# A Pilot Search for Gravitational Self-Lensing Binaries with the Zwicky
Transient Facility

Allison Crossland1,∗ Eric C. Bellm1,† Courtney Klein2 James R. A. Davenport1
Thomas Kupfer3 Steven L. Groom4 Russ R. Laher4 Reed Riddle5 1 DIRAC Institute,
Department of Astronomy, University of Washington, 3910 15th Avenue NE,
Seattle, WA 98195, USA  2 Department of Physics and Astronomy, University of
California Irvine, CA 92697, USA  3 Hamburg Observatory, University of
Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany  4 IPAC, California
Institute of Technology, 1200 E. California Blvd, Pasadena, CA 91125, USA  5
Caltech Optical Observatories, California Institute of Technology, 1200 E.
California Blvd, Pasadena, CA 91125, USA  ∗<EMAIL_ADDRESS>†<EMAIL_ADDRESS>
###### Abstract

Binary systems containing a compact object may exhibit periodic brightening
episodes due to gravitational lensing as the compact object transits the
companion star. Such “self-lensing” signatures have been detected before for
white dwarf binaries. We attempt to use these signatures to identify detached
stellar-mass neutron star and black hole binaries using data from the Zwicky
Transient Facility (ZTF). We present a systematic search for self-lensing
signals in Galactic binaries from a subset of high-cadence ZTF data taken in
2018. We identify 19 plausible candidates from the search, although because
each candidate is observed to only brighten once, other origins such as
stellar flares are more likely. We discuss prospects for more comprehensive
future searches of the ZTF data.

##  1 Introduction

The observed population of Galactic compact objects is shaped by the methods
by which they are discovered. For binary systems with active mass transfer,
instabilities in the accretion disk or wind accretion can create high X-ray
luminosities detectable by all-sky monitors (for a recent review, see Kalemci
et al., 2022). Detached, non-interacting compact objects are expected to be
more numerous, but harder to detect. Large radial velocity shifts may be
identified in spectroscopic surveys (e.g., Thompson et al., 2019; Yi et al.,
2022; Shenar et al., 2022), although other stellar binaries may mimic these
signatures (Bodensteiner et al., 2022, and references therein). Gaia has
identified compact objects through their astrometric wobble (El-Badry et al.,
2023b, a). And microlensing can identify free-floating compact objects as the
gravity of the compact object bends the light of the background star towards
the observer (Sahu et al., 2022).

For edge-on binary systems containing a compact object and a companion star,
we may see a periodic brightening pulse due to gravitational “self-lensing” as
the compact object transits the companion (Maeder, 1973; Gould, 1995).
Multiple stellar-mass self-lensing systems have been discovered with Kepler
(Kruse & Agol, 2014; Kawahara et al., 2018; Masuda et al., 2019)111Some
authors have interpreted periodic temporal signatures from the supermassive
black hole systems in AGN as self-lensing (D’Orazio & Di Stefano, 2018; Hu et
al., 2020; Ingram et al., 2021)., all of which to date contain a white dwarf
(WD). In some eclipsing WD systems, lensing has been suggested to modify the
observed eclipse depth (e.g., Steinfadt et al., 2010; Muirhead et al., 2013;
Kaplan et al., 2014), although the statistical significance has been generally
low.

However, population synthesis estimates suggest that thousands of neutron star
(NS) and black hole (BH) binaries could have detectable self-lensing signals
(Wiktorowicz et al., 2021). As the self-lensing signal from NS and BH systems
is higher amplitude than from WDs, it may be detectable at the photometric
precison available from ground-based synoptic surveys. Such surveys provide an
advantage of greater depth and temporal duration, allowing a search for self-
lensing signals from many more stars than can be observed by Kepler or TESS.

The Zwicky Transient Facility (ZTF; Bellm et al., 2019b; Graham et al., 2019)
provides a powerful dataset for such a search. ZTF is a wide-field time-domain
survey of the Northern Hemisphere sky that uses a mosaic CCD camera (Dekany et
al., 2020) mounted on the 48-inch Samuel Oschin Schmidt Telescope at Palomar
Observatory. ZTF’s high observing cadence (minutes–days), large field of view
(47 deg2), and moderate single-image depth ($m_{\rm lim}\sim 20.5$ mag)
provide an opportunity to detect relatively rare and short-lived self-lensing
pulses (Wiktorowicz et al., 2021; Nir & Bloom, 2023).

In this paper, we describe a pilot search for NS and BH self-lensing in a
small subset of continuous cadence ZTF data taken in the Galactic Plane. We
describe the ZTF dataset used for this search, outline our process of
searching for self-lensing pulses, and present final candidates from the
search (Table 1). We close by discussing the interpretation of these
candidates as well as prospects for future self-lensing searches with ZTF.

##  2 Methods and Data

###  2.1 Dataset

The Zwicky Transient Facility is a high-cadence, wide area optical time domain
survey. ZTF has conducted a variety of surveys to date (Bellm et al., 2019a).
Since we expect the duration of self-lensing pulses to be on the order of
minutes to hours, for our initial search we focused on a subset of “continuous
cadence” data collected during August of 2018 (Kupfer et al., 2021). The goal
of these observations was to identify short-timescale variability through
near-continuous cadence observations of a small set of high stellar density
fields. In the August 2018 campaign, fourteen ZTF fields were observed. Fields
were observed in pairs, with each pair observed for two nights. During
observations, ZTF generally alternated between the two fields over a period of
about three hours, taking 30 second exposures at each position in the ZTF $r$
band.

The images were reduced by ZTF’s automated data processing pipelines (Masci et
al., 2019). The ZTF data system produces PSF photometry catalogs for each
direct (unsubtracted) image; these catalogs are then matched to deeper source
catalogs from the coadded ZTF reference image to produce object light curves,
grouped by the ZTF field, readout channel, and filter. From the 14 fields we
have light curves for 117,483,764 sources. We performed analysis both on HDF5
“matchfiles” produced by the IPAC pipelines as well as in a version ingested
in the Astronomical eXtensions for Spark (AXS) system (Zečević et al., 2019).

###  2.2 Candidate Detection

Our initial categorization of these data was motivated by a search for stellar
flares (Klein et al., 2020). As the search algorithm can identify a range of
outburst shapes, we recognized that the raw flare candidates might also
include self-lensing signals.

The initial flare candidates were identified using the FINDflare change-point
algorithm (Chang et al., 2015). We defined a continuous light curve as a
series of observations with a gap between observations no larger than 30
minutes. We treated each continuous section of the light curve separately and
disregarded any epoch with a ZTF catflag $>$ 0\. Our average light curve
duration was 2.35 hours. Due to short time scales observed, we did not worry
about the effects of periodic variability.

To detect spikes in flux we used the FINDflare criteria of:

$x_{i}-x_{L}>0$ (1) $\frac{|x_{i}-x_{L}|}{\sigma_{L}}>3$ (2)
$\frac{|x_{i}-x_{L}+w_{i}|}{\sigma_{L}}>1$ (3)

where $x_{i}$ is the psfflux of a given epoch, $x_{L}$ is the median value of
flux in the continuous light curve, $\sigma_{L}$ is the standard deviation of
the flux in the continuous light curve, and $w_{i}$ is the psffluxerr of a
given epoch. When three consecutive observations meet these criteria it was
counted as a flare candidate. This search produced 15,262 flare candidates.
Due to the duration of the continuous-cadence light curve segments we were
limited to identifying one flare per segment. We did not impose any selection
criteria based on position in the color-magnitude diagram or detection by
Gaia.

