11institutetext: Institute of Astronomy, KU Leuven, Celestijnenlaan 200D,
B-3001 Leuven, Belgium
Department of Astrophysics, IMAPP, Radboud University Nijmegen, P.O. Box 9010,
6500 GL Nijmegen, The Netherlands
Max Planck Institute for Astronomy, Koenigstuhl 17, 69117 Heidelberg, Germany
11email<EMAIL_ADDRESS>
# Probing the temperature gradient in the core boundary layer of stars with
gravito-inertial modes

The case of KIC 7760680

M. Michielsen 11 C. Aerts 112233 D. M. Bowman 11112233

(Received 17 November 2020; accepted 8 April 2021)

###### Abstract

Aims. We investigated the thermal and chemical structure in the near-core
region of stars with a convective core by means of gravito-inertial modes. We
determined the probing power of different asteroseismic observables and
fitting methodologies. We focus on the case of the B-type star KIC 7760680,
rotating at a quarter of its critical rotation velocity.

Methods. We computed grids of 1D stellar structure and evolution models for
two different prescriptions of the temperature gradient and mixing profile in
the near-core region. We determined which of these prescriptions is preferred
according to the prograde dipole modes detected in 4 yr Kepler photometry of
KIC 7760680. We considered different sets of asteroseismic observables and
compared the outcomes of the regression problem for a $\chi^{2}$ and a
Mahalanobis distance merit function, where the latter takes into account
realistic uncertainties for the theoretical predictions and the former does
not.

Results. Period spacings of modes with consecutive radial order offer a better
diagnostic than mode periods or mode frequencies for asteroseismic modelling
of stars revealing only high-order gravito-inertial modes. We find KIC 7760680
to reveal a radiative temperature gradient in models with convective boundary
mixing, but less complex models without such mixing are statistically
preferred for this rotating star, revealing extremely low vertical envelope
mixing.

Conclusions. Our results strongly suggest the use of measured individual
period spacing values for modes of consecutive radial order as an
asteroseismic diagnostic for stellar modelling of B-type pulsators with
gravito-inertial modes.

###### Key Words.:

Asteroseismology – convection – stars: oscillations (including pulsations) –
stars: interiors – Methods: Statistical – techniques: photometric

##  1 Introduction

Asteroseismology, the study of stellar interiors by means of detected
identified oscillation modes (Aerts et al., 2010), has two major aims. The
first goal is to provide masses, radii, and ages of stars relying on current
knowledge of stellar structure, providing these three quantities with far
better precision than is available from single-epoch data of surface
properties (such as spectroscopy or interferometry). The second goal is to
improve the input physics of stellar interiors so as to achieve better models
for stellar evolution. The need for better models is obvious when it comes to
angular momentum transport across stellar evolution (Gehan et al., 2018; Aerts
et al., 2019) and mixing that guides the change in the mass fractions of the
chemical species (Salaris & Cassisi, 2017). Here our aim is to probe the
temperature and chemical structure in the deep interior of stars and to
improve our models of those quantities, since these are aspects of the stellar
structure that propagate throughout and influence the star’s evolution.

Although angular momentum and element transport are two different physical
phenomena, they are intimately related, and as such they are treated by
similar diffusion coefficients in stellar evolution codes (Maeder, 2009). To
date, asteroseismology has mainly been used to evaluate and improve the
transport of angular momentum (e.g. Fuller et al., 2019; Eggenberger et al.,
2019a, b; den Hartogh et al., 2020; Deheuvels et al., 2020) and to assess the
level of chemical mixing (e.g. Deheuvels et al., 2016; Buldgen et al., 2019;
Bellinger et al., 2019; Angelou et al., 2020), with a specific focus on red
giants and on stars of low mass. These studies are all based on acoustic or
mixed modes excited by envelope convection. In the current work we rely on
gravito-inertial modes in a rotating star of intermediate mass with a well-
developed convective core and a radiative envelope. These modes are excited by
a heat mechanism (Pamyatnykh, 1999) and have frequencies that are similar to
the rotation frequency of the star, such that the modes’ spin parameters
$s=2\Omega/\omega>1$, with $\Omega$ and $\omega$ the rotation and oscillation
frequencies. In this case the Coriolis and buoyancy forces act together as
restoring forces of the oscillations. These oscillations correspond to
gravito-inertial modes (see Aerts et al., 2019, Fig. 5). In particular, we
focus on a B-type pulsator. The extent of and mass in the rotating core-
boundary layers are critical unknowns for this type of stars in the current
theory of stellar structure, yet these properties drive their evolution. For
such stars, the asteroseismic modelling cannot rely on a perturbative
treatment of the Coriolis force, in contrast to the case of acoustic or mixed
modes in slowly rotating stars (see e.g. Ballot et al., 2010, for the limits
of the perturbative method).

Despite their interest from a stellar evolution and chemical yield point of
view, few B-type stars with gravito-inertial modes have been modelled
asteroseismically, and the focus has been on exceptionally slow rotators
(Moravveji et al., 2015; Wu et al., 2020). On the other hand, fast rotators
offer the more interesting case to probe stellar interiors with the aim of
improving stellar evolution theory (Moravveji et al., 2016; Buysschaert et
al., 2018; Szewczuk & Daszyńska-Daszkiewicz, 2018). One of the reasons for the
limited number of modelled cases is that only the Kepler space telescope
(Borucki et al., 2010) offered sufficiently long time-series photometry to do
so, and massive stars were avoided by it (see Bowman, 2020, for a recent
review). While time series covering a few months allow us to detect high-order
gravito-inertial modes in main-sequence stars and to identify their mode
wavenumbers $(l,m)$ from period spacing patterns (Degroote et al., 2010b;
Pápics et al., 2012; Kallinger et al., 2017), only the 4 yr Kepler light
curves of B-type pulsators lead to sufficient frequency precision to
discriminate among various profiles representing convective boundary mixing
(hereafter CBM; Pedersen et al., 2018). Another reason for the scarcity of
asteroseismically modelled B-type pulsators is that few of those observed by
Kepler have delivered suitable modes to perform modelling of their interiors.
Finally, the methodology used to perform the modelling in the case of gravito-
inertial modes is more complex than the framework based on modes in the
perturbative regime of rotation (see Aerts, 2021, for a review of the various
modelling regimes).

Several prescriptions to describe the temperature gradient, $\nabla_{T}(r)$,
and the diffusion coefficient defining the local chemical mixing profile in
the convective core boundary layers, $D_{\rm mix}(r)$, exist. We consider two
main options: firstly convective penetration with an adiabatic temperature
gradient, $\nabla_{\rm ad}$, and a constant mixing profile in the core
boundary layers, following Zahn (1991), and secondly diffusive overshooting
with a radiative temperature gradient, $\nabla_{\rm rad}$, and an
exponentially decaying CBM profile, following Freytag et al. (1996) and Herwig
(2000). Here we are concerned with convective boundary layers adjacent to a
convective core in a main-sequence star. We take a data-driven approach based
on observed gravito-inertial modes, which are excellent probes of the near-
core regions deep inside stars, rather than relying on (uncalibrated)
numerical simulations of these regions.

In a proof-of-concept study, Michielsen et al. (2019) showed that observed
gravito-inertial modes hold the potential to discriminate between a radiative
and an adiabatic temperature gradient, along with assessment of the mixing
profile in the core boundary layers, provided that the mass, evolutionary
stage, internal rotation, and metallicity are known. In this work we improve
this initial theoretical study and apply it to one of the rotating B-type
pulsators with a long series of prograde dipole gravito-inertial modes of
consecutive radial order, KIC 7760680. Our aim is to assess the temperature
and chemical gradients of the core boundary layers in this star from
observations, but also to investigate the impact of various choices of
asteroseismic observables in the modelling process on the inferred outcome. We
do not focus on the observational aspects of the detected gravito-inertial
modes because the frequency uncertainties are typically two orders of
magnitude below those stemming from unknown input physics of stellar models
(see Aerts et al., 2018). We thus consider the case of KIC 7760680 by relying
on its observational properties derived previously by Pápics et al. (2015),
and briefly recall the observational input of the modelling in the next
section.

##  2 The gravito-inertial B-type pulsator KIC 7760680

We have opted to use the slowly pulsating B star KIC 7760680 as a case study
for this paper. Our reasons for this choice are twofold. First, this is the
only star rotating at a considerable fraction of its critical rotation rate
(about 26%, Moravveji et al., 2016) with an extensive identified mode period
pattern (36 prograde dipole modes of consecutive radial order). Second, this
star has a remarkably low level of inferred mixing in its envelope compared to
slower rotating B-type pulsators, while one would expect its rotation to
induce strong mixing beyond the convective core. The radial shear-induced
mixing in differentially rotating massive stars is about three to ten orders
of magnitude higher according to theoretical predictions than the envelope
mixing found by Moravveji et al. (2016). We refer to Fig. 6 in Meynet & Maeder
(2000), Fig. 3 in Mathis et al. (2004), and Figs. 15 and 16 in Decressin et
al. (2009) for various examples. An explanation for this low envelope mixing
proposed by Moravveji et al. (2016) is that the star is nearly a rigid-body
rotator.

Following up on the best model parameters for the star found by Moravveji et
al. (2016) listed in Table 1, we assess whether a more structured CBM profile
is meaningful and leads to a better fit to the star’s observed oscillation
modes. Moravveji et al. (2016) considered $\nabla_{\rm rad}$ in the CBM
region, limited the parameter estimation to the use of a $\chi^{2}$ merit
function, and performed mode frequency fitting. Given the mathematical
challenges connected to it, no precision estimation for the parameters in
Table 1 was done in this previous study. In the case of strong correlations
among the parameters to estimate, as is the case here, the very small grid
step size listed in Table 1 tends to be an underestimation of the true
uncertainties because this neglects the parameter correlation structure. We
aim to improve upon this omission of error estimation, while searching for a
more accurate interior model of the star. This improvement concerns several
aspects, namely the use of more optimized asteroseismic observables, a more
appropriate merit function taking the correlation structure of the fitting
problem into account, and a more structured $D_{\rm mix}(r)$ profile compared
to the initial study by Moravveji et al. (2016). Another reason to choose this
star as a test case is that it has an excellent asteroseismic calibration of
its near-core rotation frequency. Moravveji et al. (2016) derived a value of
$0.4805$ d-1 under the assumption of rigid rotation, which corresponds to
$26$% of its critical Roche rotation frequency. To reduce dimensionality, we
thus fixed this value of the rotation frequency.

