MNRAS 000,1{10 (2019) Preprint 6 May 2019 Compiled using MNRAS LATEX style file v3.0 A seismic scaling relation for stellar age Earl Patrick Bellinger1? 1SAC Postdoctoral Research Fellow, Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Denmark Accepted XXX. Received YYY; in original form ZZZ ABSTRACT A simple solar scaling relation for estimating the ages of main-sequence stars from asteroseismic and spectroscopic data is developed. New seismic scaling relations for estimating mass and radius are presented as well, including a purely seismic radius scaling relation (i.e., no dependence on temperature). The relations show substantial improvement over the classical scaling relations and perform similarly well to grid- based modeling. Key words: asteroseismology { stars: solar-type, fundamental parameters, evolution −2 1 1 INTRODUCTION R ν ∆ν T 2 ' max eff (4) A solar scaling relation is a formula for estimating some R νmax; ∆ν Teff; unknown property of a star from observations by scaling where ± Hz, ∆ ± Hz (Hu- from the known properties of the Sun. These relations have νmax; = 3090 30 µ ν = 135:1 0:1 µ ber et al. 2011), and T ± K(Prˇsa et al. the form eff; = 5772:0 0:8 2016). These relations are useful thanks to the exquisite pre- Y Ö X Pi ' i (1) cision with which asteroseismic data can be obtained. For a Y X typical well-observed solar-like star, ∆ν and ν can be mea- i ;i max sured with an estimated relative error of only 0.1% and 1%, where Y is some property we wish to estimate, such as the respectively (see, e.g., Figure 5 of Bellinger et al. 2019). radius of the star. The quantity Y is the corresponding These relations can be analytically derived from the property of the Sun (e.g., the solar radius), the vector X fact that ∆ν scales with the mean density of the star and contains measurable properties of the star (e.g., its effective νmax scales with the acoustic cut-off frequency (Ulrich 1986; temperature and luminosity), the vector X contains the Brown et al. 1991; Kjeldsen & Bedding 1995; Stello et al. corresponding solar properties, and P is some vector of ex- 2008; Kallinger et al. 2010). These relations are not perfectly ponents. An example scaling relation for estimating stellar accurate, however (e.g., Stello et al. 2009a; Belkacem et al. radii can be derived from the Stefan{Boltzmann law as: 2011; Gaulme et al. 2016; Themeßl et al. 2018; Ong & Basu −2 1 R Teff L 2 2019), especially when it comes to evolved stars, which has ' (2) resulted in several suggested corrections (White et al. 2011; R Teff; L Sharma et al. 2016; Guggenberger et al. 2016, 2017). where R is the radius, Teff the effective temperature, and L In recent years, there has been a great push for im- the luminosity. proved determination of stellar properties|and particularly In the era of space asteroseismology, relations known stellar ages, for which asteroseismology is uniquely capable. as the seismic scaling relations have enjoyed wide usage. Besides their intrinsic interest, knowledge of the ages of stars They are used to estimate the unknown masses and radii of is useful for a broad spectrum of activities in astrophysics, stars by scaling asteroseismic observations with their helio- ranging from charting the history of the Galaxy (e.g., Miglio seismic counterparts. These observations include the average et al. 2013; Bland-Hawthorn & Gerhard 2016; Casagrande frequency spacing between radial mode oscillations of con- et al. 2016; Silva Aguirre et al. 2018; Spitoni et al. 2019) to secutive radial order|the large frequency separation, ∆ν| understanding the processes of stellar and exoplanetary for- and the frequency at maximum oscillation power, νmax. From mation and evolution (e.g., Maeder & Mermilliod 1981; Lin these and the observed effective temperature, one can esti- et al. 1996; Bodenheimer & Lin 2002; Soderblom 2010; Winn mate the stellar mass M and radius R of a star via: & Fabrycky 2015; Nissen et al. 2017; Bellinger et al. 2017). 