###  2.3 Temporal Profile Modeling

To distinguish stellar flares from candidate self-lensing pulses, we used
nonlinear least-squares to fit the temporal profile of the candidates
identified by findFlare with three analytic models. We fit each lightcurve
with a fast-rise, exponential decay stellar flare model (Davenport et al.,
2014), a Gaussian, and a constant. We used the Akaike information criterion
(AIC) to select lightcurves which were best-fit by the Gaussian model. This
approach yielded 1910 instances brightening pulses better fit by a symmetric,
Gaussian-like profile that we investigated as potential self-lensing signals
(Figure 1).

Figure 1: RA/Dec positions of 1910 identified bright pulses in our 2018
continuous-cadence ZTF dataset.

###  2.4 Initial Screening

We visually inspected each of the 1910 continuous-cadence light curve segments
with identified bright pulses. We identified three broad categories: noisy
data that clearly did not exhibit the desired self-lensing pulse shape,
remaining curves with asymmetric fast-rise, exponential decay shapes
characteristic of stellar flares but not excluded by our model selection
procedure (§2.3), and outbursts of interest that we analyzed further.
Approximately 90% of the 1910 candidates fell into the first two categories
and were discarded.

Next, using the NASA/IPAC Infrared Science Archive (IRSA), we more carefully
inspected the images of each source at the time of the potential pulse. Around
80 instances of data artifacts were discovered (e.g., Figure 2), and those
candidates were rejected.

Figure 2: Three examples of artifacts found in our dataset that caused
spurious brightening pulses.

Asteroids transiting a background star were another frequent contaminant
(Figure 3; see also Sorabella et al., 2023). As an asteroid passes in front of
a background star, we see a short, symmetrical brightening pulse, similar to
what we might expect of a self-lensing pulse. Using SkyBot (Berthier et al.,
2006), a solar system body identification service, and the python package
Astroquery (Ginsburg et al., 2019), we cross-matched the time and location of
each pulse in our dataset to all known asteroid positions in the solar system.
We looked for instances of an asteroid passing within 15 arcseconds of the
source location at the same time as the brightening pulse. We rejected 50
candidates coincident with an asteroid transit.

Figure 3: Consecutive images showing an asteroid transiting a star (circled in
orange).

Finally, to rule out any brightening pulses caused by hidden data artifacts,
we compared each of the remaining light curves to that of a neighboring, non-
variable star. A neighboring source exhibiting a similar pulse as our target
indicated that the pulse was caused by an artifact in the data processing or
calibration, not self-lensing. Four instances of neighbor variability were
found, and those candidates were rejected.

After these steps, we were left with 23 self-lensing candidates.

###  2.5 Mass Estimation

Given a self-lensing signal, it is possible to estimate the mass of the
compact object. We used the approximate relations of Masuda et al. (2019) for
the self-lensing pulse amplitude

$\begin{split}s_{sl}=2\left(\frac{R_{E}}{R_{\star}}\right)^{2}=7.15\times
10^{-5}\left(\frac{R_{\star}}{R_{\odot}}\right)^{-2}\left(\frac{P}{1day}\right)^{2/3}\\\
\times\left(\frac{M_{\bullet}}{M_{\odot}}\right)\left(\frac{M_{\bullet}+M_{\star}}{M_{\odot}}\right)^{1/3}\end{split}$
(4)

and duration

$\begin{split}\tau_{sl}=\frac{R_{\star}P}{\pi
a}\cdot\frac{\pi}{4}=1.8\,\mathrm{hr}\times\frac{\pi}{4}\left(\frac{P}{1day}\right)^{1/3}\\\
\times\left(\frac{M_{\bullet}+M_{\star}}{M_{\odot}}\right)^{-1/3}\left(\frac{R_{\star}}{R_{\odot}}\right)\end{split}$
(5)

where $M_{\bullet}$ is the mass of the compact object and
$M_{\star},R_{\star}$ the mass and radius of the companion star. These
approximations are valid when $R_{E}\ll R_{\star}$, where $R_{E}$ is the
Einstein radius (Wiktorowicz et al., 2021). With a well-sampled pulse duration
and amplitudes and an estimate of the companion star mass and radius, we can
eliminate the unobserved orbital period $P$ and derive the compact object
mass.

To obtain the mass and radius of the companion star, we first cross-matched
our pulse candidates to the stellar parameters catalog of Green et al. (2019),
which provides an estimated absolute magnitude of the companion assuming it is
on the main sequence. We used the empirical luminosity–mass and mass-radius
relations of Demircan & Kahraman (1991) to convert the absolute luminosity
$L_{\star}$ into $M_{\star}$ and $R_{\star}$:

$M_{\star}=\left(\frac{L_{\star}}{1.18L_{\odot}}\right)^{0.27}M_{\odot}$ (6)

and

$R_{\star}=\begin{cases}1.06R_{\odot}\left(\frac{M_{\star}}{M_{\odot}}\right)^{0.945},&M_{\star}<1.66M_{\odot}\\\
1.33R_{\odot}\left(\frac{M_{\star}}{M_{\odot}}\right)^{0.555},&M_{\star}>1.66M_{\odot}\end{cases}$
(7)

Given the coarseness of our assumptions about the companion star, these mass
estimates are primarily useful in identifying pulses which would be
implausible for stellar-mass compact objects. We rule out four more objects
based on improbable mass estimates, and are left with 19 candidates (Figure 4
and Appendix). Table 1 provides the resulting mass estimates.

We evaluated whether the companion star would fill the Roche Lobe given the
binary component masses and orbital periods. If so, we would expect temporal
signatures of active accretion such as flickering that would mask the self-
lensing signal and thus disfavor a self-lensing interpretation of the observed
isolated pulse. Only one candidate had a Roche filling factor larger than than
one: ZTF J185515.77$+$275426.63, with $R_{\star}/R_{\mathrm{RL}}=1.46$.

###  2.6 Box Least Squares and Phase folding

As a final step in the data analysis process, we looked for periodicity in the
full ZTF light curve for each candidate. In addition to the small amount of
continuous-cadence data from ZTF (§2.1), there is more sparsely sampled data
spanning a much longer time range. Finding a periodic flux increase in the
full, sparsely-sampled light curve would be strong evidence for a self-lensing
origin. To search for this periodicity, we estimated the pulse period by
computing a Box Least Squares Periodogram with the Astropy python package
(Astropy Collaboration et al., 2022) and phase folded the full light curve
using the estimated period. No clear pulse period was determined for any of
our sources using this method.

