A comparison of the dynamical and model-derived parameters of the pulsating eclipsing binary KIC9850387
AAstronomy & Astrophysics manuscript no. KIC9850387 © ESO 2021February 23, 2021
A comparison of the dynamical and model-derived parameters ofthe pulsating eclipsing binary KIC9850387
S. Sekaran , A. Tkachenko , C. Johnston , , and C. Aerts , , Instituut voor Sterrenkunde (IvS), KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgiume-mail: [email protected] Department of Astrophysics, IMAPP, Radboud University Nijmegen, NL-6500 GL, Nijmegen, the Netherlands Max Planck Institute for Astronomy, Koenigstuhl 17, 69117 Heidelberg, GermanyReceived December 17, 2020; accepted February 18, 2021
ABSTRACT
Context.
One-dimensional (1D) stellar evolutionary models incorporate interior mixing profiles as a simplification of multi-dimensional physical processes that have a significant impact on the evolution and lifetime of stars. As such, the proper calibrationof interior mixing profiles is required for the reconciliation of observational parameters and theoretical predictions. The modellingand analysis of pulsating stars in eclipsing binary systems that display gravity-mode (g-mode) oscillations allows for the preciseconstraints on the interior mixing profiles through the combination of spectroscopic, binary and asteroseismic obervables.
Aims.
We aim to unravel the interior mixing profile of the pulsating eclipsing binary KIC9850387 by comparing its dynamicalparameters and the parameters derived through a combination of evolutionary and asteroseismic modelling.
Methods.
We created a grid of stellar evolutionary models using the stellar evolutionary code mesa and performed an isochrone-cloud(isocloud) based evolutionary modelling of the system. We then generated a grid of pulsational models using the stellar pulsationcode gyre based on the age constraints from the evolutionary modelling. Finally, we performed asteroseismic modelling of theobserved (cid:96) = (cid:96) = ff erent combinations of observational constraints, merit functions, andasteroseismic observables to obtain strong constraints on the interior properties of the primary star. Results.
Through a combination of asteroseismic modelling and dynamical constraints, we found that the system comprises two main-sequence components at an age of 1 . ± . ff ect due to the systematicfrequency o ff set of the theoretical modes from the nearest observed modes. We also found evidence of rotational splitting in the formof a prograde-retrograde dipole g mode doublet with a missing zonal mode, implying an envelope rotational frequency that is threetimes higher than the core rotational frequency and about 20 times slower than the orbital frequency, but we note that this result isbased completely on the rotational splitting of a single dipole mode. Conclusions.
We find that the dynamical parameters and the parameters extracted from the asteroseismic modelling of period-spacingpatterns are only barely compliant, reinforcing the need for homogeneous analyses of samples of pulsating eclipsing binaries that aimto calibrating interior mixing profiles.
Key words. stars: individual: KIC9850387 – binaries: eclipsing – stars: oscillations – stars: fundamental parameters – stars: evolution– asteroseismology
1. Introduction
Stellar evolutionary models are the realisation of our understand-ing of stellar structure and evolution, and they are the foundationfor more complex models and simulations of stellar populations,clusters, and galaxies (Salaris & Cassisi 2005). As such, any in-accuracies in the mathematical description or calibration of thevarious input physics that are included in these models inevitablypropagate to the results of studies that rely on them as input. Inaddition, the one-dimensional (1D) nature of stellar evolutionarymodels necessitates the inclusion of parametric simplificationsof multi-dimensional physical processes, such as the 1D approx-imation of turbulent convection through the incorporation of themixing length theory (Böhm-Vitense 1958).Stellar evolutionary models are therefore limited by theframework of these simplifications, complicating the task of im-proving these models. However, it has become increasingly clear in recent decades that one of the dominant sources of uncertaintyregulating the evolutionary pathways of stars is the amount andprescriptions of the various mixing processes occurring in stel-lar interiors due to their ability to modify the amount of fuelavailable for nuclear fusion (Maeder 2009; Brott et al. 2011;Langer 2012; Meynet et al. 2013). The influence of these mech-anisms is particularly significant for the evolution of stars thatare born with or that develop a convective core on the main se-quence ( M (cid:38) .
15 M (cid:12) ): Uncertainties in the calibration and im-plementation of these mixing processes propagate to later evo-lutionary stages, most ubiquitously to the red-giant branch forintermediate-mass stars with 1 .
15 M (cid:12) (cid:46) M (cid:46) . (cid:12) (Con-stantino et al. 2015; Silva Aguirre et al. 2020) and to the pre-supernova stage for the massive stars (Martins et al. 2015; Bow-man 2020).Internal chemical mixing is a result of independent or cou-pled physical mechanisms operating in di ff erent regions of the Article number, page 1 of 21 a r X i v : . [ a s t r o - ph . S R ] F e b & A proofs: manuscript no. KIC9850387 stellar interior (see Salaris & Cassisi 2017 for a review) and canbe broadly divided into two classes for the M (cid:38) .
15 M (cid:12) regime:core-boundary mixing (CBM) and envelope mixing. CBM is ancollective term that includes all prescriptions for the transport ofmaterial from the convective core into the radiative envelope (orvice versa), such as convective entrainment (Meakin & Arnett2007; Cristini et al. 2019), convective penetration and overshoot-ing (Viallet et al. 2015). Of these various prescriptions, it is con-vective penetration and overshooting that are most commonlyimplemented in most 1D stellar evolutionary codes (Viallet et al.2015), and both are referred to in the literature as ’overshooting.’Both of these prescriptions are based on the traversing of convec-tive fluid parcels beyond the limits set by the Schwarzchild orLedoux criterion due to their inertia. In convective penetration(also known as ’step overshooting’; Zahn 1991), the region inwhich these fluid particles traverse is subjected to the adiabatictemperature gradient, resulting in an extended region aroundthe convective core being instantaneously mixed and modify-ing the thermal structure of the star. In convective overshoot-ing (also known as ’di ff usive exponential overshooting’; Freytaget al. 1996; Herwig 2000), the extended region is subjected tothe radiative temperature gradient, and hence only the chemi-cal structure of star is changed. In both types of CBM regions,di ff erent functional forms of temperature gradient have been im-plemented, but distinguishing between them observationally isnon-trivial (Michielsen et al. 2019, Michielsen et al. 2020, sub-mitted).The net e ff ect of overshooting is an increase in the convec-tive core mass, and hence for the star to appear more luminous(i.e. brighter). This e ff ect is therefore degenerate with the stellarmass, and classical observables would not be able to distinguishbetween both types of overshooting. However, the analyses ofstars that show gravity-mode (g-mode) pulsations (e.g. Pedersenet al. 2018; Michielsen et al. 2019) have demonstrated that thetype and amount of CBM can be constrained by modelling themasteroseismically. This has been demonstrated by a number ofobservational studies of g-mode pulsators (e.g. Moravveji et al.2015, 2016; Buysschaert et al. 2018; Walczak et al. 2019; Wu &Li 2019; Fedurco et al. 2020).Envelope mixing, similar to CBM, is an umbrella term thatincludes a number of rotational and pulsational mechanisms (seeAerts 2021 for a detailed discussion), including meridional cir-culation, hydrodynamical instabilities, magnetorotational insta-bilities and internal gravity waves. Envelope mixing in 1D stel-lar models is typically implemented by means of di ff usive mix-ing coe ffi cients in the transport equations for the regions outsideof the convective core and CBM zones. There are notable ex-ceptions, such as the advective implementation in the Geneva(Ekström et al. 2012) and CESTAM (Marques et al. 2013) stel-lar evolutionary codes. Due to this implementation, the envelopemixing is artifically decoupled from the CBM. It was noted byMoravveji et al. (2015, 2016) that the inclusion of envelope mix-ing significantly improved the fit of the g-mode pulsational fre-quencies and therefore they confirm that such mixing is an im-portant component that should be included in 1D stellar models.Due to the sensitivity of g modes to stellar interior mixingprofiles, the analysis of g-mode pulsating stars allows for thecalibration of these interior mixing profiles. For pulsating F-typestars, these modes are low-frequency pulsations that are excitedby the convective flux-blocking mechanism (Guzik et al. 2000;Dupret et al. 2005). At high radial order, the g modes that havethe same spherical harmonic degree (cid:96) and azimuthal order m (seeAerts et al. 2010 for a complete description) have the character-istic of being equally spaced in period in the non-rotating ap- proximation. This gives rise to the so-called asymptotic approx-imation of the period spacing (Tassoul 1980): Π (cid:96) = Π √ (cid:96) ( (cid:96) +
1) ; (1)where, Π = π (cid:32)(cid:90) r r N drr (cid:33) − . (2)In these equations, r is the distance from the stellar centre, N isthe Brunt-Väisälä frequency and r and r are the radial bound-aries of the g-mode propagation cavity in the star. Deviationsfrom this equidistant spacing of g modes are a result of either1) mode trapping due to the near-core chemical gradient (Miglioet al. 2008), or 2) near-core rotation, which introduces a slopeinto the pattern due to the pulsational periods being a ff ected bythe Coriolis force (Bouabid et al. 2013). The morphology of theobserved g-mode period-spacing patterns can therefore be linkedto mixing and rotational characteristics of the g mode propaga-tion cavity (e.g. Van Reeth et al. 2015a,b, 2016; Ouazzani et al.2017, 2020; Li et al. 2018, 2019a, 2020b).The potential for g modes in unravelling the interior mixingprofiles in stars is unfortunately hampered by the highly corre-lated nature of the various free parameters used in the evolution-ary models. Aerts et al. (2018) proposed a forward modellingscheme for the purposes of taking these correlations, as well asuncertainties due to the imperfections in the input physics of theequilibrium models, into account when performing asteroseis-mic modelling. In addition, restricting the ranges of the multi-dimensional parameter space would significantly alleviate thesedegeneracies in stellar modelling, resulting in more precise andaccurate solutions.The largest number and most stringent of constraints to limitthe parameter space can be obtained when studying double-linedeclipsing binary systems. One can extract the surface metal-licities, e ff ective temperatures, masses, radii, and surface grav-ities and rotational frequencies of the individual components.In addition to all of these parameters, binarity also enforces aco-evolutionary scenario, demanding equal ages of the individ-ual components. These systems have been used to calibrate theamount of CBM included in stellar evolutionary models, testingeither the convective penetration prescription (e.g. Ribas et al.2000; Torres et al. 2014; Tkachenko et al. 2014b; Stancli ff eet al. 2015; Claret & Torres 2016), the convective overshoot-ing prescription (e.g. Tkachenko et al. 2014a; Stancli ff e et al.2015; Claret & Torres 2017, 2018, 2019; Constantino & Bara ff e2018; Johnston et al. 2019a,b) or both (e.g. Claret & Torres 2017;Tkachenko et al. 2020). A secondary advantage of studying suchsystems is the ability to compare three di ff erent sets of parame-ters: the dynamical and spectroscopic parameters, those derivedfrom the evolutionary models, and those derived from the pulsa-tional models.There have been a total of 34 eclipsing binary systems withg-mode period-spacing patterns discovered so far in the Kepler data (Li et al. 2020a), and only three have been modelled (Zhanget al. 2018; Guo & Li 2019; Zhang et al. 2020), none of whichfrom the viewpoint of assessing internal mixing. As such, it isthe goal of our study to add to the dearth of literature on theevolutionary and asteroseismic modelling of g-mode pulsatorsin eclipsing binary systems. In this paper, we present the evolu-tionary and asterosesismic modelling of the pulsating eclipsing
Article number, page 2 of 21. Sekaran et al.: Evolutionary and asteroseismic modelling of the pulsating eclipsing binary KIC9850387 binary KIC9850387 that was identified as the most promisingcandidate in terms of g-mode asteroseismic potential in Sekaranet al. (2020), hereafter Paper I. It is a short period ( P orb = .
