Gyro-Kinematic Ages for around 30,000 Kepler Stars
Yuxi, Ruth Angus, Jason L. Curtis, Trevor J. David, Rocio Kiman
DDraft version February 4, 2021
Typeset using L A TEX twocolumn style in AASTeX63
Gyro-Kinematic Ages for around 30,000 Kepler Stars
Yuxi(Lucy) Lu ,
1, 2
Ruth Angus ,
2, 3, 1
Jason L. Curtis , Trevor J. David ,
2, 3 and Rocio Kiman
2, 4, 5 Department of Astronomy, Columbia University, 550 West 120 th Street, New York, NY, USA American Museum of Natural History, Central Park West, Manhattan, NY, USA Center for Computational Astrophysics, Flatiron Institute, 162 5 th Avenue, Manhattan, NY, USA Department of Physics, Graduate Center, City University of New York, 365 5th Ave, New York, NY 10016, USA Department of Physics and Astronomy, Hunter College, City University of New York, 695 Park Avenue, New York, NY 10065, USA
Submitted to
The Astronomical Journal
ABSTRACTEstimating stellar ages is important for advancing our understanding of stellar and exoplanet evo-lution and investigating the history of the Milky Way. However, ages for low-mass stars are hard toinfer as they evolve slowly on the main sequence. In addition, empirical dating methods are difficult tocalibrate for low-mass stars as they are faint. In this work, we calculate ages for Kepler F, G, and cru-cially K and M dwarfs, using their rotation and kinematic properties. We apply the simple assumptionthat the velocity dispersion of stars increases over time and adopt an age–velocity–dispersion relation(AVR) to estimate average stellar ages for groupings of coeval stars. We calculate the vertical velocitydispersion of stars in bins of absolute magnitude, temperature, rotation period, and Rossby numberand then convert velocity dispersion to kinematic age via an AVR. Using this method, we estimate gyro-kinematic ages for 29,949 Kepler stars with measured rotation periods. We are able to estimateages for clusters and asteroseismic stars with an RMS of 1.22 Gyr and 0.26 Gyr respectively. Withour
Astraea machine learning algorithm, which predicts rotation periods, we suggest a new selectioncriterion (a weight of 0.15) to increase the size of the McQuillan et al. (2014) catalog of Kepler rotationperiods by up to 25%. Using predicted rotation periods, we estimated gyro-kinematic ages for starswithout measured rotation periods and found promising results by comparing 12 detailed age–elementabundance trends with literature values.
Keywords:
Stellar ages, Stellar kinematics, Stellar rotation INTRODUCTIONThe age of a star is one of its most important, yet dif-ficult to determine, quantities. Stellar ages are useful inmany fields and at many geometric scales. On a smallscale, the age of exoplanets can be inferred from theirhost stars, and can be used to study how the propertiesof exoplanets change over time (e.g., David et al. 2020).On the galactic scale, the ages of stars can be used tounderstand how the Milky Way formed and is evolving(Spina et al. 2018; Bedell et al. 2018; Ness et al. 2019).However, age is not a directly measurable physical quan-tity, but rather an estimation of a star’s evolutionary
Corresponding author: Yuxi (Lucy) [email protected] state. The main source of uncertainty comes from thefact that the observable features of stars—their lumi-nosities and temperatures—change very slowly while onthe main sequence. This is particularly true of low-mass K and M dwarfs. As a result, even when measuredprecisely, these observables cannot tightly constrain theages of most main-sequence stars. In general, most avail-able dating methods cannot estimate ages for M dwarfs(for a detailed review of these methods and their ap-plication areas, see Soderblom 2010). These low-massstars are faint, which makes observations hard, and theyevolve extremely slowly on the main sequence. In addi-tion, stellar evolution models are often poorly calibratedfor these stars. The M dwarfs with the most accurateand precise age measurements are those in open clusters,or with a binary companion that can be independently a r X i v : . [ a s t r o - ph . S R ] F e b Lu et al. dated. However, this only applies to a small number ofstars, and getting precise and accurate ages for old fieldM dwarfs remains challenging. .Isochrone fitting is currently the most productivemethod to infer ages for individual field stars (e.g., Nord-str¨om et al. 2004; Buder et al. 2019; Berger et al. 2020).Although it works well on open clusters that contain apopulation of stars of the same age, this method hastrouble estimating precise ages for individual stars. Thedensity of the isochrones around the zero-age main se-quence is extremely high as stars evolve slowly on themain sequence, so this degeneracy of the model pre-vents isochrone fitting from estimating accurate and pre-cise ages for stars that have not yet gone through ∼ gy-rochronology (Barnes 2003, 2007). Stars spin down overtime due to the loss of angular momentum from magne-tized winds (e.g., Kawaler 1988; Weber & Davis 1967;Schatzman 1962). As a result, the rotation period ofa star can be used to probe its age (Skumanich 1972).Empirical gyrochronology makes the simple assumptionthat the rotation period and age of a star are correlatedand it requires nearly no assumptions about the physicsinvolved. One can then estimate the age of a star aslong as it has a measured rotation period, and temper-ature, mass, or color. Alternatively, there are physics-based semi-empirical methods (e.g., Matt et al. 2015;van Saders & Pinsonneault 2013). Such an approachhas been used to age-date cool dwarf stars with medianstatistical uncertainties on the order of 10% (Claytoret al. 2020), although large systematic uncertainties per-sist in all gyrochronology models (e.g., Curtis et al. 2019,2020). All empirical and semi-empirical methods needto be calibrated with data for benchmark stars. Suchbenchmarks include members of open clusters, individ-ual stars characterized with asteroseismology or inter- Some progress for a small amount of M dwarf with high resolutionspectroscopic data has been made using chemo-kinematic ages(Veyette & Muirhead 2018), where they estimated M dwarf agesbetween 4-9 Gyr with uncertainties