# Pulsating subdwarf B stars observed with K2 during Campaign 7 and an
examination of seismic group-properties.

M. D. Reed1,2, A. Slayton1, A. S. Baran1,2,3, J. H. Telting2,4,5, R. H.
Østensen1,2,6, C. S Jeffery7, M. Uzundag8,9, S. Sanjayan2,10
1Department of Physics, Astronomy and Materials Science, Missouri State
University, 901 S. National, Springfield, MO 65897, USA
2ARDASTELLA Research Group, Institute of Physics, Pedagogical University of
Cracow, ul. Podchora̧ży ch 2,30-084 Kraków, Poland
3 Embry-Riddle Aeronautical University, Department of Physical Science,
Daytona Beach, FL 32114, USA
4Nordic Optical Telescope, Rambla José Ana Fernández Pérez 7, 38711 Breña
Baja, Spain
5 Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120,
DK-8000 Aarhus C, Denmark
6Recogito AS, Storgaten 72, N-8200 Fauske, Norway
7Armagh Observatory and Planetarium, College Hill, Armagh BT61 9DG, N. Ireland
8 Instituto de Física y Astronomía, Universidad de Valparaíso, Gran Bretaña
1111, Playa Ancha, Valparaíso 2360102, Chile
9European Southern Observatory, Alonso de Cordova 3107, Santiago, Chile
10Centrum Astronomiczne im. Mikołaja Kopernika, Polskiej Akademii Nauk, ul.
Bartycka 18, 00-716 Warszawa, Polska<EMAIL_ADDRESS>
(Accepted Received )

###### Abstract

We report the discovery of four new pulsating subdwarf B (sdBV) stars from
Campaign 7 of the Kepler spacecraft’s K2 mission. EPICs 215776487, 217280630,
218366972, and 218717602 are all gravity (g)-mode pulsators and we also detect
two pressure (p)-mode pulsations in EPIC 218717602. We detect asymptotic
$\ell\,=\,1$ sequences in all four stars, allowing us to identify nearly all
of the g modes. We detect evenly-spaced frequency multiplets in EPIC 218717602
from which we determine a rotation period near seven days. Spectroscopic
observations determine that EPIC 218366972 is in a 5.92 d binary with most
likely a white dwarf companion of canonical mass while the others have no
detected companions. As we detect no multiplets in EPIC 218366972, it is added
to the growing list of subsynchronously rotating stars. With 40 Kepler-
detected sdBV stars and a growing number of TESS publications, we update an
examination of the group properties to provide direction for models. We notice
a correlation between effective temperature and period of maximum pulsation
amplitude, at least for g-mode pulsations, and update the previously-observed
effective temperature – rotation period relation.

###### keywords:

Stars: oscillations – Stars: subdwarfs

##  1 Introduction

Kepler space telescope data have been fundamental in advances in
asteroseismology. The original mission (K1) concluded with four years of
nearly-uninterrupted single-instrument data. No comparable data set has ever
previously been obtained. Kepler data do not suffer from daytime gaps nor
differing instrumentation or observing sites trying to use multiple Earth
longitudes to obtain complete coverage. There are no atmospheric effects and
no Earth-orbiting complications (such as the South Atlantic Anomaly). The
follow-on mission, K2, has similarly obtained heretofore unobtainable data
sets, though of shorter (typically about 80 days) duration. The K2 mission
observed 20 fields in the ecliptic plane, ingeniously using solar pressure to
aid with pointing stability (Howell et al., 2014).

Those data have resulted in nearly 1,600 publications on astrophysics
unrelated to Kepler’s primary mission of discovering planets. An important
area of contribution is asteroseismology. Asteroseismology uses pulsations to
discern stellar structure and evolution. Prior to Kepler, horizontal branch
asteroseismology, where we can explore compact, evolved cores undergoing
helium fusion, was more effort than result. There were debates whether
oscillations had been detected at all in red clump stars (solar-like
oscillations).Amongst hot extreme horizontal branch stars, observations had
done little to identify modes and hence constrain models.

Significant effort was expended trying to identify pulsation modes in
pulsating sdB (sdBV) stars using follow-up longer-duration photometry (e.g.
Reed et al., 2007), multicolour photometry (e.g. Randall et al., 2006), and
time-resolved spectroscopy (e.g. Baran et al., 2010), with limited success. In
large part, this was due to ground-based observations’ low-duty-cycle
(typically $<30$% coverage) or limited duration on larger facilities. Kepler
and K2’s unique data sets allowed observers to finally fully exploit these
stars’ pulsations with seismic analyses.

Subdwarf B stars pulsate in both pressure (p) and gravity (g) modes; with the
hotter stars primarily p-mode dominated and the cooler stars predominantly
g-mode-dominated pulsators. Typical p modes have periods of a few minutes with
amplitudes rarely above 10 parts-per-thousand (ppt) while g modes have longer
periods, typically 1-4 hours, with slightly lower amplitudes. The variable
star classifications for sdBV stars are V361 Hya for p mode pulsators
(Kilkenny et al., 1997), V1093 Her for g mode pulsators (Green et al., 2003),
and DW Lyn (Schuh et al., 2006) for hybrid pulsators which pulsate in both p
and g modes. These classifications have become less distinct as the majority
of Kepler-observed sdBV stars have both types of pulsations. DW Lyn as a
hybrid has two strong p-mode pulsations and only one g-mode pulsation yet for
most of the Kepler-observed sdBV stars, the situation is reversed with more g
modes observed than p modes. To distinguish which type of pulsation is
dominant (in amplitude and number of detected pulsation perodicities), in this
paper we use p+g and g+p for p and g-mode dominated hybrid pulsators,
respectively. For a review of sdB and sdO stars, see Heber (2016), which
includes some early Kepler results.

Seismic discoveries using K1 and K2 data obtained for sdBV stars include mode
identifications using asymptotic g-mode-overtone period spacings (Reed et al.,
2011) and rotationally-induced frequency multiplets (Baran et al., 2012).
These two methods have provided around two thousand identified modes; to date,
_every_ Kepler-observed sdBV star which has been analyzed has had the majority
of its periodicities associated with pulsation modes (where $n$ represents
radial overtones, $\ell$ the number of surface nodes, and $m$ the azimuthal
surface nodes). The most recent review of Kepler results for sdBV stars is
Reed et al. (2018) and we examine group properties in §4.

In this paper we analyze four sdBV stars discovered during K2’s Campaign 7
(C7) and place them in context with what has been detected so far. EPIC
215776487 (2MASS 19413850-2333426, GALEX J194138.5-233342) has $K_{M}=16.3$,
EPIC 217280630 (GALEX J191534.6-205107) has $K_{M}=16.3$, EPIC 218366972
(2MASS 19345376-1855522, GALEX J193453.7-185552) has $K_{M}=15.9$, and EPIC
218717602 (2MASS 19334689-1817137, GALEX J193346.9-181713) has $K_{M}=15.8$

These four stars were part of our K2 Guest Observer program which observed
nine stars during C7, with these four pulsating. None of these stars were
previously known to pulsate, making them new discoveries.

In §4 we will discuss how these stars fit with what has been learned from
Kepler and published TESS (Transiting Exoplanet Survey Satellite Ricker et
al., 2016) data to date with a brief review. As this is our 47th paper using
Kepler data, and with recent TESS publications by Charpinet et al. (2019);
Reed et al. (2020), and Sahoo et al. (2020), it is a good time to take an
updated examination of group properties.

##  2 Spectroscopic observations and results

As part of our follow-up spectroscopic survey (Telting et al., 2014), low-
resolution spectra (R $\sim$ 2000) have been collected for EPIC 215776487,
EPIC 217280630, EPIC 218366972, and EPIC 218717602 using the 2.56m Nordic
Optical Telescope (NOT) with ALFOSC, grism #18 and a 0.5 arcsec slit. We used
CCD#14 giving an approximate wavelength range of 3450 – 5350Å, and a
resolution based on the width of arc lines of 2.2Å. Exposure times of 900 s
were used for all spectra.

The spectra were reduced and analyzed in the same way. Standard reduction
steps within IRAF include bias subtraction, removal of pixel-to-pixel
sensitivity variations, optimal spectral extraction, and wavelength
calibration based on helium arc-lamp spectra. The target spectra and the mid-
exposure times were shifted to the barycentric frame of the solar system.
Radial velocities (RVs) were derived with the FXCOR package in IRAF. The RVs
were adjusted for the position of the target in the slit, judged from slit
images taken just before and after the spectral exposure.

The spectra of all four targets have the characteristic appearance of single
sdB stars, for which we cannot exclude binarity with companions of much lower
luminosity, such as main-sequence M stars or white dwarfs. For EPIC 217280630
no 2MASS magnitudes are listed. For the other 3 targets there is no clear near
infrared excess observed (from 2MASS), again indicating single stars.
Nevertheless, for EPIC 218366972 we detect clear RV variability (see below).

###  2.1 EPIC 215776487, EPIC 217280630, and EPIC 218717602

For EPIC 215776487, EPIC 217280630, and EPIC 218717602, we determined the RV
with the average spectrum of each target as a template spectrum. We also used
those average spectra to derive the atmospheric parameters of the stars, and
we list these parameters in Table 1. For this purpose we used the fitting
procedure of Edelmann et al. (2003) with the metal-line blanketed local
thermodynamic equilibrium (LTE) models of solar composition described in Heber
et al. (2000).

