# The GW Vir instability strip in the light of new observations of PG 1159
stars.
Discovery of pulsations in the central star of Abell 72 and variability of RX
J0122.9$-$7521

Paulina Sowicka Nicolaus Copernicus Astronomical Center, Polish Academy of
Sciences, ul. Bartycka 18, PL-00-716, Warszawa, Poland  Gerald Handler
Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, ul.
Bartycka 18, PL-00-716, Warszawa, Poland  David Jones Instituto de Astrofísica
de Canarias, E-38205 La Laguna, Tenerife, Spain  Departamento de Astrofísica,
Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain  Nordic Optical
Telescope, Rambla José Ana Fernández Pérez 7, 38711, Breña Baja, Spain  John
A. R. Caldwell  McDonald Observatory, 82 Mt. Locke Road, McDonald Observatory,
TX 79734, USA  Francois van Wyk  South African Astronomical Observatory, PO
Box 9, Observatory, 7935 Cape, South Africa  Ernst Paunzen Department of
Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University,
Kotlářská 2, Brno, Czech Republic  Karolina Bąkowska Institute of Astronomy,
Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University,
ul. Grudziądzka 5, 87-100 Toruń, Poland  Luis Peralta de Arriba Centro de
Astrobiología (CAB), CSIC-INTA, Camino Bajo del Castillo s/n, 28692 Villanueva
de la Cañada, Madrid, Spain  Isaac Newton Group of Telescopes, E-38700 Santa
Cruz de La Palma, La Palma, Spain  Lucía Suárez-Andrés Isaac Newton Group of
Telescopes, E-38700 Santa Cruz de La Palma, La Palma, Spain  Klaus Werner
Institut für Astronomie und Astrophysik, Kepler Center for Astro and Particle
Physics, Eberhard Karls Universität, Sand 1, D-72076 Tübingen, Germany  Marie
Karjalainen Astronomical Institute, Czech Academy of Sciences, Fričova 298,
25165, Ondřejov, Czech Republic  Daniel L. Holdsworth Jeremiah Horrocks
Institute, University of Central Lancashire, Preston, PR1 2HE, UK

(Accepted September 19, 2023)

###### Abstract

We present the results of new time series photometric observations of 29 pre-
white dwarf stars of PG 1159 spectral type, carried out in the years
$2014-2022$. For the majority of stars, a median noise level in Fourier
amplitude spectra of $0.5-1.0$ mmag was achieved. This allowed the detection
of pulsations in the central star of planetary nebula Abell 72, consistent
with g-modes excited in GW Vir stars, and variability in RX J0122.9$-$7521
that could be due to pulsations, binarity or rotation. For the remaining stars
from the sample that were not observed to vary, we placed upper limits for
variability. After combination with literature data, our results place the
fraction of pulsating PG 1159 stars within the GW Vir instability strip at
36%. An updated list of all known PG 1159 stars is provided, containing
astrometric measurements from the recent Gaia DR3 data, as well as information
on physical parameters, variability, and nitrogen content. Those data are used
to calculate luminosities for all PG 1159 stars to place the whole sample on
the theoretical Hertzsprung-Russell diagram for the first time in that way.
The pulsating stars are discussed as a group, and arguments are given that the
traditional separation of GW Vir pulsators in “DOV” and “PNNV” stars is
misleading and should not be used.

PG 1159 stars (1216), Pulsating variable stars (1307), Stellar pulsations
(1625), Non-radial pulsations (1117), Stellar evolution (1599), CCD
photometry(208), Hertzsprung Russell diagram(725), Post-asymptotic giant
branch(1287)

††facilities: GTC (OSIRIS), SAAO: Radcliffe, Elizabeth (SHOC), ING:Newton
(WFC), Struve (ProEM), Danish 1.54m Telescope (DFOSC), McGraw-Hill
(Andor)††software: Astropy (Astropy Collaboration et al., 2013, 2018, 2022),
ccdproc (Craig et al., 2017), dustmaps (Green, 2018), matplotlib (Hunter,
2007), numpy (Harris et al., 2020), pandas (Wes McKinney, 2010; pandas
development team, 2020), Period04 (Lenz & Breger, 2005), scipy (Virtanen et
al., 2020)

##  1 Introduction

Pre-white dwarf stars of PG 1159 spectral type (named after the prototype, PG
1159$-$035, Green & Liebert 1979) are important to study in the context of
stellar evolution, as they are supposed main progenitors of H-deficient white
dwarfs (WDs). PG 1159 stars populate the GW Vir instability strip, together
with central stars of planetary nebulae with C-rich Wolf-Rayet spectra
([WC]-types, exhibiting He, C, and O lines in emission, Crowther et al. 1998),
and [WC]-PG1159 stars, so-called transition objects (Leuenhagen et al., 1993;
Toalá et al., 2015). PG 1159 stars exhibit a broad absorption “trough” made by
He II at $4686\,$Å and adjacent C IV lines (see, e.g., Fig. 2 from Werner &
Rauch 2014), and typically have He-, C- and O-rich atmospheres, but notable
variations in He, C, and O abundances were found from star to star (e.g.,
Dreizler & Heber 1998, Werner 2001). Other groups of (pre-) white dwarf stars
also show He II and C IV lines – while the O(He) stars show significantly less
carbon than PG 1159 stars (up to 3% in their atmospheres, Reindl et al. 2014),
the limit to distinguish between PG 1159 stars and DO white dwarfs is model-
dependent – Werner et al. (2014) adopted C/He up to 9% (by mass) for DO stars.

Their formation history involves either a single star evolution scenario – a
“born-again” episode (a Very Late Thermal Pulse – VLTP, or a Late Thermal
Pulse – LTP; PG 1159-hybrid stars experience an AGB Final Thermal Pulse –
AFTP), or binary evolution – binary white dwarf merger (Werner et al., 2022b;
Miller Bertolami et al., 2022). Only some stars within the GW Vir instability
strip show pulsations, a striking difference from the other two classical
white dwarf instability strips (DAV and DBV), which are believed to be pure
(e.g., see Fontaine & Brassard 2008). The GW Vir pulsations are due to
nonradial g-modes, where the main restoring force is gravity (buoyancy),
driven by the $\kappa-\gamma$ mechanism associated with the partial ionization
of the K-shell electrons of carbon and/or oxygen in the envelope. The
pulsations typically are of short period (between 300 s and about 6000 s) and
low amplitude (typically 1 mmag – 0.15 mag, Córsico et al. 2019).

A current hypothesis, based on combined photometric and spectroscopic
observations, states that there is a clear separation within PG 1159 stars:
all N-rich (about 1% atmospheric N/He abundance) PG 1159 stars are pulsators,
while all N-poor ones (below about 0.01% N/He) do not pulsate (Dreizler &
Heber, 1998; Sowicka et al., 2021). Since N is a tracer of the evolutionary
history, an important conclusion follows: the pulsating and non-pulsating PG
1159 stars have different evolutionary histories, and it seems necessary that
a star undergoes a VLTP in order to develop pulsations. Recently, considerable
progress has been made in the study of PG 1159 stars’ atmospheric structure,
composition, and evolution through optical and ultraviolet spectroscopy and
advancement in non-LTE model atmospheres, as well as in probing their
interiors through asteroseismology with space-based observations (e.g., the
TESS mission (Ricker et al., 2015) observed several already known GW Vir
stars, Córsico et al. 2021). In the light of these findings, it is important
to further test this hypothesis on a larger sample of PG 1159 stars, by
obtaining high-quality, high-speed photometric observations aimed at detecting
low-amplitude pulsations if present, as well as spectroscopic observations
capable of detecting the nitrogen lines.

The number of known PG 1159 stars has increased in recent years, both due to
the detection of pulsations typical for GW Vir stars in new photometric
surveys (e.g., TESS, Uzundag et al. (2021, 2022), confirmed by spectroscopy),
and classification of targets of spectroscopic surveys (e.g., the most recent
discoveries with HET, Bond et al. 2023). Currently, 67 PG 1159 stars are
known111Based on a list from Werner & Herwig (2006), updated by us., including
hybrid-PG 1159 stars (whose atmospheres have traces of hydrogen). While these
stars lay within the GW Vir instability strip, some of them were either never
checked for (or reported) variability, or the quality of previous observations
was not sufficient to detect low amplitude pulsations. They also could have
been observed when beating between closely spaced modes was destructive and
pushed the observed amplitudes below the detection threshold. Moreover, some
of these objects have temporally highly variable pulsation spectra (Ciardullo
& Bond, 1996). Therefore, it is worth re-observing those stars in different
observing cycles to look for photometric variability. To date, there was no
extensive and systematic photometric survey for variability among those stars
since the works of Grauer et al. (1987b), Ciardullo & Bond (1996), and
González Pérez et al. (2006).

The aim of the work presented in this paper is to obtain new photometric
observations of a selected sample of PG 1159 stars to find new pulsators (or
candidates) and put limits on non-variability. We also provide the most up-to-
date list of PG 1159 stars and their properties from the Gaia mission and
follow-up works. Finally, we place the PG 1159 stars on the theoretical
Hertzsprung-Russell (HR) diagram
($\log{}L_{\star}/L_{\odot}-\log{}T_{\mathrm{eff}}$) and discuss the
implications of our findings.

##  2 Photometric observations

We selected a sample of PG 1159 stars for a survey of variability carried out
in the years $2014-2022$ with a network of telescopes, covering both
hemispheres. The selection was based on only one criterion: a given star was
included in our target list if it was never observed photometrically with time
resolution sufficient for the detection of GW Vir pulsations, or was
classified as non-variable, but either the reported detection limits could
have been improved by new observations or were not provided by the previous
authors. The top panel in Figure 1 shows the brightness distribution of
observed stars. The observing plan assumed the acquisition of observing blocks
lasting at least one hour per target. The following telescopes and instruments
were used for observations:

Figure 1: Top: Gaia magnitude distribution of the observed sample of 29 PG
1159 stars. Bottom: Distribution of the median noise level achieved in the
survey.

  * •

DFOSC at the 1.54-m Danish Telescope at ESO (DK):
The 1.54-m Danish Telescope located at La Silla Observatory was equipped with
the Danish Faint Object Spectrograph and Camera (DFOSC; Andersen et al. 1995).
DFOSC uses a 2K$\times$2K thinned Loral CCD chip with a Field of View (FOV) of
13.7$\times$13.7 arcmin. No filter was used. Six stars were observed with this
telescope.

  * •

OSIRIS at 10.4-m Gran Telescopio Canarias (GTC):
The 10.4-m Gran Telescopio Canarias (GTC) is located at the Observatorio del
Roque de los Muchachos (ORM, La Palma) and is equipped with Optical System for
Imaging and low-Intermediate-Resolution Integrated Spectroscopy (OSIRIS; Cepa
1998). OSIRIS consists of a mosaic of two CCDs of 2048$\times$4096 pixels each
and has an unvignetted FOV of 7.8$\times$7.8 arcmin. Either no filter or a
Sloan r’ filter was used. We used 2$\times$2 binning and a standard readout
time of about 23 seconds. Eleven stars were observed with this telescope.

  * •

WFC at 2.54-m Isaac Newton Telescope (INT):
The 2.54-m Isaac Newton Telescope (INT) is located at the Observatorio del
Roque de los Muchachos (ORM, La Palma) and is equipped with the Wide Field
Camera (WFC; Walton et al. 2001), an optical mosaic camera mounted in the
prime focus. WFC consists of four thinned EEV 2k$\times$4k CCDs. Because the
readout time of the whole CCD mosaic is rather long, we used it in windowing
mode – for a FOV of 5$\times$5 arcmin (910$\times$910 pixels) the readout time
was 6 seconds in the slow (less noisy) mode. No binning was used. We used a
Harris V filter. Five stars were observed with this telescope.

  * •

ProEM at the 2.1-m Otto Struve Telescope (MD):
The 2.1-m Otto Struve Telescope is located at McDonald Observatory, and is
equipped with ProEM, which is a frame-transfer CCD detector with optional
electron-multiplication with high frame-rate, optimized for high-speed time-
series photometry (providing effectively zero readout time). The CCD has
1024$\times$1024 pixels and a FOV of 1.6$\times$1.6 arcmin. We used 4$\times$4
binning for an effective plate scale of 0.36 arcsec pixel-1. We used a BG40
filter. Nine stars were observed with this telescope.

  * •

Andor at the 1.3-m McGraw-Hill Telescope (MDM):
The 1.3-m McGraw-Hill Telescope is located at the MDM Observatory, on the
south-west ridge of Kitt Peak in Arizona. It was equipped with Andor Ikon
DU937_BV CCD camera, which was used in Frame Transfer mode and 4$\times$4
binning. We used a BG38 filter. One star was observed with this telescope.

  * •

SHOC at the SAAO 1.9-m Telescope and 1.0-m Telescope (SA19, SA10):
The telescopes are located at the Sutherland station of the South African
Astronomical Observatory (SAAO), and are equipped with one of the Sutherland
High Speed Optical Cameras (SHOC; Coppejans et al. 2013). SHOC 1 and 2 are
high-speed cameras operating in frame-transfer mode for visible wavelength
range that have an electron-multiplying (EM) capability222The EM mode has not
been used for observations presented in this work.. The imaging area of the
detectors is 1024$\times$1024 pixels, which corresponds to a FOV of
2.79$\times$2.79 arcmin for the 1.9-m telescope with the focal reducer, and
2.85$\times$2.85 arcmin for the 1.0-m telescope. A selection of amplifiers can
be used, each resulting in a different gain setting, as well as binning and
readout speed. The slowest readout speed was usually chosen, resulting in the
lowest readout noise. Binning was determined by the observer to match the
observing conditions and especially avoid undersampling of the Point Spread
Function (PSF). Observations were done without a filter. Four stars were
observed with these telescopes.

