patterns needed to measure internal rotation. They found that, in general,
tidal synchronism occurs for orbital periods less than 10 d, with the
exception of 3 stars for which an internal rotation rate was sub-synchronous.
They conjectured that tidally excited modes may be able to transfer internal
angular momentum to the orbit, in a novel process they dubbed inverse tides.

###  6.2 Tidally tilted pulsators

When I discovered the roAp stars and proposed the oblique pulsator model
(Kurtz, 1982, Section 2.12.2), the clue to oblique pulsation was in frequency
multiplets split by exactly the rotation frequency. In slowly rotating stars,
rotationally perturbed multiplets, such as dipole triplets and quadrupole
quintuplets, have frequency splittings that differ from the rotation frequency
by a factor of $(1-C_{n,\ell})$, where $C_{n,\ell}$ is the Ledoux constant,
which is not zero. The splitting of the multiplet components by exactly the
rotation frequency showed that the mode axis was inclined to the rotation axis
as a consequence of the strong, roughly dipolar magnetic field. This suggested
that other perturbations from spherical symmetry should also compete with
rotation for the pulsation axis selection, and that tidal deformation is an
excellent candidate to do that.

Forty years later, obliquely pulsating close binary stars have now been
discovered where the pulsation axis coincides with the tidal axis. The first,
HD 74423 (Handler et al., 2020), is composed of two similar mass $\lambda$ Boo
stars (Section 2.12.3) in a 1.6-d orbit where one star is a $\delta$ Sct star
pulsating in a single mode which shows 11 frequencies in the FT split by
exactly the orbital frequency – the signature of oblique pulsation. A detailed
analysis found a zonal ($m=0$) distorted dipole pulsation mode with much
higher amplitude on the $L_{1}$ side facing the companion; this led to the
star being called a single-sided pulsator.

[] Tidally tilted pulsators Obliquely pulsating stars in close binaries where
the tidal axis is the pulsation axis. Single-sided pulsators Tidally tilted
pulsators where the tidal distortion leads to pulsation predominantly on
either the $L_{1}$ or $L_{3}$ side.

The second is CO Cam (Kurtz et al., 2020), a marginal Am $\delta$ Sct star in
a 1.27-d orbit with an undetected G main-sequence companion. It pulsates in
four p modes with the pulsation axis aligned with the tidal axis, and three g
modes that do not show this alignment. Pulsation modeling suggests that one of
the p modes is the fundamental mode and that the others have a mixed p mode–g
mode character. Mode identification independent of modeling is difficult
because of the tidal distortion of the observable atmospheric layer. The best
models require some core overshoot of 1% to 2% of the pressure scale height;
models with no overshoot do not match the observed frequencies, hence there is
an asteroseismic constraint on core mixing in this close tidal binary.

The third is TIC 63328020 (Rappaport et al., 2021), a $\delta$ Sct star with a
1.1-M⊙ companion in a 1.11-d orbit. There are two pulsation modes, where the
primary one is a sectoral dipole ($\ell=1,m=|1|$) mode. This star does not
show the single-sided pulsation nature of the previous two, hence we refer to
the class as tidally tilted pulsators with the term singled-sided pulsator
reserved for those with much higher amplitudes on one side of the star –
either the $L_{1}$ or $L_{3}$ side.

Fuller et al. (2020) discussed the theory of tidal trapping, based on the
above three tidally tilted pulsators. They examined how the tidal distortion
affects the mode frequencies; how it causes alignment of the pulsation axis
with the tidal axis; how it traps modes, largely confining them to the tidal
poles or equator; and how it amplifies the mode amplitude at the tidal poles
where the modes propagate closer to the stellar surface. Study of the orbital
variations of HD 74423, CO Cam, and TIC 63328020 showed that they partially
fill their Roche lobes to varying degree, such that with only three examples
it is not clear why their single-sided pulsations differ.

The tidally tilted pulsators show asteroseismic promise. The oblique pulsation
character helps with mode identification, a requisite for asteroseismology,
even with the tidal distortion. The coupling of the surface p modes to
interior g modes in a tidally distorted star requires further development of
the models. The first three were discovered because their FTs are particularly
simple, with only one or a few modes excited. Most $\delta$ Sct stars show
many excited modes (Section 2.1), and in the cases of tidally tilted pulsators
with many modes, there is complexity to be disentangled in the FT. For
relatively slowly rotating stars, the tool to do that is the échelle diagram,
and it is probable that many more of these stars will emerge from the TESS
data. Further developments to the theory are also needed, including
incorporating the Coriolis and centrifugal forces, and possibly unanticipated
surprises as this new behavior is explored.

There is related behavior in other stars, e.g., the series of p modes
separated by the orbital frequency, but which are not harmonics of that
frequency, in the heartbeat star KIC 4544587 (Hambleton et al., 2013), and in
the eclipsing binary U Gru (Bowman et al., 2019b). For the latter, Bowman et
al. (2019b) discussed tidally excited and perturbed modes, and also single-
sided mode trapping. It is clear that the interaction of pulsation modes and
tidal distortion in close binaries offers new asteroseismic inference, with a
range of behavior culminating in the tidally tilted pulsators. The problem of
mode selection and excitation is not solved for these stars, but then, it is
not solved for any $\delta$ Sct stars. At least in this case of close
binaries, when the mode frequencies are harmonics of the orbital frequency,
there is a clear coupling to the binarity that can be exploited. An
observational goal is to find tidally tilted pulsators in eclipsing binaries
where the eclipses add further measures of mass and radius, and the changes in
the pulsation mode amplitude and phase during eclipse offers tomography of the
mode geometry.

Yu et al. (2021) have theoretically studied the case of dynamical tides in
close double degenerate white dwarfs with orbital periods less than an hour,
in extreme cases less than 10 min. These stars are, e.g., the double-
degenerate interacting AM CVn stars, which are precursors of white dwarf
mergers (Section 5.1), Supernovae Ia, and are a source of gravitational waves
that LISA is expected to detect. Yu et al. predicted that coupling of g modes
and p modes can produce photometric variations up to 1%, which can be larger
than the ellipsoidal variations. Furthermore, the pulsations generated by the
dynamical tide have a phase shift of about $50^{\circ}$ to the orbital
variation, which will allow them to be distinguished from the ellipsoidal
variability. Even for faint white dwarfs, TESS observations offer the
possibility to detect a 1% effect, hence we can look forward to observations
that will enable asteroseismic application of this new theory.

### Summary: Asteroseismology of close binary stars

  1. 1.

Eclipsing binaries with pulsating components provide multiple measures of
stellar fundamental parameters. Together they constrain mass, radius, age, and
luminosity to the highest precision for application in exoplanet and galactic
archeology studies.

  2. 2.

The phase modulation (PM) technique has opened up unexplored parameter space
in the orbital periods of binary stars. It allows orbital solutions for binary
stars without the need of spectroscopy to obtain radial velocities, saving
vast amounts of telescope time and astronomers’ time.

  3. 3.

Doppler beaming also allows the measurement of radial velocities without the
need of spectra.

  4. 4.

Heartbeat stars and tidally tilted pulsators provide observations to test
theories of stellar tides and their interaction with pulsation, and with
rotational and orbital angular momentum. They test mode trapping, tidal
distortion of pulsation frequency and amplitude, and pulsation mode axis
selection.

  5. 5.

The heartbeat stars support the thesis of tidal locking where feedback among
orbital, rotational, and stellar evolution changes keep the pulsation
frequency resonant with the orbital frequency.

##  7 Epilogue – past and future

I am fortunate that my career has spanned the time from the beginning of
helioseismology in the 1960s through to this blossoming of asteroseismology
today. This review has highlighted the results for stars across the HR Diagram
(except for the solar-like oscillators and red giants, as a consequence of
other recent reviews of those iconic asteroseismic targets). All of this work
uses asteroseismology to see inside the stars. Fundamental and other stellar
parameters are extracted from asteroseismic inference: mass, radius, age,
metallicity, atomic diffusion, luminosity, distance, magnetic fields, interior
rotation, angular momentum transfer, convective overshoot, core burning stage.
In the spirit of Howard Florey, asteroseismologists are delighted to be
exploiting stellar pulsation in more depth than previously possible, and to be
discovering new kinds of pulsation behavior, all improving our understanding
of stellar structure and evolution, a bedrock of astrophysics, since stars are
the luminous tracers of the universe.

Space photometry and ground-based large scale surveys have created an
explosion of asteroseismic data. Big data techniques and machine learning are
already active in exploring and categorizing the huge data sets, and this will
expand in the future. Human labor cannot sift the large numbers of light
curves and FTs as computers can. This is particularly true of the ground-based
surveys such as ZTF and VRO, the latter of which will produce 20 Tb of data
per night! These ground-based surveys will enable and expand asteroseismology
of many classes of stars discussed in this review, such as white dwarfs, Blue
Large Amplitude Pulsators, EL CVn binaries, Cepheids, RR Lyr stars, and
Stochastic Low Frequency O and B stars.