For EPIC 215776487 we obtained nine useful spectra between Oct 2016 and Aug
2017, and achieved a signal-to-noise (S/N) level between 24 and 60 with median
S/N=34. The average spectrum has S/N$\approx$100\. The median RV error is
$8.4\,{\rm km^{\cdot}s^{-1}}$ and the RV root-mean-square (RMS) of the
individual spectra around the average velocity is $7.1\,{\rm
km^{\cdot}s^{-1}}$.

For EPIC 217280630 we obtained 10 useful spectra between Jun 2016 and Aug
2017, and achieved a S/N level between 29 and 57 depending on observing
conditions, with median S/N=46. The average spectrum has S/N$\approx$110\. The
median RV error is $9.9\,{\rm km^{\cdot}s^{-1}}$ and the RV RMS of the
individual spectra around the average velocity is $10.5\,{\rm
km^{\cdot}s^{-1}}$.

For EPIC 218717602 we obtained eight useful spectra between Oct 2016 and Aug
2017, and achieved a S/N level between 44 and 74, with median S/N=68. The
average spectrum has S/N$\approx$150\. The median RV error is $5.3\,{\rm
km^{\cdot}s^{-1}}$ and the RV RMS of the individual spectra around the average
velocity is $7.8\,{\rm km^{\cdot}s^{-1}}$.

We conclude that for these three targets our RV measurements are consistent
with single stars.

Table 1: Results of spectral analysis. Errors on the final digits are given in parentheses. Star |  $T_{\rm eff}$ (K) |  $\log g$ (cgs) | $\log\left(\frac{N_{He}}{N_{H}}\right)$
---|---|---|---
EPIC 215776487 | 27860(160) | 5.45(2) | -2.718(38)
EPIC 217280630 | 22770(150) | 5.01(2) | -2.104(77)
EPIC 218366972 | 28160(110) | 5.44(2) | -2.862(28)
EPIC 218717602 | 24470(160) | 5.17(2) | -2.633(61)

###  2.2  The orbit of EPIC 218366972

For EPIC 218366972 we obtained 12 useful spectra between Oct 2016 and Oct
2017, and achieved a S/N level between 34 and 71, with a median S/N=51. The
average spectrum has S/N$\approx$140\. For determining the RVs we have first
used the average spectrum as a cross-correlation template, and subsequently a
spectral model (as in Table 2) as a template. The median RV error is
$7.7\,{\rm km^{\cdot}s^{-1}}$.

We find significant RV variations, and list the orbital solution obtained
while assuming a circular orbit, in Table 2. With an orbital-velocity
amplitude of $66.3\,{\rm km^{\cdot}s^{-1}}$, an orbital period of 5.92 days,
and assuming that the companion is an unseen white dwarf, we derive the
following constraints from the mass function. For a canonical mass of the
subdwarf of 0.47 M⊙ (Van Grootel et al., 2014) the minimum mass of the WD
companion is 0.58 M⊙. For an ensemble of a canonical-mass sdB with a
canonical-mass WD (0.6 M⊙) the orbital inclination is 79o. Assuming a mass of
0.3 M⊙ for the sdB the minimum mass of the WD is still 0.48 M⊙, and the
minimum orbital separation is 12.6 R⊙. Using canonical masses and $i=90^{o}$,
the separation is 14.3R⊙. At that separation and using R${}_{\rm sdB}=0.2$R⊙,
eclipses would occur for $i>89^{0}$. No eclipses are observed in the K2 data
(§3.3) indicating $i<89^{o}$.

Table 2: Results of spectral binary analysis for EPIC 218366972. Solution of
RV wrt mean spectrum
---
Amplitude | $64.01\pm 3.09\,{\rm km^{\cdot}s^{-1}}$
Period |  $5.9190\pm 0.0021\,$d
Reduced $\chi^{2}$ | 1.408
RMS | $8.8\,{\rm km^{\cdot}s^{-1}}$
Solution of RV wrt model template
System velocity | $38.12\pm 3.51\,{\rm km^{\cdot}s^{-1}}$
Amplitude | $66.28\pm 3.43\,{\rm km^{\cdot}s^{-1}}$
Period |  $5.9218\pm 0.0026$ d
Reduced $\chi^{2}$ | 0.73
RMS | $8.8\,{\rm km^{\cdot}s^{-1}}$
Solution of RV wrt model template, with all RV errors set to $9.7\,{\rm
km^{\cdot}s^{-1}}$
System velocity | $38.20\pm 3.15\,{\rm km^{\cdot}s^{-1}}$
Amplitude | $70.95\pm 4.03\,{\rm km^{\cdot}s^{-1}}$
Period |  $5.9187\pm 0.0024$ d
Reduced $\chi^{2}$ | 1.00
RMS | $7.9\,{\rm km^{\cdot}s^{-1}}$

To determine the atmospheric parameters of EPIC 218366972, we shifted all
individual spectra to remove the orbital velocities, and made an average
orbit-corrected spectrum, to which we applied the same modelling procedure as
described above for the other stars. The fit results are presented in Table 1.

Figure 1: Fourier transforms of EPIC 215776487, EPIC 217280630, EPIC 218366972, and EPIC 218717602. Horizontal (blue on-line) lines indicate the detection threshold. Table 3: Period list for EPIC 215776487. †: This frequency was fitted using data from days 25-40 of the run only with $\sigma\,=\,0.086$ ppt. ID | Freq | Per | Amp | S/N | $\ell$ | $n$ | $\frac{\Delta P}{P}$
---|---|---|---|---|---|---|---
|  $\mu$Hz | Sec | ppt |  |  |  | %
f1 | 110.384 (8) | 9059.31 (67) | 0.27 (3) | 7.3 | 1 | 33 | 3.2
f2 | 136.606 (10) | 7320.30 (51) | 0.23 (3) | 6.3 | 1 | 26 | 0.03
f3 | 171.452 (13) | 5832.53 (46) | 0.16 (3) | 4.4 | 1 | 20 | -1.6
f4 | 197.027 (6) | 5075.45 (16) | 0.35 (3) | 9.8 | 1/2 | 17/32 | -7.7/-5.3
f5 | 214.565 (10) | 4660.59 (22) | 0.22 (3) | 6.2 | 2 | 29 | 4.2
f6 | 221.276 (10) | 4519.24 (21) | 0.22 (3) | 6.2 | 2 | 28 | 5.1
f7 | 258.647 (8) | 3866.28 (12) | 0.27 (3) | 7.6 | 1 | 12 | 3.3
f8 | 348.718 (12) | 2867.65 (10) | 0.19 (3) | 5.3 | 1 | 8 | -0.5
f9 | 372.872 (7) | 2681.88 (5) | 0.32 (3) | 9.0 |  |  |
f10 | 390.700 (11) | 2559.51 (7) | 0.20 (3) | 5.7 | 1 |  |
f11 | 419.783 (7) | 2382.18 (4) | 0.32 (3) | 9.0 | 1/2 | 6/13 | 3.2/8.4
f12† | 1005.921 (71) | 994.11 (7) | 0.41 (7) | 4.8 |  |  |
Table 4: Period list for EPIC 217280630. † Values for f6 are from Lorentzian fitting. ID | Freq | Per | Amp | S/N | $\ell$ | $n$ | $\frac{\Delta P}{P}$
---|---|---|---|---|---|---|---
|  $\mu$Hz | Sec | ppt |  |  |  | %
f1 | 60.259 (8) | 16549.39 (2.18) | 0.144 (23) | 6.2 | 1 | 80 | -3.7
sA | 65.295 (12) | 15315.08 (2.81) | 0.140 (22) | 3.9 | 1 | 70 | 1.1
f2 | 75.454 (62) | 13253.04 (10.84) | 0.390 (22) | 11.0 | 1 | 63 | 6.6
f3 | 147.652 (65) | 6772.68 (2.96) | 0.184 (22) | 7.1 | – | – | –
f4 | 156.544 (47) | 6387.98 (1.91) | 0.178 (22) | 6.1 | 1 | 28 | -4.3
f5 | 161.638 (54) | 6186.66 (2.06) | 0.153 (22) | 5.9 | 1 | 27 | -1.4
f6† | 285.85 (18) | 3498.31 (2.23) | 0.107 | – | 1 | 13 | 2.7

##  3 K2 observations and pulsation analyses

Campaign 7 spanned 82 days between 4 October and 26 December, 2015. Our data
are short-cadence observations, with integration times of 58.85 seconds which
were downloaded from MAST as pixel files. Fluxes were extracted using aperture
photometry and spacecraft artefacts were removed using our custom process
described in Baran et al. (2017); Ketzer et al. (2017). Temporal spectra
(Fourier transforms, FTs; Fig. 1) were produced to examine the pulsations and
sliding FTs (SFTs) were produced to examine the time-stability of the
pulsations. The 1.5/T frequency resolution of these data is $2.14\,\mu$Hz and
to make it unlikely any peak in the FT is due to random noise requires a S/N
of $4.2\sigma$, where $\sigma$ is the average level of the FT. As low
frequency noise is more difficult to remove, $\sigma$ were calculated in
frequency regions nearly devoid of pulsations and linearly interpolated
between, where pulsations occur.

