#  zoomies: A tool to infer stellar age from vertical action in Gaia data

Sheila Sagear Center for Computational Astrophysics, Flatiron Institute, 162
5th Avenue, Manhattan, NY, USA  Department of Astronomy, University of
Florida, 211 Bryant Space Science Center, Gainesville, FL 32611  Adrian M.
Price-Whelan Center for Computational Astrophysics, Flatiron Institute, 162
5th Avenue, Manhattan, NY, USA  Sarah Ballard Department of Astronomy,
University of Florida, 211 Bryant Space Science Center, Gainesville, FL 32611
Yuxi (Lucy) Lu American Museum of Natural History, Central Park West,
Manhattan, NY, USA  Ruth Angus American Museum of Natural History, Central
Park West, Manhattan, NY, USA  Center for Computational Astrophysics, Flatiron
Institute, 162 5th Avenue, Manhattan, NY, USA  David W. Hogg Center for
Cosmology and Particle Physics, Department of Physics, New York University,
726 Broadway, New York, NY 10003, USA  Max-Planck-Institut für Astronomie,
Königstuhl 17, D-69117 Heidelberg, Germany  Center for Computational
Astrophysics, Flatiron Institute, 162 5th Avenue, Manhattan, NY, USA

###### Abstract

Stellar age measurements are fundamental to understanding a wide range of
astronomical processes, including galactic dynamics, stellar evolution, and
planetary system formation. However, extracting age information from Main
Sequence stars is complicated, with techniques often relying on age proxies in
the absence of direct measurements. The Gaia data releases have enabled
detailed studies of the dynamical properties of stars within the Milky Way,
offering new opportunities to understand the relationship between stellar age
and dynamics. In this study, we leverage high-precision astrometric data from
Gaia DR3 to construct a stellar age prediction model based only on stellar
dynamical properties; namely, the vertical action. We calibrate two distinct,
hierarchical stellar age–vertical action relations, first employing
asteroseismic ages for red giant branch stars, then isochrone ages for main-
sequence turn-off stars. We describe a framework called zoomies based on this
calibration, by which we can infer ages for any star given its vertical
action. This tool is open-source and intended for community use. We compare
dynamical age estimates from zoomies with ages derived from other techniques
for a sample of open clusters and main-sequence stars with asteroseismic age
measurements. We also compare dynamical age estimates for stellar samples from
the Kepler, K2, and TESS exoplanet transit surveys. While dynamical age
relations are associated with large uncertainty, they are generally mass-
independent and depend on homogeneously measured astrometric data. These age
predictions are uniquely useful for large-scale demographic investigations,
especially in disentangling the relationship between planet occurrence,
metallicity, and age for low-mass stars.

Stellar kinematics (1608), Stellar ages (1581), Galaxy dynamics (591),
Exoplanets (498)

††facilities: Gaia, Kepler, K2, TESS††software: numpy (Harris et al., 2020),
matplotlib (Caswell et al., 2024), astropy (Robitaille et al., 2013;
Collaboration et al., 2018, 2022), numpyro (Bingham et al., 2019; Phan et al.,
2019), gala (Price-Whelan, 2017; Price-Whelan et al., 2020), agama (Vasiliev,
2019)

##  1 Introduction

The Gaia data releases (Prusti et al., 2016; Vallenari et al., 2023) have
revolutionized our understanding of galactic dynamics, along with stellar and
planetary astronomy within a broader galactic context. The motions of stars
within the galaxy encode age information: generally speaking, older stars
exhibit higher velocity dispersion or vertical action as a population (Wielen,
1977; Rocha-Pinto et al., 2004; Almeida-Fernandes & Rocha-Pinto, 2018).
Historically speaking, low-mass dwarfs have been especially useful in
calibrating stellar age and galactic kinematics, by virtue of their long
lifetimes (Reid et al., 1995; Faherty et al., 2009; Kiman et al., 2019; Angus
et al., 2020; Popinchalk et al., 2021).

Understanding the ages of stars is key to understanding not only how stars
themselves evolve, but also how galaxies evolve and how planetary systems
behave over time. Stellar ages are particularly interesting for constraining
the occurrence rate and evolution of planetary systems over time. Transit
surveys enabled by NASA’s Kepler Mission (Borucki et al., 2010) and the
Transiting Exoplanet Survey Satellite (TESS, Ricker et al. 2014) have
discovered thousands of exoplanets in the past decade (Guerrero et al., 2021).
The number of exoplanets known today has not only enabled the study of
individual planets and systems, but also their ensemble properties. Here Gaia
has also played a pivotal role by improving constraints on stellar properties
(e.g. Fulton & Petigura 2018; Berger et al. 2020a, 2023) as well as linking
planet occurrence and galactic position or kinematics (Winter et al., 2020;
Dai et al., 2021; Chen et al., 2021a; Longmore et al., 2021; Bashi & Zucker,
2021; Zink et al., 2023). It is as-yet uncertain whether correlations between
planet occurrence and galactic position are due to the galactic environment
directly (e.g. Cai et al. 2017; Ndugu et al. 2022) or reflect a separate,
underlying relationship between planet occurrence and stellar metallicity or
age (e.g. Nielsen et al. 2023).

Investigating how the orbits of planets evolve over time is key to fully
understanding how they form, the dependence of planet properties on stellar
properties (such as metallicity), and prospects for habitability. However,
this investigation likely hinges on our knowledge of stellar age, which is
notoriously difficult to constrain. A precise understanding of stellar age is
even more elusive for the predominant exoplanet hosts, FGK-type dwarfs and
(especially) M dwarfs. Small, rocky, habitable-zone exoplanets are abundant
around the lowest-mass stars (e.g. Mulders et al. 2015) and relatively easy to
observe using transit surveys due to their relatively large planet radius-
stellar radius ratio (Tarter et al., 2007). Unfortunately, it is difficult to
constrain ages for the lowest-mass stars because their lifetimes are often
longer than the age of the universe (Choi et al., 2016). Isochrone grids are
additionally an unreliable source of age measurements, as they often fail to
reproduce empirical constraints for M dwarfs due to the presence of strong
magnetic fields and starspots (Feiden & Chaboyer, 2012; Boyajian et al., 2012;
Mann et al., 2019).

Though measuring ages for low-mass stars is uniquely challenging, measuring
ages for stars of any mass is non-trivial. Stellar ages cannot be directly
measured; they can only be inferred using techniques such as comparing
measured properties to isochrone models and asteroseismology (e.g. Karataş et
al. 2005; Lebreton & Montalbán 2009; Silva Aguirre et al. 2015) or using
empirical relations such as gyrochronology and age-activity relations (e.g.
Barnes 2007; Soderblom et al. 1991; for review, Soderblom 2010). Recent
studies such as Lu et al. (2021) also combine such empirical methods with
certain stellar kinematic information. These quantities are associated with
relative ages with varying levels of precision (e.g. Epstein & Pinsonneault
2013; Tayar et al. 2022), but no method can be accurately applied to wide
ranges of stellar mass.