We attempted to determine whether the non-detection of periodicity could be
used to rule out self-lensing as the explanation for these pulses, or whether
ZTF did not have observations at the relevant periods. Using the relations in
§2.5, we derived best-fit orbital periods and uncertainties given the observed
continuous-cadence pulse amplitudes and durations, and inferred stellar masses
and radii (Table 1). We constructed a grid of potential orbital periods
ranging from plus to minus three times the standard deviation of the best-fit
period222For one sources, ZTF J194939.66+211830.55, our Monte Carlo estimate
of the period uncertainty yielded a range too large for a practical search due
to a larger uncertainty in its absolute magnitude.. We used period bins of
one-third the pulse duration. For each test period, we phase-folded the ZTF
lightcurve omitting the night of continuous-cadence observations and counted
the number of observations which fell into the phase bin defined by the start
time and duration of the detected pulse. If one or more ZTF observations
occurred during the pulse phase, we ruled out self-lensing at that orbital
period. Dividing the number of periods not ruled out by the total number of
test periods provides an estimate of the probability that the ZTF data support
a self-consistent self-lensing explanation for the data (Table 1).

Seven of our nineteen self-lensing candidates are ruled out entirely by this
procedure–ZTF would have detected self-lensing signatures at any of the
plausible orbital periods. Most of the remaining candidates are unlikely to be
due to self-lensing: only two candidates have more than 50% of their plausible
orbital periods allowed by the ZTF non-detection of periodicity.

Figure 4: ZTF lightcurves of two self-lensing candidates. The top panel shows
the full ZTF light curve history of the source while the bottom panel shows
the identified pulse.

##  3 Discussion

From our initial dataset of 1910 light curves, we identify 19 candidate self-
lensing pulses not caused by data artifacts, asteroid transits, or repeated
stellar flares. They also show reasonable estimates for the masses of the
compact object and companion star (Table 1). However, after analysis of all
available ZTF data, they do not show a periodic brightening pulse as we would
expect to see for a self-lensing system, and in most cases ZTF observations
rule out periodic self-lensing pulses for the majority of plausible orbital
periods. We consider two other possible explanations for these brightening
episodes.

###  3.1 Single Stellar Flares

The most probable explanation for these pulses is a stellar flare event.
Stellar flares are extremely common, particularly of the low-mass stars that
would produce the largest self-lensing signals. Most stellar flares exhibit a
characteristic light curve shape, with a fast rise and slow decay. While we
have attempted to remove flares with the model selection procedure described
in §2.3 and additional vetting of pulse shape and outburst history (§2.4),
many of our remaining candidates still exhibit a degree of asymmetry and out-
of-pulse variability that suggest a flare origin. Stellar flares exhibit a
wide range of morphologies as well; high-precision TESS data suggest that as
many as 9% of stellar flares can be fit by Gaussian profiles (Howard &
MacGregor, 2022). Lightcurve asymmetry and other flare characteristics can be
identified with metrics in our continuous-cadence data but would be more
challenging to discern at sparser cadences. One possibility would be to assess
the likelihood of observing a single large flare in our ZTF lightcurves using
empirical flare frequency distributions for the inferred spectral type of each
star.

###  3.2 Microlensing by Free-Floating Planets

Isolated, short-duration pulses may also be due to microlensing by free-
floating planetary-mass objects (e.g., Sumi et al., 2011; Mróz et al., 2017,
2018). In such cases the unknown distance between the background source and
the lens is not negligible, and the lens mass cannot be determined directly.
Microlensing events due to lensing by terrestrial-mass objects with short
durations comparable to these ZTF pulses have been reported (e.g., Mróz et
al., 2020). In such cases, finite-source effects may be detected (e.g., Witt &
Mao, 1994). We defer detailed analysis and microlensing model fitting of these
events to future work.

##  4 Conclusion

In this pilot search for non-interacting Galactic compact binaries with the
Zwicky Transient Facility, we searched for self-lensing signatures in order to
detect these otherwise hidden objects. Although we found no clear-cut cases of
periodic self-lensing signatures in this initial dataset, this study lays the
groundwork for larger searches of the complete ZTF lightcurves. Population
synthesis suggests thousands of detached compact binaries could be discovered
in ZTF (Wiktorowicz et al., 2021), although our study highlights that a range
of contaminants, particularly image artifacts, stellar flares, and
microlensing by free-floating planets, will complicate such a search.
Nevertheless, a systematic search of the ZTF data would provide an opportunity
to probe an otherwise inaccessible compact binary population and improve our
understanding of binary and stellar evolution.

## Acknowledgements

We thank Eric Agol and Matthew Middleton for useful discussions.

Support for this work was provided in part by the Research Corporation Scialog
grant #24221. ECB acknowledges support by the NSF AAG grant 1812779 and grant
#2018-0908 from the Heising-Simons Foundation.

This research has made use of IMCCE’s SkyBoT VO tool.

This research has made use of the NASA/IPAC Infrared Science Archive, which is
funded by the National Aeronautics and Space Administration and operated by
the California Institute of Technology.

Based on observations obtained with the Samuel Oschin Telescope 48-inch and
the 60-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-2034437 and a collaboration including Caltech,
IPAC, the Weizmann Institute of Science, the Oskar Klein Center at Stockholm
University, the University of Maryland, Deutsches Elektronen-Synchrotron and
Humboldt University, the TANGO Consortium of Taiwan, the University of
Wisconsin at Milwaukee, Trinity College Dublin, Lawrence Livermore National
Laboratories, and IN2P3, France. Operations are conducted by COO, IPAC, and
UW.

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Lightcurves of all self-lensing candidates in Table 1. For each source, the
top panel shows the full ZTF lightcurve and the lower is zoomed on the pulse
detected in continuous-cadence data. Data points with pipeline flags
corresponding to imaging artifacts are marked with a cross and excluded from
analysis.

