Asteroseismic masses, ages, and core properties of γ Doradus stars using gravito-inertial dipole modes and spectroscopy
Joey S. G. Mombarg, Timothy Van Reeth, May G. Pedersen, Geert Molenberghs, Dominic M. Bowman, Cole Johnston, Andrew Tkachenko, Conny Aerts
MMNRAS , 1–14 (2018) Preprint 21 February 2019 Compiled using MNRAS L A TEX style file v3.0
Asteroseismic masses, ages, and core properties of γ Doradus starsusing gravito-inertial dipole modes and spectroscopy
J. S. G. Mombarg (cid:63) , T. Van Reeth , , , M. G. Pedersen , G. Molenberghs , ,D. M. Bowman , C. Johnston , A. Tkachenko and C. Aerts , Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium Sydney Institute for Astronomy (SIfA), School of Physics, University of Sydney, New SouthWales 2006, Australia Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000 Aarhus C, Denmark I-BioStat, Universeit Hasselt, Martelarenlaan 42, B-3500 Hasselt, Belgium I-BioStat, KU Leuven, Kapucijnenvoer 35, B-3000 Leuven, Belgium Department of Astrophysics, IMAPP, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands
Accepted 2019 February 15. Received 2019 February 2019; in original form 2018 November 25
ABSTRACT
The asteroseismic modelling of period spacing patterns from gravito-inertial modes in starswith a convective core is a high-dimensional problem. We utilise the measured period spacingpattern of prograde dipole gravity modes (acquiring Π ), in combination with the effectivetemperature ( T eff ) and surface gravity (log g ) derived from spectroscopy, to estimate the fun-damental stellar parameters and core properties of 37 γ Doradus ( γ Dor) stars whose rotationfrequency has been derived from
Kepler photometry. We make use of two 6D grids of stel-lar models, one with step core overshooting and one with exponential core overshooting, toevaluate correlations between the three observables Π , T eff , and log g and the mass, age, coreovershooting, metallicity, initial hydrogen mass fraction and envelope mixing. We providemultivariate linear model recipes relating the stellar parameters to be estimated to the three ob-servables ( Π , T eff , log g ). We estimate the (core) mass, age, core overshooting and metallicityof γ Dor stars from an ensemble analysis and achieve relative uncertainties of ∼
10 per centfor the parameters. The asteroseismic age determination allows us to conclude that efficientangular momentum transport occurs already early on during the main sequence. We find thatthe nine stars with observed Rossby modes occur across almost the entire main-sequencephase, except close to core-hydrogen exhaustion. Future improvements of our work will comefrom the inclusion of more types of detected modes per star, larger samples, and modelling ofindividual mode frequencies.
Key words: asteroseismology – methods: statistical – stars: fundamental parameters – stars:interiors – stars: oscillations
The photometric data provided by space-based missions such asCoRoT (Auvergne et al. 2009) and
Kepler (Koch et al. 2010) haveheralded a new era for asteroseismology. Here, we are concernedwith gravito-inertial asteroseismology, i.e., the study of gravitymodes (g modes) in rotating intermediate-mass stars. Such modesare subject to both the Coriolis force and buoyancy as restoringforces. While CoRoT data led to the first discoveries of period spac-ings of gravity-mode pulsators in the core-hydrogen burning phase(Degroote et al. 2010; Pápics et al. 2012), it did not allow the identi-fication of the angular degree of these detected oscillations withoutambiguity. Secure mode identification had to await nearly unin- (cid:63)
Contact: [email protected] terrupted time series photometry with at least a factor ten longertime base, such as that assembled with the
Kepler space telescope(Borucki et al. 2010). Meanwhile, lots of progress has been madeon the observational side in this topic over the past few years withfirm detections of period spacing patterns reported in Pápics et al.(2014, 2015); Kurtz et al. (2014); Saio et al. (2015); Van Reeth et al.(2015a); Schmid et al. (2015); Van Reeth et al. (2016); Murphy et al.(2016); Guo et al. (2016); Ouazzani et al. (2017); Saio et al. (2017,2018); Szewczuk & Daszyńska-Daszkiewicz (2018) and Li et al.(2019).As shown in these papers, we have now reached the stagewhere tens of gravity-mode frequencies have been measured in thesepulsators with sufficient precision to identify their mode degree fromthe 4-year nominal
Kepler light curves and hence to start testing andimproving stellar structure theory of intermediate-mass stars. © 2018 The Authors a r X i v : . [ a s t r o - ph . S R ] F e b J. S. G. Mombarg et al.
Forward modelling applications to a few stars have shown theneed of core overshooting and envelope mixing in both B-type andF-type gravity-mode pulsators, in order to be able to explain themeasured mode trapping properties (Moravveji et al. 2015, 2016;Schmid & Aerts 2016). Moreover, the near-core rotation rates de-rived from the g modes (Van Reeth et al. 2016; Ouazzani et al. 2017;Van Reeth et al. 2018) have revealed shortcomings in stellar evolu-tion theory in terms of angular momentum transport, both duringthe core-hydrogen burning phase (Rogers 2015; Aerts et al. 2017;Townsend et al. 2018; Ouazzani et al. 2018) and in the red giantphase (Mosser et al. 2015; Eggenberger et al. 2017; Gehan et al.2018) – see Aerts et al. (2019) for a recent extensive review.Since the evolution of a star is greatly affected by the stellarrotation profile (Maeder 2009), the opportunity to calibrate theo-retical stellar models from empirically derived angular momentumdistributions in stellar interiors at different evolutionary stages is ofmajor importance. In order to compute these angular momentumdistributions, the mass and radius of the star as a whole and ofits convective core have to be estimated, as well as the star’s age.Given that forward asteroseismic modelling of intermediate-massstars is a high dimensional problem (+6D, Aerts et al. 2018), arobust statistical methodology is needed.In this paper, we explore the feasibility of estimating the mostimportant stellar parameters, i.e., (core) mass, core overshooting,initial hydrogen mass fraction, metallicity, the amount of envelopemixing, and age (or age-proxy) of γ Doradus (henceforth γ Dor)stars. While these pulsators of spectral type late-A to early-F werealready known to have g modes excited by a flux blocking mech-anism from ground-based studies (e.g., Kaye et al. 1999; Guziket al. 2000; Dupret et al. 2005; Cuypers et al. 2009; Bouabid et al.2013), the derivation of g-mode period spacings had to await thelong-term uninterrupted
Kepler data (e.g., Bedding et al. 2015; VanReeth et al. 2015a,b).Van Reeth et al. (2016, 2018) and Ouazzani et al. (2017) havedeveloped methods to infer the near-core rotation frequency fromthe slope of the measured period spacing pattern of γ Dor stars.These authors applied their methods to ensembles of 40 and 4 stars,respectively, of such gravito-inertial pulsators. Here, we considerthe sample by Van Reeth et al. (2016) because 37 of these starshave been monitored with high-resolution spectroscopy (Tkachenkoet al. 2013) and the asymptotic period spacing from their progradesectoral dipole modes, effective temperatures, surface gravities andmetallicity have been determined in a homogeneous way.We explore the modelling capacity of the combined seismicparameter for dipole modes (in the non-rotating limit), Π , andthe spectroscopically derived effective temperature, T eff and surfacegravity, log g , as a major simplification of typical forward modellingthat is based on the fitting of all the individual g-mode frequencies.In our approach, we first derive correlations between the the seis-mic parameter Π , the effective temperature and surface gravity onone hand, and the correlation between these three observables andthe stellar parameters varied in two 6D stellar model grids on theother hand. Previous studies in the literature have derived correla-tions between the stellar mass, metallicity, and step overshootingfor low-order modes in β Cep stars (e.g., Briquet et al. 2007; Wal-czak et al. 2013) and for high-order g-modes in a