The data were reduced using the following procedures. For data from DK we
applied standard IRAF routines for all reduction steps. We extracted bad
columns and hot pixels from the night’s bias frames and flat fields, then
cleaned the images for bad and hot pixels after the basic reduction steps
(bias subtraction, dark and flat correction). As the last step, we checked for
intensity gradients in the x and y directions (which sometimes occur in the
presence of a bright Moon) and removed them, if necessary. The data from all
the other instruments were reduced using standard Astropy (Astropy
Collaboration et al., 2013, 2018, 2022) ccdproc (Craig et al., 2017) routines
consisting of bias subtraction, dark correction (only for observations with
ProEM), flat-field and gain correction. Then, we performed aperture photometry
using our own photometry pipeline with the use of adaptive circular apertures
with sizes scaled to the seeing conditions for each frame (Sowicka et al.,
2018, 2021) with a scaling factor determined for each star and run. Comparison
stars were chosen (wherever possible) such that they were brighter than the
target and close to it, isolated and outside any faint nebulae, and when the
target was the brightest in the field, an “artificial” comparison star
comprising the summed flux from up to three available comparison stars was
used. Because our target stars usually are much hotter than the available
comparison stars, the differential light curves were corrected for
differential color extinction by fitting a straight line to a Bouguer plot
(differential magnitude vs. air mass). In the final step, we cleaned the light
curves by removing outliers ($3.5\sigma$ clipping) and parts of data with bad
quality (e.g., observations through thick clouds). We also inspected our
differential magnitudes plotted against FWHM measurements to make sure that
there is no correlation introduced by our photometry procedure. The constancy
of the comparison stars was checked by examining differential light curves
when more than one comparison star could be used. In case of fields with only
a single comparison star, we looked up their Gaia
$G_{\mathrm{BP}}-G_{\mathrm{RP}}$ colors, transformed these to
$V-I_{c}$333https://gea.esac.esa.int/archive/documentation/GDR3/Data_processing/chap_cu5pho/cu5pho_sec_photSystem/cu5pho_ssec_photRelations.html,
and those to $B-V$ (Caldwell et al., 1993). In that way, and with a rough
correction for interstellar reddening, we inferred that none of the single
comparison stars had $(B-V)_{0}<0.7$ and hence none of them lies in a
$\kappa$-driven instability strip.

In this work, we present the results for a sample of 29 PG 1159 stars that are
not surrounded by bright planetary nebulae. The list of targets, observing log
and information on the scaling factor used in the photometry procedure is
given in Table 1. The light curves are presented in Figure 2.

Table 1: Log of photometric observations

Name | Equip. | Observer | Date | Filter | $t_{\mathrm{exp}}$ | Scale | $\Delta T$ | $\Delta f$ | Med. noise
---|---|---|---|---|---|---|---|---|---
|  |  | +UTC Start |  | (s) | factor |  | (d-1) | (mmag)
BMP 0739$-$1418 | DK | EP | 2014-12-26T05:48:30 | no filter | 30 | 1.5 | 2.39 hr | 10.06 | 0.31 |
H1504+65 | GTC | SA | 2016-03-09T02:07:53 | Sloan r | 6 | 1.5 | 48 min | 30.00 | 0.42 |
HS 0444+0453 | DK | EP | 2014-12-26T03:44:44 | no filter | 20 | 1.2 | 1.81 hr | 13.24 | 0.52 |
HS 0704+6153 | GTC | SA | 2016-03-09T22:30:36 | Sloan r | 10 | 1.5 | 48 min | 30.34 | 0.53 |
HS 1517+7403 | MD | GH | 2016-05-24T02:59:58 | BG40 | 10 | 0.9 | 1.51 hr | 15.88 | 0.61 |
MCT 0130$-$1937 | SA19 | PS | 2014-12-05T19:15:59 | no filter | 10 | 0.9 | 2.30 hr | 10.45 | 0.83 |
PG 1151$-$029 | INT | NH | 2016-03-29T21:44:06 | Harris V | 10 | 1.5 | 1.45 hr | 16.61 | 0.57 |
PG 1520+525 | MD | GH | 2016-05-30T02:47:51 | BG40 | 15 | 1.2 | 1.15 hr | 20.87 | 0.66 |
PN A66 (Abell) 21 | DK | EP | 2015-02-10T02:01:46 | no filter | 40 | 1.2 | 1.16 hr | 20.71 | 2.02 |
GTC | SA | 2016-03-08T22:26:12 | Sloan r | 6 | 1.2 | 50 min | 29.04 | 0.44 |
PN A66 (Abell) 72 | SA10 | FW | 2022-10-07T18:35:41 | no filter | 25 | 1.8 | 3.05 hr | 7.87 | 1.29 |
2022-10-08T18:01:35 | no filter | 30-35 | 1.5 | 2.04 hr | 11.76 | 1.58 |
PN IsWe 1 | INT | MK+Students | 2016-10-19T04:11:53 | Harris V | 5 | 0.9 | 1.94 hr | 12.40 | 0.81 |
PN Jn 1 | INT | LSA, PSh | 2016-12-12T20:47:54 | Harris V | 10 | 1.2 | 3.01 hr | 7.98 | 0.81 |
MD | JC | 2017-08-16T07:54:09 | BG40 | 10 | 1.2 | 3.75 hr | 6.40 | 0.30 |
MD | JC | 2017-08-17T09:37:05 | BG40 | 10 | 1.2 | 1.93 hr | 12.41 | 0.35 |
INT | DJ | 2017-08-28T01:45:39 | Harris V | 5 | 1.5 | 3.71 hr | 6.47 | 0.41 |
INT | DJ | 2017-08-30T03:03:13 | Harris V | 10 | 1.2 | 2.64 hr | 9.09 | 0.45 |
PN Lo (Longmore) 3 | DK | EP | 2015-02-10T00:41:13 | no filter | 40 | 1.2 | 1.16 hr | 20.65 | 2.40 |
RX J0122.9$-$7521 | SA19 | PS | 2014-12-04T19:05:30 | no filter | 10 | 0.9 | 1.95 hr | 12.32 | 0.49 |
2014-12-09T18:32:43 | no filter | 10 | 0.9 | 2.42 hr | 9.92 | 0.46 |
SDSS J000945.46+135814.4 | GTC | SA | 2017-12-06T22:18:56 | no filter | 10 | 1.2 | 58 min | 24.63 | 2.55 |
SDSS J001651.42$-$011329.3 | SA19 | PS | 2017-12-06T22:18:55 | Sloan r | 20 | 0.9 | 1.71 hr | 14.04 | 1.58 |
SDSS J055905.02+633448.4 | GTC | SA | 2017-09-15T04:09:52 | Sloan r | 20 | 1.2 | 59 min | 24.43 | 0.98 |
SDSS J075540.94+400918.0 | GTC | SA | 2016-03-06T23:31:33 | Sloan r | 15 | 1.2 | 57 min | 25.21 | 0.59 |
SDSS J093546.53+110529.0 | DK | EP | 2015-01-02T06:26:02 | no filter | 30 | 0.9 | 1.55 hr | 15.45 | 1.37 |
MD | JC | 2017-05-06T03:00:17 | BG40 | 30 | 1.2 | 2.92 hr | 8.23 | 1.85 |
GTC | SA | 2018-08-14T14:45:58 | Sloan r | 20 | 1.2 | 57 min | 25.05 | 0.86 |
SDSS J102327.41+535258.7 | INT | LPA | 2016-02-03T02:40:54 | Harris V | 20 | 1.2 | 2.52 hr | 9.54 | 1.58 |
SDSS J105300.24+174932.9 | MD | JC | 2017-05-02T03:34:16 | BG40 | 20 | 0.9 | 2.27 hr | 10.56 | 0.78 |
MD | JC | 2017-05-07T03:08:52 | BG40 | 22 | 0.9 | 3.04 hr | 7.89 | 1.91 |
GTC | SA | 2017-12-29T02:27:34 | Sloan r | 10 | 1.2 | 1.05 hr | 22.96 | 0.57 |
MDM | KB | 2019-04-24T04:19:17 | BG38 | 30 | 1.5 | 1.99 hr | 12.09 | 0.79 |
MDM | KB | 2019-04-25T03:36:34 | BG38 | 30 | 1.2 | 1.82 hr | 13.20 | 1.55 |
MDM | KB | 2019-04-26T02:54:19 | BG38 | 30 | 1.2 | 4.00 hr | 6.01 | 1.11 |
SDSS J121523.09+120300.8 | DK | EP | 2015-04-14T02:32:52 | no filter | 40 | 0.9 | 1.26 hr | 19.04 | 2.97 |
GTC | SA | 2018-01-17T06:01:53 | Sloan r | 20 | 1.2 | 1.13 hr | 21.17 | 0.95 |
SDSS J123930.61+244321.7 | INT | PS, MT | 2016-03-11T00:46:00 | Harris V | 20 | 0.9 | 2.10 hr | 11.42 | 1.24 |
SDSS J134341.88+670154.5 | MD | GH | 2016-05-26T02:57:55 | BG40 | 20 | 1.5 | 59 min | 24.56 | 1.11 |
2016-05-29T02:47:32 | BG40 | 20 | 1.2 | 1.50 hr | 16.00 | 0.92 |
SDSS J141556.26+061822.5 | MD | JC | 2017-05-05T05:14:23 | BG40 | 30 | 1.2 | 5.20 hr | 4.62 | 0.69 |
SDSS J144734.12+572053.1 | MD | GH | 2016-05-28T02:46:07 | BG40 | 30 | 1.2 | 1.59 hr | 15.08 | 2.25 |
SDSS J191845.01+624343.7 | MD | JC | 2017-05-08T07:08:02 | BG40 | 30 | 0.9 | 4.15 hr | 5.78 | 1.05 |
Sh 2$-$68 | GTC | SA | 2016-04-23T04:45:44 | Sloan r | 10 | 1.5 | 58 min | 24.81 | 0.58 |
Sh 2$-$78 | GTC | SA | 2016-04-24T04:27:31 | Sloan r | 15 | 1.2 | 1.24 hr | 19.30 | 0.75 |

Note. — SA – Support Astronomer; Students – Rosa Clavero, Francisco Galindo,
Bartosz Gauza; GTC – GTC+OSIRIS; DK – DK+DFOSC; MD – McDonald 2.1-m+ProEM; MDM
– MDM 1.3-m+Andor; SA19 – SAAO 1.9-m+SHOC; SA10 – SAAO 1.0-m+SHOC; INT –
INT+WFC.

We refer to the central stars using the PN designations throughout the paper.

##  3 Frequency analysis

The light curves prepared in the previous step were the subject of frequency
analysis. We used Period04 (Lenz & Breger, 2005) to calculate Fourier
amplitude spectra for each star and run separately, up to the corresponding
Nyquist frequency. The Fourier amplitude spectra are shown in Figure 2. The
frequency range for which our survey is sensitive to varies from star to star.
The length of observations varied from slightly below an hour to a few hours,
resulting in poor frequency resolution for the shortest ones (based on Loumos
& Deeming (1978) criterion of $\Delta{f}=1.0/\Delta{T}$ for only the detection
of modes444We note that for a correct determination of amplitudes and phases
the criterion is $\Delta{f}>1.5/\Delta{T}$.). For each Fourier amplitude
spectrum, we calculated the median noise level, as well as our detection
threshold (dashed line in Fig. 2), adopted as amplitude ratio of $S/N\geq{4}$
(Breger et al., 1993). Table 1 includes the length of observations,
corresponding frequency resolution, and median noise level in the Fourier
spectra for all observed targets.

Figure 2: Light curves and their respective Fourier amplitude spectra of the
survey targets. Plots for different stars are separated with horizontal lines.
Light curves: Note different scales. Fourier spectra: they were calculated up
to their respective Nyquist frequencies, but are plotted until 1200 d-1.
Dashed lines show the detection threshold of S/N$\geq{4}$. Note different
scales. Figure 2: Continued. Figure 2: Continued. Figure 2: Continued. Figure
2: Continued.

##  4 Survey results

The bottom panel of Figure 1 presents a histogram of the number of stars vs.
the median noise level in the Fourier spectrum. In cases when the same star
was observed multiple times, the lowest achieved level was taken. For the
majority of observed stars, we reached a noise level of about 1 mmag or below.
Grauer et al. (1987b) and Ciardullo & Bond (1996) reported their threshold for
non-variable targets as the maximum amplitude in the Fourier spectra and
reached values of $2.4-2.7$ and $2.4-5.3$ mmag, respectively. Inspection of
Figure 2 shows that our results are comparable to theirs, while our sample
covered fainter stars ($15.4-18.6$ mag in $Gaia$, see the top panel of Fig.
1). This allowed us to discover pulsations in the central star of planetary
nebula Abell 72, and variability in RX J0122.9$-$7521\. In addition to that,
five objects from our sample could also be variable, but need follow-up
observations for eventual confirmation. The majority of our sample did not
show any variability consistent with GW Vir pulsations, and in those cases we
put upper limits on non-variability. Each Fourier spectrum was also inspected
for the presence of short-period $\epsilon$-driven modes (Córsico et al.,
2009). No sign of such modes with periods shorter than about $200$ s
(frequencies above 400 d-1 ) was found in any of the stars.

##  5 Comments on selected stars

Below, we comment on stars that showed peaks of interest in the Fourier
spectrum. While most of our observations turned out to be non-detections, we
have to mention one caveat. Pulsating PG 1159 stars are known for their
variable pulsation spectra, even on a night-to-night basis. This is often
caused by the interference between closely spaced modes, which occasionally
becomes destructive and pushes the amplitudes of the modes below the detection
threshold. Possible nonlinear mode coupling could have the same effect (e.g.,
Vauclair et al., 2011). Non-detections because of those reasons could be
avoided by observing the targets on multiple nights over the visibility
period. While this was the case for eight targets, we were not able to acquire
multiple runs for the remaining sample, and this has to be kept in mind
regarding our non-detections.

###  5.1 Pulsator – PN Abell 72 555We refer to the central stars using the PN
designations throughout the paper.

The central star of the planetary nebula Abell 72 was observed in October 2022
over two consecutive nights. The light curves and Fourier amplitude spectra
are presented in Fig. 3. We detected significant peaks reaching amplitudes on
the order of 10 mmag in the nightly Fourier amplitude spectra, on both nights
located in the same frequency range, consistent with g-mode pulsations seen in
GW Vir stars. We classify Abell 72 as a multiperiodic pulsator and
observations on a longer time base are needed to resolve its pulsation modes.