between 2-3.5 Gyr. ferometry, and binary systems. Unfortunately, the gy-rochronology relations remain uncalibrated for M dwarfsdue to the difficulties in getting ages for suitable bench-marks.Huge efforts and progress have been made to cal-ibrate an empirical gyrochronology relation for FGKstars (e.g., Barnes 2007; Mamajek & Hillenbrand 2008;Meibom et al. 2009; Barnes 2010; Garc´ıa et al. 2014;Matt et al. 2015; Angus et al. 2015; van Saders et al.2016; Curtis et al. 2020). Large surveys such as Kepler(Borucki et al. 2010), Gaia (Gaia Collaboration et al.2016, 2018) and TESS (Ricker et al. 2015) are expandingthe calibration sample to an extraordinary level; how-ever, calibrating gyrochronology for M dwarfs with thesesamples will remain challenging for the following rea-sons: • Open clusters are generally young because theydissolve in the Milky Way on a timescale of ∼ Therefore, open clusters cannot yet provide theprecise ages and rotation periods for old M dwarfsneeded to fully calibrate gyrochronology (at leastnot with the current generation of telescopes). • Asteroseismic stars with measurable asteroseismicsignals are generally massive and/or evolved. Itis extremely hard to detect asteroseismic signalsin low-mass stars that are on the main sequencemostly due to the low amplitudes (e.g., Rodr´ıguezet al. 2016). While asteroseismology can be usefulfor calibrating gyrochronology for F and G dwarfsand subgiants, it cannot yet provide precise agesfor M dwarfs. • Binary systems where the age of one of the stars isdetermined by other methods can be useful bench-marks. For example, binaries where one star canbe age-dated with asteroseismology or white dwarfcooling show promise for calibrating gyrochronol-ogy. However, such systems are relatively rare. Although Ag¨ueros et al. (2018) did contribute a few early Mdwarfs in the 1.4 Gyr NGC 752 cluster, the mid-to-late M dwarfsremained out of reach. yro-Kinematic Ages for Kepler Stars hot in the first place (Bird et al. 2013). Itis still debatable which theory is correct but it is possi-ble that both of these mechanisms play significant rolesin shaping the kinematic of the stars (e.g., Binks et al.2020). There is a long history of studying the kinematicproperties of stars in the solar neighborhood; for ex-ample, Str¨omberg (1946) analyzed the velocity of 444stars within 20 pc of the Sun and found stars with high-est galactic latitude have the highest velocity dispersion.Further observations (e.g., Nordstr¨om et al. 2004; Holm-berg et al. 2007, 2009; Aumer & Binney 2009; Yu &Liu 2018; Ting & Rix 2019) confirmed older stars ex-hibit higher velocity dispersions, especially in the verti-cal direction relative to the galactic plane. This relationconnecting the age and velocity dispersion is called theage–velocity dispersion relation (AVR).We can group stars with similar stellar properties(e.g., temperature and rotation period, which we assumemeans they also have similar ages), and calculate the ve-locity dispersion within each group. Applying an AVR,we can then estimate the kinematic age of a group ofstars based on its vertical velocity dispersion. We referto this approach as gyro-kinematic age-dating.One of the advantages of using kinematics to deter-mine ages for stars is the fact that the underlying physicsis simple: kinematic ages are independent of any stellarphysics and only rely on the simple assumption that thevertical velocity dispersion of stars increase over time. Aspecific AVR does have to be adopted, but Martig et al.(2014) suggested the disk heating mechanism would re-sult in a simple power-law relation for the AVRs so theimpact of this model choice and any bias that it intro-duces is simple and relatively easy to unpack. As a re-sult, kinematic ages are useful for calibrating other em-pirical age-dating methods and providing insights intostellar physics.However, there are caveats to using this method to cal-ibrate empirical relations like gyrochronology. Firstly, kinematic ages are averages for a population of stars.Instead of estimating the true individual ages, we areinferring the expected ages based on the kinematic prop-erties of likely-coeval stars. As a result, kinematic agescannot be calculated for individual stars this way. More-over, other sample selection biases could also affect theages. For example, the orientation of the Kepler fieldwould favor stars closer to the galactic plane since Ke-pler observed nearby stars at a low galactic latitude.This could potentially introduce bias in our age measure-ment. Despite this limitation, kinematic ages can stillbe used to calibrate gyrochronology (Angus et al. 2020)and provide another perspective to study the physicsdriving stellar evolution.Here, we present a catalog of gyro-kinematic ages for29,949 Kepler stars with rotation periods measured byMcQuillan et al. (2014), Garc´ıa et al. (2014), and San-tos et al. (2019). The column description for the gyro-kinematic ages catalog with 29,949 stars with measuredrotation periods is shown in table 1.We also investigate the potential for estimating agesfor Kepler stars with predicted rotation periods using
Astraea (Lu et al. 2020). Astraea is a tool to pre-dict rotation periods from stellar properties such as lu-minosity, temperature and R var . With these predictedperiods, we then estimate gyro-kinematic ages and cal-culate the individual age–element abundance trends forsolar twins.In Section 2, we describe the data and method used toestimate gyro-kinematic ages. In Section 3, we comparethe gyro-kinematic ages with isochrone ages from Bergeret al. (2020), M dwarfs in white dwarf binary systems(Kiman et al. 2020), asteroseismic ages (Silva Aguirreet al. 2017), and open cluster ages (Curtis et al. 2020).In Section 4.1, we discuss the limitations and discrepan-cies between gyro-kinematic ages and ages estimated byother methods. We also investigate the age–abundancetrends found for solar twins in Section 4.2, using agesfrom predicted periods and abundances from APOGEEDR16 (Majewski et al. 2017; Ahumada et al. 2020), andwe compare our results to those of Bedell et al. (2018).We conclude in Section 5. DATA & METHODS2.1.
