# Discovery, TESS Characterization, and Modeling of Pulsations in the
Extremely Low Mass White Dwarf GD 278

Isaac D. Lopez High Point University, Department of Physics, One University
Parkway, High Point, NC 27268, USA  Department of Astronomy & Institute for
Astrophysical Research, Boston University, 725 Commonwealth Ave., Boston, MA
02215, USA  J. J. Hermes Department of Astronomy & Institute for Astrophysical
Research, Boston University, 725 Commonwealth Ave., Boston, MA 02215, USA
Leila M. Calcaferro Grupo de Evolución Estelar y Pulsaciones, Facultad de
Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Paseo
del Bosque s/n, 1900, La Plata, Argentina  Instituto de Astrofísica La Plata,
CONICET-UNLP, Paseo del Bosque s/n, 1900, La Plata, Argentina  Keaton J. Bell
NSF Astronomy and Astrophysics Postdoctoral Fellow  DIRAC Institute,
Department of Astronomy, University of Washington, Seattle, WA 98195, USA
Adam Samuels Department of Astronomy & Institute for Astrophysical Research,
Boston University, 725 Commonwealth Ave., Boston, MA 02215, USA  Zachary P.
Vanderbosch Department of Astronomy, University of Texas at Austin, Austin,
TX-78712, USA  Alejandro H. Córsico Grupo de Evolución Estelar y Pulsaciones,
Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La
Plata, Paseo del Bosque s/n, 1900, La Plata, Argentina  Instituto de
Astrofísica La Plata, CONICET-UNLP, Paseo del Bosque s/n, 1900, La Plata,
Argentina  Alina G. Istrate Department of Astrophysics/IMAPP, Radboud
University, PO Box 9010, 6500, GL Nijmegen, The Netherlands

(Accepted October 4, 2021)

###### Abstract

We report the discovery of pulsations in the extremely low mass (ELM), likely
helium-core white dwarf GD 278 via ground- and space-based photometry. GD 278
was observed by the _Transiting Exoplanet Survey Satellite_ (_TESS_) in Sector
18 at a 2-min cadence for roughly 24 d. The _TESS_ data reveal at least 19
significant periodicities between $2447-6729$ s, one of which is the longest
pulsation period ever detected in a white dwarf. Previous spectroscopy found
that this white dwarf is in a 4.61 hr orbit with an unseen $>$0.4 $M_{\odot}$
companion and has ${T}_{\mathrm{eff}}=9230\pm 100$ K and $\log{g}=6.627\pm
0.056$, which corresponds to a mass of $0.191\pm 0.013$ $M_{\odot}$. Patterns
in the TESS pulsation frequencies from rotational splittings appear to reveal
a stellar rotation period of roughly 10 hr, making GD 278 the first ELM white
dwarf with a measured rotation rate. The patterns inform our mode
identification for asteroseismic fits, which unfortunately do not reveal a
global best-fit solution. Asteroseismology reveals two main solutions roughly
consistent with the spectroscopic parameters of this ELM white dwarf, but with
vastly different hydrogen-layer masses; future seismic fits could be further
improved by using the stellar parallax. GD 278 is now the tenth known
pulsating ELM white dwarf; it is only the fifth known to be in a short-period
binary, but is the first with extended, space-based photometry.

asteroseismology — Galaxy: stellar content — stars: individual: GD 278
(WDJ013058.07+532139.71) — stars: oscillations — stars: white dwarfs

††journal: ApJ††facilities: _Gaia_ , Struve, _TESS_ ††software: IRAF (Tody,
1986), barycorrpy (Kanodia & Wright, 2018), lightkurve (Lightkurve
Collaboration et al., 2018), Pyriod (Bell, 2020), MESA (Paxton et al., 2019),
GYRE (Townsend & Teitler, 2013), matplotlib (Hunter, 2007), numpy (Harris et
al., 2020)

##  1 Introduction

White dwarf stars represent the final evolutionary state of main-sequence
stars with masses $\lesssim$ 8-10 $M_{\odot}$, and are the end stages for
$\sim$98% of all stars in the Milky Way. The observed mass distribution of
spectroscopically confirmed hydrogen-atmosphere (DA) white dwarfs from the
Sloan Digital Sky Survey (SDSS) peaks around 0.6 $M_{\odot}$, with smaller
distributions within both tail ends, the smaller of which peaks around 0.4
$M_{\odot}$ (e.g., Kepler et al. 2007; Tremblay et al. 2011). The lower mass
limit for isolated white dwarfs is $\sim$0.4 $M_{\odot}$(Kilic et al., 2007),
constrained by the age of the Milky Way. Therefore, the progenitors of
extremely low mass (ELM; $\lesssim$ 0.3 $M_{\odot}$) white dwarfs must have
been giant stars that experienced enhanced mass loss due to a close companion,
the first episode of which occurred on the first ascent of the red-giant
branch (Marsh et al., 1995; Driebe et al., 1998). This process can occur in a
close binary, where the progenitor star transfers mass either via Roche lobe
overflow (Iben & Tutukov, 1986) or a common-envelope event (Paczynski, 1976;
Iben & Livio, 1993).

DAV stars (also called ZZ Ceti stars) are variable white dwarfs that undergo
pulsations in a narrow temperature range where a convection zone forms, from
roughly 12,500 K to 10,000 K in canonical-mass (0.6 $M_{\odot}$) white dwarfs.
These global, non-radial $g$-mode pulsations are driven by partially ionized
regions that form within their atmospheres. These pulsations manifest as
optical variations of up to 30% and offer the unique opportunity to probe
their interiors via asteroseismology (Winget & Kepler, 2008; Fontaine &
Brassard, 2008; Althaus et al., 2010; Córsico et al., 2019).

Pulsations have previously been observed in nine ELM white dwarfs (Hermes et
al., 2012, 2013a, 2013b; Bell et al., 2015, 2017, 2018; Pelisoli et al.,
2018b; Guidry et al., 2021), including one with a millisecond pulsar companion
(Kilic et al., 2015, 2018). These discoveries effectively extend the DAV
instability strip to lower surface gravities and cooler temperatures (6.0
$\lesssim\log g\lesssim$ 6.8 and 7800 K $\lesssim T_{\rm eff}\lesssim$ 10,000
K). This low-mass extension likely also contains contaminants, especially
metal-poor A stars (Pelisoli et al., 2018a, b), in contrast to the space
occupied purely by classical DAVs at higher masses (e.g., Bell et al. 2017).

Most of the known radial-velocity-confirmed ELM white dwarfs were discovered
by the ELM Survey (Brown et al., 2010, 2012, 2013, 2016, 2020; Kilic et al.,
2011, 2012; Gianninas et al., 2015), a targeted spectroscopic search for ELM
white dwarfs, which has discovered 98 double-white-dwarf binaries within the
SDSS footprint. An additional eight ELM white dwarfs were reported by
Kosakowski et al. (2020), as part of an extension of the ELM Survey into the
southern hemisphere. This southern extension relies on photometry from the VST
ATLAS and SkyMapper surveys, and also employs a _Gaia_ astrometry-based
selection, which chooses candidates from a unique region of parallax-magnitude
space occupied by known ELM white dwarfs.

The population of ELM white dwarfs represent a collection of objects that
contribute to a steady foreground of gravitational waves. The sensitivity of
the future _Laser Interferometer Space Antenna_ (_LISA_) observatory is
expected to be set by the noise floor from unresolved double-white-dwarf
binaries (Amaro-Seoane et al., 2017). Identifying and characterizing compact
binary systems in the Milky Way will help characterize the noise floor that
may otherwise impede on _LISA_ ’s ability to detect gravitational waves.

The field of white dwarf astronomy is currently undergoing a transformation
thanks to space-based missions such as _Gaia_ (Gaia Collaboration et al.,
2016) and the _Transiting Exoplanet Survey Satellite_ (_TESS_ ; Ricker et al.
2015). With the release of _Gaia_ DR2, the number of white dwarfs we now know
about has exploded from $\sim$33,000 to $\sim$260,000 (Gentile Fusillo et al.,
2019), including a large catalog of candidate ELM white dwarfs (Pelisoli &
Vos, 2019).

_TESS_ has completed its initial 2-year mission to observe $\sim$85% of the
celestial sphere in search of small planets orbiting nearby stars.
Observations were split over twenty-six 24$\times$96 square degree sectors,
each of which were observed for roughly 27 days. _TESS_ delivered full-frame
images at a 30-min cadence, as well as 2-min cadence data for select objects,
as part of their Guest Investigator and Director’s Discretionary Targets
programs. The 2-min cadence observations give the opportunity for continuous
monitoring of bright white dwarfs ($V\lesssim$ 16.5 mag), which both provide
much needed follow-up observations and, due to the near 27-day continuous
observations, give excellent opportunities to unambiguously determine periods
without aliasing and with precision sufficiently high for reliable
asteroseismology.