Figure 3: Light curves and Fourier amplitude spectra of two SAAO observing
runs on the central star of planetary nebula Abell 72. The Fourier amplitude
spectra were calculated up to the Nyquist frequency, but are shown up to 1200
d-1. Insets show a zoomed in view into the frequency range of detected
pulsations. Dashed lines show the detection threshold of S/N$\geq{4}$. Note
the same scales for the light curves and Fourier spectra.

###  5.2 Candidates

  * •

HS 0444+0453
There is an interesting, but statistically insignificant ($S/N=3.3$) peak
around 45.7 d-1 (period of about 1890 s). If confirmed, it fits within the
observed period range of GW Vir pulsators.

  * •

HS 1517+7403
There are two statistically significant peaks: 17.5 and 40.5 d-1 (S/N = 5.8
for both, periods of 4945 and 2133 s, respectively). Such long periods are
usually found in GW Vir central stars of planetary nebulae, but no nebula
around HS 1517+7403 has been reported. Given the short duration of the single
run available and that only a single comparison star could be used, it is not
clear whether these peaks are due to pulsations of the target.

  * •

PN IsWe 1
The highest, possibly unresolved peak at 49 d-1 has S/N=4.4 and corresponds to
a period of about 30 min (1764 s). While such pulsation periods are observed
in GW Vir stars, observations on a longer time base are necessary for
confirmation.

  * •

PN Jn 1
Ciardullo & Bond (1996) observed the star twice, obtaining peaks of maximum
amplitude of 2.4 and 4.0 mmag in the Fourier amplitude spectra. They did not
detect significant peaks (reaching 99% confidence level) but two candidates:
540.5 and 538.5 $\mu{}$Hz (46.70 and 46.53 d-1 , respectively), and as a
result did not claim the detection of pulsations in the central star of
planetary nebula Jn 1. González Pérez et al. (2006) observed the star once and
did not find the peaks tentatively detected by Ciardullo & Bond (1996),
instead they found a barely significant peak at 2200 $\mu{}$Hz (190.1 d-1 ).
Nevertheless, they claimed discovery of pulsations on that basis, but called
for more observations to confirm their findings. We observed Jn 1 a total of
five times in three different runs, and achieved very good median noise levels
of 0.30–0.81 mmag. In none of the runs we saw signs of peaks previously
reported or strong peaks occurring in more than one of our own runs (e.g., a
peak at $~{}100$ d-1 with $S/N=4.3$ only in McDonald–1 run). We thus conclude
that there is no convincing evidence that Jn 1 pulsates, and that it requires
observations of similar quality to our first McDonald run for eventual
confirmation.

  * •

RX J0122.9$-$7521
RX J0122.9$-$7521 was observed twice in December 2014. The light curves and
Fourier amplitude spectra are presented in Fig. 4. We detected a significant
peak in the Fourier amplitude spectra of both nights, located at the same
frequency of about 35 d-1 and reaching an amplitude of $4-5$ mmag. RX
J0122.9$-$7521 was also observed by TESS in Sectors 1, 13, 27, and 28. The
same frequency as in our ground-based data is present in the TESS observations
(34.78 d-1 ). With $T_{\mathrm{eff}}=180000$ K that would make RX
J0122.9$-$7521 the hottest known variable/pulsating PG 1159 star. We further
discuss this star in Sect. 10.1.

Figure 4: Light curves and Fourier amplitude spectra of two SAAO observing
runs on RX J0122.9$-$7521\. The Fourier amplitude spectra were calculated up
to the Nyquist frequency, but are shown up to 1200 d-1. Insets show a zoomed
in view into the frequency range of the detected variations. Dashed lines show
the detection threshold of S/N$\geq{4}$. Note the same scales for the light
curves and Fourier spectra.

  * •

SDSS J102327.41+535258.7
One suitable comparison star was used. There are two peaks: 15.7 and 30.4 d-1
with $S/N$ of 4.3 and 4.2, which correspond to periods of 92 and 47 min (5507
and 2839 s), respectively.

  * •

SDSS J105300.24+174932.9
There are two interesting, but insignificant ($S/N=3.7$ and 3.1) peaks at
251.9 and 281.5 d-1 , respectively, in the first McDonald run. In the second
run, five days later, a peak in the same frequency region is present (at about
257 d-1 ), but due to higher noise the signal (if real) has only slightly
higher amplitude than the highest noise peaks. In the remaining runs we were
not able to reach a better noise level than in the first McDonald run, except
the GTC run (that was too short).

###  5.3 Nonpulsators

  * •

MCT 0130$-$1937
There is a significant low-frequency trend in the light curve (around 10 d-1)
that is likely not intrinsic to the star, particularly because there was only
one comparison star available that was fainter than the target.

  * •

PG 1151$-$029
There is a significant low-frequency trend in the light curve, corresponding
to a peak around 20 d-1), that is likely caused by sky transparency variations
that night.

  * •

PG 1520+525
There is a significant peak ($S/N=4.33$) at 725 d-1. To assess whether this
peak is real, we calculated differential light curves between the target and
two different comparison stars, as well as between these comparison stars, and
then computed the FTs of these light curves. The aforementioned peak showed up
only in the difference of (target$-$comparison star 1), and not, as expected
if the target was variable, also in the difference of (target$-$comparison
star 2). We conclude that, even though formally significant in that
differential light curve, this peak is not intrinsic to PG 1520+525.

  * •

SDSS J000945.46+135814.4
The observing run is too short (hence too low frequency resolution) to decide
whether the signals around 50 d-1 may be intrinsic to the star.

  * •

SDSS J093546.53+110529.0
In the FT from DK there is a peak around 16 d-1 that is due to a low-frequency
trend in the light curve and not present in the runs from the other two
instruments, one of which has a much lower noise level. We therefore conclude
that this signal is not due to pulsations from the target.

  * •

SDSS J134341.88+670154.5
The second run shows a significant low-frequency trend around 20 d-1 that is
likely due to variable sky conditions and not intrinsic to the star.

  * •

SDSS J141556.26+061822.5
There is a low-frequency peak at 9.46 d-1 with $S/N=5.8$. This peak
corresponds to a period of about 2.5 h (9000 s), which is too long for GW Vir
pulsations. This peak is also present in differential light curves between the
target and either of the comparison stars. Therefore, the peak might be
intrinsic to the target, but of different origin than GW Vir pulsations, e.g.,
rotation, binarity, or spots.

##  6 Impurity of GW Vir instability strip

Previous observations showed that only about 50% of stars within the GW Vir
instability strip pulsate (see, e.g., Fig. 2 in Uzundag et al. 2022). A
significant number of PG 1159 stars were discovered since then, and with our
new results, we can re-determine the current occurrence rate for PG 1159
stars. In Table 2 we listed the physical parameters of all known PG 1159 stars
with updated information about their variability. For a total of 67 PG 1159
stars, 24 stars are confirmed as pulsating, which corresponds to 36%. Still,
the majority of PG 1159 stars within the instability strip are found to be
non-pulsators. Sowicka et al. (2021) recently showed that there was a clear
separation between N-rich ($\approx 1\%$ N/He) pulsators and N-poor ($<0.01\%$
N/He) non-pulsators. We therefore also listed the N abundance, where
available, in Table 2. To date, only 26 PG 1159 stars have published N
abundances.

Table 2: Properties of PG 1159 stars

Name | $T_{\mathrm{eff}}$ | $\log{g}$ | PN | Puls. | N | BC | $\log{L_{\star}}$ | Ref.
---|---|---|---|---|---|---|---|---
| (K) | (cm s-2) |  |  |  | (mag) | L⊙ |
BMP J0739$-$1418 | 120000 | 6.0 | yes | NOP | N-poor | -7.269 | $3.48^{+0.11}_{-0.12}$ | W+2023
FEGU 248$-$5 | 160000 | 6.5 | yes | NVD | N-poor | -8.039 | $3.89^{+0.11}_{-0.10}$ | W+2023
H1504+65 | 200000 | 8.0 | no | NOP | N-poor | -8.700 | $2.167^{+0.062}_{-0.064}$ | W+2004a, WD2005, WR2015, NW2004
HE 1429$-$1209 | 160000 | 6.0 | no | yes | no lit. data | -8.039 | $3.46^{+0.11}_{-0.12}$ | W+2004b
HS 0444+0453 | 90000 | 7.0 | no | NOP | no lit. data | -6.429 | $1.28^{+0.14}_{-0.16}$ | D1999
HS 0704+6153 | 75000 | 7.0 | no | NOP | N-poor | -5.832 | $1.03^{+0.17}_{-0.20}$ | DH1998
HS 1517+7403 | 110000 | 7.0 | no | NOP | N-poor | -7.045 | $1.84^{+0.10}_{-0.12}$ | DH1998
HS 2324+3944 | 130000 | 6.2 | no | yes | no lit. data | -7.460 | $3.390^{+0.092}_{-0.084}$ | F+2010, S+1999, C+2021
MCT 0130$-$1937 | 90000 | 7.5 | no | NOP | N-poor | -6.429 | $1.42^{+0.14}_{-0.16}$ | W+2004c, WR2014
NGC 246 | 150000 | 5.7 | yes | yes | N-poor | -7.855 | $3.786^{+0.081}_{-0.084}$ | W+2005, CB1996
NGC 650 | 140000 | 7.0 | yes | NOP | no lit. data | -7.658 | $3.27^{+0.67}_{-0.36}$ | NS1995
NGC 6852 | 150000 | 6.0 | yes | yes | no lit. data | -7.855 | $2.93^{+0.34}_{-0.26}$ | K. Werner, GP+2006
NGC 7094 | 110000 | 5.7 | yes | yes | N-poor | -7.045 | $3.83^{+0.10}_{-0.12}$ | F+2010, S+2007
PG 0122+200 | 80000 | 7.5 | no | yes | N-rich | -6.043 | $1.20^{+0.17}_{-0.19}$ | WR2014, F+2007
PG 1144+005 | 150000 | 6.5 | no | yes | N-rich | -7.855 | $3.13^{+0.10}_{-0.10}$ | W+2005, W+2016, S+2021
PG 1151$-$029 | 140000 | 6.0 | no | NOP | no lit. data | -7.658 | $2.471^{+0.097}_{-0.091}$ | W+2004c
PG 1159$-$035 | 140000 | 7.0 | no | yes | N-rich | -7.658 | $2.596^{+0.085}_{-0.086}$ | W+2005, W+2016, C+2008, O+2022
PG 1424+535 | 110000 | 7.0 | no | NOP | N-poor | -7.045 | $1.838^{+0.092}_{-0.115}$ | W+2005, W+2015
PG 1520+525 | 150000 | 7.5 | yes | NOP | N-poor | -7.866 | $2.591^{+0.081}_{-0.087}$ | W+2005, W+2016
PG 1707+427 | 85000 | 7.5 | no | yes | N-rich | -6.243 | $1.47^{+0.15}_{-0.17}$ | W+2005, W+2015, H+2018, K+2004
PN A66 21 | 140000 | 6.5 | yes | NOP | no lit. data | -7.658 | $2.118^{+0.088}_{-0.086}$ | W+2004c
PN A66 43 | 110000 | 5.7 | yes | yes | N-rich | -7.045 | $3.69^{+0.10}_{-0.12}$ | F+2010, V+2005
PN A66 72 | 170000 | 6.5 | yes | yes | N-rich | -8.212 | $3.35^{+0.11}_{-0.12}$ | B+2023
PN IsWe 1 | 90000 | 7.0 | yes | NOP | no lit. data | -6.429 | $1.34^{+0.14}_{-0.16}$ | D1999
PN Jn 1 | 150000 | 6.5 | yes | NOP | no lit. data | -7.855 | $2.687^{+0.097}_{-0.095}$ | RW1995
PN K 1$-$16 | 160000 | 5.8 | yes | yes | no lit. data | -8.039 | $3.601^{+0.083}_{-0.088}$ | W+2010, G+1987, C+2021
PN Kn 12 | 170000 | 6.5 | yes | NVD | no lit. data | -8.212 | $3.20^{+0.36}_{-0.28}$ | B+2023
PN Kn 61 | 170000 | 6.5 | yes | yes | N-rich | -8.212 | $3.54^{+0.37}_{-0.27}$ | DM+2015, B+2023, S+2023
PN Kn 130 | 170000 | 6.5 | yes | NVD | N-poor | -8.212 | $3.40^{+0.13}_{-0.13}$ | B+2023
PN Lo 3 | 140000 | 6.3 | yes | NOP | no lit. data | -7.658 | $3.08^{+0.15}_{-0.14}$ | W+2004c
PN Lo 4 | 170000 | 6.0 | yes | yes | N-poor | -8.212 | $3.65^{+0.18}_{-0.13}$ | W+2010, BM1990
PN Ou 2 | 170000 | 6.5 | yes | NVD | no lit. data | -8.212 | $2.28^{+0.46}_{-0.22}$ | B+2023
PN VV 47 | 130000 | 7.0 | yes | NOP | no lit. data | -7.460 | $2.04^{+0.11}_{-0.10}$ | RW1995
RL 104 | 80000 | 6.0 | no | NVD | N-rich | -6.046 | $3.17^{+0.15}_{-0.18}$ | W+2022
RX J0122.9$-$7521 | 180000 | 7.5 | no | NOP | no lit. data | -8.389 | $2.958^{+0.067}_{-0.071}$ | W+2004c
RX J2117.1+3412 | 170000 | 6.0 | yes | yes | no lit. data | -8.212 | $3.394^{+0.067}_{-0.071}$ | W+2005, V+2002, C+2021
SALT J172411.7$-$632147 | 160000 | 6.5 | no | yes | N-poor | -8.039 | $3.01^{+0.11}_{-0.12}$ | J+2023
SALT J213742.6$-$382901 | 180000 | 7.0 | no | yes | N-rich | -8.376 | $3.04^{+0.17}_{-0.13}$ | J+2023
SDSS J000945.46+135814.4 | 120000 | 7.5 | no | NOP | no lit. data | -7.279 | $2.49^{+0.44}_{-0.25}$ | K+2016
SDSS J001651.42$-$011329.3 | 120000 | 5.5 | no | NOP | no lit. data | -7.269 | $3.19^{+0.22}_{-0.18}$ | H+2006
SDSS J034917.41$-$005919.3 | 90000 | 7.5 | no | yes | no lit. data | -6.429 | $1.33^{+0.19}_{-0.17}$ | H+2006, W+2012
SDSS J055905.02+633448.4 | 110000 | 7.5 | no | NOP | no lit. data | -7.050 | $1.49^{+0.17}_{-0.17}$ | W+2014
SDSS J075415.11+085232.1 | 120000 | 7.0 | no | yes | no lit. data | -7.269 | $1.70^{+0.58}_{-0.29}$ | W+2014, K+2014
SDSS J075540.94+400918.0 | 100000 | 7.6 | no | NOP | no lit. data | -6.764 | $1.62^{+0.19}_{-0.18}$ | H+2006
SDSS J093546.53+110529.0 | 100000 | 7.6 | no | NOP | no lit. data | -6.764 | $1.47^{+0.17}_{-0.17}$ | H+2006
SDSS J102327.41+535258.7 | 110000 | 7.6 | no | NOP | no lit. data | -7.050 | $2.15^{+0.23}_{-0.21}$ | H+2006
SDSS J105300.24+174932.9 | 100000 | 7.0 | no | NOP | no lit. data | -6.762 | $1.59^{+0.13}_{-0.14}$ | W+2014
SDSS J121523.09+120300.8 | 100000 | 7.6 | no | NOP | no lit. data | -6.764 | $1.65^{+0.25}_{-0.21}$ | H+2006
SDSS J123930.61+244321.7 | 100000 | 7.5 | no | NOP | no lit. data | -6.764 | $1.64^{+0.25}_{-0.21}$ | W+2014
SDSS J134341.88+670154.5 | 100000 | 7.6 | no | NOP | no lit. data | -6.764 | $1.44^{+0.12}_{-0.14}$ | H+2006
SDSS J141556.26+061822.5 | 120000 | 7.5 | no | NOP | no lit. data | -7.279 | $1.83^{+0.15}_{-0.13}$ | W+2014
SDSS J144734.12+572053.1 | 100000 | 7.6 | no | NOP | no lit. data | -6.764 | $1.58^{+0.19}_{-0.17}$ | H+2006
SDSS J152116.00+251437.5 | 140000 | 6.0 | no | NOP | no lit. data | -7.658 | $3.11^{+0.42}_{-0.29}$ | W+2014
SDSS J155610.40+254640.3 | 100000 | 5.3 | no | NVD | no lit. data | -6.762 | $3.23^{+0.40}_{-0.37}$ | R+2016
SDSS J163727.03+485355.2 | 100000 | 7.5 | no | NVD | no lit. data | -6.764 | $1.86^{+0.39}_{-0.22}$ | K+2016
SDSS J191845.01+624343.7 | 100000 | 7.2 | no | NOP | no lit. data | -6.762 | $1.65^{+0.14}_{-0.15}$ | W+2014
SDSS J212531.92$-$010745.8 | 100000 | 7.5 | no | NOP | no lit. data | -6.764 | $2.54^{+0.41}_{-0.25}$ | K. Werner
Sh 2$-$68 | 84000 | 7.2 | no | NOP | no lit. data | -6.205 | $1.70^{+0.17}_{-0.16}$ | G+2010
Sh 2$-$78 | 120000 | 7.5 | yes | NOP | no lit. data | -7.279 | $1.79^{+0.11}_{-0.11}$ | D1999
TIC 95332541 | 100000 | 7.5 | no | yes | N-poor | -6.764 | $2.14^{+0.12}_{-0.13}$ | U+2021, R+2023
TIC 333432673 | 120000 | 7.5 | no | yes | no lit. data | -7.279 | $1.924^{+0.082}_{-0.095}$ | U+2021
TIC 403800675 | 110000 | 7.5 | no | yes | no lit. data | -7.050 | $1.73^{+0.10}_{-0.12}$ | U+2022
TIC 1989122424 | 110000 | 7.5 | no | yes | no lit. data | -7.050 | $1.29^{+0.11}_{-0.13}$ | U+2022
WD J070204.29+051420.56 | 100000 | 7.5 | no | NVD | N-poor | -6.764 | $1.63^{+0.12}_{-0.14}$ | R+2023
NGC 6765 | - | - | yes | NVD | no lit. data | - | - | NS1995
PG 2131+066 | 95000 | 7.5 | no | yes | N-rich | - | - | WR2014, K+1995
RX J0439.8$-$6809 | 250000 | 8.0 | no | NOP | N-poor | - | - | WR2015