The photometric space missions have been designed for exoplanet studies, and
they have been successful beyond expectation, with exoplanets now known in
their thousands, soon to be tens of thousands. For these missions we
asteroseismologists initially rode on the coattails of the exoplanet
community, but now that the usefulness of asteroseismology to exoplanets is
known, asteroseismology is built in to the science case for future missions.
Nevertheless, hot stars are not the main hosts of exoplanets, and they are not
expected to have inhabited planets, even if habitable planets orbiting them
should be found, for the obvious reason of their short lifetimes. Hot main-
sequence stars are also rarer as a consequence of the initial mass function
and those short lifetimes. But, as this review has shown, the massive upper
main-sequence stars, the white dwarfs and subdwarfs, the most luminous blue
giants and supergiants, and the classical pulsators, are all of interest
asteroseismically.

Most of the variability in hot pulsators comes from variations in $T_{\rm
eff}$. As a simple consequence of their spectral energy distributions, the hot
stars have much larger pulsation amplitudes in the UV and Blue. This is easy
to see by picturing the changes in a blackbody curve with changes in
temperature for stars for which the flux peaks in the UV or blue. The TESS
bandpass is roughly $0.6-1.0$ $\upmu{m}$, that of Kepler was $0.4-0.9$
$\upmu{m}$, filters with broadly white bandpasses, but towards the redder end
of the spectrum. That is the right choice for the study of exoplanet transits
and of the stochastic pulsators such as the solar-like stars and red giants.
But to obtain much higher signal-to-noise for the hotter stars, a blue filter
is needed. A dedicated space mission, which could be called the Blue
Asteroseismic Survey Satellite (BASS), is needed to build on the revolutionary
asteroseismic results obtained from the exoplanet surveys.

Asteroseismology informs our understanding in hotter stars:

  * •

of stellar structure and evolution, including interior rotation, angular
momentum transfer, and mixing;

  * •

of the most massive stars, precursors to core-collapse supernovae, black holes
and neutron stars, gamma ray bursters, observable gravitational wave events,
and sources of much of the chemical evolution of the galaxy.

in cooler stars:

  * •

of galactic archeology;

  * •

of the fundamental parameters – mass, radius, and age – needed to characterize
exoplanet host stars in our search for habitable planets;

  * •

of red giant evolutionary core-burning state;

  * •

of the post-main-sequence giant to red giant stages, including interior
rotation and magnetic fields.

in compact stars:

  * •

of mass, radius, temperature, luminosity, age, rotation, the extent of the
non-degenerate surface layers in white dwarfs;

  * •

of physics under extreme conditions in white dwarfs and subdwarfs, the end
stages of the evolution of 97% of all stars;

  * •

of double and single degenerate binary stars, precursors of distance scale
Supernovae Ia, and further sources of chemical evolution.

in close binary stars:

  * •

of tides, tidal distortion of pulsation modes, and of mode trapping;

  * •

of orbital and rotational angular momentum evolution and exchange;

  * •

of orbital circularization.

We have indeed found Eddington’s “appliance” to “pierce the outer layers of a
star and test the conditions within”. He would have been amazed and gratified,
as are we.

## DISCLOSURE STATEMENT

The author is not aware of any affiliations, memberships, funding, or
financial holdings that might be perceived as affecting the objectivity of
this review.

## ACKNOWLEDGMENTS

I thank Professors Conny Aerts and Rob Kennicutt for their many scientific
comments and their guidance in the preparation of this review. I also thank
Professor Simon Jeffery for providing and discussing the pulsation HR Diagram
shown in Figure 1, modified for this review.

##  8 Appendix: Introduction to the supplementary materials

In writing this review there were many personal stories of my own and
historical stories of other researchers that I wished to tell. Some of those
are in the main text, and others are here in this supplement. These stories
give a sense of the human side to the research, especially for those entering
the field and for those from other fields who are reading the review out of
general interest. In addition, for many sections of the review I wanted to
provide more depth for the specialist. Those additional details are in the
sections of this supplement. The review stands on its own. For those who enjoy
the stories and want the additional details, I recommend when reading the main
text that you take an excursion to this supplement directly from the links
provided.

##  Appendix A OBAF stars

###  A.1 The naming of the SPB stars and discovery of the pulsation driving
mechanism in B stars

####  A.1.1 The naming of the SPB stars

At a stellar pulsation meeting in 1990 in Bologna, Italy, Christoffel Waelkens
gave a talk about a new class of pulsating late-B stars that he proposed to
name the Slowly Oscillating B stars, in analogy with the rapidly oscillating
Ap stars that I had previously named. In the question time at the end of his
talk, I commented that his class would come to be known by its acronym, and
that he really did not want to be studying SOBs. He and some of the audience
expressed some confusion about this English colloquialism, but after it was
discreetly explained, he changed the name to Slowly Pulsating B (SPB) stars
(Waelkens et al., 1990; Waelkens, 1991).

####  A.1.2 The pulsation driving mechanism in B stars

When I was a student at the University of Texas in the early 1970s, I
presented my first professional talk at the December 1973 AAS meeting in
Tucson, Arizona. It was a 10-min talk on the evolved metallic-lined stars, the
$\delta$ Del stars (now called $\rho$ Pup stars) (Kurtz, 1976). My talk was in
a session where all other talks were about numerous ideas to explain pulsation
driving in $\beta$ Cep stars – then a mystery. Not one of those ideas was
conclusive, and, as it turned out later, not one was correct. At the end of my
talk there were no questions from the audience, so the chair of the session,
Martin Schwarzschild, asked forcefully and loudly, in his way, “WHAT are these
stars REASONS to PULSATE?” I thought I knew the answer to his question, but
since one of my undergraduate textbooks had been his Structure and Evolution
of the Stars (Schwarzschild, 1958), I thought to myself, “if he doesn’t know,
then what I am thinking cannot be right”. He terrified me, so I diverted the
question to the theoreticians in the audience who had given the talks on
$\beta$ Cep driving mechanisms. One of them gently answered, “Martin, they are
in the instability strip”, to which he softly replied, “Oh. I missed that.” I
understood, on meeting him in later years, that he was just being kind in
providing a poor graduate student giving his first professional talk with a
question.

The problem of driving in $\beta$ Cep and SPB stars was that the
$\kappa$-mechanism needs abundant ions to provide sufficient driving to
overcome damping in other layers of a star, and the prime drivers for many
pulsating stars are H and He, the most abundant elements (as Cecilia Payne had
shown). H and He do not work for the B stars as their ionization zones, where
the driving occurs, are too close to the surface of the star, so that there is
insufficient mass in overlaying layers to cause recompression. Cox et al.
(1992) and Moskalik & Dziembowski (1992) were among the first to show that a
bump in opacity from heavy elements – the Z-bump – primarily from Fe and Ni,
could drive pulsations among the B stars. After decades, the driving mechanism
for these hot main-sequence stars had been found. See Gautschy & Saio (1996),
Pamyatnykh (1999) and Miglio et al. (2007) for further discussion of the
driving mechanism and the instability strips of B stars.

###  A.2 The discovery of the $\gamma$ Dor stars

Michel Breger was a pioneer in the study of $\delta$ Sct stars (e.g., Breger
1969), and under his supervision I also began studying them in the early
1970s. Using ground-based differential photometry, we saw low frequency excess
in the amplitude spectra of the stars we studied. Because we were using at
least two comparison stars for each $\delta$ Sct star studied, we did not
understand the origin of this low-frequency excess, and dismissed it as
probably an artifact caused by differential extinction, given that our
comparison stars had different temperatures to the $\delta$ Sct stars. We know
now that he and I were probably seeing evidence of g modes in $\delta$ Sct
stars, but we did not recognize this at the time.

The history of the discovery of g mode pulsation in A and F stars is
interesting. Variability in $\gamma$ Doradus itself was first noted by Cousins
& Warren (1963)999Alan Cousins published in MNRAS from 1923 to 2001, the year
of his death. This 74-year publishing span may be a record. who had used the
star as a comparison star for $\beta$ Dor (a Cepheid), but noticed an excess
scatter in the $\gamma$ Dor measurements. Cousins et al. (1989) then noted
that $\gamma$ Dor has a period close to 0.74 d. Cousins (1992) reported the
analysis of data gathered over years at the South African Astronomical
Observatory (SAAO) by Fred Marang, Francois van Wyk, Bob Stobie, and himself.
It was not the custom at that time to include these others as co-authors,
although they were acknowledged. Alan Cousins spoke independently to Luis
Balona and to me about the frequency analysis of his data. I failed to
disentangle the amplitude spectra, but Luis Balona did so successfully, and
was acknowledged in Cousins’ paper. Cousins (1992) announced that $\gamma$ Dor
has two pulsation frequencies, $1.32098$ d-1 and $1.36354$ d-1. He noted that
these are too low in frequency for a $\delta$ Sct star, and he speculated
about possible spots or tidal distortion as an explanation for the two
frequencies. It was not understood at that time that these two frequencies are
from g modes.

