# Short-term variability and mass loss in Be stars
VI. Frequency groups in $\gamma$ Cas detected by TESS

Jonathan Labadie-Bartz,1 Dietrich Baade,2 Alex C. Carciofi,1 Amanda Rubio,1
Thomas Rivinius,3 Camilla C. Borre,4 Christophe Martayan3 and Robert J.
Siverd5
1Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Universidade de
São Paulo, Rua do Matão 1226, Cidade Universitária, 05508-900 São Paulo, SP,
Brazil
2European Organisation for Astronomical Research in the Southern Hemisphere
(ESO), Karl-Schwarzschild-Str. 2, 85748 Garching b. München, Germany
3European Organisation for Astronomical Research in the Southern Hemisphere
(ESO), Casilla 19001, Santiago 19, Chile
4Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus
University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark
5Gemini Observatory/NSF’s NOIRLab, 670 N. A’ohoku Place, Hilo, HI, 96720, USA
E-mail<EMAIL_ADDRESS>(JLB)E-mail<EMAIL_ADDRESS>(DB)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

In photometry of $\gamma$ Cas (B0.5 IVe) from the SMEI and BRITE-Constellation
satellites, indications of low-order non-radial pulsation have recently been
found, which would establish an important commonality with the class of
classical Be stars at large. New photometry with the TESS satellite has
detected three frequency groups near 1.0 ($g1$), 2.4 ($g2$), and 5.1 ($g3$)
d-1, respectively. Some individual frequencies are nearly harmonics or
combination frequencies but not exactly so. Frequency groups are known from
roughly three quarters of all classical Be stars and also from pulsations of
$\beta$ Cep, SPB, and $\gamma$ Dor stars and, therefore, firmly establish
$\gamma$ Cas as a non-radial pulsator. The total power in each frequency group
is variable. An isolated feature exists at 7.57 d-1 and, together with the
strongest peaks in the second and third groups ordered by increasing frequency
($g2$ and $g3$), is the only one detected in all three TESS sectors. The
former long-term 0.82 d-1 variability would fall into $g1$ and has not
returned at a significant level, questioning its attribution to rotational
modulation. Low-frequency stochastic variability is a dominant feature of the
TESS light curve, possibly caused by internal gravity waves excited at the
core-envelope interface. These are known to be efficient at transporting
angular momentum outward, and may also drive the oscillations that constitute
$g1$ and $g2$. The hard X-ray flux of $\gamma$ Cas is the only remaining major
property that distinguishes this star from the class of classical Be stars.

###### keywords:

Stars: massive - stars: mass loss - stars: emission-line, Be - stars:
oscillations - stars: individual: $\gamma$ Cassiopeiae, HR 2817, HD 71042, HD
144555, HD 156172

††pubyear: 2020††pagerange: Short-term variability and mass loss in Be stars
VI. Frequency groups in $\gamma$ Cas detected by TESS–B.4

##  1 Introduction

The eponymous emission lines of classical Be stars (for a broad review see
Rivinius et al., 2013) form in a rotationally supported equatorial disk that
unequivocally consists of material ejected from the star itself. Although Be
stars rotate closer to the critical rate than all other stars on and near the
main sequence, additional energy and angular momentum are required to lift
matter into an orbit. Under the influence of viscous friction (Haubois et al.,
2012; Rímulo et al., 2018) and radiative ablation (Kee et al., 2018), Be disks
decay on time-scales of months to years. Much of the necessary regular
replenishment of the disk seems to take place in the form of stellar
outbursts. From space photometry, evidence has been found (Baade et al.,
2018b, a; Baade & Rivinius, 2020) that the superposition of non-radial
pulsation (NRP) modes can lead to non-linear coupling, resulting in enhanced
amplitudes that are much larger than the sum of the individual mode
amplitudes. In this way, enough energy may be available and released for the
ejection of matter into the disk. A small part of the ejecta accumulates the
necessary specific angular momentum through viscous redistribution while most
of the rest falls back to the star (Okazaki et al., 2002; Haubois et al.,
2012).

The largest sample of Be stars with time-series photometry from space is
currently being assembled by the TESS satellite (described in Sect. 3.1). A
first reconnaissance analysis by Labadie-Bartz & Carciofi (2020) found a
considerable diversity of variability types (see also Balona & Ozuyar, 2020).
One of them is the presence of distinct frequency groups, here understood as
an assembly of frequencies (3 or more) that are usually no more than a few
tenths of d-1 from each other. They occur in about three quarters of all
classical Be stars. Their fraction drops from more than 80% among early-type
Be stars to less than 60% at late types. However, the decline is also a matter
of instrumental sensitivity, as the photometric amplitudes fall sharply
towards the late portion of the B spectral type (e.g. Saio et al., 2007).

Frequency groups in Be stars have also been detected with all other stellar
space photometers that have observed Be stars, namely MOST (Walker et al.,
2005), SMEI (Goss et al., 2011), CoRoT (Semaan et al., 2018), Kepler (Kurtz et
al., 2015), and BRITE-Constellation (Baade et al., 2018a). In the majority of
cases, three main groups are found: one at very low frequencies (hereafter
$g0$, $\sim 0.05$ d-1), one centered roughly between 0.5 and 3 d-1 (hereafter
$g1$), and the third one (hereafter $g2$) located at approximately twice the
frequencies of the middle group and often very crudely twice as wide as the
middle one. Additional frequency groups may also exist, typically near integer
multiples of $g1$ and $g2$, and usually with relatively lower amplitudes.
During outbursts of Be stars, the appearance of frequency groups can change
drastically (Huat et al., 2009).

Pápics et al. (2017) and Baade et al. (2018a) have suggested, and partly
illustrated by examples, that, in Be stars, the central group ($g1$) may
consist of genuine pulsation frequencies, the low group ($g0$) may be mainly
populated by difference frequencies of the middle group, and the high group
($g2$) may be primarily comprised of sum frequencies and first harmonics, also
of the middle group. Frequency groups have also been reported for $\beta$ Cep,
SPB, and $\gamma$ Dor stars (McNamara et al., 2012) so that they are not
specific to Be stars and not even restricted to B-type stars. Recently, Saio
et al. (2018b) developed the idea that frequency groups form when rapid
rotation displaces the eigenfrequencies of sectoral modes such as to coincide
- and resonate - with combination and/or harmonic frequencies of the base
modes.

This paper reports the detection of frequency groups in the prominent Be star
$\gamma$ Cas. Sect. 2 introduces $\gamma$ Cas and similar stars and discusses
the spectroscopic and photometric history of the system up until the epoch of
TESS. The TESS satellite and its observations, and the extraction of data and
their processing are discussed in Sect. 3. The time-series analysis (TSA) of
TESS data is performed in Sect. 4, Sect. 5 discusses the results and places
them in context with comparisons to (recent) historical data for $\gamma$ Cas
and other Be stars, and Sect. 6 summarizes the conclusions.

##  2 $\gamma$ Cas

###  2.1 Previous observations and $\gamma$ Cas-like stars

Since $\gamma$ Cas (27 Cas, HR 264, HD 5394, HIP 4427, TIC 51962733; B0.5 IVe)
was the first Be star discovered (Secchi, 1866), it was only natural to
consider it the prototypical Be star. However, Harmanec (2002) compiled
several observations that make the prototype character of $\gamma$ Cas less
obvious but by no means implausible. $\gamma$ Cas could be a relatively
extreme representative of its cohort, not least because hardly any other Be
star has been observed as intensively as $\gamma$ Cas has. More challenging to
$\gamma$ Cas’s status were (i) the ubiquity of low-order NRPs in Be stars
(Rivinius et al., 2003; Semaan et al., 2018), which, in $\gamma$ Cas, remained
undetected for a long time (Borre et al., 2020), and (ii) the presence in
$\gamma$ Cas of a thermal X-ray flux at a level intermediate between that from
shocks in winds of luminous OB stars and that from accretion in high-mass
X-ray binaries but exhibiting a hardness without equivalent in other early-
type stars (Smith et al., 2016). These X-ray properties are shared by a small
but still growing fraction of Be stars (Nazé et al., 2020a).

The apparent restriction to Be stars of such X-ray properties suggests that
the circumstellar disk plays a role in their formation. Smith et al. (2016)
describe the notion, developed in several earlier papers by the same authors
and their collaborators, that the X-rays are due to the interplay between two
magnetic fields, one in the disk and one in the photosphere which, however,
are not directly detected. The stellar one was proposed to manifest itself
indirectly by rotational modulation of the stellar brightness. The
corresponding 0.82 d-1 frequency was detectable for a decade but disappeared
later (Henry & Smith, 2012; Borre et al., 2020). The field in the disk was
assumed to lead to magneto-rotational instability (MRI) so that gas parcels
may fall back to the star, on the way to the photosphere are accelerated by
the postulated stellar magnetic field, and finally release their gravito-
kinetic energy as X-rays. Recent observations of the $\gamma$ Cas-like star
$\pi$ Aqr showed that a strong long-term increase in the H $\alpha$ line
emission was not accompanied by a similar change in the X-ray flux (Nazé et
al., 2019).

In the picture of Smith et al. (2016) for $\gamma$ Cas, the 0.82 d-1 rotation
rate is important for two reasons. First, it is very nearly the star’s
critical rotation frequency which Smith et al. invoke to activate effective
local envelope dynamos. Second, the highest possible stellar rotation rate is
needed to explain the rate of propagation of migrating subfeatures in spectral
lines as corotation of exophotospheric cloudlets with the photosphere. It
should be noted that several of the data strings discussed by Smith et al. in
this regard hardly cover a single rotation period and even less. If these
features are due to traveling waves such as high-order NRP and possess their
own prograde angular velocity, the frequency window of the possible stellar
rotation rate obviously becomes much wider. The model does not state which
physical quantity is advected by the rotation and leads to the photometric
variability nor does it explain the high phase coherence.

Near the main sequence, $\sim$10% of all massive OBA stars possess large-scale
magnetic fields (e.g. Grunhut et al., 2017; Sikora et al., 2019). The most
notable exception to this rule are classical Be stars for which a survey of 85
objects did not yield a single detection (down to a detection limit of $\sim$
50 – 100 Gauss; Wade et al., 2016). This is generally understood as an
incompatibility of a stable Keplerian disk with large-scale magnetic fields of
$\sim$100 Gauss or stronger (Owocki, 2006; ud-Doula et al., 2018). Because the
small-scale magnetic fields invoked for X-ray production in $\gamma$ Cas-like
stars (i.e. classical Be stars with hard thermal X-ray flux) are too ill
constrained by observations, it is not known whether they would lead to a
similarly prohibitive magnetic coupling between star and disk.

Furthermore, the current magnetic concept does not incorporate the presence of
a companion star that was discovered in some of these so-called $\gamma$ Cas-
like stars, including $\gamma$ Cas itself (Nemravová et al., 2012) and $\pi$
Aqr (Bjorkman et al., 2002). The companions have not so far been seen directly
in any part of the spectrum, but have been inferred from radial velocity
measurements of the Be star or its disk. Under the assumption that the
secondaries are compact objects, some of the X-ray observations have been
attributed to accretion (Hamaguchi et al., 2016; Postnov et al., 2017;
Tsujimoto et al., 2018). A more Be-star-specific model was recently developed
by Langer et al. (2020) which explains the X-ray flux by the interaction of
the fast wind of a helium star with the Be disk. While the Smith et al. model
predicts that all (early-type) Be stars may develop $\gamma$ Cas-like X-ray
activities, the Be + helium-star and Be + compact object models are obviously
only applicable if the assumed companion exists. Such companions may be
numerous but are not easy to detect (Wang et al., 2018; Klement et al., 2019;
Langer et al., 2020).

The first possible evidence of NRP in $\gamma$ Cas was presented by Ninkov et
al. (1983) and later ascertained by Yang et al. (1988) using additional
observations. It consisted of multiple narrow features traversing photospheric
line profiles from the most negative to the most positive velocities. The
inferred azimuthal mode orders, $|m|$, were $\sim 12\pm 4$. From a short
series of spectra, Horaguchi et al. (1994) confirmed the presence of the line-
profile variability a few years later. While rotational Doppler spreading
enhances the spectroscopic visibility of high-order NRP modes, their
photometric effects are strongly diluted in integral light. Low-order modes
($m\approx-2$) that are typical of many other (not quite as hot) Be stars
(Rivinius et al., 2003; Labadie-Bartz et al., 2017; Semaan et al., 2018) were
not reported. Furthermore, ground-based photometry did not indicate
multiperiodicity (Henry & Smith, 2012), the tell-tale signature of NRP.
Therefore, Borre et al. (2020) analyzed medium-high-cadence photometry by
several satellites with a time span of 8+4 years (16 years in total). The
observations confirmed that the 0.82 d-1 variability (Smith et al., 2016) has
faded below detectability even from space. Instead, variability with
frequencies of 1.25 and 2.48 d-1 was found, thereby probably re-uniting
$\gamma$ Cas with its non-radially pulsating brethren because both frequencies
are too high to be due to any rotational modulation. However, with just two
frequencies, one of which (1.25 d-1) was only marginally detected, the
evidence for NRP was not overwhelmingly broad, especially since it could not
be confirmed that 2.48 d-1 is associated with the B-star primary and not with,
e.g., the rotation of its invisible companion.

