Asymmetric drift in MaNGA: Mass and radially-dependent stratification rates in galaxy disks

We measure the age-velocity relationship from the lag between ionized gas and stellar tangential speeds in ~500 nearby disk galaxies from MaNGA in SDSS-IV. Selected galaxies are kinematically axisymmetric. Velocity lags are asymmetric drift, seen in the Milky Way's (MW) solar neighborhood and other Local Group galaxies; their amplitude correlates with stellar population age. The trend is qualitatively consistent in rate (d(sigma)/dt) with a simple power-law model where sigma is proportional to t^b that explains the dynamical phase-space stratification in the solar neighborhood. The model is generalized based on disk dynamical times to other radii and other galaxies. We find in-plane radial stratification parameters sigma_(0,r} (dispersion of the youngest populations) in the range of 10-40 km/s and 0.2<b_r<0.5 for MaNGA galaxies. Overall b_r increases with galaxy mass, decreases with radius for galaxies above 10.4 dex (M_solar) in stellar mass, but is ~constant with radius at lower mass. The measurement scatter indicates the stratification model is too simple to capture the complexity seen in the data, unsurprising given the many possible astrophysical processes that may lead to stellar population dynamical stratification. Nonetheless, the data show dynamical stratification is broadly present in the galaxy population, with systematic trends in mass and density. The amplitude of the asymmetric drift signal is larger for the MaNGA sample than the MW, and better represented in the mean by what is observed in the disks of M31 and M33. Either typical disks have higher surface-density or, more likely, are dynamically hotter (hence thicker) than the MW.


INTRODUCTION
The dynamical structure of stellar populations in the MW disk is characterized by the age-velocity-dispersion relation (AVR), whereby older disk stars are dynamically hotter than their younger cohort, and hence have greater vertical extent.Knowledge of this relationship stems back nearly a century to observations by Strömberg (1925) of the varying scale-heights of stars of different types in the solar neighborhood (and substantially updated and refined by e.g., Edvardsson et al. 1993;Dehnen & Binney 1998;Nordström et al. 2004;Holmberg et al. 2007;Sharma et al. 2021).
AVR is a dynamical stratification of stars in the phase space of position and velocity, and as such AVR has direct relevance to understanding how stellar disks form and evolve (Seabroke & Gilmore 2007;Aumer & Binney 2009).The processes leading to this stratification have been debated for over 70 years.They include dynamical ★ E-mail: mab@astro.wisc.eduheating from two-body scattering of stars in dynamically cold stellar disks with in-plane resonances (spiral arms and bars) or giant molecular clouds (Spitzer & Schwarzschild 1951, 1953;Kokubo & Ida 1992;Carlberg et al. 1985); stirring by compact halo objects, satellites or mergers (Toth & Ostriker 1992;Walker et al. 1996;Huang & Carlberg 1997;Abadi et al. 2003;Benson et al. 2004;House et al. 2011;Helmi et al. 2012;Few et al. 2012;Ruiz-Lara et al. 2016;Pinna et al. 2019); or a relic of a dynamically hotter distribution of bulk velocities of star-forming clouds at earlier epochs (Brook et al. 2004;Bournaud et al. 2009;Forbes et al. 2012, commonlly referred to as gas cooling).Recent models often include both these 'heating' and 'cooling' scenarios in the formation of disk dynamical stratification (e.g., Martig et al. 2014b;Navarro et al. 2018;Bird et al. 2021;McCluskey et al. 2023).Whether the AVR represents gas cooling, dynamical heating, or both, very basic questions remain as to whether AVRs exist broadly in the external galaxy population, and if so, how do they compare to what we observe in the MW?
Measuring AVR in resolved stellar populations currently is limited to galaxies in the Local Group (LG Collins et al. 2011;Dorman et al. 2015;Beasley et al. 2015;Bhattacharya et al. 2019;Quirk et al. 2019Quirk et al. , 2022)).Based on a compilation of eight LG galaxies, only two of which have stellar masses above 10 10 M ⊙ , Leaman et al. (2017) have suggested there is an interplay between heating mechanisms that correlates with galaxy mass.It is also abundantly clear from multi-color images of edge-on galaxies that star-formation and young stellar populations are confined mostly to disk mid-planes at least in more massive spirals, while older stellar populations are evident in thicker disks that extend beyond the central dust layer.In lower mass disks (galaxies with   < 120 km s −1 , or below about log M ★ /M ⊙ < 10.3 ± 0.1 dex in stellar mass, scaling M ★ from the MW), however, the situation appears to be different at least insofar as dust lanes (and therefore one may presume star-forming regions) are not as thinly distributed (Dalcanton et al. 2004).Progress is needed beyond these qualitative statements and AVR measurements solely in the LG.
In principle AVR can be measured directly in integrated starlight with ample spectral resolution to measure integrated absorption linewidths in dynamically cold systems, and ample signal-to-noise (SNR) to disentangle different population components.Unfortunately, the necessary telescopes and instrumentation for such measurements is limited, and hence so too are the available data.However, it is possible to use velocities alone to estimate stellar dispersions by employing the well-known lag between the tangential speed of stars and the circular speed of the potential.This lag is known as asymmetric drift (AD).In integrated star-light, the challenge has still been to determine the stellar kinematics of multiple age components.Shetty et al. (2020) pioneered such a method in a small sample of galaxies with optical integral-field spectroscopy (IFS).The results show great promise but the method is computationally expensive and requires exquisite SNR, as do other methods employing higher moments of the velocity field (Coccato et al. 2018;Poci et al. 2019).
In this paper we step back to measure AD in single (light-weighted mean) stellar components.We take advantage of very large IFS surveys to leverage the dynamic range in mean stellar population ages between different galaxies and different regions within galaxies.We infer AVR from stellar population age and AD measurements in the context of a simple, time-continuous stratification model.In so doing we recognize two basic assumptions inherent to this approach, namely (i) that the stratification process between galaxies and different regions of individual galaxies are the same; and (ii) that the stratification rate is continuous in time.We expect these assumptions are at best approximations, so we explore radial and mass dependencies of AVR model parameters to test them.
This paper uses optical IFS of nearby galaxies from the MaNGA survey (Bundy et al. 2015;Yan et al. 2016b) in SDSS-IV (Blanton et al. 2017).MaNGA uses fiber integral-field units (Drory et al. 2015) with the BOSS spectrographs (Smee et al. 2013) on the Sloan 2.5m telescope (Gunn et al. 2006).The MaNGA observing strategy, target selection, spectrophotometric calibration and data reduction pipeline are described in Law et al. (2015), Wake et al. (2017), Yan et al. (2016a), and Law et al. (2016), respectively.Salient features of the spectroscopic data are discussed in relevant sections below.We take advantage of existing kinematic measurements of stars and gas from the Data Analysis Pipeline (DAP; Westfall et al. 2019;Belfiore et al. 2019) to select targets and define kinematic geometries.
The paper is structured as follows.The MaNGA data and sample selection is presented in Section 2. Measurements that define the kinematics to compute asymmetric drift and spectral indices to estimate stellar populations ages in integrated starlight are described in Section 3 and Appendix A. The definition for asymmetric drift is found in Section 3.1.A correction to the gas tangential speed, due to its finite pressure support, to estimate the circular speed of the potential is detailed in Section 3.3; notably, this method does not rely on direct measurements of the gas dispersion in low signal-to-noise data well below the instrumental spectral resolution.A robust and reproducible method to determine stellar population ages based on D n (4000) is defined in Section 3.4, with further details provided in Appendix B.
A simple power-law stratification model is motivated in Section 4, starting with the relation of AD to in-plane components of the stellar velocity dispersion ellipsoid (SVE; 4.1), with supporting derivations found in Appendix C. The parameters of this stratification model are the radial components of the initial velocity dispersion at zero age ( 0, ) and the power-law stratification exponent   .In this context, analysis of our age and asymmetric drift measurements are presented in Section 5. Corrections for the different light-weighting for stellar population ages and kinematics have been extensively modeled, with modest corrections estimated and applied, as summarized in Section 5.1 and detailed in Appendix D. Application of the stratification model to the data, and resulting values for the stratification parameters as a function of radius and mass can be found in Sections 5.3 and 5.4 respectively.We examine the impact of bars on the derived stratification parameters in Sections 5.3 as well.For unbarred galaxies, the covariance between the stratification parameters and correlation with other physical parameters, such as radial light concentration, are found in Section 5.5.Finally, in Section 6 we make a comparison of the stratification parameters derived for MaNGA galaxies with those for the three most massive galaxies in the Local Group: M31, MW, M33.Our methods and findings are consolidated in a summary, Section 7.Where applicable we adopt a flat ΛCDM cosmology with Ω  = 0.3 and  0 = 70 km s −1 Mpc −1 .All magnitudes are in the AB system.

Data
The kinematic and spectrophotometric measurements used in this paper are based on the final SDSS-IV Data Release 17 (DR17; Abdurro'uf et al. 2022) products for MaNGA described in more detail in papers describing the DAP Westfall et al. (2019); Belfiore et al. (2019).The spectral resolution is ∼67 km s −1 ().The median spatial resolution is 1.4 arcsec, varying as a function of mass given the MaNGA sample selection (Wake et al. 2017); 75% of the sample has a resolution between 1.1 and 2.1 arcsec, while 90% of the sample has a resolution between 0.8 and 3.4 arcsec.Most relevant for the current analysis is that the distribution of apparent half-light radii is independence of mass.Given the extent of the analysis involved, however, the sample is limited to the set of galaxies processed in SDSS-IV Data Release 14 (Abolfathi et al. 2018), which consists of 2744 unique observations of galaxies with MaNGA IFUs.Because we are interested in measuring kinematics in specific geometric regions of each galaxy (see Section 3) we eschew the Voronoi-binned data and use only the spaxel-based DAP data products, which for DR17 correspond to the data-type SPX-MILESHC-MASTARSSP.For this data-type the stellar and ionized gas kinematics were determined with pPXF (Cappellari 2017) and a hierarchically-clustered (Westfall et al. 2019, Sec. 5)  .The 798 kinematically regular galaxies (selected as described in text) are marked for the final subset of 497 galaxies (dark points), 192 galaxies excluded for being too inclined (red points), and 109 galaxies excluded for having too low S/N in either the stellar or ionized gas velocity fields (yellow points).In the middle panel the 497 galaxies are color coded by Sersic index as given in the key.Galaxies with strong bar morphology are plotted with the same color-coding in the right panel.Two parallel dashed lines delineate the blue cloud, green valley and red sequence.
of relying only on velocities (versus higher moments) for our analysis is that the details of the stellar template constructions are relatively unimportant.We also use the DAP's measure of D n (4000) (as defined by Balogh et al. 1999) and ionized gas fluxes.In addition to the DAP, we made use of the photometry (we adopt Petrosian magnitudes) and Sersic index provided by the NASA-Sloan Atlas (NSA, Blanton et al. 2011).

