Data-Driven Ages from Asteroseismology for FGK Dwarfs

Introduction

Stellar ages are vital for unraveling the formation history of our Galaxy, as well as stars and their planetary systems therein. The fidelity to which we can interrogate these processes is limited by stellar ages. Large stellar samples with age measurements are often derived from isochrones, which are less reliable for stars that evolve more slowly, e.g., FGK dwarfs. Isochrone-derived ages for these stars have uncertainties commensurate with the ages themselves due to the isochrones’ tightly packed tracks in this region of the Hertzsprung-Russell (HR) diagram [51]. Other techniques have advantages and shortcomings that make them more competitive for age-dating some stars over others. Gyrochronology, which infers stellar ages based on their expected rotation spin-down, is relatively powerful at measuring the ages of younger stars, but less effective for stars without spots and stars older than the spin-down stall [55, 6, 19, 12] (Lu et al. 2024). Lithium abundance has been shown to correlate with stellar age as Li is transported and burned within stars’ hot convective zones over its first few hundred Myr, but older than the lithium depletion boundary at ∼500 Myr, there is no longer enough Li with which to mark stellar age [22, 66]. One of the most effective age-dating methods is by tagging stars based on membership in co-natal populations (e.g., clusters, associations). However, this method is limited to stars that can be traced back to an open cluster, which becomes increasingly difficult on Gyr timescales as stars interact with the Galactic potential and other stars, thereby losing their birth kinematic information [20]. Thanks to Gaia DR3 [61], it is now also possible to infer dynamical ages on a population level using stellar kinematics, with dynamically hotter stars being generally older [50, 53]. However, meaningful dynamical ages remain difficult to obtain for individual stars.

Asteroseismology is another stellar age measurement technique. It is capable of measuring individual stellar ages to within 10-15% precision [31, 57, 15], and is not limited to specific age ranges, unlike other methods, e.g., gyrochronology [6, 12] (Lu et al. 2024). Asteroseismic ages are measured from internal stellar oscillations, the frequencies and modes of which map to the interior sound speed of the star and allow us to deduce many of its properties, including interior temperature and pressure, mass, and age [1]. Specifically, ages can be deduced from asteroseismic measurements in two ways. Ideally, precisely identifying the modes associated with the measured frequencies allows the small and large frequency separations ($\delta\nu$ and $\Delta\nu$, respectively) to be resolved. $\delta\nu$ is particularly sensitive to age because it probes the hydrogen mass fraction in the core as stars burn their hydrogen fuel and decrease in density, while $\Delta\nu$ maps more strongly to mass, although both separations encode information about the other parameter (for more background, we refer the reader to Figure 1.11 and Section 1 of Aerts et al. [1], as well as Valle et al. [60]).

However, $\delta\nu$ is more difficult to obtain than $\Delta\nu$, and while Kepler was able to obtain $\delta\nu$ for some main sequence stars (e.g., Aguirre et al. [2]), most asteroseismic ages of main sequence Kepler stars have come from another method of using asteroseismology to deduce ages, which is to combine stellar evolution models with two key global parameters: $\nu_{\max}$ (the frequency of maximum amplitude of the pressure mode) and $\Delta\nu$ (the large frequency separation). These parameters, derived from time-series photometry, are related via scaling relations [59, 8] to stellar radius, mass, and density, with which stellar evolution models can produce an age estimate.

Asteroseismology is most effective for stars with high signal-to-noise, high-precision light curves sampled with high cadence over a long baseline. With its high-precision photometry and high cadence over nearly four years, the NASA Kepler mission has enabled asteroseismology for tens of thousands of stars, most of which are evolved giants [23, 46, 47]. Kepler has also enabled asteroseismology for the brightest main-sequence and sub-giant stars, but because the internal oscillation frequencies of small stars are significantly more difficult to detect than those of giant stars, the sample of Kepler FGK dwarfs with asteroseismic ages is small. Serenelli et al. [54] produced ages for 415 dwarf and sub-giant FGK stars, while Aguirre et al. [2] produced ages for 66 main-sequence FGK stars. Since most exoplanet hosts stars are FGK dwarfs, expanding the sample of stars with asteroseismic measurements to the dwarf regime will be invaluable for investigating planet formation and evolution.

In 2027, ESA will launch the PLAnetary Transits and Oscillations of stars (PLATO) space telescope. PLATO will have 24 ‘normal’ cameras with a cadence more than twice as fast as that of Kepler’s short cadence (25 seconds, compared to 58.5 seconds), in addition to two ‘fast’ cameras with an even faster cadence of 2.5 seconds [48, 49]. Together with a selection function that favors stars brighter than those observed by Kepler, this will enable PLATO to measure precise solar-like oscillations for 245,000 FGK dwarf stars [40, 49, 11, 24].