In this paper, we present the discovery of pulsations in the ELM white dwarf
GD 278 via high-speed optical photometry. We also present frequency analysis
of the _TESS_ observations, including asteroseismic modeling of the frequency
solution we obtain. The rest of this paper is organized as follows: Section 2
describes how GD 278 was selected for follow up and describes the
observations. Section 3 presents the frequency analysis of the _TESS_ data.
Asteroseismic modeling of GD 278 is presented in Section 4. A discussion and
conclusions from this discovery are given in Section 5.

##  2 Observations

GD 278 (WDJ013058.07+532139.71, $G$ = 14.9 mag) was selected for follow up
observations from the Gentile Fusillo et al. (2019) _Gaia_ DR2 catalog of
white dwarfs based on its location on the Hertzsprung-Russell (HR) diagram
relative to other known pulsating ELM white dwarfs and on its intrinsic
variability, which we estimated from its empirically determined $G$-band flux
uncertainty (Guidry et al., 2021). Objects that are intrinsically variable
(e.g. pulsators, cataclysmic variables, eclipsing binaries) may have
anomalously large flux uncertainties at a given magnitude. With this technique
we estimate GD 278 to be among the top 1% most variable white dwarfs within
200 pc, prompting us to obtain high-speed photometry. Shortly after our
increased interest in GD 278, it was published as an ELM white dwarf in a
4.61-hr single-lined spectroscopic binary by the ELM Survey (Brown et al.,
2020).

Figure 1: High-speed photometry taken with the 2.1 m Otto Struve telescope at
McDonald Observatory, identifying pulsations on 2019 August 2 (left) and
confirming them on 2019 September 29 (right) in the extremely low-mass white
dwarf GD 278.  Figure 2: An 8-day sample _TESS_ photometry of the new
pulsating extremely low mass white dwarf GD 278, which was observed in Sector
18. The 2-minute-cadence data have been smoothed with a 7-point moving
average.

###  2.1 McDonald Observations

Discovery high-speed photometry was obtained with the 2.1 m Otto Struve
telescope at McDonald Observatory near Fort Davis, Texas. Observations were
obtained over three nights in 2019, all using the frame-transfer ProEM
instrument and a red-cutoff BG40 filter to reduce sky noise. Exposure times
were 5 s on 2019 August 2 and 15 s on 2019 September 29 and 2019 November 6.
Skies were clear with 1–2″ seeing, with the exception of the 2019 November
data, which were affected by poor weather.

Bias, dark, and dome-flat frames were also obtained at the beginning of each
night. We reduced the science frames using standard IRAF packages and
procedures (Tody, 1986). Flux values of the target star were divided by a
bright comparison star in order to perform differential photometry. Data from
the McDonald observations are shown in Figure 1.

Highly discrepant points due to bad weather have been clipped by hand, and the
times have been barycentric corrected using the software routine `barycorrpy`
(Kanodia & Wright, 2018).

###  2.2 _TESS_ Observations

Based in part on our detection of variability from McDonald Observatory, we
submitted a TESS Director’s Discretionary Time (DDT) proposal (PI: Lopez) to
secure 2-min photometry of candidate variable ELM white dwarfs. This included
GD 278 (TIC 308292831, $T$=14.8 mag), which was observed by _TESS_ in Sector
18 (2019 November 2–27) by Camera 2, temporarily overlapping with our data
obtained at McDonald Observatory on 2019 November 6. _TESS_ delivered a
$11\times 11$ target pixel file for GD 278 at a 2-min cadence, with the first
mid-exposure time ($t_{0}$) starting at BJD 2458790.66104. Data was accessed
and downloaded using the Python package `lightkurve` (Lightkurve Collaboration
et al., 2018). We tested different apertures for light curve extraction, but
found no improvement over the 4-pixel aperture determined by the Science
Processing Operations Center (SPOC) pipeline, which we adopt for our analysis.

An 8-day sample of the GD 278 _TESS_ photometry is presented in Figure 2. For
visual clarity, the data has been smoothed with a 7-point moving average.
Crowding is a factor for the large _TESS_ plate scale (the SPOC pipeline
estimates that 42% of the flux in the chosen aperture is coming from GD 278),
but this amplitude correction from flux dilution has been applied to our final
adopted PDCSAP light curve.

##  3 Analysis

###  3.1 Frequency Solution of the _TESS_ Observations

GD 278 exhibits an irregular pulsation pattern with up to 25$\%$ peak-to-peak
brightness variations in both the McDonald and the _TESS_ light curves. In
order to determine the frequency components contributing to the observed light
curve, we employed the Python package
`Pyriod`111http://www.github.com/keatonb/Pyriod, a period detection and
fitting software that utilizes a Lomb-Scargle periodogram (Bell, 2020). In
Table 1, we list 19 frequencies ($f_{a},f_{b},...,f_{s}$) that we detect with
significance in the _TESS_ data, which we interpret as independent pulsation
modes occurring within the white dwarf GD 278. Properties of these modes that
we include in Table 1 are their periods, frequencies, amplitudes, phases
(relative to the $t_{0}$ in Section 2.2), and associated uncertainties. The
dominant frequency $f_{a}$ is located at 190.36 $\mu$Hz and has an amplitude
6.32 times the significance threshold (described below). We find one
combination frequency within this set, occurring at the location
$f_{a}+f_{c}$, which is likely not a natural oscillation but rather a
nonlinear artifact (e.g., Wu 2001).

Table 1: The 19 significant pulsation periods/frequencies, one nonlinear combination frequency ($f_{a}+f_{c}$), and the orbital frequency ($f_{\mathrm{orb}}$) detected within the GD 278 _TESS_ data, along with their measured amplitudes, phases (relative to BJD 2458790.66104), and associated errors. Each detection has an amplitude above the 0.1% False Alarm Probability threshold of 0.34%.  ID |  Period (s) |  Frequency ($\mu$Hz) |  Amplitude (%) | Phase
---|---|---|---|---
$f_{a}$ | 5253.23 $\pm$ 0.19 | 190.359 $\pm$ 0.007 | 2.15 $\pm$ 0.06 | 0.503 $\pm$ 0.004
$f_{b}$ | 3365.56 $\pm$ 0.14 | 297.127 $\pm$ 0.012 | 1.25 $\pm$ 0.06 | 0.836 $\pm$ 0.007
$f_{c}$ | 4028.83 $\pm$ 0.21 | 248.211 $\pm$ 0.013 | 1.19 $\pm$ 0.06 | 0.508 $\pm$ 0.007
$f_{d}$ | 5814.8 $\pm$ 0.7 | 171.975 $\pm$ 0.021 | 0.71 $\pm$ 0.06 | 0.528 $\pm$ 0.013
$f_{e}$ | 4628.8 $\pm$ 0.5 | 216.039 $\pm$ 0.022 | 0.67 $\pm$ 0.06 | 0.798 $\pm$ 0.013
$f_{f}$ | 5550.6 $\pm$ 0.8 | 180.161 $\pm$ 0.025 | 0.61 $\pm$ 0.06 | 0.681 $\pm$ 0.015
$f_{g}$ | 6295.4 $\pm$ 1.0 | 158.846 $\pm$ 0.026 | 0.58 $\pm$ 0.06 | 0.206 $\pm$ 0.015
$f_{h}$ | 4428.5 $\pm$ 0.5 | 225.812 $\pm$ 0.028 | 0.53 $\pm$ 0.06 | 0.012 $\pm$ 0.017
$f_{i}$ | 5198.3 $\pm$ 0.8 | 192.370 $\pm$ 0.029 | 0.53 $\pm$ 0.06 | 0.986 $\pm$ 0.017
$f_{j}$ | 4259.1 $\pm$ 0.5 | 234.79 $\pm$ 0.03 | 0.50 $\pm$ 0.06 | 0.470 $\pm$ 0.018
$f_{k}$ | 5896.9 $\pm$ 1.0 | 169.58 $\pm$ 0.03 | 0.48 $\pm$ 0.06 | 0.982 $\pm$ 0.018
$f_{l}$ | 5135.1 $\pm$ 0.8 | 194.74 $\pm$ 0.03 | 0.47 $\pm$ 0.06 | 0.063 $\pm$ 0.019
$f_{m}$ | 4756.5 $\pm$ 0.7 | 210.24 $\pm$ 0.03 | 0.47 $\pm$ 0.06 | 0.440 $\pm$ 0.019
$f_{n}$ | 2447.86 $\pm$ 0.18 | 408.52 $\pm$ 0.03 | 0.46 $\pm$ 0.06 | 0.086 $\pm$ 0.019
$f_{o}$ | 6729.0 $\pm$ 1.8 | 148.61 $\pm$ 0.04 | 0.38 $\pm$ 0.06 | 0.865 $\pm$ 0.024
$f_{p}$ | 4959.6 $\pm$ 1.0 | 201.63 $\pm$ 0.04 | 0.37 $\pm$ 0.06 | 0.483 $\pm$ 0.024
$f_{q}$ | 4093.8 $\pm$ 0.7 | 244.27 $\pm$ 0.04 | 0.37 $\pm$ 0.06 | 0.115 $\pm$ 0.024
$f_{r}$ | 2658.51 $\pm$ 0.28 | 376.15 $\pm$ 0.04 | 0.35 $\pm$ 0.06 | 0.028 $\pm$ 0.026
$f_{s}$ | 3591.3 $\pm$ 0.5 | 278.45 $\pm$ 0.05 | 0.34 $\pm$ 0.06 | 0.664 $\pm$ 0.026
$f_{a}+f_{c}$ | 2280.15 $\pm$ 0.15 | 438.567 $\pm$ 0.028 | 0.54 $\pm$ 0.06 | 0.816 $\pm$ 0.017
$f_{\mathrm{orb}}$ | 16611 $\pm$ 8 | 60.20 $\pm$ 0.03 | 0.46 $\pm$ 0.06 | 0.091 $\pm$ 0.019
Figure 3: Periodogram of the GD 278 _TESS_ data in black. In grey is the
periodogram of the same data after prewhitening by the frequencies in Table 1.
The red dashed line represents the 0.1% False Alarm Probability threshold,
which we estimate to be 0.34%. Peaks above this line have a $<$0.1% chance of
being caused by random noise, and thus represent intrinsic signals from the
star.

