# Discovery of a double detonation thermonuclear supernova progenitor

Thomas Kupfer Department of Physics and Astronomy, Texas Tech University, PO
Box 41051, Lubbock, TX 79409, USA  Evan B. Bauer Center for Astrophysics
$\mid$ Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA  Jan van
Roestel Division of Physics, Mathematics and Astronomy, California Institute
of Technology, Pasadena, CA 91125, USA  Eric C. Bellm DIRAC Institute,
Department of Astronomy, University of Washington, 3910 15th Avenue NE,
Seattle, WA 98195, USA  Lars Bildsten  Kavli Institute for Theoretical
Physics, University of California, Santa Barbara, CA 93106, USA  Department of
Physics, University of California, Santa Barbara, CA 93106, USA  Jim Fuller
Division of Physics, Mathematics and Astronomy, California Institute of
Technology, Pasadena, CA 91125, USA  Thomas A. Prince  Division of Physics,
Mathematics and Astronomy, California Institute of Technology, Pasadena, CA
91125, USA  Ulrich Heber Dr. Karl Remeis-Observatory & ECAP, Astronomical
Institute, Friedrich-Alexander University Erlangen-Nuremberg (FAU),
Sternwartstr. 7, 96049 Bamberg, Germany  Stephan Geier  Institut für Physik
und Astronomie, Universität Potsdam, Haus 28, Karl-Liebknecht-Str. 24/25,
D-14476 Potsdam-Golm, Germany  Matthew J. Green  Department of Astrophysics,
School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel
Shrinivas R. Kulkarni Division of Physics, Mathematics and Astronomy,
California Institute of Technology, Pasadena, CA 91125, USA  Steven Bloemen
Department of Astrophysics/IMAPP, Radboud University Nijmegen, P.O. Box 9010,
6500 GL Nijmegen, The Netherlands  Russ R. Laher IPAC, California Institute of
Technology, 1200 E. California Blvd, Pasadena, CA 91125, USA  Ben Rusholme
IPAC, California Institute of Technology, 1200 E. California Blvd, Pasadena,
CA 91125, USA  David Schneider  Dr. Karl Remeis-Observatory & ECAP,
Astronomical Institute, Friedrich-Alexander University Erlangen-Nuremberg
(FAU), Sternwartstr. 7, 96049 Bamberg, Germany

(Received June 1, 2019; Revised January 10, 2019)

###### Abstract

We present the discovery of a new double detonation progenitor system
consisting of a hot subdwarf B (sdB) binary with a white dwarf companion with
an $P_{\rm orb}$=76.34179(2) min orbital period. Spectroscopic observations
are consistent with an sdB star during helium core burning residing on the
extreme horizontal branch. Chimera light curves are dominated by ellipsoidal
deformation of the sdB star and a weak eclipse of the companion white dwarf.
Combining spectroscopic and light curve fits we find a low mass sdB star,
$M_{\rm sdB}=0.383\pm 0.028$ M⊙ with a massive white dwarf companion, $M_{\rm
WD}=0.725\pm 0.026$ M⊙. From the eclipses we find a blackbody temperature for
the white dwarf of $26,800$ K resulting in a cooling age of $\approx$25 Myrs
whereas our MESA model predicts an sdB age of $\approx$170 Myrs. We conclude
that the sdB formed first through stable mass transfer followed by a common
envelope which led to the formation of the white dwarf companion $\approx$25
Myrs ago.

Using the MESA stellar evolutionary code we find that the sdB star will start
mass transfer in $\approx$6 Myrs and in $\approx$60 Myrs the white dwarf will
reach a total mass of $0.92$ M⊙ with a thick helium layer of $0.17$ M⊙. This
will lead to a detonation that will likely destroy the white dwarf in a
peculiar thermonuclear supernova. PTF1 J2238+7430 is only the second confirmed
candidate for a double detonation thermonuclear supernova. Using both systems
we estimate that at least $\approx$1 % of white dwarf thermonuclear supernovae
originate from sdB+WD binaries with thick helium layers, consistent with the
small number of observed peculiar thermonuclear explosions.

Eclipsing binary stars(444) — White dwarf stars(1799) — Close binary
stars(254) — B subdwarf stars(129)

††journal: ApJL††facilities: PO:1.2m (PTF), PO:1.2m (ZTF), Hale (DBSP),
ING:Herschel (ISIS), Keck:I (HIRES), Keck:II (ESI), Hale (Chimera)††software:
Gatspy (VanderPlas & Ivezić, 2015; Vanderplas, 2015), FITSB2 (Napiwotzki et
al., 2004), LCURVE (Copperwheat et al., 2010), emcee (Foreman-Mackey et al.,
2013), MESA (Paxton et al., 2011, 2013, 2015, 2018, 2019), Matplotlib (Hunter,
2007), Astropy (Astropy Collaboration et al., 2013, 2018), Numpy (Oliphant,
2015), ISIS (Houck & Denicola, 2000), MAKEE
(https://sites.astro.caltech.edu/~tb/makee/)

##  1 Introduction

Most hot subdwarf B stars (sdBs) are core helium burning stars with masses
around 0.5 M⊙ and thin hydrogen envelopes (Heber 1986, 2009, 2016). A large
number of sdB stars are in close orbits with orbital periods of $P_{\rm
orb}$$<10$ days (Napiwotzki et al., 2004; Maxted et al., 2001), with the most
compact systems reaching orbital periods of $\lesssim 1$ hour (e.g. Vennes et
al. 2012; Geier et al. 2013; Kupfer et al. 2017a, b, 2020a, 2020b). The only
way to form such tight binaries is orbital shrinkage through a common envelope
phase followed by the loss of angular momentum due to the radiation of
gravitational waves (Han et al., 2002, 2003; Nelemans, 2010).

