11institutetext: Institute of Astronomy, KU Leuven, Celestijnenlaan 200D,
B-3001 Leuven, Belgium
11email<EMAIL_ADDRESS>
# Towards a systematic treatment of observational uncertainties in forward
asteroseismic modelling of gravity-mode pulsators

Dominic M. Bowman  Mathias Michielsen

(Received 6 July 2021 / accepted 22 Sept 2021)

###### Abstract

Context. In asteroseismology the pulsation mode frequencies of a star are the
fundamental data that are compared to theoretical predictions to determine a
star’s interior physics. Recent significant advances in the numerical,
theoretical and statistical asteroseismic methods applied to main sequence
stars with convective cores have renewed the interest in investigating the
propagation of observational uncertainties within a forward asteroseismic
modelling framework.

Aims. We aim to quantify the impact of various choices made throughout the
observational aspects of extracting pulsation mode frequencies in main
sequence stars with gravity modes.

Methods. We use a well-studied benchmark slowly pulsating B star, KIC 7760680,
to investigate the sensitivity of forward asteroseismic modelling to various
sources of observational uncertainty that affect the precision of the input
pulsation mode frequencies.

Results. We quantify the impact of the propagation of the observational
uncertainties involved in forward asteroseismic modelling. We find that one of
the largest sources of uncertainty in our benchmark star is in the manual
building of period spacing patterns, such that the inclusion of a potentially
ambiguous pulsation mode frequency may yield differences in model parameters
of up to 10% for mass and age depending on the radial order of the mode.

Conclusions. We conclude that future asteroseismic studies of main sequence
stars with a convective core should quantify and include observational
uncertainties introduced by the light curve extraction, iterative pre-
whitening and the building of period spacing patterns, as these propagate into
the final modelling results.

###### Key Words.:

asteroseismology – stars: early-type – stars: oscillations – stars: evolution
– stars: rotation – stars: fundamental parameters – stars: massive

##  1 Introduction

In recent decades, asteroseismology has become an established method to
accurately probe the interior physics of stars from quantitative comparison of
their observed pulsation frequencies with those predicted by theoretical
models of stellar structure and evolution (Chaplin & Miglio, 2013; Hekker &
Christensen-Dalsgaard, 2017; Bowman, 2020; Aerts, 2021). Across the
Hertzsprung–Russell (HR) diagram, stars are able to excite a variety of
different types of pulsation modes, which are standing waves with properties
defined by stellar structure. Stellar pulsations are typically grouped into
pressure (p) modes and gravity (g) modes based on their dominant restoring
forces being the pressure force and buoyancy, respectively (Aerts et al.,
2010). However, for rotating stars, the Coriolis force is also important,
which gives rise to gravito-inertial modes (Aerts et al., 2019). Stars can be
approximated as spheres, so stellar pulsations are typically described using
spherical harmonics. The radial order, $n$, defines the radial geometry of a
pulsation mode, and the angular degree, $\ell$, and the azimuthal order, $m$,
together define the surface geometry. Identified observed pulsation mode
frequencies in terms of $\\{n,\ell,m\\}$ are then quantitatively compared to a
grid of predicted frequencies calculated from models via forward asteroseismic
modelling (Aerts, 2021).

In main sequence late-B stars, which are born with a convective core (i.e.
$3\lesssim M\lesssim 8$ M⊙), an opacity mechanism operating in the Z bump is
efficient in exciting high radial order (asymptotic) g modes with periods of
order days (Dziembowski et al., 1993; Townsend, 2005; Walczak et al., 2015),
which defines the group of slowly pulsating B (SPB) stars (Waelkens, 1991).
Such a coherent pulsation excitation mechanism is predicted to produce
pulsation modes with essentially infinite lifetimes from the observer’s
perspective. The modes are observed as periodic variations in a star’s surface
flux and can be extracted by means of Fourier analysis (Aerts et al., 2010).
In reality, however, the majority of coherent pulsators are not observed to
have perfectly periodic pulsation modes (see e.g. Bowman et al. 2016; Van
Beeck et al. 2021), which introduces additional uncertainties in the practical
analysis of a pulsating star’s light curve.

In the last decade significant advances in our understanding of main sequence
B-type stars have been made because of asteroseismology. A veritable
revolution in the amount and quality of time series data assembled by the
CoRoT (Auvergne et al., 2009), Kepler/K2 (Borucki et al., 2010; Koch et al.,
2010; Howell et al., 2014), BRITE (Weiss et al., 2014, 2021) and now TESS
(Ricker et al., 2015) space missions have provided the necessary continuous,
high-precision and long-term data for detailed asteroseismic modelling studies
(Aerts, 2015; Bowman, 2020). Light curves from space missions have driven
increasingly more sophisticated g-mode modelling methods to be developed,
which have tested the impact of rotation, interior mixing, convection and
opacity tables on the resultant stellar pulsation frequencies (Moravveji et
al., 2015, 2016; Buysschaert et al., 2018; Szewczuk & Daszyńska-Daszkiewicz,
2018; Wu & Li, 2019; Mombarg et al., 2019; Szewczuk et al., 2021; Pedersen et
al., 2021; Michielsen et al., 2021). Recently, Aerts et al. (2018) introduced
the Mahalanobis distance (MD) within a g-mode forward asteroseismic modelling
framework to appropriately deal with the strong correlations and degeneracies
among model parameters, because it takes theoretical uncertainties into
account. Such a modelling setup has been recently applied to 26 SPB stars
observed by the Kepler mission and revealed a large range in the amount of
envelope mixing (Pedersen et al., 2021). Furthermore, the differences and
advantages of using the MD compared to the $\chi^{2}$ merit function in the
context of g-mode asteroseismology have been extensively demonstrated by
Michielsen et al. (2021).

From the observational perspective, the methods of how one extracts pulsation
frequencies and incorporates the various sources of observational uncertainty
at the different stages of analysing time series photometry sometimes yields
significantly different results (see e.g. Handler et al. 2019). This can then
propagate into the process of mode identification in g-mode pulsators relying
on the morphology of period spacing patterns. Such patterns are built to
satisfy the requirement of consecutive radial order modes of the same angular
degree and azimuthal order from a list of extracted pulsation frequencies, but
typically less than 10% of the total number of extracted pulsation frequencies
are used in the resultant period spacing pattern (see e.g. Pápics et al. 2014,
2015, 2017; Christophe et al. 2018). In some cases multiple g-mode period
spacing patterns for a star are plausible and it is not clear which of them
are optimal for modelling (see e.g. Szewczuk & Daszyńska-Daszkiewicz 2018;
Szewczuk et al. 2021; Pedersen et al. 2021).

In this paper, motivated by the recent modelling study of Michielsen et al.
(2021) of the SPB star KIC 7760680, we explore the implications of various
choices within the process of extracting its pulsation mode frequencies and
g-mode period spacing pattern. In section 2 we justify our choice of KIC
7760680, and in section 3 we describe the different approaches to extracting
and analysing light curves and in the building of g-mode period spacing
patterns. In section 4 we demonstrate the impact of these different approaches
on the resultant best-fit model parameters when performing forward
asteroseismic modelling, and we conclude in section 5.

##  2 Benchmark star: KIC 7760680

Within forward asteroseismic modelling of main-sequence B-type stars pulsating
in high-radial order g modes (i.e. SPB stars), the requirement is to extract
and identify pulsation mode frequencies from a time series (e.g. light curve).
Such stars have typically been found to exhibit prograde ($m=1$) or zonal
($m=0$) dipole ($\ell=1$) modes spanning from a few to tens of consecutive
radial order modes. Although a non-negligible fraction of these stars have
dipole retrograde ($m=-1$) patterns, or those formed of quadrupole ($\ell=2$)
modes (Degroote et al., 2010; Pápics et al., 2011, 2014, 2015, 2017;
Christophe et al., 2018; Szewczuk et al., 2021; Pedersen et al., 2021). The
choices of how to extract and identify the pulsation mode frequencies,
however, can differ in the literature.

We note it is also necessary to include additional constraints in forward
asteroseismic modelling to delimit the large parameter space within the HR
diagram, such as spectroscopic estimates of the effective temperature ($T_{\rm
eff}$), metallicity ($[{\rm M}/{\rm H}]$), and luminosity ($L$) or surface
gravity ($\log\,g$). Not only does this drastically reduce the number of
stellar structure models needed, but the best asteroseismic model is typically
selected from within the spectroscopic error box in the HR diagram (see e.g.
Moravveji et al. 2015, 2016; Michielsen et al. 2021). In this paper, we do not
investigate the propagation of uncertainties arising from the choices in setup
and determination of spectroscopic parameters (see e.g. Tkachenko et al.
2020). The fractional uncertainties of pulsation mode frequencies are
typically several orders of magnitude smaller compared to spectroscopic
parameters in pulsating early-type stars, and are used to fine-tune the
physics of the best model.

To perform a quantitative investigation of the impact of choices in the
extraction of g-mode period spacing patterns, we selected the SPB star KIC
7760680 because:

  * •

it was observed continuously during the 4-yr Kepler mission, hence its long
light curve has high resolving power for extracting individual frequencies;

  * •

it is a bright star ($V=10.3$ mag) and thus its light curve has high
photometric precision;

  * •

it has precise spectroscopic parameters available and the star is known to be
single (Pápics et al., 2015; Gebruers et al., 2021); and

  * •

it has a long series of consecutive radial order g modes with high pulsation
mode amplitudes.

Moreover, KIC 7760680 was recently the subject of detailed modelling efforts
(Moravveji et al., 2016; Pedersen et al., 2021; Michielsen et al., 2021). Most
importantly, these modelling efforts have all implicitly relied on the period
spacing pattern extracted in the discovery paper by Pápics et al. (2015). The
known spectroscopic parameters of KIC 7760680 are given in Table 1.

Table 1: Spectroscopic parameters for KIC 7760680 determined by Pápics et al.
(2015).

$T_{\rm eff}$ | $\log\,g$ | $v_{\rm micro}$ | $v\,\sin\,i$ | $[{\rm M}/{\rm H}]$
---|---|---|---|---
(K) | (dex) | (km s-1) | (km s-1) | (dex)
$11650\pm 210$ | $3.97\pm 0.08$ | $0.0^{+0.6}_{-0.0}$ | $61.5\pm 0.5$ | $0.14\pm 0.09$

##  3 From a telescope to a period spacing pattern

Figure 1: Schematic overview of the various steps involved in observational
aspects prior to forward asteroseismic modelling of g modes in main-sequence
stars. The steps denoted in green are those we assess the impact of in this
work.

