11institutetext: Institute of Astronomy, KU Leuven, Celestijnenlaan 200D,
3001, Leuven, Belgium
11email<EMAIL_ADDRESS>22institutetext: Department of
Astrophysics, IMAPP, Radboud University Nijmegen, PO Box 9010, 6500 GL
Nijmegen, The Netherlands 33institutetext: Max Planck Institut für Astronomie,
Königstuhl 17, 69117 Heidelberg, Germany 44institutetext: IRAP, Université de
Toulouse, CNRS, UPS, CNES, 14 avenue Édouard Belin, F-31400 Toulouse, France
55institutetext: Center for Interdisciplinary Exploration and Research in
Astrophysics (CIERA), Northwestern University, 2145 Sheridan Road, Evanston,
IL 60208, USA

# Age uncertainties of red giants due to cumulative rotational
mixing of progenitors calibrated by asteroseismology ††thanks: The full Table
A.1 is only available in electronic form at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-
bin/qcat?J/A+A/

D. J. Fritzewski 11 C. Aerts 112233 J. S. G. Mombarg 44 S. Gossage 55 T. Van
Reeth 11

###### Abstract

Context. Galactic archaeology largely relies on precise ages of distant
evolved stars in the Milky Way. Nowadays, asteroseismology can deliver ages
for many red giants observed with high-cadence, high-precision photometric
space missions such as CoRoT, Kepler, K2, TESS, and soon PLATO.

Aims. Our aim is to quantify age uncertainties of currently slowly-rotating
red giants due to the cumulative effect of their fast rotation during core-
hydrogen burning. Their rotation in earlier evolutionary phases caused mixing
of elements resulting in heavier helium cores and the prolongation of their
main sequence lifetime. These rotational effects are usually ignored when age-
dating red giants, despite our knowledge of fast rotation for stars with
$M\geq 1.3$ M⊙.

Methods. We use a sample of 490 F-type gravito-inertial pulsators
($\gamma\,$Doradus stars) with precise asteroseismic estimates of their
internal rotation rate from Kepler asteroseismology and with luminosity
estimates from _Gaia_. For this sample, which includes stars rotating from
nearly zero to about 60 % of the critical rate, we compute the cumulative
effect on the age in their post-main sequence evolution caused by rotational
mixing on the main sequence. We use stellar model grids with different
physical prescriptions mimicking rotational mixing to assess systematic
uncertainties on the age.

Results. With respect to non-rotating models, the sample of 490
$\gamma\,$Doradus stars, as red giant progenitors, reveals age differences up
to 5 % by the time they start hydrogen-shell burning when relying on the
theory of rotationally induced diffusive mixing as included in the MIST
isochrones. Using rotational mixing based on an advective-diffusive approach
including meridional circulation leads to an age shift of 20 % by the time of
the tip of the red giant branch.

Conclusions. Age-dating of red giants is affected by the cumulative effect of
rotational mixing during the main sequence. Such rotationally-induced age
shifts should be taken into account in addition to other effects if the aim is
to perform Galactic archaeological studies at the highest precision.

###### Key Words.:

Asteroseismology – Stars: rotation – Stars: fundamental parameters – Stars:
interiors – Stars: evolution – Stars: oscillations (including pulsations)

##  1 Introduction

Transport processes in stellar interiors have a significant impact on the
global properties and the chemical evolution of stars across the entire mass
range (e.g., Martins & Palacios, 2013; Pedersen et al., 2021; Lagarde et al.,
2017, for high-, intermediate-, and low-mass stars, respectively).
Asteroseismology has pinpointed two dominant tightly connected effects of
chemical mixing due to the accumulation over the core hydrogen burning main-
sequence phase: a considerable increase in the main-sequence duration and a
heavier helium core at the end of the main sequence (up to a factor two more
massive than without envelope mixing, see Johnston, 2021; Pedersen, 2022). In
this work, we consider the former for intermediate-mass stars from the
viewpoint of age-dating red giant stars for Galactic archaeology.

Unravelling the formation and evolutionary history of the Milky Way has many
facets. One important element concerns chemical tagging of stars dispersed
throughout the galaxy (e.g., De Silva et al., 2015; Schiavon et al., 2017;
Vitali et al., 2024). Other aspects involve determination of the spatial
structure and evolution of stellar populations, moving groups, and streams in
the Milky Way’s bulge, bar, thin, and thick discs (e.g., Bovy et al., 2016;
Rojas-Arriagada et al., 2017; Silva Aguirre et al., 2018; Helmi et al., 2018).
Since about a decade, asteroseismology of red giants offers high-precision
ages for these old bright stars across the Milky Way (see the extensive
studies by Miglio et al., 2013; Chiappini et al., 2015; Montalbán et al.,
2021; Hon et al., 2021; Wang et al., 2023). Age-chemo-kinematic properties of
red giants indicate efficient radial migration in the thin disc, different
chemo-dynamical histories of the thick and thin discs, and changes in the star
formation after the formation of the thick disc (Miglio et al., 2021). The age
determinations of these populations of red giants in the Galaxy are of
critical importance to ensure a proper interpretation of the archaeological
sites of the Milky Way. While various core boundary mixing prescriptions are
nowadays being considered for the core helium burning phase of red giants
(e.g., Noll et al., 2024), almost none of the constructed models take into
account the previous long history of these stars during the core hydrogen
phase, determining the size and mass of the helium core. Yet, the effective
chemical mixing in the progenitor stars needs to be treated with great care
(Montalbán et al., 2013).

In this work, we focus on well-studied intermediate-mass gravity-mode
pulsators on the main sequence. These $\gamma\,$Doradus ($\gamma\,$Dor) stars
are the progenitors of red giants with a mass above about 1.3 M⊙. Starting
from the measured internal rotation rates of a population of $\gamma\,$Dor
stars covering the mass range from 1.3 M⊙ to 2 M⊙, we aim to study the arising
systematic age uncertainties of red giants, when ignoring the rotation of
their progenitors in their core hydrogen burning phase.

For red giants of lower mass, it is justified to ignore the mixing caused by
rotation, because their dwarf progenitors experience efficient rotational
slow-down due to magnetic braking (Barnes, 2007), as confirmed by
gyrochonology studies of old open clusters based on high-precision space
photometry (Meibom et al., 2011a, b, 2015; Barnes et al., 2016). Even if
magnetic braking weakens during the second half of the main sequence (van
Saders et al., 2016; Hall et al., 2021), stars born with a mass below 1.2 M⊙
are slow rotators from early on in their evolution, their rotation period
being far longer than their critical break-up period. In addition, the
rotational evolution of cool stars is only a function of mass (and
metallicity), thus suppressing possible age spreads due to rotation. Hence, we
only consider red giants with a mass above 1.2 M⊙ in this work and study the
impact of their progenitors’ rotation on their age-dating for the evolutionary
phases beyond the main sequence.

##  2 Sample selection and global stellar parameters

A large homogeneous sample of 611 $\gamma$ Dor stars with high-precision
measurement of their near-core rotation frequency, $\Omega_{\rm rot}$, from
Kepler space photometry (Borucki et al., 2010) was presented in Li et al.
(2020). The other asteroseismic observable deduced from their detected series
of identified low-degree modes of consecutive radial order is the buoyancy
travel time, $\Pi_{0}$, defined as

$\Pi_{0}\equiv 2\pi^{2}\left(\int_{r_{1}}^{r_{2}}\frac{N(r)}{r}{\rm
d}r\right)^{-1},$ (1)

where $N(r)$ is the Brunt-Väisälä frequency and $r_{1}$ and $r_{2}$ are the
inner and outer position of the mode propagation cavity (Aerts et al., 2010).
This observable is a measure of the size and shape of the gravity-mode cavity,
which changes during the evolution of the star. The chemical gradient profile
notably changes as the stars’ convective cores shrink throughout the main
sequence. The resulting change in the shape of $N(r)$ has been measured from
gravito-inertial asteroseismology based on 4-year Kepler light curves (e.g.,
Aerts et al., 2021).