We are motivated by the prospect of conducting large-scale demographic studies
on how the orbital dynamics and properties of exoplanets change over time. We
now have high-precision astrometric information from Gaia homogeneously
derived for virtually every nearby star. With stellar parallax, proper motion,
and radial velocity measured by Gaia along with some assumed potential model
of the Milky Way, we can compute a star’s galactic orbit. It has been
established for decades that there is a relationship between the velocity
dispersion of stars and age (the age–velocity dispersion relation, or AVR)
(e.g. Strömberg 1946; Wielen 1977), and dynamical ages have shown potential in
calibrating ages even for low-mass dwarf stars (e.g. Veyette & Muirhead, 2018;
Newton et al., 2018; Lu et al., 2023, 2024). We leverage this nearly stellar-
mass-independent galactic orbit information to construct an age–galactic orbit
relation based on a calibration sample of stars with independently inferred
ages.

We note that there is as yet no definite consensus on the dominant cause of
the AVR. The AVR may be primarily driven by the orbital heating histories of
stars, such that the orbits of older stars have been kinematically heated over
time (Spitzer & Schwarzschild, 1953). The effect of disk heating on the AVR
has been studied through various N-body and cosmological simulations (Hänninen
& Flynn, 2002; Holmberg et al., 2007; House et al., 2011; Aumer et al., 2016;
Grand et al., 2016). Alternatively, the AVR may be observed as a consequence
of the birthplaces of stars in the galaxy over time. Older stars may have been
born kinematically hotter than younger stars, which form with a smaller
velocity dispersion at later times, following a so-called “upside-down” and
“inside out” galactic assembly process (Bird et al., 2013, 2021). Indeed,
there is evidence to suggest that both mechanisms may contribute significantly
to the observed AVR (McCluskey et al., 2024). We acknowledge that if disk
heating is the primary driver of the AVR, then mapping stellar age to galactic
orbits will be weakly dependent on stellar mass. Even in this case,
dynamically-derived stellar ages will still be far less dependent on stellar
mass than e.g. ages from isochrones or asteroseismology. To account for this
lack of consensus in the primary cause of the AVR, we call the dynamical age
model “generally mass-independent”.

Though we cannot directly measure the ages of stars, these dynamical age
estimates provide homogeneous age predictions for hundreds of millions of
stars with a wide range of stellar masses and properties. Because these age
predictions depend only on dynamical data and the physical assumptions of
galactic dynamics, they can be measured independently from other stellar
properties such as metallicity and stellar rotation. Such age measurements may
enable us to disentangle the relationship between stellar age, metallicity,
and other processes of interest (such as exoplanet orbital dynamics).

Due to systematic offsets between different age-dating methods, it is
preferable that age predictions for an entire sample of interest are
determined using the same method. Most samples of homogeneously inferred
stellar ages are small, especially for main-sequence stars; for example, there
are only $\mathcal{O}(10^{2})$ main-sequence stars with precise asteroseismic
age measurements (Huber et al., 2013; Silva Aguirre et al., 2015). Most
stellar age prediction methods also depend strongly on stellar mass, further
limiting the available age samples within a certain stellar mass bin (Barnes,
2007; Soderblom, 2010). This method of deriving dynamical stellar ages is
particularly useful because it can be used to generate age predictions for all
stars with Gaia radial velocities in a homogeneous way for a wide range of
stellar masses.

In this work, we introduce a fully dynamical stellar age–vertical action
relation calibrated on independently-measured ages of red giant branch and
main sequence turn-off stars. We release the software used in this analysis as
an open-source dynamical age prediction tool called zoomies. We validate and
compare dynamical age predictions for stars with independent asteroseismic
ages, open clusters, and the large stellar samples of the Kepler, K2 and TESS
missions. This paper is organized as follows: in Section 2 we describe the
stellar samples used in calibration and validation. In Section 3 we describe
our methods, including calculating dynamical properties, hierarchical model
construction, and model fitting. In Section 4, we discuss internal validation
checks on the dynamical age model. In Section 5, we introduce the model as an
open-source age prediction tool and discuss dynamical age predictions for
external stellar samples. In Section 6, we discuss how our assumptions may
affect our results, and we conclude in Section 7.

##  2 Data

###  2.1 Calibration Samples

We construct and calibrate two models to predict stellar age from Galactic
orbital quantities. We employ two samples of independently measured stellar
ages to calibrate the models (the Calibration Samples). We use a set of 20,917
red giant branch stars with asteroseismic ages from Stokholm et al. (2023)
(hereafter the red-giant calibration sample, or “RGB” sample). We use only the
stars from Stokholm et al. (2023) that have radial velocities measured with
Gaia. We separately use a set of 33,715 main sequence turn-off stars from the
APOGEE DR17 sample with ages derived with the Bayesian isochrone-fitting code
StarHorse (Queiroz et al., 2023) (hereafter the main-sequence turn off
calibration sample, or the “MSTO” sample). We take only stars labeled as main
sequence turn-off stars by Queiroz et al. (2023) and filter on the included
age warning flag “age_inout” indicating significant differences between input
and output $\mathrm{log}$ $g$, $T_{\textrm{eff}}$, and median metallicity.
Using these two stellar age calibration samples, we calibrate two distinct
dynamical age models; with this, we demonstrate that all dynamical age models
depend on the properties of the calibration sample, especially their $\ln
J_{z}$ distribution (see Figure 4).

###  2.2 Prediction Sample

The Prediction Samples refer to the stellar samples we generate age
predictions for. This potentially includes all Gaia stars with radial velocity
measurements. In this paper, we perform a preliminary analysis using the
subset of Kepler, K2, and TESS main-sequence target stars with Gaia DR3 radial
velocity measurements. In Figure 1, we present a visual comparison of the RGB
Sample, MSTO Sample, and the Prediction Sample including all stars with Gaia
DR3 radial velocities. In the left panel, we show a color-magnitude diagram
(CMD) of the three samples with Gaia $B_{p}-R_{p}$ color and absolute Gaia (G)
magnitude calculated from the G-band mean magnitude and the Gaia parallax. In
the right panel, we compare the vertical action distributions for each of the
three samples.

Whether stellar age estimates are generated with model-dependent, empirical,
or kinematically motivated methods, these ages will be dependent on the
calibration sample used. Different age predictions will invariably be offset
according to the samples used to calibrate the age prediction model. The
dynamical age predictions we present are no different; the age–action
relations will be best calibrated in the regions of $\ln J_{z}$ where the
calibration samples lie. Unless the prediction sample of interest is
comparable to the calibration sample in $\ln J_{z}$ space, dynamical ages
should generally be treated as relative, rather than absolute, age
predictions. We show in the right panel of Figure 1 that the $\ln J_{z}$
distribution of the least restrictive prediction sample (all stars with Gaia
radial velocities) differs significantly from the $\ln J_{z}$ distribution of
both the RGB and MSTO samples; in such cases, we warn that stellar ages should
be interpreted more conservatively as relative ages. Even so, understanding
relative stellar ages is useful in many contexts (e.g. for investigating the
evolution of planetary systems over time), especially since dynamical ages are
measured independently of most stellar properties and are available for very
large samples of stars. We include more detailed discussions of this effect in
Section 5.2.

