pages=-,link,pagecommand=

11institutetext: Observatoire de Genève, Chemin Pegasi 51, 1290 Versoix,
Switzerland
11email<EMAIL_ADDRESS>
# The impact of convective criteria on the properties of massive stars

Y. Sibony  C. Georgy  S. Ekström  G. Meynet

###### Abstract

Context. Libraries of stellar models computed with either the Ledoux or the
Schwarzschild criterion to determine the sizes of convective regions are
available in the literature. It is still not clear, however, which of these
two criteria should be used, although many works have been devoted to that
question in the past.

Aims. In the framework of the evolution of single rotating stars, we study the
differences between models computed with Ledoux and Schwarzschild criteria on
the internal structure, evolutionary track in the Hertzsprung-Russell diagram
(HRD), lifetimes, evolution of the surface abundances and velocities, and
masses of the He and CO cores. We investigate the consequences on the nature
of the supernova (SN) progenitors and the type of SN events, as well as on the
stellar yields of light elements. We also study the impact on the outputs of
population synthesis models.

Methods. Models with initial masses between 7 and 120 M⊙ at solar metallicity
($Z$=0.014) and with an initial rotation equal to 0 or 0.4 times the critical
velocity at the zero-age main sequence were computed with either the
Schwarzschild or the Ledoux criterion until the end of the C-burning phase.

Results. Models with initial masses between 15 and 32 M⊙ computed with the
Schwarzschild criterion show larger intermediate convective zones attached to
the H-burning shell than models computed with the Ledoux criterion. Their CO
cores and outer convective zones in the red supergiant (RSG) phase are also
smaller. This impacts many outputs of stars during the core He-burning phase.
Schwarzschild models have smaller CO cores and outer convective zones in the
RSG phase, and their blue-to-red supergiant ratio is much higher than for
Ledoux models. They also produce longer crossings of the Hertzsprung gap and
favour blue loops. The upper luminosity of RSGs is little affected by the
change in the convective criterion. The maximum luminosity of RSG progenitors
for type II-P SN events is lowered from 5.2 to 4.95 when the Ledoux criterion
is used instead of the Schwarzschild criterion in non-rotating models. The
Schwarzschild criterion predicts longer-lasting, less nitrogen-enriched, and
faster-rotating Cepheids. Rotational mixing tends to decrease the differences
between Schwarzschild and Ledoux models.

Conclusions. The results of this paper can be used as first guidelines to set
up observational programs that may help to distinguish between these two model
families.

###### Key Words.:

Convection – Stars: evolution, massive, rotation

##  1 Introduction

Numerous difficulties arise when stars are modeled. The treatment of
convection has long been one of these difficulties. The classical way of
dealing with convection in one-dimensional (1D) stellar evolution codes is the
following: 1) find the boundaries of the convective zone by applying an
instability criterion, 2) empirically increase the size of (some of) the
convective zones (a process often called “overshooting”), and 3) compute a
thermal gradient to be applied inside the convective zone. For the first
point, two criteria are widely used in the literature to determine the
stability of a thermally stratified medium with the gravity as the restoring
force: the so-called Schwarzschild criterion, which writes (for stability
against convection)

$\nabla_{\text{rad}}<\nabla_{\text{ad}},$ (1)

where $\nabla_{\text{rad}}=\left(\frac{\text{d}\ln T}{\text{d}\ln
P}\right)_{\text{rad}}$ is the radiative thermal gradient, and
$\nabla_{\text{ad}}=\left(\frac{\text{d}\ln T}{\text{d}\ln
P}\right)_{\text{ad}}$ is the adiabatic thermal gradient; and the Ledoux
criterion, which writes (also for stability)

$\nabla_{\text{rad}}<\nabla_{\text{ad}}+\frac{\varphi}{\delta}\nabla_{\mu},$
(2)

where $\varphi=\left(\frac{\partial\ln\rho}{\partial\ln\mu}\right)_{T,P}$,
$\delta=-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{T,\mu}$, and
$\nabla_{\mu}=\left(\frac{\text{d}\ln\mu}{\text{d}\ln P}\right)$. It is
possible that some layers inside the star are Ledoux-stable but Schwarzschild-
unstable. In case of a thermally dissipative medium, however, Kato (1966)
showed that Eq. (2) is reduced to Eq. (1) due to the onset of oscillatory
convection. For this reason, the Schwarzschild criterion is most of the time
preferred to the Ledoux criterion in stellar evolution computations. Another
option is to apply a partial mixing of the layers encompassed by the two
criteria. This is called “semiconvection” (see Langer et al., 1983). In this
framework, models computed with the Ledoux criterion correspond to a totally
inefficient semiconvection, while models computed with the Schwarzschild
criterion correspond to an infinitely efficient semiconvection.