In a recent remarkably thorough study of the $\gamma$ Cas-like stars $\pi$ Aqr
and BZ Cru, Nazé et al. (2020b) found strong spectroscopic and photometric
evidence of high-order prograde tesseral NRP modes ($l\sim 6\pm 2,|m|\sim
2^{+2}_{-1}$) with frequencies that fall into the range of p modes ($\sim$ 7 –
12 d-1). These rapid line profile variations in $\pi$ Aqr were first reported
in Rivinius et al. (2003), and were later estimated to correspond to an
$l=-m=5$ prograde mode in Peters & Gies (2005). $\pi$ Aqr was observed during
very different states of the disk but the spectroscopic signature of the
migrating subfeatures was unaffected. Therefore, the variability is stellar,
and the range in velocity of these features shows that they are associated
with the Be star, not its companion, which is less easily deduced from
photometry only. Nazé et al. do not link the high-order non-radial pulsations
to the X-ray properties of $\gamma$ Cas-like stars but insist that the - in
grayscale plots of difference spectra - very similar variability on comparable
time-scales to $\gamma$ Cas itself is the manifestation of corotating
exophotospheric cloudlets. Interestingly, the authors report that the
difference between two frequencies in $\pi$ Aqr near 11.8 d-1 corresponds to
the 3.1 yr interval between two outbursts with amplitude $\sim$0.25 mag during
the long course of which some pulsation amplitudes changed. This is similar to
what has been described by Baade et al. (2018b, a) for other Be stars and can
involve similarly long time-scales (Baade et al., 2018c). It would be the
first such case linked to p mode pulsations. However, Nazé et al. caution that
partial obscuration of the stellar disk by the circumstellar disk may be the
cause.

Balona & Ozuyar (2020) have proposed another model with magnetic fields that
are too weak for detection with current means. The model is suggested for Be
stars at large and assumes a tilted dipole field that causes matter ejected by
reconnection events from magnetic spots to accumulate at two opposite
locations in the equatorial plane. Depending on the relative strength of these
two clouds, variations with the stellar rotation frequency or its first
harmonic may occur. Because the trapped matter is in all coordinates only
loosely locked and its amount will not be constant, coherent variations are
not predicted. The model may not easily explain the mostly sinusoidal shape of
the light curves of many periodic Be stars.

Although Balona & Ozuyar do not point this out, the phenomenon they describe
seems to have its spectroscopic counterpart in Štefl frequencies (Štefl et
al., 1998; Baade et al., 2016). Their signature is the variability in anti-
phase with respect to each other of double emission peaks in the extreme wings
of He i and other lines and occurs in phases of enhanced mass loss when there
is enough gas to produce helium line emission. Štefl frequencies are typically
$\sim 10$% lower than that of the dominating spectroscopic quadrupole $g$
mode. Therefore, Štefl frequencies are thought to form in newly ejected matter
orbiting in the inner disk that is not yet equalized in azimuth. They would,
therefore, be unrelated to the actual stellar rotation but to orbital periods
in the disk. However, they would be asymptotic tracers of the critical stellar
rotation rate, as it sets an upper limit for the Štelf frequency. For the same
reasons as in the magnetic model of Balona & Ozuyar, phase coherent
variability is not expected. In spite of the large physical differences
between the two competing explanations of the Štefl frequencies, the expected
high similarity of the photometric behavior seems to suggest that photometry
alone is not likely to be able to distinguish between them. However, this is
only true if the discussion focuses on just one frequency. For a more complete
understanding of Be stars, the full ensemble of frequencies needs to be
considered.

One important point which plain rotating models necessarily fail to account
for is the large-scale, large-amplitude photospheric velocity field that is
particularly prominent at low inclinations. However, this characteristic line-
profile variability is well described by non-radial g mode pulsation (Maintz
et al., 2003).

###  2.2 Recent spectroscopic and photometric context

The viscous decretion disk (VDD; Lee et al., 1991; Carciofi, 2011) model
successfully describes the majority of observed properties of Be star disks
and their variability with time, including the regions in which certain
observables are formed and the associated time-scale and magnitude of changes
in these observables during epochs of disk build up or dissipation (Haubois et
al., 2012; Haubois et al., 2014). Of particular relevance to the present study
of $\gamma$ Cas, the visible continuum arises in the inner-most disk (with
about 1 – 2 stellar radii) due to an increased continuum emission by the
ejecta, mostly due to hydrogen free-bound recombination and also electron
scattering, and can change rapidly in response to a change in the environment
near to the star. By contrast, the H $\alpha$ line is formed over a large
region of the disk where viscous time-scales are much longer, and small
additions of material (relative to the already large amount of H
$\alpha$-emitting material) contribute little to the overall line strength.

The H $\alpha$ emission has been gradually and steadily trending upward from
about 2001 until the epoch of TESS, growing in the absolute value of the
equivalent width (EW) from $\sim$ 25 Å to 37 Å and in the flux ratio of
emission-peak to continuum (E/C) from $\sim$ 3.5 to 5.5 (Pollmann, 2009;
Nemravová et al., 2012; Borre et al., 2020). However, a disk has existed for
decades even prior to 2001 (Doazan et al., 1983; Harmanec, 2002). This trend
has continued beyond the TESS observations reported in this paper as seen in
recent spectra in the Be Star Spectra Database (BeSS111http://basebe.obspm.fr;
Neiner et al., 2011). In other words, the disk of $\gamma$ Cas was strong and
apparently growing during the TESS observations, having been built up over
decades, and the amount of material in the disk at the time of the TESS
observations was higher than during the period studied by Borre et al. (2020).
In a strong and dense disk such as this the photometric sensitivity à la VDD
to additional matter injected may be (partly) saturated when the scattering of
stellar photons may increase underproportionally (Ghoreyshi et al., 2018).
This would (possibly strongly) dilute the photometric signature of any
discrete or short-lived episodes of enhanced mass ejection, provided the
geometry of the inner disk is not significantly altered. However this effect
would not drastically reduce the variability that arises in the photosphere
(e.g. pulsation).

##  3 Observations and data reduction

###  3.1 The TESS observatory

The four identical 10 cm cameras of the Transiting Exoplanet Survey Satellite
(TESS; Ricker et al., 2015) cover a combined field of view of 24∘ $\times$ 96∘
that extends from the ecliptic plane to one of its poles. With 13 steps in
ecliptic longitude, one hemisphere is observed in one year. Depending on
ecliptic latitude, stars are observed in up to 13 such sectors, each spanning
$\sim$27.4 d in time.

The focal-plane array of each TESS camera comprises a 2 $\times$ 2 mosaic of
CCDs with 2048 $\times$ 2048 pixels each and records light over the range 600
to 1050 nm, which includes the conventional $I$ and $Z$ and portions of the
$V$, $R$, and $Y$ bands. 90% of the flux contained in the point-spread
function (PSF) is ensquared within 4 $\times$ 4 pixels, each of which measure
0.35 arcmin on a side.

Stars brighter than $\sim$7.5 in $I_{\rm C}$ saturate the pixel at the center
of the PSF on the TESS CCDs (but precise light curves are still produced for
stars much brighter than this; see Sect. 3.3). For optimal targets, the noise
floor is approximately 60 ppm h-1. Photon noise dominates down to $I_{\rm C}$
$\approx$ 13.5 mag. Owing to its very wide and elongated orbit, TESS can
perform uninterrupted observations for more than 300 h, is not plagued by
near-Earth radiation belts, and suffers only little from terrestrial and lunar
stray light (which is the main source of systematic trends). The Full Frame
Images (FFIs) for the entire CCD are available at a 30-minute cadence from
which light curves can be extracted for any star observed by TESS, and a
selection of targets are also observed in a 2-minute cadence mode.

###  3.2 TESS observations of $\gamma$ Cas

The ranges in Barycentric Julian Date (BJD), the TESS sectors, the CCDs, and
the number of data points used for time-series analysis (TSA) are provided in
Table 1. $\gamma$ Cas was observed in two consecutive sectors (50.3 days;
Sectors 17 and 18), followed by a 141-day gap and was again observed in Sector
24 for an additional 26.5 days. This makes it impossible to trace any
variations with the binary orbital period of 203.55 d (Nemravová et al.,
2012). $\gamma$ Cas was not among the targets pre-selected for 2-minute
cadence observations, so the full frame images with 30-minute cadence,
available for download at MAST222https://mast.stsci.edu/, were instead used
(TESS Input Catalog ID: TIC 51962733).

Table 1: TESS observation of $\gamma$ Cas.  BJD start | BJD end | Sector | Camera | CCD | nobs
---|---|---|---|---|---
2458764.69 | 2458789.69 | 17 | 2 | 1 | 1070
2458790.67 | 2458815.02 | 18 | 2 | 2 | 1017
2458955.80 | 2458982.26 | 24 | 4 | 4 | 1225

###  3.3 Data extraction and processing

For each sector, a target pixel file from the TESS FFIs centered on $\gamma$
Cas was downloaded with TESScut333https://mast.stsci.edu/tesscut/ (Brasseur et
al., 2019). The size in pixels was chosen to adequately capture the extent of
the bloom columns, which span up to 101 rows (20 $\times$ 130 pixels for
Sectors 17 and 18, and 20 $\times$ 150 pixels for Sector 24). Although
$\gamma$ Cas is very bright and highly saturates the CCDs, it is still
possible to extract a reliable, high-quality light curve. Two different
methods are viable. Since the TESS CCDs are charge conserving, an aperture
mask that encloses all charge-containing pixels is one option. Alternatively,
a light curve can be extracted from the PSF wings, excluding saturated pixels
and columns (i.e. ‘halo photometry’). Both extraction methods were performed,
and are further described in Appendix A. The resulting light curves are very
similar in that they contain the same features. However, the version extracted
using all charge-containing pixels is of higher quality (having higher signal-
to-noise and being less affected by systematics such as scattered light) and
is adopted for the remainder of this work. Both versions of the light curve
are compared in Appendix A.

The resulting light curve is shown in the top panel of Fig. 1. In order to
facilitate the extraction of periodic signals, each sector of data was
detrended to remove low frequency signals with $f<0.2$ d-1. The red curve
shown in the upper panel of Fig. 1 represents these low frequency signals,
which are subtracted to produce the detrended version of the light curve that
is shown in the second panel of Fig. 1. This detrended version is used to
calculate the amplitude spectra and periodic signals discussed throughout this
work. Because data for each sector must be extracted separately, it is
difficult to determine if trends with time-scales similar to the sector
duration ($\sim$25 days) are astrophysical or systematic in nature. This is
discussed further in Secs. 2.2 and 5.2 and Appendix A, especially in regards
to the apparent brightening trend in Sector 18.

Within 20 arcmin from $\gamma$ Cas, SIMBAD (Wenger et al., 2000) lists two
stars with known magnitude brighter than 10 mag: the A5 star BD+59 148 ($V$ =
9.76 mag, distance: 708") and the F3 star BD+59 158 ($V$ = 9.81 mag, 738").
Both of these sources are sufficiently distant from $\gamma$ Cas and the bloom
columns caused by saturation that they do not contribute flux to the pixels in
the aperture masks used for either version of light curve extraction. This was
verified visually for all sectors.

Figure 1: Top: TESS light curve of $\gamma$ Cas showing Sectors 17 and 18
(left) and 24 (right) in black, with low frequency trends traced in red. There
is a 141 day gap between Sectors 18 and 24. Sector 18 data are vertically
offset from their mean to appear continuous with Sector 17, while Sectors 17
and 24 have their mean value subtracted. Second from top: The light curve
after removing low frequency trends with time-scales greater than five days.
The remaining panels show the wavelet analysis of the detrended light curve.
Note that the color scale in the third, fourth, and fifth panels from the top
are independent of each other, in order to see the relatively low amplitudes
of the two higher frequency groups. However, the color scale is consistent
across the full time baseline. Horizontal dashed lines mark the location of
the strongest signal in each of the four groups. Hashed regions are gaps in
the observations.

