11institutetext:  Institute of Astronomy, KU Leuven, Celestijnenlaan 200D,
B-3001 Leuven, Belgium
email<EMAIL_ADDRESS>22institutetext: Department of Astrophysics,
IMAPP, Radboud University Nijmegen, PO Box 9010, 6500 GL, Nijmegen, The
Netherlands 33institutetext: Max Planck Institute for Astronomy, Koenigstuhl
17, 69117, Heidelberg, Germany 44institutetext: Guest Researcher, Center for
Computational Astrophysics, Flatiron Institute, 162 Fifth Ave, New York, NY
10010, USA 55institutetext: I-BioStat, Universiteit Hasselt, Martelarenlaan
42, B-3500 Hasselt, Belgium 66institutetext: I-BioStat, KU Leuven,
Kapucijnenvoer 35, B-3000 Leuven, Belgium

# Astrophysical properties of 15062 Gaia DR3 gravity-mode pulsators

Pulsation amplitudes, rotation, and spectral line broadening††thanks: The data
file with the Gaia identification, fundamental parameters, dominant frequency
and its amplitude, and the line broadening values for all 15062 stars in the
studied sample are available electronically at the CDS via anonymous ftp to
cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-
bin/qcat?J/A+A/???/A??

C. Aerts 11223344 G. Molenberghs 5566 J. De Ridder 11

(Received December 17, 2022; Accepted February 8, 2023)

###### Abstract

Context. Gravito-inertial asteroseismology saw its birth thanks to high-
precision CoRoT and Kepler space photometric light curves. So far, it gave
rise to the internal rotation frequency of a few hundred intermediate-mass
stars, yet only several tens of these have been weighed, sized, and age-dated
with high precision from asteroseismic modelling.

Aims. We aim to increase the sample of optimal targets for future gravito-
inertial asteroseismology by assessing the properties of 15062 newly found
Gaia DR3 gravity-mode pulsators. We also wish to investigate if there is any
connection between their fundamental parameters and dominant mode on the one
hand, and their spectral line broadening measured by Gaia on the other hand.

Methods. After re-classifying about 22% of the F-type gravity-mode pulsators
as B-type according to their effective temperature, we construct histograms of
the fundamental parameters and mode properties of the 15062 new Gaia DR3
pulsators. We compare these histograms with those of 63 Kepler bona fide class
members. We fit errors-in-variables regression models to couple the effective
temperature, luminosity, gravity, and oscillation properties to the two Gaia
DR3 parameters capturing spectral line broadening for a fraction of the
pulsators.

Results. We find that the selected 15062 gravity-mode pulsators have
properties fully in line with those of their well-known Kepler analogues,
revealing that Gaia has a role to play in asteroseismology. The dominant
g-mode frequency is a significant predictor of the spectral line broadening
for the class members having this quantity measured. We show that the Gaia
vbroad parameter captures the joint effect of time-independent intrinsic and
rotational line broadening and time-dependent tangential pulsational
broadening.

Conclusions. While the Gaia mission was not designed to detect non-radial
oscillation modes, its multitude of data and homogeneous data treatment allow
us to identify a vast amount of new gravity-mode pulsators having fundamental
parameters and dominant mode properties in agreement with those of such Kepler
bona fide pulsators. This large new sample of Gaia DR3 pulsators can be
followed up with dedicated high-precision photometric or high-resolution
spectroscopic instruments to embark upon asteroseismic modelling.

###### Key Words.:

Asteroseismology – Methods: Statistical – Astronomical Data Bases – Stars:
Rotation – Stars: Interiors – Stars: oscillations (including pulsations)

##  1 Introduction

The Gaia space mission of the European Space Agency (Gaia Collaboration et
al., 2016b) is currently revolutionising the entire field of astrophysics.
Even though Gaia is in the first place an astrometric mission, it also
delivers the largest homogeneous survey of broad-band photometric and medium-
resolution spectroscopic data (Gaia Collaboration et al., 2016a). While the
Gaia mission was not at all designed to deliver input for the research field
of asteroseismology (Aerts et al., 2010), it does contribute important
information for that recent emerging topic within stellar astrophysics.
Indeed, aside from stellar luminosities deduced from the high-precision
parallaxes (Bailer-Jones et al., 2018), the Gaia instrumentation also delivers
years-long photometric time-series data at milli-magnitude (mmag) precision in
the Gaia G band. Although these Gaia G light curves are only sparsely sampled,
they do allow to populate a wide range of stellar variability classes (see
Eyer & Mowlavi, 2008, for a description of the “variability tree”). In
particular, Gaia data allows us to study the classes of pulsating variables
(cf., Aerts et al., 2010, Chapter 2) with unprecendented numbers of
membership. Rimoldini et al. (2022) classified more than 12 million variables,
among which RR Lyr stars (Clementini et al., 2022), Cepheids (Ripepi et al.,
2022), young stellar objects (Marton et al., 2022), and long period variables
(Lebzelter et al., 2022).

In this work, we focus on stars observed by Gaia and classified from its Data
Release 3 (DR3 Gaia Collaboration et al., 2021, 2022b) as gravity-mode (g-mode
hereafter) pulsators by Coordination Unit 7 treating variable stars (Holl et
al., 2018; Gaia Collaboration et al., 2019; Eyer et al., 2022). In their Gaia
DR3 Performance Verification Paper (PVP), Gaia Collaboration et al. (2022a,
hereafter termed Paper I) assigned the stars we revisit in the present study
to the classes of the Slowly Pulsating B (SPB stars, Waelkens, 1991; De Cat &
Aerts, 2002) or $\gamma\,$Doradus stars ($\gamma\,$Dor, Kaye et al., 1999;
Handler, 1999). These main-sequence g-mode pulsators are the best laboratories
for asteroseismic probing of the deep interior of dwarfs with a mass between
1.3 M⊙ and 9 M⊙ (Aerts, 2021, for a general review of asteroseismology of such
stars). By now, hundreds of single and binary dwarfs with a convective core
have their rotation measured just outside their core from series of
consecutive radial-order dipole g-mode oscillations (Kurtz et al., 2014;
Triana et al., 2015; Keen et al., 2015; Saio et al., 2015; Van Reeth et al.,
2016; Moravveji et al., 2016; Murphy et al., 2016; Pápics et al., 2017; Zwintz
et al., 2017; Van Reeth et al., 2018; Li et al., 2019, 2020; Sekaran et al.,
2021). Both the $\gamma\,$Dor and SPB pulsators reveal time-dependent spectral
line variations due to the tangential velocity fields at the stellar surface
(Aerts et al., 1999; De Cat & Aerts, 2002; Aerts et al., 2004; De Cat et al.,
2006). These pulsators are intermediate-mass dwarfs in the core-hydrogen
burning phase without strong stellar winds.

The Gaia DR3 light curves analysed in Paper I resulted in an order-of-
magnitude increase in the population of the two classes of non-radial g-mode
main-sequence pulsators. The position of these new candidate SPB and
$\gamma\,$Dor pulsators in the Hertzsprung-Russell diagram was compared with
theoretically predicted instability strips, each of which based on the
dominant excitation mechanism for one particular choice of input physics,
leading to coherent g modes with infinite lifetime in Paper I. It was found
that many of the Gaia g-mode pulsators occur outside the borders of such
instability strips for these two classes of g-mode pulsators. This was
ascribed to inaccuracies in the Gaia effective temperature, their fast
rotation and/or different input physics or (past) binarity not treated in
instability predictions, in addition to too low opacities of heavy elements
such as iron and nickel in the excitation layers as is well-known from
previous excitation studies of SPB pulsators (Moravveji, 2016; Daszyńska-
Daszkiewicz et al., 2017; Szewczuk & Daszyńska-Daszkiewicz, 2017). Moreover,
aside from coherent eigenmodes with long lifetime driven by opacity layers or
at the bottom of the outer convective envelope, internal gravity waves with
short lifetimes excited at the interface between the convective core and/or
the convective outer envelope and radiative zones have been suggested from
multi-dimensional hydrodynamical simulations mimicking dwarfs in the
considered mass regime (Rogers et al., 2013; Grassitelli et al., 2015b, a;
Edelmann et al., 2019; Horst et al., 2020). All these predicted g modes and
internal gravity waves act together in the stellar interior and those reaching
the stellar surface with detectable amplitude give rise to complex light
curves and time-dependent spectral line profile variations. This is in
agreement with modern time-resolved space photometric large surveys delivering
$\mu$mag precision and highlighting a continuous coverage of observed
intermediate-mass pulsating dwarfs along the main sequence (Uytterhoeven et
al., 2011; Bowman et al., 2019; Pedersen et al., 2019; Antoci et al., 2019;
Bowman et al., 2020; Balona & Ozuyar, 2020, and Paper I), after earlier
similar findings from high-resolution ground-based time-resolved spectroscopy
mentioned above.

Here, we study the astrophysical properties of the new Gaia DR3 g-mode
pulsators found in Paper I. We consider all these pulsators having their
luminosity, effective temperature, and gravity determined by the Gaia Data
Processing Analysis Consortium (DPAC, Gaia Collaboration et al., 2016a, 2021).
We compare the properties of these Gaia DR3 g-mode pulsators with those of the
sample of 63 best-known bona fide g-mode pulsators observed with the NASA
Kepler space telescope, for which high-resolution follow-up spectroscopy was
assembled and interpreted. These Kepler and spectroscopic data led to the
identification of dipole g modes of consecutive radial order and to
asteroseismic modelling for these 63 dwarfs (Mombarg et al., 2021; Pedersen et
al., 2021).

In the current follow-up study of Paper I we consider the amplitude of the
dominant frequency in the Gaia G band and relate it to the fundamental
parameters for both the Gaia DR3 and Kepler g-mode pulsators. Moreover, we
consider the sub-samples of Gaia DR3 g-mode pulsators for which an estimation
of the spectral line broadening is available from Gaia’s Radial Velocity
Spectrometer (RVS) within the large homogeneous Gaia DR3 data set (Creevey et
al., 2022; Frémat et al., 2022). Our general aim is to search for
relationships between the fundamental parameters and pulsational properties of
g-mode pulsators. In particular, we investigate if there is any connection
between the properties of the dominant g mode of the stars and their rotation
and/or spectral line broadening. So far, similar studies have been hampered by
small sample sizes (Aerts et al., 2014a) or by separate and/or inhomogeneous
treatment of statistical modelling based on the observables deduced from
photometric and spectroscopic data (Simón-Díaz et al., 2017; Burssens et al.,
2020). Even though the Gaia photometric and spectroscopic instruments were not
designed to study non-radial oscillations nor optimised to deduce the line
broadening of stars hotter than 7 000 K, DR3 does provide unprecedentedly
large samples of homogeneously treated g-mode pulsators in terms of their line
broadening and fundamental parameters compared to those available in the
literature prior to Gaia DR3.

We discuss the sample selection for the current paper in Sect. 2 and consider
the fundamental parameters and dominant variability characteristics of 15062
g-mode pulsators in Sect. 3 and Sect. 4, respectively. Section 5 focuses on
the astrophysical interpretation of the measured spectral line broadening of
the sample, based on the method of errors-in-variables and on multi-variable
regression models constructed via the technique of backward selection. We
discuss our findings and conclude in Sect. 6.

##  2 Sample selection

Paper I resulted in 106 207 Gaia DR3 main-sequence pulsators of spectral types
O, B, A, or F fullfilling four criteria: 1) their Gaia DR3 G light curve
consists of at least 40 epochs; 2) their dominant cyclic frequency (denoted
here as $\nu$) in the Gaia G light curve occurs in the range $[0.7,25]$ d-1;
3) this frequency $\nu$ differs from any of the instrumental frequencies 4, 8,
12, 16, 20, and 24 d-1 by more than 0.05 d-1; 4) $\nu$ has a false alarm
probability in the definition by Baluev (2008) below 0.001. Despite these
already strict selection rules, additional restrictions on the frequency
interval to which $\nu$ had to belong for each of the four considered
pulsation classes were imposed in Paper I to beat instrumental effects in
Fourier space, because they occur at mmag level and intervene with the signal
of non-radial oscillations also occuring at such level for dwarf stars of
intermediate mass.

The following four classes of pulsators were considered in Paper I:
$\beta\,$Cep stars, Slowly Pulsating B (SPB) stars, $\delta\,$Sct stars, and
$\gamma\,$Dor stars (cf. Aerts et al., 2010, Chapter 2). We refer to Paper I
and its literature references for the details of the additional selection
rules imposed upon $\nu$ based on common knowledge of the pulsational
properties for these four well-known classes of variables, but recall here
that the dominant modes of $\beta\,$Cep and $\delta\,$Sct stars are p modes
with observed frequencies typically above 3 d-1, while SPB and $\gamma\,$Dor
stars have dominant g modes with observed frequencies mostly below 3 d-1,
except for the fastest rotators.