Table 5: Period list for EPIC 218366972. † frequencies were Lorentzian fitted. ⋆ indicates mode identifications which are less certain. ID | Freq | Per | Amp | S/N | $\ell$ | $n$ | $\frac{\Delta P}{P}$
---|---|---|---|---|---|---|---
|  $\mu$Hz | Sec | ppt |  |  |  | %
sA | 119.945 (10) | 8337.14 (71) | 0.203 (26) | 4.0 | 1/2 | 29/53 | -0.5 / -6.0
sB | 127.638 (11) | 7834.66 (66) | 0.191 (26) | 3.8 | 1 | 27 | 2.8
sC | 136.773 (11) | 7311.39 (61) | 0.183 (26) | 3.7 | 1/2 | 25/46 | -2.1 / -2.9
f04 | 141.552 (07) | 7064.53 (37) | 0.297 (28) | 5.9 | 1 | 24 | 1.3
f05 | 152.926 (4) | 6539.13 (17) | 0.548 (28) | 11.3 | 1 | 22 | -4.4
f06 | 159.091 (8) | 6285.72 (32) | 0.267 (28) | 5.8 | 1/2 | 21/39 | -3.6 / 0.3
sD | 165.698 (10) | 6035.08 (37) | 0.126 (28) | 4.0 | 1 | 20 | -1.7
f08 | 172.580 (4) | 5794.43 (12) | 0.591 (28) | 12.5 | 1 | 19 | 4.1
f09 | 178.679 (10) | 5596.61 (30) | 0.212 (28) | 4.5 |  |  |
f10 | 181.005 (2) | 5524.70 (5) | 1.398 (28) | 29.6 | 1 | 18 | -1.5
f11 | 188.826 (3) | 5295.88 (8) | 0.766 (26) | 16.2 |  |  |
f12 | 209.051 (3) | 4783.53 (8) | 0.597 (26) | 13.2 | 1 | 15 | 8.4
f13 | 221.076 (7) | 4523.34 (14) | 0.309 (26) | 6.8 | 1/2 | 14/27 | 6.5 / 3.0
f14 | 236.030 (8) | 4236.75 (15) | 0.252 (26) | 5.8 | 1/2 | 13/25 | -5.7 / 8.3
f15† | 244.74 (23) | 4086.1 (3.8) | 0.16 (–) | – | 2 | 24 | 5.0
f16 | 249.782 (1) | 4003.48 (1) | 2.742 (26) | 62.3 | 1 | 12 | 3.0
f17 | 254.659 (2) | 3926.82 (4) | 0.832 (26) | 18.9 | 2 | 23 | -2.2
f18 | 264.374 (5) | 3782.51 (7) | 0.439 (26) | 18.9 | 2 | 22 | -0.2
f19 | 267.029 (2) | 3744.92 (3) | 1.020 (28) | 10.0 | 1 | 11 | 1.8
f20 | 275.184 (6) | 3633.94 (8) | 0.370 (28) | 23.2 | 2 | 21 | -1.2
f21 | 278.242 (4) | 3593.99 (5) | 0.637 (28) | 14.4 |  |  |
f22 | 278.543 (3) | 3590.10 (4) | 0.769 (28) | 17.5 |  |  |
f23† | 283.21 (26) | 3531.0 (3.2) | 0.13 (–) | – |  |  |
f24 | 286.534 (3) | 3489.98 (4) | 0.743 (28) | 18.1 | 1/2 | 10/20 | -2.0 / 1.0
f26 | 289.181 (1) | 3458.04 (2) | 1.538 (28) | 37.5 | 1 | 10 | -10.5
f27 | 313.565 (2) | 3189.14 (2) | 1.023 (28) | 25.8 | 1/2 | 9/18 | -0.3 / -3.4
sE | 326.205 (12) | 3065.56 (12) | 0.168 (26) | 4.3 |  ⋆2 | 17 | 12.7
f29 | 330.446 (12) | 3026.21 (11) | 0.177 (26) | 4.6 |  ⋆2 | 17 | -14.1
f30 | 341.948 (4) | 2924.42 (3) | 0.452 (28) | 11.6 | 1 | 8 | 1.1
f31 | 342.134 (5) | 2922.83 (4) | 0.489 (28) | 12.5 | 1 | 8 | 0.5
sF | 360.526 (16) | 2773.73 (12) | 0.129 (26) | 3.9 |  |  |
f33 | 377.650 (10) | 2647.95 (7) | 0.188 (25) | 6.0 | 1 | 7 | -1.9
f34 | 424.882 (13) | 2353.59 (7) | 0.151 (25) | 4.5 |  |  |
f35 | 472.046 (12) | 2118.44 (5) | 0.165 (25) | 5.2 | 1 | 5 | 0.9
f36 | 541.427 (9) | 1846.97 (3) | 0.241 (26) | 7.2 | 1 | 4 | -0.3
sG | 613.733 (13) | 1629.37 (4) | 0.146 (25) | 3.9 |  |  |
sH | 651.577 (12) | 1534.74 (3) | 0.164 (25) | 3.9 |  |  |
sI | 660.887 (16) | 1513.12 (4) | 0.137 (25) | 4.3 |  |  |
sJ | 661.036 (17) | 1512.78 (4) | 0.111 (25) | 4.0 |  |  |
sK | 708.144 (12) | 1412.14 (2) | 0.158 (25) | 4.1 |  |  |
Table 6: Period list for EPIC 218717602. † f18 was nlls fitted using only the first 10 days of data with $\sigma\,=\,0.07$ ppt. ID | Freq | Per | Amp | S/N | $\ell$ | $n$ | $\frac{\Delta P}{P}$
---|---|---|---|---|---|---|---
|  $\mu$Hz | Sec | ppt |  |  |  | %
f01 | 103.904 (2) | 9624.29 (14) | 1.376 (27) | 40.5 | 1 | 33 | 0.1
f02 | 123.749 (1) | 8080.89 (8) | 1.689 (27) | 49.7 |  |  |
f03 | 124.653 (6) | 8022.30 (40) | 0.293 (27) | 8.6 | 1 | 27 | -8.7
f04 | 128.692 (6) | 7770.48 (39) | 0.371 (27) | 11.0 | 1 | 26 | -4.4
f05 | 132.902 (8) | 7524.35 (41) | 0.264 (27) | 7.8 | 1/2 | 25/46 | 2.1/-4.7
f06 | 142.845 (1) | 7000.58 (4) | 2.400 (28) | 70.6 | 1 | 23 | 3.0
f07 | 143.045 (1) | 6990.82 (6) | 1.760 (28) | 51.8 | 1 | 23 | -0.7
f08 | 148.202 (5) | 6747.56 (22) | 0.431 (27) | 12.7 | 1 | 22 | 6.9
f09 | 160.668 (5) | 6224.03 (18) | 0.451 (27) | 13.4 | 1 | 20 | 7.9
f10 | 168.133 (1) | 5947.67 (2) | 3.710 (27) | 112.4 | 1 | 19 | 2.9
f11 | 176.074 (1) | 5679.44 (3) | 1.989 (27) | 60.3 | 1 | 18 | 1.0
f12 | 187.938 (1) | 5320.91 (3) | 2.127 (27) | 64.5 |  |  |
f13 | 216.857 (2) | 4611.33 (4) | 1.121 (27) | 35.0 | 1 | 14 | -4.9
f14 | 261.915 (9) | 3818.03 (14) | 0.227 (27) | 7.1 | 1 | 11 | -6.4
f15 | 280.983 (9) | 3558.93 (15) | 0.169 (27) | 5.4 | 1 | 10 | -4.9
f16 | 327.077 (11) | 3057.39 (10) | 0.129 (18) | 4.8 | 1 | 8 | 4.5
f17 | 327.867 (10) | 3050.02 (10) | 0.135 (18) | 5.0 | 1 | 8 | 1.7
f18† | 7876.05 (14) | 126.967 (2) | 0.368 | 5.3 |  |  |
f19 | 8110.90 (7) | 123.290 (1) | 0.122 (43) | 4.5 |  |  |

Three of the four stars are exclusively g mode pulsators with only EPIC
218717602 having two p-mode periodicities. Most of the pulsations were
amplitude and phase stable, in which case we prewhitened them using non-linear
least-squares (nlls) fitting (e.g. Reed et al., 2004). Otherwise, we either
Lorentzian fitted the FTs, using the Lorentzian widths as an estimator of
frequency uncertainty only (e.g. Reed et al., 2014) or prewhitened smaller
sections of data when the pulsations were significantly above the noise.
Pulsations we detected are supplied in Tables 3 through 6. When the amplitudes
were sufficiently stable for prewhitening, we include the fitting error in the
tables, otherwise we do not.

While C7 spanned 82 days, to detect rotationally-split frequency multiplets
usually requires two rotations within the observations. For stars where we do
not detect frequency multiplets, we presume a spin period $>45$ days. Of the
18 K1-observed sdBV stars, which had multiple years of data, eight (44%) had
spin periods $\geq 45$ days.

###  3.1 Pulsation analyses of EPIC 215776487

A total of 12 periodicities were detected above the detection threshold (shown
as a horizontal blue line in Fig. 1) between 110 and $1006\,\mu$Hz (994 and
9058 s). These were all stable in amplitude/frequency and so were nlls fitted.
Their frequencies, periods, amplitudes, and S/N are provided in Tab. 3.