Figure 1: Left: Color-magnitude diagram of the RGB calibration sample (red),
MSTO calibration sample (blue), and a representative prediction sample (gray).
The representative prediction sample consists of the first 100,000 stars
returned from a Gaia search excluding objects with no radial velocity
measurements and $\varpi\leq 0$. Right: Vertical action distributions for the
RGB (red), MSTO (blue) and representative Gaia sample (gray).

##  3 Methods

We construct a hierarchical Bayesian model to predict stellar age from
Galactic orbital quantities. We first determine which Galactic orbital
quantities to use. The ideal orbital quantity should be one that can be
calculated homogeneously for a wide range of stars (given an assumed Milky Way
potential model), and one that strongly correlates with stellar age. We could
construct an age–vertical velocity ($v_{z}$) relation, but this would
introduce a dependence on the temporal position of each star throughout its
orbit. Any stars with wide galactic orbits in the $z$-direction that happen to
be near the edge of their orbits would have a vertical velocity close to zero,
which may unfairly bias their dynamical age prediction.

One alternative metric is an integral of motion that encodes time-independent
information about a star’s galactic orbit. Given a static potential model of
the galaxy, the action–angle variables are integrals of motion that describe
an object’s orbit. Instead of vertical velocity, we elect to use an integral
of motion that describes the extent of a star’s orbit independently from time:
the vertical action, $J_{z}$. $J_{z}$ is the integral of the vertical galactic
position $z$ vs. the vertical velocity $v_{z}$. $J_{z}$ labels a star’s
galactic orbit, no matter where a star happens to be in its orbit at the
current moment. The vertical velocity $v_{z}$ varies with time on the orbit,
where $J_{z}$ (along with $z_{max}$ and other orbital constants of motion)
would not. The vertical action is defined as

$J_{z}=\frac{1}{2\pi}\oint{\mathrm{d}z\,v_{z}}$ (1)

where $z$ describes the vertical position and $v_{z}$ describes the vertical
velocity of the object, and the integral is over the path of one full vertical
orbit. Since $J_{z}$ is independent of time, we contend that using $J_{z}$
will produce a tighter age-galactic motion relation than using the vertical
velocity alone. However, the $J_{z}$ computations depend on the assumed
galactic potential model and quality of the Gaia kinematic data in hand. We
discuss the implications of considering only $J_{z}$ in the age–action model
(and not $J_{r}$, the radial action, or $J_{\phi}$, the azimuthal action) in
Section 6.2.

###  3.1 Calibrating the age–action relation

In this section, we describe the process of constructing and calibrating the
age–action relation. We use an axissymetric potential model of the Milky Way
to compute the actions, implemented in the Python package gala (Price-Whelan,
2017; Price-Whelan et al., 2020). This potential model includes a spherical
galactic nucleus and bulge, a 3-component sum of Miyamoto-Nagai galactic disks
(Miyamoto & Nagai, 1975) representing an approximation to the potential from
an exponential disk (Smith et al., 2015), and a spherical Navarro-Frenk-White
dark matter halo model (Navarro et al., 1996). The disk model follows the
galactic rotation curve from Eilers et al. (2019), and the vertical structure
is set by the shape of the phase-space spiral in the solar neighborhood from
Darragh-Ford et al. (2023).

When calibrating the age–$\ln J_{z}$ model, we assume that the dynamical ages
of stars are independent from stellar mass. We return to this assumption and
discuss its implications in Section 6.1. For each star in each calibration
sample, we first calculate the logarithm of the vertical action $\ln J_{z}$
given this Milky Way potential model. We calculate $\ln J_{z}$ from the Gaia
proper motion, radial velocity, and using inverse parallax as the distance
estimate. We use the gala software to make these calculations. We take the
initial conditions given by the Gaia data and integrate each star’s orbit
through time within the galactic potential model. We numerically integrate the
orbits to calculate actions using the Staeckel Fudge implementation in the
Agama software (Vasiliev, 2019). In both calibration samples, the errors in
$\ln J_{z}$ are much smaller than the intrinsic variance in $\ln J_{z}$ (the
average $\ln J_{z}$ uncertainty in both the RGB and MSTO samples is
approximately 0.05, compared to a mean intrinsic variance of approximately
1.5), so we neglect the errors in $\ln J_{z}$.

Next, we construct a hierarchical Bayesian model to describe the age–$\ln
J_{z}$ relation. In this model, we wish to describe the log of the intrinsic
variance (the squared uncertainty) in the age direction,
$\mathrm{ln}V_{\tau}$, as a free parameter. Though we intend to construct a
hierarchical model to predict stellar age given $\ln J_{z}$ (in other words,
constructing a $P(\tau|\ln J_{z})$ model), fitting a model to the age–$\ln
J_{z}$ relation proves difficult when incorporating $\mathrm{ln}V_{\tau}$.
Because the age–$\ln J_{z}$ relation is steep and has large scatter in the age
direction, the value of the $\ln V$ free parameter balloons and fails to
converge. To counteract this, we instead consider the $\ln J_{z}$ vs. age
relation, $P(\ln J_{z}|\tau)$, where $\mathrm{ln}V_{\mathrm{ln}J_{z}}$
describes the intrinsic variance in the $\ln J_{z}$ direction. Once the $P(\ln
J_{z}|\tau)$ model is calibrated, we obtain $P(\tau|\ln J_{z})$ using a
process we describe in detail in Section 3.2.

Since we calibrate the model on samples of main sequence turn-off stars and
red giant branch stars, the number density of stars along the age axis will
reflect these samples. For example, the MSTO stellar sample has a high number
density near 10 Gyr (see Figure 4, bottom left panel), which would unfairly
bias the age prediction towards 10 Gyr when applied to other populations of
stars with different number density distributions, such as main-sequence FGKM
dwarfs. To counteract this effect, we estimate the stellar number density
along the age axis $\lambda(\tau)$ as part of the age–action model, and we
marginalize out $\lambda(\tau)$ when applying the model to other stars. This
process enables us to apply only the slope and variance of the age–action
model to other stellar samples, without also applying the calibration sample’s
number density distribution. Accounting for the number density in this way
produces a more generally applicable dynamical age prediction model.