##  4 Time-series analysis

A frequency analysis was performed for the entire set of TESS observations,
and for each sector separately. The Period04 package (Lenz & Breger, 2005) was
used, and the amplitude spectra are shown in Fig. 2 in black.

Because of the short TESS observing baseline, the window function includes
significant side-lobes (Fig. 2, inset), which overlap when multiple signals
exist at similar frequencies (as is the case with the frequency groups). This
can cause spurious peaks in the periodogram with power comparable to real
signals. Pre-whitening against each signal is a more reliable approach for
determining the underlying frequencies relative to the Lomb-Scargle analysis
without pre-whitening. However, some degree of spurious peaks is unavoidable
given the nature of variability in $\gamma$ Cas.

To determine the individual frequencies present in the data, the Vartools
light curve analysis software (Hartman, 2012) with the Lomb-Scargle routine
(Zechmeister & Kürster, 2009; Press et al., 1992) was used to detect and
iteratively pre-whiten the data against each recovered frequency up to a false
alarm probability (FAP) of 10-2 (in a similar fashion as Rivinius et al.,
2016), and is also shown in Fig. 2 as vertical red lines (again for the full
light curve and each sector separately). The frequency, phase, amplitude, SNR,
FAP and group membership are printed in Table 2, with the phase being measured
relative to the time of the first observation of the data analyzed (BJD =
2458764.69 for Sector 17 and the full light curve, 2458790.67 for Sector 18,
and 2458955.80 for Sector 24). The noise floor is frequency dependent (i.e.
the frequency spectrum has a red noise component), and a threshold of four
times the noise level (further discussed and defined in Sec. 4.3) is used to
determine the signals reported in Table 2.

Both methods, with and without pre-whitening, are generally consistent in
their identification of the dominant signals. Whenever specific values for
frequency or amplitude are used in the following, they are from the pre-
whitening analysis of the combined light curve extracted from all count-
containing pixels unless otherwise stated. Formal error estimates are given
for the frequency values, according to the method of Montgomery & O’Donoghue
(1999), which assumes Gaussian uncorrelated noise. These, however, represent
lower limits for the following reasons. All but the highest frequency signals
detected in each light curve exist in densely populated groups, where there is
significant overlap between the window function of adjacent signals, adding
some degree of additional uncertainty in the values of the recovered frequency
and amplitude as first explored in Loumos & Deeming (1978). The variable
amplitude of the groups further complicates this. In terms of error
estimation, the frequency groups, although astrophysically real, can be
considered as a form of correlated noise, meaning that error estimates
computed analytically under the assumption of Gaussian uncorrelated noise will
be too low. For these reasons, in general the most relevant quantities here
are not necessarily the frequency and amplitude of specific frequencies, but
more so the location, strength, width, structure, and variability of the
groups themselves. The most notable exceptions to this are the strongest
signal at $f=2.4796(1)$ d-1, and the isolated high frequency signal at
$f=7.571(1)$ d-1, which are introduced below.

###  4.1 Frequency groups

There are two main frequency groups. The first (group 1 = $g1$) is centered
around 0.998 d-1 (the strongest signal being at $f=1.1346(6)$ d-1) and the
second group, $g2$, is centered around 2.38 d-1 (with its strongest signal at
$f=2.4796(1)$ d-1). A third group ($g3$) centered near 5.07 d-1 is also
present (the strongest signal being at $f=5.0559(5)$ d-1), as is an isolated,
single signal at $f=7.571(1)$ d-1 ($g4$)444Although only one signal is
detected in this ‘group,’ for the sake of consistency in nomenclature this is
hereafter referred to as $g4$.. These groups were identified visually from the
amplitude spectrum, and are consistent with the dominant peaks identified
through standard frequency analysis methods (i.e. those reported in Table 2).
The centers of $g1$, $g2$, and $g3$ reported above are the average of the
frequencies (weighted by amplitude) reported for the ‘All Sectors’ light curve
listed in Table 2.

$g3$ appears much narrower than $g1$ and $g2$ (although this could be a matter
of instrumental sensitivity), with a width of approximately 0.04 d-1 (taken to
be the difference between the highest and lowest frequency signals for the
full light curves shown in Table 2), compared to widths of $\sim$0.3 d-1 for
$g1$ and $\sim$0.7 d-1 for $g2$. Note, however, that this convention likely
underestimates group widths, as the wings at the edges of the groups
(especially in $g1$ and $g2$) are not accounted for. There are no obvious
harmonic relationships between the strongest signals of these groups. $g2$ is
centered around slightly more than twice the frequency of $g1$, and $g3$ is
slightly higher than twice the frequency of $g2$. The signal at 7.571(1) d-1
likewise is not obviously harmonically related to any other frequencies, but
is 1.4975(2) times the frequency of the signal at $f=5.0559(5)$ d-1, and is
very close to the sum of the strongest signals in $g2$ and $g3$, differing by
0.036(1) d-1. It is however possible for more complex combination frequencies
to manifest in the power spectrum of pulsators, including classical Be stars
(e.g. Kurtz et al., 2015). Partly owing to a dearth of well-defined
frequencies that stand out in $g1$ and $g2$, no such attempt was made to
robustly test combination frequencies in the TESS data for $\gamma$ Cas. There
are no signals detected at frequencies higher than 7.6 d-1; with the 30-minute
cadence, the Nyquist frequency is 24 d-1.

In addition to the clearly dominant signal of 2.4796(1) d-1, there are two
other frequencies in $g2$ that seem to stand out, these being the second and
third highest amplitude signals detected in the amplitude spectrum for the
full light curve (see Table 2). While these signals are closely spaced,
differing by 0.0581(4) d-1, and are not totally resolved in the individual
sectors, each sector has signals that appear to correspond to one or both of
these frequencies (and they are also present in the sum and product of the
amplitude spectra of all 3 sectors). These two signals are marked with short
black vertical lines in Fig. 2.

Because of the short observational baselines, and the fact that data reduction
must be done separately for each sector, there is significant difficulty in
measuring signals with periods greater than $\sim$10 days in TESS data. Other
studies find a low frequency group ($g0$) in a large fraction of Be stars,
typically located around 0.05 d-1 (see Sec. 1), which can neither be confirmed
nor ruled out in the present analysis of $\gamma$ Cas. Likewise, variability
associated with the orbital period of 203.55 d for $\gamma$ Cas is
inaccessible with TESS. However, the overall brightening trends seen in
Sectors 17 and 18 (Fig. 1) may be real (see Sec. 5.2).

Figure 2: A representation of the power spectrum computed from a classical
Lomb-Scargle treatment (black) and through applying the Lomb-Scargle technique
with iterative pre-whitening (red vertical lines). The dotted green vertical
lines mark the historical frequencies at 0.82 d-1, 1.25 d-1, and 2.48 d-1, and
the two short vertical black lines mark the location of the second and third
strongest signals in $g2$. The blue shaded regions indicate the frequency
groups (but do not quantitatively reflect their widths as defined in Sec.
4.1). The green curve shows the detection threshold used for the signals given
in Table 2, as defined in Sections 4.1 and 4.3. From top to bottom, the panels
show Sectors 17, 18, 24 and the full light curve, respectively. The lower-most
panel presents the multiplication and mean (with inverted amplitude) of the
periodograms from Sectors 17, 18 and 24. The window function for each is
centered at 6 d-1, and with an arbitrary vertical scale. There are no signals
detected beyond 8 d-1. The grey bar from 0 to 0.2 d-1 identifies the region
where low frequency trends were removed prior to the frequency analysis.

###  4.2 Group variability

The position of the groups does not vary significantly between Sectors 17, 18,
and 24, but their strength and/or their composite frequencies do (with the
exception of $g4$). The most notable aspects of this variation are the
increase in strength of $g1$ from Sector 17 to 18, followed by a decrease
between Sectors 18 and 24, and the decrease in strength of $g2$ from Sector 17
to Sector 18 followed by a return to near its original strength in Sector 24.
The most significant signal in $g3$ (near 5 d-1) decreases in amplitude by
$\sim$40% from Sector 17 to 18, and also increases back towards its original
strength in Sector 24. The signal near 7.6 d-1 ($g4$) may decrease in
amplitude slightly after Sector 17, but this is less certain and of a lower
level compared to the variability of the other groups.

In an attempt to coarsely visualize where the variability does or does not
appear coherent, the lower panel of Fig. 2 compares the product of the
amplitude spectra of Sectors 17, 18, and 24 to their sum. The product
emphasizes Lomb-Scargle features that are persistent in time, i.e., are strong
at all times and do not significantly shift in frequency. This view of the
amplitude spectrum illustrates a few aspects of the data. Especially in
regards to $g1$ and $g2$, the densely populated groups and relatively wide
window function lead to a situation where the amplitude spectrum within a
group never approaches zero. In $g1$ there are no signals that clearly stand
out above the ‘continuum’ of the group. In $g2$, the three strongest signals
are apparent in both the sum and product amplitude spectrum. Likewise, in $g3$
and $g4$, the structure appears the same in the sum and product versions,
suggesting stability of the frequencies (but not necessarily the amplitudes)
of these signals.

A wavelet analysis was performed to determine how the power of the detected
frequencies varies with a higher temporal resolution compared to simply
computing the power spectrum separately for the different sectors. The Python
package scaleogram555https://github.com/alsauve/scaleogram was used for this
purpose. The wavelet analysis is shown in Fig. 1. The variability in $g1$,
$g2$, and $g3$ is apparent. However, the structure of the variable group power
has some complexity. With many frequencies in $g2$, a complex beating pattern
in the light curve can manifest in the wavelet plot as a variable group
amplitude without a well defined period. The same logic may apply to the
variable amplitude of $g1$, however there is an absence of any coherent
dominant signals. The structure of $g3$ is relatively simple. In the amplitude
spectrum of the full light curve, $g3$ is made up of two signals spaced by
0.0351(1) d-1. The isolated signal at $f=7.571(1)$ d-1 is generally constant
from sector to sector, but appears mildly variable in strength on shorter
time-scales. This may simply be an artifact owing to its low amplitude, or any
even lower amplitude signals below the detection threshold in the vicinity.

###  4.3 Stochastic variability

In a study of 70 OB stars observed with TESS, each member of the sample
exhibits a certain level of stochastic variability that manifests as red noise
in their photometric amplitude spectra (Bowman et al., 2020). $\gamma$ Cas is
no exception to this rule. Fig. 3 shows the amplitude spectrum of $\gamma$ Cas
on a log-log scale, along with the red noise profile as defined below. The red
noise profile is then multiplied by 4 which is used as a threshold666This
threshold, determined from the combined light curve, is also applied to each
individual sector. for the signals reported in Table 2, above which the
signals identified through pre-whitening are shown. The morphology of the
amplitude spectrum carries information about the physical origin of the
stochastic variation. Following the prescription of Bowman et al. (2019a, b,
2020), in order to parameterize this signal for convenient comparison to
recent models, a functional fit of the form (Bowman et al., 2020, their
equation 2)

$\alpha(\nu)=\frac{\alpha_{0}}{1+(\frac{\nu}{\nu_{char}})^{\gamma}}+C_{w},$
(1)

is applied, where $\alpha_{0}=0.136$ ppt is the amplitude at $\nu=0$ d-1,
$\gamma=2.958$ is the logarithmic amplitude gradient, $\nu_{char}=3.618$ d-1
is the characteristic frequency of stochastic variability, and $C_{w}=0.012$
ppt is the constant white noise component. The above function with these
values is shown as the solid red line in Fig. 3. These values for these
parameters are the result of a Markov chain Monte Carlo (MCMC) fit following
the method of Bowman et al. (2020). Nazé et al. (2020c) fitted the same
function but deduced fairly different parameter values. Several circumstances
probably contributed to the difference: Nazé et al. (2020c) applied a much
simpler signal-extraction method, which clips much of the valid photometric
signal (see Appendix A), used Sectors 17 and 18 only (i.e., without Sector
24), joining them in a different fashion (in their light curve, there is a
large jump between the two sectors), and apparently did not remove trends and
frequency groups prior to the red-noise measurements (only the strongest
signal each in $g2$ and $g3$ were removed). The consequences of this are their
higher values for $\alpha_{0}$ (0.31 ppt) and $C_{w}$ (0.025 ppt), and lower
values for $\gamma$ (1.82) and $\nu_{char}$ (1.15 d-1).