Note. — Properties of PG 1159 stars. Bold–this work. NOP – Not Observed to
Pulsate, NVD – No Variability Data available. The last three stars either lack
Gaia measurements or Teff and $\log{g}$ determinations and were excluded from
the analysis. References: BM1990 – Bond & Meakes (1990), B+2023 – Bond et al.
(2023), CB1996 – Ciardullo & Bond (1996), C+2008 – Costa et al. (2008), C+2021
– Córsico et al. (2021), DH1998 – Dreizler & Heber (1998), D1999 – Dreizler
(1999), DM+2015 – De Marco et al. (2015), F+2007 – Fu et al. (2007), F+2010 –
Friederich et al. (2010), G+1987 – Grauer et al. (1987a), GP+2006 – González
Pérez et al. (2006), G+2010 – Gianninas et al. (2010), H+2006 – Hügelmeyer et
al. (2006), H+2018 – Hoyer et al. (2018), J+2023 – Jeffery et al. (2023),
K+1995 – Kawaler et al. (1995), K+2004 – Kawaler et al. (2004), K+2014 –
Kepler et al. (2014), K+2016 – Kepler et al. (2016), NS1995 – Napiwotzki &
Schoenberner (1995), NW2004 – Nagel & Werner (2004), O+2022 – Oliveira da Rosa
et al. (2022), RW1995 – Rauch & Werner (1995), R+2016 – Reindl et al. (2016),
R+2023 – Reindl et al. (2023), S+1999 – Silvotti et al. (1999), S+2007 –
Solheim et al. (2007), S+2021 – Sowicka et al. (2021), S+2023 – Sowicka et al.
in prep. (2023), U+2021 – Uzundag et al. (2021), U+2022 – Uzundag et al.
(2022), V+2002 – Vauclair et al. (2002), V+2005 – Vauclair et al. (2005),
W+2004a – Werner et al. (2004a), W+2004b – Werner et al. (2004b), W+2004c –
Werner et al. (2004c), W+2005 – Werner et al. (2005), WD2005 – Werner & Drake
(2005), W+2010 – Werner et al. (2010), W+2012 – Woudt et al. (2012), W+2014 –
Werner et al. (2014), WR2014 – Werner & Rauch (2014), W+2015 – Werner et al.
(2015), WR2015 – Werner & Rauch (2015), W+2016 – Werner et al. (2016), W+2022
– Werner et al. (2022a), W+2023 – Weidmann et al. (2023)

##  7 Properties of all known PG 1159 stars

Thanks to the Gaia mission (Gaia Collaboration et al., 2016, 2023) the
community received precise measurements of positions and distances of more
than 1 billion stars. For the first time, consistent distance measurements
became available for almost the entire sample of PG 1159 stars777With the
exception of two stars without sufficient Gaia data, which are listed at the
bottom of Table 3. In Table 3 we compiled available Gaia DR3 information for
PG 1159 stars: identifiers, positions, Gaia $G$ magnitudes and parallaxes with
geometric distances determined by Bailer-Jones et al. (2021). We also list the
corresponding reddening E(B-V) at these distances, determined from the 3D
reddening map of Green et al. (2018) (Bayestar17) using the Python package
dustmaps. Even though a newer version of Bayestar is available (Bayestar19,
Green et al. 2019), it did not cover the distances of all the stars in our
sample, hence we used the Bayestar17 reddening map for all but six stars.
Those six stars were not covered because of declination south of
$-30^{\circ}$. For these cases, we used the 2D dust maps of Schlafly &
Finkbeiner (2011) and Schlegel et al. (1998) (SFD), which are equivalent to
Bayestar in terms of units. We did not take into account the reddening by the
surrounding planetary nebulae in the case of PG 1159 stars being the central
stars of planetary nebulae. We also listed the RUWE (Renormalized Unit Weight
Error) coefficient for each star, and marked in bold values higher than the
canonical 1.4, which might either suggest an unreliable astrometric solution
(in a few cases that corresponds with a large parallax error) or be a hint
towards binarity. In the final column of Table 3 we put a remark for non-
single stars (e.g., known or suspected binaries/triples) and a subclass of so-
called “hybrid”-PG 1159 stars (exhibiting traces of hydrogen in the
atmosphere).