Figure 2: KS test (left panel) and echelle diagrams (right two panels) for
$\ell\,=\,1$ and 2 asymptotic sequences of EPIC 215776487, respectively. In
the echelle diagrams, black circles indicate periods that match the
$\ell\,=\,1$ sequence, blue triangles match the $\ell\,=\,2$ sequence, black
circles with a blue surround match both sequences, and red points do not fit
either sequence.

For g modes in sdB stars, typical $\ell=1$ asymptotic period sequences have
been found to have spacings ($\Pi_{\ell=1}$) near 250 s, and even a cursory
differencing of EPIC 215776487’s periods reveals similar spacings. In usual
fashion, we do a Kolomogorov-Smirnof (KS) test, which can reveal commonly
spaced periods. Very surprisingly for EPIC 215776487 there is no signature
trough near 250 s to indicate the sequence (left panel of Fig. 2). Another
tool used to discover asymptotic period sequences is an echelle diagram. We
produced one with a spacing of 250 s and the sequence appeared (middle panel
of Fig. 2). Using that as our guide, we found eight periods that are part of
the $\ell=1$ sequence and these modes indicate a period spacing of
$247.3\,\pm\,0.4\,s$. From the $\ell=1$ sequence, an $\ell=2$ sequence can be
calculated from the relation $\Delta\Pi_{\ell=2}=\Delta\Pi_{\ell=1}/\sqrt{3}$.
Two periods were found to fit the $\ell=2$ sequence and two of the $\ell=1$
periods could also fit the $\ell=2$ sequence, and are marked accordingly in
Tab. 3. All but two of the frequencies fit these sequences. f9 does not fit
either sequence, though it is one of the highest-amplitude periodicities.
Perhaps this is a trapped mode, though we cannot confirm this as our series
are not contiguous in $n$, which would be a necessary condition for finding
trapped modes. There are no evenly-split frequencies indicative of
rotationally-split multiplets and so we cannot determine a rotation period for
EPIC 215776487. It is likely longer than our sensitivity, which is about 45 d.

Figure 3: KS test (left panel) and $\ell=1$ echelle diagram (right panel) for
EPIC 217280630. Point shapes and colours (on-line only) as in Fig. 2.

###  3.2 Pulsation analyses of EPIC 217280630

We only detect six frequencies in EPIC 217280630’s data set and two of these
are very low near 60 and 75 $\mu$Hz, which must be very close to the acoustic
cut-off. All except for f6 were nlls fitted. A KS test has a significant
trough just past 200 s and the echelle diagram confirms that sequence (both
shown in Fig. 3). An additional low-amplitude peak (labeled sA in Tab. 4,
which is obvious in the FT, below the detection threshold, but fits the
asymptotic sequence is included in both the figure and the table for EPIC
217280630. Linear regression determines $\Pi_{\ell=1}=207.56\,\pm\,0.26\,s$
for EPIC 217280630, with only f3 not fitting the $\ell=1$ sequence. Like EPIC
215776487, this non-sequence periodicity has a fairly high amplitude (the
second highest) and so could represent a trapped $\ell=1$ mode, though we have
no way to confirm this. The sequence has only one contiguous pair, and so
there is no way to search for trapped modes. The period spacing of 207 s is
extreme for a cool (coolest of these four stars), purely-g-mode pulsator.
There are two other sdBV stars with small $\Pi_{\ell=1}$ values, but they are
much hotter, p+g hybrid pulsators. There are no indicators of rotationally-
induced frequency multiplets which we interpret as a long ($>45\,d)$ rotation
period. However, it could also indicate an orientation where the $m\neq 0$
components are of lower amplitude, and therefore undetected.

Figure 4: Same as Fig. 3 for EPIC 218366972.

###  3.3 Pulsation analyses of EPIC 218366972

EPIC 218366972 is the richest pulsator of the group. We detect 36
periodicities above the detection limit with another 11 _suspected_. The
strongest trough in the KS test (left panel of Fig. 4) is near 250 s, though
it is not especially significant compared to others. As $\ell=1$ modes have
the least geometric cancellation (Pesnell, 1985) and therefore typically
higher amplitudes, we did an amplitude cut at 0.5 ppt and, with just those
periods, the trough became significant.The echelle diagram (right panel of
Fig. 4), spaced at 254 s, easily shows that sequence, however EPIC 218366972
has a significant “hook” feature at lower radial overtones. This has been seen
in several other sdBV stars (Baran & Winans, 2012). We calculated a period
spacing of $254.95\,\pm\,0.50\,s$ with linear regression above the hook, and
then linearly fitted the hook feature for inclusion in Tab. 5. We calculated
where $\ell=2$ sequence periodicities should occur from the $\ell=1$ sequence
and those that match are labelled as such in Tab. 5. Many of the $\ell=1$
modes also match the $\ell=2$ sequence, and since we cannot distinguish
between them, we label them with both. An intriguing feature is how linear the
periods look below the turn of the “hook” feature ($<4\,000$ s). A KS test of
just those periods reveals a broad peak with a central value of 271 s and
folding across that period produces an echelle with seven periods in line. As
such, an alternative view would be that there is a sequence with periods
$>3\,450$ s with an asymptotic spacing of 254.95 s and another at 271 s up to
and including f26 (3 458 s).

Surprisingly for so many frequencies there are no obvious rotationally-induced
frequency multiplets. If EPIC 218366972 were tidally locked, a rotation period
of 5.9 d would be a frequency of about $2.0\mu$Hz and $\ell=1$ modes would
have a separation of half that near $1.0\mu$Hz. There are no frequency
splittings near those values or multiples. The closest we find to a multiplet
would be a possible triplet of f15-f16-f17, which are split by 4.95 and
4.88$\mu$Hz, respectively. However, those three periods also fit asymptotic
sequences, making that a more likely fit. As such, we do not detect any
frequency multiplets in EPIC 218366972. In this case, binarity indicates an
orientation favorable for viewing multiplets, and so we can safely assume that
the rotation period is $>45$ d.

Figure 5: Same as Fig. 3 for EPIC 218717602.

We can also use the FT to search for a signal from Doppler boosting caused by
the binary motion. Following the method of Telting et al. (2012) and using
K$\,=\,67$ km$\cdot$s-1, we calculate a Doppler boosting signal of 0.3 ppt at
$1.96\,\mu$Hz. The final step in our lightcurve processing removes trends
$>1.5$ days, and so we examined a lightcurve without this processing step.
Unfortunately, the unprocessed lightcurve has low-frequency noise greater than
0.3 ppt and so we cannot detect the binary signal using Doppler boosting.
There is also no indication of eclipses in the lightcurve.

###  3.4 Pulsation Analyses of EPIC 218717602

We were able to prewhiten 19 frequencies for EPIC 218717602 and, as is obvious
just by looking at the FT, most of these readily fit into an $\ell=1$
asymptotic sequence. The KS test shows a very deep trough near 265 s, which is
easily reproduced in the echelle diagram (both are shown in Fig. 5). There is
a slight hook feature, but a linear regression finds a period spacing of
$263.15\,\pm\,0.48\,s$. This sequence includes all but four of the g modes. Of
those, there are two pairs (f02, f03 & f16,f17) separated by $0.84\pm
0.08\mu$Hz. If $\ell=1$ (as f03, f16, and f17 fit the sequence) then this
would be a rotation period of about 7 d. These are not the highest amplitude
pulsations in EPIC 218717602 and so the multiplet detection is not secure.
EPIC 218717602 also has two p-mode frequencies, with f18 only appearing at the
beginning of the run. However, without any observational evidence for mode
identifications, they do not tell us much. Only that another quite cool sdB
star has p modes (see Tab. 7 for others).

##  4 Examining the group

Kepler’s original mission (K1) observed 18 sdBV stars (Østensen et al., 2010,
2011) and K2 observed 139 of our proposed sdB targets in short-cadence mode
during its 20 campaigns. Many K1-observed sdBV stars have over three years, or
about 1.5 million data points, of observations, while K2 only observed
individual fields for roughly 80 days, resulting in about 110 000 data points
per target. While this vast wealth of data takes quite some time to process
(particularly K2 stars for which we have to begin with pixel files), first-
look analyses anticipates about 50 pulsators from K2. To date a total of 34 of
the roughly 69 Kepler-observed sdBV stars have been analyzed and published
(including the four in this paper), so there is still some ways to go.
Additionally, TESS has now completed its two year main mission, during which
it observed about 1 000 of our proposed sdB targets, few of which have been
examined as of this writing. As such, it seems a good point to examine
progress and compare and review what has been detected.