In Fig. 1 we illustrate the various steps that are involved up until forward
asteroseismic modelling of g-mode pulsators, but the implementation of these
steps varies within the literature. In the following subsections we describe
each step in more detail and highlight the choices that are typically made.

###  3.1 Light curve extraction

For targets from the nominal Kepler mission, there are different data products
publicly available via the Mikulski Archive for Space Telescopes (MAST)111MAST
website: https://archive.stsci.edu/. Each Kepler target was assigned a postage
stamp of pixels to download, such that image reduction and light curve
extraction was performed on ground. Two types of light curves from these
postage stamps were extracted for each target. The first data product are
light curves extracted using simple aperture photometry (SAP), and the second
are those from the Kepler mission pipeline (PDC-SAP; Jenkins et al. 2010). The
Kepler mission pipeline produces light curves that are noise optimised, which
means that the selected apertures are relatively small to minimise the
contribution of the sky background. However, smaller apertures are more prone
to target drift and thermal recovery of the Kepler satellite after pointing,
which can produce long-period trends in the extracted time series.

However, few studies have investigated the differences in g-mode pulsation
frequencies obtained from different light curve extractions. Tkachenko et al.
(2013) identified instrumental frequencies in a sample of $\gamma$ Dor stars
observed by Kepler by comparing the analysis of PDC-SAP and customised light
curves, and demonstrated the importance of creating optimised light curves for
a subsequent frequency analysis. The customised light curve and period spacing
pattern extractions for the SPB star KIC 7760680 by Pápics et al. (2015) were
later used by Moravveji et al. (2016), Pedersen et al. (2021) and Michielsen
et al. (2021) for modelling purposes. Yet, these studies assumed the
observations to be free of any systematic observational uncertainties and
biases in how the light curve and pattern were extracted. Furthermore, the
method of Pápics et al. (2015) extracted and estimated the uncertainties of
pulsation mode frequencies assuming they were all completely independent,
which is an idealistic scenario.

We show the SAP and PDC-SAP light curves from data release (DR) 25 of the
Kepler mission pipeline and the Pápics et al. (2015) light curve of KIC
7760680 in Fig. 2. We refer to these as the LCSAP, LCPDC, and
LC${}_{\textup{\smash{Pápics}}}$ light curves, and consider them as three
representative light curves of KIC 7760680 that one may consider to analyse
for the purpose of forward asteroseismic modelling. The LCSAP and LCPDC light
curves were downloaded from MAST and converted to have flux units of
magnitudes. The customised LC${}_{\textup{\smash{Pápics}}}$ light curve was
extracted directly from the Kepler postage stamps using a larger aperture mask
and underwent additional de-trending using a low-order polynomial (Pápics et
al., 2015). For consistency, we convert the LC${}_{\textup{\smash{Pápics}}}$
light curve to have flux units of magnitudes. All three light curves are
normalised to have a median flux of zero.

Figure 2: 1-yr excerpts of the LCSAP, LCPDC and
LC${}_{\textup{\smash{Pápics}}}$ light curves of KIC 7760680, in which the
high-amplitude pulsations can be seen.

The LCSAP, LCPDC and LC${}_{\textup{\smash{Pápics}}}$ light curves of KIC
7760680 shown in Fig. 2 look nearly identical. The statistical properties
including the total number of data points, the total time span, the maximum
and minimum fluxes, and the standard deviation of the fluxes are given in
Table 2. The start and end time stamps of all three light curves (i.e. BJD
2454953.538827 and 2456424.001160) are exactly the same yielding a total time
span of 1470.462 d. However, the number of data points in the
LC${}_{\textup{\smash{Pápics}}}$ light curve is slightly larger compared to
the LCSAP and LCPDC light curves. The differences in the minimum, maximum and
standard deviation of the flux values of each light curves are because of the
different data reduction steps taken in extracting each light curve. We plot
the spectral window of the LCPDC and LC${}_{\textup{\smash{Pápics}}}$ light
curves in Fig. 3, but not of the LCSAP light curve because it is identical to
that of the LCPDC light curve. As shown in Fig. 3, the difference in the
spectral windows of the LCPDC and LC${}_{\textup{\smash{Pápics}}}$ light
curves are notable at the $\sim$0.5% level in the $\pm 1$ d-1 frequency range.

Table 2: Statistical properties of the complete 4-yr LCSAP, LCPDC and LC${}_{\textup{\smash{Pápics}}}$ light curves of KIC 7760680. | $N_{\rm data}$ | max(flux) | min(flux) | std(flux)
---|---|---|---|---
|  | (mmag) | (mmag) | (mmag)
LCSAP | $65263$ | $24.7056$ | $-26.0299$ | $9.2244$
LCPDC | $65263$ | $27.0814$ | $-23.9249$ | $9.1888$
LC${}_{\textup{\smash{Pápics}}}$ | $65898$ | $24.9922$ | $-24.9852$ | $9.1340$
Figure 3: Spectral windows of the LCPDC and LC${}_{\textup{\smash{Pápics}}}$
light curves of KIC 7760680. The difference between the spectral windows of
these light curves is caused by difference in the number of data points (i.e.
duty cycle) in the LC${}_{\textup{\smash{Pápics}}}$ light curve given that the
start and end time stamp for all light curves are the same.

###  3.2 Iterative pre-whitening

After a light curve has been extracted, the next step is to extract
significant and independent pulsation mode frequencies using iterative pre-
whitening. The premise of iterative pre-whitening in asteroseismology of
coherent pulsators assumes that each individual pulsation mode within a time
series can be represented by a (co)sinusoid with a constant amplitude, $A$,
frequency, $\nu$, and phase, $\phi$. Thus, the Fourier decomposition of a time
series reveals the individual pulsation mode frequencies. It is more common
within this framework to extract pulsation mode frequencies iteratively by
sequentially extracting them in order of decreasing amplitude, and optimising
their parameters using a least-squares fit to the light curve with the
equation:

$\Delta\,m=\sum_{i}^{N}A_{i}\,\cos\left(2\pi\nu_{i}\left(t-t_{0}\right)+\phi_{i}\right)\leavevmode\nobreak\
,$ (1)

where $t$ is the time with respect to a zero-point $t_{0}$. In this way,
optimising $N$ cosinusoids (each with a frequency, $\nu_{i}$, amplitude,
$A_{i}$, and phase, $\phi_{i}$) sequentially using Eqn. (1) allows one to
subtract the least-squares model from the light curve, and find the next
frequency using an iterative approach. This methodology is the backbone of the
commonly-used Period04 software package designed for this specific purpose
(Lenz & Breger, 2004, 2005).

We note that there is some variance in the literature on how the Fourier
transform of the input time series is calculated. For example, Period04 and
some studies utilise the Discrete Fourier Transform (DFT) for unevenly-sampled
data (Deeming, 1975; Kurtz, 1985) to directly calculate a frequency spectrum
(see e.g. Lenz & Breger 2004; Kurtz et al. 2015; Bowman 2017). Whereas other
codes prefer (a variant of) the Lomb-Scargle algorithm (Scargle, 1982; Press &
Rybicki, 1989) for calculating the periodogram of a light curve (see e.g.
Degroote et al. 2009; Antoci et al. 2019; Van Beeck et al. 2021). In this work
we utilize the former method, such that our results are fully consistent and
verifiable with the Period04 software package.

####  3.2.1 Fitting of sinusoids with least-squares

It has recently been shown that SPB stars exhibit amplitude and frequency
modulation and long-period beating of closely-spaced frequencies (Van Beeck et
al., 2021). Furthermore, SPB stars also exhibit stochastically-excited gravity
waves, which lead to damped modes with finite lifetimes (Bowman et al., 2019a,
b; Neiner et al., 2020). Therefore not all signals extracted using iterative
pre-whitening represent independent and perfectly periodic pulsation modes. In
all implementations of iterative pre-whitening, the light curve is decomposed
into harmonic terms, which are then interpreted as either pulsation modes or
not (see e.g. Poretti et al. 2009; Chapellier et al. 2011).

Some studies that utilise iterative pre-whitening perform a least-squares fit
in which the frequency is kept fixed to the Fourier frequency taken from the
amplitude spectrum. In this implementation the assumption is that all
independent frequencies are well described by the corresponding Fourier
frequency and have uncorrelated uncertainties. This can lead to a large number
of extracted frequencies, especially when combined with a non-conservative
significance criterion (see e.g. Pápics et al. 2012; Balona 2014).

A multi-cosinusoid non-linear least-squares fit using Eqn. (1), in which
$\nu_{i}$, $A_{i}$ and $\phi_{i}$ are all free parameters both during and at
the end of iterative pre-whitening, has the advantage of taking the covariance
of pulsation mode frequencies, amplitudes and phases into account and allows
for the calculation of correlated uncertainties (see section 3.2.3). This is
typically done as a second step such that the fixed-frequency least-squares
solution is used as input to the multi-cosinusoid non-linear least-squares
fit, which outputs optimised frequencies, amplitudes and phases and
uncertainties determined from the error matrix (Lenz & Breger, 2005; Kurtz et
al., 2015; Bowman, 2017). Frequency and phase are highly correlated in a
cosinusoid model, and a non-linear least-squares fit of hundreds of
cosinusoids can become an unwieldy numerical problem and lead to ill-
conditioned covariance and error matrices. Thus, an appropriate compromise is
to perform a multi-cosinusoid non-linear least squares fit after all spurious
and instrumental frequencies have been removed (Aerts et al., 2010; Bowman,
2017; Van Beeck et al., 2021).

####  3.2.2 Optimum significance criterion

The most commonly used criterion is the empirically-derived signal-to-noise
(S/N) criterion of Breger et al. (1993), which defines significant frequencies
to have an amplitude S/N $\geq 4$ in the frequency spectrum. However, it is
based on ground-based data of p-mode pulsators, such that its applicability to
space photometry of g-mode pulsators is questionable. Some studies advocate
for a higher threshold for space telescope data because there is an increased
likelihood of classifying instrumental frequencies as astrophysical (Kuschnig
et al., 1997; Baran et al., 2015; Zong et al., 2016). Whereas others advocate
for S/N $\geq 3$ when extracting g modes based on knowledge from the expert
user (e.g. Li et al. 2019). Statistical investigations of light curves from
space telescopes using $p$-values over-estimate the number significant
frequencies compared to a S/N criterion (e.g. Degroote et al. 2009; Blomme et
al. 2011; Pápics et al. 2012). Moreover, the results of five different
iterative pre-whitening strategies applied to 38 Kepler SPB stars revealed
large differences in the number of significant frequencies (Van Beeck et al.,
2021).