SdB binaries with white dwarf (WD) companions which exit the common envelope
phase at $P_{\rm orb}$$\lesssim$2 hours will reach contact while the sdB is
still burning helium (Bauer & Kupfer, 2021). Due to the emission of
gravitational waves the orbit of the binary will shrink until the sdB fills
its Roche Lobe at a period of $\approx 30-100$ min, depending on the
evolutionary stage and envelope thickness of the hot subwarf (e.g. Savonije et
al. 1986; Tutukov & Fedorova 1989; Tutukov & Yungelson 1990; Iben & Tutukov
1991; Yungelson 2008; Piersanti et al. 2014; Brooks et al. 2015; Neunteufel et
al. 2019; Bauer & Kupfer 2021).

The known population of sdB + WD binaries consists mostly of systems with
orbital periods too large to start accretion before the sdB turns into a WD
(Kupfer et al., 2015). Currently only four detached systems with a WD
companion are known to have $P_{\rm orb}$$<2$ hours (Vennes et al., 2012;
Geier et al., 2013; Kupfer et al., 2017a, b; Pelisoli et al., 2021). Just
recently Kupfer et al. (2020a, b) discovered the first two Roche lobe filling
hot subdwarfs as part of a high-cadence Galactic Plane survey using the Zwicky
Transient Facility (Kupfer et al., 2021). Both systems can be best explained
as Roche Lobe filling sdOB stars which have started mass transfer to a WD
companion. The light curves in both systems show deep eclipses from an
accretion disk. Due to their high effective temperatures, both sdOB stars are
predicted to be in a short lived phase where the sdOB undergoes residual
hydrogen shell burning.

The most compact known sdB binary where the sdB is still undergoing core-
helium burning is CD–30∘11223\. The binary has an orbital period $P_{\rm
orb}$=70.5 min and a high mass WD companion ($M_{\rm WD}\approx 0.75$ M⊙;
Vennes et al. 2012; Geier et al. 2013). The sdB in CD–30∘11223 will begin
transferring helium to its WD companion in $\approx 40$ Myr when the system
has shrunk to an orbital period $P_{\rm orb}$$\approx$40 min. After the WD
accretes $\approx$0.1 M⊙, helium burning is predicted to be ignited unstably
in the accreted helium layer on the WD surface (Brooks et al., 2015; Bauer et
al., 2017). This could either disrupt the WD even when the mass is
significantly below the Chandrasekhar mass, a so-called double detonation
supernova (e.g. Livne 1990; Livne & Arnett 1995; Fink et al. 2010; Woosley &
Kasen 2011; Wang & Han 2012; Shen & Bildsten 2014; Wang 2018) or just detonate
the He-shell without disrupting the WD which results in a faint and fast .Ia
supernova with subsequent weaker He-flashes (Bildsten et al., 2007; Brooks et
al., 2015). Therefore, systems like CD–30∘11223 are predicted to be either the
progenitors for double detonation thermonuclear supernovae or perhaps faint
and fast .Ia supernovae that do not disrupt the WD.

De et al. (2019, 2020) presented the discovery of a sample of calcium-rich
transients consistent with a thick helium shell double detonation on a sub-
Chandrasekhar-mass WD (Polin et al., 2019, 2021). The majority of these
transients are located in old stellar populations with only a small sub-sample
found in in star forming environments.

The question remains just how common systems like CD–30∘11223 are. To address
this question we have conducted a search for (ultra-)compact post-common
envelope systems using the Palomar Transient Factory (PTF; Law et al. 2009;
Rau et al. 2009) and subsequently the Zwicky Transient Facility (ZTF; Graham
et al. 2019; Masci et al. 2019) based on a color selected sample from Pan-
STARRS data release 1. The PTF used the Palomar 48′′ Samuel Oschin Schmidt
telescope to image up to $\approx 2000$ deg2 of the sky per night to a depth
of R${}_{\rm mould}\approx 20.6$ mag or $g^{\prime}\approx 21.3$ mag. PTF was
succeeded by the Zwicky Transient Facility which started science operation in
March 2018 using the same telescope but a new camera with a field-of-view of
$47$ deg2. Here we report the discovery of a new thermonuclear supernova
double detonation progenitor system consisting of an sdB with a WD companion:
PTF1 J223857.11+743015.1 (hereafter PTF1 J2238+7430) with orbital period of 76
min showing similar properties to CD–30∘11223.

##  2 Observations

###  2.1 Photometry

As part of the Palomar Transient Factory (PTF), the Palomar 48-inch (P48)
telescope imaged the sky every night. The reduction pipeline for PTF applies
standard de-biasing, flat-fielding, and astrometric calibration to raw images
(Laher et al., 2014). Relative photometry correction is applied and absolute
photometric calibration to the few percent level is performed using a fit to
SDSS fields observed in the same night (Ofek et al., 2012). The lightcurve of
PTF1 J2238+7430 has 144 epochs, with good photometry in the Rmould band with a
typical uncertainty of 0.01-0.02 mag. The majority of observations were
conducted during the summer months June - August 2013 and 2014 and the cadence
is highly irregular, ranging from a few minutes to years. The object was also
observed as part of the Zwicky Transient Facility (ZTF) public survey (Graham
et al., 2019; Bellm et al., 2019). Image processing of ZTF data is described
in full detail in Masci et al. (2019). We extracted the light curve from ZTF
data release 6 which consists of 34 observations in ZTF-$r$ taken randomly
over $\approx$ 1.5 years between August 2018 and November 2019.