We can potentially use the forthcoming PLATO data releases as training sets to infer ages for even more stars using their spectra. Specifically, the intersection of PLATO targets with APOGEE spectra will enable the training of a model on stellar spectra and asteroseismic labels that can then infer labels given other spectra. Previous studies have used data-driven methods to extend stellar age measurements to larger samples using spectroscopic information [5, 32, 18, 58] (Almanach et al. 2026). Of these, studies using The Cannon include Ness et al. [44], which used a data-driven model from The Cannon to infer stellar masses and ages for Kepler red giant stars with APOGEE spectra, and Lu et al. 2022, which used The Cannon and APOGEE spectra to relate the $\alpha$-abundances of giant stars throughout the Milky Way to their ages. In this study, we train The Cannon on APOGEE spectra and asteroseismic labels of stars from the Serenelli et al. [54] sample, and then demonstrate that the labels – including age – inferred by the model from the APOGEE spectra of other stars are both accurate and precise.

This manuscript is organized as follows. In § 2, we provide an overview of The Cannon. In § 3, we describe the sample used to train The Cannon, conduct cross-validation (3.1), and verify that the model is actually learning information about the asteroseismic parameter $\Delta\nu$ and stellar age (3.2). In § 4, we describe our results, including the table of inferred labels for our Kepler-APOGEE test set (4.1) and full APOGEE test set (4.2) and their respective uncertainties (4.3). We also compare our results to an asteroseismic Kepler gold standard sample (4.4). In § 5, we discuss how this study can be applied to upcoming PLATO asteroseismic observations of FGK dwarf stars. Finally, we conclude in § 6.

The Cannon

The Cannon [43] (Ho et al. 2017) is a data-driven framework for predicting stellar parameters from stellar spectra. In this framework, a model is first trained on a set of labeled stars with well-characterized parameters and spectra. With this trained model in hand, as long as a star has spectra, we can use the model to infer its parameters (otherwise known as “labels”) for this star. For this work, we use The Cannon 2 [14], which features faster optimization and more complex flux modeling than the original software. The Cannon is comprised of a “training phase” (during which the regression model is trained on training labels and spectra) and a “test phase” (during which the trained model takes in spectra from the test sample in order to predict labels for those stars).

Training phase

The Cannon flux model can be described as:

where $l_n$ is the vector of labels for each star, $v$ is the vectorizer, and $\theta_{\lambda}$ is the vector of coefficients at each pixel. The vectorizer can be as simple as $\sum l_n$, but for this work we find that linear terms alone are insufficient, so we choose $v$ to be a second-order polynomial expansion of $l_n$. The noise term is drawn from a Gaussian centered at zero with variance of $\sigma_{\lambda,n}^{2} + s_{\lambda}^{2}$, where $\sigma_{\lambda,n}$ is the pixel-wise, per-star flux uncertainty for each training set spectrum and $s_{\lambda}^{2}$ is the model intrinsic pixel-wise scatter (this is reported by The Cannon upon training the model as s2). $\theta_{\lambda}$ is fit such that the following log-likelihood function is optimized:

Thus, at each wavelength pixel $\lambda$, the best-fit model coefficients and intrinsic scatter are those that maximize the likelihood:

Our model employs six labels that comprise $l_n$: $T_{\mathrm{eff}}$, $\log g$, [Fe/H], [Mg/H], age, and $\Delta\nu$. We find that $\nu_{\max}$ contributes relatively little new information compared to the complexity it adds. Therefore, we omit it from our label space. Since we use a quadratic model in this work, this results in 28 features (one scalar normalization feature, six linear features, six squared features, and 15 cross-terms). For this study, in which our label space consists of only two elemental abundances, we opt not to add regularization to the model.

Test phase

After the model is trained, the optimized $\theta_{\lambda}$ and $s_{\lambda}^{2}$ are used to infer the labels $l_n$ of the test sample. The labels are predicted by taking the minimum of the negative log-likelihood functions, as follows:

$l_n$ can then be fed back into the trained model to produce an inferred flux model for each wavelength pixel $\lambda$, and each stellar spectrum $n$.

Training Sample

We construct our training set from the APOKASC catalog of FGK dwarf stars from Serenelli et al. [54], hereafter S17. The APOKASC catalog is comprised of stars observed with Kepler and analyzed by the Kepler Asteroseismology Science Consortium (KASC), as well as with the Apache Point Observatory Galactic Evolution Experiment (APOGEE) spectrograph, which provides high-resolution ($R \sim22{,}500$) spectra in the $H$-band (1.51–1.7 $\mu$m). The stellar parameters used in S17 came from the Sloan Digital Sky Survey (SDSS-IV)/APOGEE Data Release 13 (DR13; [3]), but in the era of SDSS-V/Milky Way Mapper (MWM; [13, 25, 56, 65, 28]), we use updated $T_{\mathrm{eff}}$, $\log g$, and chemical abundances from SDSS-V Data Release 19 (DR19; [41]). For our training spectra, we use stacked spectra from the APOGEE spectra provided under the Astra framework in SDSS-V/MWM.