To assess the validity of the S/N $\geq 4$ criterion in g-mode
asteroseismology, we create two sets of 1000 synthetic light curves. We use
the long-cadence time stamps of KIC 7760680, and calculate white Gaussian
noise typical of stars with Kepler instrumental magnitudes (Kp) of 10 and 12
mag in the two sets. We calculate the amplitude spectra of each light curve up
to the Nyquist frequency (i.e. 24.45 d-1) and extract the S/N value of the
highest amplitude (noise) peak as the ratio of the peak’s amplitude to the
local noise level using a 1-d-1 frequency window in the residual amplitude
spectrum. The histograms for the resultant S/N values from the 1000 synthetic
light curves of each brightness subset are shown in Fig. 4.

The similarities of the two histograms in Fig. 4 demonstrate how the S/N of
noise peaks in a Kepler light curve are not wholly sensitive to the shot noise
level as set by the brightness of the star. There are 339 and 359 light curves
in the Kp 10 and 12 mag subsets whose tallest (noise) peak satisfies S/N $\geq
4$, respectively. To reduce the false alarm probability of extracting noise
peaks as significant frequencies to 10 in 1000 (i.e. 1%), one would need to
impose a S/N threshold of 4.52 and 4.60 for the Kp 10 and 12 mag sets of
synthetic light curves, respectively222See also similar tests by Zong et al.
(2016) based on short-cadence Kepler data of pulsating sub-dwarf stars..
Frequencies with $3.5<$ S/N $<4.0$ extracted from Kepler photometry are
indistinguishable from high-amplitude noise, and those between $4.0\leq$ S/N
$\leq 4.6$ have a modest chance of being noise peaks. Hence as a conservative
significance threshold we advocate using S/N $\geq 4.6$ within an iterative
pre-whitening methodology for Kepler data.

Figure 4: Histogram showing the signal-to-noise ratio (S/N) of the highest
amplitude frequency extracted from 1000 synthetic light curves containing only
white noise consistent with Kepler targets of 10 and 12 mag.

####  3.2.3 Comparison of extracted frequencies

With all this is mind, we calculate the amplitude spectra of the LCSAP, LCPDC,
and LC${}_{\textup{\smash{Pápics}}}$ light curves of KIC 7760680 by means of a
DFT (Kurtz, 1985). All three light curves have the same cadence333Strictly
speaking, we mean the same average cadence, given that Kepler data have a
periodic Nyquist frequency (see Murphy et al. 2013). and total time span, so
all have the same Nyquist frequency (24.4662 d-1) and Rayleigh resolving power
(0.00068 d-1). We use a frequency oversampling factor of 20 when calculating
the amplitude spectra. The three amplitude spectra of the LCSAP, LCPDC and
LC${}_{\textup{\smash{Pápics}}}$ light curves are shown in Fig. 5. We plot the
pair-wise difference of each amplitude spectrum for the low-frequency regime
in Fig. 5. By design, the Kepler pipeline aims to remove much of the long-
period systematics from the LCSAP light curve when producing the LCPDC light
curve, as shown in Fig. 5. There is also a dramatic reduction in the low-
frequency noise (i.e. $\nu<0.1$ d-1) in the LC${}_{\textup{\smash{Pápics}}}$
light curve, which is primarily the result of the larger pixel mask (Pápics et
al., 2015). To emphasise this effect, we also include the logarithmic
amplitude spectra of all three light curves in Fig. 5. The reduction in noise
in the high frequency regime (i.e. $\nu>10$ d-1) is also significant, with the
shot noise being on average $13\%$ lower in the amplitude spectrum of
LC${}_{\textup{\smash{Pápics}}}$ compared to LCPDC light curve.

Figure 5: Top panel: amplitude spectra of the LCSAP, LCPDC and
LC${}_{\textup{\smash{Pápics}}}$ light curves of KIC 7760680, in which the low
frequency g-mode pulsation modes can be seen. Middle panel: pair-wise
difference in amplitude spectra shown in the top panel. Bottom panel:
logarithmic amplitude spectra up to the Kepler long cadence Nyquist frequency
of 24.4662 d-1.

Figure 6: Residual light curves and amplitude spectra of the LCSAP (top),
LCPDC (middle) and LC${}_{\textup{\smash{Pápics}}}$ (bottom) light curves of
KIC 7760680, after the nine highest-amplitude frequencies have been removed
during iterative pre-whitening. The full 4-yr light curves are shown but only
a small frequency region containing the $\ell=1$ period spacing pattern is
shown in the amplitude spectra for visibility. The nine frequencies extracted
from each light curve (shown as dashed vertical lines) are given in Table 3.
Note the difference in ordinate-axis scale.

The differences among the three amplitude spectra in Fig. 5 are maximal at the
location of the g mode frequencies (i.e. $0.6\leq\nu\leq 1.1$ d-1), which are
caused by the interference of the spectral windows of multiple g modes. We
extract the dominant pulsation mode, optimise its parameters using a non-
linear least-squares fit to each light curve separately and provide the
resultant frequencies, amplitudes and phases in Table 3. We also provide the
S/N value calculated using a window of 1 d-1 centred on the extracted
frequency to estimate of the local noise level in the residual amplitude
spectrum. From a comparison of the dominant pulsation mode (i.e. frequency
label $1$ in Table 3), the resultant frequencies, amplitudes and phases are
consistent within $1\sigma$ across all three light curves.

Table 3: Single-cosinusoid non-linear least-squares solutions for the first nine iterations for the three light curves of KIC 7760680, such that each frequency is extracted and optimised independently. The zero point of the time scale is $t_{0}=2455688.770$ BJD. | Freq. label | Frequency | Amplitude | Phase | S/N
---|---|---|---|---|---
|  | (d-1) | (mmag) | (rad) |
LCSAP | 1 | $0.8628724\pm 0.0000010$ | $10.658\pm 0.030$ | $0.4982\pm 0.0028$ | $141.19$
LCPDC | 1 | $0.8628723\pm 0.0000010$ | $10.652\pm 0.029$ | $0.4981\pm 0.0027$ | $141.11$
LC${}_{\textup{\smash{Pápics}}}$ | 1 | $0.8628727\pm 0.0000010$ | $10.665\pm 0.028$ | $0.4992\pm 0.0027$ | $146.34$
LCPDC | 2 | $0.8385280\pm 0.0000014$ | $5.451\pm 0.020$ | $-2.2739\pm 0.0037$ | $104.20$
LCPDC | 2 | $0.8385282\pm 0.0000014$ | $5.457\pm 0.020$ | $-2.2739\pm 0.0036$ | $104.27$
LC${}_{\textup{\smash{Pápics}}}$ | 2 | $0.8385273\pm 0.0000013$ | $5.448\pm 0.019$ | $-2.2751\pm 0.0035$ | $108.12$
LCSAP | 3 | $0.8948497\pm 0.0000015$ | $3.695\pm 0.014$ | $2.2767\pm 0.0039$ | $98.81$
LCPDC | 3 | $0.8948486\pm 0.0000014$ | $3.699\pm 0.014$ | $2.2774\pm 0.0037$ | $99.28$
LC${}_{\textup{\smash{Pápics}}}$ | 3 | $0.8948487\pm 0.0000013$ | $3.693\pm 0.012$ | $2.2735\pm 0.0033$ | $103.42$
LCSAP | 4 | $0.8143166\pm 0.0000024$ | $1.915\pm 0.012$ | $-1.0916\pm 0.0064$ | $61.81$
LCPDC | 4 | $0.8143177\pm 0.0000022$ | $1.915\pm 0.011$ | $-1.0907\pm 0.0059$ | $62.12$
LC${}_{\textup{\smash{Pápics}}}$ | 4 | $0.8143186\pm 0.0000019$ | $1.913\pm 0.010$ | $-1.0872\pm 0.0051$ | $64.99$
LCSAP | 5 | $0.9921895\pm 0.0000027$ | $1.509\pm 0.011$ | $1.6430\pm 0.0071$ | $58.97$
LCPDC | 5 | $0.9921889\pm 0.0000024$ | $1.519\pm 0.010$ | $1.6422\pm 0.0063$ | $59.63$
LC${}_{\textup{\smash{Pápics}}}$ | 5 | $0.9921894\pm 0.0000020$ | $1.515\pm 0.008$ | $1.6434\pm 0.0052$ | $62.47$
LCSAP | 6 | $0.9133795\pm 0.0000037$ | $1.022\pm 0.010$ | $1.8320\pm 0.0097$ | $44.69$
LCPDC | 6 | $0.9133824\pm 0.0000032$ | $1.023\pm 0.009$ | $1.8296\pm 0.0085$ | $44.99$
LC${}_{\textup{\smash{Pápics}}}$ | 6 | $0.9133808\pm 0.0000025$ | $1.015\pm 0.007$ | $1.8362\pm 0.0067$ | $47.40$
LCSAP | 7 | $0.9318204\pm 0.0000055$ | $0.661\pm 0.010$ | $-0.6734\pm 0.0145$ | $31.42$
LCPDC | 7 | $0.9318298\pm 0.0000047$ | $0.662\pm 0.008$ | $-0.6730\pm 0.0125$ | $31.64$
LC${}_{\textup{\smash{Pápics}}}$ | 7 | $0.9318241\pm 0.0000035$ | $0.669\pm 0.006$ | $-0.6675\pm 0.0094$ | $34.01$
LCSAP | 8 | $0.9678508\pm 0.0000061$ | $0.582\pm 0.009$ | $-1.3675\pm 0.0160$ | $29.83$
LCPDC | 8 | $0.9678491\pm 0.0000052$ | $0.585\pm 0.008$ | $-1.3487\pm 0.0137$ | $30.08$
LC${}_{\textup{\smash{Pápics}}}$ | 8 | $0.9678499\pm 0.0000038$ | $0.586\pm 0.006$ | $-1.3552\pm 0.0099$ | $32.17$
LCSAP | 9 | $0.7942562\pm 0.0000067$ | $0.512\pm 0.009$ | $0.5684\pm 0.0178$ | $27.61$
LCPDC | 9 | $0.7942562\pm 0.0000056$ | $0.516\pm 0.008$ | $0.5502\pm 0.0149$ | $27.93$
LC${}_{\textup{\smash{Pápics}}}$ | 9 | $0.7942592\pm 0.0000041$ | $0.509\pm 0.005$ | $0.5706\pm 0.0108$ | $29.53$

We continue extracting significant frequencies using iterative pre-whitening
for each of the three light curves, and treat each extracted and optimised
cosinusoid as independent of those that came before it, in order of decreasing
amplitude (i.e. we perform a single-cosinusoid non-linear least squares fit at
each iterative pre-whitening stage). We use amplitude S/N $\geq 4.6$ as our
significance criterion. The results of this single-cosinusoid non-linear least
squares optimisation are shown in Table 3, for which only the first nine
iterations are shown for clarity. We emphasise that the non-linear least-
squares fit of an individual cosinusoid at a single pre-whitening iteration
uses the residual light curve at that specific iteration (i.e. previous
frequencies with larger amplitudes have already been subtracted from the light
curve). The residual light curves and amplitude spectra after extracting the
nine highest-amplitude frequencies are shown in Fig. 6. The residual light
curves after so few iterations already show significant differences, despite
the optimised cosinusoid parameters all being within $1\sigma$ of one another.

