11institutetext:  Instituut voor Sterrenkunde (IvS), KU Leuven,
Celestijnenlaan 200D, B-3001 Leuven, Belgium
11email<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

# Detection of period-spacing patterns due to the gravity modes of rotating
dwarfs in the TESS southern continuous viewing zone

S. Garcia Detection of period-spacing patterns due to the gravity modes of
rotating dwarfs in the TESS southern continuous viewing zoneDetection of
period-spacing patterns due to the gravity modes of rotating dwarfs in the
TESS southern continuous viewing zone T. Van Reeth Detection of period-spacing
patterns due to the gravity modes of rotating dwarfs in the TESS southern
continuous viewing zoneDetection of period-spacing patterns due to the gravity
modes of rotating dwarfs in the TESS southern continuous viewing zone J. De
Ridder Detection of period-spacing patterns due to the gravity modes of
rotating dwarfs in the TESS southern continuous viewing zoneDetection of
period-spacing patterns due to the gravity modes of rotating dwarfs in the
TESS southern continuous viewing zone A. Tkachenko Detection of period-spacing
patterns due to the gravity modes of rotating dwarfs in the TESS southern
continuous viewing zoneDetection of period-spacing patterns due to the gravity
modes of rotating dwarfs in the TESS southern continuous viewing zone L.
IJspeert Detection of period-spacing patterns due to the gravity modes of
rotating dwarfs in the TESS southern continuous viewing zoneDetection of
period-spacing patterns due to the gravity modes of rotating dwarfs in the
TESS southern continuous viewing zone C. Aerts Detection of period-spacing
patterns due to the gravity modes of rotating dwarfs in the TESS southern
continuous viewing zoneDetection of period-spacing patterns due to the gravity
modes of rotating dwarfs in the TESS southern continuous viewing zone2233

(Accepted November 22, 2021)

###### Abstract

Context. The theory of stellar evolution presents shortcomings when confronted
with asteroseismic probes of interior physical properties. The differences
between observations and theory are often great because stellar models have
mainly been calibrated from observables connected to the surface of stars.
Period-spacing patterns caused by gravity modes are a particularly powerful
asteroseismic tool that are useful for probing the near-core rotation and
mixing of chemical elements in main-sequence stars with convective cores.

Aims. We aim to compose a catalog of intermediate-mass stars in the Transiting
Exoplanet Survey Satellite (TESS) southern continuous viewing zone (CVZ) to
reveal period-spacing patterns caused by gravity modes for use in future
asteroseismic modeling.

Methods. TESS full frame images (FFI) were inspected to select stars of
intermediate- and high-mass using color-magnitude criteria. Light curves were
extracted from custom masks per star, adopting stringent constraints on the
aperture masks and contamination. The extracted light curves were subject to
iterative prewhitening to detect gravity modes. We developed a method relying
on the assumption that period spacings are an approximately linear function of
the mode periods to build a template pattern. This template was used to
extract the patterns and their uncertainties, relying on a bootstrap approach.

Results. Our TESS catalog of high-quality period-spacing patterns is the first
of its kind and contains 140 gravity-mode patterns in 106 $\gamma\,$Dor stars
and two slowly pulsating B-type (SPB) stars. Half of these patterns contain
seven or more measured mode periods and the longest pattern contains 20 modes.
We provide the community with a convenient software tool to search for period-
spacing patterns and to process the extracted light curves.

Conclusions. Our catalog offers a fruitful starting point for future gravity-
mode asteroseismology of rotating dwarfs with convective cores in the southern
hemisphere.

###### Key Words.:

Asteroseismology – Waves – Stars: Rotation – Stars: Interiors – Stars:
oscillations (including pulsations) – Stars: catalog

##  1 Introduction

The theory of stellar structure and evolution is well established and capable
of describing the different stages of a star throughout its life in general
terms (e.g., Kippenhahn et al., 2012). However, the theory is mostly
calibrated to the surface properties of stars, such as their surface
gravities, surface chemical compositions, surface rotations, and effective
temperatures. Today, advances in asteroseismology and the advent of high-
precision space photometry from telescopes such as CoRoT (Auvergne et al.,
2009), Kepler (Koch et al., 2010) and TESS (Ricker et al., 2015) allow us to
probe stellar interiors with a precision that cannot be reached from
extrapolating surface quantities (e.g., Hekker & Christensen-Dalsgaard, 2017;
García & Ballot, 2019; Bowman, 2020; Aerts, 2021, for recent reviews).
Asteroseismic modeling based on space photometry has revealed large
discrepancies between observations and the theory of stellar structure and
evolution, such as in the transport of angular momentum (e.g., Aerts et al.,
2019).

Gravity modes (hereafter g modes) are stellar oscillations that have buoyancy
as their dominant restoring force and have optimal probing power in the near-
core regions of main-sequence stars (e.g., Aerts et al., 2018). They are
detected in main-sequence stars with a convective core and a radiative
envelope known as $\gamma$ Doradus ($\gamma$ Dor) stars (Kaye et al., 1999)
and SPB stars (Waelkens, 1991), which have masses from 1.3 to 1.9 $M_{\odot}$
and 3 to 9 $M_{\odot}$, respectively (e.g., Aerts et al., 2010, Chapter 2, for
more properties). To detect and accurately measure the frequencies of their
individual g modes, which have periodicities of the order of days, high-
precision long-term uninterrupted observations are needed. These requirements
are met for time-series photometry from space missions such as Kepler and
TESS, which allows us to detect g-mode period spacing patterns. The first such
detection in a dwarf was made for the SPB star HD 50230 from CoRoT space
photometry by Degroote et al. (2010). The five-month CoRoT light curve was
sufficient to detect a period-spacing pattern of eight dipole g modes thanks
to this star’s very slow rotation, also justifying asteroseismic modeling
having ignored the Coriolis acceleration in the pulsation computations (Wu &
Li, 2019).

Gravity-mode period spacing patterns, describing the difference in periods of
g modes with an identical spherical degree, $\ell,$ and azimuthal order, $m,$
while having a consecutive radial order, $n$, are a powerful asteroseismic
tool (Aerts et al., 2010, for a detailed theoretical derivation). This tool
allows us to probe the near-core regions of main-sequence stars with
convective core and a radiative envelope. As shown by Shibahashi (1979) and
Tassoul (1980), g-mode periods $P_{n\ell}$ are equidistant for a spherical
chemically homogeneous non-rotating star when considering the asymptotic
regime, that is with $2\pi/P_{n\ell}\ll N$ with $N$ the buoyancy frequency:

$P_{nl}=\frac{\Pi_{0}}{\sqrt{\ell(\ell+1)}}\,(n+\epsilon_{g}),$ (1)

with

$\Pi_{0}=\frac{2\pi^{2}}{\int_{r_{1}}^{r_{2}}N(r)\,r^{-1}\,\mathrm{d}r},$ (2)

where $\epsilon_{g}$ is a phase term which depends on the boundaries, $r_{1}$
and $r_{2}$, of the g-mode propagation cavity. Gradients in the stellar
chemical composition profile cause mode trapping, which introduces wave-like
behavior and periodic dips also known as buoyancy glitches in the period-
spacing patterns. As shown by Miglio et al. (2008), the amplitude and location
of these modulations in the pattern depend on the steepness and the location
of the chemical gradients inside the star.

For a rotating star, the Coriolis acceleration and the difference between the
corotating reference frames and the inertial ones of the observer shift the
g-mode frequencies quite drastically. As a result, the observed period
spacings reveal a decreasing trend for prograde ($m>0$) and zonal ($m=0$)
modes when plotted as a function of increasing mode period as observed in an
inertial frame of reference. For retrograde modes ($m<0$), on the other hand,
an overall increase in the observed spacings of modes with increasing
pulsation period occurs (e.g., Bouabid et al., 2013; Van Reeth et al., 2015a,
b; Ouazzani et al., 2017). Schmid & Aerts (2016) investigated the limit of the
rotation frequency at which the Coriolis acceleration can still be treated
perturbatively and found this approximation already breaks down for rotation
frequencies above roughly $\sim\\!0.1\,$d-1, which is the case for almost all
intermediate-mass dwarfs (cf. Aerts et al., 2017).

A strong magnetic field in the stellar interior requires the Lorentz force to
be included in the pulsation computations. This modifies the morphology of the
observed period spacing pattern by introducing a saw-tooth modulation of the
period spacing values as the consecutive pulsation periods increase (Prat et
al., 2019, 2020; Van Beeck et al., 2020). Moreover, coupling between inertial
modes in the rotating core and g modes in the envelope occurs and may cause
dips superposed to buoyancy glitches at particular mode periods in the spacing
diagram (Ouazzani et al., 2020; Saio et al., 2021; Lee, 2021). This implies
that interpretations of buoyancy glitches from mathematical analytical
descriptions ignoring the Coriolis acceleration (e.g., Cunha et al., 2019) can
only be applied to slowly-rotating non-magnetic stars. The majority of g-mode
pulsators require the modeling and interpretation of the observed period
spacing patterns to be done numerically, based on the inclusion of the
Coriolis (and perhaps Lorentz) force when solving the pulsation equations (cf.
Townsend & Teitler, 2013; Townsend et al., 2018).

After the initial discovery by Degroote et al. (2010), it took another five
years before period-spacing patterns were detected from four-year Kepler light
curves in several hundreds of $\gamma$ Dor stars (e.g., Van Reeth et al.,
2015b; Li et al., 2019, 2020) and several dozens of SPB stars (e.g., Pápics et
al., 2017; Pedersen et al., 2021; Szewczuk et al., 2021). These patterns have
been used to measure the near-core rotation rates of all these stars (e.g.,
Van Reeth et al., 2016; Christophe et al., 2018a; Van Reeth et al., 2018; Li
et al., 2019, 2020; Takata et al., 2020; Pedersen et al., 2021) and place
constraints on the chemical transport processes that take place in the deep
stellar interior (e.g., Mombarg et al., 2020, 2021; Pedersen et al., 2021).
From the point of view of improving stellar evolution theory, dwarfs are the
most interesting targets as they still have their evolved stages ahead of them
and uncertainties in the transport processes are cumulative over time.
Moreover, g modes are potentially excited along the main sequence in various
instability regions for stars born with a convective core covering a broad
mass range (Aerts et al., 2010, Chapter 3). This is why we focus our work on
dwarfs covering spectral types between O and F.

To date, the photometric observations obtained with the nominal Kepler mission
cover the longest time base and are more precise than the observations from
any other high-cadence space-photometry mission. Hence, most breakthroughs in
g-mode asteroseismology of dwarfs were achieved thanks to Kepler observations
so far. Here, we exploit the asteroseismic potential of the ongoing TESS space
mission and compare it to that of Kepler. The TESS extended mission is
gradually providing data of progressively higher frequency resolution and has
opened the door to analyzing numerous stars located in regions of the sky
other than the Kepler field of view and in different metallicity regimes. In
this work, we present results based on the first full year of uninterrupted
TESS monitoring, to evaluate its capacity for the g-mode asteroseismology of
rotating dwarfs. Future works will involve the addition of data from the
extended TESS mission to the stars in our current catalog to increase their
capacity for asteroseismic modeling.

Our study is aimed at identifying new $\gamma$ Dor or SPB stars that have been
observed by TESS in the southern continuous viewing zone (CVZ) to build the
first TESS catalog of high-quality g-mode period-spacing patterns for such
pulsators. The southern CVZ was observed uninterruptedly during the first year
of the nominal TESS mission, with a 24° square field-of-view centered at the
Southern ecliptic pole rotating about every 27 d. This long observation period
has provided light curves with a nominal frequency resolution of about
$0.003\,\mathrm{d}^{-1}$.

The paper is organized as follows. In Section 2, we describe our criteria for
selecting O/B- and A/F-type stars in the TESS southern CVZ. In Section 3, we
describe our method used to extract light curves from the TESS full frame
images, including our data analysis treatments to detrend and optimize the
extracted light curves to search for g modes. In Section 4, we discuss the
frequency extraction from the light curves and our posterior analysis. Section
5 describes our method for finding period-spacing patterns. Finally, we
discuss our final catalog of g-mode pulsators with period-spacing patterns in
Section 6.