We use these two powerful asteroseismic observables, $\Omega_{\rm rot}$ and
$\Pi_{0}$, characterising the deep stellar interior of $\gamma\,$Dor stars to
estimate the stellar masses and radii from grid modelling. Such type of
asteroseismic modelling requires additional constraints to break degeneracies
and to find unique solutions. Since we wish to apply ensemble asteroseismic
grid modelling, we need to use a homogeneous set of global stellar parameters.
In contrast to Li et al. (2020) who used Kepler input catalogue values, we
extract stellar parameters from _Gaia_ Data Release 3 (DR3, Gaia Collaboration
et al., 2023b; Creevey et al., 2023), which are appropriate for F-type
$\gamma\,$Dor pulsators as shown by Aerts et al. (2023). This yields
astrophysical parameters based on the _Gaia_ General Stellar Parametrizer from
Photometry (GSP-Phot) for 536 of the 611 stars. Six of these objects are
removed as they are known binaries with pulsating components so their
parameters in the Gaia data might be inaccurate.

Figure 1: Comparison between luminosity values obtained from the _Gaia_
absolute $G$ magnitude (photometric luminosity) and the model dependent _Gaia_
-FLAME. The dashed line is the bisector.

Mombarg et al. (2021) have shown the benefit of using the _Gaia_ stellar
luminosity ($L/L_{\sun}$), along with $\Pi_{0}$, the surface gravity ($\log
g$) and the effective temperature ($T_{\rm eff}$), in asteroseismic grid
modelling of identified prograde dipole modes for a sample of 37 $\gamma\,$Dor
stars with high-resolution high signal-to-noise spectroscopy. Due to the very
uncertain $\log g$ estimates, we perform grid modelling from the method by
Mombarg et al. (2019) but based only on the three observables $\Pi_{0}$,
$T_{\rm eff}$, and $L/L_{\odot}$.

At least two ways occur to obtain the stellar luminosity from _Gaia_ DR3.
First the Gaia astrophysical parameters table includes the luminosity estimate
from the Final Luminosity Age Mass Estimator (FLAME, Creevey et al., 2023).
Secondly, we calculate it from the observed $G$-band magnitude using the
parallax, extinction estimate, and bolometric correction (see Creevey et al.,
2023, for the _Gaia_ bolometric correction computation). The former method
relies on the same input parameters as the latter, but includes a rather
sophisticated scheme of placing these values on theoretical isochrones to
deduce $\log\,L/L_{\odot}$. These isochrones are calibrated from slowly
rotating benchmark stars, while $\gamma\,$Dor stars tend to be faster
rotators. Hence the FLAME luminosities, while seemingly precise, may suffer
from systematic uncertainties. In order to assess this, we compute the
luminosities without relying on stellar models. The downside of this method is
that we need an estimate of the interstellar reddening (which we take from
Gaia DR3). Yet, for most stars this reddening is small
($E(G_{\mathrm{BP}})-E(G_{\mathrm{RP}})<0.25$ mag) as they are positioned away
from the Galactic plane and typically nearby at a distance $d<1.5$ kpc.
Comparing the two estimates allows us to evaluate the involved uncertainties.

Figure 1 provides a comparison between both values and reveals that they are
in good agreement with each other. The scatter of $\pm 0.1$ dex around the
line of unity is expected because the photometric luminosities include fewer
and independently treated variables whereas FLAME treats all variables in a
global framework with additional model constraints. As seen in Fig. 1, we find
some obvious outliers to the generally tight correlation, with the photometric
luminosities maximally up to a factor of five higher than the FLAME values.

To further clean our sample, we investigate the correlation between the
effective temperatures given by Li et al. (2020) from the Kepler Input
Catalogue and those in Gaia DR3. These temperatures are mostly in agreement.
However, 19 stars have an effective temperature outside the $\gamma\,$Dor
range, rather pointing to them being early A-type or B-type pulsators ($T_{\rm
eff}>8500$ K). We use the same approach as Aerts et al. (2023) to reclassify
these gravity-mode pulsators as Slowly Pulsating B stars.

Even in this cleaned sample some stars exhibit larger $\Pi_{0}$ values than
those expected for $\gamma$ Dor stars (Van Reeth et al., 2016). Since we will
employ grid-based asteroseismic modelling, we limit the sample to stars with
$\Pi_{0}<6500$ s. This value might exclude some of the youngest stars but it
ensures that the modelled stars fall onto the used grid. Our final sample
includes 490 stars with homogeneously deduced astrophysical parameters from
Gaia DR3 and 4-year Kepler light curves.

We show the final sample in the observational Hertzsprung-Russell diagram
(HRD) in Fig. 2 along with the measured $\Pi_{0}$. Further, we show the
theoretical $\gamma$ Dor instability strips from Dupret et al. (2005). The
lower edge of the observed instability strip agrees very well with theoretical
predictions (with a mixing-length parameter $\alpha_{\mathrm{MLT}}=2$),
whereas the upper edge is at higher luminosities (cf. also Mombarg et al.,
2024a). These three instability regions were calculated for only one choice of
input physics and for one excitation mechanism. Many Gaia DR3 candidate
$\gamma$ Dor stars are observed at higher luminosities (e.g. Gaia
Collaboration et al., 2023a) as we also find here.

Figure 2: Hertzsprung-Russell diagram of our final sample containing 490
$\gamma$ Dor stars. The colour-coding gives the $\Pi_{0}$ which is used as an
input parameter in the grid modelling. The solid lines indicate the
theoretical $\gamma$ Dor instability strip from Dupret et al. (2005) with a
mixing-length parameter $\alpha_{\mathrm{MLT}}=2$, while the dashed and dotted
lines show the position of the instability strip for
$\alpha_{\mathrm{MLT}}=1.5$ and $\alpha_{\mathrm{MLT}}=1$, respectively.

##  3 Asteroseismic parameter estimation

Our nominal asteroseismic modelling purpose for this work is to estimate each
star’s mass ($M$) and evolutionary stage represented by the core hydrogen mass
fraction ($X_{c}$). In order to achieve this, our approach consists of an
MCMC-based grid search for each of the 490 stars in our final sample which we
model independently as it concerns field stars not belonging to an
astrophysical ensemble. We used the measured $\Pi_{0}$ along with Gaia DR3
$L/L_{\sun}$ and $T_{\rm eff}$ estimates as inputs to place the stars on a
stellar evolution grid.

The grid consists of quasi-randomly sampled 1-dimensional (1D) Modules for
Experiment in Stellar Astrophysics (MESA r11701, Paxton et al., 2011, 2013,
2015, 2018, 2019) models computed by Mombarg et al. (2021), to which we refer
for the details of the input physics. In terms of element transport, we recall
here that the overall chemical mixing caused by various active physical
phenomena (such as rotational mixing, internal gravity wave mixing, magnetic
diffusion, etc.) is simplified in this grid of 1D models in terms of 1)
instantaneous full convective mixing in the core relying on the mixing-length
theory (Böhm-Vitense, 1958), 2) diffusive exponentially decaying overshooting
with one free parameter ($f_{\rm ov}\in[0.01,0.03]$) in the transition layer
between the convective core and the radiative envelope, and 3) constant
envelope mixing described by a free parameter. This approach allows us to
assess the level of mixing in these three zones without having to rely on
uncalibrated theories of element transport, because these 1D approximations of
inherently 3D phenomena result in unsuitably spiky mixing profiles throughout
the stellar interior making pulsation computations unreliable (cf. Aerts,
2021, for more detailed motivations of this commonly adopted approach in
asteroseismic modelling of stars with a convective core). The covered mass
range of the grid is $[1.3,2.0]\,$M⊙, while the metallicity $Z$ varies between
0.011 and 0.023 following the measured values for the sub-sample of 37 best
characterised $\gamma\,$Dor stars with high-precision spectroscopy and its
homogeneous analysis (Van Reeth et al., 2015).