The periodogram of the _TESS_ data is shown in black in Figure 3, which was
used as input to determine the periods in Table 1. In grey we show a
periodogram of the _TESS_ data after prewhitening by the list of adopted
frequencies in Table 1. The red dashed line represents the 0.1% False Alarm
Probability threshold; peaks above this value have a low probability ($<$0.1%)
of resulting from random noise and are likely intrinsic to the star. We
bootstrapped the value of this threshold by keeping the time sampling the same
but each time randomly selecting fluxes from the set of the flux values. After
$10^{4}$ iterations, the False Alarm Probability threshold was taken to be
value of the tenth highest amplitude within the ensemble of periodograms, and
has a value of 0.34%.

In some cases, we chose not to consider certain peaks due their proximity to
other peaks with very high amplitudes that we took to be independent modes.
For example, we do not consider the two peaks located at 186.4 ${\rm\mu}$Hz
and 188.1 ${\rm\mu}$Hz, due to their proximity to the peak located at 190.36
${\rm\mu}$Hz. This mode is one of the lowest-frequency pulsations in GD 278,
which may lose phase coherence in the same way that lower-frequency modes do
in normal-mass DA white dwarfs (Hermes et al., 2017; Montgomery et al., 2020).
Thus, these residual peaks to slightly lower frequencies from $f_{6c}$ may not
be independent modes themselves, but rather artifacts from the phase
incoherence and amplitude variations of the high-amplitude pulsation.

###  3.2 Simultaneous McDonald and _TESS_ Observations

The 2019 November data from McDonald Observatory were obtained simultaneously
while GD 278 was being observed by _TESS_. We show the 8.16 hr of overlapping
data in Figure 4. Time is given in _TESS_ Barycentric Julian Date (TBJD),
which is the Julian Date at the Solar System barycenter, offset by 2457000.0.
The higher sampling rate (15 s) and signal-to-noise of the ground-based
observations reveal shorter-timescale variations that are missed by the 2-min
cadence _TESS_ data, especially the double-peaked feature located at roughly
1739.86 TBJD in Figure 4.

Figure 4: Top: Simultaneous McDonald (15-s exposures) and _TESS_ (120-s
exposures) observations of GD 278, captured on 2019 November 8. A
representative error bar is shown in the top left corner of each plot. Bottom:
Both data sets overplotted. The McDonald data (red) have been resampled to the
same 2-min cadence as the _TESS_ data (blue).

For better comparison, we have resampled the McDonald observations at the same
2-min cadence of _TESS_. These data are plotted over each other in bottom
panel of Figure 4 to highlight that the shortest-period features would be
better resolved with future _TESS_ observations obtained in the 20-second-
cadence mode. It is possible that our frequency solution in Table 1 lacks some
of the higher-frequency components that contribute to the peaky, nonlinear
light curve shape resolved by the McDonald observations. However, such modes
are not independent pulsations and should not significantly affect our
asteroseismic analysis in Section 4.

###  3.3 Orbital Period Measurement

In addition to pulsations, we also recover the spectroscopically determined
orbital period of 4.6092 $\pm$ 0.0048 hr from Brown et al. (2020) in the
_TESS_ photometry. We detect a significant signal at 60.21 $\pm$ 0.03 $\mu$Hz,
which provides an independent measurement of the orbital period from the
_TESS_ data of 4.6142 $\pm$ 0.0022 hr, in good agreement with the period
determined from spectroscopy and caused by the 209.1 $\pm$ 5.1 km s-1 radial-
velocity variability of the ELM white dwarf. This photometric signal, with an
amplitude in the _TESS_ bandpass of 0.46 $\pm$ 0.06%, may be caused by Doppler
beaming of the white dwarf (Shporer et al., 2010; Hermes et al., 2014a).

Table 2: We identify common frequency spacings that may reveal the identity (especially the $\ell$ values) for eight different modes in GD 278; each multiplet is illustrated in Figure 5 and Figure 6. In each subset we have attempted to identify the $m=0$ central component critical for asteroseismic analysis; we use a dagger symbol (†) to identify uncertain $m=0$ identifications. If a frequency present appears near where we might expect a rotationally split multiplet component but does not exceed the 0.34% significance threshold, we include it here but mark it with an asterisk (∗).  ID |  $\ell$ |  $m$ |  Period (s) |  Frequency ($\mu$Hz) | Amp. (%) | Splitting ($\mu$Hz)
---|---|---|---|---|---|---
$f_{1a}$ | 1 |  $-$1 | 5135.1 $\pm$ 0.8 | 194.74 $\pm$ 0.03 | 0.47 |
$f_{1b}$ | 1 | 0 |  4756.5 $\pm$ 0.7 | 210.24 $\pm$ 0.03 | 0.47 | 15.50 $\pm$ 0.06
$f_{1c}$ | 1 | 1 | 4428.5 $\pm$ 0.5 | 225.812 $\pm$ 0.028 | 0.53 | 15.57 $\pm$ 0.06
$f_{2a}$ | 1 |  $-$1 | 4259.1 $\pm$ 0.5 | 234.79 $\pm$ 0.03 | 0.50 |
$f_{2b}$ | 1 | 0 |  4028.83 $\pm$ 0.21 | 248.211 $\pm$ 0.013 | 1.19 | 13.42 $\pm$ 0.04
$f_{2c}^{*}$ | 1 | 1 | 3815.8 $\pm$ 0.7 | 262.07 $\pm$ 0.05 | 0.32 | 13.85 $\pm$ 0.06
$f_{3a}$ | 1 |  $-$1 | 3365.56 $\pm$ 0.14 | 297.127 $\pm$ 0.012 | 1.25 |
$f_{3b}^{*}$ | 1 | 0† |  3226.1 $\pm$ 0.6 | 309.98 $\pm$ 0.06 | 0.24 | 12.85 $\pm$ 0.07
$f_{4a}$ | 1 | 0† |  2658.51 $\pm$ 0.28 | 376.15 $\pm$ 0.04 | 0.35 |
$f_{4b}^{*}$ | 1 | +1 | 2569.0 $\pm$ 0.4 | 389.26 $\pm$ 0.06 | 0.26 | 13.10 $\pm$ 0.10
$f_{5a}$ | 1 |  $-$1 | 2447.86 $\pm$ 0.18 | 408.52 $\pm$ 0.03 | 0.46 |
$f_{5b}$ | 1 | 0 |  2367.4 $\pm$ 0.21 |  |  |
$f_{5c}^{*}$ | 1 | 1 | 2292.0 $\pm$ 0.3 | 436.29 $\pm$ 0.06 | 0.26 | 2 $\times$ (13.89 $\pm$ 0.09)
$f_{6a}$ | 2 |  $-$2 | 6729.0 $\pm$ 1.8 | 148.61 $\pm$ 0.04 | 0.38 |
$f_{6b}$ | 2 |  $-$1 | 5896.8 $\pm$ 1.1 | 169.58 $\pm$ 0.03 | 0.48 | 20.97 $\pm$ 0.07
$f_{6c}$ | 2 | 0 |  5253.2 $\pm$ 0.19 | 190.359 $\pm$ 0.007 | 2.15 | 20.78 $\pm$ 0.04
$f_{6d}^{*}$ | 2 | 2 | 4270.8 $\pm$ 0.8 | 234.15 $\pm$ 0.05 | 0.32 | 2 $\times$ (21.89 $\pm$ 0.05)
$f_{7a}$ | 2 |  $-$2 | 6295.4 $\pm$ 1.0 | 158.846 $\pm$ 0.026 | 0.58 |
$f_{7b}$ | 2 |  $-$1 | 5550.6 $\pm$ 0.8 | 180.161 $\pm$ 0.025 | 0.61 | 21.32 $\pm$ 0.05
$f_{7c}$ | 2 | 0 |  4959.6 $\pm$ 1.0 | 201.63 $\pm$ 0.04 | 0.37 | 21.46 $\pm$ 0.07
$f_{7d}^{*}$ | 2 | 1 | 4473.1 $\pm$ 0.9 | 223.56 $\pm$ 0.05 | 0.32 | 21.93 $\pm$ 0.09
$f_{7e}$ | 2 | 2 | 4083.5 $\pm$ 0.7 | 244.89 $\pm$ 0.04 | 0.37 | 21.33 $\pm$ 0.09
$f_{8a}$ | 2 |  $-$1 | 5814.8 $\pm$ 0.7 | 171.975 $\pm$ 0.021 | 0.71 |
$f_{8b}$ | 2 | 0† |  5198.3 $\pm$ 0.8 | 192.370 $\pm$ 0.029 | 0.53 | 20.39 $\pm$ 0.05
$f_{8c}$ | 2 | +1 | 4628.8 $\pm$ 0.5 | 216.039 $\pm$ 0.022 | 0.68 | 23.67 $\pm$ 0.05