High-cadence observations were conducted using the Palomar 200-inch telescope
with the high-speed photometer CHIMERA (Harding et al., 2016) which is a
2-band photometer which uses frame-transfer, electron-multiplying CCDs to
achieve 15 ms dead time covering a $5\times 5$ arcmin field of view.
Simultaneous optical imaging in two bands is enabled by a dichroic beam
splitter centered at 567 nm. Data reduction was carried out with the ULTRACAM
pipeline (Dhillon et al., 2007) customized for CHIMERA. All frames were bias-
subtracted and flat-fielded. 1300 observations in $g^{\prime}$ and
$r^{\prime}$ with a 5 sec exposure time were obtained on 2017-07-26 and 2700
observations in $g^{\prime}$ and $i^{\prime}$ with a 4 sec exposure time were
obtained on 2017-12-14.

Figure 1: Left panel: Phase folded at $P_{\rm orb}$=$76.341750$ min ZTF and
PTF light curve for PTF1 J2238+7430. Right panel: Radial velocity plotted
against orbital phase for PTF1 J2238+7430. The RV data were phase folded with
the orbital period and are plotted twice for better visualization. The
residuals are plotted below.

###  2.2 Spectroscopy

Optical spectra were obtained with the Palomar 200-inch telescope and the
Double-Beam Spectrograph (DBSP; Oke & Gunn 1982) using a low resolution mode
($R\sim 1500$). 31 consecutive exposures were obtained on 2017-05-25 and
2017-05-29 and 15 consecutive exposures were obtained on 2017-05-25 using a
180 sec exposure time. Each night an average bias and normalized flat-field
frame was made out of 10 individual bias and 10 individual lamp flat-fields.
To account for telescope flexure, an arc lamp was taken at the position of the
target after each observing sequence. For the blue arm, FeAr and for the red
arm, HeNeAr arc exposures were taken. Both arms of the spectrograph were
reduced using a custom PyRAF-based pipeline
111https://github.com/ebellm/pyraf-dbsp(Bellm & Sesar, 2016). The pipeline
performs standard image processing and spectral reduction procedures,
including bias subtraction, flat-field correction, wavelength calibration,
optimal spectral extraction, and flux calibration.

Additionally PTF1 J2238+7430 was also observed with the William Herschel
Telescope (WHT) and the ISIS spectrograph (Carter et al., 1993) using a medium
resolution mode (R600B grating, $R\approx 2500$). 10 consecutive exposures
with an exposure time of 180 sec were obtained on 2017-07-26. 10 bias frames
were obtained to construct an average bias frame and 10 individual lamp flat-
fields were obtained to construct a normalized flat-field. CuNeAr arc
exposures were taken before and after the observing sequence to correct for
instrumental flexure. One dimensional spectra were extracted using optimal
extraction and were subsequently wavelength and flux calibrated.

To obtain high-resolution spectra, PTF1 J2238+7430 was observed with
Keck/HIRES and Keck/ESI. We obtained 5 consecutive exposures with Keck/HIRES
on 2017-08-14 and 2017-08-30 as well as 14 consecutive exposures with Keck/ESI
on 2018-07-20. ThAr arc exposures were taken at the beginning of the night.
The spectra were reduced using the
MAKEE222https://sites.astro.caltech.edu/~tb/makee/ pipeline following the
standard procedure: bias subtraction, flat fielding, sky subtraction, order
extraction, and wavelength calibration.

Figure 2: Left panel: Fit of synthetic LTE models to the hydrogen Balmer lines
of a coadded DBSP spectrum. The normalized fluxes of the single lines are
shifted for better visualisation. Right panel: Fits of $v_{\rm rot}\sin{i}$ to
the helium lines seen in the HIRES and ESI spectra. The atmospheric parameters
were fixed to the values derived from the WHT and DBSP spectra.

##  3 Orbital and atmospheric parameters and light curve fitting

As evident in Fig. 1 PTF1 J2238+7430 shows strong periodic ellipsoidal
variability in its light curve at $P_{\rm orb}$ = 76.341750(1) min. This
variability is caused by the tidal deformation of the sdB primary under the
influence of the gravitational force of the companion. We use the PTF and the
ZTF lightcurve with its multi-year baseline and the Chimera light curves to
derive the orbital period of the systems. The analysis was done with the
Gatspy module for time series analysis which uses the Lomb-Scargle
periodogram333http://dx.doi.org/10.5281/zenodo.14833 (VanderPlas & Ivezić,
2015). The error was derived from bootstrapping.

Radial velocities were measured by fitting Gaussians, Lorentzians, and
polynomials to the hydrogen and helium lines to cover continuum, line, and
line core of the individual lines using the FITSB2 routine (Napiwotzki et al.,
2004). The procedure is described in full detail in Geier et al. (2011). We
fitted the wavelength shifts compared to the rest wavelengths using a
$\chi^{2}$-minimization. Assuming circular orbits, a sine curve was fitted to
the folded radial velocity (RV) data points (Fig. 1).