Thanks to a generous time allocation from the NOT, we have been able to obtain
spectra of all of our Kepler targets (Telting et al., 2014). We do this both
to constrain binarity and so that we are fitting the same resolution spectra
to the same atmospheric model grids. While there may be systematics between
model grids or differing instrumentation in data from multiple sources, our
single-sourced _relative_ values should be accurate. We refer readers to any
of the references provided in Tab. 7 for details on the spectra and their
processing. Table 7 lists seismic, spectroscopic, and orbit-rotation
properties of 38 published (or _in press_) Kepler-observed sdBV stars, one
blue horizontal branch (BHB) star with closely related pulsation properties,
and four TESS-observed sdBV stars. Rotation is deduced strictly from pulsation
frequency multiplets while the binary period may be deduced from RV or
photometric variations. To date, from K1 there were two stars analyzed for
which multiplets were not detected while K2 has many. From K1, we know that
rotation periods typically span tens of days, and so any periods longer then
about 45 d (44% of K1 stars) would not likely be resolved during K2
observations. Each TESS sector of observations spans about 26 days, and so
rotation periods longer than about 12 days (68% of K1+K2 stars) would likely
not be detected. In this “group” summary, we only consider sdBV stars observed
during K1, K2, and TESS missions and excluding atypical pulsation types (e.g.
Jeffery et al., 2017). This sample should include stars with similar bulk
physical properties that have observations obtained in a roughly-homogeneous
manner capable of providing mode identifications. Two exceptions to this are
the sdB+WD binaries Feige 48 (Reed et al., 2012) and KPD 1930$+$2752 (Reed et
al., 2011) included only in Fig. 11 which have binary periods under one day
and rotation periods derived from ground-based pulsation data.

The real revolution from Kepler data is the ability to observationally
correlate periodicities with pulsation modes (mode identifications). The main
tools which have been used are asymptotic g-mode period spacings, which
provide $\ell$ and relative $n$ values; rotationally-induced frequency
multiplets, which provide $\ell$, $m$, and can provide relative $n$ if several
multiplets of the same degree are detected; and g-mode frequency multiplets
splittings, which have relative spacings dictated by the Ledoux constant
(Ledoux, 1951), and this can provide $\ell$ values.

These identifications are invaluable for constraining stellar structure
models, from which we discern internal physics. Important pulsation properties
include the smoothness of asymptotic sequences (e.g. Reed et al., 2014), which
describe less stratified transition regions than expected (Constantino et al.,
2015); mode trapping (e.g. Østensen et al., 2014), which has now been
associated with convective core overshoot (Guo & Li, 2018; Ostrowski et al.,
2021); both p and g mode overtone spacings (Reed et al., 2011; Baran et al.,
2012), which describe the resonant cavity; and frequency multiplets, which
provide information on rotation, including differential rotation, and for
stars in binaries, which all indicate subsyncrhonous rotation for close
binaries (periods under 10 days, e.g. Telting et al., 2012), with the
exception of the 3 hour binary 2M1938+4603 (Østensen et al., 2010),
constraining synchronization time scales for post-common-envelope (PCE)
binaries (e.g. Pablo et al., 2012). Hybrid pulsators allow for radial scrutiny
as g modes probe deeper than p modes (Charpinet et al., 2014), with the latter
being mostly envelope (defined as above the He/H transition).

Figure 6 shows the locations of the pulsators in Table 7 in a Kiel diagram.
Included in the figure are non-pulsators observed during K1 (Østensen et al.,
2010, 2011) and sample zero-age helium main-sequence (ZAEHB) and evolutionary
tracks (Reed et al., 2004). It is well-known that a larger fraction of sdB
stars are observed to pulsate in g modes than p modes, or equivalently, cooler
sdB stars are more likely to pulsate at observable amplitudes than hotter
ones. Therefore it is not too surprising that most of the Kepler-observed
stars are g+p. Commensurate with that is very few of the non-pulsators
observed by Kepler are cooler sdB stars. There is only one below 26 000 K
while above 30 000 K 26 of the 28 (93%) K1-observed stars are not observed to
pulsate.

Figure 6: Kiel diagram of Kepler and TESS-observed sdB stars. Colours and
symbols are indicated within the figure: Non-pulsators are not shaped for
binary information. The dashed-dotted line indicates the zero-age helium main-
sequence (ZAEHB) for a core mass of $0.50M_{\odot}$ with varying envelope
masses (two are labeled on the plot) and the dashed line indicates the ZAEHB
for a fixed envelope mass of $2\times 10^{-4}M_{\odot}$ and varying core
masses (three are labeled on the plot). The solid lines shows evolution for a
core mass of 0.5M⊙ and an envelope mass of 0.001M⊙. Errorbars for the average
of the errors from Table 7 are indicated in the lower right.

###  4.1 Hybrid pulsators

While models have largely been successful in predicting where p-mode
pulsations should occur in the Kiel diagram (contours in Fig. 7), there has
been difficulty getting g-mode instabilities up to observed effective
temperatures (Jeffery & Saio, 2006a, b; Hu et al., 2009; Bloemen et al.,
2014). Prior to Kepler observations, it was presumed that p-mode pulsations
occurred in hotter sdB stars, g-mode pulsations occurred in cooler sdB stars,
and rare hybrid pulsators would inhabit the temperature boundary. However, as
can be seen in Figs. 6 and 7, hybrid pulsators occur across nearly the full
range of temperatures, including the hottest and third coolest stars in our
sample. In Tab. 7 the transition from p+g to g+p occurs at 29 000 – 30 000 K.
The coolest p+g star and the hottest g+p star both have $T_{\rm
eff}\,=\,29\,600\,K$.

Figure 7: Kiel diagram showing p-mode instability contours (reproduced from
Charpinet et al., 2001) with points and average errorbars from Figure 6.

###  4.2 Pulsators in binaries

Table 7 separates the pulsators by binary type. All of our sdB binaries with
white dwarf or M-dwarf companions are g+p, though we know this is not a unique
feature. Ground-based observations have observed p mode sdBV stars with both
types of companions. All save one of our Kepler/TESS-observed sdBV stars with
F/G companions are p+g pulsators. That sole g+p pulsator (CD$-$28∘ 1974),
observed with TESS, is also the hottest g+p star. Perhaps these trends are
indicative of formation channels, as the sdB+WD/dM binaries would have
experienced at least one common-envelope (CE) phase (Han et al., 2002, 2003)
while the sdB+F/G binaries would likely have had their envelopes stripped via
Roche-lobe-overflow (RLOF) (e.g. Vos et al., 2017). There could also be an
observational bias in that cooler sdB stars are fainter than their F/G
companions, making them harder to detect. This could likely be answered by
examining our full set (K1, K2, and TESS) of non-pulsators which were
initially selected using GALEX observations. If there were a reasonable number
of GALEX-selected sdB stars below 30 000 K with F/G companions, then we can
rule out an observational selection effect. This should be revealed in our
forthcoming K2 summary paper when pulsators and non-pulsators are compared.

Slightly more than half of our pulsators show no indication of a companion.
This means there were no indications of spectral lines from a companion, no RV
variations outside of standard deviations, and no photometric variations which
could be produced by the reflection effect, ellipsoidal variations, Doppler
boosting, or phase-induced pulsation aliasing caused by light-travel across an
orbit. SdB formation channels include stripping giant stars at the tip of the
red giant branch via RLOF or CE ejection, or by merging two helium white
dwarfs (WDs). The result of the different evolutionary paths is that the
envelope-stripping ones produce sdB stars with masses sharply peaked at the
so-called “canonical” value of $0.47M_{\odot}$ or slightly less while merged
WDs can have a broad range of masses from 0.4 to over $0.7M_{\odot}$ (See Fig.
12 of Han et al., 2003). GAIA data should help answer this question as it will
detect astrometric binaries and, in many cases, determine masses when combined
with spectroscopic and/or spectral energy distribution data.

Figure 8: Kiel diagram showing $\ell=1$ g-mode period spacings,
$\Pi_{\ell=1}$. Colour codings provided in the figure and different point
types are for clarity only.

###  4.3 Observational correlations and trends

Observations are meant to provide constraints and direction to models, from
which we determine the physics of stars. Observations we list in Table 7
include spectroscopic ones, $T_{\rm eff}$, $\log g$ and binarity, and seismic
ones, pulsation type, $P_{\rm Amax}$ (period of highest amplitude),
$\Pi_{\ell=1}$ (g-mode asymptotic period spacing for $\ell=1$), and rotation
period (from frequency multiplets).

Figure 9: Comparing $T_{\rm eff}$ to $P_{\rm Amax}$. The four p+g pulsators
have their g-mode $P_{\rm Amax}$ connected to their p-mode $P_{\rm Amax}$ by
dotted lines. Note that CD$-$28∘ 1974 has had its point shifted down 0.02 dex
for clarity. Point shapes and colours (on-line only) are indicated in the
figure.

Figure 8 is a Kiel diagram where point colours indicate detected
$\Pi_{\ell=1}$. There are no obvious trends. The longer period spacings are
mostly clustered in the middle (around $T_{\rm eff}=27\,000\,$K and $\log
g=5.35$), the median values (near $\Pi{\ell=1}=250-260\,s$) extend from
$T_{\rm eff}=23\,000\,$K to $T_{\rm eff}=30\,000\,$K, and the lower period
spacings have tendencies to be towards the extremes; $\log g>5.5$ or $T_{\rm
eff}<27\,000\,K$, which also includes lower $\log g$. Clearly asymptotic
period spacings depend on something other than temperature or gravity.

Figure 9 shows the period of the highest-amplitude pulsation ($P_{\rm Amax}$)
in each star with $T_{\rm eff}$. Point type and colour indicate binary
information. As previously noted, the sdB+F/G binaries are in the hottest
group of stars. The p+g-mode pulsators show $P_{\rm Amax}$ for both types of
pulsations, connected by dotted lines. Even though the $\log$ scale compresses
the ordinate, there is a clear trend of $P_{\rm Amax}$ towards longer periods
in cooler stars. This trend seems clear in the g/g+p region but less so in the
p/p+g stars. If we ignore the sdB+F/G stars, the p mode periods show a clear
trend in the same direction as the g modes, whereas if we include those, there
is little correlation and even a slightly reversed trend.