Our method will not only require calibrating the age–action relationship, but
also the calibration sample’s stellar number density in age, $\lambda(\tau)$.
To estimate this, we include a Poisson likelihood component in the model
likelihood function, similar in structure to the one used by Everall et al.
(2022) in a different context. If the number density profile of stars along
the age axis is $\lambda(\tau)$, we take the probability of drawing a
population of stellar ages $\tau_{i}$ from $\lambda(\tau)$ to be the Poisson
likelihood function (derived in Lombardi et al. 2013; Everall & Das 2020):

$\ln\mathcal{L_{\mathrm{Poisson}}}=\sum_{i=1}^{N}\ln\lambda(\tau_{i})-\int{\lambda(\tau)\mathrm{d}\tau}$
(2)

In the hierarchical model, we describe the number density
$\mathrm{ln}\lambda(\tau)$ as a third-degree spline defined between 0 and 14
Gyr. We add the Poisson likelihood factor into the complete model likelihood
function. By fitting $\lambda(\tau)$ of the calibration sample in the
age–action model, we can be relatively confident that all other model
parameters describing the age–$\ln J_{z}$ relation are not influenced by the
calibration sample’s non-uniform $\lambda(\tau)$. The $\lambda(\tau)$
distribution is specific to each calibration sample and is dependent on
stellar mass, selection functions, and any number of other factors; we do not
want the $\lambda(\tau)$ distribution of the calibration sample to affect the
prediction sample’s dynamical age estimates. By fitting the calibration
sample’s $\lambda(\tau)$ distribution and ignoring it when using the age–$\ln
J_{z}$ model to generate age predictions for other stars, we minimize the risk
of the calibration sample’s unique $\lambda(\tau)$ distribution affecting
dynamical age estimates for other stars.

Finally, we model the $\ln J_{z}$ –age slope as a monotonic quadratic spline
(a quadratic spline that must increase at each successive knot value).

We summarize the model fixed and free parameters here. The knot locations of
the $\tau(\mathrm{ln}J_{z})$ monotonic spline, $t_{\tau(\mathrm{ln}J_{z})}$,
are fixed at every $3.75$ Gyr between ${-1}$ and $14$ Gyr. The knot locations
of the $\mathrm{ln}\lambda(\tau)$ third-degree spline,
$t_{\mathrm{ln}\lambda(\tau)}$, are fixed at every $1$ Gyr between ${-1}$ and
$14$ Gyr. The model free parameters include the knot values of the
$\tau(\mathrm{ln}J_{z})$ monotonic spline
$\mathcal{S}(t_{\tau(\mathrm{ln}J_{z})})$, the knot values of the
$\mathrm{ln}\lambda(\tau)$ third-degree spline
$\mathcal{S}(t_{\mathrm{ln}\lambda(\tau)})$, and the intrinsic variance
$\mathrm{ln}V_{\mathrm{ln}J_{z}}$. We show the graphical model representation
of this model in Figure 2. A list of the model free and fixed parameters is
provided in Table 1. The model parameters associated with
$\mathrm{ln}\lambda(\tau)$ which are ignored when using the model to generate
age predictions for other stars are marked with “$[\star]$”.

Table 1: Dynamical age–action model parameters

Parameter | Prior | Description
---|---|---
$\mathrm{ln}{V_{\mathrm{ln}J_{z}}}$ | $\mathcal{N}(9,5)$ | Log of intrinsic variance
$V_{\mathrm{ln}J_{z}}$ | Deterministic | Intrinsic variance
$t_{\tau(\mathrm{ln}J_{z})}$ | Fixed | Knot vector for $\tau(\mathrm{ln}J_{z})$ spline, where $t_{\tau(\mathrm{ln}J_{z})}=(t_{\tau(\mathrm{ln}J_{z}),1}...t_{\tau(\mathrm{ln}J_{z}),5})$
$\mathcal{S}(t_{\tau(\mathrm{ln}J_{z})})$ | $\mathcal{U}(0,5)$ | Knot values for $\tau(\mathrm{ln}J_{z})$ spline, defined at each point in the knot vector $t_{\tau(\mathrm{ln}J_{z})}$
$t_{\mathrm{ln}\lambda(\tau)}[\star]$ | Fixed | Knot vector for $\mathrm{ln}\lambda(\tau)$ spline, where $t_{\mathrm{ln}\lambda(\tau)}=(t_{\mathrm{ln}\lambda(\tau),1}...t_{\mathrm{ln}\lambda(\tau),15})$
$\mathcal{S}(t_{\mathrm{ln}\lambda(\tau)})[\star]$ | $\mathcal{U}(-10,15)$ | Knot values for $\mathrm{ln}\lambda(\tau)$ spline, defined at each point in the knot vector $t_{\mathrm{ln}\lambda(\tau)}$
$ln\lambda(\tau)[\star]$ | Observed | Log of stellar number density
$\tau$ | Observed | Stellar age
$\mathrm{ln}J_{z,pred}$ | Observed | Predicted $\ln J_{z}$

Note. — The priors listed for the spline knot values
$\mathcal{S}(t_{\tau(\mathrm{ln}J_{z})})$ and
$\mathcal{S}(t_{\mathrm{ln}\lambda(\tau)})$ apply independently for each knot.
A graphical model representation of the model is shown in Figure 2.

Note. — $[\star]$: We fit $\lambda(\tau)$ in the dynamical age model, but we
do not apply $\lambda(\tau)$ to generate dynamical age predictions for the
prediction sample. We ignore these parameters when we apply the calibrated
model to a prediction sample of stars.

Figure 2: Graphical model representation of the hierarchical Bayesian stellar
age–$\ln J_{z}$ model. $t_{\tau(\mathrm{ln}J_{z})}$ and
$t_{\mathrm{ln}\lambda(\tau)}$ refer to the fixed spline knot vectors for the
$\tau(\mathrm{ln}J_{z})$ and $\mathrm{ln}\lambda(\tau)$ splines, respectively,
and $\mathcal{S}(t_{\tau(\mathrm{ln}J_{z})})$ and
$\mathcal{S}(t_{\mathrm{ln}\lambda(\tau)})$ represent the associated knot
value free parameters. A list of the model parameters and their priors is
given in Table 1. As in Table 1, “$[\star]$” denotes a stellar number density
model parameter that we ignore when we apply the calibrated model to a
prediction sample of stars.

We generate two mean posterior age–Jz relations: one using the red-giant
branch star calibration age sample and one using the main-sequence turn off
star calibration age sample. For each calibration sample, we sampled the free
parameters using No U-Turn Sampling (Homan & Gelman, 2014) with the Python
package numpyro (Phan et al., 2019; Bingham et al., 2019) with chains of 2,000
draws each. We use 2,000 tuning steps for each fit. We calculate the Gelman-
Rubin $\hat{r}$ statistic for each model free parameter to check for
convergence. All free parameters converged with $\hat{r}<1.05$ for both the
RGB and MSTO models.

###  3.2 Generating stellar age predictions

Figure 3: Illustration of the conditional probabilities $P(\ln J_{z}|\tau)$
and $P(\tau|\ln J_{z})$ in relation to the joint probability $P(\tau,\ln
J_{z})$. The blue heatmap represents the joint probability $P(\tau,\ln J_{z})$
and shows a simulated stellar sample in $\tau$–$\ln J_{z}$ space generated
from the mean posterior dynamical age model. The pink and red highlighted
horizontal regions correspond to the conditional probability $P(\ln
J_{z}|\tau)$, shown on the top panel as probability density functions in the
$\ln J_{z}$ dimension. The light and dark green highlighted vertical regions
correspond to the conditional probability $P(\tau|\ln J_{z})$, shown on the
right panel as probability density functions in the $\tau$ dimension. The
$P(\tau|\ln J_{z})$ probability density functions are analogous to stellar age
predictions based on $\ln J_{z}$ from Gaia.