74 d)eclipsing binary in a near-circular orbit with an intermediate-mass primary star and a solar-like secondary star. The primarystar displayed remarkable (cid:96) = (cid:96) =
2. Isochrone-cloud construction and fitting
In Paper I, a full observational analysis of KIC9850387 wasperformed, resulting in the extraction of the orbital and funda-mental parameters of the system as well as of the individualcomponents (listed in Table 1). The (cid:46)
1% errors on many ofthese parameters are not unusual for detached eclipsing systems:The individual analyses of several tens of such systems (listedin DEBCat, Southworth 2015) boast similar precisions. Usingthese parameters, we can calculate evolutionary models that bestmatch our observations. To achieve this goal, we constructedisochrone-clouds (hereafter isoclouds) as detailed in Johnstonet al. (2019b). Each isocloud is comprised of isochrones gener-ated from equivalent evolutionary phase (EEP) tracks (see Dot-ter 2016 for a full description) generated from single-star main-sequence evolutionary tracks computed with the stellar evolu-tionary code mesa (version 10348; Paxton et al. 2011, 2018).These evolutionary tracks are generated using the same inputphysics as Johnston et al. (2019b), fixing the initial helium abun-dance Y ini at 0.274 as per the cosmic B-star standard (Nieva &Przybilla 2012), fixing α MLT at 1.8 (Joyce & Chaboyer 2018),and using an initial metallicity ( Z ini ) of 0.010 to match the spec-troscopic metallicity derived for the primary star.We adopted the Ledoux criterion to position the convec-tive boundary and di ff usive exponential overshooting as the Table 1.
Fundamental parameters of KIC9850387 presented in Paper Ithat were used in our evolutionary and asterosesimic modelling.
Primary Secondary M [M (cid:12) ] 1.66 + . − . + . − . R [R (cid:12) ] 2.154 + . − . + . − . T e ff [K] 7335 + − + − log g [dex] 3.993 + . − . + . − . l r + . − . + . − . [M / H] [dex] -0.11 + . − . – f rot, surf [d − ] 0.122 + . − . – Notes.
The errors quoted are based on 68% HPD intervals.Descriptions of the various parameters in this table:
Primary and Secondary– M : Mass – R : Equivalent radius (the radius that each star would have if it wasa perfect sphere) – log g: Logarithm of the surface gravity – T e ff : E ff ective temperature – l r : Light ratio of the star with respect to the total flux – [M / H]: Global metallicity – f rot, surf : The surface rotational frequency of the primary star Table 2.
Ranges of M ini , f ov and log D mix used to create evolutionarytracks. Min Max Step M ini [ M (cid:12) ] 0.80 2.00 0.05 f ov D mix Notes. f ov and D mix are fixed at 0 for evolutionary tracks with M ini < . CBM prescription according to Freytag et al. (1996) and Herwig(2000). Restricting to one type of CBM is justified because aster-oseismic analyses of a sample of A / F-type γ Dor pulsators haveshown the results using this CBM to be indistinguishable fromconvective penetration (Mombarg et al. 2019). This prescriptionassumes an exponential decrease of the mixing e ffi ciency withdistance from the convective core as convective fluid parcelspropagate further away from the core into the radiative zone.The amount of di ff usive mixing in the CBM region [ D CBM ( r )]is calculated using the following equation: D CBM ( r ) = D exp (cid:34) − r − r ) f ov H p (cid:35) . (3)Here, the reference radius r = r cc − f H p , where r cc is theradius of the convective core according to the Ledoux crite-rion, f = .
002 and H p is the local pressure scale height. D is the amount of di ff usive mixing within the inner edge of theconvective-core boundary at the reference radius r and f ov isthe length scale over which the di ff usive exponential overshoot-ing applies.The prescriptions of Johnston et al. (2019b) include rangesfor f ov and the amount of di ff usive envelope mixing D mix , takenfrom the asteroseismic calibrations of Aerts (2015); Bowman(2020) and Moravveji et al. (2015, 2016) respectively. Morespecifically, the di ff usive envelope mixing adopted in Johnstonet al. (2019b) takes the form of D mix ∝ ρ − / , which is an approx-imation of the mixing due to internal gravity waves taken from2D hydrodynamical simulations (Rogers & McElwaine 2017)and implemented in mesa by Pedersen et al. (2018). However,these parameters have been calibrated using g-mode pulsators,and are therefore only appropriate for stars that develop a sub-stantial convective core over the main sequence ( M ini (cid:38) . (cid:12) as noted by Dotter 2016). The low dynamical mass ofthe secondary ( M s = . + . − . M (cid:12) ) necessitates the calcula-tion of evolutionary tracks of stars below this limit. Choi et al.(2016) computed a grid of solar-calibrated stellar models andisochrones but only include rotational mixing in their tracks with M ini ≥ .
15 M (cid:12) to reflect the slow rotational frequencies ob-served in low-mass stars. Considering that rotational mixing isthe only form of envelope mixing present in their tracks, it can beconcluded that they e ff ectively set the di ff usive envelope mixingat zero for their evolutionary tracks with M ini < .
15 M (cid:12) . In ad-dition, a significant portion of the envelope of low-mass stars isconvective, implying that mixing in the radiative region outsideof the core would have a minimal impact on their evolution. Wetherefore fix both f ov and D mix at 0 for our evolutionary tracksfor M ini < .
15 M (cid:12) . The ranges of M ini , f ov and log D mix used tocreate our evolutionary tracks are listed in Table 2. It should benoted that our use of an upper bound of 4.0 for log D mix is atyp-ically high for intermediate-mass g-mode pulsators (e.g. Mom- Article number, page 3 of 21 & A proofs: manuscript no. KIC9850387 barg et al. 2019 had used an upper bound of 1.0 for log D mix ).Such high values allow for the testing of the existence of tidalmixing mechanisms in the stellar interior.An isochrone is traditionally constructed from evolutionarymodel tracks with the same input physics and the same free-parameter (e.g. f ov and D mix ) values. However, due to our choiceof fixing f ov and log D mix at 0 for M ini < .