For our MCMC sampling, we use flat priors in mass ($M$), core-hydrogen content
($X_{c}$), and core overshoot ($f_{\mathrm{ov}}$) while the prior in
metallicity ($Z$) is taken from a Gaussian distribution centred on
$Z_{0}=0.014$ ($\sigma=0.01$), given that the stars are young from a Galactic
perspective and hence are expected to have near Solar metallicity. This is
indeed found from various studies based on high-precision spectroscopic
analyses done for various subsets of the brighest $\gamma\,$Dor stars in the
sample, as summarised by Gebruers et al. (2021). Moreover, given its inferior
effect on forward asteroseismic modelling based on the fitting of actual
identified mode frequencies (Aerts et al., 2018) compared to the four unknowns
$(M,Z,f_{\rm ov},X_{c})$ found by Mombarg et al. (2021), and to reduce
dimensionality, we fix the envelope mixing to the lowest value of
$D_{\mathrm{mix}}=1\,$cm2 s-1 in the grid. This effectively comes down to the
assumption that rotational mixing is mainly active in the shear layer above
the convective core but has negligible effect in the outer envelope of
rotating F-type stars. We come back to this assumption in Sect. 6, where we
assess the effect of higher levels of rotational mixing outside the core
boundary layer on asteroseismic parameter determinations. The merit function
for the grid evaluation is the Euclidean distance between the grid points and
the measurements.

For each star the grid search results in a distribution of $M$, $X_{c}$,
$f_{\mathrm{ov}}$, and $Z$. The latter is restricted by the chosen prior but
still varies across the whole range of the model grid for some stars, showing
the need for these variable in the parameter estimation based on the selection
of the best models. Based on the MCMC chains, we subsequently determine the
best values and uncertainties for the stellar radius ($R$), the stellar age
($t$), and the mass of the convective core ($M_{\mathrm{cc}}$) by extracting
those quantities from the model grid.

Before discussing the resulting distributions, we take a look at how well the
grid modelling recovers the input parameters. Similarly to the above mentioned
determined properties such as the stellar radius, we determined the values and
uncertainties of $L/L_{\sun}$, $T_{\mathrm{eff}}$, and $\Pi_{0}$ from the best
fitting model.

Figure 3: Correlations between observed properties (input values) and the
recovered values (model). From left to right, we show $\log\,L/L_{\sun}$,
$T_{\mathrm{eff}}$, and $\Pi_{0}$. All three properties have outlying values
to a perfect recovery. Notably, outliers in one panel are not necessarily
outliers in other panels. To show this property, we highlight the three most
outlying points (difference divided by measurement uncertainty) in each panel
with colour and indicate the same stars with coloured outlines in the other
two panels. Outliers in luminosity are shown in orange, outliers in effective
temperature in blue, and outliers in $\Pi_{0}$ in green. The shift towards
intermediate $\Pi_{0}$ is discussed in the text.

Figure 3 shows the correlations between the input and recovered values. We
find the majority of the recovered values to be in agreement with the input.
In particular, the stellar luminosity was well retrieved. Yet, all three
parameters have a varying level of outlying data points that could typically
not be mapped to the grid during the modelling. However, we note that only one
star is an outlier in all three parameters. In most cases a single input
parameter fell outside the grid space while the others gave good constraints,
indicating possibly an inaccurate measurement of the one, outlying parameter.

A notably exception to the random outliers is the systematic shift towards
intermediate $\Pi_{0}$ in the retrieved values. Stars for which a lower
$\Pi_{0}$ was recovered than measured (i.e. stars to the right of the line of
unity) have typically measured a very high value that would indicate young
stars. Their position in the HRD (Fig. 2), places them well above the zero-age
main sequence (ZAMS). Hence, these stars are either still on the pre-main
sequence or well evolved with an inaccurate $\Pi_{0}$ measurement. Our grid
concerns only the main sequence evolution which results in the latter
assumption in the modelling. Revisiting these stars would be of great interest
to either uncover populations of young stars among the _Kepler_ $\gamma$ Dor
stars or to rectify their asymptotic period-spacing. Stars to the left of the
line of unity seem to be not as evolved in the HRD as indicated by their
$\Pi_{0}$, yet we find no global reasoning that could be applied to all stars
to explain their data. However, we note that stars for which we could not
recover the luminosity and effective temperature are typically found at the
edge of the instability strip in the HRD.

Quantitatively, the luminosity shows the strongest correlation between input
and recovered values with a Pearson correlation $r=0.99$. The effective
temperature is not that well recovered with $r=0.53$. Given the often
inaccurate effective temperatures from the _Gaia_ catalogue this is not
surprising. Despite the obvious horizontal band in the $\Pi_{0}$ comparison,
we find $r=0.80$, due to the well recovered truly evolved stars with a low
$\Pi_{0}$.

Figure 4: Distributions of the grid modelling results. Left: Asteroseismic
mass against the normalised core hydrogen mass fraction $X_{c}^{\prime}$. We
find the majority of stars to have masses between 1.4 and 1.7 $M_{\odot}$ and
to be in the first part of their main sequence evolution with
$X_{c}^{\prime}>0.3$. Note that the $X_{c}^{\prime}$-axis is inverted to
represent the time evolution from left to right. Right: Radius distribution of
our sample. We find only a few evolved stars and most stars have a radius
between 1.5 and 2.2 $R_{\sun}$ as expected in this mass regime for main
sequence stars.

The asteroseismic masses, normalised core hydrogen mass fraction
($X_{c}^{\prime}\equiv X_{c}/X_{\mathrm{ini}}$), and radii are shown in Fig.
4. We find the majority of the stars to have $1.4\lesssim M/M_{\sun}\lesssim
1.7$ (see also Mombarg et al., 2024a) and are a significant fraction into
their main sequence evolution but not near their end ($X_{c}^{\prime}>0.3$).
The mean age of our sample is 1.3 Gyr and we find indeed very few young stars
(see the discussion above). The more massive stars in our sample tend to be
most evolved. Overall, these distributions are in agreement with the position
of the $\gamma$ Dor instability strip (cf. Fig. 2) that is centred on these
masses and narrows for evolved stars. Corresponding to the age and mass
distributions, the stellar radii are distributed as expected for main sequence
$\gamma$ Dor stars with the majority in the range $1.5\lesssim
R/R_{\sun}\lesssim 2.5$. However, the few evolved stars near the end of their
main sequence life time have larger radii.

Our $\gamma\,$Dor sample composed in the previous section is the most
homogeneous data set of red giant progenitors with high-precision
asteroseismic observables on the main sequence, covering all possible rotation
rates (cf. Fig. 6 in Aerts, 2021). As we are interested in the influence of
rotation on stellar evolution, we now explore their second asteroseismic
parameter, the near-core rotation rate $\Omega_{\mathrm{rot}}$. From Fig. 5,
we find a decreasing near-core rotation rate with age as previously seen in Li
et al. (2020) (as a function of $\Pi_{0}$). Asteroseismology applied to stars
with a convective core across stellar evolution has shown that quasi-rigid
rotation in the radiative envelope is an excellent approximation for the
considered mass regime (Aerts et al., 2019; Aerts, 2021), particularly for
$\gamma\,$Dor stars (Van Reeth et al., 2018; Li et al., 2020; Saio et al.,
2021) and can be modelled with an efficient angular momentum transport
throughout the star (Mombarg, 2023; Moyano et al., 2024).

Relying on the asteroseismic mass and radius, we estimate the current ratio
between the rotational velocity and the critical velocity
($v/v_{\mathrm{crit}}\ $) adopting the Keplerian approximation111For this
estimation, we use the mass and radius from the non-rotating asteroseismic
models. as used in the MESA Isochrones & Stellar Tracks (MIST, Dotter, 2016;
Choi et al., 2016) models, which rely on the MESA version published by Paxton
et al. (2015). We find that very few stars in our sample exceed
$v/v_{\mathrm{crit}}\ $$=0.6$ (Fig. 5). Indeed the 95th percentile of the data
is at $v/v_{\mathrm{crit}}\ $$=0.53$, while 68 % of the stars are slower
rotators with $0.15<$ $v/v_{\mathrm{crit}}\ $$<0.39$. The upper boundary of
$v/v_{\mathrm{crit}}\ $increases with age but we observe only one $\gamma$ Dor
star with a near-critical rotation. In contrast to the trend of the near-core
rotation, we find only a slight upward trend for the critical rotation rate
with age while the majority of stars can be found at a constant rate
$v/v_{\mathrm{crit}}\ $$\approx 0.25$. This result for field stars is in
agreement with the recent findings for the young open cluster NGC 2516 (Li et
al., 2023) in which the fastest $\gamma$ Dor stars rotate at 0.4
$v/v_{\mathrm{crit}}\ $.