Point 2 was introduced in stellar evolution codes as a necessity to reproduce
some observational features, such as the width of the main sequence (MS) of
open clusters (see e.g. Maeder & Mermilliod, 1981; Maeder & Meynet, 1989;
Martinet et al., 2021). Different implementations for this additional mixing
(co)exist in stellar evolution codes: penetrative overshoot (e.g. Maeder &
Mermilliod, 1981; Zahn, 1991), diffusive overshoot (e.g. Freytag et al., 1996;
Herwig et al., 2006), entrainment (Meakin & Arnett, 2007; Cristini et al.,
2017; Scott et al., 2021), or more recently deduced from 2D fully compressible
time-implicit simulations (Baraffe et al., 2023). A thorough discussion of
these various implementations and their link to hydrodynamics simulation can
be found in Viallet et al. (2015).

Finally, point 3 requires the computation of a thermal gradient to be applied
inside the convective regions. For deep convection, the timescale for
radiative exchanges between the convective cells and the environment are
longer than the convective instability timescale, and the process is very
close to being adiabatic, thus the adiabatic gradient $\nabla_{\text{ad}}$ can
be applied. In convective regions closer to the surface, more complex theories
need to be used. A usual choice in this case is the mixing-length theory
(Böhm-Vitense, 1958).

To obtain a complete picture of convection in stellar interiors, 3D
hydrodynamics simulations are required. Convection in stars is very turbulent:
the estimated Reynolds number is as high as $10^{9}$ (Arnett & Meakin, 2016).
The mass flux is therefore highly asymmetric, requiring a multi-dimensional
treatment. Efforts to do this have been made during the past decade for
various physical environments (e.g. Meakin & Arnett, 2007; Cristini et al.,
2017; Woodward et al., 2015; Jones et al., 2017; Horst et al., 2021). However,
all of these simulations were limited to a small region and/or a short time
compared to the size and lifetime of a star. These results can nevertheless be
used to build new algorithms that can obtain a better agreement between the 1D
stellar evolution codes and the 3D convection simulations. Some attempts have
been made (e.g. Scott et al., 2021), but these procedures are not yet fully
mature and still need to be improved for routine use in stellar evolution
calculations.

It is not clearly established which of the criteria for convection should be
used in stellar evolution computations. Moreover, when the Ledoux criterion is
used, the efficiency of semiconvection is not known either. Since the
pioneering works of Stothers & Chin (1973, 1975, 1976), very little progress
has been made. Lawlor et al. (2015) simulated and compared the evolution and
supernova (SN) light curves of Population III stars computed with either
criterion and found Schwarzschild stars to be much larger and cooler than
their Ledoux counterparts. Chun et al. (2018) calibrated the mixing-length
parameter with red supergiants (RSGs) of different metallicities and predicted
a decrease in the hydrogen content of stellar envelopes with increasing
metallicity when using the Schwarzschild criterion, but not the Ledoux
criterion. Schootemeijer et al. (2019) computed grids of Ledoux models at the
metallicity of the Small Magellanic Cloud (SMC; $Z=0.002$) and found that more
efficient semiconvection increases the ratio of blue to red supergiants.
Kaiser et al. (2020) computed non-rotating models of 15, 20, and 25 M⊙ at
solar metallicity to study the relative impact of various convective
parameters. They found the strength of convective boundary mixing to be more
effective than the choice of criterion for convection, except for determining
the initial location of intermediate convective zones. Anders et al. (2022)
performed a 3D hydrodynamical simulation of a convective zone adjacent to a
semiconvective region. They reported that both criteria gave the same results
when the evolutionary timescale exceeds the convective-overturn timescale
(e.g. during the main sequence), but that differences can emerge when the
evolution occurs on a rapid timescale (e.g. the Hertzsprung gap is crossed).
We pursue our efforts and to try to determine which criterion reproduces the
observed features of massive stars better (Saio et al., 2013a; Georgy et al.,
2014b, 2021). To do this, we provide the first detailed comparison of
Schwarzschild and Ledoux criterion models over a wide mass range at solar
metallicity, including the effects of rotation.