We show all the optimised frequencies within the low-frequency regime at the
end of iterative pre-whitening in Fig. 7. Although there are some significant
frequencies above 2.5 d-1, the vast majority of significant frequencies lie
within the $0.6\leq\nu\leq 1.1$ and $1.6\leq\nu\leq 2.0$ d-1. As can be seen
in Fig. 7, the significant frequencies in high-amplitude g-mode pulsators
commonly appear in frequency groups, which can be explained by combination
frequencies (Kurtz et al., 2015). In this setup of treating each cosinusoid as
independent of those that came before it, the respective frequencies and
phases extracted from the three different light curves in the first $\sim$ 50
iterations are consistent within $2\sigma$. At this stage of the iterative
pre-whitening the extracted frequencies have $6<{\rm S/N}<10$. However, after
about 50 iterations the frequency discrepancies are larger, and in some cases
are as large as $7\sigma$. Furthermore, the extracted lists of frequencies
from the three light curves are not the same, because the amplitude S/N $\geq
4.6$ criterion yields a different number (and extracted order) of frequencies
for each light curve because of its underlying noise profile in the Fourier
domain.

Figure 7: Amplitude spectra of the optimised frequencies for KIC 7760680 using
the single-cosinusoid solution for each of the LCSAP, LCPDC and
LC${}_{\textup{\smash{Pápics}}}$ light curves. The amplitude spectrum (grey)
is calculated from the LC${}_{\textup{\smash{Pápics}}}$ light curve, such that
low-amplitude modes only become significant above the noise after high-
amplitude modes have been extracted during pre-whitening. Frequencies
extracted below $\nu\leq 0.1$ d-1 are dominated by instrumental systematics.

####  3.2.4 Multi-cosinusoid non-linear least-squares fitting

As a comparison, we also perform another iterative pre-whitening setup and
extract significant frequencies using the LC${}_{\textup{\smash{Pápics}}}$
light curve of KIC 7760680 and a multi-cosinusoid non-linear least-squares fit
with $N$ cosinusoids (cf. Eqn. 1) at each stage of the pre-whitening. In this
second setup, we also use amplitude S/N $\geq 4.6$ as our significance
criterion. The results from this multi-cosinusoid non-linear least-squares fit
setup are shown in Table LABEL:table:_multi-freq, in which the last column
provides the frequency difference between the single- and multi-cosinusoid
non-linear least-squares solutions, expressed in terms of the frequency
uncertainty, $\sigma_{\nu}$, of the multi-cosinusoid solution. We note that
some frequency labels are missing (e.g. 10 and 11), which is because they were
identified as being within $2.5/\Delta\,T$ of a higher amplitude frequency
hence were discarded and not included in the ultimate multi-cosinusoid fit
owing to their potential spurious nature (see Loumos & Deeming 1978). Clearly,
the resultant frequencies are discrepant in some cases as much as 15$\sigma$,
but in others less than 1$\sigma$. The differences are not systematic with
amplitude. The standard deviation of these frequency discrepancies is
4.2$\sigma$.

Although it is more computationally demanding, we conclude that such a
numerical setup is important when extracting g-mode frequencies from space
telescope light curves, since it includes the covariance of the frequencies,
amplitudes and phases of all significant pulsation modes as opposed to only
single pulsation mode being treated as independent of all others. Our
conclusions are two-fold:

  1. 1.

the light curve extraction has a significant impact on the pulsation mode
frequencies of a g-mode pulsator at the level of the spectral windows being
sufficiently different enough to produce different pulsation mode frequencies,
which gets systematically worse for lower amplitude pulsation modes;

  2. 2.

the difference in extracted frequencies from a single-cosinusoid and multi-
cosinusoid non-linear least-squares fit (cf. Eqn. 1) for a given light curve
is representative of additional random uncertainty, which in some cases is
much larger than the formal frequency precision.

The reason for the former conclusion can be understood as the analysis of a
somewhat different data set yielding slightly different results. The reason
for the latter conclusion is that the multi-cosinusoid non-linear least-
squares fit includes the covariance matrix for all frequencies, amplitudes and
phases in the determination of uncertainties.

###  3.3 Building period spacing patterns

In this section we investigate the uncertainties in g-mode period spacing
patterns considering two aspects: (i) the propagation of multi-cosinusoid non-
linear least-squares uncertainties on the resultant period spacing pattern;
and (ii) subjective selection from a list of extracted significant
frequencies.

####  3.3.1 Statistical scatter in period spacing patterns

Figure 8: Comparison of different period spacing patterns for KIC 7760680,
including the solution from Pápics et al. (2015) based on the
LC${}_{\textup{\smash{Pápics}}}$ light curve. The multi-cosinusoid non-linear
least-squares fitted to the LCSAP, LCPDC and LC${}_{\textup{\smash{Pápics}}}$
light curves are also shown on top of the amplitude spectrum in the top
panels. The top-right panel is a zoom-in of the long-period regime. The
resultant period spacing patterns of these four solutions are shown in the
middle panel, with the residual of our patterns with that of Pápics et al.
(2015) shown in the bottom panel. Note that the formal uncertainties on the
data in the middle panel are smaller than the symbol sizes. Figure 9:
Resultant period spacing patterns arising from randomly perturbing pulsation
mode frequencies within their 1$\sigma$ uncertainties 100 times based on the
patterns shown in Fig. 8 as input. The colour scheme is the same as in Fig. 8.

To demonstrate the impact of a multi-cosinusoid non-linear least-squares fit,
we compare the period spacing pattern extracted by Pápics et al. (2015)
assuming independent frequencies, with those resulting from our fits for the
three LCSAP, LCPDC and LC${}_{\textup{\smash{Pápics}}}$ light curves in Fig.
8. The differences among the three non-linear least-squares solutions is
between $1-2\sigma$ in frequency. However, we find a much larger discrepancy
between the solution of Pápics et al. (2015) and that of our fits to the
LC${}_{\textup{\smash{Pápics}}}$ light curve, which is between
$0.1\leq\sigma\leq 13$ and has a median value of $4.2\sigma$. This propagates
into a median uncertainty of order 10 sec in the period spacing diagram, but
the discrepancy is largest for the low amplitude modes (i.e. pulsation periods
$>1.28$ d). The main source of this uncertainty is that Pápics et al. (2015)
treated each frequency as independent during their iterative pre-whitening
procedure, with their uncertainties determined using the idealistic formulae
of Montgomery & O’Donoghue (1999), which assume parameters to be uncorrelated
(see Appendix A of Breger et al. 1999). Therefore, the (lack of such a) fit
represents an additional source of random uncertainty of, on average,
4.2$\sigma$ in frequency, and at least 10 sec in $\Delta$P values in the
period spacing pattern, which is systematically larger for lower amplitude
modes.

As a second exercise, we generate three sets of 100 perturbed period spacing
patterns by randomly perturbing the three LCSAP, LCPDC and
LC${}_{\textup{\smash{Pápics}}}$ patterns in Fig. 8 within their $1\sigma$
frequency uncertainties. The resultant 100 period spacing patterns are shown
in Fig. 9. We note that each set of 100 perturbed patterns are statistically
significant and valid solutions, as they were all generated from the same
input and $1\sigma$ uncertainties. Clearly, the propagation of the $1\sigma$
uncertainties in this Monte Carlo style sampling leads to a ‘cloud’ of period
spacing pattern solutions, whose scatter of up to 100 sec in the $\Delta$P
ordinate axis of Fig. 9 is a measure of the true uncertainty for KIC 7760680.
We note that the scatter in Fig. 9 is largest for the LCSAP light curve and
smallest for the LC${}_{\textup{\smash{Pápics}}}$ light curve, which
demonstrates the importance of optimising the light curve extraction
methodology for g-mode asteroseismology.

####  3.3.2 Subjective component of building patterns

When manually building a g-mode period spacing pattern, one selects
frequencies to maximise the number of consecutive radial orders, whilst
adhering to some prior knowledge of what the average period spacing ought to
be for a given star. This often leads to an inside-out approach to building
patterns, such that the spacing between a few high-amplitude modes is
replicated towards shorter and longer periods. This methodology was applied by
Pápics et al. (2015) when extracting the prograde dipole period-spacing
pattern of KIC 7760680. This approach, however, is reliant on the expectation
that period spacing patterns are smooth and periodic. It has recently been
shown that resonances between overstable convective modes or inertial modes in
the cores of massive stars with g modes in the envelope can create sharp dips
in period spacing patterns at different periods dependent on the rotation rate
(Ouazzani et al., 2020; Saio et al., 2021; Lee, 2021). Also, the effect of a
strong magnetic field within the convective cores of main sequence stars is
predicted to perturb period spacing patterns in a mode-specific way (Prat et
al., 2019, 2020; Van Beeck et al., 2020). Therefore it is timely to revisit
the extraction of g-mode period spacing patterns in KIC 7760680 allowing for
such morphological effects.