The parameters derived here for $\gamma$ Cas are typical for the sample of
Bowman et al. (2020), where it was determined that internal gravity waves
excited by core convection are the most likely physical explanation for this
feature given the values and ranges of $\alpha_{0}$, $\gamma$, and
$\nu_{char}$ for the sample (Rogers et al., 2013; Edelmann et al., 2019; Horst
et al., 2020; Ratnasingam et al., 2020). The red noise characteristics of
$\gamma$ Cas also lend themselves to the same conclusion, since internal
gravity waves are predicted to have values of $0.8\,\leq\gamma\,\leq\,3$, and
still contribute power to the frequency spectrum even out towards 100 d-1.
Although Bowman et al. (2020) prefer the internal gravity wave explanation, it
remains plausible that sub-surface convection zones can produce similar
observables in OB stars (Lecoanet et al., 2019, although it seems difficult
for sub-surface convection zones to produce an amplitude excess out to the
observed relatively high frequencies). Furthermore, there are no classical Be
stars among the 15 B dwarf stars in the Bowman et al. (2020) sample, and the
mean $v\,sin\,i$ value of the 15 B dwarfs is only 40 km s-1 (with a maximum of
162 km s-1) – far lower than typical values of the near-critically rotating Be
stars. It is therefore not a foregone conclusion that internal gravity waves
are a ubiquitous feature of classical Be stars, as they appear to be for the
more modestly rotating collection of Bowman et al. (2020).

Figure 3: Amplitude spectrum of $\gamma$ Cas. The green curve is identical to
what is plotted in Fig. 2 for the combined light curve, but on a log-log
scale. The black curve is computed in the same way, but after pre-whitening
against the signals in the frequency groups, which is then fit with Equation 1
(solid red line). The white noise floor is marked by a horizontal dashed line.
The red noise profile is then multiplied by a factor of 4 (dotted red line),
which is used as the threshold above which the periodic signals identified
through iterative pre-whitening are considered to be significant (which are
given in Table 2). Shaded rectangles approximately mark the frequency groups.
Table 2: The frequencies above four times the red noise level are listed here,
ordered by frequency group and amplitude. The SNR is calculated in the
standard way in the vartools package by comparing the Lomb-Scargle power of a
given peak to the mean value of the periodogram after applying iterative
5-sigma clipping. The logarithm of the formal false alarm probability (FAP) is
also shown. The information in this table is from iterative pre-whitening of
the light curve extracted from all count-containing pixels. Formal error
estimates for the frequency determined according to Montgomery & O’Donoghue
(1999) are given in braces in units of the last significant digit. However,
these should be understood as lower limits (see Section 4). Errors are not
provided for amplitude or phase, because the variability of the signals makes
such determinations difficult and somewhat arbitrary.

Group | Freq. | Amp. | Phase | SNR | log(FAP)
---|---|---|---|---|---
| (d-1) | (ppt) |  |  |
Sector 17
$g1$ | 1.069(7) | 1.04 | -0.037 | 329.2 | -15.8
$g1$ | 1.008(7) | 1.00 | -0.114 | 344.2 | -13.0
$g1$ | 0.585(9) | 0.82 | -0.337 | 189.5 | -10.5
$g1$ | 0.90(1) | 0.74 | -0.189 | 152.5 | -10.2
$g1$ | 1.26(1) | 0.72 | -0.465 | 147.0 | -10.2
$g1$ | 0.80(1) | 0.64 | 0.100 | 109.2 | -9.4
$g2$ | 2.472(2) | 3.65 | 0.129 | 5224.6 | -85.8
$g2$ | 2.323(3) | 2.27 | -0.082 | 1934.1 | -52.5
$g2$ | 2.375(7) | 1.03 | 0.348 | 355.1 | -12.8
$g2$ | 2.194(9) | 0.85 | 0.324 | 226.4 | -10.8
$g2$ | 2.113(9) | 0.79 | -0.499 | 176.8 | -10.2
$g2$ | 2.44(1) | 0.76 | 0.263 | 161.8 | -9.9
$g2$ | 2.27(1) | 0.69 | 0.384 | 124.6 | -9.8
$g2$ | 1.95(1) | 0.60 | -0.339 | 106.8 | -10.0
$g2$ | 2.07(1) | 0.60 | 0.225 | 102.2 | -10.3
$g2$ | 2.63(1) | 0.59 | 0.083 | 99.6 | -8.8
$g2$ | 2.32(1) | 0.49 | -0.093 | 59.4 | -7.4
$g3$ | 5.051(7) | 1.04 | 0.124 | 325.1 | -12.1
$g3$ | 5.09(1) | 0.48 | 0.052 | 53.9 | -7.5
$g3$ | 4.86(3) | 0.24 | -0.449 | 13.5 | -4.3
$g3$ | 5.02(3) | 0.23 | 0.355 | 12.5 | -3.8
$g4$ | 7.56(2) | 0.37 | -0.028 | 26.6 | -5.6
Sector 18
$g1$ | 1.115(4) | 1.85 | 0.125 | 1309.9 | -38.1
$g1$ | 0.967(8) | 1.00 | -0.179 | 274.1 | -13.5
$g1$ | 1.253(8) | 0.96 | 0.241 | 242.2 | -14.7
$g1$ | 0.788(8) | 0.96 | 0.487 | 239.7 | -16.2
$g1$ | 0.85(1) | 0.75 | 0.468 | 126.6 | -10.9
$g1$ | 1.03(1) | 0.72 | 0.136 | 108.4 | -11.3
$g2$ | 2.485(3) | 2.56 | -0.326 | 2176.9 | -54.9
$g2$ | 2.286(6) | 1.24 | 0.066 | 498.6 | -19.6
$g2$ | 2.417(8) | 0.98 | 0.117 | 260.5 | -14.2
$g2$ | 2.18(1) | 0.80 | 0.002 | 160.8 | -11.6
$g2$ | 2.06(1) | 0.64 | 0.009 | 99.6 | -9.4
$g2$ | 2.68(1) | 0.57 | 0.445 | 72.3 | -8.6
$g2$ | 2.13(1) | 0.50 | 0.187 | 45.3 | -7.5
$g2$ | 2.50(1) | 0.47 | 0.366 | 37.1 | -7.9
$g3$ | 5.05(1) | 0.64 | 0.235 | 100.0 | -10.0
$g3$ | 5.14(2) | 0.33 | -0.175 | 17.2 | -4.7
$g3$ | 5.09(3) | 0.25 | -0.477 | 12.1 | -3.2
$g4$ | 7.56(2) | 0.31 | -0.381 | 16.7 | -4.5

Group | Freq. | Amp. | Phase | SNR | log(FAP)
---|---|---|---|---|---
| (d-1) | (ppt) |  |  |
Sector 24
$g1$ | 1.148(8) | 0.80 | -0.006 | 354.0 | -19.4
$g1$ | 0.816(9) | 0.69 | -0.085 | 265.3 | -16.9
$g2$ | 2.479(1) | 3.69 | 0.176 | 7640.2 | -134.1
$g2$ | 2.314(4) | 1.42 | 0.305 | 1134.1 | -38.9
$g2$ | 2.277(5) | 1.25 | 0.024 | 920.6 | -35.3
$g2$ | 2.411(6) | 0.94 | -0.067 | 515.3 | -22.2
$g2$ | 2.619(9) | 0.73 | 0.329 | 285.2 | -17.3
$g2$ | 2.57(1) | 0.61 | -0.135 | 194.8 | -15.0
$g2$ | 2.53(1) | 0.60 | 0.252 | 195.1 | -13.2
$g2$ | 1.98(1) | 0.58 | 0.000 | 163.5 | -14.1
$g2$ | 2.05(1) | 0.58 | -0.344 | 166.8 | -15.9
$g3$ | 5.053(7) | 0.91 | -0.133 | 476.7 | -23.1
$g3$ | 5.09(1) | 0.44 | 0.283 | 89.9 | -10.3
$g3$ | 5.13(2) | 0.24 | -0.385 | 24.5 | -5.4
$g3$ | 5.19(3) | 0.19 | 0.193 | 15.6 | -4.1
$g4$ | 7.57(2) | 0.29 | -0.343 | 36.1 | -5.9
All Sectors
$g1$ | 1.1346(6) | 0.71 | 0.245 | 572.6 | -21.8
$g1$ | 1.0206(7) | 0.62 | 0.133 | 709.6 | -25.5
$g1$ | 0.8099(8) | 0.59 | -0.350 | 373.0 | -16.1
$g2$ | 2.4796(1) | 3.67 | -0.218 | 13244.3 | -256.9
$g2$ | 2.3244(3) | 1.30 | -0.400 | 2356.9 | -68.0
$g2$ | 2.2663(4) | 1.01 | 0.442 | 1289.5 | -40.7
$g2$ | 2.4161(5) | 0.95 | -0.483 | 487.1 | -19.0
$g2$ | 2.2902(7) | 0.69 | 0.255 | 476.6 | -19.9
$g2$ | 2.5421(8) | 0.56 | -0.085 | 348.7 | -16.1
$g2$ | 2.6331(8) | 0.55 | 0.034 | 297.8 | -14.8
$g2$ | 2.1187(9) | 0.53 | 0.041 | 390.1 | -16.1
$g2$ | 1.9452(9) | 0.52 | -0.142 | 297.6 | -14.2
$g2$ | 2.3607(9) | 0.50 | -0.257 | 577.8 | -21.6
$g3$ | 5.0559(5) | 0.90 | -0.382 | 934.0 | -28.9
$g3$ | 5.091(1) | 0.45 | 0.319 | 269.8 | -14.1
$g4$ | 7.571(1) | 0.30 | -0.442 | 103.4 | -8.1

##  5 Discussion

Besides the stochastic variability discussed in Sec. 4.3 and emphasized in
Fig. 3, the frequency groups $g1$ and $g2$ are the dominant features in the
TESS light curve of $\gamma$ Cas. These complexes are much wider than the core
of the window function (Fig. 2) and, therefore, unambiguously identify
$\gamma$ Cas as a multi-mode non-radial pulsator (since it is not plausible
that all of the periodic photometric signals are circumstellar or directly
related to rotation considering the time-scales and lack of exact harmonics).
The visualisations of the variability as two sets of many individual
frequencies (Fig. 2) with strong mutual beat patterns in light curves (Fig. 1,
top panel) and as wavelet transforms of four frequency ranges with strong
amplitude variations (Fig. 1, bottom panels) are complementary presentations
of the same behavior (Rivinius et al., 2016). The considerable variability in
the total power of each frequency group revealed by Fig. 1 (bottom panels) is
matched by changes in the appearance of the light curve (top panels). Because
this activity does not seem to be periodic (although a time-scale of $\sim 5$
d may be indicated for $g2$), more than two frequencies must be involved,
provided that the frequencies are constant.

Considering the wavelet analysis in Fig. 1, the variations in power of the
groups are not globally correlated. Although $g3$ and $g4$ appear at first
glance to be qualitatively different from $g1$ and $g2$, the former pair being
much narrower and relatively simple, this could just be a matter of reduced
photometric amplitude with increasing frequency, and the noise floor of the
data. Although $g3$ only has two confidently detected frequencies in the
framework of the amplitude spectrum of the full light curve, its variability
seen in the wavelet analysis of Fig. 1 suggests a higher degree of complexity.
If $g3$ only contained these two frequencies, a beat period of 28.5 days is
expected. While this seems to be approximately realized in the TESS data ($g3$
is stronger in the first half of each $\sim$27 day-long sector, and weaker in
the second), $g3$ also appears to be variable on shorter time-scales, which
could be an artifact of its somewhat low amplitude or intrinsic variability in
the amplitudes of one or both main frequencies, or it could reflect the
presence of additional nearby frequencies with low amplitudes, analogous to
the structure of $g2$. Fig. 3 hints at the latter, where there seems to be
structure above the Fourier ‘continuum’ in the vicinity of $g3$ that extends
beyond the two frequencies identified in Fig. 2 and Table 2. Comparative
conjectures concerning $g4$ are even more difficult, again owing to its low
amplitude. Despite this, it is reasonable to claim that $g4$ is located at
approximately three times the frequency of the strongest signal in $g2$ (and
about 1.5 times the frequency of the strongest signal in $g3$), that $g4$ does
not appreciably vary in frequency or amplitude from sector to sector, and that
there does not seem to be any further structure besides the single detected
peak above the noise level in Fig. 3.

