11institutetext: Institute of Astrophysics, FORTH, Dept. of Physics,
University of Crete, Voutes, University Campus, GR-71003 Heraklion, Greece
11email<EMAIL_ADDRESS>22institutetext: Argelander-Institut für
Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
33institutetext: Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69,
53121 Bonn, Germany 44institutetext: School of Physics and Astronomy, Monash
University, VIC 3800, Australia 55institutetext: Institut d’Astrophysique de
Paris, CNRS-Sorbonne Université, 98 bis boulevard Arago, F-75014 Paris, France
66institutetext: Niels Bohr International Academy, Niels Bohr Institute,
Blegdamsvej 17, DK-2100, Copenhagen, Denmark 77institutetext: DARK, Niels Bohr
Institute, University of Copenhagen, Jagtvej 128, DK-2200, Copenhagen, Denmark
88institutetext: Department of Physics and Astronomy, Seoul National
University, Gwanak-gu, Seoul, 151-742, Republic of Korea

# Stripped-envelope stars in different metallicity environments

II. Type I supernovae and compact remnants

David R. Aguilera-Dena 112233 Bernhard Müller 44 John Antoniadis 113322
Norbert Langer 2233 Luc Dessart 55 Alejandro Vigna-Gómez 6677 Sung-Chul Yoon
88

Stripped-envelope stars can be observed as Wolf-Rayet (WR) stars, or as less
luminous hydrogen-poor stars with low mass loss rates and transparent winds.
Both types are potential progenitors of Type I core-collapse supernovae (SNe).
We use grids of core-collapse models obtained from helium stars at different
metallicities to study the effects of metallicity on the transients and
remnants these stars produce. We characterise the surface and core properties
of our core collapse models, and investigate their explodability employing
three criteria. In cases where explosions are predicted, we estimate the
ejecta mass, explosion energy, nickel mass and neutron star (NS) mass.
Otherwise, we predict the mass of the resulting black hole (BH). We construct
a simplified population model, and find that the properties of SNe and compact
objects depend strongly on metallicity. Ejecta masses and explosion energies
for Type Ic SNe are best reproduced by models with $Z=0.04$ which exhibit
strong winds during core helium burning. This implies that either their mass
loss rates are underestimated, or that Type Ic SN progenitors experience mass
loss through other mechanisms before exploding. The distributions of ejecta
masses, explosion energies and nickel mass for Type Ib SNe are not well
reproduced by progenitor models with WR mass loss, but are better reproduced
if we assume no mass loss in progenitors with luminosities below the minimum
WR star luminosity. We find that Type Ic SNe become more common as metallicity
increases, and that the vast majority of progenitors of Type Ib SNe must be
transparent-wind stripped-envelope stars. We find several models with pre-
collapse CO-masses of up to $\sim 30\,\mathrm{M}_{\odot}$ may form $\sim
3\,\mathrm{M}_{\odot}$ BHs in fallback SNe. This may carry important
consequences for our understanding of SNe, binary BH and NS systems, X-ray
binary systems and gravitational-wave transients.

###### Key Words.:

Stars: massive – Stars: Wolf-Rayet – Supernovae: general

##  1 Introduction

Most massive stars are thought to be members of binary or multiple systems
that eventually interact via mass transfer (e.g. Sana & Evans 2011; Sana et
al. 2012, 2014; Moe & Di Stefano 2017). Such binary interaction is expected to
at least partially remove the hydrogen envelope (e.g. Podsiadlowski et al.
1992; Yoon et al. 2010; Gilkis et al. 2019; Langer et al. 2020; Laplace et al.
2020; Klencki et al. 2020, 2021; Sen et al. 2021), leaving behind a stripped-
envelope star. Stripped-envelope stars differ substantially from their
hydrogen-rich counterparts in their observational properties, their evolution,
and the possible outcomes that they lead to. The most massive and luminous
among them are observed as classical Wolf-Rayet (WR) type stars. These objects
are distinguished by very strong mass-loss rates and optically-thick winds,
which lead to their characteristic emission line-dominated spectra, with
either strong nitrogen (WN subclass), carbon (WC subclass) or oxygen features
(WO subclass, see e.g. Crowther 2007 for a review). It has been empirically
found that there is a metallicity-dependent threshold luminosity above which
stripped-envelope stars may become observable as WR stars. (Shenar et al.
2020). Stars below this luminosity limit are thought to have weaker, optically
thin winds (Vink 2017; Sander & Vink 2020). Such stars are also UV-bright, but
optically-faint, which renders them difficult to observe (Wellstein et al.
2001; Götberg et al. 2017).

In Aguilera-Dena et al. (2021), hereafter referred to as Paper I, we presented
a grid of detailed one-dimensional evolutionary models of stripped-envelope
stars, and proposed a method to characterise the minimum WR luminosity as a
function of metallicity. We then used stellar evolution models to explore the
influence of metallicity during late evolutionary phases of stripped-envelope
stars. We found that the properties of our models depend sensitively on the
metallicity-dependent mass loss. Combining our theoretical and numerical
findings, we constructed a model population of stripped-envelope stars,
finding that the populations of WR and transparent-wind stripped-envelope
stars are shaped by a combined effect of the minimum WR luminosity, and the
strength of their mass loss, both of which change as a function of metallicity
at different rates.

Stripped-envelope stars are also progenitors of hydrogen-poor supernovae (SNe,
of Type IIb, Ib and Ic), which are characterised by very weak or absent
hydrogen features in their spectra near maximum light (e.g. Filippenko 1997).
Envelope stripping has been found to have a strong influence in the evolution
of massive stars, and in the properties that determine what kind of SN
explosion they lead to (e.g. Woosley 2019; Schneider et al. 2021; Laplace et
al. 2021). With this in mind, in this paper we aim to study the effect of
metallicity on stripped-envelope stars through the study of hydrogen-poor SNe,
which can be observed in detail at much larger distances than their
progenitors. These transients are relatively frequent in the local Universe.
Most common among them are the “normal” Type IIb, Type Ib and Type Ic;
accounting for around 30% of all nearby core-collapse SNe (Cappellaro et al.
1999; van den Bergh et al. 2005; Prieto et al. 2008; Smartt et al. 2009; Li et
al. 2011; Shivvers et al. 2017). Less frequent sub-classes of stripped-
envelope SNe include Type Ic-BL SNe, often found in association with long
gamma-ray bursts (Hjorth & Bloom 2012), and Type I superluminous SNe (SLSNe;
Perley et al. 2016; Japelj et al. 2016; Schulze et al. 2018), typically found
in environments of lower metallicity (Lunnan et al. 2014). A common origin for
these transients has been proposed, that relies in an evolutionary channel
that is distinct from the ordinary stripped-envelope SNe (Aguilera-Dena et al.
2018, 2020).

The absence of hydrogen lines in the spectra of ordinary Type Ib and Type Ic
SNe indicates that their progenitors have completely lost their hydrogen-rich
envelope by the time they collapse (Dessart et al. 2011; Hachinger et al.
2012). Both SN types have light curves that are often indistinguishable from
each other (Drout et al. 2011). Type Ib SNe however are characterised by
helium lines, which are absent in Type Ic SNe. The spectra of Type Ib SNe are
consistent with stripped-envelope progenitors that end their evolution with a
helium envelope that is rich in nitrogen. Conversely, Type Ic SNe originate in
stripped-envelope stars that have lost this layer through winds, and have
exposed the products of helium burning (e.g. Dessart et al. 2012, 2020).
Modjaz et al. (2011) found that Type Ib and Type IIb SNe typically originate
in environments that are less metal-rich than those of Type Ic SNe. These
properties indicate that metallicity-dependent wind mass loss is a key factor
in the late evolution of their progenitors.

While massive WR stars are often regarded as progenitors of hydrogen-poor SNe
(Gaskell et al. 1986), obervationally-inferred ejecta masses (Taddia et al.
2018; Barbarino et al. 2021) are lower than typical WR masses (e.g. Sander et
al. 2019). Rather, their observed light-curves and spectra are better
reproduced by explosion models of relatively low-mass helium stars (e.g.
Dessart et al. 2011, 2020). Some SNe have been found to have properties that
suggest WR stars as more likely progenitor candidates (e.g. Gal-Yam et al.
2022), but their properties are unusual compared to the bulk of the population
of stripped-envelope SNe.

Several studies of evolutionary channels for interacting binaries as likely
progenitors of Type I core-collapse SNe have also been carried out in recent
years. Several approaches have been taken, including detailed binary evolution
(e.g. Wellstein & Langer 1999; Yoon et al. 2010, 2017), rapid binary
population synthesis (e.g. Zapartas et al. 2017; Kruckow et al. 2018; Vigna-
Gómez et al. 2018), and detailed evolutionary calculations of single (e.g.
Arnett 1974; Woosley et al. 1995; McClelland & Eldridge 2016; Yoon 2017;
Woosley 2019; Ertl et al. 2020) and binary helium stars (e.g. Dewi et al.
2002; Dewi & Pols 2003; Tauris et al. 2015). Evolutionary channels for Type I
core-collapse SNe have been widely found in these studies, but explaining the
rate, distribution of ejecta mass and SNe types remains challenging.

In a recent study, Yoon (2017) found that an increase in the metallicity
dependent mass-loss rate that is observed to in carbon-rich WC stars, could
account for the formation of the faintest WC and WO type stars in our Galaxy.
The detailed SN light curve and spectral models of Dessart et al. (2020)
confirm that an increase in the metallicity-dependent mass loss rate of WR
stars can lead to the production of Type Ic SNe, and results in a sharp
dichotomy in the chemical compositions and a change of trend in the initial-
to-final mass relation of progenitors of Type Ic and Type Ib SNe.

Unlike the case of Type II SNe, where several progenitors have been directly
identified in observations, most attempts to recognise the progenitor systems
to type Ibc SNe in archival data of their fields have been unsuccessful,
setting only upper limits (Smartt 2009). So far, only two progenitor
identifications for type Ib SNe (Cao et al. 2013; Kilpatrick et al. 2021) and
a candidate for a type Ic SN (Kilpatrick et al. 2018) exist. Non-detections
place constrains on the progenitor stars to these explosions (and the binary
companions, if present), that suggest them to likely be hot and compact (Yoon
et al. 2012). The lack of hydrogen in their spectra also sets a stringent
constraint on the pre-collapse surface hydrogen abundance of less than
$0.001\,\mathrm{M}_{\odot}$ (Dessart et al. 2011). With this in mind, two main
evolutionary channels have been proposed to explain the inferred abundances of
these SNe: (i) Single massive stars becoming stripped via strong stellar winds
(e.g. Georgy et al. 2009), and (ii) binary stars that have been stripped due
to interaction (Podsiadlowski et al. 1992; Langer 2012).

Evidently, variations in surface abundances and metallicity in WR mass loss
have a very strong impact on the observed properties of WR stars, as well as
in SNe that might result from them. Motivated by this, we study the evolution
of helium stars as a proxy for stripped-envelope stars including wind mass
loss, in a similar fashion to Yoon (2017) and Woosley (2019). We extend the
initial conditions to cover different metallicity environments, focusing on
high metallicities, which are not often addressed in the literature, but that
can account for changes in properties of populations of WR stars and Type I
core-collapse SNe. We also pay particular attention to the lower luminosity
limit of WR stars through the method found in Paper I, and explore its effect
in the properties of stripped-envelope SNe.

We have divided paper as follows: In Sect. 2 we briefly describe the models of
stripped-envelope stars at the pre-collapse stage that we employ, and the
methods we employ to analyse them. In Sect. 3 we present our main results and
in Sect. 4 we discuss our predictions for SN and compact-object populations.
We finalise our paper with a discussion in Sects. 5 and conclusions in Sect.
6.

##  2 Method

###  2.1 Pre-supernova evolution of stripped-envelope stars

To quantify the effect of metallicity on populations of stripped-envelope SNe,
we study the structure and composition of grids of stellar evolutionary
calculations of helium stars at core collapse, defined as the time where the
iron core reaches an infall velocity of more than 1000 km s-1. These core-
collapse stellar structure models correspond to the end-points of the
evolutionary calculations presented in Paper I. Here, we give a short
description of the models, but refer the reader to Paper I for more details on
the numerical and physical parameters employed in the stellar evolution
calculations. The grids of core collapse models of non-rotating helium stars
models were computed using the Modules for Experiments in Stellar Astrophysics
(MESA) code (Paxton et al. 2011, 2013, 2015, 2018), version 10398.

The grids of evolutionary calculations analysed here correspond to seven
metallicities, from 0.01 to 0.04 in steps of 0.005, scaled from solar
abundances found by Grevesse et al. (1996). Each grid has models with initial
masses between 1.5 and 70 $\,\mathrm{M}_{\odot}$ in steps of 0.5
$\,\mathrm{M}_{\odot}$. However, only the evolutionary sequences with an
initial mass above $4.0-4.5$ $\,\mathrm{M}_{\odot}$ (depending on metallicity)
were calculated up to core collapse. Evolutionary sequences with lower initial
masses either failed to converge to core collapse due to the presence of off-
centre carbon and neon burning layers that are numerically difficult to
resolve, or do not experience to core collapse at all.

The initial models of these calculations were generated through pre-main
sequence models of standard composition that were evolved with artificial
mixing and no mass loss until core-hydrogen depletion. After that, they were
relaxed until reaching thermal equilibrium, without mass loss, but with
standard mixing. This guarantees that the surface abundances of our models
correspond to CNO equilibrium abundances, with enhanced nitrogen and reduced
amounts of carbon and oxygen.