Atmospheric parameters such as effective temperature, Teff, surface gravity,
$\log{g}$, helium abundance, $\log{y}=\log\frac{n(He)}{n(H)}$, and projected
rotational velocity, $v_{\rm rot}\sin{i}$, were determined by fitting the
rest-wavelength corrected average DBSP, ISIS and HIRES spectra with metal-
line-blanketed LTE model spectra (Heber et al., 2000). Teff and $\log{g}$ were
derived from the Balmer and helium lines from the ISIS and DBSP spectra
whereas $\log{y}$ and $v_{\rm rot}\sin{i}$ were measured with the HIRES
spectra. High-resolution echelle spectra are not well suited to measure Teff
and $\log{g}$ because the broad hydrogen absorption lines span several
individual echelle orders and merging of the echelle spectra could introduce
systematic errors. The full procedure is described in detail in Kupfer et al.
(2017a, b). PTF1 J2238+7430 shows typical Teff, $\log{g}$, and $\log{y}$ and
$v_{\rm rot}\sin{i}$=185$\pm$5 ${\rm km}\,{\rm s}^{-1}$. The rotational
velocity is consistent with a tidally locked sdOB star (see Sec. 4.1). Figure
2 shows the main Balmer and helium lines with the best fit to the data. Table
1 summarizes the atmospheric and orbital parameters.

Table 1: Overview of the measured and derived parameters for PTF1 J2238+7430

Right ascension | RA [hrs] | 22:38:57.11
---|---|---
Declination | Dec $[^{\circ}]$ | +74:30:15.1
Magnitudeb |  $g$ [mag] | 15.244$\pm$0.023
Parallaxa |  $\varpi$ [mas] | $1.0001\pm 0.0225$
Distance |  $d$ [kpc] | $1.00\pm 0.03$
Absolute Magnitude | $M_{\rm g}$ [mag] | $4.40\pm 0.20$
(reddening corrected)
Proper motiona (RA) |  $\mu_{\alpha}$cos$(\delta)$ [mas yr-1] | $0.344\pm 0.056$
Proper motiona (Dec) |  $\mu_{\delta}$ [mas yr-1] | $-1.833\pm 0.051$
Atmospheric parameters of the sdB
Effective temperaturec | Teff [K] | 23 600$\pm$400
Surface gravityc | $\log{g}$ | 5.42$\pm$0.06
Helium abundanced | $\log{y}$ |  $-$2.11$\pm$0.03
Projected rotational velocityd |  $v_{\rm rot}\sin{i}$ [${\rm km}\,{\rm s}^{-1}$] | 185$\pm$5
Orbital parameters
|  $T_{0}$ [BMJD UTC] | 57960.47584170(3)
Orbital period |  $P_{\rm orb}$ [min] | 76.341750(1)
RV semi-amplitude |  $K$ [${\rm km}\,{\rm s}^{-1}$] | $378.0\pm 3.7$
System velocity |  $\gamma$ [${\rm km}\,{\rm s}^{-1}$] | $-6.2\pm 2.14$
Binary mass function |  $f_{\rm m}$ [M⊙] | 0.0597$\pm$0.0020
Derived parameters
Mass ratio | $q=\frac{M_{\rm WD}}{M_{\rm sdB}}$ | $0.528\pm 0.020$
sdB mass |  $M_{\rm sdB}$ [M⊙] | $0.383\pm 0.028$
sdB radius |  $R_{\rm sdB}$ [R⊙] | $0.190\pm 0.003$
WD mass |  $M_{\rm WD}$ [M⊙] | $0.725\pm 0.026$
WD radius |  $R_{\rm WD}$ [M⊙] | $0.0109^{+0.0002}_{-0.0003}$
WD blackbody temperature | Teff [K] | $26,800\pm 4600$
Orbital inclination |  $i$ [∘] | $88.4^{+1.6}_{-3.3}$
Separation |  $a$ [R⊙] | $0.615\pm 0.010$
Roche filling factor |  $R_{\rm sdB}$/$R_{\rm Rochelobe}$ | $0.951\pm 0.010$

a from Gaia eDR3 (Gaia Collaboration et al., 2016, 2021)

b from PanSTARRS DR1 (Chambers et al., 2016)

c adopted from from DBSP and ISIS

d adopted from ESI and HIRES

Figure 3: Chimera light curves un-binned (grey) and binned (black) shown
together with the LCURVE fits (red) observed optical SDSS bandpasses. The
lower two panels show the region when the WD is being eclipsed by the sdB. The
blue solid curve marks the same model without eclipses of the WD. The lower
panels show the region when the white dwarf is being eclipsed. Lower left
panel: $g^{\prime}$ light curve, Lower right panel: $r^{\prime}$ light curve