Figure 10: Comparing $T_{\rm eff}$ to rotation periods. Arrows indicate lower
limits and dotted lines connect rotation periods that are different for p and
g modes. Open symbols indicate rotation was determined from p-mode multiplets.
Point shapes and colours (on-line only) are the same as Fig. 9.

###  4.4 Rotation

It was pointed out in Reed et al. (2014) that there was a trend for cooler
stars to have longer rotation periods. With more pulsators studied, this is
re-examined in Fig. 10. There are four differential rotators (indicated by
points joined by a line) and K2 and TESS data often did not detect multiplets,
which means either the pulsations are pole-on, or we only have a lower limit
on rotation. We presume the latter, as shown by arrows in Fig. 10, as pole-on
orientations are unlikely. Two exceptions may be K10001893 and K8302197 for
which no multiplets were detected in over 1 000 days of K1 data, indicating
either a pole-on viewing angle, or extremely slow rotation ($>715$ days). Open
points in Fig. 10 indicate rotation was determined from p-mode multiplets. It
has been observed that in radially differential rotators (Foster et al., 2015,
and below) p modes indicate faster rotation than g modes. Rather than a
correlation, we can now only state that below 24 000 K rotation is only slower
than 45 days and above 32 500 K rotation is only faster than 25 days. There
are no binary stars in our sample cooler than 26 000 K and no sdB+dM or sdB+WD
binaries hotter than 32 000 K.

Figure 11: Comparing orbital to rotation periods. Arrows indicate lower limits
and dotted lines connect rotation periods that are different for p and g
modes. Open points indicate rotation was determined from p-mode multiplets.
Point shapes and colours (on-line only) are the same as Fig. 9. Diagonal line
indicates tidally-locked rotation. The figure includes the sdB+WD binaries
Feige 48 (Reed et al., 2012) and KPD 1930$+$2752 (Reed et al., 2011) which
have periods less than one day even though neither was observed by Kepler. We
also include the sdB+dM binary AA Dor which is not a pulsator but has a
spectroscopically determined rotation period (Vuckovic et al., 2016).

The effect of binarity on rotation is investigated in Fig. 11. The black
diagonal line indicates tidally-locked rotation. Interesting features in Fig.
11 include that _all_ space-based-observed sdB+WD stars are primarily g-mode
pulsators that rotate subsynchronously. As there are only eight such systems,
and g-mode pulsations occur more often than p-mode ones, this may be a
selection effect. We include the two p-mode sdB+WD stars Feige 48 (Reed et
al., 2012) and KPD 1930$+$2752 (Reed et al., 2011) (from ground-based data) as
they are the shortest-period sdB+WD binaries which also have frequency
multiplets to indicate rotation period. KPD 1930$+$2752 is tidally locked
while Feige 48 is nearly so. All the g-mode sdB+WD binaries have very similar
rotation periods, though four of those only have lower limits on rotation. The
sdB+dM binaries all have short binary periods, and this is almost certainly a
selection effect. They are usually detected by the so-called “reflection
effect” where the sdB stars heat one side of the dM stars, causing brightness
variations with the orbital period. This effect will not be detected if the
binary separation is too large and the dM will not be observed as the sdB star
far outshines it (e.g. §3 of Reed & Stiening, 2004). The sdB+dM binaries in
our sample are tidally locked until the binary period exceeds $\sim$0.25 d
then they all rotate subsynchronously. The sdB+F/G stars in our sample rotate
commensurate with their apparently single sdB counterparts in the same
temperature range (Fig. 10) but supersynchronous to their long-period orbits.

Figure 12: Comparing envelope to interior rotation periods for stars where
multiplets have been detected in both g and p modes. PG 0048 and E211779126
only have lower limits for their g-mode pulsations. Errorbars are omitted when
smaller than the points. Point shapes and colours (on-line only) are the same
as Fig. 9.

Figure 12 shows the eight stars where multiplets have been detected in both p
and g modes, or there were sufficient pulsations that they should have been,
providing a lower limit. It is expected that p-mode pulsations mostly sample
the envelope while g-mode pulsations sample deeper into the star (Charpinet et
al., 2014). To date there have been five that appear to be radially-
differential rotators (Baran et al., 2017; Foster et al., 2015; Reed et al.,
2019; Reed et al., 2020; Baran et al., 2019) and three that likely rotate as
solid-bodies (Baran et al., 2012; Kern et al., 2017; Reed et al., 2019). There
are no obvious correlations with binarity as half of the apparently single and
sdB+F/G stars rotate differentially and half rotate like solid bodies. Both of
the sdB stars with dM companions rotate differentially. In all cases of
differential rotation, p-mode pulsations indicate faster rotation than g-mode
pulsations, with the possible exception of KIC 10139564 (Baran et al., 2012)
which has $1\sigma$ errorbars that overlap the solid-body rotation line.

##  5 Discussion

###  5.1 Campaign 7 Pulsators

In this paper we analyzed four newly-discovered sdBV stars and add them to the
growing number of sdBV stars with a large fraction of identified pulsation
modes. We detect asymptotic g-mode period spacing sequences in all four, with
three having $\Pi_{\ell=1}$ values right near 250 s and the other near to 200
s. Only three other sdBV stars have such unusually short asymptotic spacings;
two are hot p+g pulsators and the other another cool g-mode-only pulsator. So
the extremely low asymptotic spacings only occur for stars at both extremes.
However, stars with similar physical properties to each group also have
“normal” spacings near to 250 s and so this mystery has yet to be solved.

Three of the stars in C7 are apparently single, while we have discovered EPIC
218366972 to have RV variations, indicative of a white dwarf companion in a
5.92 d orbit. Somewhat surprisingly we do not detect any multiplets in EPIC
218366972 and so presume the rotation period to be $>45$ d. Only in EPIC
218717602 do we detect multiplets which indicate a rotation period near 7 d.
We could have anticipated detecting two sdBV stars with multiplets based on
the 56% detection rate from K1 for periods under 45 days.

###  5.2 The Group of Kepler and TESS-observed Pulsating Subdwarf B Stars

We completed ensemble analyses so that modelers may use this information to
deduce physical processes within these stars. In our analyses we have only
included Kepler and TESS-observed sdBV stars as it provides a roughly
homogeneous observing sample with the highest quality data. In examining the
group, we think we have revealed some underlying relationships which may prove
fruitful for model study.

####  5.2.1 Pulsations

As has been known for some time, pulsators and non-pulsators overlap in the
Kiel diagram (Figs. 6 and 7), with a higher percentage of cool sdBV stars
pulsating with g modes. It has been observed that hotter stars favour p modes
and cooler ones favour g modes, and we also find this. Of the 43 stars in our
sample, the dividing line between p+g and g+p pulsators is 29 000-30 000 K.
The p/p+g pulsators cluster around the central instability contour indicating
that pulsation driving models are likely reasonably accurate. Driving models
for g-mode pulsations have been less certain (which is why instability
contours are not included in Fig. 7) as the temperature range seems strongly
dependent on the amount of enhancement of iron-group elements in the driving
region (e.g. Jeffery & Saio, 2006b; Hu et al., 2009). More problematic are the
hybrid pulsators. Originally discovered in a limited mid-temperature range
bordering p and g pulsations, Kepler and TESS observations detect them from 22
800 to 36 300 K. 49% of the stars in our sample are hybrid pulsators.
Pulsation driving models will need to account for the two hybrids with p modes
completely outside those instability contours and g-mode pulsations that span
from 22 800 to 36 300 K.

Our sample has seven stars (16%) with $T_{\rm eff}$ below 25 000 K and none
are in known binaries. Similarly there are no PCE (sdB+WD/dM) binaries with
$T_{\rm eff}$ above 31 500 K in our sample (also seven stars), though at least
two are known from ground-based observations; the sdB+dM binary PG 1336-018
(Kilkenny et al., 1998) and the sdB+WD binary KPD 1930+2752 (Billéres et al.,
2000). This poses two questions; Does binarity shut off pulsation driving
below 25 000 K? And does PCE binarity adversely effect pulsation driving above
31 500 K? It is likely too early to draw conclusions (particularly on the hot
side, where two systems are known) but this will be worth watching as the
sample is completed for K2-discovered pulsators and increased with TESS data.

Of the relationships we examined, the one with the clearest correlation
appears to be between $T_{\rm eff}$ and PAmax which shows a clear trend for
cooler stars to have longer periods (lower frequencies) of their highest-
amplitude pulsations. Reed et al. (2020) noted two g-mode outliers below the
trend that both have indications of smaller-than-canonical masses. If the
$T_{\rm eff}$–PAmax relationship is indicative of the resonant cavity size,
then stars along this relationship would likely have a common convective core
size (inner boundary of the resonant cavity) with varying envelope
thicknesses. Thicker envelopes equate to lower $T_{\rm eff}$, a larger
resonant cavity, and therefore longer PAmax. Another outlier (E248368658)
above the trend could indicate a higher-massed core.

