feature	B	>50	13.2	4.6	−7.1	13.2	0.72	0.22	−0.03	0.05	0.05	0.23	0.01
Feature	C	[10, 35]	14.0	5.1	44.0	66.2	3.01	4.17	−0.20	0.19	0.43	0.29	0.45
Feature	C	[35, 50]	13.4	5.5	15.8	27.2	1.21	0.81	−0.13	0.06	0.26	0.15	0.22
Feature	C	>50	13.6	4.7	18.3	16.6	1.10	0.26	−0.11	0.04	0.22	0.08	0.21
Feature	D	[10, 35]	15.7	4.5	49.7	104.0	2.72	5.64	−0.32	0.21	0.46	0.28	0.65
Feature	D	[35, 50]	18.0	6.7	34.7	37.3	1.44	0.96	−0.28	0.14	0.31	0.12	0.54
Feature	D	>50	17.9	6.5	57.6	45.2	1.49	0.55	−0.21	0.11	0.38	0.06	0.47
Full	A	[10, 35]	17.4	4.4	−5.9	5.1	−0.26	0.36	0.24	0.12	0.04	0.69	0.00
Full	A	[35, 50]	17.1	4.8	−20.2	10.5	0.12	0.32	0.21	0.12	0.04	0.40	0.00
Full	A	>50	15.8	5.6	−43.8	23.8	0.23	0.32	0.21	0.13	0.05	0.32	0.00
Full	B	[10,35]	15.8	4.7	−6.7	6.0	−0.33	0.50	0.03	0.09	0.03	0.69	0.02
Full	B	[35, 50]	15.1	5.8	−8.7	11.4	0.39	0.46	0.00	0.07	0.06	0.36	0.01
Full	B	>50	11.4	5.0	−4.7	13.1	0.79	0.24	−0.02	0.04	0.08	0.23	0.03
Full	C	[10, 35]	12.1	4.0	160.5	177.1	9.11	10.04	−0.30	0.22	0.57	0.21	0.60
Full	C	[35, 50]	12.2	6.4	27.7	69.3	1.58	2.20	−0.16	0.10	0.44	0.16	0.42
Full	C	>50	10.7	4.6	26.7	27.0	1.27	0.42	−0.12	0.04	0.36	0.10	0.31
Full	D	[10, 35]	13.4	3.6	265.2	375.5	18.06	24.56	−0.36	0.22	0.66	0.18	0.71
Full	D	[35, 50]	16.2	6.9	158.5	164.1	4.13	3.77	−0.38	0.22	0.59	0.17	0.69
Full	D	>50	16.1	7.6	109.5	99.4	2.18	1.17	−0.26	0.15	0.55	0.10	0.60

Note. Simple stellar library algorithms, Column (1), are described in Section 6.2. The "full" algorithm using template set B is identified as "Simple SL" in Tables 5 and C3. Column definitions and mock numbers are the same as in Table C1, with the addition of the template set in Column (2) and the referencing of "simple" rather than "realistic" mocks.

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Figure C2 shows two instances of failed and successful fits of the simple stellar library algorithm using template set C, where the failed case is characterized by a high yfrac value > 0.85.

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Table C3 presents metrics using "realistic" mocks for the best simple stellar library algorithm ("full" with template set B), the two-step stellar library algorithm (2-Step 1-Bin SL), and our final two-step stellar library algorithm that constrains the relative weight of young and old stellar components in three stellar temperature bins (2-Step 3-Bin).

Table C3.  Stellar Library Algorithm Performance Metrics Based on Realistic Mock Spectra

Algorithm	δVmock	ΔVy		${\rm{\Delta }}{V}_{o}$		$\delta {V}_{\mathrm{Algo}}/\delta {V}_{\mathrm{mock}}$		${\rm{\Delta }}{\mathtt{yfrac}}$		Failure Fraction
	Range	Med	σMAD	Med	σMAD	Med	σMAD	Median	σMAD	Cat	Flip	High
	($\mathrm{km}\,{{\rm{s}}}^{-1}$)	($\mathrm{km}\,{{\rm{s}}}^{-1}$)		($\mathrm{km}\,{{\rm{s}}}^{-1}$)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)
Simple SL	[10,35]	10.66	2.26	−4.68	5.19	−0.14	0.43	0.02	0.07	0.00	0.57	0.00
Simple SL	[35,50]	6.20	3.52	−4.34	7.61	0.74	0.25	0.01	0.04	0.00	0.26	0.00
Simple SL	>50	4.07	2.63	−0.86	8.30	0.91	0.15	−0.02	0.02	0.04	0.22	0.00
2-Step 1-Bin SL	[10,35]	12.06	3.43	−4.52	4.86	−0.18	0.41	0.10	0.11	0.00	0.60	0.00
2-Step 1-Bin SL	[35,50]	6.39	3.88	−3.54	7.40	0.75	0.25	0.01	0.05	0.00	0.30	0.00
2-Step 1-Bin SL	>50	3.62	2.35	1.14	5.46	0.96	0.09	−0.02	0.02	0.02	0.17	0.00
2-Step 3-Bin SL	[10,35]	8.46	8.87	−4.06	4.75	0.41	0.54	−0.04	0.03	0.02	0.36	0.04
2-Step 3-Bin SL	[35,50]	8.93	8.73	−17.1	15.54	0.41	0.39	−0.05	0.03	0.02	0.26	0.07
2-Step 3-Bin SL	>50	8.11	7.80	−25.8	27.23	0.54	0.36	−0.07	0.04	0.06	0.19	0.05

Note. Algorithms, Column (1), are described in Sections 6.2 and 6.4. The remaining columns are defined identically as in Table C1.

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