TESS has now observed $\gamma$ Doradus itself in 6 Sectors, and I have
analyzed those data to compare with the early work. Stunningly, there are two
principal pulsation modes in the TESS data with frequencies of $1.32098$ d-1
and $1.36378$ d-1, the same as Cousins’ values within the uncertainties. That
is remarkable, given that he and his collaborators were working with sparse
ground-based data spanning years, giving amplitude spectra with complex, and
confusing, spectral windows. In a further study of $\gamma$ Dor, Balona et al.
(1994a) correctly suggested that $\gamma$ Dor may be an example of a new class
of pulsating star, then Balona et al. (1994b) suggested that $\gamma$ Dor is
the best example of that new class. That was the discovery paper of the class
of $\gamma$ Dor stars, a class that was later defined by Kaye et al. (1999).

###  A.3 Frequency variability in two $\delta$ Sct stars

It is not surprising to find frequency variability in $\delta$ Sct stars,
given their complex behavior with many modes, both radial and nonradial,
simultaneously excited. This is illustrated here with two, of many, cases:
that of 4 CVn for which Breger et al. (2017) analyzed over 800 nights of
ground-based photometry obtained over more than 40 yr; and that of KIC 5950759
(Bowman et al., 2021) for which 4 yr of Kepler data were analyzed.

Breger et al. (2017) noted the complexity of the FT of 4 CVn and concentrated
their attention on higher amplitude modes. They found both frequency and
amplitude variations that are correlated from mode-to-mode over the decades,
although the signs and amplitudes of the variations differ. The dominant modes
studied were nonradial dipole and quadrupole modes for which, interestingly,
they found frequency changes of opposite sign for modes of order $+m$ and
$-m$. Since the frequency separation of those modes is a measure of rotation
in the pulsation cavity, their result suggests internal changes in the
rotation of 4 CVn over the 40 yr of observations.

Bowman et al. (2021) combined photometry for KIC 5950759 from both ground-
based WASP observations and a 4-yr Kepler data set, along with spectroscopy to
determine fundamental parameters. KIC 5950759 is unusual: it is a high
amplitude $\delta$ Sct (HADS) star (meaning $\Delta V>0.3$ mag peak-to-peak),
and it has low metallicity, hence may also be an SX Phe star (Population II
$\delta$ Sct star). Its FT shows an abundance of harmonics and combination
frequencies for the two highest amplitude frequencies, which were identified
as the radial fundamental and first-overtone modes. Evolutionary models put
the star in the contraction phase at the end of the terminal-age main-sequence
(TAMS), but the measured period changes for both modes, $\dot{P}/P=10^{-6}$
yr-1, are two orders of magnitude higher than expected from evolution.

Asteroseismic frequency changes have great promise for measuring stellar
evolution in real time. But we must identify and separate all the cases of
frequency changes arising from other causes to realize that promise.

###  A.4 New techniques to determine internal rotation

Takata et al. (2020a) developed a new, model-independent diagnostic of
internal rotation in $\gamma$ Dor stars using prograde sectoral g modes.
Instead of modeling the slopes in the $\Delta P-P$ diagram, they plotted the
pulsation frequencies, $\nu$, versus the square root of the frequency
differences, $\sqrt{\Delta\nu}$, then used that to make mode identification,
determine the average internal rotation rate, and measure $\Pi_{0}$ ($P_{0}$
in their notation). In a test of a few cases, they showed that the rotation
rates are consistent with those found for the same stars previously in
multiple other studies. This new method will be useful for large ensembles of
$\gamma$ Dor and SPB stars expected from TESS data where series of prograde
sectoral g modes frequencies are found. Following Van Reeth et al. (2016),
Takata et al. extended the method to model r modes, too, with improved
estimates of the average internal rotation and of $\Pi_{0}$.

In an interesting development, Ouazzani et al. (2020) and Saio et al. (2021)
independently examined the interaction of core inertial modes with the
surrounding radiative envelope. Inertial modes occur where the Coriolis force
is primary restoring force; they have frequencies between zero and twice the
rotation frequency of the cavity in which they are trapped, in this case the
stellar core. These modes in the rotating convective core can resonantly
couple with g modes (hence gravito-inertial modes) propagating above that
core. This gives an observational measure of the rotation of the convective
core, something the g modes by themselves cannot provide, as they are
evanescent in convection zones.

Ouazzani et al. provided a theoretical exposition for this new connection, and
Saio et al. applied the method to 16 $\gamma$ Dor stars. They found a
distinctive dip in the $\Delta P-P$ diagram, which occurs where there is a
resonance at a spin parameter $2f/\nu\sim 8-11$, where $f$ is the rotation
frequency of the convective core and $\nu$ is a g mode pulsation frequency in
the co-rotating frame. Allowing the core to rotate differentially to the
surrounding envelope in their models, they found for the 16 stars nearly
uniform rotation, with less evolved stars having slightly higher core rotation
rates. Ouazzani et al. found in their models multiple resonances possible,
although only one dip in the the $\Delta P-P$ diagram has yet been observed
for each star.

In light of these results, note that KIC 11145123 (Hatta et al., 2019, Section
2.4), with a core rotation rate 6 times the envelope rate, differs from the 16
$\gamma$ Dor stars studied by Saio et al. (2021). However, KIC 11145123
rotates very slowly; even at 6 times the envelope rotation rate the core
rotation period is still only $\sim$17 d, whereas all 16 stars studied by Saio
et al. have core rotation periods in the $0.4-0.8$ d range.

###  A.5 Two examples of atomic diffusion in asteroseismic models.

Deal et al. (2017) first noted the importance of including atomic diffusion in
asteroseismic modeling of exoplanet host stars, which they found increased the
age determined in the exoplanet host star 94 Cet by few percent. They noted
that the effect on age could be larger for hotter stars and for lower
metallicity. It is clear that the inclusion of atomic diffusion for
asteroseismic modeling is important for stars with masses greater than 1.4 M⊙.

Mombarg et al. (2020) added atomic diffusion to models for asteroseismic
inference. They found a better match to the observations in the case of the
$\delta$ Sct star KIC 11145123 (Section 2.4) with atomic diffusion than
without it, although in another case the atomic diffusion did not provide
better models. For KIC 11145123 they also examined whether atomic diffusion
might explain its abnormal abundances. Spectroscopic analysis by Takada-Hidai
et al. (2017) found that the star is metal weak, $[{\rm Fe/H}]=-0.71$. From
its high radial velocity and position above the galactic plane, they suggested
that it is a blue straggler that has undergone mass accretion from a
companion. Mombarg et al. (2020) revisited this problem, and using distance
and proper motion from Gaia found the star to be 260 pc away from the galactic
plane with a peculiar velocity of 72 km s-1, suggesting it might be a runaway
star. They attempted to model the surface abundances with atomic diffusion and
had some success, but could not reproduce the metal deficiency starting with a
solar abundance model. KIC 11145123 still has mysteries to solve.

###  A.6 Evidence of weak spots in upper main-sequence stars

Degroote et al. (2011) presented evidence of photometric variation due to
spots in a B star, HD 174648, observed by CoRoT. From a sample of about 90 000
stars in the Kepler, K2 and TESS data, Balona (2021) suggested that about
$30-40$% of upper main-sequence stars in the Kepler data show evidence of
rotational variation. Li et al. (2020b) examined a sample of 611 $\gamma$ Dor
stars, and, using a more stringent criterion than Balona, found 9.5% of those
stars to have rotational variations. The origin of spots in upper main-
sequence stars is not yet known. Cantiello & Braithwaite (2019) examined the
possibility that weak magnetic fields of order of a few Gauss could be
generated by dynamos in the surface convection zones of A stars; perhaps those
could generate spots. Whatever the resolution of this problem of origin, the
evidence is good that there are weak spots (compared to the Sun) that cause
rotational light variations in upper main-sequence stars. Since the rotation
periods for many of these stars are in the same range as the g mode and r mode
pulsation periods, care is necessary to identify correctly the surface
rotation signal. The importance of doing so is that the spots and r modes
measure the surface rotation period, which can then be compared to the deep
near-core period obtainable from asteroseismology using the g modes. Li et al.
(2020b) did just that, using period spacings of g modes to examine near core
rotation rates, and spot variations, where identified, to measure surface
rotation rates.

###  A.7 Be star history

Angelo Secchi ($1818-1878$) was an Italian Jesuit priest and astronomer.
Following the Roman revolution of 1848, he spent two years at Stonyhurst
College in Lancashire – not far from my own UK institution – before becoming
the director of the Roman College Observatory on his return in 1850, a
position he held until his death in 1878. In 1870 when Italy was unified and
the final papal state, which included Rome, was incorporated in the Kingdom of
Italy, Secchi refused to shift his allegiance (of course!) from Pope Pius IX
to King Victor Emmanuel II. He was illustrious by then and remained director
of the observatory by the expedient of threatening to leave Italy and take up
one of offers he had elsewhere in Europe. The Royal government chose not to
interfere. The observatory was nationalized on Secchi’s death in 1878, but was
resurrected as the Vatican Observatory in 1891. Secchi is most famous for his
work in spectroscopy and his first introduction of his five spectral classes,
later superseded by the Harvard classes we use now. One of his first
discoveries after installing a new spectroscope was finding that $\gamma$ Cas
(classified B0IVe now) had a “very beautiful luminous line” (Secchi, 1866)
which he identified as the Fraunhofer F-line (H$\beta$); he further noted that
in addition to that “shiny band” there are others. He concluded that “This
star therefore has a spectrum opposite that of the ordinary type of white
stars.” This was the discovery of emission lines in stars, and, in particular,
was the first B emission line star of a class now known as Be stars.