Convection was modeled through mixing length theory (Böhm-Vitense 1958), with
$\alpha_{\mathrm{MLT}}=2.0$. We adopted the Ledoux criterion for convection,
and we modeled semiconvection using $\alpha_{\mathrm{SC}}=1.0$. To calculate
the energy generation rate and changes in chemical composition, we use the
approx21 nuclear network in MESA, which includes 21 species from 1H to 56Ni,
and neutrons.

To avoid convergence issues, we employ MESA’s mlt++ (Paxton et al. 2013), and
exclude radiative acceleration in the envelope by setting the velocity to 0 in
layers with $T>10^{8}$ K during the late evolution. We use the mass loss rates
for WN and WC stars of Yoon (2017), who adapts the observed relations of each
class, as measured by Hainich et al. (2014) and Tramper et al. (2016),
respectively. The mass loss rates are given by

$\dot{\text{M}}_{\text{WN}}=f_{\text{WR}}\left(\frac{\text{L}}{\,{\rm
L}_{\odot}}\right)^{1.18}\left(\frac{\text{Z}_{\mathrm{init}}}{\,{\rm
Z}_{\odot}}\right)^{0.6}10^{-11.32}\frac{{\text{M}_{\odot}}}{\text{yr}},$ (1)

for Y=1-Z${}_{\text{init}}$, and by

$\dot{\text{M}}_{\text{WC}}=f_{\text{WR}}\left(\frac{\text{L}}{\,{\rm
L}_{\odot}}\right)^{0.85}\left(\frac{\text{Z}_{\mathrm{init}}}{\,{\rm
Z}_{\odot}}\right)^{0.25}\text{Y}^{0.85}10^{-9.2}\frac{{\text{M}_{\odot}}}{\text{yr}}$
(2)

for Y$<$0.9. The intermediate range is linearly interpolated between the two
regimes, using

$\dot{\text{M}}=(1-x)\dot{\text{M}}_{\text{WN}}+x\dot{\text{M}}_{\text{WC}},$
(3)

with $x=(1-\text{Z}_{\text{init}}-\text{Y})/(1-\text{Z}_{\text{init}}-0.9)$.
We set $f_{\text{WR}}$=1.58, and scale these equations assuming $\,{\rm
Z}_{\odot}=$0.02.

###  2.2 Explodability criteria and observable supernova properties

To assess the explodability of the core-collapse models, we employ four
different tests. First, we measure the so-called compactness parameter at core
collapse, as proposed by O’Connor & Ott (2011). It is defined as

$\xi_{M}=\frac{M/\,\mathrm{M}_{\odot}}{R/1000\ \mathrm{km}},$ (4)

and evaluated at mass coordinate 2.5 $\,\mathrm{M}_{\odot}$ (and therefore
labelled as $\xi_{2.5}$). A high value of this quantity is indicative of a
lower probability for a successful SN within the neutrino-driven explosion
paradigm. However, a clear boundary between successful and failed SN explosion
cannot be drawn from this one-parameter check alone.

To complement this, we employ the two-parameter criterion proposed by Ertl et
al. (2016). They propose an explodability test based on a relation between
proxies of the core mass and the core density gradient of stars at collapse
via the following parameters:

$M_{4}=M(s=4)/\,\mathrm{M}_{\odot},$ (5)

the mass coordinate at the location where the specific entropy $s$ is equal to
4; and

$\mu_{4}=\left.\frac{\mathrm{d}m/\,\mathrm{M}_{\odot}}{\mathrm{d}r/1000\
\mathrm{km}}\right|_{s=4}.$ (6)

The former is thought to be a good proxy of the core mass, while the latter is
a measure of the steepness of the density gradient at the same location. Ertl
et al. (2016) provide a test using a combination of these two values to
determine the final fate of a stellar model.

As a final approach we use the semi-analytical model proposed by Müller et al.
(2016), and the updated version of the same model presented by Mandel & Müller
(2020), which includes the effect of fallback onto the newly formed compact
object. These semi-analytical models were constructed within the context of
the neutrino-driven SN explosion mechanism, and were calibrated against
results of 3D numerical simulations. These models also yield predictions for
several SN and compact-object observables, such as neutron star (NS) and black
hole (BH) gravitational masses, explosion energy and nickel mass produced in
the resulting explosions. Using the predictions for explosion energy and
remnant mass, we also calculate the kick velocity imparted on the newly-formed
compact object following Vigna-Gómez et al. (2018). If the initial energy of
the explosion is small, then the asymmetric inner ejecta is expected to fall
back completely. This results in a weak explosion launched when the initial
ejecta become subsonic. In this case, the transport of energy by the sound
pulse becomes decoupled from the transport of matter. The sound pulse quickly
becomes spherical and any explosion asymmetry becomes attenuated. Therefore,
in cases where explosion energies are low, the kick velocity is set to 0 km
s-1, as it is expected to be very small.

The model of Müller et al. (2016) is a set of equations that treats the pre-
explosion and explosion phases of a core collapse event in a simplified form.
The model takes the density, chemical composition, binding energy, sound speed
and entropy profiles of a stellar model at core collapse, and computes the
amount of mass that is accreted by the proto-NS, and the location of the
formed shock as a function of time. It then computes the location of the
heating region behind the shock, and estimates if and when the neutrino
heating conditions become sufficient to trigger an explosion. If the gain
region acquires enough energy to produce an explosion, then it calculates the
evolution of the NS mass and the explosion energy until they settle to their
final state. During the explosion phase, the model accounts for energy input
by neutrinos, and energy loss and gain by nuclear disassociation and explosive
nuclear burning.

In the version of the model presented by Mandel & Müller (2020), the shock is
allowed to propagate further out even if the NS reaches the maximum mass
threshold due to ongoing accretion after an explosion has been triggered. The
shock is left to evolve until it either exceeds escape velocity (resulting in
the termination of accretion onto the remnant) or is attenuated to a weak
sound pulse that ejects a small part of the envelope.

These two explosion models depend on several variable parameters that affect
the explodability of each stellar model, and the properties of the successful
SN explosions. The main variable parameters in the explosion model are
$\alpha_{\mathrm{out}}$, the volume fraction occupied by neutrino-driven
outflows far away from the gain radius; $\alpha_{\mathrm{turb}}$, a correction
to the expansion of the shock radius due to turbulent stresses;
$\beta_{\mathrm{expl}}$, the shock compression ratio during the explosion
phase; $\zeta$, the efficiency factor for the conversion of accretion energy
into neutrino luminosity; $\tau_{1.5}$, the cooling timescale for a 1.5
$\,\mathrm{M}_{\odot}$ NS, and the maximum gravitational mass of a NS.

We calculate the explosion properties of our models with a large range of
values for all parameters. In the main text, we present the outcome of these
calculations using a fiducial set of parameters. They correspond to
$\alpha_{\mathrm{out}}=0.4$, $\alpha_{\mathrm{turb}}=1.18$,
$\beta_{\mathrm{expl}}=4$, $\zeta=0.75$ and $\tau_{1.5}=1.2$. We set the
maximum NS gravitational mass to $2.05\,\mathrm{M}_{\odot}$. This choice
corresponds to a lower value of $\alpha_{\mathrm{out}}$ and $\zeta$, compared
with the standard values used by Müller et al. (2016). We use this particular
combination of parameters, as it more closely resembles the outcome expected
from employing the Ertl et al. (2016) explodability criterion. We discuss the
impact of parameter variation, as well as the impact of the binding energy
that is used to calculate the outcome of the explosions in Appendix B. The
models from Set B, presented in Appendix B.2, correspond to those used by
Antoniadis et al. (2022).

###  2.3 Population synthesis models

To gain a better understanding of how the measurable parameters of Type I SN
explosions evolve as a function of metallicity, we have generated populations
of helium stars in a similar way as described in Paper I. We randomly sample
main sequence stars from a Salpeter initial mass function (IMF Salpeter 1955),
and use relation between zero-age main-sequence (ZAMS) and helium-core masses
from Woosley (2019), given by

$\displaystyle\mathrm{M}_{\mathrm{He,ini}}$ $\displaystyle\approx$
$\displaystyle\left\\{\begin{array}[]{cc}0.0385\
\mathrm{M}_{\mathrm{ZAMS}}^{1.603},&\mathrm{if}\
\mathrm{M}_{\mathrm{ZAMS}}<30\,\mathrm{M}_{\odot}\\\
0.5\textrm{M}_{\mathrm{ZAMS}}-5.87\,\mathrm{M}_{\odot},&\textrm{if}\
\mathrm{M}_{\mathrm{ZAMS}}\geq 30\,\mathrm{M}_{\odot}\\\ \end{array}\right..$
(9)

We count the number of transients generated, independently of the lifetime of
helium stars, assuming a constant star-formation rate, and convolve this with
our results from Sect. 3.2. Counting every transient generated from this
sampling, assuming that they are all stripped, and then looking at the
relative distribution of their properties, is equivalent to assuming that the
stripping probability is the same at each mass, and ignores the contribution
of Type Ib and Type Ic SNe that are generated through other channels. While
the assumptions in this population model are an oversimplification, they allow
us to crudely estimate how the fate of stripped-envelope stars change across
cosmic time.

A caveat in our calculations is that our models with initial masses below
$4-4.5$ $\,\mathrm{M}_{\odot}$ do not reach core collapse, since they ignite
nuclear burning above a degenerate core and experience convergence problems.
Therefore, we cannot assess the minimum mass at which stars produce SNe, and
stars with masses smaller than our least massive core collapse model that
produce SNe cannot be analyzed using the Müller et al. (2016) model; which
does not allow us to infer their explosion properties. To account for the
number of successful supernovae regardless of the results from our
evolutionary calculations, we follow Chanlaridis et al. (2022) and set the
lower limit in initial helium-core mass for core collapse at 3
$\,\mathrm{M}_{\odot}$, which they find to be metallicity independent if no
core overshooting is included during helium burning. This is further justified
since stripped-envelope stars of this mass are always below
$\mathrm{L}^{\mathrm{tau}}_{\mathrm{min,WN}}$, implying that their mass loss
rates are very small during core helium burning (although they may experience
intense mass loss after core helium depletion, see Chanlaridis et al. (2022)).
Consequently the evolution of stripped-envelope stars near the boundary
between core collapse progenitors and WDs or thermonuclear explosions is
similar, independent of their initial chemical composition. We estimate the
distribution of observables in this regime by extrapolating our finding for
higher-mass stars. More specifically, NS gravitational masses are assigned a
random value between 1.22 and 1.3 $\,\mathrm{M}_{\odot}$, explosion energies
are in the range of 0.5-0.8$\times 10^{51}$ erg, and nickel masses are in the
range 0.04-0.15 $\,\mathrm{M}_{\odot}$. Kick velocities are assigned through
these values, according to Eq. 2 of Mandel & Müller (2020), and ejecta mass is
inferred by extrapolating the initial mass–final mass relation, and removing
the corresponding baryonic mass that was lost to the formation of the NS (see
below for further justification).

##  3 Impact of metallicity on Type I supernovae and compact remnants

In this Section, we study the effect of metallicity on the structure and
chemical composition of stripped-envelope stars at the end of their evolution,
and how it affects the transients that they produce. To do this, we
characterize the final outputs of the models described Paper I. To remain
consistent in our notation, in Table 1 we recall the definitions of key
quantities from Paper I, which are important for the analysis of core collapse
models.

Table 1: Notation definitions. Symbol | Definition
---|---
$\mathrm{L}^{\mathrm{tau}}_{\mathrm{min,WN}}$ | Minimum luminosity above which a stellar model is classified as a WN star (see Paper I)
$\mathrm{L}^{\mathrm{tau}}_{\mathrm{min,WC}}$ | Minimum luminosity above which a stellar model is classified as a WC star (see Paper I)
$\mathrm{L}^{\mathrm{evo}}_{\mathrm{min,WC}}$ | Minimum luminosity at which a WC is produced by winds in our evolutionary calculations
$\mathrm{L}^{\mathrm{evo}}_{\mathrm{min,WO}}$ | Minimum luminosity at which a WO is produced by winds in our evolutionary calculations
$\mathrm{t}_{\mathrm{total}}$ | Total lifetime of a helium star from core helium ignition to core collapse
$\mathrm{t}_{\mathrm{He-N}}$ | Lifetime of a helium star spent with an outer helium-nitrogen shell
$\mathrm{t}_{\mathrm{He-C}}$ | Lifetime of a helium star spent with an outer helium-carbon shell
$\mathrm{t}_{\mathrm{He-O}}$ | Lifetime of a helium star spent with oxygen surface abundance higher than 0.05 after core helium depletion

To determine the stellar models for which our wind prescriptions are accurate,
we employ the analytical method derived in Paper I to find the metallicity-
dependent transition luminosity between WR stars, and their low mass
counterparts, which have optically thin winds and lower mass loss rates.
Namely, we set these luminosities as

$\mathrm{L}^{\mathrm{tau}}_{\mathrm{min,WN}}=6.85\times
10^{4}\left(\frac{Z}{0.02}\right)^{-0.71}\,{\rm L}_{\odot}.$ (10)

For WC-type stars, the fit to the observations takes the form

$\mathrm{L}^{\mathrm{tau}}_{\mathrm{min,WC}}=7.51\times
10^{3}\left(\frac{Z}{0.02}\right)^{-1.55}\,{\rm L}_{\odot}.$ (11)

These luminosities are converted to stellar mass estimates using the mass-
luminosity relation of Langer (1989). The explosion properties of stripped-
envelope stars with luminosities below the minimum WR luminosities are also
shown, but the implications of lower mass loss rates on transparent-wind
stripped-envelope stars are considered a posteriori.