15 M (cid:12) , we e ff ectivelyhave to fuse the isochrones created from the M ini ≤ .
10 M (cid:12) and M ini ≥ .
15 M (cid:12) evolutionary tracks together: A single isochronetherefore has an f ov and log D mix value of 0 for 0 .
80 M (cid:12) ≤ M ini ≤ .
10 M (cid:12) and a fixed combination of f ov and log D mix as per theranges listed in Table 2 for 1 .
15 M (cid:12) ≤ M ini ≤ .
00 M (cid:12) , with atotal of 1000 datapoints with 0 . ≤ M ini ≤ .
00. Similarly, eachisocloud, which is a combination of all isochrones at an age τ for all combinations of f ov and log D mix in our grid, exhibits thesame behaviour. Each of our isoclouds comprise a total of 72 000datapoints: 8 f ov values × D mix values × T e ff and log g within their 1 σ observational errors ina Monte-Carlo framework for 1000 iterations, and retained thebest-fitting isocloud in each iteration. Due to the fact that an iso-cloud is e ff ectively a set of curves, we calculated the reduced χ ( χ ) value for each of the 72 000 datapoints in an isocloudwith respect to the primary and secondary T e ff and log g (i.e.144 000 total χ values), and took the sum of the 50 smallest χ values for both the primary and secondary star (i.e. the sumof 100 χ values across both components) as our goodness-of-fit metric. In addition, to account for the fact that the stellarmodels are in much better agreement with dynamical parame-ters for low-mass stars (e.g. Chaplin et al. 2014; Higl & Weiss2017), we chose to bias the isocloud fitting towards the sec-ondary component. This was done by decreasing the errors onthe secondary parameters by a factor of five, e ff ectively weight-ing the secondary parameters five times more heavily than theprimary ones. We then sample this distribution of the 1000 best-fitting isoclouds and determined the evolutionary parameters ( τ , M , f ov , log D mix , the central hydrogen fraction X c , and the in-ferred convective-core mass M cc ) based on the intersection of thedynamical T e ff and log g constraints with the isocloud param-eters corresponding to the 95% highest posterior density (HPD) Table 3.
Evolutionary model parameters of KIC9850387 extracted fromisochrone-cloud fitting, based on the intersection of the dynamical T e ff and log g constraints with the isocloud parameters corresponding to the95% HPD interval of the Monte-Carlo age distribution ( τ MC = . + . − . Gyr).
Primary Secondary τ [Gyr] 2 . ± . . ± . M [M (cid:12) ] 1 . ± . . ± . X c . ± . . ± . f ov . ± .
02 –log D mix . ± . M cc [M (cid:12) ] 0 . ± .
03 – We refer to the T e ff determined spectroscopically by fixing the log g at the dy-namical value, as detailed in Section 3.3 of Paper I, as the ’dynamical’ T e ff . interval of the Monte-Carlo age distribution, corresponding to τ MC = . + . − . Gyr. These parameters are listed in Table 3, andthe log T e ff − log g (or Kiel) diagram displaying our isocloud fit-ting results is shown in Figure 1. The large asymmetry in the er-rors of τ MC is a consequence of older isoclouds covering a largerfraction of the T e ff − log g parameter space at higher masses,as the varying amounts of core-boundary and envelope mix-ing included in the models cause the corresponding evolution-ary tracks to diverge more strongly with increasing age. Theseisoclouds therefore start to overlap more greatly at older ages,allowing for a greater range in the upper age bound. It shouldbe noted that the isocloud diverges into three distinct sub-cloudsdue to the logarithmic scaling of the log D mix values in our grid.As noted in Paper I, there is a slight discrepancy between theevolutionary and dynamical masses of the primary star. How-ever, our isoclouds were calculated from evolutionary tracks us-ing a fixed Z ini = . . (cid:46) Z ini (cid:46) . Z ini as another free parameter leads to ad-ditional degeneracies (e.g. between Z ini and M , and between Z ini and f ov ). These degeneracies would have further propagated toour subsequent asteroseismic analyses, and as such we chose notto include Z ini as a free parameter for our study.It can be seen in Table 3 that both f ov and D mix are uncon-strained in our fit, although log D mix > M cc is well constrained, and ourvalue of M cc / M ∗ = . ± .
02 is compatible with the range(0 . (cid:46) M cc / M ∗ (cid:46) . γ Dor starsby Mombarg et al. (2019).
3. Asteroseismic modelling
Our asteroseismic modelling is based on two di ff erent sets of as-teroseismic observables: the asymptotic period spacing Π (de-scribed in Eqs. (1) and (2) and the individual period-spacing val-ues ( ∆ P (cid:96) ) of the (cid:96) = (cid:96) = f rot, core = . − . We aretherefore able to estimate the observational values of Π from Π
0, obs (cid:39) ∆ P (cid:96) ∗ √ (cid:96) ( (cid:96) +
1) in the approximation of a non-rotatingstar. The determination of Π is more complicated for fast rota-tors (e.g. Van Reeth et al. 2016, 2018; Li et al. 2018, 2019a,b,2020a,b) as this requires an estimation of the slope of the periodspacing pattern, which is absent for KIC9850387. We are there-fore able to perform asteroseismic modelling using the quantity Π directly, without estimation of the rotation frequency, throughEqs. (1) and (2), by comparing our Π
0, obs estimates with val-ues predicted from the non-rotating stellar evolutionary models.One of the advantages of isocloud fitting (see Section 2) is that itprovides age constraints for our stellar evolutionary models. Wetherefore restricted our evolutionary models to the ages that were It should be noted that we use a 95% HPD only for our isocloud fit based onmodel-independent dynamical constraints, but 68% confidence and HPD inter-vals for the model-dependent evolutionary and asteroseismic parameters pre-sented in the rest of this paper. This was done to ensure su ffi cient baseline cov-erage of the parameter space of the stellar structural models. Article number, page 4 of 21. Sekaran et al.: Evolutionary and asteroseismic modelling of the pulsating eclipsing binary KIC9850387 . . . . . log ( T eff [K]) . . . . . . . . . . l og ( g [ c g s ] ) PrimarySecondary . . . . .
995 3 . . . . . Fig. 1.
Isocloud fit to the components of KIC9850387 on the log T e ff − log g diagram. The red regions correspond to the best-fitting isocloud,and the pink regions represent the isoclouds in the 95% HPD interval of the fit. The black regions on the isocloud represent the log T e ff − log g values corresponding to the dynamical masses of the individual components. The grey curves are mesa evolutionary tracks with f ov = .
005 andlog D mix = . f ov = .
04 and log D mix = . f ov = .
04 and log D mix = . (cid:12) ) indicated at the ZAMS of each track. The inset plots are magnified regions around the position of the primary (top) and secondary(bottom) component. within the 95% HPD interval of the Monte Carlo isocloud-fittingage distribution ( τ MC = . + . − . Gyr).In order to obtain theoretical ∆ P (cid:96) , values, one would have tocalculate pulsational models. These models require stellar struc-tural models (i.e. a model describing the temperature, density, . . . . . (cid:12) ]050010001500200025003000 N u m b e r o f v a l u e s Fig. 2.
Example of the distribution of M values in the posterior distri-bution of models selected through our Monte-Carlo fitting framework.The vertical dashed line represents the median value and the verticalsolid lines represent the upper and lower bounds of 68% HPD of the M values. chemical and mixing profiles from the centre to the surface ofthe star) as an input. Using our isocloud-fitting age constraint, wecomputed a grid of stellar structural models with masses between1.2 M (cid:12) to 2.0 M (cid:12) in steps of 0.02 in X c as an input to computepulsational mode predictions in the adiabatic framework usingthe stellar pulsation code gyre (revision 5.2; Townsend & Teitler2013; Townsend et al. 2018). We then use the theoretical pulsa-tional frequencies of the zonal (cid:96) = (cid:96) = Π .For both approaches, our fitting methodology involves theperturbation of the Π
0, obs and observational ∆ P (cid:96) values of theperiod-spacing patterns ( ∆ P (cid:96) , obs ) within their respective 1 σ ob-servational errors in a Monte-Carlo framework for 10000 itera-tions, and then retaining the parameters of the best-fitting modelin each iteration. To determine the best-fitting model within thisframework, we used two di ff erent merit functions: 1) the χ and 2) the Mahalanobis Distance (MD). As discussed in detailin Aerts et al. (2018), the MD is a maximum-likelihood pointestimator that takes into account uncertainties in the theoreticalasteroseismic predictions and that treats correlations in the freeparameters of our stellar model grid appropriately by incorporat-ing its covariance matrix ( V ) into the distance calculation:MD = ( Y theo − Y obs ) T ( V + Λ ) − ( Y theo − Y obs ) . (4)In this equation, Y theo and Y obs are the vectors representing thetheoretical asteroseismic predictions and asteroseismic observ-ables that are being compared, and the matrix Λ is a diagonal ma- Article number, page 5 of 21 & A proofs: manuscript no. KIC9850387 Π , o b s [ s ] χ MD Y = (Π , ‘ =1 ) , Full Y = (Π , ‘ =1 ) , Spectroscopic Y = (Π , ‘ =1 ) , Dynamical Y = (Π , ‘ =2 ) , Full Y = (Π , ‘ =2 ) , Spectroscopic Y = (Π , ‘ =2 ) , Dynamical Y = (Π , ‘ =1 , Π , ‘ =2 ) , Full Y = (Π , ‘ =1 , Π , ‘ =2 ) , Spectroscopic Y = (Π , ‘ =1 , Π , ‘ =2 ) , Dynamical Π , obs − Π , theo [s] Fig. 3.