Given a value of $\ln J_{z}$, we use the calibrated age–vertical action
relation to predict an age probability distribution. When using the mean
posterior age–$\ln J_{z}$ model to predict ages for stars with $\ln J_{z}$, we
ignore the number density in age. In other words, we consider only the mean
and intrinsic variance of the age–$\ln J_{z}$ relation, and we do not consider
the number density in age of the calibration samples when generating stellar
age predictions.

Due to the calibration samples’ large scatter on the age axis, we construct
the age-action model to find $P(\ln J_{z}|\tau)$, as described in Section 3.1.
In order to use this model to predict age from $\ln J_{z}$, we adjust this
model to find $P(\tau|\ln J_{z})$. We invoke Bayes’ theorem:

$P(\tau|\ln J_{z})=\frac{P(\ln J_{z}|\tau)P(\tau)}{P(\ln J_{z})}$ (3)

Because we have taken the step of fitting $\lambda(\tau)$ of the calibration
sample, we can safely assume that the rest of the model parameters described
in Section 3.1 (namely, $\mathcal{S}(t_{\mathrm{ln}J_{z}(\tau})$ and
$\mathrm{ln}V_{\mathrm{ln}J_{z}}$) are sufficiently independent from the
stellar number density of the calibration sample. Therefore, we take the
stellar age prior $P(\tau)$ to be uniform here. Marginalizing over $\tau$
allows us to calculate the marginal probability $P(\ln J_{z})$:

$P(\ln J_{z})=\int P(\ln J_{z}|\tau)P(\tau)\mathrm{d}\tau$ (4)

In Figure 3, we show a visual illustration of the conditional probabilities
$P(\tau|\ln J_{z})$ and $P(\ln J_{z}|\tau)$ in terms of the joint probability
$P(\tau,\ln J_{z})$. We illustrate using a simulated stellar sample generated
from the mean posterior dynamical age model with the stellar number density
$\lambda(\tau)$ marginalized out.

To draw the mean posterior dynamical age model in age–$\ln J_{z}$ space, we
first generate a test grid $\tau_{\textrm{grid}}$ of possible stellar ages
from 0 to 14 Gyr, arbitrarily spaced by 0.014 Gyr. We then evaluate the
$\tau(\ln J_{z})$ monotonic spline at each point on $\tau_{\textrm{grid}}$
using the mean posterior $\mathcal{S}(t_{\tau(\mathrm{ln}J_{z})})$ knot
values. We take the $\ln J_{z}$ values on the mean posterior spline
corresponding to $\tau_{\textrm{grid}}$ to be $\ln J_{z\textrm{grid}}$. The
mean posterior age–$\ln J_{z}$ splines evaluated over $\tau_{\textrm{grid}}$
(in other words, $\tau_{\textrm{grid}}$ vs. $\ln J_{z\textrm{grid}}$) are
shown in red in the left panels of Figure 4.

To calculate $P(\tau|\ln J_{z})$ for a star of interest with a given $\ln
J_{z}$, we evaluate Equations 3 and 4. We evaluate the integral in Equation 4
numerically with Simpson’s rule across $\ln J_{z\textrm{grid}}$ to generate a
stellar age probability distribution across $\tau_{\textrm{grid}}$. With this,
we are able to take any given $J_{z}$ value and generate a predicted dynamical
age probability distribution, evaluated across $\tau_{\textrm{grid}}$.

In Figure 4, we present a comparison of the age–$\ln J_{z}$ model fit between
the RGB calibration sample and MSTO sample. On the left panels, we show both
calibration samples in (asteroseismic or isochrone) age vs. $J_{z}$ space with
the best-fit model over-plotted. On the right panels, we show generated age
prediction distributions for a set of test $\ln J_{z}$ values for both models.
We demonstrate that the analogous test age predictions differ, especially for
large $J_{z}$, due to the $J_{z}$ distributions of the calibration sample.
However, both models agree in terms of relative age (i.e. when stars are
ranked in order of youngest to oldest).

Figure 4: Top left: 2-D histogram of the RGB calibration sample in Stokholm et
al. (2023) asteroseismic age vs. $\ln J_{z}$ space. The mean posterior
$\tau$–$\ln J_{z}$ monotonic spline is overplotted, $\pm$ the square root of
the best-fit intrinsic variance. Bottom left: 2-D histogram of the MSTO
calibration sample in Queiroz et al. (2023) isochrone ages vs. $\ln J_{z}$
space. The mean posterior $\tau$–$\ln J_{z}$ monotonic spline is overplotted,
$\pm$ the square root of the best-fit intrinsic variance. Top right: Generated
age prediction distributions for a set of test $\ln J_{z}$ values using the
RGB-calibrated model. Bottom right: Generated age prediction distributions for
the same set of test $\ln J_{z}$ values using the MSTO-calibrated model.

##  4 Validation

###  4.1 Internal Model Validation

Figure 5: Left: 2-D histogram of the red giant calibration sample, and the
mean (ridgeline) of the model fit (red). Right: Predicted density of stellar
ages from model fit (red). While we fit and can re-generate the stellar number
density along the age axis according to the calibration sample, we ignore this
number density when predicting stellar ages, effectively using only the
age–$J_{z}$ spline model represented in red.

We internally validate the model using a $k$-fold cross-validation method. We
split each calibration sample randomly into 10 equal subsamples. We designate
one subsample as the “withheld” subsample, and the other nine subsamples
combined as the “validation” sample. We calibrate the dynamical age model
using the validation sample. We then evaluate the calibrated model’s goodness
of fit using the withheld subsample, with

$\chi=\frac{\tau-\tau_{pred}}{\sqrt{V+\sigma_{\tau}^{2}}}$ (5)

where $\tau$ represents the stellar ages for the withheld sample,
$\tau_{pred}$ represents the mean dynamical age predictions for the withheld
sample, V is the mean posterior intrinsic variance, and $\sigma_{\tau}$
represents the stellar age error for the withheld sample. We repeat this
process 10 times, where each of the 10 subsamples in turn becomes the withheld
sample. We verify that $\sigma_{\chi}$ (the standard deviation of $\chi$ for
each of the ten iterations) is close to 1.

We additionally verify that from the calibrated dynamical age model, we are
able to reproduce both the age–action trend and the stellar number density of
the calibration sample. A visualization of this verification is shown in
Figure 5. On the left panel, we show $\ln J_{z}$ vs. age for the RGB
calibration sample as a 2-D histogram with arbitrary bins, along with the
best-fit age–$J_{z}$ model shown in red. The color corresponds to the number
density of stars appearing in each bin. On the right panel, we show the number
density in each bin predicted by our model. The dynamical age model encodes
information about both the age–action trend and the calibration sample number
density; however, in predicting ages for stars given $\ln J_{z}$, we do not
consider the number density factor $\lambda(\tau)$.