After removing 7 stars from the S17 catalog with reported ages of -99 Gyr (denoting stars for which an age was unable to be obtained) and another 8 stars without APOGEE spectra from SDSS-V/MWM DR19, we have exactly 400 stars remaining. We then remove stars that are at the base of the red giant branch (RGB) based on the SDSS-V/MWM APOGEE Stellar Parameter and Chemical Abundances Pipeline (ASPCAP; [41]) $T_{\mathrm{eff}}$ and $\log g$, in order to better scope our model for age inference of main sequence Sun-like stars. After excluding these early RGB stars (specifically, stars with $\log g < 3.8$ and $T_{\mathrm{eff}} < 5250$ K), we have 335 stars remaining in our training sample.

We next apply continuum normalization on our training set spectra. As done in Behrnard et al. (2025), we begin by removing the gaps between the APOGEE chips at 15790–15870 Å and 16420–16490 Å. We then compute the normalized flux as:

where $\lambda_{0}$ is the starting point wavelength of the smoothing window, $\sigma_{\lambda}$ is the flux uncertainty, and the weights $w_{\lambda}$ are:

with the Gaussian smoothing window, $L$, set to 10 Å. In this manner, weights $w_{\lambda}$ are calculated for every wavelength pixel in each spectrum. We set the flux errors to be very large (aka 999) at all bad pixel values (where the flux or flux uncertainties are zero or not finite). Finally, we clip all flux uncertainty values to a minimum of 0.005. This is common among SNR $>200$/pix APOGEE targets, whose flux uncertainties are underestimated ([39, 7]).¹

Cross-validation

To validate our training set, we perform k-fold cross-validation. We choose the number of folds as $k=4$. In doing so, we split the sample into four nearly equal-sized folds (three folds of 84 stars and one fold of 83 stars), and rotate each fold once as the test set and three times as part of the training set. For each of the four test sets, we independently train one model. For each star in a given test set, its corresponding model (trained on the other three-quarters of sample) is applied to the star’s ASPCAP ($T_{\mathrm{eff}}$, $\log g$, [Fe/H], and [Mg/H]) and S17 (age and $\Delta\nu$) labels in order to produce a model spectrum. For each star, then, we can compute the $\chi^{2}$ as

where $f_{\lambda}$ is the model spectrum and $\bar{f}_{\lambda}$ is the continuum-normalized APOGEE spectrum.

From this first pass of 4-fold cross-validation, we notice high $\chi^{2}$ for several stars in our sample and manually vet their spectra. We uncover five multi-lined spectroscopic binaries: four SB2s (KIC 9025370, KIC 9693187, KIC 6145937, and KIC 9787965) and one SB3 (KIC 10610027). Additionally, we identify two stars with abnormally poor continuum normalization (KIC 7976303 and KIC 10454113). For both stars, poor skyline subtraction results in visually apparent skylines around

1524 nm. We removed these seven stars from our training set in order to ensure that the trained model does not bear the imprint of double- or triple-lines (or the missed skylines). We checked the remaining highest $\sim15\%$ $\chi^{2}$ stars for fast rotators, whose higher magnetic activity might be responsible for smearing the spectra and inflating the $\chi^{2}$, and found none.

After removing the putative SB2s and SB3, as well as the two stars with poor skyline subtraction, we are left with 328 stars in our final training sample. Our training set is provided in Table 1. We perform 4-fold cross-validation once more on this final training set and report the results in Table 2. We compare these predictions, which represent the moderately degraded capabilities of The Cannon (because the model is trained on 75% rather than 100% of the training set), to their corresponding ASPCAP and S17 labels in Figure 1. In each box in Figure 1, we note the RMS scatter between the APOKASC-derived labels and the Cannon-derived labels, as well as the intrinsic scatter among the APOKASC label. If the former is less than the latter, it means that there is a 1:1 correlation between the two label values that cannot be explained by the intrinsic scatter in the training set. We demonstrate that we stay under this intrinsic scatter for all parameters. However, we note that our age and $\Delta\nu$ predictions are poorer for older stars ($\gtrsim8$ Gyr) with larger $\Delta\nu$ values and systematically pulls them to smaller values. This is because our training set includes relatively fewer stars with these properties. We anticipate that more stars with these properties will be provided by future asteroseismology surveys (e.g., PLATO).