Sample Selection
We expect AD to be measurable in all galaxies with detectable dense gas, but if significant kinematic irregularities or asymmetries are present, this likely prevents the measurement of an accurate AD signal.The DAP kinematics were used to select galaxies on the basis of the regularity of both stellar and ionized gas velocity fields.
To make this assessment for such a large galaxy sample, and in order to be quantitative, we fit simple kinematic models to all 2744 galaxies in the DR14 MaNGA sample in an automated way.The kinematic fitting is based on the scheme developed by Westfall et al. (2011) and Andersen & Bershady (2013) using a single, inclined disk model with smooth axisymmetric tangential motion and a radial trend described by  () =  rot tanh(/ℎ rot ).The fitting solves for the geometric and kinematic parameters.The scheme is extended here to fit ionized gas and stellar velocity fields simultaneously with the same geometry (center, inclination, position angle) while allowing for the rotation curve parameters ( rot , ℎ rot ) and systemic velocity to be independent.We perform 12 velocity-field fits for each galaxy: (1) four fits are permutations of fixing and freeing the center and inclination and (2) each of these four fits is done when only fitting the ionized gas data, only fitting the stellar data, and fitting both tracers simultaneously.Fits with fixed geometric parameters adopt photometric values parameters provided by the NSA.The purpose of the multiple fitting permutations is to assess the stability of both the kinematic and geometric parameters constrained by different sets of tracers.Based on visual inspection of a subset of cases for a range of variation in fitted parameters we assessed quantitative limits to derive a kinematically regular sample.We find it is useful to limit variation between permutations and tracers on the systemic velocity, kinematic center, position angle, and inclination.Our AD- analysis is of galaxies that satisfy the following set of constraints on the results from the velocity field fitting: (i) When the two velocity fields are fitted simultaneously, the differences between the systemic velocities fit to the ionized gas and stellar velocity fields must be within ∼ 7 km s −1 (absolute value) for all four of the fit permutations.
(ii) In the two simultaneous fits to the ionized gas and stellar data with the dynamical center left free, the dynamical center must be within half of the imposed bounding box on the morphological center.The bounding box is set to ±(1,2,2,3,3) arcsec for IFUs with (19,37,61,91,127) fibers respectively.
(iii) For all the relevant fit permutations, the difference between the ionied-gas-only and stellar-only fitted position angles must be less than 15 • .
(iv) In all six fits when the inclination is freely fit, the best-fitting kinematic inclination must be in the range 15 • <  kin < 80 • .The difference between the ionized-gas-only and stellar only kinematic inclinations must be less than 20 • both with and without a fixed center.When both tracers are fit simultaneously, the kinematic inclination must also be within 20 • of photometric inclination derived from the standard formula using  = 1 − / and an intrinsic oblateness  0 = 1/8 characteristic of spiral galaxies (e.g., Padilla & Strauss 2008) . The simultaneous application of all these cuts yields a sample of 798 galaxies with regular kinematics.Visual inspection of these galaxies resulted in a further culling to eliminate 192 objects that appear too inclined for reliable kinematics and colors due to uncertainties in internal extinction corrections.This visual inspection removes a subset of galaxies with a median inclination of ∼ 66 • , including all galaxies with photometric inclinations above 70 • .The same visual inspection also identified an additional 109 objects for removal because they had either too low in continuum or line S/N to have adequately sampled stellar or ionized gas velocity fields, re-  spectively, despite the fitting process described above.The following analysis uses the remaining 497 galaxies (18% of the parent sample).
The distribution of this sample in relation to the parent sample in the  −  -  color-magnitude diagram (CMD) is shown in Figure 1.Unsurprisingly, highly-inclined systems are preferentially at redder colors and lower absolute magnitude due to uncorrected internal-extinction.The low-S/N galaxies tend to be found at the peripheries of the CMD distribution at low luminosity or red color.Examples of the galaxy morphology are shown in Figure 2 and 3 for a range of Sersic index, unbarred and barred (based on our visual inspection).These Figures shows that in our sample the high-index galaxies still have disks, concluding that the high-index values are due to the light-weighted nature of the one-zone Sersic model, i.e., the high Sersic indices are a measure of the profile shape of the inner light concentration.
Figure 1 also shows that barred and unbarred galaxies in our sample have similar distributions in color total stellar mass.Despite the kinematics of the barred galaxies having sufficient regularity to meet our sample selection criteria, the bar perturbations to the velocity field are evident, particularly at small radii in Figure 3. Since our measurements avoid the inner regions due to beam smearing, described in the following section, a priori it was not clear if the bar perturbations would be problematic to our analysis.For example, Stark et al. (2018) found inner position-angle twists that appeared to be associated with bars typically occurred within  50 /2, where  50 is the half-light radius; this radius is within the smallest radial bin we consider (Section 3).We therefore initially retain the barred sample with the intent to determine if there are systematic differences in the stratification properties in barred and unbarred galaxies.In the end, our expectations turn out to be too optimistic; we eliminate barred galaxies (roughly a quarter of the sample) in our tertiary analysis (Section 5.3.2 and onward).

KINEMATIC AND SPECTROPHOTOMETRIC MEASUREMENTS
This section defines asymmetric drift (3.1), and then steps through our measurement process of defining radial apertures and a common weighting of the data (3.2),pressure corrections (3.3) to the gas tangential speed, and stellar population ages (3.4) based on the D n (4000) spectral index.

Asymmetric drift
Asymmetric drift (  ) is defined as the lag between the tangential speed of the stars ( , * ) and the circular speed of the potential:   ≡   −  , * .We define an effective velocity dispersion  AD as the quadrature difference between the above quantities: The two quantities  AD and   are tightly correlated.In our data the correlation is characterized well by a slope of 1.5 between log   versus log  AD , with a zeropoint of   / AD ∼ 1/4 when  AD = 100 km s −1 , i.e., where   ∼ 25 km s −1 .Median values of  AD and   for our sample are 70 and 16.5 km s −1 , respectively.For the purpose of this work we adopt the ionized gas tangential velocity ( , ) as a proxy for   and determine if any correction for pressure support is warranted.This correction is in the same sense as asymmetric drift, i.e.,  , <   , but arises for different physical reasons.For cool and cold gas, corrections for pressure support are minimal (of order a few km s −1 ) and difficult to determine, and hence typically are ignored (e.g., Swaters 1999).For the warm ISM component, traced by ionized gas, Andersen et al. (2006) have shown that the median H line-width is ∼ 18 km s −1 in late-type spirals; this amounts to a negligible correction to  AD for   as small as 10 to 20 km s −1 (e.g., Westfall et al. 2011).This is consistent with the km s −1 -level agreement between HI and H rotation curves for galaxies in the DiskMass Survey (Bershady et al. 2010), as shown in Martinsson et al. (2013).However, the MaNGA sample is more heterogeneous and displays a larger range of   .Recent comparisons of CO and H tangential velocities for a sample including MaNGA galaxies suggest modest corrections to the ionized galaxies velocities are needed in some cases where the gas dispersion is high (Su et al. 2022).We develop and implement a correction for the ionized gas velocities in Section 3.3.