Data
To construct the gyro-kinematic age catalog, westarted with the rotation period catalog from McQuillanet al. (2014) with 34,030 measured rotation periods andadded an extra 4,637 stars from Garc´ıa et al. (2014) and The code is available at https://github.com/lyx12311/Astraea
Lu et al.
Table 1.
Catalog description of the 29,949 gyro-kinematic ages with measured rotation periods.Column Unit Description kepid
Kepler ID
Prot days rotation period
Prot err days error on rotation period source id
Gaia DR2 Source ID ra deg right ascension from Gaia DR2 ra error deg error on right ascension from Gaia DR2 dec deg declination from Gaia dec error deg error on declination from Gaia DR2 all vz km s − vertical velocity from radial velocity or Angus et al. (in prep) vz err all km s − error on vertical velocities vel dis km s − velocity dispersion vel dis err km s − uncertainty on velocity dispersion due to uncertainties on stellar parameters kin age Gyr gyro-kinematic ages kin age err
Gyr error on gyro-kinematic ages combining error from AVR fits and unvcertainties on the stellar parameters Ro Rossby number teff
K temperature abs G mag absolute magnitude from Gaia DR2
Note —This table is published in its entirety in a machine-readable format in the online journal.
Santos et al. (2019), which, combined, contain 38,667stars. We then accessed their Gaia data from the pub-licly available
Kepler – Gaia
DR2 cross-matched catalogproduced with a 1 (cid:48)(cid:48) search radius. We calculated effective temperatures ( T eff ) and abso-lute Gaia G -band magnitudes ( M G ) from these data.First, we accounted for reddening and extinction frominterstellar dust for each star using the Bayestar dustmap implemented in the dustmaps Python package(Green 2018), and astropy (Astropy Collaborationet al. 2013; Price-Whelan et al. 2018). We then usedGaia DR2 photometric color, ( G BP − G RP ) , to estimateeffective temperatures for the stars in our sample, usingthe calibrated relation in Curtis et al. (2020). We calcu-lated the Rossby number ( R o ) using convective overturntime ( τ ) defined in equation 11 in Wright et al. (2011)and R o = P rot / τ .We cross-matched our rotation sample with the spec-troscopic catalog produced by the Large Sky Area Multi-Object Fibre Spectroscopic Telescope (LAMOST; Cuiet al. 2012a) to obtain radial velocity (RV) measure-ments. 14,328 stars in our sample had RVs from eitherLAMOST DR5 Cui et al. (2012b); Xiang et al. (2019)or Gaia DR2. For stars with RVs, we calculated thevertical velocity, v z , using astropy (Astropy Collabora-tion et al. 2013; Price-Whelan et al. 2018). However, for Available at gaia-kepler.fun stars without RV measurements, we inferred their ver-tical velocities from their Gaia proper motions and 3Dpositions. Because stars in the Kepler field are locatedat low Galactic latitudes, their vertical velocities can beinferred precisely from their proper motions with a typ-ical precision of around 10% (Angus et al, in prep). Weexcluded 5,379 stars hotter than 8000 K and stars with | v z | >
100 km s − . This left us with 29,949 stars withmeasured rotation periods and vertical velocities.In addition to the primary photometric, spectroscopic,and kinematic data from Gaia, LAMOST, and Kepler,we also used multiple age catalogs in order to test ourresults. We used the isochrone age catalog from Bergeret al. (2020), the Kepler LEGACY asteroseismic age cat-alog from Silva Aguirre et al. (2017), the open clustercatalog from Curtis et al. (2020), and white dwarf cool-ing ages for two white dwarf–M dwarf binaries from Ki-man et al. (2020), calculated using wdwarfdate . Indi-vidual element abundances from APOGEE DR16 (Ma-jewski et al. 2017; Ahumada et al. 2020) were also usedto compare the individual element abundance trendswith stellar age found for solar twins (Bedell et al. 2018).2.2.
Methods
To obtain the gyro-kinematic ages for these 29,949stars, we calculated the vertical velocity dispersion ( σ v z )for stars in bins of M G , T eff , rotation period ( P rot ), and Available at https://github.com/rkiman/wdwarfdate yro-Kinematic Ages for Kepler Stars R o ), which we refer to as MTPR fromnow on. For each star, we calculated σ v z by takingthe median absolute deviation (MAD) of vertical ve-locities of a group of stars that are close to that starin MTPR space. We then multiply the MAD by 1.5 toapproximate the standard deviation of vertical veloci-ties, assuming the underlying distribution is GaussianRousseeuw & Croux (1993). We opted for MAD as itreduces sensitivity to outliers compared to the standarddeviation.We then converted σ v z to gyro-kinematic age usingthe AVR for metal-rich stars ([Fe/H] > − . < T eff < < log g < σ v z ) space to re-estimatethe AVR for metal rich stars since they did not providethe intercept value for the formula. We fit for α and β in the relation: ln age = β ln ( σ v z + α ) and found α =-2.80 ± β = 1.58 ± ∼
20% of the stars in our sample with LAMOST spec-tra have metallicities [Fe/H] < − . N stars that are closest to the target star (near-est neighbor method), and by 2) constructing a 4D binin phase-space with the target star in the center and se-lecting all the stars within a bin of a set size (binningmethod).To test these two methods, we generated mock ver-tical velocities with the semi-empirical gyrochronologymodel described in Spada & Lanzafame (2020) and triedto recover the result. We first generated ages from thegyrochronology model, using the temperatures and ro-tation periods from the 29,949 stars with measured rota-tion periods. We then converted these ages into σ v z usingthe AVR in Yu & Liu (2018). We generated mock verti-cal velocities by drawing from a Gaussian with a meanvelocity of 0 km s − and an uncertainty of σ v z . Linearinterpolation/extrapolation was used to calculate theages of stars between/outside the Spada & Lanzafame(2020) model grid points. Figure 1 shows the simulatedvelocity dispersion in temperature versus rotation space. Figure 1.