###  5.1 Stochastic variation, internal gravity waves, angular momentum
transport, mode excitation, and Rossby waves

Especially since internal gravity waves excited by core convection in
differentially rotating massive stars are predicted to be efficient at
transporting angular momentum from the core outwards (Aerts et al., 2019; Lee
& Saio, 2020), this is an important topic to consider in Be stars, whose
envelopes are somehow made to spin rapidly. If, as seems likely for many other
OB stars, a spectrum of internal gravity waves is the origin of the bulk of
the stochastic variability in $\gamma$ Cas (Sec. 4.3), this can aid in
transporting angular momentum to the envelope, which would otherwise spin down
as the star evolves and the envelope expands (and loses angular momentum to
winds and mainly to the disk in the case of Be stars). If angular momentum is
transported to the outer layers of an already rapidly rotating star, then
outbursts may be a way to remove this excess angular momentum, thus allowing
the star to avoid super-critical rotation (Krtička et al., 2011; Georgy et
al., 2013; Rímulo et al., 2018; Baade & Rivinius, 2020).

A complication in the picture of the amplitude spectrum of $\gamma$ Cas (and
the majority of all early-type Be stars) is the presence of periodic signals
in the standard g mode regime (namely, $g1$ and $g2$). However, this
delineation is blurred by rapid rotation. With a spectral type of B0.5 IVe,
$\gamma$ Cas is too hot for the $\kappa$ mechanism to drive g modes in the
standard view of low frequency pulsation in SPB stars ($f\lesssim 4$ d-1), yet
the bulk of its periodic signals lie in this regime. Global Rossby waves (r
modes) are expected and observed in moderately to rapidly rotating stars (Van
Reeth et al., 2016; Saio et al., 2018a), which tend to form frequency groups
just below the rotation frequency (for $m=1$) and/or just below twice the
rotation frequency (for $m=2$). In all the examples of Saio et al. (2018a),
including one Be star, the $m=1$ group is both predicted and observed to be
significantly stronger than the $m=2$ group. In the case of $\gamma$ Cas, $g2$
is stronger than $g1$ (or even in the most conservative estimates they are of
similar strength if the strongest signals in $g2$ are discounted), suggesting
that $g1$ and $g2$ are not simply the manifestations of two groups of global r
modes of $m=1$ and $m=2$. This bears resemblance to the Be star analyzed in
Saio et al. (2018a), KIC 6954726, which has a first frequency group centered
near 0.9 d-1 that is consistent with r modes of $m=1$, and a second group
centered near 1.7 d-1 (and of similar strength to the first group) that is
inconsistent with $m=2$ r modes (both in location and relative strength). As
discussed in Rivinius et al. (2016), this star was undergoing a quasi-
continuous series of small outbursts at the time of the Kepler observations so
that circumstellar activity likely contributes to aspects of the amplitude
spectrum. r modes are necessarily retrograde in the co-rotating frame of the
star, and thus do not positively contribute to the outward angular momentum
transport (and in fact can slightly decrease the angular momentum in near-
surface layers).

It is possible that low-frequency g modes are excited by resonant coupling
with internal gravity waves excited by core convection (Lee & Saio, 2020). The
role of observable periodic signals excited in this way were explored for the
Be star HD 51452 (B0 IVe), observed with the CoRoT satellite (Neiner et al.,
2012). Like other early-type Be stars, the most prominent periodic signals are
at frequencies between 0.5 – 3 d-1, and are inconsistent with rotation and p
mode oscillations. The proposed origin of these signals are stochastic
gravito-inertial modes that are excited by core convection and modified by the
Coriolis acceleration, in addition to possible Rossby waves (r modes) driven
by the Coriolis acceleration. HD 51452 is also mentioned in Saio et al.
(2018a), where the authors suggest odd and even r modes of $m=1$ as sufficient
to explain the observed frequency groups without the need to invoke internal
gravity waves driven by core convection as an excitation mechanism.

A recent study of the Be star HD 49330 (B0.5 IVe) by Neiner et al. (2020)
demonstrates all the salient points of this section. HD 49330 has similarities
to $\gamma$ Cas, being of the same spectral type and having frequencies in
both the g and p mode regimes, although its disk is far weaker and less
persistent compared to $\gamma$ Cas. In an effort to understand the observed
amplitude spectrum of HD 49330 and its correlations with outbursts observed by
CoRoT (Huat et al., 2009; Floquet et al., 2009), Neiner et al. (2020) modeled
various aspects of the system and found that stochastically excited internal
gravity waves act to transport angular momentum to the surface layers, which
then become destabilized and oscillate in stochastically driven g modes,
whereupon outbursts can be triggered. Neiner et al. (2020) conclude that the
$\kappa$ mechanism drives the observed p modes, but cannot drive the observed
g modes. Instead, the g modes are excited by stochastic internal gravity
waves.

The stochastic low frequency excess observed in the amplitude spectrum of
$\gamma$ Cas likely reflects the presence of internal gravity waves driven at
the inner boundary of the radiative zone. This spectrum of internal gravity
waves can excite g mode pulsation similar to the case of HD 49330, which can
explain the presence of $g1$ and $g2$ in $\gamma$ Cas. At the same time, r
modes may (and probably do) exist which can also contribute to the observed
frequency groups. Circumstellar activity may also cause photometric
variability in the vicinity of $g1$ and $g2$. The $\kappa$ mechanism is likely
the driving force behind the higher frequency signals in $g3$ and $g4$ (under
the assumption they are p modes). Given that the disk of $\gamma$ Cas was
growing at the time of the TESS observations, it is possible that all three of
these processes (stochastically driven g modes, r modes, and circumstellar
activity) are contributing to $g1$ and $g2$. Detailed modeling and
spectroscopic identification of the line profile variability associated with
these frequencies is needed to determine the physical processes responsible
for, and their relative contribution to, the observed amplitude spectrum of
$\gamma$ Cas.

###  5.2 Mass ejection and comparison to other early-type Be stars

The signals recovered in the TESS data of $\gamma$ Cas are perfectly ordinary
compared to those found in large samples of classical Be stars (Labadie-Bartz
& Carciofi, 2020; Balona & Ozuyar, 2020). There are a number of different
types of variability seen in the brightness of Be stars, including frequency
groups as well as individual isolated frequencies, stochastic variability, and
longer time-scale (a few days and longer) changes in the mean brightness. In
some Be stars monitored with the BRITE-Constellation satellites, mean
brightness variations by up to a few per cent and on time-scales of weeks to
months have been observed at times when the amplitude of $g1$ modes was
temporarily increased. In some instances, the growth in amplitude appeared
larger, and was more peaked in time, than would follow from the mere temporary
agreement in phase (beating) of frequencies and can be due to non-linear
amplification (Baade et al., 2016, 2018b; Baade & Rivinius, 2020). The
proposed explanation of the variable brightness (Haubois et al., 2012;
Ghoreyshi et al., 2018) is in terms of discrete mass-loss events that are
driven by the non-linear amplitude superposition of several NRP modes and lead
to an increased continuum emission by the ejecta. However, if the matter is
ejected into the line of sight, the total brightness can be reduced.

In spite of the - in comparison to BRITE and SMEI (e.g. Borre et al., 2020) \-
much shorter time baseline of less than a month (mostly one TESS sector),
about one-quarter of early-type Be stars observed by TESS (B0 – B3) show
variability in their mean brightness on time-scales of a few days or longer
and with amplitudes typically of a few percent (Labadie-Bartz & Carciofi,
2020; Labadie-Bartz et al., 2020). TESS data for four early-type Be stars are
shown in Appendix B to illustrate this point and serve as a comparison to
$\gamma$ Cas. In most, but not all, cases of days-long net brightening events,
they are accompanied by increased amplitudes of one or more of the main
frequency groups (mostly $g1$ or $g2$). By analogy to the much slower and
often higher-amplitude events that are well understood to trace the build up
of the circumstellar disk (e.g. Baade et al., 2016, 2018b; Labadie-Bartz et
al., 2018), these more rapid changes in mean brightness are likely the result
of varying amounts of material being lifted above the photosphere. This leads
to the notion that, as the result of the temporary phase-coherent combination
of multiple NRP modes, active Be stars can quasi-permanently (when averaged
over months or years) eject matter that can be picked up by viscosity to build
a disk. The process may be stochastic as already discussed by Kurtz et al.
(2015).

The TESS light curve of $\gamma$ Cas does not clearly invite for a similar
conclusion because the variations in mean brightness do not include clearly
defined extrema (the mean brightness may also be affected by the signal and
background extraction). Accordingly, the time intervals of temporary in-phase
superposition of multiple NRP modes that stand out as bright blobs in the
wavelet transform but are also visible in the light curve (see bottom and top
panels, respectively, of Fig. 1) do not clearly correlate with changes in the
mean brightness.

The lack of such a correlation neither invalidates the interpretation given
above of the cases presented in Appendix B nor is it expected that the
correlation is seen in TESS data of a given Be star at all times or in all Be
stars. In fact, the saturation effect described in Sect. 2.2 even implies that
in stars with a very dense inner disk small mass-ejection events may become
photometrically very insignificant. At the time of the TESS observations, the
disk of $\gamma$ Cas was clearly very strongly developed. Furthermore, there
can also be extended time intervals with significant mass loss but so small
changes in its rate that the light curve is smooth over the duration of a
single TESS sector (Labadie-Bartz et al., 2017; Rímulo et al., 2018). Any
superimposed discrete mass-loss events may also be so separated in time that a
single TESS sector does not capture one of them. Nevertheless, it is well
possible that the longer term increase in brightness seen in Sectors 17 and 18
reflects enhanced mass loss rates over this time period. However, this is more
ambiguous than the cases discussed in Appendix B, where the evidence for brief
mass ejection events is less hampered by a strong disk that has been growing
for decades.

###  5.3 Historical and new photometric frequencies

There is no clear indication in TESS Sectors 17, 18, or 24 of the $1.215811\pm
0.000030$ d period (0.82250(2) d-1 frequency) found by Henry & Smith (2012)
with an initial amplitude of $\sim$5-7 ppt and confirmed by Borre et al.
(2020) to have faded and eventually dropped below detectability. In the
combined TESS data, the closest signal appears at 0.8099(8)d-1 and is the
third strongest signal in $g1$ with an amplitude of 0.59 ppt. Formally, the
0.82 d-1 frequency has not, therefore, returned although, in view of the
challenges discussed above of accurately and consistently measuring a
frequency in a highly variable frequency group, the uncertainty may be larger
than the numbers suggest. Smith et al. (2016) do not identify the physical
property that is supposedly rotationally modulated with 0.82 d-1, and they do
not explain the phase coherence reported by Smith et al. (2006) which is not
an obvious property of weak, stochastically generated magnetic structures.
Therefore, there is no reason to pick a convenient frequency in $g1$ and call
it the stellar rotation frequency. In fact, from an investigation of TESS
light curves of 15 $\gamma$ Cas stars including $\gamma$ Cas itself, Nazé et
al. (2020c) do not report a single rotation frequency.

As Borre et al. (2020) found, the 0.82 d-1 frequency seems to have given way
to a newly developing variability with frequency 2.47936(2) d-1 (which falls
into $g2$) and time-dependent amplitudes between 1.5 and 4.5 mmag (ranging
between about 1 – 3 mmag in the red filter of BRITE and 3 – 5 mmag in the blue
filter). This amplitude difference considering the red and blue filters is
expected for pulsation in hot stars, as photometric amplitudes are higher at
shorter wavelengths. TESS found the latter of these signals still present in
all sectors and at 2.4796(1) d-1 in their combination. In agreement with Borre
et al. (2020) and the group behavior described above, the amplitude was
variable between 2.6 and 3.7 ppt but was at all times the largest of all
individual signals (Table 2). Borre et al. (2020) determined that the 0.82 d-1
and 2.48 d-1 frequencies are not 1:3 harmonics, having a conservatively
estimated ratio of 3.0158(1). The TESS value for the latter frequency is in
agreement with this conclusion. The 1.25 d-1 variability tentatively
identified by Borre et al. (2020) is not confirmed by TESS (see Fig. 2). The
detection was declared tentative because the feature appeared surrounded by
others in the power spectrum. In this region, TESS now found group $g1$. The
strongest signal in $g3$, 5.0559(5) d-1, seen by TESS is also apparent in the
blue-filter BRITE data at a similar amplitude (about 1 ppt, Borre et al.,
2020). But, owing to its low SNR and proximity to an integer multiple of 1
d-1, the signal could not be unambiguously attributed to $\gamma$ Cas. Since
TESS does not suffer from daily aliases, this problem is avoided and the 5.06
d-1 signal tentatively reported in Borre et al. (2020) is confirmed.