We have subdivided this Section into the following parts: in Sect. 3.1 we
present the distribution of final masses and surface chemical compositions in
our models. In Sect. 3.2 we characterize the internal structure of our models
at core collapse and apply several tests to assess which models are more
likely to produce SNe, and which ones are more likely to form BHs as a result
of core collapse.

###  3.1 Pre-collapse masses and chemical compositions of Type I supernova
progenitors

The pre-collapse masses of our helium-star models, presented in Fig. 1, are
determined by an interplay between their lifetime, and the intensity of their
winds. A change in the trend in their final mass occurs between around 7 and
11 $\,\mathrm{M}_{\odot}$, indicated by dots in Fig. 1, and it is also
observed in the models of Yoon (2017) and Woosley (2019). As discussed in
Paper I, it corresponds to the metallicity-dependent transition mass between
stars that lose mass as WNs for most of their lives, and those that spend a
significant time as WC and WO stars with stronger mass-loss rates. The ejecta
mass of SNe can be estimated, to first order, from Fig. 1, but a thorough
analysis of this is presented in Sects. 3.2 and 4.

Figure 1: Final mass (at core-collapse) as a function of initial mass for
helium star models of different metallicities. The dots on each line indicate,
for each metallicity, the initial helium star mass at which the transition
between WN- and WC-type mass loss is observed to occur in the models. The
rhombi indicate the value of the minimum luminosity WN-type models, obtained
Eq. 10, and using the mass-luminosity relation of Langer (1989). Helium stars
below this limit do not have optically thick WR winds, and have overestimated
mass loss rates.

The type of SN that helium stars may produce not only depends on the final or
ejecta mass, but also on the mass and composition of their envelope (Dessart
et al. 2020). According to Hachinger et al. (2012), approximately only 0.01
$\,\mathrm{M}_{\odot}$ of helium in the envelope of a hydrogen-poor star at
core collapse is enough to produce detectable helium lines in the SN spectrum.
This would render all of our models Type Ib SNe progenitor candidates.
However, Dessart et al. (2011, 2012) determined that the spectral type is not
only sensitive to helium mass, but also to the envelope composition at core-
collapse and the mixing that takes place during the SN explosion. Furthermore,
the model spectra of Dessart et al. (2020), produced by progenitors similar to
ours, show that Type Ic SNe spectra can be produced by helium stars if their
mass loss is strong enough to remove the helium-nitrogen envelope. Thus, there
is a sharp transition between the evolution of progenitors of Type Ib and Ic
SNe, which roughly corresponds to the transition between WN- and WC-type
stars. A peculiar large-scale mixing of nickel and helium might make some WN-
type progenitors appear as Type Ic SNe. While helium-deficient WR stars can
only produce Type Ic SNe, the complexity of 3D mixing and of the excitation of
HeI lines by non-thermal processes does not exclude the possibility that some
helium-rich WR progenitors appear as helium-deficient, Type Ic SNe.

Figure 2: Final mass of helium in the envelope, as calculated by Eq. 12, as a
function of initial mass for helium stars of different metallicities. The dots
on each line indicate, for each metallicity, the initial helium star mass at
which the transition between WN- and WC-type mass loss, is observed to occur
in the models. The rhombi indicate the value of the minimum initial mass above
which helium stars are observable as WN-type stars, obtained Eq. 10. Helium
stars below this limit do not have optically thick WR winds, and have
overestimated mass loss rates.

Figure 3: Surface mass fractions of 4He, 12C, 14N and 16O of our helium star
models at core-collapse, as a function of initial mass, for different
metallicities. The dots on each line indicate, for each metallicity, the
initial helium star mass at which the transition between WN- and WC-type mass
loss, is observed to occur in the models.

Fig. 2 shows the remaining mass of helium in the envelope as a function of
initial mass, at the time of core collapse. We calculate this quantity as

$\mathrm{M}_{\mathrm{He}}=\int_{\mathrm{T}<10^{8}\
\text{K}}\mathrm{Y}(m)\,\mathrm{d}m,$ (12)

where, by only considering the regions with temperature smaller than $10^{8}$
K, we exclude the helium formed in the core during collapse, thereby only
accounting for the envelope content. Note that this is different from the
envelope mass (i.e. the total mass above the carbon-oxygen core), since only
the mass in the form of helium is accounted for.

As shown in Fig. 2, our helium star models sharply transition from having
abundant helium in their envelope, to having only very little. Most models
that lose mass as WN stars throughout their evolution have between 0.8 and 1.4
$\,\mathrm{M}_{\odot}$ of helium in their envelopes. On the other hand, the
amount of helium in the envelope converges at a value of around 0.2
$\,\mathrm{M}_{\odot}$ for models with initial masses large enough to produce
WC-type stars (denoted by dots in the Figure), with very few models reaching
core collapse between these two regimes. The position where this transition
occurs depends on metallicity, but models with intermediate helium masses
occur at all metallicities, and may lead to SNe with “intermediate” spectra
between a Type Ib and a Type Ic (see model he9 from Dessart et al. 2020). We
highlight, however, that the exact spectral properties of a Type I SN depend
on nickel mixing, and some WN-type stars may also produce Type Ic SNe.

Contrary to the convective helium-burning cores of hydrogen-rich stars, the
convective cores of helium stars shrink in size as they evolve. As their
convective cores decrease in mass, a smooth composition gradient develops in
the formerly convective region, followed by a sharp transition between the
helium and nitrogen envelope and the layers enriched with carbon and oxygen.
The layers directly above the helium-free core later become part of a
convective helium burning shell, which can grow in mass due to the smooth
composition gradient left behind by the gradual retreat of the convective
helium core. This means that most of the helium is stored in the nitrogen-rich
envelope, and stars that lose these layers have a similarly low amount of
helium at core collapse.

Total helium mass is, however, not the only factor that determines whether a
star explodes as a Type Ib or a Type Ic SN. Fig. 3 shows the behavior of the
surface abundances of He, C, N and O in our core-collapse models as a function
of initial mass. As can be seen, there is a clear dichotomy in surface
chemical composition. In particular, we find a steep change in N abundance,
but also a gradual reduction of He and an increase of C and O as a function of
initial mass, transitioning from CNO equilibrium abundances to the abundances
that result from helium burning. Our models have a sharp divide between these
two populations of helium stars, with no intermediate cases. Surface nitrogen
abundance has the sharpest drop since it burns at a lower temperature than
helium, leading to a sharp transition in its abundance between the initially
convective region and the envelope. The absence of nitrogen-rich layers
corresponds to the transition between Type Ib and the intermediate Type Ibc
and Type Ic spectra in Dessart et al. (2020). We therefore use that as the
dividing line between the two cases in the following sections.

As initial mass increases, there is a trend for the carbon abundance to
initially increase, and then decrease again. A similar trend is observed for
surface oxygen, except the trend has a local maximum and minimum before
continuing to increase for the highest final masses. Above
$\sim$20$\,\mathrm{M}_{\odot}$, final surface abundances seem to depend mostly
on initial mass. This implies that this quantity is determined early in the
evolution, and leads to small variations in the surface abundances for models
of similar final mass, independent of metallicity.

###  3.2 Properties of supernova explosions from stripped-envelope stars

Since the final masses and surface compositions of models of stripped-envelope
stars with similar initial masses change with metallicity, the relative rate
of different SN types coming from them is also metallicity-dependent. This is
a consequence of how the SN type depends on surface chemical composition
(Dessart et al. 2020). However, SN rates also depend on the number of stars
that actually become SNe, instead of collapsing into a black hole directly
(Patton & Sukhbold 2020). The change in surface chemical composition with mass
and metallicity has been addressed in Sect. 3.1. The latter issue is discussed
here. First, we discuss the carbon-oxygen core masses and central carbon
content at the end of helium burning in our models. Then, to gauge the
explodability of our core collapse models, we employ the four methods
described in Sect. 2. The results from these analyses are discussed here, and
a different visualisation of the outcome of these tests is presented in the
Appendix.

Carbon core mass and the mass fraction of carbon at the end of helium burning
are quantities which will influence the subsequent carbon and neon burning
stages, and final fates of helium stars of different helium masses and
metallicities (Chieffi & Limongi 2020; Patton & Sukhbold 2020; Sukhbold &
Adams 2020). Lower core carbon abundances at the end of helium burning result
in weaker carbon burning, which does not require convection to transport the
energy generated away from the burning region. This will impact the structure
of the core, making it more compact, resulting in a higher gravitational
binding energy, and therefore making SN explosions less likely to take place
as metallicity increases (Sukhbold & Woosley 2014). We find that our helium
star models of the same initial mass will progressively have less massive
carbon-oxygen cores as metallicity increases, as shown in the top panel of
Fig. 4. More massive carbon-oxygen core masses will in turn result in a lower
central carbon mass fraction at the end of helium burning, as shown in the
bottom panel.

The threshold mass above which models spend some time as WC-type stars before
core collapse corresponds to the change in the trend of final mass as a
function of initial helium mass (see Fig. 1), and this division is accompanied
by different trends in carbon-oxygen core masses and central carbon mass
fractions at the end of helium burning as a function of initial mass. This is
broadly consistent with the results of Schneider et al. (2021), but
demonstrates that the dependence of WR winds on metallicity and surface
composition (or WR type) have a strong impact on the core evolution of helium
stars.

Figure 4: Top: Carbon-oxygen core mass at the end of helium burning, as a
function of initial helium star mass. Bottom: Central carbon abundance at the
end of helium burning, as a function of carbon-oxygen core mass. Thin lines
represent models where core carbon burning occurs convectively, whereas thick
lines represent models where carbon burning occurs radiatively. The dots on
each line indicate, for each metallicity, the initial helium star mass at
which the transition between WN- and WC-type mass loss, is observed to occur
in the models. The rhombi indicate the value of the minimum initial mass above
which helium stars are observable as WN-type stars, obtained Eq. 10. Helium
stars below this limit do not have optically thick WR winds, and have
overestimated mass loss rates.

Figure 5 shows the compactness parameter at core collapse of our models, as a
function of both initial helium star mass and final mass. Because helium stars
experience shrinking of their convective core during helium burning, a helium
star of a given initial mass will often have a lower compactness than a
hydrogen rich, non-stripped star whose ZAMS mass correspond to the same helium
core mass at the beginning of core helium burning, and the first sharp
increase in the value of $\xi_{2.5}$ will occur at a corresponding lower ZAMS
mass than for stars with a hydrogen envelope (Schneider et al. 2021; Laplace
et al. 2021).

We observe a general trend for features in the behaviour of compactness as a
function of initial helium star mass, such as peaks and valleys, to be
displaced to higher initial helium star masses with increasing metallicity,
correlated with the drop in final mass for similar initial masses, as shown in
the bottom panel of Fig. 5.

Figure 5: Compactness parameter as a function of initial mass (top) and final
mass (bottom) for helium star models of different metallicities. Circles
represent models with final surface abundances that correspond to Type Ib SN
progenitors, whereas crosses represent models with surface abundances that
correspond to Type Ic SN progenitors. The dashed horizontal line at
$\xi_{2.5}$ = 0.35 divides the majority of exploding models, from those that
produce BHs.

Although the compactness parameter alone is not enough to determine whether a
stellar model at core collapse will lead to a successful neutrino-driven SN
explosion or not, we find that most cases with $\xi_{2.5}<0.35$ are predicted
to explode according to the Ertl et al. (2016) test, as well as with the
Müller et al. (2016) test (see Fig. 15). At every metallicity, we find that
most helium star models with initial mass below $\sim$ 35 - 40
$\,\mathrm{M}_{\odot}$ will produce successful explosions, with a few
exceptions located mostly at the peak in $\xi_{2.5}$, found at a final mass of
around 8 $\,\mathrm{M}_{\odot}$. The similarity in $\xi_{2.5}$ for models with
similar final masses is due to the fact that the final mass is roughly
determined before the beginning of carbon burning. The remaining lifetime is
too short for stars to lose a significant amount of mass, and since core
structure is mostly determined by the mass of the carbon oxygen core, features
such as the transition between models with convective and radiative core
carbon burning occur at similar final masses, correlated to the first peak in
$\xi_{2.5}$ (Sukhbold & Woosley 2014). However, variations occur between
models with similar masses at different metallicities, likely related to
differences in their core composition and mass, and although subtle, they have
an effect in the outcome of SN and compact object populations that result from
our models (see Sect. 4).

An interesting feature in the behaviour of $\xi_{2.5}$ appears at final masses
above 15 $\,\mathrm{M}_{\odot}$, where we find a region in which two solutions
to the compactness parameter appear. The two branches of values of the
compactness parameter models in this region correspond to values of
$\xi_{2.5}$ between 0.4 - 0.6 and 0.6 - 0.8. All values are above the limit of
$\xi_{2.5}\sim 0.35$ where we expect to find exploding models, and all models
are predicted to form BHs without SNe, according to both the Ertl et al.
(2016) and the Müller et al. (2016) tests with our choice of parameters.
However, variations in the values of parameters in the Müller et al. (2016)
model can lead to SN explosions in this regime, and some of these progenitor
models are expected to produce fallback SNe and BHs in the lower mass gap,
when analysed with the Mandel & Müller (2020) model (see Fig. 16). Because all
of the explosions predicted in this regime are powered by fallback, we refer
to this region as the “island of fallback explodability” (see Antoniadis et
al. 2022, for a discussion). The exact number of fallback SNe, as well as
their properties, depend sensitively on the model parameters. However, every
set of parameters we explored produces at least some fallback SNe in this
regime (see Appendix B). Explosions could also be facilitated in this regime
by the presence of strong rotation (e.g. Aguilera-Dena et al. 2020), which can
produce successful SN explosions even when neutrino emission is not enough to
revive the SN shock (e.g. Summa et al. 2018). However, rotation is not
expected to be high in the cores of stripped-envelope stars formed by case A
and case B mass transfer due to the loss of angular momentum during this phase
(Yoon et al. 2010).