To model the lightcurves obtained with CHIMERA we used the LCURVE code
(Copperwheat et al., 2010). We use a Roche geometry, and the free parameters
in our fit are: the phase ($t_{0}$), the scaled radii ($r_{1,2}$), the mass
ratio $q$, the inclination $i$, secondary temperature $T_{\rm WD}$, and the
velocity scale ($\mathrm{[}K+K_{\rm WD}]/\sin i$). We use a passband-dependent
gravity-darkening law and use a gravity darkening value ($y_{g,r}$) from
Claret & Bloemen (2011) and find $\beta=0.425$ for $g^{\prime}$, $\beta=0.395$
for $r^{\prime}$, and $\beta=0.37$ for $i^{\prime}$. We assume an uncertainty
of $0.03$ on the value and use a Gaussian prior. We use fixed limb darkening
coefficients ($a_{\mathrm{1}}$, $a_{\mathrm{2}}$, $a_{\mathrm{3}}$,
$a_{\mathrm{4}}$) taken from Claret & Bloemen (2011). We use
$a_{\mathrm{1}}=0.82,a_{\mathrm{2}}=-0.65,a_{\mathrm{3}}=0.55$, and
$a_{\mathrm{4}}=-0.19$ for $g^{\prime}$,
$a_{\mathrm{1}}=0.81,a_{\mathrm{2}}=-0.89,a_{\mathrm{3}}=0.79$, and
$a_{\mathrm{4}}=-0.27$ for $r^{\prime}$, and
$a_{\mathrm{1}}=0.78,a_{\mathrm{2}}=-1.01,a_{\mathrm{3}}=0.91$, and
$a_{\mathrm{4}}=-0.31$ for $i^{\prime}$. We also model the relativistic
beaming ($F$) as in Bloemen et al. (2011). We calculate the beaming parameters
by assuming a blackbody spectrum and using the effective wavelength of the
$g^{\prime}$, $r^{\prime}$, and $i^{\prime}$ filters. We find $F=1.80$ for
$g^{\prime}$, $F=1.57$ for $r^{\prime}$, and $F=1.46$ for $i^{\prime}$. The
full approach is also described in Kupfer et al. (2017a, b, 2020a, 2020b) and
Ratzloff et al. (2019). In addition, we add a 2nd order polynomial to correct
for any long timescale trends which are the result of a changing airmass over
the course of the observations. The best value of $\chi^{2}$ for this model
was 1350 for 1300 data points for the g-band light curve which includes also a
weak eclipse of the hot WD. Although the eclipse is weak ($\leq 1$ %; Fig. 3),
the $\chi^{2}$ for the non-eclipsing solution is 1400 which is statistically
significantly worse compared to the solution with the weak eclipse. We use the
MCMC sampler emcee (Foreman-Mackey et al., 2013) to determine the best-fit
values and uncertainty on the parameters. Figure 3 shows the Chimera light
curves with the best fitted model. The lower panels are zoomed in around the
region when the WD is being eclipsed.

##  4 Results

###  4.1 System parameters

Although, PTF1 J2238+7430 is a single-lined binary we can derive system
parameters using the combined results from the light curve analysis with
results from the spectroscopic fitting. Parameters derived in this way by a
simultaneous fit to the Chimera light curves are summarized in Table 1. The
given errors are all $95\,\%$ confidence limits.

We find that PTF1 J2238+7430 consists of a low mass sdB with a high-mass WD
companion. We derive a mass ratio $q=M_{\rm sdB}/M_{\rm WD}=0.528\pm 0.020$, a
mass for the sdB $M_{\rm sdB}=0.383\pm 0.028$ M⊙, and a WD companion mass
$M_{\rm WD}=0.725\pm 0.026$ M⊙. PTF1 J2238+7430 is found to be eclipsing at an
inclination angle of $i=88.4^{+1.6}_{-3.3}\,{}^{\circ}$ which allows us to
measure the radius and the black-body temperature of the WD companion. We
determine a black-body temperature of $26,800\pm 4600$ K for the WD and a
radius of $R_{\rm WD}=0.0109^{+0.0002}_{-0.0003}$ R⊙. The radius was found to
be $<5\%$ above the zero-temperature value and is fully consistent with
predictions from Romero et al. (2019) for carbon-oxygen core white dwarfs.

Zahn (1977) predicted that the sdBs in close sdB binaries with orbital periods
below $\approx 0.3$ days should be synchronized to the orbit. More recently,
Preece et al. (2018) found that only the most compact sdB binaries should be
synchronized. From the system parameters we find that the sdB would have a
projected rotational velocity $v_{\rm rot}\sin{i}$$=181\pm 6$ ${\rm km}\,{\rm
s}^{-1}$ if synchronized to the orbit. The measured $v_{\rm
rot}\sin{i}$$=185\pm 5$ ${\rm km}\,{\rm s}^{-1}$ is consistent with a
synchronized orbit.

We calculate the absolute magnitude ($M_{\rm g}$) of PTF1 J2238+7430 using the
visual PanSTARRS g-band magnitude g=15.244$\pm$0.023 mag and the parallax from
Gaia eDR3 (Gaia Collaboration et al., 2016, 2021). Because the object is
located near the Galactic Plane, significant reddening can occur. Green et al.
(2019) present updated 3D extinction maps based on Gaia parallaxes and stellar
photometry from Pan-STARRS 1 and 2MASS444http://argonaut.skymaps.info/ and
find towards the direction of PTF1 J2238+7430 an extinction of $E(g-r)=0.24\pm
0.03$ at a distance of $1.00$ kpc; this results in a total extinction in the
g-band of $A_{\rm g}=0.84\pm 0.11$ mag, and with the corrected magnitude, we
find an absolute magnitude of $M_{\rm g}=4.40\pm 0.20$ mag consistent with a
hot subdwarf star (Geier et al., 2019).

###  4.2 Comparison with Gaia parallax

To test whether our derived system parameters are consistent with the parallax
provided by Gaia eDR3, we compared the measured parameters from the light
curve fit to the predictions using the Gaia parallax. The approach follows a
similar strategy as described in Ratzloff et al. (2019) and Kupfer et al.
(2020a). Using the absolute magnitude $M_{\rm g}=4.40\pm 0.20$ mag, we find a
luminosity of $L=11.5\pm 3.0$ L⊙ using a bolometric correction
$BC_{\mathrm{g}}=-2.30$ mag derived for our stellar parameters from the MESA
Isochrones & Stellar Tracks (MIST; Dotter 2016; Choi et al. 2016; Paxton et
al. 2011, 2013, 2015, 2018). Using the Stefan-Boltzmann law applied to a black
body ($L=4\sigma\pi R_{\rm{sdB}}^{2}T_{\rm{eff}}^{4}$), we can solve for the
radius of the sdBs, and combined with $R_{\rm{sdB}}^{2}=GM_{\rm{sdB}}/g$, we
can solve for mass of the sdBs:

$M_{\rm{sdB}}=\frac{L_{\rm{sdB}}10^{\log(g)}}{4\pi\sigma GT_{\rm{eff}}^{4}}$
(1)

Using these equations we find $M_{\rm sdB}=0.39\pm 0.10$ M⊙ and $R_{\rm
sdB}=0.17\pm 0.03$ R⊙. Although the error bars are rather large, this result
is in agreement with the results from the light curve and spectroscopic fits.