Figure 5: Rotational properties of the sample stars in the context of their
evolutionary stage with the age represented by the normalised core hydrogen
mass fraction ($X_{c}^{\prime}$). We note the inverted x-axis. Top: The near-
core rotation rate decreases with age as angular momentum is redistributed in
the growing star. Bottom: The fraction of the critical rotation rate is mostly
constant for the majority of stars with $0.1<$ $v/v_{\mathrm{crit}}\ $$<0.6$.

The results from our grid modelling, as well as the input parameters can be
found in the online Table 1. Further, we provide $v/v_{\mathrm{crit}}\ $and
the derived MIST ages at certain evolutionary points.

##  4 Age-dating the population of successors using MIST models

The main aim of this work is to provide age uncertainties at late stages of
stellar evolution by comparing non-rotating and rotating stellar evolution
tracks while keeping the other aspects of the microscopic and macroscopic
input physics the same. In order to make this comparison, we rely on the often
used MIST stellar evolution tracks and isochrones (Dotter, 2016; Choi et al.,
2016). Using precomputed grids of isochrones to age-date evolved stars is a
standard approach in stellar evolution and Galactic archaeology studies.

The original database of MIST models in Choi et al. (2016) has been extended
to include rotation velocities ranging from zero velocity (no rotation) to 90
% of the Keplerian critical velocity ($v_{\rm crit}$) for increasing steps of
10 % in $v/v_{\rm crit}$ by Gossage et al. (2019). The entire MIST database is
based on the same input physics for all aspects of the models. It thus allows
us to assess the impact of rotation and its induced physical effects from
grid-based modelling. The theory of shellular rotation and its accompanying
rotational mixing and gravity darkening adopted in the MIST models is
described extensively in Paxton et al. (2015), to which we refer for details.
We use the rotating MIST models from Gossage et al. (2019) to estimate the
effect of rotation on the ages of stars in the mass range $M\in[1.3,2.0]\,$M⊙
for their evolved evolutionary phases lying ahead by using the population of
asteroseismically studied $\gamma\,$Dor main-sequence stars.

Figure 6: Distributions of rotation rates as fraction of the critical rotation
($v/v_{\mathrm{crit}}\ $). The black distribution shows the back-traced
initial rotation (at ZAMS). It is compared to the current distribution
(orange) as derived from the asteroseismic grid modelling. The stars exhibit
slightly larger fractions of the critical rotation velocity with respect to
their initial rotation, as expected from the population seen in Fig. 5.

Given the variation of the radius as a star evolves during the main sequence,
the critical rotational velocity and hence $v/v_{\mathrm{crit}}\ $change as
the core hydrogen burning progresses. For each star we choose a family of
tracks based on the maximum likelihood point estimator of the asteroseismic
mass. Within this family, consisting of ten tracks
($0.0\leq$$v/v_{\mathrm{crit}}\ $$\leq 0.9$ at ZAMS), we calculate
$v/v_{\mathrm{crit}}\ $at the grid point closest to the current core hydrogen
mass fraction. Finally, we select the track closest to the observed
$v/v_{\mathrm{crit}}$. Figure 6 shows the distribution of the back-traced
(ZAMS) $v/v_{\mathrm{crit}}\ $in comparison to the observed current
distribution. The current distribution (as observed from the asteroseismic
modelling) is broader and shifted slightly to faster rotation as it can also
be seen from Fig. 5, when interpreted as an evolutionary sequence.

Once having selected the MIST track that comes closest to the asteroseismic
$v/v_{\mathrm{crit}}\ $and age per star, its ages for future evolutionary
stages can easily be acquired by simply considering the equivalent
evolutionary point222The EEPs describe the stellar evolution in a time-
independent way to simplify the comparison between models with different
physical properties (see Dotter, 2016, for more details). (EEP) on the MIST
track. We consider the age effects at the terminal-age main sequence (TAMS,
EEP 454), the tip of the red giant branch (TRGB, EEP 605), the zero-age
Helium-burning stage (ZAHeB, EEP 631), and the terminal-age Helium-burning
stage (TAHeB, EEP 707). For each of these evolutionary points, we also extract
the corresponding age from the non-rotating track of the same mass as the
reference value to estimate the effect of internal rotation, by expressing it
as a fraction of the stellar age for the non-rotating case. Since rotation
introduces extra mixing in the stellar interior, we expect the ages of the
stars in evolved phases resulting from the rotating MIST tracks to be above
those of the non-rotating tracks.

##  5 Age shifts in successor populations due to main-sequence rotation

The cores of most single stars born with $M\geq 1.2\,$M⊙ generally rotate fast
during core hydrogen burning because these stars are not slowed down by
magnetic braking. As proven by asteroseismology of an ensemble of
intermediate-mass field stars, their cores only slow down considerably between
the TAMS and the red giant phase (Fig. 6 in Aerts, 2021). While it may be
justified to ignore the effects of rotation and internal mixing in stellar
evolution models of intermediate-mass stars after the TAMS, this is not a good
approximation during the initial 90 % of their evolution (Johnston, 2021).
Despite this fact, age-dating of red giants from asteroseismology is usually
done by ignoring the cumulative effect of the internal mixing caused by
rotation in the progenitor phases (e.g., Martig et al., 2015; Ness et al.,
2016; Aguirre Børsen-Koch et al., 2022). Moreover, the core overshooting
description used to mimic shear mixing in the core boundary layer is often
frozen to just one value in the models used to perform the age-dating of red
giants, instead of allowing it to vary during the long main-sequence phase to
mimic time-dependent rotational shear mixing as done in this work.

While the latest age-dating methods, such as the one developed by Aguirre
Børsen-Koch et al. (2022), can accommodate rotating models, its application to
populations of red giants usually ignores the progenitor’s rotational mixing,
underestimating in this way the age uncertainties. Recent precision
estimations for the ages of red giants with the best asteroseismology
available quote $\sim\\!\,$11 % age uncertainty when ignoring the rotation as
of the ZAMS (Montalbán et al., 2021). Large population studies relying on a
variety of CoRoT, Kepler, K2, and TESS data quote larger uncertainties up to
some 25 % (Miglio et al., 2021; Stokholm et al., 2023; Willett et al., 2023).

Figure 7: Relative ages between the rotating and non-rotating MIST tracks
($t_{\mathrm{r}}/t_{\mathrm{nr}}$) against stellar mass for two evolutionary
points (TAMS left and TAHeB right). The colour-coding gives the ZAMS
$v/v_{\mathrm{crit}}\ $value. Both distributions are qualitatively similar
although at the TAHeB the median value evolves to larger relative ages.

Figure 7 shows the expected relative age differences at the TAMS and the end
of the core Helium burning stage (TAHeB) from the rotating and non-rotating
MIST models when we propagate the population of $\gamma\,$Dor stars forward
towards these evolutionary phases. It can be seen that the age differences
between rotating and non-rotating models reach up to 4.5 % at TAMS for the
fastest rotators but remain modest (below 1 %) for the majority of stars for
the physics of rotational mixing adopted in the MIST models.

More specifically, we find a distinct upper envelope of the age spread with
stellar mass among the successors of the lowest-mass $\gamma\,$Dor stars. The
least massive evolved stars resulting from the main-sequence sample ($M\simeq
1.3\,M_{\sun}$) are hardly affected by the additional rotational mixing, given
that their measured rotation rates are close to those of a non-rotating star.
Starting from $M=1.4\,M_{\sun}$, the progenitor stars rotate typically between
5 % and 20 % of the critical rotation, which adds $\leq 1$ % to the stellar
age by the end of the main sequence. For more massive progenitor pulsators
($M\geq 1.5\,M_{\sun}$) the age difference is purely a function of the
rotation rate, that is, the distribution of age differences follows the
observed distribution of the asteroseismic rotation rates and reaches $\sim$
5 %. We point out that, as of $M>1.7\,M_{\sun}$, the sample size of the
$\gamma$ Dor population is small and may not be representative for all stars
of such mass.

The distribution stays very similar at later evolutionary stages, although by
the end of core Helium burning the overall age-spread between slowly and
faster rotating $\gamma$ Dor stars increases as seen from the right panel of
Fig. 7. Notably the fastest rotating and most massive star exhibits a smaller
relative age at TAHeB compared to the TAMS, contrary to other stars in the
sample. The results for the two intermediate stages are shown in Appendix C.