Simulated velocity dispersion for 29,949 starswith measured rotation periods based on the semi-empiricalgyrochronology model described in Spada & Lanzafame(2020) and the AVR described in Yu & Liu (2018).
To establish which of these two methods provides themost accurate and precise gyro-kinematic ages, we sim-ulated the velocity dispersions of stars and attempted
Lu et al. to recover those dispersions. Figure 2 shows the recov-ery results after optimizing the bin size and number ofnearest neighbors by minimizing χ . The minimum re-duced χ values were 0.82 for the binning method and1.46 for the nearest stars method. Figure 2 shows thesimulated velocity dispersions, compared with the mea-sured velocity dispersions for the nearest neighbor andbinning methods. The top panel shows the results us-ing the nearest neighbor method, and indicates thatthis method introduces bias in the recovered velocitydispersions: it over-predicts at low velocity dispersionand under-predicts at high velocity dispersion. This iscaused by the boundaries: for stars at the edges of theperiod distribution, their nearest neighbors are all onone side. As a result, the velocity dispersions of starswith the longest rotation periods will be calculated us-ing stars that, on average, are younger than them andtherefore have smaller velocity dispersions. Similarlythe velocity dispersions of stars with the shortest ro-tation periods are over-estimated because their nearestneighbors all have longer rotation periods and are olderthan they are.The binning method does not experience bias at theboundaries. However, the precision of the estimatedvelocity dispersion is poor in sparsely populated areas,where there are few stars in each bin. Therefore, we alsoset a minimum and maximum number of stars for eachbin, in which we decrease/increase the bin size by 10%each time if the stars exceed the maximum/minimumnumber until the number of stars is between the mini-mum and maximum. The lower limit ensures there areenough stars in each bin so the velocity dispersion cal-culated is relatively precise, and the upper limit ensuresthat we are not including stars with a large range of ages.Because it is less biased, we decided to use the binningmethod to estimate gyro-kinematic ages for these starsin MTPR space. The gyro-kinematic ages we infer withthis method depends on the choice of bin size, so we op-timized the bin size using a sample of measured stellarages. 2.2.1. Optimizing the Bin Size
In this section, we describe the process of optimizingthe bin size in MTPR space, as well as the minimumand maximum numbers of stars in each bin in order toachieve the best agreement with the stars with knownage measurements from asteroseismology and isochronefitting.We first performed χ tests with respect to stars withasteroseismic ages from the Kepler LEGACY sample(Silva Aguirre et al. 2017). This LEGACY sample in-cludes 66 main-sequence stars analyzed with seven dif- htp Figure 2.
Recovery velocity dispersion of mocked data (1)for using the binning method and the nearest stars method.The optimized number of nearest stars and bin size in T eff and P rot gave a minimum χ of 0.82 and 1.46, respectively. Thenearest star method recovers poorly at the boundaries (whenthe velocity dispersion is extreme), and therefore, we decidedto use the binning method to calculate gyro-kinematic ages. ferent stellar models. The uncertainties for these aster-oseismic ages were calculated by taking the range of theresults from these seven different models. After cross-matching the asteroseismic stars with our star sample,we found 13 stars in common. We then excluded starsthat appeared to be experiencing weakened magneticbraking (van Saders et al. 2016). At Ro ≈
2, the mag-netic braking mechanism of stars is thought to lose ef-ficiency and, as a result, stars stop spinning down. Al-though our gyro-kinematic dating method is not techni-cally incompatible with stars that are no longer spinningdown, it can only calculate accurate ages for stars thatare well-represented in our data set, and it is not clear yro-Kinematic Ages for Kepler Stars stardate software package (Angus et al. 2019b,a). Weexcluded seven stars with predicted rotation periods dif-fering by more than five days from their measured ro-tation periods, leaving us with six stars. Although it isbeyond the scope of this paper to investigate weakenedmagnetic braking from a kinematic age standpoint, wenote that a preliminary examination indicates that theasteroseismic stars removed from our sample show goodage agreement with the gyro-kinematic ages of stars intable 2 of McQuillan et al. (2014), which include starswith low measurement confidence that could be goingthrough magnetic braking.To find the optimum bin size, we performed a 10 × × ×