###  4.3 Kinematics of the binary systems

We find that PTF1 J2238+7430 has evolved from a $\approx$2 M⊙ star (see Sect.
5.2), and we expect the system is a member of a young stellar population.
Using the proper motion from Gaia eDR3 (Gaia Collaboration et al., 2016, 2018,
2021), the distance and the systemic velocities (see Tab. 1) we calculate the
Galactic motion for PTF1 J2238+7430.

We employed the approach described in Odenkirchen & Brosche (1992) and Pauli
et al. (2006). As in Kupfer et al. (2020a), we use the Galactic potential of
Allen & Santillan (1991) as revised by Irrgang et al. (2013). The orbit was
integrated from the present to 3 Gyr into the past. We find that the binary
moves within a height of 200 parsec of the Galactic equator and with very
little eccentricity between 9 and 10 kpc from the Galactic center. From the
Galactic orbit we conclude that PTF1 J2238+7430 is a member of the Galactic
thin disk population consistent with being member of a young stellar
population.

Figure 4: Visualization of the proposed evolutionary pathway for PTF1
J2238+7430. The red box marks the current evolutionary phase. Each
evolutionary phase is numbered according to their order in the evolution and
the direction of the sequence is marked with arrows.

Figure 5: Left panel: Predicted evolution based on the MESA model for the PTF1
J2238+7430 system. The current observed $\log{g}$ and Teff and error bars for
the system are shown in red. The dashed curve shows the evolution the star
would follow in isolation, while the solid curve shows the trajectory it
follows due to encountering the Roche limit, depicted by the gray shaded
region in the inset. Right panel: Future evolution of the system until the
helium ignites.

##  5 Predicted Evolution of the binary system

###  5.1 Formation of the sdB + WD system

Ruiter et al. (2010) found that the dominant way to form compact double
carbon-oxygen core WDs is through stable mass transfer which forms the sdB
followed by a phase of unstable mass transfer which forms the white dwarf
companion. They present a specific example which starts with a $2.88$ M⊙ and
$2.45$ M⊙ binary pair. In PTF1 J2238+7430 weak eclipses of the WD companion
imply a blackbody temperature of $26\,800\pm 4600$ K. From the blackbody
temperature we can estimate the cooling age and find a cooling time of
$\approx$25 million years, significantly shorter than the predicted current
age of the sdB of $\approx$170 million years (see Sec. 5.2). Therefore, we
predict that the sdB was formed first, and we propose the following
evolutionary scenario (illustrated in Fig. 4) for PTF1 J2238+7430 which
explains all observational properties and is similar to the scenario discussed
in Ruiter et al. (2010).

The system started as a $\approx 2$ M⊙ main sequence star (see Sect. 5.2)
which will become the sdB, and a slightly lower mass companion with an orbital
period of a few weeks. The sdB progenitor evolves first and starts stable mass
transfer to the companion star. At the end of that phase the sdB has formed
with the observed mass of $\approx 0.4$ M⊙ and the orbital periods has
substantially widened consistent with the first stable RLOF channel described
in Han et al. (2002, 2003). The companion star has accreted $\approx 1.7$ M⊙
of material from the sdB progenitor and turned into a $\approx$3.5–4 M⊙ star
which will then evolve off the main sequence and overflow its Roche Lobe while
the sdB star is still burning helium. Due to the large mass ratio at this
point, mass transfer will be unstable and initiate a common envelope. The CE
phase could happen either during the RGB or AGB phase of the secondary
depending on the binary separation at that point. In either case it would
leave a compact binary with a massive WD and an sdB at an orbital period of
$\approx 86$ minutes. The observed high WD mass of $0.725\pm 0.026$ M⊙ is
consistent with the evolution from an intermediate mass main sequence star
(Cummings et al., 2018). The final phase of unstable mass transfer happened
$\approx 25$ million years ago, after which the WD cooled to its currently
observed temperature while gravitational wave radiation decreased the orbital
period to the currently observed period of $76$ minutes. As also discussed in
Ruiter et al. (2010), there could exist a substantial fraction of compact
sdB+WD binaries where the sdB was formed first through stable mass transfer.