Within the sample of 106 207 candidate pulsators assigned to the four
pulsation classes in Paper I, those having frequencies above the spin
frequency of the Gaia satellite are most affected by mmag-level instrumental
effects, which may lead to spurious frequencies unrelated to the star. For
this reason, we focus the current work on the two classes of main-sequence
pulsators having their dominant frequency well below the 4 d-1 spin frequency
of the Gaia satellite. For now, with the relatively sparse DR3 light curves,
this is the best approach to study the astrophysical properties of the Gaia
DR3 g-mode pulsators without being contaminated by spurious instrumental
frequencies.

Appendix B of Paper I discussed the results for the dominant frequency $\nu$
in the Gaia DR3 light curves of the 63 bona fide g-mode pulsators (26 SPB and
37 $\gamma\,$Dor stars) whose entire amplitude spectrum is known with a level
of precision better than about $10^{-6}$ d-1 in cyclic frequency and a few
$\mu$mag in amplitude (Van Reeth et al., 2015; Pedersen et al., 2021). Aerts
et al. (2021) relied on the mode identification for all these stars’ detected
and identified dipole modes of consecutive radial order to deduce the
convective and wave Rossby numbers for these best-known Kepler g-mode
pulsators, covering the mass range from 1.3 M⊙ up to about 9 M⊙. All these 63
pulsators have a dominant Gaia G amplitude, $A_{\nu}$, below 35 mmag and their
dominant frequency occurs in the interval $\nu\in[0.7,3.2]$ d-1 (see Figs B.1
and B.2 in Paper I). Some of the new g-mode pulsators identified from Gaia DR3
in Paper I have higher dominant frequencies. Moreover, some of the Gaia DR3
g-mode pulsators have frequencies that are hard to unravel from the Gaia
instrument frequencies caused by the spinning of the satellite.

Guided by Figs. B.1 and B.2 in Paper I summarising the dominant frequency and
amplitude for the 63 bona fide Kepler g-mode pulsators, we further apply a
fifth and sixth constraint in this work, in addition to the selecton criteria
of Paper I mentioned above, namely we demand that 5) $\nu\in[0.7,3.2]$ d-1 and
6) $A_{\nu}\leq 35\,$mmag. These two extra restrictions are applied to the SPB
and $\gamma\,$Dor stars assigned to those two g-mode pulsator classes in Paper
I. This is to ensure that we are dealing with non-radial oscillations rather
than satellite frequencies. Moreover, we restrict these two samples to those
pulsators having a measurement of $\log\,L$, $\log\,T_{\rm eff}$, and
$\log\,g$ in the DR3 gspphot tables. We use those values in order to maximise
the sample size of g-mode pulsators treated in one homogeneous way by DPAC
routines, given that we need to cover temperatures from $\sim\\!6\,500\,$K all
the way up to 25 000 K (see Paper I for details and Gaia Collaboration et al.,
2022b; Creevey et al., 2022).

A continuous coverage of pulsating B, A, and F stars along the main sequence
was found in Paper I, in agreement with Kepler and TESS results (e.g., Balona
& Ozuyar, 2020). Since the variability classification used in Paper I relied
only on the Gaia G-band DR3 light curves, it cannot distinguish between B- and
F-type pulsators without spectroscopic information (cf., Pedersen et al.,
2019; Gebruers et al., 2021). On the other hand, the Kepler data clearly
revealed that $\gamma\,$Dor and SPB pulsators have different astrophysical and
pulsational properties (Van Reeth et al., 2015; Saio et al., 2018; Pedersen et
al., 2021). Thus, we wish to treat them as two separate classes. We do so by
relying on the Gaia DR3 effective temperature to reclassify the g-mode
pulsators. Following the upper limit in effective temperature from the
instability predictions by Xiong et al. (2016) for $\gamma\,$Dor stars as
treshold, we use $T_{\rm eff}=8500\,$K to distinguish between $\gamma\,$Dor
and SPB candidates. In practice, we reclassify all $\gamma\,$Dor candidates as
SPB if their effective temperature is above 8500 K and, vice versa, we re-
assign all SPB stars with a temperature below 8500 K as $\gamma\,$Dor
pulsator. This leads to a re-classification of 3244 $\gamma\,$Dor candidates
as new SPB pulsators based on their Gaia DR3 effective temperature. This re-
assignment gives a fractional memberships of 29% SPB and 71% $\gamma\,$Dor
pulsators, which is fully in line with a Salpeter-type initial mass function
(IMF, Salpeter, 1955) for the typical masses of 1.6 M⊙ for $\gamma\,$Dor stars
(Mombarg et al., 2021) and of 4 M⊙ for SPB stars (Pedersen, 2022b). As we
discuss later in the paper, another choice of the treshold temperature to
distinguish the two classes does not impact any of the results.

A critical aspect of the current study compared to other surveys of g-mode
pulsating dwarfs is that all the DR3 data and observables were obtained in one
homogeneous way in terms of data selection and analysis. This is in contrast
to the treatment of ground-based photometry and spectroscopy obtained for the
much smaller dedicated asteroseismology samples for these two classes so far.
While Gaia DR3 can only deliver the dominant mode at this stage, it provides
by far the largest homogeneous survey of $\gamma\,$Dor and SPB pulsators to
date.

##  3 Fundamental parameters of the g-mode pulsators in the two samples

Figure 1:  Histograms (normalised to 100% occurrence) of the gspphot values
for $T_{\rm eff}$ for 11636 Gaia DR3 $\gamma\,$Dor stars (left, grey) and 3426
SPB stars (right, cyan). The width of the bars is according to the average
error (150 K for $\gamma\,$Dor and 400 K for the SPB stars). Asteroseismic
values obtained from Kepler data are shown for 37 $\gamma\,$Dor (grey,
hatched) and 26 SPB stars (black cross-hatched), respectively. For the right
panel, 31 SPB stars with a temperature above 22 000 K in the Gaia DR3 sample
were omitted for visibility reasons.

Figure 2:  Same as Fig. 1 but for $\log\,g$.

Figure 3:  Same as Fig. 1 but for $\log\,(L/{\cal L_{\odot}})$.

Figures 1, 2, and 3 show the histograms of the effective temperature, surface
gravity, and luminosity for all 15062 Gaia DR3 g-mode pulsators in our two
samples taken from the gspphot tables, in comparison with those quantities
deduced for the 63 bona fide pulsators. For the latter, we took the high-
precision values for these quantities from detailed asteroseismic modelling of
their interior based on numerous identified dipole g modes by Pedersen (2022b,
26 SPB stars) and Mombarg et al. (2021, 37 $\gamma\,$Dor stars). Both these
asteroseismic studies followed the methodology in Aerts et al. (2018) for the
modelling of the internal stucture of these stars. In order to be able to
compare the distributions of the samples with vastly different numbers of
stars, we show the normalised histograms as percentages of the entire sample
population. For the $\gamma\,$Dor stars, the distributions of the effective
temperature ($T_{\rm eff}$) and luminosity ($\log(L/{\cal L_{\odot}}$), with
${\cal L}_{\odot}$ the solar luminosity) agree remarkably well between the 37
bona fide versus 11636 Gaia DR3 pulsators, revealing Gaia’s power to deduce
these two fundamental parameters for large samples of such hot F-type
pulsators. The gravities are somewhat lower than the asteroseismic values.
Comparing the Gaia radii deduced from the DR3 $T_{\rm eff}$ and
$\log\,(L/{\cal L_{\odot}})$ values shown in the left panel of Fig. 4 reveals
that they are entirely compatible with the asteroseismic distribution, keeping
in mind that the latter only consists of 37 pulsators.

Figure 4:  Same as Fig. 1 but for the radius of the stars deduced from
$\log\,(L/{\cal L_{\odot}})$ and $T_{\rm eff}$.

As for the SPB stars, the Gaia DR3 sample is concentrated in the lower part of
the SPB instability strip as already found in Paper I. This paper also
reported on possible systematic biases in the astrophysical parameters of hot
stars as derived from the Gaia DR3 gspphot tables, more particularly shifts to
lower temperatures due to poorly estimated reddening. Since we only work with
the dominant frequency, we cannot exclude that some of the (reclassified) cool
SPB stars are actually early A-type stars with rotational modulation. The
majority of 3426 SPB stars in our sample have relatively low Gaia luminosities
compared to the 26 SPB stars in the asteroseismic sample. These two aspects
combined limit the power of Gaia DR3 to estimate radii based on the gspphot
tables for the entire class of SPB stars, yet the radius distribution of the
3426 SPB stars is compatible with the one of the 26 bona fide SPB stars (Fig.
4, right panel).

Overall, the distributions for the three fundamental parameters $T_{\rm eff}$,
$\log\,g$, and $\log\,(L/{\cal L}_{\odot})$ of the Gaia $\gamma\,$Dor and SPB
stars are in good agreement with the asteroseismic values of the bona fide
Kepler pulsators in these two classes, keeping in mind that the Gaia sample of
SPB stars mainly contains cool class members. We conclude that Gaia DR3
gspphot values reveal proper distributions of radii for g-mode pulsators from
their luminosity and effective temperature estimates when compared with the
radii of the 63 bona fide pulsators from the best asteroseismic model for each
of them.

##  4 Pulsational properties of the dominant g modes

Figure 5:  Same as Fig. 1 but for the dominant frequency in the Gaia G light
curve.

Figure 6:  Same as Fig. 1 but for the amplitude of the dominant frequency in
the Gaia G light curve. For the bona fide $\gamma\,$Dor and SPB pulsators, we
computed the amplitude by fitting a harmonic signal to the Gaia time series
using the main frequency derived from the 4-yr Kepler light curve.

As already emphasised in Paper I, Gaia has good capacity to detect non-radial
oscillation modes in main-sequence stars. The two Gaia g-mode pulsator samples
treated here result from quite strict selection rules on the Gaia G
photometric light curves, yet they are already an order of magnitude larger
than the corresponding Kepler samples. Despite the sparse Gaia sampling, it is
to be anticipated that many more g-mode pulsators will be selected once the
DR4 and DR5 data sets will become available.

Figure 5 shows the distributions for the dominant frequency in the DR3 Gaia G
band light curves. Overplotted are the distributions for the dominant
frequency deduced from the far higher-precision and 4-yr uninterrupted high-
cadence Kepler light curves taken from Van Reeth et al. (2015) and Pedersen et
al. (2021) for the $\gamma\,$Dor and SPB stars, respectively. We recall that
we used the dominant g-mode frequency range covered by these two samples of
bona fide g-mode pulsators as selection criterion, to restrict the Gaia DR3
samples to pulsators adhering to this same appropriate frequency range. Hence
it is built in that we find compatible ranges. Yet, also the distributions are
reasonably well in agreement between the Gaia DR3 and Kepler pulsators,
keeping in mind the small samples sizes for the latter.

Figure 6 shows the histograms for the amplitude of the dominant frequency
found in the Gaia G light curve. These bona fide pulsators did not survive our
six selection criteria, mainly because they have fewer than 40 epochs in Gaia
DR3 and/or their dominant frequency did not meet the FAP criterion. In order
to be able to compare the amplitudes between the Gaia DR3 and Kepler
pulsators, and to exclude instrumental effects for the bona fide pulsators, we
computed their Gaia G amplitude by imposing their dominant frequency found in
their Kepler light curve onto the Gaia G data, irrespective of the number of
epochs in the latter and their FAP value. Both histograms in Fig. 6 visualise
the current detection threshold to find g modes in the Gaia G light curves. It
is seen that DR3 allows us to detect g-mode frequencies with an amplitude
above 4 mmag. The Kepler data delivered g modes with far lower amplitudes as
seen in the histogram, because the mission was designed to assemble $\mu$mag-
precision uninterrupted 30-min cadence photometry for exoplanet hunting
(Borucki et al., 2010) and for asteroseismology (Gilliland et al., 2010). We
find that the g-mode amplitude distributions for Gaia DR3 and Kepler hardly
overlap for the class of the SPB pulsators as most of the 26 bona fide Kepler
SPB stars have low dominant amplitudes outside Gaia’s reach.