Kepler slowlypulsating B-type star (Moravveji et al. 2015). In Section 2, we in-vestigate correlations between mass, central and initial hydrogenfraction, metallicity, the amount of envelope mixing, and the massand radius of the convective core, by means of linear multivariateregression to investigate the correlations between the parameters ina simple manner. We do this for both a step and an exponential core overshooting formalism since it was recently shown that g modespotentially allow for these two overshooting prescriptions to be dis-tinguished (Pedersen et al. 2018). In particular, we also investigatehow well a benchmark model – based on one of these two coreovershooting prescriptions – can be approximated by a model basedon the other overshooting prescription for the mass range of γ Dorstars in Section 3.Armed with the knowledge of the correlation structure in thetwo 6D model grids, we explore the capacity of these diagnostics ( Π , T eff , log g ) for parameter estimation from seismic modelling,using the methodology based on maximum likelihood estimation de-veloped in Aerts et al. (2018). The results of our forward modellingare presented in Section 4. In Section 5, we describe our methodol-ogy of determining uncertainties from ensemble modelling and weconclude in Section 6. In the framework of the traditional approximation of rotation (TAR;Eckart 1960; Unno et al. 1989), the period of a high-order g-modecan be well approximated by P co ≈ Π (cid:112) λ l , m , s ( n g + α g ) , (1)with λ the eigenvalue of the Laplace tidal equation, depending onthe mode geometry (spherical degree l and azimuthal order m ) andthe spin parameter s = f co / f rot (cf., Townsend 2003; Bouabidet al. 2013; Van Reeth et al. 2018). The phase term α g depends onthe stellar structure and Π ≡ π (cid:18)∫ gc Nr d r (cid:19) − . (2)Here, the quantity N is the Brunt-Väisälä frequency and N / r is inte-grated over the gravity-mode cavity indicated as ‘gc’. An example ofsuch a cavity is shown in Fig. A1 in the Appendix. The asymptoticperiod spacing of g modes, i.e., the difference in period betweentwo modes of consecutive radial order and same angular degree, inthe case of a chemically homogeneous, non-magnetic star is definedin the corotating frame as ∆ P co ≈ Π (cid:112) λ l , m , s . (3)Depending on the nature of the mode and on the value of the spinparameter, the value of λ l , m , s can be approximated by simple ana-lytical expression (cf., Townsend 2003; Saio et al. 2017, Fig. A1).The spin parameter for the gravito-inertial modes and Rossby modesof the stars in our sample ranges from 1 to 30 (cf., Fig. 2 of Aertset al. 2017). Given this broad range, it is not obvious to resort toanalytical approximations for λ l , m , s in the case of γ Dor stars. Thismotivated Van Reeth et al. (2016) to work with numerical solutionsof the Laplace tidal equation.Comparison between the measured and theoretically predictedgravity-mode periods requires the transformation of the periodsto an inertial frame of reference. In the case of a uniform stellarrotation with frequency f rot , the periods of the oscillation modes inan inertial frame can be computed as P inert = f co + m f rot , (4) MNRAS , 1–14 (2018) steroseismic modelling of γ Dor stars where f co is the mode frequency in the corotating frame. We adoptthe convention that prograde modes correspond to m > 0.In the forward modelling applied here, we rely on the obser-vational estimates of Π and of f rot as determined by Van Reethet al. (2016, 2018). We therefore compute the quantity Π fromtheoretical stellar evolution models. The diagnostic power of theBrunt-Väisälä frequency and the Π of g modes are well knownand have already been exploited asteroseismically since ground-based multi-site white dwarf asteroseismology (Winget et al. 1991;Brassard et al. 1992). The g modes of these compact objects haveperiods of order a few to ten minutes, such that beating patternscan be covered in observing runs lasting several weeks. Moreover,the rotation period of such stars is of order days, such that therotational effect on the pulsation modes can be treated from a per-turbative approach. The models and g modes in white dwarfs arequite different from those of young stars, hence the parameters thatcan be deduced from them, as well as the mode trapping propertiesdiffer accordingly. From an observational point of view, the applica-tion and exploitation of g-mode asteroseismology to core-hydrogenburning stars had to await long-term uninterrupted Kepler pho-tometry, given that the modes have periodicities of order days andbeating patterns of years. Moreover, the g modes and rotation ratesin these stars have similar periodicities, such that rotation cannot betreated perturbatively, except for a few ultra-slow rotators. This re-quires dedicated asteroseismic modelling tools suitable to interpretthe measured frequencies. Here, we explore and apply aspects ofthe methodology developed specifically for gravito-inertial modesby Aerts et al. (2018). We first highlight relevant properties of theasteroseismic grids upon which we rely and subsequently exploitthe probing power of the three observables ( Π , T eff , log g ). Van Reeth et al. (2016) computed two extensive grids of non-rotating stellar modes with the MESA stellar evolution code (r7385,Paxton et al. 2018, and references therein). These models vary instellar mass ( M (cid:63) ), metallicity ( Z ), diffusive envelope mixing ( D mix ,constant throughout the radiative zone), the extension of the coreovershoot region α ov / f ov (i.e., for step/exponential overshoot, ex-pressing the mean free path of a convective fluid element in a ra-diative region in terms of the local pressure scale height), the initial( X ini ) and normalised central hydrogen content ( X (cid:48) c = X c / X ini ). Thelatter is a proxy for the evolutionary stage, hence for the age of themodel. In this paper, these two grids of evolution models were ex-tended to lower mass compared to Van Reeth et al. (2016) – a lowerlimit of 1.2 M (cid:12) instead of 1.4 M (cid:12) . For the models with a convectiveenvelope, we did not consider any undershooting, because this isnot important to assess the probing power of high-order g modes.In Table 1, an overview is given of the parameters varied acrossthe two grids and respective step sizes. The efficiency of convection,in our model grids parameterized by the mixing length parameter α MLT , was kept fixed at 1.8 throughout the grids. As this was donewithout further discussion in Van Reeth et al. (2016), we point outhere that this parameter does influence the three quantities Π , T eff ,and log g . This is particularly the case for the lower-mass stars inthe model grids, since these have the larger convective envelopeand are hence affected more by the treatment of convection thanthe higher-mass stars. Using table 2 from Viani et al. (2018) withthe appropriate α MLT value, we made an estimate for the range ofthe mixing length parameter for the stars in our sample, resulting in α MLT ∈ [ . , . ] . Varying α MLT between 1.5 and 1.9, we find thatonly for the most evolved low-mass stars do the differences in T eff Parameter Lower boundary Upper boundary Step size
Step overshoot M (cid:63) (cid:12) (cid:12) (cid:12) Z D mix [ cm s − ] -1 0 1 α ov X ini X (cid:48) c Exponential overshoot M (cid:63) (cid:12) (cid:12) (cid:12) Z D mix [ cm s − ] -1 1 1 f ov X ini X (cid:48) c Table 1.
Range of the model grids for which Π , T eff , and log g have beencomputed. become comparable to typical observational uncertainties, while for Π and log g the differences are always negligible. We illustrate thisin Figs C1–C3 in Appendix C for a low-mass and high-mass γ Dormodel. We do stress that α MLT has a major influence in forwardmodelling based on fitting individual g-mode frequencies (Aertset al. 2018, table 2), but that this dependency plays an inferior rolecompared to the other parameters varied in our grids for forwardmodelling based on Π .For the modelling done here, we also transformed to the param-eter X (cid:48) c = X c / X ini ∈ [ , ] , i.e., to the fraction of the main-sequenceduration, as a proxy for the age, rather than using X c itself as inVan Reeth et al. (2016). In this way, the span of X (cid:48) c in the grids isthe same for all model tracks ranging from the zero-age main se-quence (ZAMS; X (cid:48) c =
1) to the terminal-age main sequence (TAMS; X (cid:48) c = X (cid:48) c > .