Table 1: Stellar parameters of the best asteroseismic model of KIC 7760680
derived by Moravveji et al. (2016).

Parameter | value | grid step size
---|---|---
$M_{\rm ini}$ [ M⊙] | 3.25 | 0.05
$Z_{\rm ini}$ | 0.020 | 0.001
$f_{\rm CBM}$ | 0.024 | 0.001
log($D_{\text{env}}$) | 0.75 | 0.25
$X_{c}$ | 0.503 | 0.001

Notes. From top to bottom, the parameters are initial mass, metallicity, CBM
parameter, envelope mixing, and central hydrogen content.

The Kepler light curve of this pulsator and its Fourier transform have already
been discussed in great detail in Pápics et al. (2015), so we do not repeat
this information here for brevity. In practice, we use the period values of
the 36 prograde dipole modes listed in Table 1 of Pápics et al. (2015) as
asteroseismic input for our modelling, $\textbf{Y}^{\text{obs}}$ composed of
$\text{Y}_{i}^{\text{Obs}}$ with $i=1,\ldots,36$. Pápics et al. (2015) list
the formal errors from a least-squares fit to the light curve, after pre-
whitening all the modes with amplitudes above four times the local noise
level. Schwarzenberg-Czerny (1991) showed that the formal least-squares errors
are statistically equivalent to the errors derived in the Fourier domain if
one takes into account the variance in the data, the signal-to-noise ratio,
and the total time base of the data set. However, the author noted that a
correction factor to the formal errors is required to take into account the
correlated nature of the data, following their covariance structure. In line
with the method in Schwarzenberg-Czerny (1991), we multiply the errors listed
in Table 1 of Pápics et al. (2015) by a correction factor of four, as was done
for the modelling performed by Moravveji et al. (2016). This leads to errors
on the mode periods lower than $\sim 10\,$s, while the time resolution
restriction stemming from the light curve that may have affected the choice of
frequencies during the pre-whitening process amounts to $\sim 60\,$s.

As additional input to guide the modelling, we rely on
T${}_{\text{eff}}=11650\pm 210$K, log $g$=3.97 $\pm$ 0.08 dex, and [M/H]=0.14
$\pm$ 0.09 dex, again taken from Pápics et al. (2015). We also rely on
$\log\,(L/L_{\odot})=2.19\pm 0.06$ based on Gaia DR2 astrometry deduced by
Pedersen et al. (2020).

##  3 Chemical mixing profiles

(a) Near-core region with radiative temperature gradient

(b) Near-core region with temperature gradient according to Péclet number

Figure 1: Radial profiles of a 3 M⊙ star halfway through the main sequence
with a central hydrogen content $X_{c}=0.4$. The top panels show the
temperature gradients. The middle panels show the structure (solid) and
composition (dashed) components of the Brunt-Väisälä profile, as well as the
shape of the mixing profiles, divided in convective core (grey), near-core
mixing (blue), and diffusive mixing in the outer radiative envelope (green).
The bottom panels show the mode inertia of two g-modes with different radial
orders.

In the diffusion approximation, the transport equation describing the rate of
change in the mass fraction of each individual chemical element $i$ in the
considered chemical mixture, denoted here as $X_{i}$, reads

$\displaystyle{\frac{\partial X_{i}(r)}{\partial t}}=\displaystyle{{\cal
R}_{i}(r)}+\displaystyle{\frac{1}{\rho(r)r^{2}}\frac{\partial}{\partial
r}\left[\rho(r)r^{2}D_{\rm mix}(r)\frac{\partial X_{i}(r)}{\partial
r}\right],}$ (1)

with ${\cal R}_{i}(r)$ the change due to nuclear reactions and $D_{\rm
mix}(r)$ the local mixing coefficient. In this work this mixing coefficient in
the convective boundary layers is implemented as extended convective
penetration described by Michielsen et al. (2019), with an extra diffusive
mixing profile in the radiative envelope due to internal gravity waves as
deduced by Rogers & McElwaine (2017), following Pedersen et al. (2018). The
mixing profile outside of the regions determined to be convective by the
Ledoux criterion can hence be described by

$\displaystyle D_{\text{CBM}}(r)$ $\displaystyle=D_{0}$
$\displaystyle\text{for}\quad r_{0}<r<r_{\rm cp},$ (2) $\displaystyle
D_{\text{CBM}}(r)$ $\displaystyle=D_{0}\exp{\left(\frac{-2(r-r_{\rm
cp})}{f_{\rm CBM}H_{\rm p{}}}\right)}$ $\displaystyle\text{for}\quad r_{\rm
cp}<r<r_{\rm CBM},$ (3) $\displaystyle D_{\text{env}}(r)$
$\displaystyle=D_{\text{env}}\frac{\rho(r_{\rm CBM})}{\rho(r)}$
$\displaystyle\text{for}\quad r_{\rm CBM}<r,$ (4)

with $r_{0}=r_{\rm cc}-f_{0}H_{\rm p{}}$ being the radius where the transition
between core mixing and core boundary mixing is made, $r_{\rm cc}$ being the
radius of the convective core as obtained from the Ledoux criterion applied
for mixing length theory (MLT; Böhm-Vitense, 1958), and $H_{\rm p{}}$ the
local pressure scale height; $D_{0}$ is the mixing coefficient at $r_{0}$, and
is hence influenced by the choice of $f_{0}$. The edge of the step-like CBM in
the transition region is given by $r_{\rm cp}=r_{0}+\alpha_{\rm CBM}H_{\rm
p{}}$. Beyond this value the mixing profile takes on an exponentially decaying
function with free parameter $f_{\rm CBM}$ until the core boundary transition
region reaches the envelope of the star, whose bottom is situated at $r_{\rm
CBM}$ and where the envelope mixing becomes active rather than CBM. This
corresponds to the radius where the convective boundary mixing coefficient
becomes smaller than $D_{\text{env}}$, which stands for the mixing at the
bottom of the radiative envelope. The latter mixing coefficient takes over as
of this position towards the surface of the star. The free parameters
$\alpha_{\rm CBM}$ and $f_{\rm CBM}$ determine the extent of the step and
exponentially decaying profile part of $D_{\rm mix}(r)$ in the core boundary
region, respectively.

Employing this prescription for the mixing profile, two cases are considered
regarding the temperature gradient in the core boundary transition region. The
first one (seen in Fig. 1(a)) adopts the radiative temperature gradient,
$\nabla_{\text{T}}=\nabla_{\text{rad}}$, outside the convective core
($r>r_{0}$). The second one (seen in Fig. 1(b)) assumes that the overshooting
material influences the entropy stratification in the transition layers, as
proposed by Zahn (1991). Instead of a transition in temperature gradient based
on the mass coordinate in the convective boundary mixing region used in
Michielsen et al. (2019), we opt for a transition based on the Péclet number,
as this is more physically motivated (see e.g. Viallet et al., 2015, for an
overview of the different convective regimes). The Péclet number is defined as
the ratio of the advective transport rate to the diffusive transport rate,
such that the flow can be considered diffusive when Pe $\ll$1 and adiabatic
when Pe$\gg$1\. Bridging these two regimes was implemented by making a gradual
transition based on the Péclet number, in terms of a factor
$h=(\log(\text{Pe})+2)/4$. This factor was constructed to create a small
transition region between the regimes instead of introducing a discontinuity
at the interface. This implementation causes Pe=1 to correspond to the
temperature structure being halfway between $\nabla_{\text{ad}}$ and
$\nabla_{\text{rad}}$:

$\begin{array}[]{ll}\nabla_{\text{T}}=\nabla_{\text{ad}}&\text{for}\ \
\text{Pe}>100,\\\
\nabla_{\text{T}}=h\nabla_{\text{ad}}+(1-h)\nabla_{\text{rad}}&\text{for}\ \
0.01<\text{Pe}<100,\\\ \nabla_{\text{T}}=\nabla_{\text{rad}}&\text{for}\ \
\text{Pe}<0.01.\end{array}$ (5)

One of our goals is to answer whether the 36 detected dipole gravito-inertial
modes of KIC 7760680 allow us to distinguish between these two options of
temperature gradients in the transition layer between its convective core and
radiative envelope, adopting the observational diagnostics of the star from
Pápics et al. (2015).

##  4 Computation of theoretical mode frequencies

###  4.1 Stellar equilibrium models

Two grids of stellar models were computed as input for pulsation computations.
These two grids provide equilibrium models with the same input physics, except
for the adopted temperature gradient in the core boundary transition region.
One grid adopts the radiative temperature gradient in that transition zone and
is termed the radiative grid. For the other grid we adopt the temperature
gradient according to Eq. 5 and we call it the Péclet grid. Both grids are
identical in terms of the free parameter ranges listed in Table 2. The ranges
of the mass and central hydrogen fraction were guided by the earlier study of
Moravveji et al. (2016), while the metallicity range relied on the
spectroscopic results taken from Pápics et al. (2015).

The grids were computed using the stellar evolution code MESA (Paxton et al.,
2011, 2013, 2015, 2018, 2019) version r12115. The MESA models contain the
standard chemical mixture of OB stars in the solar neighbourhood deduced by
Nieva & Przybilla (2012) and Przybilla et al. (2013) and make use of the OP
opacity tables (Seaton, 2005). An Eddington grey atmosphere is used as
atmospheric boundary condition. To set the initial mass fractions ($X_{\rm
ini},Y_{\rm ini},Z_{\rm ini}$) we rely on the initial metallicity $Z_{\rm
ini}$ as derived from the spectroscopic measurements derived by Pápics et al.
(2015), and vary it along its uncertainty range. The initial helium fraction
is then determined by adopting an enrichment law $Y_{\rm ini}=Y_{p}+(\Delta
Y/\Delta Z)Z_{\rm ini}$, where the primordial helium abundance, $Y_{p}$, and
the galactic enrichment ratio, $\frac{\Delta Y}{\Delta Z}$, come into play.
There is currently no consensus on the value of $\frac{\Delta Y}{\Delta Z}$
(see e.g. Verma et al., 2019, and references therein). Therefore, we have
taken $Y_{p}=0.2465$, as determined by Aver et al. (2013), and we required
that the enrichment ratio was able to reproduce the mass fractions of the
adopted chemical mixture ($X$=0.71, $Y$=0.276, $Z$=0.014) derived by Nieva &
Przybilla (2012). This led us to adopt $\Delta Y/\Delta Z$=2.1. After $Y_{\rm
ini}$ is determined according to this enrichment law, $X_{\rm ini}$ is set
following $X_{\rm ini}=1-Y_{\rm ini}-Z_{\rm ini}$. We refer to Nsamba et al.
(2020) for an in-depth comparison between this approach of using an enrichment
ratio, and including the initial helium abundance as a free variable.