##  5 Properties of spectral line broadening

Aside from photometric and astrometric data, the Gaia satellite also delivers
spectroscopic data. Its spectrometer RVS has a median resolving power of 11
500 and is sensitive to the wavelength range from 846 to 870 nm. While it was
built with the primary goal to measure the radial velocity of as many Gaia
sources as possible (Katz et al., 2022), we use the RVS data to study the
spectral line broadening of g-mode pulsators. Our aim is to investigate if,
aside from rotational line broadening, there is any connection between the
overall line broadening, the fundamental parameters, and the oscillation
properties for the two large Gaia DR3 samples of g-mode pulsators, as
suggested previously based on line-profile simulations (Aerts et al., 2009;
Aerts & Rogers, 2015). While RVS on average provides a resolving power of only
$\sim\\!26\,$km s-1, non-radial oscillations generate variations in the width
and the skewness of spectral lines (Aerts & De Cat, 2003) and these may affect
the way that the line broadening values were determined (we refer to Frémat et
al., 2022, for a detailed description omitted here).

Line-profile variations caused by the g modes of $\gamma\,$Dor and SPB stars
occur at the level of several to tens of km s-1 in the centroid of the line
(e.g., Aerts et al., 1999; De Cat et al., 2000; Mathias et al., 2001; De Cat &
Aerts, 2002; Mathias et al., 2004; De Cat et al., 2006). High-resolution time-
series spectroscopy of bright g-mode pulsators offers a powerful tool to
identify the spherical wavenumbers $(l,m)$ of the dominant oscillation mode(s)
provided that the oscillation cycle is well covered (Briquet et al., 2003; De
Cat et al., 2005). Such applications couple the velocity field, computed from
the theory of non-radial oscillations, to the observed line-profile variations
to infer the radial and tangential components of the velocity vector due to
each non-radial oscillation mode (Aerts et al., 2010, Chapter 6). This
requires the spectroscopy to have high resolving power and signal-to-noise
ratio (SNR) (typically above 50 000 and 300, respectively; Aerts & De Cat,
2003).

In the absence of high-quality spectroscopy, or in the case where only a few
snapshot spectra are available, line-profile modelling by means of the proper
time-dependent pulsational velocity field is impossible. In such a case, it is
customary to approximate the overall line broadening due to oscillations and
rotation together by a single time-independent function called macroturbulence
(Simón-Díaz et al., 2010). Even though its functional form assumes a
symmetrical line profile (cf. Aerts et al., 2014b, Fig. 9), the
macroturbulence correlates strongly with quantities representing the line-
profile variability (Simón-Díaz et al., 2017), such as the velocity moments
(Balona, 1986; Aerts et al., 1992).

Fitting time-resolved line-profile variations due to oscillations or spots
artificially with a symmetrical macroturbulent profile leads to time-
variability in the macroturbulence which is in excellent agreement with the
mode frequencies or rotation periods of intermediate-mass dwarfs (Aerts et
al., 2014b). This suggests that macroturbulence is merely a downgraded (and
often poor) time-independent symmetrical simplification of the true spectral
line profiles caused by oscillations and/or spots. Nevertheless, in the
absence of time-resolved spectroscopy, it is a sensible approach to fit the
line profiles of snapshot spectra with a synthetic time-independent
macroturbulent broadening profile, particularly for large surveys of stars
such as offered by Gaia.

The g modes have dominant tangential displacements, implying that their
velocity at the limb of the star dominates the detected line-profile
variability. Yet, a common attitude in the literature has been to rely on the
ad-hoc assumption that the radial and tangential components of the
macroturbulent broadening profile are equal (Simón-Díaz & Herrero, 2014). This
is the reason why unrealistic, often supersonic, values for the macroturbulent
surface velocities are obtained. This in turn affects the estimation of the
surface rotation (Aerts et al., 2014b). For that reason, it is essential to
estimate the surface rotation first, independently from the macroturbulent
broadening (Serebriakova et al., 2023).

In the following sections we investigate Gaia’s capacity to shed light on the
astrophysical cause(s) of its measurements of the spectral line broadening
from RVS. We do so for the two classes of $\gamma\,$Dor and SPB stars, whose
velocity field at the stellar surface due to their non-radial oscillations is
dominantly tangential (De Cat & Aerts, 2002; Aerts et al., 2004).

###  5.1 Gaia DR3 line broadening parameters

While Gaia’s medium-resolution RVS was not built to assess line broadening by
stellar oscillations combined with rotation, it offers unprecedently large
stellar samples analysed with a common methodology. We test the behaviour of
line broadening measured with RVS with respect to the acting velocity fields
at the stellar surface, where we know that our two samples undergo the joint
effect of time-independent rotational and time-dependent pulsational line
broadening. To do so, we rely on two Gaia DR3 parameters offered as
measurement of spectral line broadening: vbroad (Frémat et al., 2022) and
vsini-esphs (Creevey et al., 2022). Frémat et al. (2022) already made a
careful study of these two quantities to interpret spectral line broadening
for more than 33 million stars having $T_{\rm eff}\in[3.1;14.5]$kK. This range
fully encapsulates the one of our $\gamma\,$Dor sample and largely overlaps
with the SPB sample.

Figure 7:  Comparison between Gaia DR3 measurements of vsini-esphs and vbroad
for the 100 $\gamma\,$Dor (grey triangles) and 156 SPB (cyan squares) stars
having both quantities available. The full dotted line indicates the bisector
while the coloured dashed lines represent the best linear regression models
for both samples.

The Gaia DR3 parameter vbroad captures the overall line broadening after
deconvolving the spectra with Gaia’s RVS instrument Along-Scan Line Spread
Function (Sartoretti et al., 2022). This parameter vbroad thus includes the
joint effect of all possible astrophysical causes that give rise to spectral
line broadening, such as oscillations, rotation, spots, turbulent convective
velocities, etc. The catalogued value of vbroad is the median value obtained
by the Multiple Transit Analysis (MTA) over at least six valid transits. The
corresponding catalogued uncertainty is the standard deviation with respect to
this median value. This implies that the uncertainty may be a measure of the
line-profile variability, because its range captures the line broadening
occurring at minimally six different epochs aside from the contribution of the
noise. For the current study, we have vbroad measurements for 1775 of the
11636 $\gamma\,$Dor stars and for 190 of the 3426 SPB pulsators.

Another estimate of the RVS line broadening denoted as vsini-esphs is obtained
by the so-called Extended Stellar Parametrizer module developed within the
Astrophysical ParameterS Inferences System APSIS (Creevey et al., 2022). APSIS
is able to treat the parameters of hot stars and delivers vsini-esphs as an
intermediate data product approximating time-independent rotational
broadening. Its computation was done on the basis of the averaged values of BP
and RP and the averaged RVS spectrum. Although this implies the variability to
be filtered out from the quantity vsini-esphs to some level, it still contains
a contribution from the oscillations (cf., De Pauw et al., 1993, for the
theoretical expression of the line width due to non-radial oscillations).
Nevertheless, vsini-esphs is a cleaner measurement of the time-independent
projected surface rotation velocity than vbroad when rotation dominates the
spectral line broadening, as is the case for g-mode pulsators. The error
published for vsini-esphs is an approximation of the statistical error and
does not represent a measurement of the time-variable line broadening like the
standard deviation for vbroad. The quantity vsini-esphs deduced by APSIS
results from an optimal RVS and BP/RP treatment for stars having a $T_{\rm
eff}$ above 7500 K, while it relies on BP/RP alone for all stars with $T_{\rm
eff}>7000\,$K that were not observed by RVS. Values for vsini-esphs are
available for 384 of the 11636 $\gamma\,$Dor stars and for 1104 of the 3426
SPB stars.

The wavelength coverage of RVS was constructed so as to achieve optimal
radial-velocity data for a broad range of stellar populations. Here we rely on
its data for the purpose of studying spectral line broadening, for which the
RVS wavelength domain is not optimal. This is particularly so for the hottest
stars under study here. As a consequence, some of the vbroad and vsini-esphs
measurements have large uncertainties. It is therefore essential to include
these uncertainties, aside from the values themselves, in any proper
astrophysical interpretation. We refer to Frémat et al. (2022) for a detailed
and nuanced discussion about the quality of the vbroad and vsini-esphs
measurements deduced from template spectra relying on the Gaia $T_{\rm eff}$
estimates. In particular, Frémat et al. (2022) discussed in detail the
correlations between these two parameters for stars covering a broad range of
magnitudes and temperatures. Of particular relevance for the current work is
Fig. 16 in Frémat et al. (2022), illustrating two Hertzsprung-Russell diagrams
with density maps as a function of median vbroad values. That figure clearly
reveals high vbroad values along the upper main sequence, where the
$\gamma\,$Dor and SPB pulsators are situated (encompassing the p-mode
dominated class of $\delta\,$Sct stars not treated here due to the much higher
risk that their dominant frequency has an instrumental origin, as explained in
Sect. 2). The figure reveals higher vbroad values for hotter stars but Frémat
et al. (2022) do not provide any formal quantitative comparisons between
vbroad and the stellar parameters.

We show the 100 $\gamma\,$Dor and 156 SPB pulsators having a measurement of
both vbroad and vsini-esphs in Fig. 7, along with their uncertainties. It can
be seen that the overall range of the two quantities is rougly the same for
these $\gamma\,$Dor and SPB stars. For each of the stars in Fig. 7, the two
plotted quantities have similar yet not always equal values according to the
uncertainty estimates. We remind that similarities between the two samples as
a whole also occur for their ranges of the dominant mode amplitude and mode
frequency (cf. Figs 5 and 6). It is then a natural question whether the
oscillation properties cause the time-variability measured by vbroad and its
standard deviation. On the other hand, we investigate whether the decrease of
observed mode amplitude for faster rotators, as found in Paper I for the
dominant p modes of the Gaia DR3 $\delta\,$Sct stars, also occurs for g-mode
pulsators.

In what follows we offer regression models accomodating errors-in-variables,
which allows one to handle different measurements of the same astrophysical
quantity (here the overall time-averaged spectral line broadening) having both
star-specific and measurement-specific errors. These errors must be propagated
properly when constructing the regression models and interpreting their
outcome. This has been used in the context of Gaia data before, for example in
a comparison between asteroseismic and astrometric parallaxes following DR1
(De Ridder et al., 2016). We first provide a general description of the
methodology. Subsequently we apply it to the sample of the 37 bona fide
$\gamma\,$Dor stars. For all these 37 stars we also have, in addition to their
Gaia DR3 data, estimates of their “true” pulsational and rotational line
broadening deduced from one homogeneous treatment of high-resolution high
signal-to-noise ground-based spectroscopy taken with one instrument (such
homogeneous spectroscopic information is not available for all 26 bona fide
SPB pulsators). We use the results obtained for the 37 bona fide $\gamma\,$Dor
stars to treat the Gaia DR3 g-mode samples optimally with the aim to interpret
their spectral line broadening properties.

###  5.2 Errors-in-variables model

Let us denote two observed quantities by $Y_{i}$ and $X_{i}$ and their true
but unknown values by $Y_{i}^{\ast}$ and $X_{i}^{\ast}$. The errors-in-
variables model is then specified by:

$\displaystyle Y_{i}$ $\displaystyle=$ $\displaystyle
Y_{i}^{\ast}+\varepsilon_{Yi},$ (1) $\displaystyle X_{i}$ $\displaystyle=$
$\displaystyle X_{i}^{\ast}+\varepsilon_{Xi},$ (2) $\displaystyle
Y_{i}^{\ast}$ $\displaystyle=$
$\displaystyle\beta_{0}+\beta_{1}X_{i}^{\ast}+\varepsilon_{i},$ (3)

with $i$ indexing the stars in a sample. Here, $\beta_{0}$ and $\beta_{1}$ are
fixed but unknown regression coefficients to be estimated from the data. The
measurement error variances $\mbox{var}(\varepsilon_{Yi})=\sigma^{2}_{Yi}$ and
$\mbox{var}(\varepsilon_{Xi})=\sigma^{2}_{Xi}$ are obtained from the
observations. The residual error component $\varepsilon_{i}$, with variance
$\sigma^{2}$, quantifies imperfection in the regression relationship.

If $Y_{i}^{\ast}$ and $X_{i}^{\ast}$ would be nearly identical, then
$\beta_{0}$ is expected to be close to 0 and $\beta_{1}$ would be close to 1.
If the regression relationship in Eq. (3) is very precise relative to the
measurement error, then $\sigma^{2}$ would be near 0. Expressions (1)–(3)
yield the mean and variance relationships:

$\displaystyle E(Y_{i})$ $\displaystyle=$
$\displaystyle\beta_{0}+\beta_{1}X_{i},$ (4) $\displaystyle\mbox{Var}(Y_{i})$
$\displaystyle=$
$\displaystyle\beta_{1}^{2}\sigma^{2}_{Xi}+\sigma^{2}_{Yi}+\sigma^{2}.$ (5)

Assuming (approximate) normality, a fully parametric specification follows,
thus enabling maximum likelihood estimation:

$Y_{i}\sim
N(\beta_{0}+\beta_{1}X_{i},\beta_{1}^{2}\sigma^{2}_{Xi}+\sigma^{2}_{Yi}+\sigma^{2}).$
(6)

The SAS procedure NLMIXED (SAS Institute Inc., 2014) was used for the maximum
likelihood estimation.