As can be seen in Figs 8 and 9, the high-amplitude modes between $0.98\leq
P\leq 1.23$ d have a relatively small uncertainty region, with their formal
and Monte Carlo style sampling of the true uncertainty being
$\sigma\left(\Delta{\rm P}\right)\simeq 10$ sec. For periods outside this
range, much larger Monte Carlo uncertainties of order 100 sec are found. The
inclusion of high-amplitude modes between $0.98\leq P\leq 1.23$ d as members
of the g-mode period spacing pattern is trivial; there are few alternatives.
However, the choices from outside this range become more challenging. For
example, Fig. 8 shows that some neighbouring frequencies with higher
amplitudes were not selected to be part of the pattern by Pápics et al. (2015)
for morphology reasons.

Figure 10: Top panel: zoom-in of the residual amplitude spectrum after the
first nine pulsation mode frequencies have been removed to illustrate the
choice of frequencies to include as part of a variant period spacing pattern.
The location of two of the nine previously-extracted mode frequencies are
within this period range and are denoted by dashed vertical green lines. Solid
blue lines denote frequencies that have been chosen as part of one of the five
variant period spacing patterns. Middle panel: the resultant period spacing
patterns of our optimised LC${}_{\textup{\smash{Pápics}}}$ pattern is shown in
green, and the additional five variant period spacing patterns are shown in
blue. Bottom panel: residuals of the additional patterns with that of our
optimised LC${}_{\textup{\smash{Pápics}}}$ benchmark pattern. The period
regime of the top panel is denoted as the shaded grey region in the middle and
bottom panels.

To test the limitations of this inside-out approach when building period
spacing patterns, we construct different patterns from the extracted list of
significant frequencies using the multi-cosinusoid non-linear least-squares
fit solution described in section 3.2.3 and provided in Table
LABEL:table:_multi-freq. We restrict ourselves to only the frequency list from
the LC${}_{\textup{\smash{Pápics}}}$ light curve, because it is the highest
quality. We also restrict ourselves to choosing different patterns from modes
within in the low-frequency regime, because this is where the mode density is
high and other significant frequencies are potential pattern members. This
means that our choices of which significant frequencies to include and exclude
within each different pattern are from the period range of $1.23\leq P\leq
1.47$ d. In the case of KIC 7760680, the low-amplitude g modes with periods
above 1.27 d only become significant after several high-amplitude modes have
been removed during iterative pre-whitening. In Fig. 10, we show a zoom-in of
the residual amplitude spectrum after the first nine pulsation mode
frequencies have been removed.

We build five ‘variant’ period spacing patterns by selecting different
significant frequencies to those of Pápics et al. (2015) from the list of
significant frequencies in Table LABEL:table:_multi-freq and show them in Fig.
10. We label five locations in Fig. 10 at which a choice must be made on which
frequency to include. For example, in variant patterns 1 and 2, we select the
other frequencies from an apparent triplet of independent modes. In variant
patterns 3 and 4, we select adjacent significant frequencies to those selected
by Pápics et al. (2015). Finally, in variant pattern 5, we intentionally
select an ‘incorrect’ frequency at the end of the pattern to test the
modelling sensitivity to the inclusion of such a low-amplitude long-period
mode. For each of the five variant patterns, we also create a counterpart
pattern for which the potentially ambiguous frequency is omitted entirely.
Some of the five variant patterns may seem more appealing as solutions and
others do not because they introduce glitches in the pattern morphology (e.g.
variant pattern 3). This is because period spacing values are correlated with
the period values so small differences in period can produce large differences
in period spacings. The residuals between each of the five variant patterns
and that of the optimised LC${}_{\textup{\smash{Pápics}}}$ solution is much
larger than those in Fig. 9 at locations where an alternative frequency has
been introduced.

##  4 Modelling Results

Recently, Michielsen et al. (2021) revisited the modelling of the SPB star KIC
7760680 using the MD as well as a classical $\chi^{2}$ merit function to
extract properties including mass, age, metallicity, convective boundary and
envelope mixing, using the period spacing pattern of Pápics et al. (2015).
Forward asteroseismic modelling is at least a 7D minimisation problem, with
the interior rotation rate being one of these parameters (Aerts et al., 2018).
Precise rotation rates of main-sequence stars can be measured from the tilt in
g-mode period spacing patterns (Aerts, 2021). Hence the determination of the
rotation frequency a priori from the g-mode period spacing pattern reduces the
dimensionality of the subsequent modelling. This motivated Michielsen et al.
(2021) to fix the rotation frequency of KIC 7760680 and assume a uniform
interior rotation profile with a rotation frequency of 0.4805 d-1, with this
value being determined from the previous modelling of Moravveji et al. (2016).
As our study is the observational counterpart to their theoretical study, we
also assume the same rotation frequency as a fixed input in forward
asteroseismic modelling and the exact same numerical setup as employed by
Michielsen et al. (2021) for comparison purposes.

We use the ‘radiative’ grid setup of Michielsen et al. (2021), which includes
stellar structure and evolution models calculated using MESA (r12115; Paxton
et al. 2011, 2013, 2015, 2018, 2019). We also make use of the nested-parameter
setup of Michielsen et al. (2021), such that fitting can be done using 4, 5 or
6 free parameters whilst penalising for the increased number of free
parameters. In the 4-parameter setup, the free parameters are stellar mass
($M$), metal mass fraction ($Z$), core hydrogen content ($X_{c}$) and envelope
mixing ($\log\,D_{\rm env}$) using the expected profile for internal gravity
waves (i.e. $D_{\rm env}\propto\rho^{-1}$) as predicted based on
hydrodynamical simulations by Rogers & McElwaine (2017). In the 5-parameter
setup, the addition of core-boundary mixing (CBM) using a diffusive
exponential prescription ($f_{\rm CBM}$) is included. In the 6-parameter
setup, a second CBM parameter is included, specifically the extent of a step
profile above the convective core ($\alpha_{\rm CBM}$). The reason for
including $f_{\rm CBM}$ and $\alpha_{\rm CBM}$ as separate parameters is to
include a larger range and diversity of core masses in the model grid, which
g-modes directly probe (Michielsen et al., 2019, 2021).

From prior testing, the MESA model grid of Michielsen et al. (2021) was
computed to finely sample the possible parameter space of KIC 7760680 based on
its g-mode period spacing pattern, far beyond its 3$\sigma$ spectroscopic
error box in the HR diagram. Mass values range between $2.8\leq M\leq 3.7$ M⊙
in steps of 0.1 M⊙, core hydrogen content ranges between $0.30\leq X_{\rm
c}\leq 0.60$ in steps of 0.02 and three metallicity values of $Z=$ 0.015,
0.019 and 0.023 are included, which covers the spectroscopic uncertainties,
with the corresponding helium mass fraction calculated using an enrichment law
for each $Z$ value and imposing $X+Y+Z=1$. Interior mixing parameters include:
$0.000\leq f_{\rm CBM}\leq 0.030$ in steps of 0.005, $0.00\leq\alpha_{\rm
CBM}\leq 0.30$ in steps of 0.05, and $0.0\leq\log\left(D_{\rm env}\right)\leq
2.0$ in steps of 0.5. For each MESA model, the corresponding adiabatic
prograde dipole g-mode frequencies using the pulsation code GYRE (v5.2;
Townsend & Teitler 2013) were calculated assuming the pre-determined rotation
frequency of 0.4805 d-1 and uniform interior rotation. We refer the reader to
Michielsen et al. (2021) for full details of the MESA grid calculation and the
GYRE calculations.

The method of forward asteroseismic modelling involves using a maximum
likelihood estimator and a quantitative fit of observables to the theoretical
counterparts from the grid of models, which in this case is the pulsation
periods within a g-mode period spacing pattern. Recent forward asteroseismic
modelling studies have moved beyond a simplistic $\chi^{2}$ merit function in
favour of using the MD (Aerts et al., 2018), which allows non-linear
correlations between fitting parameters and observables to be included, as
well as incorporating theoretical uncertainties in the form of the full
variance-covariance matrix for the observables and the theoretical predictions
in addition to observational uncertainties. In the following subsections, we
employ the same methodology of Michielsen et al. (2021) and perform
quantitative fitting between observed pulsation periods to those predicted
from a grid of stellar structure models using the MD as merit function. In
each instance, for the purpose of making our results comparable to those of
Michielsen et al. (2021), the best model within the grid is defined as that
with the lowest MD value from within the 3$\sigma$ spectroscopic error box of
KIC 7760680 (cf. Table 1), and the same grid of models is used.

###  4.1 Modelling perturbed patterns

In this section, we empirically test if the quality of the extracted light
curve of KIC 7760680 and the scatter within the resultant period spacing
pattern clouds are large enough to impact the resultant best fitting model
parameters. The g-mode period spacing pattern of Pápics et al. (2015) is
compared to the 100 perturbed patterns derived from the same light curve (i.e.
LC${}_{\textup{\smash{Pápics}}}$) from Fig. 9. Furthermore, the differences
between the three sets of 100 perturbed patterns for the LCSAP, LCPDC and
LC${}_{\textup{\smash{Pápics}}}$ perturbed patterns provide an insight on the
impact of the light curve quality and effect of a multi-cosinusoid non-linear
least-squares fit on the model parameters.

####  4.1.1 Scatter in best-fitting models

The first question is whether the quality of the light curves and the
resultant g-mode period spacing pattern(s) have large enough uncertainties to
produce a significantly different set of best model parameters. We employ the
4-, 5- and 6-parameter nested model selection methodology of Michielsen et al.
(2021) for each pattern from the three sets of 100 perturbed patterns for KIC
7760680 and determine the best model from within the grid in each case. Our
results are summarised as histograms in Fig. 11 for each setup, with the
parameters for each of the models specified in Table 4. All 300 perturbed
patterns return the same best model denoted as ‘4A’ in the 4-parameter setup.
Interestingly, this is the same best model as found by Michielsen et al.
(2021) when using the pattern of Pápics et al. (2015) and the 4-parameter
setup. Hence when only considering a 4-parameter setup (i.e. $M$, $Z$, $X_{c}$
and $\log\,D_{\rm env}$) the impact of the quality of extracted light curve
and the observational uncertainties can be considered small compared to the
theoretical uncertainties in the case of KIC 7760680, such that they do not
significantly influence the selection of the best model in our MD framework.