To determine where the transition from core to near-core mixing is made, we
fixed $f_{0}=0.005$, with the exception of setting $f_{0}=0$ for the models
where both $\alpha_{\rm CBM}$ and $f_{\rm CBM}$ are equal to zero as there is
no CBM region for this case. The mixing length parameter was set to
$\alpha_{\text{mlt}}=2.0$ in the mixing length theory as developed by Cox &
Giuli (1968). We used the Ledoux criterion for convection without allowing for
semi-convection, since this form of slow mixing does not occur in the presence
of convective boundary mixing (see e.g. Kaiser et al., 2020), which is
included in the vast majority of models in our grids. Appendix A provides the
link to the detailed MESA setup.

Table 2: Parameter ranges of each of the two grids of equilibrium models constituting of 117600 grid points used for the asteroseismic modelling of KIC 7760680. Parameter | lower boundary | upper boundary | step size
---|---|---|---
$M_{\rm ini}$ [ M⊙] | 2.8 | 3.7 | 0.1
$Z_{\rm ini}$ | 0.015 | 0.023 | 0.004
$\alpha_{\rm CBM}$ | 0 | 0.3 | 0.05
$f_{\rm CBM}$ | 0 | 0.03 | 0.005
log($D_{\text{env}}$) | 0 | 2 | 0.5
$X_{c}$ | 0.3 | 0.6 | 0.02

###  4.2 Pulsation computations

To model the 36 consecutive dipole g modes detected in KIC 7760680, the
pulsation mode properties of the MESA equilibrium models were computed
employing the stellar oscillation code GYRE (Townsend & Teitler, 2013;
Townsend et al., 2018), version 5.2. We computed the dipole g-modes for a
fixed rotation frequency of 0.4805 d-1, assuming rigid rotation as discussed
previously. We relied on the traditional approximation of rotation (TAR,
following the implementation by Townsend et al., 2018) and adopted the
adiabatic approximation to compute the dipole gravito-inertial mode
frequencies. The mode inertias of the considered g-modes are dominant near the
core of the star, as illustrated in Fig. 1. Hence the adiabatic approximation
is sufficient for our modelling work, whilst non-adiabatic effects mainly
become important in the outer stellar envelope.

Fig. 1 reveals the difference in temperature gradient between two models from
the two MESA grids. Although it is relatively small and leaves the composition
term of the Brunt-Väisälä frequency
(N${}_{\text{composition}}=\frac{g^{2}\rho}{P}\nabla_{\mu}$) unchanged, it
does change the thermal structure term
(N${}_{\text{structure}}=\frac{g^{2}\rho}{P}(\nabla_{\text{ad}}-\nabla_{\text{T}}$)),
which causes a different mode trapping. This is illustrated for the gravito-
inertial modes with radial orders 25 and 45.

It is this type of difference in the mode kernels that brings about the
potential to assess the temperature gradient in an observed star, provided
that it delivers suitable modes to make this assessment. It was shown by
Moravveji et al. (2016, see their Fig. 10) that KIC 7760680 is a suitable
candidate to offer this opportunity.

The final GYRE inlist is provided through the link in Appendix A and provides
us with a list of theoretically predicted prograde dipole mode frequencies,
$\textbf{Y}^{\text{Theo}}$ composed of $\text{Y}_{j}^{\text{Theo}}$, where $j$
stands for the radial order, for each of the equilibrium models in the two
grids.

##  5 Modelling approaches

The asteroseismic modelling of a star involves various aspects, some of
statistical nature and others connected with astrophysical considerations. The
problem comes down to an estimation effort to assess the free parameters
covered in the two model grids listed in Table 2. We achieve this by relying
on the list of observed gravito-inertial mode frequencies, or any other
observables derived from them and denoted as $\textbf{Y}^{\text{Obs}}$ with
$\text{Y}_{i}^{\text{Obs}}$ labelled as $i=1,\ldots,36$, where the radial
order of the observed modes is unknown a priori. These observables are matched
to a list of theoretically predicted analogues $\textbf{Y}^{\text{Theo}}$,
with $\text{Y}_{j}^{\text{Theo}}$ to be computed from a list of model
frequencies of radial order $j=1,\ldots,100$. There are various aspects of
this matching process that have to be considered. Ultimately it comes down to
solving regression problems, where various choices can be made for the
observables to match and the merit functions to minimize. We discuss each of
these aspects in the following subsections, noting here that we do not use the
spectroscopic and astrometric information as observables in the fitting
procedure, but use the measured spectroscopic abundances to determine the
range of initial metal and helium mass fractions.

In the case of gravito-inertial modes of B-type pulsators with 4 yr Kepler
light curves, we are in the situation where non-seismic observables from
spectroscopy or astrometry, such as $T_{\rm eff}$, $\log\,g$, or
$\log\,(L/L_{\odot})$ have relative observational uncertainties that are
typically two orders of magnitude or more above those of the measured
frequencies (see Table 1 in Aerts et al., 2019). For this reason, these
classical observables are not included in the merit functions, but rather
serve to eliminate stellar models in the two grids having values of these non-
seismic observables outside the observed regime (see e.g. Buysschaert et al.,
2018).

###  5.1 Model selection criterion

The asteroseismic modelling involves maximum likelihood estimation based on
the chosen observables and their theoretically predicted counterparts computed
from astrophysical models. The modelling may involve nested or non-nested
statistical models (see Claeskens & Hjort, 2008). We are dealing with both
cases. Firstly a comparison between the performance of models within one grid
of equilibrium models where one or both of the CBM parameters are fixed at
zero ($\alpha_{\rm CBM},f_{\rm CBM}$, or both) versus a free parameter to
estimate. Such a comparison constitutes a nested model comparison allowing the
most appropriate statistical model to be selected. We are doing this model
selection as a test to evaluate whether the added complexity in the core
boundary transition region of the models improves the fit to the data more
than the entailed punishment by the selection criterion for having additional
free parameters. Secondly a comparison between the performance of the
radiative versus Péclet grids while estimating the same free parameters for
both grids deals with a model selection based on non-nested models as only the
input physics changes, but not the degrees of freedom.

In the case of model selection for nested statistical models, we use the
Akaike Information Criterion corrected for small sample size (AICc, Claeskens
& Hjort, 2008, Chapter 2). This type of model selection rewards fit quality,
but penalizes complexity. Moreover, the AICc is an appropriate criterion to
evaluate nested models with a limited number of observables yet considerable
dimensionality of the free parameter vector, as we have here (36 modes to
estimate six free parameters, see Table 2). The AICc is defined as

$\text{AICc}=-2\ln{\cal L}+\frac{2kN}{N-k-1},$ (6)

with $N$ and $k$ being the number of observables and free parameters,
respectively, and ${\cal L}$ being the likelihood of a stellar model.

In our framework of asteroseismic modelling, $N=36$ or 35 depending on whether
we choose to fit mode frequencies, periods, or period spacings. The number of
free parameters $k$ is according to the parameters varied in the grids and is
4, 5, or 6 in our case, which means that ($M_{\rm ini},Z_{\rm ini},\alpha_{\rm
CBM},f_{\rm CBM},\log(D_{\text{env}}),X_{c}$) are used to assess the general
performance of the two grids, and the same list but with only one or neither
of ($\alpha_{\rm CBM},f_{\rm CBM}$) included to test if increased CBM
complexity performs better or worse within one grid. In this latter
application, we make use of the property that two nested models A and B can be
compared in performance through their difference in AICc; if
$\Delta\text{AICc}=\text{AICc}_{\text{A}}-\text{AICc}_{\text{B}}>2$, model B
is favoured over model A, with the evidence being (very) strong if
$\Delta\text{AICc}>6$ (10) (Kass & Raftery, 1955; Claeskens & Hjort, 2008).
These regimes are derived from the value of $2\ln[P(D|A)/P(D|B)]$, with
$P(D|X)$ the probability of observing $D$ given model $X$. A difference in
$\Delta\text{AICc}$ of 6 or 10 implies that the probability of reproducing the
data better with model A than with model B is roughly 95% or 99.3%,
respectively.

###  5.2 Merit functions: Mahalanobis distance versus $\chi^{2}$

Uncalibrated physical ingredients required as input physics of stellar
evolution models are usually dealt with via the inclusion of free parameters.
Often these ingredients are correlated. Thus, the systematic uncertainties
originating from these limitations in the model input physics cause a
variance–covariance structure in the theoretically predicted observables.
This, as well as non-linear correlations among the observables and hence in
the overall correlated nature of the estimation problem to solve, has to be
addressed in the regression problem.

Since the beginnings of space asteroseismology, Bayesian inferences have been
developed and applied in the modelling of solar-like oscillations of low-mass
stars (e.g. Appourchaux et al., 2009; Serenelli et al., 2013; Silva Aguirre et
al., 2013; Metcalfe et al., 2014; Silva Aguirre et al., 2015, 2017). In such
applications the Coriolis acceleration is ignored in the prediction of the
oscillation frequencies. The asteroseismic observables of rotating stars with
a convective core and gravito-inertial modes demand the inclusion of the
Coriolis acceleration in a non-perturbative way, implying that the g-mode
frequencies are strongly dependent on the rotational frequency. Moreover, the
observables are also correlated in a non-linear way, for example the modes are
often strongly trapped in the near-core region (Pápics et al., 2017). In
addition, in the mass regime under study here, several of the free parameters
of the equilibrium models to be estimated are also strongly correlated,
following tight and sometimes non-linear relationships, such as between mass
and metallicity, convective-core-mass and mass, and convective-core-mass and
age (e.g. Maeder, 2009).