Spaxel-based measurements in radial bins
The inclined-disk fits used to select our sample also provide the rotation amplitude ( rot ), the scale of the inner-rise of the rotation curve (ℎ rot ), and the kinematic position angle and inclination.The tanh rotation curve model is a rather simplistic description of the tangential motion, but it serves well to characterize the velocity gradients in the data, particularly at small radii.We measure the AD signal and other quantities directly from the data, examples of which are shown for AD in Shetty et al. (2020).
To deproject the observed spaxels to their relative radial and azimuthal positions on the galaxy we adopt the mean of the photometric and kinematic inclination, and the kinematic position angle.We use spaxels within ±40 • of the kinematic major axis (in the disk plane) to keep corrections for azimuthal projection to below 25% while having a sufficiently broad wedge to sample an ample number of spaxels in each data-cube.The observed, line-of-sight radial velocities  los for each spaxel are deprojected in the usual manner to derive the deprojected tangential speed  = ( los −  sys )/(cos  sin ), where  sys is the systemic velocity of each galaxy, and  is the azimuthal angle in the disk plane for each spaxel.
We then compute weighted moments with 5 clipping of the distributions of kinematic measurements made for individual spaxels, grouped in nested radial rings, 2 arcsec in width.The choice of width minimizes spatial covariance in data-cubes grided at 0.5×0.5 arcsec 2 from data sampled with 2 arcsec diameter fibers.An example of our binning and spaxel-based computations is given in Appendix A and Figure A1.We determine the error-weighted mean of  2 AD computed spaxel by spaxel, as defined in Equation 1.We stress this is not the same as taking the difference of the square of the mean stellar and gas velocities, i.e., ⟨ 2 , − 2 , * ⟩ is not the same as ⟨ 2 , ⟩ − ⟨ 2 , * ⟩.The former, in addition to being the correct formulation, has the practical advantage of enabling robust statistics and allowing for assessment of the spaxel-level distributions for measurements in the rare cases where the error-weighted mean of  2 AD is negative.These negative values are discarded from further analysis, and hence is a 'survival' bias.However, this 'survival bias' is inconsequential to the results of Points are color-coded by estimated gas velocity dispersion, given in the key, and assigned according to line-ratios as described in the text.Black dotted and dashed lines represent demarcations of HII-like line-ratios from Kauffmann et al. (2003) and Kewley et al. (2001) respectively; solid green and red lines demark dynamically cold, intermediate and hot ionization regions as measured by Law et al. (2021b).Right panels: Fractional corrections to the measured asymmetric drift  AD for gas pressure-support as a function of the quadrature difference between the corrected and the measured gas tangential speed, radius (normalized by  50 ) and stellar mass.Uncertainties include a full propagation of errors from gas and stellar velocities, line-ratios, and the population variance (as measured by Law et al. 2021b) in the inferred   for a given set of line-ratios.The horizontal dashed line represents our threshold of a <30% correction for inclusion in our remaining analysis.
our analysis because the number of negative measurements are small (a few % of the total).
We also compute the mean D n (4000) values as well as emissionline ratios for [O III]/H, [N II]/H and [S II]/H for the corresponding spaxels in each ring.We adopt D n (4000) as an age proxy, and present its conversion and calibration to a light-weighted mean age in Section 3.4.We use the emission-line flux ratios in Section 3.3 to estimate a correction to the gas velocity for pressure support.Since the   kinematics are the paramount measure, often with the largest errors, and to avoid bias with different weightings, all mean values (specifically, emission-line ratios in Section 3.3 and stellar population ages in Section 3.4) are error-weighted using the inverse variance of   .
The total number of usable independent measurements from this sample is 2877 -on average five to six measures per galaxy.We present results in five discrete radial bins between 0 < / 50 < 3.5 with upper limits in / 50 of 0.5, 1., 1.75, 2.5, and 3.5; the median values of / 50 for data in these bins are 0.4, 0.8, 1.3, 2.1 and 2.85.The number of measurements peaks in the second and third bins.Numbers drop in the inner bin because we exclude measurements for  < 4" based on the recommendations from Law et al. (2021a) to avoid impact from beam-smearing.The median value of  50 is 7.1 arcsec; at 0.6  50 roughly half the sample has a radius ∼4 arcsec.In the outer two bins the survey design coverage limits the radial extent sampled.However, the distribution for apparent  50 (in arcsec) has no trend in mass so the sampling in mass is uniform across radial bins.
The preponderance of measurements are at  > ℎ rot , where the rotation curves are relatively flat.For our sample we find a median ℎ rot / 50 of 0.6.The distribution of this ratio as a function of stellar mass is shown in Figure 4; the median drops smoothly from 0.7 to 0.45 in mass bins from 6.8 × 10 9 to 1.4 × 10 11 M ⊙ with a scatter (67% CL) of 0.18 in any mass interval.A similar trend with rotation speed has been observed by Andersen & Bershady (2013).The trend is in the anticipated sense that more massive galaxies have more concentrated mass profiles, as indicated by the increasing Sersic index with mass.This trend is, however, stronger for mass than for the Sersic index.Only the inner-most radial bin has some cases where  ≤ ℎ rot , but these are for lower-mass galaxies that have more slowly rising rotation curves so that in any event the radial velocity gradients within measurement annuli are small.Dalcanton & Stilp (2010) have developed an elegant formalism for describing the anticipated lag between   and  , due to pressure support in the gas otherwise moving in circular orbits in the galactic potential.Su et al. (2022) show that this formalism does an excellent job of correcting H tangential speeds to match that of the cold gas in disks.Referring to equation 11 of Dalcanton & Stilp (2010), this lag depends on the radial gradients of the gas velocity dispersion and density.In the context of pressure support in a multi-phase medium, the relevant gas density is the total gas density, while the gas dispersion is the radial component for the observed tracer (in our case H from the ionized gas component, whose dispersion we assume is isotropic).

Corrections for gas dispersion
When both the gas dispersion and density profiles can be described as radial exponential profiles with scale lengths of ℎ  and ℎ  , respectively, then (2) Since we do not have gas density profiles for individual galaxies (see Westfall et al. 2014), we leverage the results of Bigiel & Blitz (2012) who find that ℎ  ∼ 0.61  25 in nearby spiral galaxies, with modest scatter ( 25 is defined as the radius of the 25 mag arcsec −2 -band isophote).For an exponential stellar disk obeying Freeman's law (Freeman 1970) with scale-length ℎ  one finds ℎ  ∼ 1.88 ℎ  ∼ 1.13  50 .(Using  50 runs the risk of under-estimating ℎ  for more bulge dominated systems where  50 /ℎ  < 1.67, but the effect is minimal in practice.)In contrast, there is little evidence for significant radial gradients in the ionized gas velocity dispersion   outside of the inner regions where rotation curves steeply rise (Andersen et al. 2006;Martinsson et al. 2013).Beam-smearing may be a significant contributor to the measured increase sometimes seen in   at small radii (Law et al. 2021a), and our analysis here will work outside of this radial domain.It is therefore reasonable to assume that ℎ  ≪ ℎ  so we adopt 2 (3) The amplitude of the correction to  AD can be estimated from equations 1 and 3.In our analysis, we use the emission line-ratios to estimate our gas line-widths since almost all of our lines are well below the instrumental resolution of the data.An added advantage of this method is the minimization of the bias inherent in measuring linewidths at low-to-moderate S/N (Law et al. 2021a).This estimation is based on the tight correlation found by Law et al. (2021b)  The result of this inversion of the Law et al. (2021b) correlation, applied to all of our measurements at all S/N, is shown in Figure 5, left-hand panels.Note that the preponderance of measurements are dynamically cold or HII-like (75% based on d  2 and 90% based on d 2 , noting that the dynamically cold demarcation for [N II]/H is essentially identical to Kauffmann et al. (2003)), and only 8% are dynamically hot.The derived   values are qualitatively reasonable; quantitatively they agree with the measured   to within 1% in the median, but with a 40% dispersion dominated by the kinematic measurements errors and corrections.The advantage of the line fluxratio inferred values is that   remains well defined at small linewidths where, in the presence of measurement error, the correction for the instrumental resolution to the measured kinematics sometimes lead to undefined values (see Westfall et al. 2019;Law et al. 2021a;Chattopadhyay et al. 2024, in the specific context of MaNGA data).
The fractional value of this correction to  AD is shown in Figure 5 as a function of radius and mass.The median correction is < 3%; corrections for 67/90% of the corrections are below 5/15%.We apply these corrections to derive  AD,c .There are modest trends in the correction increasing with radius and decreasing with stellar mass, as expected from Equation 3, a relatively flat radial profile and mass-dependence for   .We remove 124 measurements (4%) with corrections > 30% from further analysis, but this does not preferentially bias against mass, radius, or ionization state.Four objects are removed from the sample, but these only had one or two initial measurements.

Ages
We adopt an empirical approach to estimate light-weighted mean ages for MaNGA spectra based on D n (4000): Age estimates remain as closely linked to the data as possible and they are easily reproducible.Model estimates are used to calibrate our relation between D n (4000) and age.
Light-and mass-weighted mean ages have been estimated for MaNGA spectra by Sánchez et al. (2016a,b, 2018, Pipe3D), Goddard et al. (2017); Parikh et al. (2018, Firefly) and Lu et al. (2023,   2 Adopting ℎ  /ℎ  as large as 0.2 only changes the results of the correction by a few percent. .9 1 pPXF) based on full-spectrum fitting to stellar populations synthesis (SPS) models.There are systematic differences in age estimates between models, the origin of which are not easily understood.Here we adopted the Pipe3D values for purposes of calibration, but again we emphasize that we do so in a way that is transparent and easily convertible to other calibrations.These SPS estimates rely on spectra that are spatially averaged to reach minimum S/N requirements.Unfortunately the spatial apertures do not match the geometric regions we require to measure kinematics and hence these data products cannot be used directly in this analysis.This is secondary benefit of using D n (4000) that can be readily measured in the specific geometric regions of galaxies where we probe kinematics.
Figure 6 shows the correlation of D n (4000) with stellar population age.Data includes all spatial apertures defined by Pipe3D for all galaxies in our sample with age estimates.We exclude 14% of the apertures with the largest errors in D n (4000) (> 0.004), which appear to have uncertain age estimates based on their lack of correlation with D n (4000).Pipe3D does not measure D n (4000) so we measured this quantity from the corresponding spectra, consistent with the definition from the DAP.These apertures cover a broad range in D n (4000) and age that spans our full data.The spaxels also sample the full radial range of our data.
As shown in Appendix B, it is not possible to adopt a simple model (SSP or single star-formation history) to match the trends between age and D n (4000).This illustrates the well-known need for fitting SPS models to the data.We take advantage of this fitting, and convert D n (4000) into an age using the thirdorder polynomial function shown in Figure 6 with x=D n (4000): log(age) = −5.295+ 21.9 − 10.875 2 + 1.86753 .This function traces the measurements well with little (< 10%) systematic trends as a function of D n (4000).The formal random errors in age based on errors in D n (4000) alone are much smaller than the width of the distribution.The 67% CL in log(age) is ±0.13 dex (equivalent to a rms of 35% in age), fairly constant with D n (4000).The residuals illustrated in Figure 6 are computed in the orthogonal direction to the fit and are factor of 2.3 smaller than this.There are residuals with metallicity and extinction (Appendix B) qualitatively consistent with expectations: When Pipe3D estimates higher than average extinction or metallicity at a given D n (4000) the corresponding Pipe3D age is lower.These residuals are consistent with the observed distribution about our polynomial fit.Future work might leverage the Balmer decrement and Lick indices to make suitable corrections if systematics in the model calibration warrant.
The result of applying the conversion from D n (4000) to age is shown in Figure 7 at 0.8  50 .We refer to the light-weighted mean age derived from D n (4000) as  LW .Galaxies are color-coded by their total stellar mass.Not surprisingly, galaxies with larger mass index tend to have larger D n (4000) values consistent with older and more metal rich populations.These galaxies also have larger  AD .