###  5.2 Future evolution

To understand the future evolution of the system we employed MESA version
12115 (Paxton et al., 2011, 2013, 2015, 2018, 2019). Bauer & Kupfer (2021) use
MESA models to show that sdB stars with mass $M\lesssim 0.47$ M⊙ can descend
from either lower-mass main sequence progenitors that ignite central He
burning via an off-center degenerate He flash ($M_{\rm ZAMS}\lesssim 2.3$ M⊙),
or they can descend from higher-mass main sequence progenitors that ignite He
at the center under non-degenerate conditions ($M_{\rm ZAMS}\gtrsim 2.3$
M⊙).555The precise value of the progenitor $M_{\rm ZAMS}$ for which He
ignition conditions change depends somewhat on metallicity and overshoot
(Ostrowski et al., 2021), but generally lies between about 2.0 and 2.3 M⊙.
They show that these scenarios lead to different H envelope structures that
influence the subsequent radius evolution of the sdB star, with stars
descended from higher mass progenitors having more compact envelopes and
correspondingly higher $\log{g}$ values, as shown in the top panels of figure
5 in Bauer & Kupfer (2021). The measured $\log{g}$ for PTF1 J2238+7430
requires a relatively extended envelope with a radius that requires that the
sdB star descended from the lower-mass channel with a progenitor mass around 2
M⊙. We find that our best matching MESA model for the measured $\log{g}$ and
Teff of this system is a 0.41 M⊙ sdB model descended from a 2.14 M⊙ main
sequence star that ignited the He core via a degenerate He-core flash. This
model has a sharp transition from the He core to an H envelope with solar
composition. When He ignites, we remove most of the envelope, leaving a thin H
envelope layer of $10^{-3}$ M⊙ so that the subsequent sdB evolution track
matches the observed $\log{g}$ and Teff of PTF1 J2238+7430. Figure 5 shows the
$\log{g}$–Teff evolution of this MESA model, where it approaches the current
observed state of PTF1 J2238+7430 after $\approx$170 Myr of evolution, and
will encounter its Roche lobe and begin transferring mass soon after.

We model the future binary evolution of this system with a 0.75 M⊙ WD
companion using the MESA binary capabilities. The WD model is constructed with
a C/O core using the make_co_wd test case from MESA, rescaled to a mass of
0.75 M⊙, and cooled to the current observed temperature before initializing it
into the MESA binary model at the currently observed orbital period with the
sdB model. The sdB is currently observed at 95 % Roche Lobe filling and will
continue to spiral in due to gravitational wave radiation. In our model the
sdB will soon fill its Roche lobe and start to donate its hydrogen rich
envelope in six million years at a low rate of $\lesssim 10^{-10}$ M⊙ yr-1
(see Bauer & Kupfer, 2021, for a detailed oberview). Because of the large
initial radius of the H envelope, mass transfer will proceed at this low rate
for $\approx$50 Myr before the H envelope is exhausted and the He core is
finally exposed at a much more compact radius. While the sdB is still helium
core burning $\approx$60 Myr from today, the sdB will begin to donate helium
rich material onto the WD at the expected rate of $\approx$1–3$\times 10^{-8}$
M⊙ yr-1, as shown in Figure 5. A helium rich layer will slowly build up for
$10$ million years, reaching a critical mass of $0.17$ M⊙, after which the
MESA WD model experiences He ignition in the accreted envelope. At this point
the binary has an orbital period of $\approx 10$ min. The sdB has been
stripped down to a mass of $0.25$ M⊙, and the WD has a total mass of $0.92$
M⊙.

Our MESA model predicts that at this point the accreting WD will experience a
thermonuclear instability that will lead to a detonation that will likely
destroy the WD in a thermonuclear supernova (Woosley & Kasen, 2011; Bauer et
al., 2017). Our MESA model for the WD accretor includes the NCO reaction chain
as in Bauer et al. (2017), and this governs ignition in the accreted He
envelope. Because this ignition mechanism is initiated by electron captures on
14N, it occurs at a density above $\rho=10^{6}\,\rm g\,cm^{-3}$ where a
detonation is likely to form (Woosley & Weaver, 1994; Woosley & Kasen, 2011).
The structure of our MESA model at the point of detonation is very similar to
the model for CD–30∘11223 in Bauer et al. (2017), which includes a more
detailed discussion of detonation formation under these ignition conditions.At
the time of the thermonuclear supernova, the sdB remnant has an orbital
velocity of $911$ ${\rm km}\,{\rm s}^{-1}$ and will be released as a hyper-
runaway star exceeding the escape velocity of the Galaxy (Bauer et al., 2019;
Neunteufel, 2020; Neunteufel et al., 2021; Liu et al., 2021). Fig. 4
illustrates the evolutionary sequence proposed for PTF1 J2238+7430.

##  6 Supernova rate estimate

Models of thermonuclear supernovae in WDs with thick ($\gtrsim 0.1$ M⊙) helium
shells indicate that they will yield transients classified as peculiar Type I
supernovae (Polin et al., 2019; De et al., 2019). PTF1 J2238+7430 together
with CD-30∘11223 therefore mark a small sample of double detonation peculiar
thermonuclear supernova progenitors. Using both systems we can estimate a
lower limit of thermonuclear supernovae originating in compact hot subdwarf +
WD binaries where the sdB donates helium rich material during helium core
burning. Both systems will have an age of $\approx$500 Myrs at the time of the
helium shell detonation and are located within 1 kpc. Because of their young
age, we compare the rate of these double detonation progenitors to the
supernova Ia rate as a function of star formation. Under the assumption that
these systems typically have an age of $\approx$500 Myrs at time of explosion
we find a lower limit of double detonation explosions of $\frac{2}{500}$
kpc-2Myr-1 from the two known systems. We can compare that to the local star
formation rate of $10^{-3}$ M⊙kpc-2yr-1 which leads to a double detonation
rate of $\approx 4\times 10^{-6}$ yr-1. Sullivan et al. (2006) found a
supernova Ia rate of $3.9\pm 0.7\times 10^{-4}$ SNe yr-1 (M⊙ yr-1)-1 of star
formation. With a Galactic star formation rate of $\approx$1M⊙ yr-1, we find
that the rate at which peculiar thermonuclear supernovae with thick $\approx
0.15$ M⊙ helium shells occur in star forming galaxies could be at least 1 % of
the type Ia supernova rate. This is in reasonable agreement with the presently
observed low rate of thick helium shell detonations. We note that
thermonuclear supernovae with thick helium layers are likely to produce a
transient that would be classified as a peculiar SN Ia with lower luminosities
and redder color compared to ordinary SN Ia (Polin et al., 2019).