The non-linear correlations among the free parameters and observables, along
with the inclusion of the uncertainties for the theoretical predictions in the
estimation problem lend itself to use a more appropriate merit function
instead of the traditional $\chi^{2}$ (whose results we include in Appendix B
for comparison). Here we take the full variance–covariance structure among the
observables and the theoretical predictions into account, thereby mitigating
the aforementioned shortcomings. This is achieved by building a regression
model via the use of the so-called Mahalanobis distance (see Aerts et al.,
2018, for its application to asteroseismic modelling),

$\displaystyle\text{MD}_{j}=\left(\textbf{Y}^{\text{theo}}_{j}-\textbf{Y}^{\text{obs}}\right)^{T}\left(V+\Sigma\right)^{-1}\left(\textbf{Y}^{\text{theo}}_{j}-\textbf{Y}^{\text{obs}}\right),$
(7)

where $\textbf{Y}^{\text{obs}}$ is the vector of observables,
$\textbf{Y}^{\text{theo}}_{j}$ is the corresponding vector of predicted values
in gridpoint $j$ of the grids of equilibrium models, $V$ is the
variance–covariance matrix of $\textbf{Y}^{\text{theo}}$ due to the unknown
physical ingredients in the input physics of the equilibrium models, and
$\Sigma$ is the variance due to the measurement errors of
$\textbf{Y}^{\text{obs}}$. A major advantage of this approach is that it
allows the inclusion of theoretical uncertainties for the predictions
$\textbf{Y}^{\text{theo}}$ used in the modelling.

The likelihood function corresponding to the Mahalanobis distance to be used
in the statistical model selection is assumed to adhere to normality (Aerts et
al., 2018, for arguments why this is a good approach). Using the likelihood
function of the multivariate normal distribution, we find the likelihood of
parameters $\boldsymbol{\theta}=(\theta^{1},\theta^{2}...\theta^{k})$ given
the observed data $\mathbf{D}$,

$\displaystyle\cal L(\boldsymbol{\theta}|\mathbf{D})=$
$\displaystyle\exp\left(\vphantom{-}\frac{1}{2}\left(\vphantom{\left(\sum\right)}\ln(|V+\Sigma|)+k\ln(2\pi)\right.\right.$
$\displaystyle+\left.\left.\left(\textbf{Y}^{\text{theo}}-\textbf{Y}^{\text{obs}}\right)^{T}\left(V+\Sigma\right)^{-1}\left(\textbf{Y}^{\text{theo}}-\textbf{Y}^{\text{obs}}\right)\right)\right)$
$\displaystyle=$
$\displaystyle\exp\left(-\frac{1}{2}\left(\ln(|V+\Sigma|)+k\ln(2\pi)+\text{MD}\right)\right),$
(8)

which is proportional to the probability density function

$\displaystyle\cal L(\boldsymbol{\theta}|\mathbf{D})\propto
P(\mathbf{D}|\boldsymbol{\theta}).$ (9)

We can write the AICc in this case as

$\text{AICc}=\ln(|V+\Sigma|)+k\ln(2\pi)+\text{MD}+\frac{2kN}{N-k-1}.$ (10)

To determine the uncertainty region of the best solution, we use the
likelihood in combination with Bayes’ theorem, stating that the probability of
a parameter $\theta^{m}$ occurring in the interval —$\theta^{m}_{a}$,
$\theta^{m}_{b}$— is given by

$\displaystyle P(\theta^{m}_{a}<\theta^{m}<\theta^{m}_{b}|\mathbf{D})=$
$\displaystyle\frac{\sum_{i}^{q}P(\mathbf{D}|\boldsymbol{\theta}_{i})P(\boldsymbol{\theta}_{i})}{\sum_{j}^{Q}P(\mathbf{D}|\boldsymbol{\theta}_{j})P(\boldsymbol{\theta}_{j})}$
$\displaystyle=$
$\displaystyle\frac{\sum_{i}^{q}P(\mathbf{D}|\boldsymbol{\theta}_{i})\prod_{l}^{k}P({\theta}_{i}^{l})}{\sum_{j}^{Q}P(\mathbf{D}|\boldsymbol{\theta}_{j})\prod_{l}^{k}P({\theta}_{j}^{l})}.$
(11)

Here the index $j$ is summed over all $Q$ equilibrium models in the grid that
are consistent within 3$\sigma$ of the spectroscopic observables, and index
$i$ is summed over the $q$ models with the highest likelihood so that
$P(\theta^{m}_{a}<\theta^{m}<\theta^{m}_{b}|\mathbf{D})=0.95$. Sometimes only
one equilibrium model in the grid fulfils the requirement to reach this
probability of 0.95. In that case we consider the uncertainty region to span a
smaller area than the one defined by the grid step. This leaves us with the
step sizes of the grid in each of its dimensions as an upper limit of the
uncertainty region in that particular case.

###  5.3 Observables to fit

We investigate which is the optimal set of observables to be used in the
stellar modelling procedure. In addition to just the mode periods, we also
investigate the period spacing values (i.e. the differences in period between
two pulsation modes of consecutive radial order, $\Delta P_{n}\equiv
P_{n+1}-P_{n}$). The best of these sets of observables is determined from the
condition numbers of the variance–covariance matrices $V+\Sigma$ following the
Mahalanobis distance definition.

In general, the condition number $\kappa$ of a matrix $A$ is defined as the
ratio of its maximum to minimum eigenvalue,

$\kappa(A)=\frac{|\lambda_{max}(A)|}{|\lambda_{min}(A)|},$ (12)

and gives a handle on how well- or ill-conditioned the matrix is with respect
to the inversion to be computed. A rule of thumb is that $\log(\kappa)$ gives
an estimate of the number of digits of accuracy lost in addition to the loss
due to arithmetic methods. Given this, we search for the vector of
asteroseismic observables that gives the lowest condition numbers.

###  5.4 Constructing the theoretical period spacing pattern

When modelling an observed pattern of gravito-inertial modes of consecutive
radial order, the theoretical mode periods are typically matched to the
observed ones, as inspired by asymptotic relations for gravito-inertial modes
(Aerts et al., 2010). However, different choices can be made to determine the
starting point to build the mode pattern. Three such choices are compared in
modelling performance in this paper. In the first we begin matching mode
periods starting from the theoretical period that is closest to the highest-
frequency mode detected in the observed pattern. A second option is to build
the pattern starting from the mode with the highest observed amplitude. The
third method is to match each observed mode period to its best matching
theoretical counterpart, and adopt the longest sequence of consecutive modes
that we get in this way. In the case of multiple mode series with the same
length, a final pattern selection is made based on the best match between
theory and observations. We henceforth refer to these three options of pattern
construction as highest frequency, highest amplitude, and longest sequence,
respectively.

##  6 Modelling results

###  6.1 Selected observables to fit

When calculating the condition number of the variance–covariance matrices
according to Eq. 12, we find values on the order of $10^{6}$ when we solely
consider the observed mode periods to fit. The variance–covariance matrices
for this case, which express the coverage of the mode periods across the model
grids, are shown for the two grids, and for the three ways of constructing the
period spacing pattern in Figs. 19, 20, 21, 22, 23, and 24. Variances in the
theoretical mode period predictions across the entire grids reach up to
$\sim$3400 s in this case. This is about 50 times larger than the
observational uncertainties for the mode periods due to the time resolution
and noise properties added together.

On the other hand, the variance–covariance matrices for the case of
theoretically predicted period spacing values shown in Figs. 25, 26, 27, 28,
29, and 30 cover up to $\sim$400 s in variance for the theoretical predictions
of the spacings, which is about 20 times larger than the formal uncertainties
on the observed period spacing values quoted by Pápics et al. (2015) and
Moravveji et al. (2016). When we fit the period spacing values in the detected
pattern instead of the periods, the condition numbers significantly improve,
yielding $\kappa(A)\sim 10^{3}$. We therefore conclude that the period
spacings are the optimal set of observables to fit for our case study of KIC
7760680.

The results for both sets of observables are very similar for this star in
terms of the correlations and constraints on the optimal solutions. The
correlation structures of the radiative grid are shown in Figs. 2 and 3 for
the two sets of observables. While the correlation structures in these figures
are similar, the condition numbers for the cases of the period spacing values
as observables are far lower than those for the periods as observables, and
hence constitute the best modelling setup in terms of numerical stability.

The better suitability of the period spacing values to fit as observables
compared to the mode periods themselves also has advantages from the input
physics point of view. The mode periods are subject to systematic uncertainty
connected to various choices for the chemical mixture and opacities in B-type
pulsators, as illustrated in Moravveji et al. (2015, their Fig. 3). This
systematic uncertainty decreases drastically when considering mode period or
mode frequency differences. Similarly, dipole gravito-inertial modes of
$\gamma\,$Doradus pulsators computed from equilibrium models with atomic
diffusion are globally shifted compared to those without taking atomic
diffusion and radiative levitation into account (Mombarg et al., 2020). The
mode period shift due to atomic diffusion is illustrated graphically in Fig. 5
of Aerts (2021).

We note that Moravveji et al. (2016) fitted the mode frequencies in their
modelling of KIC 7760680. Just as for the case of considering mode periods,
their procedure makes the modelling result more subject to uncertainties in
the input physics of the equilibrium models than our new approach in this
work. This is particularly so because these authors did not include the
covariance among the free parameters to estimate in their modelling. The
results in our paper improve the asteroseismic modelling of KIC 7760680 on
these two fronts, also providing precision estimation that was not attempted
in the first modelling efforts for this g-mode pulsator.

Figure 2: Correlation plot for the radiative grid, using periods in a
Mahalanobis distance merit function. The theoretical pulsation pattern was
constructed according to the longest sequence method. The 50% best models are
shown, colour-coded according to the log of their merit function value (at
right). The figures on the diagonal show binned parameter distributions, and
the larger panel at the top right shows a Hertzsprung–Russell diagram with the
1 and 3$\sigma$ spectroscopic $T_{\rm eff}$ and astrometric luminosity error
boxes. Figure 3:  Correlation plot (see Fig. 2) for the radiative grid, using
period spacings in a Mahalanobis distance merit function. The theoretical
pulsation pattern was constructed according to the longest sequence method.
Figure 4:  Correlation plot (see Fig. 2) for the Péclet grid, using period
spacings in a Mahalanobis distance merit function. The theoretical pulsation
pattern was constructed according to the longest sequence method. Figure 5:
Correlation plot (see Fig. 2) for the Péclet grid, using period spacings in a
Mahalanobis distance merit function. The theoretical pulsation pattern was
constructed according to the highest amplitude method. Figure 6:  Correlation
plot (see Fig. 2) for the Péclet grid, using period spacings in a Mahalanobis
distance merit function. The theoretical pulsation pattern was constructed
according to the highest frequency method.