Relating 𝜎 AD to 𝜎 𝑟
To relate our measure of AD to a stratification model requires associating  AD with components of the stellar velocity dispersion ellipsoid.Since the radial component of the stellar velocity ellipsoid,   , is typically the largest, and like  AD probes in-plane motions, for simplicity we relate  AD to   .Starting with Equation C5 in Appendix C we adopt reasonable values for the ellipsoid flattening and tilt, respectively, of   /  = 0.6 (e.g., Pinna et al. 2018;Nitschai et al. 2021) and an intermediate value of  = 0.5, which ranges from 0 to 1, to evaluate [(  /  ) 2 − 1] − 1/2 = −0.82 and arrive at: for a general Sersic profile of index   where   ∼ 2  − 1/3 (see, e.g., Graham et al. 2005).The term [(  /  ) 2 − 1] has a viable range between -1 and 0. The partial derivative for a tanh rotation-curve model, evaluated with equation C6, is a subdominant term for / 50 > 0.25 (Figure C1).Adopting the median values of ℎ rot / 50 = 0.6 (see Figure 5) and an exponential light profile (  = 1),  AD /  = 1.6 +0.2 −0.1 at  50 , where the nominal values assumes   /  = 0.6 and  = 0.5.The range of values encapsulates 0.3 <   /  < 1 and 0 <  < 1.A similar computation for 1 <   < 5 yields  AD /  = 1.8 +0.4  −0.1 at  50 .The weak dependence of this -ratio on   can be seen immediately by substituting in the approximation for   into the second term of Equation 4. In the range 0.5 < / 50 < 3 for an exponential profile  AD /  can be well approximated as a linear function of r given as  AD /  = 0.85 + 0.75(/ 50 ).This approximation is useful for relating  AD to components of the stellar velocity ellipsoid, but we use equation 4 for our analysis.We also apply the measured ℎ rot and  50 values for our sample.In the context of the model there is at most 10% systematic uncertainty in this scaling for a plausible range of  and   /  = 0.6 at a given radius.

Stratification model parameterization
The stratification of disk stars in phase-space may be either a relic of disk-settling, i.e., gas-cooling (Brook et al. 2004;Bournaud et al. 2009), or the result of a secular dynamical process of disk heating (Spitzer & Schwarzschild 1951, 1953).Possibly it is a combination of both (Bird et al. 2013;Martig et al. 2012Martig et al. , 2014a,b),b).Here we wish merely to have a simple model that describes the present-day stratification that (a) reasonably describes what we see in the Milky Way's solar neighborhood; (b) can be generalized to other radii and galaxies on basic astrophysical grounds; and (c) this generalization can be tied as directly as possible to observable quantities.
For this reason, we turn to the long-standing disk-heating model proposed by several authors (Wielen 1977;Binney et al. 2000;Aumer & Binney 2009;Aumer et al. 2016) who posed that for isotropic scattering, independent of stellar orbit,  ( 2 )/ =   where  is the diffusion coefficient inversely proportional to V. From this one finds that with  = 1/3 and parameters for the solar cylinder of  0 = 6 km s −1 and  = 0.05 Gyr.These values for  0 and  are consistent with dispersions for the cool atomic and molecular gas layer and the dynamical time-scale of the disk at the solar circle, respectively.We use the latter fact to make a general model applicable at all radii by adopting i.e.,  dyn () is 1/4 of the orbital period at radius . 3  In the case where the perturbers are highly concentrated to the disk mid-plane (e.g., giant molecular clouds), it might well be argued that the vertical oscillation period about the disk mid-plane,   (), would be better suited to equate with  in Equations 5 and 6.For nearly circular orbits in an axisymmetric potential the vertical oscillation period, , can be found from the second derivative of the potential with respect to height.This yields   () = 2/ = √︁ 2/  0 (), where  0 () is the disk mid-plane density at radius  (Binney & Tremaine 2008).
As it so happens at the solar circle,   () and  dyn () are nearly the same even though their scalings with radius are different.If we assumes a constant mass scale-height with radius, and that dynamical mass surface-density scales with surface-brightness, then for the MW (using parameters adopted in Section 6.1.2) we find  dyn /  = 1±0.1,i.e., unity to within 10%, in the range 0.7 < / 50 < 2.7.This range contains 83% of our measurements.In our smallest and largest radial bins the difference between the two time-scales is < 35%.The uncertainties in estimating mass scale-heights and mass surfacedensities (particularly when atomic and molecular gas mass surfacedensities are not available) to determine  0 and hence   are likely larger than the differences between  dyn () and the actual   () values at any given radius of interest.In contrast, the circular speed is reliably measured in our data.For these reasons we opt to retain  dyn () rather than   for our reference time-scale ().Wielen (1977) pointed out that another reasonable assumption is for the diffusion coefficient  to be constant, yielding a power-law index  = 0.5 rather 0.33.Numerous theoretical predictions and observational analyses of stellar ages and kinematics in the MW Consequently for our analysis we specifically use our conversion  AD to   , and in fitting trends of   with age and radius, we allow both the amplitude (  ,0 ) and index (  ) to be model parameters: where  dyn is defined by Equation 6.We make no assumptions about   ,   ,0 , their variation between galaxies, or their trends with radius.We stress that this stratification model says nothing about time's arrow and simply is a convenient way to predict the relation between stellar population age and kinematics (i.e., their AVR) in today's galaxies if their stratification process is similar to that of the MW.In a large sample of galaxies, we might expect the AVR in some cases is dominated by one or more impulsive events ex situ to the disk, e.g., mergers; in these cases our stratification model may be inadequate to describe well the data.This is something we can test.We also adopt the light-weighted mean age ( LW ) for , which carries an implicit assumption that the light-weighted age as determined from SPS (continuum shape and absorption-line equivalent widths) corresponds to the characteristic asymmetric drift signal measured from the stellar kinematics.We address low-level systematics inherent in this assumption below.

Systematics between light-weighted ages and kinematics
While we measure light-weighted ages and kinematics from fullspectrum fitting over the same range in wavelength, the effective light-weighting for stellar age and stellar kinematics (here, velocities) are not necessarily identical.This might come about, for example, because the relevant line-strengths that contain the kinematic signal do not scale with age in the same way as the continuum level.Further, the stellar population ages are, by definition, referenced specifically to some wavelength or wavelength range (we chose the -band to be near the mid-point in the spectral range), and, unlike the kinematics, do not depend in detail on the wavelength distribution of relative line-strengths.These differences may lead to subtle systematic mismatches between characteristic velocities and ages.We addressed this by creating mock spectra with star-formation histories and rates of dynamical stratification that appear to span what is observed in our data, as described in Appendix D. We find that in the presence of dynamical stratification there are systematic differences between recovered velocities and velocities expected from   () given the recovered light-weighted age: pPXF tends to overestimate ages and velocities (and hence underestimate  AD ).The amplitude of the systematics depends primarily on   and D n (4000), peaking at the lower values of D n (4000) ∼ 1.3 for the fastest rotation speeds.While the correction can be as much as 30%, the mean correction is 11% at  50 , increasing on average with radius from 8% to 14% between 0.25 < / 50 < 3. The typical uncertainty in this correction increases from 2% to 4% over the same radial range, as illustrated in the right-hand map of Figure D1.In summary, the corrections are quite modest and well determined.They are applied to the data, based on the   and D n (4000)values of each radial bin in each galaxy, in all subsequent analysis.

Stratification model applied to MaNGA data
Figure 7 shows the trends of  AD at 0.8  50 with D n (4000), lightweighted mean age, local dynamical time, and the ratio of lightweighted mean age to the local dynamical time.Notable are the opposite trends of the two time-scales with  AD and stellar mass.On the one hand stellar age correlates with total mass, while at a given radial scale the increase in the rotation speed in more massive galaxies decreases the dynamical time.These two factors are both at play in the correlation of  AD with log(1 +  LW / dyn ) that we use to constrain the stratification model.We measure  dyn directly from the data, finding values of 41 ± 14 Myr at / 50 = 0.8, with the range enclosing 67% of the sample in the radial bin.Points represent measurements from individual galaxies at a particular radius, color-coded by stellar mass as given in the key.Solid grey curves represent the best-fitting stratification model to the ensemble of galaxies at each radius.Dashed and dotted curves correspond to stratification models with   fixed to 0.33 and 0.5, respectively.

Fitting to galaxy ensembles
There are two ways we are able to fit the stratification model to the data.The first is to aggregate measurements across all galaxies.This has the advantage of maximizing the number of measurements, and therefore is conducive to breaking down the data by radius, total galaxy mass, or both.The potential pitfall is the inherent assumption that the stratification parameters, even at a given radius and galaxy mass, are the same for different galaxies.Variations in both the amplitude (  ,0 ) and heating index (  ) between galaxies may wash out the correlation between log(1 +  LW / dyn ) and  AD , possibly yielding best-fitting values for   that are systematically too low and values for   ,0 that are too high.
Figure 8 shows the result of applying the stratification model to our data as an ensemble.The results are shown to fits done in different radial bins, adopting an exponential light profile for the disk.Median dynamical times and ranges are given for each radial bin.We use the scaling from   to  AD given in Section 3.1 (equation 4), and compute   () at each radius according to equation 7. We then find   ,0 and   at each radius that minimizes  2 defined using the differences between the measured and predicted asymmetric drift ( AD ); we define  2 as the orthogonal distance to the model line defined by Equation 7(essentially an orthogonal least-squares formulation in the presence of heteroscedastic errors in both variables Akritas & Bershady 1996).A minimum  2 value is found numerically, by brute force, with the help of a zoom-in function for added precision once the global minimum is found.This 'best fit' model (in a population sense) is represented as the thick grey line in Figures 8.Alternatively, by fixing   to historically preferred values for the MW of 0.33 and 0.5, we can repeat the exercise to find the corresponding best values of   ,0 at each radius.These are shown as dashed and dotted lines in Figures 8.In a  2 sense these fixed-  models do not do as well.

Fitting to individual galaxies
We also fit the stratification model in a second way, namely to data for individual galaxies.We refer to this as the 'indy' method.Again, we adopt an exponential light profile for the disk.In this case there are significant limitations in the number of measurements per fit as well as the dynamic range in both log(1+ LW / dyn ) and  AD .Conveniently, most galaxies exhibit significant radial population gradients.
In contrast with the ensemble method, the inherent assumptions with the indy method are that the stratification parameters are the same across radius, while assumptions about the impact of variations in the merger history, or satellite and halo populations are not an issue as they are in the ensemble method.
In practice, we find there are diminishing returns to deriving reliable stratification parameters when the range in the sampled timescale (Δ log(1+ LW / dyn )) is within a factor of a few of the combined errors in  = log(1 +  LW / dyn ) and  = log  AD .Defining the latter two errors as   and   , and the mean of the combined errors across the measurements for an individual galaxy as ⟨( 2  +  2  ) 1/2 ⟩, in our subsequent analysis we restrict our consideration to stratification parameters derived from indy fitting to galaxies with We also require there to be at least 4 radial measurements per galaxy. 4his reduce the available sample to 221 galaxies (145 unbarred).The results of these fits, unexpectedly, yield noisy stratification parameter estimates for any given galaxy (see Section 5.5 and associated Figures below), but their statistical averages, as discussed below, provide useful constraints on these parameters.