###  A.8 More about roAp stars

Ryabchikova et al. (2007) derived the variation of pulsation amplitude and
phase in 10 roAp stars from line bisector studies, showing how the pulsation
amplitude rises with atmospheric height, accompanied by pulsation phase
changes. Freyhammer et al. (2009) studied the radial velocity pulsation and
rotation variations in the singly-periodic roAp star HD 99563, with
contemporaneous data from the 8-m VLT and Subaru telescopes obtained over the
full 2.91-d rotation cycle. From a study of spectral lines of Ca, Eu, Nd, and
the core of H$\alpha$, they found pulsation amplitude increasing with
atmospheric height, also accompanied by phase changes. They found that the
single pulsation mode is a distorted dipole mode, and that the distortion
appears to be different for spectral lines of different ions, meaning that the
geometry of the pulsation mode varies with atmospheric depth, consistent with
the findings of Kochukhov (2006) for HR 3831 and Kurtz & Holdsworth (2020) for
HD 6532.

Deal et al. (2021) examined the fundamental properties of 14 roAp stars using
both interferometry and asteroseismology with a grid of models. Interestingly,
they found that roAp stars are systematically less massive and older than non-
pulsating Ap stars, a result that may inform the still problematic question of
pulsation driving in these stars. They also constrained their models better
with the additional knowledge of the large separation, $\Delta\nu$, for two
stars, HR 1217 and $\alpha$ Cir. While the magnetic field in roAp stars
significantly perturbs the mode frequencies, it does so almost equally for all
modes, so that the large separation is still an important asteroseismic
constraint. Deal et al. found normal, solar metallicities for those two stars,
which when contrasted with the stunning overabundances of the rare earth
elements in the atmosphere is an important observational constraint on atomic
diffusion.

Detailed asteroseismology of the deep interior of roAp stars will be
difficult. The high overtone magneto-acoustic modes are most sensitive to the
outer envelope, and the gravito-inertial g modes that so beautifully
illuminate our view of the cores of $\gamma$ Dor and SPB stars (section 2.5)
are not observed. Murphy et al. (2020c) found in magnetic models that some low
overtone g modes can be excited in Ap stars, but high-overtone g modes are
heavily damped in stars with field strengths stronger than $1-4$ kG, as is
typical in Ap stars. They also showed from models that field strengths greater
than about 1 kG strongly damp low-overtone $\delta$ Sct p modes, which, if
they were observed, could inform the conditions somewhat deeper than the high-
overtone modes typically observed.

####  A.8.1 Frequency variability in roAp stars

Holdsworth (2021) reviewed Kepler observations of roAp stars. Only 14 roAp
stars – both known and newly discovered – were observed by Kepler and all show
some frequency variability (as do many $\delta$ Sct stars, Section 2.2). That
frequency variability is typically non-cyclic, as is the case for 16 yr of
ground-based observations of the roAp star HR 3831 (Kurtz et al., 1997). As
with frequency variability in many classes of pulsating stars, this may be
evolutionary weather (Section 2.2). Ap stars in general are seldom found in
close binaries (Schöller et al., 2012), so the frequency variability does not
have a contribution from orbital motion (Section D.1).

####  A.8.2 TESS and roAp stars

TESS observed 960 stars classified as Ap in 2-min cadence in all 13 sectors of
cycle 1 to determine the fraction of Ap stars that are roAp stars, and to find
new roAp stars for further analysis. In addition, a team searched the light
curves of over 50 000 stars hotter than $T_{\rm eff}=6000$ K for the signature
of roAp pulsation in the FTs (Holdsworth et al., 2021). Together with earlier
examinations of Ap stars in subsets of TESS cycle 1 sectors by Balona et al.
(2019) and Cunha et al. (2019), Holdsworth et al. (2021) found 12 new roAp
stars and further analyzed 44 previously known ones. There are, as of their
writing, 88 known roAp stars. It had previously been suspected that roAp stars
only appeared rare because their pulsation amplitudes were too low to be
detected in ground-based observations. The TESS data show that they are truly
rare, finding only 3% of the Ap stars observed to be roAp stars. The reason
for this low incidence is not known.

The TESS data yielded the longest pulsation period in an roAp star, 25.8 min,
and the shortest, 4.7 min, extending the period range, hence radial overtones
for these stars. The relation between rotation period and roAp pulsations is
not yet known, hence the increased sample provides more data to examine this.
Holdsworth et al. (2021) found that 68% of the 56 roAp stars with TESS data
show rotational modulation from spots. This is useful for precise
determination of the rotation period, but it is only for stars with $P_{\rm
rot}\stackrel{{\scriptstyle{<}}}{{{\scriptstyle\sim}}}22$ d, as a consequence
of the 27-d TESS sectors.

Mathys et al. (2020) hypothesized, then showed that Ap stars with TESS data
that do not show rotational modulation due to spots are almost all super-
slowly-rotating Ap (ssrAp) stars, which they defined to be magnetic Ap stars
with rotation periods longer than 50 d. Of the 54 ssrAp stars they found in
cycle 1 data and 68 in cycle 2 data (Mathys, Kurtz & Holdsworth, in press), at
most a few can be pole-on, hence they significantly increased the number of
ssrAp stars known. These stars have magnetic field strengths in the range
$0-10$ kG typical of Ap stars, and they have rotation periods known, in some
cases, to be greater than a century. Hence they address the question of
magnetic braking. Interestingly, 22% of 54 ssrAp stars in cycle 1, and 19% of
the 69 ssrAp stars in cycle 2, are roAp stars (or candidates), an apparently
significantly larger fraction than the 3% found for the whole 960 Ap stars
sample. This thus begins to address the relation between rotation and
pulsation in roAp stars.

[] ssrAp stars Super slowly rotating Ap stars with rotation periods greater
than 50 d up to centuries.

A prime reason for finding more roAp stars is to find stars with rich FTs with
alternating even and odd degree modes that readily give mode identification
for asteroseismic modeling. The finding of new roAp stars also leads to
objects with higher amplitudes suitable for future high time resolution
spectroscopic study with large telescopes. Holdsworth et al. (2021) provide
first analyses of the roAp stars with TESS data, characterizing their
pulsation and rotation. As in the study of HD 86181 (Shi et al., 2021), many
stars have rotationally split multiplets useful for constraints on mode
geometry, and some stars have multiple modes suitable for future asteroseismic
modeling to determine fundamental parameters such as mass, age, luminosity,
and overall metallicity in the pulsation cavity. Determining the latter may
inform the question of the depth of the extreme peculiarities seen in the
atmosphere, and that is relevant to theoretical calculations of atomic
diffusion timescales.

##  Appendix B Pulsating white dwarf stars

###  B.1 The Chandrasekhar – Eddington debate

It is less than 100 years ago that in 1930 Subrahmanyan Chandrasekhar arrived
by ship from Chennai (then Madras) to begin his postgraduate studies at
Cambridge University. On the sea voyage he spent his time calculating the
maximum mass a white dwarf star can have considering the quantum mechanical
Pauli exclusion principle and including, for the first time, Special
Relativity. He was a 20-year-old prodigy at the time. During the early 1930s
Chandra derived the complete solution to the problem of a relativistically
degenerate star. He famously presented this work publicly at the Royal
Astronomical Society January 1935 meeting at Burlington House. He was
surprised to have his work dismissed in a following talk by Professor Arthur
Eddington (whose quote from The Internal Constitution of the Stars opened this
review). Eddington did not believe Chandrasekhar’s results on white dwarf
stars, and such was Eddington’s stature that Chandrasekhar’s result was not
widely accepted by astronomers for decades. It was, however, accepted at the
time by physicists, notably Niels Bohr and Wolfgang Pauli. While Eddington and
Chandrasekhar had a generally friendly personal relationship, the acrimonious
debate on the nature of white dwarfs appeared to have overtones: of age and
authority (Professor – student), and race and nationality (it was a time of
Empire); see Wali (1982) for a thorough discussion of this confrontation.
Chandrasekhar was worn down by the debate, when he knew his results were
correct, so he published a textbook (Chandrasekhar, 1939) on stellar structure
(which I used as an undergraduate and still have on my bookshelf) and moved on
to other fields. His recognition for the derivation of the Chandrasekhar limit
of $1.44$ M⊙ for the maximum mass of a white dwarf was awarded the Nobel Prize
in Physics in 1983.