Best-fitting Π values of our Π -based asteroseismic modelling, based on three di ff erent grid setups (full, spectroscopic and dynamical).The left panel shows the results of using a χ merit function and the right panel shows the results of using an MD merit function. The verticaldashed black line represents the zero-point of the di ff erence between the observational and theoretical Π (i.e. Π , obs − Π , theo = trix with squared observational errors as each of the diagonal ele-ments. The free parameters to be estimated, determining the the-oretical asteroseismic predictions, are M , X c , f ov and log D mix .The MD takes uncertainties in the theoretical asteroseismic pre-dictions due to the imperfect input physics of the stellar struc-tural models in the grid into account. It does so via the matrix V ,where we assume that its components are well described by thevariance induced by the whole range of the four free parametersdefining the theoretical asteroseismic predictions. Π modelling results We performed our Π modelling by restricting the size of ourgrid based on varying types of observational constraints. Dueto the tight constraints imposed by the dynamical parameters,a grid setup created by restricting the parameter space based onthe dynamical M , T e ff and log g would necessarily be very small,comprising only a few tens of models in total. In order to inves-tigate the e ff ects of di ff erent levels of parameter space restrictionon our results, we chose to construct a third grid setup of inter-mediate detail and size between the full evolutionary grid andthe small asteroseismic grid based on the dynamical constraints.To that end, we obtained a second set of T e ff and log g constraintsby performing atmospheric parameter determination for the pri-mary star of KIC9850387 (as detailed in Section 3.3 of Paper I)but ignoring any binary or eclipse information (i.e. treating thesystem as if it was a single star and performing a classical atmo-spheric analysis).Using these ‘pseudo-single-star’ spectroscopic and dynam-ical parameters, we created three grid setups: 1) The full grid,as the name suggests, is the grid resulting from the 95% HPDinterval of the Monte-Carlo age distribution of our isocloud fit (as described in Section 2) without any other constraints; 2) Thespectroscopic grid is a subset of the full grid based on the 3 σ in-tervals of the ‘pseudo-single-star’ spectroscopic T e ff and log gvalues; and 3) the dynamical grid is a subset of the full gridbased on the 3 σ intervals of the dynamical M , T e ff and log g val-ues. The models in the spectroscopic and dynamical grids have6965 K ≤ T e ff ≤ . ≤ log g ≤ .
13 (spectroscopicgrid), and 1 .
63 M (cid:12) ≤ M ini ≤ .
70 M (cid:12) , 7081 K ≤ T e ff ≤ . ≤ log g ≤ . σ intervals to construct our gridsto ensure that the grids are su ffi ciently large and to introduce suf-ficient variance into the grids, which as mentioned significantlyimpacts our MD calculations.For each grid setup, we fitted the Π , obs estimates for eachof the dipole ( (cid:96) =
1) and quadrupole ( (cid:96) =
2) modes separately,as well as combined into a single vector with two components,resulting in three di ff erent solutions. More specifically, for eachof our grid setups (1,2 and 3), we used three di ff erent sets of ob-servables and predictions in independent fits: a) Y = ( Π , (cid:96) = ),b) Y = ( Π , (cid:96) = ), and c) Y = ( Π , (cid:96) = , Π , (cid:96) = ). The results foreach parameter setup using either the χ or the MD merit func-tions are listed in Table 4 (full grid), Table 5 (spectroscopic grid)and Table 6 (dynamical grid). Comparisons of the χ and theMD results in terms of the best-fitting Π values and positionson Kiel diagrams are shown in Figures 3 and 4 respectively. Theerrors are based on 68% HPD intervals of the Monte-Carlo pa-rameter distributions (e.g. see Figure 2). This allows for propercomparison with the evolutionary modelling and dynamical re-sults, which are based on 68% confidence intervals.It should be noted that in some cases, the models that wereselected in every single Monte-Carlo iteration had the same val-ues for the free parameters, due to the limited resolution of Article number, page 6 of 21. Sekaran et al.: Evolutionary and asteroseismic modelling of the pulsating eclipsing binary KIC9850387 . . . . . . . . . . . . . . l og ( g [ c g s ] ) χ . . . . . . . MDlog ( T eff [K]) Fig. 4.
Positions of the best-fitting models from Π -based asteroseismic modelling on Kiel diagrams. The main plots show the results of usingthe whole grid in the fit, while the inset plots show the results of applying the ‘pseudo-single-star’ spectroscopic T e ff and log g (solid box) anddynamical M , T e ff and log g (dotted box) constraints. The observed positions of the star according to the spectroscopic and dynamical parametersare represented by black ’X’ symbols and stars respectively. The 1 σ and 3 σ spectroscopic and dynamical error bars are represented with straightend-caps and ball end-caps respectively. The Y obs = ( Π , (cid:96) = ), Y obs = ( Π , (cid:96) = ) and Y obs = ( Π , (cid:96) = , Π , (cid:96) = ) solutions for the di ff erent grid subsetsare represented by the same colours and symbols as in Figure 3. The left panel shows the results of using a χ merit function and the right panelshows the results of using an MD merit function. The error bars on the asteroseismic solutions are based on 68% HPD intervals of the Monte Carloparameter distributions. The solid and dashed grey curves are the same as in Figure 1. the grid and its subsets. In these cases, we adopt upper- andlower-bound errors on the estimated parameters ( M , X c , f ov andlog D mix ) that are of a single step size in the grid (i.e. 0.5 M (cid:12) in M , 0.005 in f ov and 0.5 in log D mix ) and propagate these errors tothe inferred parameters ( R , T e ff , log g, M cc and τ ). These errors(hereafter single-grid-step errors) should be interpreted as a con-servative upper limit on the precision of the extracted parametersbased on the resolution of our grid. Parameters for which thesesingle-grid-step errors are assumed are indicated by ∗ next to thevalues in the tables.It can be seen that there is generally little di ff erence in theresults regardless of whether a χ or the MD merit function isused, regardless of the grid setup or Y configurations. This wasan expected result, as the χ and MD merit functions convergefor vectors of unit or near-unit length (due to the term ( V + Λ ) − used in the computation of the MD, see Eq. 4). The small minimaof the respective merit functions shown in Tables 4 to 6 are dueto the near-perfect agreement (sub-second di ff erences in the bestcases) of the theoretical predictions with the perturbed observ-ables in the Monte-Carlo framework, given that we fitted onlyone observable by the means of four free parameters. In addi-tion, there is little di ff erence in the results for the three di ff er-ent Y configurations. This is also unsurprising, as it was alreadynoted in Paper I that the calculated Π values for each of thesemodes agree within 2 σ . The fits using Y obs = ( Π , (cid:96) = , Π , (cid:96) = )are in agreement with those derived from evolutionary modelling(cf. Table 3). Overall, our results of purely asteroseismic mod-elling with Y = ( Π ) provide weaker constraints on the externalproperties ( M , R , T e ff and log g), superior constraints on the inte- rior properties ( M cc , f ov and D mix ) and similar constraints on theevolutionary stage ( X c and τ ) when compared to evolutionarymodelling (cf. Tables 3 and 4). This showcases the superiorityof g-mode asteroseismic modelling over evolutionary modellingwhen it comes to constraining interior properties.The fits using the spectroscopic and dynamical grids (Ta-bles 5 and 6) resulted in superior constraints on most parametersthan using either purely asteroseismic or evolutionary modellingwhen using both the χ and MD merit functions. However, thebest-fitting Π values and errors are almost identical regardlessof the grid subset used in the fit (Figure 3), demonstrating thevalue of asteroseismic observational constraints on parameter es-timation due to its degenerate nature. It can also be seen that thedynamical grid results cluster at the edge of the grid (see in-set plot demarcated by a dotted box in Figure 4), indicating thatthe minima of the respective merit functions are outside of therange of the dynamical grid. When compared to the full grid re-sults, the spectroscopic and dynamical grid results also trendedtowards lower f ov and D mix values.As mentioned previously, the period-spacing patterns ofKIC9850387 are relatively flat (i.e. do not display significantdips), implying either a high amount of mixing at the bottom ofthe radiative envelope or a young evolutionary stage (i.e. high X c values). In general, the envelope mixing levels in intermediate-mass g-mode pulsating stars are hard to infer, as investigated andargued by Mombarg et al. (2019). Their sample study utilisedprinciple component analysis to deduce an expectation value of D mix = s − for the 37 pulsators in their study, which ismuch lower than the wide range of values covered by B-type pul- Article number, page 7 of 21 & A proofs: manuscript no. KIC9850387
Table 4.
Results of asteroseismic modelling with Y = ( Π ) using the full grid of pulsational models, with either a χ or MD merit function. χ MD Y obs = ( Π , (cid:96) = ) Y obs = ( Π , (cid:96) = ) Y obs = ( Π , (cid:96) = , Π , (cid:96) = ) Y obs = ( Π , (cid:96) = ) Y obs = ( Π , (cid:96) = ) Y obs = ( Π , (cid:96) = , Π , (cid:96) = ) M [M (cid:12) ] 1 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . f ov . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . log D mix . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . X c . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . R [R (cid:12) ] 2 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T e ff [K] 6934 + − + − + − + − + − + − log g [dex] 4 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . M cc [ M (cid:12) ] 0 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . τ [Gyr] 1 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . χ | MD min . · − . · − . · − . · − . · − . · − Notes.
The table is divided into two, with the estimated parameters ( M , X c , f ov and log D mix ) presented in the top four rows and the inferredparameters ( R , T e ff , log g, M cc and τ ) in the bottom five rows. The errors quoted are based on 68% HPD intervals of the Monte Carlo parameterdistributions. The minimum values of each merit function across all of the Monte Carlo iterations for each parameter setup are quoted in the bottomrow of the table. Table 5.