The histograms in Fig. 8 shows the evolution of the relative age distributions
at the four considered evolutionary stages. As discussed above, the TAMS
distribution shows the smallest relative age differences with a strong peak in
the smallest bin, while the distributions are somewhat wider at later stages.

Figure 8: Histogram of the relative ages between rotating and non-rotating
models at different evolutionary stages. The majority of red-giant ages are
underestimated by a few percent under the assumption of non-rotating models.
The TRGB distribution is indistinguishable from the distribution at ZAHeB and
hence not visible.

Our results reflect that rotational mixing is an important transport process
throughout the long main sequence evolution, which lasts about 90 % of the
entire lifetime of the star before it turns into a red giant and evolves
further towards the white dwarf stage. Nevertheless, additional transport
processes occurring specifically during the red giant phase (such as
thermohaline mixing) and not considered here should also be taken into account
in the determination of ages for the highest-precision Galactic archaeology
based on surface abundances and asteroseismology together (Lagarde et al.,
2017).

##  6 Additional systematic uncertainties

###  6.1 For the asteroseismic parameters

Recent computational work by Mombarg et al. (2024a) has shown that it is in
principle possible to perform asteroseismic modelling of intermediate-mass
stars from rotating stellar models. The 1D models delivered by Mombarg et al.
(2022) circumvent the spiky internal mixing profiles that occur when using the
1D prescriptions from Heger et al. (2000), causing inaccuracies in the
computation of oscillation modes. Mombarg et al. (2022) rely on a 1D formalism
of Zahn (1992) for the implementation of rotational mixing in their models,
while this is done self-consistently in 2D by Mombarg et al. (2023, 2024b). In
the 1D case, the rotational profile used for computing the efficiency of
rotational mixing comes from models computed with the 2D ESTER evolution code
(Espinosa Lara & Rieutord, 2013; Rieutord et al., 2016). The 2D-to-1D ESTER-
informed mixing profile adopted in MESA evolution models brings a potential
improvement of major importance in asteroseismic modelling of intermediate-
mass stars with respect to the simplistic 1D treatment of chemical mixing by
means of a constant and ad hoc core overshooting description as done in the
literature (Aerts, 2021) and adopted in Sect. 2.

Here, we provide grid-based modelling for our $\gamma\,$Dor sample by using a
2D-to-1D model grid from Mombarg et al. (2024a), which differs from the
previously employed MESA grid by the inclusion of angular momentum transport
mechanisms. This rotational mixing replaces the constant envelope mixing
($D_{\mathrm{mix}}$) in the above used models. The angular momentum transport
is implemented as diffusive processes and includes dynamical and secular shear
instability, Eddington-Sweet circulation, Solberg-Høiland instability,
Goldreich-Schubert-Fricke instability, and a Spruit-Tayler dynamo for angular
momentum transport via magnetic torques. We refer the reader to Heger et al.
(2000) and the MESA instrumentation papers (Paxton et al., 2011, 2013, 2015,
2018, 2019) for details on these processes and their implementation in MESA.
Mombarg (2023) calibrated the angular momentum transport from several slowly
rotating, well-studied $\gamma$ Dor stars with known near-core and surface
rotation rates. Their new model grid is regularly sampled with $M\in[1.3,2.0]$
($\Delta M=0.1\,M_{\sun}$), $f_{\mathrm{ov}}\in[0.005,0.035]$ ($\Delta
f_{\mathrm{ov}}=0.005$), $\Omega/\Omega_{\mathrm{crit}}\in[0.05,0.6]$
($\Delta\Omega/\Omega_{\mathrm{crit}}=0.05$), and fixed a metallicity
$Z=0.014$. The grid is more sparsely sampled than the non-rotating grid used
above.

We applied these rotating models to our sample in the same way as outlined in
Sects. 2-5 but included the measured asteroseismic near-core rotation rate as
an additional input constraint. The stellar parameters obtained from this
model are comparable to those from our previous analysis. On a star-to-star
basis, shifts in the core hydrogen mass fraction are often compensated by
changes in the core-overshoot and stellar mass. In combinations this results
only in slight changes the evolution of the stellar properties with age as
discussed in Fig. 5. Further, the initial distribution of rotation rates is
similar to our previous results (for a more detailed analysis of the initial
rotation rates we refer the reader to Mombarg et al., 2024a).

With only minor shifts in the overall asteroseismic stellar parameters
compared to our initial model, in which the mixing due to rotation was
modelled with additional core overshoot, we do not find significant difference
for the MIST-based ages at later evolutionary stages with respect to the
analysis above. Some stars have slightly different relative ages at the four
considered evolutionary points, yet the overall distribution hardly changes.
This illustrates that mimicking element transport processes by a fudge factor
in the overshoot zone is a valid approach, as long as the chemical mixing can
be described diffusively (cf., Aerts, 2021).

###  6.2 From evolution models with meridional circulation

In contrast to the MIST models computed with the code MESA, which adopts a
diffusive approach and treats the rotationally induced mixing and angular
momentum processes as described in Heger et al. (2000), other stellar
evolution models rely on a diffusive-advective treatment. Such a treatment
allows for the inclusion of meridional circulation, often implemented
following the theory by Maeder & Zahn (1998). Here we rely on the publicly
available Geneva models computed by Georgy et al. (2013), in order to assess
the systematic age uncertainty caused by the different input physics to treat
rotation.

Ouazzani et al. (2019) have shown from asteroseismology of $\gamma\,$Dor stars
that their diffusive-advective models computed with the CESTAM code, which
also relies on the theory by Maeder & Zahn (1998), need fixes in terms of
angular momentum transport. Better agreement between models and asteroseismic
data has been achieved by Moyano et al. (2023, 2024) from the inclusion of
internal magnetism in the Geneva code, in addition to meridional circulation
and shear mixing. However, neither these new Geneva models nor the CESTAM ones
are publicly available. For this reason, we assess the age uncertainty due to
absence or inclusion of meridional circulation by comparing the results from
the MIST models with those derived from the SCYLIST Geneva models made
publicly available by Georgy et al. (2013). The use of these two sets of
isochrones and stellar tracks is common to age-date stars.

It has already been shown that the inclusion of meridional circulation can
result in significant differences of the evolutionary tracks of rotating
models beyond the TAMS (Martins & Palacios, 2013). The influence of rotation
on luminosity and temperature are analysed in detail in Choi et al. (2016) and
Gossage et al. (2018). We deduce from Fig. 20 of Choi et al. (2016) that the
ages between rotating MIST models and Geneva models are quite different for
stars with $M<3\,\mathrm{M}_{\sun}$, the mass range considered in this work.

For the models with a mass of 1.7 M☉, which is the lowest mass in the publicly
available grid of rotating Geneva models computed by Georgy et al. (2013), we
find a relative age difference of 10 % for $v/v_{\mathrm{crit}}\ $$=0.2$ at
the TRGB compared to the non-rotating model, while this difference increases
to 20 % for $v/v_{\mathrm{crit}}\ $$=0.4$. These values are significantly
larger than in the MIST case, for which we found 1 % and 2 %, respectively.
When comparing ages obtained from the MIST models to these from the Geneva
models, we find a very similar TRGB age in the non-rotating case, albeit a
smaller TAMS age. Yet, in the Geneva case the main sequence life time is
extended by $\sim\\!30\,\%$ for $v/v_{\mathrm{crit}}\ $$=0.4$, resulting in a
large relative age difference at the TRGB.

We conclude that different input physics for angular momentum and/or element
transport, notably the inclusion of meridional circulation due to rotation or
not, may add significant systematic uncertainties in red giant ages when
taking into account the rotation in their progenitor phases.

###  6.3 Sample selection

For this study, we employed the largest homogeneous sample of $\gamma$ Dor
stars with $\Pi_{0}$ and $\Omega_{\rm rot}$ estimates available in the
literature, in order to assess the influence of rotation on the age
distribution of red giants. Despite being the largest such sample, it is still
small compared to the overall stellar population and selection effects might
influence our results.