10 grid search with temperatures between50 K and 500 K, and M G , log (rotation period), andlog ( R o ) between 0.1 dex and 0.5 dex. We comparedgyro-kinematic ages with the asteroseismic ages to calcu-late χ at each point in this grid. We took this approachto keep computation time low. This yielded a bin sizeof M G = 0.46 mag, log (period) = 0.14 dex, log ( R o )= 0.14 and temperature = 150 K with a χ of 0.05. Wethen estimated the uncertainty on these gyro-kinematicages by performing bootstrapping on the parameters weused ( M G , rotation periods, R o , temperature, and ver-tical velocity) with 50 random samplings and computedthe standard deviation of the resulting gyro-kinematicages as the uncertainty. In order to make sure 50 sam-pling is enough, we also estimated the uncertainties us-ing 100 random samplings for ten random stars and theuncertainty did not change.Next, we performed similar χ tests for Kepler starswith isochrone ages. Berger et al. (2020) produced a cat-alog of stellar properties for 186,301 Kepler stars basedon Gaia data, with median isochrone age uncertaintiesof 2.71 Gyr (54%). To optimize the bin size with re-spect to these isochronal ages, we first excluded starsolder than 10 Gyr, where there is an overdensity thatis likely due to systematic errors in the model. Starswith age uncertainties greater than 3 Gyr were also ex-cluded, which removes most of the K and M dwarfs from our comparison sample. After excluding these stars, wewere left with 8,958 stars with isochrone ages with mea-sured rotation periods to compare gyro-kinematic ageswith, in which most of these stars are subgiants and F/Gdwarfs.We then minimized the χ by performing a similargrid search as done for the asteroseismic stars; we alsochanged the minimum and maximum number of stars al-located to each bin. The large sample size of isochroneages allow us to determine the requisite minimum andmaximum number of stars in each bin. As mentioned inthe previous section, the lower limit for the number ofstars in each bin ensures there are enough stars to calcu-late precise velocity dispersion, and the upper limit en-sures that we are not including stars with a large rangeof ages. The number density of the stars in the MTPRspace indicates the rate at which these stars move inthe phase space, in which a higher density region cor-responds to a slower evolution rate and vice versa. Asa result, setting a fixed number of stars in each bin en-sures that stars in each bin are of a similar age. Inother words, the bin size grows in regions of parameterspace where isochrones are spaced farther apart (e.g., onthe subgiant branch) and shrinks where isochrones aretightly packed (e.g., on the main sequence). If the num-ber of stars exceeded the minimum or maximum limit,the bin was reduced/increased by 10% until the numberof stars in each bin was within the limits or the bin sizehad changed by more than 100% from its original binsize. The minimum and maximum number of stars ineach bin were optimized to be 15 and 30, respectively.The optimized bin size for the isochrone ages was M G = 0.5 mag, log( P rot ) = 0.1 dex, R o = 0.4, and T eff =500 K, with a reduced χ of 0.8; the bin sizes for M G and log( P rot ) are on par with those found for the aster-oseismic stars, whereas the R o and T eff bins are muchlarger. These large bin sizes are concerning, so we alsocalculated χ for the isochrone ages using the optimizedbin size from the asteroseismic stars and obtained a χ value of 1.02. Since this χ value was similar to the minimized χ for the isochrone ages, we decided to usethe bin size from the asteroseismic stars to determine allthe gyro-kinematic ages since the asteroseismic ages aremore precise.We then estimated the uncertainties on the gyro-kinematic ages with the bootstrapping method de-scribed in the previous paragraphs but with the opti-mized bin size. This yielded a median age uncertaintyfrom bootstrapping to be 1.30 Gyr across all ages.We also added the uncertainties from theAVR fit. In which we calculated the uncer-tainty, using error propagation, to be σ ln age = Lu et al. (cid:112) (ln σ v z σ β ) + ( σ α ) + (( β/σ v z ) σ σ vz ) , in which σ α , σ β , and σ σ vz are uncertainties on α , β , and σ v z , re-spectively. The uncertainty for the vertical velocitydispersion is the bootstrapping uncertainties. Thisyields a median uncertainty from the AVR to be 1.76Gyr across all ages.Figure 3 shows the 2D projections of the stars inMTPR space colored by their gyro-kinematic ages. Therotation period and R o versus T eff space (top and mid-dle panels) shows that stellar rotation slows down asstars age, as expected. Converting rotation period toRossby number flattens the upward trend seen in thecoolest stars in the top panel. This conversion effec-tively transforms the data to a shape that is more com-patible with the square-shaped bins we use to calculatedgyro-kinematic ages. Including Rossby number as anadditional dimension in the binning process improvesage accuracy by ∼ < T eff < RESULTS3.1.
Comparing Gyro-Kinematic Ages with BenchmarkAges
After optimizing the bin size for gyro-kinematic pro-cedure, we calculated gyro-kinematic ages for membersof stars clusters and for two white dwarf–M dwarf bi-naries. For the former, we used the Curtis et al. (2020)catalog which is an assembly of rotation periods froma variety of benchmark clusters, including the Pleiades(120 Myr; Rebull et al. 2016), Praesepe (670 Myr; Dou-glas et al. 2017, 2019), Hyades (730 Myr; Douglas et al.2016, 2019), NGC 6811 (1 Gyr; Meibom et al. 2011;Curtis et al. 2019), NGC 752 (1.4 Gyr; Ag¨ueros et al.2018), NGC 6819 (2.5 Gyr; Meibom et al. 2015), andRuprecht 147 (2.7 Gyr; Curtis et al. 2020). This catalogprovides Gaia DR2 data and classifies subsets of eachcluster as “benchmark rotators” if they satisfy criteriafor single-star membership outlined in Appendix A.4 ofthat paper.Kiman et al. in prep. studied the ages of low-massstars using white dwarf–M dwarf binary systems. Theydeveloped wdwarfdate , an open source Python codewhich estimates the age of a white dwarf from an ef-fective temperature and a surface gravity in a Bayesianframework. The white dwarf age is the sum of its cool- https://wdwarfdate.readthedocs.io/en/latest Figure 3.