Same as Table 4, but for a grid subset based on the 3 σ interval of the ‘pseudo-single-star’ spectroscopic T e ff and log g values (seefootnote). χ MD Y obs = ( Π , (cid:96) = ) Y obs = ( Π , (cid:96) = ) Y obs = ( Π , (cid:96) = , Π , (cid:96) = ) Y obs = ( Π , (cid:96) = ) Y obs = ( Π , (cid:96) = ) Y obs = ( Π , (cid:96) = , Π , (cid:96) = ) M [M (cid:12) ] 1 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . f ov . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . log D mix . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . X c . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . R [R (cid:12) ] 2 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T e ff [K] 7181 + − + − + − + − + − + − log g [dex] 4 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . M cc [ M (cid:12) ] 0 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . τ [Gyr] 1 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . χ | MD min . · − . · − . · − . · − . · − . · − Notes.
Grid subset based on 6965 K ≤ T e ff ≤ . ≤ log g ≤ . Table 6.
Same as Table 4, but for a grid subset based on the 3 σ interval of the dynamical M , T e ff and log g values (see footnote). χ MD Y obs = ( Π , (cid:96) = ) Y obs = ( Π , (cid:96) = ) Y obs = ( Π , (cid:96) = , Π , (cid:96) = ) Y obs = ( Π , (cid:96) = ) Y obs = ( Π , (cid:96) = ) Y obs = ( Π , (cid:96) = , Π , (cid:96) = ) M [M (cid:12) ] ∗ . + . − . ∗ . + . − . ∗ . + . − . ∗ . + . − . ∗ . + . − . ∗ . + . − . f ov ∗ . + . − . ∗ . + . − . ∗ . + . − . ∗ . + . − . ∗ . + . − . ∗ . + . − . log D mix . + . − . ∗ . + . − . ∗ . + . − . . + . − . ∗ . + . − . ∗ . + . − . X c . + . − . ∗ . + . − . ∗ . + . − . . + . − . ∗ . + . − . ∗ . + . − . R [R (cid:12) ] 2 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T e ff [K] 7462 + − + − + − + − + − + − log g [dex] 3 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . M cc [ M (cid:12) ] 0 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . τ [Gyr] 1 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . χ | MD min . · − . · − . · − . · − . · − . · − Notes.
The ∗ indicate the estimated parameters for which single-grid-step errors are assumed. Grid subset based on 1 .
63 M (cid:12) ≤ M ini ≤ .
70 M (cid:12) ,7081 K ≤ T e ff ≤ . ≤ log g ≤ . sators with much bigger convective cores (Aerts 2021, Table 1).Here, our results imply an evolved star with flat patterns reveal-ing an above-average level of mixing for F-type g-mode pul-sators ( D mix ∼ s − ). In fact, given the flat period-spacingpatterns, this value should be seen as a lower limit since higherenvelope mixing will not change the already flat spacing patternsand does not change the global parameters of the stars (Mombarget al. 2019).The minima of the merit functions listed in Tables 4, 5, and6 reveal that using just one observable to estimate the four freeparameters ( M , X c , f ov and log D mix ) is not very constraining asthere are too many degrees of freedom. It is therefore of inter-est to fit the individual ∆ P (cid:96) values of the observed dipole andquadrupole modes. ∆ P (cid:96) modelling results We performed our ∆ P (cid:96) modelling in an identical way to our Π modelling, based on the three grid subsets (full, spectroscopicand dynamical), using three parameter setups ( Y = ( ∆ P (cid:96) = , i ), Y = ( ∆ P (cid:96) = , j ) and Y = ( ∆ P (cid:96) = , i , ∆ P (cid:96) = , j ), where i and j are theindices of the ∆ P (cid:96) values of the corresponding period-spacingpatterns), and computed the same two merit functions ( χ andMD). These results are listed in Table 7 (full grid), Table 8 (spec-troscopic grid) and Table 9 (dynamical grid). Comparisons ofthe χ and the MD results in terms of the best-fitting ∆ P (cid:96) val-ues, period-spacing patterns and positions on Kiel diagrams areshown in Figures 5, 6 and 7 respectively.During our best solution and precision estimation, we en-countered a number of solutions whose precision could not beassessed, because the same grid point was selected in everyMonte-Carlo iteration (as described in Section 3). These so-called ’single-grid-point’ solutions occur more frequently for the χ (7 / / Y obs vector. In these cases, we once again as-signed single-grid-point errors to the estimated parameters andpropagated them to the inferred parameters. Similar to our purelyasteroseismic Π -based results, ∆ P (cid:96) -based modelling resulted inweaker constraints on the external properties ( M , R , T e ff andlog g) and superior constraints on the interior properties ( M cc , f ov and D mix ) when compared to evolutionary modelling.The e ff ect of the choice of merit function is demonstrated byFigure 5. The MD solutions o ff er a larger variety of appropriatesolutions according the various MD minima, because this met-ric allows for theoretical uncertainty in the modelling. Since the χ is a specific solution of the regression based on the assump-tion that there are no theoretical imperfections, while ignoringcorrelations among the observables and parameters, its best so-lutions always occur in the MD solution space as well, but of-ten at much higher MD values. This is graphically illustrated inFigure 5, where the broad coverage of the best allowed periodspacing values selected by the MD is due to the allowance ofuncertainty in the theoretical models via the variance covered bythe entire grid. For example, the MD value of the model from thedynamical grid (Table 9) with the lowest χ value ( χ = χ and MD solutions from the dynami-cal grid, demanding compliance with both the ∆ P (cid:96) values of thedipole and quadrupole modes, binary and the spectroscopy, arevery similar. The MD methodology is sensitive to the model-independent constraints imposed upon the problem set, as it al- lows for the theory of the used grid to be imperfect at the levelcaptured in the variance-covariance matrix induced by the freeparameter ranges. Nevertheless, both metrics end up with an al-most equal solution when demanding compliance with all ob-servational constraints at the 3 σ level. The overall best solutioncombining all of the asteroseismic and dynamical constraints isindicated by the MD solution in boldface in Table 9.Similar to our Π -based results, our dynamical grid re-sults imply above-average levels of envelope mixing, and inthe Y = ( ∆ P (cid:96) = , i ) and Y = ( ∆ P (cid:96) = , i , ∆ P (cid:96) = , j ) cases we find highvalues of D mix (cid:39)
25 cm s − compared to those derived for sin-gle γ Dor pulsators by (Mombarg et al. 2019). While this mayindicate extra tidal mixing, casting doubt on the use of eclipsingbinaries as testbeds of stellar structural and asteroseismic theoryof single stars, several slowly-rotating single stars also displaynear-flat period-spacing patterns (Li et al. 2019a) as was foundfor KIC9850387. Hence we posit that the estimated level of en-velope mixing could still be due to intrinsic non-tidal elementtransport mechanisms.
4. Investigating high-frequency modes
As noted in Paper I, a handful of independent high-frequency modes were observed in the frequency spectrum ofKIC9850387. It was also stated that no frequency splitting orcharacteristic spacing was observed. Having obtained theoreti-cal models that explain our observed ∆ P (cid:96) values, we test thatclaim by computing theoretical frequencies of the (cid:96) = (cid:96) = − to 13.4 d − frequency range for the stellarstructural models within the overall best MD solution indicatedin bold in Table 9. It was found that only two or three frequen-cies were obtained per structural model within the consideredrange, and the modes associated with these frequencies havevarying (cid:96) values, with modes of certain (cid:96) values not occurringin the observed range. For example, the pulsational model withthe closest match between the theoretical and observational fre-quencies, with parameters M = .
70 M (cid:12) , X c = . f ov = . D mix = .
5, consists an (cid:96) = (i.e. n p = n g =
3) at a frequency of 13.22 d − and a low-order (cid:96) = n g =
5) at a frequency of 12.93 d − (see Figure 8, where weshow all modes occurring in the 12.0 d − to 14.0 d − frequencyrange).It can be seen that the theoretical modes are o ff set fromthe observed modes. The same phenomenon was also observedfor the F-type binary pulsator KIC10080943 (Schmid & Aerts2016). Following these authors, we also posit that this o ff setis due to a so-called surface e ff ect (see Ball 2017 for a re-cent review) that is commonplace in solar-like stars due to un-certainties in the physical descriptions of the near-surface con-vection and non-adiabatic e ff ects in the outer envelope. Indeed,while intermediate-mass stars such as the primary component ofKIC9850387 have thin convective envelopes, it has been shownthat time-dependent convection theory is required to model theoscillations properly (see Dupret et al. 2005). The mesa and gyre codes do not include a treatment of time-dependent convection.Hence, frequency shifts between the theoretical predictions andthe observed modes are expected given that the best models werefitted to the g modes. Their mode energy is determined by thephysics in the deep adiabatic interior of the star, where the ap-proximation of time-independent convection is appropriate. A mixed mode is a type of pulsational mode with p-mode character in the thinconvective envelope and g-mode character in the radiative interior. See e.g. Aertset al. (2010) for more information.