###  3.4 Possible Multiplet Mode Identification

We find evidence for patterns in the observed periodogram that suggest common
frequency spacings that could arise from rotational splittings if the ELM
white dwarf is rotating at roughly 10 hr, which would cause $\ell=1$ mode
splittings of $\approx 13.7$ ${\rm\mu}$Hz and $\ell=2$ mode splittings of
$\approx 21.5$ ${\rm\mu}$Hz (e.g., Aerts 2021).

In total, we identify eight mode splittings in the GD 278 _TESS_ periodogram:
five that are consistent with $\ell=1$ modes and three $\ell=2$ modes. In
Table 2, we list 24 individual observed pulsation periods that make up eight
groups, one for each multiplet. Also listed in Table 2 are the assumed $\ell$
and $m$ numbers, periods, frequencies, amplitudes, and the amount adjacent
modes within a group are split by. We observe weighted mean rotational
splittings of roughly 14.0 ${\rm\mu}$Hz for the $\ell=1$ modes and 21.5
${\rm\mu}$Hz for the $\ell=2$ modes, both consistent with an overall rotation
rate of roughly 10 hr.

The six modes with an asterisk have amplitudes less than the significance
threshold; although these amplitudes lie below the significance threshold, we
relax this threshold based on their location near predictions from common
frequency splittings (e.g., Winget et al. 1991; Dunlap et al. 2010).

We show an illustration of the five $\ell=1$ splittings in Figure 5. The
location of the modes are indicated by the red tick marks, while the red
dashed line represents the 0.1% False Alarm Probability threshold at 0.34%
amplitude. In grey, we plot the periodogram of the original _TESS_ dataset.
Over this periodogram, we plot in black a periodogram of the _TESS_ data,
after prewhitening by all periods listed in Table 2, not including the group
of split modes being shown in a given subplot. In some cases ($f_{3}$ and
$f_{4}$) we only identify two components, so it is not possible to
unambiguously determine which of the two components corresponds to the central
($m=0$) mode. There is therefore ambiguity on which modes should be considered
the $m=0$ mode used for the asteroseismic analysis in Section 4.

We repeat this illustration for the three identified $\ell=2$ modes in Figure
6, this time indicating the location of the modes by blue tick marks. One of
the $\ell=2$ modes ($f_{8}$) only has three identified components, which again
leads to ambiguity in determining which of the three components corresponds to
the $m=0$ mode.

Figure 5: Possible evidence for five $\ell=1$ mode splittings, identified
within the GD 278 _TESS_ periodogram and listed in Table 2. Red tick marks
indicate the frequencies that we propose form rotationally split multiplets,
consistent with an overall 10 hr rotation rate. For each mode we show in grey
the prewhitened _TESS_ light curve, subtracted of all frequencies listed in
Table 1 other than those indicated by the red points, and show the resulting
periodogram in black. The red dashed lines mark the 0.34% significance
threshold.  Figure 6: Similar to Figure 5, except for the three $\ell=2$ mode
splittings identified identified within the GD 278 _TESS_ periodogram and
listed in Table 2. Blue tick marks indicate the frequencies that we propose
form the rotationally split multiplets, also consistent with 10 hr rotation.

##  4 Asteroseismic Modeling

The $m=0$ modes we identify in Table 2 serve as the baseline for our
asteroseismic investigation into GD 278, offering us five likely $\ell=1$
modes and three likely $\ell=2$ modes observed in this pulsating white dwarf.
For completeness, our full asteroseismic analysis explores solutions using all
12 different possible combinations of identified $m=0$ modes from the
ambiguous $\ell=1$ modes of $f_{3}$ and $f_{4}$, as well as the ambiguous
$\ell=2$ mode of $f_{8}$.

###  4.1 Initial Expectations from MESA Modeling

As an initial pass at guiding expectations, we generated a model of the ELM
white dwarf GD 278 using the stellar evolution code MESA version r12115
(Paxton et al., 2011, 2013, 2015, 2018, 2019). Our input model to generate a
0.19 $M_{\odot}$ ELM white dwarf followed the inlists first described by
Istrate et al. (2016b), which evolve a close binary and include element
diffusion and rotational mixing. In short, we start with a 1.1 $M_{\odot}$,
$Z$ = 0.0142 metallicity ZAMS progenitor in a 2.90-day binary with a 1.4
$M_{\odot}$ companion and evolve the system through mass-loss until the
remnant is a 0.19 $M_{\odot}$ ELM white dwarf; further details are discussed
in Istrate et al. (2016b).

We evolved this ELM white dwarf model through its many diffusion-induced CNO
flashes (e.g., Althaus et al. 2001; Panei et al. 2007; Althaus et al. 2013)
until it arrived at the final cooling track, and paused the evolution at 9180
K. Here our 0.1904 $M_{\odot}$ model has a surface gravity of $\log{g}$ =
6.644, a hydrogen-layer mass of $M_{H}=8.2\times 10^{-4}M_{\star}$, a radius
of $R_{\star}=0.03443R_{\odot}$, and a luminosity of $L_{\star}=7.56\times
10^{-3}L_{\odot}$. We artificially smoothed the Brunt-Väisälä frequency from
this static model with a built-in weighted smoothing in MESA using eight cells
(rather than the default two cells), in order to avoid any numerical noise
dominating our resultant pulsation frequencies.

We took the output of this smoothed MESA model and pulsated it using GYRE
(Townsend & Teitler, 2013), to generate an output set of expected non-radial
pulsation modes. We did not attempt to generate a grid of models to explore to
perform a full asteroseismic fit, but first sought to understand the possible
radial order of the observed modes in GD 278 from the TESS observations. We
compare the observed periods to the theoretical MESA predictions in Table 3,
and especially note that these are not fits but rather the output of a model
selected to match the spectroscopy of GD 278. As a reminder, the atmospheric
parameters from spectroscopy by Brown et al. (2020) include the 3D corrections
of Tremblay et al. (2015) and find ${T}_{\mathrm{eff}}=9230\pm 100$ K and
$\log{g}=6.627\pm 0.056$, which corresponds to a mass of $0.191\pm 0.013$
$M_{\odot}$.

Table 3: Properties of the theoretical mode periods ($\Pi^{T}$) of a representative MESA model we obtained for GD 278 defined by the spectroscopic parameters from Brown et al. (2020): $M=0.1904$ $M_{\odot}$, $M_{H}=8.2\times 10^{-4}M_{\star}$, ${T}_{\mathrm{eff}}$= 9180 K, $L_{\star}=7.56\times 10^{-3}L_{\odot}$, $\log{g}$= 6.644, and $R_{\star}=0.03443R_{\odot}$. The observed periods identified in Table 2 are also included ($\Pi^{O}$). This model has $1/\chi^{2}=0.0024$ (see Equation 1).  ID |  $\Pi^{O}$ (s) |  $\Pi^{T}$ (s) |  $\ell$ |  $k$ |  $\delta\Pi$ (s)
---|---|---|---|---|---
$f_{1}$ | 4756.5 | 4761.8 | 1 | 56 |  $-$5.3
$f_{2}$ | 4028.83 | 4020.8 | 1 | 47 | 8.0
$f_{3}$ | 3226.1 | 3204.6 | 1 | 37 | 21.5
$f_{4}$ | 2658.51 | 2637.7 | 1 | 30 | 20.8
$f_{5}$ | 2367.4 | 2395.0 | 1 | 27 |  $-$27.6
$f_{6}$ | 5253.2 | 5270.2 | 2 | 108 |  $-$17.0
$f_{7}$ | 4959.6 | 4932.8 | 2 | 101 | 26.8
$f_{8}$ | 5198.3 | 5221.8 | 2 | 107 |  $-$23.5

This MESA model features an overall $\ell=1$ asymptotic mean period spacing of
roughly 83 s, and an $\ell=2$ mean period spacing of roughly 48 s. The
observed period range in Table 3 implies that the $\ell=1$ modes have radial
orders ranging from $27<k<56$, with $\ell=2$ modes of especially high radial
order, ranging from $101<k<108$.