Name | Absolute | Pulse | Peak Time | Star Mass | Star Radius | Compact Object | Inferred Period | Fraction
---|---|---|---|---|---|---|---|---
| Magnitude | Duration | (MJD) | ($M_{\odot}$) | ($R_{\odot}$) | Mass ($M_{\odot}$) | (days) | Periods
|  | (minutes) |  |  |  |  |  | Allowed
ZTF J182258.03+090638.69 | 8.80${}^{+0.15}_{-0.10}$ | 18.4 | 58333.26 | $0.35\pm 0.01$ | $0.39\pm 0.01$ | $2.93\pm 0.21$ | $0.56\pm 0.02$ | 0.0
ZTF J182644.38+083602.22 | 11.35${}^{+0.30}_{-0.25}$ | 23.9 | 58334.27 | $0.18\pm 0.01$ | $0.22\pm 0.01$ | $1.63\pm 0.24$ | $4.09\pm 0.32$ | 0.13
ZTF J183450.80+124708.39 | 13.10${}^{+0.20}_{-0.25}$ | 25.3 | 58334.21 | $0.12\pm 0.01$ | $0.14\pm 0.01$ | $0.51\pm 0.07$ | $5.72\pm 0.43$ | 0.31
ZTF J185020.78+271940.09 | … | 13.4 | 58337.32 | … | … | … | … | …
ZTF J185349.93+270237.63 | 10.55${}^{+0.25}_{-0.20}$ | 26.3 | 58338.23 | $0.23\pm 0.01$ | $0.26\pm 0.02$ | $2.73\pm 0.33$ | $5.03\pm 0.32$ | 0.11
ZTF J185515.77+275426.63 | 2.35${}^{+0.47}_{-0.75}$ | 28.1 | 58338.31 | $1.74\pm 0.36$ | $1.81\pm 0.21$ | $59.50\pm 14.08$ | $0.38\pm 0.05$ | 0.0
ZTF J185533.82-035452.51 | 2.55${}^{+0.50}_{-0.60}$ | 50.3 | 58348.25 | $1.65\pm 0.27$ | $1.70\pm 0.20$ | $56.09\pm 13.72$ | $2.44\pm 0.32$ | 0.0
ZTF J190041.86-032337.74 | 6.62${}^{+0.38}_{-0.38}$ | 52.9 | 58348.31 | $0.60\pm 0.06$ | $0.65\pm 0.06$ | $10.36\pm 1.95$ | $9.53\pm 0.97$ | 0.05
ZTF J190822.05+094118.41 | 10.25${}^{+0.60}_{-0.47}$ | 41.2 | 58334.23 | $0.24\pm 0.03$ | $0.28\pm 0.04$ | $1.99\pm 0.56$ | $11.81\pm 1.98$ | 0.32
ZTF J191032.52+282444.42 | 10.08${}^{+0.37}_{-0.43}$ | 34.3 | 58337.24 | $0.25\pm 0.03$ | $0.29\pm 0.03$ | $2.02\pm 0.47$ | $6.20\pm 0.79$ | 0.14
ZTF J191320.55+304759.30 | 10.05${}^{+0.30}_{-0.25}$ | 64.6 | 58345.27 | $0.26\pm 0.02$ | $0.29\pm 0.02$ | $1.83\pm 0.27$ | $37.01\pm 2.97$ | 0.83
ZTF J191752.11+345400.39 | 11.25${}^{+0.37}_{-0.40}$ | 12.1 | 58345.30 | $0.19\pm 0.02$ | $0.22\pm 0.02$ | $2.26\pm 0.46$ | $0.67\pm 0.08$ | 0.0
ZTF J192607.74+252059.72 | 5.80${}^{+1.10}_{-0.90}$ | 29.4 | 58338.30 | $0.74\pm 0.18$ | $0.79\pm 0.19$ | $20.80\pm 10.17$ | $1.80\pm 0.82$ | 0.0
ZTF J194043.20+280959.74 | … | 21.4 | 58337.22 | … | … | … | … | …
ZTF J194939.66+211830.55 | 7.00${}^{+1.50}_{-2.17}$ | 48.0 | 58343.20 | $0.55\pm 0.39$ | $0.60\pm 0.40$ | $8.50\pm 12.87$ | $7.65^{\dagger}$ | …
ZTF J195500.02+044612.08 | 10.70${}^{+0.25}_{-0.25}$ | 10.6 | 58340.30 | $0.22\pm 0.01$ | $0.25\pm 0.02$ | $2.28\pm 0.29$ | $0.31\pm 0.02$ | 0.0
ZTF J200714.10+152533.71 | 11.28${}^{+0.42}_{-0.38}$ | 32.7 | 58344.26 | $0.19\pm 0.02$ | $0.22\pm 0.02$ | $1.32\pm 0.27$ | $8.30\pm 0.95$ | 0.27
ZTF J201150.62+195603.90 | 9.25${}^{+0.10}_{-0.10}$ | 56.3 | 58343.24 | $0.31\pm 0.01$ | $0.35\pm 0.01$ | $1.54\pm 0.08$ | $12.40\pm 0.36$ | 0.74
ZTF J201356.50+164231.85 | 10.25${}^{+0.20}_{-0.20}$ | 6.7 | 58343.30 | $0.24\pm 0.01$ | $0.28\pm 0.01$ | $6.12\pm 0.60$ | $0.14\pm 0.01$ | 0.0
Table 1: ZTF self-lensing candidates identified by their sexigesimal RA and
Dec. Absolute magnitude estimates are taken from Green et al. (2019); two
candidates were not cataloged. †The orbital period of ZTF J194939.66+211830.55
was poorly constrained.


# The implications of large binding energies of massive stripped core collapse
supernova progenitors on the explosion mechanism

Dmitry Shishkin Department of Physics, Technion, Haifa, 3200003, Israel;
<EMAIL_ADDRESS><EMAIL_ADDRESS>Noam Soker
Department of Physics, Technion, Haifa, 3200003, Israel;
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(January 2023)

###### Abstract

We examine the binding energies of massive stripped-envelope core collapse
supernova (SECCSN) progenitors with the stellar evolution code mesa, and find
that the jittering jets explosion mechanism is preferred for explosions where
carbon-oxygen cores with masses of $\gtrsim 20M_{\odot}$ collapse to leave a
neutron star (NS) remnant. We calculate the binding energy at core collapse
under the assumption that the remnant is a NS. Namely, stellar gas above mass
coordinate of $\simeq 1.5-2.5M_{\odot}$ is ejected in the explosion. We find
that the typical binding energy of the ejecta of stripped-envelope progenitors
with carbon-oxygen core masses of $M_{\rm CO}\gtrsim 20M_{\odot}$ is $E_{\rm
bind}\gtrsim 2\times 10^{51}{~{}\rm erg}$. We claim that jets are most likely
to explode such cores as jet-driven explosion mechanisms can supply high
energies to the explosion. We apply our results to SN 2020qlb, which is a
SECCSN with a claimed core mass of $\simeq 30-50M_{\odot}$, and conclude that
the jittering jets explosion mechanism best accounts for such an explosion
that leaves a NS.

stars: jets – stars: massive – supernovae: general – supernovae: individual:
SN 2020qlb

##  1 INTRODUCTION

The binding energy of a core collapse supernova (CCSN) progenitor plays a
crucial role in determining the explosion outcome, like explosion energy and
remnant mass. Typical explosion energies are estimated to be in the range of
$E_{\rm exp}\simeq 10^{50}-10^{52}{~{}\rm erg}$ (e.g., Yang & Chevalier 2015;
Utrobin et al. 2015; Gal-Yam 2019; Burrows & Vartanyan 2021a). The binding
energy of the most massive pre-collapse cores have similar or even larger
values than these typical explosion energies (e.g., Pejcha & Thompson 2015;
Bruenn et al. 2016; Chan et al. 2020; Wang et al. 2022; Burrows et al. 2020).
The explosion mechanism should both overcome the binding energy and account
for the explosion energy (radiation + final kinetic energy of the ejecta).

Two theoretical explosion mechanisms of non-rotating (or slowly rotating) pre-
collapse cores utilize the gravitational energy of the collapsing core to
power CCSNe. These are the delayed-neutrino explosion mechanism (e.g., Bethe &
Wilson 1985; Ertl et al. 2016; Burrows et al. 2020; Bruenn et al. 2020; Bollig
et al. 2021; Burrows & Vartanyan 2021b; Zha et al. 2023) and the jittering-
jets explosion mechanism (e.g., Soker 2010; Papish & Soker 2011; Gilkis &
Soker 2015; Soker 2019, 2022a). Studies show that the maximum energy that the
delayed neutrino mechanism can supply to overcome the binding energy of the
ejecta is $E_{\nu}^{\rm max}\simeq 2\times 10^{51}{~{}\rm erg}$, resulting in
maximum explosion energies (after removing the binding energy) of $E_{\rm
exp}\simeq 2\times 10^{51}{~{}\rm erg}$ (e.g., Fryer et al. 2012; Ertl et al.
2016; Sukhbold et al. 2016; Gogilashvili et al. 2021). In addition, the
delayed neutrino mechanism has problems in producing the observed amount of
${}^{56}{\rm Ni}$, in particular in stripped-envelope CCSNe (SECCSNe; for a
recent study see Sawada & Suwa 2023)