3 −4 3 Unlike with mass and radius, there is no scaling relation M νmax ∆ν Teff 2 ' (3) for stellar age. Instead, a multitude of methods have been M ν ; ∆ν T max eff; developed for matching asteroseismic observations of stars to theoretical models of stellar evolution (e.g., Brown et al. ? E-mail: [email protected] 1994; Pont & Eyer 2004; Stello et al. 2009b; Gai et al. 2011; © 2019 The Authors 2 E. P. Bellinger Bazot et al. 2012; Metcalfe et al. 2014; Lebreton & Goupil Spectral Type 2014; Lundkvist et al. 2014; Silva Aguirre et al. 2015, 2017; FG K Bellinger et al. 2016; Angelou et al. 2017; Rendle et al. 2019), which then yields their ages. Scaling relations are attractive resources because they 500 can easily and immediately be applied to observations with- out requiring access to theoretical models. In this paper, I seek to develop such a relation to estimate the ages of main sequence stars, as well as to improve the scaling relations for 1000 estimating their mass and radius. The strategy is as follows. It is by now well-known that the core-hydrogen abundance (and, by proxy, the age) of a Hz] μ [ main-sequence star is correlated with the average frequency spacing between radial and quadrupole oscillation modes max 2000 ν 1.5 (e.g., Christensen-Dalsgaard 1984; Aerts et al. 2010; Basu & 1.4 Chaplin 2017). This spacing is known as the small frequency 1.3 separation and is denoted by δν. The diagnostic power of δν 1.2 is owed to its sensitivity to the sound-speed gradient of the 1.1 stellar core, which in turn is affected by the mean molec- 1 ular weight, which increases monotonically over the main- 5000 0.9 sequence lifetime as a byproduct of hydrogen{helium fusion. 0.8 Here I formulate new scaling relations that make use of this spacing δν, and I also add a term for metallicity. Rather 7000 6000 5000 than by analytic derivation, I calibrate the exponents of this relation using 80 solar-type stars whose ages and other pa- Teff [K] rameters have been previously determined through detailed fits to stellar evolution simulations. Finally, I perform cross- Figure 1. Locations of the stars used in this study in the νmax{ validation to estimate the accuracy of the new relations. Teff diagram. The lines in the background are theoretical stellar evolution tracks of the indicated masses computed with MESA (Paxton et al. 2011, 2013, 2015, 2018). The zero-age main se- quence is indicated with a dashed line; stars evolve upward and 2 DATA to the right. The background color indicates spectral type. The I obtained spectroscopic and asteroseismic measurements of position of the Sun is indicated with the solar symbol ( ). solar-like stars from the Kepler Ages (Silva Aguirre et al. 2015; Davies et al. 2016) and LEGACY (Lund et al. 2017; Silva Aguirre et al. 2017) samples. These stars were observed for suitable choices of the powers P = »α; β; γ; δ; ¼. Note the by the Kepler spacecraft during its nominal mission (Borucki metallicity is first exponentiated. The uncertainties on all et al. 2010). These stars are all main-sequence or early sub- solar quantities are propagated except in the case of metal- giant stars with no indications of mixed modes. Their posi- licity, where there is no agreed upon uncertainty (e.g., Berge- mann & Serenelli 2014). From an analysis of solar data tions in the νmax{Teff plane are shown in Figure1. The large and small frequency separations of these stars range from 38 (Davies et al. 2014) one can find δν = 8:957 ± 0:059 µHz. ± to 180 µHz and from 2.8 to 13 µHz, respectively. For the solar age I use τ = 4:569 0:006 Gyr (Bonanno & The ages, masses, and radii of these stars are taken from Fr¨ohlich 2015). To give a concrete example, in order to re- Bellinger et al.(2019) as derived using the Stellar Parame- cover the classical radius scaling relation (Equation4), i.e., ters in an Instant (SPI, Bellinger et al. 2016) pipeline. The Y = R, we have P = »1; −2; 0; 1/2; 0¼. SPI method uses machine learning to rapidly compute stellar I now seek to find the exponents for scaling relations that best match to the literature values of radius, mass, and ages, masses, and radii of stars by connecting their observa- 2 tions to theoretical models. For the present study, I selected age. I define the goodness-of-fit χ for a given vector P as 80 of these stars having δν measurements with uncertainties 2 Õ Yˆ − Y smaller than 10% and ν measurements with uncertainties χ2 = i i (6) max σ smaller than 5%. i i where Yˆi is the literature value of the desired quantity (e.g., age) for the ith star, Y is the result of applying the scaling 3 METHODS i relation in Equation5 with the given powers P, and the Here I will detail the construction of the new scaling rela- uncertainties σ are the measurement uncertainty of Yˆ . Given tions and the procedure for their calibration. The scaling this function, I use Markov chain Monte Carlo (Foreman- relations shown in Equations3 and4 can be written more Mackey et al. 2013; Foreman-Mackey 2016) to determine the generically as follows. For a given quantity Y (and corre- posterior distribution of P for each of Y = R; M; τ. Finally, I sponding solar quantity Y ), use the mean-shift algorithm (Fukunaga & Hostetler 1975; Pedregosa et al.