Extension to multiple predictors $X_{1i},\dots,X_{pi}$ is straightforward,
upon replacing Eq. (6) by:

$Y_{i}\sim
N\left(\beta_{0}+\sum_{j=1}^{p}\beta_{j}X_{ji},\sum_{j=1}^{p}\beta_{j}^{2}\sigma^{2}_{Xji}+\sigma^{2}_{Yi}+\sigma^{2}\right),$
(7)

with obvious notation. It is convenient to write Eq. (7) in vector notation
as:

$Y_{i}\sim
N(\mbox{\boldmath$X$}_{i}^{\prime}\,\mbox{\boldmath$\beta$},\mbox{\boldmath$\beta$}^{\prime}\,\Sigma_{x,i}\,\mbox{\boldmath$\beta$}+\sigma^{2}_{Yi}+\sigma^{2}),$
(8)

where $\mbox{\boldmath$\beta$}=(\beta_{0},\beta_{1},\dots,\beta_{p})^{\prime}$
and $\Sigma_{x,i}$ is a diagonal matrix with
$(0,\sigma^{2}_{X1i},\dots,\sigma^{2}_{Xpi})^{\prime}$ along the diagonal.

Model-based prediction of $Y_{i}^{\ast}$ and its standard deviation can be
expressed as:

$\displaystyle\widehat{Y}_{i}^{\ast}$ $\displaystyle=$
$\displaystyle\widehat{\beta}_{0}+\sum_{j=1}^{p}\widehat{\beta}_{j}X_{ji},$
(9) $\displaystyle\widehat{\mbox{s.d.}}(\widehat{Y}^{\ast}_{i})$
$\displaystyle=$
$\displaystyle\sqrt{\widehat{\mbox{\boldmath$\beta$}}^{\prime}\Sigma_{x,i}\widehat{\mbox{\boldmath$\beta$}}+\mbox{\boldmath$x$}_{i}^{\prime}\widehat{\mbox{var}}(\widehat{\mbox{\boldmath$\beta$}})\mbox{\boldmath$x$}_{i}+\sigma^{2}_{Yi}+\widehat{\sigma}^{2}},$
(10)

where the unknown parameters have been replaced by their data-based estimates.
Expressions (9)–(10) can be used to assess the quality of the model fit. The
second term under the square root in Eq. (10) takes the uncertainty in the
estimated regression coefficient into account.

###  5.3 Spectral line broadening for the 37 bona fide $\gamma\,$Dor stars

Figure 8: Quality of predictions of vbroadH, by vbroadG (upper panel), vsiniH
(middle panel), and the set of covariates (lower panel) for the 37 bona fide
Kepler $\gamma\,$Dor stars. The vertical bars are defined by the predicted
value $\pm$ its standard deviation, based on the errors-in-variables models in
Table 2.

For the 37 bona fide $\gamma$ Dor stars, we now add three more quantities to
the dominant frequency $\nu$ from Kepler photometry and the Gaia DR3 values
for $\log\,T_{\rm eff}$, $\log\,g$, $\log\,(L/{\cal L}_{\odot}$), and
$A_{\nu}$. Following their discovery as g-mode pulsators in the Kepler data,
Tkachenko et al. (2013) set up a ground-based spectroscopic campaign with the
HERMES spectrograph attached to the 1.2m Mercator telescope situated at La
Palma Observatory, Spain (Raskin et al., 2011). HERMES has a spectral
resolution of 85000 and covers wavelengths from 377 to 900 nm. The HERMES
spectra allowed Van Reeth et al. (2015) to deduce the overall spectral line
broadening, here denoted as vbroadH, for the 37 stars and to unravel it into
separate components stemming from time-independent rotational broadening
($v\sin\,i_{\rm H}$) and a broadening component due to the joint effect of
microturbulence and the oscillation modes at the particular epoch of the
observed spectrum. Microturbulence represents an artificial Gaussian line
broadening needed to bring observed spectral lines into agreement with line
predictions from 1-dimensional atmospheric models. This small broadening is
needed to take into account the occurrence of small-scale turbulent motions in
the line forming region that are not included in atmosphere models. On the
other hand, the velocities due to non-radial g-mode oscillations result in
time-dependent line broadening. In the case of $\gamma\,$Dor stars, their
joint net effect in the line-of-sight is of similar order of a few km s-1
(Aerts et al., 2004; De Cat et al., 2006). We therefore take the Gaussian line
broadening determined by Van Reeth et al. (2015) as a good approximation for
the overall pulsational broadening and denote it as voscH.

The two quantities $v\sin\,i_{\rm H}$ and voscH were derived from the observed
spectra after ensuring that none of the 37 stars are spectroscopic binaries.
In practice, Van Reeth et al. (2015) found the following ranges for these two
parameters for the sample of 37 stars: voscH $\in[2.1;4.7]\,$km s-1 and vsiniH
$\in[11;170]\,$km s-1. The values and ranges reveal that this sample of bona
fide $\gamma\,$Dor stars consists of slow to moderate rotators (compared to
their breakup velocity) and that their rotational velocity is typically an
order of magnitude larger than their tangential g-mode and microturbulent
velocity together, where we recall that both quantities are integrations
across the visible stellar surface in the line-of-sight. Since these two
velocity components influence the width of spectral lines added in quadrature,
these ranges show that it is extremely challenging to unravel pulsational from
rotational broadening, even for high-resolution spectroscopy (cf., Aerts et
al., 2004, their Fig. 8). Moreover, since snapshot spectra cannot deliver
proper time-dependent line-profile variations and only the line broadening is
deduced, assuming a symmetrical line while ignoring its true shape, the
relative uncertainties for voscH are considerable.

We use the 37 measured values for vbroadH, $v\sin\,i_{\rm H}$, and voscH to
interpret the Gaia DR3 measurements of the overall line broadening (denoted as
vbroadG for the bona fide $\gamma\,$Dor stars), realising that the RVS
resolving power is in principle insufficient to unravel pulsational from
rotational line broadening. DR3 delivered vbroadG for 27 of the bona fide
$\gamma\,$Dor stars. We used these values to treat the following questions:

  1. 1.

Is the quantity vbroadG obtained by Gaia RVS different from the independently
obtained higher-precision quantities vbroadH or vsiniH?

  2. 2.

Can the variability in vbroadH measured from HERMES for the sample of the 37
bona fide $\gamma\,$Dor stars be predicted by the Gaia DR3 covariates
$\log\,T_{\rm eff}$, $\log\,g$, $\log\,(L/{\cal L}_{\odot}$), $\nu$, and
$A_{\nu}$ and if so with what kind of quality?

Answering these two questions will help to find an astrophysical
interpretation of Gaia’s vbroad values for the two new large samples of g-mode
pulsators without being able to rely on measurements of line broadening
deduced from high-resolution spectroscopy as we only have it available from a
homogeneous data analysis for the 37 bona fide $\gamma\,$Dor pulsators.

Table 1: Estimates (and standard errors) for the model parameters of the errors-in-variables model in Eq. (6) fitted to the bona fide $\gamma$ Dor stars, for four combinations of $X$ and $Y$, where the HERMES quantities are available for all 37 stars and vbroadG for 27 of them.  | $X$ | vbroadG | vbroadH | vsiniH | vsiniH
---|---|---|---|---|---
| $Y$ | vbroadH | vbroadG | vbroadH | vbroadG
Effect | Par. | Estimates (s.e.)
Intercept | $\beta_{0}$ | 4.8(2.4) |  $-$2.5(2.5) | 0.53(0.97) |  $-$2.3(2.4)
Slope | $\beta_{1}$ | 1.02(0.04) | 0.94(0.04) | 0.99(0.02) | 0.94(0.04)
Res. var. | $\sigma^{2}$ | 0.0000(0.0002) | 0.0000(0.0000) | 0.0000(0.0000) | 0.0000(0.0000)
Effect | Par. | 95% confidence intervals
Intercept | $\beta_{0}$ | [$-$0.16;9.74] | [$-$7.63;2.55] | [$-$1.44;2.49] | [$-$7.19;2.56]
Slope | $\beta_{1}$ | [0.94;1.11] | [0.86;1.02] | [0.95;1.03] | [0.86;1.01]

To tackle the first question, we fit the statistical model in Eqn. (6) for
four combinations of $X$ and $Y$. The parameter estimates and statistical
properties of the regression models are presented in Table 1. We find that the
residual variances $\sigma^{2}$ are all extremely close to zero. None of the
intercepts are significantly different from zero, and none of the slopes are
significantly different from unity. This implies that all three quantities are
essentially equal to each other, within the uncertainty limits specified by
the measurement errors. The fractions of the variance explained by each of the
four models range from 94% to 100%.

Given that vbroadG is missing for 10 of the 37 $\gamma\,$Dor stars, the models
involving this variable were refitted after multiple imputation (Molenberghs &
Kenward, 2007) to examine the potential impact of missingness on the results.
This well-known statistical technique was only recently introduced in
astrophysics for the treatment of missing data, such as the multivariate
stellar astrophysics study relating nine measured quantities to surface
nitrogen by Aerts et al. (2014a) and the time-series analysis of visual
binaries by Claveria et al. (2019). The method was applied here as follows.
First, based on a so-called imputation model, 100 copies of each missing
values are drawn from the predictive distribution of what is missing, given
what is observed. Second, each so-completed dataset is then analysed with the
model that would be used had the data been complete. Third, the 100 results
combined into a single result using appropriate combination rules. Results
were qualitatively very similar to those reported in Table 1, offering
confidence that missingness does not play an important role in the
relationships offered in Table 1. For this reason and simplicity, we proceed
with the results presented in Table 1.

Following up on the study by Van Reeth et al. (2015), we conclude from the
bona fide $\gamma\,$Dor g-mode pulsators that single epoch spectra, even if of
high resolution and high SNR, cannot distinguish the overall line broadening
from the line broadening caused by just rotation when working with a fudge
parameter relying on the assumption of a time-independent symmetrical line
profile. Given this spectral line modelling limitation, we find that Gaia RVS
delivers good approximate values for spectral line broadening compared to
those deduced from snapshot high-resolution spectroscopy for early F-type
stars. Yet the uncertainties deduced from the HERMES spectra are lower,
because its better suitable spectral coverage includes more spectral lines
whose shape is determined by the temperature rather than pressure broadening.

To address the second question, we examine the effect of the Gaia DR3
variables $\log\,(L/{\cal L}_{\odot}$), $\log\,T_{\rm eff}$, $\log\,g$,
$A_{\nu}$, and $\nu$ on the independently obtained parameter vbroadH. This can
be done with or without adding vbroadG and with or without adding voscH to the
set of predictors. We proceed by backward selection, starting with the full
set of predictors and then progressively removing the one with the highest
$p$-value, until only significant effects remain (i.e., all $p\leq 0.05$). In
both versions with vbroadG included in the predictor set, this is the only one
remaining after model selection, and we recover the result already reported in
Table 1, as expected.

Table 2: Estimates (standard errors) for the model parameters of the errors-in-variables model for vbroadH, fitted to the bona fide $\gamma$ Dor stars, based on backward selection from a set of predictors. Effect | Par. | Est. (s.e.) |  $p$-value | 95% conf. int.
---|---|---|---|---
Intercept | $\beta_{0}$ | 95(4) |  | [86;104]
$100\cdot(\log\,T_{\rm eff}\\!-\\!3.85)$ | $\beta_{1}$ |  $-$28(5) |  $<$0.0001 | [$-$37;$-$18]
$\log\,g-4.0$ | $\beta_{2}$ | 60(14) | 0.0001 | [32;88]
$\nu-1.75$ | $\beta_{3}$ | 57(6) |  $<$0.0001 | [44;70]
Res. var. | $\sigma^{2}$ | 70(152) | 0.3623 | [$-$238;378]

When vbroadG cannot be considered, as is the case for the majority of Gaia
targets, the following insignificant predictors are removed by means of
backward selection: first $A_{\nu}$, second $\log\,(L/{\cal L}_{\odot}$), and
third voscH. Since the latter variable is removed, whether or not it is
included among the predictors to select from does not matter. Hence, only one
additional model is obtained, the fit of which is presented in Table 2. This
model explains about 58% of the variance present in vbroadH via the effective
temperature, surface gravity, and dominant frequency as covariates, which are
all delivered by Gaia DR3. We note that the ranges of the covariates are
$[3.83;3.87]$ for $\log\,T_{\rm eff}$, $[3.71;4.48]$ for $\log\,g$, and
$[0.78;3.01]$ d-1 for $\nu$, which is why they were linearly transformed as
displayed in Table 2 to stabilise the model fit optimally.