The above limits on the explosion energies of the delayed neutrino mechanism
come either from direct calculations (e.g., Bollig et al. 2021; Gogilashvili
et al. 2021) or by scaling to observed CCSNe, such as the Crab nebula and SN
1987A (e.g., Sukhbold et al. 2016). We do note that some studies do attribute
larger explosion energies to the delayed neutrino mechanism, e.g., Fryer et
al. (2018) who base their explosion energies on Fryer et al. (2012). Fryer et
al. (2012) adopt the convection-enhanced neutrino-driven paradigm and assume
that the explosion energy is equal to the energy stored in the convective
region at the time of collapse, and derive a maximum available energy of
$3-4\times 10^{51}{~{}\rm erg}$. However, they derive these high values of
explosion energies with progenitor masses of $\lesssim 20M_{\odot}$ only,
while we focus on more massive progenitors. We therefore consider here the
results of the later studies that we quoted above that give the limit of
$E_{\nu}^{\rm max}\simeq 2\times 10^{51}{~{}\rm erg}$. In any case, this
discussion suggests that we should take this upper limit of the explosion
energy in the delayed neutrino mechanism with caution.

The jittering jets explosion mechanism, on the other hand, can account for
much larger explosion energies (e.g., Gilkis et al. 2016; Soker 2022b). The
magneto rotational explosion mechanism that works only for rapidly rotating
pre-collapse cores can also account for large explosion energies (e.g.,
LeBlanc & Wilson 1970; Khokhlov et al. 1999; López-Cámara et al. 2013; Wheeler
et al. 2015; Bromberg & Tchekhovskoy 2016; Kuroda et al. 2020; Gottlieb et al.
2022c, a; Fujibayashi et al. 2022; Powell et al. 2022) because the newly born
neutron star (NS) launches fixed-axis jets. Whether the explosion is by
jittering jets (most CCSNe according to that model; e.g., Soker 2022b) or by
fixed-axis jets, the jets operate in a negative feedback cycle, i.e., the jet
feedback explosion mechanism (e.g., Soker 2016a). As well, the jets can
influence the direction of later jets (e.g., Papish & Soker 2014a; Gottlieb et
al. 2022b). Even if the stochastically accreted mass has sub-Keplerian angular
momentum, it might still form an accretion belt that can launch jets (e.g.,
Schreier & Soker 2016; Garain & Kim 2023).

The above discussion implies that CCSNe with large kinetic energies of $E_{\rm
exp}\gtrsim 2\times 10^{51}$, e.g., SN 2020jfo (Ailawadhi et al. 2023 –
$E_{\rm exp}=2.9\times 10^{51}{~{}\rm erg}$, but see Teja et al. 2022 for a
lower estimated energy); SN 2020qlb (West et al. 2023 – $E_{\rm exp}=20\times
10^{51}{~{}\rm erg}$); SN 2012au (Pandey et al. 2021 – $E_{\rm
exp}=4.8-5.4\times 10^{51}{~{}\rm erg}$) require jets to drive the explosion.

In this study we examine the interesting case of the hydrogen-poor and super-
energetic CCSN SN 2020qlb. West et al. (2023) estimate the kinetic energy of
SN 2020qlb as $\simeq 20\times 10^{51}{~{}\rm erg}$ and suggest that a
magnetar supplies the large amount of energy of the ejecta. They also provide
fitting parameters for the magnetar model (e.g., Maeda et al. 2007; Kasen &
Bildsten 2010; Woosley 2010; Metzger et al. 2015; Nicholl et al. 2017; Gomez
et al. 2022) and estimate the ejecta plus the remnant mass to be $M_{\rm
ej}+M_{\rm rem}\simeq{\rm few}\times 10M_{\odot}$. We consider a stripped
envelope supernova, i.e., the envelope mass is very low. Therefore, the pre-
collapse core mass is the sum of the ejecta mass, the remnant mass, and a
small amount of mass carried as energy by neutrinos: $M_{\rm core}\simeq
M_{\rm ej}+M_{\rm rem}\simeq{\rm few}\times 10M_{\odot}$.

It seems nonetheless, that there are two reasons why the explosion of SN
2020qlb must be driven by jets and not by the delayed neutrino mechanism. The
first one is that the formation of an energetic magnetar must be accompanied
by the launching of even more energetic jets at the explosion itself and
possibly after the explosion as well (e.g. Soker 2016b, 2017; Soker & Gilkis
2017; Shankar et al. 2021; Soker 2022c, b). The second reason is the new
finding of the present study, where by a stellar evolutionary code (section 2)
we show (section 3) that the binding energies of such massive cores are above
the energies the delayed neutrino mechanism can supply (Janka, 2012; Soker,
2022b). In section 4 we summarize our results and discuss their implications
in the context of the jet feedback explosion mechanism.

##  2 Numerical Scheme

We use the stellar evolution code mesa (Paxton et al. 2010, 2013, 2015, 2018,
2019) to simulate the structure of $52$ CCSN progenitor models, all with
initial metalicity of $z=0.02$. We base our numerical routine on the
‘20M_pre_ms_to_core_collapse’ example from mesa r22.05.1, but simulate the
evolution using version r15140. Starting from the zero age main sequence
(ZAMS), we evolve the star until center He depletion (when helium abundance in
the center is either $\simeq 1\%$ or $\simeq 5\%$), at which point we
numerically remove the hydrogen-rich envelope, leaving only the He core. We
then evolve the star until core-collapse. We introduce several changes to the
example routine to both adhere to our requirements, i.e., having at least 21
isotopes and having sufficiently high resolution (Shishkin & Soker, 2021), and
to fit mesa version r15140. We expand on the numerical scheme in appendix A.

By varying the ZAMS stellar mass, the wind parameters, and the exact time when
we numerically remove the envelope we obtain stripped-envelope CCSN
progenitors, i.e., hydrogen-poor stellar progenitors, with varying mass of a
carbon-oxygen (CO) core, $M_{\rm CO}\simeq 4-35M_{\odot}$. This mass range of
SECCSNe corresponds to ZAMS stellar masses range of $\simeq 20-82M_{\odot}$.
In some cases the core contains several solar masses of helium, while in other
cases the core has a much lower helium mass, depending on the above
parameters.

Because the inner part of the core collapses to form a NS or a black hole (BH)
remnant, only the binding energy of the outer core is relevant to our study.
We calculate the binding energy of the outer core, $E_{\rm bind}(r)$, by
integrating over the sum of the internal energy and gravitational energy from
the surface down to the mass coordinate that separates the ejecta and the
final remnant $m=M_{\rm in}$, i.e., the inner boundary of the ejecta. We
calculate the binding energy of the ejecta for two values of this mass
coordinate $M_{\rm in}=1.5~{},2.5M_{\odot}$, because we consider cases where
the remnant is a NS.

We refer to the latest evolution point we simulate in the stellar evolution as
collapse. This point of collapse must adhere to an iron core more massive than
$M_{\rm Fe}>1.4$, where we assume collapse is imminent. This value of iron
core mass is more or less when the iron core mass reaches its maximum value
(as at the onset of collapse the iron disintegrates). About half of
simulations reach infall velocities at the edge of the iron core of $v_{\rm
fe,infall}>100{~{}\rm km\;s^{-1}}$. Other simulations encounter numerical
difficulties and we had to terminate them at somewhat earlier times.