99. Intotal, the step overshoot grid contains 1530 evolutionary tracks with819 774 stellar models and the exponential overshoot grid contains2295 evolutionary tracks with 1 300 590 stellar models.For each stellar model in the grids, Π was computed, alongwith the effective temperature T eff and surface gravity log g , aswell as the mass M cc and radius R cc of the convective core. Werefer to Table D1 in Appendix D, where we assembled the measuredvalues of Π from Kepler data as determined by Van Reeth et al.(2018) under the assumption of rigid rotation. For the stars withboth gravito-inertial prograde dipole modes and Rossby modes,these are improved values compared to those in Van Reeth et al.(2016). The observational estimate of Π for all these stars wasdeduced along with estimation of the near-core rotation frequency f rot from the slope of the measured period spacing patterns. Therelative observational errors for Π range from 0.2 to 26 per centfor the various sample stars. Below, we compare these observableswith the theoretical predictions for Π computed for each of the gridmodels. We recall explicitly that Van Reeth et al. (2016) carefullyomitted modes that are trapped to estimate Π and f rot from the data(cf., their figs 4 and 9). These two observed quantities, along withthe mode identification, were derived using the TAR to compute λ l , m , s with the pulsation code GYRE (Townsend & Teitler 2013;Townsend et al. 2018) for each of the models in the two extensive6D grids, which cover the relevant parameter ranges of γ Dor stars.In this way, our observational estimation of Π and f rot does notdepend on particular choices of these model parameters.We recall that mixing in the radiative zone had to be introduced MNRAS000
99. Intotal, the step overshoot grid contains 1530 evolutionary tracks with819 774 stellar models and the exponential overshoot grid contains2295 evolutionary tracks with 1 300 590 stellar models.For each stellar model in the grids, Π was computed, alongwith the effective temperature T eff and surface gravity log g , aswell as the mass M cc and radius R cc of the convective core. Werefer to Table D1 in Appendix D, where we assembled the measuredvalues of Π from Kepler data as determined by Van Reeth et al.(2018) under the assumption of rigid rotation. For the stars withboth gravito-inertial prograde dipole modes and Rossby modes,these are improved values compared to those in Van Reeth et al.(2016). The observational estimate of Π for all these stars wasdeduced along with estimation of the near-core rotation frequency f rot from the slope of the measured period spacing patterns. Therelative observational errors for Π range from 0.2 to 26 per centfor the various sample stars. Below, we compare these observableswith the theoretical predictions for Π computed for each of the gridmodels. We recall explicitly that Van Reeth et al. (2016) carefullyomitted modes that are trapped to estimate Π and f rot from the data(cf., their figs 4 and 9). These two observed quantities, along withthe mode identification, were derived using the TAR to compute λ l , m , s with the pulsation code GYRE (Townsend & Teitler 2013;Townsend et al. 2018) for each of the models in the two extensive6D grids, which cover the relevant parameter ranges of γ Dor stars.In this way, our observational estimation of Π and f rot does notdepend on particular choices of these model parameters.We recall that mixing in the radiative zone had to be introduced MNRAS000 , 1–14 (2018)
J. S. G. Mombarg et al. in addition to core overshooting in models of B-type pulsators to fitthe frequencies of trapped g modes (Moravveji et al. 2015, 2016).In Fig. 1, the evolution of Π as a function of the stellar age is illus-trated, where either X ini , Z , α ov / f ov or D mix is being varied, whilethe three remaining parameters are kept fixed, for masses 1.3, 1.6,and 1.9 M (cid:12) . This gives a good visual representation of the depen-dencies of Π on these four model parameters in the grids. It can beseen that there is no unique monotonic relation between mass, ageand Π because the core overshooting, metallicity and initial hydro-gen (in this descending order of importance) do have an effect largerthan the typical measurement uncertainty of Π deduced from the Kepler light curves (cf., Aerts et al. 2018, for a thorough discussionof theoretical uncertainties on g-mode frequencies). The envelopemixing does not influence the period spacing values as much as theother three parameters. This is entirely as expected, given that theg modes mainly probe the core overshooting, while they are hardlyaffected by the envelope mixing. However, the mixing at the inter-face between the overshoot zone and the bottom of the radiativeenvelope is responsible for the details of the mode trapping and hasto be considered when performing frequency fitting as opposed tomatching of the asymptotic period spacing (cf., Moravveji et al.2016; Van Reeth et al. 2016; Pedersen et al. 2018). Π Following problem set 3 in Aerts et al. (2018), we construct astatistical model for Π based on the stellar input parameters of themodel grids by performing a linear multivariate regression, adoptingthe following form Π = β + β α ov + β D mix + β M (cid:63) + β Z + β X ini + β X (cid:48) c + (cid:15), (5)where (cid:15) is the residual and the coefficients β i are computed accord-ing to an ordinary least-squares regression, β = ( X (cid:62) X ) − X (cid:62) Y . (6)Here, Y is a vector with length equal to the number of grid points i = , . . . , q , containing all values of Π , i and X = (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) α ov , D mix , M (cid:63), Z X ini , X (cid:48) c , α ov , D mix , M (cid:63), Z X ini , X (cid:48) c , ... ... ... ... ... ... ... α ov , q D mix , q M (cid:63), q Z q X ini , q X (cid:48) c , q (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) . (7)For the step overshoot grid, q =
819 774, while for the exponentialovershoot grid, q = (cid:15) captures the non-linearity in Π as it oc-curs in the model grids. We point out that both the coefficients β i and (cid:15) depend on the choices of microphysics that went into thecomputation of the stellar models. This concerns opacity tables,the equation of state, nuclear reactions and chemical mixtures. Theeffect of these choices on g-mode pulsation frequencies cannot berepresented by a few simple model parameters, as discussed in de-tail in sections 2 to 4 in Aerts et al. (2018). That paper’s table 2 alsoincludes a detailed quantification and hierarchical ordering of theeffect of the choices of micro- and macro-physics when performingforward asteroseismic modelling of main-sequence stars.Once a value for each of the six input parameters is chosen, themass and radius of the convective core (denoted as M cc and R cc ,respectively) then follow. The computation of M cc and R cc in oursetup comes through the estimation of the core boundary mixingproperties, of which the parameters f ov and D mix are simple repre-sentations. With Eq. 5, we represent the Π from the grid models Step overshoot Exponential overshoot β ( Offset ) ± ± β ( α ov ) -258.2 ± ± β ( D mix ) -53.0 ± ± β ( M (cid:63) ) ± ± β ( Z ) ± ± β ( X ini ) ± ± β ( X (cid:48) c ) ± ± β ( M cc ) ± ± β ( R cc ) -24236.95 ± ± Table 2.
Regression coefficients from an ordinary least squares fit for thebest statistical model in Eq. (8) according to the BIC for Π (s).Step overshoot Exponential overshoot β ( Offset ) ± ± β ( α ov ) -393.9 ± ± β ( D mix ) - -0.48 ± β ( M (cid:63) ) ± ± β ( Z ) -87369.35 ± ± β ( X ini ) -6879.91 ± ± β ( X (cid:48) c ) -792.5 ± ± β ( M cc ) -11630.93 ± ± β ( R cc ) ± ± Table 3.
Same as Table 2, but for a linear regression in the form of Eq. (8)for T eff (K). Step overshoot Exponential overshoot β ( Offset ) ± ± β ( α ov ) -0.2478 ± ± β ( D mix ) -0.0015 ± ± β ( M (cid:63) ) -0.2440 ± ± β ( Z ) -1.22 ± ± β ( X ini ) ± ± β ( X (cid:48) c ) -0.0080 ± ± β ( M cc ) -3.592 ± ± β ( R cc ) ± ± Table 4.
Same as Table 2, but for a linear regression in the form of Eq. (8)for log g ( g given in cm − s − ). by a multivariate linear statistical estimation, with the aim to havea fast tool to compute model properties for parameter sets that fallwithin the range of those of the grids but do not coincide with anactual grid point. The fit in Eq. 5 also highlights the correlationsamong the six parameters that should be kept in mind when tryingto fit observed values of Π . Since the fit for Π from Eq. 5 is anapproximation, there is no longer a unique correspondence betweenthe six input parameters and the values for M cc and R cc for param-eter combinations that do not coincide with a grid model. For thisreason, we also pay specific attention to M cc and R cc .Some of the models in the grids have a growing convective coreas the hydrogen-burning progresses, while others have a shrinkingconvective core, where the transition occurs roughly around a birthmass of 1.6 M (cid:12) (cf., Fig. 3.6 in Aerts et al. 2010). This is illustratedfor our grids of models in Fig. 2 in terms of the mass and size ofthe convective core. The phenomenon of a shrinking or growingcore lies at the basis of the different correlation structure seen in thedifferent morphologies of the trends for different masses in Fig. 1 interms of the stellar mass, because the mode cavities are influencedby it during the evolution of the star. Moreover, the extent of theovershoot zone affects the mode trapping and shrinkage or growthof the convective core is influenced by the core overshooting. Asummary representation of the maximal growth of the convective MNRAS , 1–14 (2018) steroseismic modelling of γ Dor stars Figure 1.