Figure 6: Summary of the key parameters obtained from the core collapse models
of helium stars through the explosion model of Müller et al. (2016), as a
function of initial mass (left) and final (right) mass. We include the NS
gravitational mass obtained for core collapse models that successfully
explode, the BH mass for models that directly collapse (which corresponds to
their final mass), the explosion energy and the nickel mass. Black dashed
lines indicate the division between Type Ib and Type Ic progenitors, and the
hashed region corresponds to areas that our model grid does not cover. The
horizontal, dashed line corresponds to the minimum mass above which models
reach core collapse as a WC- or WO-type star, and models to the right of this
line are therefore expected to be Type Ic SN progenitors. Figure 7: Same as
Fig. 6, but using the explosion model of Mandel & Müller (2020), which
includes the effect of fallback.

For the fiducial set of parameters we employed (see Sect 2.2), the
explodability, and the estimates for explosion energies, NS gravitational
masses, BH masses and nickel masses obtained by analysing our models with the
Müller et al. (2016) and Mandel & Müller (2020) explosion models are
summarized in Figs. 6 and 7, respectively. The black line in the Figures
indicates the value of the minimum mass above which models evolve as WC-type
stars, which we take as a threshold below which successfully exploding stars
are expected to be observable as Type Ib SNe, and above as Type Ic. We find
that most of these quantities are comparable to those inferred from stripped-
envelope SN observations (e.g. Taddia et al. 2018; Barbarino et al. 2021), but
discuss this in more detail in Sect. 4.

As Ertl et al. (2020) had already found with their models with enhanced mass
loss, the initial (helium) masses of stars located in the so called “islands
of explodability” become displaced if wind mass loss rates are different.
However, as Figs. 6 and 7 suggest, final mass is not the only defining factor
that determines the explodability of a model and the properties it will have
upon explosion. A region at around final masses of $\sim$8
$\,\mathrm{M}_{\odot}$ where low mass BHs can form exists for most
metallicities, but the location and extent of this region varies significantly
as a function of metallicity. There is a tendency for the highest final mass
where explosions take place to increase with decreasing metallicity (see Fig.
6 and Sect. 4). This implies that explosions with high ejecta masses are more
likely to occur in low metallicity environments. Such explosions also tend to
be more energetic. However, the exact morphology and location of these regions
depends on the choice of parameters in the Müller et al. (2016) model. Since
the maximum ejecta mass of Type Ic SNe decreases in high metallicity
environments, and the minimum mass of BHs that come from these very massive
stars decreases twofold, both because of the increase in mass loss but also
due to this effect.

As shown in Fig. 7, the number and the mass of models that experience fallback
also varies as a function of metallicity, but it is a relatively small number.
Many fallback SNe are predicted to be more energetic explosions (with energies
ranging from $6\times 10^{51}$) erg, and up to a few times $10^{52}$ erg),
with higher nickel masses than the rest of the sample (often in excess of 0.2
$\,\mathrm{M}_{\odot}$). Including the effect of fallback in our model does
not change the predictions of the properties of the explosions we calculated
without including fallback, but produces models that successfully explode in
places where explosions were not originally expected. However, this depends on
the input parameters of the model, and we may underestimate transients that
originate from progenitors of lower mass, which are likely more numerous in
the Universe (see Appendix B).

NS gravitational masses predicted for our models range between 1.32 and 2.03
$\,\mathrm{M}_{\odot}$. It is noteworthy that we observe a tight correlation
between compactness and NS mass (see Fig. 17), and that the smallest NS mass
is larger than the least massive known NS (Antoniadis et al. 2016). We
attribute this to the lack of core-collapse models below 4
$\,\mathrm{M}_{\odot}$ in our grid. The maximum NS mass we find is close to
the maximum known NS mass (Cromartie et al. 2020), and it is also close to the
limit of 2.05 $\,\mathrm{M}_{\odot}$ that is imposed in the Müller et al.
(2016) model. Exploring the effect of variations on this maximum mass in the
resulting SN parameters might help understand the origin of the distribution
of the NS mass distribution, but is beyond the scope of this work.

The real distribution of compact object masses cannot be directly drawn from
our single stellar models. One reason for this, in particular for BH masses,
is that the range of final masses in our models, which will roughly correspond
to the range in BH masses, is metallicity dependent. Another reason is that we
do not have models in the range of (pulsational) pair instability, which will
in turn determine the maximum mass of BHs. Furthermore, binarity might have a
role in determining the distribution of BH masses in the Universe beyond that
of creating stripped-envelope stars. However, this distribution has been
computed by other authors in different works (Farmer et al. 2020; Woosley &
Heger 2021), and it is particularly relevant at metallicities lower than that
of our models. It is believed that low-metallicity environments are more
efficient at producing the binary BHs we observe as mergers today (e.g.
Neijssel et al. 2019).

Regardless, we can draw some conclusions from the distributions in Figs. 6 and
7. For instance, a lower mass BH population comes for models with final masses
of around 6–9 $\,\mathrm{M}_{\odot}$. This distribution will dominate the BH
mass distribution, as it is favored by the IMF, and the number and mass of
said BHs depends on metallicity. The BHs in this regime produced at low
metallicity will peak at the final masses of low metallicity models since they
correspond to stars of lower initial mass. Similarly, the most massive BHs
formed by helium stars will be formed at low metallicity, regardless of
whether the maximum BH mass depends or not in metallicity. In the case where
the effect of fallback is considered, we also expect a population of mass gap
BHs to appear.

The explosion energies predicted for our models range from about 0.5 $\times
10^{51}$ erg, up to about 5.6 $\times 10^{51}$ erg; and the kinetic energy is
correlated with the final mass (see Fig. 17), likely due to its dependence on
the binding energy of the progenitor at core collapse. We find that the most
energetic explosions correspond to those models that have higher values of
$\xi_{2.5}$, which also correlates with high values of $\mu_{4}\mathrm{M}_{4}$
(in the regime where models explode), implying that high energy explosions
come from models that are on the verge between a successful explosion, and
collapsing into a BH. Progenitors that are predicted to form a low mass BH,
but explode nonetheless due to fallback accretion according to the Mandel &
Müller (2020) model have extremely high energies that reach up to $\sim
1.2\times 10^{52}$ erg. Such explosions, however, come from very high mass
stars, and thus might form a rare sub-class of Type Ic SN.

The nickel masses that we predict are predominantly in the range of 0.025–0.15
$\,\mathrm{M}_{\odot}$; somewhat lower than what is inferred for most stripped
envelope SNe (e.g. Taddia et al. 2018; Barbarino et al. 2021, but see Sect.
5.3.3 for a discussion about potential biases). These nickel yields have a
large spread as a function of initial and final helium star mass, but they are
correlated very tightly with $\mu_{4}$, which may reflect the size of the
region where nickel is synthesized during the explosion. Fallback SNe found in
our models are predicted to have higher nickel masses, in the range of 0.2–0.3
$\,\mathrm{M}_{\odot}$, with the exception of a model that produces $\sim$0.45
$\,\mathrm{M}_{\odot}$.

In Fig. 8 we present the ratio of explosion energy to ejecta mass,
$\mathrm{E}/\mathrm{M}$, that we find using the Mandel & Müller (2020) model.
We find that most models have a value of this quantity that ranges between 3
and $8\times 10^{50}\ \mathrm{erg}/\,\mathrm{M}_{\odot}$. Models that are
predicted to produce fallback SNe are found to have only large values of this
quantity, while ordinary SNe cover the whole range. This is similar to the
value of $5\times 10^{50}\ \mathrm{erg}/\,\mathrm{M}_{\odot}$ used in Dessart
et al. (2020), that was found to reproduce the light curves of Type I SNe, but
we predict a significant variation of this parameter, allowing for SNe to have
varying light curve properties depending on their progenitor.

The narrow range of values $\mathrm{E}/\mathrm{M}$, along with the narrow
range in nickel masses predicted for the majority of our SN models, regardless
of whether they are Type Ib or Ic, is consistent with the fact that the light
curves of Type Ib and Ic SNe are often indistinguishable from each other
(Drout et al. 2011).

Figure 8: Ratio of explosion energy to ejecta mass inferred using the Mandel &
Müller (2020) model,as a function of final mass. The horizontal line at
$5\times 10^{50}\ \mathrm{erg}/\,\mathrm{M}_{\odot}$ shows the value of
$\mathrm{E}/\mathrm{M}$ used in Dessart et al. (2020) to produce light curves
of Type I SNe.

We highlight, however, that the uncertainties in the predictions of
explodability, explosion energy, nickel mass and remnant mass are more
uncertain in BH forming SNe than in NS forming explosions. However, we find
that the trends and correlations we present are robust, but vary depending on
the choice of parameters (see Appendix B). Ejecta masses predicted from our
models are found to be larger at low metallicity for a fixed initial mass.
This fact, combined with the behaviour of the transition mass between
nitrogen- and helium-rich progenitors, and nitrogen-poor ones, implies that
the maximum ejecta mass at which Type Ib SNe are observed will be larger at
low metallicities. The maximum ejecta mass at which Type Ic SNe can be
observed in different metallicity environments is also expected to decrease at
high metallicity, due to the fact that the maximum final mass at which
explosions take place decreases with increasing metallicity.

All of these results correspond purely to the outcome of our models, but do
not correspond to number of observable events, or the real distribution of SN
observables. An estimate of how these distributions will change is estimated
in the next Section through a simplified population model.

##  4 Type I supernova and compact object populations

In this Section, we discuss the properties of Type I SN and compact remnants
that result from the simple population synthesis model described in Sect. 2.3.
In Sect. 4.1 we present the resulting compact object populations, and in Sect.
4.2 we discuss the resulting distribution of SN observables.

###  4.1 Compact object populations

Figure 9: Top: Distribution of NS gravitational masses that result from
analyzing our core-collapse models with the Mandel & Müller (2020) model,
weighted by the IMF from Salpeter (1955), compared to the observed
distribution inferred by (Antoniadis et al. 2016). The average value of the
distributions are represented by vertical lines at the top of the Figure. The
lowest mass neutron stars correspond to progenitors outside of our model grid,
and their resulting mass has been assumed to be between 1.22 and 1.3
$\,\mathrm{M}_{\odot}$. Bottom: Mean value (solid lines) and standard
deviation (hatched, shaded regions) of the total distributions, and the
distributions separated by SN type. The mean value (black dashed line) and
standard deviation (grey dashed lines) of the first NS mass peak of the
observed distribution of (Antoniadis et al. 2016) is shown for comparison.

The resulting distributions of NS masses, presented in Fig. 9, depend only
weakly on metallicity. We find that the average NS mass is around 1.42
$\,\mathrm{M}_{\odot}$, but the distributions are skewed towards slightly
lower NS masses as metallicity increases. The distributions resemble the low
mass peak in the distribution of millisecond pulsars inferred by Antoniadis et
al. (2016), but their contribution to the high mass peak is negligible. These
distributions are not affected by our implementation of fallback, but we do
not rule out that some massive NS may be formed by late fallback in models
where we do not predict it with our formulation. We also find that the lowest
mass NSs in our populations are formed in Type Ib SNe, whereas the most
massive components originate in Type Ic SNe, as shown in the lower panel of
Fig. 9 (see Fig. 17). This is due to the correlation between $\xi_{2.5}$ and
resulting NS mass. Type Ib SNe are produced in models whose final masses are
predicted to be below the first compactness peak, and most of them have low
values of $\xi_{2.5}$ due to their smaller final mass. Progenitors of Type Ic
SNe, on the other hand, have a larger scatter of values of $\xi_{2.5}$ since
many of them come from models in the region where core carbon burning
transitions from radiative to convective, and have a larger scatter in
$\xi_{2.5}$ (Sukhbold & Woosley 2014), and consequently in NS mass.
Alternatively, the calculations presented in Sect. B.1 have a few cases where
high mass NSs are produced in low energy explosions, that likely do not
receive sizable kicks. In this scenario, although rare, these systems may be
favourably created in binary systems that statistically are much more likely
to stay bound after explosion, therefore biasing the sample of NSs in binary
systems.

The distribution of BH masses that results from our calculations, shown in
Fig. 10, has a stronger metallicity dependence. The peak that corresponds to
the high mass BHs, with masses larger than $\sim 10\,\mathrm{M}_{\odot}$,
comes from the models in our grid with the highest initial masses. This peak
tends to be at a higher mass for lower metallicities, owing both to the
decrease in mass loss, and to the decrease in final mass of the location of
the threshold between models that form BHs and models that form SNe.

Figure 10: Distribution of BH masses at different metallicities, that result
from analyzing our model with the Mandel & Müller (2020) model, weighted by
the IMF from from Salpeter (1955).

The more abundant intermediate mass peak, located at
5–8$\,\mathrm{M}_{\odot}$, corresponds to models that form BHs in the first
high compactness peak. Given our assumptions on the population, we expect that
BHs of this mass should be much more frequent than the high mass models.