Eddington believed that all stars must end their lives as white dwarfs, hence
his antipathy to Chandrasekhar’s mass limit. In the 1930s the two other end
states for stars, neutron stars and black holes, were yet to be understood,
and it was only in the mid-1960s to early 1970s that neutron stars and black
holes were observationally discovered. Initially, it was thought that only
stars with masses less than $1.44$ M⊙ could become white dwarfs, but we now
know that mass loss – primarily in winds and shell loss episodes during late
thermal pulses of Asymptotic Giant Branch stars, but also by other mechanisms
– allows about 97% of all stars to end up as white dwarfs as their final
stages. So Eddington’s visceral desire that all stars end as white dwarfs was
almost correct, albeit for wrong reasons and incorrect arguments.
Chandrasekhar was right. The 3% of stars that Eddington was not correct about
produce the spectacular physics: of neutron stars; of stellar-mass black holes
– the seeds of supermassive black holes, hence galactic jets and quasars; and
of merging neutron stars and black holes – the sources of detectable
gravitational waves.

###  B.2 ZZ Cet stars (DAV)

The vast majority of white dwarfs show H in their atmospheres, hence are
spectroscopically classified DA. Their temperatures range from 170 000 K down
to around 4500 K for the coolest white dwarfs known, where that lower limit is
set by the age of the galaxy; the first white dwarfs that formed when the
galaxy was young have not yet had time to cool any further. Asteroseismically,
the most abundant pulsating white dwarfs are the DAV ZZ Cet stars. If we could
observe white dwarfs to the distances we can for brighter stars, white dwarfs
would be the most abundant pulsating stars observed in the galaxy.

It is now more than 50 years since the discovery of the first pulsating white
dwarf star, HL Tau 76 (Landolt, 1968). In that time there have been excellent
asteroseismic successes in the studies of these stars. Particularly, the
spacing of the g mode periods in white dwarfs yield a precise mass measure. A
mass-radius relation then leads to a radius determination, somewhat dependent
on the metallicity. Forward modeling leads to asteroseismic luminosity and
temperature estimates. Given radius, temperature, and luminosity, an
asteroseismic distance can be calculated and compared with Gaia distances,
with good agreement.

[] ZZ Cet stars DAV white dwarfs; g mode pulsators. $T_{\rm eff}$ of
$10\,400-12\,400$ K; periods of $100-1400$ s. Typical mass is around $0.6$ M⊙.

Deviations from regularity in the period spacing in white dwarfs give measures
of the extent of the outer, non-degenerate layers where the modes are trapped.
The precise period spacings also give constraints on atomic diffusion (Section
2.7), an important mechanism of stratification in white dwarfs. The primary
asteroseismic data in white dwarfs are, therefore, g mode period spacings with
mode identifications. A robust method to identify those modes comes from
recognition of rotationally-split multiplets – typically triplets for dipole
modes and quintuplets for quadrupole modes. A collateral, important benefit is
the ability to determine the rotation period, hence angular momentum for the
pulsation cavity, the outer envelope.

White dwarf studies continue from the ground, outpacing what has yet been
accomplished with Kepler and TESS data. An outstanding new initiative is the
exploitation of data from the Zwicky Transient Facility (ZTF). The ZTF (Bellm
et al., 2019) is a northern analogue to the Vera Rubin Observatory (VRO),
albeit in a much smaller telescope (1-m aperture, as opposed to 8-m). Among
the numerous, broad science objectives of the ZTF are stellar variability
studies (Graham et al., 2019, section 8). With a shortest cadence of 38.3 s,
the ZTF will discover abundant pulsating white dwarfs, the majority of which
will be DAV stars. It will, of course, find vast numbers of other types of
pulsating stars – possibly mostly $\delta$ Sct stars – from which automated
searches will seek, find, and classify objects for followup, much of which
will be ground-based, but some of which will be with TESS, PLATO, and their
successors in space.

[] ZTF Zwicky Transient Facility VRO Vera Rubin Observatory; previously known
as the Large Synoptic Survey Telescope (LSST)

An stunning example of this potential is the discovery of a 6.94-min rotation
period in ZTF J190132.9+145808.7, a white dwarf with a mass of nearly the
Chandrasekhar mass so that it almost as small as the Earth’s Moon ($R\sim
2100$ km), with a magnetic field in the $600-900$ MG range (Caiazzo et al.,
2021). This star is probably the result of a merger of two white dwarfs with
orbital angular momentum being transferred to rotation in the remnant, hence
the very short rotation period. This is not the shortest known white dwarf
rotation period, however. Reding et al. (2020) discovered in Kepler data the
fastest rotating, isolated white dwarf star, EPIC 228939929. This star has a
rotation period of 5.28 min seen in rotational light variations caused by a
spot associated with a 5 MG magnetic field.

The ZTF discovered ZTF J190132.9+145808.7, and ground-based followup, both
photometric and spectroscopic, led to the discoveries and conclusions
discussed above. Large Scale Surveys have been spectacularly successful in
pulsating star research, OGLE and ASAS being prime examples. But those did not
have the short cadence capability of the ZTF, hence that facility leads in the
detection of pulsating, rotating and eclipsing compact stars, where the
periods are as short as minutes. TESS will provide long duration, precise
light curves for some of these objects with a 20-s cadence, in the best cases,
and 2-min cadence in many others.

Hermes et al. (2017b) studied 27 ZZ Cet stars observed by the Kepler K2
mission, with spectra obtained for all 27. They discovered that pulsation
periods shorter than about 800 s are stable over the 90-d time span of the K2
data, but periods longer than 800 s have a Lorentzian profile in the FT
reminiscent of the stochastic pulsators, such as solar-like stars and red
giants, implying modes lifetimes of only a few days to weeks, similar to the
theoretical damping times for longer period g modes in white dwarfs. This is
an interesting new discovery that is yet to be understood or exploited. Hermes
et al. measured the internal rotation of 20 of the 27 stars from the frequency
splitting of dipole triplets, doubling the number of DAV stars for which
asteroseismic rotation rates are known. The rotation periods are as short as
hours, but typically just over a day. This is far slower than would be the
case if the red giant core conserved angular momentum as it contracted with
evolution, supporting strong angular momentum transfer between core and
envelope in red giant stars. Asteroseismology of mixed modes in red giants
show their cores to be rotating about 10 times faster than their envelopes
(Beck et al. 2012; Mosser et al. 2012), giving periods in agreement with the
white dwarf rotation periods (Aerts et al., 2019).

Hermes et al. also showed that the DAV stars observed by Kepler have the same
correlation of pulsation period with temperature previously known from ground-
based studies: the weighted mean pulsation period (WMP) decreases from the
blue to red edges of the instability strip. The WMP (which could have been
called $P_{\rm max}$) is similar to $\nu_{\rm max}$, the frequency of maximum
pulsation amplitude in stochastic pulsators. In red giants $\nu_{\rm max}$
decreases with increasing luminosity because of decreasing sound speed and
increasing radius (hence the Leavitt Law – the PL relation); in DAV white
dwarfs $P_{\rm max}$ increases with decreasing temperature because the surface
convection zone deepens as the stars cool.

Pulsation in DAV stars is driven by convection in the surface H ionization
zone (Brickhill 1991; Wu & Goldreich 1999). As white dwarfs cool and cross the
ZZ Cet instability strip at around $13\,000$ K (depending on mass), excitation
in the relatively thin convection zone drives pulsations in low radial
overtone modes (of which the dipole and quadrupole modes are observable) with
periods in the $100-300$ s range; the mode frequencies for these are stable.
Since white dwarfs evolve simply by cooling, as the star cools across the
instability strip, the surface convection zone deepens and the periods become
longer. Stars in the middle of the strip have the highest amplitudes, periods
longer than 300 s, and many combination frequencies resulting from nonlinear
interactions among the modes caused by changes in the convection zone during
the pulsation (Brickhill, 1992). Montgomery (2005) showed how these
nonlinearities can be used to measure the depth of the convection zone and
constrain mode identification, a critical precursor for asteroseismic
inference. These stars in the middle of the instability strip also have modes
that show amplitude and frequency variability, unlike the hotter ZZ Cet stars
which have stable modes. Finally, as the stars approach the red edge of the
DAV instability strip around $11\,000$ K, increased radiative damping
overwhelms the convective driving (Luan & Goldreich, 2018).