###  4.2 Comparison with Asteroseismic RGB Ages

Figure 6: A 2-D histogram of the Pinsonneault et al. (2018) sample of red-
giant-branch stars in asteroseismic age-$\ln J_{z}$ space. The RGB-calibrated
dynamical ages for this sample are designated with the red line and shaded
region. The solid red line represents the statistical mode of each dynamical
age probability distribution associated with each star in the Pinsonneault et
al. (2018) sample, and the shaded region represents $68\%$ of the distribution
mass around the mode. The 2-D histogram is row-normalized to reduce the visual
effect of the stellar density distribution on the $\tau$ axis. The posterior
predictions based on dynamics alone are both truncated and poorly constrained,
but with this comparison we demonstrate that our purely dynamical age
predictions are consistent with independent age measurements.

We validate our model-predicted ages against another independent sample of red
giant branch stars with asteroseismic age predictions. In Figure 6, we show a
column-normalized heatmap of stellar ages from the APOKASC catalog
(Pinsonneault et al., 2018), which derived spectroscopic and asteroseismic
ages for red giant stars. The Pinsonneault et al. (2018) sample closely
matches the RGB calibration sample in terms of stellar mass and temperature
distribution, but have ages determined independently of dynamics. We
calculated vertical actions for the stars in the Pinsonneault et al. (2018)
sample that have Gaia DR3 radial velocities. With Figure 6, we demonstrate
that our dynamical age predictions are consistent in age–$\ln J_{z}$ space
with the Pinsonneault et al. (2018) sample, where the $\ln J_{z}$ distribution
is comparable to our calibration sample. The age predictions based on dynamics
alone are both truncated and poorly constrained, but we seek to demonstrate
that our dynamical age predictions are consistent with independent age
measurements.

##  5 Results

We publish the dynamical age model as an open-source, user-friendly tool
called zoomies111www.github.com/ssagear/zoomies (Sagear & Price-Whelan, 2024).
This tool takes an input of Gaia data for any star with a Gaia radial velocity
and outputs a predicted age distribution. We use this tool to generate
dynamical age predictions for several samples of stars.

We note that dynamical age predictions generated by this tool are dependent on
the precision and accuracy of ages in the calibration samples, as seen in the
differing age probability distributions given $\ln J_{z}$ associated with each
calibrated model, shown in Figure 4. Indeed, the uncertainty in the age-$\ln
J_{z}$ relation contributes significantly to the wide tails of the age
probability distributions (the median age uncertainty for the RGB sample from
Stokholm et al. (2023) $\sigma_{\tau}\approx 1$ $\mathrm{Gyr}$, and for the
MSTO sample from Queiroz et al. (2023) $\sigma_{\tau}\approx 0.5$
$\mathrm{Gyr}$). Age estimates generated for stellar populations that are
similar to the calibration sample (in terms of $M_{\star}$ and $\ln J_{z}$)
may be considered in an absolute sense. However, for stellar populations that
differ significantly from the calibration sample, it may be more useful to
consider the generated age estimates as relative ages, e.g. in comparing older
vs. younger groups within a larger population. zoomies offers functionality
for users to calibrate a dynamical age model using any calibration samples
with externally measured ages and Gaia astrometric data; the precision of the
resulting age estimates will depend on the age uncertainties in the chosen
calibration sample and the associated scatter along the $\ln J_{z}$ axis. In
the following sections, we compare and discuss our dynamical age estimates for
external stellar samples.

###  5.1 Comparison with Open Cluster ages

Figure 7: Violin plot of selected open cluster dynamical ages. Each cluster
segment is labelled according to the open cluster it represents. Each segment
represents the age probability distribution for the median $\ln J_{z}$ value
for cluster members (with arbitrary width), according to the RGB-calibrated
dynamical age model. The position of each cluster segment on the x-axis
represents the cluster age reported by Dias et al. (2021). The one-to-one
relation is shown with the red dotted line.

We generate dynamical age predictions for open cluster members with
independently measured cluster ages. We use the cluster membership lists and
ages presented by Dias et al. (2021) to identify individual members of each
cluster. We select eight open clusters to present in this paper, and we
include an analysis of a random sample of the clusters in Dias et al. (2021)
in Appendix A.

We select the eight clusters in Figure 4 with the following method: first, we
bin the cluster ages determined by Dias et al. (2021) from 0 to 14 Gyr with an
arbitrary width of 1 Gyr. We do this to select an evenly distributed sample of
young and old open clusters. Next, we take the cluster in each bin that has
the most members, ensuring that each cluster has more than 100 members at this
stage. For each cluster, we omit the members that have a membership
probability less than 0.5. We then download the Gaia astrometric data for the
remaining cluster members, vetting on the availability of Gaia data (parallax
$>0$ and radial velocity measured). We present the selected clusters, the
number of remaining members and their ages in Table 2. We perform the same
analysis on a random sample of clusters and include this analysis in Appendix
A.

Using the Gaia data for this sample, we calculate $\ln J_{z}$ for each star
according to the method described in Section 3. For each cluster, we take the
median $\ln J_{z}$ across all cluster members and generate the corresponding
dynamical age probability distribution. We take this age distribution to be
the representative age distribution for the entire cluster.

In Figure 7, we show the age distributions for each selected cluster as a
violin plot. The age distributions for each labelled cluster are represented
vertically, and the x-axis represents the independently measured cluster age
from Dias et al. (2021). The dashed red line represents the one-to-one
relation. Though our dynamical age distributions are not as precise as the
measured cluster ages, we demonstrate that the relative age predictions for
open clusters scale appropriately using only dynamical data.

Table 2: Open cluster sample parameters from Dias et al. (2021). $N$ is the number of cluster members used in analysis after vetting based on availability of Gaia parallax and radial velocity, and membership probability $>=0.5$ according to Dias et al. (2021). $\tau_{Cluster}$ is the independently measured cluster age from Dias et al. (2021).  Cluster | $N$ | $\tau_{Cluster}$
---|---|---
|  | (Gyr)
NGC 2477 | 421 | 0.94
NGC 7789 | 639 | 1.62
NGC 6819 | 132 | 2.63
Trumpler 5 | 192 | 3.46
Melotte 66 | 69 | 4.49
NGC 188 | 84 | 6.19
Collinder 261 | 105 | 7.90
Berkeley 17 | 25 | 9.98

###  5.2 Kepler, K2, and TESS main samples

Figure 8: Left: Combined dynamical age distributions (calibrated with the MSTO
sample) for the KIC sample, partitioned by $T_{\textrm{eff}}$ (from Mathur et
al. 2017). Right: $\ln J_{z}$ distribution for the entire KIC sample (green)
compared to the $\ln J_{z}$ distributions for the RGB and MSTO calibration
samples (red and blue, respectively).

In this Section, we investigate dynamical ages derived with zoomies for
several samples of stars from large exoplanet transit surveys. We have
confined ourselves in this manuscript to a description of the zoomies
machinery, and leave its detailed application to exoplanetary studies to
future work. However, we comment upon large-scale trends, and emphasize the
potential usefulness of dynamical ages to the exoplanet community.