There is a gap between these two regions, also present in the results of other
groups (e.g., Ertl et al. 2020; Woosley et al. 2020), but the precise width
and location of this gap is found to be metallicity dependant. Extrapolating
from our results might suggest predicting the location and shape of this gap
is a difficult (but necessary) effort. The location of this gap at some
metallicity may be masked by BHs coming from different environments, so
establishing a metallicity behaviour for this gap, and convolving it with our
knowledge of the metallicity evolution of galaxies of different redshifts, is
key to properly interpret the observations of BH masses in the Universe, which
will become more numerous in the future, with the advent of ongoing and future
gravitational-wave observatories.

Our models show also an apparent dearth of BHs above a maximum mass, but this
limit is artificial and corresponds only to the maximum mass of the models in
our grid.

Fallback supernovae produce a low mass peak in the distribution, predicted to
be at around 3 $\,\mathrm{M}_{\odot}$, and generally less populated than the
other two. However, this is sensitive to the number of fallback SNe produced,
which may vary in reality. If fallback SNe tend to appear in a certain range
of masses, they can create patterns in the distribution of high mass BHs,
which manifests itself as valleys where fewer BHs of a certain mass are found,
since they instead inhabit the lowest mass peak. If this is indeed the case,
then the need arises to study this effect in more detail, to be able to better
interpret the distributions of BH masses that will become available from
future observations of binary BH mergers. The formation of such systems may
help explain the asymmetric compact object mergers that have been recently
detected by LIGO (Antoniadis et al. 2022).

A final trend that is noteworthy in the BH mass distribution is a change in
the slope of the distribution of BHs above a certain mass, also found by
Woosley et al. (2020), caused by the change in the implemented mass loss rate
from WN-type mass loss to WC-type mass loss; which corresponds to the change
in trend in final mass shown in Fig. 1. If this is resolved with the
increasing number of gravitational wave sources observed, it could potentially
aid in constraining the mass loss rates of very massive stars.

###  4.2 Distribution of supernova properties

Figure 11: Cumulative distribution functions of ejecta masses, explosion
energies, nickel masses and kick velocities, separated for the full sample of
models and separated between Type Ib and Ic SNe that result from analyzing our
core-collapse models with the Mandel & Müller (2020) model, weighted by the
IMF from Salpeter (1955). Different color lines represent different
metallicities. Observed distributions of ejecta masses, explosion energies and
nickel masses are drawn from (Taddia et al. 2018) for Type Ib, IIb (grouped
with Type Ib) and Ic SNe, and additionaly from Barbarino et al. (2021) for a
larger sample of Type Ic SNe. The average values of each distribution are
indicated with vertical lines at the top of each panel.

We present the obtained distributions for ejecta masses, explosion energies,
kick velocities and nickel masses for Type Ib and Ic SNe, as cumulative
distribution functions in Fig. 11. Observed distributions obtained for Type Ib
SNe are shown in comparison with the observed sample of Type Ib and Type IIb
SNe from Taddia et al. (2018), and Type Ic SNe are compared to the previous
sample, combined with that from Barbarino et al. (2021), who performed a
similar study, focused on Type Ic SNe. We compare our results to observations
only after separating them by SN Type in hopes that the results of Taddia et
al. (2018) and Barbarino et al. (2021) are representative for the
distributions of explosion properties of each SN Type, but refrain from
comparing them to the combined sample since the observed SN properties do not
reflect the relative occurrence of different SN Types.

Many trends become apparent in our estimation of the distribution of SN
observables. The predicted distribution of ejecta masses of Type Ic SNe
resembles the distributions we obtain at high metallicity, but likely the
metallicity range that corresponds well with the observed distribution is much
too high to explain most Type Ic SNe. We also find that, regardless of the
metallicity, no Type Ic SNe are produced within our models with ejecta masses
lower than about 2 $\,\mathrm{M}_{\odot}$.

The predicted distribution of ejecta masses of Type Ib SNe is skewed to
considerably lower values than the observations. These results, however, have
been taken at face value from the outcome of our stellar evolution tracks, and
many of them overestimate the mass loss rates of the progenitors of Type Ib
SNe, since they have luminosities below
$\mathrm{L}_{\mathrm{min,WN}}^{\mathrm{tau}}$. To remedy this, we repeat our
calculation using the initial mass as a better proxy of the final mass, but
retaining the same distribution of NS masses. This results in the distribution
shown in Fig. 12, yielding a better fit to the data.

Figure 12: Cumulative distribution of ejecta masses for Type Ib SNe (top) and
for all stripped-envelope SNe (bottom) that result from our models, but
corrected for the strong variation in wind between WR stars and low-mass
helium stars. Here, ejecta masses for stars with initial mass above the
minimum mass of WN-type stars are drawn from our model results, and for the
rest we assume no mass loss, but the same NS mass as predicted from its
original model.

As expected, we find that the ejecta masses of stripped-envelope SNe, both in
Type Ib and Type Ic, decrease as metallicity increases. As shown by the mean
values and standard deviations of the distributions in Fig. 13, ejecta masses
for both type of SNe decrease steeply as a function of metallicity, but even
applying a correction for the progenitors of Type Ib SNe with very low mass,
the range at which these events are found cannot be reproduced from our
models.

Figure 13: Mean values and standard deviation of ejecta masses for both
explosion energies, nickel masses and kick velocities, from the full sample as
well as for Type Ib and Type Ic SNe individually, at different metallicities,
that result from analyzing our core-collapse models with the Mandel & Müller
(2020) model, weighted by the IMF from Salpeter (1955).

We find the average explosion energies estimated for our models ranges between
about 1.3 and 1.8 $\times 10^{51}$ erg. We find that this average decreases as
a function of metallicity. The average values of Type Ib SNe tend to be below
the observed average energy at every metallicity, whereas our Type Ic SN
models have larger average energies than the observed samples. The explosion
energies of Type Ib SNe vary relatively little as a function of metallicity,
whereas those of Type Ic SNe tend to decrease quite drastically with
increasing metallicity. The energies estimated for Type Ib SNe are
considerably lower, but we have not corrected these values for the different
wind mass loss rates for low-mass helium stars. The energies predicted for
Type Ic SNe at high metallicity are in good agreement with the observed
values, although the number of explosions below 1051 erg are underestimated.

We find that the amount of 56Ni synthesized in SNe decreases as metallicity
increases. The distributions of 56Ni mass peak at around 0.05
$\,\mathrm{M}_{\odot}$ and few cases are observed with nickel masses above 0.1
$\,\mathrm{M}_{\odot}$. For both SN types, we find that the amount of nickel
synthesized during the explosion is found to decrease with increasing
metallicity. The values we find are systematically lower than the observed
values, which also have a much broader distribution, and are found to have
closer to 0.15 $\,\mathrm{M}_{\odot}$ of nickel on average.

The distribution of kick velocities obtained from our models peaks at around
300–400 km s-1, and we find that kick velocities tend to decrease with
increasing metallicity. We also find that Type Ic SNe tend to produce SNe with
considerably larger kicks, on average, than those coming from Type Ib SNe.

Except for the emergence of a new, low-mass peak in the BH mass distribution,
fallback SNe are found to be too rare to have a percievable effect on the
distribution of SN observables in large samples. This is due to the fact that
these events are predominantly originating in very high mass stars, which are
therefore very rare.

Another immediate consequence of the metallicity dependence of stellar winds
is an increase of rates of SNe Type Ic with respect to SNe Type Ib for higher
metallicity. We estimate the number ratio of Type Ic SNe to Type Ib SNe as a
function of metallicity, and find it to be strongly metallicity dependent, as
shown in Fig. 14.

Figure 14: Number ratios of different quantities as a function of metallicity,
as obtained from our model population. Top: Number ratio of Type Ib SNe to
Type Ic SNe, $\langle\mathrm{N}_{\mathrm{Ic}}$ / N${}_{\mathrm{Ib}}\rangle$,
compared with the observed ratio of Type Ib SNe to Type Ic SNe
$\langle\mathrm{N}_{\mathrm{Ic}}/\mathrm{N}_{\mathrm{Ib}}\rangle_{\mathrm{obs}}$
and of Type Ib SNe plus Type IIb to Type Ic SNe
$\langle\mathrm{N}_{\mathrm{Ic}}/(\mathrm{N}_{\mathrm{Ib}}+\mathrm{N}_{\mathrm{IIb}})\rangle_{\mathrm{obs}}$
in the local Universe, inferred from the relative rates of Shivvers et al.
(2017). Bottom: Number ratio of BHs to SN explosions
$\langle\mathrm{N}_{\mathrm{BH}}$ / N${}_{\mathrm{SN}}\rangle$ and number
ratio of Type Ib SNe with WR progenitors to Type Ib SNe with transparent wind
helium star progenitors $\langle\mathrm{N}_{\mathrm{Ib\ from\ WR}}$ /
N${}_{\mathrm{Ib\ from\ TWHS}}\rangle$.

According to our models, the number ratio of Type Ic to Type Ib SNe should
increase as a function of metallicity, starting from approximately 0.2 at a
metallicity of 0.01, to about 0.7 at a metallicity of 0.04. This rising trend
is mainly due to the fact the initial helium mass threshold for the production
of a Type Ic supernova decreases with increasing metallicity. We compare our
calculation to the observed ratio of stripped envelope SNe using the observed
rates of different SN types in the local Universe from Shivvers et al. (2017).
We find that the value estimated from our models around solar metallicity is
between the observed ratio of Ic to Ib in the local Universe, which represents
an upper limit to our model; and the ratio of Type Ic SNe and the rest of
stripped-envelope SNe (Types Ib and IIb), which is a more realistic quantity
to compare our results to.

Two final quantities that we derive from our population model, and present in
the right y-axis of Fig. 14, are the relative ratio of BHs to SNe, and the
relative ratio of Type Ib SNe that have WR type progenitors to those that have
transparent wind helium stars as progenitors (i.e. stars that have
luminosities below $\mathrm{L}_{\mathrm{min,WN}}^{\mathrm{tau}}$.

We find that, as metallicity increases, the number of massive stars that
produce BHs tends to decrease. At a metallicity of 0.01, we find that around
16% of massive stars produce BHs, whereas the number decreases to about 8% in
the metallicity range 0.025 – 0.04. This is due to the reduction in final mass
as a function of initial helium star mass (see Fig. 1), combined with the
location of the islands of explodability (e.g. Fig. 5). On the other hand, as
metallicity increases, the number of Type Ib SNe that have WR stars as
progenitors tends to increase. This result is non trivial because, even though
the minimum luminosity at which we expect stripped-envelope stars to be WRs
decreases as a function of metallicity, so does the minimum luminosity at
which our models produce WC-type stars, meaning that even though there are
more WN-type stars at lower luminosities, the threshold at which they explode
as Type Ic SNe also decreases. In the metallicity range of our models, this
number tends to increase, but the behaviour is likely different in
metallicities beyond those in our grids. However, the absolute value of this
ratio at all metallicities in our grid is below 18%, implying that most Type
Ib SNe originate from low mass helium stars with transparent winds, and not WR
stars. The number is likely to be even lower at metallicities corresponding to
environments like the SMC and below, and this trend should be reflected in the
methods that are employed in the search of direct imaging of Type Ib SN
progenitors.

##  5 Discussion

The results presented in this work were derived from evolutionary calculations
of helium stars with metallicities in the range 0.01 – 0.04, and their
resulting core collapse models. The aim of studying helium stars in this
metallicity range is to explore a possible evolutionary channel that can help
explain the observed prevalence of Type Ic SNe in high metallicity
environments (Modjaz et al. 2011), and address the discrepancy between
predicted final masses of stripped-envelope SN progenitors (e.g. Woosley et
al. 1993; Meynet & Maeder 2005; Yoon et al. 2010; Georgy et al. 2012), and the
distribution of ejecta masses observed in SNe, particularly of Type Ic (e.g.
Taddia et al. 2018; Barbarino et al. 2021).

We divide our core collapse models into possible progenitors of either Type Ib
or Type Ic SNe according to whether their surface has a helium abundance of
$\mathrm{Y}=1-Z_{\mathrm{init}}$ and high nitrogen abundance, or whether it
has decreased helium abundance, and is instead enriched with the products of
helium burning, respectively. This dichotomy has been found to be at the heart
of the differences in spectral features of Type I core collapse SNe by Dessart
et al. (2020).

We then analyze our results according to the explodability tests of O’Connor &
Ott (2011), Ertl et al. (2016) and Müller et al. (2016), the latter including
the effect of fallback following Mandel & Müller (2020). The latter
explodability test consists of a parametrized, semi-analytical model of the
explosion, which yields predictions of NS gravitational masses, ejecta masses
and nickel masses in successfully exploding models. We also estimate the
distribution of kick velocities following Mandel & Müller (2020). Finally, we
construct a simple synthetic population of stripped envelope SNe, to contrast
our results with observations.

In this Section, we discuss and interpret the results obtained from analyzing
our core collapse models and the resulting population, as well as the
uncertainties that might have an impact in our results.

###  5.1 Uncertainties in core collapse stellar models

Some uncertainties in our models are inherent to our physical and numerical
assumptions, and others stem from poorly constrained physical parameters. Many
are discussed in detail in Paper I, but here we list a few, paying particular
attention to those that affect our models at core collapse.