Radial trends
The immediate impression from Figure 8 is that there is a significant trend with radius of increasing overall amplitude of  AD at a given log(1 +  LW / dyn ), as well as a flattening of the correlation between these two quantities.The derived model parameters   ,0 and   are plotted in Figure 9 (left-hand panel) as a function of radius adopting an exponential light profile for the disk (equation 4). 5 We clip the data at 2.5 about the best fit; reducing the clipping does not change the qualitative trend, but the results are considerably noisier.Confidence intervals (ensemble method) and weighted errors (indy method) are computed on the clipped sample.The stratification amplitude   ,0 rises with with radius, between 15 and 40 km s −1 .The stratification IFU, and roughly half and 20% of sample galaxies observed with the 37-and 61-fiber IFUs, respectively.We do not find systematic differences in the stratification parameters of the remaining galaxies as a function of IFU size. 5The results adopting the generalized Sersic profile and Sersic indices for each galaxy yielded nearly identical results, so we do not report them here.
MNRAS 000, 1-20 ( 2023) Asymmetric drift in MaNGA disks 11 to data from all galaxies in a given radial bin.The black curve represents when both parameters free (dotted lines give 67% CL uncertainties on combined parameters).Best fitting values for   ,0 for   fixed to values of 0.33 and 0.5 are shown as orange and teal lines and dots, respectively.The inset plots the trend of reduced- 2 ,  2  , for these different cases ( 2  for   = 0.5 is off scale), as well as the unbarred sample shown in the right panel.Right: Best-fitting models to all unbarred galaxies are shown in green (grey repeats results for all galaxies from the left panel).Red data points are averages for fits to individual galaxies, binned by the mean radius of the measurements for a given galaxy, as described in the text.
index   also declines with radius in the range 0.4 <   < 0.2.The results for fixed   = 0.33 and 0.5 serve to depress best-fitting values for   ,0 and also keep these values nearly constant with radius; this simply illustrates well the covariance between the two stratification parameters.

Barred and unbarred samples
Barred galaxies comprise ∼23% of our sample.It is reasonable to expect that in radial regions where bars are present, our formulation for inferring velocity dispersion from asymmetric drift may break down.Bars are due to resonances that perturb orbits far from circular motion and can lead to gas shocks that strongly differentiate between (collisional) gas and (collisionless) stellar radial and tangential motions.
In the right-hand panel of Figure 9, we recompute the best-fitting stratification parameters as a function of radius for the ensemble method, this time excluding galaxies with visually identified bars (green curves).The result leads to a smoother and steeper trend in   with radius, peaking at 0.5.Commensurately, the amplitude of   ,0 decreases.There is no change outside of / 50 = 1.75, as one might expect since bars do not modulate orbits at large radii.Trends with radius in the stratification parameters fit only to barred galaxies are not plotted; we find they are very erratic, leading, for example, to strongly negative values of   at intermediate radius.For this reason we focus our remaining analysis on the unbarred sample.

Ensemble versus individual galaxy fitting methods
The right-hand panel of Figure 9 also compares results for unbarred galaxies for the ensemble and indy methods (the latter are shown as red points).For the indy method we compute a weighted average of the stratification parameters for individual galaxies in radial bins according to the mean radius of the measurements for that galaxy.As a result, the radial bins in the indy-method averages are over a broader range of radii than for the ensemble-method.Given the MaNGA target selection (Wake et al. 2017), there is also a slight increase in the mean mass at larger mean radii; the effect is less than 0.1 dex, but primarily there are no galaxies below 10 10 M ⊙ in the largest indy radial bin.The agreement in   ,0 is remarkably good.The trend with   is somewhat flatter for the indy-method results, but still consistent within the uncertainties with the ensemble method; the flattening may be due to the wider radial coverage in the indy method.Overall, this comparison appears to indicate that any systematics due to the assumptions described in Section 5.2 are not particularly strong.

Correlation with galaxy stellar mass
Another immediate impression from Figure 8 is that at any given radius there is a gradient in total galaxy mass along the trend-line between the  AD and log(1+ LW / dyn ).This differentiation weakens at larger radius where the slope also appears to lessen.It is reasonable to wonder if, or to what extent a mass-dependence in both quantities drives the correlation between these two parameters.That is to say,

*
.5 1 1.5 2 2.5 3 Figure 10.Stratification parameters   ,0 and   versus radius scaled to  50 for three bins in stellar mass (solar units) for unbarred galaxies.Radial bins correspond to those in Figures 8 and 9. Best parameters and 67% CL from fitting the model to the galaxy ensemble are shown as green polygons.Open points in red represent weighted averages of the best model parameters for fits to individual galaxies, binned by the mean radius of the measurements for a given galaxy, as described in the text.Error bars represent the error in the weighted mean (y), and the width of the radial bin encompassing 67% of the measurements (x).
it is conceivable that there exist primary causal links between mass and both mean stellar population age (as a reflection of trends in starformation history with halo mass) and stellar velocity dispersion (a proxy for mass in dispersion-dominated systems), the result of which is an apparent correlation between  AD and log(1 +  LW / dyn ) that is not driven by a stratification process.Even in systems that are not dispersion-dominated (such as galaxy disks), it is reasonable to expect that more massive galaxies will tend to have more massive disks with larger velocity dispersions, and hence larger  AD .
Alternatively, we suggest this mass-gradient manifests in the data as a reflection of a general age-velocity relationship in stellar populations where (i) more massive galaxies tend to have more evolved (older and more metal rich) stellar populations (i.e., the trends in the CMD); and (ii) within galaxies the inner regions tend to have older stellar populations (so-called inside out growth).Equivalently: latetype galaxies tend to be less massive and have smaller bulges (and more modest population gradients), while early-type galaxies tend to be more massive and have larger bulges (and more pronounced populations gradients transitioning from bulge to disk).These trends reflect the distribution of, e.g., D n (4000), within any given galaxy.The correlation between  AD and log(1 +  LW / dyn ), in this picture, represents a general age-velocity relationship in stellar populations with all disk galaxies.The fact that there is general agreement between the stratification parameters in the ensemble and indy methods (Figure 9) supports this picture.
To further demonstrate that this correlation is not driven entirely by the correlation of mean stellar population age with galaxy stellar mass, we can also divide our sample by mass into three equal bins by number of galaxies (and roughly equal number of individual measurements) and repeat the fitting exercise.The mass boundaries are log M ★ /M ⊙ = 10.4 and 10.8.The median mass in each bin is 10.2, 10.6 and 11.0 in the log, with bins widths of 0.9, 0.4, and 0.5 dex6 .The results, as a function of both mass and radius, are illustrated in Figure 10 for both ensemble and indy methods.Again, there are clearly non-zero values for   in all mass bins and at most radii, further supporting a stratification model driving a general agevelocity relationship in stellar populations.It is also interesting to note that the stratification parameters at  =  50 are very similar in all mass intervals, despite the different radial trends.This may suggest a self-similar stratification mechanism at this radius across galaxy masses.

Stratification rate versus stellar mass
While the trends in the stratification parameters with radius appear somewhat more erratic when divided by mass (due to smaller subsamples), the general trends with radius in   ,0 and   seen for the and   versus galaxy total stellar mass for all radii combined.Small circles represent results for the indy method (individual galaxies), color-coded by their Sersic index as given in the key.
Open circles represent galaxies that do not have sufficient dynamic range in  LW / dyn (Equation 8).Light grey, dark grey and black lines represent the weighted standard deviation, error in the mean and mean for the indy method measurements.Large, filled circles represent the best fit values for the ensemble method.
full sample are evident for the intermediate and higher-mass subsamples based on results for the ensemble method: the stratification index (  ) is lower at larger radii.In contrast, there is little evidence for a radial gradient in the stratification parameters for log M ★ /M ⊙ < 10.4.This suggests the stratification process may be different in lower-mass galaxies.The results for the radial trends from the indy method are more difficult to assess given their limited dynamic range in mean radius, but the results do suggest that the stratification index (  ) increases with galaxy mass.The mass-dependence of   is seen more clearly when we recompute the marginal trend in mass alone by fitting to all radii in each mass bin (ensemble method) or combining all galaxies of a given mass for the indy method.The results shown in Figure 11  between  AD and log(1 +  LW / dyn ) is driven by a stratification process whose index (  ) increases with mass.

Discussion
Overall the stratification index (  ) appears to be lower for the ensemble method than the indy method.Our preference here is for the indy method values: Because the effective radial range of the indy measurements is modest, systematics are also likely modest.The ensemble method, as already noted, may tend to bias toward lower   and higher   ,0 if the population exhibits a significant range of stratification.We find no appreciable change in minimum reduced  2 ( 2  ) or sample rejection fraction (with iterative clipping) in the ensemble method when dividing the sample by mass.This suggests additional astrophysical scatter or the limited efficacy of the simple model to explain age-velocity relations in the galaxy population.
We explored if the dependence on stratification parameters was better defined by the Sersic index (as a measure of stellar mass concentration).Figure 11 also serves to show that trends in the stratification parameters with mass are mimicked by trends of the Sersic index with mass.Given the range of the stratification parameters at a given mass and the strong correlation of Sersic index with mass, we have been unable to find any additional significant dependence on the Sersic index.
As one might anticipate from Equation 7, and as noted above (Figures 9) through 11, there is a strong covariance between   ,0 and   .This is best seen in the stratification parameter values determined for individual galaxies, as shown in Figure 12.The slope of the correlation between log   ,0 and   is well described by the characteristic value of − log(1+/ dyn ) = −1.5, as seen from inspection of Figure 8, and as it should be; the correlation is slightly steeper(shallower) for the higher(lower) mass galaxies, also consistent with the trends in the log(1 + / dyn ) distributions as a function of mass as seen in this figure.