The _Kepler_ field contains mostly older field stars (Berger et al., 2020), as
also seen from Fig 4. Hence, we are missing the youngest $\gamma$ Dor stars in
our analysis and our sample might not represent the full variety of $\gamma$
Dor stars. Future, large-scale studies similar to Li et al. (2020) with stars
observed with the Transiting Exoplanet Survey Satellite (TESS) might provide
insights into a younger population of $\gamma$ Dor stars since the TESS
continuous viewing zone contains a much younger population of stars than the
_Kepler_ field (Avallone et al., 2022).

Aerts et al. (2023) present distributions of stellar parameters for a much
larger sample of 11 636 $\gamma$ Dor star candidates identified from _Gaia_
photometry (Gaia Collaboration et al., 2023a). Our distributions of stellar
parameters ($T_{\mathrm{eff}}$, $\log L/L_{\sun}$, $R/R_{\sun}$) are in
qualitative agreement with these results. The majority of these stars are
currently being analysed in detail from high-cadence TESS space photometry to
confirm their $\gamma\,$Dor nature (Hey & Aerts, in prep.) but we do not yet
possess asteroseismic parameters such as $\Pi_{0}$ and $\Omega_{\mathrm{rot}}$
for them. The stars from this sample (next to stars in the TESS continuous
viewing zones) are prime targets for future population studies with TESS.

###  6.4 Insights from more massive stars

Since we focused our analysis on $\gamma$ Dor stars, our sample has a very
limited mass-range. The results of our grid modelling place most of the stars
in the range 1.4 M${}_{\odot}\leq$ M $\leq 1.7\,$M⊙, which coincides with the
centre of the $\gamma$ Dor instability strip. As seen from Fig. 5 the stars in
our sample are mostly moderate rotators while higher-mass stars often rotate
at higher fractions of the critical velocity. Similar to the sample of
$\gamma$ Dor stars, Pedersen et al. (2021) homogeneously analysed a sample of
26 slowly pulsating B (SPB) stars from the Kepler mission with asteroseismic
forward modelling.

We placed the updated results for this sample from Pedersen (2022) on the MIST
tracks in a similar fashion to our analysis above. Since we have only a small
sample, we cannot deduce a statistically meaningful result. Yet, we find a
similar relative age difference of $2-3\,\%$ for the majority of slowly
rotating stars ($v/v_{\mathrm{crit}}\ $$\leq 0.5$). However, near-critical
rotation is a common phenomenon among B stars (Townsend et al., 2004) and some
stars in the sample of Pedersen (2022) rotate with $v/v_{\mathrm{crit}}\
$$>0.7$. For these stars the main sequence lifetime is extended by up to 50 %
at $M\approx 4\,M_{\sun}$. While this is a higher mass than for most red giant
stars, we have seen that the prolonged main sequence for the more massive
$\gamma$ Dor stars is mainly a function of the rotation rate and not so much
of the mass. Hence, the distribution of the rotation rates of lower-mass SPB
stars might be directly transferable to a distribution of age uncertainties
for the mass regime $[2,4]\,$M⊙.

##  7 Conclusions

Based on 490 $\gamma$ Dor stars with homogeneously deduced astrophysical
parameters from Gaia DR3 and 4-year Kepler light curves, we characterise the
population of red giant progenitors with masses in the range $1.3\leq
M/M_{\sun}\leq 2.0$. We employ asteroseismic grid modelling and obtain masses,
the current core hydrogen mass fraction, and radii for these stars. Together
with the asteroseismic near-core rotation rate, we adopt a common approach to
age-date stars and place the sample stars on rotating MIST evolutionary
tracks.

The current population of $\gamma$ Dor stars show moderate rotation with
$0.15\lesssim$ $v/v_{\mathrm{crit}}\ $$\lesssim 0.4$ for the majority of
stars. When comparing the influence of this rotation to age-dating at later
stages of their evolution, we find a modest increase of the age. However, the
fastest rotators can be up to 5 % older when leaving the main sequence.
Typical fast rotation ($v/v_{\mathrm{crit}}\ $$\sim\\!0.4$) increases the main
sequence age by 2 % when relying on the transport processes adopted in the
MESA stellar evolution code, but can reach up to 20% when relying on models
including meridional circulation.

Given the unknown rotation history of isolated red giants in the Milky Way,
the age spreads caused by their rotating progenitors as presented here should
always be considered in the uncertainty analysis for (asteroseismic) red giant
ages. Effects of core boundary mixing during core helium burning are nowadays
taken into account in the construction of red giant models calibrated by
asteroseismology (Constantino et al., 2015; Bossini et al., 2017; Noll et al.,
2024). Yet the rotational mixing during the long main-sequence phase is
usually ignored, while it has a major impact on the size and mass of the
helium core by the TAMS (Johnston, 2021; Pedersen, 2022). As we have shown
here the rotation during the main sequence also impacts the age by the time
the star is a red giant. Our work shows the importance of taking into account
transport processes in the stellar interior during the entire main sequence
when age-dating red giants for high precision Galactic archaeological studies.
Asteroseismology of young and medium-aged open clusters with a variety of
metallicities and both main sequence and red giant pulsators among their
members offers the best way to calibrate internal mixing and angular momentum
transport processes across the entire nuclear evolution phases of
intermediate-mass stars (see Fritzewski et al., 2024; Li et al., 2023, for
initial attempts of such modelling).

###### Acknowledgements.

The authors appreciate the swift report and encouraging words from the
anonymous referee. CA is grateful to Melissa Ness and Danny Horta for pleasant
and valuable discussions on Galactic archaeology. The research leading to
these results has received funding from the KU Leuven Research Council (grant
C16/18/005: PARADISE). CA also acknowledges financial support from the
European Research Council (ERC) under the Horizon Europe programme (Synergy
Grant agreement N∘101071505: 4D-STAR). While partially funded by the European
Union, views and opinions expressed are however those of the author(s) only
and do not necessarily reflect those of the European Union or the European
Research Council. Neither the European Union nor the granting authority can be
held responsible for them. JSGM acknowledges funding from the French Agence
Nationale de la Recherche (ANR), under grant MASSIF (ANR-21-CE31-0018-02).
This research has made use of NASA’s Astrophysics Data System Bibliographic
Services and of the SIMBAD database and the VizieR catalogue access tool,
operated at CDS, Strasbourg, France. This work has made use of data from the
European Space Agency (ESA) mission _Gaia_ (https://www.cosmos.esa.int/gaia),
processed by the _Gaia_ Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the _Gaia_ Multilateral Agreement.
Software: This work made use of Topcat (Taylor, 2005). This research made use
of the following Python packages: IPython (Pérez & Granger, 2007); MatPlotLib
(Hunter, 2007); NumPy (Harris et al., 2020); Pandas (McKinney, 2010); SciPy
(Virtanen et al., 2020).