####  5.1.1 Uncertainties in the mass at core collapse

Our evolutionary calculations are modeled using the empirical mass loss rates
suggested by Yoon (2017). However, stars with luminosities below
$\mathrm{L}_{\mathrm{min,WN}}^{\mathrm{tau}}$ are believed to lose mass at a
much lower rate. The precise mass loss rates of these stars are unknown, but
the simulations of Vink (2017) suggest that the mass loss rates of such stars
are orders of magnitude below those of WR stars.

Since we extrapolate WR mass loss rates into this regime, the final masses of
models with luminosities below $\mathrm{L}_{\mathrm{min,WN}}^{\mathrm{tau}}$
are overestimated. Furthermore, the core evolution of these models does not
correspond to that of helium stars of the appropriate final mass. This likely
has no effect in whether they explode or not, since models with luminosities
below $\mathrm{L}_{\mathrm{min,WN}}^{\mathrm{tau}}$ have low initial masses,
and are expected to have low values of $\xi_{2.5}$, leading to successful SN
explosions. The approach we take to overcome this limitation in our population
models is to assume that the helium stars that produce SNe in this regime have
final masses that can be approximated by their initial masses, since they have
very small mass loss rates. We further justify this assumption a posteriori by
arguing that this makes our estimates for ejecta masses resemble the observed
distribution more closely. For these stars, we assume that the NS
gravitational mass is 1.22–1.3 $\,\mathrm{M}_{\odot}$, and we assume that the
minimum mass at which stars explode as SNe is 3 $\,\mathrm{M}_{\odot}$,
following Chanlaridis et al. (2022).

The mass loss rates of WR stars are also uncertain. Recent theoretical
estimates of WR winds (Sander et al. 2020; Sander & Vink 2020) suggest that WR
winds have a weaker dependence on metallicity, and that WC star mass loss
rates are lower than the empirically obtained values of Tramper et al. (2016).
This would imply that the final masses of WC stars in our models are
underestimated, as well as those of models with high Z.

The masses of our models at core collapse may also be overestimated due to the
implementation of MESA’s mlt++, which enhances the efficiency of energy
transport by convection, and results in more compact stars that show very
little or no envelope inflation (e.g. Sanyal et al. 2015). The mass loss rates
of WR stars should not be affected by this assumption, since they only depend
on the luminosity and metallicity of helium stars, which are well constrained
by our models. However, if they exceed the Eddington limit in their envelopes
by a large amount, it may lead to episodic mass loss (e.g. Owocki 2015).
Furthermore, if inflation causes helium stars to expand significantly beyond
their initial radius, it might trigger a second episode of mass transfer with
their companion, which will also reduce their final mass.

Another uncertainty in our models comes from ignoring the main sequence and
envelope stripping phases, and modelling directly fully stripped helium stars.
As discussed in detail in Paper I, envelope stripping is not always efficient
in removing all of the hydrogen from the surface of a star. Therefore, they
are likely to spend a fraction of their helium burning lifetime with a
hydrogen envelope of varying mass, which can result in higher final and ejecta
masses than those predicted in our models.

####  5.1.2 Uncertainties in the core structure

In addition to the uncertainties introduced in our modelling by ignoring the
main sequence evolution of the progenitors of stripped-envelope stars, and by
their mass loss rates; other issues affect the accuracy of numerical
simulations of massive star evolution, which are not unique to helium stars,
but affect stellar evolution models in general. Numerical resolution and the
choice of nuclear network are known to lead to different core structures at
the core collapse, which in turn may affect the explodability of stellar
models (Farmer et al. 2016). We use a nuclear network with only 21 isotopes,
which leads to uncertainties in the final core structure of our models. We
have made this choice for practical reasons, since we wanted to cover a large
mass and metallicity parameter space. We argue that, while the exact masses at
which models transition from convective to radiative nuclear burning is
uncertain due to our choice in nuclear network, our calculations cover a broad
enough range that this uncertainty will likely not affect the main conclusions
we draw from our results.

The efficiency of the 12C($\alpha$,$\gamma$)16O reaction in particular remains
uncertain (deBoer et al. 2017), and can have a large effect on the evolution
and outcome of stripped-envelope stars, as it may influence the amount of
carbon remaining at the end of helium burning, which in turn influences
whether carbon burning proceeds radiatively or convectively (Imbriani et al.
2001). This is known to have a large effect on the explodability of massive
stars (Sukhbold & Woosley 2014; Patton & Sukhbold 2020; Schneider et al.
2021), as well as on the upper limit for BH formation (Farmer et al. 2019,
2020). The effect of these uncertainties is beyond the scope of this work, but
they can have an effect on the exact predictions of our analysis. However, we
argue that because of the size of our model grids, the effect of metallicity
dependent mass loss on the evolution of helium stars explored in this work
will have a larger impact than that of the uncertainties inherent to stellar
evolution models.

The efficiency of convection and the presence of overshooting in helium stars
are also uncertain. Varying these parameters may lead to convective helium-
burning cores of different mass. This will result different lifetimes, and
lead to significant differences in evolution, particularly in the lower mass
regime. Overshooting in low mass helium stars will most significantly affect
the locations of the boundaries in initial mass that lead to different final
outcomes Chanlaridis et al. (2022). This uncertainty is perhaps the largest in
our population models, as it might imply that the minimum mass at which stars
become core collapse SNe is dependent on metallicity. We believe that
variations in this limit will likely have an effect on real populations of
stripped-envelope SNe, but the conclusions derived from our simplistic model
will remain broadly valid. However, we highlight the need for a more detailed
population synthesis model of stripped-envelope stars, that additionally
includes the efficiency of stripping as a function of both mass and
metallicity.

####  5.1.3 Determination of supernova properties

Using the Müller et al. (2016) SN explosion model, we estimate properties of
the resulting explosions of individual models. These properties are relatively
uncertain on their own, and accurate estimations of the explosion outcome can
only be more accurately determined through detailed 3D simulations. However,
this model can yield predictions in much shorter timescales, and predicts
trends in the behaviour of large sets of models, revealing the consequences of
the assumptions made to produce them. These explosion models, however, depend
on a series of free parameters (see Sect. 2, and Appendix B) and variations in
their value produce different results.

We have performed several tests to characterize the effect of variations of
these parameters. We find that, within a reasonable range of values, the
outcome of the models is broadly similar. The distributions produced by
simulating SNe explosions with a different set of parameters changes the
individual values of the predicted outcomes, and the range where explosions
are expected to occur, so we chose the set of parameters that more closely
resembles the predictions of the Ertl et al. (2016) explodability test.

###  5.2 Comparison to previous stellar evolution models

Numerical studies to investigate evolutionary channels for the progenitors of
stripped envelope SNe have been carried out extensively in the context of
single star evolution (e.g. Meynet & Maeder 2003; Georgy et al. 2009, 2012;
Chieffi & Limongi 2013), massive stars in binaries (e.g. Podsiadlowski et al.
1992; Yoon et al. 2017; Schneider et al. 2021; Laplace et al. 2021), and
through the evolution of helium stars (e.g. Dessart et al. 2020; Ertl et al.
2020).

Single star models where wind stripping is the only component to produce
hydrogen free stars at core collapse require very large initial masses
($\gtrsim 25\,\mathrm{M}_{\odot}$), which may be in conflict with the rate of
stripped-envelope SNe, and predict high final masses, which is in conflict
with the observed distribution of ejecta masses (e.g. Yoon 2015). Binary
stellar models are much less restrictive in the minimum mass that is required
to produce stripped-envelope stars, but often grids of binary models have very
few suitable progenitors for Type Ic SNe (e.g. Yoon et al. 2010, 2017).

Yoon (2017) suggested that modifying the mass loss rates of WR stars might
contribute to produce models in the right mass and compositon range, and these
models were found to reproduce the spectral properties of both Type Ib and
Type Ic SNe by Dessart et al. (2020). The models in this work have very
similar properties and evolution as those from Dessart et al. (2020), but
extend the applicability of their predictions by covering a larger mass and
metallicity range, and analysing their explodability properties.

We find that the core properties of our models at core collapse are consistent
with previous models of stripped envelope stars coming from binary
calculations, and –like other single helium star models (e.g Woosley 2019;
Ertl et al. 2020)– form a distinct family of transient progenitors with
properties that differ from those obtained for single stars: stripped-envelope
stars tend to have lower compactness, lower central carbon mass fraction at
the end of helium burning, and lower carbon-oxygen core masses than stars with
similar main sequence masses that did not become stripped (Woosley 2019;
Laplace et al. 2021; Schneider et al. 2021).

Studying several metallicities, however, allows us isolate the effect of mass
loss in core and surface properties of helium stars. We find that the effect
of distinct mass loss in the WN and WC phases can have a strong effect on the
precollapse properties of WR stars. Most notably, we find features that were
either absent or not reported in previous models: compactness parameters of
helium stars with final masses above $\sim$16-25 $\,\mathrm{M}_{\odot}$
bifurcate, creating two families of solutions that have typical compactness
parameters of $\sim$ 0.4–0.6 and $\sim$ 0.6–0.8. Explodability tests (O’Connor
& Ott 2011; Ertl et al. 2016; Müller et al. 2016) predict these models to form
BHs with approximately the final stellar mass, but using the model of Mandel &
Müller (2020), we find that many of the low compactness models are candidates
for forming fallback SNe (i.e. a fallback island of explodability). These
models may be similar to the successfully exploding models of M¿60
$\,\mathrm{M}_{\odot}$ of Ertl et al. (2016), but the connection between them
and the reason for the bifurcation are unclear.

###  5.3 Transients and compact objects

####  5.3.1 Explodability of stripped-envelope stars

We find that the distribution of $\xi_{2.5}$ is similar for helium stars of
different metallicities that have similar final masses. Differences between
the distributions of $\xi_{2.5}$ as a function of final mass, however, likely
arise due to differences in CO core masses and central carbon abundances at
the end of helium burning, and result in variations in the core structure.
These discrepancies are reflected in the different values of $M_{4}$ and
$\mu_{4}$ that we find for stars of similar final mass, as well as in the
results obtained from applying the Müller et al. (2016) and Mandel & Müller
(2020) model to our stellar models at core collapse (see Fig. 17).

Difference in core structure in models of equal final mass imply that the
ejecta mass and remnant mass distribution at different metallicities will be
affected by more than just a displacement in the final mass distribution of
models (see Figs. 6 and 7. The imprint of these differences is summarized in
Figs. 11 and 10, and are further discussed below.

####  5.3.2 Ejecta masses of stripped-envelope supernovae

Many of our predictions are agnostic to the assumptions and uncertainties
inherent in the SN model. In particular, the masses of models that explode or
implode are in agreement with the predictions obtained from the Ertl et al.
(2016) test. This suggests that the distributions of BH masses and of SN
ejecta masses are roughly independent of the way that they were obtained, and
only reflect the metallicity dependence of WR mass loss.

We find that the distribution of ejecta masses for Type Ic SNe is reproduced
by our models, preferentially at very high metallicity (0.03 – 0.04). This
suggests several interesting possibilities for the progenitors of these SNe.
Modjaz et al. (2011) found that most Type Ic are indeed produced in
environmnets of supersolar metallicity, but it’s unlikely that the typical
metallicity of their progenitors is twice solar and above. Instead, this might
reflect that the mass loss rates inferred for either WN-type stars, WC-type
type stars, or both, are larger than what is reported in the literature, in
stark contrast with the predictions of Sander et al. (2020) and Sander & Vink
(2020).

Another possibility is that stripped-envelope stars experience short episodes
of mass loss at the end of their lives, likely originating from their
proximity to the Eddington limit. Another explanation is that, since many of
them are likely to be formed in binaries, they may experience episodes of mass
transfer after they have been stripped, and therefore reach the end of their
lives with lower masses. Because helium stars lose mass at such a high rate,
both their radii and their luminosities decrease during their core helium
burning lifetime. This implies that, even if they are in a close binary,
helium stars will not interact with their companions during core helium
burning, and can only interact in later stages if helium shell burning drives
a radial expansion.

We also find that the ejecta masses of our models, even at the highest
metallicities in our grid, fail to reproduce the lowest observed ejecta
masses. This phenomenon might be explained by the reasons stated above, or it
might be caused by a larger fraction of their progenitors experiencing late
time fallback during explosion, leading to the formation of either massive NSs
or low mass BHs, and low ejecta mass explosions (although fallback SNe are not
predicted in the relevant mass regime by Ertl et al. 2020, but see Vigna-Gómez
et al. 2021 and Appendix B).

The observed distribution of Type Ib SNe is closer to the results from our
models at low metallicity, but the predicted distributions of ejecta masses
for Type Ib SNe obtained by taking the output from the models at face value
are overestimated. Helium stars with initial masses below or near the minimum
mass at which stripped-envelope stars are observable as WN-type stars lose
less mass than their WN counterparts. Through our simplistic population model,
we find that most progenitors of Type Ib SNe will in fact not be WR stars, but
rather helium stars with transparent winds that lie below this limit. We
propose this as an explanation to the stark discrepancy between the ejecta
masses of Type Ib SNe predicted from our models, and the observed sample from
Taddia et al. (2018), who find Type Ib SNe with significantly larger ejecta
masses than those that result from our models, and find that correcting for
this produces a distribution of ejecta masses that is more resemblant to the
observed distribution. However, Type Ib SNe progenitors that come from stars
that are stripped of their hydrogen envelope after a significant fraction of
their helium burning lifetimes has taken place, may also contribute to
increasing the number of Type Ib progenitors with high ejecta mass, and to
increase the scatter in the distribution.