As the DAV white dwarfs cool even further towards the red edge of the
instability strip, some of them undergo outbursts, a remarkable new phenomenon
first discovered by Bell et al. (2015) in a 1.5-yr Kepler light curve of KIC
452982; Bell et al. (2017) listed six DAVs that outburst. They found that the
outbursts release $10^{26}-10^{27}$ J causing an increase in brightness of the
star by about 15%, recur irregularly on a timescale of days, and cause the
pulsation mode amplitudes and frequencies to vary. Luan & Goldreich (2018)
suggested that the outbursts are connected with resonant three-mode coupling
among overstable parent modes and (otherwise) damped daughter modes. This is
potentially testable with studies of combination terms in the frequencies of
outbursting DAV stars. This outburst phenomenon is now also seen in as-yet
unpublished TESS data, including a star that is also an eclipsing binary with
an orbital period of 1.5 hr. It will be interesting to see whether binarity
has an importance in the outbursting DAV phenomenon, and whether the stars
have any relation to cataclysmic variables. Large numbers of outbursting DAV
stars are expected to be discovered with the ZTF and VRO surveys, to be
followed up from the ground and with short-cadence TESS, and later PLATO,
observations. Since the outbursts last for $4-24$ hr, they may have been
present in ground-based studies, but were lost in the data processing which
has to correct for daily artifacts. Some reprocessing of ground-based
multisite campaigns may resolve this issue, but it is the long time spans of
the space data that will illuminate this phenomenon.

Bognár et al. (2020) reported on TESS first year observations for 18
previously known ZZ Cet stars, comparing the TESS results with ground-based
results, with some increase in the number of detected modes for 8 of the
stars. The other 10 did not show detectable pulsations from a combination of
reduced amplitude because of the 2-min cadence, field crowding because of the
relative large TESS pixels (21 arcsec), and low count rates for the faster
stars. This provides some caution for future use of TESS for faint objects,
although the more recent 20-s cadence is a great benefit for white dwarf
studies.

###  B.3 GW Vir stars (DOV)

The GW Vir stars are the hottest class of pulsating white dwarfs and pre-white
dwarfs. They are DO white dwarfs, classified so by the presence of HeII lines
in their spectra; they are H deficient, and rich in C and O. They are driven
by the $\kappa$-mechanism where the variable opacity comes from partial
ionization of C and O in the upper layers of the stars. The prototype GW Vir,
also known as PG 1159, is one of the best-studied of all pulsating white
dwarfs. Winget et al. (1991) found over 100 mode frequencies, determining the
mass to high precision, the rotation period, and limiting the magnetic field
strength. Costa et al. (2008) extended that to nearly 200 mode frequencies,
improving the mass determination and magnetic field limit, and measuring the
extent of the mode trapping region. Both of these studies used data from the
Whole Earth Telescope (WET). No GW Vir stars were observed by the Kepler
mission, but now 6 known GW Vir and PNNV stars have been observed by TESS
(Córsico et al., 2021). That work includes PNNV that are also Wolf-Rayet (WC)
stars, which have similar pulsation characteristics to the GW Vir stars. These
stars will become even hotter as they contract, reaching $T_{\rm
eff}>200\,000$ K before evolving down the white dwarf cooling tracks. The
study of Córsico et al. (2021) confirmed, with more mode frequencies, earlier
ground-based studies of some stars, determined the rotation period in one
star, and showed that the mass determinations from period spacings are robust,
and that the asteroseismic distances agree well with Gaia distances. These
hottest pre-white dwarfs are in transition from being the cores of red giants
to spending the rest of time (unless the proton has a finite lifetime) on the
white dwarf cooling tracks, eventually becoming black dwarfs.

[] GW Vir stars include DOV white dwarfs; g mode pulsators. $T_{\rm eff}$ of
$75,000-100\,000$ K; periods of $300-2600$ s.

Planetary Nebulae Nuclei Variables (PNNV); g mode pulsators. $T_{\rm eff}$ of
$100\,000-200\,000$ K; periods of $400-6000$ s.

###  B.4 V777 Her stars (DBV)

The DBV stars show He I lines and little or no H in their spectra. Winget
(1982) predicted the existence of this class theoretically, then Winget et al.
(1982) successfully discovered the prototype, GD 358 = V777 Her. Hermes et al.
(2017c) reported a study of rotation in the hottest ($T>30\,000$ K) known DBV
star, PG 0112+104 using a 79-d Kepler K2 data set. This star shows a clear,
stable 10.17-hr rotational light curve from a spot (or spots) that is
reminiscent of the typical rotational light curves of the magnetic spotted
$\alpha^{2}$ CVn Ap stars. This was the first known spotted, pulsating white
dwarf star. There are 26 detected pulsation frequencies from 11 independent
dipole and quadrupole modes, which can be used to probe radial differential
rotation, since the dipole and quadrupole modes have different pulsation
cavities; that study is yet to be done. The low amplitudes suggest that more
such stars will be found in TESS data, with surface spots, and dipole and
quadrupole modes that will provide better understanding of the radial
dependence of rotation in white dwarfs, as has been done in many main-sequence
stars (see Aerts et al. 2019).

[] V777 Her stars DBV white dwarfs; g mode pulsators. $T_{\rm eff}$ of
$22\,000-32\,000$ K; periods of $120-1080$ s.

Bell et al. (2019) have reported on a first study of a DBV star with 2-min
cadence TESS data showing the benefit of the long data sets for stars with
stable frequencies. The best asteroseismic temperatures from their models were
somewhat discordant with the spectroscopic temperature; further modeling
should resolve that. With 20-s TESS we can expect significant new constraints
on white dwarf structure.

###  B.5 Pulsating Extremely Low Mass Variable (ELMV) white dwarf stars

Low mass (LM) white dwarfs are now known in relatively large numbers as a
consequence of targeted surveys. They are generated when mass transfer in
binary evolution exposes the core of a red giant before the He flash. There is
a subset these stars at Extremely Low Mass (ELM) with $M<0.18$ M⊙. Hermes et
al. (2012) discovered the first pulsating ELM (ELMV), SDSS
J184037.78+642312.3, a DAV star with a mass of only $0.17$ M⊙. Adiabatic
pulsation analysis showed that these stars pulsate in g modes that primarily
probe the core region, unlike higher mass DAV stars where the g modes have
pulsation cavities largely confined to the non-degenerate atmosphere. Althaus
et al. (2013) calculated evolutionary models and determined mass and cooling
age for 58 ELM stars, three of which pulsate in g modes.

[] Low mass (LM) white dwarfs He core white dwarfs with
$M\stackrel{{\scriptstyle{<}}}{{{\scriptstyle\sim}}}0.43$ M⊙; remnant red
giant cores stripped prior to the He flash. Extremely Low mass variables
(ELMV) He core, H atmosphere white dwarfs with $M<0.18$ M⊙ with surface
temperatures in the range $7800-10\,000$ K; g mode pulsators with periods of
$100-6300$ s.

Córsico & Althaus (2016) performed a non-adiabatic pulsation analysis of
binary evolutionary sequences for ELMV stars, starting with a 1.0-M⊙ main-
sequence star with a 1.4-M⊙ neutron star companion. They found large numbers
of excited p modes and g modes driven by the $\kappa-\gamma$ mechanism, i.e.
excitation by time variable opacity enhanced by radial depth dependence of the
adiabatic exponent in the ionization zone. They also found some short period g
modes to be unstable via the $\epsilon$-mechanism providing driving in the H
shell burning zone, as there is sufficient residual H in ELM white dwarfs for
some nuclear energy generation. That these ELM white dwarfs can still sustain
some H fusion via the p-p chain lengthens their cooling time by orders of
magnitude compared to slightly more massive white dwarfs with $M>0.18-0.20$ M⊙
where multiple CNO flashes burn off the residual H quickly. Calcaferro et al.
(2017) performed asteroseismic analyses for 9 ELMV stars using those models.
Each star has only a few mode frequencies, whereas the models suggest many
modes should be excited. Forward modeling gave masses consistent with
spectroscopic masses.

###  B.6 Blue Large Amplitude Pulsators (BLAPs)

BLAPs are a newly discovered class of pulsators found in both the OGLE-IV and
ZTF surveys. These stars are stripped red giant cores that pulsate with
periods in the range $180-2400$ s, with amplitudes of a few percent. While
there are only 18 currently known, they seem to split into two groups of lower
($4.2\leq\log g\leq 4.6$) and higher ($5.3\leq\log g\leq 5.7)$ surface
gravity. Byrne & Jeffery (2020) and Byrne et al. (2021) have modeled their
evolutionary stage, concluding that they are pre-ELM H-shell burning white
dwarfs. The pulsations are driven by the Fe-peak opacity enhanced by the
radiative levitation of Fe for stars in both gravity ranges, so they are
likely the same phenomenon. Although some BLAPs overlap with the sdBV
pulsators (Section 4.2) in the HR Diagram, they have larger amplitudes than
the sdBVs, hence are not confused with them. There is discussion about the
nomenclature for this class, which could therefore change. They are beautiful
example of a new class of a late evolutionary stellar evolution stage where
asteroseismology will inform our understanding of red giant cores. It is
expected that many more of these stars will be found in large surveys, such as
the ZTF and VRO.

[] Blue Large Amplitude Pulsators (BLAPs) pre-ELM H-shell burning white
dwarfs; g mode pulsators with periods of $180-2400$ s.