Our stellar model includes rotation, with the initial rotation velocity set
such that the star is synchronized with the initial orbital period. Moreover,
we take into account the effect of tides and spin-orbit coupling. Given this,
we note that this final MESA model has a 11.6-hr surface rotation period.
Interestingly, the model is rotating marginally differentially, a consequence
imprinted in the CNO flashing episodes (see Section 4.1.1 in Istrate et al.
2016b). If the rotational splittings we identify in Section 3.4 are correct,
this model predicts rotation at almost exactly the same rate as observed in GD
278.

We can also estimate the expected distance to this white dwarf by comparing
the luminosity of the model ($L_{\star}=0.00756$ $L_{\odot}$) to the _Gaia_
apparent magnitude ($G$=14.88 mag). Using the 3D reddening map of (Green et
al., 2019), we find that the effects of reddening do not become an issue at
the coordinate of GD 278 until $>$200 pc, so we assume no reddening
coefficient. We use the bolometric correction of (Choi et al., 2016) for an
object with ${T}_{\mathrm{eff}}$= $9250$ K and $\log{g}$ = 6.5. We obtain a
seismic distance of 104.4 pc. This is significantly discrepant with the
parallax distance determined from Gaia eDR3 of $151.19\pm 0.78$ pc (Bailer-
Jones et al., 2021), suggesting our model is underluminous compared to that
needed to explain the measured Gaia parallax.

###  4.2 Asteroseismic Fits with LPCODE

We further analyzed the identified periods observed in GD 278 using the low-
mass white dwarf models of Calcaferro et al. (2018). In short, the models
expand upon the ELM white dwarf models described in Althaus et al. (2013)
generated using the LPCODE stellar evolution code. This code computes in
detail the complete evolutionary stages that lead to the white dwarf
formation, allowing the study of the white dwarf evolution consistently with
the predictions of the evolutionary history of the progenitors (see, Althaus
et al., 2005, 2013, 2015, for details). For ELM white dwarfs, LPCODE computes
initial models by mimicking the binary evolution of progenitor stars (Althaus
et al., 2013). Adiabatic pulsation periods for non-radial dipole ($\ell=1$)
and quadrupole ($\ell=2$) $g$-modes were computed employing the adiabatic
version of the LP-PUL pulsation code (see, Córsico & Althaus, 2006, for
details). The models of low-mass white dwarfs we employ in this work include
not just canonically thick hydrogen layers, but also include new evolutionary
sequences of thinner envelopes (Calcaferro et al., 2018).

We searched a grid of ELM white dwarf models for the best-fit solution in a
broad range of effective temperatures ($8500-9800$ K), overall mass
($0.15-0.44$ $M_{\odot}$), and hydrogen envelope thicknesses in the interval
$-5.8\lesssim\log(M_{\rm H}/M_{\star})\lesssim-1.7$ (depending on the stellar
mass). Eventually we restricted our attention to the region within roughly
4$\sigma$ of the overall mass determined from the spectroscopic parameters
($0.15-0.24$ $M_{\odot}$).

Using this grid of models, we searched the 12 different periods solutions that
arise from the ambiguity of the $f_{3}$, $f_{4}$, and $f_{8}$ splittings
(Section 3.4), fitting only the $m=0$ modes. We explored best-fit models for
each of the 12 period solutions, both restricted to the narrow range around
the spectroscopic solution and across the broader range of overall stellar
mass and effective temperature. Our best-fit models were optimized to minimize
a quality function ($\chi^{2}$) that considers the differences between the
observed pulsation periods, $\Pi^{O}$, and theoretical pulsation periods,
$\Pi^{T}$, and is defined as

$\chi^{2}\left(M_{\star},{T}_{\mathrm{eff}},M_{H}\right)=\frac{1}{N}\sum_{i=1}^{N}\left(\Pi^{O}_{i}-\Pi^{T}_{i}\right)^{2}$
(1)

where $M_{\star}$ is the stellar mass, ${T}_{\mathrm{eff}}$ is the effective
temperature, $M_{H}$ is the mass fraction of the hydrogen envelope, and $N$ is
the number of observed pulsation periods. For this procedure, we fix the
values of $\ell$ at the outset, according to the identifications in Table 2.

Unfortunately, there is not a global minimum preferred fit within the
1$\sigma$ bounds provided by the spectroscopy of Brown et al. (2020). For all
12 period solutions, the ELM white dwarf model with the formal best fit to the
observed $m=0$ pulsation periods occurs with a stellar mass of 0.239
$M_{\odot}$, an effective temperature between $9680-9710$ K, and a hydrogen-
layer mass of roughly $M_{H}=3.6\times 10^{-3}M_{\star}$.

As all 12 period solutions return similar (though not identical) quality maps,
we restrict further discussion to only the solution defined explicitly in
Table 2. The quality of the fit to the observed periods for this solution
across all effective temperatures and stellar masses is shown in Figure 7.
Each point ($M_{\star}$, $T_{\rm eff}$) in the map corresponds to a hydrogen-
layer mass ($M_{\rm H}$/$M_{\star}$) that maximizes the value of $1/\chi^{2}$
for that stellar mass and effective temperature. The green box represents the
1$\sigma$ bounds of the mass and effective temperature obtained from
spectroscopy. For this model, the formal best (global) solution has
$1/\chi^{2}$ = 0.0154, with an overall mass of 0.239 $M_{\odot}$, a hydrogen-
layer mass of $M_{H}=3.55\times 10^{-3}M_{\star}$, and an effective
temperature of 9698 K. Notably, this model is $>$3$\sigma$ in tension with
both the observed spectroscopic effective temperature and mass.

Similar to Section 4.1, we estimate the distance to this white dwarf by
comparing the seismic luminosity of the global best-fit model
($L_{\star}=0.00645$ $L_{\odot}$) to the _Gaia_ apparent magnitude. We obtain
a seismic distance of 96.4 pc for this model. Again, this value is
significantly discrepant with the $151.19\pm 0.78$ pc parallax distance
(Bailer-Jones et al., 2021).

Figure 7: Map of the inverse of the quality function ($1/\chi^{2}$) as a
function of the stellar mass ($M_{\star}$) and effective temperature
(${T}_{\mathrm{eff}}$), considering the eight $m=0$ modes identified in bold
in Table 2. Each point on the map corresponds to a hydrogen envelope mass
($M_{H}$) that maximizes the value of $1/\chi^{2}$. The green box represents
the GD 278 spectroscopic mass and effective temperature, and the corresponding
$1\sigma$ uncertainties, from Brown et al. (2020).

If we restrict our fits and require them to be within a 1$\sigma$ box
consistent with the spectroscopic parameters, the best-fit model occurs at an
overall mass of 0.1822 $M_{\odot}$ (with $\log{g}$ = 6.727) and an effective
temperature of 9226 K, with an extremely thin hydrogen-layer mass of
$M_{H}=3.08\times 10^{-6}M_{\star}$, more than three orders of magnitude
thinner than the global best-fit model, as well as the exploratory MESA model
in Section 4.1. This restricted solution has a goodness of fit value of
$1/\chi^{2}$ = 0.0111. Our period solution and radial orders of this
restricted best-fit are described in Table 4. The observed period range in
Table 4 implies that the $\ell=1$ modes have radial orders of $22<k<46$, with
$\ell=2$ modes ranging from $83<k<88$. For this model, with
$L_{\star}=0.00612$ $L_{\odot}$, we find a seismic distance of 93.9 pc.