##  2 Data set

To select our sample of stars, we started from the TESS Input Catalog (TIC)
version 8 (Stassun et al., 2019) and reduced it to the TESS southern CVZ by
imposing an ecliptic latitude $\beta\leq-72$°. To exclude extended objects and
keep only point-like sources, we used the TIC’s flag Objtype=star. Stars
likely to be white dwarfs or giants were identified with the TIC’s flag
wdflag=1 and lumclass=GIANT, respectively, and excluded from the sample. The
first flag represents a cut in absolute Gaia magnitude and Gaia color
($G_{BP}-G_{RP}$) while the second flag represents a cut in radius (calculated
from Gaia parallaxes) and $T_{\text{eff}}$. We refer to Stassun et al. (2019)
for a description of the TIC flags.

To narrow down our sample of stars to candidates of spectral type F and
hotter, that is the most likely g-mode pulsators, we used 139 TESS O/B-type
stars selected manually by Pedersen et al. (2019) and 616 Kepler A/F-type
$\gamma\,$Dor pulsators taken from Tkachenko et al. (2013); Li et al. (2020).
We placed those stars in a color-magnitude diagram and used them to define two
pairs of color-magnitude cuts that enclose 95% of these bona fide O/B- and
A/F-type stars. By applying these pairs of cuts to our TESS sample, we
extracted all the O/B- and A/F-type candidate pulsators of interest. We
calculated their absolute magnitudes as

$M=m-5\log(d)+5,$ (3)

where $m$ is the apparent magnitude in a given passband. To obtain the
distance $d$, we used the Bayesian distance estimate from Bailer-Jones et al.
(2018) reported in the TIC. To ensure reliable distances, we used stars with a
positive Gaia parallax of relative error of less than 25% that passed both
astrometric and photometric criteria given by Eqs. (1) and (2) in Arenou et
al. (2018), namely, with TIC flag gaiaqflag=1. To minimize the effect of
extinction, we used the 2MASS infrared bands $J$, $H,$ and $K$ and adopted the
cuts listed in Table 1. Figure 1 shows these cuts in $K$ and $J-K$ as straight
lines and our sample in the background. A/F-type candidates correspond to
stars in the top-left quadrant delineated by the red straight lines minus the
O/B-type candidates that correspond to stars in the top-left quadrant
delineated by the cyan straight lines. The final candidates are obtained after
an analogous additional selection in a color-magnitude diagram based on $H$
and $J-H$. Table 1 and Figure 1 do not consider corrections for extinction.
The potential contamination by cooler stars that are not expected to pulsate
in g modes will be treated in Section 4, based on the frequency analysis
results.

Figure 1: 2MASS color-magnitude diagram showing the pairs of cuts in absolute magnitude and color defining candidate g-mode pulsators of spectral type O/B (cyan straight lines) and A/F (red straight lines) in $K$ and $J-K$. The cyan and red circles are bona fide O/B- and A/F-type stars, respectively, while the retained dwarfs from the TIC in the southern CVZ are plotted in the background. The side histograms show the distribution of stars; the pairs of cuts enclose 95% of the respective bona fide stars. Table 1: Pairs of cuts in absolute magnitude and color in the 2MASS system (uncorrected for extinction) used to define O/B- and A/F-type candidates. The pairs of $K$ and $J-K$ are displayed in Figure 1. Spectral type candidate | $K$ | $J-K$ | $H$ | $J-H$
---|---|---|---|---
O/B | 1.429 | 0.06 | 1.55 | 0.045
A/F | 2.300 | 0.30 | 2.35 | 0.240

Finally, to favor a high signal-to-noise ratio (S/N) and non-contaminated flux
in the light curves, we limited our sample further to stars with apparent TESS
magnitude brighter than 15 (uncorrected for extinction) and situated at least
2 arcsec apart from other stars in the TIC. Our selected sample consists of
345 O/B-type candidates and 9369 A/F-type candidates in the TESS southern CVZ,
all from our Galaxy.

##  3 Our TESS data reduction pipeline

Figure 2: Square 20-by-20 pixels around TIC 38845463, TESS sector 1. The color
bar is common to the three panels and shows a logarithmic scale of the flux in
electrons per second. The red circle represents the target star while white
circles indicate TIC neighboring stars down to four TESS magnitudes fainter
with respect to the target star. The symbol sizes are inversely proportional
to the TESS magnitude. Declination grids are $2^{\prime}$ apart. Left: Median
image of the target pixel file. Middle: Panel with the final aperture mask
(red shade) and background mask (gray shade) overplotted. The aperture mask
results from the threshold parameter $n=20$ (see text for explanation). Right:
Best fit based on the left panel used to estimate the level of contamination
in the aperture mask due to flux from neighboring stars. The image was modeled
as six 2-D Gaussian functions plus a 2-D plane. See Section 3.1 for further
details.

Figure 3: Our star sample without constraints on the aperture mask size (7385
sources, gray histogram) and after imposing a minimum size of 9 pixels (2162
sources, blue histogram). Other constraints described in Section 3.1 apply to
both histograms. Left: Sample distribution of the median aperture mask sizes,
calculated per star over all TESS sectors. Middle: Sample distribution of the
TESS magnitudes. The dashed red line marks a magnitude value of 13.5. Right:
Median contamination level caused by neighboring stars, calculated per star
over all TESS sectors. The dashed orange line marks a contamination of 5%.

We searched sectors 1 to 13 of TESS for long-cadence (i.e., 27.4 minutes)
full-frame images available in the Mikulski Archive for Space Telescopes and
used the TESScut API (Brasseur et al., 2019) to download, for every star in
our sample, a 20X20 pixel image with the target star at the center. These
images are known as target pixel files and contain the flux information for
all available time stamps. The 20X20 pixel size was chosen such that the
target pixel file contains both the flux of the target star and the flux of
the representative background around it; a typical example is shown in Figure
2 in which the middle panel shows a background mask 11 times larger than the
aperture mask. Light curves were extracted from the target pixel files using
aperture photometry, as explained in Section 3.1, while the background and
systematic effects were corrected using two standard statistical methods, that
is, a principal component analysis (PCA) and a linear regression, as further
detailed in Section 3.2. The Python package Lightkurve (Lightkurve
Collaboration et al., 2018) was used during the reduction.

###  3.1 Aperture and background masks

To define the aperture mask of a star, we used the median frame of all the
target pixel files (left panel in Figure 2) and selected all pixels with a
flux count larger than the median flux plus $n$-times the standard deviation.
Those pixels are our first estimate of the aperture mask. The standard
deviation was robustly estimated as 1.4826 times the median absolute deviation
(Ruppert, 2011). To reduce the contamination from nearby stars falling into
the aperture mask, we used the increasing values of $n=5.0,7.5,10,15,20,30,$
and $40$ for the threshold in standard deviation to shrink the aperture mask
until the target star was the only bright star contained within it by at least
4 TESS magnitudes. The target and neighboring stars with apparent TESS
magnitudes $m_{\textrm{TESS}}^{\textrm{target}}$ and $m_{\textrm{TESS}}$,
respectively, within the aperture mask thus follow the condition:

$m_{\textrm{TESS}}-m_{\textrm{TESS}}^{\textrm{target}}\geq 4,$ (4)

ensuring that the flux of individual fainter stars contributes no more than
approximately $0.25\%$ of the total flux within the aperture mask. For cases
where the resulting aperture mask consists of disjointed sets of pixels, only
the set containing the target star is kept. Finally, to help prevent flux from
neighboring stars leaking into the aperture mask, pixels showing an increase
in flux in any direction away from the target star are removed from the mask
and used as its edge. The background mask was defined in the same way as the
first estimate of the aperture mask but selecting the pixels below a threshold
with $n=3$, thus ensuring a minimum two-standard-deviation flux gap between
the aperture mask and the background mask. A typical example of both final
apertures is shown in the middle panel of Figure 2.

To estimate the level of contamination in the aperture mask due to the flux of
neighboring stars, we calculate the ratio of this flux to that of the target
star. To obtain such fluxes, all stars complying with Eq. (4) were modeled as
2D Gaussian functions and fitted to the median image of the target pixel file
along with a 2D plane to account for the background flux. The Gaussian
functions were centered at the location of each star, had all the same
standard deviation, and their relative amplitudes kept the same relation as
the fluxes from the stars. Fluxes were converted from TESS magnitudes using
the TASOC Photometry pipeline (Handberg et al., 2021). The right panel in
Figure 2 shows an example of this fit.

We rejected the aperture mask (together with the target pixel file) when the
final mask contained stars that do not comply with Eq. (4). To avoid both
corrupted and bleeding pixels, we also rejected masks that had pixels with
null counts or that were too elongated (i.e., with fewer than four rows or
columns, while the other is at least 14-pixels long). To average out the
stochastic noise of individual pixels, we only kept aperture masks with at
least nine pixels, as shown in the left panel of Figure 3. After careful
assessment, our sample consisted of 2162 stars with a flux contamination due
to neighboring stars smaller than 5%, as shown in the right panel in Figure 3.
The middle panel in Figure 3 shows that our light curve extraction is
consistent with previous data pipelines for which the extraction is considered
to be trustworthy only for stars brighter than TESS magnitude of 13.5 (e.g.,
Handberg et al., 2019; Huang, 2020; Caldwell et al., 2020).

Because of our stringent constraints on the aperture mask, not all TESS
sectors yielded a satisfactory mask for a given star. We therefore only kept
stars with aperture masks found in at least 11 of the 13 TESS sectors, as
shown by the dashed line in Figure 4. The 11 sectors are not necessarily
consecutive. After this cut, our sample consists of 1967 stars. Figure 4 also
shows a higher level of contamination when the aperture mask is found in fewer
numbers of TESS sectors, indicating that stars in a crowded field are more
prone to fail the requirements of the aperture mask selection.

Figure 4: Blue histogram in the bottom shows the number of TESS sectors per
star with a satisfactory aperture mask. The dark-gray histogram shows the mean
contamination in each bin of the blue histogram. The dashed red line shows the
cut for stars with at least 11 TESS sectors (a total of 1967 stars).

###  3.2 Light curve extraction and correction

To remove part of the systematic flux variability in the extracted light
curves, we used the data quality flags provided in the headers of the target
pixel file111Descriptions about TESS quality flags can be found in the section
“Data Product Overview” in https://outerspace.stsci.edu/display/TESS. (Twicken
et al., 2020) and removed the data from the time stamps affected by loss of
fine pointing222Flagged as attitude tweak, coarse pointing and desaturation
event., thermal transients333Flagged as safe mode., Earth pointing, and other
specific effects444Flagged as manual exclude. (e.g., coronal mass ejections).
We then extracted the light curves using simple aperture photometry with the
aperture masks we constructed as described in Section 3.1. An example is shown
in Figure 5, where the gaps in the data correspond to the removed time stamps.
We noted that the use of the quality flags according to the TESS release notes
did not cover all systemics present in the light curves and proceeded to
manually remove time intervals (common to all stars) which still were
significantly affected by systematic effects (e.g., telescope jittering,
thermal transients, and loss of fine pointing). Such time intervals are
present in sectors 1 to 8, as listed in Table 2 and shown in red in Figure 5.

Figure 5: Uncorrected light curves of TIC 30192406, showing in red the time
intervals that have been excluded for all stars according to Table 2. Sector 1
shows an example of jittering of the satellite. Sector 2 shows an example of
scattered sunlight reflected by the Earth or the Moon. Sector 3 shows an
example of systematic flux variability caused by the periodic re-pointing of
the camera. Figure 6: First seven normalized principal components (PC; columns
of the matrix $\mathbf{U}$) from the background mask of TIC 374944608, sector
2. The PCs are displayed in ascending order with the first normalized PC at
the bottom and manually set 0.25 units apart from each other for a better
visualization. Only black normalized PCs have a level of scattering $<10^{-4}$
(see Section 3.2) and were used for the detrending of the light curve. Figure
7: Final light curve for TIC 374944608 as derived from our developed pipeline
discussed in Section 3.