Evolution of Π along the main sequence for changing metallicity (top left), amount of core overshooting (top right), the amount of mixing inthe radiative zone (bottom left) and initial hydrogen mass fraction (bottom right), for a 1.3-, 1.6- and 1.9-M (cid:12) star (lightest colour to darkest). The blue linescorrespond to an exponential overshooting prescription and the red lines to a step overshooting prescription, both of which consider the radiative temperaturegradient in the overshoot zone. When a parameter is not being varied, it is set at Z = . α ov = . f ov = . D mix = . s − and X ini = . core mass during the hydrogen burning, for the entire mass range ofthe grids and for the two descriptions of the overshooting, is shownin Fig. 3. The gain in core mass gradually decreases as the birthmass increases. It disappears for the most massive stellar modelsin our grid, as their convective core never grows, but only shrinksafter birth. Moreover, the evolution of the core mass is stronglydependent on the value of the overshoot parameter.The results in Figs 2 and 3 imply that the mass and size of theconvective core correlate with Π and that the correlation structureis different for step and exponential overshoot. In order to investi-gate these dependencies, we add these two quantities to the linearregression model, Π = β + β α ov + β D mix + β M (cid:63) + β Z + β X ini + β X (cid:48) c + β M (cid:48) cc + β R (cid:48) cc + (cid:15), (8)where M (cid:48) cc is expressed in terms of M (cid:63) and R (cid:48) cc is expressed interms of R (cid:63) . The core boundary is defined by the Ledoux criterionand we defined the core radius R cc as the point where the Brunt-Väisälä frequency becomes larger than a small threshold N > − rad s − . With this set up, we test how a multivariate linearmodel captures the influence of the core mass and size on Π . Theresults are listed in Table 2 for both grids.We evaluate the capacity of the multivariate linear model inEq. (8) by inspecting the square-root of the residual sum of squaresof the fit, averaged over all grid points, denoted here as (cid:112) (cid:104) RSS (cid:105) .We find a value of 252 s for step overshoot and 253 s for exponentialovershoot. As these (cid:112) (cid:104)
RSS (cid:105) values are comparable with typicalmeasurement uncertainties for Π (Table D1), we conclude that Π can generally be well approximated by a multivariate linear statisti-cal model up to the level of the measurement uncertainties. We dokeep in mind that the approximation works less well near the TAMS,where non-linearity of Π is larger than for the other evolutionary phases (cf., Fig. 1). Non-linear multivariate regression to approxi-mate Π (and T eff and log g further on) is beyond the scope of thispaper, but may be interesting to consider as an improvement of ourcurrent work for future stellar modelling of near-TAMS pulsators.Both overshooting prescriptions suggest that the amount ofmixing in the radiative zone, D mix , has only a modest effect on Π , as already reflected in Fig. 1 (see also table 2 in Johnston et al.2019). However, we expect D mix to have some effect on Π (VanReeth et al. 2015b) and mainly on the mode trapping, as was shownfrom forward modelling of g modes in the Kepler
B-type pulsatorKIC 10526294 (Moravveji et al. 2015). For this reason, we appliedthe principle of ‘backward selection’, where one eliminates one-by-one the least significant β -parameter and test if the simpler statisticalmodel is more appropriate (cf., Aerts et al. 2014, where this isexplained in more detail and was applied to a similar problem).A formal way to deduce which statistical model is the more ap-propriate one is the Bayesian Information Criterion (BIC). Amongmany other statistical tests, the BIC corrects for the complexity ofa statistical model by applying a penalty involving the degrees offreedom (Claeskens & Hjort 2008) rather than just using the RSSfor the model selection. Here, we make an application, usingBIC ≡ q ln (cid:18) RSS q (cid:19) + k ln ( q ) , (9)where k is the number of free parameters and q is the number ofgrid points. Starting with a statistical model described by Eq. (8),we compute the corresponding β values and their p -value, which isdefined in the case q (cid:29) k as, p = [ − Φ (| t |)] , (10)where Φ (| t |) is the cumulative standard normal distribution, i.e.,the integral of the standard normal density between −∞ and | t | . Its MNRAS000
B-type pulsatorKIC 10526294 (Moravveji et al. 2015). For this reason, we appliedthe principle of ‘backward selection’, where one eliminates one-by-one the least significant β -parameter and test if the simpler statisticalmodel is more appropriate (cf., Aerts et al. 2014, where this isexplained in more detail and was applied to a similar problem).A formal way to deduce which statistical model is the more ap-propriate one is the Bayesian Information Criterion (BIC). Amongmany other statistical tests, the BIC corrects for the complexity ofa statistical model by applying a penalty involving the degrees offreedom (Claeskens & Hjort 2008) rather than just using the RSSfor the model selection. Here, we make an application, usingBIC ≡ q ln (cid:18) RSS q (cid:19) + k ln ( q ) , (9)where k is the number of free parameters and q is the number ofgrid points. Starting with a statistical model described by Eq. (8),we compute the corresponding β values and their p -value, which isdefined in the case q (cid:29) k as, p = [ − Φ (| t |)] , (10)where Φ (| t |) is the cumulative standard normal distribution, i.e.,the integral of the standard normal density between −∞ and | t | . Its MNRAS000 , 1–14 (2018)
J. S. G. Mombarg et al.
Figure 2.
Evolution from near-ZAMS to TAMS of the mass (top panel)and radius (middle panel) fraction of the convective core for different stellarmasses. The evolution of the absolute radius of the convective core is givenin the bottom panel. The tracks are for X ini = . Z = . D mix = . s − and f ov = . argument t is inverse of the relative uncertainty on β , where theabsolute uncertainties are computed by taking the square-root ofthe diagonal elements of the variance-covariance matrix V ( β ) = ( X (cid:62) X ) − σ , (11)in which σ = q − ( k + ) ( Y − Xβ ) (cid:62) ( Y − Xβ ) , (12)where for our grids q (cid:29) k . Next, we compute a new statistical modelwhere the β i with the highest p -value from the previous model isomitted, and the new BIC is evaluated. This process is repeateduntil the BIC of the new model is higher than the previous one.This previous model is then selected to be the optimal statisticalmodel. We refer the interested reader to Aerts et al. (2018) for moredetails on the statistical framework discussed here. For the statisticalmodels for Π , none of the parameters can be omitted according tothe BIC criterion. T eff and log g For 37 of the γ Dor stars with identified dipole modes in the sam-ple from Van Reeth et al. (2016), a measurement of T eff , log g ,and [M/H] has been obtained from high-resolution spectroscopy.Uncertainties for [M/H] are large ( ∼
100 per cent) (fig. 2 and ta-ble 5 in Van Reeth et al. 2015a), but the spectroscopic T eff andlog g measurements have relative average precisions of ∼ Figure 3.
The maximum difference in mass fraction of the convective coreas a function of stellar mass for different overshooting prescriptions. Forvisibility purposes, the data points corresponding to exponential overshoothave been shifted by 0.01 M (cid:12) along the abscissa axis. and ∼ T eff and log g are comparable with the av-erage precision on Π (for the stars that have spectroscopic data)these observables are good candidates to lift part of the degeneracybetween Π and the stellar parameters. For the statistical mod-els of T eff , the values of (cid:112) (cid:104) RSS (cid:105) are 233 and 229 K for step andexponential overshoot, respectively. For log g , the statistical modelhas (cid:112) (cid:104) RSS (cid:105) = .
05 dex for both overshooting prescriptions. For T eff this is larger than the typical uncertainty on the observed value mea-sured by Van Reeth et al. (2015a), while for log g it is significantlybetter than the observed uncertainties. As shown in Fig. 4, there isa correlation between Π and T eff from near-ZAMS ( X (cid:48) c ∼ . X (cid:48) c ∼ X ini , Z , α ov / f ov or D mix while keeping the other parameters fixed and do this for 1.3,1.6 and 1.9 M (cid:12) .Analogous to the procedure developed for Π above, wesearched for an optimal linear multivariate regression model for T eff and log g . Again, we start with a statistical model as describedin Eq. (8) and perform backward selection to see if any redundantparameters can be eliminated, by minimizing the BIC. For expo-nential overshoot we find that D mix has little effect on T eff , but BICincreases when this parameter is not taken into account, suggestingit is not a redundant parameter. In the case of step overshoot, adecrease in BIC of suggests the model without D mix is statisticallyfavoured. The results of these linear regressions for T eff are listed inTable 3. For the statistical models of log g , we find that none of theparameters may be omitted according to the BIC (Table 4). MNRAS , 1–14 (2018) steroseismic modelling of γ Dor stars Figure 4.
Correlation between the effective temperature T eff and the seismic parameter Π along the main sequence (starting at X (cid:48) c = .
99) for changingmetallicity (top left), amount of core overshooting (top right), the amount of mixing in the radiative zone (bottom left) and initial hydrogen mass fraction(bottom right), for a 1.3- (lightest colours), 1.6- and 1.9- M (cid:12) (darkest colours) star. The blue lines correspond to an exponential overshooting prescription andthe red lines to step overshooting prescription, both of which consider the radiative temperature gradient in the overshoot zone. When a parameter is not beingvaried, it is set at the values X ini = . Z = . D mix = . s − , α ov = .