Table 3: Astrometric properties of PG 1159 stars

Name |  Gaia ID | RA | Dec. |  Gaia G | $\varpi{}_{\mathrm{Gaia}}$ |  $\sigma_{\varpi_{\mathrm{Gaia}}}$/$\varpi{}$ | rgeo | E(B-V) | RUWE | Remarks
---|---|---|---|---|---|---|---|---|---|---
|  | (deg) | (deg) | (mag) | (mas) | ($\%$) | (kpc) | (mag) |  |
BMP J0739$-$1418 | 3030005560828868096 | 114.96064 |  $-14.30718$ | 15.61 | $0.458\pm 0.042$ | 9 | $2.10_{-0.19}^{+0.18}$ | $0.258\pm 0.021$ | 1.041 |  |
FEGU 248$-$5 | 5594969135329315072 | 115.59902 |  $-32.79746$ | 17.00 | $0.528\pm 0.052$ | 10 | $1.90_{-0.16}^{+0.19}$ | 0.944aaReddening from SFD | 0.997 |  |
H1504+65 | 1645296216119116928 | 225.54006 | +66.20535 | 16.29 | $2.156\pm 0.051$ | 2 | $0.47_{-0.11}^{+0.12}$ | $0.0144\pm 0.0028$ | 1.050 |  |
HE 1429$-$1209 | 6324298665725984512 | 218.08641 |  $-12.38006$ | 16.01 | $0.441\pm 0.054$ | 12 | $2.16_{-0.22}^{+0.23}$ | $0.101575\pm 0.000099$ | 1.012 |  |
HS 0444+0453 | 3281864642080410112 | 071.76880 | +04.97804 | 16.23 | $2.271\pm 0.062$ | 3 | $0.441_{-0.014}^{+0.011}$ | $0.0473\pm 0.0033$ | 0.985 |  |
HS 0704+6153 | 1099093607199220096 | 107.38536 | +61.80533 | 16.98 | $1.643\pm 0.074$ | 5 | $0.615_{-0.030}^{+0.033}$ | $0.0413\pm 0.0050$ | 0.974 |  |
HS 1517+7403 | 1697669356564165632 | 229.19327 | +73.86865 | 16.63 | $1.319\pm 0.061$ | 5 | $0.781_{-0.037}^{+0.031}$ | $0.0278\pm 0.0024$ | 0.963 |  |
HS 2324+3944 | 1923253820774422272 | 351.81644 | +40.02323 | 14.77 | $0.702\pm 0.034$ | 5 | $1.400_{-0.054}^{+0.074}$ | $0.1343\pm 0.0013$ | 1.112 | hybrid |
MCT 0130$-$1937 | 5140121722033618560 | 023.16399 |  $-19.36138$ | 15.76 | $2.395\pm 0.066$ | 3 | $0.414_{-0.011}^{+0.010}$ | $0.0297\pm 0.0021$ | 1.283 |  |
NGC 246 | 2376592910265354368 | 011.76385 |  $-11.87198$ | 11.80 | $1.799\pm 0.079$ | 4 | $0.538_{-0.017}^{+0.020}$ | $0.04481\pm 0.00092$ | 1.530 | triple |
NGC 650 | 406328443354164480 | 025.58192 | +51.57541 | 17.42 | $0.294\pm 0.203$ | 69 | $3.7_{-1.5}^{+2.8}$ | $0.1431\pm 0.0073$ | 1.727 |  |
NGC 6852 | 4237745794618477440 | 300.16337 | +01.72801 | 17.91 | $0.39\pm 0.12$ | 30 | $3.0_{-0.9}^{+1.1}$ | $0.1083\pm 0.0042$ | 1.017 |  |
NGC 7094 | 1770058865674512896 | 324.22072 | +12.78859 | 13.52 | $0.604\pm 0.034$ | 6 | $1.607_{-0.076}^{+0.092}$ | $0.12600\pm 0.00046$ | 0.970 | hybrid |
PG 0122+200 | 2786529465445503488 | 021.34385 | +20.29910 | 16.75 | $1.641\pm 0.080$ | 5 | $0.618_{-0.032}^{+0.042}$ | $0.0396\pm 0.0018$ | 0.982 |  |
PG 1144+005 | 3795664157996369024 | 176.64674 | +00.20928 | 15.16 | $0.802\pm 0.058$ | 7 | $1.220_{-0.076}^{+0.085}$ | $0.02041\pm 0.00080$ | 1.088 |  |
PG 1151$-$029 | 3601781534594624000 | 178.56280 |  $-03.20143$ | 16.07 | $1.060\pm 0.063$ | 6 | $0.938_{-0.046}^{+0.060}$ | $0.0382\pm 0.0040$ | 1.044 |  |
PG 1159$-$035 | 3600841623951744640 | 180.44149 |  $-03.76130$ | 14.69 | $1.691\pm 0.064$ | 4 | $0.585_{-0.021}^{+0.020}$ | $0.0241\pm 0.0031$ | 1.129 |  |
PG 1424+535 | 1605381435770077312 | 216.48109 | +53.25704 | 15.88 | $1.771\pm 0.041$ | 2 | $0.566_{-0.011}^{+0.012}$ | $0.0126\pm 0.0016$ | 1.033 |  |
PG 1520+525 | 1595941441250636672 | 230.44399 | +52.36779 | 15.55 | $1.295\pm 0.041$ | 3 | $0.783_{-0.030}^{+0.027}$ | $0.0256\pm 0.0029$ | 1.045 |  |
PG 1707+427 | 1355161726346266112 | 257.19864 | +42.68358 | 16.65 | $1.402\pm 0.052$ | 4 | $0.733_{-0.026}^{+0.032}$ | $0.0477\pm 0.0012$ | 1.002 |  |
PN A66 21 | 3163546505053645056 | 112.26128 | +13.24679 | 15.93 | $1.689\pm 0.069$ | 4 | $0.584_{-0.021}^{+0.024}$ | $0.0318\pm 0.0013$ | 1.086 |  |
PN A66 43 | 4488953930631143168 | 268.38446 | +10.62340 | 14.66 | $0.458\pm 0.033$ | 7 | $2.09_{-0.11}^{+0.12}$ | $0.1946\pm 0.0087$ | 1.038 | hybrid |
PN A66 72 | 1761341417799128320 | 312.50856 | +13.55817 | 16.01 | $0.548\pm 0.064$ | 12 | $1.84_{-0.21}^{+0.18}$ | $0.06740\pm 0.00065$ | 1.042 |  |
PN IsWe 1 | 250358801943821952 | 057.27473 | +50.00410 | 16.47 | $2.350\pm 0.057$ | 2 | $0.424_{-0.009}^{+0.010}$ | $0.197\pm 0.045$ | 0.903 |  |
PN Jn 1 | 2871119705335735552 | 353.97219 | +30.46843 | 16.00 | $1.011\pm 0.065$ | 6 | $0.982_{-0.059}^{+0.071}$ | $0.0900\pm 0.0044$ | 1.120 |  |
PN K 1$-$16 | 2160562927224840576 | 275.46708 | +64.36482 | 14.98 | $0.589\pm 0.035$ | 6 | $1.737_{-0.092}^{+0.090}$ | $0.0388\pm 0.0035$ | 1.102 |  |
PN Kn 12 | 1823929193070538624 | 300.84391 | +21.59786 | 18.44 | $0.33\pm 0.17$ | 49 | $3.5_{-1.1}^{+1.5}$ | $0.2839\pm 0.0071$ | 0.961 |  |
PN Kn 61 | 2052811676760671872 | 290.41223 | +38.31588 | 18.25 | $0.14\pm 0.11$ | 80 | $5.9_{-1.8}^{+2.4}$ | $0.1327\pm 0.0056$ | 0.986 | binary? |
PN Kn 130 | 1941078175572093696 | 348.27200 | +45.43838 | 16.54 | $0.497\pm 0.054$ | 11 | $2.115_{-0.27}^{+0.27}$ | $0.1826\pm 0.0022$ | 1.051 |  |
PN Lo 3 | 5509004952576699904 | 108.70594 |  $-46.96087$ | 16.74 | $0.467\pm 0.074$ | 16 | $2.10_{-0.27}^{+0.30}$ | 0.172aaReddening from SFD | 1.787 |  |
PN Lo 4 | 5414927915911816704 | 151.44074 |  $-44.35931$ | 16.59 | $0.330\pm 0.052$ | 16 | $3.06_{-0.40}^{+0.60}$ | 0.147aaReddening from SFD | 1.039 |  |
PN Ou 2 | 430204780732841600 | 007.73643 | +61.40952 | 19.27 | $0.77\pm 0.22$ | 28 | $1.59_{-0.38}^{+0.82}$ | $0.3860\pm 0.0072$ | 1.009 |  |
PN VV 47 | 936605992140011392 | 119.46507 | +53.42137 | 17.06 | $1.065\pm 0.079$ | 7 | $0.985_{-0.076}^{+0.076}$ | $0.0371\pm 0.0050$ | 1.008 |  |
RL 104 | 180006683580428928 | 067.56196 | +40.40398 | 13.71 | $0.947\pm 0.021$ | 2 | $1.020_{-0.020}^{+0.025}$ | $0.3071\pm 0.0035$ | 0.964 |  |
RX J0122.9$-$7521 | 4637921057358156416 | 020.72372 |  $-75.35420$ | 15.38 | $1.196\pm 0.035$ | 3 | $0.830_{-0.024}^{+0.023}$ | 0.053aaReddening from SFD | 1.089 |  |
RX J2117.1+3412 | 1855295171732158080 | 319.28448 | +34.20766 | 13.02 | $1.991\pm 0.035$ | 2 | $0.4986_{-0.0094}^{+0.0082}$ | $0.0600\pm 0.0021$ | 0.948 |  |
SALT J172411.7$-$632147 | 5910236846008692352 | 261.04877 |  $-63.36322$ | 16.59 | $0.585\pm 0.063$ | 11 | $1.78_{-0.19}^{+0.18}$ | 0.065aaReddening from SFD | 0.936 |  |
SALT J213742.6$-$382901 | 6585736932806500736 | 324.42712 |  $-38.48355$ | 16.95 | $0.538\pm 0.087$ | 16 | $1.94_{-0.26}^{+0.36}$ | 0.036aaReddening from SFD | 1.061 |  |
SDSS J000945.46+135814.4 | 2767982864653184640 | 002.43941 | +13.97065 | 18.07 | $0.31\pm 0.17$ | 55 | $2.6_{-0.7}^{+1.3}$ | $0.0829\pm 0.0041$ | 0.989 |  |
SDSS J001651.42$-$011329.3 | 2541718902258404736 | 004.21425 |  $-01.22487$ | 16.75 | $0.273\pm 0.079$ | 29 | $3.36_{-0.60}^{+0.81}$ | $0.06729\pm 0.00039$ | 0.995 |  |
SDSS J034917.41$-$005919.3 | 3251245339191040256 | 057.32256 |  $-00.98874$ | 17.80 | $1.15\pm 0.12$ | 10 | $0.85_{-0.08}^{+0.12}$ | $0.1274\pm 0.0061$ | 1.048 |  |
SDSS J055905.02+633448.4 | 286746241613044096 | 089.77088 | +63.58012 | 18.59 | $0.98\pm 0.16$ | 16 | $1.06_{-0.14}^{+0.17}$ | $0.1606\pm 0.0025$ | 1.078 |  |
SDSS J075415.11+085232.1 | 3145662944130394496 | 118.56299 | +08.87560 | 19.08 | $0.57\pm 0.23$ | 39 | $1.8_{-0.5}^{+1.2}$ | $0.0321\pm 0.0065$ | 0.961 |  |
SDSS J075540.94+400918.0 | 920621124593362816 | 118.92053 | +40.15497 | 17.80 | $0.95\pm 0.13$ | 13 | $1.14_{-0.13}^{+0.19}$ | $0.0572\pm 0.0041$ | 0.968 |  |
SDSS J093546.53+110529.0 | 589674614326779136 | 143.94384 | +11.09133 | 17.75 | $1.10\pm 0.14$ | 13 | $0.96_{-0.10}^{+0.13}$ | $0.0411\pm 0.0030$ | 0.985 |  |
SDSS J102327.41+535258.7 | 851812381256776832 | 155.86423 | +53.88297 | 17.92 | $0.50\pm 0.11$ | 23 | $2.03_{-0.37}^{+0.47}$ | $0.0280\pm 0.0028$ | 0.977 |  |
SDSS J105300.24+174932.9 | 3982986781494206080 | 163.25103 | +17.82578 | 16.76 | $1.429\pm 0.076$ | 5 | $0.714_{-0.034}^{+0.040}$ | $0.0190\pm 0.0016$ | 0.987 |  |
SDSS J121523.09+120300.8 | 3908341899157118080 | 183.84614 | +12.05020 | 18.14 | $0.75\pm 0.14$ | 19 | $1.44_{-0.23}^{+0.35}$ | $0.03492\pm 0.00078$ | 0.928 |  |
SDSS J123930.61+244321.7 | 3959650269965155584 | 189.87752 | +24.72270 | 18.30 | $0.69\pm 0.16$ | 23 | $1.51_{-0.26}^{+0.37}$ | $0.0405\pm 0.0090$ | 0.975 |  |
SDSS J134341.88+670154.5 | 1672427588951276800 | 205.92436 | +67.03180 | 17.13 | $1.455\pm 0.055$ | 4 | $0.707_{-0.026}^{+0.023}$ | $0.0284\pm 0.0014$ | 0.937 |  |
SDSS J141556.26+061822.5 | 3673120627847661184 | 213.98441 | +06.30622 | 17.44 | $1.04\pm 0.13$ | 12 | $1.00_{-0.11}^{+0.14}$ | $0.0303\pm 0.0055$ | 1.102 |  |
SDSS J144734.12+572053.1 | 1613731019696686208 | 221.89206 | +57.34807 | 18.03 | $0.835\pm 0.092$ | 11 | $1.23_{-0.13}^{+0.21}$ | $0.0342\pm 0.0082$ | 1.028 |  |
SDSS J152116.00+251437.5 | 1270099761612163328 | 230.31665 | +25.24375 | 17.87 | $0.26\pm 0.11$ | 43 | $4.4_{-1.4}^{+2.1}$ | $0.0386\pm 0.0066$ | 1.058 | hybrid |
SDSS J155610.40+254640.3 | 1220049614357436544 | 239.04334 | +25.77784 | 17.91 | $0.086\pm 0.098$ | 115 | $7.5_{-3.0}^{+3.3}$ | $0.0630\pm 0.0077$ | 0.981 | binary? |
SDSS J163727.03+485355.2 | 1410694377877399552 | 249.36262 | +48.89866 | 18.35 | $0.57\pm 0.11$ | 19 | $2.01_{-0.36}^{+0.84}$ | $0.0261\pm 0.0055$ | 1.018 |  |
SDSS J191845.01+624343.7 | 2240494910007892608 | 289.68757 | +62.72883 | 17.58 | $0.970\pm 0.074$ | 8 | $1.111_{-0.079}^{+0.101}$ | $0.0262\pm 0.0020$ | 1.012 |  |
SDSS J212531.92$-$010745.8 | 2686081102494206080 | 321.38303 |  $-01.12941$ | 17.54 | $0.35\pm 0.11$ | 32 | $2.9_{-0.7}^{+1.3}$ | $0.03730\pm 0.00084$ | 1.034 | binary |
Sh 2$-$68 | 4276328581046447104 | 276.24337 | +00.85976 | 16.40 | $2.446\pm 0.059$ | 2 | $0.405_{-0.010}^{+0.010}$ | $0.622\pm 0.048$ | 1.088 | hybrid |
Sh 2$-$78 | 4506484097383382272 | 285.79198 | +14.11631 | 17.61 | $1.43\pm 0.10$ | 7 | $0.696_{-0.044}^{+0.059}$ | $0.3160\pm 0.0085$ | 1.045 |  |
TIC 95332541 | 2997192526074656640 | 090.68749 |  $-13.85096$ | 15.32 | $2.593\pm 0.043$ | 2 | $0.3845_{-0.0050}^{+0.0055}$ | $0.0575\pm 0.0049$ | 1.023 |  |
TIC 333432673 | 2950907725113997312 | 100.31517 |  $-13.69000$ | 15.21 | $2.552\pm 0.043$ | 2 | $0.3892_{-0.0054}^{+0.0054}$ | $0.119\pm 0.018$ | 1.093 |  |
TIC 403800675 | 3486203758501245440 | 179.36518 |  $-28.06384$ | 16.16 | $1.875\pm 0.062$ | 3 | $0.535_{-0.018}^{+0.019}$ | $0.0591\pm 0.0040$ | 1.004 |  |
TIC 1989122424 | 6462935326662402944 | 319.40996 |  $-55.46694$ | 16.75 | $1.471\pm 0.062$ | 4 | $0.688_{-0.026}^{+0.022}$ | 0.058aaReddening from SFD | 0.987 |  |
WD J070204.29+051420.56 | 3128765207057429504 | 105.51783 | +5.23904 | 14.98 | $3.089\pm 0.053$ | 2 | $0.3228_{-0.0057}^{+0.0053}$ | $0.0472\pm 0.0015$ | 1.091 |  |
NGC 6765 | 2039515046435901440 | 287.77732 | +30.54545 | 17.60 | $0.276\pm 0.078$ | 28 | $4.0_{-1.0}^{+1.5}$ | $0.1505\pm 0.0023$ | 1.002 |  |
PG 2131+066 | - | - | \-  | - | - | - | - | - | - | binary |
RX J0439.8$-$6809 | - | - | \-  | - | - | - | - | - | - |  |

Note. — Astrometric properties of PG 1159 stars from Gaia DR3 (Gaia
Collaboration et al., 2023). Geometric distances (rgeo) are from Bailer-Jones
et al. (2021), E(B-V) from Bayestar17 (Green et al., 2018) except for targets
with Dec. south of $-30^{\circ}$ where SFD maps (Schlafly & Finkbeiner, 2011;
Schlegel et al., 1998) were used. Uncertainties in E(B-V) are calculated as
half of the difference between values at 16th and 84th percentile. The last
three objects were excluded from the analysis because of the lack of either
Gaia measurements, or $T_{\mathrm{eff}}$ and $\log{g}$.

##  8 PG 1159 stars on the Hertzsprung-Russell diagram

PG 1159 stars plotted in the surface gravity-effective temperature diagram
$\log{g}-\log{}T_{\mathrm{eff}}$ (also called the Kiel diagram) cluster
horizontally along the lines of constant $\log{g}$, and vertically along the
lines of constant $\log{}T_{\mathrm{eff}}$ (see Figure 5). The reasons are the
current sensitivity of spectroscopic observations (large uncertainties, for
some PG 1159 stars even $\pm 0.5$ cm s-2 in $\log{g}$) and availability of
advanced model atmospheres for these extremely hot stars, with the latter
usually provided in grids with a step of $\log{g}=0.5$ cm s-2 and
$T_{\mathrm{eff}}=10000~{}\mathrm{K}$.