The remaining systematic variability (e.g., the gradual increase of flux in
sector 2 due to scattered light or the rapid decrease of flux at the beginning
of sector 8 as illustrated in Figure 5) and background flux were removed using
a linear regression of the light curve against the flux variability present in
the background mask defined in Section 3.1. We started this process by
extracting a light curve using aperture photometry from each of the background
pixels (an average of 330 pixels per pixel target file in our sample).
Subsequently, we applied a PCA to these light curves to capture their dominant
flux variability (see Feinstein et al., 2019). We let $\mathbf{B}$ be the
matrix whose columns are the light curves extracted from the background
pixels, its singular value decomposition is

$\mathbf{B}=\mathbf{U}\textbf{S}\textbf{V}^{T}\;,$ (5)

where $\mathbf{U}$ and V are orthonormal matrices and S is a diagonal matrix
with entries $s_{i}>s_{i+1}$. The principal components of $\mathbf{B}$ are
given by the columns of $\mathbf{US}$, which are ordered by their contribution
to the background flux variability. No universal number $k$ of principle
components can be used to estimate background flux variability because
different TESS sectors are subjected to different systematic effects. Since
the first principal components capture most of the systematic variability
while subsequent ones are affected by an increasing amount of white noise (see
Figure 6 for an example), we used the level of scatter in the normalized
principal component (columns of $\mathbf{U}$) as a criterion to determine $k$.
The level of scatter was determined by the median of the moving variances with
a window length $W=16$ hours. Denoting the elements in a column of
$\mathbf{U}$ by $f(t)$, where the time $t$ represents a cadence, the moving
variance at a cadence $t$ was computed as

$\sigma^{2}_{L}(t)=\frac{1}{N-1}\sum^{N}_{|t_{n}-t|<W}\left(f(t_{n})-\overline{f(t_{n})}\right)^{2}\;,$
(6)

where $\overline{f(t_{n})}$ is the mean of $f(t_{n})$ within the window
$|t_{n}-t|<W$. The $W=16$ hours was chosen to yield a large number of short
light curve segments while being shorter than the typical period of g modes.
We found that the level of scatter in the columns of $\mathbf{U}$, that is,
$\@vec{u_{i}}$, increased with $i$ as expected, since increasing values of $i$
had less systematic signal and more white noise. The level of scatter
converged to a value of order $10^{-3}$ for $i\gtrsim 6$ regardless of the
TESS sector. We therefore used the principal components with a level of
scatter $<10^{-4}$ to represent the systematic variability present in the
background. The median number of principal components used by our method is
$k=4$. To prevent further injection of white noise into the reduced light
curves, we used a maximum of seven principal components.

Once the number $k$ of principal components was determined, we created the
regressor matrix $\mathbf{X,}$ using the principal components as its columns
and added a constant column to account for a constant level of background.
Subsequently, we performed the following linear regression, assuming that the
model fits the data within Gaussian uncertainties:

$\@vec{Y}=\mathbf{X}\@vec{w}+\varepsilon,$ (7)

where $\@vec{Y}$ represents the uncorrected light curve of the target star,
$\@vec{w}$ contains the regression coefficients, and $\varepsilon$ represents
the noise. The corrected light curve was then computed as
$\@vec{Y}-\mathbf{X}\@vec{w}$. These corrected light curves were then
normalized by subtracting their mean flux and then dividing them by it. Values
further away than 5$\sigma$ were treated as outliers and removed (e.g., spikes
due to cosmic rays). Finally, we stitched together the normalized light curves
from each TESS sector as shown in Figure 7.

##  4 Frequency analysis from iterative prewhitening

We analyzed the light curves resulting from our developed data analysis
pipeline following a procedure of iterative prewhitening. Van Beeck et al.
(2021) has offered an extensive description of five different prewhitening
methods applied to g-mode pulsators, relying on various regression techniques
and stopping criteria. Since Van Beeck et al. (2021) developed their
methodology for g modes in SPB stars and four-year Kepler light curves, their
paper is highly relevant for our TESS work as well. We refer to that paper for
a detailed description, as well as an elaborate comparative study of the
efficiency of these five methods. Here, we rely on a method using the same
frequency resolution restriction, a stopping criterion based on the amplitudes
of the modes, and a nonlinear optimization to achieve the final regression
result, as such methods were found to be the most powerful procedures based on
the assumption of a harmonic fit to the light curve by Van Beeck et al. (2021,
see methods 2 and 3). We provide a summary of the adopted procedure, where
frequencies were extracted in a 5-step process from the stitched light curves
as the one shown in Figure 7.

Step 1: We computed a Lomb-Scargle periodogram of the light curve using a
10-fold oversampled frequency range (compared to $T^{-1}$ with $T$ the time
span of the light curve) from zero up to the Nyquist frequency. The frequency
with highest amplitude is selected and a harmonic fit with that frequency to
the light curve is determined, using linear regression. The best-fitting
harmonic function is subtracted from the light curve and the process is
iteratively repeated on the residual light curve until the selected frequency
has an amplitude with $\mathrm{S/N}<4$. The noise level is calculated as the
mean amplitude of the periodogram within a symmetric frequency window of size
$1\ \mathrm{d}^{-1}$ centered on the selected frequency. An example of the
periodogram for TIC 38845463 is visualized in panel A of Figure 11 discussed
further in the text. In order to find candidates with g-mode period-spacing
patterns, we only kept stars with at least ten significant potential g-mode
frequencies, that is, those with pulsations of periods longer than 0.25 days.
This restriction, illustrated in Figure 8, leaves us with a sample of 369
candidate stars.

Figure 8: Number of potential g-mode pulsations with $\mathrm{S/N}\geq 4$ per
star. Frequencies were deduced from Step 1 described in Section 4. Our sample
of 1967 stars has 1108 stars that are not in compliance with our Step 1
criterion and those are not plotted. The dashed red line shows the cut used to
select rich enough period-spacing pattern candidates, namely, stars with at
least ten significant mode periods. This resulted in 369 candidate pulsators.

Step 2: Unresolved frequencies were removed by requiring a conservative
difference of at least $2.5\times T^{-1}$ between extracted frequencies
(Loumos & Deeming, 1978). In case of unresolved frequencies, we kept the one
with the largest amplitude.

Step 3: All accepted frequencies were optimized simultaneously using a
Levenberg-Marquardt algorithm to perform a non-linear regression, with the
output of the linear regression as initial input guesses. Uncertainties in the
parameters were calculated following Montgomery & O’Donoghue (1999) and
corrected for their possible correlated nature following Schwarzenberg-Czerny
(2003). Frequencies whose corresponding amplitudes were consistent with zero
within three standard deviations were considered as insignificant and
rejected.

Step 4: To minimize the influence of the spectral window convolved with
dominant frequencies in the periodogram during the iterative prewhitening
process, we used the amplitude criterion developed by Van Reeth et al.
(2015a):

$\alpha\leq\frac{A_{f}}{A_{loc}}\leq\frac{1}{\alpha}\,,$ (8)

where $A_{f}$ and $A_{loc}$ are the optimized amplitudes and the amplitudes
from the original Lomb-Scargle periodogram, respectively. This constraint is
independent of the S/N and helps to avoid spurious frequency detections that
can occur for space time-series data like Kepler and TESS with
$\mathrm{S/N}>4$ (e.g., Zong et al., 2016; Baran & Koen, 2021; Bowman &
Michielsen, 2021). Moreover, Van Beeck et al. (2021) have shown criteria based
on mode amplitudes to work better than just using S/N as a stop criterion. In
practice, we used $\alpha=0.75,$ which was found to work optimally for TESS
light curves by Antoci et al. (2019).

Step 5: Combination frequencies were not considered as independent mode
frequencies to hunt for period-spacing patterns. Such combination frequencies
were identified through the following equation:

$\left|f_{k}-\left(n_{i}f_{i}+n_{j}f_{j}\right)\right|\leq\varepsilon\,,$ (9)

where we adopt the terminology of Degroote et al. (2009), meaning that $f_{i}$
and $f_{j}$ are the parent frequencies, $n_{i}$ and $n_{j}$ are integer
combination coefficients, $f_{k}$ is the combination frequency, and
$\varepsilon$ is the threshold tolerance. We selected the 20 highest-amplitude
frequencies per star as parent frequencies and searched for combinations of up
to two parents, which leads to $|n_{i}|+|n_{j}|\leq 2$.

Given the large number of frequencies per star (we note that Figure 8 only
counts g modes and ignores p modes), a linear combination of frequencies is
likely to occur close to another independent frequency in the data without
having to be a combination frequency (Pápics, 2012; Kurtz et al., 2015). This
is illustrated in Figure 9 by the background count level marked with a gray
shade while genuine combination frequencies correspond to the excess above
such level. We therefore took three times that background level (gray dashed
line in Figure 9) as a 67% probability of being a genuine combination
frequency. Such a probability corresponds to $\varepsilon=0.0002$ (red
vertical line in Figure 9) and is consistent with the threshold tolerance
reported by Li et al. (2019).

Figure 9: Histogram following Eq. (9) for the frequencies of stars in our
g-mode sample. The gray shade marks the background level that represents a
random match among combination frequencies and the horizontal dashed gray line
marks 3 times that level. The vertical red shows the intercept of the dashed
gray line and the distribution; it marks $\varepsilon=0.0002$ according to Eq.
(9). Frequencies occurring to the left of this line have a 67% probability of
corresponding to a genuine combination frequency. Figure 10: Number of
frequencies remaining in our sample after each step in Section 4 is applied.
The pie chart is based on the frequencies of the 369 stars after step 1 of our
frequency analysis. The total number of frequencies is 10927. Figure 11: Best-
fit period-spacing pattern for TIC 374944608. Plots generated with the
interactive code FLOSSY. A : Lomb-Scargle periodogram (solid black line),
observed periods (vertical dotted black lines), best-fit linear template
(vertical red lines). Observed periods used for the fit are indicated with
black triangles at the top. B : Deviations from the linear pattern (i.e.,
difference between the periods indicated by the black triangles and the red
vertical lines). C : Period spacing as a function of mode period. Both black
and white circles are the observations. The red markers are the best-fit
linear pattern with slope $\alpha$. Note that the fit is performed on $P$, not
on $\Delta P$, and missing mode periods in the observations create
artificially larger $\Delta P$ values (white circles). D : échelle diagram.
The black circles are the periods used for the fit. The size of the circles is
proportional to the amplitude of the amplitudes in the periodogram. The red
markers are the best-fit linear pattern. The supplementary material contains a
version of this figure for every pattern in Table LABEL:Tab:results.

Figure 10 shows the impact on the number of independent mode frequencies after
each step of the frequency analysis. As an example, the final accepted
frequencies of TIC 374944608 (light curve in Figure 7) are indicated with
vertical dotted lines in Figure 11 (to be discussed below). We checked the 369
light curves of our sample after the frequency analysis and removed eclipsing
binaries. These were studied further in the separate paper by IJspeert et al.
(2021). We also inspected the light curves for cases where systematic flux
variability persisted after our data reduction pipeline (see Section 3) and
removed these from the sample. This inspection was done by eye, narrowing down
our sample of period-spacing pattern candidates to 304 stars.

##  5 Period-spacing pattern search

Figure 12: Values of the cost function around the best-fit solution for TIC
374944608 shown in Figure 11. Plots generated with the interactive code
FLOSSY. Top: Correlation plots. Bottom: Minimum value of the cost function S
as a function of the template period-spacing parameters. The supplementary
material contains a version of this figure for every pattern in Table
LABEL:Tab:results.