KIC	$T_{\mathrm{eff}}$ [K]	$\log g$ [dex]	[Fe/H] [dex]	[Mg/H] [dex]	Age [Gyr]	$\Delta\nu$ [$\mu$Hz]
1435467	6111	3.83	0.08	0.01	2.60	70.56
2010607	6102	3.57	0.06	-0.00	2.75	42.48
2450729	5825	3.90	-0.23	-0.18	6.03	61.91
2837475	6209	3.74	-0.03	-0.06	1.13	75.72
2849125	5959	3.51	0.30	0.22	2.58	41.32
2852862	6058	3.63	-0.08	-0.10	2.26	53.56
2991448	5615	3.97	-0.09	-0.04	6.88	61.73
2998253	6108	4.03	0.08	0.01	2.62	87.23
3112152	5946	3.84	0.00	-0.00	3.78	65.13
3112889	6035	3.77	-0.23	-0.24	3.63	53.71

Metric	$T_{\mathrm{eff}}$ [K]	$\log g$ [dex]	[Fe/H] [dex]	[Mg/H] [dex]	Age [Gyr]	$\Delta\nu$ [$\mu$Hz]
RMSE	$126\pm19$	$0.11\pm0.02$	$0.06\pm0.00$	$0.06\pm0.00$	$1.41\pm0.16$	$14.7\pm1.3$
$\mu$	$5889\pm14$	$3.85\pm0.01$	$-0.00\pm0.02$	$-0.02\pm0.02$	$4.29\pm0.09$	$65.6\pm0.56$
$\sigma$	$219\pm14$	$0.13\pm0.01$	$0.17\pm0.00$	$0.16\pm0.01$	$1.80\pm0.01$	$14.0\pm1.0$

Since this study is largely motivated by the need for stellar ages at scale, we compute the relative age error from four-fold cross-validation for each star in our training set. We report the median relative fractional age error to be 20.0%. That is, for the typical star in our training set, its predicted age from The Cannon was 20% away from the age measured by APOKASC.

Age information validation

We have demonstrated that The Cannon can predict labels (including ages) of FGK dwarfs in a self-consistent manner. However, we must also show that these age predictions are not merely driven by the relation between Fe and Mg abundances and age, independent of the asteroseismic parameter, $\Delta\nu$. On average, younger stars tend to be $\alpha$-poor, while older stars tend to be more $\alpha$-rich ([26] (Lu et al. 2026)). Therefore, our first test is to isolate only the young, $\alpha$-rich stars from our training sample and check that our model is learning information from the input spectra, rather than simply

following the expected broad trend of increasing $\alpha$ enhancement with increasing stellar age. We define young, $\alpha$-rich stars as age $\le6$ Gyr and [Mg/Fe] $\ge0.1$ [37, 67, 36], where [Mg/Fe] = [Mg/H] − [Fe/H] (since chemical abundances in dex units correspond to log-space). In Figure 1, we draw orange circles around these young, $\alpha$-rich stars and demonstrate that they follow the same 1:1 relation as the rest of the training set under 4-fold cross-validation. As with the broader cross-validation test, we show that even for this subset of our training sample, the intrinsic scatter for every APOKASC label is larger than the scatter of its RMS with respect to The Cannon’s predicted labels.

We additionally apply a test commonly used in Galactic archaeology studies, in which the training sample is divided into six mono-abundance bins and then cross-validated. This is done to verify that our ability to predict the stellar age is not simply driven by learning the abundance trends encoded in the labels. If The Cannon is truly learning no age information outside of its dependence on iron and $\alpha$-abundances, then some bins will exhibit uniformly random scatter rather than a correlation between the Cannon and APOKASC parameters. As shown in Figure 2, however, this is not the case. We quantify this as done in Figure 1 by comparing the RMS scatter between the Cannon- and APOKASC-derived age for each mono-abundance bin to the intrinsic scatter for the corresponding APOKASC-derived parameter for that bin. For all mono-abundance bins, the RMS scatter is smaller than the intrinsic scatter, confirming that we are indeed learning asteroseismic information. As in 4-fold cross-validation, The Cannon under-predicts ages of older stars. This is most apparent for the highest [Mg/Fe] mono-abundance bins, although this is likely due to higher [Mg/Fe] bins containing more older ($\ge6$ Gyr) stars.

Results

Kepler-APOGEE test sample

Our test sample is comprised of all Kepler stars with APOGEE spectra, culled to resemble the training set in ASPCAP $T_{\rm eff}$, $\log g$, [Fe/H], and [Mg/H] space. The initial test set of Kepler stars with APOGEE spectra contains 6131 stars. We then remove stars that lie beyond the minimum or maximum $T_{\rm eff}$, $\log g$, [Fe/H], or [Mg/H] of our training sample, resulting in 3812 stars. We restrict our test sample to be contained within the $T_{\rm eff}$, $\log g$, [Fe/H], and [Mg/H] parameter space spanned by our training set, i.e., the minimum and maximum limits:

1. 5176 $\le T_{\rm eff} \le$ 6395 K

2. 3.46 $\le\log g \le$ 4.41 dex

3. −0.79 $\le$ [Fe/H] $\le$ 0.51 dex

4. −0.72 $\le$ [Mg/H] $\le$ 0.46 dex

We make an additional cut to the test set based on the model spectral fit $\chi^2$, which is calculated for each test star by generating a model spectrum using the inferred labels and comparing it to the input continuum-normalized spectrum. We omit stars where the $\chi^2$ is greater than 100,000 [7, 10], which corresponds to a reduced $\chi^2$ threshold of approximately 13.5 (with 7409 pixels and 6 fitted parameters, there are 7403 degrees of freedom). This removes two more stars from our sample, KIC 3341831 and KIC 6467256, leaving a final catalog of 3810 stars. Table 3 reports the label value (and uncertainties, see next section) from The Cannon for each star in the final test sample, along with the reduced $\chi^2$. In Figure 3, we show the test sample (circles) overlaid by the training sample (squares) in ASPCAP $T_{\rm eff}$ and $\log g$ space, color-coded by stellar ages as predicted by The Cannon.

KIC	SDSS ID	$T_{\mathrm{eff}}$ [K]	$\log g$ [dex]	[Fe/H] [dex]	[Mg/H] [dex]	Age [Gyr]	$\Delta\nu$ [$\mu$Hz]	$\chi_{\mathrm{red}}^{2}$
12689153	67743667	5776 $\pm$ 31	4.17 $\pm$ 0.04	-0.04 $\pm$ 0.03	-0.03 $\pm$ 0.03	5.60 $\pm$ 0.42	116.2 $\pm$ 0.8	1.45
12735580	67800894	5894 $\pm$ 31	3.90 $\pm$ 0.05	0.17 $\pm$ 0.03	0.08 $\pm$ 0.03	3.90 $\pm$ 0.42	70.2 $\pm$ 0.8	2.15
12735793	67800901	5843 $\pm$ 31	4.09 $\pm$ 0.04	-0.10 $\pm$ 0.03	-0.10 $\pm$ 0.03	5.47 $\pm$ 0.42	99.7 $\pm$ 0.8	1.59
12736465	67800824	5971 $\pm$ 31	3.90 $\pm$ 0.06	-0.03 $\pm$ 0.03	-0.11 $\pm$ 0.03	3.41 $\pm$ 0.42	71.7 $\pm$ 0.8	1.21
12737160	67800533	5760 $\pm$ 31	4.02 $\pm$ 0.04	0.17 $\pm$ 0.03	0.10 $\pm$ 0.03	5.09 $\pm$ 0.42	71.6 $\pm$ 0.8	1.96
12783625	67800923	6035 $\pm$ 31	3.98 $\pm$ 0.05	0.08 $\pm$ 0.03	-0.02 $\pm$ 0.03	2.91 $\pm$ 0.42	80.0 $\pm$ 0.8	1.72
12784610	67800812	5684 $\pm$ 31	4.08 $\pm$ 0.04	-0.04 $\pm$ 0.03	-0.04 $\pm$ 0.03	6.34 $\pm$ 0.42	105.0 $\pm$ 0.8	2.05
12833827	67800943	5309 $\pm$ 31	3.89 $\pm$ 0.04	-0.08 $\pm$ 0.03	0.00 $\pm$ 0.03	7.99 $\pm$ 0.42	54.9 $\pm$ 0.8	2.02
12834387	67800878	6251 $\pm$ 31	3.92 $\pm$ 0.05	-0.00 $\pm$ 0.03	-0.05 $\pm$ 0.03	1.20 $\pm$ 0.42	86.1 $\pm$ 0.8	1.82
12834614	67801064	6138 $\pm$ 31	3.62 $\pm$ 0.05	-0.01 $\pm$ 0.03	-0.06 $\pm$ 0.03	1.70 $\pm$ 0.42	50.4 $\pm$ 0.8	1.40

Full APOGEE test sample

The test sample can be further expanded to include all stars visited by APOGEE within our training label space, regardless of membership in the Kepler Input Catalog (KIC). After removing APOGEE stars with SNR$\le$200 and whose inferred spectra from The Cannon have a $\chi^2\ge$100,000, we are left with an expanded test set of 48,828. These stars’ predicted $T_{\rm eff}$, $\log g$, [Fe/H], [Mg/H], age, and $\Delta\nu$, along with the reduced $\chi^2$ of their inferred spectra, are shown in Table 4.