Table 4: Properties of the best-fit asteroseismic model for GD 278 using the LPCODE evolution code, restricted to within 1$\sigma$ of the spectroscopic parameters, where all modes are assumed to be the $m=0$ components identified in Table 2. The characteristics of this best-fit model are as follows: $M_{\star}=0.182\ M_{\odot}$, $M_{H}=3.1\times 10^{-6}M_{\star}$, ${T}_{\mathrm{eff}}$= 9226 K, $L_{\star}=6.12\times 10^{-3}L_{\odot}$, $\log{g}$= 6.727, $R_{\star}=0.0306R_{\odot}$, and $1/\chi^{2}$ = 0.0111.  ID |  $\Pi^{O}$ (s) |  $\Pi^{T}$ (s) |  $\ell$ |  $k$ |  $\delta\Pi$ (s)
---|---|---|---|---|---
$f_{1}$ | 4756.5 | 4760.13 | 1 | 46 |  $-$3.63
$f_{2}$ | 4028.83 | 4043.10 | 1 | 38 |  $-$14.27
$f_{3}$ | 3226.1 | 3238.14 | 1 | 31 |  $-$12.04
$f_{4}$ | 2658.51 | 2644.92 | 1 | 25 | 13.59
$f_{5}$ | 2367.4 | 2355.23 | 1 | 22 | 12.17
$f_{6}$ | 5253.2 | 5253.85 | 2 | 88 |  $-$0.65
$f_{7}$ | 4959.6 | 4956.15 | 2 | 83 | 3.45
$f_{8}$ | 5198.3 | 5194.41 | 2 | 87 | 3.89

This thin-hydrogen-layer solution at 0.182 $M_{\odot}$ is the most commonly
occurring restricted best-fit (in eight of 12 cases), and in all eight cases
implies an $\ell=1$ mean period spacing of roughly 100.0 s. In the other four
cases, the best restricted fit (within 1$\sigma$ of the spectroscopic
parameters) occurs with models featuring an overall mass of 0.192 $M_{\odot}$,
effective temperatures ranging from $9340-9360$ K, and a thicker hydrogen-
layer mass of $M_{H}=2.38\times 10^{-4}M_{\star}$. With the slightly higher
overall mass, the $\ell=1$ mean period spacing for these models is roughly
91.4 s. The goodness-of-fit parameter for the 0.192 $M_{\odot}$ solution is
worse than the 0.182 $M_{\odot}$ solution, with $1/\chi^{2}$ = 0.0093.
However, it is most proximate to the spectroscopic values without invoking an
artificially stripped hydrogen-layer mass that is much thinner than what is
expected from stellar evolution (Panei et al., 2007; Steinfadt et al., 2010).

Given that our absolute best-fit model across all parameters disagrees
significantly with the spectroscopic parameters, we are reluctant to use our
asteroseismic analysis to adopt an overall stellar model. This could implicate
our mode identifications and our assumption of a roughly 10-hr rotation period
in Section 3.4 as being systematically incorrect. We note that some of our
splittings within the same multiplet are inconsistent; for example, the three
modes in $f_{2}$ have frequency spacings that are formally $>$$4\sigma$
discrepant. However, the vast majority of rotational multiplets identified in
Table 2 have frequency separations consistent to within $2\sigma$, and the
implied rotation rate agrees for both the $\ell=1$ and $\ell=2$ modes,
independently.

It is also possible that even if we have correctly identified all eight $m=0$
modes present, the relatively long periods in GD 278 have such high radial
order that none of the modes are particularly sensitive to interior structure
transitions. These long-period modes may not lend themselves to a strongly
favored, unique global solution, as seen in some higher-mass DAVs (e.g.,
Giammichele et al. 2017).

###  4.3 Search for a Mean Period Spacing

Based on some of the ambiguities in the asteroseismic analysis, we sought to
constrain at least the mean density of the ELM white dwarf GD 278 by trying to
measure and then analyze the observed mean period spacing, which approximates
the overall mean density and thus overall mass of the star using the $g$-mode
pulsations present (Althaus et al., 2010).

First, we attempted to search for a roughly constant period spacing in our
observations. Since we cannot know the radial order ($k$) of the modes a
priori, we first searched for any sign in the data for recurring period
spacings. The long-period pulsations present ($>2300$ s) imply relatively high
radial orders ($k>20$), so the period spacing between modes should be
asymptotic and nearly constant (Tassoul, 1980).

Figure 8: Period transform of the GD 278 Lomb-Scargle periodogram shown in
Figure 3. Unfortunately, we cannot see any clear recurrent period spacings in
the periodogram itself.  Figure 9: We searched for a constant period spacing
among all detected significant pulsation frequencies listed in Table 1 using
three tests: an Inverse Variance test (top panel), a Kolmogorov-Smirnov test
(middle panel), and a Fourier Transform test (bottom panel). The strongest
signals emerge at roughly 104 s, 84 s, 116 s, and 72.5 s, and are put in
context in Figure 10.

We initially searched for patterns in the period spacing in the data itself,
by taking a period transform of the period transform; in short, we inverted
the Lomb-Scargle periodogram in Figure 3 so that it was in period and not
frequency, and then took the period transform of this period transform (e.g.,
Winget et al. 1991) within the range of all pulsation power ($2000-8000$ s).
This is shown in Figure 8. Unfortunately this did not reveal any dominant
peaks, so was not valuable for determining recurrent period spacings in the
data itself.

As a next step, we searched for a constant period spacing using three
additional tests (e.g., Calcaferro et al. 2017b). In short, we ignored our
mode identifications and searched all significant frequencies given in Table 1
using a Kolmogorov-Smirnov test (K-S test; Kawaler 1988), an Inverse Variance
test (I-V test; O’Donoghue 1994), and the Fourier Transform test (F-T test;
Handler et al. 1997). The results of the three tests are shown in Figure 9.
There are some suggestive indications of constant period spacings at roughly
72.5 s, 84 s, 104 s, and 116 s; most prominently at 84 s.

Figure 10: Average of the computed dipole period spacings for $\ell=1$ modes
as a function of effective temperature for various cooling models of ELM white
dwarfs (Calcaferro et al., 2018) with masses between $0.15-0.44$ $M_{\odot}$
and canonically thick hydrogen envelopes. Plotted over these models are the
two possible values of the constant period spacing for GD 278 found from our
analysis in Figure 9 (roughly 84 s and 104 s, horizontal lines), as well as
bounds from the spectroscopic effective temperature and uncertainties
(vertical cyan lines). For white dwarfs $<$0.2 $M_{\odot}$ we expect mean
period spacings $\overline{\Delta\Pi}>80$ s.

To put these mean period spacings into context, we show in Figure 10 a series
of expected period spacings as a function of effective temperature for a suite
of ELM white dwarf models with masses between $0.15-0.44$ $M_{\odot}$
(described in Section 4.2). All models have the thickest possible (canonical)
hydrogen-layer masses: sequences with $M_{\star}<0.18\ M_{\odot}$ are
characterized by $M_{H}\approx 10^{-2}M_{\star}$ (since these model ELM white
dwarfs do not undergo CNO flashes), while those with $M_{\star}>0.18\
M_{\odot}$ have $M_{H}<2\times 10^{-3}M_{\star}$. The effective temperature
constraints from spectroscopy are shown as vertical cyan lines.

Figure 10 shows that the 116 s possible period spacing is not within the range
expected for $\ell=1$ modes of ELM white dwarfs. The roughly 104 s, 84 s, and
72.5 s period spacings would imply values of the overall stellar masses lower
than 0.14 $M_{\odot}$, roughly 0.20 $M_{\odot}$, and roughly 0.24 $M_{\odot}$,
respectively. The two $\ell=1$ mean period spacings in Figure 9 at 104 s and
84 s are marked with horizontal orange lines. We also repeated the exercise
shown in Figure 9 on the smaller list of eight likely $m=0$ modes from Table
2, which includes a mix of $\ell=1$ and $\ell=2$ modes. We do not visualize
this result but these tests show slight preference for possible mean period
spacings at roughly 62 s and 72 s, which fall at an unexpected range based on
previous spectroscopy of GD 278, so we do not discuss those tests further.

The strongest response to the K-S test in Figure 9 occurs at roughly 84 s,
which is very similar to the $\ell=1$ mean period spacing of roughly 82 s that
we expect from our MESA modeling described in Section 4.1 for a white dwarf of
the given spectroscopic parameters and a canonically thick hydrogen layer.
However, the ambiguity of this signal and the mode identification presented in
Table 2 precludes closing the chapter on asteroseismic investigation of GD 278
until we can find a significantly best-fit solution. Improved fits are likely
to come from utilizing the precise parallax distance from Gaia, which we leave
to future work.

##  5 Discussion and Conclusions

We discovered GD 278 as part of a search of the Gentile Fusillo et al. (2019)
_Gaia_ DR2 catalog of white dwarfs for pulsating ELM white dwarfs. It was
selected for its position on the _Gaia_ color-magnitude diagram relative to
other known pulsators and on its intrinsic variability, which we inferred from
its _Gaia_ $G$-band flux uncertainty. This method for selecting candidate
variable stars was described further by Guidry et al. (2021), who used the
method to confirm 32 new DAVs and one new pulsating ELM white dwarf.

GD 278 stood out as a likely variable candidate and we confirmed its
pulsations from McDonald Observatory in 2019, before it was announced as a
bona fide ELM white dwarf in a 4.61-hr single-lined binary by Brown et al.
(2020), selected from an existing catalog of bright white dwarfs (Raddi et
al., 2017). Our ground-based discovery of pulsations allowed us to obtain
2-min cadence observations with TESS, obtaining the first space-based
photometry of a pulsating ELM white dwarf. This revealed the richest set of
oscillation periods ever obtained for an ELM white dwarf.