Figure 5: Positions of PG 1159 stars in the surface gravity-effective
temperature $\log{g}-\log{}T_{\mathrm{eff}}$ diagram. Star symbols represent
pulsating PG 1159 stars, circles – non-variable, and squares – with no
reported photometric observations. Stars with planetary nebulae are marked
with dashed circles. A typical error bar is shown in the bottom left corner.
Multiple stars overlap in this diagram, which can be seen as wider and darker
borders of symbols. See text for details. Lines represent evolutionary tracks
from Miller Bertolami & Althaus (2006): solid – VLTP (from left to right,
final masses: 0.870, 0.741, 0.664, 0.609, 0.584, 0.565, 0.542, 0.530, 0.515
M⊙), single dashed line – LTP (0.512 M⊙).

For the discussion in the context of asteroseismology it is useful to place
the PG 1159 stars into the theoretical Hertzsprung-Russell diagram
($\log{}L_{\star}/L_{\odot}-\log{}T_{\mathrm{eff}}$). However, this requires
the knowledge of stellar luminosities and effective temperatures. Derivation
of stellar luminosities is especially challenging, because it relies on
knowing the total bolometric flux of a given star. There are many ways to
tackle this difficult problem. One solution is based on determining the
Spectral Energy Distribution (SED) by fitting model atmospheres to broadband
photometric magnitudes (see, e.g., Uzundag et al. 2022). For such hot stars as
PG 1159s, the UV photometry and a grid of model atmospheres covering those
short wavelengths is essential and, to date, not available for the whole
sample of PG 1159 stars.

Another method is to derive the bolometric luminosities, either from
mathematical prescriptions, or from apparent magnitudes using bolometric
corrections (BCs). In the second case, the observed apparent magnitudes are
converted to absolute magnitudes in a given passband $b$ using a distance
modulus $DM$ (a logarithmic measure of the distance to the star):

$m_{b}=M_{b}+DM,$ (1)

where $m_{b}$ is the apparent magnitude and $M_{b}$ is the absolute magnitude
in the passband $b$. Incorporating the definition of the absolute magnitude
gives:

$DM=m_{b}-M_{b}=5\log{}_{10}\frac{d}{\mathrm{(10pc)}},$ (2)

where $d$ is distance in parsecs. With the correction for interstellar
absorption between the object and observer, the absolute magnitude in a
passband $b$ can be derived from:

$M_{b}=m_{b}-DM-A_{b},$ (3)

where $A_{b}$ is the extinction in a passband $b$. Then the bolometric
magnitude is

$M_{\mathrm{bol}}=M_{b}+BC_{b},$ (4)

where $M_{\mathrm{bol}}$ is absolute bolometric magnitude and $BC_{b}$ is
bolometric correction in a given passband, a quantity dependent not only on
the photometric passband used in observations, but also the theoretical
stellar spectrum used in calculation of the correction (different sets of
effective temperature, surface gravity, and metallicity will give different BC
values). In the case of extremely hot stars such as pre-white dwarfs of the PG
1159 type, this requires using models including non-local thermodynamic
equilibrium (non-LTE) effects. Finally, the absolute bolometric magnitude
$M_{\mathrm{bol}}$ of a star of a bolometric luminosity $L_{\star}$,
referenced to the Sun, is given by:

$-2.5\log_{10}\frac{L_{\star}}{L_{\odot}}=M_{\mathrm{bol}}-M_{\mathrm{bol},\odot},$
(5)

where $M_{\mathrm{bol},\odot}=4.74$ is the absolute bolometric magnitude of
the Sun, and $L_{\odot}=3.828\times 10^{33}\,\mathrm{erg}\,\mathrm{s}^{-1}$ is
the absolute bolometric luminosity of the Sun888IAU Resolution 2015 B2.

For white dwarf stars, the first commonly used/tabulated BC values were
compiled by Bergeron et al. (1995) for hydrogen- and helium-rich white dwarf
model atmospheres, but for a small grid covering surface gravity (only
$\log{g}=8$ cm s-2 ) and effective temperature (up to $100000\,\mathrm{K}$
only for DA white dwarfs). This work was expanded by Holberg & Bergeron
(2006), who provided an extensive grid for both DA and DB white dwarfs999The
DA grid covered $T_{\mathrm{eff}}=2500$ K to 150000 K and $\log{g}$ = 7.0 to
9.0 cm s-2 , while the DB grid covered $T_{\mathrm{eff}}=3250$ K to 150000 K
and $\log{g}$ = 7.0 to 9.0 cm s-2 .. The latter work is regularly updated on
line101010https://www.astro.umontreal.ca/b̃ergeron/CoolingModels/ . In the
most up-to-date version of the tables, models of Bédard et al. (2020), which
include non-LTE effects, are used at the highest effective temperatures.
Unfortunately, no bolometric corrections have ever been extensively compiled
for PG 1159 stars. Some PG 1159 stars had bolometric corrections estimated for
the purpose of deriving luminosities for asteroseismic modeling (Uzundag et
al. 2021 list three previously used values), but no tabulated prescription has
ever been provided.

We calculated the luminosities of PG 1159 stars based on currently available
data. We used the distances and interstellar reddening values described in
Section 7. The reddening for each star was converted to extinction using the
reddening law of Fitzpatrick (2004) with $R_{V}=3.1$. Gaia magnitudes were
converted to $V$ using the following prescription111111Gaia DR3 documentation.
We note that a few objects were slightly outside the range of applicability
for this relationship.:

$\displaystyle G-V$
$\displaystyle=-0.02704+0.01424(G_{\mathrm{BP}}-G_{\mathrm{RP}})$
$\displaystyle-0.2156(G_{\mathrm{BP}}-G_{\mathrm{RP}})^{2}$
$\displaystyle+0.01426(G_{\mathrm{BP}}-G_{\mathrm{RP}})^{3}.$

We used tabulated bolometric corrections for pure-helium model atmospheres
(DB) provided online on the aforementioned website by the Montreal group. As
the bolometric correction primarily depends on the effective temperature and
because there are no bolometric corrections computed with proper models for PG
1159 stars, we used those models as the best approach currently available. The
tabulated values do not cover surface gravities below $\log{g}=7.0$ cm s-2 ,
therefore the ones for $\log{g}=7.0$ cm s-2 were used for matching effective
temperatures. The values for effective temperatures over 150000 K were
extrapolated to higher effective temperatures for a given $\log{g}$. The
linear extrapolation was done in $\log{T_{\mathrm{eff}}}$ vs. BC space using
interp1d class from the scipy sub-package interpolate and
“fill_value=‘extrapolate’ ”, using the available BC values for
$T_{\mathrm{eff}}$ in the range $75000-150000$ K for a given $\log{g}$. Table
2 lists the physical properties and chosen BC values for each PG 1159 star in
the sample. Then, the luminosities were calculated following equations 1 – 5
and are also listed in Table 2 with uncertainties. The errors were propagated
the following way: a) for $DM$ using asymmetric errors from Table 3, b) for
E(B-V) using symmetric errors from Table 3, c) for $BC_{V}$ using asymmetric
errors adopted as the $BC_{V}$ values $\pm 10000$ K for each object, d) for
$G$, $G_{\mathrm{BP}}$, and $G_{\mathrm{RP}}$ magnitudes the symmetric errors
were calculated as $1.09\cdot G/\mathrm{SNR}$, where SNR is roughly
phot_g_mean_flux_over_error121212https://dc.zah.uni-
heidelberg.de/gaia/q3/cone/info#note-e (example for $G$).

Figure 6 shows positions of PG 1159 stars in the theoretical Hertzsprung-
Russell diagram ($\log{}L_{\star}/L_{\odot}-\log{}T_{\mathrm{eff}}$). For
illustration purposes, the blue dotted lines represent theoretical blue edges
for $l=1$ and $l=2$ modes from Gautschy et al. (2005), but the blue edge is
composition dependent – “fuzzy” (Quirion et al., 2007), and with the red
dotted lines we show the presently observed red edges.

Figure 6: Positions of PG 1159 stars in the theoretical Hertzsprung-Russell
diagram ($\log{}L_{\star}/L_{\odot}-\log{}T_{\mathrm{eff}}$). Star symbols
represent pulsating PG 1159 stars, circles – non-variable, and squares – with
no reported photometric observations. Stars with planetary nebulae are marked
with dashed circles. N-rich PG 1159 stars are shown with filled green symbols,
while N-poor ones with filled black symbols. Lines represent evolutionary
tracks from Miller Bertolami & Althaus (2006): solid – VLTP (from left to
right, final masses: 0.870, 0.741, 0.664, 0.609, 0.584, 0.565, 0.542, 0.530,
0.515 M⊙), single dashed line – LTP (0.512 M⊙). For illustration purposes, the
blue dotted lines represent theoretical blue edges for $l=1$ and $l=2$ modes
from Gautschy et al. (2005), but the blue edge is composition dependent. The
red dotted lines indicate estimated observed red edges, beyond which no GW Vir
star has to date been reported.

##  9 Discussion

The number of pulsating PG 1159 stars increased to 24 objects with our
discovery of pulsations in Abell 72. The main observational challenge in the
detection or confirmation of variability in those stars lies in two main
areas. Firstly, the amplitudes of the g-mode pulsations are quite low. While
Abell 72 showed pulsation amplitudes of up to 10 mmag, PG 1144+005, on the
other hand, showed a highly variable (between consecutive nights) Fourier
spectrum with amplitudes ranging from 3 to 6 mmag (Sowicka et al., 2021). This
requires reaching a noise level below 1 mmag for a significant detection
(assuming S/N$\geq$4), which is a challenging task for these faint stars. We
were not able to reach noise levels below 1.5 mmag for Longmore 3, SDSS
J000945.46+135814.4, SDSS J001651.42$-$011329.3, SDSS J102327.41+535258.7, and
SDSS J144734.12+572053.1. Another challenge is the aforementioned change in
amplitude spectra for some stars, between observing seasons or even
consecutive nights. It is therefore always possible that the star is observed
in a temporary state where the pulsations destructively interfere. We aimed at
obtaining more than one run for each star in the sample with a sufficient
quality, but this was only possible for eight stars.

Our results allowed us to update the fraction of PG 1159 pulsators in the GW
Vir instability strip. While previous works quoted values of about $20-50\%$,
but including not only PG 1159 stars but also the other stars populating the
GW Vir instability strip, we obtain 36% using only stars of PG 1159 spectral
type. Our fraction is consistent with previous estimates and shows that only
about 1/3 of PG 1159 stars within the GW Vir instability strip are observed to
vary.

In this context, it is interesting to see how the variability compares to the
nitrogen abundance observed in PG 1159 stars, in the light of the nitrogen
dichotomy (N-rich pulsators, N-poor nonpulsators, found by Dreizler & Heber
1998) that appears to hold. While the majority of those stars do not have a
determination of their atmospheric nitrogen abundance available in the
literature, there are a few stars that may not fit this hypothesis. The most
recent analysis of the pulsating central star of NGC 246 by Löbling (2018)
implies sub-solar N abundance. SALT J172411.7$-$632147 is a N-poor pulsator
reported by Jeffery et al. (2023). New spectra of TIC 95332541 analyzed by
Reindl et al. (2023) revealed that it is another N-poor pulsator. Longmore 4
is a known pulsator, and does not show N in a number of medium-resolution
spectra. It is interesting in the context of the outbursts that it exhibits,
temporarily changing its spectral type from PG 1159 to [WCE] (Werner et al.,
1992; Bond, 2014). RL 104 is also an interesting object, as it is N-rich and
claimed to have evolved from a binary merger scenario, but to date was not
observed photometrically.

With such a sample tested for variability, we placed the PG 1159 stars in the
theoretical Hertzsprung-Russell diagram. We determined luminosities following
the procedure described in Sect. 8. We plotted them against available
evolutionary tracks for PG 1159 stars. In general, very good agreement between
the evolutionary tracks and positions of PG 1159 stars was obtained. The
majority of the stars are within the evolutionary tracks for typical PG 1159
masses ($0.5-0.6$ M⊙). Only one star is found beyond 0.87 M⊙ – H1504+65.
Nevertheless, a few shortcomings of our attempt need to be noted. The
distances from Gaia for some stars have large uncertainties due to large
relative errors of parallaxes. In Table 3 we marked 14 stars whose relative
parallax errors exceed 20%. Four of them are confirmed or suspected binaries,
therefore their determined positions might be uncertain. It is worth comparing
the distances determined using different (independent) methods, e.g., using
planetary nebulae line strengths, but this is out of the scope of this work.
Uzundag et al. (2021) quoted available in the literature values of bolometric
correction for PG 1159 stars for three objects: PG 1159$-$035
(T${}_{\mathrm{eff}}=140000$ K, $\log{g}=7.0$ cm s-2 ): $BC=-7.6$, RX
J2117+3142 (T${}_{\mathrm{eff}}=170000$ K, $\log{g}=6.0$ cm s-2 ): $BC=-7.95$,
and PG 2131+066 (T${}_{\mathrm{eff}}=95000$ K, $\log{g}=7.5$ cm s-2 ):
$BC=-6.0$. They interpolated those values to obtain $BC=-7.05$ for TIC
95332541 and TIC 333432673 (T${}_{\mathrm{eff}}=120000$, $\log{g}=7.5$),
assuming only the dependence on the effective temperature. We investigated the
difference between tabulated BC for DA and DB models. For Teff and $\log{g}$
of PG 1159$-$035 (the only exact match with tabulated values), we found BC of
$-7.964$ and $-7.658$ for DA and DB models, respectively. The value for the DB
model atmosphere agrees well with the quoted value of $BC=-7.6$. We also
checked how the BC value from DB table changes with $\log{g}$ for a given
temperature. For T${}_{\mathrm{eff}}=140000$ K and $\log{g}=7.0,7.5,8.0,8.5$
cm s-2 we obtained $BC=-7.658,-7.668,-7.676,-7.681$, respectively. Therefore,
we do not expect significant interpolation errors in the parameter space of
interest.