###  6.2 Theoretical mode period pattern construction

The Mahalanobis distance (MD) yields the same best model regardless of the
method used for the theoretical pattern construction in the radiative grid.
The MD and AICc values for this grid change slightly (Table 4) due to the
changed correlation structure, as can be seen in the variance–covariance
matrices in Figs. 25, 26, and 27, but they are relatively small differences.
The highest amplitude method is slightly disfavoured to compose the pattern
compared to the other two in terms of their MD values.

Although the global morphology of the correlation plots remains rather similar
in the Péclet grid (Figs. 4, 5, and 6), the selected best model does change
based on the way the theoretical pattern is constructed, as can be seen in the
summary plot in Fig. 9 and in Table 4. Looking at the differences in AICc
values, the model returned using the longest sequence method is preferred over
the other two methods, and the distributions of the MD values are in general
shifted in favour of it (Fig. 7). This method searches for a closely matching
sequence of frequencies to start building the pattern from, as opposed to the
other methods that match just one frequency and build the pattern from there
onward. It thus comes as no surprise that this method yields better matching
patterns overall.

Figure 7:  Distribution of the Mahalanobis distances using $\Delta P$ as
observables to fit. The inset shows the cumulative distribution function of
the 20 best models in the grids.

###  6.3 Radiative grid versus Péclet grid

When comparing the best models of the radiative and Péclet grid, the AICc
cannot be coupled to a hypothesis test to determine how strongly one of these
two astrophysical models is preferred because we are not dealing with
evaluating nested regression models. We have an equal number of free
parameters to fit for both of the grids. However, we can still rank the models
with the same free parameters based on their AICc values. We find that the
radiative grid has systematically lower AICc values than the Péclet grid, both
for the best models (Table 4) and for the distribution of the MD values as a
whole (Fig. 7). Without being able to make firm statistical conclusions at the
level of a significant difference, we find that the radiative grid provides a
better fit to the observations.

In the next section we also consider nested regression models for both the
radiative and Péclet grids. These models point out that, statistically
speaking, the least complex grid without CBM is preferred (Table 7). From this
point of view, the treatment of the temperature gradient in the CBM region
becomes irrelevant since the presence of such a region altogether is
statistically less favourable.

Figure 8:  Period spacing patterns of the observations, of the best models of
the two grids, and of the nested grid where no CBM is included in the models,
as extracted by the longest sequence method. The formal errors on the
observations are smaller than the symbol sizes. The vertical bars in the
bottom left corner show the maximum considered uncertainty for the theoretical
predictions approximated by the variance–covariance matrix of that particular
grid. The middle and bottom panels show the relative difference in period
spacing and period, respectively, between the observation and the model. The
narrow grey areas indicate the formal 1$\sigma$ observational uncertainty from
Pápics et al. (2015), including the correction factor of four.

Figure 9: Parameters of the selected best model with their uncertainty ranges
indicated by the black lines. If no uncertainty was found through Bayes’
theorem, its upper limit is indicated by the dashed grey lines. The blue and
red symbols indicate the results for the radiative and Péclet grid,
respectively, while the labels on the y-axis indicate how the theoretical
frequency pattern was constructed. The top row of figures represents the full
grids, while the rows underneath represent the nested grids with fewer
parameters.

###  6.4 Nested statistical regression models

To verify if the better fit quality outweighs the penalties for increased
model complexity, we compare the AICc values of the best regression models
based on the full grid with six free parameters (Table 4) to those based on
partial grids with five free parameters where either $\alpha_{\rm CBM}$ (Table
5) or $f_{\rm CBM}$ (Table 6) is fixed at 0. Additionally, we compare all of
these regression model fits to the results from the partial grid composed of
stellar models without any presence of CBM having just four free parameters
(Table 7). Since the discovery of their period spacing patterns (Degroote et
al., 2010a), slowly rotating B-type g-mode pulsators have been shown in the
literature to have good solutions for a very small amount of CBM (Wu et al.,
2020). This was our motivation for these tests on nested statistical models.

On the other hand, mixing in the radiative envelope was found to be a
necessary ingredient for these stars (e.g. Moravveji et al., 2015). However,
in none of these studies with asteroseismic modelling was a fixed zero CBM
considered in a way that takes proper account of a smaller number of degrees
of freedom in the regression, nor were covariances due to theoretical
uncertainties considered in the fitting problem. In particular, Moravveji et
al. (2016) did not consider models without CBM in their modelling of KIC
7760680.

The added complexity in the regression models is more heavily penalized
statistically than rewarded by the better quality of the fit. We consistently
find $\Delta\text{AICc}>10$ when comparing models with five free parameters to
those with six, or when comparing four free parameters to five. This implies a
strong statistical preference for the less complex regression models in the
case of this pulsator with exceptionally low mixing beyond its convective
core. This capacity of the best models with and without CBM is illustrated in
Fig. 8, where we have indicated the maximum variance in the theoretical
predictions according to the complete radiative and Péclet grids.

We point out that a different conclusion would be reached if we were to rely
on the $\chi^{2}$ merit function, as is usually done in the literature of
asteroseismic modelling. Appendix B offers all the results of the modelling of
the prograde dipole modes in KIC 7760680 when ignoring any variance–covariance
in the theoretical predictions (i.e. pretending that the theory adopted in the
stellar models is fully correct) as was done by Moravveji et al. (2016). Under
this (incorrect) assumption, the quality of the fit increases by adding the
complexity of a CBM layer with two free parameters to the models. This is
assessed quantitatively by comparing the minima listed in Tables 8, 9, 10, and
11 with those in Tables 4, 5, 6, and 7. It is visually shown in Fig. 8 if one
ignores the theoretical uncertainties whose maxima are indicated by the
coloured lines in the bottom left of the upper panel. A firm conclusion
irrespective of the choice of merit function, even in the case where any
theoretical uncertainties are ignored, is that the radiative grid delivers the
better regression solution.

We conclude that if we allow for theoretical uncertainties in the regression
problem at the level of the variance–covariance covered by the model grids,
the prograde dipole modes of KIC 7760680 do not provide statistical evidence
of the occurrence of CBM in this star, which has an exceptionally low mixing
beyond the convective core. For the models without CBM, an increase in the
uncertainties on the inferred model parameters occurs compared to the
situation where more complex models with CBM are considered. This can be seen
by the increased uncertainty of the solutions for the case of the decreasing
number of free parameters in Fig. 9.

###  6.5 Bootstrapping observed frequencies

In this paper we have been relying on the observational input from Pápics et
al. (2015), but we wondered how the frequency resolution of the data may have
impacted the selection of modes during the extraction and identification of
period spacing patterns via iterative pre-whitening (see Sect. 2). This is
particularly important for modes of rather low amplitude as choices in the
method of iterative pre-whitening can impact the resultant period spacing
pattern. To assess if and how our asteroseismic modelling might depend on the
robustness of an extracted period spacing pattern, we considered the
statistical resampling method of bootstrapping where we replaced one of the
modes in the observed period spacing pattern by another value, relying on the
frequency resolution of the light curve, R=$1/\Delta T$=0.00068d-1.

We perturbed different observed frequencies within the pattern one at a time
by increasing or decreasing its value by R. We performed this experiment for a
mode at high period (at 1.45261d), a trapped mode (at 1.24544d), a non-trapped
mode (at 1.117507d), and for a mode at low period (at 0.90147d) within the
extracted pattern from Pápics et al. (2015). For each of these cases we then
repeated the whole analysis for these patterns based on the one perturbed mode
frequency instead of the one from Pápics et al. (2015). This gives us an idea
of how sensitive the asteroseismic modelling solutions are to an individual
frequency being different in the pattern. Such a sensitivity analysis is
standard in modern statistics and is of particular relevance in the case of a
trapped gravito-inertial mode leading to a dip in the period spacing pattern.

The results of remodelling all these bootstrapped patterns are shown in Fig.
10. The best solutions sometimes deviate from the one found for the
unperturbed pattern, but for the cases where this occurs the uncertainty range
almost always includes the unperturbed solution. Our modelling results hence
reveal to be mostly robust against the replacement of one mode in the pattern
according to the frequency resolution. We recall that this resolution is a
gross overestimation of the actual frequency uncertainties, but it must be
kept in mind in the context of a pre-whitening procedure (Pápics et al.,
2014).

Figure 10: Same as Fig. 9, but the different lines per extraction method are
for each of the different perturbed patterns from the bootstrapping; the upper
lines are for the unperturbed pattern.

##  7 Discussion and conclusion

In this work we investigated the relative importance of various aspects when
performing forward asteroseismic modelling of gravito-inertial modes. It is
important to not only consider which physical prescriptions or free parameters
are used in the computational setup, but to also take different methods for
the observational input of the regression problem into account. We highlighted
the impact of choices in the procedures via the case study of KIC 7760680,
which is the B-type star with the largest number of identified prograde dipole
gravito-intertial modes and the best assessed near-core rotation rate among
all the published main-sequence gravito-inertial mode pulsators to date
(Pápics et al., 2015; Moravveji et al., 2016).

We focused our study on the question of whether it is possible to infer the
most likely temperature gradient in the near-core boundary layer from
identified high-order gravito-inertial modes. This was done by considering two
model grids based on two different prescriptions for this temperature
gradient, following the potential capacity of dipole g-mode asteroseismology
published in the twin papers by Pedersen et al. (2018) and Michielsen et al.
(2019). While doing so, we considered two different merit functions to solve
the regression problem, following Aerts et al. (2018). One of these is
$\chi^{2}$ while the other is the Mahalanobis distance taking into account the
variance–covariance properties of the theoretical predictions, including the
entire correlation structure of the regression problem at hand. Moreover, we
assessed the quality of different sets of observables to perform the
regression based on the comparison between observed and theoretically
predicted pulsation patterns.

In terms of a comparison between the two merit functions, we found that using
the Mahalanobis distance not only changes the preferred model parameters as
compared to $\chi^{2}$, but it constrains the optimal solution better.
Irrespective of the merit function, we found a clear preference for the model
grid based on the radiative temperature gradient in the small core-boundary
layer for KIC 7760680 in models that include this layer, instead of a
gradually changing temperature gradient between the adiabatic and radiative
gradient based on the Péclet number. We emphasise that this preference for a
radiative temperature gradient in the small core-boundary layer has been
obtained for just this one rotating B-type pulsator, and it should not be
considered a general result for all B stars given the immense diversity in
observed period spacing patterns and the large range of CBM and envelope
mixing levels for 26 such stars found by Pedersen et al. (2021), as summarized
in Aerts (2021, Table 1). The summary of all available g-mode asteroseismology
of B stars has so far revealed that KIC 7760680 has exceptionally low envelope
mixing among those stars rotating at a considerable fraction of the critical
rotation rate. We also found that models without any convective penetration or
overshooting are statistically preferred for this rotating star, as shown by
our evaluation of nested regression models.