##  3 Results

###  3.1 Binding Energy towards Collapse

As the stellar core evolves towards collapse and nucleosynthesis of heavier
elements takes place, the core becomes denser and the binding energy of the
inner layers of the star increases. We demonstrate this for one stellar model
of carbon-oxygen core mass (mainly oxygen) of $M_{\rm CO}=13.2M_{\odot}$ in
Fig 1 where we present the star at three times: at center oxygen depletion, at
center silicon depletion, and at collapse. We present the composition of the
main isotopes by lines with different colors and the binding energy $E_{\rm
bind}(m)$ by the black lines. Here $E_{\rm bind}(m)$ is the binding energy
(gravitational + internal) of the envelope laying above mass coordinate $m$.
Relevant to this study is the binding energy of the ejecta, which is the mass
above mass coordinate $m=M_{\rm in}\simeq 1.5,~{}2.5M_{\odot}$. The mass
$M_{\rm in}$ is the baryonic mass of the NS remnant (the corresponding final
gravitational masses will be $\simeq 1.35,~{}2.1M_{\odot}$). We mark these two
masses by the vertical lines. We see that at collapse the binding energy
$E_{\rm bind}(M_{\rm in})$ is larger than at earlier times.

Figure 1: Abundances and the binding energy as function of mass coordinate at
three times for a SECCSN (hydrogen-poor) progenitor model with carbon-oxygen
core mass of $M_{\rm CO}^{\rm collapse}=13.2M_{\odot}$, which corresponds to a
ZAMS mass of $M_{\rm ZAMS}\approx 40M_{\odot}$. The colored step-like lines
are the abundances according to the inset in the lower panel. The black and
smoothly varying lines represent $E_{\rm bind}(m)$, which is the binding
energy of the envelope laying above mass coordinate $m$. The three panels
present these quantities at three different times: center oxygen depletion
(upper panel, binding energy by the black solid-line), center silicon
depletion (middle panel; binding energy by the black dashed-line), core
collapse (lower panel; binding energy by black dotted-line). Note that the
middle panel contains the binding energy at the three times to allow for
comparison. The two vertical lines mark the mass coordinates $m=M_{\rm
in}=1.5M_{\odot}$ and $m=M_{\rm in}=2.5M_{\odot}$. Helium that appears only at
collapse results from disintegration of iron.

In Fig. 2 we present composition and binding energy for a model with a much
more massive core of $M_{\rm CO}=27M_{\odot}$. There are two qualitative
differences between this model and the one we present in Fig. 1. The first
qualitative difference is that the binding energy at collapse is somewhat
smaller than at the earlier time that we present in the figure. The
explanation to the decreasing binding energy shortly before collapse is that
the envelope expands starting from deep in the oxygen burning shell and
outwards. We find (by drawing the density profiles) that moving from the upper
to the middle panel of Fig. 2 the density from $m\simeq 10M_{\odot}$ and
outward decreases, reducing the binding energy. This mass coordinate is deep
inside the shell where oxygen (teal line) burns to S+Si (yellow line).

Figure 2: Similar to Fig. 1 but for a more massive core of $M_{\rm CO}^{\rm
collapse}=26.5M_{\odot}$, which corresponds to a ZAMS mass of $M_{\rm
ZAMS}\approx 65M_{\odot}$. Note that the left vertical axis is scaled
differently than in Fig. 1.

The second qualitative difference comes from the much higher binding energy of
the ejecta of the descendant CCSN of the more massive model, i.e., $E_{\rm
bind}(M_{\rm in})\gtrsim 2\times 10^{51}{~{}\rm erg}$. The implication is that
we do not expect that the neutrino driven explosion mechanism can account for
explosions of such cores. We argue that jets explode these cores. We leave the
discussion of this point, as well as our view that jets also explode cores
with lower binding energy, to section 4, where we also refer to the claim of a
very massive core of SN 2020qlb (West et al., 2023). We first find the range
of such high-binding-energy cores.

###  3.2 High-binding-energy cores

We search for the mass range of cores that have binding energies at collapse
of $E_{\rm bind}(M_{\rm in})\gtrsim 2\times 10^{51}{~{}\rm erg}$. We present
the results in Fig. 3. We present the binding energy for an inner ejecta mass
coordinate of $M_{\rm in}=2.5M_{\odot}$ (upper panel) and $M_{\rm
in}=1.5M_{\odot}$ (lower panel). We focus on the binding energy of these two
mass coordinates $M_{\rm in}=1.5-2.5M_{\odot}$ as the iron core masses at
collapse falls within this mass range. The horizontal line at $2\times
10^{51}{~{}\rm erg}$ is the approximate energy above which we do not expect
that neutrino heating by itself can explode the core. In appendix B we provide
linear fits to the binding energy at collapse as function of CO core mass for
these two mass coordinates $M_{\rm in}=1.5,~{}2.5M_{\odot}$ (table B.1).

Figure 3: The binding energies of the simulated models as a function of the
carbon-oxygen core mass nearing collapse (vertical axis). The panels show the
final binding energy at two mass coordinates: $M_{\rm in}=2.5M_{\odot}$ (top)
and $M_{\rm in}=1.5M_{\odot}$ (bottom). The red data points (filled circles
and stars at the outer panels) are stripped-envelope (SE) models (SECCSNe),
whilst purple empty-circles data points are models from Shishkin & Soker
(2022) that have hydrogen rich envelopes. Red stars are extrapolated data
points, as explained in appendix B. The horizontal line at $2\times
10^{51}{~{}\rm erg}$ denotes the binding energy above which we do not expect
the neutrino delayed explosion mechanism to explode the core.

We simulated $52$ stripped-hydrogen envelope cases. In $27$ cases the cores
reach collapse as we present by the red circles in the figure. In $25$ cases
cases the numerical code encountered difficulties and we had to stop the
simulation before reaching collapse. In these cases we extrapolate from an
evolutionary time before collapse to the collapse time as we explain in
appendix B (red stars in the figure). We also include in the figure the
binding energies of models with hydrogen-rich envelope that we take from
Shishkin & Soker (2022), as we mark by open purple circles. The values we
calculate for the binding energies of these core-collapse progenitors are
similar to those of other studies (e.g., Wang et al. 2022; Burrows et al.
2020).

From Fig. 3 (with a more rigorous derivation in appendix B) we draw our
conclusion that in cases where the inner mass of $M_{\rm in}=2.5M_{\odot}$ of
the core collapses to form a NS, the delayed neutrino mechanism cannot explode
cores with masses of $M_{\rm CO}\gtrsim 15M_{\odot}$ (or maybe rarely do so).
For $M_{\rm in}=1.5M_{\odot}$ we find this limit to be $M_{\rm CO}\gtrsim
13M_{\odot}$.