Gaia DR3 Source ID	SDSS ID	$T_{\rm eff}$ [K]	$\log g$ [dex]	[Fe/H] [dex]	[Mg/H] [dex]	Age [Gyr]	$\Delta\nu$ [$\mu$Hz]	$\chi^2_{\rm red}$
420868586978944768	117524371	5943 $\pm$ 31	4.04 $\pm$ 0.04	-0.02 $\pm$ 0.03	-0.00 $\pm$ 0.03	3.77 $\pm$ 0.42	85.2 $\pm$ 0.8	2.06
421036129367476352	117526412	6016 $\pm$ 31	3.87 $\pm$ 0.04	-0.33 $\pm$ 0.03	-0.34 $\pm$ 0.03	4.69 $\pm$ 0.42	61.4 $\pm$ 0.8	2.31
421238332125586816	117528719	6027 $\pm$ 31	3.79 $\pm$ 0.05	0.06 $\pm$ 0.03	-0.02 $\pm$ 0.03	2.93 $\pm$ 0.42	58.1 $\pm$ 0.8	1.77
4111233547185089536	129044157	5584 $\pm$ 31	4.13 $\pm$ 0.04	-0.13 $\pm$ 0.03	0.04 $\pm$ 0.03	8.60 $\pm$ 0.42	78.9 $\pm$ 0.8	2.67
5337389398147162112	129049943	6255 $\pm$ 31	3.98 $\pm$ 0.04	-0.15 $\pm$ 0.03	-0.25 $\pm$ 0.03	2.83 $\pm$ 0.42	84.7 $\pm$ 0.8	3.37
2075118195014471680	129051766	6102 $\pm$ 31	3.81 $\pm$ 0.05	-0.07 $\pm$ 0.03	-0.15 $\pm$ 0.03	3.20 $\pm$ 0.42	69.6 $\pm$ 0.8	1.83
2101443973710952448	129055079	5827 $\pm$ 31	3.81 $\pm$ 0.04	-0.13 $\pm$ 0.03	-0.19 $\pm$ 0.03	3.47 $\pm$ 0.42	49.5 $\pm$ 0.8	2.23
120903986512700160	129058214	5769 $\pm$ 31	3.85 $\pm$ 0.05	0.03 $\pm$ 0.03	0.02 $\pm$ 0.03	5.67 $\pm$ 0.42	64.5 $\pm$ 0.8	1.53
6036578840946969600	129058425	6387 $\pm$ 31	3.99 $\pm$ 0.04	-0.45 $\pm$ 0.03	-0.53 $\pm$ 0.03	1.28 $\pm$ 0.42	66.5 $\pm$ 0.8	3.11
120953464536688128	129059538	6064 $\pm$ 31	4.00 $\pm$ 0.05	-0.24 $\pm$ 0.03	-0.23 $\pm$ 0.03	4.73 $\pm$ 0.42	78.1 $\pm$ 0.8	1.48

Predicted label uncertainties

In order to quantify the uncertainties on our inferred test set labels, we employ the procedure used in [7], in which the inter-observation scatter introduced by different APOGEE observations is estimated for 500 stars with multiple APOGEE visits that lie within the training sample parameter space. Together with the intrinsic model scatter from The Cannon, this observational uncertainty amounts to the overall uncertainty per parameter per star in our test sample. While the model uncertainties for each of the two spectra $A$ and $B$, $\sigma_A$ and $\sigma_B$, are readily provided by the covariance matrices from The Cannon, we must fit the observational uncertainty using the following equation for each parameter:

where $l_A$ and $l_B$ are the inferred label for each of the two spectra $A$ and $B$, $\sigma_A$ and $\sigma_B$ are the scatter from The Cannon model estimated during the test phase, and $\sigma_{\mathrm{inflate}}$ is the inter-observation scatter that we are trying to fit, such that $Z_{A,B}$ for the 500 stars follows a unit normal distribution. We therefore fit for a $\sigma_{\mathrm{inflate}}$ for each label, and report the test sample uncertainties in Table 3 as the quadrature sum of $\sigma_{\mathrm{inflate}}$ and the scatter for each star from the model covariance matrix (analogous to $\sigma_A$ and $\sigma_B$). As in Behmard et al. [7], the global $\sigma_{\mathrm{inflate}}$ contributes the bulk of the total error budget, compared to the Cannon model error, which varies from star to star.

One implication of a nearly constant age uncertainty is that the fractional age uncertainty varies almost entirely directly with age. While the majority (72%) of stars in our test sample have age predictions under 10% fractional age uncertainty, stars younger than 4.2 Gyr will by construction have fractional age uncertainties larger than 10% (and stars younger than 2.1 Gyr will have fractional age uncertainties larger than 20%, and so on). However, we should stress that a better measure of this method’s efficacy at age inference is the relative fractional age error from cross-validation compared to the APOKASC ages, which was reported to be 20% in § 3.1.