Article number, page 9 of 21 & A proofs: manuscript no. KIC9850387 . . . . . . . P ‘ , o b s [ d ] χ MD Y = (∆ P ‘ =1 ) , Full Y = (∆ P ‘ =1 ) , Spectroscopic Y = (∆ P ‘ =1 ) , Dynamical Y = (∆ P ‘ =2 ) , Full Y = (∆ P ‘ =2 ) , Spectroscopic Y = (∆ P ‘ =2 ) , Dynamical Y = (∆ P ‘ =1 , ∆ P ‘ =2 ) , Full Y = (∆ P ‘ =1 , ∆ P ‘ =2 ) , Spectroscopic Y = (∆ P ‘ =1 , ∆ P ‘ =2 ) , Dynamical ∆ P ‘, obs − ∆ P ‘, theo [s] Fig. 5.
Same as Figure 3, but representing the results of ∆ P (cid:96) -based asteroseismic modelling. χ Obs ‘ = 1 , Full ‘ = 1 , Spectroscopic ‘ = 1 , Dynamical ‘ = 2 , Full ‘ = 2 , Spectroscopic ‘ = 2 , Dynamical ‘ = 1 & ‘ = 2 , Full ‘ = 1 & ‘ = 2 , Spectroscopic ‘ = 1 & ‘ = 2 , Dynamical . . . . . . . . P [d] MD ∆ P [ s ] Fig. 6.
Best-fitting (cid:96) = (cid:96) = ∆ P (cid:96) -based asteroseismic modelling. The top panel shows the results of usinga χ merit function and the bottom panel shows the results of using an MD merit function. The pulsational model with the closest match between thetheoretical and observational frequencies can only potentiallyexplain two out of the four observational frequencies. In orderto explain more of the observational behaviour, we hypothesisethat the pair of observed high-amplitude modes at 13.21 d − and 13.25 d − displayed in Figure 8 is part of a rotationally split mul-tiplet. However, the closest theoretical frequency is the (cid:96) = − , and rotational splitting of such a modewould result in a quintuplet. Therefore, it is more likely that thedoublet is a rotationally-split prograde-retrograde ( m = Article number, page 10 of 21. Sekaran et al.: Evolutionary and asteroseismic modelling of the pulsating eclipsing binary KIC9850387 . . . . . . . . . . . . . . l og ( g [ c g s ] ) χ . . . . . . . MDlog ( T eff [K]) Fig. 7.
Same as Figure 4, but representing the results of ∆ P (cid:96) -based asteroseismic modelling. m = −
1) doublet with a missing zonal ( m =
0) mode. This phe-nomenon is a result of cancellation e ff ects (Aerts et al. 2010)due to the near 90 ◦ inclination of the system, and has been ob-served previously in hybrid p- and g-mode pulsators in the massrange of KIC9850387, most notably in Kurtz et al. (2014) wheremultiple well-resolved complete multiplets were reported.As such, we further restricted our pulsational models to thosethat included (cid:96) = − to 13.4 d − fre-quency range. We then determined the model with the smallestdi ff erence in frequency between the (cid:96) = − ). This bestmodel has parameters M = .
65 M (cid:12) , X c = . f ov = . D mix = .
5, and three modes were obtained within thisrange: The fundamental radial mode (i.e. n p = (cid:96) =
0) at a fre-quency of 12.86 d − , a low-order octupole g mode (i.e. n g = (cid:96) =
3) at a frequency of 12.96 d − and the g mode (i.e. n g = (cid:96) =
1) at a frequency of 13.18 d − . These theoretical mode pre-dictions, along with the observed modes, are displayed in Figure9. It should be noted that modes with (cid:96) > ff ects (Aerts et al. 2010), although there have been claims tothe contrary (see e.g. Baran & Østensen 2013). As such, we hy-pothesise that the observed mode at 12.92 d − is the fundamen-tal radial mode. The fundamental radial mode is o ff set from thenearest observed mode by ∼ − while the dipole g modeis o ff set from the nearest observed mode by ∼ − . The val-ues of these frequency o ff sets are smaller than those observedfor KIC10080943 (Schmid & Aerts 2016), who were unable toidentify the degree of the p modes for KIC10080943.We then use this model providing the optimal fit to both theg modes in the deep stellar interior and the observed p modes tocalculate the rotational frequency implied by the splitting of the (cid:96) = (cid:96) + f nlm given by the following equation (see e.g. Aerts et al. 2010): f nlm = f nl + m (1 − C nl ) f rot, puls . (5)In this equation, f nl is the unperturbed frequency of the zonalmode, f rot, puls is the rotational frequency about the pulsationalaxis and C nl is the Ledoux constant (Ledoux 1951). We com-puted it for the best model and obtained a value of C n, (cid:96) = = . (cid:96) = mode. It canbe inferred from the figure that the high-order g modes have thehighest probing power in the near-core region, and the radial andg modes have the highest probing power in the stellar envelope.Using Eq. (5) and f nl = . · (13 . + . = .
23 d − (the midpoint of the doublet), we obtained a value of f rot, puls = . ± . − . This value is 3.2 times that of f rot,core = . ± . − reported by Li et al. (2020a), anda factor of 7.2 times lower than the surface rotational frequency f rot, surf = . + . − . d − reported in Paper I. In that paper, thesurface rotation frequency was derived from the projected ro-tational velocity ( v rot sin i orb ) determined through spectral linemodelling, relying on the orbital inclination ( i orb ) and radius ofthe primary star ( R p ) determined through eclipse modelling, as-suming aligned pulsational and orbital axes. This would implya sharp decrease in the rotational frequency from the surfaceto the envelope, and a more gradual decrease from the enve-lope to the near-core region (as displayed in Figure 11). How-ever, the surface rotational frequency deduced from v rot sin i orb is necessarily an overestimation of the true surface rotationalrate as v rot sin i orb was used as a proxy for the total veloc-ity broadening required to fit the spectral lines. The contribu-tion from asymmetric line-profile variations due to the oscil-lation was ignored (as was mentioned in Paper I). Assumingthat the envelope rotates rigidly (i.e. f rot, surf = f rot, puls ), we ob-tain v rot sin i orb = . ± .
005 km s − . Keeping all other atmo- Article number, page 11 of 21 & A proofs: manuscript no. KIC9850387
Table 7.
Same as Table 6, but representing the results of our asteroseismic modelling with Y = ( ∆ P (cid:96) ) using the full grid of pulsational models. χ MD Y = ( ∆ P (cid:96) = , i ) Y = ( ∆ P (cid:96) = , j ) Y = ( ∆ P (cid:96) = , i , ∆ P (cid:96) = , j ) Y = ( ∆ P (cid:96) = , i ) Y = ( ∆ P (cid:96) = , j ) Y = ( ∆ P (cid:96) = , i , ∆ P (cid:96) = , j ) M [M (cid:12) ] ∗ . + . − . ∗ . + . − . ∗ . + . − . . + . − . ∗ . + . − . ∗ . + . − . f ov ∗ . + . − . ∗ . + . − . ∗ . + . − . . + . − . ∗ . + . − . ∗ . + . − . log D mix ∗ . + . − . ∗ . + . ∗ . + . − . . + . − . ∗ . + . ∗ . + . − . X c ∗ . + . − . ∗ . + . − . ∗ . + . − . . + . − . ∗ . + . − . ∗ . + . − . R [R (cid:12) ] 3 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T e ff [K] 6434 + − + − + − + − + − + − log g [dex] 3 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . M cc [ M (cid:12) ] 0 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . τ [Gyr] 1 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . χ | MD min
187 292 333 19 20 153
Table 8.
Same as Table 7, but for a grid subset based on the 3 σ interval of the ‘pseudo-single-star’ spectroscopic T e ff and log g values (seefootnote). χ MD Y = ( ∆ P (cid:96) = , i ) Y = ( ∆ P (cid:96) = , j ) Y = ( ∆ P (cid:96) = , i , ∆ P (cid:96) = , j ) Y = ( ∆ P (cid:96) = , i ) Y = ( ∆ P (cid:96) = , j ) Y = ( ∆ P (cid:96) = , i , ∆ P (cid:96) = , j ) M [M (cid:12) ] 1 . + . − . . + . − . ∗ . + . − . ∗ . + . − . . + . − . ∗ . + . − . f ov . + . − . . + . − . ∗ . + . − . ∗ . + . − . . + . − . ∗ . + . − . log D mix . + . − . ∗ . + . ∗ . + . − . ∗ . + . − . ∗ . + . ∗ . + . X c . + . − . . + . − . ∗ . + . − . ∗ . + . − . . + . − . ∗ . + . − . R [R (cid:12) ] 1 . + . − . . + . − . . + . − . ∗ . + . − . . + . − . . + . − . T e ff [K] 7351 + − + − + − + − + − + − log g [dex] 4 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . M cc [ M (cid:12) ] 0 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . τ [Gyr] 1 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . χ | MD min
243 399 342 11 17 93
Notes.
Grid subset based on 6965 K ≤ T e ff ≤ . ≤ log g ≤ . Table 9.