We employ the Kepler Input Catalog (KIC) (Brown et al., 2011) together with
the Gaia-Kepler cross-match222https://gaia-kepler.fun/ to identify sources
with a Gaia radial velocity measurement. We generate dynamical age predictions
according to the methods described in Section 3 for 100,885 Kepler stars for
which we can measure a vertical action. Figure 8 shows combined histograms for
Kepler stellar age predictions, binned by $T_{\textrm{eff}}$. For the Kepler
stellar sample, we have employed the $T_{\textrm{eff}}$ values presented in
the Kepler Data Release 25 (DR25) catalog (Mathur et al., 2017). We obtain the
data set from the NASA Exoplanet Archive (Kepler Mission, 2019)333Accessed on
2023-12-05. We first draw a randomly-chosen 100 points from the stellar age
probability distribution generated by zoomies for each star, before combining
these probability samples for all stars within a given $T_{\textrm{eff}}$
range. As we expect, we find that the hottest stars are more likely to have
younger predicted ages, while age predictions for the coolest stars are more
uniformly distributed between young and old. We note that the majority of the
Kepler stellar samples does not match the mass range of our calibration
samples, and so emphasize that these age predictions should be carefully
considered as relative age predictions rather than absolute ages.

Figure 9: A 2-D histogram of the Mathur et al. (2023) sample of Kepler main-
sequence stars in gyrochronological age-$\ln J_{z}$ space. The MSTO-calibrated
dynamical ages for this sample are designated with the green line and shaded
region. The solid green line represents the statistical mode of each dynamical
age probability distribution associated with each star in the Mathur et al.
(2023) sample, and the shaded region represents $68\%$ of the distribution
mass around the mode. The 2-D histogram is row-normalized to reduce the visual
effect of the stellar density distribution on the $\tau$ axis.

We also present a preliminary comparison of our age predictions with the
Mathur et al. (2023) sample of Kepler gyrochronological ages. Mathur et al.
(2023) measured gyrochronological ages for around 55,000 main-sequence stars
using Kepler. In Figure 9, we show a comparison of the gyrochronological ages
from Mathur et al. (2023) with the analogous dynamical age estimates from the
MSTO-calibrated model. We consider the MSTO-calibrated age predictions for the
Kepler sample because the Kepler sample $\ln J_{z}$ distribution matches the
$\ln J_{z}$ distribution of the MSTO sample more closely than the RGB sample
(see Figure 8, right panel).

While the scatter is large for $\tau>4$ Gyr in age-action space, the positive
relationship between $\ln J_{z}$ and age can be seen for young ages in Figure
9. As we expect, the slope of the dynamical age relations matches the slope
seen in age–action space for the Mathur et al. (2023) Kepler gyrochronology
sample to first order. With this comparison, we demonstrate that dynamical
ages generated with this method are consistent with gyrochronological ages of
main-sequence Kepler stars primarily for $\tau<4$ Gyr.

Figure 10:  Top Left: Combined dynamical age probability distributions for
Kepler (solid), K2 (dashed), and TESS (dotted) stellar samples with
$T_{\textrm{eff}}$ between 5500 and 5700 K. Bin widths arbitrarily set to 0.5
Gyr. Top Right: Same as top left, but for stars with $T_{\textrm{eff}}$
between 4200 and 4500 K. Bottom Left: Combined $\ln J_{z}$ KDEs for the
Kepler, K2 and TESS stars with $T_{\textrm{eff}}$ between 5500 and 5700 K.
Bottom right: Same as bottom left, but for stars with $T_{\textrm{eff}}$
between 4200 and 4500 K.

We compare the ages of the Kepler, K2 and TESS survey samples. We repeat the
process described for the Kepler sample for the K2 and TESS target stars with
positive parallax measurements and radial velocities from Gaia. For a
consistent comparison, we compare stellar ages within the same selected ranges
of $T_{\textrm{eff}}$. For the K2 stellar sample, we employ the
$T_{\textrm{eff}}$ values presented in the K2 Ecliptic Plane Input Catalog
(Huber et al., 2016), and for the TESS stellar sample, we employ the
$T_{\textrm{eff}}$ values presented in the TESS Input Catalog (TIC) 8 (Stassun
et al., 2019). We present the combined age distributions for each survey
sample, along with the vertical action distributions, in Figure 10 for two
selected temperature ranges. K2 stars appear slightly older on average than
Kepler stars. Differences in the age distributions between the transit survey
samples originate from different mission designs: the surveys’ different
selection functions, combined with different pointing strategies, contribute
to differences in the average observed stellar populations’ kinematics. While
the original Kepler mission’s field of view was a small stationary area
pointing near, but not directly on, the galactic plane (Gilliland et al.,
2011), the K2 mission observed several target fields along the ecliptic across
a period of several years (Howell et al., 2014). Yet, both missions use the
same instrument and bandpass. This may cause different distributions in
vertical action, as seen in Figure 10. In contrast, TESS’s wide, near-infrared
bandpass and its pointing throughout the celestial sphere (Ricker et al.,
2014) may be reflected in the difference in age and vertical action
distributions between hotter and cooler TESS stars, compared to the Kepler and
K2 samples.

##  6 Discussion

In the following subsections, we return to our assumptions about the dynamical
age model’s dependence on stellar mass and discuss the limitations of our
model related to sample distributions in action space. We also discuss the
implications of these results on planet occurrence.

###  6.1 Stellar Mass Dependence

As discussed in Section 1 and Section 3, we assume in our dynamical age
relation that the kinematic galactic motion of stars are mass-independent,
consistent with an “upside-down” and “inside-out” model of galactic assembly
(Bird et al., 2013, 2021). Alternatively, it is possible that the AVR is
driven primarily by orbital heating (e.g. Spitzer & Schwarzschild 1953). If
stellar kinematic properties are determined primarily by orbital heating, the
dynamical age model we construct will not be entirely mass-independent and may
produce less accurate age predictions for stars with masses significantly
different from our calibration sample (i.e. the lowest-mass stars). However,
we contend that in this case, this dynamical age relation is still far less
dependent on stellar mass than other age determination methods, such as
isochrone fitting and asteroseismic ages, especially when interpreting age
predictions as relative ages for low-mass stars. In order to avoid any mass
dependence in a dynamical age model, a large calibration sample of low-mass
stars with relatively confident age measurements is needed.

###  6.2 Impact of the full orbital action distribution on age predictions

Here we discuss the extent to which one may apply the dynamical age model to
prediction samples that do not closely match the calibration sample in action
space. In selecting a calibration sample, we necessarily assume that the $\ln
J_{z}$ distribution of the calibration sample matches that of the prediction
sample of interest. However, we selected the RGB and MSTO samples primarily
because of the accuracy and precision of their ages determined with
asteroseismology and isochrone-fitting, respectively (not necessarily because
their $\ln J_{z}$ distributions match the $\ln J_{z}$ distributions of our
prediction samples). In many cases, sufficiently large calibration samples
with accurate, independently-measured ages that also match the $\ln J_{z}$
distribution of the prediction sample are not available. This is particularly
likely for the lowest-mass stars, which also happen to be the most prominent
exoplanet hosts.