Local Group age-velocity dispersion relations
There can now be found in the literature measurements of stellar velocity dispersion and age for Local Group galaxies to directly compare our results.Here we limit our comparison to three galaxies with comparable range of mass as our MaNGA sample, namely (in order of decreasing mass) M31, the Milky Way and M33.We adopt total stellar masses of 1.04 ± 0.05 × 10 11 M ⊙ for M31 from a compilation by Mutch et al. (2011); 5.9 ± 1.3 × 10 10 M ⊙ for the MW from Licquia & Newman (2016); and 4.5 ± 1.5 × 10 9 M ⊙ for M33 from Corbelli (2003).The next three subsections details how  AD ,  50 ,   and   ,0 are derived for each galaxy.

M31
We use the measurements of stellar age and kinematics from two comprehensive studies of M31 by Dorman et al. (2015) and Quirk et al. (2019).From Dorman et al. (2015) we estimate  LOS from their Figures 16 at radii of 10 and 15 kpc, and also from Figure 12, where we adopt a characteristic radius of 12.5 kpc.Parenthetically we note that inside of 8 kpc ∼ 1.25 50 the age-velocity dispersion abruptly changes, referring to Dorman et al. (2015)'s Figure 16.While there may remain a clear distinction in the kinematics of the youngest and older populations, the intermediate and older populations are indistinguishable in terms of dispersion alone.
To convert the line-of-sight velocity dispersions ( LOS ) to  AD we first scale  LOS to   .When the rotation curve is nearly flat and the epicycle approximation (equation C2) ∼ 1/2, we can write [(sin 2 () + cos 2 ()/2) sin 2 () +  2 cos 2 ()].Looking at the field coverage on M31 in Dorman et al. (2015)'s analysis, a reasonable estimate the characteristic on-sky azimuthal angle sampled in this work is  = 45 • , which corresponds to  ∼ 77 • for a disk inclination of  = 77 • .For  = 0.6 we find   = 1.03  LOS .
To convert from   to  AD using equation 4 requires knowing the radial range of their measurements with respect to M33's halflight radius.For this estimate we use -band measurements and light profile decomposition from Courteau et al. (2011), yielding bulge Sersic index of   = 2.2±0.3 and effective radius   = 1.0±0.2kpc, a disk scale-length of 5.3 ± 0.5 kpc, and a disk-to-total luminosity ratio of 0.75 ± 0.04.Numerically integrating the light profile we find  50 (M31) = 6.5 +1 −3 kpc.For an exponential disk we then have 1 <  AD /  < 2.6 in the radial range of interest between 1.5 < / 50 < 2.3.Quirk et al. (2019) directly measure asymmetric drift (  ) which we convert to  AD adopting   = 250 km s −1 .We adopt their   values derived using HI-envelope estimates of the circular speed.It is interesting to note that   < 0 for their youngest stellar bin (main sequence stars with ages of 30 Myr).They attribute this to star-forming regions originating from dense molecular clouds that are dynamically colder than the ambient atomic and molecular gas.Interestingly, Shetty et al. (2020) also see evidence for such behavior in their multi-age analysis of asymmetric drift of MaNGA galaxies, albeit for a slightly older age stellar population and with respect to the ionized gas tangential speed.However, the corresponding dispersions for the same age population from Dorman et al. (2015) seem, on the face of it, to be inconsistent with this hypothesis for M31.It may be that velocity shear and molecular clouds lead to very rapid initial heating of the in-plane dispersion, as suggested by Kokubo & Ida (1992).Resolving this tension no doubt will yield a better understanding of heating processes in disks, particularly at early ages.For the purposes of establishing the age-velocity dispersion relation for M31, we exclude the Quirk et al. ( 2019)   value for the youngest age in our analysis.The resulting data are plotted in Figure 13, where there is good agreement between the Dorman et al. (2015) and Quirk et al. (2019) values.A simple linear regression yields   = 0.28 ± 0.03 and   ,0 = 26 ± 3 km s −1 .

Milky Way
We use the measurements of stellar age and velocity dispersion for the solar neighborhood from the Geneva-Copenhagen survey (Nordström et al. 2004;Holmberg et al. 2007Holmberg et al. , 2009)).Specifically, we adopt the data as presented in Figure 31 of Nordström et al. (2004) for the radial component dispersions, noting Seabroke & Gilmore (2007)'s caution against interpreting dispersion values for in-plane components.
To scale   to  AD we estimate  50 by adopting estimates from Licquia & Newman (2016) of a disk-to-total luminosity ratio of 0.84 ± 0.03 and a disk scale-length of 2.7 ± 0.2 kpc.We adopt a bulge Sersic index of 1.7 and a half-light radius of 1 kpc.Numerically integrating the light profile we find  50 (MW) = 3.85 ± 0.5 kpc.This yields  AD /  ∼ 2.5 at the solar circle, which we take to be at a radius of 8 kpc.The resulting data are plotted in Figure 13.We neglect the errors associated with these measurements since they are small and relatively uniform.A simple linear regression yields   = 0.32 ± 0.03 (essentially identical to values found by Nordström et al. (2004)) and   ,0 = 9 ± 1 km s −1 .
We also use more recent measurements of   by Mackereth et al.
(2019) that cover a range of radii based on APOGEE-2 DR 14 and Gaia DR2 data (see references therein).

M33
We use measurements from Beasley et al. (2015) for star-cluster ages and velocity dispersions and adopt their inclination of  = 56 • for M33's disk.To convert the line-of-sight velocity dispersions ( LOS ) to  AD we first scale  LOS to   .Adopting the same approach as for M31, we find the characteristic in-plane projection angle  ∼ 61 • for M33, so that for  = 0.6 we have   = 1.12LOS .To convert from   to  AD using equation 4 requires knowing the radial range of their measurements with respect to M33's half-light radius.Simon et al. (2006) estimate the disk scale-length as 1.36 ± 0.06 kpc in the -band (adopting a distance of 800 kpc), which agrees well with the earlier -band measurements of Regan & Vogel (1994).Regan & Vogel (1994) tabulate scale-lengths in a wide range of passbands, indicating significant color-gradients (the disk appears larger at bluer wavelengths).However, there are some discrepancies in the tabulated -band value from Guidoni et al. (1981) and the original work.To match to the -band estimates for M31, we adopt Regan & Vogel (1994)'s -band measurements with an uncertainty bracketed by their -band value and the original -band value from Guidoni et al. (1981), corrected to the same distance as Simon et al. (2006).Assuming a pure exponential disk we arrive at  50 (M33) = 2.4 +0.5 −0.1 kpc.This places the  LOS measurements from Beasley et al. (2015) to fall between 1 < / 50 < 1.65, yielding the corresponding scaling of 1.6 <  AD /  < 2.2.The resulting data are plotted in Figure 13.A simple linear regression yields   = 0.32 ± 0.06 and   ,0 = 15 ± 4 km s −1 .
More recent work by Quirk et al. (2022) based on photometric identification and spectroscopic measurements of field star kinematics in M33 shows little indication for the dramatic increase in asymmetric drift or velocity dispersion with stellar population age.I.e., their data is consistent with a nearly flat AVR.The origin of the differences between these two studies is unclear, but it is relevant to note the related work by Gilbert et al. (2022) who find M33 has a bimodal velocity distribution in its red-giant branch stellar population.Gilbert et al. (2022) identify the hotter component as a compact halo population; when it is included in Quirk et al. (2022)'s analysis, the AVR is no longer flat and more closely resembles the result from Beasley et al. (2015).Certainly in our MaNGA data no such distinction in populations is made.However, we do see evidence for some MaNGA galaxies of similar mass as M33 having flat AVR, and indeed one might argue the distribution in   is bimodal below log M ★ /M ⊙ ℎ −2 70 = 10.4.For present comparative purposes with MaNGA we continue with the results based on star-cluster measurements from Beasley et al. (2015).

Comparison between MaNGA and the Local Group
Figure 14 shows the stratification parameters for the MaNGA sample as a function of radius (i) for the indy-method results divided into three mass bins; and (ii) for the ensemble method for all masses combined.Keeping in mind that MaNGA measurements are based on asymmetric drift, these are compared to values derived for the three massive galaxies in the Local Group.
There are clearly different radial ranges being probed in the two samples, with larger radii (with respect to the half-light radius) for the Local Group.Nonetheless, there is reasonable overlap near 1.5 / 0.5 for the indy method, while the ensemble method extends over the full range sampled in the Local Group.The MW appears to be an outlier in having unusually low   ,0 .While   ,0 is a particularly large extrapolation of the stratification model based on the data for the MW compared to M31 and M33, as seen in Figure 13, the same Figure does show that overall  AD is low for the MW at comparable values of stellar age/ dyn .M33 and M31 appear to follow the trend of increasing   ,0 with radius reasonably well.Both of these galaxies appear to have had significant interactions over the last 2-3 Gyr (Putman et al. 2009;D'Souza & Bell 2018).In light of this, and the fact that   ,0 is a model extrapolation for all data considered here,   ,0 may not represent the true birth dispersion of stars.The characteristic stratification index values (  ) for the Local Group are quite uniform, and while they do not show any clear mass trend, they are within the range of values and the radial trend seen in the MaNGA data.