In Fig. 4 we present a more detailed binding energy profile of the pre-
collapse stripped-envelope models that we simulated. When we take the lowest
binding energy of the inner core at the onset of collapse we find the lower
limit of core masses that the neutrino-driven explosion cannot account for to
be $M_{\rm CO}\simeq 15M_{\odot}$. However, due to possible uncertainties in
the calculations by mesa, evident by the spread of points in Fig. 3 and the
different binding energies of similar core mass models in Fig. 4, we
conservatively take the limit at $M_{\rm CO}\simeq 20M_{\odot}$. Namely, we
argue that jets are responsible for the explosions of all cores of masses
$M_{\rm CO}\gtrsim 20M_{\odot}$.

Figure 4: Upper panel: The binding energies of the simulated stripped-envelope
models (vertical axis) as a function of the mass coordinate (horizontal axis)
at collapse. The color of each line signifies the final CO core mass at
collapse, scaled with the color bar on the right. Black dots mark the mass
coordinate where there is a steep decrease in the density inside the core.
Gray squares denote the edge of the iron core. Lower panel: The binding energy
for hydrogen-rich envelope models from Shishkin & Soker (2022). The grey
horizontal line marks more or less the upper binding energy that the neutrino
delayed mechanism can account for.

In table B.1 we provide the linear fit parameters for the binding energy at
collapse for the edge of the iron core (gray squares in the figure) and the
binding energy curve break (black circles). The binding energy curve break is
point we refer to as separating the inner core from the outer core, and is the
point of lowest binding energy for most simulated cases that reached collapse.

##  4 Discussion and Summary

We simulated the evolution of $52$ massive SECCSN progenitor models
corresponding to ZAMS masses of $20\lesssim M_{\rm ZAMS}\lesssim 82$. We
removed the entire hydrogen-rich envelope, and calculated the binding energy
just before core collapse. The final core mass depends on the ZAMS mass and on
the mass loss parameter (appendix C). We present the structure of the pre-
collapse progenitor for two cases in Figs. 1 and 2. We find that to a fare
accuracy we can linearly fit the binding energy of these stripped-envelope
progenitors to the CO core mass $M_{\rm CO}$ (Fig. 3 and table B.1).

We present our main results in Fig. 3. In those figures the horizontal gray
line represents the approximate maximum energy that the neutrino-driven
mechanism can supply, $E_{\nu}^{\rm max}=2\times 10^{51}{~{}\rm erg}$. We find
that the binding energy calculated at $M_{\rm in}=1.5M_{\odot}$ and $M_{\rm
in}=2.5M_{\odot}$, of progenitors with a carbon-oxygen core mass of $M_{\rm
CO}\gtrsim 13M_{\odot}$ and $M_{\rm CO}\gtrsim 15M_{\odot}$, respectively, are
larger than $E_{\nu}^{\rm max}$. Namely,

$E_{\nu}^{\rm max}\lesssim\begin{cases}&E_{\rm bind,1.5}\quad{\rm for}\quad
M_{\rm CO}\gtrsim 13M_{\odot}\\\ &E_{\rm bind,2.5}\quad{\rm for}\quad M_{\rm
CO}\gtrsim 15M_{\odot}.\end{cases}$ (1)

We account for the uncertainty of our simulations and parameter choices by
setting a higher upper limit of $\simeq 20M_{\odot}$ for the cores that the
neutrino explosion mechanism might explode. The main conclusion is that the
delayed neutrino explosion mechanism is unlikely to explode stars with a core
mass of $M_{\rm CO}\gtrsim 20M_{\odot}$. The jittering jets explosion
mechanism, on the other hand, has no limiting explosion energy in these ranges
as it is fueled by accretion onto the compact remnant (e.g., (Gilkis et al.,
2016; Soker & Gilkis, 2017)).

Let us apply our results to a specific SECCSN. In a recent paper West et al.
(2023) deduce that SN 2020qlb had an explosion energy of $\simeq 20\times
10^{51}{~{}\rm erg}$ and estimate the progenitor pre-explosion mass, ejecta
plus remnant mass, to be $M_{\rm ej}+M_{\rm rem}\simeq 30-50M_{\odot}$.
According to our results the binding energy alone of such cores is $E_{\rm
bind}>3\times 10^{51}{~{}\rm erg}$. We therefore conclude that jets must have
exploded SN 2020qlb. Jets can also supply the kinetic energy of the ejecta.
Namely, jet-driven explosions might make the magnetar powering less critical
or not needed at all (although a magnetar might be present). If indeed a
magnetar, i.e., a rapidly rotating magnetized NS, was formed, then most likely
the explosion was via jittering jets. The reason is that an explosion driven
by a fixed-axis jets, like if the core is rapidly rotating, will not expel
much mass from the equatorial region, which in turn is accreted by the newly
formed central object. Therefore, the final mass of the remnant will be large
and the remnant will be a BH (see discussion in Soker 2022b). We note that
strong fixed-axis jets might remove some mass even from the equatorial plane,
e.g., Gottlieb et al. (2022b). However, they assumed relativistic jets
launched by an already formed BH of $4M_{\odot}$. The jittering jets explosion
mechanism asserts that fixed-axis jets from a NS cannot remove mass from the
equatorial plane, at least for several seconds as 2D and 3D simulations show
(Papish & Soker 2014b; Papish & Soker 2014a). The jets might remove mass from
the equatorial plane at later times. However, by that time the central star
has already grown to a BH by accreting from the equatorial plane. Note that
the jets that the papers above simulated are sub-relativistic, initial jet
velocity of $10^{5}{~{}\rm km}{~{}\rm s}^{-1}$. This implies that per unit
energy the jets have higher momentum than what relativistic jets have.

On a large scope, our study adds to the growing evidence pointing to the major
roles that jets play in the explosion, as well as pre-explosion and post-
explosion, of CCSNe (for a recent review see Soker 2022b).

## Acknowledgments

We thank an anonymous referee for helpful comments. This research was
supported by a grant from the Israel Science Foundation (769/20).

## Data availability

The data underlying this article are available in Zenodo, at
https://doi.org/10.5281/zenodo.7529670.

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##  Appendix A Numerical prescription details

Our numerical scheme files (‘inlists’) are a modified version of the
‘20_pre_ms_to_cc’ mesa version r22.05.1 ‘test_suite’ example. We adapted this
example to run on mesa version r15140 and incorporated certain parameters
(e.g., overshooting and mesh resolution) according to our previous works
(Shishkin & Soker, 2021, 2022). The full ’inlists’ that we used are available
online111Zenodo: Modified inlists to reproduce the models. Also included a
full simulation list with key parameters and the simulated models at several
different time points.. Here we mention some of the important parameters.

In a similar fashion to our previous papers which focus on the convective
profile of the inner layers of massive stars (Shishkin & Soker, 2021, 2022),
we use the exponential overshooting prescription (Herwig, 2000) with
symmetrical (both ‘bottom’ and ‘above’) and uniform (all burning regions)
settings and $f=0.01,f_{0}=0.004$ parameters.

We chose the Henyey scheme (Henyey et al., 1965) for mixing length theory
(MLT, Böhm-Vitense 1958) with $\alpha_{\rm MLT}=1.5$. We also enable the
Ledoux criterion (Ledoux, 1947) and set thermohaline option to ‘Kippenhahn’
with ‘thermohaline_coeff = 1’ alongside ‘$\rm alpha\\_{semiconvection}=0.01$.

We make use of the ’Dutch’ wind loss scheme (e.g., Vink et al. 2001; Nugis &
Lamers 2000), and vary the scaling factor (along with the initial mass) to
achieve different core masses.