Same as Table 7, but for a grid subset based on the 3 σ interval of the dynamical M , T e ff and log g values (see footnote). χ MD Y = ( ∆ P (cid:96) = , i ) Y = ( ∆ P (cid:96) = , j ) Y = ( ∆ P (cid:96) = , i , ∆ P (cid:96) = , j ) Y = ( ∆ P (cid:96) = , i ) Y = ( ∆ P (cid:96) = , j ) Y = ( ∆ P (cid:96) = , i , ∆ P (cid:96) = , j ) M [M (cid:12) ] ∗ . + . − . ∗ . + . − . ∗ . + . − . ∗ . + . − . ∗ . + . − . ∗ . + . − . f ov ∗ . + . − . ∗ . + . − . ∗ . + . − . ∗ . + . − . ∗ . + . − . ∗ . + . − . log D mix ∗ . + . − . ∗ . + . − . ∗ . + . − . ∗ . + . − . . + . − . . + . − . X c ∗ . + . − . ∗ . + . − . ∗ . + . − . ∗ . + . − . . + . − . ∗ . + . − . R [R (cid:12) ] 2 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . T e ff [K] 7434 + − + − + − + − + − + − log g [dex] 3 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . M cc [ M (cid:12) ] 0 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . τ [Gyr] 1 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . χ | MD min
597 1207 927 2403 2937 7555
Notes.
The overall best solution of our asterosesimic analyses is indicated in bold. Grid subset based on 1 .
63 M (cid:12) ≤ M ini ≤ .
70 M (cid:12) , 7081 K ≤ T e ff ≤ . ≤ log g ≤ . spheric parameters identical, it was found that the syntheticspectrum generated using a v rot sin i orb = − and includ-ing a pulsational velocity broadening in the form of macrotur-bulence following Aerts et al. (2009, 2014), requires v macro =
15 km s − . Such a profile provides a qualitatively similar fit to the observed profile than the best-fitting synthetic spectrum with v rot sin i orb =
13 km s − and v macro = − found in Paper I, asshown in Figure 12. Therefore, we conclude that the asteroseis-mic and spectroscopic data are consistent with a rigidly rotatingenvelope rotating ∼ Article number, page 12 of 21. Sekaran et al.: Evolutionary and asteroseismic modelling of the pulsating eclipsing binary KIC9850387 . . . . . − ]0100200300400500600700 A m p li t ud e [ pp m ] Obs ‘ = 2 ‘ = 3 ‘ = 4 Fig. 8.
Pulsational model with the closest frequency match betweenthe observed and theoretical modes in the high-frequency regime ofKIC9850387. The vertical dotted green lines represent the orbital har-monics. the convective core. The near-core region and envelope of theprimary star rotate sub-synchronously with respect to the binaryorbit by factors ∼
69 and ∼
22, respectively.In the above, we have assumed aligned pulsa-tional and orbital axes. Allowing for misalignment (i.e. f rot, puls = f rot, surf sin (90 ◦ − i o ff set ), where i o ff set is the angle be-tween the pulsational and orbital axes), we obtain i o ff set = ± ◦ .This would imply that the axes are near orthogonal to each other.While this could be an example of a so-called tidally-trapped orsingle-sided pulsation (Handler et al. 2020; Kurtz et al. 2020),the lack of correlation between these high-frequency modesand the orbital frequency makes this unlikely. Moreover, fasterenvelope than core rotation has been detected in several otherpulsating close binaries (e.g. KIC819776, Sowicka et al. 2017,and the inner binary of the triple system HD201433, Kallingeret al. 2017).One of the potential mechanisms for this e ff ect is the ’in-verse’ tidal mechanism (Fuller 2020), where tidal interactionwith unstable pulsation modes can transfer energy and angu-lar momentum in a manner that forces the star away from syn-chronicity. The di ff erential rotation of the star, coupled with theasynchrononicity of the surface and envelope rotation with re-spect to the orbit ( f orb = .
364 d − ) noted in Paper I, seems toreinforce this argument. However, the non-detection of tidallyexcited or perturbed pulsations (reported in Paper I), and the factthat close binaries with synchronous surface rotation but asyn-chronous core rotation exist (e.g. KIC819776, Sowicka et al.2017) cast doubt on this possibility, though does not rule it out.The second scenario in which a faster envelope than core rota-tion can develop is by angular momentum transport by internalgravity waves (Rogers 2015), which reinforces our use of mix-ing profiles calibrated by the theoretical simulations of internalgravity waves in our evolutionary models (see Section 2).
5. Discussion and conclusions
Gravity-mode period-spacing series found in the primary of theeclipsing binary KIC9850387 allowed for high precision esti-mation of the stellar parameters of this intermediate-mass F- . . . . . − ]0100200300400500600700 A m p li t ud e [ pp m ] Obs ‘ = 0 ‘ = 1 ‘ = 2 ‘ = 3 ‘ = 4 Fig. 9.
Pulsational model with the closest frequency match between thetheoretical (cid:96) = − and 13.25 d − . The vertical dotted green lines represent theorbital harmonics. . . . . . . R/R ? R o t a t i o n a l K e r n e l ‘ = 1 , n g = 39 ‘ = 0 , n p = 1 ‘ = 1 , n g = 1 Fig. 10.
Rotational kernels of the high- and low-order modes of the best-fitting pulsational model as a function of the fractional radius R / R (cid:63) . type pulsator. We coupled the period spacing patterns of iden-tified dipole and quadrupole modes with an evolutionary and as-teroseismic modelling-based analysis by comparing the obser-vationally and theoretically derived parameters of this star. Toachieve this goal, we computed a grid of evolutionary modelswith 0 .
80 M (cid:12) ≤ M ini ≤ .
00 M (cid:12) with a range of f ov and D mix values typical for this type of pulsator (Mombarg et al. 2019),aside from allowing for a broader range of envelope mixing lev-els to investigate the potential influence of tidal mixing mecha-nisms in the envelope. We then performed isocloud fitting in aMonte-Carlo framework similar to Johnston et al. (2019a), andobtained the evolutionary parameters based on the intersection ofthe dynamical T e ff and log g constraints of each component withthe isocloud parameters corresponding to the 95% HPD intervalof the Monte Carlo age distribution ( τ MC = . + . − . Gyr).We exploited the slow-rotating nature of the primary star ofKIC9850387 by estimating Π directly from the mean period- Article number, page 13 of 21 & A proofs: manuscript no. KIC9850387 . . . . . . R/R ? . . . . . . . f r o t [ d − ] CoreEnvelopeSurface
Fig. 11.
Core (taken from Li et al. 2020a, based on the slope of the g-mode period-spacing pattern), envelope (from the rotational splitting ofthe g mode) and surface (from spectroscopic line broadening ignoringpulsational velocity broadening) rotational frequencies ( f rot ) as a function of the fractional radius ( R / R (cid:63) ) of the primarycomponent of KIC9850387. The core and envelope rotationalfrequencies are positioned at the maxima of the rotational ker-nels of the dipole n g =
39 and n g = ∆ P (cid:96) of the (cid:96) = (cid:96) = Π values extracted from our gridsof evolutionary models. Additionally, we modelled the individ-ual ∆ P (cid:96) values of the observed dipole and quadrupole period-spacing patterns by constructing theoretical period-spacing pat-terns based on non-rotating stellar structural models. We usedtwo di ff erent merit functions ( χ and MD) and tested di ff erentsetups based on the imposition of ‘pseudo-single-star’ spectro-scopic and dynamical constraints. It was found that our astero-seismic modelling provided stronger constraints on the interiorproperties ( M cc , f ov and D mix ) of the primary than the evolution-ary modelling, demonstrating the probing power of g-modes. Wealso found an overall agreement between the asteroseismic andevolutionary modelling results.Our results reinforce the claim of main-sequence binaryevolutionary stage made in Paper I, contradicting the conclu-sion of Zhang et al. (2020) that the system comprises two pre-main-sequence components. Our best-fitting models allowed forstrong constraints on the parameters describing the interior mix-ing profile of the star, comprising a low amount of exponentially-decaying core overshooting ( f ov = . D mix =
25 cm s − ) for this type of pulsator(Mombarg et al. 2019). These findings led to precise constraintson the evolutionary stage of the primary ( X c, p = . ± .
02) andevolutionary modelling allowed for constraints on the secondary( X c, s = . ± . . ± . M cc = . ± .
01, whichis within range of expectation values reported for single F-typeg-mode pulsators by Mombarg et al. (2019). . . . . . . ˚ A]0 . . . . . . . N o r m a li s e d F l u x Fig. 12.