Frequencies persistently present during the TESS observations were the
strongest feature in $g2$ at 2.4796(1) d-1 (and, more tentatively, at
2.3244(3) and 2.2663(4) d-1) and the two strongest ones in $g3$ at 5.0559(5)
and 5.091(1) d-1 as well as the seemingly isolated frequency at 7.571(1) d-1
(see Table 2 and Fig. 2). Of these variabilities, the last one had the most
constant amplitude. Nazé et al. (2020c) recently analyzed identical TESS data,
but only for Sectors 17 and 18 and extracting the signal in a much less
sophisticated way than described in Appendix A. They found frequencies of
2.480, 5.054, and 7.572 d-1 but do not give error estimates or amplitudes.

Frequency group $g1$, which in $\gamma$ Cas includes the 0.82 and 1.25 d-1
frequencies, is, in other Be stars, near its lower limit often home to the
Štefl frequencies (Sect. 2.1) that are thought to trace activity in the
innermost disk. In the latter, only in Sector 18 a feature at 1.115(4) d-1
stands out above the remainder of $g1$ (Table 2). The apparent lack of Štefl
frequencies in photometry, however, does not exclude ongoing star-to-disk mass
transfer. Because Štefl frequencies may arise in newly ejected matter not yet
in circularized orbits and/or not yet homogeneously distributed in azimuth,
the indicated high inner-disk density could accelerate this process and reduce
its amplitude. Frequent small events would further equalize the matter density
in azimuth and circularize the orbits more quickly. The photometric signature
of Štefl frequencies is also more likely to manifest in systems with higher
inclination angles, whereas the intermediate inclination angle of $\gamma$ Cas
($\sim 42^{\circ}$, Stee et al., 2012) can be a hindrance to their detection.
Furthermore, such circumstellar signals could exist among $g1$ and/or $g2$,
but their identification as such would require spectroscopic confirmation.

Yang et al. (1988) reported periods between 7000 and 10000 s for the traveling
subfeatures in spectral lines (high-order NRP modes). The corresponding
frequency range from 8.6 to 12.3 d-1 in the TESS amplitude spectra is devoid
of significant features, indicating long-term amplitude variations or
cancellation of high-spatial-frequency signals in integral-light data.

##  6 Conclusions

The presence of frequency groups proves that $\gamma$ Cas pulsates in low-
order NRP modes because frequency groups have been found and attributed to
low-order NRP in many Be stars, in $\beta$ Cep and SPB stars, and even outside
the B-star domain in $\gamma$ Dor stars (cf. Introduction). Another
commonality suggested by the TESS observations is the involvement of NRPs in
several discrete mass-loss events within a month in some Be stars (Appendix
B). At the time of the TESS observations, the strongly developed disk of
$\gamma$ Cas probably diluted the photometric evidence of individual events
while the mean star-to-disk mass-transfer rate must have been high to let the
disk grow even further. Considering all other available observed properties,
these findings reduce to a minimum the qualitative difference between $\gamma$
Cas and the class of classical Be stars as a whole. This concerns the
properties of central stars and circumstellar disks alike as well as the
physical processes governing them. Nazé et al. (2020c) recently arrived at a
similar conclusion for a sample of 15 Be stars sharing the X-ray properties of
$\gamma$ Cas.

The establishment of low-order NRP also reinforces the spectroscopic
identifications of higher-order NRP (Yang et al., 1988) that have also been
seen in other Be (Vogt & Penrod, 1983; Kambe et al., 1997) and non-emission-
line B stars (Smith, 1985). It is, therefore, very difficult to instead
imagine undetectable magnetic fields producing azimuthally roughly equidistant
super-photospheric cloudlets (Smith et al., 2016). Even more challenging is it
to anchor anything in a stellar atmosphere such that it does not show
detectable phase wobble. If the principle is that same objects require same
explanations, $\gamma$ Cas is to be explained by what explains other Be stars.
Since most other Be stars do not require an explanation of hard X-rays,
because they do not emit them, and there is nothing in the standard toolbox
for single Be stars suggesting that X-rays should arise anyway, the X-ray
properties of $\gamma$ Cas need to be explained by something that is not
needed to understand the basics of Be stars. A good candidate is the presence
of helium-star companions (Langer et al., 2020, and references therein) that
are tracers of an important formation channel of Be stars.

The picture conveyed by Appendix B is that, in their equatorial regions, some
Be stars are continuously bubbling like some hot springs on Earth. The part
made visible by TESS goes down to small events of up to a week (very few
stellar rotations) in duration. In addition, there can be sometimes cyclically
repeating major events and a putative floor of activity powered by high-order
NRPs and turbulence. The relative contributions to the total mass loss to the
disk are unknown. In $\gamma$ Cas, the star-to-disk mass-transfer rate seems
to have increased in recent years because the profile of the emission in H
$\alpha$ has slowly grown in EW and E/C and has changed from flat-topped to
peaked, indicating an overall growth in disk mass and an increased density in
the outer disk (Labadie-Bartz et al. in prep).

The 2.48 d-1 and 5.06 d-1 signals have probably persisted from 2015 (Borre et
al., 2020) through 2020 (TESS). The 0.82 d-1 frequency found between 1997 and
2011 by Henry & Smith (2012) has not returned in 2019/2020. It falls into $g1$
which exhibited amplitude variability over the TESS timeline. Nevertheless, at
$\sim 6.5$ mmag, the initial amplitude of the 0.82 c/d variability was large.
This frequency seems too high for the Štefl frequency of $\gamma$ Cas which is
at the extreme high end of possible stellar rotational frequencies (Henry &
Smith, 2012). Like 2.48 d-1, it is likely just one of several frequencies in
its parental group, and there is no reason to single it out for specific
interpretative purposes. The two magnetic models for Be stars zoom into just
one frequency and, therefore, are not likely to capture the full picture. Most
notably, they do not seem to offer provisions for the (non-linear) coupling of
several frequencies to initiate mass-loss events. The case of $\pi$ Aqr (Nazé
et al., 2020b) may show that $\gamma$ Cas-like stars are not exempt from this.
0.82 d-1 and 2.48 d-1 are confirmed by Borre et al. (2020) to be not obviously
numerically related despite being nearly at a 1:3 ratio.

The lack of phase coherent variability in $g1$ seems in agreement with the
nondetection of spectroscopic signatures of low-order NRP modes in $\gamma$
Cas. On the other hand, the yearslong presence at high amplitude of the 0.82
c/d frequency in $g1$ indicates that the nature of $g1$ may be more mixed than
it appeared at the time of the TESS observations.

The amplitude spectrum of $\gamma$ Cas contains frequencies well above the
conventional g mode domain. This includes $g3$ near 5.1 d-1 and the seemingly
solitary feature at 7.57 c/d. The latter should be outside the domain of
rotationally split low-order g modes. Because $\gamma$ Cas is close to the hot
limit of the $\beta$ Cephei instability strip, it may be a hybrid $\beta$
Cep/Be(SPB) pulsator (Moździerski et al., 2019). This property is shared by
about one-quarter to one-third of all Be stars in the $\beta$ Cephei domain
(Labadie-Bartz & Carciofi, 2020; Balona & Ozuyar, 2020). Like other early-type
Be stars with pulsation in the traditional g mode domain, it remains to be
explained what mechanism(s) act to drive these modes. Excitement by internal
gravity waves is one attractive possibility (Neiner et al., 2020). Angular-
momentum transport by gravity waves may aggravate the possible surface-
angular-momentum crisis of Be stars (Krtička et al., 2011; Rímulo et al.,
2018). Mass loss driven by NRP would avoid this problem, so that NRP modes
might couple such as to enable mass loss (Baade & Rivinius, 2020). r modes may
also contribute to these lower frequency groups.

## Acknowledgements

The authors are grateful for the comments from the anonymous referee, which
have improved the manuscript. This paper includes data collected by the TESS
mission, which are publicly available from the Mikulski Archive for Space
Telescopes (MAST) and at the Space Telescope Science Institute (STScI). STScI
is operated by the Association of Universities for Research in Astronomy,
Inc., under NASA contract NAS 5–26555. Funding for the TESS mission is
provided by NASA’s Science Mission directorate. This research has made use of
NASA’s Astrophysics Data System (ADS, Kurtz et al., 2000), the SIMBAD database
(Wenger et al., 2000), operated at CDS, Strasbourg, France and the BeSS
database, operated at LESIA, Observatoire de Meudon, France:
http://basebe.obspm.fr. J.L.-B. acknowledges support from FAPESP (grant
2017/23731-1). A.C.C. acknowledges support from CNPq (grant 307594/2015-7) and
FAPESP (grant 2018/04055-8). Funding for the Stellar Astrophysics Centre is
provided by The Danish National Research Foundation (Grant agreement no.:
DNRF106). This research made use of Astropy,777http://www.astropy.org a
community-developed core Python package for Astronomy (Astropy Collaboration
et al., 2013, 2018). This work makes use of observations from the LCOGT
network.

## Data Availability

All TESS data are available publicly. There are many ways to access the TESS
data products, the most straightforward of which is via the MAST website
(https://mast.stsci.edu/). The extracted light curves used in this work will
be provided upon request to the corresponding author.