###  B.7 DQV, hot DAV, and Ultra-massive DAV stars

After the main classes, DOV, DBV, and DAV, there are three additional classes
of variable white dwarfs. The DQ white dwarfs, which have H and He deficient,
C rich atmospheres; the hot-DAV white dwarfs with temperatures at the hot end
of the DBV instability strip; and the ultra-massive (UM) DAV white dwarfs
which have excellent potential to study crystallization in detail (De Gerónimo
et al., 2019). While pulsation remains a possible explanation for the DQV
stars, evidence is strong that their variability is rotational, arising from a
spot or spots in the presence of a strong magnetic field (Williams et al.,
2016). In that case, no asteroseismic inference is possible for these stars,
but their rapid rotation (periods in the range minutes to 10s of minutes)
carries information about their origin and precursors. Those are thought to be
either AGB single stars or mergers of double-degenerate binary stars. In the
AGB scenario the spread in rotation rates for DQV stars must be a function of
internal angular momentum transfer in the precursor during its evolution, and
magnetic braking once the white dwarf is formed. TESS is targeting these stars
to build up a broader pictures of their variability.

[] DQV white dwarfs White dwarfs with H and He deficient, C rich atmospheres.
It is uncertain whether their variability is pulsational or rotational. Hot
DAV white dwarfs H atmosphere white dwarfs at the hot edge of the DB
instability strip. UM DAV white dwarfs Ultra-Massive DAV white dwarfs with
masses in the range $1.10-1.29$ M⊙. Strongly crystallized C-O or O-Ne cores.

The hot DAV stars are H atmosphere stars at the hot edge of the DB (helium
atmosphere) instability strip. Shibahashi (2005) predicted their existence,
following which three cases were found (Kurtz et al. 2013; Kurtz et al. 2008).
Some uncertainty whether these stars constitute a new class of pulsating white
dwarfs persists, but the recent discovery of a fourth such star, HE 1017–1352,
by Romero et al. (2020) strengthens the case. For the latter star there are
TESS data for which the pulsation signal, while not at the 1/1000 level of
False Alarm probability, is consistent with the robust detections from ground-
based photometry. The lower significance of the TESS signal is a result of the
lack of blue light in the TESS passband, which demonstrates why ground-based
photometric observations of hot pulsating white dwarfs are still needed in the
space photometry era.

The adiabatic pulsation properties of the UM DAV ($1.10\leq M\leq 1.29$ M⊙)
white dwarfs were studied in depth theoretically by De Gerónimo et al. (2019),
who found the cores to be highly crystallized and suggested that g mode period
spacings may be able to discern stars with C-O cores from those with O-Ne
cores, an intriguing prospect.

###  B.8 EL CVn binary systems

Maxted et al. (2013) discovered a new kind of pulsator in a ELM white dwarf
star in an eclipsing binary, 1SWASP J024743.37-251549.2. This pulsator is a
pre-He white dwarf, the core of a stripped red giant star, in orbit with an A
or F star. It pulsates in mixed modes, which have g mode character in the core
and p mode character in the outer layers, hence they probe the entire star.
Maxted et al. (2014) found 17 of these systems and named the class after the
brightest of those, EL CVn. Rappaport et al. (2015) reported on two of the
Kepler systems (out of 13 found in Kepler data) with mass and radius
determinations, finding thermal bloating of the pre-white dwarf to the size of
a typical red dwarf, up to 14 times what the cold degenerate radius would be.
van Roestel et al. (2018) found 36 EL CVn systems in a machine learning search
of 800 000 stars with the ZTF, and more than 100 of these systems are now
known from a variety of surveys. They found the pre-He white dwarfs to have
temperatures of $7900-17\,000$ K, while their AF companions have
$6600-10\,000$ K. They also found a secure mass range of $0.12-0.28$ M⊙, but
possibly as high as $0.5$ M⊙, and radii of $0.17-0.65$ R⊙ – very bloated for
white dwarfs.

[] EL CVn binaries Bloated pre-He white dwarfs in short-period ($<3$ d) orbits
about an A or F star that in some cases is also a $\delta$ Sct star.

Lagos et al. (2020) showed that EL CVn stars are almost all in hierarchical
triples with the orbital period of the eclipsing pair less than 3 d. The
pulsation signal, if any, from the LM or ELM white dwarf is strongly diluted
by the light of the AF primary, which is in many cases a $\delta$ Sct star.
Wang et al. (2020) analyzed two EL CVn systems with TESS data finding
pulsations in both the $\delta$ Sct primary and the pre-He white dwarf. The
frequency ranges of these two types of variables are mostly separated in
frequency space, so both can independently be studied asteroseismically.

The EL CVn stars are important for understanding how close binaries with white
dwarfs form. This is a field that has developed in the last 10 years from
ground-based wide-field survey data, primarily SuperWASP101010https://wasp-
planets.net, and now is benefiting from long duration light curves from space
data. At present only two EL CVn systems have pulsating white dwarf
components, but with the expansion of discovery these systems, particularly by
ZTF and VRO, asteroseismic study will complement inference from orbital
analyses.

##  Appendix C Extreme Horizontal Branch stars: The hot subdwarf O and B
variables

In the early 1950s Jesse Greenstein – the founder of the astronomy program at
CalTech – was observing white dwarfs, sdB, and sdO stars with the coudé
spectrograph on the then-largest optical telescope, the Palomar 200-inch
(5-m). At IAU Symposium 3 held in Dublin in September 1955 he commented, “As
far as I know, light variations have not been found in the white dwarfs”
(Greenstein, 1957). He finished his conference report by noting, “Observations
designed to detect possible short-period light variations [in white dwarfs and
subdwarfs] would be desirable.” Greenstein recognized then that the lack of
detection of photometric variations could well be that the amplitudes were
just too small for the precision then obtainable. He was correct. A decade
later in the 1960s variations began to be found in white dwarfs and subdwarfs.
Landolt (1968) discovered pulsation for the first time in a white dwarf, HL
Tau 76. For the subdwarf stars the discovery of the class of p mode sdBV
pulsators now known as V361 Hyi (or EC 14026) stars only came in the late
1990s, more than 40 years after Greenstein’s suggestion.

In a case of theory leading observational discovery (as for the DBV white
dwarfs; Section B.4), Charpinet et al. (1996) predicted sdB stars to pulsate
from models. They were then independently discovered the following year in the
Edinburgh-Cape Blue Object Survey (Kilkenny et al. 1997, Koen et al. 1997,
Stobie et al. 1997, O’Donoghue et al. 1997). Just a few years later Green et
al. (2003) serendipitously discovered a new class of sdBV g mode pulsators now
known as V1093 Her stars. For unknown reasons the incidence of pulsation for
the V361 Hyi stars is only about 10%, while that for the V1093 Her is about
75%. Both classes have pulsation driven by the $\kappa$-mechanism operating on
Fe and Ni opacity (Charpinet et al., 1996) where abundances of those elements
have been enhanced in the driving zone by radiative levitation (Charpinet et
al. 2001, Jeffery & Saio 2006). Bloemen et al. (2014) performed detailed
calculations of both radiative levitation and gravitational settling for 9
elements showing that Fe and Ni are enhanced by factors up to 60 and 4000,
respectively, in the driving zone, naturally accounting for both the p mode
and g mode pulsations and the extents of the instability strips.

[] V361 Hyi stars subdwarf B variables, sdBV stars, pulsating in p modes with
periods of $60-800$ s. sdBVr stars
(r for rapid). $28\,000\leq T_{\rm eff}\leq 45\,000$ K. V1093 Her stars sdBV
stars pulsating in g modes with periods of $0.5-4$ hr. sdBVs stars
(s for slow). $23\,000\leq T_{\rm eff}\leq 30\,000$ K. DW Lyn stars hybrid
sdBV stars pulsating in both p modes and g modes.
sdBVrs stars. V366 Aqr stars sdBV stars with enhanced Sr, Y, Zr pulsating in g
modes. $T_{\rm eff}\sim 36\,000$ K.

Asteroseismology of sdBV stars has provided the fundamental parameters mass
and radius, as well as structural parameters – the envelope mass, the core
mass and the core composition – illuminating their evolutionary origins. Their
masses are concentrated near 0.47 M⊙, supporting the stripped red giant origin
for these stars. Asteroseismology of the much rarer sdOV stars is still
developing (Van Grootel et al., 2019). Kilkenny et al. (2017) and Kilkenny et
al. (2019) discovered an sdOV star, EC 03089-6421, with $T_{\rm eff}=52\,000$
K and $M\sim 0.1$ M⊙, pulsating with very short periods of 27 s, 31 s, and 34
s. The pulsations in this star are potentially driven by the
$\epsilon$-mechanism and it may be a progenitor of an ELM white dwarf. Finding
pulsations driven by the $\epsilon$-mechanism is a long-standing goal, with
previous claims or suggestions that have not been confirmed. It will be
exciting if models can confirm $\epsilon$-mechanism excitation in EC
03089-6421, giving a direct asteroseismic constraint on the nuclear energy
generation in this star.