2D projections of 29,949 stars in MTPR space col-ored by gyro-kinematic ages. The rotation period/ R o slowsdown as a star ages and the evolved and low-mass stars arethe oldest stars in our catalog. yro-Kinematic Ages for Kepler Stars To determine the main sequence lifetime of the progen-itor star, the progenitor’s mass is estimated from theWD mass using an initial-to-final mass relation (IFMR;Cummings et al. 2018), and then the progenitor lifetimeis adopted from a MESA isochrone (MIST; Dotter 2016).We found two Kepler stars that have measured rota-tion periods and are comoving with a white dwarf (KIC11075611 and KIC 12456401). We age-dated the whitedwarfs with wdwarfdate , using the spectroscopic mea-surements for effective temperature and surface gravityfrom Gentile Fusillo et al. (2019). We do not have a largesample of these stars but these binaries will be morepowerful in the future when a larger sample is available.In order to obtain the gyro-kinematic ages for starsin clusters, for each target star in the cluster, we cal-culated the velocity dispersion of the target star usingthe optimal bin size determined in the previous sectionat the T eff , M G , P rot , and R o for the target star as thebin center. We then calculated the age using the Yu &Liu (2018) AVR. The gyro-kinematic cluster ages anduncertainties were then calculated by taking the meanand standard deviation of the gyro-kinematic ages of allthe stars in the cluster.Figure 4 compares the gyro-kinematic ages for thesebenchmarks with their literature ages. In order to bettercompare gyro-kinematic ages, which are average ages,with isochrone ages, which are individual ages, we alsotook the average of the isochrone ages with the sameoptimized bin in MTPR space to get rid of outliers andapplied extreme deconvolution ( XD ), which is useful fordisentangling underlying distributions from noisy data.This method finds the underlying density distributionfrom noisy, heterogeneous and incomplete data using aGaussian mixture model approach (Bovy et al. 2011).Without applying extreme deconvolution, the high levelof scatter caused by large age uncertainties prevents ameaningful visual comparison between gyro-kinematicand isochrone ages. In other words, the comparison plotlooks like a scatter plot with little correlation. Extremedeconvolution predicts the noise-free data and providesa guide for the eye, allowing a more meaningful com-parison to be drawn. The black dots in Figure 4 show500 samples from the Gaussian that best describes theunderlying noise-free distribution, estimated using ex-treme deconvolution, for comparison against isochroneages. The root mean square (RMS) and median abso- ∼ bergeron/CoolingModels/ lute deviation (MAD) between the gyro-kinematic agesand benchmark ages are RMS = 1.22 Gyr, MAD = 1.08Gyr for the clusters, and RMS = 0.26 Gyr, MAD = 0.20Gyr for asteroseismic stars.In general, the gyro-kinematic ages agree wellwith other benchmark ages. There are two obvi-ous disagreements—a deviation between gyro-kinematicages and ages of very young clusters ( < DISCUSSION & FUTURE WORK4.1.
Advantages & Limitations
Using gyro-kinematic ages comes with several caveats: • Average ages — gyro-kinematic ages do not nec-essarily reflect the actual age of each individualstar. The gyro-kinematic age of each star is as-signed according to the kinematics (i.e., verticalvelocity dispersion) of stars with similar rotationperiods, temperatures, Rossby numbers, and lu-minosities. This means that, by design, gyro-kinematic ages vary smoothly across MTPR space,with a characteristic length-scale set by the binsize. This means the gyro-kinematic age for a starthat has abnormal stellar properties or has fol-lowed an atypical path of stellar evolution (e.g.,a blue straggler or tidally-synchronized binary)will most likely be incorrect. For example, a starthat spins anomalously rapidly compared to anal-ogous stars of a similar age will have an under-estimated age from gyro-kinematics. Althoughthis caveat should be acknowledged when usinggyro-kinematic ages, this feature is shared by allage-dating techniques. Relations that are cali-0
Lu et al.
Figure 4.
Comparison of gyro-kinematic ages with literature ages for benchmarks, including stars with isochrone ages (Iso;Berger et al. 2020), Kepler LEGACY stars characterized with asteroseismic ages (AS; Silva Aguirre et al. 2017), members ofstar clusters (Curtis et al. 2020), and white dwarf–M dwarf binaries (WD; Kiman et al. 2020). We overestimate ages of veryyoung cluster likely due to the presence of older tight binaries contaminating the reference bins used to calculate the velocitydispersion used to assign their gyro-kinematic ages. The black dots show the results for the isochrone stars, where we haveapplied extreme deconvolution to aid the visual interpretation. The systematic offset between isochrone ages and gyro-kinematicages could be due to bias in the binning method and/or an offset between the age scales for the MIST isochrone models (Choiet al. 2016) used in Berger et al. (2020) and our adopted AVR. RMS = 1.22 Gyr, MAD = 1.08 Gyr for the clusters, and RMS= 0.26 Gyr, MAD = 0.20 Gyr for asteroseismic stars. Another version of this graph without applying extreme deconvolutionon the isochrone ages comparison is shown in figure A.1 in the Appendix section. brated using data with some amount of intrinsicscatter (e.g., caused by stars that still exhibit theirinitial conditions) that is not captured by mea-surement uncertainties (as is true of all stellar ro-tation data) will only reflect the average behaviourof stars. • Uncertainties —It is hard to estimate the true un-certainties for gyro-kinematic ages. We have takeninto account the formal uncertainties on the stel- lar parameters (temperature, absolute magnitude,rotation period, Rossby number, and vertical ve-locity). We tested how the bin size might affectthe gyro-kinematic ages by perturbing the bin sizein each stellar parameter space 10 times by up to20% and calculated the standard deviation of theages. For example, the optimized bin size is [ M G ,period, R o , temperature] = [0.46 dex, 0.14 dex,0.14, 150 K], and we tested how much the tem-perature bin size might have affected the results yro-Kinematic Ages for Kepler Stars • Ages for rapid rotators and old G stars —Aspointed out previously (e.g., Section 3), we werenot able to estimate accurate ages for youngrapidly rotating stars most likely because tight bi-naries are contaminating their MTPR bins. InSections 1 and 2, we also pointed out that wewould not be able to estimate ages for stars withRossby numbers greater than (cid:38)