## References

  * Aerts et al. (2019) Aerts C., Mathis S., Rogers T. M., 2019, ARA&A, 57, 35
  * Astropy Collaboration et al. (2013) Astropy Collaboration et al., 2013, A&A, 558, A33
  * Astropy Collaboration et al. (2018) Astropy Collaboration et al., 2018, AJ, 156, 123
  * Baade & Rivinius (2020) Baade D., Rivinius T., 2020, in Neiner C., Weiss W. W., Baade D., Griffin R. E., Lovekin C. C., Moffat A. F. J., eds, Proceedings of the conference Stars and their Variability Observed from Space. pp 35–38
  * Baade et al. (2016) Baade D., et al., 2016, A&A, 588, A56
  * Baade et al. (2017) Baade D., et al., 2017, in Second BRITE-Constellation Science Conf.: Small satellites – big science, Proc. Polish Astron. Soc., Vol. 5, eds. K. Zwintz and E. Poretti. pp 196–205 (arXiv:1611.01113)
  * Baade et al. (2018a) Baade D., et al., 2018a, in Wade G. A., Baade D., Guzik J. A., Smolec R., eds, BRITEning up the Be Phenomenon Vol. 8, 3rd BRITE Science Conference. pp 69–76 (arXiv:1708.08413)
  * Baade et al. (2018b) Baade D., et al., 2018b, A&A, 610, A70
  * Baade et al. (2018c) Baade D., et al., 2018c, A&A, 620, A145
  * Balona & Ozuyar (2020) Balona L. A., Ozuyar D., 2020, MNRAS, 493, 2528
  * Bernhard et al. (2018) Bernhard K., Otero S., Hümmerich S., Kaltcheva N., Paunzen E., Bohlsen T., 2018, MNRAS, 479, 2909
  * Bjorkman et al. (2002) Bjorkman K. S., Miroshnichenko A. S., McDavid D., Pogrosheva T. M., 2002, ApJ, 573, 812
  * Borre et al. (2020) Borre C. C., et al., 2020, A&A, 635, A140
  * Božić et al. (1999) Božić H., Ruždjak D., Sudar D., 1999, A&A, 350, 566
  * Bowman et al. (2019a) Bowman D. M., et al., 2019a, Nature Astronomy, 3, 760
  * Bowman et al. (2019b) Bowman D. M., et al., 2019b, A&A, 621, A135
  * Bowman et al. (2020) Bowman D. M., Burssens S., Simón-Díaz S., Edelmann P. V. F., Rogers T. M., Horst L., Röpke F. K., Aerts C., 2020, A&A, 640, A36
  * Brasseur et al. (2019) Brasseur C. E., Phillip C., Fleming S. W., Mullally S. E., White R. L., 2019, Astrocut: Tools for creating cutouts of TESS images (ascl:1905.007)
  * Carciofi (2011) Carciofi A. C., 2011, in Neiner C., Wade G., Meynet G., Peters G., eds, IAU Symposium Vol. 272, Active OB Stars: Structure, Evolution, Mass Loss, and Critical Limits. pp 325–336 (arXiv:1009.3969), doi:10.1017/S1743921311010738
  * Doazan et al. (1983) Doazan V., Franco M., Rusconi L., Sedmak G., Stalio R., 1983, A&A, 128, 171
  * Edelmann et al. (2019) Edelmann P. V. F., Ratnasingam R. P., Pedersen M. G., Bowman D. M., Prat V., Rogers T. M., 2019, ApJ, 876, 4
  * Floquet et al. (2009) Floquet M., et al., 2009, A&A, 506, 103
  * Georgy et al. (2013) Georgy C., Ekström S., Granada A., Meynet G., Mowlavi N., Eggenberger P., Maeder A., 2013, A&A, 553, A24
  * Ghoreyshi et al. (2018) Ghoreyshi M. R., et al., 2018, MNRAS, 479, 2214
  * Goss et al. (2011) Goss K. J. F., Karoff C., Chaplin W. J., Elsworth Y., Stevens I. R., 2011, MNRAS, 411, 162
  * Grunhut et al. (2017) Grunhut J. H., et al., 2017, MNRAS, 465, 2432
  * Hamaguchi et al. (2016) Hamaguchi K., Oskinova L., Russell C. M. P., Petre R., Enoto T., Morihana K., Ishida M., 2016, ApJ, 832, 140
  * Harmanec (2002) Harmanec P., 2002, in Tout C. A., van Hamme W., eds, Astron. Soc. Pacific Conf. Ser. Vol. 279, Exotic Stars as Challenges to Evolution. p. 221
  * Hartman (2012) Hartman J., 2012, VARTOOLS: Light Curve Analysis Program, Astrophysics Source Code Library (ascl:1208.016)
  * Haubois et al. (2012) Haubois X., Carciofi A. C., Rivinius T., Okazaki A. T., Bjorkman J. E., 2012, ApJ, 756, 156
  * Haubois et al. (2014) Haubois X., Mota B. C., Carciofi A. C., Draper Z. H., Wisniewski J. P., Bednarski D., Rivinius T., 2014, ApJ, 785, 12
  * Henry & Smith (2012) Henry G. W., Smith M. A., 2012, ApJ, 760, 10
  * Horaguchi et al. (1994) Horaguchi T., et al., 1994, PASJ, 46, 9
  * Horst et al. (2020) Horst L., Edelmann P. V. F., Andrássy R., Röpke F. K., Bowman D. M., Aerts C., Ratnasingam R. P., 2020, A&A, 641, A18
  * Huat et al. (2009) Huat A.-L., et al., 2009, A&A, 506, 95
  * Kambe et al. (1997) Kambe E., Hirata R., Ando H., Cuypers J., Katoh M., Kennelly E. J., Walker G. A. H., Tarasov A. E., 1997, ApJ, 481, 406
  * Kee et al. (2018) Kee N. D., Owocki S., Kuiper R., 2018, MNRAS, 474, 847
  * Klement et al. (2019) Klement R., et al., 2019, ApJ, 885, 147
  * Krtička et al. (2011) Krtička J., Owocki S. P., Meynet G., 2011, A&A, 527, A84
  * Kurtz et al. (2000) Kurtz M. J., Eichhorn G., Accomazzi A., Grant C. S., Murray S. S., Watson J. M., 2000, A&AS, 143, 41
  * Kurtz et al. (2015) Kurtz D. W., Shibahashi H., Murphy S. J., Bedding T. R., Bowman D. M., 2015, MNRAS, 450, 3015
  * Labadie-Bartz & Carciofi (2020) Labadie-Bartz J., Carciofi A. C., 2020, in Neiner C., Weiss W. W., Baade D., Griffin R. E., Lovekin C. C., Moffat A. F. J., eds, Proceedings of the conference Stars and their Variability Observed from Space. pp 137–145
  * Labadie-Bartz et al. (2017) Labadie-Bartz J., et al., 2017, AJ, 153, 252
  * Labadie-Bartz et al. (2018) Labadie-Bartz J., et al., 2018, AJ, 155, 53
  * Labadie-Bartz et al. (2020) Labadie-Bartz J., Carciofi A. C., de Amorim T. H., Rubio A., Luiz A., Ticiani dos Santos P., Thomson-Paressant K., 2020, arXiv e-prints, p. arXiv:2010.13905
  * Langer et al. (2020) Langer N., Baade D., Bodensteiner J., Greiner J., Rivinius T., Martayan C., Borre C. C., 2020, A&A, 633, A40
  * Lecoanet et al. (2019) Lecoanet D., et al., 2019, ApJ, 886, L15
  * Lee & Saio (2020) Lee U., Saio H., 2020, MNRAS, 497, 4117
  * Lee et al. (1991) Lee U., Osaki Y., Saio H., 1991, MNRAS, 250, 432
  * Lenz & Breger (2005) Lenz P., Breger M., 2005, CoAst, 146, 53
  * Loumos & Deeming (1978) Loumos G. L., Deeming T. J., 1978, Ap&SS, 56, 285
  * Maintz et al. (2003) Maintz M., Rivinius T., Štefl S., Baade D., Wolf B., Townsend R. H. D., 2003, A&A, 411, 181
  * McNamara et al. (2012) McNamara B. J., Jackiewicz J., McKeever J., 2012, AJ, 143, 101
  * Montgomery & O’Donoghue (1999) Montgomery M. H., O’Donoghue D., 1999, DSSN, 13, 28
  * Moździerski et al. (2019) Moździerski D., et al., 2019, A&A, 632, A95
  * Nazé et al. (2019) Nazé Y., Rauw G., Smith M., 2019, A&A, 632, A23
  * Nazé et al. (2020a) Nazé Y., Motch C., Rauw G., Kumar S., Robrade J., Lopes de Oliveira R., Smith M. A., Torrejón J. M., 2020a, MNRAS, 493, 2511
  * Nazé et al. (2020b) Nazé Y., Pigulski A., Rauw G., Smith M. A., 2020b, MNRAS, 494, 958
  * Nazé et al. (2020c) Nazé Y., Rauw G., Pigulski A., 2020c, MNRAS, 498, 3171
  * Neiner et al. (2011) Neiner C., de Batz B., Cochard F., Floquet M., Mekkas A., Desnoux V., 2011, AJ, 142, 149
  * Neiner et al. (2012) Neiner C., et al., 2012, A&A, 546, A47
  * Neiner et al. (2020) Neiner C., Lee U., Mathis S., Saio H., Lovekin C. C., Augustson K. C., 2020, arXiv e-prints, p. arXiv:2007.08977
  * Nemravová et al. (2012) Nemravová J., et al., 2012, A&A, 537, A59
  * Ninkov et al. (1983) Ninkov Z., Yang S., Walker G., 1983, HvaOB, 7, 167
  * Okazaki et al. (2002) Okazaki A. T., Bate M. R., Ogilvie G. I., Pringle J. E., 2002, MNRAS, 337, 967
  * Owocki (2006) Owocki S., 2006, in Kraus M., Miroshnichenko A. S., eds, Astronomical Society of the Pacific Conference Series Vol. 355, Stars with the B[e] Phenomenon. p. 219
  * Pápics et al. (2017) Pápics P. I., et al., 2017, A&A, 598, A74
  * Peters & Gies (2005) Peters G. J., Gies D. R., 2005, in Ignace R., Gayley K. G., eds, Astronomical Society of the Pacific Conference Series Vol. 337, The Nature and Evolution of Disks Around Hot Stars. p. 294
  * Pigulski & Pojmański (2008) Pigulski A., Pojmański G., 2008, A&A, 477, 917
  * Pollmann (2009) Pollmann E., 2009, BeSN, 39, 32
  * Postnov et al. (2017) Postnov K., Oskinova L., Torrejón J. M., 2017, MNRAS, 465, L119
  * Press et al. (1992) Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P., 1992, Numerical recipes in C. The art of scientific computing
  * Ratnasingam et al. (2020) Ratnasingam R. P., Edelmann P. V. F., Rogers T. M., 2020, MNRAS, 497, 4231
  * Ricker et al. (2015) Ricker G. R., et al., 2015, JATIS, 1, 014003
  * Rímulo et al. (2018) Rímulo L. R., et al., 2018, MNRAS, 476, 3555
  * Rivinius et al. (2003) Rivinius T., Baade D., Štefl S., 2003, A&A, 411, 229
  * Rivinius et al. (2013) Rivinius T., Carciofi A. C., Martayan C., 2013, A&ARv, 21, 69
  * Rivinius et al. (2016) Rivinius T., Baade D., Carciofi A. C., 2016, A&A, 593, A106
  * 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. (2007) Saio H., et al., 2007, ApJ, 654, 544
  * Saio et al. (2018a) Saio H., Kurtz D. W., Murphy S. J., Antoci V. L., Lee U., 2018a, MNRAS, 474, 2774
  * Saio et al. (2018b) Saio H., Bedding T. R., Kurtz D. W., Murphy S. J., Antoci V., Shibahashi H., Li G., Takata M., 2018b, MNRAS, 477, 2183
  * Secchi (1866) Secchi A., 1866, AN, 68, 63
  * Semaan et al. (2018) Semaan T., Hubert A. M., Zorec J., Gutiérrez- Soto J., Frémat Y., Martayan C., Fabregat J., Eggenberger P., 2018, A&A, 613, A70
  * Sikora et al. (2019) Sikora J., Wade G. A., Power J., Neiner C., 2019, MNRAS, 483, 2300
  * Smith (1985) Smith M. A., 1985, ApJ, 297, 206
  * Smith et al. (2006) Smith M. A., Henry G. W., Vishniac E., 2006, ApJ, 647, 1375
  * Smith et al. (2016) Smith M. A., Lopes de Oliveira R., Motch C., 2016, AdSpR, 58, 782
  * Stee et al. (2012) Stee P., et al., 2012, A&A, 545, A59
  * Tsujimoto et al. (2018) Tsujimoto M., Morihana K., Hayashi T., Kitaguchi T., 2018, PASJ, 70, 109
  * Van Reeth et al. (2016) Van Reeth T., Tkachenko A., Aerts C., 2016, A&A, 593, A120
  * VanderPlas & Ivezić (2015) VanderPlas J. T., Ivezić Ž., 2015, ApJ, 812, 18
  * VanderPlas et al. (2012) VanderPlas J., Connolly A. J., Ivezic Z., Gray A., 2012, in Proceedings of Conference on Intelligent Data Understanding (CIDU. pp 47–54 (arXiv:1411.5039), doi:10.1109/CIDU.2012.6382200
  * Vogt & Penrod (1983) Vogt S. S., Penrod G. D., 1983, ApJ, 275, 661
  * Wade et al. (2016) Wade G. A., Petit V., Grunhut J. H., Neiner C., MiMeS Collaboration 2016, in Sigut T. A. A., Jones C. E., eds, Astronomical Society of the Pacific Conference Series Vol. 506, Bright Emissaries: Be Stars as Messengers of Star-Disk Physics. p. 207
  * Walker et al. (2005) Walker G. A. H., et al., 2005, ApJ, 635, L77
  * Wang et al. (2018) Wang L., Gies D. R., Peters G. J., 2018, ApJ, 853, 156
  * Wenger et al. (2000) Wenger M., et al., 2000, A&AS, 143, 9
  * Yang et al. (1988) Yang S., Ninkov Z., Walker G. A. H., 1988, PASP, 100, 233
  * Zechmeister & Kürster (2009) Zechmeister M., Kürster M., 2009, A&A, 496, 577
  * ud-Doula et al. (2018) ud-Doula A., Owocki S. P., Kee N. D., 2018, MNRAS, 478, 3049
  * Štefl et al. (1998) Štefl S., Baade D., Rivinius T., Stahl O., Wolf B., Kaufer A., 1998, in Bradley P. A., Guzik J. A., eds, Astronomical Society of the Pacific Conference Series Vol. 135, A Half Century of Stellar Pulsation Interpretation. p. 348

##  Appendix A Light curve extraction from TESS images

At an $I$-band magnitude around 2.4, $\gamma$ Cas heavily saturates the TESS
detectors. However, since primarily the core of the point spread function
(PSF) and bloom columns are affected, valid brightness measurements can be
extracted from the non-saturated pixels in the PSF wings (i.e. halo
photometry). Alternatively, a much larger aperture mask can be used if it
includes all charge-containing pixels. The TESS CCDs are designed to conserve
charge even for highly saturated objects. That is, excess charges from
saturated pixels flow to neighboring unsaturated pixels along the column and
can be collected and measured from them. Tests carried out prior to launch
demonstrated this for stars at least as bright as 4th magnitude in the TESS
band (according to the TESS instrument handbook
v0.1888https://archive.stsci.edu/missions/tess/doc/TESS_Instrument_Handbook_v0.1.pdf).
Using an aperture mask that includes all pixels that collect charge from the
incident flux of a very bright star (e.g. $\gamma$ Cas) is therefore a valid
method for extracting a light curve provided that the brightness limit at
which charge is no longer conserved does not significantly interfere with the
astrophysical signals.