We calibrate the dynamical age model in this work on $J_{z}$ , which
represents one slice of a three dimensional projection of action space,
including $J_{z}$ , $J_{r}$, and $J_{\phi}$. We contend that the best way to
mitigate the effects of prediction sample that do not match the calibration
sample is to build a joint age–action model that considers all three actions.
We will address this in a future work, as an age prediction method based on
all orbital actions should produce more precise ages. However, short of
building a three-dimensional age–action model, one workaround with the current
1D method is to re-sample or re-weight the prediction sample in action space
to more closely match the calibration sample.

###  6.3 Planet Occurrence and Dynamical Ages

The investigation of planet formation and evolution is only one application of
demographic stellar age constraints. While we aim for `zoomies` to be broadly
useful the astronomical community interested in stellar ages in the Milky Way,
we focus here on its potential use for studying exoplanet host stars. Gaia has
enabled exoplanet study to occur within a new galactic context (see e.g.
Winter et al. 2020; Dai et al. 2021; Chen et al. 2021a; Longmore et al. 2021;
Bashi & Zucker 2021; Zink et al. 2023). In this Section, we describe the
relative strength of dynamically-derived stellar ages for investigations of
the (on average) $\mathcal{O}10^{5}-10^{6}$ stars monitored by space-based
transit surveys.

Folding together both isochrone or spectroscopic age constraints together with
Gaia parallaxes or kinematic measurements, the evolution of planetary systems
over time has already been seen (Berger et al., 2023; Chen et al., 2021b). For
example, the radius distribution of planets shifts as stars age (Berger et
al., 2020b), and there exists an apparent change in transit multiplicity as
well (Chen et al., 2021b). Zink et al. (2023) found that raw occurrence is
affected by galactic height (as determined by Gaia parallaxes), one of many
proxies for stellar age. We consider the dynamical ages for the population of
stars observed by Kepler. Nominally, we might expect to see some trend with
age– though dynamical age dating involves much larger uncertainty than other
methods.

The degree to which the dynamical age relation we developed is constraining at
the population level for the Kepler sample is shown in Figure 11. We overplot
the dynamical age distributions for the entire sample of Kepler Objects of
Interest (KOI), versus KIC field stars with no transiting planets. We do see a
visible offset in the same direction as Chen et al. (2021b): that is, host
stars to transiting planets are, on average, younger than field stars. This
figure is meant to convey the relative strength of dynamical age information
alone: fully disambiguating the ages of field stars vs. planet hosts would
require many additional steps (given that the population of field stars in
fact hosts planets too, though not in a transiting geometry). In this sense,
the KIC sample is contaminated with planet hosts and not representative of
stars that don’t have planets. Even given this degree of contamination, Figure
11 shows to first order the extent to which dynamical information alone can be
useful at a demographic level.

Figure 11: Combined dynamical age distributions for the entire KIC catalog
(green) and all KOIs (red).

##  7 Conclusions

We summarize our results and work as follows:

  * •

We calibrate a flexible model to predict stellar ages given the vertical
actions of stars using two independent calibration samples: one based on
asteroseismic red giant branch star ages and one using isochrone main-sequence
turn-off star ages.

  * •

We compare these dynamical age predictions with an external sample of red
giant branch stars with asteroseismic and spectroscopic ages and find general
agreement, though our age predictions are less precise (as expected).

  * •

We compare our dynamical age predictions with measured ages for open clusters
and main-sequence stars with gyrochronologcal ages and again find that our
ages agree with other independent methods, at least in relative age ranking.

  * •

We produce age predictions for the Kepler, K2, and TESS main samples and show
that the stars observed by these surveys have naturally different galactic
kinematics and intrinsic age distributions.

  * •

We release our model and the age prediction tools used in this analysis in an
open-source, user-friendly software package called zoomies (Sagear & Price-
Whelan, 2024).

The magnitude and precision of Gaia data that has become available in recent
years allows us to generate homogeneous, mass-independent relative age
predictions using only astrometric data and a potential model of the Milky
Way. This newly enables large-scale age analyses on a wide range of stellar
masses, even the lowest-mass stars. These dynamical ages can be useful for any
investigation involving stellar populations, and especially for demographic
studies of the lowest-mass exoplanet hosts and the dynamical evolution of
their planetary systems. In a future work, we plan to combine orbital
eccentricity information for planet-hosting stars (e.g. Van Eylen et al.,
2019; Sagear & Ballard, 2023) with homogeneous stellar metallicity information
(e.g. Anderson et al., 2021) to disentangle the underlying relationship
between planet occurrence, orbital dynamics, metallicity, and age.

We thank Jamie Tayar and Zachary Claytor for careful readings of this
manuscript and thoughtful suggestions to improve this work. We also thank
Christopher Lam, Quadry Chance, Natalia Guerrero, Dana Yaptangco, Nazar
Budaiev, and Soichiro Hattori for helpful discussions about this analysis.

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. This paper includes data
collected by the Kepler, K2, and TESS missions and obtained from the MAST data
archive at the Space Telescope Science Institute (STScI). The specific
observations analyzed can be accessed via https://doi.org/10.17909/T9059R
(catalog https://doi.org/10.17909/T9059R) (KIC),
https://doi.org/10.17909/T93W28 (catalog https://doi.org/10.17909/T93W28)
(EPIC), and https://doi.org/10.17909/fwdt-2x66 (catalog
https://doi.org/10.17909/fwdt-2x66) (TIC). STScI is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS5–26555. Support to MAST for these data is provided by the NASA
Office of Space Science via grant NAG5–7584 and by other grants and contracts.
This research has made use of the NASA Exoplanet Archive, which is operated by
the California Institute of Technology, under contract with the National
Aeronautics and Space Administration under the Exoplanet Exploration Program.

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##  Appendix A Open Cluster Age Comparisons

Figure 12: Age probability distributions for randomly selected open clusters
from the Dias et al. (2021) sample. 15 clusters with $0<\tau<2$ Gyr and 25
clusters with $2<\tau<14$ Gyr according to Dias et al. (2021) were randomly
selected to present a mix of young and old clusters. Each subplot shows the
age probability distributions for each star associated with each cluster, with
color denoting the membership probability according to Dias et al. (2021). The
red vertical lines and shaded regions mark the cluster ages and $1\sigma$
uncertainties (respectively) presented in Dias et al. (2021).

We present an extension of the open cluster dynamical age comparison described
in Section 5.1. We consider the open clusters listed in Dias et al. (2021)
with more than 100 listed members. Since the vast majority of open clusters in
Dias et al. (2021) have $\tau<1$ Gyr, we randomly select 15 clusters with
$0<\tau<2$ Gyr and 25 clusters with $2<\tau<14$ Gyr according to Dias et al.
(2021), with uniform probability within each bin. For each star associated
with each chosen open cluster, we calculate $\ln J_{z}$ and the corresponding
dynamical age probability distribution using the RGB-calibrated dynamical age
model. In each subplot of Figure 12, we show the probability distribution of
each star associated with each cluster (teal), along with the cluster age and
$1\sigma$ age uncertainty presented in Dias et al. (2021) with the red lines
and shaded regions, respectively.