However, mode identification of the frequencies in Table 2 was a particular
challenge, with rotational splitting exceeding the frequency differences
between adjacent radial overtones. One method for identifying the spherical
degree ($\ell$) is to look for triplets or quintuplets of modes with
frequencies consecutively split by equal amounts. For GD 278, we identified
five possible $\ell=1$ splittings and three possible $\ell=2$ splittings, many
overlapping with one another. The $\ell=1$ modes we observe have a mean
frequency separation of $\approx$$14.0$ ${\rm\mu}$Hz, and the $\ell=2$ modes
are separated from one another by $\approx$$21.5$ ${\rm\mu}$Hz; both are
consistent with an overall rotation period of roughly 10 hr. This is the first
direct estimate of the rotation rate of an ELM white dwarf. The role of
rotation is likely vital for understanding the frequent occurrence of heavy
metals in the atmospheres of ELM white dwarfs (Hermes et al., 2014b; Istrate
et al., 2016b, a).

We recover the $4.6092\pm 0.0048$ hr orbital period from the spectroscopy of
Brown et al. (2020) in the _TESS_ photometry, finding a marginal but
significant variation at $4.6142\pm 0.0022$ hr. The 4.61-hr orbital period of
GD 278 is very close to the $4.5912\pm 0.0012$ hr orbital period of SDSS
J184037.78+642312.3, which was the first ELM white dwarf discovered to exhibit
pulsations (Hermes et al., 2012). That work notes that pulsating ELM white
dwarf with a rotation period of 4.6 hr would have $\ell=1$ modes split into
components separated by 30 ${\rm\mu}$Hz, but a 4.6 hr orbital period is likely
not short enough for the white dwarf to have synchronized with the orbital
period through tidal dissipation (Fuller & Lai, 2013). Models by Istrate et
al. (2016b) show that low-mass white dwarfs can have rotation rates several
times faster than their orbital period upon arrival on the cooling track.

We detected a significant pulsation mode in GD 278 at $6729.0\pm 1.8$ s, which
is so far the longest pulsation period detected in a white dwarf. Previously,
SDSS J222859.93+362359.6 had the longest detected pulsation period at
$6234.9\pm 6.0$ s. That object also happens to be the coolest known pulsating
ELM white dwarf with ${T}_{\mathrm{eff}}=7870\pm 120$ K (Hermes et al.,
2013b). However, extensive monitoring of SDSS J222859.93+362359.6 does not
show radial-velocity variability. Unless this object is in an orbit with a low
inclination, it may not actually be an ELM white dwarf (Bell et al., 2017).

We see marginal evidence from a K-S test of all pulsation modes present that
there is a roughly 84 s mean period spacing, which is what we expect from
asteroseismic models of a roughly 0.20 $M_{\odot}$ ELM white dwarf computed by
both MESA and the LPCODE stellar evolution codes, assuming the star has a
thick hydrogen layer. While we do not adopt a formally best-fit asteroseismic
fit with our exploration of a grid of models using LPCODE, we do find that
there are asteroseismic solutions for our TESS observations of GD 278 with
parameters that are consistent with those found from spectroscopy (Brown et
al., 2020), however further inferred luminosities/distances do not agree.
Given the large number of variables that lead to these conclusions, it is
difficult to explain the discrepancies.

With continued monitoring of GD 278, it may be possible to measure rates of
change of the significant frequencies within the star, which in general are
related to the cooling rate of the white dwarf and the rate of change of the
white dwarf radius (Winget & Kepler, 2008). There are no significant
constraints yet on rates of period change for ELM white dwarfs, but these
rates of period change would be incredibly valuable at distinguishing whether
the ELM white dwarf has undergone CNO flashes in its past (Calcaferro et al.,
2017a). Unfortunately, GD 278 will not be observed by _TESS_ in Cycle 4, but
hopefully it will again in a future extended mission. Starting with Cycle 5,
full frame images will be saved every 300 s, enabling asteroseismic study of
even more bright ELM white dwarfs.

We would like to thank Dave Doss, John Kuehne, Coyne Gibson, and the other
staff at McDonald Observatory for their support while observing. We would also
like to thank the anonymous referee for their helpful feedback. I.D.L.
acknowledges support by NSF grant AST-1812874 and by the Massachusetts Space
Grant Consortium (MASGC). We acknowledge salary and travel support through
TESS Guest Investigator Program 80NSSC20K0592, as well as K2 Guest Observer
Program 80NSSC19K0162. L.M.C. and A.H.C. acknowledge the AGENCIA through the
Programa de Modernización Tecnológica BID 1728/OC-AR, and by the PIP
112-200801-00940 grant from CONICET. K.J.B. is supported by the National
Science Foundation under Award AST-1903828. This work has made use of data
from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This paper includes data
collected by the _TESS_ mission, which are publicly available from the
Mikulski Archive for Space Telescopes. Funding for the _TESS_ mission is
provided by the NASA Science Mission Directorate.

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$\mathsf{dtMPDA}$ $\mathcal{S}$, we show that all simple $\mathsf{TCWs}$ in
$\mathsf{STCW}(\mathcal{S},k)$ have bounded split-width. Actually, we will
prove a slightly more general result. We first identify some properties
satisfied by all simple $\mathsf{TCWs}$ generated by a $\mathsf{dtMPDA}$, then
we show that all simple $\mathsf{TCWs}$ satisfying these properties have
bounded split-width. Considering $Y=X\cup\\{\zeta\\}$,

######  Lemma .

A $k$-round well-timed simple $\mathsf{TCW}$ using $n>1$ stacks has split-
width at most $K=\max(kn+2(kn-1)|Y|,(k+2)(2|Y|+1))$.

Notice that if we have only one stack ($n=1$) hence also one round ($k=1$) the
bound $4|Y|+2$ on split-width was established in Claim 3.3.

###### Proof.

Again, the idea is to play the split-game between _Adam_ and _Eve_. _Eve_
should have a strategy to disconnect the word without introducing more than
$K$ blocks. The strategy of _Eve_ is as follows: Given the $k$-round word $w$,
_Eve_ will break it into $n$ split-$\mathsf{TCWs}$. The first
split-$\mathsf{TCW}$ only has stack 1 edges, and the second has stack edges
corresponding to stack 2, etc. up to the last split-$\mathsf{TCW}$ for stack
$n$. Each split-$\mathsf{TCW}$ will have at most $k|Y|$ holes and can now be
dealt with as we did in the case of TPDA. The only thing to calculate is the
number of cuts required in isolating each split-$\mathsf{TCWs}$, which we do
in the rest of the proof below.

##### Obtaining the $n$ split-$\mathsf{TCWs}$ containing only stack $p$ edges

Since we are dealing with $k$-round $\mathsf{TCWs}$, we know that the stack
operations follow a nice order : stacks 1, …, $n$ are operated in order $k$
times. More precisely, by ($\mathsf{T}^{\prime}_{1}$), the set $V$ of points
of the simple $\mathsf{TCW}$ $\mathcal{V}$ can be partitioned into
$V=V_{1}^{1}\uplus\cdots\uplus V_{1}^{n}\uplus\cdots\uplus
V_{k}^{1}\uplus\cdots\uplus V_{k}^{n}$ such that
$V_{1}^{1}<\cdots<V_{1}^{n}<\cdots<V_{k}^{1}<\cdots<V_{k}^{n}$ and
$V_{1}^{p}\cup\cdots\cup V_{k}^{p}$ contains all and only stack operations
from stack $p$. _Eve_ ’s strategy is to separate all these sets, i.e., cut
just after each $V_{i}^{p}$. This results in $kn$ blocks. This will not
disconnect the word if there are edges with clock timing constraints across
blocks, i.e., from some $V_{i}^{p}$ to some other block $V_{j}^{q}$ on the
right with $p\neq q$.

Fix some clock $x\in Y$ and assume that there are timing constraints which are
checked in block $V_{j}^{q}$ and are reset in some block on the left. All
these crossing over timing constraints for clock $x$ come from some block
$V_{i}^{p}$ containing the last reset block of clock $x$ on the left of
$V_{j}^{q}$. With at most two cuts, we detach the consecutive sequence of
resets $R_{j}^{q}(x)$ which are checked in $V_{j}^{q}$. Doing this for every
clock $x\in Y$ and every block $V_{j}^{q}$ except $V_{1}^{1}$, we use at most
$2(kn-1)|Y|$ cuts. So now, we have $kn+2(kn-1)|Y|$ blocks.