In response to the second question, we find the dominant g-mode frequency
$\nu$ to be a significant predictor of the high-resolution spectroscopic line
broadening, alongside the temperature and gravity of the star. This offers the
opportunity to predict the line broadening for all the Gaia $\gamma\,$Dor
stars without a Gaia measurement of the line broadening if these three
covariates are available, as is the case for the majority of the Gaia DR3
$\gamma\,$Dor stars. Of course it should be born in mind that some predictors
exhibit mild to strong correlation, given their astrophysical meaning. In the
particular application of the bona fide $\gamma\,$Dor stars, the strongest
correlation among the predictors is the one between $\log\,(L/{\cal
L}_{\odot})$ and $\log\,g$, namely -0.70. The correlation between
$\log\,T_{\rm eff}$ and $A_{\nu}$ is 0.40, while $\nu$ correlates equally with
$\log\,T_{\rm eff}$ and with $\log\,g$ with a moderate value of 0.32. All
other correlations are much smaller. Hence, the regression coefficients in a
model with multiple predictors should be interpreted as the effect of change
by one unit in a predictor, while all others remain constant, in this case for
the three surviving predictors $\log\,T_{\rm eff}$, $\log\,g$, and $\nu$. A
graphical perspective on the predictions for vbroadH from vbroadG, vsiniH, and
the three covariates is shown in Fig. 8, using Eqns. (9)–(10).

###  5.4 Results for the Gaia DR3 $\gamma\,$Dor pulsators

Figure 9:  Gaia DR3 measurements of vbroad versus each of the five covariates
as indicated for the 1775 $\gamma\,$Dor (grey triangles) and 190 SPB (cyan
squares) stars having these quantities available. The lower right panel shows
the standard deviation of vbroad as a function of the dominant g-mode
amplitude. When invisible, the errors are smaller than the symbol sizes.

Armed with the knowledge that vbroadG and vbroadH are equal for the 37 bona
fide $\gamma\,$Dor stars to within their measurement uncertainties from high-
resolution and Gaia RVS spectra, and with the predictive model for these
quantities given in Table 2, we now look at the sample of 11 636 Gaia DR3
$\gamma\,$Dor stars. For all of those, full information is available on
$\log\,T_{\rm eff}$, $\log\,g$, $\log\,(L/{\cal L}_{\odot})$, $\nu$, and
$A_{\nu}$, deduced in one homogeneous way from Gaia DR3 following Paper I. For
100 of these stars, both vbroadG (as of now again simplified to vbroad) and
vsini-esphs are recorded. These are the so-called completers of the Gaia DR3
$\gamma\,$Dor stars. For 1775 $\gamma\,$Dor stars there is a measurement on
vbroad but not on vsini-esphs and for 384 vsini-esphs information is available
but vbroad is missing. For the remaining 9577 stars both of these line
broadening quantities are missing.

Table 3: Estimates (standard errors) for the model parameters of the errors-in-variables model, relating vbroad to vsini-esphs, on the 100 completers within the Gaia DR3 set of $\gamma$ Dor stars.  Effect | Par. | Estimate (s.e.) |  $p$-value | 95% conf. int.
---|---|---|---|---
Intercept | $\beta_{0}$ |  $-$5(3) |  | [$-$12;2]
Slope | $\beta_{1}$ | 0.90(0.05) |  $<$0.0001 | [0.81;0.99]
Res. var. | $\sigma^{2}$ | 0(4) | 0.98 | [$-$7;7]

When considering the 100 completers only, we again conclude that vbroad and
vsini-esphs are identical within the bounds specified by the measurement
errors, given that the slope parameter is roughly equal to unity and the
residual variance is not different from zero (see Table 3). Figure 7 already
showed that the two variables vbroad and vsini-esphs are similar for the
$\gamma\,$Dor stars with both estimates, while graphically revealing the
different meaning of their uncertainty regions. The grey dashed line in that
figure represents the regression model in Table 3.

For the full Gaia DR3 $\gamma\,$Dor data set, the relationship between vbroad
and vsini-esphs can also be assessed using all stars after performing multiple
imputation. For this application we drew ten imputations using information on
vbroad, vsini-esphs, their standard errors, and the covariates $\log\,T_{\rm
eff}$, $\log\,g$, $\log\,(L/{\cal L}_{\odot})$, $\nu$, and $A_{\nu}$. Given
that less than 1% of stars have complete information, the relationship found
from multiple imputation is relatively different from the one found for the
100 completers. This is not surprising given the relatively large
uncertainties of the two broadening parameters and the fact that they are
rather weakly correlated with other information in the dataset. Moreover, the
very large fraction of incomplete information destabilizes the inference from
multiple imputation. These issues taken together suggest that the results of
multiple imputation are too unstable to provide trustworthy results. For these
reasons, further analysis will be based on completers only for each of the
regression applications discussed below.

We now turn to the relationship between vbroad and the predictor variables,
applying backward selection to the errors-in-variables model for the 1775
$\gamma\,$Dor stars having this quantity and the covariates $\log\,T_{\rm
eff}$, $\log\,g$, $\log\,(L/{\cal L}_{\odot}$), $\nu$, and $A_{\nu}$ (whose
regression coefficients we denote as
$\beta_{1},\beta_{2},\beta_{3},\beta_{4},\beta_{5}$, respectively). It turns
out that all these covariates are significant except the amplitude of the
dominant frequency, which has a $p-$value of 0.0662 and is thus borderline
significant. This is why we present the regression models with and without
this covariate in Table 4. Both these models explain 42% of the variance in
the measurements of vbroad. It can be seen in Table 4 that keeping $A_{\nu}$
in the model does not alter the regression coefficients of the other four
covariates. We show the measurements of vbroad as a function of each of the
covariates in Fig. 9.

Table 4: Estimates (standard errors) for the parameters of the errors-in-variables model for vbroad for the 1775 Gaia DR3 $\gamma\,$ Dor stars with measured values for this quantity, based on backward selection from the set of listed predictors. Both models explain 42% of the variance in vbroad.  Effect | Par. | Estimate (s.e) |  $p$-value | 95% conf. int.
---|---|---|---|---
With $A_{\nu}$
Intercept | $\beta_{0}$ |  $-$478(262) |  | [$-$993;$-$36]
$\log\,T_{\rm eff}$ | $\beta_{1}$ | 238(87) | 0.0065 | [67;409]
$\log\,g$ | $\beta_{2}$ |  $-$100(19) |  $<$0.0001 | [$-$138;$-$63]
$\log\,(L/{\cal L}_{\odot}$) | $\beta_{3}$ |  $-$45(15) | 0.0022 | [$-$74;$-$16]
$\nu$ | $\beta_{4}$ | 44(1) |  $<$0.0001 | [41;46]
$A_{\nu}$ | $\beta_{5}$ |  $-$192(104) | 0.0662 | [$-$397;13]
Res. var. | $\sigma^{2}$ | 397(20) |  $<$0.0001 | [358;437]
Without $A_{\nu}$
Intercept | $\beta_{0}$ |  $-$478(260) |  | [$-$988;$-$32]
$\log\,T_{\rm eff}$ | $\beta_{1}$ | 237(87) | 0.0062 | [68;407]
$\log\,g$ | $\beta_{2}$ |  $-$100(19) |  $<$0.0001 | [$-$138;$-$63]
$\log\,(L/{\cal L}_{\odot}$) | $\beta_{3}$ |  $-$45(15) | 0.0023 | [$-$73;$-$16]
$\nu$ | $\beta_{4}$ | 44(1) |  $<$0.0001 | [41;47]
Res. var. | $\sigma^{2}$ | 395(20) |  $<$0.0001 | [356;435]

As discussed in Sect. 5.1, vsini-esphs is based on the averaged BP/RP (and
averaged RVS spectrum when available) and therefore has smaller uncertainty
than vbroad whose uncertainty interval represents the time-dependent line
broadening covered by at least six snapshot spectra. By construction, vsini-
esphs is expected to be a better representative of the time-independent
surface rotation velocity of the star than vbroad. Indeed, the latter quantity
approximates the overall time-dependent spectral line broadening due to
various phenomena acting together because it was computed as the median value
from individual transits taken at different epochs and treated as such by the
MTA.

To test if vsini-esphs and vbroad indeed capture different astrophysical
information, we repeat the same backward model selection for vsini-esphs,
considering the same covariates for the 384 $\gamma\,$Dor stars having a
measurement of vsini-esphs. This leads to the successive removal of $A_{\nu}$,
$\log(L/{\cal L}_{\odot})$, and $\log\,T_{\rm eff}$ as being insignificant.
The coefficients of the resulting regression model are listed in Table 5,
while the plots of the measurements of vsini-esphs as a function of each of
the covariates are included in Appendix A (Fig. 10, to be compared with Fig.
9). We find that vsini-esphs does not depend on the effective temperature and
the luminosity, while the surface gravity and dominant frequency remain
significant covariates. These two covariates offer the same dependence for
vsini-esphs as for vbroad, that is lower $\log\,g$ (a more evolved star) and
higher $\nu$ give larger line broadening. As far as $\nu$ is concerned, this
is well understood in terms of an asteroseismic interpretation and in
agreement with the findings based on the HERMES spectroscopy by Van Reeth et
al. (2015) for the bona fide $\gamma\,$Dor stars. Indeed, a higher dominant
g-mode frequency in the inertial frame of an observer corresponds to a faster
rotating star (Van Reeth et al., 2015, 2016; Pápics et al., 2017, for
galleries of Kepler light curves and frequency spectra as a function of
rotation frequency). Hence, higher asteroseismic $\nu$ is a signature of
faster stellar rotation and thus of larger line broadening, irrespective of
whether one considers vsini-esphs or vbroad.

While the resulting regression model for vsini-esphs of 384 class members
explains only 5% of the variance in that quantity, it is 42% for vbroad of the
1775 stars having this quantity. Thus the time-independent projected
rotational velocity represented by vsini-esphs of the $\gamma\,$Dor stars is
independent of their effective temperature and luminosity, while inversely
proportional to (but only weakly dependent on) their gravity. On the other
hand, the time-dependent quantity vbroad does connect to the effective
temperature of the $\gamma\,$Dor stars, such that the hotter the star, the
larger vbroad. Our astrophysical interpretation of these findings connects
well to the excitation mechanisms and to the level of line broadening found
for $\gamma\,$Dor stars in the literature. Indeed, since the Kepler data
allowed for detailed asteroseismic modelling, we know that the dominant modes
of the bona fide g-mode pulsators are dipole prograde modes and that these
stars occupy a narrow range in mass, namely $[1.3;1.9]\,$M⊙, while they cover
the entire main sequence (Mombarg et al., 2019, 2021). This pulsation class
thus has stars with quite a broad range of $\log\,g$ and radii (cf. Fig. 4).
The variability in $\log\,T_{\rm eff}$ and $\log\,g$ revealed among the class
members is thus mainly a signature of evolutionary status.

The regression models for vbroad and vsini-esphs reveal more evolved stars to
have larger spectral line broadening, while maximal time-dependent line
broadening occurs for $T_{\rm eff}$ between 6500 and 7500 K (cf. the grey
triangles in the upper left panel of Fig. 9). This is precisely the
temperature range where Grassitelli et al. (2015a) found a maximal effect of
turbulent pressure in the stellar envelope of evolved A- and F-type dwarfs,
offering an additional mechanism to excite high-order eigenmodes in such
objects, aside from the classical $\kappa$ mechanism being active in the
hotter $\gamma\,$Dor stars and flux blocking at the bottom of the convective
envelope causing such g modes in the cool class members (Guzik et al., 2000;
Dupret et al., 2005; Xiong et al., 2016). In addition, Tkachenko et al. (2020)
already showed that ignoring the turbulent pressure in stellar atmosphere
models affects estimation of microturbulent broadening and results in an
overestimation of the effective temperature at a few % level. Moreover, the
authors found this effect to get worse as the star evolves, that is for
decreasing $\log\,g$. We thus conclude to have found observational evidence
from Gaia DR3 vbroad measurements that time-dependent macroturbulent spectral
line broadening in these stars is connected with their excited g modes and/or
surface gravity, in addition to surface rotation. The amplitude limitation
from Gaia DR3 and the comparative distributions of the dominant amplitudes and
frequencies between the Gaia DR3 and bona fide $\gamma\,$Dor pulsators (cf.
left panels of Figs 5 and 6) suggest that the detected dominant frequencies
are due to large-scale (i.e., low-degree) gravito-inertial modes. The
interplay of the dominant g mode with the rotation of the star, along with
variability in $\log\,T_{\rm eff}$ and $\log\,g$ due to poor treatment of
turbulent pressure in the line-forming region, explain the overall spectral
line broadening estimates from Gaia DR3.