Related to the $T_{\rm eff}$–PAmax relationship, we anticipated finding a
similar relationship with mean period spacings, $\Pi_{\ell=1}$, but do not. A
recent study by Uzundag et al. (2021) examined $\Pi_{\ell=1}$ in models with a
narrow range of total mass and two convective core masses. They note that
$\Pi_{\ell=1}\propto\bar{g}^{-1}R_{\star}/\left(R_{\star}-\,R_{core}\right)$.
From this, they found agreement with previous studies (e.g. Castellani et al.,
1985; Constantino et al., 2015; Ostrowski et al., 2021), that smaller
convective cores have smaller $\Pi_{\ell=1}$ and stars with larger convective
cores, and consequently lower $\log g$, have larger $\Pi_{\ell=1}$. Those
results could be indicative of why Fig. 8 shows no obvious patterns.

####  5.2.2 Rotation

We examined the correlation between rotation period and $T_{\rm eff}$ (Fig.
10), as noted by Reed et al. (2014), and find that it only seems to hold at
extreme values of $T_{\rm eff}$. In our sample there are no fast-rotating cool
nor slowly-rotating hot sdBV stars with no known binaries at all in our cool
sample and no PCE (sdB+WD/dM) binaries in our hot sample. The mid-$T_{\rm
eff}$ star E217280630 has a rotation period of 7 days which is in a range
dominated by sdB+WD binaries. It would be worth obtaining additional
observations to search for a white dwarf companion.

Unfortunately there is currently little hope for additional constraints on
those stars with only lower limits on their rotation periods. No ground-based
observations have resolved multiplets in g-modes of sdBV stars and
transparency variations, which occur on similar time scales to those pulsation
periods, make it extremely difficult even to observe g-mode sdBV pulsations
without space-bourne telescopes. As K1 observations show, only long-duration
space-based observations are likely to fully resolve frequency multiplets and
provide rotation periods.

Patterns do emerge when comparing orbital to rotation periods of known binary
systems. First, all PCE stars in our sample with binary periods longer than
0.3 days rotate subsynchronously while those shorter than 0.3 days are
synchronized.

All of our sdB+WD systems with measured rotation periods are between 28 and 51
days. There are two known sdB+WD systems with shorter binary periods and also
rotation periods under one day (Fig 11). While 44% of our sample only have
lower limits near 45 days, we would speculate that sdB+WD systems, which have
undergone two common-envelope phases, have a common PCE initial rotation
period near 45 days which then evolves towards shorter periods.

We see a very similar pattern in our sample of sdB+dM binaries. Four stars
have periods near 10 days, three stars are tidally locked with periods under
0.3 days, and two are in between and differentially rotating with the envelope
spinning faster. This has two possibilities; either the rotation of these
binaries, which have only had one common-envelope epoch, is correlated with
the binary period, or these PCE systems also have a common initial rotation
period which evolves to shorter periods. We think the latter is the most
likely explanation, particularly since the differential rotators have faster-
spinning envelopes which are likely being spun up by the companions, through
tidal interactions. Furthermore, the dissipation of tidal energy can be
achieved in the form of pulsations.

Our sample of sdB+F/G stars are only on the hot end and rotate faster than the
average but not dissimilar to single sdB stars in the same temperature range.
It is presumed their temperatures are related to thinner envelopes yet
material deeper within (g mode) sdB stars is correlated with slower rotation.
As these stars likely formed via RLOF perhaps that mechanism produces faster
rotators, or conversely, RLOF does not slow rotation during mass loss like the
CE mechanism does. A larger sample of sdB+F/G stars may answer this question.

It is more difficult to find meaning with differential rotation (Fig. 12).
Both of the sdB stars with dM companions for which we have both p and g-mode
multiplets have differential rotation. It is unlikely a common property of all
sdB stars with dM companions as two sdB stars with dM companions in Fig. 11
have g-mode multiplets indicating they are tidally locked. As all the sdB+dM
rotation periods are short, TESS data will likely expand this sample. Our
sample of sdB+F/G stars and those without known companions shows a mixture. We
would encourage investigations to determine if any of these systems have
short-period companions, as difficult as that may be. Only by ruling out
binarity with periods up to ten days can we determine if differential rotation
is inherent to a property unrelated to binarity. If it is not related to
binarity, then a companion “spinning up” sdB stars cannot be the cause and it
must therefore be related to mass loss near the tip of the red giant branch.

The above relationships likely relate to mass loss and angular momentum
transport during their formation mechanisms. Another piece of evidence is the
comparison between binary and rotation periods. Most PCE sdB stars in our
sample rotate subsynchronously to their orbit unless it is less than about
half a day. This information should be useful in modeling PCE binaries.

###  5.3 The Future

Our examination of the group properties includes an intermediate sample from
K2 and a very preliminary sample from TESS. There are over a dozen suspected
pulsators observed during K2 which have yet to be thoroughly analyzed and TESS
has now observed thousands of sdB stars, with perhaps a couple hundred new
pulsators. Those data should produce a statistically significant sample from
which to examine pulsation properties. However, even with an expanded sample,
rotation is likely to be problematic as K1 showed us that we really need
continuous data for more than a year before we are likely to resolve frequency
multiplets.

We did not examine individual periods of the stars in our ensemble in our
analyses. So in addition to the properties discussed in this paper, we
encourage modelers to examine properties which include trapped modes, radial
indices where asymptotic sequences begin and end, the “hook’ feature in
asymptotic sequences, and the standard deviation in $\Pi$ (e.g. Constantino et
al., 2015). Additionally, it may well be worth a look at other “hook”-feature
pulsators to see if they too have a different linear asymptotic period
sequence shortward of the bend, as EPIC 218366972 does.

Separate from pulsations, GAIA is providing reliable parallaxes (Bailer-Jones
et al., 2018; Gaia Collaboration et al., 2021), from which distances, radii,
and masses can be determined (e.g. Kilkenny et al., 2019; Baran et al., 2019;
Reed et al., 2020). In combination with the remaining K2 sdBV stars, and the
many yet to be discovered by TESS, powerful observational tools are becoming
available. We are likely on the brink of understanding the underlying
relationships hinted at in this paper.

Data Availability Statement: The data underlying this article are available in
the Mikulski Archive for Space Telescopes (MAST). https://archive.stsci.edu/
Data obtained with the Nordic Optical Telescope is available after a one-year
proprietary period. http://www.not.iac.es/archive/

ACKNOWLEDGMENTS: Funding for this research was provided by the National
Science Foundation grant #1312869 and NASA grant 14-K2GO2-2-0047 as part of
the K2 guest observer program. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the authors and do not
necessarily reflect the views of the National Science Foundation or NASA. JAC
was funded by the Missouri Space Grant, which is funded by NASA. ASB
gratefully acknowledges financial support from the Polish National Science
Centre under projects No. UMO-2017/26/E/ST9/00703 and
NOU̇MO-2017/25/B/ST9/02218. This paper includes data obtained by the _Kepler_
mission. Funding for the _Kepler_ mission is provided by the NASA Science
Mission directorate. Data presented in this paper were obtained from the
Mikulski Archive for Space Telescopes (MAST). STScI is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA
Office of Space Science via grant NNX13AC07G and by other grants and
contracts.

The spectroscopic observations used in this work were obtained with the Nordic
Optical Telescope at the Observatorio del Roque de los Muchachos and operated
jointly by Denmark, Finland, Iceland, Norway, and Sweden.