225 and f ov = . Π at constant T eff seen in the tracks of the 1.3 M (cid:12) models results from thetransition from pre-MS to the MS. These models are already on the main-sequence track and are burning hydrogen in their convective cores. A major uncertainty in evolution theory of star born with a convec-tive core is the efficiency of the mixing inside the core overshootregion. This translates into the question of the functional prescrip-tion of the overshooting. It was recently shown that g modes havethe potential to unravel the most appropriate shape, or at least todiscriminate between an exponential and a step overshooting pre-scription (Pedersen et al. 2018).Here, we compare theoretical models with step versus expo-nential overshooting prescriptions for the mass range of the γ Dorstars to test if we can make a distinction between our two grids ofmodels. We do this by applying the method of parameter estimationand model selection described in Aerts et al. (2018, problem 2).For both overshooting prescriptions, we chose a benchmark modelfrom one grid and fit the corresponding Π , effective temperatureand surface gravity to the models in the other grid. Adopting thenotations in Aerts et al. (2018), we use the subscript to indicatethe ‘observational’ values (BM) and a superscript to distinguishbetween a step (s) and an exponential (e) overshooting prescription.We then compute the optimal (independent) parameters θ = ( M (cid:63) , X (cid:48) c , α ov / f ov , Z , X ini , D mix ) , (13)for both benchmark models according to θ ( e ) = arg r (cid:48) e min i = (cid:20)(cid:16) Y ( e ) i − Y ( s ) BM (cid:17) (cid:62) (cid:16) ˆ V ( e ) (cid:17) − (cid:16) Y ( e ) i − Y ( s ) BM (cid:17)(cid:21) , (14) θ ( s ) = arg r (cid:48) s min j = (cid:20)(cid:16) Y ( s ) j − Y ( e ) BM (cid:17) (cid:62) (cid:16) ˆ V ( s ) (cid:17) − (cid:16) Y ( s ) j − Y ( e ) BM (cid:17)(cid:21) , (15)where the index i = , . . . ,
819 774 runs over all stellar modelsin the grid with the exponential overshooting prescription and theindex j = , . . . , θ indicates thecorresponding grid) and Y i / j = (cid:169)(cid:173)(cid:171) Π , i / j T eff , i / j log g i / j (cid:170)(cid:174)(cid:172) . (16)Moreover, we define the matrix V ( Y ) = q (cid:48) − q (cid:48) (cid:213) k = (cid:0) Y k − ¯ Y (cid:1) (cid:0) Y k − ¯ Y (cid:1) (cid:62) , (17)taking into account the variance across the grid, where ¯ Y is themean value of Y in the grid. We choose eight benchmark γ Dormodels; a young low-mass star, an old low-mass star, a young high-mass star and an old high-mass star, with Z = . X ini = . α ov ( f ov ) = . ( . ) and D mix = . s − , all for both over-shooting prescriptions. In Table 5 we list the maximum likelihoodestimate (MLE) from using an incorrect overshooting prescriptionfor each of these eight benchmark models. For all of these models,the resulting θ from fitting ( Π , T eff , log g ) does not change dras-tically, compared to the step size of the grids for M (cid:63) and X (cid:48) c . Thegrids contain too few points in the other parameters to draw a con-clusion on how robust the MLE is for these parameters. However, MNRAS000
819 774 runs over all stellar modelsin the grid with the exponential overshooting prescription and theindex j = , . . . , θ indicates thecorresponding grid) and Y i / j = (cid:169)(cid:173)(cid:171) Π , i / j T eff , i / j log g i / j (cid:170)(cid:174)(cid:172) . (16)Moreover, we define the matrix V ( Y ) = q (cid:48) − q (cid:48) (cid:213) k = (cid:0) Y k − ¯ Y (cid:1) (cid:0) Y k − ¯ Y (cid:1) (cid:62) , (17)taking into account the variance across the grid, where ¯ Y is themean value of Y in the grid. We choose eight benchmark γ Dormodels; a young low-mass star, an old low-mass star, a young high-mass star and an old high-mass star, with Z = . X ini = . α ov ( f ov ) = . ( . ) and D mix = . s − , all for both over-shooting prescriptions. In Table 5 we list the maximum likelihoodestimate (MLE) from using an incorrect overshooting prescriptionfor each of these eight benchmark models. For all of these models,the resulting θ from fitting ( Π , T eff , log g ) does not change dras-tically, compared to the step size of the grids for M (cid:63) and X (cid:48) c . Thegrids contain too few points in the other parameters to draw a con-clusion on how robust the MLE is for these parameters. However, MNRAS000 , 1–14 (2018)
J. S. G. Mombarg et al.
Figure 5.
Difference between Π in the step overshoot grid ( Π ( s ) ) andthe exponential overshoot grid ( Π ( e ) ), for pairs of models with the sameparameters M (cid:63) , X (cid:48) c , D mix , Z , X ini , varied across the grid. An α ov in onegrid corresponds to 10 f ov in the other grid. The pairs of models are equallydistributed across both grids and are arbitrarily labelled from 1 to 3541.When M (cid:63) ≥ . (cid:12) the maximum difference is comparable with a typicalobservational uncertainty on Π . For visual aid, a dashed line is plotted atzero difference. even though there is scatter in α ov / f ov , D mix , Z and X ini , similarmasses and age-proxies are found.Furthermore, we note that for α ov = f ov , the MLE agreeswell with the input parameters (see Fig. 5). This scaling with afactor 10 between α ov and f ov is a commonly used rule-of-thumbwhen comparing step overshoot and exponential overshoot. In Fig. 5,the difference in Π is plotted between two models, each from adifferent grid, that have the same parameters ( X (cid:48) c within 0.001)where α ov = f ov . This scaling rule does give similar values for Π within the parameter space of γ Dor stars, compared to typicaluncertainties on this seismic parameter, as the difference is lessthan 60 s for masses above 1.4 M (cid:12) . The deviation between the twogrids at the lower masses is a result from the fact that this factor10 is not exact and also mass dependent (11.36 ± Y ( obs ) can be matched in the other grid. This can be seen in the lastthree columns of Table 5. Π , T eff , AND log g We aim to test the feasibility of estimating (some of) the six param-eters in the two grids for all γ Dor stars in the sample of Van Reethet al. (2016) with Π deduced from prograde dipole modes in anautomated way. We recall that these measured values of Π were de-rived along with estimation of the near-core rotation frequency (VanReeth et al. 2016, 2018). Our aim is to test if Π , along with spectro-scopic measurements of T eff and log g are sufficient to estimate M (cid:63) , X (cid:48) c , α ov / f ov , and possibly Z , X ini and/or D mix from an ensemble of γ Dor stars (see Fig. 6, illustrating the 2D projected 1 σ errors on theobserved parameters of each star). We use an automated grid-basedapproach, keeping in mind the uncertainties of individual theoret-ically computed g-mode frequencies due to unknown aspects ofthe input physics of the stellar models, as assessed by Aerts et al.(2018). These authors describe a new method for forward seismic modelling, where the correlation between the observables is takeninto account. This method is based on the Mahalanobis distancewhich is defined in our 3D case as D M , j = (cid:16) Y ( th ) j − Y ( obs ) (cid:17) (cid:62) ( Λ + V ) − (cid:16) Y ( th ) j − Y ( obs ) (cid:17) . (18)The matrix Λ = diag ( σ Π , σ T eff , σ g ) takes the uncertainties ofthe observed quantities into account and the (co-)variance matrix V is computed according Eq. (17) for both grids.We first get MLEs for all stars in our sample for whichspectroscopic data is available by minimizing the Mahalanobisdistance across each of the grids. As has already been demonstratedwith the statistical models presented in Section 2, some parametersare more strongly correlated with the observables than others. Wetherefore assess whether it is possible to reduce the dimensionof our forward modelling problem from a principal componentanalysis (PCA). For each star, the best 10 per cent of the modelsis selected and the basis of this 6D solution space is redefinedin terms of its principal components (PCs). These PCs are theeigenvectors of the correlation matrix and the correspondingnormalised eigenvalues give the percentages of the total variancethat is captured by each PC. As a rule-of-thumb, the number of PCsselected is the number of PCs which is needed to describe at least80 per cent of the total variance (Jolliffe 2002). In this way, we findthat four PCs are sufficient for all the 37 stars in the sample.The PCs that can be dropped are predominantly connected with D mix and X ini . Therefore, we wish to remove these two parametersfrom the grid to reduce the dimensionality, by fixing a value forthem. In order to assign the most appropriate fixed value, we con-sider each of the six combinations of these two parameters availablein the grids and determine the MLE for all stars in our sample. For thefour estimated parameters, the average standard deviations betweenthese six MLEs is found to be (cid:104) σ M (cid:63) (cid:105) = .
06 M (cid:12) , (cid:104) σ X c (cid:105) = . (cid:104) σ α ov (cid:105) = .
061 and (cid:104) σ Z (cid:105) = . (cid:104) σ M (cid:63) (cid:105) = .