####  5.3.3 Explosion energies, nickel yields and kicks

The values predicted for SN explosion energies, nickel yields and kick
velocities from our helium star models rely on the Müller et al. (2016) model.
We have calibrated the input parameters of the model to agree with the
predictions from the explodability test of Ertl et al. (2016), and to produce
explosion energies and values of $\mathrm{E}/\mathrm{M}$ in a range that we
found reasonable.

For many of the models, we find explosion energies that are larger than those
predicted by the more detailed study of Ertl et al. (2020), even when only
neutrino-driven explosions are considered. However, recent 3D simulations of
SN explosions from massive progenitors (e.g. Powell & Müller 2020) have found
that producing SNe with energies in excess of about $3\times 10^{51}$ erg is
plausible. The highest energies in our models come from fallback SNe. The
models where we predict fallback SNe to take place are typically of higher
mass, so a larger energy is required to unbind the envelope of the progenitor.
As the fallback prescription of Mandel & Müller (2020) has been calibrated
based on 3D fallback simulations of relatively weak explosion, the
extrapolation to the very energetic fallback supernovae predicted in this
study is quantitatively uncertain, and the largest explosion energies, which
well exceed any 3D models of neutrino-driven models, may be overestimated. On
the other hand, late time fallback accretion provides an additional energy
source which allows explosions to occur, even after the central proto-NS has
reached its maximum mass. This is not included in our analytic treatment.

Furthermore, the nickel masses in fallback explosions are more uncertain than
the energetics. Whereas the transfer of energy from the engine to the ejecta
in a fallback explosion can be approximated analytically in a quasi-spherical
picture, the amount of Ni that survives fallback will depend sensitively on
mixing instabilities, for which we presently do not have an effective
treatment in our analytic supernova models.

Regardless of the high energies that some individual models reach, the
predicted distribution of energies in our population models does not contain
many of these very high energy explosions, as they originate predominantly in
high mass models. The energy distribution we obtain for Type Ib SNe at Z=0.01
seems to be in rough agreement with the observed sample of Taddia et al.
(2018), although it is skewed to slightly lower energies (see Fig. 11). A
possible explanation for this behaviour might be that the progenitor mass of
the majority of our models is underestimated, since they correspond to helium
stars below the minimum WN luminosity limit. A larger final masses will result
in larger carbon oxygen core masses at core collapse, which is, in turn,
likely to shift the distribution to higher energies. Furthermore, progenitors
of Type Ib SNe that come from stars that were not fully stripped after the
main sequence may also result in more energetic explosions since they are
expected to have higher final mass, and higher core mass. This conjecture can
be tested when accurate progenitor models of Type Ib SNe become available.

The energy distribution of Type Ic SNe on the other hand has the opposite
effect. Since most of the progenitors of Type Ic SNe at low metallicity come
from stars of higher mass (and are also therefore more rare), their energy
distribution we obtain is skewed to much larger energies. It is only at higher
metallicity that we find a broad energy distribution that also includes lower
energy explosions, and resembles the observed distribution more closely.

Like ejecta masses, explosion energies of Type Ic SNe are only reproduced by
very high metallicity models, and even at our highest metallicity, we find
that the SNe with the lowest kinetic energies in the observed sample are
missing from our predictions. This supports our claim that Type Ic SN
progenitors likely experience additional mass loss, either due to winds or to
a different effect that acts on shorter timescales.

It must be noted, however, that ejecta masses and explosion energies inferred
from light curves of stripped-envelope SNe are degenerate (i.e., several
combinations of these two parameters can produce identical light-curves. See
Dessart et al. 2020). However, comparing the value of
$\mathrm{E}/\mathrm{M}_{\mathrm{ej}}$ predicted by our models to the observed
one results in a distribution that is similar from those obtained in
observations. We highlight, however, that there is a large scatter in the
value of $\mathrm{E}/\mathrm{M}_{\mathrm{ej}}$ resulting from our models. The
distribution of this quantity ranges from about $0.3\times 10^{50}\
\mathrm{erg}/\,\mathrm{M}_{\odot}$ in models with the lowest masses,
monotonically increasing with initial mass up to about 2 in the models with
the highest initial mass, while some models with initial masses in the range
of 20–40 $\,\mathrm{M}_{\odot}$ reach slightly larger values, as well as those
that are predicted to become fallback SNe.

Nickel masses in our models tend to be considerably lower than those found by
Taddia et al. (2018) and Barbarino et al. (2021) at every metallicity. Our
population model predicts that the median nickel mass of Type Ib SNe is about
a factor of 3 lower than that of the analysed sample. In Type Ic SNe, the
difference is less stark, about a factor 2, but is still widely
underestimated. We find that nickel mass has a tendency to decrease with
metallicity.

The nickel mass in the observed samples is obtained by means of Arnett’s rule
(Arnett 1974). It has been found that this estimate overestimates the nickel
mass produced in stripped-envelope SNe of all types (Dessart et al. 2016;
Afsariardchi et al. 2021), which may help explain the discrepancy between our
estimations and the observed sample. As with explosion energies and ejecta
masses, in the case of Type Ib SNe, an additional cause for the prediction of
low nickel masses might be that their final mass is underestimated, since we
find that most Type Ib SNe are originated in stars with luminosities below the
minimum WR luminosity, and therefore have overestimated mass loss rates.

Kick velocities are computed following Vigna-Gómez et al. (2018). They depend
on the explosion energy and the remnant mass (as well as in the mass enclosed
between the acceleration region and the point where the shock exceeds the
escape velocity of the material below it), and are therefore correlated (see
Fig. 17). The distribution of kick velocities in our models resembles that
from Mandel & Müller (2020), and resembles the velocity distribution of
pulsars (Hobbs et al. 2005). However, we find that the distribution has a
tendency to decrease with increasing metallicity, likely as a consequence of
the decrease in mass of SN progenitors.

####  5.3.4 Neutron star and black hole mass distributions

We find that Type Ic SNe produce more massive NSs than Type Ib SNe. In
agreement with Ertl et al. (2020) and Schneider et al. (2021), we find that NS
gravitational mass is correlated to $\xi_{2.5}$ (see Fig. 17). This implies
that the prediction is also independent of the SN explosion model, since most
Type Ic progenitors come from stars that are more massive than those of Type
Ib SNe at the time of core collapse, and also tend to have a larger value of
$\xi_{2.5}$.

Variations in the locations of the “islands of explodability” as a function of
metallicity are clearly reflected in different BH mass distributions. Our
results resemble those of Woosley et al. (2020), but we find that variations
in metallicity result in more drastic changes in the locations of the peaks
and valleys of the BH mass spectrum.

The emergence of a low mass peak at around $3\,\mathrm{M}_{\odot}$ in some of
our grids arises from fallback SNe that result from some high mass models with
relatively low values of $\xi_{2.5}$. The intermediate mass peak in our
distributions becomes displaced from around $9\,\mathrm{M}_{\odot}$ at Z=0.01
to around $6\,\mathrm{M}_{\odot}$ at Z=0.04, and has a structure that results
from the variations in core properties as a function of mass in models in this
regime, consequence of the transition between convective and radiative core
carbon burning (Sukhbold & Woosley 2014).

The peak mass in the distribution that corresponds to high mass BHs, above
$10\,\mathrm{M}_{\odot}$, also decreases as a function of metallicity. Due to
variations in explodability that arise as a function of metallicity, even in
models with the same final mass, it is enough to bring this peak from about
$15\,\mathrm{M}_{\odot}$ in the Z=0.01 grid, down to about
$11\,\mathrm{M}_{\odot}$ at Z=0.04. The highest mass BH at each metallicity
corresponds to the highest mass model, and carries no information about
whether the location of the BH mass gap depends on metallicity.

This behaviour reflects the need to take metallicity dependence of BH mass
production into account when generating predictions for the BH mass
distributions in the Universe, and when interpreting the findings of the
LIGO/VIRGO collaboration (e.g. Fishbach et al. 2021) and future gravitational
wave observatories.

####  5.3.5 The metallicity evolution of progenitors of stripped-envelope
supernovae

We find that the number ratio of Type Ib SNe to Type Ic SNe increases as a
function of metallicity due to the metallicity dependent mass loss rates of WR
stars. Our findings are in broad agreement with the relative rates found in
the local Universe (Shivvers et al. 2017), and with the fact that Type Ic SNe
are found in environments of higher metallicity than other core collapse SNe
(Modjaz et al. 2011). According to our analysis in Paper I, stars in our
sample that evolve to become progenitors of Type Ic SNe will be luminous
enough to be observable as WC-type stars, meaning that they are unaffected by
the uncertainty in the mass loss rates of low luminosity helium stars.

On the other hand, we find that although the relative number of Type Ib SNe
with transparent wind stripped-envelope stars as progenitors relative to those
that have WR progenitors both tends to increase as a function of metallicity,
it is below 0.2 even at the highest metallicity in our grid, and about
0.06–0.08 at approximately solar metallicity. Even though the value of the
minimum luminosity of WR stars as a function of metallicity is approximate,
this implies that, in our simplistic model, the overwhelming majority of
progenitors of Type Ib SNe are in fact not WR stars, but rather stripped-
envelope stars with transparent winds, and luminosities below the WR
threshold. This result is further backed by the fact that, making this
assumption, the distribution of ejecta masses we obtain for Type Ib SNe
resembles the observed distribution more closely.

Because all of our explodability tests roughly coincide in their results, the
inferred values of ratios presented in Fig. 14 are independent of our SN
calculations, and their uncertainties stem from uncertainties in our
determination of the minimum WR luminosity, the minimum mass at which helium
stars produce core-collapse SNe and the stripping efficiency of stars as a
function of metallicity.

The number of SNe coming from low luminosity helium stars is not expected to
be affected by a reduction in the mass loss rate of their progenitor, since
their low initial mass also implies a low value of $\xi_{2.5}$. Therefore, we
argue that our calculations on the relative rate of SN types and the spectral
class of the progenitors of Type Ib SNe (Fig. 14) are not dependent on the
effect of decreasing winds for helium stars with luminosities below the
minimum WR luminosity, but they are uncertain inasmuch as the minimum WR
luminosity and the minimum mass at which stars produce SNe are uncertain.

Although WR stars are not very luminous in optical bands, since they are
compact and hot, the problem becomes more dramatic for low mass stripped-
envelope stars, as they do not show strong emission lines, and are less
luminous and comparably compact. The fact that we predict that most Type Ib
SNe to form in transparent wind helium stars might be the reason why many
searches for archival images of their progenitors in SN explosion sites have
been unfruitful (e.g. Smartt 2009). During the core helium burning phase, low
mass helium stars with transparent winds are not very bright in optical
wavelengths (Götberg et al. 2018), where the searches for their progenitors
have taken place. Models predict that, below a certain mass, helium stars
expand during the late phases of their evolution. This expansion is caused by
a combination of inflation and the presence of a helium burning shell (e.g.
Yoon et al. 2010; Woosley 2019). This implies that the temperatures of helium
stars after core helium depletion will decrease, leading to a larger
brightness in the optical band (Yoon et al. 2012; Kim et al. 2015). The
transition mass at which expansion becomes important cannot be directly
assessed from our models, since it requires a more detailed treatment of the
envelope (i.e. without the use of mlt++). However, our current models indicate
that this is metallicity dependent, and may have consequences for the fate of
these stars, particularly if they are in binaries (Yoon et al. 2010; Vigna-
Gómez et al. 2021).

Direct observations have been so far limited by the fact that most of these
explosions have not been detected in the nearby Universe. Regardless, new
atmosphere and stellar evolution models of low mass stripped-envelope stars
are required to constrain their properties at core collapse. Their mass loss
rates and the response of their envelopes to expansion (both due to inflation
and helium shell burning) as a function of metallicity, are paramount to
understand current and future identifications of progenitors of Type Ib SNe,
and currently available upper limits on their optical brightness. Together
with direct observations of stripped-envelope stars may reveal phenomena that
occur in the final few thousand years of the evolution of stripped-envelope
stars, which are so far poorly constrained.

###  5.4 Implications for double compact object assembly and gravitational-
wave sources

Massive binaries are believed to be the progenitors of double compact object
mergers and gravitational-wave sources. Moreover, interacting massive binaries
are believe to be common in the field (e.g., Sana et al. 2012), and these
interactions usually result in mass transfer episodes and subsequent envelope
stripping (e.g. Bhattacharya & van den Heuvel 1991; Tutukov & Yungelson 1993;
Belczynski et al. 2016). Additionally, wind stripping can also occur in
massive binaries, particularly at high metallicities (e.g. Georgy et al.
2012). Therefore, the evolution of stripped helium stars play a fundamental
role in the assembly and characterisation of double compact objects.

####  5.4.1 Compact object formation and islands of explodability

The predictions of the properties of a compact object remnant is a difficult
problem. The main question, in the context of gravitational-wave sources, is:
what is the mass of a remnant given a stellar progenitor? While for decades
the answer seemed to be that increasing masses lead to increased remnants
(e.g. Fryer et al. 2012), more recent and more detailed studies suggest that
the structure of the star at core collapse plays a crucial and complex role
(e.g. O’Connor & Ott 2011; Ertl et al. 2016; Patton & Sukhbold 2020; Ertl et
al. 2020). One of the main features that has emerged from the discovery of
“islands of explodability” can be interpreted as models that are similar in
mass at core collapse but lead to significantly different compact object
remnants, the most extreme case scenario being a light neutron star and a
heavy black hole. Additionally, it represents regions on the progenitor mass
space that are very discontinuous in the remnant mass space. According to our
results, the picture may be further complicated by the presence of fallback
SNe resulting from the evolution of very massive stars (see Fig. 16), that
lead to the formation of remnants with very small masses. This probably has
important implications in the rates of double compact objects. If these
islands of explodability are real, i.e., not a numerical artifact, that would
decrease rates of binary black hole mergers and likely increase the rate of
neutron star/black hole mergers (modulo disruption via neutron star natal
kicks).