For the regression problem based on the Mahalanobis distance, the use of
different sets of quantities to be compared with the respective observables
can heavily influence the condition number of the used variance–covariance
matrix. In particular, the way the theoretical pulsation quantities are used
deserves attention when adopting the Mahalanobis distance as merit function
since it impacts the structure of the variance–covariance matrix to be
inverted in the regression. The effects for the regression based on a
$\chi^{2}$ merit function are smaller since in that case we ignore the
covariance among the theoretical prediction or even the theoretical
uncertainty as a whole. We should find a compromise between the best numerical
stability and the use of maximum observable information in the definition of
the regression problem. The application of singular value decomposition or
principle component analysis as adopted for some of the modelling efforts of
low-mass stars (e.g. Angelou et al., 2017) or intermediate-mass pulsators
(Mombarg et al., 2019) might be a useful additional or complementary approach
to the one we adopted here for g-mode asteroseismology of B stars.

Table 3: Stellar parameters of the best asteroseismic model of KIC 7760680 obtained in this work for the cases without and with CBM. Parameter | Models w/o CBM | Models with CBM
---|---|---
$M_{\rm ini}$ [ M⊙] | 3.3 | 3.5
$Z_{\rm ini}$ | 0.019 | 0.023
$\alpha_{\rm CBM}$ | (…) | 0.2
$f_{\rm CBM}$ | (…) | 0.0
log($D_{\text{env}}$) | 1.5 | 0.0
$X_{c}$ | 0.44 | 0.44

The best model parameters we found for the cases with and without CBM are
provided in Tables 4 and 7 and are listed in Table 3. Overall, the model
parameters obtained by Moravveji et al. (2016) listed in Table 1 are outside
of the uncertainties on the model with CBM, but within those of the
statistically preferred model without CBM. The well-known mass–metallicity and
mass–CBM degeneracies for pulsators in this mass range (Moravveji et al.,
2015, their Fig. 5) are the basis of the uncertainty regions of the estimated
parameters, whose projection in one dimension are shown graphically in Fig. 9.
The solved regression problem based on the Mahalanobis distance merit function
properly takes into account these correlations and places the solution based
on the $\chi^{2}$ as obtained by Moravveji et al. (2016) within the
uncertainty regions of the estimated parameters. This is remarkably consistent
given that we used a non-flat $D_{\rm mix}(r)$ profile and performed the
fitting from the $\Delta$P values instead of the mode frequencies as used by
Moravveji et al. (2016). Our less refined grid step implied a major reduction
in the analysis time of the problem and allowed us to treat the primary goal
set for this paper, which was to investigate if we can distinguish between a
different thermal and chemical structure in the CBM region while having fixed
the shape of the mixing profile as diffusive core overshooting in the core-
boundary layer, and as being due to wave mixing in the envelope. With this
choice of core overshooting, which is often taken in the literature, we found
that a radiative temperature gradient in the CBM region gives a better fit to
the data than the models with a temperature gradient based on the Péclet
number. However, the presence of such a CBM region altogether turned out to be
statistically disfavoured by the AICc for this particular star rotating at
about a quarter of its critical rate.

Finally, it is noteworthy that the best model with CBM listed in Table 3 falls
within the $3\sigma$ error box of the spectroscopic $T_{\rm eff}$ and
astrometrically derived $\log(L/L_{\odot})$ as shown in Fig. 3, while these
observables were not used in the asteroseismic modelling process. This a
posteriori evaluation of the asteroseismic solution is reassuring as there was
no need to reject models based on these non-asteroseismic observables.

Overall, we conclude that choices in the modelling methodology should be made
with great care and consideration. The optimal modelling setup might vary from
star to star because some pulsators reveal signatures of trapped modes, while
others do not (Pápics et al., 2017). A fully automated modelling strategy may
therefore be premature for ensemble modelling of gravito-inertial B-type
pulsators covering the entire instability strip of such stars (see Pedersen et
al., 2020), particularly as our current case study was done for a star with
very high-precision observables and an exceptionally low level of mixing in
the core-boundary layer and stellar envelope compared to most other gravito-
inertial pulsators of this kind among the B-type stars (Pedersen et al.,
2021).

###### Acknowledgements.

The authors are grateful to the MESA and GYRE developers teams led by Bill
Paxton and Rich Townsend for their efforts and for releasing their software
publicly; this study would not have been possible without their codes. The
authors thank Geert Molenberghs for his helpful advice on the statistical
aspects of the study, Joey Mombarg for the fruitful discussions, and Cole
Johnston for his insights in theoretical pulsation pattern construction. We
thank the referee for the comments and suggestions towards improving the
manuscript. The research leading to these results has received funding from
the Research Foundation Flanders (FWO) by means of a PhD scholarship to MM
under project No. 11F7120N and a senior postdoctoral fellowship to DMB with
grant agreement No. 1286521N, from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement no. 670519: MAMSIE to CA), and from the KU Leuven Research Council
(grant C16/18/005: PARADISE).

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##  Appendix A MESA and GYRE Inlists

The example MESA and GYRE inlists used for this work are available from the
MESA Inlists section of the MESA Marketplace:
cococubed.asu.edu/mesa_market/inlists.html.

##  Appendix B Reduced $\chi^{2}$ as merit function

###  B.1 Model selection criterion

The most commonly used merit function to assess the goodness of fit in forward
asteroseismic modelling of SPB pulsators is a $\chi^{2}$, which is based on a
Euclidian distance. Denoting a set of uncorrelated observables as
$\textbf{Y}^{\text{Obs}}$ with errors according to a normal distribution with
variance $\sigma^{2}_{\text{Y}^{\text{Obs}}}$, and the set of corresponding
values predicted by a theoretical model as $\textbf{Y}^{\text{Theo}}$, we
write the reduced $\chi^{2}$ score as

$\displaystyle\chi^{2}_{\text{red}}=\frac{1}{N-k}\sum^{N}_{i}\left(\frac{\text{Y}^{\text{Theo}}_{i}-\text{Y}^{\text{Obs}}_{i}}{\sigma_{\text{Y}^{\text{Obs}}_{i}}}\right)^{2},$
(13)

where $N$ is the number of observables and $k$ is the number of free
parameters. In general, our MD formulation used in the main part of the paper
can be connected to the $\chi^{2}_{\text{red}}$ merit function by ignoring
uncertainties on the theoretical predictions of the observables and by
ignoring correlations among the observables. Ignoring theoretical
uncertainties comes down to ignoring the covariance structures shown in
Appendix D. We provide the $\chi^{2}_{\text{red}}$ results here to assess the
differences in the solutions to the regression problem by making these
simplifications, following the earlier assessment of theoretical uncertainties
in Aerts et al. (2018). The modelling procedures applied to the g-mode
frequencies of rotating SPB stars in the literature so far consider the mode
frequencies as independent observables (Moravveji et al. 2016; Szewczuk &
Daszyńska-Daszkiewicz 2018), such that using a $\chi^{2}_{\text{red}}$ is
meaningful. Nevertheless, we show the results obtained by using this
simplified merit function for two cases, of mode periods and of mode period
spacings, where the latter in principle demands using a formalism suitable for
correlated data such as the MD. This allows the reader to assess the impact
induced by adopting a simplified $\chi^{2}_{\text{red}}$ for applications to
correlated data, while also ignoring theoretical uncertainties.

Using $\chi^{2}_{\text{red}}$ as the merit function, the AICc reduces to

$\text{AICc}=\chi^{2}_{\text{red}}+\frac{2kN}{N-k-1}.$ (14)

###  B.2 Observables to fit

When considering $\chi^{2}$ as the merit function, there is no
variance–covariance matrix whose numerical stability needs to be taken into
account through condition numbers. We simply compare the use of different sets
of observables and look at the correlation plots and modelling results to
assess their use.

Figure 11:  Correlation plot for the radiative grid, using periods in a
$\chi^{2}$ merit function. The theoretical pulsation pattern was constructed
according to the longest sequence method. The 50% best models are shown,
colour-coded according to the log of their merit function value (at right).
The figures on the diagonal show binned parameter distributions, and the
larger panel at the top right shows an Hertzsprung–Russell diagram with the 1
and 3$\sigma$ spectroscopic $T_{\rm eff}$ and astrometric luminosity error
boxes.

The correlation plots show a better constrained region with optimal solutions
when selecting the period spacings as observables instead of the periods
themselves. This can be seen comparing Figs. 11 and 12 for the radiative grid,
or Figs. 13 and 14 for the Péclet grid.

Figure 12:  Correlation plot (see Fig. 11) for the radiative grid, using
period spacings in a $\chi^{2}$ merit function. The theoretical pulsation
pattern was constructed according to the longest sequence method. Figure 13:
Correlation plot (see Fig. 11) for the Péclet grid, using periods in a
$\chi^{2}$ merit function. The theoretical pulsation pattern was constructed
according to the longest sequence method. Figure 14:  Correlation plot (see
Fig. 11) for the Péclet grid, using period spacings in a $\chi^{2}$ merit
function. The theoretical pulsation pattern was constructed according to the
longest sequence method.

Apart from the correlation structure, the actual $\chi^{2}$ values themselves
are also lower when fitting period spacings. This is no surprise since the
mode periods have much smaller relative errors, of order 0.003% or even
smaller in the case of KIC 7760680 (Pápics et al. 2015), than the propagated
errors on $\Delta P$, which are about 10 to 20 times larger.

###  B.3 Theoretical mode period pattern construction

Although the different methods used to construct the theoretical frequency
pattern can influence the distribution of the $\chi^{2}$ values (Fig. 15), the
best model in the grid and its $\chi^{2}_{\text{red}}$ value remain identical
when considering the full grids of models (Table 8). The optimal solution
slightly changes for some of the nested statistical models (Tables 11, 9, and
10), but this change in most cases only amounts to one grid step-size for one
parameter. Additionally, the differences in the correlation plots are minimal,
and the same global morphology applies (Figs. 12, 16, and 17).