##  10 One or two GW Vir instability domains?

The establishment of the PG 1159 spectral class (e.g., see Werner 1992)
occurred subsequently to the discovery of pulsations in PG 1159$-$035 (McGraw
et al., 1979) itself. At the time when the pulsating PG 1159 stars emerged as
a new group of pulsators (e.g., Bond et al. 1984), they were considered the
hottest subgroup of the helium-rich DO white dwarf stars (Wesemael et al.,
1985). For that reason, and for the similarity with the designations of the
groups of pulsating white dwarfs already known (DAV and DBV) the PG 1159
pulsators were dubbed “the DOVs”.

However, the second pulsating star of the PG 1159 spectral type discovered was
located in a planetary nebula (Grauer & Bond, 1984) and subsequent searches
(e.g., Ciardullo & Bond 1996) revealed several of these “Planetary Nebula
Nucleus Variables” (PNNVs). Even though it was realized that the “DOVs” were
likely just the same type of pulsating stars, but in a more advanced
evolutionary stage than the “PNNVs”, the two groups were historically often
separated. The main reason for this separation was that one group of pulsators
are surrounded by nebulae whereas the others were not and that one group has
significantly longer pulsation periods than the other. Furthermore,
theoretical computations (e.g., Córsico et al. 2006, cf. Fig. 6) show that the
blue edge of the instability strip intersects with the evolutionary tracks of
pre-white dwarf stars in such a way that many of them leave the strip during
their evolution and later re-enter it, giving the impression of two separated
instability regions.

Quirion et al. (2007) and Fontaine & Brassard (2008) argued, mostly on a
theoretical basis, that this separation should be dropped, and that all hot
pulsating pre-white dwarf stars should be called “the GW Vir stars”131313GW
Vir is the variable star designation of PG 1159$-$035, Kholopov et al.
(1985).. This was motivated by the fact that the pulsational driving mechanism
of all GW Vir stars is the same, that stars with a pure DO spectral type are
not known to pulsate141414The PG 1159 spectral class had meanwhile been
established as a separate group, and we recall that some pulsating pre-white
dwarfs are of [WCE] or [WCE/PG 1159] spectral types., and that not all stars
classified as “PNNV” even possess a detected planetary nebula. To this it can
be added that there are other intrinsically variable central stars of
planetary nebulae (e.g., Handler et al. 2013) that do not pulsate at all,
which is why a designation “PNNV” is equally misleading as is “DOV”.

Schoenberner & Napiwotzki (1990) showed that some PNNs are spectroscopically
indistinguishable from the white dwarfs similar to PG 1159$-$035, and assigned
them all to “PG 1159” type. PG 1159 subclasses introduced by Werner (1992) did
not take into account the presence or absence of a nebula, treating the PG
1159 spectral class as a whole. Therefore, not only is the pulsation driving
mechanism the same for those stars, but they also share some spectroscopic
properties representative of the whole class.

Moreover, the commonly used surface gravity-effective temperature diagram
presented in Fig. 5 shows that it is impossible to separate the two groups in
the $\log{g}-\log{}T_{\mathrm{eff}}$ plane – stars with planetary nebulae are
found throughout the whole GW Vir instability strip. This refutes the argument
that the PNNVs usually have much lower surface gravities, as no strict
boundary can be placed in such a plane.

In the following, we examine the question whether these two groups are
distinct, or should be distinguished, from an observational point of view. The
top panel in Figure 7 shows the ranges of pulsation periods observed in
pulsating pre-white dwarf stars (of PG 1159 spectral type) versus stellar
radius (as derived from Fig. 6 and the Stefan-Boltzmann law).

Figure 7: Top: The period ranges of pulsating pre-white dwarf stars (only of
PG 1159 spectral type) versus stellar radius (horizontal bars). Bottom: The
period ranges of pulsating pre-white dwarf stars (only of PG 1159 spectral
type) versus pulsation constant (horizontal bars). Objects surrounded by a
planetary nebula are denoted with grey-dotted bars. The object marked with an
ellipse is RX J0122.9$-$7521 (see Sect. 10.1).

Several things are noteworthy in Fig. 7. First of all, there is a clear
overlap between the objects with and without a planetary nebula, already
suggesting these two groups are not distinct. Second, an obvious trend as
already noticed by others earlier is visible, namely that the larger, less
evolved objects have longer pulsation periods.

To look into this in some more detail, the bottom panel in Fig. 7 compares the
pulsation periods with the pulsation constant
$Q=P\sqrt{\rho_{\ast}/\rho_{\odot}}$, a measure of the radial overtone of the
excited pulsation modes. There is a slight trend that the longer the pulsation
period, the smaller is the pulsation constant. In other words, the more
evolved a pulsating pre-white dwarf, the higher radial overtones of the
gravity modes are excited. This is consistent with the theoretical expectation
that with progressing evolution the pulsational driving region becomes located
closer to the stellar surface (see Gautschy et al. 2005).

We therefore conclude that the distinction between “DOVs” and “PNNVs” is,
according to current knowledge, artificial and based on selection effects and
hence should not be used. All pulsating pre-white dwarf stars oscillating in
gravity modes excited by the $\kappa-\gamma$ mechanism due to ionization of
carbon and oxygen should henceforth be called “GW Vir stars”.

###  10.1 The case of RX J0122.9$-$7521

In Sect. 5.2 we reported the detection of variability of RX J0122.9$-$7521 and
mentioned that it would be the hottest GW Vir pulsator. However, we are
reluctant to claim the firm detection of pulsation for this star, for several
reasons.

Although its 41-min period fits in the range of pulsation periods observed in
GW Vir stars (Althaus et al., 2010), we detected only a single period that
could therefore in principle be of a different origin, like rotation,
binarity, or spots (Reindl et al., 2021). Furthermore, in Fig. 7, the period
of this star is rather long with respect to stars with similar radii.

RX J0122.9$-$7521 lies outside the theoretical blue edge of GW Vir instability
strip, and Quirion et al. (2004) unsurprisingly did not find an asteroseismic
model with unstable periods in this star. Werner (1995) reported the detection
of nitrogen in its spectrum.

In any case, time-resolved spectroscopy or high signal-to-noise photometry
would be needed to establish the cause of the variability of RX
J0122.9$-$7521.

##  11 Summary and conclusions

We obtained new photometric observations of 29 PG 1159 stars. Over 86 hours of
time-series photometry were collected in the years $2014-2022$ using
telescopes of different sizes, ranging from 1.0-m to 10.4-m, and located in
both hemispheres. For the majority of stars we achieved a median noise level
in Fourier amplitude spectra in the range $0.3-1.0$ mmag, which allowed us to
discover multiperiodic pulsations in the central star of planetary nebula
Abell 72, and variability in RX J0122.9$-$7521 that could be due to
pulsations, binarity or rotation. Five stars showed interesting peaks but
require follow-up observations for confirmation. For the remaining stars our
observations put limits on nonvariability. As a result, we derived the
fraction of pulsating PG 1159 stars within the GW Vir instability strip – 36%.

In the light of N dichotomy in PG 1159 stars, we compared the new variability
results with the literature data on N abundances for those stars and
identified objects that could be culprits for this hypothesis: NGC 246, SALT
J172411.7$-$632147 and TIC 95332541 may be N-poor pulsators. Longmore 4 is
probably a N-poor pulsator, but temporarily changes its spectral type from PG
1159 to [WCE] during outbursts.

Taking advantage of the currently available data, we used distances derived
from $Gaia$ parallaxes, interstellar extinction from 3D reddening maps, and
bolometric correction values from DB tables, to derive luminosities and place
the PG 1159 stars in the theoretical Hertzsprung-Russell diagram. Regardless
of the possible caveats of our approach, all stars align well with the PG 1159
evolutionary tracks from Miller Bertolami & Althaus (2006).

Finally, we derived radii and pulsation constants for known pulsators, and
plotted them against period ranges observed in those stars to further argue
against the distinction between “DOVs” and “PNNVs”, and suggested using only
the “GW Vir” designation for all stars belonging to that family of pulsating
white dwarfs.

##  12 Author contributions

PS with GH and DJ applied for observing time. PS, GH, DJ, JC, FvW, EP, KB, LP,
LSA, and MK observed the targets. DLH extracted single FITS files from data
cubes of the SA19+SHOC run. PS did the data reduction for all targets except
those observed with DK+DFOSC (EP reduced those data). PS also did photometry
and frequency analysis, as well as compiled astrometric parameters of PG 1159
stars and bolometric corrections, derived luminosities, placed the sample in
the HR diagrams, and computed radii and pulsation constants. GH and DJ
supervised the work. KW provided parts of the data included in Table 2. PS
wrote the text with contribution from GH and feedback from co-authors.

We thank Philip Short, Nicholas Humphries, and Martha Tabor who contributed to
the observations. This research was supported in part by the National Science
Foundation under Grant No. NSF PHY-1748958 and by the Polish National Center
for Science (NCN) through grants 2015/18/A/ST9/00578 and 2021/43/B/ST9/02972.
MK acknowledges the support from ESA-PRODEX PEA4000127913. This paper uses
observations made at the South African Astronomical Observatory (SAAO). Based
on observations made with the Gran Telescopio Canarias (GTC), installed in the
Spanish Observatorio del Roque de los Muchachos of the Instituto de
Astrofísica de Canarias, in the island of La Palma. Based on observations with
the Isaac Newton Telescope operated by the Isaac Newton Group at the
Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de
Canarias on the island of La Palma, Spain. This paper includes data taken at
The McDonald Observatory of The University of Texas at Austin. Data were
obtained (in part) using the 1.3 m McGraw-Hill Telescope of the MDM
Observatory. 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.