06 M (cid:12) , (cid:104) σ X c (cid:105) = . (cid:104) σ f ov (cid:105) = . (cid:104) σ Z (cid:105) = . D mix and X ini can be set to the fixed values 1 . s − and 0.71, respectively.The results of the parameters estimations from MLE are listedin Table D1 for both overshooting prescriptions. The correspondingfractional mass and fractional radius of the convective core of thebest model are also listed. For M (cid:63) , X (cid:48) c , M (cid:48) cc and R (cid:48) cc (fractional massand radius of the convective core), the grid is comprised of a suffi-ciently large number of points to make a comparison. However, forthese parameters there is no obvious discrepancy reported betweenthe two overshooting prescriptions for any given star.The relation between the near-core rotation frequency f rot andthe evolutionary state of the stars in our sample has previously beeninvestigated by Aerts et al. (2017, their fig. 1), where the spectro-scopic log g was used as a proxy for the evolutionary stage. In Fig. 7,log g is replaced by the MLE of X (cid:48) c for both overshooting prescrip-tions. We colour-code for the measured near-core rotation rate, forwhich the relative uncertainty is typically a few per cent. From thesemore precise estimates of the proxy for stellar age, the data revealthat the cores of intermediate-mass A and F stars spin down as thestars evolve.Ouazzani et al. (2018) report that faster rotators are less mas-sive and younger than the slow rotators. This is consistent withour findings that the two aforementioned faster rotating near-ZAMSstars are less massive than the slower rotating stars with roughlythe same X (cid:48) c , as shown in the top row of Fig. 7. As the γ Dor insta-bility region is relatively small (Bouabid et al. 2013), it is possible
MNRAS , 1–14 (2018) steroseismic modelling of γ Dor stars M (cid:63) [ M (cid:12) ] X (cid:48) c α ov f ov D mix [cm − s − ] Z X ini Π [s] T eff [K] log g BM 1.30 0.90 0.150 - 1.0 0.014 0.71 4327 6669 4.33GRID 1.35 0.92 - 0.0075 10.0 0.014 0.73 4330 6678 4.33BM 1.30 0.90 - 0.0150 1.0 0.014 0.71 4352 6669 4.33GRID 1.25 0.88 0.225 - 1.0 0.014 0.69 4343 6657 4.33BM 1.30 0.10 0.150 - 1.0 0.014 0.71 3140 6158 4.04GRID 1.35 0.10 - 0.0010 10.0 0.018 0.69 3147 6186 4.04BM 1.30 0.10 - 0.0150 1.0 0.014 0.71 3182 6104 4.01GRID 1.25 0.10 0.225 - 1.0 0.014 0.69 3179 6099 4.02BM 2.00 0.90 0.150 - 1.0 0.014 0.71 5551 9217 4.29GRID 2.00 0.90 - 0.0150 0.1 0.014 0.71 5535 9216 4.29BM 2.00 0.90 - 0.0150 1.0 0.014 0.71 5531 9218 4.29GRID 2.00 0.90 0.150 - 1.0 0.014 0.71 5551 9217 4.29BM 2.00 0.10 0.150 - 1.0 0.014 0.71 4172 7100 3.69GRID 1.80 0.17 - 0.0225 10.0 0.010 0.71 4166 7088 3.69BM 2.00 0.10 - 0.0150 1.0 0.014 0.71 4210 6957 3.64GRID 1.90 0.14 0.300 - 1.0 0.010 0.73 4221 6979 3.63
Table 5.
The MLEs for eight different benchmark models (BM) using a grid with the incorrect overshooting prescription, i.e., step overshoot models are fittedto an exponential overshoot grid and vice versa.
Figure 6.
2D projected 1 σ error boxes of the observable ( Π , T eff , log g ) for the 37 stars in our sample for which spectroscopic measurements are available. for stars to be born on the ZAMS outside of the instability regionand later evolve into it, and vice versa. Therefore, it is expectedthat high-mass γ Dor stars are slow rotators since they are closerto the TAMS than low-mass stars. Yet, the fact we do not observeany evolved fast rotators might also be caused by an observationalselection bias, because the structure of a TAMS star combined withfast rotation creates a dense, and typically unresolvable spectrum of pulsation modes (e.g., Buysschaert et al. 2018). Nevertheless, thefact that the fastest rotators in our sample are all found to be rela-tively young suggests that the transport of angular momentum fromthe core to the envelope must be efficient already early on in thestellar evolution. Given that the convective core size decreases asthe star evolves (cf., Fig. 2), our findings suggest that the efficiencyof angular momentum transport is somehow connected to the con-
MNRAS000
MNRAS000 , 1–14 (2018) J. S. G. Mombarg et al. vective core, as already hinted at by Aerts et al. (2019). So far, theindividual roles of gravito-inertial and Rossby modes in angularmomentum transport remain unclear. As can been seen in Fig. 7, wefind stars with observed Rossby modes (plotted as triangles) to besituated across the main sequence, except close to the TAMS.In the second row of Fig. 7 we plot the MLEs of the overshootversus X (cid:48) c and colour-code by the rotation rate to investigate if ahigher overshooting estimate corresponds with faster rotation at theinterface of the convective core and the radiative envelope. For bothovershooting prescriptions, the MLE of the overshooting shows nocorrelation between the amount of core overshooting mixing andthe near-core rotation frequency. This result is complementary tothe conclusion by Aerts et al. (2014) that the surface nitrogen excessof a sample of OB stars, which demands efficient mixing in the stel-lar envelope, is not correlated with their surface rotation frequencybut rather with the frequency of the pressure mode with dominantamplitude. Taking the results of both studies together suggest thatcore boundary mixing may have various causes, such as rotationalmixing (e.g., Brott et al. 2011) or pulsational mixing induced bywaves (e.g., Rogers & McElwaine 2017), but that its shape remainsto be determined. In our modelling, these various kinds of mixingwere coded with two functional shapes with a free parameter, i.e., ascore overshooting and envelope mixing, without specifying the un-derlying physical phenomenon. The estimated parameters of thesetwo prescriptions, f ov (or α ov ) and D mix , do not reveal what causesthe near-core mixing, but only provide a rough level of mixing re-quired to explain the g-mode behaviour. We find that we cannotpinpoint particular values for our ensemble of stars, suggesting thatdifferent physical phenomena may lie at the basis of the overall levelof near-core mixing. This is in agreement with the findings for themore massive B-type pulsators (Aerts 2015).In the last two rows of Fig. 7 the fractional mass and radiusof the convective core are placed on the ordinate axis. We see thatestimation of the convective core mass and size is well achieved,despite the absence of predictive power for the core overshoot pa-rameter. This is the same conclusion as found by Johnston et al.(2019) for three gravity-mode pulsators in close binaries and pointsto a strong probing power of dipole gravity modes (through Π )to assess convective core properties. When we compare the fourstars closest to ZAMS, we notice that the two stars with a lower f rot have more massive and larger convective cores than the two fasterrotating stars. However, the overlapping confidence intervals do notallow us to claim a significant dichotomy with certainty. Determining uncertainty intervals from the distributions of the Ma-halanobis distance of each individual star (Aerts et al. 2018; problem1) is not meaningful because of the large span of these intervals inthe grids. This should not come as a surprise, given that one is thendetermining an estimator based on a single star, i.e., a sample of sizeone. A convenient way to deal with this consists of considering thesample of S =
37 as an ensemble. This is taken to mean a collectionof stars that, while having star-specific characteristics and henceparameters θ s , makes up a sufficiently ‘natural’ family fulfilling thesame underlying theory of stellar structure.Evidently, when a sample of size S is available, the averageof the parameters θ s , say θ , can be estimated much more preciselythan the parameter of an individual star. If we think of a given starparameter θ s as being decomposed into θ s = θ + t s , with t s the star-specific deviation around θ , then we can use the precision of (cid:98) θ ,the MLE, as a measure of precision for the entire sample. Precisely,we write the likelihood for the entire sample as L = S (cid:214) s = ( π ) P / | V s | − / × exp (cid:26) − (cid:2) Y s ( θ s ) − Y ∗ s (cid:3) (cid:62) V − s (cid:2) Y s ( θ s ) − Y ∗ s (cid:3)(cid:27) , (19)with P = dim ( Y ) . In our application, S =
37 and P =
3. The kernelof the log-likelihood is (cid:96) = − S (cid:213) s = ln | V s | − S (cid:213) s = (cid:2) Y s ( θ s ) − Y ∗ s (cid:3) (cid:62) V − s (cid:2) Y s ( θ s ) − Y ∗ s (cid:3) . (20)Consider now the deviance function for the k th component of θ D ( h ) = − S (cid:213) s = (cid:2) Y s ( θ s ) − Y ∗ s (cid:3) (cid:62) V − s (cid:2) Y s ( θ s ) − Y ∗ s (cid:3) (21) + S (cid:213) s = (cid:104) Y s ( θ ( k , h ) s ) − Y ∗ s (cid:105) (cid:62) V − s (cid:104) Y s ( θ ( k , h ) s ) − Y ∗ s (cid:105) , with θ ( k , h ) s = θ s , except in the k th component, where we put θ ( k , h ) s , k = θ s , k + h , where h is a (discrete) perturbation. We thensearch for both the negative and positive h values, say h L and h U ,that satisfy D ( h L ) = D ( h U ) = .