Figure 15:  Distribution of the reduced $\chi^{2}$, using $\Delta P$ as
observables to fit. The inset shows the cumulative distribution function of
the 20 best models in the grids. Figure 16:  Correlation plot (see Fig. 11)
for the radiative grid, using period spacings in a $\chi^{2}$ merit function.
The theoretical pulsation pattern was constructed according to the highest
amplitude method. Figure 17:  Correlation plot (see Fig. 11) for the radiative
grid, using period spacings in a $\chi^{2}$ merit function. The theoretical
pulsation pattern was constructed according to the highest frequency method.

###  B.4 Nested statistical regression models

Investigating the best regression models with varying complexity in the CBM of
the models reveals a different trend than for the Mahalanobis distance.
Comparing the best models of the full grids with six free parameters (Table 8)
to those of the partial grids with five (Tables 9 and 10) or four free
parameters (Table 11), reveals that the more complex models are generally
preferred over the simpler ones. In contrast to when using the Mahalanobis
distance, the increased fit quality plays a dominant role over the penalty for
the increased model complexity. The main reason for this is that no
uncertainty is propagated for the theoretical predictions when using a
$\chi^{2}$ merit function.

When comparing the different partial grids we find a clear preference for the
models with an exponentially decaying CBM coefficient. This is in agreement
with the results of Moravveji et al. (2016). Using a $\chi^{2}$ merit function
the authors found that an exponentially decaying CBM prescription outperformed
a step-like mixing coefficient in the CBM region. We thus recover the previous
result under the restricted assumption that the theoretical model predictions
are error-free.

###  B.5 Radiative grid versus Péclet grid

According to the $\chi^{2}_{\rm red}$ values the radiative grid gives a better
fit to the observations than the Péclet grid. Although there is a clear
difference between the best models of the two grids (Table 8), the
distribution of their $\chi^{2}_{\text{red}}$ values does not differ too much
(Fig. 15). However, a different conclusion arises when using periods as the
observables without considering the period spacings. Specifically, the
$\chi^{2}_{\text{red}}$ distributions are again similar between the two grids,
but the preferred best model stems from the Péclet grid (Figs. 18 and 8).

We note that the best solutions according to the Mahalanobis distance are not
among the best solutions delivered by the $\chi^{2}_{\text{red}}$ metric
(Table 4). On the other hand, the best models according to the
$\chi^{2}_{\text{red}}$ still score among the better models following the
Mahalanobis distance (listed in Table 8), although they are rated
significantly worse than the best ones for that metric.

Even though the conclusions of the analysis of this star with the
$\chi^{2}_{\text{red}}$ merit function would not have been influenced by the
way we construct the theoretical pulsation pattern, the choice of observables
(mode periods versus period spacings) changes the answer to the question of
which temperature gradient is preferred when only looking at the best point
estimator, without taking into account any precision estimation.

Figure 18:  Distribution of the reduced $\chi^{2}$, using periods as
observables to fit. The inset shows the cumulative distribution function of
the 20 best models in the grids.

##  Appendix C Best models

Tables listing the parameters of the best fit models.