We point out that the regression model for vbroad in Table 4 explains 42% of
the measured variance in the spectral line broadening, while it was 58% for
the bona fide pulsators. Both these results are readily understood given that
we are dealing with multiperiodic g-mode pulsators. Indeed, $\gamma\,$Dor
pulsators have tens of high-order low-degree g modes active simultaneously,
whichever of the three excitation mechanisms is dominant (Van Reeth et al.,
2015). The line broadening captures the collective effect of all these modes
together (Aerts et al., 2009). Yet, the frequencies of the excited g modes in
addition to the dominant one were not included in the regression model,
because the Gaia light curves currently do not provide sufficient data to
unravel the multiperiodic oscillations active in these stars. While the
frequencies and amplitudes of the second strongest variability signal were
determined in Paper I, it was found that quite a number of those frequencies
cannot be distinguished from frequencies above 3 d-1 that may result from
instrumental effects. That is why we did not use these secondary frequencies
from DR3. It is to be anticipated that improved regression models explaining a
higher fraction of the variance in the spectral line broadening will become
possible from DR4 and particularly DR5, because the longer time base and
doubling of the number of epochs in the Gaia photometry will allow to unravel
several additional g-mode frequencies, particularly when combined with
additional light curves dedicated to asteroseismology as illustrated from
combined Hipparcos and TESS or ground-based data (cf. Waelkens et al., 1998;
De Cat et al., 2007; Cuypers et al., 2009). Yet, Gaia’s sampling is too sparse
to deliver all the modes active in these multiperiodic g-mode pulsators, while
they do contribute to the overall broadening of the spectral lines (cf.
Mathias et al., 1994, for the theoretical expression of the spectral line
width due to multiperiodic non-radial oscillations). The fraction of the
variance explained by regression models relying on the fundamental parameters
and the significant frequencies in Gaia light curves will therefore always be
limited, even for the bona fide class members. In this sense, the 42% reached
for the model in Table 4 is high.

Table 5: Estimates (standard errors) for the parameters of the errors-in-variables model for vsini-esphs for the 384 Gaia DR3 $\gamma\,$ stars with measured values for this quantity, based on backward selection from a set of predictors. Effect | Par. | Estimate (s.e.) |  $p$-value | 95% conf. int.
---|---|---|---|---
Intercept | $\beta_{0}$ | 198(56) |  | [88;308]
$\log\,g$ | $\beta_{2}$ |  $-$37(15) | 0.0161 | [$-$66;$-$7]
$\nu$ | $\beta_{4}$ | 17(6) | 0.0047 | [5;28]
Res. var. | $\sigma^{2}$ | 599(152) | 0.0002 | [296;901]

###  5.5 Results for the Gaia DR3 SPB pulsators

We now repeat the same analyses for the 3426 new Gaia DR3 SPB stars. Among
these, both vbroad and vsini-esphs are recorded for 156 stars. For 34 of them
there is a measurement on vbroad but not on vsini-esphs, while for 948 vsini-
esphs is available but vbroad is missing. For the remaining 2288 SPB stars
both of these are missing. In line with the arguments provided in Section 5.4,
attention is restricted to an analysis of the completers to test relationships
for vsini-esphs and vbroad.

Table 6: Estimates (standard errors) for the model parameters of the errors-in-variables model, relating vbroad to vsini-esphs, on 156 completers within the SPB sample.  Effect | Par. | Estimate (s.e.) |  $p$-value | 95% conf. int.
---|---|---|---|---
Intercept | $\beta_{0}$ | -11(5) |  | [$-$20;$-$2]
Slope | $\beta_{1}$ | 1.06(0.05) |  $<$0.0001 | [0.97;1.15]
Res. var. | $\sigma^{2}$ | 25(30) | 0.41 | [$-$35;84]

The results of the test whether or not the two measures for the spectral line
broadening are equal are given in Table 6 and shown graphically in Fig. 7
(cyan symbols). Just like for the Gaia DR3 $\gamma\,$Dor sample, we again have
a relationship that is consistent with the hypothesis that vbroad and vsini-
esphs are identical, keeping in mind the errors for vsini-esphs and the
standard deviation for vbroad.

Next, the relationship between vbroad and the candidate predictors is examined
relying on the 190 SPB stars having these data, once again applying backward
selection to the errors-in-variables model. All covariates are again
significant. The regression model in Table 7 explains 21% of the variance in
vbroad. Backward selection applied to vsini-esphs for the 1104 SPB stars with
this quantity reveals $T_{\rm eff}$ and $\nu$ to be significant predictors,
with a regression model explaining 8% of the variance. For the regression
coefficient of $\nu$, we assign the same astrophysical interpretation as
before that higher $\nu$ corresponds to faster surface rotation following the
SPB studies by Pápics et al. (2017); Pedersen et al. (2021); Pedersen (2022a).

As for the role of the effective temperature in vbroad and vsini-esphs, Pápics
et al. (2017) already found evidence that hotter SPB stars are more massive
and tend to rotate faster. The $T_{\rm eff}$ dependence revealed is thus in
first instance a dependence on stellar mass rather than stellar evolution as
we found for the $\gamma\,$Dor stars. This result is as expected, given that
SPB stars cover a factor 3 in mass, from 3 M⊙ to 9 M⊙. Pedersen et al. (2021)
placed the observed properties of the 26 bona fide Kepler SPB stars included
here and those studied from the ground by De Cat & Aerts (2002) into the
context of stellar evolution theory. This showed large diversity of $\nu$ and
$A_{\nu}$ in terms of the spectroscopic $\log\,T_{\rm eff}$ and $\log\,g$, as
well as $\log(L/{\cal L}_{\odot})$ from Gaia DR2. This diversity was
interpreted as due to the range in mass and rotation rate, the latter covering
from almost zero to almost critical rotation for the Kepler sample (Aerts et
al., 2021; Pedersen, 2022a). Despite the limited predictive power of
regression model in Table 7, it reveals that larger line broadening occurs for
hotter and/or more evolved SPB stars with higher dominant g-mode frequencies
(cf. Fig. 9). This highlights that hotter younger stars have faster rotation,
shifting their g modes further into the gravito-inertial regime of the
frequency spectrum than those of slower rotators (see Aerts et al., 2019, for
a discussion of the various frequency regimes of waves connected to the
dominant restoring forces).

Table 7: Estimates (standard errors) for the model parameters of the errors-in-variables model for vbroad of the 190 SPB stars having a measurement for this quantity, based on backward selection from the set of predictors. The model explains 21% of the variability in the data. Effect | Par. | Estimate (s.e.) |  $p$-value | 95% conf. int.
---|---|---|---|---
Intercept | $\beta_{0}$ |  $-$2764(643) |  | [$-$4033;$-$1495]
$\log\,T_{\rm eff}$ | $\beta_{1}$ | 963(219) |  $<$0.0001 | [532;1395]
$\log\,g$ | $\beta_{2}$ |  $-$213(51) |  $<$0.0001 | [$-$312;$-$113]
$\log(L/{\cal L}_{\odot})$ | $\beta_{3}$ |  $-$109(42) | 0.0105 | [$-$192;$-$26]
$\nu$ | $\beta_{4}$ | 34(8) | 0.0001 | [19;50]
$A_{\nu}$ | $\beta_{5}$ |  $-$1492(715) | 0.0384 | [$-$2903;$-$81]
Res. var. | $\sigma^{2}$ | 1909(300) |  $<$0.0001 | [1317;2501]
Table 8: Estimates (standard errors) for the model parameters of the errors-in-variables model for vsini-esphs based on the 1104 SPB stars in the sample having a measurement for this quantity, based on backward selection from a set of predictors. The model explains 8% of the variability in the data. Effect | Par. | Estimate (s.e.) |  $p$-value | 95% conf. int.
---|---|---|---|---
Intercept | $\beta_{0}$ |  $-$573(319) |  | [$-$1204;58]
$\log\,T_{\rm eff}$ | $\beta_{1}$ | 157(80) | 0.0496 | [0;315]
$\nu$ | $\beta_{4}$ | 20(6) | 0.0016 | [8;32]
Res. var. | $\sigma^{2}$ | 745(154) |  $<$0.0001 | [442;1049]

The luminosity, $\log(L/{\cal L}_{\odot})$, now also has a significant
contribution as predictor for vbroad with a $p-$value of 0.0105. The model
reveals that less luminous SPB stars have higher line broadening but its
regression coefficient is not very accurate. Moreover, the sample of SPB stars
with a measurement of vbroad is an order of magnitude smaller than for the
$\gamma\,$Dor stars and is skewed towards low-mass class members, limiting
this interpretation to only a small part of the SPB instability region. This
is graphically illustrated in Fig. 9, where trends reveal that more evolved
and more luminous SPB stars have higher vbroad but are not well enough
represented in membership to have an equally important effect on the
regression model than the cool class members. Moreover, the luminosity panel
in Fig. 9 reveals more of a quadratic than linear trend for the SPB stars. As
already highlighted above and unlike for the $\gamma\,$Dor stars, the
luminosity of SPB pulsators is mostly determined by their mass rather than by
their evolutionary stage as for the $\gamma\,$Dor stars. This, along with the
relatively large scatter for $\log\,g$ and for the effective temperature, as
well as the lower fraction of the variance explained by the linear regression
model for vbroad, makes the distillation of a simple astrophysical
interpretation for vbroad more difficult for SPB stars. This conclusion is by
itself fully in line with the diversity in pulsational behaviour occurring in
the Kepler sample of bona fide SPB pulsators (Pedersen et al., 2021; Pedersen,
2022b).

Finally, we stress that time-dependent macroturbulent spectral line broadening
due to the g modes of SPB pulsators has already been found in several of the
brightest class members (Aerts et al., 2014b), with values in agreement with
those we find here for the new faint Gaia DR3 class members. Moreover, the
density of modes excited by the $\kappa\,$mechanism peaks in the lower part of
the instability strip, near 13 000 K (Pápics et al., 2017). These pulsators
have a mass regime where the interpretation of turbulent pressure exciting
extra high-order g modes in addition to the classical $\kappa\,$mechanism does
not hold (Grassitelli et al., 2015b). Macroturbulence in these pulsators has
already been established as a time-independent downgraded quantity
representing their dominant tangential pulsational velocity by Aerts et al.
(2009) and Aerts et al. (2014a).

##  6 Discussion and conclusions

Thanks to the homogeneous treatment of its multitude of observations and its
large scale survey capacity, the Gaia mission has its role to play for
gravito-inertial asteroseismology. First of all, its photometric light curves
allow to discover thousands of new g-mode pulsators belonging to the classes
of the F-type $\gamma\,$Dor stars or SPB-type stars. Secondly, its broadening
parameter vbroad contains astrophysical information on stellar oscillations
having mmag-level observed amplitudes. We found those results after
reassigning a fraction of 22% of the $\gamma\,$Dor candidates as SPB pulsators
according to their effective temperature being above 8500 K, a property not
taken into account in the variability classifications used in Paper I.

We find the two samples of new Gaia DR3 g-mode pulsators to have similar
fundamental parameters than those of bona fide class members, although the
Gaia SPB pulsators only cover the cooler and less massive class members. We
studied the astrophysical properties of the new $\gamma\,$Dor and SPB
pulsators from regression models built upon the principle of errors-in-
variables, with their fundamental parameters and dominant oscillation
properties as predictors of the overall spectral line broadening. The Gaia DR3
quantity vsini-esphs offers a good estimate of the overall time-independent
spectral line broadening, reflecting that the surface rotation of the stars in
our samples is the dominant line broadening mechanism. All regression models
revealed the dominant g-mode frequency to be a significant predictor of the
Gaia DR3 vbroad parameter and its standard deviation, which together represent
the overall time-dependent spectral line broadening.