There is an intriguing class of heavy metal sdBV stars, the V366 Aqr stars,
with only three members known. They have temperatures in the range of the V361
Hyi p mode pulsators, yet they pulsate in g modes. Strikingly, they show
overabundances of Ge, Sr, Y, and Zr, the latter up to $10^{4}$ solar. The
standard Fe-Ni opacity driving mechanism does not explain how sdB stars this
hot pulsate in g modes, rather than p modes. The $\epsilon$-mechanism has been
proposed, but ruled out for the third member, PHL 417, by Østensen et al.
(2020) based on the stability of the highest amplitude mode. Hence, the
driving mechanism for the V366 Aqr stars is not yet known.

Lynas-Gray (2021) reviewed asteroseismology of hot subdwarf stars, discussing
individually many of the sdBV stars observed by the main Kepler and K2
missions. The TESS mission is now observing many more of these stars, and
studies are in progress. Uzundag et al. (2021) introduced about 40 sdBV stars
observed by TESS in 2-min and 20-s cadence and analyzed 5 V1093 Her stars
pulsating exclusively in g modes. No rotational multiplets were found among
primary dipole modes, and the main asteroseismic inference was that standard
convective core mixing provides good agreement with the mean period spacings.

That is not in agreement with Charpinet et al. (2019) who analyzed a single
TESS sdBV star with detailed asteroseismic inference. They found a He core
mass larger than in standard models in this star and three others previously
analyzed. They concluded that further understanding of the physical processes
that govern the core structure are needed. They also found a precise mass of
$0.39\pm 0.01$ M⊙, which is lower than the mean sdB mass of $0.47$ M⊙,
suggesting that the star evolved from a red giant that had not ignited He in a
core flash. They derived radius, luminosity and a double-layered He and H
envelope structure as a consequence of gravitational settling. Their best
models have a core with $0.20\pm 0.01$ M⊙ with 43% of the He already burned.
They found a rough estimate of the core O mass fraction, which ultimately may
constrain the ${}^{12}{\rm C}(\alpha,\gamma)^{16}{\rm O}$ reaction rate. These
results illustrate the power of asteroseismology for sdBV stars.

Silvotti et al. (2007) used an $O-C$ analysis of pulsation timing (Section
D.1) for the sdBV star V391 Peg, which shows both p modes and g modes. They
found a cyclic variation in the pulsation frequencies and inferred the
presence of a planet with a minimum mass of 3.2 MJup in a 3.2-yr orbit. They
inferred that the planet would have been about 1 AU from the progenitor star
when it was on the main-sequence, suggesting that planets orbiting at that
distance may survive the red giant stage, with implications for the fate of
Earth. Later, Silvotti et al. (2018) found with more extensive data that the
presence of the planet is uncertain, since the oscillatory variations in two
of the frequencies may not be consistent.

The rewards from detecting planets orbiting pulsating stars are great, but
this star is an additional warning about the interpretation of pulsational
frequency variability (Evolutionary weather; Sections 2.2). That is also true
for attempts to measure evolutionary frequency changes. Zong et al. (2016a)
give an explanation for these shorter time-scale frequency variations in the
sdB star KIC 10139564. They find amplitude and frequency modulations that are
the clear signature of three-mode non-linear resonant coupling. Zong et al.
(2016b) also find similar nonlinear mode couplings in the DBV white dwarf KIC
8626021, for which they emphasize the complications for trying to measure
white dwarf cooling times.

The number of known subdwarfs is increasing dramatically, as with other
classes of asteroseismic targets. Geier (2020) has provided a catalog of
nearly 6000 hot subdwarfs. There is also a contentious possible new class of
subdwarfs, the sdA stars. Kepler et al. (2016) recently classified a group of
stars as sdA based only on their hydrogen dominated spectra and surface
gravities, $5.5\leq\log g\leq 6.5$, showing they lie below the main-sequence.
Kepler et al. proposed that many of them are ELM white dwarfs (Section B.5),
but Hermes et al. (2017a) strongly disputed that, and their status is yet to
be clarified.

##  Appendix D New techniques: FM, PM, and sNa

###  D.1 The frequency modulation (FM) and phase modulation (PM) techniques

It has long been recognized that stable pulsation frequencies can be used as
frequency standards to study orbital motion in pulsating stars. Traditionally,
this was exploited in $O-C$ diagrams, where, typically, the observed ($O$)
time of pulsation maximum of a star minus a chosen computed ($C$) value was
plotted against time. A straight line in this plot means the period is
constant over the duration of observations; any other shape indicates
variations in the period. For example, for steady evolutionary period changes
a parabola is expected in the diagram, whereas for orbital motion with a
circular orbit a sinusoid is expected. Information about the orbit can be
extracted, just as is done using spectroscopy to obtain orbital radial
velocities.

In examining large numbers of FTs for stars in the Kepler data with 4 yr of
observations, I recognized that we were seeing stars where all pulsation
frequencies were appearing as equally spaced multiplets. That could happen
asteroseismically if all modes were nonradial with a full complement of
azimuthal $m$ modes excited. But the patterns did not look like that. The
theoretical explanation was presented by Shibahashi & Kurtz (2012), with a
first application to the multiply-eclipsing quintuplet system HD 181469 (KIC
4150611), which has a $\delta$ Sct/$\gamma$ Dor star as the largest mass
component (Hełminiak et al., 2017). We called the method FM, for frequency
modulation, and further development (Shibahashi et al., 2015) showed how
radial velocities can be obtained without spectra, which are exceedingly
expensive in terms of telescope time. This means binary star orbits can now be
completely solved with only photometric time series for pulsating stars with
stable pulsation frequencies. Such stars are common and the method is being
exploited, particularly for $\delta$ Sct stars.

However, many $\delta$ Sct stars show non-periodic amplitude modulation
(Bowman et al., 2016), and that broadens and distorts the peaks in the FTs
which interferes with measuring the frequency multiplets produced by FM. A
solution to that problem was developed by Murphy et al. (2014, 2016b). It is a
version of the $O-C$ diagram. As binary stars orbit their barycentre, the
light travel time from the star to Earth varies, causing the pulsation phase
to vary, thus they called the technique PM, for phase modulation. That can be
used to extract the variable time delays and a plot of those against time in a
new kind of $O-C$ diagram. They showed that all of the information about a
binary star orbit that is available from spectroscopic radial velocities is
available from PM. Stars with one pulsating component are, therefore, called
PM1 (in analogy to SB1, single-lined spectroscopic binaries) and stars with
two pulsations components are similarly PM2. Murphy et al. (2018) then
examined over 2000 main-sequence A and F stars, discovering 341 non-eclipsing
binary stars, 24 of which were PM2, tripling the number of stars on this
region of the main-sequence with orbital solutions.

The PM method is an application of pulsating stars to the related field of
study of the orbits of binary stars, hence it provides fundamental data, such
as masses, rather than asteroseismic inference on the interiors of those
stars. The method is not affected by amplitude modulation of the pulsations,
since that does not affect the pulsation phase; in this way it is superior to
the FM method. It does require binning the data into time chunks – Murphy et
al. (2018) used 10-d sections of the light curves for this – which puts a
lower limit of 20 d on the orbital period determined. FM uses the entire data
set, hence has better frequency resolution, and can thus explore periods
shorter than 20 d. PM is also automated, whereas FM needs hands-on reductions.
With an additional 10 binary systems Murphy et al. (2020a) used the PM method
to explore the orbits of 351 binary stars with orbital periods up to 1500 d in
the Kepler data – largely unexplored parameter space since, for geometric
reasons, such long periods are unusual in eclipsing binaries. The 1500-d
limit, is, of course, a consequence of the duration of the main Kepler
mission. Future, longer duration photometric missions will lengthen the
orbital periods that can be studied, filling in the gap between eclipsing
binary and visual binary studies.

The FM and PM methods are complementary. They obviate the need for telescope
time to obtain spectra over full orbits, hence save vast amounts of money and
astronomers’ time. Murphy et al. (2018) estimated that their study of 341
binaries saved about 100 yr of medium to large telescope time! The FM and PM
methods provide binary star fundamental parameters to complement, and to
check, asteroseismic determinations in the same stars. And they show
asteroseismologists the patterns expected in FTs from orbital motions, so that
those frequency multiplets are not mistaken for rotationally split frequency
multiplets.

###  D.2 Super-Nyquist Asteroseismology (sNa)

In examining the frequency multiplets in Kepler data that led to the FM and PM
techniques, I noticed that some pulsating stars show evidence of frequency
triplets indicative of orbital motion, but only for some frequencies, not all.
Murphy et al. (2013) showed that this is the result of the periodic modulation
of the observing time by the light travel time effect of the Kepler satellite
orbit around the Sun. The Kepler mission had cadences that were equally spaced
in time on board, thus not equally spaced in barycentric time. That lifts the
Nyquist ambiguity in the FT, greatly extending the number of stars that can be
studied asteroseismically in Kepler long-cadence data. An example application
is the discovery by Hey et al. (2019) of 6 new roAp stars with correctly
identified pulsation frequencies in the super-Nyquist frequency range in the
Kepler data. Shibahashi & Murphy (2018) have proposed how the technique may be
applied to data from the future PLATO mission.

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