Figure 11: Histograms showing the number of times each best model was returned from the three sets of 100 perturbed g-mode period spacing patterns derived from the LCSAP, LCPDC and LC${}_{\textup{\smash{Pápics}}}$ light curves. The parameters of the models are given in Table 4. Solutions based on a 4-, 5- and 6-parameter setup are shown in the top, middle and bottom panels, respectively. The model names in red (i.e. 4A, 5C and 6A) are the best models determined by Michielsen et al. (2021) using the g-mode period spacing pattern from Pápics et al. (2015). Table 4: Parameters of the best models for the 4-, 5- and 6-parameter setups given in Figs 11 and 12. | $M$ | $Z$ | $\log(D_{\rm env})$ | $\alpha_{\rm CBM}$ | $f_{\rm CBM}$ | $X_{\rm C}$
---|---|---|---|---|---|---
| (M⊙) |  |  |  |  |
4A | 3.4 | 0.019 | 1.5 | – | – | 0.44
5A | 3.2 | 0.015 | 0.0 | – | 0.020 | 0.46
5B | 3.4 | 0.023 | 0.0 | – | 0.005 | 0.40
5C | 3.4 | 0.023 | 1.0 | – | 0.020 | 0.46
5D | 3.5 | 0.023 | 0.5 | – | 0.020 | 0.46
5E | 3.4 | 0.019 | 0.5 | – | 0.010 | 0.42
5F | 3.3 | 0.023 | 0.5 | – | 0.005 | 0.40
6A | 3.5 | 0.023 | 0.0 | 0.20 | 0.000 | 0.44
6B | 3.4 | 0.023 | 0.5 | 0.05 | 0.000 | 0.40
6C | 3.4 | 0.019 | 0.5 | 0.20 | 0.000 | 0.46
6D | 3.0 | 0.019 | 0.0 | 0.25 | 0.010 | 0.46
6E | 3.3 | 0.023 | 0.0 | 0.05 | 0.005 | 0.40
Figure 12: Top panel: best fitting period spacing patterns for best models
returned using the 5- and 6-parameter setup in Fig. 11. The parameters of the
models are given in Table 4. The optimised pattern based on the
LC${}_{\textup{\smash{Pápics}}}$ light curve (cf. Fig. 8) is included to guide
the eye. Bottom panel: residuals between each best-fitting theoretical pattern
and the optimised pattern based on the LC${}_{\textup{\smash{Pápics}}}$ light
curve.

However, this is not the case when using the 5-parameter setup (i.e. addition
of $f_{\rm CBM}$ as a fitting parameter), as can be seen in Fig. 11. Since g
modes are particularly sensitive to the size of the convective core, the
addition of $f_{\rm CBM}$ as a free parameter allows for a more diverse range
of theoretical patterns to fit to the observations. For example, larger CBM in
the form of $f_{\rm CBM}$ directly modifies the size of the ‘dips’ in a period
spacing pattern (Miglio et al., 2008), hence allowing for more flexibility in
fitting the morphology of a period spacing pattern. Our modelling results in
the 5-parameter setup, summarised in Fig. 11, show that most of the perturbed
patterns prefer a different best model to that of Michielsen et al. (2021).
The difference in the best-fitting parameters may seem small at first glance,
but it is important to remember that the MD is maximum-likelihood point
estimator and the uncertainty region for parameters of a given solution depend
on the best model and the underlying distribution in MD values for each fit
(Michielsen et al., 2021; Mombarg et al., 2021). Therefore, these results
demonstrate two conclusions when using the common 5-parameter setup in forward
asteroseismic modelling: (i) the use of a multi-cosinusoid non-linear least-
squares fit in determining which frequencies are extracted and included in
g-mode period spacing pattern impacts the resultant best model parameters; and
(ii) the quality of the light curve impacts the morphology of the period
spacing pattern (both in the periods and their uncertainties) such that
different best-fitting model parameters are found.

It is typical to use the 5-parameter setup (i.e. including only a single
parameter, such as $f_{\rm CBM}$ or $\alpha_{\rm CBM}$ to control the amount
of CBM) in modelling studies of stars with g-mode period spacing patterns
(Moravveji et al., 2015, 2016; Szewczuk & Daszyńska-Daszkiewicz, 2018; Mombarg
et al., 2019; Pedersen et al., 2021). However, Michielsen et al. (2021) went
beyond such a setup and allowed for an additional parameter, such that the
combination of both $f_{\rm CBM}$ and $\alpha_{\rm CBM}$ controlled the total
amount and shape of CBM. Interestingly, in this 6-parameter setup, the vast
majority of best fitting models from our three sets of 100 perturbed patterns
return the exact same model as found by Michielsen et al. (2021) using the
g-mode period spacing pattern of Pápics et al. (2015). This demonstrates that
the addition of a second parameter in modelling CBM in main sequence g-mode
pulsators is more robust against unaccounted for uncertainties in the
observations. In other words, in a 5-parameter modelling setup, the
theoretical model grid does not contain the same flexibility to take into
account the shape of the observed period spacing pattern and its
uncertainties, such that observed periods perturbed within their uncertainties
return different results, which are of order 10% for mass. However, in the
6-parameter setup, the theoretical model grid is more robust and provides more
consistent fits that can better handle potential discrepancies that may or may
not have been introduced during the light curve reduction and iterative pre-
whitening.

The theoretical period spacing patterns for the best fitting models are shown
in Fig. 12. It is clear that the variance in the residual theoretical patterns
with the optimised observed pattern in Fig. 12 is more than 200 sec, which is
larger than the $\sim 50$ sec variance among the perturbed observed patterns
in Fig. 9. Therefore, the theoretical uncertainties associated with different
input physics in stellar models are dominant in forward modelling. Yet, the
observational uncertainties are not negligible and contribute to the scatter
in Fig. 12, specifically because the perturbed locations and depths of dips in
the observed patterns are sufficient to sometimes return a different best
model. We note that largest theoretical uncertainties are associated with
short-period low-radial order g modes, whereas the largest observational
uncertainties are associated with the low-amplitude large-radial order g
modes.

####  4.1.2 Impact on inferred core masses

The second question is whether the observed scatter in the best fitting models
and resultant theoretical period spacing patterns (cf. Fig. 12) for the 5- and
6-parameter solutions produces a significant difference in the inferred core
mass compared to the uncertainty region for an individual model. For the both
the 5- and 6-parameter setups, we determine the core mass for each of the
three sets of 100 perturbed period spacing patterns from the LCSAP, LCPDC and
LC${}_{\textup{\smash{Pápics}}}$ light curves and show them in Fig. 13. We use
two formulations for the inferred core mass: (i) the mass contained within the
region that satisfies the Ledoux criterion; and (ii) that of (i) and also the
mass within the overlying CBM region.

In Fig. 13 we demonstrate the larger scatter in the inferred core mass values
(both with and without the inclusion of the CBM region) in the 5-parameter
setup compared to the 6-parameter setup for the three sets of 100 perturbed
patterns. For example, in the 6-parameter setup, more than 65% of the 100
LCSAP and almost all of the LCPDC and LC${}_{\textup{\smash{Pápics}}}$
patterns have the same inferred core mass when including or excluding the CBM
region. On the other hand, in the 5-parameter setup, there is less agreement.
The inclusion of the CBM region when determining the core mass has a much
larger effect when using the 5-parameter setup. For example, the majority of
the LC${}_{\textup{\smash{Pápics}}}$ patterns prefer a core mass of
approximately 0.58 M⊙ including the CBM region, but the LCSAP and LCPDC
patterns prefer core masses in excess of 0.75 M⊙. We note that the masses
among the best models range between $3.0\leq M\leq 3.5$ M⊙, such that the
scatter in the fractional core masses in the 5-parameter setup are also much
larger compared to the 6-parameter setup.

To judge whether the width of the distribution in core mass is significant, we
compare it to the uncertainty for the inferred core mass from an individual
model. For simplicity, we focus on the modelling results based on the 100
perturbed patterns from only the LC${}_{\textup{\smash{Pápics}}}$ light curve,
because they are superior to the LCSAP and LCPDC solutions. From the 100
perturbed patterns based on the LC${}_{\textup{\smash{Pápics}}}$ light curve,
we find that the 2$\sigma$ core mass uncertainties range between 0.01 and 0.1
M⊙. This corresponds to a fractional uncertainty up to 20% dependent on the
model. On the other hand, when looking at the 6-parameter setup, the
distribution in inferred core masses is much narrower and the uncertainties
are much smaller as well. In most of the 100 perturbed patterns based on the
LC${}_{\textup{\smash{Pápics}}}$ light curve, the uncertainty on the core mass
in the 6-parameter is much smaller than the grid step size, such that only the
best model is returned within its own 2$\sigma$ confidence interval. Hence the
fractional uncertainties on core mass in the 6-parameter setup are of order 1%
compared to the 20% in the 5-parameter setup. This further demonstrates the
robustness of the 6-parameter setup and the MD modelling framework against
potentially unaccounted for observational uncertainties.

Figure 13: Histograms for the inferred core masses of the three sets of 100
perturbed period spacing patterns for the LCSAP, LCPDC and
LC${}_{\textup{\smash{Pápics}}}$ light curves. The top and bottom panels show
the distribution for the 5- and 6-parameter setups. Two forms of the inferred
core mass are provided, one defined as the mass within the region that
satisfies the Ledoux criterion, and the second which additionally includes the
mass within the core-boundary mixing (CBM) region as well.

####  4.1.3 Parameter confidence intervals

Following Michielsen et al. (2021), we also calculate confidence intervals for
model parameters in each setup using Bayes’ theorem and the distribution of
likelihoods returned from the forward asteroseismic modelling results. Figure
14 shows the value of each parameter and its 2$\sigma$ confidence interval for
the 5- and 6-parameters for the three best fitting (i.e. lowest MD value)
perturbed period spacing patterns from the set of 100 for the
LC${}_{\textup{\smash{Pápics}}}$ light curve. Similarly to Michielsen et al.
(2021), if the application of Bayes’ theorem returns no other models within
the 2$\sigma$ confidence interval based on the model’s likelihood, the
uncertainty is taken to be smaller than the grid step size for a given
parameter.

For the 5-parameter setup, Fig. 14 shows that the confidence interval for each
individual parameter is quite different from one pattern to another, but
generally consistent within 2$\sigma$, but the inferred core masses generally
are not (cf. Fig. 13). Whereas for the 6-parameter setup, the best three
models all return the same model parameter values, and have parameter
uncertainties smaller than the step size of the grid. Our results indicate
that asteroseismic modelling studies of g-mode period spacing patterns are
most robust against unaccounted for observational uncertainties when using the
6-parameter setup in a MD framework.