To search for period-spacing patterns in our sample, we fitted the following
template to the list of periods of each star (Li et al., 2019):

$P_{i}=\sum_{j=0}^{i-1}\Delta P_{j}+P_{0}=\Delta
P_{0}\frac{(1+\alpha)^{i}-1}{\alpha}+P_{0}\;,$ (10)

where $\Delta P_{j}=P_{j+1}-P_{j}$ is the local period spacing and $\Delta
P_{0}$ is the period spacing at a reference period $P_{0}$. This period
pattern allows for a linear change $\alpha\equiv\mathrm{d}(\Delta
P)/\mathrm{d}P$ of the local period spacing caused by stellar rotation
(Ouazzani et al., 2017; Christophe et al., 2018b; Li et al., 2019). The
template depends on the three parameters $\\{P_{0},\Delta P_{0},\alpha\\}$. To
account for the amplitude of the individual periods and the local size of the
period spacing, we fitted Eq. (10) to the observations, by minimizing the
following custom cost function:

$S\left(P_{0},\Delta P_{0},\alpha\right)=\sum_{i=1}^{n}\frac{A_{i}}{A_{\rm
max}}\frac{\left(P_{\,i}^{\mathrm{\;obs}}-{P}_{i}\right)^{2}}{\sigma_{i}^{2}+\Delta{P}_{i}^{2}}\;,$
(11)

where $P_{i}$ is the estimated period closest to the observed pulsation period
$P^{\mathrm{\;obs}}_{i}$, $\Delta P_{i}$ is the estimated local period
spacing, $A_{i}$ is the observed amplitude corresponding to $P_{i}^{\rm obs}$,
$A_{\rm max}$ is the maximum observed amplitude, and $\sigma_{i}$ is the
observed period uncertainty. Rather than minimizing the square of the absolute
differences $(P_{i}^{\rm obs}-P_{i})^{2},$ we minimize the relative
differences $(P_{i}^{\rm obs}-P_{i})^{2}/\Delta P^{2}_{i}$. In this way,
period mismatches are more strongly penalized when they are large compared to
the local period spacing. The addition of $\sigma_{i}^{2}$ in the denominator
serves to limit the penalization when the local period spacing is comparable
to the observational period uncertainty, $\sigma_{i}$. The extra weight,
$A_{i}/A_{\rm max}$, serves to penalize a pattern more strongly when it
mismatches the higher-amplitude mode periods. The minimization of the cost
function $S$ was done with the quasi-Newton method L-BFGS-B (Byrd et al.,
1995) implemented in the Python module Scipy (Virtanen et al., 2020).

To find the location of the patterns in the periodogram as well as the initial
guesses $\mathbf{\theta}^{\textrm{\;init}}=\\{P_{0}^{\textrm{\;init}},\Delta
P_{0}^{\textrm{\;init}},\alpha^{\textrm{\;init}}\\}$ for the fit, we used two
diagnostic plots to cover both the cases of rapid and slow rotators. Slow
rotators show an approximately constant period spacing. Their period-spacing
pattern can therefore be identified in an échelle diagram, where the period is
plotted as a function of the period modulo $\Delta P$. In such a case, g modes
of a given angular degree roughly form vertical ridges (analogous to the
acoustic modes in the case of solar-like oscillations). On the other hand,
rapid rotators show a period spacing that depends approximately linearly on
the mode period (Van Reeth et al., 2016; Ouazzani et al., 2017). Therefore,
their period-spacing pattern can be easier identified in a plot of $\Delta P$
as a function of period. For each star, we also complemented such two plots
with a periodogram where observed and template periods were overplotted.

To facilitate the exploration of the parameter space, we developed the
interactive tool FLOSSY555https://github.com/IvS-KULeuven/FLOSSY, a Python
utility that allows a user to efficiently browse the periodogram of a large
number of stars and visualize the period-spacing patterns by displaying the
period échelle diagram and period-spacing plot at each location in the
periodogram. FLOSSY also overplots Eq. (10) in the aforementioned plots with
customized parameters $\\{P_{0},\Delta P_{0},\alpha\\}$. The latter can be
modified on the fly, along with the number of mode periods to fit. Figure 11
shows part of FLOSSY’s output, as well as the best-fit pattern for TIC
374944608.

We used FLOSSY to manually select the $\mathbb{\theta^{\textrm{\;init}}}$ for
every candidate period-spacing pattern found from the list of mode periods per
star. In doing so we considered the parameter space
$|P_{0}-P_{0}^{\textrm{\;init}}|\leq\delta P/2$, $100\ \textrm{s}\leq\Delta
P_{0}\leq 4000\ \textrm{s}$ and, $-0.3\leq\alpha\leq 0.3$, where
$P_{0}^{\textrm{\;init}}\in\\{P^{\mathrm{\;obs}}_{i}\\}$ and $\delta P$
corresponds to the resolution set in Step 2 of the procedure discussed in
Section 4. To ensure that we found a global minimum, we computed $S$ around
the best-fit solution in a radius of 400 s for $P_{0}$, 40 s for $\Delta
P_{0},$ and 0.05 units for $\alpha$. Those values for $S$ are shown in Figure
12, which is also an output of FLOSSY.

To estimate uncertainties for the detected period-spacing pattern, we computed
the 68% confidence interval of the parameters using a bootstrap method with
non-parametric residual resampling in the periodogram. We generated 10000
datasets of the same size as the original one. Subsequently, we minimized Eq.
(11) in each of these datasets using as initial guess the same
$\mathbf{\theta}^{\textrm{\;init}}$ as in the original best fit. The
confidence intervals were then determined as the 16% and 84% quantiles of the
bootstrap distribution for the parameters $\mathbf{\theta}$. As an example,
Figure 13 shows the bootstrap distribution of $\alpha$ for the pattern found
in TIC 374944608.

Figure 13: Confidence interval (CI) for $\alpha$ following a bootstrap
residual procedure for TIC 374944608. The meaning of the various vertical
dashed lines are indicated in the legend.

##  6 Catalog of g-mode pulsators in the TESS southern CVZ with identified
period-spacing patterns

Our final catalog of g-mode pulsators revealing period-spacing patterns
consists of 108 bright dwarfs in the TESS southern CVZ. These stars revealed a
total of 140 resolved period-spacing patterns listed in Table
LABEL:Tab:results. Each of these patterns are shown in the supplementary
material in the same format as Figures 11 and 12. Stars in our catalog have
apparent TESS magnitudes between 7.5 and 12, with a median of about 10; the
star TIC 350144657 is an exception with an apparent TESS magnitude of about
6.9. The contamination of light curves, due to the flux from neighboring
stars, is $<2\%$ thanks to our stringent requirements on the aperture mask
described in Section 3.1. Figure 14 shows the distributions of brightness and
contamination in the catalog. Only two stars, TIC 177162802 and TIC 375038081,
are candidates to be SPB stars, as determined by the color-magnitude selection
done in Section 2, while the other members of the catalog are $\gamma$ Dor
stars. Figure 15 shows a Gaia color-magnitude diagram that compares our
catalog to the 611 $\gamma$ Dor stars with detected period-spacing patterns
found by Li et al. (2020) in 4-year Kepler light curves. We find that our
catalog stars occur on the hotter end of the Kepler $\gamma$ Dor stars.

Figure 14: Final catalog. Top: Apparent TESS magnitude. Bottom: Median
contamination level in the aperture mask caused by flux of neighboring stars.
Figure 15: Gaia color-magnitude diagram showing our sample (red) and the 611
$\gamma$-Dor stars with period-spacing patterns found by Li et al. (2020) in
Kepler data (yellow). Stars marked with blue circles are SPB candidates.
Background stars correspond to the TESS southern CVZ. Magnitudes are not
corrected for extinction

Out of the 140 period-spacing patterns, 93 have a downward slope ($\alpha<0$)
and 47 have an upward slope ($\alpha>0$). The former are prograde or zonal g
modes while the latter are retrograde g modes or Rossby modes (Van Reeth et
al., 2016). The averaged period-spacing value per pattern, $\langle\Delta
P\rangle$, is $\sim 110$ s. The shortest pattern contains four measured
pulsation periods, more than half of the patterns contain more than 12 and the
longest pattern contains 20. In 26% of the stars we detected two or three
patterns. When multiple patterns are detected in a star, this allows for a
better constraint of the stellar interior from asteroseismic modeling (Aerts
et al., 2018). Furthermore, 29% of our catalog stars are hybrid pulsators,
meaning that they also exhibit p-mode pulsations, therefore, providing us with
a means to probe the outer stellar envelope and allowing for a differential
study of the star.

Figure 16 shows the distributions of the pattern parameters. Typical
uncertainties for the pattern parameters reported in Table LABEL:Tab:results
are 43 s for $P_{0}$, 13 s for $\Delta P_{0}$ and 0.006 for $\alpha$. The
pattern slopes fulfill $|\alpha|\leq 0.1$ for 88% of the catalog stars, while
the tails of the distribution reach $|\alpha|\sim 0.2$. Using the empirical
relations in Li et al. (2020), we can estimate the near-core rotation to be
$<1.68\,\textrm{d}^{-1}$ for 86% of the stars in our sample, with a few stars
reaching up to about 2.86 $\textrm{d}^{-1}$. We also made use of the Figure 8
in Li et al. (2020) to define the empirical cut delineating regimes of dipole
and quadrupole modes in Figure 17 (dashed blue line). This suggests that 13 of
our prograde patterns have $\ell=2,$ while the rest of them have $\ell=1$. We
noted that because pulsations in $\gamma$ Dor stars are sensitive to
metallicity, the empirical estimates above drawn using Li et al. (2020) remain
to be confirmed. Since the nominal Kepler field-of-view was in the northern
hemisphere, we cannot directly cross-validate our TESS southern CVZ results
with those from Kepler.

Figure 16: Characterization of the best-fit patterns in our sample (see also
Table LABEL:Tab:results). The stacked histograms show retrograde modes in red
and prograde modes in cyan. The vertical orange and blue lines are the median
of the retrograde and prograde distributions, respectively. A : Slope
$\alpha\equiv\mathrm{d}\Delta P/\mathrm{d}P$. B : Mean period $\langle
P\rangle$. C : Mean period spacing $\langle\Delta P\rangle$. D : Span of the
overtones. Figure 17: $P$-$\alpha$ relation for g-mode pulsators in our
catalog. The dashed blue line is the empirical cut from Figure 8 in Li et al.
(2020) that separates prograde $\ell=1$ g modes (below) from $\ell=2$ (above).
The median of the uncertainties reported in Table LABEL:Tab:results has been
plotted as a typical uncertainty for $\alpha$. Uncertainties in $\langle
P\rangle$ are smaller than the symbol size.

We found a positive correlation for prograde modes with $l=1$ between the
parameters $\langle P\rangle$ and $\alpha$ with a Pearson correlation
coefficient of 0.67 and a $p$-value of $10^{-9}$, while prograde modes with
$l=2$ show the same Pearson correlation coefficient but with a $p$-value of
0.016. Since $\langle P\rangle$ is a proxy for the evolutionary stage, these
correlations reveal that the near-core rotation rate of the stars slows down
as they evolve, implying that an efficient angular momentum transport
mechanism must be at work as already found in the literature (cf. Aerts et
al., 2019, for a review). We did not search for a correlation in retrograde
modes because their parameter $\alpha$ is less sensitive to the star’s local
rotation rate. For those, a more precise analysis involving the traditional
approximation of rotation (TAR; Eckart, 1960) is necessary and will be
addressed in a future paper. Furthermore, the range in overtones in the
patterns is a proxy for the radial order $n$ of the g modes. The exact value
of $n$, $\ell,$ and $m$ can only be identified from asteroseismic modeling,
for example based on the TAR as applied in Van Reeth et al. (2016). The mode
identification and asteroseismic modeling based on the TAR will be addressed
in a future paper dedicated to the ensemble of stars in our catalog of g-mode
pulsators, relying on the pattern properties deduced in this work.