The split-$\mathsf{TCW}$ $\mathcal{V}^{q}$ for stack $q$ consists of the reset
blocks of the form $R_{j}^{q}(x)$ ($1\leq j\leq k$, $x\in Y$) together with
the split blocks $W_{j}^{q}=V_{j}^{q}\setminus\bigcup R_{i}^{p}(y)$ where the
union ranges over $i\geq j$, $p\neq q$ and $y\in Y$. The split-$\mathsf{TCW}$
$\mathcal{V}^{q}$ is disconnected from the other split-$\mathsf{TCWs}$
$\mathcal{V}^{p}$ with $p\neq q$. Moreover, we can check that, by removing the
reset blocks $R_{i}^{p}(y)$ with $p\neq q$, we have introduced at most
$k^{\prime}|Y|$ holes in $W_{1}^{q},\cdots,W_{k}^{q}$, where $k^{\prime}=k$ if
$q<n$ and $k^{\prime}=k-1$ if $q=n$. So the number of blocks of
$\mathcal{V}^{q}$ is at most $k|Y|+k+k^{\prime}|Y|\leq k(2|Y|+1)$.

Notice that if $k=1$ and $n>1$ then we have at most $2|Y|+1$ blocks in
$\mathcal{V}^{q}$ with $q<n$ and $|Y|+1$ blocks in $\mathcal{V}^{n}$.

We apply now the TPDA game on each split-$\mathsf{TCW}$ $\mathcal{V}^{q}$
using at most $4|Y|+2$ extra cuts. We obtain a split bound of
$K=\max(kn+2(kn-1)|Y|,(k+2)(2|Y|+1))$. ∎

Having established a bound on the split-width for $\mathsf{dtMPDA}$ restricted
to $k$ rounds, we now discuss the construction of tree automata
$\mathcal{B}^{K,M}_{\textsf{valid}}$ and $\mathcal{A}^{K,M}_{\mathcal{S}}$
when the underlying system is a $\mathsf{dtMPDA}$. As a first step, we keep
track of the current round (context) number in the finite control. This makes
sure that the tree automaton only accepts runs using at most $k$-rounds. The
validity check (and hence, $\mathcal{B}^{K,M}_{\textsf{valid}}$) is as before.
The only change pertains to the automaton that checks correctness of the
underlying run, namely, $\mathcal{A}^{K,M}_{\mathcal{S}}$, as we need to
handle $n$ stacks (and $k$ rounds) instead of the single stack.

### Tree Automata Construction for MultiStack and Complexity

Given a $\mathsf{dtMPDA}$ $\mathcal{S}=(S,\Sigma,\Gamma,X,s_{0},F,\Delta)$, we
first construct a $\mathsf{dtMPDA}$ $\mathcal{S}^{\prime}$ that only accepts
runs using at most $k$-rounds. The idea behind constructing
$\mathcal{S}^{\prime}$ is to easily keep track of the $k$-rounds by
remembering in the finite control of $\mathcal{S}^{\prime}$, the current round
number and context number. The initial states of $\mathcal{S}^{\prime}$ is
$(s_{0},1,1)$. Here 1,1 signifies that we are in round 1, and context 1 in
which operations on stack 1 are allowed without changing context. The states
of $\mathcal{S}^{\prime}$ are $\\{(s,i,j)\mid 1\leq i\leq k,1\leq j\leq n,s\in
L\\}$.

Assuming that $(s,\gamma,a,\mathsf{op},R,s^{\prime})\in\Delta$ is a transition
of $\mathcal{S}$, then the transitions $\Delta^{\prime}$ of
$\mathcal{S}^{\prime}$ are as follows:

  1. (1)

$((s,i,j),\gamma,a,\mathsf{op},R,(s^{\prime},i,j))\in\Delta^{\prime}$ if
$\mathsf{op}$ is one of $\mathsf{nop}$, $\downarrow^{j}_{c}$ or
$\uparrow^{j}_{c\in I}$,

  2. (2)

$((s,i,j),\gamma,a,\mathsf{op},R,(s^{\prime},i,h))\in\Delta^{\prime}$ if $j<h$
and $\mathsf{op}$ is one of $\downarrow^{h}_{c}$ or $\uparrow^{h}_{c\in I}$,

  3. (3)

$((s,i,j),\gamma,a,\mathsf{op},R,(s^{\prime},i+1,h))\in\Delta^{\prime}$ if
$h<j$ and $\mathsf{op}$ is one of $\downarrow^{h}_{c}$ or $\uparrow^{h}_{c\in
I}$.

The final states of $\mathcal{S}^{\prime}$ are of the form $\\{(s,i,j)\mid
s\in F\\}$. It can be shown easily that accepting runs of
$\mathcal{S}^{\prime}$ correspond to accepting $k$-round bounded runs of
$\mathcal{S}$.

Now, given the $\mathsf{dtMPDA}$ $\mathcal{S}^{\prime}$, the tree automaton
$\mathcal{B}^{K,M}_{\textsf{valid}}$ that checks for validity and
realizability are exactly as in Theorem 4.

The only change pertains to the automaton that checks correctness of the
underlying run. The tree automaton for the underlying system,
$\mathcal{A}^{K,M}_{\mathcal{S}}$ stores the set $E^{s}$ of pairs $(i,j)$ of
left endpoints of _distinct_ blocks such that there is at least one push-pop
edge pertaining to stack $s$ from $\mathsf{Block}(i)$ to $\mathsf{Block})(j)$.
The other change is in the transitions of $\mathcal{A}^{K,M}_{\mathcal{S}}$:
the transitions $\mathop{\mathsf{Add}_{i,j}^{\curvearrowright
I}}((i,a^{1})\oplus(j,a^{2}))$ in Table 4 will now differ, since this checks
the correctness of well-nesting of stack edges when we have only one stack.
This transition will be replaced with
$\mathop{\mathsf{Add}_{i,j}^{\curvearrowright
I,s}}((i,a^{1})\oplus(j,a^{2}))$, which also talks about the particular stack
$s$. Thus, instead of one set $E$ as in the case of TPDA, we require $n$
distinct sets to ensure well-nesting for each stack edge. Secondly, we need to
ensure that the $k$-round property is satisfied. Rather than doing this at the
tree automaton level, we have done it at the $\mathsf{dtMPDA}$ level itself,
by checking this in $\mathcal{S}^{\prime}$. This blows up the number of
locations by $nk$, the number of stacks and rounds. Thus, the number of states
of the tree automaton $\mathcal{A}^{K,M}_{\mathcal{S}}$ that checks
correctness when the underlying system is a $k$-round $\mathsf{dtMPDA}$ is
hence $\leq(nk|\mathcal{S}|)^{{\mathcal{O}}(K)}\cdot
2^{{\mathcal{O}}(nK^{2}(|X|+1))}$, where $K=(4nk+4)(|X|+2)$.

######  Proposition .

Let $\mathcal{S}$ be a $k$-round multistack timed automaton of size
$|\mathcal{S}|$ (constants encoded in unary) with $n$ stacks and set of clocks
$X$. Then, we can build a tree automaton $\mathcal{A}^{K,M}_{\mathcal{S}}$ of
size $(nk|\mathcal{S}|)^{{\mathcal{O}}(K)}\cdot
2^{{\mathcal{O}}(nK^{2}(|X|+1))}$ such that
$\mathcal{L}(\mathcal{A}^{K,M}_{\mathcal{S}}){=}\\{\tau{\in}\mathcal{L}(\mathcal{A}^{K,M}_{\textsf{valid}}){\mid}\llbracket\tau\rrbracket\in\mathsf{STCW}(\mathcal{S})\\}$.

Using arguments similar to Theorem 5, we obtain

######  Theorem .

Checking emptiness for $k$-round $\mathsf{dtMPDA}$ is decidable in ExpTime.

##  7\. Conclusion

The main contribution of this paper is a technique for analyzing timed systems
via tree automata. This is a new approach, which is quite different from all
existing approaches, most of which go via the region/zone based techniques for
timed automata. The hardest part of our approach, i.e., checking realizability
is oblivious of the underlying timed system and hence we believe that this
technique can be applied uniformly to a variety of timed systems. While, we
have made a few simplifying assumptions to best illustrate this method, our
technique can easily be adapted to remove many of these. For instance,
diagonal constraints of the form $x-y\in I$ can be handled easily by adding
matching edges. For a constraint $x-y\in[1,5]$, we add an edge between the
last reset of $x$ and last reset of $y$ with $[1,5]$ interval. Thus, we can
check the diagonal constraint at the time of last reset.

As future work, we would also like to extend our results to other restrictions
for $\mathsf{dtMPDA}$ such as bounded scope and phase. This would only require
us to prove a bound on the split-width and modify the system automaton
appropriately to handle the abstract behaviors generated by such systems. Our
techniques could also be applied to the more general model [14] of recursive
hybrid automata. Another interesting future work is to use our technique to go
beyond reachability and show results on model checking for timed systems.
While model-checking against untimed specifications is easy to obtain with our
approach, the challenge is to extend it to timed specifications. We also see a
strong potential to investigate the emptiness problem for classes of
alternating timed automata and hybrid automata. At the very least, we would
obtain new width-based under-approximations which may lead to new decidability
results.

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