Figure 14: Parameters and 2$\sigma$ confidence intervals for the three best
fitting LC${}_{\textup{\smash{Pápics}}}$ perturbed period spacing patterns
shown as filled circles and solid lines, respectively. If a confidence
interval calculation using Bayes’ theorem did not return any additional
models, an upper limit estimate is taken to be the grid step size for a given
parameter, which is indicated by the dashed grey lines.

###  4.2 Modelling subjective patterns

In this subsection, we investigate the impact of manually selecting
(ambiguous) frequencies as part of a period spacing pattern on the modelling
results. We define a benchmark period spacing pattern for comparison purposes.
Since it was demonstrated that the period spacing pattern was far more robust
and less prone to random frequency uncertainties caused by choices in the
light curve extraction and pre-whitening, we choose the pattern optimised
using a multi-cosinusoid non-linear least-squares fitted to the
LC${}_{\textup{\smash{Pápics}}}$ light curve, which is shown in Figs 8 and 10.
As in the previous section, the rotation frequency is kept fixed for all of
the subsequent modelling setups. Since the changes in individual frequencies
among the variant patterns are extremely small, we checked that the same
rotation frequency was derived from the tilt of the g-mode period spacing
pattern for all patterns within its uncertainties. A summary of the modelling
results for the benchmark solution is shown in Fig. 15, in which the resultant
2D and 1D parameter probability distributions are provided, as well as the
distribution of all models in the HD diagram colour-coded by their MD value.
We note that the distribution of best models may appear sporadic, but this is
a result of theoretical uncertainties contained within the modelling grid
(Michielsen et al., 2021).

Figure 15: Summary figure for the modelling results of the benchmark period
spacing pattern, which is the Pápics et al. (2015) pattern optimised using a
multi-cosinusoid non-linear least-squares fitted to the
LC${}_{\textup{\smash{Pápics}}}$ light curve (cf. Fig. 8). The cyan boxes in
the HR diagram denote the 1- and 3-$\sigma$ spectroscopic error box for KIC
7760680.

As discussed in section 3.3.2, we construct five variant patterns each
containing only a single different frequency compared to the benchmark
pattern, as shown in Fig. 10. For each of these variant patterns, we also
create a counterpart in which the alternative frequency is simply omitted,
which is referred to as the ‘B’ pattern. This means that in the modelling of a
‘B’ pattern, there is no enforcement of consecutive radial order for where the
ambiguous frequency was omitted. An observed ‘B’ pattern could potentially
thus be best fitted by a theoretical pattern that contains none or multiple
radial orders within the gap where the ambiguous frequency was omitted. This
is different to the corresponding ‘A’ pattern for which the best fitting
theoretical pattern is forced to consist of 36 consecutive radial orders.
Finally we include a ‘robust’ pattern in which only unambiguous periods were
included between $0.92<P<1.22$ d. For each variant pattern, we perform the
same modelling setup as the benchmark model, with specific discussion for each
case below. Summary figures for the modelling of variant patterns are shown in
Figs 17 and 18, and derived parameter uncertainties using Bayes’ theorem are
shown in Fig. 16.

Figure 16: Parameters and 2$\sigma$ confidence intervals for benchmark,
variant and robust period spacing patterns are shown as filled circles and
solid lines, respectively. If a confidence interval calculation using Bayes’
theorem did not return any additional models, an upper limit estimate is taken
to be the grid step size for a given parameter, which is indicated by the
dashed grey lines.

####  4.2.1 Variant patterns 1, 1B, 2 and 2B

In variant patterns 1 and 2, we select the middle and left-most frequencies
from an apparent triplet of independent modes whereas in the benchmark pattern
the right-most frequency was selected. This means that variant pattern 1
includes 0.8046355 d-1 instead of 0.8029739 d-1, and variant pattern 2
includes 0.8064031 d-1 instead of 0.8029739 d-1 as shown in Fig. 10. Since
variant pattern 2 is made from the same triplet, there is no equivalent B
pattern, since it would be the same as variant pattern 1B. Visually, the
morphology of variant pattern 1 appears just as reasonable as the benchmark
pattern: it is relatively smooth and periodic. Whereas the inclusion of the
left-most frequency in the triplet in variant pattern 2 introduces a
noticeable glitch between $1.24<P<1.26$ d.

The summary figure for the modelling results of variant pattern 2 is shown in
Fig. 17. The parameters and their uncertainties for variant patterns 1, 1B and
2 are shown in Fig. 16. In the case of variant patterns 1 and 2, the best
models within the grid are no longer within the 3$\sigma$ spectroscopic error
box, and neither is the best model from variant pattern 1B. However, this may
be a chance circumstance owing to the sampling of the grid parameters given
that the distribution of best models in the HR diagram and individual
parameters and uncertainties are similar to the benchmark model. In all cases,
regardless of where the model with the lowest MD is located in the HR diagram,
the best model is always selected from within the 3$\sigma$ spectroscopic
error box following Michielsen et al. (2021). The inclusion of an alternate
frequency (variant patterns 1 and 2) or omitting the frequency (variant
pattern 1B) yields parameters that are mostly consistent within 2$\sigma$ to
the benchmark model, which is shown in Fig. 16. For example, the $X_{\rm c}$
value of variant pattern 2 is 0.38 compared to 0.44 in the benchmark model,
but the 2$\sigma$ confidence interval for $X_{\rm c}$ in variant pattern 1
includes all the other values. This is also the case for $\alpha_{\rm CBM}$
and $f_{\rm CBM}$. Therefore, the inclusion of an alternate frequency and
subsequent modelling of the different period spacing pattern at this specific
location of a dip in the pattern of KIC 7760680 is sufficient to produce
different confidence interval in some variant patterns, but that the
parameters themselves are generally consistent.

####  4.2.2 Variant patterns 3 and 3B

In variant pattern 3, we alternatively select an adjacent frequency, 0.7335336
d-1 instead of 0.7313646 d-1, as illustrated in Fig. 10. Visually it is easy
to see that this alternate frequency does not yield as satisfying a period
spacing pattern, since it introduces a noticeable glitch between $1.36<P<1.38$
d. The summary figure for the modelling results of variant patterns 3 and 3B
are shown in Fig. 17. The parameters and their uncertainties for variant
patterns 3 and 3B are shown in Fig. 16. In the case of modelling variant
pattern 3, the parameters and their uncertainties are generally consistent
with the benchmark model at the 2$\sigma$ level. But a comparison of variant
pattern 3B reveals that it is discrepant with the benchmark model, despite the
solution being well constrained individually. Therefore, we conclude that it
is preferable to include a potentially ambiguous frequency in an unbroken
pattern of consecutive radial order g modes rather than omit it entirely and
fit two parts of a broken pattern.

####  4.2.3 Variant patterns 4 and 4B

In variant pattern 4, we alternatively select an adjacent frequency, 0.7609956
d-1 instead of 0.7599243 d-1 as illustrated in Fig. 10. The two frequencies
involved in variant pattern 4 are special in this study because they both
arise from an apparent doublet in the amplitude spectrum and are only
separated by 1.6 times the Rayleigh resolution. Therefore, according to our
iterative pre-whitening setup of keeping the highest amplitude frequency
within 2.5 times the Rayleigh, only one of them was included in our list of
frequencies in Table LABEL:table:_multi-freq. However, they statistically have
the same amplitude: $0.06\pm 0.01$ mmag.

The summary figure for the modelling results of variant patterns 4 and 4B are
shown in Fig. 18. The parameters and their uncertainties for variant patterns
4 and 4B are shown in Fig. 16. The exact same best model in the case of
variant pattern 4 is returned as the benchmark model. This is not surprising
since the alternative mode has only a very small (i.e. $\delta\,\nu=0.001$
d-1) frequency difference to the benchmark pattern. This implies that small
frequency shifts and discarded spurious and unresolved frequencies according
to the Loumos & Deeming (1978) do not strongly impact the modelling results.
However, in the case of variant pattern 4B, in which neither frequency is
included, much larger parameter uncertainties are returned. Hence, similar to
the results based on modelling variant pattern 3B, we conclude it is
preferable to not omit ambiguous frequencies as members of patterns. Moreover,
this exercise is direct evidence of why high-precision space photometry of
order 1 yr or longer is beneficial for forward asteroseismic modelling.
Otherwise, with degraded frequency resolution from short light curves,
incorrect or unresolved frequencies could be chosen.

####  4.2.4 Variant patterns 5 and 5B

Finally, in variant pattern 5 we intentionally select an ‘incorrect’ frequency
at the end of the pattern to test the modelling sensitivity to the inclusion
of such a low-amplitude long-period mode. This means we include 0.6827688 d-1
instead of 0.6846335 d-1, as illustrated in Fig. 10. The summary figure for
the modelling results of variant patterns 5 and 5B are shown in Fig. 18. The
parameters and their uncertainties for variant patterns 5 and 5B are shown in
Fig. 16. In this scenario, the impact on the modelling results is negligible;
the same best model in the case of variant patterns 5 and 5B as the benchmark
model are returned. This is interesting since such a frequency is the lowest
radial order within a pattern for KIC 7760680, hence the tradeoff between
increased probing power from additional radial orders and decreased precision
because of its relatively large frequency uncertainty does not impact the
modelling results. Or in other words, the inclusion of this particular
frequency contributes to the goodness-of-fit metric in terms of the resultant
MD value of the best model, but the model selection itself is not impacted by
omitting this frequency entirely.

####  4.2.5 Robust pattern

Our investigation of testing the impact of different choices in building
period spacing patterns from a list of significant pulsation frequencies has
revealed that the same best model is not always returned, even if the
alternative frequency only differs by 0.002 d-1. In particular, a low envelope
mixing value of $\log\,D_{\rm env}=0.0$ is always returned for KIC 7760680,
but $X_{\rm c}$ ranges between 0.30 and 0.54. On the other hand, the 2$\sigma$
confidence intervals in Fig. 16 show that most solutions are generally
consistent with each other. This may lead one to wonder how well constrained
the solution space is if only the robust part of the period spacing pattern is
used in modelling, that is only include periods in a pattern for which there
is little-to-no ambiguity in which frequencies to include. In such a setup we
restrict ourselves to only including high-amplitude modes with periods between
$0.92<P<1.22$ d where the density of observed pulsation frequencies in the
amplitude spectrum is much lower. For completeness, we also exclude the
highest frequency from the pattern since its amplitude is quite low. This
leaves us with 16 consecutive radial mode frequencies compared to the 36 in
the benchmark and variant patterns (and 35 in the counterpart B patterns)
discussed in this work.