SUMMARY AND CONCLUSION
In this paper we have explored the age-velocity dispersion relation of galaxy disks in the MaNGA survey.Out of a sample of ∼ 2800 galaxies, we have selected a subset of ∼ 500 with regular kinematics indicative of rotating disks that have good tracers of tangential speed for both ionized gas and stars.We deemed these galaxies had suitably axisymmetric motions to characterize gas tangential speeds that could be related to the circular speed of the potential with modest corrections for pressure support.We have estimated and applied that correction to the gas kinematics from a relation between emissionline flux ratios and gas velocity dispersion calibrated at very high signal-to-noise Law et al. (2021b).We calibrated a proxy for light-weighted mean ages,  LW , for unresolved stellar populations based on the D n (4000) spectral index.This estimator is both reliable and observationally robust, i.e., it can be measured directly in individual spectra and repeatably in other data-sets.Galaxies in our sample contain stellar populations with  LW between 0.2 and 10 Gyr.
We used a light-weighted measurement of the stellar tangential speed to measure asymmetric drift with respect to the ionized gas.The asymmetric drift signal scales with the amplitude of the stellar velocity dispersion ellipsoid, but as a measure itself asymmetric drift depends only on velocities.Using asymmetric drift as a proxy for the stellar velocity dispersion alleviates the need to calibrate dispersion measures that extend well below the instrumental resolution.We showed that the asymmetric drift is proportional to in-plane radial component   to within an estimatable factor of order unity.We have also provided this proportionality.Values for  AD for our sample range from 20 to 300 km s −1 , which correspond to   of roughly half these values.
We combined the sample age and dispersion measures to determine a mean age-velocity dispersion relation for the ensemble as well as for individual galaxies.Ensemble measurements were made at five characteristic radii with median values between 0.4 and 2.85  50 , and in three bins of mass centered at log M ★ /M ⊙ = 10.2, 10.6 and 11.0.There is a modest bias between the light-weighting of the mean stellar population age and asymmetric drift signals.We have modeled this bias and corrected for it.
To characterize the AVR in the MaNGA sample, we have applied a simple power-law relation,  ∝   , between velocity dispersion, , and the age of the stellar population, scaled by the local dynamical time measured for each galaxy at each radius.The two parameters of this model are the radial velocity dispersion of the stellar population and young ages,   ,0 and the power-law index (or stratification rate)   .We find that barred galaxies in our sample yield erratic results for the derived stratification parameters, despite our visual selection for axisymmetric velocity fields.Consequently, we have removed barred galaxies (23% of our sample) from tertiary analysis.
This analysis shows that stratification rates for unbarred galaxies tend to decrease with radius and increases with stellar mass, with values in the range 0.2 <   < 0.5.These values of   bracket the theoretical expectations for dynamical disk heating found in the literature.The values of   ,0 are anti-correlated, and are in the range of 10 km s −1 <  0, < 25 km s −1 .Lower-mass galaxies (log M ★ /M ⊙ < 10.4) appear to have little trend of their stratification parameters with radius.This transition is very close (in stellar mass) to that noted by Dalcanton et al. (2004) for galaxies to exhibit thin dust lanes.It is possible that changes in the mid-plane density of star-forming clouds at this mass transition play an important role in driving different trends in dynamical stratification; it is equally plausible that the inside-out mass-assembly and star-formation quenching processes evident in higher-mass galaxies are less pronounced at lower mass as a consequence.
A comparison of the stratification parameters of MaNGA galaxies to those derived from literature measurements for the three most massive galaxies in the Local Group (M33, the MW, and M31) show broadly consistent radial and mass trends.Hence dynamical stratification in MaNGA galaxies is broadly consistent with what we know about stratification in the Local Group.The stratification rate   values for MaNGA galaxies are well within the range measured for the Milky Way, noting that for in-plane motions  is smaller (and  0 is larger) compared to vertical motions (Nordström et al. 2004;Holmberg et al. 2007Holmberg et al. , 2009)).However, the MW appears to have a particularly small value of   ,0 , and it is at the extrema of the range of values computed for galaxies of comparable mass in the MaNGA sample.
The correspondence of the best-fitting values in integrated starlight with those for resolved stellar populations in the Local Group indicates the method of using AD in integrated star-light is reliable.Nonetheless, we do find that the simple stratification model does not capture well the observed dispersion in the data, and we find modest systematics between fitting ensembles of galaxies compared to individual galaxies.This suggests that a richer analysis of high SNR data that is able to resolve the AVR in integrated star-light at specific locations within galaxies (e.g., Shetty et al. 2020) will lead to progress in our understanding of the dynamical stratification of stellar populations.

APPENDIX C: DERIVATION OF RELATION BETWEEN 𝜎 AD AND COMPONENTS OF THE STELLAR VELOCITY ELLIPSOID
We derive an equation for asymmetric drift that can be related to observables in the MaNGA data set for Sersic density profiles and a generalized rotation curve.With several more assumptions and formalisms, we express this relation entirely in terms of factors modulating a single component of the stellar velocity dispersion ellipsoid (SVE).We choose here   .We derive a specific case where the density profile is exponential and the rotation curve is well described by a radial tanh function.In what follows we define the SVE vertical to radial axis ratio as  ≡   /  .While specific examples are common in the literature for application to resolved stellar kinematics of the Milky Way's solar neighborhood and several notable cases for application to external galaxies (e.g., Gerssen et al. 1997Gerssen et al. , 2000;;Shapiro et al. 2003;Noordermeer et al. 2008;Weĳmans et al. 2008;Herrmann & Ciardullo 2009;Westfall et al. 2011;Gerssen & Shapiro Griffin 2012;Westfall et al. 2014) these derivations often differ in their assumptions about which terms can be ignored and are less general.
The quantity  AD (equation 1; asymmetric drift) are related to components of the SVE and derivatives of density and velocity distribution functions via integral moments of the Collisionless Boltzman Equation (CBE).The rudiments of such a relation are found in Binney & Tremaine (2008).For tractable application to observational data, simplifying assumptions are required.As commonly done, we assume the galaxy stellar dynamics are roughly in steady state, azimuthally symmetric, and have no radial streaming motion.This allows us to eliminate terms such that the integral of the radialvelocity moment (  ) of the CBE yields (e.g.Binney & Tremaine 2008, eq. 4.227): The first two terms can be consolidated in terms of, e.g.,   , by combining the integrals of the   and     moments of the CBE to derive the epicyle approximation: The third term of equation C1 can be parsed by assuming the radial dependence of the tracer density () is proportional to the ratio of the stellar surface-density (Σ) and the vertical thickness (ℎ  ):  ∝ Σ/ℎ  .Dimensionally, we expect  ∝ (/ℎ) 2 , where  is a component of the SVE and ℎ is a characteristic length-scale in that dimension.From this one finds, e.g.,  ∝ (  /ℎ  ) 2 and the familiar scaling   ∝ √︁ Σℎ  .Indeed, for galaxies with disks described by exponential light profiles Martinsson et al. (2013) showed that   ∝ exp(−/2ℎ  ).It would also be reasonable to expect   to have a similar radial dependence, as seen in the Milky Way (Lewis & Freeman 1989).This happenstance is equivalent to  or the relevant length-scales being constant with radius -a further assumption that we adopt.Generalizing for a Sersic surface-density profile of index   where Σ() = Σ 50 exp(−  ((/ 50 ) 1/  − 1)) and   ∼ 2  − 1/3, the argument of the derivative in the third term of equation C1 can then be written as a quantity proportional to exp(−2  ((/ 50 ) 1/  − 1)).The result can be expressed as ) This reduces to  2  (2/ℎ  ) for   = 1, i.e., an exponential disk.The last term represents the tilt in the SVE with respect to a cylindrical coordinate system referenced to the disk.As shown by Amendt & Cuddeford (1991), the derivative can be approximated as where 0 ≤  ≤ 1 and the extrema correspond to cylindrical and spherical orientation, respectively.These substitutions yield where the logarithmic derivative of the tangential speed can be evaluated directly from the data or a parametric fit to the data.For example, for a rotation curve well described by a model where  () =  rot tanh(/ℎ rot ) we find  ln    ln  =  sech() csch() (C6) where  = /ℎ rot .We adopt this form in our analysis here, which is suitable at all radii in a galaxy where the radial profile of disk mass and tangential velocity are well characterized by a Sersic profile of constant thickness and tanh, respectively.The dominant term in equation C5 is given by equation C3 for radii above ∼ 0.25  50 for the full range of Sersic index and rotation curves in our sample.However this term is only weakly dependent on the Sersic index.The different terms in equation C5 are illustrated in Figure C1.

APPENDIX D: SYSTEMATICS BETWEEN LIGHT-WEIGHTED AGES AND KINEMATICS
To quantify potential mismatch between light-weighted ages and kinematics from full-spectrum fitting over the same range in wavelength, we created 5 × 10 4 mock spectra spanning a wide range of star-formation histories (SFH; ()) and rates of dynamical stratification that appear to span what is observed in our data.The mock spectra are composites of solar metallicity SSPs (MIUS-CAT, Vazdekis et al. 2012), rendered at the MaNGA spectral resolution, weighted according to the SFH, and each velocity broadened and shifted according to their age and the dynamical stratification model.We adopt the power-law dynamical stratification for   exactly of the form of equation 7 with 0.05 <   < 0.5 and 5 <   ,0 (km s −1 ) < 50.Model dynamical times ranged from 0.013 to 0.13 Gyr with a median of 0.042 Gyr, closely matching the observed median dynamical time of 0.035 Gyr, ranging from 0.02 Gyr at 0.25  50 to 0.05 Gyr at 1.25  50 (90% of the sample has dynamical times between 0.011 and 0.077 Gyr over this radial range).The velocity scale of each simulation is set by  2  = 35 2 +  2 AD ( = 14Gyr) where  AD = 1.62   (i.e.,  =  50 , although this choice is not important).The   zeropoint value of 35 km s −1 is introduced to ensure the oldest stellar populations retain some rotational support as part of the galaxy disk population.SSPs are broadened by   () and shifted in velocity by ( 2  −  AD () 2 ) 1/2 , following equation 1.Following studies of disk galaxies, e.g., the Milky Way (Pilyugin & Edmunds 1996), we take the star-formation rate () to be smooth and of the form: with  the age since the first stars formed.While actual star-forming histories are not always smooth and in cases include bursts, modeling such stochasticity is beyond the current scope.The above () consists of a rising rate up to  top , and either a flat or exponentially declining rate thereafter.We match the observed range in D n (4000) and age by letting 0.5 <  top (Gyr) < 8, −0.5 <  −1  (Gyr −1 ) < 1, and allowing formation epochs to span from 2.1 and 10.05 Gyr after the Big Bang.This corresponds to formation redshifts of 0.3 <  < 3.While this range extends to recent times, keep in mind that these simulations are meant to mimic specific locations within galaxy disks, the outskirts of which may begin to form relatively late.Taking the age of the universe to be 14 Gyr, some simulation are still in the rising portion of their SFH today.The simulation values considered are for the current epoch ( = 0), suitable for comparison to the MaNGA observations of galaxies with modest look-back times under 1 Gyr.
To match simulation analysis to our data analysis we use full spectrum fitting with pPXF, with no regularization, to recover the velocities and light-weighted ages.The same SSPs that form the mocks are adopted as templates.This eliminates the effect that template mismatch might have on the measured kinematics and ages; in the case of velocities we find template mismatch is a very modest effect (< 6 km s −1 ; Shetty et al. 2020).
We quantify the systematic in terms of the ratio of the measured  AD from pPXF ( AD (pPXF)) to the quantity estimated from the model   () at the light-weighted age derived from the pPXF fitting ( AD (model[age pPXF ])).In this way we estimate the correction for  AD appropriate for the pPXF-measured age.
The amplitude of the  AD systematic is shown in Figure D1 as a bivariate function of   and D n (4000)(middle), and compared to the distribution of data in our sample at all radii (left).The middle map serves as the look-up table to correct  AD as a function of the observed D n (4000) and   derived from equation 3.
This paper has been typeset from a T E X/L A T E X file prepared by the author.