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Table 7: Spectroscopic and pulsation properties of published Kepler and TESS-observed sdBV stars organized by binary type, pulsation type, and then decreasing effective temperature. Column 1 provides the KIC, EPIC identifications, column 2 other identifications, column 3 provides the binary status where sdB is listed for single stars, sdB+WD for those with white dwarf companions, sdB+dM for those with M-dwarf main sequence companions and sdB+F or sdB+G for those with main sequence F or G companions. Column 4 provides the pulsation type where p are p-mode only pulsators, p+g are predominantly p mode hybrid pulsators, g are g-mode only pulsators, and g+p are predominantly g mode hybrid pulsators. Column 5 lists the g-mode $\ell=1$ asymptotic period spacing, Column 6 the maximum amplitude and Column 7 the pulsation period of maximum amplitude. Columns 8 and 9 provide spectroscopic properties. Column 10 provides the binary period, column 11 provides the rotation rate and if two values are given, the first is derived from p-mode multiplets (for the envelope) and the second from g-mode multiplets (for deeper interior), and column 12 has references. The references are 1: Østensen et al. (2014), 2: Reed et al. (2020) 3: Baran et al. (2012), 4: Baran et al. (2017), 5: Foster et al. (2015), 6: Sahoo et al. (2020), 7: Charpinet et al. (2011), 8: Reed et al. (2011), 9: Uzundag et al. (2017), 10: Kern thesis 11: Kern et al. (2017), 12: Silvotti et al. (2019), 13: Bachulski et al. (2016), 14: this work, 15: Baran et al. (2015), 16: Ketzer et al. (2017), 17: Charpinet et al. (2019), 18: Reed et al. (2014), 19: Reed et al. (2018), 20: Reed et al. (2019), 21: Reed et al. (2020), 22: Telting et al. (2012); Kern et al. (2018), 23: Telting et al. (2014), 24: Østensen et al. (2014), 25: Reed et al. (2016), 26: Baran et al. (2016), 27: Østensen et al. (2010), 28: Baran et al. (2019), 29: Baran et al. (2018), 30: Pablo et al. (2012), 31: Baran & Winans (2012), 32: Østensen et al. (2012), 33: Jeffery et al. (2017) Notes: ♢ indicates stars with pulsation results only from the one month survey data. ⋆ No multiplets were detected for these stars which means spin periods longer than the observations, or that the pulsation axis is very nearly pole-on. In these cases, a lower limit of the spin period is provided. † RV excesses have been measured for these stars (Sanjayan, S. et al. _in press_). K1718290 is listed as a blue horizontal branch (BHB) star.  Kepler ID | Other | Binary | Type | $\Pi_{\ell=1}$ | $A_{\rm max}$ | $P_{\rm Amax}$ | $T_{\rm eff}$ | $\log g$ | Porbit | Pspin | Ref.
---|---|---|---|---|---|---|---|---|---|---|---
|  | Status |  | (s) | (ppt) | ($\mu$Hz) | (K/1 000) | (dex, cgs) | (d) | (d) |
K2991276 |  | sdB | p | – | 2.25 | 8201.2 | 33.9 (2) | 5.82 (4) | – | 6.3 | 1
E248411044 | UY Sex | sdB | p | – | 4.89 | 7038.163 | 33.03 (20) | 5.88 (1) | – | 24.6 (3.5) | 2
K10139564 |  | sdB | p+g | 207/310 | 7.95 | 5760.2 | 31.86 (0.13) | 5.67 (3) | – | 25.6 (1.8)/23.12 (62) | 3
E211779126 | 2M 0856+1701 | sdB | g+p | 256 | 0.795 | 266.5 | 28.54 (8) | 5.39 (1) | – | $16/>45^{\star}$ | 4
K3527751 |  | sdB | g+p | 266.4 (2) | 7.11 | 255.7 | 27.82 (16) | 5.35 (3) | – | 15.3 (7)/42.6 (3.4) | 5
T169285097 | SB 815 | sdB | g+p | 265.09 (6) | 1.642 (21) | 258.1879 (29) | 27.20 (55) | 5.39 (10) | – | $>12^{\star}$ | 6
K5807616♢ | KPD1943+4058 | sdB | g+p | 242.12 (62) | 0.142 | 167.8 | 27.1 | 5.51 | – | – | 7,8
K10001893 |  | sdB | g+p | 268.0(5) | 1.162 | 274.3 | 26.7 | 5.3 | – | $>715^{\star}$ | 9
K2569576 | NGC 6791 B3 |  †sdB | g+p | 252.27 (66) | 3.601 | 198.4 | 24.25 (46) | 5.17 (5) | – | 64.5 (8.2) | 10
K2697388 |  | sdB | g+p | 240.06 (19) | 33.29 | 156.4 | 23.39 (12) | 5.29 (2) | – | 41.9 (3.6)/52.8 (9.3) | 11
E220641886 | HD 4539 | sdB | g+p | 256.5 | 0.80 | 83.4 | 22.80 (16) | 5.20 (2) | – | $>45^{\star}$ | 12
E212707862 |  | sdB | g | 252.6 (1.1) | 0.434 | 296.9 | 28.30 (16) | 5.48 (3) | – | 80 | 13
E215776487 |  | sdB | g | 247.8 (1.5) | 0.489 | 197.0 | 27.86 (16) | 5.45 (2) | – | – | 14
K8302197 |  | sdB | g | 258.61 (62) | 0.87 | 187.0 | 27.45 (20) | 5.44 (3) | – | $>715^{\star}$ | 15
E203948264 |  | sdB | g | 261.3 (1.1) | 0.722 | 203.5 | 26.76 (61) | 5.26 (9) | – | 45.9 (8) | 16
T457168745 | PG0342+026 | sdB | g | 232.25 (30) | 0.819 (21) | 219.274 (6) | 26.0 (1.1) | 5.59 (12) | – | $>12^{\star}$ | 6
T67584818 | SB 459 | sdB | g | 259.16 (56) | 1.72 (5) | 207.314 (9) | 24.9 (5) | 5.35 (10) | – | $>12^{\star}$ | 6
E218717602 |  | sdB | g | 263.55 (61) | 3.71 | 168.1 | 24.47 (16) | 5.17 (2) | – | – | 14
K2437937 | NGC 6791 B5 |  †sdB | g | 248.9 (1.3) | 1.03 | 240.5 | 23.84 (68) | 5.31 (9) | – | 91.3 (14.1) | 10
T278659026 | EC 21494-7018 | sdB | g | 196.8 | 0.3184 (43) | 249.269 (3) | 23.72 (23) | 5.65 (3) | – | $>12^{\star}$ | 17
E217280630 |  | sdB | g | 207.35 (21)/195.28 (39) | 0.39 | 75.5 | 22.77 (15) | 5.01 (2) | – | 7 | 14
K10670103 |  | sdB | g | 251.6 (2) | 13.99 | 138.1 | 21.49 (54) | 5.14 (5) | – | 88 (8) | 18
K1718290 |  | BHB | g | 276.3 (1) | .268 | 91.9 | 21.80 (14) | 4.67 (3) | – | 96.5 | 33
E211823779 |  | sdB+F1V | p | – | 2.052 | 7131.9538 (5) | 36 | 6 | – | 11.5 (8) | 19
E211938328 | LB 378, EGGR 266 | sdB+F6V | p | – | 2.00 | 9648.77 | 32 | 5.8 | 635 (146) | 21.5 (6) | 18
E212508753 | PG 1315-123 | sdB+G | p+g | 236.5 (1.3) | 1.79 | 8116.1/497.62 | 36.23 (71) | 5.61 (9) | $>100$ | 15.83 (19)/16.18(57) | 20
E220614972 | PG 0048+091 | sdB+G | p+g | 207.45 (40) | 1.792 | 5339.2/443.7 | 32.46 (27) | 5.77 (6) | $>100$ | 4.39 (48)/$>45^{\star}$ | 20
T13145616 | CD$-$28∘ 1974 | sdB+F/G | g+p | 268.85 (32) | 1.894 (91) | 469.215 (12) | 29.60 (38) | 5.55 (9) | $>1\,000$ | $>12^{\star}$ | 21
K11558725 |  | sdB+WD | g+p | 244.45 (32) | 0.95 | 274.64187 (2) | 27.91 (32) | 5.41 (1) | 10.055 (5) | 44 | 22
K7668647 | FBS1903+432 | sdB+WD | g+p | 248 | 0.42 | 194.5 | 27.7 (3) | 5.50 (3) | 14.174 (4) | 50.5 (5) | 23
K10553698 |  | sdB+WD | g+p | 263.15 | 1.367 | 202.0 | 27.42 (0.29) | 5.44 (24) | 3.387 (14) | 42.9 | 24
E218366972 |  | sdB+WD | g | 254.95 (50) | 2.742 | 249.8 | 28.16 (11) | 5.44 (2) | 5.92 (1) | $>45^{\star}$ | 14
E201206621 | PG 1142-037 | sdB+WD | g | 267.9 (10) | 0.34 | 137.38 | 27.954 (54) | 5.32 (1) | 0.54109 (2) | $>45^{\star}$ | 25
E211696659 |  | sdB+WD | g | 227.05 (56) | 0.188 | 233.41 | 27.57 (30) | 5.70 (3) | 3.1604 (15) | 28.4 (1.4) | 17
K7664467 |  | sdB+WD | g | 263 | 0.495 | 247.0 | 27.44 (12 ) | 5.38 (2) | 1.5591 (6) | 35.1 (6) | 26
Table 8: continued Kepler ID | Other | Binary | Type | $\Pi_{\ell=1}$ | $A_{\rm max}$ | $P_{\rm Amax}$ | $T_{\rm eff}$ | $\log g$ | Porbit | Pspin | Ref.
---|---|---|---|---|---|---|---|---|---|---|---
E246683636 | V1405 Ori | sdB+dM | p+g | 225 | 4.534 | 4703.538 | 31.4 (2) | 5.47 (4) | 0.398023 (0.3) | 0.555 (29)/4.3 | 2
K9472174♢ | 2M1938+4603 | sdB+dM | p+g | 255.63 (30) | 3.01 | 3712.369 | 29.6 | 5.42 | 0.1258 | 0.1258 | 8,27
E246387816 | EQ Psc | sdB+dM | g+p | 233 | 2.21 | 495.8 | 28.69 (5) | 5.64 (1) | 0.80083 (1) | 9.4 | 28
E228755638 | HW Vir | sdB+dM | g+p | – | 0.100 | 309.2 | 28.07 (5) | 5.51 (1) | 0.117 | 0.117 | 29
K11179657 |  | sdB+dM | g+p | 259.6 (14) | 1.66 | 186.5 | 26 | 5.14 | 0.394 | 7.2 | 30,31
E246023959 | PHL 457 | sdB+dM | g+p | 259 (2) | 1.96 | 265.5 | 26.69 (6) | 5.31 (1) | 0.3128903 (4) | 2.5/4.6 | 28
K2991403 |  | sdB+dM | g | 268.52 (74) | 1.07 | 334.8 | 27.3 | 5.43 | 0.443 | 10.46 | 30,31
K2438324 | NGC 6791 B4 | sdB+dM | g | 236.2 (2.1) | 1.40 | 216.3 | 27.10 (82) | 5.69 (10) | 0.398 | 9.21 (18) | 10,31