## References

  * Althaus et al. (2010) Althaus, L. G., Córsico, A. H., Isern, J., & García-Berro, E. 2010, A&A Rev., 18, 471, doi: 10.1007/s00159-010-0033-1
  * Andersen et al. (1995) Andersen, J., Andersen, M. I., Klougart, J., et al. 1995, The Messenger, 79, 12
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Astropy Collaboration et al. (2022) Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167, doi: 10.3847/1538-4357/ac7c74
  * Bailer-Jones et al. (2021) Bailer-Jones, C. A. L., Rybizki, J., Fouesneau, M., Demleitner, M., & Andrae, R. 2021, AJ, 161, 147, doi: 10.3847/1538-3881/abd806
  * Bédard et al. (2020) Bédard, A., Bergeron, P., Brassard, P., & Fontaine, G. 2020, ApJ, 901, 93, doi: 10.3847/1538-4357/abafbe
  * Bergeron et al. (1995) Bergeron, P., Wesemael, F., & Beauchamp, A. 1995, PASP, 107, 1047, doi: 10.1086/133661
  * Bond (2014) Bond, H. E. 2014, AJ, 148, 44, doi: 10.1088/0004-6256/148/3/44
  * Bond et al. (1984) Bond, H. E., Grauer, A. D., Green, R. F., & Liebert, J. W. 1984, ApJ, 279, 751, doi: 10.1086/161943
  * Bond & Meakes (1990) Bond, H. E., & Meakes, M. G. 1990, AJ, 100, 788, doi: 10.1086/115560
  * Bond et al. (2023) Bond, H. E., Werner, K., Jacoby, G. H., & Zeimann, G. R. 2023, MNRAS, 521, 668, doi: 10.1093/mnras/stad524
  * Breger et al. (1993) Breger, M., Stich, J., Garrido, R., et al. 1993, A&A, 271, 482
  * Caldwell et al. (1993) Caldwell, J. A. R., Cousins, A. W. J., Ahlers, C. C., van Wamelen, P., & Maritz, E. J. 1993, South African Astronomical Observatory Circular, 15, 1
  * Cepa (1998) Cepa, J. 1998, Ap&SS, 263, 369, doi: 10.1023/A:1002144913887
  * Ciardullo & Bond (1996) Ciardullo, R., & Bond, H. E. 1996, AJ, 111, 2332, doi: 10.1086/117967
  * Coppejans et al. (2013) Coppejans, R., Gulbis, A. A. S., Kotze, M. M., et al. 2013, PASP, 125, 976, doi: 10.1086/672156
  * Córsico et al. (2006) Córsico, A. H., Althaus, L. G., & Miller Bertolami, M. M. 2006, A&A, 458, 259, doi: 10.1051/0004-6361:20065423
  * Córsico et al. (2009) Córsico, A. H., Althaus, L. G., Miller Bertolami, M. M., González Pérez, J. M., & Kepler, S. O. 2009, ApJ, 701, 1008, doi: 10.1088/0004-637X/701/2/1008
  * Córsico et al. (2019) Córsico, A. H., Althaus, L. G., Miller Bertolami, M. M., & Kepler, S. O. 2019, A&A Rev., 27, 7, doi: 10.1007/s00159-019-0118-4
  * Córsico et al. (2021) Córsico, A. H., Uzundag, M., Kepler, S. O., et al. 2021, A&A, 645, A117, doi: 10.1051/0004-6361/202039202
  * Costa et al. (2008) Costa, J. E. S., Kepler, S. O., Winget, D. E., et al. 2008, A&A, 477, 627, doi: 10.1051/0004-6361:20053470
  * Craig et al. (2017) Craig, M., Crawford, S., Seifert, M., et al. 2017, astropy/ccdproc: v1.3.0.post1, v1.3.0.post1, Zenodo, doi: 10.5281/zenodo.1069648
  * Crowther et al. (1998) Crowther, P. A., De Marco, O., & Barlow, M. J. 1998, MNRAS, 296, 367, doi: 10.1046/j.1365-8711.1998.01360.x
  * De Marco et al. (2015) De Marco, O., Long, J., Jacoby, G. H., et al. 2015, MNRAS, 448, 3587, doi: 10.1093/mnras/stv249
  * Dreizler (1999) Dreizler, S. 1999, Reviews in Modern Astronomy, 12, 255
  * Dreizler & Heber (1998) Dreizler, S., & Heber, U. 1998, A&A, 334, 618
  * Fitzpatrick (2004) Fitzpatrick, E. L. 2004, in Astronomical Society of the Pacific Conference Series, Vol. 309, Astrophysics of Dust, ed. A. N. Witt, G. C. Clayton, & B. T. Draine, 33, doi: 10.48550/arXiv.astro-ph/0401344
  * Fontaine & Brassard (2008) Fontaine, G., & Brassard, P. 2008, PASP, 120, 1043, doi: 10.1086/592788
  * Friederich et al. (2010) Friederich, F., Rauch, T., Werner, K., Koesterke, L., & Kruk, J. W. 2010, in American Institute of Physics Conference Series, Vol. 1273, 17th European White Dwarf Workshop, ed. K. Werner & T. Rauch, 231–234, doi: 10.1063/1.3527810
  * Fu et al. (2007) Fu, J. N., Vauclair, G., Solheim, J. E., et al. 2007, A&A, 467, 237, doi: 10.1051/0004-6361:20066295
  * Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1, doi: 10.1051/0004-6361/201629272
  * Gaia Collaboration et al. (2023) Gaia Collaboration, Vallenari, A., Brown, A. G. A., et al. 2023, A&A, 674, A1, doi: 10.1051/0004-6361/202243940
  * Gautschy et al. (2005) Gautschy, A., Althaus, L. G., & Saio, H. 2005, A&A, 438, 1013, doi: 10.1051/0004-6361:20042486
  * Gianninas et al. (2010) Gianninas, A., Bergeron, P., Dupuis, J., & Ruiz, M. T. 2010, ApJ, 720, 581, doi: 10.1088/0004-637X/720/1/581
  * González Pérez et al. (2006) González Pérez, J. M., Solheim, J. E., & Kamben, R. 2006, A&A, 454, 527, doi: 10.1051/0004-6361:20053468
  * Grauer & Bond (1984) Grauer, A. D., & Bond, H. E. 1984, ApJ, 277, 211, doi: 10.1086/161685
  * Grauer et al. (1987a) Grauer, A. D., Bond, H. E., Green, R. F., & Liebert, J. 1987a, in IAU Colloq. 95: Second Conference on Faint Blue Stars, ed. A. G. D. Philip, D. S. Hayes, & J. W. Liebert, 231–236
  * Grauer et al. (1987b) Grauer, A. D., Bond, H. E., Liebert, J., Fleming, T. A., & Green, R. F. 1987b, ApJ, 323, 271, doi: 10.1086/165824
  * Green (2018) Green, G. 2018, The Journal of Open Source Software, 3, 695, doi: 10.21105/joss.00695
  * Green et al. (2019) Green, G. M., Schlafly, E., Zucker, C., Speagle, J. S., & Finkbeiner, D. 2019, ApJ, 887, 93, doi: 10.3847/1538-4357/ab5362
  * Green et al. (2018) Green, G. M., Schlafly, E. F., Finkbeiner, D., et al. 2018, MNRAS, 478, 651, doi: 10.1093/mnras/sty1008
  * Green & Liebert (1979) Green, R. F., & Liebert, J. W. 1979, in IAU Colloq. 53: White Dwarfs and Variable Degenerate Stars, ed. H. M. van Horn, V. Weidemann, & M. P. Savedoff, 118
  * Handler et al. (2013) Handler, G., Prinja, R. K., Urbaneja, M. A., et al. 2013, MNRAS, 430, 2923, doi: 10.1093/mnras/stt092
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2
  * Holberg & Bergeron (2006) Holberg, J. B., & Bergeron, P. 2006, AJ, 132, 1221, doi: 10.1086/505938
  * Hoyer et al. (2018) Hoyer, D., Rauch, T., Werner, K., & Kruk, J. W. 2018, A&A, 612, A62, doi: 10.1051/0004-6361/201732401
  * Hügelmeyer et al. (2006) Hügelmeyer, S. D., Dreizler, S., Homeier, D., et al. 2006, A&A, 454, 617, doi: 10.1051/0004-6361:20064869
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
  * Jeffery et al. (2023) Jeffery, C. S., Werner, K., Kilkenny, D., et al. 2023, MNRAS, 519, 2321, doi: 10.1093/mnras/stac3531
  * Kawaler et al. (1995) Kawaler, S. D., O’Brien, M. S., Clemens, J. C., et al. 1995, ApJ, 450, 350, doi: 10.1086/176145
  * Kawaler et al. (2004) Kawaler, S. D., Potter, E. M., Vučković, M., et al. 2004, A&A, 428, 969, doi: 10.1051/0004-6361:20041475
  * Kepler et al. (2014) Kepler, S. O., Fraga, L., Winget, D. E., et al. 2014, MNRAS, 442, 2278, doi: 10.1093/mnras/stu1019
  * Kepler et al. (2016) Kepler, S. O., Pelisoli, I., Koester, D., et al. 2016, MNRAS, 455, 3413, doi: 10.1093/mnras/stv2526
  * Kholopov et al. (1985) Kholopov, P. N., Samus, N. N., Kazarovets, E. V., & Perova, N. B. 1985, Information Bulletin on Variable Stars, 2681, 1
  * Lenz & Breger (2005) Lenz, P., & Breger, M. 2005, Communications in Asteroseismology, 146, 53, doi: 10.1553/cia146s53
  * Leuenhagen et al. (1993) Leuenhagen, U., Koesterke, L., & Hamann, W. R. 1993, Acta Astron., 43, 329
  * Löbling (2018) Löbling, L. 2018, Galaxies, 6, 65, doi: 10.3390/galaxies6020065
  * Loumos & Deeming (1978) Loumos, G. L., & Deeming, T. J. 1978, Ap&SS, 56, 285, doi: 10.1007/BF01879560
  * McGraw et al. (1979) McGraw, J. T., Starrfield, S. G., Liebert, J., & Green, R. 1979, in IAU Colloq. 53: White Dwarfs and Variable Degenerate Stars, ed. H. M. van Horn, V. Weidemann, & M. P. Savedoff, 377
  * Miller Bertolami & Althaus (2006) Miller Bertolami, M. M., & Althaus, L. G. 2006, A&A, 454, 845, doi: 10.1051/0004-6361:20054723
  * Miller Bertolami et al. (2022) Miller Bertolami, M. M., Battich, T., Córsico, A. H., Althaus, L. G., & Wachlin, F. C. 2022, MNRAS, 511, L60, doi: 10.1093/mnrasl/slab134
  * Nagel & Werner (2004) Nagel, T., & Werner, K. 2004, A&A, 426, L45, doi: 10.1051/0004-6361:200400079
  * Napiwotzki & Schoenberner (1995) Napiwotzki, R., & Schoenberner, D. 1995, A&A, 301, 545
  * Oliveira da Rosa et al. (2022) Oliveira da Rosa, G., Kepler, S. O., Córsico, A. H., et al. 2022, ApJ, 936, 187, doi: 10.3847/1538-4357/ac8871
  * pandas development team (2020) pandas development team, T. 2020, pandas-dev/pandas: Pandas, latest, Zenodo, doi: 10.5281/zenodo.3509134
  * Quirion et al. (2004) Quirion, P. O., Fontaine, G., & Brassard, P. 2004, ApJ, 610, 436, doi: 10.1086/421447
  * Quirion et al. (2007) —. 2007, ApJS, 171, 219, doi: 10.1086/513870
  * Rauch & Werner (1995) Rauch, T., & Werner, K. 1995, in White Dwarfs, ed. D. Koester & K. Werner, Vol. 443, 186, doi: 10.1007/3-540-59157-5_202
  * Reindl et al. (2016) Reindl, N., Geier, S., Kupfer, T., et al. 2016, A&A, 587, A101, doi: 10.1051/0004-6361/201527637
  * Reindl et al. (2014) Reindl, N., Rauch, T., Werner, K., Kruk, J. W., & Todt, H. 2014, A&A, 566, A116, doi: 10.1051/0004-6361/201423498
  * Reindl et al. (2021) Reindl, N., Schaffenroth, V., Filiz, S., et al. 2021, A&A, 647, A184, doi: 10.1051/0004-6361/202140289
  * Reindl et al. (2023) Reindl, N., Islami, R., Werner, K., et al. 2023, arXiv e-prints, arXiv:2307.03721, doi: 10.48550/arXiv.2307.03721
  * Ricker et al. (2015) Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003, doi: 10.1117/1.JATIS.1.1.014003
  * Schlafly & Finkbeiner (2011) Schlafly, E. F., & Finkbeiner, D. P. 2011, ApJ, 737, 103, doi: 10.1088/0004-637X/737/2/103
  * Schlegel et al. (1998) Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525, doi: 10.1086/305772
  * Schoenberner & Napiwotzki (1990) Schoenberner, D., & Napiwotzki, R. 1990, A&A, 231, L33
  * Silvotti et al. (1999) Silvotti, R., Dreizler, S., Handler, G., & Jiang, X. J. 1999, A&A, 342, 745, doi: 10.48550/arXiv.astro-ph/9811178
  * Solheim et al. (2007) Solheim, J. E., Vauclair, G., Mukadam, A. S., Janulis, R., & Dobrovolskas, V. 2007, A&A, 468, 1057, doi: 10.1051/0004-6361:20077364
  * Sowicka et al. (2018) Sowicka, P., Handler, G., & Jones, D. 2018, MNRAS, 479, 2476, doi: 10.1093/mnras/sty1660
  * Sowicka et al. (2021) Sowicka, P., Handler, G., Jones, D., & van Wyk, F. 2021, ApJ, 918, L1, doi: 10.3847/2041-8213/ac1c08
  * Toalá et al. (2015) Toalá, J. A., Guerrero, M. A., Todt, H., et al. 2015, ApJ, 799, 67, doi: 10.1088/0004-637X/799/1/67
  * Uzundag et al. (2022) Uzundag, M., Córsico, A. H., Kepler, S. O., et al. 2022, MNRAS, 513, 2285, doi: 10.1093/mnras/stac1027
  * Uzundag et al. (2021) —. 2021, A&A, 655, A27, doi: 10.1051/0004-6361/202141253
  * Vauclair et al. (2005) Vauclair, G., Solheim, J. E., & Østensen, R. H. 2005, A&A, 433, 1097, doi: 10.1051/0004-6361:20041999
  * Vauclair et al. (2002) Vauclair, G., Moskalik, P., Pfeiffer, B., et al. 2002, A&A, 381, 122, doi: 10.1051/0004-6361:20011483
  * Vauclair et al. (2011) Vauclair, G., Fu, J. N., Solheim, J. E., et al. 2011, A&A, 528, A5, doi: 10.1051/0004-6361/201014457
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2
  * Walton et al. (2001) Walton, N. A., Lennon, D. J., Greimel, R., et al. 2001, The Newsletter of the Isaac Newton Group of Telescopes, 4, 7
  * Weidmann et al. (2023) Weidmann, W. A., Werner, K., Ahumada, J. A., Pignata, R. A., & Firpo, V. 2023, A&A, 676, A1, doi: 10.1051/0004-6361/202346401
  * Werner (1992) Werner, K. 1992, in The Atmospheres of Early-Type Stars, ed. U. Heber & C. S. Jeffery, Vol. 401, 273, doi: 10.1007/3-540-55256-1_321
  * Werner (1995) —. 1995, Baltic Astronomy, 4, 340, doi: 10.1515/astro-1995-0309
  * Werner (2001) —. 2001, Ap&SS, 275, 27
  * Werner & Drake (2005) Werner, K., & Drake, J. J. 2005, A&A, 434, 707, doi: 10.1051/0004-6361:20042057
  * Werner et al. (1992) Werner, K., Hamann, W. R., Heber, U., et al. 1992, A&A, 259, L69
  * Werner & Herwig (2006) Werner, K., & Herwig, F. 2006, PASP, 118, 183, doi: 10.1086/500443
  * Werner & Rauch (2014) Werner, K., & Rauch, T. 2014, A&A, 569, A99, doi: 10.1051/0004-6361/201424051
  * Werner & Rauch (2015) —. 2015, A&A, 584, A19, doi: 10.1051/0004-6361/201527261
  * Werner et al. (2004a) Werner, K., Rauch, T., Barstow, M. A., & Kruk, J. W. 2004a, A&A, 421, 1169, doi: 10.1051/0004-6361:20047154
  * Werner et al. (2014) Werner, K., Rauch, T., & Kepler, S. O. 2014, A&A, 564, A53, doi: 10.1051/0004-6361/201423441
  * Werner et al. (2005) Werner, K., Rauch, T., & Kruk, J. W. 2005, A&A, 433, 641, doi: 10.1051/0004-6361:20042258
  * Werner et al. (2010) —. 2010, ApJ, 719, L32, doi: 10.1088/2041-8205/719/1/L32
  * Werner et al. (2015) —. 2015, A&A, 582, A94, doi: 10.1051/0004-6361/201526842
  * Werner et al. (2016) —. 2016, A&A, 593, A104, doi: 10.1051/0004-6361/201628892
  * Werner et al. (2004b) Werner, K., Rauch, T., Napiwotzki, R., et al. 2004b, A&A, 424, 657, doi: 10.1051/0004-6361:20041157
  * Werner et al. (2004c) Werner, K., Rauch, T., Reiff, E., Kruk, J. W., & Napiwotzki, R. 2004c, A&A, 427, 685, doi: 10.1051/0004-6361:20041165
  * Werner et al. (2022a) Werner, K., Reindl, N., Dorsch, M., et al. 2022a, A&A, 658, A66, doi: 10.1051/0004-6361/202142397
  * Werner et al. (2022b) Werner, K., Reindl, N., Geier, S., & Pritzkuleit, M. 2022b, MNRAS, 511, L66, doi: 10.1093/mnrasl/slac005
  * Wes McKinney (2010) Wes McKinney. 2010, in Proceedings of the 9th Python in Science Conference, ed. Stéfan van der Walt & Jarrod Millman, 56 – 61, doi: 10.25080/Majora-92bf1922-00a
  * Wesemael et al. (1985) Wesemael, F., Green, R. F., & Liebert, J. 1985, ApJS, 58, 379, doi: 10.1086/191046
  * Woudt et al. (2012) Woudt, P. A., Warner, B., & Zietsman, E. 2012, MNRAS, 426, 2137, doi: 10.1111/j.1365-2966.2012.21899.x