## References

  * Aerts (2021) Aerts, C. 2021, Reviews of Modern Physics, 93, 015001
  * Aerts et al. (2021) Aerts, C., Augustson, K., Mathis, S., et al. 2021, A&A, 656, A121
  * Aerts et al. (2010) Aerts, C., Christensen-Dalsgaard, J., & Kurtz, D. W. 2010, Asteroseismology, Springer-Verlag Heidelberg
  * Aerts et al. (2019) Aerts, C., Mathis, S., & Rogers, T. M. 2019, ARA&A, 57, 35
  * Aerts et al. (2023) Aerts, C., Molenberghs, G., & De Ridder, J. 2023, A&A, 672, A183
  * Aerts et al. (2018) Aerts, C., Molenberghs, G., Michielsen, M., et al. 2018, ApJS, 237, 15
  * Aguirre Børsen-Koch et al. (2022) Aguirre Børsen-Koch, V., Rørsted, J. L., Justesen, A. B., et al. 2022, MNRAS, 509, 4344
  * Avallone et al. (2022) Avallone, E. A., Tayar, J. N., van Saders, J. L., et al. 2022, ApJ, 930, 7
  * Barnes (2007) Barnes, S. A. 2007, ApJ, 669, 1167
  * Barnes et al. (2016) Barnes, S. A., Weingrill, J., Fritzewski, D., Strassmeier, K. G., & Platais, I. 2016, ApJ, 823, 16
  * Berger et al. (2020) Berger, T. A., Huber, D., van Saders, J. L., et al. 2020, AJ, 159, 280
  * Böhm-Vitense (1958) Böhm-Vitense, E. 1958, ZAp, 46, 108
  * Borucki et al. (2010) Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977
  * Bossini et al. (2017) Bossini, D., Miglio, A., Salaris, M., et al. 2017, MNRAS, 469, 4718
  * Bovy et al. (2016) Bovy, J., Rix, H.-W., Schlafly, E. F., et al. 2016, ApJ, 823, 30
  * Chiappini et al. (2015) Chiappini, C., Anders, F., Rodrigues, T. S., et al. 2015, A&A, 576, L12
  * Choi et al. (2016) Choi, J., Dotter, A., Conroy, C., et al. 2016, ApJ, 823, 102
  * Constantino et al. (2015) Constantino, T., Campbell, S. W., Christensen-Dalsgaard, J., Lattanzio, J. C., & Stello, D. 2015, MNRAS, 452, 123
  * Creevey et al. (2023) Creevey, O. L., Sordo, R., Pailler, F., et al. 2023, A&A, 674, A26
  * De Silva et al. (2015) De Silva, G. M., Freeman, K. C., Bland-Hawthorn, J., et al. 2015, MNRAS, 449, 2604
  * Dotter (2016) Dotter, A. 2016, ApJS, 222, 8
  * Dupret et al. (2005) Dupret, M. A., Grigahcène, A., Garrido, R., Gabriel, M., & Scuflaire, R. 2005, A&A, 435, 927
  * Espinosa Lara & Rieutord (2013) Espinosa Lara, F. & Rieutord, M. 2013, A&A, 552, A35
  * Fritzewski et al. (2024) Fritzewski, D. J., Van Reeth, T., Aerts, C., et al. 2024, A&A, 681, A13
  * Gaia Collaboration et al. (2023a) Gaia Collaboration, De Ridder, J., Ripepi, V., et al. 2023a, A&A, 674, A36
  * Gaia Collaboration et al. (2023b) Gaia Collaboration, Vallenari, A., Brown, A. G. A., Prusti, T., & de Bruijne, J. H. J. e. a. 2023b, A&A, 674, A1
  * Gebruers et al. (2021) Gebruers, S., Straumit, I., Tkachenko, A., et al. 2021, A&A, 650, A151
  * Georgy et al. (2013) Georgy, C., Ekström, S., Granada, A., et al. 2013, A&A, 553, A24
  * Gossage et al. (2019) Gossage, S., Conroy, C., Dotter, A., et al. 2019, ApJ, 887, 199
  * Gossage et al. (2018) Gossage, S., Conroy, C., Dotter, A., et al. 2018, ApJ, 863, 67
  * Hall et al. (2021) Hall, O. J., Davies, G. R., van Saders, J., et al. 2021, Nature Astronomy, 5, 707
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357
  * Heger et al. (2000) Heger, A., Langer, N., & Woosley, S. E. 2000, ApJ, 528, 368
  * Helmi et al. (2018) Helmi, A., Babusiaux, C., Koppelman, H. H., et al. 2018, Nature, 563, 85
  * Hon et al. (2021) Hon, M., Huber, D., Kuszlewicz, J. S., et al. 2021, ApJ, 919, 131
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90
  * Johnston (2021) Johnston, C. 2021, A&A, 655, A29
  * Lagarde et al. (2017) Lagarde, N., Robin, A. C., Reylé, C., & Nasello, G. 2017, A&A, 601, A27
  * Li et al. (2023) Li, G., Aerts, C., Bedding, T. R., et al. 2023, A&A, submitted, arXiv:2311.16991
  * Li et al. (2020) Li, G., Van Reeth, T., Bedding, T. R., et al. 2020, MNRAS, 491, 3586
  * Maeder & Zahn (1998) Maeder, A. & Zahn, J.-P. 1998, A&A, 334, 1000
  * Martig et al. (2015) Martig, M., Rix, H.-W., Silva Aguirre, V., et al. 2015, MNRAS, 451, 2230
  * Martins & Palacios (2013) Martins, F. & Palacios, A. 2013, A&A, 560, A16
  * McKinney (2010) McKinney, W. 2010, in Proceedings of the 9th Python in Science Conference, ed. S. van der Walt & J. Millman, 51
  * Meibom et al. (2011a) Meibom, S., Barnes, S. A., Latham, D. W., et al. 2011a, ApJ, 733, L9
  * Meibom et al. (2015) Meibom, S., Barnes, S. A., Platais, I., et al. 2015, Nature, 517, 589
  * Meibom et al. (2011b) Meibom, S., Mathieu, R. D., Stassun, K. G., Liebesny, P., & Saar, S. H. 2011b, ApJ, 733, 115
  * Miglio et al. (2013) Miglio, A., Chiappini, C., Morel, T., et al. 2013, MNRAS, 429, 423
  * Miglio et al. (2021) Miglio, A., Girardi, L., Grundahl, F., et al. 2021, Experimental Astronomy, 51, 963
  * Mombarg (2023) Mombarg, J. S. G. 2023, A&A, 677, A63
  * Mombarg et al. (2024a) Mombarg, J. S. G., Aerts, C., & Molenberghs, G. 2024a, A&A, in press
  * Mombarg et al. (2022) Mombarg, J. S. G., Dotter, A., Rieutord, M., et al. 2022, ApJ, 925, 154
  * Mombarg et al. (2023) Mombarg, J. S. G., Rieutord, M., & Espinosa Lara, F. 2023, A&A, 677, L5
  * Mombarg et al. (2024b) Mombarg, J. S. G., Rieutord, M., & Espinosa Lara, F. 2024b, A&A, in press, arXiv:2401.08747
  * Mombarg et al. (2021) Mombarg, J. S. G., Van Reeth, T., & Aerts, C. 2021, A&A, 650, A58
  * Mombarg et al. (2019) Mombarg, J. S. G., Van Reeth, T., Pedersen, M. G., et al. 2019, MNRAS, 485, 3248
  * Montalbán et al. (2021) Montalbán, J., Mackereth, J. T., Miglio, A., et al. 2021, Nature Astronomy, 5, 640
  * Montalbán et al. (2013) Montalbán, J., Miglio, A., Noels, A., et al. 2013, ApJ, 766, 118
  * Moyano et al. (2024) Moyano, F., Eggenberger, P., & Salmon, S. 2024, A&A, in press, arXiv:2401.05543
  * Moyano et al. (2023) Moyano, F. D., Eggenberger, P., Salmon, S. J. A. J., Mombarg, J. S. G., & Ekström, S. 2023, A&A, 677, A6
  * Ness et al. (2016) Ness, M., Hogg, D. W., Rix, H. W., et al. 2016, ApJ, 823, 114
  * Noll et al. (2024) Noll, A., Basu, S., & Hekker, S. 2024, arXiv e-prints, arXiv:2401.05519
  * Ouazzani et al. (2019) Ouazzani, R. M., Marques, J. P., Goupil, M. J., et al. 2019, A&A, 626, A121
  * Paxton et al. (2011) Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3
  * Paxton et al. (2013) Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208, 4
  * Paxton et al. (2015) Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS, 220, 15
  * Paxton et al. (2018) Paxton, B., Schwab, J., Bauer, E. B., et al. 2018, ApJS, 234, 34
  * Paxton et al. (2019) Paxton, B., Smolec, R., Schwab, J., et al. 2019, ApJS, 243, 10
  * Pedersen (2022) Pedersen, M. G. 2022, ApJ, 930, 94
  * Pedersen et al. (2021) Pedersen, M. G., Aerts, C., Pápics, P. I., et al. 2021, Nature Astronomy, 5, 715
  * Pérez & Granger (2007) Pérez, F. & Granger, B. E. 2007, Computing in Science & Engineering, 9, 21
  * Rieutord et al. (2016) Rieutord, M., Espinosa Lara, F., & Putigny, B. 2016, Journal of Computational Physics, 318, 277
  * Rojas-Arriagada et al. (2017) Rojas-Arriagada, A., Recio-Blanco, A., de Laverny, P., et al. 2017, A&A, 601, A140
  * Saio et al. (2021) Saio, H., Takata, M., Lee, U., Li, G., & Van Reeth, T. 2021, MNRAS, 502, 5856
  * Schiavon et al. (2017) Schiavon, R. P., Zamora, O., Carrera, R., et al. 2017, MNRAS, 465, 501
  * Silva Aguirre et al. (2018) Silva Aguirre, V., Bojsen-Hansen, M., Slumstrup, D., et al. 2018, MNRAS, 475, 5487
  * Stokholm et al. (2023) Stokholm, A., Aguirre Børsen-Koch, V., Stello, D., Hon, M., & Reyes, C. 2023, MNRAS, 524, 1634
  * Taylor (2005) Taylor, M. B. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 347, Astronomical Data Analysis Software and Systems XIV, ed. P. Shopbell, M. Britton, & R. Ebert, 29
  * Townsend et al. (2004) Townsend, R. H. D., Owocki, S. P., & Howarth, I. D. 2004, MNRAS, 350, 189
  * Van Reeth et al. (2018) Van Reeth, T., Mombarg, J. S. G., Mathis, S., et al. 2018, A&A, 618, A24
  * Van Reeth et al. (2016) Van Reeth, T., Tkachenko, A., & Aerts, C. 2016, A&A, 593, A120
  * Van Reeth et al. (2015) Van Reeth, T., Tkachenko, A., Aerts, C., et al. 2015, ApJS, 218, 27
  * van Saders et al. (2016) van Saders, J. L., Ceillier, T., Metcalfe, T. S., et al. 2016, Nature, 529, 181
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261
  * Vitali et al. (2024) Vitali, S., Slumstrup, D., Jofré, P., et al. 2024, A&A, submitted, arXiv:2401.02328
  * Wang et al. (2023) Wang, K., Ren, A., Andersen, M. F., et al. 2023, AJ, 166, 42
  * Willett et al. (2023) Willett, E., Miglio, A., Mackereth, J. T., et al. 2023, MNRAS, 526, 2141
  * Zahn (1992) Zahn, J. P. 1992, A&A, 265, 115