Table 4: Best fit models according to Mahalanobis distance. Grid | Observables | Pattern construction |  $M_{\rm ini}$ [ M⊙] | $Z_{\rm ini}$ | $\alpha_{\rm CBM}$ | $f_{\rm CBM}$ | log($D_{\text{env}}$) | $X_{c}$ | $\chi^{2}_{\text{red}}$ | MD | AICc(MD)
---|---|---|---|---|---|---|---|---|---|---|---
Radiative |  $\Delta$P | longest sequence | 3.5 | 0.023 | 0.2 | 0.0 | 0.0 | 0.44 | 3479 | 182 | -294.0
|  | highest amplitude | 3.5 | 0.023 | 0.2 | 0.0 | 0.0 | 0.44 | 3479 | 186 | -290.6
|  | highest frequency | 3.5 | 0.023 | 0.2 | 0.0 | 0.0 | 0.44 | 3479 | 184 | -293.6
| Period | longest sequence | 3.5 | 0.023 | 0.2 | 0.0 | 0.0 | 0.44 | 362028 | 168 | -315.8
|  | highest amplitude | 3.5 | 0.023 | 0.2 | 0.0 | 0.0 | 0.44 | 362028 | 173 | -313.4
|  | highest frequency | 3.5 | 0.023 | 0.2 | 0.0 | 0.0 | 0.44 | 362028 | 170 | -315.2
Péclet |  $\Delta$P | longest sequence | 3.4 | 0.023 | 0.05 | 0.0 | 0.5 | 0.40 | 2398 | 230 | -255.7
|  | highest amplitude | 3.2 | 0.023 | 0.15 | 0.005 | 0.0 | 0.40 | 1616 | 246 | -241.1
|  | highest frequency | 3.3 | 0.023 | 0.0 | 0.0 | 0.5 | 0.40 | 2450 | 241 | -247.3
| Period | longest sequence | 3.4 | 0.023 | 0.05 | 0.0 | 0.5 | 0.40 | 744142 | 218 | -276.5
|  | highest amplitude | 3.2 | 0.023 | 0.15 | 0.005 | 0.0 | 0.40 | 101511 | 226 | -270.6
|  | highest frequency | 3.2 | 0.023 | 0.15 | 0.005 | 0.0 | 0.40 | 101511 | 233 | -263.9
Table 5: Best fit models according to Mahalanobis distance where $\alpha_{\rm CBM}$=0 was fixed. Grid | Observables | Pattern construction |  $M_{\rm ini}$ [ M⊙] | $Z_{\rm ini}$ | $\alpha_{\rm CBM}$ | $f_{\rm CBM}$ | log($D_{\text{env}}$) | $X_{c}$ | $\chi^{2}_{\text{red}}$ | MD | AICc(MD)
---|---|---|---|---|---|---|---|---|---|---|---
Radiative |  $\Delta$P | longest sequence | 3.3 | 0.019 | (…) | 0.01 | 0.5 | 0.42 | 1335 | 171 | -307.5
|  | highest amplitude | 3.3 | 0.019 | (…) | 0.01 | 0.5 | 0.42 | 1335 | 171 | -308.1
|  | highest frequency | 3.3 | 0.023 | (…) | 0.0 | 0.5 | 0.40 | 2369 | 150 | -329.1
| Period | longest sequence | 3.4 | 0.023 | (…) | 0.02 | 1.0 | 0.46 | 464512 | 152 | -334.9
|  | highest amplitude | 3.5 | 0.023 | (…) | 0.02 | 1.0 | 0.46 | 209399 | 155 | -334.1
|  | highest frequency | 3.3 | 0.023 | (…) | 0.0 | 0.5 | 0.40 | 818181 | 147 | -340.9
Péclet |  $\Delta$P | longest sequence | 3.0 | 0.019 | (…) | 0.025 | 0.0 | 0.44 | 1714 | 170 | -310.6
|  | highest amplitude | 3.2 | 0.015 | (…) | 0.015 | 0.0 | 0.44 | 1934 | 178 | -303.7
|  | highest frequency | 3.3 | 0.023 | (…) | 0.0 | 0.5 | 0.40 | 2369 | 164 | -317.9
| Period | longest sequence | 3.3 | 0.023 | (…) | 0.03 | 0.5 | 0.46 | 450592 | 163 | -326.5
|  | highest amplitude | 3.3 | 0.023 | (…) | 0.03 | 0.5 | 0.46 | 450592 | 168 | -323.8
|  | highest frequency | 3.4 | 0.023 | (…) | 0.03 | 0.0 | 0.46 | 265614 | 171 | -319.6
Table 6: Best fit models according to Mahalanobis distance where $f_{\rm CBM}$=0 was fixed. Grid | Observables | Pattern construction |  $M_{\rm ini}$ [ M⊙] | $Z_{\rm ini}$ | $\alpha_{\rm CBM}$ | $f_{\rm CBM}$ | log($D_{\text{env}}$) | $X_{c}$ | $\chi^{2}_{\text{red}}$ | MD | AICc(MD)
---|---|---|---|---|---|---|---|---|---|---|---
Radiative |  $\Delta$P | longest sequence | 3.4 | 0.023 | 0.05 | (…) | 0.5 | 0.40 | 2227 | 112 | -355.7
|  | highest amplitude | 3.4 | 0.019 | 0.2 | (…) | 0.5 | 0.46 | 1783 | 121 | -347.6
|  | highest frequency | 3.3 | 0.023 | 0.0 | (…) | 0.5 | 0.40 | 2369 | 116 | -351.7
| Period | longest sequence | 3.4 | 0.023 | 0.05 | (…) | 0.5 | 0.40 | 758935 | 105 | -370.5
|  | highest amplitude | 3.4 | 0.019 | 0.2 | (…) | 0.5 | 0.46 | 92393 | 108 | -369.7
|  | highest frequency | 3.4 | 0.019 | 0.2 | (…) | 0.5 | 0.46 | 92393 | 110 | -367.0
Péclet |  $\Delta$P | longest sequence | 3.4 | 0.023 | 0.05 | (…) | 0.5 | 0.40 | 2318 | 130 | -344.1
|  | highest amplitude | 3.2 | 0.023 | 0.15 | (…) | 0.5 | 0.40 | 1299 | 138 | -337.2
|  | highest frequency | 3.2 | 0.015 | 0.2 | (…) | 0.0 | 0.42 | 2630 | 132 | -343.3
| Period | longest sequence | 3.2 | 0.023 | 0.25 | (…) | 0.0 | 0.40 | 363903 | 130 | -352.4
|  | highest amplitude | 3.2 | 0.023 | 0.25 | (…) | 0.0 | 0.40 | 363903 | 134 | -351.1
|  | highest frequency | 3.2 | 0.023 | 0.25 | (…) | 0.0 | 0.40 | 363903 | 140 | -345.2
Table 7: Best fit models according to Mahalanobis distance in the subgrid without CBM. Grid | Observables | Pattern construction |  $M_{\rm ini}$ [ M⊙] | $Z_{\rm ini}$ | $\alpha_{\rm CBM}$ | $f_{\rm CBM}$ | log($D_{\text{env}}$) | $X_{c}$ | $\chi^{2}_{\text{red}}$ | MD | AICc(MD)
---|---|---|---|---|---|---|---|---|---|---|---
Radiative |  $\Delta$P | longest sequence | 3.3 | 0.019 | (…) | (…) | 1.5 | 0.44 | 1816 | 84 | -380.2
|  | highest amplitude | 3.3 | 0.019 | (…) | (…) | 1.5 | 0.44 | 1816 | 92 | -373.5
|  | highest frequency | 3.2 | 0.019 | (…) | (…) | 0.0 | 0.50 | 8601 | 83 | -381.8
| Period | longest sequence | 3.4 | 0.019 | (…) | (…) | 1.5 | 0.44 | 965418 | 97 | -376.5
|  | highest amplitude | 3.1 | 0.019 | (…) | (…) | 0.5 | 0.40 | 377038 | 99 | -377.6
|  | highest frequency | 3.3 | 0.023 | (…) | (…) | 0.5 | 0.40 | 792613 | 107 | -367.9
Table 8: Best fit models according to $\chi^{2}_{\text{red}}$. Grid | Observables | Pattern construction |  $M_{\rm ini}$ [ M⊙] | $Z_{\rm ini}$ | $\alpha_{\rm CBM}$ | $f_{\rm CBM}$ | log($D_{\text{env}}$) | $X_{c}$ | $\chi^{2}_{\text{red}}$ | MD | AICc($\chi^{2}_{\text{red}}$)
---|---|---|---|---|---|---|---|---|---|---|---
Radiative |  $\Delta$P | longest sequence | 3.2 | 0.023 | 0.25 | 0.025 | 0.5 | 0.48 | 444 | 244 | 459.3
|  | highest amplitude | 3.2 | 0.023 | 0.25 | 0.025 | 0.5 | 0.48 | 444 | 247 | 459.3
|  | highest frequency | 3.2 | 0.023 | 0.25 | 0.025 | 0.5 | 0.48 | 444 | 245 | 459.3
| Period | longest sequence | 3.2 | 0.023 | 0.1 | 0.03 | 0.5 | 0.52 | 5369 | 258 | 5384.2
|  | highest amplitude | 3.2 | 0.023 | 0.1 | 0.03 | 0.5 | 0.52 | 5369 | 265 | 5384.2
|  | highest frequency | 3.2 | 0.023 | 0.1 | 0.03 | 0.5 | 0.52 | 5369 | 262 | 5384.2
Péclet |  $\Delta$P | longest sequence | 3.2 | 0.019 | 0.2 | 0.03 | 0.5 | 0.44 | 594 | 297 | 609.1
|  | highest amplitude | 3.2 | 0.019 | 0.2 | 0.03 | 0.5 | 0.44 | 594 | 308 | 609.1
|  | highest frequency | 3.2 | 0.019 | 0.2 | 0.03 | 0.5 | 0.44 | 594 | 308 | 609.1
| Period | longest sequence | 3.2 | 0.019 | 0.15 | 0.025 | 0.0 | 0.48 | 4836 | 315 | 4850.5
|  | highest amplitude | 3.2 | 0.019 | 0.15 | 0.025 | 0.0 | 0.48 | 4836 | 330 | 4850.5
|  | highest frequency | 3.2 | 0.019 | 0.15 | 0.025 | 0.0 | 0.48 | 4836 | 328 | 4850.5
Table 9: Best fit models according to $\chi^{2}_{\text{red}}$ where $\alpha_{\rm CBM}$=0 was fixed. Grid | Observables | Pattern construction |  $M_{\rm ini}$ [ M⊙] | $Z_{\rm ini}$ | $\alpha_{\rm CBM}$ | $f_{\rm CBM}$ | log($D_{\text{env}}$) | $X_{c}$ | $\chi^{2}_{\text{red}}$ | MD | AICc($\chi^{2}_{\text{red}}$)
---|---|---|---|---|---|---|---|---|---|---|---
Radiative |  $\Delta$P | longest sequence | 3.3 | 0.023 | (…) | 0.025 | 1.0 | 0.44 | 600 | 187 | 612.4
|  | highest amplitude | 3.3 | 0.023 | (…) | 0.025 | 1.0 | 0.44 | 600 | 189 | 612.4
|  | highest frequency | 3.4 | 0.023 | (…) | 0.025 | 1.0 | 0.44 | 572 | 190 | 583.8
| Period | longest sequence | 3.3 | 0.023 | (…) | 0.02 | 1.0 | 0.48 | 5511 | 196 | 5522.9
|  | highest amplitude | 3.3 | 0.023 | (…) | 0.02 | 1.0 | 0.48 | 5511 | 203 | 5522.9
|  | highest frequency | 3.3 | 0.023 | (…) | 0.02 | 1.0 | 0.48 | 5511 | 200 | 5522.9
Péclet |  $\Delta$P | longest sequence | 3.3 | 0.019 | (…) | 0.03 | 1.0 | 0.44 | 701 | 196 | 712.7
|  | highest amplitude | 3.3 | 0.019 | (…) | 0.03 | 1.0 | 0.44 | 701 | 203 | 712.7
|  | highest frequency | 3.3 | 0.019 | (…) | 0.03 | 1.0 | 0.44 | 701 | 202 | 712.7
| Period | longest sequence | 3.2 | 0.023 | (…) | 0.03 | 0.5 | 0.48 | 6122 | 212 | 6133.7
|  | highest amplitude | 3.2 | 0.023 | (…) | 0.03 | 0.5 | 0.48 | 6122 | 220 | 6133.7
|  | highest frequency | 3.2 | 0.023 | (…) | 0.03 | 0.5 | 0.48 | 6122 | 225 | 6133.7
Table 10: Best fit models according to $\chi^{2}_{\text{red}}$ where $f_{\rm CBM}$=0 was fixed. Grid | Observables | Pattern construction |  $M_{\rm ini}$ [ M⊙] | $Z_{\rm ini}$ | $\alpha_{\rm CBM}$ | $f_{\rm CBM}$ | log($D_{\text{env}}$) | $X_{c}$ | $\chi^{2}_{\text{red}}$ | MD | AICc($\chi^{2}_{\text{red}}$)
---|---|---|---|---|---|---|---|---|---|---|---
Radiative |  $\Delta$P | longest sequence | 3.4 | 0.023 | 0.0 | (…) | 1.5 | 0.42 | 779 | 133 | 790.9
|  | highest amplitude | 3.4 | 0.023 | 0.0 | (…) | 1.5 | 0.42 | 779 | 136 | 790.9
|  | highest frequency | 3.3 | 0.023 | 0.0 | (…) | 1.5 | 0.42 | 762 | 140 | 773.9
| Period | longest sequence | 3.2 | 0.019 | 0.3 | (…) | 0.5 | 0.50 | 10435 | 144 | 10447.4
|  | highest amplitude | 3.2 | 0.019 | 0.3 | (…) | 0.5 | 0.50 | 10435 | 148 | 10447.4
|  | highest frequency | 3.5 | 0.023 | 0.15 | (…) | 1.5 | 0.34 | 11681 | 140 | 11693.2
Péclet |  $\Delta$P | longest sequence | 3.2 | 0.019 | 0.25 | (…) | 1.0 | 0.42 | 751 | 143 | 763.3
|  | highest amplitude | 3.2 | 0.019 | 0.25 | (…) | 1.0 | 0.42 | 751 | 153 | 763.3
|  | highest frequency | 3.2 | 0.023 | 0.3 | (…) | 1.0 | 0.40 | 721 | 156 | 733.4
| Period | longest sequence | 3.5 | 0.019 | 0.3 | (…) | 0.5 | 0.30 | 10182 | 195 | 10193.8
|  | highest amplitude | 3.5 | 0.019 | 0.3 | (…) | 0.5 | 0.30 | 10182 | 201 | 10193.8
|  | highest frequency | 3.5 | 0.019 | 0.3 | (…) | 0.5 | 0.30 | 10182 | 204 | 10193.8
Table 11: Best fit models according to $\chi^{2}_{\text{red}}$ in the subgrid without CBM. Grid | Observables | Pattern construction |  $M_{\rm ini}$ [ M⊙] | $Z_{\rm ini}$ | $\alpha_{\rm CBM}$ | $f_{\rm CBM}$ | log($D_{\text{env}}$) | $X_{c}$ | $\chi^{2}_{\text{red}}$ | MD | AICc($\chi^{2}_{\text{red}}$)
---|---|---|---|---|---|---|---|---|---|---|---
Radiative |  $\Delta$P | longest sequence | 3.4 | 0.023 | (…) | (…) | 1.5 | 0.42 | 754 | 88 | 763.1
|  | highest amplitude | 3.4 | 0.023 | (…) | (…) | 1.5 | 0.42 | 754 | 99 | 763.1
|  | highest frequency | 3.3 | 0.023 | (…) | (…) | 1.5 | 0.42 | 737 | 96 | 746.6
| Period | longest sequence | 3.6 | 0.023 | (…) | (…) | 1.5 | 0.38 | 15639 | 127 | 15648.4
|  | highest amplitude | 3.6 | 0.023 | (…) | (…) | 1.5 | 0.38 | 15639 | 126 | 15648.4
|  | highest frequency | 3.6 | 0.023 | (…) | (…) | 1.5 | 0.38 | 15639 | 129 | 15648.4

##  Appendix D Variance–covariance matrices

Figures showing the variance–covariance matrices for the two grids, the
different pattern construction methods, and the different sets of
asteroseismic observables. Mode periods are labelled according to decreasing
period value.

Figure 19:  Variance–covariance matrix for the periods of the radiative grid,
selected using the longest sequence method. Figure 20:  Variance–covariance
matrix for the periods of the radiative grid, selected using the highest
amplitude method. Figure 21:  Variance–covariance matrix for the periods of
the radiative grid, selected using the highest frequency method. Figure 22:
Variance–covariance matrix for the periods of the Péclet grid, selected using
the longest sequence method. Figure 23:  Variance–covariance matrix for the
periods of the Péclet grid, selected using the highest amplitude method.
Figure 24:  Variance–covariance matrix for the periods of the Péclet grid,
selected using the highest frequency method. Figure 25:  Variance–covariance
matrix for the period spacings of the radiative grid, selected using the
longest sequence method. Figure 26:  Variance–covariance matrix for the period
spacings of the radiative grid, selected using the highest amplitude method.
Figure 27:  Variance–covariance matrix for the period spacings of the
radiative grid, selected using the highest frequency method. Figure 28:
Variance–covariance matrix for the period spacings of the Péclet grid,
selected using the longest sequence method. Figure 29:  Variance–covariance
matrix for the period spacings of the Péclet grid, selected using the highest
amplitude method. Figure 30:  Variance–covariance matrix for the period
spacings of the Péclet grid, selected using the highest frequency method.