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##  Appendix A Stellar Ages & Related Caveats

As introduced in Section 2.4, our age estimates are adopted from the distmass
value added catalog (Stone-Martinez et al. 2023 submitted), which utilizes a
neural network trained on the ASPCAP stellar parameters and APOKASC
asteroseismic masses to derive stellar mass for all APOGEE stars. The reported
uncertainties, accuracy, and potential caveats related to the distmass age
estimates are discussed in detail in Stone-Martinez et al. (2023, submitted).
Here, we explore and discuss how these uncertainties and our particular sample
selection may influence our age-related results. In summary, we find that
while the general trends stay the same, both the qualitative and quantitative
aspects can differ significantly particularly among the oldest stars. We
caution against drawing strong conclusions from stellar ages until more
precise ages are available for larger samples of stars. Again, we emphasize
that this will only impact our figures that include stellar age.

###  A.1 Variation in Spatial Sampling

The distmass quality cuts described in 2.4 notably result in the exclusion of
all stars with [Fe/H] $\leq-0.7$ from our sample. In Figure A.1, we
qualitatively explore where in the Galaxy this selection criterion may
influence our results. This shows the fraction of stars $f_{\rm distmass}$ in
every spatial bin ($\Delta X=\Delta Y=\Delta Z=0.5$ kpc) which survive the
additional distmass quality cut. Close to the solar neighborhood ($6\leq R\leq
12$ kpc, $|Z|\leq 1$ kpc), the majority of stars pass this additional
selection criterion ($f_{\rm distmass}\geq 97\%$). Near the center of the
Galaxy ($R\leq 5$ kpc, $|Z|\leq 1$ kpc), $f_{\rm distmass}$ is generally lower
but still exceeds $85\%$. In the outer Galaxy ($R>15$ kpc) or beyond the plane
($|Z|\geq 1$ kpc), where most of the stars are expected to be metal-poor,
there is a sharp drop in $f_{\rm distmass}$, regularly reaching fractions
lower than $<80\%$. This shows that our sample with distmass ages, while
generally a majority, does not equally represent the full sample everywhere in
the Galaxy. Particularly beyond $R>15$ kpc we caution against directly
comparing our results including age to those made with the full sample.

Figure A.1: The same as Figure 2, but colored by the fraction of stars in our
sample that survive the additional distmass quality cuts described in Section
2.4. The purpose of this figure is to qualitatively explore how the distmass
criterion might bias our results using stellar age estimates.

###  A.2 Selection of Age Catalog

Figure A.2: Systematic uncertainties in the stellar age estimates, comparing
the distmass (green and blue) and astroNN (red) catalogs across different
ranges of stellar $\log g$ (different panels). In each panel, the x-axis is
the stellar age from asteroseismology estimates (APOKASC), and the y-axis is
the predicted age recovered from the neural network methodology. The typical
uncertainty of each point is depicted in black in the upper-left of each
panel, with the asteroseismic measurements notably having larger uncertainties
at lower $\log g$. In both catalogs and across all $\log g$, young stars are
generally recovered too old, and old stars are recovered too young. However,
there are higher uncertainties in the asteroseismic measurements for the
low-$\log g$ stars, meaning the neural network age estimates may be more
precise than the asteroseismic age estimates in this range. Figure A.3: The
spatial distribution (X-Y face-on view) of stars in APOGEE for different bins
of $\log g$ (columns). Increasing the $\log g$ range of our sample, which
would minimize some of the age-related systematics, would also severely limit
the area that our sample can probe.

Neural network-derived ages like distmass heavily rely on the accuracy of the
training set labels which transfer to the model. Small changes the training
set can result in differences in the derived ages of the full training set,
which contributes to the large uncertainty values around stellar ages.

The distmass catalog publishes results from six different neural network
models. All six models were trained using the same subset of stars and the
same ASPCAP parameters as training labels, with the only difference being the
choice of APOKASC asteroseismic masses used in the training set. The APOKASC
catalog publishes asteroseismic results from three different groups (initials
“SS”, “MO” and “TW” \- see Pinsonneault et al. 2023 in prep. for details),
referred to as the “uncorrected” masses. Each set of masses also has a
“corrected” counterpart, using Gaia parallaxes to calibrate the stellar radius
derived from asteroseismology (e.g., Zinn et al., 2019). The calibrations are
dependent on stellar $\log g$ and are less reliable for stars $\log g\leq 2$,
which is why we elect to use the uncalibrated age set for our final results.
Stone-Martinez et al. 2023 in prep. goes into more detail on this motivation.

Figure A.2 compares different three different age catalogs - the uncalibrated
distmass ages used in this paper (green points), its calibrated counterpart
(blue points), and the astroNN APOGEE VAC (Leung & Bovy, 2018; Mackereth et
al., 2019) (red points) - against the APOKASC derived stellar ages on the
x-axis for the subset of stars in the distmass training set. The comparison is
divided by ranges in stellar $\log g$ (columns, increasing left to right). We
find that all three catalogs suffer a “compression” in ages compared to the
APOKASC values, with young stars tend to be assigned ages that are too old,
and old stars tend to be fit too young. All three sets of ages predict the
ages of higher $\log g$ with better accuracy and less scatter. For lower $\log
g$ stars, the calibrated distmass ages predict the young ages slightly better,
but and astroNN and the uncalibrated distmass predicts the older ages better.
For higher $\log g$ stars, astroNN predicts the young ages slightly better,
and distmass predicts the older ages better. This offset is thought to be
caused by scatter within the training set that the neural network learns from.
The asteroseismic masses are more uncertain for low $\log g$ stars, which
results in a larger mismatch between the APOKASC results and the neural
network results. The neural network essentially fits for the relation between
[C/N] abundance and stellar age, which should be consistent across a broad
range of $\log g$. Therefore, we emphasize for low $\log g$ stars, the neural
network likely fits more precise ages than available in the pure
asteroseismology results used in the training set.