For the nuclear network we use the 22 isotopes of ‘approx21_cr60_plus_co56’
(e.g., Timmes 1999), aimed at stellar evolution up to collapse. This network
includes hydrogen, $\rm He3$ and $\rm He4$ up to the heavier isotopes of $\rm
Fe52~{},~{}Fe54~{},~{}Fe56~{},~{}Co56~{},~{}Ni56~{},~{}Cr60$.

We scale mesh refinement gradually up to ‘max_dq’ values of $1d-4$ at the
later stages (from the default value of $1d-2$) to properly resolve the fine
burning features close to core collapse.

The mesa equation of state (EOS) is a blend of the OPAL (Rogers & Nayfonov,
2002), SCVH (Saumon et al., 1995), FreeEOS (Irwin, 2004), HELM (Timmes &
Swesty, 2000), PC (Potekhin & Chabrier, 2010), and Skye (Jermyn et al., 2021)
EOSs. Nuclear reaction rates are from JINA REACLIB (Cyburt et al., 2010),
NACRE (Angulo et al., 1999) and additional tabulated weak reaction rates
Fuller et al. (1985); Oda et al. (1994); Langanke & Martínez-Pinedo (2000).
Screening is included via the prescription of Chugunov et al. (2007). Thermal
neutrino loss rates are from Itoh et al. (1996). Radiative opacities are
primarily from OPAL (Iglesias & Rogers, 1993, 1996), with low-temperature data
from Ferguson et al. (2005) and the high-temperature, Compton-scattering
dominated regime by Poutanen (2017). Electron conduction opacities are from
Cassisi et al. (2007) and Blouin et al. (2020).

##  Appendix B Binding energy estimation

Because of numerical difficulties of stripped-envelope progenitors
(specifically some steep gradients) some simulations did not reach the phase
of core collapse, although they did reach oxygen depletion and/or silicon
depletion at the center. Time steps became much too short and we had to
terminate the simulations before core collapse. In these cases we estimated
the binding energy at collapse (red-stars in Fig. 3) by extrapolating the
binding energy during earlier phases using linear fits.

We made linear fits to the binding energies as function of the CO core masses
at three evolutionary phases: oxygen depletion, silicon depletion, and core
collapse. In Fig. B.1 we present these three fittings by blue, orange, and red
lines, respectively, for $M_{\rm in}=2.5M_{\odot}$ (upper panel) and $M_{\rm
in}=1.5M_{\odot}$ (lower panel). From these three lines we can find the ratio
of the binding energy at core collapse to the binding energy at oxygen
depletion and to the binding energy at silicon depletion. In cases where we
did not reach core collapse we use this ratio at the given CO core mass to
calculate the expected binding energy at core collapse. We mark these energies
by red-stars in Fig. B.1 and use them in Fig. 3. Error bars attached to the
red stars signify the $1\sigma$ intervals of the this extrapolation procedure.
We note that the CO core mass does not change much after oxygen depletion in
the non-extended helium phase simulations. The average difference between the
CO core mass at central oxygen depletion and at core collapse is $\Delta
m_{\rm CO}^{\rm core}=0.06\pm 0.33M_{\odot}$.

Figure B.1: The binding energy of the envelope above mass coordinate $M_{\rm
in}=2.5M_{\odot}$ (upper panel) and $M_{\rm in}=1.5M_{\odot}$ (lower panel) as
a function of the final carbon-oxygen core mass. The blue circles are at
central oxygen depletion ($5\%$ oxygen in the center), the orange circles are
at silicon depletion ($5\%$ silicon in the center), and red circles are at
core collapse. The three respective lines are the linear fit to the points.
Red stars are the extrapolated values for the binding energy at collapse based
on available earlier data points (oxygen depletion or silicon depletion) for
the cases that did not reach collapse (see text).

We fit the binding energy $E_{\rm bind}$ versus the CO core mass $M_{\rm CO}$
by a linear fit $E_{\rm bind}=aM_{\rm CO}+b$. In Table B.1 we list the values
of the two coefficients for the six lines (three stage for two values of the
mass that collapses to form the NS). We also list (last column) the number of
data points that were used at each fitting.

| $aM_{\rm CO}+b$ | $\rm Fit_{1.5M_{\odot}}$ | $\rm Fit_{2.5M_{\odot}}$ | $\rm Fit_{BEbreak}$ | $\rm Fit_{Fecore}$ | No. of points
---|---|---|---|---|---|---
Collapse | a $[\rm{10^{51}erg}/M_{\odot}]$ | $0.122\pm 0.024$ | $0.137\pm 0.024$ | $0.131\pm 0.022$ | $0.124\pm 0.023$ | 27
b $[\rm{10^{51}erg}]$ | $0.537\pm 0.46$ | $-0.053\pm 0.444$ | $-0.028\pm 0.421$ | $0.314\pm 0.424$
$\rm Si_{cntr}~{}depletion$ | a $[\rm{10^{51}erg}/M_{\odot}]$ | $0.132\pm 0.017$ | $0.145\pm 0.02$ | $--$ | $--$ | 43
b $[\rm{10^{51}erg}]$ | $0.159\pm 0.38$ | $-0.099\pm 0.445$ | $--$ | $--$
$\rm O_{cntr}~{}depletion$ | a $[\rm{10^{51}erg}/M_{\odot}]$ | $0.15\pm 0.017$ | $0.175\pm 0.02$ | $--$ | $--$ | 47
b $[\rm{10^{51}erg}]$ | $0.002\pm 0.394$ | $-0.447\pm 0.479$ | $--$ | $--$
Table B.1: The linear fits to the lines in Fig. B.1 and the number of data
points for each of the simulation groups: at collapse (second row), at center
silicon depletion (third row) and center oxygen depletion (bottom row). The
third and fourth columns are the fits to the binding energy $E_{\rm
bind,1.5},E_{\rm bind,2.5}$ at mass coordinates $M_{\rm in}=1.5M_{\odot}$ and
$M_{\rm in}=2.5M_{\odot}$, respectively. In the fifth column we present the
linear fit to the variation of the binding energy at the black dots in Fig. 4
with the CO core mass. In the sixth column we present the linear fit to the
variation of the binding energy at the edge of the iron core (gray squares in
Fig. 4) with the CO core mass. Linear fits are in units of energy $E_{\rm
bind}~{}[\rm 10^{51}erg]$ to CO core mass $M_{\rm CO}~{}[\rm M_{\odot}]$.
Values and errors ($2\sigma$) are in accordance with Fig. B.1.

##  Appendix C Simulations parameter space

In Fig. C.2 we present the simulations that we conducted in a three-parameters
space. As input we show the dutch wind scaling factor in mesa in the range of
$0.5<\alpha_{\rm Dutch,wind}<1$ and the zero age main sequence (ZAMS) mass in
the range of $20M_{\odot}<M_{\rm ZAMS}<82M_{\odot}$. As an output we present
the final CO core mass (mainly oxygen mass) in units of solar mass according
to the color bar.

Figure C.2: Wind (Dutch scheme) scaling factors (vertical axis) and the ZAMS
masses of the cases (horizontal axis) that we simulated, and the final CO core
mass (by color bar). We denote with a black X the cases where we extended the
He burning to a later stage before removing the hydrogen envelope.