##  Appendix A Online table

Table 1 gives an overview on the columns of the online Table.

Table 1: Overview on the columns in the online table containing the input parameters and asteroseismically determined parameters for the 490 analysed $\gamma$ Dor stars. Name | Unit | Description
---|---|---
KIC | - | KIC ID
RA | deg | Right ascension from _Gaia_ DR3
Dec | deg | Declination from _Gaia_ DR3
Lum | $L_{\sun}$ | Photometric Luminosity ($L$)
err_Lum | $L_{\sun}$ | Uncertainty on photometric Luminosity
Teff | K | Effective temperature from _Gaia_ DR3 ($T_{\mathrm{eff}}$)
err_Teff | K | Uncertainty on effective temperature
Pi0 | s | Buoyancy travel time from Li et al. (2020) ($\Pi_{0}$)
errl_Pi0 | s | Lower uncertainty on $\Pi_{0}$ from Li et al. (2020)
erru_Pi0 | s | Upper uncertainty on $\Pi_{0}$ from Li et al. (2020)
frot | d-1 | Near-core rotation rate from Li et al. (2020) ($\Omega_{\mathrm{rot}}$)
errl_frot | d-1 | Lower uncertainty on $\Omega_{\mathrm{rot}}$ from Li et al. (2020)
erru_frot | d-1 | Upper uncertainty on $\Omega_{\mathrm{rot}}$ from Li et al. (2020)
Xc’ | - | Normalised core hydrogen mass fraction ($X_{c}^{\prime}$)
errl_Xc’ | - | Lower uncertainty on $X_{c}^{\prime}$
erru_Xc’ | - | Upper uncertainty on $X_{c}^{\prime}$
age | Gyr | Asteroseismic age ($t$)
errl_age | Gyr | Lower uncertainty on age
erru_age | Gyr | Upper uncertainty on age
mass | $M_{\sun}$ | Asteroseismic mass ($M$)
errl_mass | $M_{\sun}$ | Lower uncertainty on mass
erru_mass | $M_{\sun}$ | Upper uncertainty on mass
radius | $R_{\sun}$ | Asteroseismic radius ($R$)
errl_radius | $R_{\sun}$ | Lower uncertainty on radius
erru_radius | $R_{\sun}$ | Upper uncertainty on radius
vvcrit | - | Ratio of current rotation rate to critical rotation rate ($v/v_{\mathrm{crit}}\ $)
MIST_age_TAMS | Gyr | Age of rotating MIST track at TAMS ($t_{\mathrm{r}}$)
NR_MIST_age_TAMS | Gyr | Age of non-rotating MIST track at TAMS ($t_{\mathrm{nr}}$)
TAMS_frac | - | Fraction of rotating to non-rotating age at TAMS ($t_{\mathrm{r}}/t_{\mathrm{nr}}$)
MIST_age_TRGB | Gyr | Age of rotating MIST track at TRGB ($t_{\mathrm{r}}$)
NR_MIST_age_TRGB | Gyr | Age of non-rotating MIST track at TRGB ($t_{\mathrm{nr}}$)
TRGB_frac | - | Fraction of rotating to non-rotating age at TRGB ($t_{\mathrm{r}}/t_{\mathrm{nr}}$)
MIST_age_ZAHeB | Gyr | Age of rotating MIST track at ZAHeB ($t_{\mathrm{r}}$)
NR_MIST_age_ZAHeB | Gyr | Age of non-rotating MIST track at ZAHeB $t_{\mathrm{nr}}$)
ZAHeB_frac | - | Fraction of rotating to non-rotating age at ZAHeB ($t_{\mathrm{r}}/t_{\mathrm{nr}}$)
MIST_age_TAHeB | Gyr | Age of rotating MIST track at TAHeB ($t_{\mathrm{r}}$)
NR_MIST_age_TAHeB | Gyr | Age of non-rotating MIST track at TAHeB ($t_{\mathrm{nr}}$)
TAHeB_frac | - | Fraction of rotating to non-rotating age at TAHeB ($t_{\mathrm{r}}/t_{\mathrm{nr}}$)

##  Appendix B Comparison between asteroseismically informed grid modelling
and the _Gaia_ -FLAME results

The _Gaia_ -FLAME parameters in the astrophysical parameter tables of _Gaia_
DR3 do not only include the above mentioned luminosity but also estimates of
the stellar mass, radius, and age for most of the observed stars. In Fig. 9,
we show these three parameters in comparison to our asteroseismically inferred
values. Both the masses and the radii are overall in good agreement, while the
ages estimated from both methods diverge.

Despite being in good agreement, we find that on average the asteroseismic
mass is slightly lower than the FLAME mass. This offset can mainly be
attributed to the different underlying stellar models. The outliers in the
left panel of Fig. 9 are at the same level as seen in previous comparisons in
the main text.

The radii are also in good agreement between the two methods. However, we find
more outliers with larger asteroseismic radii. These are stars are likely more
evolved in our model.

The ages are not in a good agreement between our asteroseismically informed
model and FLAME. While the ages from the FLAME pipeline are mostly constrained
between 0.5 and 2 Gyr, our results are between 0 and 3 Gyr. The colour
gradient in right panel of Fig. 9 shows that the additional constraint from
$\Pi_{0}$ has great probing power for stellar ages.

Figure 9: Comparison between asteroseismic parameter estimates and the _Gaia_
-FLAME values. Left: The stellar mass estimate from our analysis is mostly
consistent with the FLAME pipeline. However, a slight systematic offset to
lower masses in our analysis exist. Centre: Most radii agree between the two
methods, while some outliers with large radii from the asteroseismic analysis
can be found. These stars have typically a larger asteroseismic age and are
more evolved in our analysis. Right: The additional asteroseismic information
from $\Pi_{0}$ gives a good handle on the stellar age. Consequently, we
observe a large scatter and larger spread in age compared to _Gaia_ -FLAME.

##  Appendix C Age spread at different evolutionary points

We analysed the relative ages not only at the end of the core hydrogen and
core helium burning but also at the tip of the red giant branch and the
beginning of the core helium burning. Fig. 10 shows the distributions for
these two evolutionary stages. As already seen from the histogram in Fig. 8
the distributions are very similar with a very slight evolution between them.

Figure 10: Relative ages at TRGB (left) and the ZAHeB (right) comparing
rotating and non-rotating models similar to Fig 7.