Double compact object mergers with highly asymmetric mass ratios, such as the
$\approx$23.0-2.6 $\rm{M_{\odot}}$ gravitational wave event GW190814 (Abbott
et al. 2020), might be the right candidates to probe the explodability island
hypothesis. Currently, rapid population synthesis models struggle predicting
such asymmetric mergers (e.g. Zevin et al. 2020), even when considering the
possibility of explodability islands (Mandel et al. 2021). This idea is
explored further by Antoniadis et al. (2022).

####  5.4.2 High mass X-ray binaries

High mass X-ray binaries are observed systems that can potentially become
double compact objects. The outbursts of Be X-ray binaries (e.g. Rappaport &
van den Heuvel 1982), comprised of Be stars (e.g. Rivinius et al. 2013) and a
compact accretor, have been observed in the Milky Way and the Magellanic
Clouds (Reig 2011). The compact object progenitor is believed to have been
stripped in a mass transfer episode with the stellar companion (e.g.
Vinciguerra et al. 2020), and therefore likely spend a fraction of its
lifetime as a stripped-envelope star. Its behaviour will be determined by
whether or not its luminosity is above the minimum WR luminosity; i.e. if it
is a WR star or not. Furthermore, the presence of inflation will determine
whether or not binaries that include a stripped-envelope star will have
episodes of accretion, and the question remains whether this accretion will be
stable or lead to a merger.

If the star in an X-ray binary is indeed a WR, its winds are expected to widen
the orbit. Moreover, higher mass loss rates, as those expected at higher
metallicities, lead to longer period orbits. Heavier X-ray binary systems are
comprised of O-type stars with black hole companions, which have been the
archetype of interacting compact object binaries since the discovery of Cygnus
X-1 (e.g., Bowyer et al. 1965; Webster & Murdin 1972; Bolton 1972). Recently,
the inferred black hole mass of Cygnus X-1 has been corrected from $14.8\
\rm{M_{\odot}}$ to $21.2\ \rm{M_{\odot}}$ (Miller-Jones et al. 2021),
suggesting that winds from massive stars at high metallicities have been
overestimated in the last decades. The canonical evolutionary pathway in the
formation of Cygnus X-1 involves a helium-star with a stripped (initial) mass
between $22-30\ \rm{M_{\odot}}$, depending on the metallicity (Neijssel et al.
2021).

Having a better understanding of the evolution of stripped-envelope stars is
paramount to understand the observable time of X-ray binary progenitors as WR
stars (van den Heuvel et al. 2017), the evolution of the orbit and assembly of
the X-ray binary phase (e.g. Repetto et al. 2017), and the fate of the system
as a double compact object (e.g. Klencki & Nelemans 2019).

####  5.4.3 Gravitational wave sources

Most double compact objects formed in isolation are expected to engage in at
least one mass transfer episode throughout their evolution. The canonical
binary black hole formation scenario from isolated binaries includes stripping
via the common envelope phase (e.g. Belczynski et al. 2016; Stevenson et al.
2017). However, more recent studies suggest that stable mass transfer might be
frequent during binary black hole formation (Neijssel et al. 2019; Marchant et
al. 2021). The details of stripping are complex and not well understood (e.g.
Vigna-Gómez et al. 2022), but it is not unreasonable to expect that stripped
stars will (eventually) be stripped-envelope stars. The fate of a binary
system is deeply entangled with the evolution of stripped helium stars. The
post-stripping radial evolution dictates whether or not they will engage in an
additional mass transfer episode, as well as if they can merge within the age
of the Universe. Stellar winds not only widen the orbit, but directly affect
the mass of the stellar core of what will become the compact object remnant.
This combined effect suggests that higher stellar winds of stripped stars,
which are likely to widen the orbits and decrease remnant masses, increases
the coalescence time of the double compact object. Less frequent and less
massive double compact object mergers are less likely to be detected by
ground-based gravitational-wave observatories.

Progenitors of binary neutron stars (e.g. Bhattacharya & van den Heuvel 1991;
Vigna-Gómez et al. 2018) or neutron star/black hole (e.g. Lipunov et al. 1994)
binaries are also believed to host stripped stars at point in their
evolutionary history. The initially most massive star is usually stripped
first and leads to configurations such as the aforementioned X-ray binaries.
The stripping of the companion is very important in the context of
gravitational-wave sources: in principle, light stripped stars will lead to
binary neutron stars, and heavy stripped stars to neutron star/black hole
binaries. However, as previously mentioned, the mass and radial evolution of
stripped stars is determinant in the fate of double compact objects. In the
canonical formation channel of neutron star binaries (e.g. Bhattacharya & van
den Heuvel 1991; Tauris et al. 2017), light stripped stars expand and are
likely to engage in a final (stable) mass transfer episode which will further
strip the donor star and might spin-up the pulsar companion. Slightly more
massive stripped stars, which also expand, have been predicted to lead to a
common envelope which results in short orbits ($\sim$ min) and leads to fast
($\sim$ Myr) double compact object mergers (Vigna-Gómez et al. 2018; Romero-
Shaw et al. 2020).

Recently, Vigna-Gómez et al. (2021) used the results from this paper to show
that heavy stripped stars, which do not expand significantly before core-
collapse, will not have a final mass transfer episode and will form radio
quiet double compact objects. Vigna-Gómez et al. (2021) also showed that the
fate of the remnant, to become either a neutron star or a black hole, doesn’t
depends only on the mass at core collapse but also on the explosion energy. It
also showed that this formation avenue is a plausible formation channel to
form neutron stars with heavy neutron star or light black hole companions.

In the context of neutron star binaries, there is a particular scenario of
interest for low-mass stripped stars: the double-core common-envelope scenario
(Brown 1995). In this scenario, a nearly equal mass binary experiences an
early common envelope phase in which both stars are giants. The common
envelope is now comprised of both envelopes and the ejection leads to a close
binary with two stripped stars. Rapid population synthesis studies predict
that $\approx$20% of binary neutron stars formed at solar metallicity form in
this way (Vigna-Gómez et al. 2018). Moreover, this formation channel might
even be more abundant at lower metallicities.

A more exotic avenue predicts the late, double core, common envelope episode
of stripped stars (Belczyński & Kalogera 2001). This formation channel results
in very short period ($\lesssim$ few hr) binaries where both stars will
explode as Type Ic SNe. The remaining lifetime of stars that have been
stripped late is short, and they conform less than 1% of binary neutron stars
at high metallicity environments (Vigna-Gómez et al. 2018). However, they
remain interesting candidates to the models presented in this paper.

##  6 Conclusions

We have estimated the final outcome after core collapse of the models of
helium stars presented in Paper I. These models approximate the late evolution
of stripped-envelope stars with the simplifying assumption that they are fully
stripped at the onset of helium burning. We analysed the resulting core
collapse models using several explodability tests. The outcome of these tests
allowed us to constrain the observables of Type I core collapse SNe that
result from the evolution of stripped-envelope stars, and to construct a
synthetic population model of the obtained properties to compare with
observations.

These stars have a helium and nitrogen rich envelope and a convective helium
burning core enriched in carbon and oxygen rich. As they lose mass, some of
them lose their nitrogen-rich envelope due to mass loss. As we employed the
mass loss rate of Yoon (2017), this change in surface abundance is also
accompanied by an increase in mass loss rate, as they transition from WN to
WC-type WR stars, which creates a bifurcation in the resulting final masses of
our models. This dichotomy is also associated with the different spectral
properties of Type Ib and Type Ic SNe (Dessart et al. 2020). Using the method
derived in Paper I, we also recognized which stars will be observable as WRs,
and which ones will have transparent winds and low mass loss rates.

We find that the core properties of our models are different from those of
stars that do not experience stripping, but are rather consistent with recent
works of stripped-envelope stars (e.g. Woosley 2019; Ertl et al. 2020; Sander
et al. 2020; Law-Smith et al. 2020). However, we find that core properties of
stripped-envelope stars such as core compactness, carbon-oxygen core mass and
central carbon fraction after helium burning are metallicity dependent, and
are also altered by the dichotomy in mass loss rate.

We find that core properties that are related to explodability of massive
stars, such as $\xi_{2.5}$, $M_{4}$ and $\mu_{4}$, tend to have similar values
for stars of similar final mass. However, subtle differences in relative core
mass and core composition alter the explodability landscape enough to have an
effect in the resulting distribution of SN properties and remnant masses that
result from our models.

After assessing the explodability of our models, we construct a synthetic
model for the distribution of SN properties and BH masses as a function of
metallicity. We find that NSs produced by helium stars tend to have a
distribution that resembles the low mass peak of the distribution, as inferred
by Antoniadis et al. (2016), and the total distribution of NS masses decreases
only weakly with increasing metallicity. However, we find that Type Ic SNe
preferentially form more massive NSs than those that result from Type Ib SNe.

We find that although the overall structure of $\xi_{2.5}$, $M_{4}$ and
$\mu_{4}$ is similar as a function of final mass, the resulting BH mass
distributions differ. The location and width of the intermediate mass and high
mass peaks, and therefore also the mass gaps where we do not expect to find
any BHs, become displaced to lower masses as a function of metallicity. We
also find a low-mass peak in the distribution of BH masses, at around
$3\,\mathrm{M}_{\odot}$, that emerges from rare fallback SNe that come from
very high mass stars (with initial helium star masses $\gtrsim$
$50\,\mathrm{M}_{\odot}$). Convolving this behaviour with the metallicity
evolution of the Universe is therefore important to be able to draw any
conclusions from the observed distributions that will become available as the
sample of known stellar mass BHs with measured masses increases.

Using the value of the minimum luminosity of WC-type stars in Eq. 11, we find
that the progenitors of Type Ic SNe are observable as WC-type stars in the
metallicities covered in our simulations. Comparing the distribution we
predict for ejecta masses in Type I SNe to observations, we find that the
observed distribution of ejecta masses of Type Ic SNe resembles our results
most closely at very high metallicities, while they are underestimated at
$\sim\,{\rm Z}_{\odot}$ and below. This may imply that either WC mass loss
rates, WN mass loss rates, or both, are higher than what we have accounted
for. Alternatively, it might suggest that late binary interactions or eruptive
mass loss episodes are common in the later phases of evolution of these stars.
We find that the explosion energy distribution of Type Ic SNe is also better
reproduced by high-metallicity models, which we interpret as confirmation that
their progenitors experience subsequent mass loss beyond what is predicted by
WR winds.

In our simplified population model, we find that the vast majority of
progenitors of Type Ib SNe are below the minumum luminosity of WN-type stars;
i.e., they are not WR stars but low mass helium stars with transpratent winds.
Assuming that Type Ib SN progenitors have very small mass loss rates does a
better job at reproducing the distribution of observed Type Ib ejecta masses.
The distributions of explosion energy and nickel mass of Type Ib SNe are
underestimated in our model compared to the observed sample. We conclude that
this is further evidence that the progenitors of most Type Ib SNe must have
low mass loss rates. This highlights the need to characterize, both
theoretically and observationally, the properties of low mass helium stars
with optically thin winds. Such systems can be studied during their core
helium burning phase, where they are most likely to be found, but the final
years of their evolution deserve special attention in order to understand why
campaigns to directly detect Type Ib SNe progenitors have been generally
unsuccessful.

We argue that detailed modeling of stripped-envelope stars, both as individual
systems, as components in binary systems, and in rapid population synthesis
codes must include the considerations utilized in this paper to have a better
understanding of several important phenomena. SN properties and populations
depend not only on the efficiency of binarity and wind mass loss to create
stripped-envelope stars, but also in the subsequent wind mass loss of
stripped-envelope stars. The metallicity dependence of WR winds and the
existence of transparent wind helium stars with very low mass loss rates will
shape the resulting islands of explodability, and therefore the populations
and properties of Type I core collapse SNe and the mass distributions of their
remnants. Differences in their evolution and outcome will affect systems such
as X-ray binaries and gravitational wave sources.

The physics of stripped-envelope stars is still uncertain. Their mass loss
rates are unconstrained in some regimes, it is unknown if they commonly
experience outbursts or second episodes of mass transfer, and it is unknown
how common they are, and what is their mass distribution. A holistic approach,
combining theory and observations of stripped-envelope stars and stripped-
envelope SNe, can be a very powerful tool to understand many far-reaching open
questions in massive star physics.

###### Acknowledgements.

D. R. A.-D. and J. A. are supported by the Stavros Niarchos Foundation (SNF)
and the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the
2nd Call of “Science and Society” Action Always strive for excellence –
“Theodoros Papazoglou” (Project Number: 01431). B. M. acknowledges support by
ARC Future Fellowship FT160100035. S.-C. Y. has been supported by the National
Research Foundation of Korea (NRF) grant (NRF-2019R1A2C2010885). A.V.-G.
received support through Villum Fonden grant no. 29466. D. R. A.-D. would like
to acknowledge valuable discussions with Eva Laplace, Jakub Klencki and
Mathieu Renzo. This research made extensive use of NASA’s ADS and
mesa111http://mesastar.org (Paxton et al. 2011, 2013, 2015, 2018).

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