Synthetic spectral fits to a Fe I line of the disentangled primarycomponent spectrum with di ff erent compositions of rotational broaden-ing. The solid and dotted black lines represent synthetic spectra withidentical atmospheric parameter inputs but with v sin i =
13 km s − and v macro = − , and v sin i = − and v macro =
15 km s − respec-tively. We exploited the slow-rotating nature of the primary star ofKIC9850387 by estimating Π directly from the mean period-spacing ∆ P (cid:96) of the (cid:96) = (cid:96) = Π values extracted from ourgrids of evolutionary models. Additionally, we modelled the in-dividual period spacing values of the consecutive zonal dipoleand quadrupole modes via pulsation computations based on non-rotating equilibrium models. We used two di ff erent merit func-tions ( χ and MD) and tested di ff erent setups based on theimposition of ‘pseudo-single-star’ spectroscopic and dynamicalconstraints. It was found that our asteroseismic modelling pro-vided stronger constraints on the interior properties ( M cc , f ov and D mix ) of the primary than the evolutionary modelling, demon-strating the probing power of g modes. We also found an over-all agreement between the asteroseismic and evolutionary mod-elling results.We found little di ff erence in the modelling results regardlessof whether a χ or the MD merit function was used and regard-less of the grid setup or Y configurations for our Π -based mod-elling. However, the di ff erences are significant for our ∆ P (cid:96) -basedmodelling, particularly when the full grid of models is usedin the fit. The application ‘pseudo-single-star’ spectroscopic re-duces these discrepancies, and the application of dynamical con-straints eliminates them altogether. Due to the degenerate natureof the estimated stellar parameters, the best-fitting period spac-ing values and errors in the χ framework are almost identicalregardless of the grid subset used in the fit, while those in the MDframework vary significantly as expected from the constructionof this merit function.After identifying the overall best model combining the as-teroseismic information of the dipole and quadrupole modeswith the dynamical constraints, we investigated the few high-frequency modes that were identified in Paper I by calculatingthe theoretical frequencies of the (cid:96) = (cid:96) = − to 13.4 d − frequency range for our best model. Thefundamental radial mode and the g mode were found within thisfrequency range, and frequency o ff sets of 0.05 d − and 0.03 d − Article number, page 14 of 21. Sekaran et al.: Evolutionary and asteroseismic modelling of the pulsating eclipsing binary KIC9850387 between these modes and the nearest observed mode were ob-tained. We posited that these frequency shifts were due to thesurface e ff ect (e.g. Ball 2017), following the explanation of sim-ilar behaviour in the binary pulsator KIC10080943 by Schmid &Aerts (2016).We investigated the hypothesis that the observed high-amplitude frequency peaks at 13.21 d − and 13.25 d − nearthe theoretical dipole g mode are a part of a rotationally-split prograde-retrograde doublet with a missing zonal mode,as was found in several p- and g-mode hybrid pulsators (e.g.Kurtz et al. 2014). This corresponds to an envelope rotationalof 0 . ± . − that is thrice as high as the core ro-tational frequency 0 . ± . − (Li et al. 2020a). Withinthe limitations of the data, the surface rotation is compatible withthe envelope rotation. Similar behaviour has been observed forother close binaries. For single pulsators of the same mass, thefaster envelope than core rotation has been explained in terms ofangular momentum transport by internal gravity waves triggeredby the convective core (Rogers 2015). Such a mechanism mayalso be active within the primary star of this binary. However,we reinforce that these conclusions are based on the enveloperotational frequency derived from the splitting of a single fre-quency doublet with posited identification as a dipole mode, andas such is subject to uncertainty.Overall, we find that asteroseismic theory and observationsare only barely compliant, reinforcing the need for homogeneousanalyses of samples of pulsating eclipsing binaries that aim atcalibrating interior mixing profiles. Such studies would allow forthe investigation of the sources of discrepancy between the var-ious parameters, and address weaknesses in the descriptions ofangular momentum transport and interior mixing mechanisms. Acknowledgements.
The research leading to these results has received fund-ing from the Fonds Wetenschappelijk Onderzoek - Vlaanderen (FWO) un-der the grant agreements G0H5416N (ERC Opvangproject) and G0A2917N(BlackGEM), and from the European Research Council (ERC) under the Eu-ropean Union’s Horizon 2020 research and innovation programme (grant agree-ment no. 670519: MAMSIE) and from the KU Leuven Research Council (grantC16 / / mesa and gyre developmentteams for making their user-friendly software publicly available. The authorswould also like to thank the Leuven MAMSIE team for useful discussions. Lastbut not least, we would like to acknowledge R. H. D. Townsend for his commentson an earlier version of the manuscript as part of the PhD jury of SS, which ledto a more thorough investigation and subsequent refinement of the internal dif-ferential rotation results for this star. References
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Article number, page 16 of 21. Sekaran et al.: Evolutionary and asteroseismic modelling of the pulsating eclipsing binary KIC9850387
Appendix A: Correlation plots of the MD values in the period-spacing modelling . . M ini [M (cid:12) ]0 . . . . f o v . . M ini [M (cid:12) ]024 l og D m i x . . M ini [M (cid:12) ]0 . . . . X c .
000 0 .
025 0 . f ov . . . . M i n i [ M (cid:12) ] .
000 0 .
025 0 . f ov l og D m i x .
000 0 .
025 0 . f ov . . . . X c D mix . . . . M i n i [ M (cid:12) ] D mix . . . . f o v D mix . . . . X c . . X c . . . . M i n i [ M (cid:12) ] . . X c . . . . f o v . . X c l og D m i x .
00 0 .
25 0 .
50 0 .
75 1 . . . . . . . M D Fig. A.1.
Correlations between model parameters for our asteroseismic modelling with Y = ( ∆ P (cid:96) = , i ). The points in each subplot are colour-codedaccording to the MD values of the whole grid. The white circle and error bars represent the maximum-likelihood estimates and half of the 95%HPD of the Monte Carlo parameter distributions. Article number, page 17 of 21 & A proofs: manuscript no. KIC9850387 . . M ini [M (cid:12) ]0 . . . . f o v . . M ini [M (cid:12) ]024 l og D m i x . . M ini [M (cid:12) ]0 . . . . X c .
000 0 .
025 0 . f ov . . . . M i n i [ M (cid:12) ] .
000 0 .
025 0 . f ov l og D m i x .
000 0 .
025 0 . f ov . . . . X c D mix . . . . M i n i [ M (cid:12) ] D mix . . . . f o v D mix . . . . X c . . X c . . . . M i n i [ M (cid:12) ] . . X c . . . . f o v . . X c l og D m i x .
00 0 .
25 0 .
50 0 .
75 1 . . . . . . . M D Fig. A.2.
Correlations between model parameters for our asteroseismic modelling with Y = ( ∆ P (cid:96) = , j ). The points in each subplot are colour-codedaccording to the MD values of the whole grid. The white circle and error bars represent the maximum-likelihood estimates and half of the 95%HPD of the Monte Carlo parameter distributions.Article number, page 18 of 21. Sekaran et al.: Evolutionary and asteroseismic modelling of the pulsating eclipsing binary KIC9850387 . . M ini [M (cid:12) ]0 . . . . f o v . . M ini [M (cid:12) ]024 l og D m i x . . M ini [M (cid:12) ]0 . . . . X c .
000 0 .
025 0 . f ov . . . . M i n i [ M (cid:12) ] .
000 0 .
025 0 . f ov l og D m i x .
000 0 .
025 0 . f ov . . . . X c D mix . . . . M i n i [ M (cid:12) ] D mix . . . . f o v D mix . . . . X c . . X c . . . . M i n i [ M (cid:12) ] . . X c . . . . f o v . . X c l og D m i x .
00 0 .
25 0 .
50 0 .
75 1 . . . . . . . M D Fig. A.3.
Correlations between model parameters for our modelling with Y = ( ∆ P (cid:96) = , i , ∆ P (cid:96) = , j ). The points in each subplot are colour-codedaccording to the MD values of the whole grid. The white circle and error bars represent the maximum-likelihood estimates and half of the 95%HPD of the Monte-Carlo parameter distributions. Article number, page 19 of 21 & A proofs: manuscript no. KIC9850387
Appendix B: Covariance matrices used in the MD calculations in the period-spacing modelling P → ∆ P ∆ P → ∆ P ‘ = 1 0 5∆ P → ∆ P ∆ P → ∆ P ‘ = 2 0 10 20∆ P → ∆ P ∆ P → ∆ P ‘ = 1 & ‘ = 2460000470000480000490000500000 451000452000453000454000455000456000 450000460000470000480000490000500000 Fig. B.1.
Covariance matrices ( V + Λ , see Eq. 4) of the individual and combined mode-fitting setups, representing the 19 ∆ P (cid:96) = values, the 10 ∆ P (cid:96) = values of the (cid:96) = ∆ P (cid:96) = and ∆ P (cid:96) = values. Appendix C: Displacement and rotational-kernel plots for the (cid:96) = and (cid:96) = modes of KIC9850387 . . . R/R ? − ξ r ( R ) / ξ r ( R ? ) . . . R/R ? − ξ h ( R ) / ξ r ( R ? ) .
07 0 .
08 0 . R/R ? R o t a t i o n a l K e r n e l N [ µ H z ] ‘ = 1 , n g = 20 ‘ = 1 , n g = 39 ‘ = 2 , n g = 36 ‘ = 2 , n g = 46 N Fig. C.1.
Left and middle panels: Radial ( ξ r ( R )) and horizontal ( ξ h ( R )) components of the Lagrangian displacement vectors. Right panel: A plot ofthe rotational kernels and the Brunt-Väisälä frequency ( N ) in the near-core region in which the g modes have the highest probing power. Thesedisplacement vectors and kernels are of the lowest and highest radial order modes of the theoretical period-spacing pattern of our overall bestmodel ( M = .
65 M (cid:12) , X c = . f ov = .
005 and log D mix = . Appendix D: Propagation diagrams of the (cid:96) = and (cid:96) = modes of KIC9850387 . . . . . . f n ‘ [ µ H z ] . . . . . . R/R ? Fig. D.1.
Propagation diagrams showing the mode cavities, frequencies and nodes of the theoretical period-spacing pattern of our overall bestmodel ( M = .
65 M (cid:12) , X c = . f ov = .
005 and log D mix = . N ), and the (cid:96) = (cid:96) = S (cid:96) = and S (cid:96) = ) respectively. The red region represents the (cid:96) = (cid:96) = (cid:96) = (cid:96) = (cid:96) = (cid:96) ==