Figure 1 .
Figure 1.Rest-frame  −  versus   (left) and stellar mass (middle and right).The left panel shows the MPL-5 MaNGA sample of 2744 galaxies (grey points).The 798 kinematically regular galaxies (selected as described in text) are marked for the final subset of 497 galaxies (dark points), 192 galaxies excluded for being too inclined (red points), and 109 galaxies excluded for having too low S/N in either the stellar or ionized gas velocity fields (yellow points).In the middle panel the 497 galaxies are color coded by Sersic index as given in the key.Galaxies with strong bar morphology are plotted with the same color-coding in the right panel.Two parallel dashed lines delineate the blue cloud, green valley and red sequence.

Figure 2 .
Figure 2. Examples of sample galaxies as a function of increasing Sersic index (labeled, left to right) without strong bar morphology.Purple hexagons indicate the MaNGA IFU footprint, with a green horizontal bar in the top-left panel showing the angular scale of 5 arcsec.Galaxy IFU and plate numbers are marked.Stellar and gas velocity fields are shown in the middle and bottom rows.Dark-grey open circles indicate a radius of 4 arcsec, within which we do not consider kinematics measurements due to the strong impact of beam-smearing on the line-of-sight measurements.

Figure 3 .
Figure 3. Examples of sample galaxies with bars as a function of increasing Sersic index, formatted as in Figure 2.

Figure 4 .
Figure 4. Distribution of the rotation-curve rise-scale (ℎ rot ) relative to the half-light radius ( 50 ) as a function of stellar mass for galaxies in our sample.Galaxies are color-coded by the Sersic index as given in the key.The thick black line represents the median value as a function of mass.

Figure 5 .
Figure 5. Left panels: Emission-line flux ratio diagrams for H, [O III]5007, H, [N II]6583, and [S II]6716,6731 for measurements at all radii of our sample.Points are color-coded by estimated gas velocity dispersion, given in the key, and assigned according to line-ratios as described in the text.Black dotted and dashed lines represent demarcations of HII-like line-ratios from Kauffmann et al. (2003) and Kewley et al. (2001) respectively; solid green and red lines demark dynamically cold, intermediate and hot ionization regions as measured byLaw et al. (2021b).Right panels: Fractional corrections to the measured asymmetric drift  AD for gas pressure-support as a function of the quadrature difference between the corrected and the measured gas tangential speed, radius (normalized by  50 ) and stellar mass.Uncertainties include a full propagation of errors from gas and stellar velocities, line-ratios, and the population variance (as measured byLaw et al. 2021b) in the inferred   for a given set of line-ratios.The horizontal dashed line represents our threshold of a <30% correction for inclusion in our remaining analysis.
between reliably-corrected MaNGA gas line-width measurements at high S/N (>50) and their line flux ratios of, e.g., [O III]/H versus [N II]/H.Specifically we use the average of the distance estimators d  2 and d 2 (defined in Law et al. (2021b), i.e., the distance in dex from the dynamically cold ridge-line in log([O III]/H) versus log([N II]/H) or log([S II]/H, respectively), and interpolate based on the mode and width of the histograms in Figure 3 of Law et al. (2021b) for d  2 to assign a characteristic value and uncertainty, respectively for the ionized gas line-width   .

Figure 8 .
Figure8.Asymmetric drift signal,  AD , versus the ratio of stellar population light-weighted age to dynamical time for five radial bins scaled by  50 given in the top left of each panel; the median and 67% range of dynamical times are given at the bottom of each panel.Points represent measurements from individual galaxies at a particular radius, color-coded by stellar mass as given in the key.Solid grey curves represent the best-fitting stratification model to the ensemble of galaxies at each radius.Dashed and dotted curves correspond to stratification models with   fixed to 0.33 and 0.5, respectively.

Figure 9 .
Figure9.Stratification parameters   ,0 and   versus radius scaled to  50 .Radial bins correspond to Figure8.Left: Best-fitting models fit simultaneously to data from all galaxies in a given radial bin.The black curve represents when both parameters free (dotted lines give 67% CL uncertainties on combined parameters).Best fitting values for   ,0 for   fixed to values of 0.33 and 0.5 are shown as orange and teal lines and dots, respectively.The inset plots the trend of reduced- 2 ,  2  , for these different cases ( 2  for   = 0.5 is off scale), as well as the unbarred sample shown in the right panel.Right: Best-fitting models to all unbarred galaxies are shown in green (grey repeats results for all galaxies from the left panel).Red data points are averages for fits to individual galaxies, binned by the mean radius of the measurements for a given galaxy, as described in the text.

Figure 11 .
Figure11.Stratification parameters   ,0 and   versus galaxy total stellar mass for all radii combined.Small circles represent results for the indy method (individual galaxies), color-coded by their Sersic index as given in the key.Open circles represent galaxies that do not have sufficient dynamic range in  LW / dyn (Equation8).Light grey, dark grey and black lines represent the weighted standard deviation, error in the mean and mean for the indy method measurements.Large, filled circles represent the best fit values for the ensemble method.

Figure 12 .
Figure12.Covariance in stratification parameters   ,0 and   .Symbols are as in Figure11, except color-coded by total stellar mass as given in the key (log M ★ /M ⊙ ).Dashed lines correspond to the mean parameter values for the indy method averaged over all mass.

Figure 13 .
Figure 13.Trends of  AD versus the ratio of the stellar age to the local dynamical time for Local Group galaxies M31, the Milky Way and M33, constructed as described in the text.The radial range scaled to the half-light radius is indicated for each galaxy.Unweighted regressions for each galaxy are shown with dashed lines.For M31, circles represent asymmetric drift measurements from Quirk et al. (2019); three and four sided points are scaled from  LOS at 10 and 15 kpc from Dorman et al. (2015); five-pointed stars are from Dorman15's Figure 12.Data used for M33 and the MW are described in the text.

Figure 14 .
Figure 14.Asymmetric drift stratification model parameters   ,0 and   versus radius scaled to  50 .MaNGA data (polygons and open dots) are shownfor the best-fitting model with both parameters free.Grey shaded polygons represent the ensemble method for all mass.Open red, green and blue points represent the indy method for the three mass bins defined in the key.Our estimates for values for M31, MW and M33 from the literature, as described in the text, are marked as filled red circles, green diamonds and light-blue triangles, respectively.Lighter green diamonds represent values for the MW fromMackereth et al. (2019).

Figure A1 .Figure B1 .
Figure A1.Example of kinematic and spectrophotometric signal extraction for 8146-12701 in bins having 2 arcsec radial width extending between ±40 • in  about the major axis in the disk plane.In all map panels the kinematic major axis and ±40 • wedges are indicated with red lines; an example aperture of 10-12 arcsec in radius is represented by a yellow contour.Top row (left to right): maps of D n (4000), the full projected stellar and gas tangential speed ( , * sin  cos  and  , sin  cos ), and the histogram of ( 2 , −  2 , * )/cos 2  -the differences of the squared, azimuthally-deprojected gas and stellar tangential speeds for every pixel in the example bin.The red curve in the histogram panel shows the cumulative distribution; the black vertical lines gives the sample weighted mean.Bottom row (left to right): flux ratio maps of [NII] and [SII] to H, [OIII] to H; and the radial profile of the inclination-projected kinematics ( , * sin ,  , sin , and  AD sin ) in the 6 radial bins with data for this galaxy.See text for further details.

Figure C1 .
Figure C1.Ratio of  AD /  versus radius scaled by the half-light radius.Values for an exponential disk (  = 1) are given by black curves filled with grey that bound 0 <  < 1 and 0.3 <  < 1.The central, thick black curve adopts  = 0.6 and  = 0.5.Values for   = 3, (also  = 0.6 and  = 0.5) are shown as a red solid curve.Values for the term in equation C3 for   = 1, 3 are shown as black and red dotted curves, respectively.Values for the term in equation C6 (with ℎ rot / 50 = 0.6) are shown as a dashed curves.

Figure D1 .
Figure D1.Bivariate distribution as a function of rotation speed (  ) and D n (4000) of observations (number per bin, left panel) and systematics in measured  AD due to differential light-weighting of kinematics and ages from full-spectrum fitting (middle and right) as a function of rotation speed (  ) and D n (4000).Systematics are derived from simulations described in the text.The midpoint (middle panel) and half-range over the midpoint (right panel) provide robust estimates of the mean and width of the simulations distribution.
Hänninen & Flynn (2002),  AD at a median radius of 0.8  50 versus (left to right): D n (4000), light-weighted mean age ( LW ) inferred from D n (4000), local dynamical time ( dyn , see text), and the ratio of these two time scales.Each point represents a galaxy measurement in the radial range from 0.5 < / 50 < 1, color-coded by stellar mass as given in the key.solar neighborhood yield 0.2 <  < 0.6; these are discussed in our introduction with the theoretical and observational literature also well summarized byKumamoto et al. (2017)andHänninen & Flynn (2002), respectively.It is also clear from these studies that the values of  and  0 depend on the component of the velocity ellipsoid: For the MW solar neighborhood the vertical index (  ) tends to be larger than the radial , but with lower initHolmberg et al. 2009;Aumer & Binney 2009;Sharma et al. 2014;Aumer et al. 2016 Binney 2009;Sharma et al. 2014;Aumer et al. 2016).