We explicitly checked via re-analyses of all regression models that none of
the astrophysical interpretations change if we use the effective temperature
of 9500 K as treshold for the reclassifications among the $\gamma\,$Dor and
SPB candidates. Such a treshold temperature follows from instability
computations by Szewczuk & Daszyńska-Daszkiewicz (2017) for the cool border of
galactic rotating SPB stars instead of the adopted 8500 K based on the hot
border for the $\gamma\,$Dor instability strip by Xiong et al. (2016). We find
from the upper left panel of Fig. 9 that Gaia DR3 shows 9500 K to be a less
natural and more abrupt treshold temperature between the two classes than the
adopted 8500 K. Nevertheless, using 9500 K still gives compliance of class
populations with the IMF and almost all the coefficients obtained for the
regression models remain within the uncertainty ranges listed in the tables we
provided here using 8500 K as treshold.

Despite the limiting resolution of the RVS spectroscopy, the line broadening
of rotating g-mode pulsators offered by Gaia is in full agreement with results
of well-known class members observed with high-precision space photometry and
high-resolution spectroscopy. In particular, we find the dominant g-mode
frequency to be a significant predictor of the overall line broadening. This
supports earlier findings that macroturbulence is merely a simplified time-
independent approximation for the true velocity fields at the stellar surface
that cause line-profile variability for g-mode pulsators (Aerts et al.,
2014b). We conclude that the combined effect of surface rotation and
tangential velocities resulting from multiperiodic g modes can be estimated
from vbroad for the case of main-sequence stars of intermediate mass. The
regression models for vbroad are fully in line with various excitation
predictions for g modes in $\gamma\,$Dor and SPB pulsators.

Given that the regression models based on the fundamental parameters and on
the dominant pulsation mode presented in Sect. 5 explain part of the
variability of vbroad, it is sensible to also consider the standard deviation
of vbroad as measured quantity and to check if its variance can be predicted
by any of the covariates. Indeed, aside from being caused by noise, it may
partially represent the time-dependence of the line broadening. From our
regression analyses (presented in Appendix B) we conclude that the noise
contribution to the standard deviation of vbroad is dominant over intrinsic
line-profile variability for the $\gamma\,$Dor pulsators. For the SPB stars,
the standard deviation of vbroad does relate to their surface rotation,
effective temperature, and g-mode frequency at the level of 20% variance
reduction for the regression model based on these three covariates.

Finally, we conclude that our analyses of $\sim\\!15\,000$ new Gaia DR3 g-mode
pulsators bring the qualitative results on vbroad by Frémat et al. (2022) in
full agreement with our quantitative assessments on macroturbulence in g-mode
pulsators, as already suggested by the simulation study in Aerts et al.
(2009).

###### Acknowledgements.

The research leading to these results has received funding from the KU Leuven
Research Council (grant C16/18/005: PARADISE). CA and JDR acknowledge support
from the BELgian federal Science Policy Office (BELSPO) through a PRODEX grant
for the ESA space mission Gaia. CA acknowleges financial support from the
Research Foundation Flanders under grant K802922N (Sabbatical leave). CA and
GM are grateful for the kind hospitality offered by the staff of the Center
for Computational Astrophysics at the Flatiron Institute of the Simons
Foundation in New York City during their work visit in the fall of 2022\. The
authors thank the referee for the suggestion to investigate the sensitivity of
the results to the temperature treshold used to reclassify the g-mode
pulsators. They also acknowledge Dominic Bowman and Andrew Tkachenko for
valuable comments which helped to improve the manuscript.

## References

  * Aerts (2021) Aerts, C. 2021, Reviews of Modern Physics, 93, 015001
  * Aerts et al. (2021) Aerts, C., Augustson, K., Mathis, S., et al. 2021, A&A, 656, A121
  * Aerts et al. (2010) Aerts, C., Christensen-Dalsgaard, J., & Kurtz, D. W. 2010, Asteroseismology, Springer-Verlag Heidelberg
  * Aerts et al. (2004) Aerts, C., Cuypers, J., De Cat, P., et al. 2004, A&A, 415, 1079
  * Aerts & De Cat (2003) Aerts, C. & De Cat, P. 2003, Space Sci. Rev., 105, 453
  * Aerts et al. (1999) Aerts, C., De Cat, P., Peeters, E., et al. 1999, A&A, 343, 872
  * Aerts et al. (1992) Aerts, C., De Pauw, M., & Waelkens, C. 1992, A&A, 266, 294
  * Aerts et al. (2019) Aerts, C., Mathis, S., & Rogers, T. M. 2019, ARA&A, 57, 35
  * Aerts et al. (2014a) Aerts, C., Molenberghs, G., Kenward, M. G., & Neiner, C. 2014a, ApJ, 781, 88
  * Aerts et al. (2018) Aerts, C., Molenberghs, G., Michielsen, M., et al. 2018, ApJS, 237, 15
  * Aerts et al. (2009) Aerts, C., Puls, J., Godart, M., & Dupret, M. A. 2009, A&A, 508, 409
  * Aerts & Rogers (2015) Aerts, C. & Rogers, T. M. 2015, ApJ, 806, L33
  * Aerts et al. (2014b) Aerts, C., Simón-Díaz, S., Groot, P. J., & Degroote, P. 2014b, A&A, 569, A118
  * Antoci et al. (2019) Antoci, V., Cunha, M. S., Bowman, D. M., et al. 2019, MNRAS, 490, 4040
  * Bailer-Jones et al. (2018) Bailer-Jones, C. A. L., Rybizki, J., Fouesneau, M., Mantelet, G., & Andrae, R. 2018, AJ, 156, 58
  * Balona (1986) Balona, L. A. 1986, MNRAS, 219, 111
  * Balona & Ozuyar (2020) Balona, L. A. & Ozuyar, D. 2020, MNRAS, 493, 5871
  * Baluev (2008) Baluev, R. V. 2008, MNRAS, 385, 1279
  * Borucki et al. (2010) Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977
  * Bowman et al. (2019) Bowman, D. M., Aerts, C., Johnston, C., et al. 2019, A&A, 621, A135
  * Bowman et al. (2020) Bowman, D. M., Burssens, S., Simón-Díaz, S., et al. 2020, A&A, 640, A36
  * Briquet & Aerts (2003) Briquet, M. & Aerts, C. 2003, A&A, 398, 687
  * Briquet et al. (2003) Briquet, M., Aerts, C., Mathias, P., Scuflaire, R., & Noels, A. 2003, A&A, 401, 281
  * Burssens et al. (2020) Burssens, S., Simón-Díaz, S., Bowman, D. M., et al. 2020, A&A, 639, A81
  * Claveria et al. (2019) Claveria, R. M., Mendez, R. A., Silva, J. F., & Orchard, M. E. 2019, PASP, 131, 084502
  * Clementini et al. (2022) Clementini, G., Ripepi, V., Garofalo, A., et al. 2022, A&A, in press, arXiv:2206.06278
  * Creevey et al. (2022) Creevey, O. L., Sordo, R., Pailler, F., et al. 2022, A&A, in press, arXiv:2206.05864
  * Cuypers et al. (2009) Cuypers, J., Aerts, C., De Cat, P., et al. 2009, A&A, 499, 967
  * Daszyńska-Daszkiewicz et al. (2017) Daszyńska-Daszkiewicz, J., Pamyatnykh, A. A., Walczak, P., et al. 2017, MNRAS, 466, 2284
  * De Cat & Aerts (2002) De Cat, P. & Aerts, C. 2002, A&A, 393, 965
  * De Cat et al. (2000) De Cat, P., Aerts, C., De Ridder, J., et al. 2000, A&A, 355, 1015
  * De Cat et al. (2007) De Cat, P., Briquet, M., Aerts, C., et al. 2007, A&A, 463, 243
  * De Cat et al. (2005) De Cat, P., Briquet, M., Daszyńska-Daszkiewicz, J., et al. 2005, A&A, 432, 1013
  * De Cat et al. (2006) De Cat, P., Eyer, L., Cuypers, J., et al. 2006, A&A, 449, 281
  * De Pauw et al. (1993) De Pauw, M., Aerts, C., & Waelkens, C. 1993, A&A, 280, 493
  * De Ridder et al. (2016) De Ridder, J., Molenberghs, G., Eyer, L., & Aerts, C. 2016, A&A, 595, L3
  * Dupret et al. (2005) Dupret, M. A., Grigahcène, A., Garrido, R., Gabriel, M., & Scuflaire, R. 2005, A&A, 435, 927
  * Edelmann et al. (2019) Edelmann, P. V. F., Ratnasingam, R. P., Pedersen, M. G., et al. 2019, ApJ, 876, 4
  * Eyer et al. (2022) Eyer, L., Audard, M., Holl, B., et al. 2022, A&A, in press, arXiv:2206.06416
  * Eyer & Mowlavi (2008) Eyer, L. & Mowlavi, N. 2008, in Journal of Physics Conference Series, Vol. 118, Journal of Physics Conference Series, 012010
  * Frémat et al. (2022) Frémat, Y., Royer, F., Marchal, O., & Blomme, R. e. a. 2022, A&A, in press, arXiv:2206.10986
  * Gaia Collaboration et al. (2016a) Gaia Collaboration, Brown, A. G. A., Vallenari, A., Prusti, T., & de Bruijne, J. H. J. e. a. 2016a, A&A, 595, A2
  * Gaia Collaboration et al. (2021) Gaia Collaboration, Brown, A. G. A., Vallenari, A., & Prusti, T. e. a. 2021, A&A, 649, A1
  * Gaia Collaboration et al. (2022a) Gaia Collaboration, De Ridder, J., Ripepi, V., & Aerts, C. a. 2022a, A&A, in press (Paper I), arXiv:2206.06075
  * Gaia Collaboration et al. (2019) Gaia Collaboration, Eyer, L., Rimoldini, L., & Audard, M. e. a. 2019, A&A, 623, A110
  * Gaia Collaboration et al. (2016b) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., Brown, A. G. A., & Vallenari, A. e. a. 2016b, A&A, 595, A1
  * Gaia Collaboration et al. (2022b) Gaia Collaboration, Vallenari, A., Brown, A. G. A., Prusti, T., & de Bruijne, J. H. J. e. a. 2022b, A&A, in press, arXiv:2208.00211
  * Gebruers et al. (2021) Gebruers, S., Straumit, I., Tkachenko, A., et al. 2021, A&A, in press, arXiv:2104.04521
  * Gilliland et al. (2010) Gilliland, R. L., Brown, T. M., Christensen-Dalsgaard, J., et al. 2010, PASP, 122, 131
  * Grassitelli et al. (2015a) Grassitelli, L., Fossati, L., Langer, N., et al. 2015a, A&A, 584, L2
  * Grassitelli et al. (2015b) Grassitelli, L., Fossati, L., Simón-Diáz, S., et al. 2015b, ApJ, 808, L31
  * Guzik et al. (2000) Guzik, J. A., Kaye, A. B., Bradley, P. A., Cox, A. N., & Neuforge, C. 2000, ApJ, 542, L57
  * Handler (1999) Handler, G. 1999, MNRAS, 309, L19
  * Holl et al. (2018) Holl, B., Audard, M., Nienartowicz, K., Jevardat de Fombelle, G., & Marchal, O. e. a. 2018, A&A, 618, A30
  * Horst et al. (2020) Horst, L., Edelmann, P. V. F., Andrássy, R., et al. 2020, A&A, 641, A18
  * Katz et al. (2022) Katz, D., Sartoretti, P., Guerrier, A., et al. 2022, arXiv e-prints, arXiv:2206.05902
  * Kaye et al. (1999) Kaye, A. B., Handler, G., Krisciunas, K., Poretti, E., & Zerbi, F. M. 1999, PASP, 111, 840
  * Keen et al. (2015) Keen, M. A., Bedding, T. R., Murphy, S. J., et al. 2015, MNRAS, 454, 1792
  * Kurtz et al. (2014) Kurtz, D. W., Saio, H., Takata, M., et al. 2014, MNRAS, 444, 102
  * Lebzelter et al. (2022) Lebzelter, T., Mowlavi, N., Lecoeur-Taibi, I., et al. 2022, A&A, in press, arXiv:2206.05745
  * Li et al. (2019) Li, G., Bedding, T. R., Murphy, S. J., et al. 2019, MNRAS, 482, 1757
  * Li et al. (2020) Li, G., Van Reeth, T., Bedding, T. R., et al. 2020, MNRAS, 491, 3586
  * Marton et al. (2022) Marton, G., Ábrahám, P., Rimoldini, L., et al. 2022, A&A, in press, arXiv:2206.05796
  * Mathias et al. (2001) Mathias, P., Aerts, C., Briquet, M., et al. 2001, A&A, 379, 905
  * Mathias et al. (1994) Mathias, P., Aerts, C., De Pauw, M., Gillet, D., & Waelkens, C. 1994, A&A, 283, 813
  * Mathias et al. (2004) Mathias, P., Le Contel, J. M., Chapellier, E., et al. 2004, A&A, 417, 189
  * Molenberghs & Kenward (2007) Molenberghs, G. & Kenward, M. 2007, Missing Data in Clinical Studies, New York: John Wiley
  * Mombarg et al. (2021) Mombarg, J. S. G., Van Reeth, T., & Aerts, C. 2021, A&A, 650, A58
  * Mombarg et al. (2019) Mombarg, J. S. G., Van Reeth, T., Pedersen, M. G., et al. 2019, MNRAS, 485, 3248
  * Moravveji (2016) Moravveji, E. 2016, MNRAS, 455, L67
  * Moravveji et al. (2016) Moravveji, E., Townsend, R. H. D., Aerts, C., & Mathis, S. 2016, ApJ, 823, 130
  * Murphy et al. (2016) Murphy, S. J., Fossati, L., Bedding, T. R., et al. 2016, MNRAS, 459, 1201
  * Pápics et al. (2017) Pápics, P. I., Tkachenko, A., Van Reeth, T., et al. 2017, A&A, 598, A74
  * Pedersen (2022a) Pedersen, M. G. 2022a, ApJ, in press, arXiv:2208.14497
  * Pedersen (2022b) Pedersen, M. G. 2022b, ApJ, 930, 94
  * Pedersen et al. (2021) Pedersen, M. G., Aerts, C., Pápics, P. I., et al. 2021, Nature Astronomy, 5, 715
  * Pedersen et al. (2019) Pedersen, M. G., Chowdhury, S., Johnston, C., et al. 2019, ApJ, 872, L9
  * Raskin et al. (2011) Raskin, G., van Winckel, H., Hensberge, H., et al. 2011, A&A, 526, A69
  * Rimoldini et al. (2022) Rimoldini, L., Holl, B., Gavras, P., et al. 2022, A&A, in press, arXiv:2211.17238
  * Ripepi et al. (2022) Ripepi, V., Clementini, G., Molinaro, R., et al. 2022, A&A, in press, arXiv:2206.06212
  * Rogers et al. (2013) Rogers, T. M., Lin, D. N. C., McElwaine, J. N., & Lau, H. H. B. 2013, ApJ, 772, 21
  * Saio et al. (2018) Saio, H., Kurtz, D. W., Murphy, S. J., Antoci, V. L., & Lee, U. 2018, MNRAS, 474, 2774
  * Saio et al. (2015) Saio, H., Kurtz, D. W., Takata, M., et al. 2015, MNRAS, 447, 3264
  * Salpeter (1955) Salpeter, E. E. 1955, ApJ, 121, 161
  * Sartoretti et al. (2022) Sartoretti, P., Blomme, R., David, M., & Seabroke, G. 2022, Gaia DR3 documentation Chapter 6: Spectroscopy, Gaia DR3 documentation, European Space Agency; Gaia Data Processing and Analysis Consortium.
  * SAS Institute Inc. (2014) SAS Institute Inc. 2014, SAS OnlineDoc 9.4, Cary, NC: SAS Institute Inc.
  * Sekaran et al. (2021) Sekaran, S., Tkachenko, A., Johnston, C., & Aerts, C. 2021, A&A, 648, A91
  * Serebriakova et al. (2023) Serebriakova, N., Tkachenko, A., Gebruers, S., et al. 2023, A&A, submitted
  * Simón-Díaz et al. (2017) Simón-Díaz, S., Godart, M., Castro, N., et al. 2017, A&A, 597, A22
  * Simón-Díaz & Herrero (2014) Simón-Díaz, S. & Herrero, A. 2014, A&A, 562, A135
  * Simón-Díaz et al. (2010) Simón-Díaz, S., Herrero, A., Uytterhoeven, K., et al. 2010, ApJ, 720, L174
  * Szewczuk & Daszyńska-Daszkiewicz (2017) Szewczuk, W. & Daszyńska-Daszkiewicz, J. 2017, MNRAS, 469, 13
  * Tkachenko et al. (2013) Tkachenko, A., Aerts, C., Yakushechkin, A., et al. 2013, A&A, 556, A52
  * Tkachenko et al. (2020) Tkachenko, A., Pavlovski, K., Johnston, C., et al. 2020, A&A, 637, A60
  * Triana et al. (2015) Triana, S. A., Moravveji, E., Pápics, P. I., et al. 2015, ApJ, 810, 16
  * Uytterhoeven et al. (2011) Uytterhoeven, K., Moya, A., Grigahcène, A., et al. 2011, A&A, 534, A125
  * Van Reeth et al. (2018) Van Reeth, T., Mombarg, J. S. G., Mathis, S., et al. 2018, A&A, 618, A24
  * Van Reeth et al. (2016) Van Reeth, T., Tkachenko, A., & Aerts, C. 2016, A&A, 593, A120
  * Van Reeth et al. (2015) Van Reeth, T., Tkachenko, A., Aerts, C., et al. 2015, ApJS, 218, 27
  * Waelkens (1991) Waelkens, C. 1991, A&A, 246, 453
  * Waelkens et al. (1998) Waelkens, C., Aerts, C., Kestens, E., Grenon, M., & Eyer, L. 1998, A&A, 330, 215
  * Xiong et al. (2016) Xiong, D. R., Deng, L., Zhang, C., & Wang, K. 2016, MNRAS, 457, 3163
  * Zwintz et al. (2017) Zwintz, K., Van Reeth, T., Tkachenko, A., et al. 2017, A&A, 608, A103