Besides quasi-linear period-spacing patterns like the one shown in panel C of
Figure 11, where the zigzag feature is caused by missing periods in the
pattern (white circles), our catalog contains tens of patterns with zigzag
signatures that are not related to missing modes. These patterns are presented
in the supplementary material, where Figures 11 and 12 are reproduced for each
period-spacing pattern in our catalog. Such signatures have also been observed
in period-spacing patterns of SPB stars, where they are interpreted as the
result of strong envelope mixing deep inside the star. These signatures were
recently used by Pedersen et al. (2021) to constrain the internal mixing
profile in SPB stars observed by Kepler. The levels of envelope mixing found
thus far in $\gamma\,$Dor stars are far lower than those of SPB stars (cf.
Table 1 in Aerts, 2021). Our catalog presents an opportunity to further assess
and refine these recent conclusions in the literature from our TESS southern
CVZ catalog of g-mode dwarf pulsators. We also noted that many of the detected
period-spacing patterns in our catalog show residuals with a sinusoidal-like
modulation after the subtraction of the linear fit (red line in panel C of
Figure 11). This type of periodic residuals is very similar to the one found
originally for the slowly rotating CoRoT SPB HD 50230 by Degroote et al.
(2010) and allows for stringent constraints on the core overshooting and
envelope mixing.

Finally, we compared our catalog to the spectroscopic parameters published in
the GALAH Data Release (DR) 3 paper by Buder et al. (2021). This DR3 has 38
stars in common with our sample. For these 38 stars, we found the effective
temperature, surface gravity, and surface velocity deduced from line-profile
broadening to be consistent with such properties of $\gamma$ Dor stars (e.g.,
Van Reeth et al., 2015b; Li et al., 2020), that is, early F- to late A-type
main-sequence stars. This agreement in stellar parameters supports the
selection methods used throughout the current paper. The corresponding
distributions can be found in Appendix A, where we also discuss the
correlation found between the surface velocity estimate from spectral line
broadening and the average pulsation period $\langle P\rangle$. As expected
for moderate- to fast-rotating pulsators, the larger surface velocity is
accompanied by a shorter average pulsation period.

##  7 Summary

In this work, we present a new data analysis pipeline to create light curves
from TESS full frame images (FFI), with an emphasis on the search for g-mode
frequencies in intermediate- to high-mass stars. We present guidelines for
extracting light curves from unprocessed TESS images, including the selection
of aperture and background masks, the identification of time stamps affected
by systematics in sectors 1-13 of the TESS southern CVZ, and a modified PCA to
detrend the light curves. A color-magnitude criterion was presented as a
method to identify main-sequence A/F- and O/B-type star candidates. We also
introduced FLOSSY, an open source utility for inspecting periodograms of
g-mode pulsators to facilitate searches for period-spacing patterns.

Based on the light curves extracted with our pipeline, we composed the first
catalog of g-mode period-spacing patterns detected in TESS space photometry of
dwarfs having colors representative of spectral types F to O. Our catalog
contains 140 g-mode period-spacing patterns observed in 106 $\gamma$ Dor stars
and 2 SPB stars. The patterns were manually reviewed and contain g-mode
frequencies having amplitudes of S/N¿4. In a future work, we will use the
detected patterns to derive the internal rotation frequency near the
convective core of the stars, as well as the buoyancy travel time across the
stars (known as $\Pi_{0}$). These two key quantities constitute important
observables that are useful for performing asteroseismic modeling of
intermediate-mass stars (e.g., Szewczuk & Daszyńska-Daszkiewicz, 2018; Mombarg
et al., 2021; Pedersen et al., 2021). The nominal frequency resolution of
modes in the detected patterns amounts to $0.003\textrm{d}^{-1}$, following
the 352 d long TESS southern CVZ light curves. This frequency resolution can
be improved by a factor of 3 when the extended TESS Cycle 3 data will be
included in the analysis. This will also lower the noise level in the Fourier
domain and offers the future potential to detect more modes per star, as well
as more stars with g-mode patterns. The global properties of the detected
patterns occurring in our catalog are listed in Table LABEL:Tab:results, and
the patterns themselves, are shown in the supplementary material. This catalog
constitutes a base for future ensemble asteroseismic modeling of TESS g-mode
pulsators following methodologies as in Mombarg et al. (2021) or Pedersen et
al. (2021). In this way, we will be able to constrain the internal physics of
more rotating dwarfs with a convective core using the new available TESS data,
in addition to the asteroseismic modeling achieved so far for a legacy sample
of g-mode pulsators from the Kepler mission (Gebruers et al., 2021). This will
increase the number of dwarfs with such modeling and will lead to a better
understanding of the transport processes and their relationship to the
internal rotation profile of these stars.

###### Acknowledgements.

The research leading to these results has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement no. 670519: MAMSIE) and from the KU
Leuven Research Council (grant C16/18/005: PARADISE). TVR gratefully
acknowledges support from the Research Foundation Flanders under grant
agreement nr. 12ZB620N. This research has made use of the SIMBAD database,
operated at CDS, Strasbourg, France. This work made use of the Third Data
Release of the GALAH Survey (Buder et al. 2021). The GALAH Survey is based on
data acquired through the Australian Astronomical Observatory, under programs:
A/2013B/13 (The GALAH pilot survey); A/2014A/25, A/2015A/19, A2017A/18 (The
GALAH survey phase 1); A2018A/18 (Open clusters with HERMES); A2019A/1
(Hierarchical star formation in Ori OB1); A2019A/15 (The GALAH survey phase
2); A/2015B/19, A/2016A/22, A/2016B/10, A/2017B/16, A/2018B/15 (The HERMES-
TESS program); and A/2015A/3, A/2015B/1, A/2015B/19, A/2016A/22, A/2016B/12,
A/2017A/14 (The HERMES K2-follow-up program). We acknowledge the traditional
owners of the land on which the AAT stands, the Gamilaraay people, and pay our
respects to elders past and present. This paper includes data that has been
provided by AAO Data Central (datacentral.aao.gov.au). This work has made use
of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. In addition to the software
cited in the main body of the paper we have also made use of Lightkurve, a
Python package for Kepler and TESS data analysis (Lightkurve Collaboration et
al., 2018), Astropy,666http://www.astropy.org a community-developed core
Python package for Astronomy (Astropy Collaboration et al., 2013, 2018),
Matplotlib (Hunter, 2007), NumPy (Harris et al., 2020), SciPy (Virtanen et
al., 2020), pandas (Wes McKinney, 2010). Finally, we would like to thank the
anonymous referee for useful comments and suggestions.

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##  Appendix A Extended tables and figures

Table 2 lists the removed time intervals before the detrending of the light
curves described in Section 3.2. Table LABEL:Tab:results lists the best-fit
values of the parameters defined in Section 5 that characterize the 140
period-spacing patterns in our catalog. The supplementary material available
online contains a version of Figures 11 and 12 that show the periodogram and
the best-fit period-spacing pattern for all stars in Table LABEL:Tab:results.
Whenever periods are plotted with gray circles in those figures, those periods
are ignored in the fit.

Figure 18 shows a negative correlation with a Pearson correlation coefficient
of -0.34 and a p-value of 0.04 between the mean pulsation period of our
catalog stars and their surface velocity estimated from spectral line
broadening as reported in GALAH DR3 (Buder et al. 2021). In other words,
larger surface velocities are measured for stars with shorter $\langle
P\rangle$. While the measurement of the surface velocity from spectral line
broadening is complicated and can never be precise in the presence of time-
dependent gravity-mode velocity broadening (Aerts et al. 2014), this result is
consistent with the expected effect of faster rotation on the periods of the
pulsation modes (Bouabid et al. 2013). There is significant scatter in the
figure, as expected for a sample of stars seen at random inclination angles,
given the small overlap of only 38 stars between our catalog and GALAH DR3.
Moreover, the uncertainties reported by GALAH DR3 are underestimates given
that the spectral line broadening changes considerably throughout the
pulsation cycles as shown by Aerts et al. (2014), while that time dependence
has been ignored in the reported velocity estimates based on snapshot spectra.
This precludes us to connect in-depth conclusions based on this correlation in
Fig. 18. Figure 19 shows the distributions of mass, surface gravity, and
effective temperature for our catalog stars available in GALAH DR3. The ranges
of these three parameters are in agreement with those of $\gamma$ Dor stars
(Van Reeth et al. 2015b; Li et al. 2020).

Figure 18: Correlation between the surface velocity deduced from spectral line broadening as reported by GALAH DR3 and our pattern parameter $\langle P\rangle$. If a star has multiple patterns, then the pattern with the highest-amplitude pulsations was used. The plot contains the 38 stars in common in GALAH DR3 and our catalog, where we only considered stars with spectra flagged as reliable in GALAH DR3. Uncertainties in $\langle P\rangle$ are smaller than the symbol size, while those of the velocities are underestimates for reasons explained in the text. Table 2: Intervals excluded from the light curves to remove systematic flux variability due to, e.g., scattered light, telescope jittering, and loss of fine pointing. TESS sector | Excluded interval
---|---
| (BJD-2457000)
1 | (1334.8, 1335.1)
(1347.0, 1349.5)
2 | (1356.2, 1356.5)
(1361.0, 1361.3)
(1363.5, 1363.8)
(1365.9, 1366.2)
(1373.8, 1374.1)
(1375.8, 1376.0),
(1377.9, 1378.7)
3 | (1380.0, 1385.0)
(1387.6, 1387.9)
(1390.1, 1390.4)
(1392.6, 1392.9)
(1395.1, 1395.4)
(1398.6, 1398.9)
(1400.6, 1400.9)
(1402.6, 1402.9)
(1404.6, 1404.9)
(1406.1, 1406.4)
4 | (1420.0, 1427.0)
5 | (1463.0, 1465.0)
6 | (1476.0, 1479.0)
7 | (1502.5, 1506.0)
8 | (1529.5, 1533.0)