84 which is the critical value ofthe χ distribution. When making use of a sufficiently fine grid,those grid points can be chosen that satisfy the above requirementsufficiently well. The corresponding confidence interval is: [ θ s , k + h L ; θ s , k + h U ] . Because the likelihood ratio is not based on a quadratic approxima-tion, the interval is not necessarily symmetric, but it will be moresymmetric in larger samples. In Appendix B, the deviance functionis plotted as a function of the four parameters we have estimated.The grid is quite coarse for overshoot and Z to properly sample D ( h ) , hence we stay on the conservative side and pick the values of h L and h U as the smallest discrete values for which the deviancefunction is larger than 3.84. For X (cid:48) c , the confidence interval may, sta-tistically seen, contain unphysical values and therefore we truncatethe interval at the edge of the grid. The confidence interval of theovershoot is only truncated at the lower edge of the grid. For bothovershoot grids we were not able to determine h L for our ensemble,therefore the lower limit on α ov and on f ov is always the lowestvalue in the respective grid. In Appendix B, the sampling of D ( h ) is shown for the four parameters that estimated from the the MLE.This yields uncertainties of 0.1 M (cid:12) on M (cid:63) , 0.004 on Z (in thesecases symmetric), all rounded to the nearest step in the respectivegrids and 0.12 and 0.10 on X (cid:48) c for the lower and upper uncertainties,respectively. The uncertainties on M (cid:48) cc and R (cid:48) cc are defined as theminimum and maximum values found in the models that lay withinthe confidence intervals of the estimated parameters. In this paper we have explored the power of Π estimated fromprograde dipole gravito-inertial modes and in some cases also fromRossby modes, combined with spectroscopic measurements of the MNRAS , 1–14 (2018) steroseismic modelling of γ Dor stars Figure 7.
Top to bottom: MLEs for X (cid:48) c versus mass, overshoot, convective core mass and convective core radius, colour-coded by the measured near-corerotation frequency from Van Reeth et al. (2016), for all 37 stars in our sample. Left column: Step overshoot. Right column: Exponential overshoot. Stars thathave observed Rossby modes in addition to gravito-inertial prograde dipole modes are plotted as triangles.MNRAS000
Top to bottom: MLEs for X (cid:48) c versus mass, overshoot, convective core mass and convective core radius, colour-coded by the measured near-corerotation frequency from Van Reeth et al. (2016), for all 37 stars in our sample. Left column: Step overshoot. Right column: Exponential overshoot. Stars thathave observed Rossby modes in addition to gravito-inertial prograde dipole modes are plotted as triangles.MNRAS000 , 1–14 (2018) J. S. G. Mombarg et al. effective temperature and surface gravity to perform asteroseismicmodelling of γ Dor stars, which have a well-developed convectivecore. We devised recipes relating the most important fundamen-tal stellar parameters to these three observables, from multivariatelinear regression, and have shown that these linear recipes have aproper predicting capacity with respect to the measurement uncer-tainties. The recipes are advantageous in the forward modelling of γ Dor stars, as they imply a huge decrease in computation timewhile still providing good constraints for the stellar parameters inthe high-dimensional parameter space. The method of backwardselection based on the BIC reveals that, among the mass, metal-licity, central and initial hydrogen mass fraction, mass and size ofthe convective core and the amount of mixing in the radiative zone,only the latter can be ignored in the linear recipes. Previously it wasfound that fitting frequencies of individual trapped gravity modesdoes require extra mixing in the radiative envelope (Moravveji et al.2015, 2016; Schmid & Aerts 2016).The forward modelling of γ Dor stars is a cumbersome task ifthe aim is to achieve stellar parameters with a relative precision of10 per cent or better. This can only be reached when Π , T eff , andlog g are measured with high precision. This problem is illustrated inFig. 6 where the 2D projected 1 σ error boxes for Y ( obs ) are plottedfor all 37 stars in our sample. Even though we use high-precisionspectroscopy here, the error boxes are still relatively large due tothe correlated nature of the three observables.This work contains the first asteroseismic forward modellingof γ Dor stars as an ensemble, using the Mahalanobis distance asdescribed in Aerts et al. (2018). With this method, we assume thesample stars to adhere to a single underlying stellar evolution theoryand that each star with its own parameters can be seen as a deviationof an average star in the ensemble to derive meaningful uncertaintiesto go along with the MLEs of M (cid:63) , X (cid:48) c , α ov / f ov , Z , as well as the massand radius of the convective core for the 37 γ Dor stars in our sample.From PCA it was concluded that the dimensionality of this problemcould be reduced to a 4D problem when fitting ( Π , T eff , log g ) , byfixing X ini and D mix . For M (cid:63) and X (cid:48) c we find in general consistentresults between a step and an exponential overshooting prescription.We have demonstrated that linear statistical models are ableto capture the correlated nature of the three observables Π , T eff and log g , and the fundamental stellar parameters, up to the level ofthe measurement uncertainties. This finding allows us to use recipesrather than having to compute dense stellar model grids when fittingthe observables ( Π , T eff , log g ) for future applications to additional γ Dor stars.Future work will involve the addition of X ini , D mix and α MLT in the modelling of the morphology of the period spacing patternsfor all γ Dor stars with suitable mode detection and spectroscopy,rather than just Π . Especially the mixing profile in the radiativezone cannot be constrained with the observables in this paper, butdoes have a large effect on the morphology of the period spacingpattern. The dips in the pattern caused by mode-trapping decreasewhen the mixing efficiency increases as this process washes out thechemical gradient and therefore reduces the effect of mode trapping,as shown in B-type stars (Moravveji et al. 2015). As computing theindividual theoretical frequencies is a more computationally de-manding exercise, the MLEs of M (cid:63) , X (cid:48) c , α ov / f ov and Z presentedin this paper provide a starting point for refined modelling to esti-mate D mix ( r ) . The ongoing NASA TESS (Ricker et al. 2015) andupcoming ESA PLATO (Rauer et al. 2014) space missions will de-liver many new g-mode pulsators covering a mass range of 1.2 to ∼
20 M (cid:12) . Therefore, the similar analysis of potentially thousands ofnew g-mode pulsators and its extension to include binary star infor- mation (e.g., Johnston et al. 2019) with the next generation of spacetelescopes will provide an improved insight of stellar structure andevolution of stars with convective cores across the HR diagram.
ACKNOWLEDGEMENTS
We thank the referee, Prof. Hiromoto Shibahashi, for his insight-ful and detailed comments on our manuscript, which improved itsclarity and quality. We thank Bill Paxton, Rich Townsend, and theircode development team members for all their efforts put into thepublic stellar evolution and pulsation codes MESA and GYRE.The research leading to these results has received funding fromthe European Research Council (ERC) under the European Union’sHorizon 2020 research and innovation programme (grant agreementN o REFERENCES
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APPENDIX A: MODE CAVITIES IN A TYPICAL γ DorSTELLAR MODEL
Here, we show an example of the mode cavities of the gravity(g) modes and pressure (p) modes in a γ Dor star.
Figure A1.
Mode cavities of the g modes (green, n pg ∈ [− , − ] ) andp modes (grey, n pg ∈ [ , ] ) in a 1.8 M (cid:12) star at X (cid:48) c = . ( l , m ) = ( , ) .The mode frequencies ω n were computed with GYRE, adopting the TARfor a uniform rotation at 20 per cent of the critical rotation velocity in theRoche formalism. The dashed lines are placed at the mode frequencies andthe black dots mark the position of the radial nodes of each mode. The solidblack line is the Brunt-Väisälä frequency, the dotted line is the dipole-modeLamb frequency S l = . The effect of rotation is only taken into account inthe computation of the mode frequencies and not in the equilibrium model(Aerts et al. 2018, for a thorough justification of such an approach in gravity-mode asteroseismology). Figure B1. deviance function D ( h ) , where h is a ‘perturbation’ in M (cid:63) ofthe best model. The dashed grey line at 3.84 indicates the critical value ofthe χ distribution. APPENDIX B: DEVIANCE FUNCTIONS
Below we show the behaviour of the deviance functions used fordetermining the confidence intervals from ensemble modelling asdescribed in Section 5.