##  Appendix A Plots of the predictors for vsini-esphs

Figure 10:  Gaia DR3 measurements of vsini-esphs versus each of the five
covariates as indicated for the 384 $\gamma\,$Dor (grey triangles) and 1104
SPB (cyan squares) stars having these quantities available. The lower right
panel shows the standard deviation of vbroad as a function of vsini-esphs for
the 100 $\gamma\,$Dor and 190 SPB stars having these quantities available.
When invisible, the errors are smaller than the symbol sizes.

##  Appendix B Regression models for the standard deviation of vbroad

The quantity vbroad can be considered a simplified estimate of the time-
dependent second moment of a line profile (Aerts et al. 1992; De Pauw et al.
1993; Mathias et al. 1994; Briquet & Aerts 2003). It is therefore meaningful
to investigate if the standard deviation of vbroad contains information on the
time-dependent line-profile variability of the Gaia DR3 g-mode pulsators.

In order to compute backward selection regression models for this quantity, we
now add vsini-esphs as sixth covariate (with $\beta_{6}$ as notation for its
regression coefficient). In line with Simón-Díaz & Herrero (2014) and Aerts et
al. (2014b), Serebriakova et al. (2023) have found from simulations that the
projected rotational velocity should first be derived from high-resolution
spectra before any meaningful derivation of left-over additional time-
dependent spectral line broadening caused by tangential velocity fields can be
done. This is in agreement with the methodology used by Aerts et al. (2014b)
to assess the quality of estimates for time-dependent macroturbulent
broadening from the line’s moment variations in well-known B-type pulsators.

Table 9: Estimates (standard errors) for the parameters of the errors-in-variables model for the standard deviation of vbroad, measured for the 1775 Gaia DR3 $\gamma\,$Dor stars. The results are based on backward selection considering a set of 6 (top) or 5 (bottom) predictors.  Effect | Par. | Estimate (s.e.) |  $p$-value | 95% conf. int.
---|---|---|---|---
With vsini-esphs — $R^{2}=0.09$
Intercept | $\beta_{0}$ |  $-$459(146) |  | [$-$750;$-$169]
$\log\,g$ | $\beta_{2}$ | 93(29) | 0.0019 | [35;151]
$\log(L/{\cal L}_{\odot})$ | $\beta_{3}$ | 85(27) | 0.0019 | [32;138]
vsini-esphs | $\beta_{6}$ | 0.26(0.08) | 0.0018 | [0.10;0.42]
Res. var. | $\sigma^{2}$ | 1122(166) |  $<$0.0001 | [793;1452]
Without vsini-esphs — $R^{2}=0.06$
Intercept | $\beta_{0}$ |  $-$517(135) |  | [$-$781;$-$253]
$\log\,T_{\rm eff}$ | $\beta_{1}$ | 136(35) | 0.0001 | [66;205]
$\log(L/{\cal L}_{\odot})$ | $\beta_{3}$ | 11(3) |  $<$0.0001 | [6;16]
$\nu$ | $\beta_{4}$ | 8(1) |  $<$0.0001 | [6;10]
$A_{\nu}$ | $\beta_{5}$ |  $-$200(88) | 0.0234 | [$-$374;$-$27]
Res. var. | $\sigma^{2}$ | 497(17) |  $<$0.0001 | [464;529]
Table 10: Estimates (standard errors) for the parameters of the errors-in-variables model for the standard deviation of vbroad, measured for the 190 Gaia DR3 SPB pulsators. The results are based on backward selection considering a set of 6 (top) or 5 (bottom) predictors.  Effect | Par. | Estimate (s.e.) |  $p$-value | 95% conf. int.
---|---|---|---|---
With vsini-esphs — $R^{2}=0.20$
Intercept | $\beta_{0}$ |  $-$515(246) |  | [$-$1001;$-$30]
$\log\,T_{\rm eff}$ | $\beta_{1}$ | 138(61) | 0.0256 | [17;258]
$\nu$ | $\beta_{4}$ |  $-$17(6) | 0.0031 | [$-$28;$-$6]
vsini-esphs | $\beta_{6}$ | 0.39(0.06) |  $<$0.0001 | [0.26;0.52]
Res. var. | $\sigma^{2}$ | 1324(156) |  $<$0.0001 | [1016;1631]
Without vsini-esphs — $R^{2}=0.01$
Intercept | $\beta_{0}$ |  $-$515(237) |  | [$-$382;$-$48]
$\log\,T_{\rm eff}$ | $\beta_{1}$ | 140(59) | 0.0185 | [24;256]
Res. var. | $\sigma^{2}$ | 1641(169) |  $<$0.0001 | [1308;1974]

Table 9 displays the regression models for the standard deviation of vbroad
for the sample of 1775 Gaia DR3 $\gamma$ Dor stars. We perform the backward
selection twice, once with the covariate vsini-esphs and once without it. For
the model including it as a potential predictor, we remove the insignificant
contributions of the variables $\log\,g$, $A_{\nu}$, and $\log(L/{\cal
L}_{\odot})$ (in that order). Not suprisingly, vsini-esphs is a significant
predictor for the standard deviation of vbroad, along with the gravity and
luminosity of the star. For the regression model without vsini-esphs, only
$\log\,g$ is removed and the predictive power decreases from 9% to 6% of the
original variance.

The combined results in Tables 3, 4, and 9 reveal that rotational broadening
of $\gamma\,$Dor stars dominates the measurement of vbroad, while the
evolutionary status (by means of the gravity and/or effective temperature)
lies at the basis of the significant predictors of vbroad. Regression models
for the standard deviation have low predictive power and are thus harder to
interpret, in line with the caveats due to the systematic uncertainties
connected with the treatments of microturbulent broadening, $\log\,g$, and
$T_{\rm eff}$ highlighted by Tkachenko et al. (2020).

As for the SPB pulsators, Table 10 displays the regression models for the
standard deviation of vbroad from backward selection for the 190 SPB stars
with such measurements. For the model with the covariate vsini-esphs included
as a potential predictor, the order of variables to be removed was $\log\,g$,
$A_{\nu}$, and $\log(L/{\cal L}_{\odot})$. The model explains 20% of the
variance in the standard deviation of vbroad and has vsini-esphs,
$\log\,T_{\rm eff}$, and $\nu$ as significant predictors. For the model
without vsini-esphs as a potential predictor, all covariates aside from
$\log\,T_{\rm eff}$ were removed as insignificant and the predictive power
disappears. Thus, we find that the standard deviation of the vbroad
measurements of SPB stars is affected by their surface rotation and is
connected to their effective temperature and dominant g-mode frequency.

The regression models for the standard deviation of vbroad have low (SPB
stars) to almost no ($\gamma\,$Dor stars) predictive power. This is not
surprising as this standard deviation is, at best, just one single value
capturing the complicated overall time-dependent broadening determined from
only a few scans across Gaia’s field of view. Our findings suggest that the
noise contribution to the standard deviation of vbroad dominates over the one
due to the astrophysical parameters for the $\gamma\,$Dor stars.