Figure 19: Histograms for the spectroscopic stellar parameters of the 38 stars in common with our catalog having spectra flagged as reliable in the GALAH DR3 catalog by Buder et al. (2021). Table 3:  TIC stars in our catalog and their period-spacing pattern parameters. The $\langle P\rangle$ and $\langle\Delta P\rangle$ are the mean period and the mean period spacing, respectively. The $P_{0}$, $\Delta P_{0}$ and $\alpha$ are the parameters defined in Eq. (10). The column Period is the number of observed periods in the pattern. The column $n$-span is the range of overtones in the model pattern. Hybrid pulsators are marker with an “Y”. TIC | TESS | $\langle P\rangle$ | $\langle\Delta P\rangle$ | $P_{0}$ | $\Delta P_{0}$ | $\alpha$ | Periods |  $n$-span | Hybrid
---|---|---|---|---|---|---|---|---|---
| (mag) | (d) | (ks) | (d) | (ks) |  |  |  |
25152923 | 9.85 | 0.383 | 0.30 | ${0.376933}_{-0.000104}^{+0.000067}$ | ${0.2399}_{-0.0027}^{+0.0030}$ | ${0.1153}_{-0.0076}^{+0.0056}$ | 7 | 12 |
32150270 | 9.85 | 0.985 | 2.70 | ${1.00830}_{-0.00085}^{+0.00082}$ | ${2.406}_{-0.054}^{+0.056}$ | ${-0.146}_{-0.012}^{+0.012}$ | 5 | 6 |
33766642 | 10.35 | 0.472 | 0.77 | ${0.42510}_{-0.00011}^{+0.00012}$ | ${0.9255}_{-0.0024}^{+0.0026}$ | ${-0.03839}_{-0.00048}^{+0.00050}$ | 8 | 22 | Y
33766642 | 10.35 | 0.895 | 1.35 | ${0.9189}_{-0.0012}^{+0.0013}$ | ${1.254}_{-0.012}^{+0.021}$ | ${-0.0445}_{-0.0025}^{+0.0034}$ | 11 | 24 | Y
33879968 | 9.77 | 0.426 | 3.61 | ${0.3667}_{-0.0027}^{+0.0015}$ | ${4.000}_{-0.070}^{+0.000}$ | ${-0.0771}_{-0.0085}^{+0.0137}$ | 7 | 8 | Y
38515566 | 8.85 | 1.551 | 2.91 | ${1.49973}_{-0.00034}^{+0.00033}$ | ${2.9069}_{-0.0057}^{+0.0059}$ | ${0.00165}_{-0.00081}^{+0.00080}$ | 12 | 16 | Y
40335866 | 9.72 | 0.260 | 0.46 | ${0.25565}_{-0.00025}^{+0.00021}$ | ${0.4898}_{-0.0014}^{+0.0051}$ | ${-0.0982}_{-0.0022}^{+0.0046}$ | 10 | 14 | Y
40663416 | 10.07 | 0.706 | 1.98 | ${0.72518}_{-0.00071}^{+0.00042}$ | ${2.263}_{-0.089}^{+0.069}$ | ${0.175}_{-0.035}^{+0.025}$ | 5 | 5 |
41483281 | 10.25 | 0.617 | 2.76 | ${0.68534}_{-0.00044}^{+0.00034}$ | ${2.6441}_{-0.0107}^{+0.0085}$ | ${-0.01876}_{-0.00123}^{+0.00083}$ | 9 | 15 | Y
55451820 | 9.60 | 0.554 | 0.92 | ${0.53764}_{-0.00065}^{+0.00042}$ | ${1.185}_{-0.031}^{+0.033}$ | ${-0.195}_{-0.025}^{+0.022}$ | 5 | 6 |
55453219 | 8.84 | 0.396 | 0.53 | ${0.412394}_{-0.000075}^{+0.000088}$ | ${0.4289}_{-0.0033}^{+0.0045}$ | ${-0.0676}_{-0.0018}^{+0.0023}$ | 5 | 12 |
55849446 | 9.27 | 0.662 | 0.55 | ${0.670135}_{-0.000096}^{+0.000101}$ | ${0.6080}_{-0.0049}^{+0.0045}$ | ${0.0879}_{-0.0049}^{+0.0046}$ | 4 | 8 |
55849446 | 9.27 | 0.473 | 0.18 | ${0.467041}_{-0.000069}^{+0.000067}$ | ${0.19380}_{-0.00045}^{+0.00044}$ | ${-0.03616}_{-0.00056}^{+0.00045}$ | 10 | 31 |
140511383 | 11.92 | 0.436 | 0.45 | ${0.37710}_{-0.00027}^{+0.00036}$ | ${0.7114}_{-0.0100}^{+0.0087}$ | ${-0.0515}_{-0.0015}^{+0.0018}$ | 9 | 20 |
140756824 | 10.83 | 0.273 | 0.18 | ${0.27303276}_{-0.00000088}^{+0.00000097}$ | ${0.179016}_{-0.000023}^{+0.000024}$ | ${0.03393593}_{-0.00000060}^{+0.00000278}$ | 4 | 9 |
140756824 | 10.83 | 0.560 | 0.75 | ${0.56470}_{-0.00097}^{+0.00069}$ | ${0.707}_{-0.021}^{+0.029}$ | ${-0.132}_{-0.016}^{+0.022}$ | 9 | 9 |
141122677 | 10.14 | 0.426 | 0.76 | ${0.34900}_{-0.00054}^{+0.00058}$ | ${1.385}_{-0.031}^{+0.027}$ | ${-0.0944}_{-0.0032}^{+0.0036}$ | 7 | 15 |
141122677 | 10.14 | 0.746 | 1.02 | ${0.7295}_{-0.0010}^{+0.0013}$ | ${1.255}_{-0.019}^{+0.272}$ | ${-0.1666}_{-0.0031}^{+0.0868}$ | 7 | 11 |
141153472 | 9.82 | 0.890 | 2.00 | ${0.8178}_{-0.0016}^{+0.0023}$ | ${3.25}_{-0.17}^{+0.16}$ | ${-0.198}_{-0.022}^{+0.025}$ | 5 | 6 |
141154953 | 10.29 | 0.544 | 0.53 | ${0.49105}_{-0.00032}^{+0.00039}$ | ${1.141}_{-0.018}^{+0.017}$ | ${-0.1321}_{-0.0031}^{+0.0033}$ | 6 | 14 |
141479660 | 11.23 | 0.465 | 1.33 | ${0.46527}_{-0.00035}^{+0.00033}$ | ${1.3324}_{-0.0032}^{+0.0065}$ | ${-0.0390}_{-0.0011}^{+0.0016}$ | 12 | 18 | Y
141826495 | 9.73 | 0.456 | 1.01 | ${0.41665}_{-0.00096}^{+0.00061}$ | ${1.571}_{-0.035}^{+0.039}$ | ${-0.1651}_{-0.0056}^{+0.0121}$ | 10 | 12 |
141826495 | 9.73 | 0.663 | 0.55 | ${0.72258}_{-0.00014}^{+0.00013}$ | ${0.7077}_{-0.0051}^{+0.0051}$ | ${0.03109}_{-0.00084}^{+0.00083}$ | 9 | 23 |
141826495 | 9.73 | 0.723 | 0.72 | ${0.714501}_{-0.000012}^{+0.000013}$ | ${0.66689}_{-0.00072}^{+0.00069}$ | ${0.0717741}_{-0.0000034}^{+0.0000016}$ | 5 | 5 |
142083629 | 7.68 | 0.382 | 1.08 | ${0.397258}_{-0.000095}^{+0.000117}$ | ${0.843}_{-0.011}^{+0.013}$ | ${-0.1869}_{-0.0058}^{+0.0068}$ | 5 | 5 | Y
142083629 | 7.68 | 0.305 | 0.53 | ${0.260452}_{-0.000061}^{+0.000092}$ | ${0.8561}_{-0.0037}^{+0.0030}$ | ${-0.08684}_{-0.00070}^{+0.00089}$ | 8 | 13 | Y
149253072 | 10.91 | 1.034 | 1.82 | ${1.02011}_{-0.00086}^{+0.00079}$ | ${1.963}_{-0.021}^{+0.022}$ | ${-0.1185}_{-0.0096}^{+0.0081}$ | 7 | 9 |
149253072 | 10.91 | 0.554 | 1.32 | ${0.49571}_{-0.00052}^{+0.00041}$ | ${1.153}_{-0.027}^{+0.024}$ | ${0.0339}_{-0.0053}^{+0.0058}$ | 5 | 9 |
149444771 | 11.23 | 0.882 | 1.12 | ${0.83452}_{-0.00093}^{+0.00085}$ | ${1.749}_{-0.062}^{+0.067}$ | ${-0.152}_{-0.014}^{+0.013}$ | 6 | 7 |
149540525 | 8.39 | 0.675 | 0.75 | ${0.67404}_{-0.00025}^{+0.00025}$ | ${0.7590}_{-0.0040}^{+0.0043}$ | ${-0.0614}_{-0.0030}^{+0.0032}$ | 8 | 12 | Y
149540525 | 8.39 | 0.625 | 0.94 | ${0.66392}_{-0.00048}^{+0.00050}$ | ${0.7939}_{-0.0086}^{+0.0098}$ | ${-0.0444}_{-0.0019}^{+0.0021}$ | 13 | 21 | Y
149573437 | 9.48 | 0.282 | 0.50 | ${0.291346}_{-0.000025}^{+0.000025}$ | ${0.4367}_{-0.0014}^{+0.0013}$ | ${-0.0742}_{-0.0013}^{+0.0011}$ | 6 | 8 | Y
149630117 | 9.05 | 0.392 | 1.47 | ${0.41974}_{-0.00011}^{+0.00011}$ | ${1.3585}_{-0.0069}^{+0.0063}$ | ${-0.0477}_{-0.0021}^{+0.0018}$ | 5 | 8 | Y
149993830 | 8.91 | 0.551 | 0.48 | ${0.55334}_{-0.00022}^{+0.00021}$ | ${0.4892}_{-0.0064}^{+0.0073}$ | ${0.0361}_{-0.0096}^{+0.0125}$ | 6 | 8 |
150063580 | 9.91 | 0.279 | 0.32 | ${0.29271}_{-0.00016}^{+0.00015}$ | ${0.3087}_{-0.0021}^{+0.0022}$ | ${-0.00950}_{-0.00106}^{+0.00095}$ | 11 | 28 | Y
150106884 | 11.45 | 0.702 | 0.72 | ${0.74061}_{-0.00038}^{+0.00038}$ | ${0.5339}_{-0.0099}^{+0.0103}$ | ${-0.0551}_{-0.0026}^{+0.0028}$ | 8 | 18 |
150106884 | 11.45 | 0.407 | 0.71 | ${0.3886610}_{-0.0000086}^{+0.0000127}$ | ${0.85487}_{-0.00055}^{+0.00047}$ | ${-0.0915352}_{-0.0000021}^{+0.0000017}$ | 5 | 5 |
150165657 | 8.66 | 0.411 | 1.02 | ${0.44579}_{-0.00045}^{+0.00050}$ | ${1.053}_{-0.017}^{+0.020}$ | ${0.0123}_{-0.0046}^{+0.0051}$ | 11 | 13 | Y
150250236 | 9.67 | 0.799 | 1.74 | ${0.80779}_{-0.00061}^{+0.00031}$ | ${1.780}_{-0.034}^{+0.027}$ | ${0.050}_{-0.018}^{+0.013}$ | 6 | 6 |
150250236 | 9.67 | 0.535 | 0.54 | ${0.52858}_{-0.00026}^{+0.00029}$ | ${0.5881}_{-0.0045}^{+0.0039}$ | ${-0.0793}_{-0.0038}^{+0.0047}$ | 5 | 12 |
150318672 | 10.16 | 0.296 | 0.36 | ${0.298849}_{-0.000103}^{+0.000096}$ | ${0.3484}_{-0.0038}^{+0.0051}$ | ${-0.0420}_{-0.0056}^{+0.0095}$ | 6 | 8 |
150324086 | 10.13 | 0.516 | 0.66 | ${0.479533}_{-0.000063}^{+0.000125}$ | ${0.7805}_{-0.0044}^{+0.0039}$ | ${-0.0388}_{-0.0012}^{+0.0015}$ | 8 | 12 |