Although the age estimates show less scatter for higher $\log g$ stars,
redefining our sample would severely limit the spatial range of the Galaxy we
can access. Figure A.3 shows the face-on distribution of stars in the disk for
various $\log g$ limits. Above $\log g\geq 2$, the outer disk ($R\geq 20$ kpc)
and the inner disk ($R\leq 5$ kpc) disappear from the sample.

Nevertheless, we did test how redefining the sample may influence our results.
In general, the observed trends and resulting conclusions do not change,
motivating the use of the original sample for the increased spatial range. The
radial metallicity gradient as a function of age (Figure 17) differs by less
than $0.001$ dex kpc-1, and the radial age gradient (Figure 20) differs by
$0.03$ Gyr kpc-1 using the higher $\log g$ sample.

###  A.3 Alpha-Age relation for different catalogs

To demonstrate how a different selection of age catalog may influence our
results, Figure A.4 shows a recreation of Figure 25 for three different age
samples:

  * •

Test Sample A ($\log g$ range): restricting to $2.0\leq\log g\leq 2.3$:
defining a new sample with higher surface gravity

  * •

Test Sample B (alternate distmass ages): using the Gaia-corrected APOKASC ages
training set from distmass, with the original sample of $1.0\leq\log g\leq
2.0$.

  * •

Test Sample C (alternate astroNN ages): using astroNN ages instead of the
distmass catalog, with the original sample of $1.0\leq\log g\leq 2.0$.

Test Sample A (Figure A.4.b) changes our limits to $2.0\leq\log g\leq 2.3$,
shifting our sample to a different $\log g$ range where the age estimates are
expected to be more reliable, while purposely avoiding the red clump. However,
this suffers from the previously-shown lack of spatial distribution, with
significantly fewer stars beyond $R>12$ kpc and within $R<6$ kpc than the
original sample used in this paper. This test sample does include a larger
number of stars near the solar neighborhood, however. The general location and
trends across galactic location are similar, but this sample has more scatter
which leads to larger contour shapes. The bimodality within the low-$\alpha$
sequence (possible evidence for a “third infall”) around $9<R<12$ kpc is not
obvious in this sample, although is perhaps present at shorter radii ($3<R<9$
kpc).

Test Sample B uses the distmass ages using the calibrated ASPCAP training set.
The Gaia-motivated calibration is dependent on $\log g$ (e.g., Zinn et al.,
2019, Pinsonneault et al. 2023 in prep.), which results in lower $\log g$
stars assigned younger ages in the calibrated catalog and higher $\log g$
stars being assigned systematically older ages and resulting in an overall age
compression in the sample obvious is Figure A.4.c. Otherwise, the results are
generally similar, with the possible bimodality within the low-$\alpha$
sequence at $9<R<12$ kpc still present in this sample. The high-$\alpha$
sequence has more spread in age, possible due to the calibrations propagating
uncertainties in the adopted calibrations (from the $T_{\textrm{eff}}$
assumptions for example) through into the final age estimates.

Test Sample C uses astroNN ages instead of distmass in Figure A.4.d, which
makes obvious the overall “compression” of the ages caused by the systematics
discussed earlier; astroNN ages do not go as old or as young as the distmass
ages. The general shapes and trends across the Galaxy otherwise stay the same,
and the bimodality within the low-$\alpha$ sequence is not obvious in this
sample either.

We also include a reproduction of the median radial age profile (Figure 20)
and the age-metallicity relation (Figure 23) using the higher-$\log g$ Test
Sample A in Figures A.5 through A.6. As before, the trends in each plot are
similar to their low-$\log g$ counterparts, but with higher scatter, as the
range of Galaxy covered by the sample is more limited despite the lower
uncertainties in stellar ages.

###  A.4 Summary of Age Caveats

Stellar ages are one of the most powerful quantities available in modern
astronomy for revealing the history of the Milky Way, but remain challenging
to measure robustly. In this study and others, it is important to consider
age-related results with caution and understand the subsequent effects that
changes in stellar ages may have on the conclusions. While different age
catalogs show generally similar trends for the results of this paper, the
quantitative aspects can differ significantly and should be regarded with
healthy skepticism until more precise age estimates are available for large
samples of stars.

Figure A.4: Reproduction of the age-alpha relation (Figure 25) using different
samples of stellar ages.

(a) Same as Figure 25, reproduced here for ease of comparison.

(b) Same as (a), but for a sample covering a higher surface gravity range of
$2.0\leq\log g\leq 2.3$ (Test Sample A)

(c) Same as (a), but using the “calibrated” ages from distmass trained on the
Gaia-corrected APOKASC catalog (Test Sample B).

(d) Same as (a), but using stellar ages from the astroNN catalog (Test Sample
C).

Figure A.5: Reproduction of the radial age gradients (Figure 20) using
different samples of stellar ages.

(a) Same as Figure 20, reproduced here for ease of comparison.

(b) Same as (a), but using a sample covering a higher surface gravity range of
$2.0\leq\log g\leq 2.3$ (Test Sample A)

Figure A.6: Reproduction of the age-metallicity relation (Figure 23) using
different samples of stellar ages.

(a) Same as Figure 23, reproduced here for ease of comparison.

(b) Same as (a), but for a sample covering a higher surface gravity range of
$2.0\leq\log g\leq 2.3$ (Test Sample A)