Kepler LEGACY sample

So far, we have only used one sample of independently-derived asteroseismic parameters and ages for Kepler stars (Serenelli et al. [54]). Aguirre et al. [2] presented a smaller but more precisely measured Kepler asteroseismic sample called LEGACY. With only 66 stars, however, it is better suited for our purposes as a gold standard comparison sample rather than a training sample. After cross-matching the LEGACY sample with our test sample, we plot the ages of the remaining 40 stars in Figure 4 and show that, once again, the RMS scatter is smaller than the intrinsic scatter among the LEGACY ages. As previously shown in Figure 1 and Figure 2, the Cannon tends to under-predict ages of older stars in the LEGACY sample. This can be explained by a lack of training data in that stellar age regime and can be improved by a larger training sample from future surveys such as PLATO, which will observe older stars in the thick disk (Boettner et al. [11]; Rauer et al. [49]). Nevertheless, in general, together with satisfactory 4-fold cross-validation and mono-abundance test results, this demonstrates that The Cannon is robust at inferring the global asteroseismology parameter $\Delta\nu$ and stellar ages for Kepler FGK dwarfs younger than $\sim6$ Gyr.

Looking Ahead to the PLATO Data Release

Comprising 26 cameras sensitive to $13 \ge m_V \ge4$, with a field of view of $49^{\prime\prime}\times49^{\prime\prime}$ (4510x4510 pixels), ESA’s PLATO will conduct a large asteroseismology survey of potential planet hosts in the Solar neighborhood and is scheduled to launch in January 2027 (Rauer et al. [49]). With better asteroseismic parameters at scale comes better stellar ages at scale (Goupil et al. [24]), potentially enabling exoplanet demographic studies that rely on stellar ages to more exquisite precision. PLATO stands to provide a larger training sample with broader parameter space coverage than the S17 sample used in this work, enabling more precise stellar age inference for a larger label space of stars.

PLATO stars as an enhanced training set

The all-sky PLATO Input Catalog (asPIC) (Montalto et al. [40]) is a catalog of all the stars that will be targeted by PLATO’s southern and (if applicable) northern fields. It contains 2,378,177 FGK dwarfs, of which 64,105 also have APOGEE spectra through Data Release 19 of SDSS-V/MWM. Of these stars, we keep the 24,642 stars with APOGEE SNR $\ge200$, and of those stars, 23,772 lie within $3500 \le T_{\mathrm{eff}} \le7000$ K and $3.0 \le\log g \le4.8$ dex, which are pictured in Figure 5. In chemical abundance space, these stars span $-2.49 \le[\mathrm{Fe}/\mathrm{H}] \le1.01$ dex and $-2.43 \le[\mathrm{Mg}/\mathrm{H}] \le1.81$ dex. This sample would potentially be a new training set for The Cannon that is both two orders of magnitude larger than that used in this study and more comprehensive in its label space coverage (as shown in Figure 5, we will be able to probe cooler main sequence stars thanks to PLATO’s brighter targets). One could then construct a test sample by taking the subset of stars with APOGEE DR19 spectra that lie within this expanded label space, resulting in 437,767 stars for which asteroseismic-quality ages are possible, regardless of whether they will be observed by PLATO.

PLATO planets

Stellar ages are vital for probing how exoplanet populations evolve. For example, they are needed for understanding planet occurrence rates (e.g., Sayeed et al. [52]; Lam et al. [30]), planet architecture sculpting (e.g., Lam & Ballard [29]; Chen et al. [16]), and how the planet radius gap changes over time (e.g., VanEylen et al. [63]; Berger et al. [9]; Weeks et al. [64]; Chen et al. [17]). The degree of precision to which stellar ages are measured in turn affects the degree to which models of exoplanet population evolution are discernible. For example, Lam & Ballard [29] showed that models of

Gyr-timescale dynamical sculpting become discernible at ages measured to better than 10% precision.

PLATO will provide uniformly- and well-measured asteroseismic ages for thousands to tens of thousands of planet hosts [49, 11, 38]. Moreover, PLATO’s overlap with the Kepler [42] and TESS [21] fields will provide precise age measurements for a modest number of known planets. Stellar spectrum-driven label transfer of asteroseismic ages can leverage the confluence of PLATO, Kepler, and TESS to expand the sample of planets with high quality ages to demographically relevant quantities.

From our test set, there are 1092 Kepler confirmed planets² for which we readily have Cannon ages, compared to 77 Kepler planets in the PLATO input catalog, 16 Kepler planets in the S17 sample, and 5 Kepler planets in the LEGACY sample. In other words, even if PLATO were to observe the northern field for the third and fourth years of its mission, obtaining asteroseismic-quality ages for Kepler planets hosted by main sequence stars at quantities suitable for age-based demographics is not possible without the help of tools like The Cannon.