150392753 | 8.52 | 0.304 | 0.17 | ${0.296854}_{-0.000034}^{+0.000036}$ | ${0.21511}_{-0.00058}^{+0.00053}$ | ${-0.06696}_{-0.00070}^{+0.00086}$ | 11 | 22 |
150440102 | 11.72 | 0.933 | 1.38 | ${0.8630}_{-0.0014}^{+0.0016}$ | ${0.975}_{-0.060}^{+0.060}$ | ${0.068}_{-0.012}^{+0.013}$ | 6 | 11 |
150440362 | 8.02 | 0.870 | 1.31 | ${0.8724}_{-0.0014}^{+0.0016}$ | ${1.301}_{-0.029}^{+0.034}$ | ${-0.045}_{-0.017}^{+0.017}$ | 5 | 9 |
167124706 | 8.75 | 0.906 | 2.46 | ${0.80673}_{-0.00022}^{+0.00017}$ | ${2.5089}_{-0.0043}^{+0.0051}$ | ${-0.00567}_{-0.00056}^{+0.00047}$ | 11 | 16 |
167124706 | 8.75 | 1.197 | 3.12 | ${0.96407}_{-0.00025}^{+0.00026}$ | ${3.0517}_{-0.0067}^{+0.0068}$ | ${0.00343}_{-0.00037}^{+0.00038}$ | 12 | 16 |
167722437 | 8.42 | 0.385 | 0.41 | ${0.41119}_{-0.00013}^{+0.00012}$ | ${0.4293}_{-0.0050}^{+0.0048}$ | ${0.0085}_{-0.0020}^{+0.0019}$ | 8 | 16 |
176874440 | 11.10 | 1.159 | 2.50 | ${1.14639}_{-0.00034}^{+0.00024}$ | ${2.5576}_{-0.0089}^{+0.0094}$ | ${-0.0561}_{-0.0044}^{+0.0050}$ | 6 | 6 |
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176980185 | 10.86 | 0.628 | 1.00 | ${0.64161}_{-0.00039}^{+0.00024}$ | ${0.936}_{-0.016}^{+0.014}$ | ${-0.0531}_{-0.0087}^{+0.0058}$ | 7 | 9 |
177082055 | 8.28 | 1.149 | 1.78 | ${1.20087}_{-0.00025}^{+0.00030}$ | ${1.7853}_{-0.0048}^{+0.0055}$ | ${0.00010}_{-0.00074}^{+0.00086}$ | 7 | 16 | Y
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177162802 | 10.68 | 0.809 | 3.00 | ${0.6510}_{-0.0029}^{+0.0018}$ | ${1.876}_{-0.051}^{+0.062}$ | ${0.0822}_{-0.0054}^{+0.0048}$ | 12 | 17 |
177164485 | 10.41 | 0.707 | 0.94 | ${0.76156}_{-0.00027}^{+0.00019}$ | ${1.0111}_{-0.0109}^{+0.0099}$ | ${0.0153}_{-0.0019}^{+0.0017}$ | 10 | 15 |
177386428 | 9.19 | 0.269 | 0.32 | ${0.256794}_{-0.000012}^{+0.000010}$ | ${0.41500}_{-0.00042}^{+0.00052}$ | ${-0.087406}_{-0.000596}^{+0.000011}$ | 6 | 7 | Y
231084221 | 10.26 | 0.482 | 0.97 | ${0.458757}_{-0.000015}^{+0.000021}$ | ${1.04841}_{-0.00088}^{+0.00092}$ | ${-0.0382265}_{-0.0002841}^{+0.0000039}$ | 5 | 5 | Y
257721280 | 9.36 | 0.860 | 0.99 | ${0.73313}_{-0.00025}^{+0.00027}$ | ${1.0200}_{-0.0050}^{+0.0051}$ | ${-0.00252}_{-0.00047}^{+0.00046}$ | 10 | 27 |
260265631 | 10.21 | 0.666 | 1.56 | ${0.601269}_{-0.000032}^{+0.000047}$ | ${1.7389}_{-0.0021}^{+0.0018}$ | ${-0.03207}_{-0.00032}^{+0.00041}$ | 6 | 10 |
260353074 | 8.97 | 0.950 | 3.64 | ${0.8575}_{-0.0027}^{+0.0047}$ | ${2.78}_{-0.18}^{+0.12}$ | ${0.107}_{-0.020}^{+0.029}$ | 5 | 6 | Y
260373272 | 10.38 | 1.075 | 1.19 | ${0.98124}_{-0.00078}^{+0.00079}$ | ${0.934}_{-0.021}^{+0.022}$ | ${0.0312}_{-0.0032}^{+0.0031}$ | 8 | 16 | Y
260502142 | 10.58 | 0.328 | 0.32 | ${0.29556}_{-0.00018}^{+0.00023}$ | ${0.3975}_{-0.0065}^{+0.0056}$ | ${-0.0278}_{-0.0019}^{+0.0025}$ | 9 | 17 | Y
260540780 | 10.30 | 0.382 | 0.32 | ${0.399407}_{-0.000075}^{+0.000099}$ | ${0.2607}_{-0.0028}^{+0.0034}$ | ${-0.0402}_{-0.0016}^{+0.0021}$ | 8 | 11 |
260542342 | 11.80 | 0.624 | 0.71 | ${0.67111}_{-0.00033}^{+0.00067}$ | ${0.591}_{-0.016}^{+0.021}$ | ${-0.0300}_{-0.0039}^{+0.0050}$ | 6 | 13 |
262614966 | 10.83 | 0.594 | 0.66 | ${0.60134}_{-0.00078}^{+0.00044}$ | ${0.618}_{-0.011}^{+0.012}$ | ${-0.0703}_{-0.0082}^{+0.0074}$ | 8 | 12 |
270503717 | 10.79 | 0.500 | 0.53 | ${0.49714}_{-0.00012}^{+0.00016}$ | ${0.5411}_{-0.0011}^{+0.0011}$ | ${-0.0423}_{-0.0011}^{+0.0012}$ | 11 | 17 |
270503717 | 10.79 | 0.952 | 2.12 | ${0.92867}_{-0.00064}^{+0.00089}$ | ${2.181}_{-0.032}^{+0.023}$ | ${-0.0310}_{-0.0097}^{+0.0126}$ | 5 | 7 |
271639931 | 8.08 | 0.380 | 0.33 | ${0.37788}_{-0.00023}^{+0.00016}$ | ${0.3284}_{-0.0046}^{+0.0048}$ | ${-0.021}_{-0.019}^{+0.021}$ | 4 | 6 |
271721220 | 10.56 | 0.622 | 1.87 | ${0.58873}_{-0.00040}^{+0.00043}$ | ${1.803}_{-0.021}^{+0.019}$ | ${0.0249}_{-0.0060}^{+0.0069}$ | 7 | 8 |
271721220 | 10.56 | 0.627 | 1.95 | ${0.59271}_{-0.00040}^{+0.00028}$ | ${1.785}_{-0.015}^{+0.017}$ | ${0.0564}_{-0.0060}^{+0.0049}$ | 6 | 8 |
271721220 | 10.56 | 0.318 | 0.14 | ${0.307272}_{-0.000017}^{+0.000016}$ | ${0.16913}_{-0.00069}^{+0.00064}$ | ${-0.02888}_{-0.00070}^{+0.00073}$ | 9 | 13 |
271723952 | 8.40 | 0.343 | 0.17 | ${0.33569}_{-0.00014}^{+0.00010}$ | ${0.1421}_{-0.0047}^{+0.0049}$ | ${0.0418}_{-0.0098}^{+0.0093}$ | 6 | 9 |
271723952 | 8.40 | 0.703 | 0.59 | ${0.66200}_{-0.00039}^{+0.00027}$ | ${0.7548}_{-0.0082}^{+0.0085}$ | ${-0.0457}_{-0.0019}^{+0.0026}$ | 11 | 21 |
272127517 | 9.35 | 0.914 | 0.99 | ${0.87120}_{-0.00029}^{+0.00061}$ | ${1.1182}_{-0.0160}^{+0.0055}$ | ${-0.0350}_{-0.0014}^{+0.0029}$ | 11 | 19 |
279055960 | 11.02 | 0.329 | 0.24 | ${0.327252}_{-0.000043}^{+0.000036}$ | ${0.24511}_{-0.00050}^{+0.00051}$ | ${-0.05659}_{-0.00091}^{+0.00098}$ | 6 | 15 |
279360930 | 11.30 | 0.290 | 0.66 | ${0.282510}_{-0.000066}^{+0.000049}$ | ${0.7006}_{-0.0017}^{+0.0017}$ | ${-0.0651}_{-0.0025}^{+0.0020}$ | 5 | 7 | Y
279510278 | 11.44 | 0.745 | 0.72 | ${0.75334}_{-0.00021}^{+0.00023}$ | ${0.7088}_{-0.0085}^{+0.0111}$ | ${-0.0159}_{-0.0058}^{+0.0081}$ | 8 | 9 |
293221812 | 11.20 | 0.346 | 0.73 | ${0.29479}_{-0.00017}^{+0.00013}$ | ${0.7541}_{-0.0052}^{+0.0051}$ | ${-0.0054}_{-0.0013}^{+0.0012}$ | 5 | 13 |
293221812 | 11.20 | 0.709 | 2.92 | ${0.79363}_{-0.00086}^{+0.00104}$ | ${2.890}_{-0.071}^{+0.088}$ | ${-0.0038}_{-0.0079}^{+0.0096}$ | 5 | 8 |
293270956 | 11.30 | 0.626 | 0.26 | ${0.61271}_{-0.00016}^{+0.00016}$ | ${0.2731}_{-0.0020}^{+0.0019}$ | ${-0.0125}_{-0.0016}^{+0.0019}$ | 9 | 22 |
293271232 | 7.51 | 0.595 | 2.27 | ${0.56875}_{-0.00033}^{+0.00046}$ | ${2.2763}_{-0.0082}^{+0.0080}$ | ${-0.0035}_{-0.0021}^{+0.0027}$ | 10 | 11 |
293271232 | 7.51 | 0.265 | 0.23 | ${0.251493}_{-0.000093}^{+0.000147}$ | ${0.2179}_{-0.0052}^{+0.0044}$ | ${0.0121}_{-0.0038}^{+0.0048}$ | 6 | 13 |
293273274 | 11.12 | 1.009 | 1.06 | ${1.03286}_{-0.00033}^{+0.00029}$ | ${1.066}_{-0.010}^{+0.010}$ | ${0.0049}_{-0.0031}^{+0.0029}$ | 7 | 13 |
293273274 | 11.12 | 0.623 | 0.68 | ${0.61201}_{-0.00033}^{+0.00040}$ | ${0.6845}_{-0.0025}^{+0.0027}$ | ${-0.00275}_{-0.00089}^{+0.00080}$ | 8 | 28 |
293345700 | 9.95 | 0.461 | 0.26 | ${0.462564}_{-0.000028}^{+0.000034}$ | ${0.2602}_{-0.0012}^{+0.0012}$ | ${-0.0091}_{-0.0035}^{+0.0024}$ | 6 | 8 |
293974233 | 11.32 | 0.466 | 2.69 | ${0.52847}_{-0.00048}^{+0.00088}$ | ${2.679}_{-0.034}^{+0.042}$ | ${-0.0016}_{-0.0044}^{+0.0057}$ | 8 | 9 | Y
294092361 | 10.56 | 1.070 | 0.79 | ${1.06922}_{-0.00019}^{+0.00017}$ | ${0.7843}_{-0.0059}^{+0.0063}$ | ${0.0449}_{-0.0085}^{+0.0091}$ | 7 | 7 |
300033585 | 9.99 | 0.378 | 1.06 | ${0.3734484}_{-0.0000094}^{+0.0000057}$ | ${1.08201}_{-0.00023}^{+0.00020}$ | ${-0.0587525}_{-0.0001301}^{+0.0000060}$ | 6 | 8 | Y
300033857 | 9.09 | 0.537 | 0.65 | ${0.537869}_{-0.000053}^{+0.000051}$ | ${0.64448}_{-0.00068}^{+0.00071}$ | ${-0.04022}_{-0.00041}^{+0.00042}$ | 11 | 18 | Y
300033857 | 9.09 | 0.673 | 1.18 | ${0.61989}_{-0.00029}^{+0.00023}$ | ${1.482}_{-0.015}^{+0.016}$ | ${-0.0671}_{-0.0041}^{+0.0036}$ | 6 | 8 | Y
300033857 | 9.09 | 0.288 | 0.50 | ${0.276815}_{-0.000045}^{+0.000041}$ | ${0.4853}_{-0.0036}^{+0.0026}$ | ${0.0140}_{-0.0033}^{+0.0034}$ | 4 | 7 | Y
300138080 | 11.64 | 0.529 | 0.70 | ${0.44776}_{-0.00043}^{+0.00047}$ | ${0.923}_{-0.013}^{+0.011}$ | ${-0.0314}_{-0.0015}^{+0.0019}$ | 7 | 19 |