De et al. (2019) presented the discovery of peculiar Type I supernova
consistent with a thick helium shell double detonation on a sub-Chandrasekhar-
mass WD (Polin et al., 2019, 2021). However, one of the distinct differences
is that the transient occurred in the outskirts of an elliptical galaxy which
points to an old stellar population which is in disagreement with our observed
systems which represent a young population. More recently, De et al. (2020)
presented a sample of calcium rich transients originating from double-
detonations with helium shells. They find that the majority of transients are
located in old stellar populations. However, De et al. (2020) note that a
small subsample (iPTF16hgs, SN2016hnk and SN 2019ofm) were found in star
forming environments, suggesting that there is a small but likely non-zero
contribution from young systems which could potentially be related to systems
like CD-30∘11223 and PTF1 J2238+7430.

##  7 Summary and Conclusion

As part of our search for short period sdB binaries we discovered PTF1
J2238+7430 using PTF and subsequently ZTF light curves. We find a period of
$P_{\rm orb}$=76.34179(2) min. Follow-up observations confirmed the system as
an sdB with $M_{\rm sdB}=0.383\pm 0.028$ M⊙ and a WD companion with $M_{\rm
WD}=0.725\pm 0.026$ M⊙. High-speed photometry observations with Chimera
revealed a weak WD eclipse which allows us to measure the blackbody
temperature and radius of the WD. We find a temperature of $26,800\pm 4600$ K
and a radius of $R_{\rm WD}=0.0109^{+0.0002}_{-0.0003}$ R⊙ fully consistent
with cooling models for carbon-oxygen core WDs. We find a cooling age of
$\approx 25$ Myrs for the WD which is significantly shorter than our age
estimate for the sdB which is $\approx$170 Myrs. This can be explained by the
sdB forming first through stable mass transfer, followed by the WD forming
$\approx 25$ Myrs ago through a common envelope phase. This shows that
evolutionary scenarios where the sdB is formed first through stable mass
transfer must be considered for compact sdB binaries with WD companions.

We employed MESA to calculate the future evolution of the system, finding that
the sdB in PTF1 J2238+7430 will start mass transfer of the hydrogen rich
envelope in $\approx$6 Myr. In $\approx$60 Myr, after a phase of hydrogen and
helium mass transfer, the WD will build up a helium layer of 0.17 M⊙ leading
to a total WD mass of 0.92 M⊙. Our models predict that at this point the WD
will likely detonate in a peculiar thermonuclear supernova making PTF1
J2238+7430 the second known progenitor for a supernova with a thick helium
layer. Using both systems we estimate that at least 1 % of type Ia supernova
originate from compact sdB+WD binaries in young populations of galaxies with
similar star formation rates compared to the Milky Way. Although this is only
a lower limit the estimate is broadly consistent with the low number of
observed peculiar thermonuclear supernovae.

This research benefited from interactions that were funded by the Gordon and
Betty Moore Foundation through grant GBMF5076. This work was supported by the
National Science Foundation through grants PHY-1748958 and ACI- 1663688. TK
would like to thank Ylva Götberg for providing the template for Fig. 4. TK
acknowledges support from the National Science Foundation through grant AST
#2107982. DS was supported by the Deutsche Forschungsgemeinschaft (DFG) under
grants HE 1356/70-1 and IR 190/1-1. Observations were obtained with the Samuel
Oschin Telescope at the Palomar Observatory as part of the PTF project, a
scientific collaboration between the California Institute of Technology,
Columbia University, Las Cumbres Observatory, the Lawrence Berkeley National
Laboratory, the National Energy Research Scientific Computing Center, the
University of Oxford, and the Weizmann Institute of Science. Based on
observations obtained with the Samuel Oschin 48-inch Telescope at the Palomar
Observatory as part of the Zwicky Transient Facility project. ZTF is supported
by the National Science Foundation under Grant No. AST-1440341 and a
collaboration including Caltech, IPAC, the Weizmann Institute for Science, the
Oskar Klein Center at Stockholm University, the University of Maryland, the
University of Washington, Deutsches Elektronen-Synchrotron and Humboldt
University, Los Alamos National Laboratories, the TANGO Consortium of Taiwan,
the University of Wisconsin at Milwaukee, and Lawrence Berkeley National
Laboratories. Operations are conducted by COO, IPAC, and UW. Some of the data
presented herein were obtained at the W.M. Keck Observatory, which is operated
as a scientific partnership among the California Institute of Technology, the
University of California and the National Aeronautics and Space
Administration. The Observatory was made possible by the generous financial
support of the W.M. Keck Foundation. The authors wish to recognize and
acknowledge the very significant cultural role and reverence that the summit
of Mauna Kea has always had within the indigenous Hawaiian community. We are
most fortunate to have the opportunity to conduct observations from this
mountain. Some results presented in this paper are based on observations made
with the WHT operated on the island of La Palma by the Isaac Newton Group in
the Spanish Observatorio del Roque de los Muchachos of the Institutio de
Astrofisica de Canarias. 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.

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