The proof of concept demonstrated in this study will also synergize with PLATO. From the expanded label space of the PLATO training sample (Figure 5), there will be an additional 720 Kepler confirmed planets with APOGEE spectra for which we will be able to determine asteroseismic-quality ages. Additionally, while there are very few Transiting Exoplanet Survey Satellite (TESS) planet hosts with asteroseismic ages (e.g., [45]), of the 928 confirmed TESS planets³, there will be 237 such planets with host stars in the expanded APOGEE label space after training on PLATO stars. In comparison, there are 134 confirmed TESS planets in the PLATO Input Catalog (including in the northern sky, which is not yet guaranteed to be observed) and 101 planets with APOGEE spectra in the current label space (without training on PLATO stars) for which we can readily obtain high-quality age inferences.

Conclusion

In this study, we trained The Cannon on the APOGEE DR19 spectra of 328 FGK dwarfs from the Serenelli et al. [54] sample of solar-like stars observed by Kepler with asteroseismology measurements, along with key spectroscopic ($T_{\rm eff}$, $\log g$, [Fe/H], and [Mg/H]; all from APOGEE DR19) and asteroseismic (age and $\Delta\nu$) parameters (which we refer to as “labels” in our data-driven model). We validated our model using a variety of tests, including 4-fold cross-validation; isolating young, $\alpha$-rich stars; and taking mono-abundance slices of the training set. We then apply our model to APOGEE DR19 spectra and predict stellar ages for 48,828 FGK dwarfs in our training label space (3810 of which are also in the Kepler Input Catalog). We compare our inferred ages with those of an independently measured asteroseismic age sample, Kepler LEGACY ([2]), and found that the stellar ages of the cross-match of these two samples are in good agreement. This is one of the first studies to extend stellar ages to larger samples for FGK dwarfs via data-driven techniques (other studies, which leveraged gyrochronological ages, include Sayyed et al. 2025 and [62]).

The method demonstrated here will have dramatic utility with the upcoming PLATO data release ([48, 40, 49]). There are 64,105 stars in the asPIC–APOGEE DR19 cross-match, which occupy a larger label space than our current training set (notably, with better coverage of cooler stars), and 439,005 stars with APOGEE DR19 spectra that lie within this expanded label space. Within this potential test set are 1812 Kepler confirmed planets and 165 TESS candidate planets. We will be able to infer ages for these planet host stars once we train The Cannon on the PLATO data release. This will enable population-level studies of planetary system changes over Gyr timescales ([11]), thus robustly extending planet demographics studies along the age axis.

Acknowledgments

We thank Adrian Price-Whelan, Andy Casey, Catherine Manea, Luke Bouma, Quadry Chance, and the CCA Astronomical Data, CCA Spectroscopy, and CCA Nearby Universe Groups for useful discussions that have helped inform and enrich this work. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. 1842473.

Funding for the Sloan Digital Sky Survey V has been provided by the Alfred P. Sloan Foundation, the Heising-Simons Foundation, the National Science Foundation, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. SDSS telescopes are located at Apache Point Observatory, funded by the Astrophysical Research Consor tium and operated by New Mexico State University, and at Las Campanas Observatory, operated by the Carnegie Institution for Science. The SDSS web site is www.sdss.org.

SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration, including the Carnegie Institution for Science, Chilean National Time Allocation Committee (CNTAC) ratified researchers, Caltech, the Gotham Participation Group, Harvard University, Heidelberg University, The Flatiron Institute, The Johns Hopkins University, L’Ecole polytechnique fédérale de Lausanne (EPFL), Leibniz-Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Extraterrestrische Physik (MPE), Nanjing University, National Astronomical Observatories of China (NAOC), New Mexico State University, The Ohio State University, Pennsylvania State University, Smithsonian Astrophysical Observatory, Space Telescope Science Institute (STScI), the Stellar Astrophysics Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Illinois at Urbana-Champaign, University of Toronto, University of Utah, University of Virginia, Yale University, and Yunnan University.

We acknowledge that IPAC/Caltech resides on the traditional, ancestral and unceded territory of the Gabrielino/Tongva peoples, the original caretakers of this land. We respectfully recognize the Gabrielino/Tongva peoples who still reside in Tovaangar (the Los Angeles basin and South Channel Islands). We also acknowledge that the main campus of the University of Florida is located on the ancestral territory of the Potano and of the Seminole peoples. The Potano, of Timucua affiliation, lived here in the Alachua region from before European arrival until the destruction of their towns in the early 1700s. The Seminole, also known as the Alachua Seminole, established towns here shortly after but were forced from the land as a result of a series of wars with the United States known as the Seminole Wars. We, the authors, acknowledge our obligation to honor the past, present, and future Native residents and cultures of California and Florida. Finally, we acknowledge the land politically designated as New York City to be the homeland of the Lenape (Lenapehoking) who were violently displaced as a result of European settler colonialism over the course of 400 years. The Lenape are a diasporic people that remain closely connected with this land and are its rightful stewards. We also recognize that New York City has one of the largest urban Native American / Indigenous populations in the United States.