Light curves were extracted using both of these methods and compared. The
aperture masks used are shown in Figs. 4 and 5 for the methods using all
charge-containing pixels and only including non-saturated halo pixels,
respectively. For halo photometry, a threshold of 1000 $e^{-}s^{-1}$ was
applied to the calibrated flux value of the postage stamp, averaged over all
images in a sector and within a radius of 9 pixels from the central position
of $\gamma$ Cas on the detector, but all columns that included one or more
saturated pixel were avoided. A higher threshold of 3000 $e^{-}s^{-1}$ was
chosen for the aperture mask that includes saturated pixels. For reference,
the mean sky background level is 175 $e^{-}s^{-1}$, 191 $e^{-}s^{-1}$, and 448
$e^{-}s^{-1}$ for sectors 17, 18, and 24, respectively.

The resulting light curves are compared in Fig. 6, along with their respective
power spectra, computed with the timeseries.LombScargle package (VanderPlas et
al., 2012; VanderPlas & Ivezić, 2015) of Astropy (Astropy Collaboration et
al., 2013, 2018). The properties of these light curves are very similar, with
the main differences being that halo photometry results in a lower SNR and an
increase in apparent low frequency trends which are likely due to systematic
effects. After removing linear and low frequency trends (with $f\leq 0.2$
d-1), the frequency spectra of the two versions of the data are essentially
identical, in that the same signals are detected in both but with slightly
higher amplitudes in the halo photometry (typically 10% – 40% for the
strongest periodic signal).

The difference in amplitude between the two methods is most likely related to
a varying PSF of $\gamma$ Cas. The method using all relevant count-containing
pixels is not affected by a variable PSF, since the PSF is sufficiently
covered at both its widest and narrowest. However, since halo photometry only
makes use of pixels in the non-saturated PSF wings, a variable PSF can change
the fraction of the total flux from the source that falls into the halo
pixels. In effect, the variability of $\gamma$ Cas is ‘double counted’ with
halo photometry, leading to artificially enhanced amplitudes. Light variations
are recorded due to the changing source brightness, but also due to a changing
fraction of the ensquared PSF within the pixels selected. Put another way,
when the source brightens, the PSF widens and an increased fraction of the
source flux falls into the halo pixels. Therefore, the amplitudes associated
with the aperture containing all count-containing pixels is deemed to be more
reliable. The ‘breathing’ of the PSF is readily seen in a movie of the images
viewed over the duration of a given sector, provided a very high contrast
ratio is used to view the images. Situations where halo photometry may provide
more reliable results include when the bloom columns of a highly saturated
object extend past the edges of the CCD (making it impossible to capture the
total flux of the source), and when other relatively bright sources fall on or
near the bloom columns (leading to contamination).

Nazé et al. (2020c) analyzed the same TESS data (Sectors 17 and 18 only, i.e.
without Sector 24). However, they only applied aperture photometry to
thumbnail images of 50$\times$50 pixels and thereby cut off the long blooming
tails in the CCD columns. Nevertheless they find basically the same
frequencies as reported here which shows that the frequencies are not too
sensitive to the signal-extraction strategy. The amplitudes (and the signal-
to-noise ratio) should be significantly lower because much of the vertically
overflown signal was missed. Indeed, the amplitude of the strongest signal in
$g2$ in this work is 3.67 ppt, and about 1.2 ppt in Nazé et al. (2020c).

Figure 4: TESS ‘postage stamp’ showing the region of the sky around $\gamma$
Cas for Sectors 17, 18, and 24. Pixels used to extract the light curve are
marked in green. For the sake of increasing the contrast in this image, the
plotted brightness of each pixel is proportional to the square root of its
flux (averaged over all images in a given sector). Figure 5: TESS ‘postage
stamp’ in the same style as Fig. 4 showing the aperture mask used for halo
photometry, where saturated pixels (and columns containing one or more
saturated pixel) are not included. Figure 6: Left: Comparison of the light
curve for $\gamma$ Cas extracted using all charge-containing pixels (black)
and using only non-saturated halo pixels (red) for sectors 17, 18, and 24
after detrending. Right: Lomb-Scargle periodogram for the light curve
extracted using all charge-containing pixels (black) and using only non-
saturated halo pixels (inverted, red) showing the main features. Owing to the
different extraction apertures, there are slight differences in the systematic
trends in the two versions of the light curves, requiring removal of data at
slightly different times near the beginning and end of each half of each
observing sector.

##  Appendix B Mass-loss indicators in TESS observations of other early-type
Be stars

The following subsections show and discuss sample light curves for four early-
type Be stars which illustrate variability patterns representative of the
types of photometric signals seen in Be stars with TESS (Labadie-Bartz &
Carciofi, 2020). These light curves were extracted using methods similar to
that for $\gamma$ Cas (but without the complications of heavy saturation) and
preserve the slow variability that illustrates that changes in the mean
brightness and times of in-phase superposition of variabilities with several
frequencies in $g1$ or $g2$ are correlated. Because the time intervals
concerned can be as long as $\sim$5 d and several of them can occur in a
single TESS sector of $\sim 27$ d, the identification of such simultaneities
is necessarily subjective.

Globally, there is no doubt that such a correlation exists, especially since
qualitatively identical, probably cyclic correlations have been found on
larger time-scales and at much higher amplitudes (Baade et al., 2017, 2018b).
Only its prevalence cannot be quantified on the basis of the data used. As
explained in Sect. 5, a change in mean magnitude can be the result of
increased light scattering by additional matter ejected through the temporary
(non-linear) in-phase superposition of several NRP modes.

###  B.1 HD 58050 (HR 2817, OT Gem, TIC 14498757)

Figure 7: Top: Raw TESS light curve (after removing trends common to other
pixels in the vicinity of the target) of OT Gem (TIC 14498757) in black, with
the red curve showing the low frequency component. Second: TESS light curve
after removing the low frequency component. Third: Wavelet analysis showing
the variability of the main frequency groups. Fourth: Lomb-Scargle
periodogram, computed after removing the low frequency component.

Božić et al. (1999) distinguished photometrically quiet and active phases with
peak-to-peak ranges of up to 0.4 mag in $V$ and time-scales up to hundreds of
days. In observations obtained during the active phase they found cyclic
variability with a repetition time of nearly 72 d and a peak-to-peak amplitude
of $\sim 0.15$ mag. The authors attributed this variability to binarity but
did not have corroborating spectral evidence. Because of the link to enhanced
photometric activity, a relation to the beating of NRP modes is equally
plausible and supported by the triangular shape of the light curve,
characteristic of short-lived mass ejection events (Bernhard et al., 2018;
Labadie-Bartz et al., 2018)

With a peak-to-peak range of $\sim 0.06$ mag, HD 58050 is relatively active in
regards to its secular brightening and fading events compared to other Be
stars observed by TESS, but much less variable than during the active state
described by Božić et al. (1999), where the authors use a significantly longer
time baseline. The TESS light curve (Fig. 7) includes two rather strong local
maxima during which the in-phase superposition of $g2$ frequencies is
particularly clearly visible.

###  B.2 HD 71042 (TIC 52929072)

Figure 8: Same as Fig. 7, but for HD 71042 (TIC 52929072).

From observations in 3 of 5 consecutive annual seasons, Labadie-Bartz et al.
(2018) classified HD 71042 as Be star with outburst behavior and a full range
of $\sim 0.2$ mag in $V$. Contrary to the majority of Be stars, the outburst
activity mainly consisted of dimmings rather than brightenings, suggesting
that the line of sight passes through the disk. There may be a 23.5 d period
with very unusual light curve. Observed in two consecutive TESS sectors, the
light curve in Fig. 8 shows three time intervals when the mean magnitude
dropped and subsequently ascended (the observations ended before a possible
recovery after the third decline). In all three intervals, the temporary in-
phase superposition of $g1$ frequencies is prominent, and to a lesser extent
those in $g2$. This short-term variability, too, is consistent with a near-
equator-on perspective. However, there are no published spectra. The 23.5 d
period may also be present in the TESS data, although the short observational
baseline makes this uncertain.

###  B.3 HD 144555 (TIC 215511795)

Figure 9: Same as Fig. 7, but for HD 144555 (TIC 215511795).

In Fig. 9, five humps in mean magnitude appear in the light curve of HD
144555. On the ascending branch of all of them, variations with frequencies
from $g1$ are temporarily co-phased. Around TJD = 1644 (TJD = BJD - 2457000),
an additional phenomenon occurred in the form of a strong admixture of
variability with $g2$ frequencies. That multiple $g1$ as well as $g2$
frequencies are in phase both alone and together may be a coincidence. But the
probability of this happening in a single TESS sector of one month may be low.
Therefore, this phase could also be an indication of strong non-linearity as
$g1$ and $g2$ are in a crudely harmonic relation. This phenomenon is seen in
other Be stars observed by TESS. The difference between the two strongest
signals in $g1$ is 0.244(1) d-1, consistent with the amplitude of $g1$ being
modulated with a period of $\sim$4 days.

###  B.4 HD 156172 (TIC 216875138)

Figure 10: Same as Fig. 7, but for HD 156172 (TIC 216875138). Figure 11: The H
$\alpha$ and H $\beta$ lines for HD 156172 (TIC 216875138) from a single NRES
spectrum obtained on 2020 July 01. Black (red) lines show un-binned (binned)
data.

The photometric variability in Fig. 10 belongs to the most unusual ones
described to date for any suspected classical Be star. Firstly, the Lomb-
Scargle amplitude spectrum of this late O-type star contains very little power
below 4 d-1, and frequency groups $g1$ and $g2$ are not discernable in a broad
distribution of weak peaks. The strongest features appear at the unusually
high frequencies of 6.972(3) (amplitude $\sim$8 ppt) and 6.66(1) d-1
(amplitude $\sim$2 ppt), respectively. This has led to the classification of
HD 156172 as a $\beta$ Cep star by Pigulski & Pojmański (2008) who listed a
single frequency of 6.97258(3) d-1 and an amplitude of 12.4 mmag in $V$. There
is a much lower-amplitude pair of signals at frequencies of 13.947(8) and
13.63(1) d-1. The higher frequency of these is consistent with being the first
harmonic of the strongest signal at 6.972(3) d-1, and the lower-frequency
signal is equal (within errors) to the sum of the two frequencies near 7 d-1.
Secondly, the difference, 0.31(1) d-1, between these these two main
frequencies appears as the beat frequency in the wavelet transform (Fig. 10).
Since the beat frequency also exists in the amplitude spectrum (as the peak
with the fourth highest amplitude, but not necessarily Lomb-Scargle power),
there seems to be a non-linear mode interaction. Thirdly, the mean brightness
of HD 156172 varies with the same beat period and a semi-amplitude of 0.1 –
0.2%. The amplitude is tiny but clearly detectable by plain visual inspection,
and there is a well isolated peak in the unfiltered Lomb-Scargle spectrum at
0.312 d-1. The maxima coincide with times of constructive interference of the
said two frequencies (blobs in the wavelet transform).

If these frequencies correspond to p modes, HD 156172 might be one of the
first Be stars in which indications of mass-loss due to p mode pulsations is
found (see also Nazé et al., 2020b). Frequencies in the nominal p mode domain
are not uncommon in Be stars (Labadie-Bartz & Carciofi, 2020; Balona & Ozuyar,
2020). However, in HD 156172, only the absence of the mostly much stronger
$g1$ and $g2$ variabilities permitted the minute variation in mean brightness
to be detected. Generally, caution should be exercised in uncritically
adopting a p mode identification merely on the basis of the observed
frequencies because rapid rotation may greatly modify the frequencies in the
inertial frame. However, the absence of significant $g1$ and $g2$ frequencies
as possible parent frequencies may make a p mode identification safer.

Inspired by the above peculiarities, an echelle spectrum was obtained on 2020
July 01 with the NRES instrument attached to the 1-m telescope at South
African Astronomical Observatory operated by LCOGT. The H $\alpha$ and H
$\beta$ line are shown in Fig. 11. This spectrum is entirely consistent with
that of a classical Be star, and seemingly inconsistent with other types of
objects that may show hydrogen emission (i.e. interacting binaries, rapidly
rotating OB stars with strong magnetic fields, supergiants, or Herbig Ae/Be
stars). Other absorption lines are consistent with the spectral classification
of a rapidly rotating late O or early B star. A more detailed investigation of
this system is beyond the scope of this work, but is certainly warranted.