300140867 | 9.84 | 0.738 | 1.39 | ${0.7326}_{-0.0011}^{+0.0014}$ | ${1.442}_{-0.016}^{+0.026}$ | ${-0.1091}_{-0.0048}^{+0.0087}$ | 9 | 16 |
300157862 | 11.08 | 0.259 | 0.89 | ${0.22706}_{-0.00042}^{+0.00043}$ | ${0.8509}_{-0.0058}^{+0.0060}$ | ${0.0126}_{-0.0019}^{+0.0021}$ | 18 | 21 |
300653681 | 10.25 | 0.771 | 1.58 | ${0.79646}_{-0.00064}^{+0.00068}$ | ${1.656}_{-0.022}^{+0.025}$ | ${0.0335}_{-0.0057}^{+0.0064}$ | 6 | 10 | Y
306631280 | 10.60 | 0.412 | 0.33 | ${0.43612}_{-0.00025}^{+0.00026}$ | ${0.3018}_{-0.0077}^{+0.0084}$ | ${-0.0120}_{-0.0038}^{+0.0037}$ | 9 | 18 |
349092320 | 9.15 | 0.751 | 1.07 | ${0.81021}_{-0.00073}^{+0.00064}$ | ${0.967}_{-0.014}^{+0.015}$ | ${-0.0213}_{-0.0022}^{+0.0023}$ | 12 | 22 |
349096085 | 11.22 | 0.438 | 2.00 | ${0.32557}_{-0.00049}^{+0.00059}$ | ${2.391}_{-0.027}^{+0.025}$ | ${-0.0399}_{-0.0027}^{+0.0030}$ | 6 | 10 | Y
349158735 | 10.76 | 0.505 | 0.43 | ${0.51496}_{-0.00026}^{+0.00025}$ | ${0.449}_{-0.013}^{+0.015}$ | ${0.018}_{-0.011}^{+0.011}$ | 6 | 9 |
349158735 | 10.76 | 0.752 | 1.62 | ${0.70333}_{-0.00046}^{+0.00053}$ | ${1.190}_{-0.026}^{+0.024}$ | ${0.1020}_{-0.0077}^{+0.0083}$ | 6 | 7 |
349310718 | 10.52 | 0.408 | 0.22 | ${0.391297}_{-0.000025}^{+0.000020}$ | ${0.28630}_{-0.00091}^{+0.00086}$ | ${-0.04142}_{-0.00053}^{+0.00056}$ | 8 | 13 |
349410895 | 9.45 | 0.244 | 0.42 | ${0.24126}_{-0.00018}^{+0.00012}$ | ${0.4155}_{-0.0078}^{+0.0078}$ | ${0.033}_{-0.041}^{+0.039}$ | 4 | 4 |
349521873 | 8.96 | 0.804 | 1.56 | ${0.86785}_{-0.00062}^{+0.00060}$ | ${1.419}_{-0.027}^{+0.032}$ | ${-0.0247}_{-0.0042}^{+0.0048}$ | 7 | 12 |
349680479 | 7.89 | 0.276 | 0.22 | ${0.265102}_{-0.000125}^{+0.000084}$ | ${0.1336}_{-0.0033}^{+0.0043}$ | ${0.0908}_{-0.0063}^{+0.0050}$ | 7 | 13 | Y
349680479 | 7.89 | 0.350 | 0.32 | ${0.35580}_{-0.00027}^{+0.00021}$ | ${0.3792}_{-0.0085}^{+0.0082}$ | ${0.121}_{-0.013}^{+0.011}$ | 6 | 11 | Y
349680479 | 7.89 | 0.428 | 0.73 | ${0.41565}_{-0.00019}^{+0.00020}$ | ${0.8063}_{-0.0068}^{+0.0069}$ | ${-0.0741}_{-0.0066}^{+0.0063}$ | 8 | 8 | Y
349683884 | 10.07 | 0.394 | 0.83 | ${0.418680}_{-0.000055}^{+0.000062}$ | ${0.7111}_{-0.0044}^{+0.0047}$ | ${-0.0574}_{-0.0017}^{+0.0018}$ | 7 | 8 | Y
349785797 | 10.44 | 0.323 | 0.27 | ${0.29155}_{-0.00012}^{+0.00016}$ | ${0.2384}_{-0.0035}^{+0.0030}$ | ${0.0100}_{-0.0013}^{+0.0015}$ | 10 | 22 |
349832567 | 10.80 | 0.448 | 0.25 | ${0.461516}_{-0.000011}^{+0.000014}$ | ${0.20589}_{-0.00030}^{+0.00035}$ | ${-0.03877}_{-0.00023}^{+0.00027}$ | 12 | 20 |
349835272 | 9.76 | 0.535 | 3.28 | ${0.516852}_{-0.000010}^{+0.000014}$ | ${3.38344}_{-0.00090}^{+0.00095}$ | ${-0.0668691}_{-0.0000012}^{+0.0000015}$ | 4 | 4 |
349835272 | 9.76 | 0.572 | 3.23 | ${0.60230}_{-0.00059}^{+0.00054}$ | ${3.331}_{-0.017}^{+0.016}$ | ${0.0368}_{-0.0026}^{+0.0022}$ | 7 | 11 |
349902873 | 8.62 | 0.287 | 0.33 | ${0.30225}_{-0.00012}^{+0.00013}$ | ${0.2972}_{-0.0044}^{+0.0045}$ | ${-0.0227}_{-0.0028}^{+0.0029}$ | 7 | 15 | Y
350092538 | 10.12 | 0.596 | 0.26 | ${0.599127}_{-0.000033}^{+0.000058}$ | ${0.25064}_{-0.00050}^{+0.00064}$ | ${-0.01811}_{-0.00053}^{+0.00080}$ | 5 | 17 |
350144504 | 7.99 | 0.329 | 0.48 | ${0.331860}_{-0.000062}^{+0.000064}$ | ${0.4833}_{-0.0038}^{+0.0039}$ | ${0.0357}_{-0.0063}^{+0.0070}$ | 5 | 6 |
350144504 | 7.99 | 0.664 | 1.02 | ${0.66329}_{-0.00030}^{+0.00050}$ | ${1.014}_{-0.016}^{+0.022}$ | ${0.107}_{-0.021}^{+0.037}$ | 5 | 5 |
350144657 | 10.81 | 0.513 | 0.27 | ${0.509384}_{-0.000028}^{+0.000028}$ | ${0.28170}_{-0.00027}^{+0.00028}$ | ${-0.03737}_{-0.00029}^{+0.00031}$ | 15 | 25 |
350295588 | 12.02 | 0.649 | 0.63 | ${0.66124}_{-0.00035}^{+0.00044}$ | ${0.6585}_{-0.0056}^{+0.0064}$ | ${0.0267}_{-0.0029}^{+0.0032}$ | 11 | 17 |
350295588 | 12.02 | 0.889 | 0.64 | ${0.91804}_{-0.00013}^{+0.00013}$ | ${0.5348}_{-0.0064}^{+0.0063}$ | ${-0.0421}_{-0.0023}^{+0.0023}$ | 7 | 11 |
350343297 | 9.00 | 0.538 | 0.78 | ${0.50172}_{-0.00026}^{+0.00027}$ | ${0.7801}_{-0.0037}^{+0.0045}$ | ${0.0010}_{-0.0012}^{+0.0012}$ | 9 | 21 | Y
350344057 | 6.94 | 0.263 | 0.11 | ${0.264377}_{-0.000042}^{+0.000061}$ | ${0.11675}_{-0.00091}^{+0.00095}$ | ${0.0176}_{-0.0027}^{+0.0032}$ | 7 | 14 | Y
350444342 | 8.07 | 0.375 | 0.69 | ${0.34963}_{-0.00335}^{+0.00074}$ | ${0.832}_{-0.021}^{+3.168}$ | ${-0.0677}_{-0.0040}^{+0.3677}$ | 9 | 9 |
350444342 | 8.07 | 0.745 | 0.61 | ${0.80028}_{-0.00051}^{+0.00050}$ | ${0.4924}_{-0.0035}^{+0.0113}$ | ${-0.02551}_{-0.00072}^{+0.00186}$ | 20 | 32 |
350477538 | 11.34 | 0.356 | 1.01 | ${0.33391}_{-0.00017}^{+0.00035}$ | ${0.925}_{-0.024}^{+0.018}$ | ${0.046}_{-0.013}^{+0.017}$ | 4 | 5 |
350715741 | 10.87 | 0.545 | 0.59 | ${0.545268}_{-0.000052}^{+0.000116}$ | ${0.59135}_{-0.00075}^{+0.00077}$ | ${-0.04540}_{-0.00031}^{+0.00028}$ | 7 | 28 | Y
350840969 | 11.36 | 0.693 | 0.58 | ${0.70792}_{-0.00022}^{+0.00028}$ | ${0.5386}_{-0.0060}^{+0.0091}$ | ${-0.0314}_{-0.0032}^{+0.0041}$ | 6 | 13 |
358181695 | 8.82 | 0.479 | 0.53 | ${0.45686}_{-0.00039}^{+0.00030}$ | ${0.707}_{-0.013}^{+0.015}$ | ${-0.0961}_{-0.0070}^{+0.0066}$ | 8 | 12 |
364325752 | 9.98 | 0.597 | 0.28 | ${0.590531}_{-0.000034}^{+0.000050}$ | ${0.27860}_{-0.00075}^{+0.00074}$ | ${0.00057}_{-0.00085}^{+0.00149}$ | 7 | 15 |
374944608 | 9.90 | 0.439 | 0.39 | ${0.453893}_{-0.000071}^{+0.000068}$ | ${0.3450}_{-0.0012}^{+0.0013}$ | ${-0.03955}_{-0.00069}^{+0.00067}$ | 15 | 22 |
374944608 | 9.90 | 0.844 | 1.38 | ${0.85144}_{-0.00033}^{+0.00030}$ | ${1.394}_{-0.016}^{+0.017}$ | ${0.0295}_{-0.0096}^{+0.0093}$ | 5 | 6 |
375038081 | 9.31 | 1.175 | 2.31 | ${0.77596}_{-0.00121}^{+0.00076}$ | ${3.489}_{-0.018}^{+0.025}$ | ${-0.03428}_{-0.00074}^{+0.00045}$ | 11 | 34 |
381950897 | 9.56 | 0.764 | 0.54 | ${0.740466}_{-0.000038}^{+0.000044}$ | ${0.4077}_{-0.0015}^{+0.0015}$ | ${0.06492}_{-0.00090}^{+0.00092}$ | 12 | 13 |
381950897 | 9.56 | 0.479 | 0.22 | ${0.470503}_{-0.000072}^{+0.000103}$ | ${0.2440}_{-0.0025}^{+0.0019}$ | ${-0.0391}_{-0.0022}^{+0.0027}$ | 8 | 16 |
382519218 | 9.55 | 0.316 | 0.20 | ${0.3012781}_{-0.0000070}^{+0.0000086}$ | ${0.27533}_{-0.00041}^{+0.00027}$ | ${-0.05858}_{-0.00025}^{+0.00035}$ | 7 | 12 |
382519218 | 9.55 | 0.347 | 0.21 | ${0.33893}_{-0.00014}^{+0.00018}$ | ${0.1958}_{-0.0087}^{+0.0086}$ | ${0.017}_{-0.013}^{+0.014}$ | 5 | 8 |
388131027 | 8.03 | 0.947 | 1.10 | ${0.94919}_{-0.00032}^{+0.00032}$ | ${1.0951}_{-0.0045}^{+0.0051}$ | ${-0.0266}_{-0.0023}^{+0.0025}$ | 11 | 13 |
391744540 | 8.69 | 0.502 | 0.44 | ${0.46847}_{-0.00021}^{+0.00022}$ | ${0.572}_{-0.010}^{+0.010}$ | ${-0.0460}_{-0.0038}^{+0.0040}$ | 6 | 15 | Y
391744540 | 8.69 | 0.378 | 0.61 | ${0.35510}_{-0.00020}^{+0.00031}$ | ${0.694}_{-0.017}^{+0.017}$ | ${-0.0416}_{-0.0098}^{+0.0095}$ | 5 | 7 | Y
391892842 | 8.09 | 0.383 | 0.35 | ${0.40119}_{-0.00030}^{+0.00016}$ | ${0.4199}_{-0.0044}^{+0.0031}$ | ${0.0439}_{-0.0021}^{+0.0011}$ | 15 | 28 |
391894459 | 9.31 | 0.436 | 1.30 | ${0.48116}_{-0.00024}^{+0.00028}$ | ${1.291}_{-0.014}^{+0.017}$ | ${-0.0021}_{-0.0029}^{+0.0036}$ | 7 | 11 | Y
407661375 | 10.04 | 0.715 | 0.46 | ${0.73411}_{-0.00022}^{+0.00033}$ | ${0.526}_{-0.019}^{+0.019}$ | ${0.0381}_{-0.0098}^{+0.0105}$ | 5 | 10 |
Table 3: continued.