2. Since these starsare either going through weakening braking (vanSaders et al. 2016) or reaching the period detec-tion limit for inactive stars or both, they may notbe well represented in the McQuillan et al. (2014)rotation sample.However, despite these limitations, the gyro-kinematicages and other benchmark ages are mostly in agreement,and gyro-kinematic ages do have some advantages overother age-dating methods. A major advantage is thatwe are able to estimate ages for low-mass stars. Further-more, we only made two simple assumptions to obtaingyro-kinematic ages—that stars in the same bin in stel-lar parameter space (in this case, P rot , T eff , R o , and M G )are similar in age and that vertical velocity dispersionincreases over time. Both of these assumptions havebeen tested extensively by gyrochronology studies (e.g.,Barnes 2003; Curtis et al. 2020) and kinematic studies(e.g., Str¨omberg 1946; Yu & Liu 2018). The only modelwe included was the AVR, and as a result the absoluteages of these stars are subject to the accuracy of that Yu& Liu (2018) AVR, but the relative age ranking shouldbe correct.4.2. Gyro-Kinematic Ages from Predicted Periods
Most stars observed by Kepler do not have a measuredrotation period. However, it may still be possible to cal-culate their gyro-kinematic ages by predicting their ro-tation periods using our machine learning method called
Astraea (Lu et al. 2020) . Using Random Forest, this Avaliable at https://github.com/lyx12311/Astraea method predicts the rotation periods of stars observedby Kepler based on T eff , log g , luminosity, Gaia pho-tometry and its error, velocities, radius, galactic lati-tude, and photometric variability ( R var ). This is impor-tant because only one in five Kepler stars has a periodmeasured by McQuillan et al. (2014) with confidence.The remaining 80% of stars did not have detectable ormeasurable rotation periods for a few potential reasons:their variability was low-amplitude, their rotation peri-ods were very long, their variability was quasi- or aperi-odic, or they were extremely faint. In Lu et al. (2020) wedemonstrated that Astraea can precisely predict the pe-riods of stars with measured rotation periods, howeverwe did not establish whether
Astraea can predict therotation periods of stars without measured rotation pe-riods. Can we use our machine learning tool to predictthe rotation periods of stars that do not have rotationperiods measured with high confidence?One way to test this is to predict the rotation periodsof stars that were not used to train the random forestalgorithm, but that do have measured rotation periodmeasurements. McQuillan et al. (2014) included a tableof stars for which the rotation periods have low confi-dence levels, parameterized as “ w ” for weight (see theirtable 2). These stars were not used to train Astraea ,however we find that we can accurately predict manyof their rotation periods. Figure 5 shows the predic-tion results for stars with low w rotation periods fromthat table. The weights were assigned according to theheight of the primary autocorrelation function (ACF)peak with respect to the troughs on each side (local peakheight; LPH) and position in T eff –LPH–period space. Asshown in the figure, most of the stars with high weights(green or yellow colored points) lie on the equality lineand the stars whose measured and predicted rotationperiods do not agree have low weights (purple coloredpoints). This suggests Astraea is able to predict accu-rate rotation periods in cases where the signal amplitudeis above some threshold. In cases where stars have lowweights, the periods predicted with
Astraea disagreewith those measured by McQuillan et al. (2014). Thiscould be because either one or both methods are incor-rect, and we are not able to tell whether we are ableto predict accurate rotation periods for these stars. Re-gardless of the reason, we suggest using a new weight,0.15, for selecting validated rotation periods in the cat-alog described in McQuillan et al. (2014). By doing so,we are able to increase the catalog size up to 25%.Another way to test the predicted periods is to seewhether the velocity dispersion of stars increases withpredicted rotation period, M G and temperature in a waythat is similar to the trends observed in the sample of2 Lu et al.
Figure 5.
Predicted rotation periods versus measured ro-tation periods from McQuillan et al. (2014) Table 2 (i.e.,those with no significant period detection), color-coded bythe measurement confidence weight “ w ” from their paper.We are able to predict rotation periods with high “ w ” val-ues ( > stars with directly measured rotation periods. We pre-dicted the rotation periods of 45,000 stars observed byKepler, but without reported rotation periods. Thesestars were selected after applying cuts to remove sub-giants and photometric binaries, following the proceduredescribed in Lu et al. (2020). We also excluded starswith stellar parameters outside of the Kepler trainingset described in Lu et al. (2020). Only predicting pe-riods for stars within the range of the stellar param-eters of the training set is extremely important sincemost machine learning algorithms, including RandomForests used by Astraea , cannot extrapolate outside ofthe training data. Rotation periods were then predictedwith
Astraea using the absolute magnitudes, tempera-tures, radii, colors, kinematics, and the light curve vari-ability of these stars. Gyro-kinematic ages were calcu-lated with the same method described in section 2.2,using these predicted rotation periods.We find tentative evidence to suggest that many of therotation periods predicted by
Astraea are accurate, orat least correctly ranked. For example, we find that the
Astraea -predicted periods are correlated with Bergeret al. (2020) isochrone ages. As shown in the followingsection, we also find that gyro-kinematic ages calculatedfrom predicted periods can reproduce some of the age-abundance trends previously measured in a sample ofSolar twins (Bedell et al. 2018). However, although this technique shows promise, it still requires some refine-ment and optimization. A thorough quantification ofthe precision and accuracy of gyro-kinematic ages, cal-culated with predicted rotation periods, is beyond thescope of this paper and we leave that to a future exer-cise.4.3.