APPENDIX C: INFLUENCE OF THE MIXING LENGTHPARAMETER
In this appendix we present the differences in the theoretical pre-diction of Π , T eff , and log g caused by the use of a different valueof α MLT in the computation of stellar models for a high-mass andlow-mass γ Dor star.
MNRAS000
MNRAS000 , 1–14 (2018) J. S. G. Mombarg et al.
Figure B2.
Same as Fig. B1, but for X (cid:48) c . Figure B3.
Same as Fig. B1, but for overshoot.
APPENDIX D: MAXIMUM LIKELIHOOD ESTIMATESOF THE STELLAR PARAMETERS
Below we present the maximum likelihood estimates described inSections 4 and 5 for each of the five stellar parameters estimated forall the 37 stars in our sample, along with the observed value of Π . This paper has been typeset from a TEX/L A TEX file prepared by the author.
Figure B4.
Same as Fig. B1, but for Z . Figure C1.
Evolution of Π for different values of the mixing length pa-rameter α MLT . The average uncertainty on Π (in grey) for the stars in oursample is also plotted for comparison. Figure C2.
Same as Fig. C1, but for T eff .MNRAS , 1–14 (2018) steroseismic modelling of γ Dor stars KIC ID Π ( obs ) [s] M (cid:63) [ M (cid:12) ] X (cid:48) c α ov f ov Z M cc / M (cid:63) R cc / R (cid:63) KIC2710594 + − + . − . + . − . - 0.018 0.082 + . − . + . − . + . − . - 0.0300 + . − . + . − . + . − . KIC3448365 + − + . − . + . − . - 0.014 0.103 + . − . + . − . + . − . - 0.0075 + . − . + . − . + . − . KIC4846809 4144 + − + . − . + . − . - 0.010 0.109 + . − . + . − . + . − . - 0.0150 + . − . + . − . + . − . KIC5114382 + − + . − . + . − . - 0.014 0.104 + . − . + . − . + . − . - 0.0075 + . − . + . − . + . − . KIC5522154 4738 + − + . − . + . − . - 0.018 0.112 + . − . + . − . + . − . - 0.0300 + . − . + . − . + . − . KIC5708550 4709 + − + . − . + . − . - 0.018 0.110 + . − . + . − . + . − . - 0.0225 + . − . + . − . + . − . KIC5788623 3960 + − + . − . + . − . - 0.010 0.098 + . − . + . − . + . − . - 0.0075 + . − . + . − . + . − . KIC6468146 4243 + − + . − . + . − . - 0.010 0.096 + . − . + . − . + . − . - 0.0225 + . − . + . − . + . − . KIC6468987 + − + . − . + . − . - 0.018 0.103 + . − . + . − . + . − . - 0.0300 + . − . + . − . + . − . KIC6678174 4766 + − + . − . + . − . - 0.018 0.100 + . − . + . − . + . − . - 0.0150 + . − . + . − . + . − . KIC6935014 4497 + − + . − . + . − . - 0.014 0.115 + . − . + . − . + . − . - 0.0225 + . − . + . − . + . − . KIC6953103 5035 + − + . − . + . − . - 0.018 0.110 + . − . + . − . + . − . - 0.0300 + . − . + . − . + . − . KIC7023122 4780 + − + . − . + . − . - 0.018 0.116 + . − . + . − . + . − . - 0.0150 + . − . + . − . + . − . KIC7365537 4723 + − + . − . + . − . - 0.010 0.091 + . − . + . − . + . − . - 0.0225 + . − . + . − . + . − . KIC7380501 4045 + − + . − . + . − . - 0.018 0.077 + . − . + . − . + . − . - 0.0010 + . − . + . − . + . − . KIC7434470 4271 + − + . − . + . − . - 0.014 0.097 + . − . + . − . + . − . - 0.0150 + . − . + . − . + . − . KIC7583663 + − + . − . + . − . - 0.014 0.104 + . − . + . − . + . − . - 0.0075 + . − . + . − . + . − . KIC7939065 4243 + − + . − . + . − . - 0.014 0.097 + . − . + . − . + . − . - 0.0300 + . − . + . − . + . − . KIC8364249 4370 + − + . − . + . − . - 0.018 0.094 + . − . + . − . + . − . - 0.0150 + . − . + . − . + . − . KIC8375138 + − + . − . + . − . - 0.010 0.110 + . − . + . − . + . − . - 0.0300 + . − . + . − . + . − . Table D1.
MLE for each star in our sample, based on the model grids. The confidence intervals were derived from treating the sample as an ensemble. Theuncertainties on M (cid:63) and Z are 0.1 M (cid:12) and 0.004, respectively. When the confidence interval of X (cid:48) c or α ov / f ov extends beyond physical values, it is truncatedat the edge of grid as is explained in Section 5. Uncertainties on the mass and size of the convective core are derived from the extreme values from the modelsthat lay within the confidence intervals of the estimated parameters. Stars in italic have observed Rossby modes. The second column lists the observed valuesof Π computed by Van Reeth et al. (2018).MNRAS000
MLE for each star in our sample, based on the model grids. The confidence intervals were derived from treating the sample as an ensemble. Theuncertainties on M (cid:63) and Z are 0.1 M (cid:12) and 0.004, respectively. When the confidence interval of X (cid:48) c or α ov / f ov extends beyond physical values, it is truncatedat the edge of grid as is explained in Section 5. Uncertainties on the mass and size of the convective core are derived from the extreme values from the modelsthat lay within the confidence intervals of the estimated parameters. Stars in italic have observed Rossby modes. The second column lists the observed valuesof Π computed by Van Reeth et al. (2018).MNRAS000 , 1–14 (2018) J. S. G. Mombarg et al.
KIC ID Π [s] M (cid:63) [ M (cid:12) ] X (cid:48) c α ov f ov Z M cc / M (cid:63) R cc / R (cid:63) KIC8645874 4525 + − + . − . + . − . - 0.018 0.090 + . − . + . − . + . − . - 0.0150 + . − . + . − . + . − . KIC8836473 4101 + − + . − . + . − . - 0.014 0.087 + . − . + . − . + . − . - 0.0150 + . − . + . − . + . − . KIC9480469 + − + . − . + . − . - 0.014 0.121 + . − . + . − . + . − . - 0.0010 + . − . + . − . + . − . KIC9595743 4313 + − + . − . + . − . - 0.014 0.102 + . − . + . − . + . − . - 0.0075 + . − . + . − . + . − . KIC9751996 4364 + − + . − . + . − . - 0.014 0.094 + . − . + . − . + . − . - 0.0010 + . − . + . − . + . − . KIC10467146 4158 + − + . − . + . − . - 0.010 0.090 + . − . + . − . + . − . - 0.0075 + . − . + . − . + . − . KIC11080103 4752 + − + . − . + . − . - 0.018 0.109 + . − . + . − . + . − . - 0.0150 + . − . + . − . + . − . KIC11099031 5035 + − + . − . + . − . - 0.018 0.112 + . − . + . − . + . − . - 0.0300 + . − . + . − . + . − . KIC11294808 3917 + − + . − . + . − . - 0.014 0.078 + . − . + . − . + . − . - 0.0150 + . − . + . − . + . − . KIC11456474 3974 + − + . − . + . − . - 0.010 0.101 + . − . + . − . + . − . - 0.0150 + . − . + . − . + . − . KIC11721304 4356 + − + . − . + . − . - 0.014 0.108 + . − . + . − . + . − . - 0.0150 + . − . + . − . + . − . KIC11754232 4426 + − + . − . + . − . - 0.014 0.102 + . − . + . − . + . − . - 0.0075 + . − . + . − . + . − . KIC11826272 4172 + − + . − . + . − . - 0.010 0.101 + . − . + . − . + . − . - 0.0075 + . − . + . − . + . − . KIC11907454 + − + . − . + . − . - 0.014 0.088 + . − . + . − . + . − . - 0.0075 + . − . + . − . + . − . KIC11917550 4101 + − + . − . + . − . - 0.014 0.092 + . − . + . − . + . − . - 0.0010 + . − . + . − . + . − . KIC11920505 4214 + − + . − . + . − . - 0.014 0.094 + . − . + . − . + . − . - 0.0010 + . − . + . − . + . − . KIC12066947 + − + . − . + . − . - 0.010 0.065 + . − . + . − . + . − . - 0.0010 + . − . + . − . + . − . Table D1 – continued MNRAS , 1–14 (2018) steroseismic modelling of γ Dor stars Figure C3.
Same as Fig. C1, but for log g .MNRAS000