The summary figure for the modelling results of the robust pattern is shown in
Fig. 18. The parameters and their uncertainties for the robust pattern are
shown in Fig. 16. The resultant distribution of the best models in the HR
diagram for the robust pattern is different to the benchmark model and the
2$\sigma$ confidence intervals are larger, in part because of the smaller
number of observables in the fitting process: 16 mode periods instead of 36,
but also because their corresponding probing powers are quite different. With
more modes at lower radial orders in the benchmark pattern, the model
parameters, such as mass, age, metallicity, are more precisely constrained
because these modes are more sensitive to the convective core mass. Therefore,
our comparison indicates that it is more beneficial to include additional
modes at the expense of potentially larger observational uncertainties if one
can maximise the radial order range and probing power of the g-mode period
spacing pattern. The 2$\sigma$ confidence intervals for the parameters from
the robust pattern shown in Fig. 16 are relatively large and encompass all the
best models of the benchmark and variant patterns discussed in this section.
Therefore, we conclude that such a modelling strategy is the most accurate but
not as precise.

##  5 Conclusions

In this work, we use the SPB star KIC 7660680 observed by the Kepler space
telescope to test the impact of different choices and setups in the extraction
of g-mode frequencies from space photometry on the determination of the best
model parameters within a forward asteroseismic modelling framework. We use
three representative light curves, denoted as LCSAP, LCPDC and
LC${}_{\textup{\smash{Pápics}}}$, with the former two being derived from the
MAST SAP and PDC-SAP light curves, respectively, and the third being the light
curve extracted by Pápics et al. (2015) in their discovery paper of KIC
7760680. We demonstrate that the LC${}_{\textup{\smash{Pápics}}}$ light curve
is superior for a consistent frequency analysis. Hence we conclude that
forward asteroseismic modelling studies should extract an optimised light
curve, since the results of a subsequent frequency analysis become divergent
among the three LCSAP, LCPDC and LC${}_{\textup{\smash{Pápics}}}$ light
curves.

We also investigate the impact of including a multi-cosinusoid non-linear
least-squares fit during iterative pre-whitening, and conclude that a
significantly different frequency list is produced. Studies that use the
Fourier frequencies as input to forward asteroseismic modelling and estimate
observational uncertainties based on the idealistic formulae of Montgomery &
O’Donoghue (1999) are underestimating the true frequency uncertainty. The
significant differences in the resultant frequency list when a multi-
cosinusoid non-linear least-squares fit is performed are understandable since
pulsation mode frequencies are not completely independent. Such a fit and the
resultant correlated uncertainties incorporate the covariance of all the
frequencies, amplitudes and phases, with frequencies and phases being highly
correlated parameters. Therefore we conclude that such a step is necessary to
produce an accurate and self-consistent list of pulsation mode frequencies.

Based on a set of simulated light curves emulating a 4-yr Kepler light curve,
we revisit the choice of significance criterion in iterative pre-whitening.
Our simulations demonstrate S/N $\geq 4.6$ corresponds to a false alarm
probability of 1% that an extracted frequency corresponds to a white noise
peak. Zong et al. (2016) used similar simulations to show that the amplitude
S/N $\geq 4$ criterion of Breger et al. (1993) is too low for Kepler-like
light curves. Hence we advocate caution in interpreting signals obtained from
iterative pre-whitening below S/N $<4.6$ with Kepler light curves.
Furthermore, Baran et al. (2015) advise S/N $\geq 5$ in the case of light
curves from the K2 mission given their increased likelihood of containing
instrumental systematics. Such a conservative significance criterion is also
applicable to analysis of light curves from the ongoing TESS mission (see e.g.
Burssens et al. 2020; Baran & Koen 2021).

To assess the propagation of observational uncertainties in forward
asteroseismic modelling, we create three sets of 100 perturbed period spacing
patterns for each of the LCSAP, LCPDC and LC${}_{\textup{\smash{Pápics}}}$
light curves. Each set of perturbed patterns took the original pattern of
Pápics et al. (2015) as input but it is optimised using a multi-cosinusoid
non-linear least-squares fit to the respective light curve. Each of the three
LCSAP, LCPDC and LC${}_{\textup{\smash{Pápics}}}$ solutions are randomly
perturbed within their 1$\sigma$ frequency uncertainties to create the 300
perturbed patterns. Following Michielsen et al. (2021), we use 4-, 5- and
6-parameter modelling setups to determine the best model for each perturbed
pattern. We find that the 4-parameter setup is insensitive to the
observational uncertainties, but a large spread in best models and inferred
core masses are returned for the 5-parameter setup. The majority of the
returned best models in the 5-parameter setup are different to the best model
found by Michielsen et al. (2021) based on the pattern of Pápics et al.
(2015). However, in the 6-parameter setup, the vast majority of best models
are consistent. Hence we conclude that the 6-parameter modelling setup is more
robust against random observational uncertainties that may be introduced from
an imperfect light curve and pulsation mode frequency extraction through
iterative pre-whitening.

Finally, we investigate the impact of potentially ambiguous pulsation
frequencies as members of period spacing patterns. Using five examples, we
build variant period spacing patterns in which a single frequency is replaced
by an adjacent one. We also create a corresponding set of ‘B’ counterpart
patterns in which the alternative frequency is removed entirely to test if it
is preferable to simply remove an ambiguous frequency within a pattern. The
resultant best models for each of the five variant patterns and their ‘B’
counterparts demonstrate that the inclusion of a potentially ambiguous
frequency can impact the confidence intervals of the resultant model
parameters, but this depends on the frequency (i.e. radial order) of the
specific pulsation mode. Moreover, it is generally preferable to include a
potentially ambiguous frequency instead of omitting them entirely from the
pattern. Our results show that some pulsation modes have more weight than
others in determining the best model based on where they are located within a
pattern. The inclusion of specific frequencies introduce perturbations to the
morphology of a period spacing pattern, which if are sufficiently close to a
dip in the pattern caused by the chemical gradient left behind from the
shrinking core during the main sequence (see e.g. Miglio et al. 2008) yields a
significantly different best model.

Interestingly, in the case of the benchmark pattern and variant patterns 1 and
2, none of the three possible frequencies chosen from an apparent triplet
produce a period spacing pattern which is well fitted by our models. We posit
that there is missing physics, such as the presence of a magnetic field in the
deep interior of KIC 7760680 (see e.g. Prat et al. 2019, 2020; Van Beeck et
al. 2020), which is causing a potential frequency perturbation at the location
of the frequency triplet between $0.80<\nu<0.81$ d-1. In that sense, the
theoretical uncertainties associated with incomplete physics within the models
remain a dominant source of uncertainty in forward asteroseismic modelling of
main-sequence stars.

Our results are based on the in-depth analysis of a single star, but they are
extendable to other coherent g-mode pulsators since KIC 7760680 is an
excellent representative of SPB stars. The Kepler sample of SPB stars span a
diverse range of masses, ages, and rotation rates (see e.g. Szewczuk et al.
2021; Pedersen et al. 2021), but all show interesting features and structure
in their observed g-mode period spacing patterns, which models cannot fully
explain. Our sensitivity study has demonstrated that the formal precision on
pulsation frequencies obtained using exquisite Kepler space photometry whilst
taking into account various data reduction steps are small but non-negligible
compared to the larger theoretical uncertainties, and can be used to
effectively fine-tune the physics of current theoretical models. In
conclusion, even though theoretical uncertainties associated with incomplete
physics within the models dominate in forward asteroseismic modelling, a more
systematic treatment of observational uncertainties allows us to better
constrain the interior physics and determine more precise and accurate
(convective core) masses, ages, and interior rotation and mixing parameters in
excellent g-mode pulsators such as KIC 7760680.

###### Acknowledgements.

The authors are grateful to the asteroseismology group at KU Leuven for useful
discussions and the referee for constructive feedback. The research leading to
these results has received funding from the Research Foundation Flanders (FWO)
by means of a senior postdoctoral fellowship to DMB (grant agreement No.
1286521N), and a PhD scholarship to MM (grant agreement No. 11F7120N). The
authors thank the Kepler science team for the excellent data. The Kepler data
presented in this paper were obtained from the Mikulski Archive for Space
Telescopes (MAST) at the Space Telescope Science Institute (STScI), which is
operated by the Association of Universities for Research in Astronomy, Inc.,
under NASA contract NAS5-26555. Support to MAST for these data is provided by
the NASA Office of Space Science via grant NAG5-7584 and by other grants and
contracts. Funding for the Kepler/K2 mission was provided by NASA’s Science
Mission Directorate. This research has made use of the SIMBAD database,
operated at CDS, Strasbourg, France; the SAO/NASA Astrophysics Data System;
and the VizieR catalog access tool, CDS, Strasbourg, France. The computational
resources and services used in this work were provided by the VSC (Flemish
Supercomputer Center), funded by the Research Foundation Flanders (FWO) and
the Flemish Government, department EWI. The authors are grateful to the MESA
(Paxton et al., 2019) and GYRE (Townsend & Teitler, 2013) developers, in
particular B. Paxton and R. H. D. Townsend, for continually supporting the
development of state-of-the-art and open-source tools for modelling pulsating
stars. This research has made use of the PYTHON library for publication
quality graphics (MATPLOTLIB; Hunter 2007), Seaborn (Waskom, 2021), Numpy
(Oliphant, 2006; van der Walt et al., 2011; Harris et al., 2020) and Pandas
(Wes McKinney, 2010).

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##  Appendix A Extended data tables

The multi-cosinusoid non-linear least-squares solution for the
LC${}_{\textup{\smash{Pápics}}}$ light curve of KIC 7760680 is provided in
Table LABEL:table:_multi-freq.

##  Appendix B Variant pattern modelling results

The summary figures for modelling the variant period spacing patterns are
shown in Figs 17 and 18.

Figure 17: Summary figures for the results from modelling variant period
spacing patterns 1, 1B, 2, 3 and 3B.

Figure 18: Summary figures for the results from modelling variant period
spacing patterns 4, 4B, 5, 5B, and the robust pattern.