Detailed age-element abundance trends withgyro-kinematic ages from predicted periods
To explore the validity of our gyro-kinematic ages, cal-culated using predicted rotation periods, we investigatedthe age-abundance trends that have been found for Solartwins. Detailed age-chemical abundance trends measurethe time-dependent chemical evolution of the gas in theMilky Way disk. These trends offer strong empiricalconstraints on nucleosynthesis processes, as well as themixing of star forming gas in the disk. We wanted tocompare our results with those found by Bedell et al.(2018) to see whether we can recover the same age-abundance trends using gyro-kinematic ages. If so, gyro-kinematic ages could provide a new avenue for Galac-tic chemical evolution studies. To provide a meaningfulcomparison to the Bedell et al. (2018) results, we focusedon Solar twins.We cross-matched our catalog with APOGEE DR16(Ahumada et al. 2020; Majewski et al. 2017) using thesame criteria described in Bedell et al. (2018) to selectour Solar twin samples (log g within 0.1 dex, [Fe/H]within 0.3 dex, and temperature within 100 K of theSun), and found 108 stars. We then calculated gyro-kinematic ages using Astraea -predicted periods to fur-ther validate our method and to test whether gyro-kinematics could be useful for future age–abundancestudies. We used predicted rather than measured rota-tion periods as most stars in our Solar twin sample didnot have measured rotation periods. Only 12 stars withvery similar ages (1-3 Gyr) have gyro-kinematic agesfrom measured periods, and such a narrow age rangemeans these stars were not able to provide constrainedage–element abundance trends.To measure the trends, we excluded any stars with agyro-kinematic upper age limit greater than 8 Gyr, asthese stars could be stars from the high- α disk and ex-hibit a different trend to those from the low- α disk (Be-dell et al. 2018). Figure 6 shows the comparison betweenthis work and 79 solar twins from Bedell et al. (2018).We calculated the age–abundance trends by performinglinear fits using numpy.polyfit , and the uncertaintieswere estimated from the covariance matrix. In general,our trends are very similar with those from Bedell et al.(2018) within the uncertainties. The ones that do not yro-Kinematic Ages for Kepler Stars CONCLUSIONIn this paper, we created a gyro-kinematic catalogcontaining 29,949 stars with measured rotation periodsfrom McQuillan et al. (2014), Garc´ıa et al. (2014), andSantos et al. (2019). We obtained the ages by binningthe stars in MTPR space with an optimized bin size of[ M G , period, R o , temperature] = [0.46 dex, 0.14 dex,0.14, 150 K]. These bin sizes produced the smallest av-erage χ value with respect to the LEGACY asteroseis-mic stars from Silva Aguirre et al. (2017). We also foundthe optimum minimum and maximum number of starsin each bin to be 15 and 30 stars, respectively, by min-imizing the average χ respect to isochrone ages fromBerger et al. (2020). We optimized the min/max num-ber of stars with respect to isochrone ages since we didnot have enough stars with asteroseismic ages to con-strain these two parameters. If the number of stars ex-ceeded the minimum or maximum limit, the bin wasreduced/increased by 10% until the number of stars ineach bin was within the limit or the bin size had changedmore than 100% from its original bin size. We estimatedthe uncertainties for these ages by perturbing the stellarparameters ( M G , rotation period, R o , temperature, andvertical velocity) within their errors via bootstrappingwith a sample size of 50.We compared the gyro-kinematic ages with otherbenchmark ages (isochrone ages from Berger et al. 2020,asteroseismic ages from Silva Aguirre et al. 2017, clusterages from Curtis et al. 2020, and WD–M dwarf binaryages from Kiman et al. 2020). Aside from the offsetbetween ages from this work and isochrone ages, gyro-kinematic ages agree well with other benchmark ages.We calculated the RMS and MAD between the gyro-kinematic ages and the benchmark ages and obtaineda RMS = 1.22 Gyr, MAD = 1.08 Gyr for the clusters,and RMS= 0.26 Gyr, MAD = 0.20 Gyr for asteroseis-mic stars. The offset between the isochrone ages andgyro-kinematic ages could be caused by systematic er-rors in the stellar evolution models used to calculate theisochrone ages or the AVR we used to calculate gyro-kinematic ages.In section 4, we also estimated gyro-kinematic agesfor Kepler stars with rotation periods predicted from Astraea . We did this so we could test whether or not theRandom Forest method described in Lu et al. (2020) isable to accurately predict periods. We first predicted ro-tation periods for stars included in table 2 of McQuillanet al. (2014), and we found we were able to predict starswith weight, w > w Gaia
Multilateral Agreement.R.A. acknowledges support from NASA award80NSSC20K1006.Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST) is aNational Major Scientific Project built by the ChineseAcademy of Sciences. Funding for the project hasbeen provided by the National Development and ReformCommission. LAMOST is operated and managed by theNational Astronomical Observatories, Chinese Academyof Sciences.This research has also made use of NASA’s Astro-physics Data System, and the VizieR (Ochsenbein et al.2000) and SIMBAD (Wenger et al. 2000) databases, op-erated at CDS, Strasbourg, France.
Facilities:
Gaia , Kepler , LAMOST, APOGEE
Software:
Astropy (Astropy Collaboration et al.2013; Price-Whelan et al. 2018), Numpy (Oliphant 2006),4
Lu et al.
Figure 6.
Individual age–element abundance trend with gyro-kinematic ages for 108 solar twins. The black dots and black linesare the results from Bedell et al. (2018), and the red dots and red lines are results from this work. The shaded area representsthe 1- σ uncertainty from the fit, estimated using the covariance matrix. In general, the abundance–age trends agree with thosefrom Bedell et al. (2018) except for ones with large scatter; e.g., Cr, Co, and Cu Astraea (Lu et al. 2020), Pandas (pandas developmentteam 2020; Wes McKinney 2010), Matplotlib (Hunter 2007), Jupyter Notebook (Kluyver et al. 2016), wdwarf-date (Kiman et al. 2020)APPENDIXA.
Additional figures
Figure A.1 shows the same comparison between benchmark ages and gyro-kinematic ages without applying extremedeconvolution to the isochrone ages comparison. REFERENCES
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