In Sect. 2, we present the main physical ingredients of the stellar models. We
discuss the impact of changing the convective criterion on the cores and
convective structures of stars in Sect. 3. Section 4 presents the impact on
various aspects of post-MS stellar evolution. We compare the properties of
stars at the end of their evolution in Sect. 5. We present models for
population synthesis in Sect. 6. We compare our results to previous works on
the subject and to an observed population of evolved massive stars in Sect. 7.
The discussion and conclusions are given in Sect. 8.

##  2 Ingredients of the models

The models computed with the Schwarzschild criterion were presented in Ekström
et al. (2012). The convective regions are determined starting from the centre
of the star and going outwards. In each shell of the model, the adiabatic and
the radiative gradient are computed and compared, which determines whether the
shell is to be radiative or convective. During central H- and He-burning, the
size of the core is artificially increased by a length corresponding to 10% of
the local pressure scale height at the edge of the formal core (overshooting).
This is not done for subsequent burning phases or for convective shells. In
the internal convective zones, the thermal gradient is imposed to be
adiabatic, and the chemical species are instantaneously mixed (so that the
chemical composition of the convective zones is always homogeneous). In the
envelope, the classical mixing-length theory (Böhm-Vitense, 1958) is used to
obtain the thermal gradient, with a mixing-length parameter $\alpha=1.6$. For
models more massive than 40$\,M_{\odot}$, the mixing-length parameter is set
to $1$, and the mixing-length is computed according to the density scale
height instead of the pressure scale height. Moreover, the turbulent pressure
is accounted for by adding an acoustic flux term in the mixing-length
formulation (see Maeder, 1987).

The Ledoux models are new models that are computed with the same physical
ingredients as the Schwarzschild models, except that the $\mu$-gradient is
computed and accounted for when a shell is to be determined as radiative or
convective. All the other ingredients are kept the same. In particular, the
treatment of overshooting and of mixing is the same, and the mass-loss
prescriptions are the same as well.

Rotation is also treated in the same way in both sets of models. The full
advection-diffusion equation for the transport of angular momentum is solved
during the main sequence (see e.g. Eggenberger et al., 2008, for more
details). The horizontal diffusion coefficient that intervenes in this
formalism was proposed by Zahn (1992), and the diffusion coefficient
associated with the shear mixing was proposed by Maeder (1997). All the
rotating models shown in this paper have an initial rotation rate
$\upsilon_{\text{ini}}/\upsilon_{\text{crit}}=0.4$, where the critical
velocity $\upsilon_{\text{crit}}$ is computed as in Georgy et al. (2011, see
their definition of $\upsilon_{\text{crit,1}}$).

Mass-loss is applied to all of our models according to different recipes,
which are summarised here (see also Ekström et al., 2012). For non-Wolf-Rayet
(WR) stars hotter than $\log(T_{\text{eff}})=4.0$, the mass-loss rates by Vink
et al. (2000, 2001) are applied. In this case, a correction factor accounting
for the effects of rotation on the mass loss is also applied (see e.g. Georgy
et al., 2011). For models up to 9$\,M_{\odot}$ and when
$\log(T_{\text{eff}})<3.8$, the Reimers (1977) rates are used, with the
parameter $\eta=0.6$. For more massive models, the mass-loss rate is a linear
fit of observational data from Sylvester et al. (1998) and van Loon et al.
(1999, see also ). When the surface abundance of H drops below $X_{s}<0.3$ and
the effective temperature is above $T_{\text{eff}}>4.0$, we switch to WR-kind
mass-loss rates (Nugis & Lamers, 2000; Gräfener & Hamann, 2008), unless they
are lower than the Vink et al. (2000) rates. Finally, we apply the de Jager et
al. (1988) rates when the above recipes are not applicable.

For massive RSGss, some layers in the external layers, very close to the
surface, reach luminosities far above the Eddington luminosity. To facilitate
the computation in this case, we therefore increase the mass-loss rate as
described above by a factor of $3$ when the over-Eddington luminosity is
overcome by a factor of $5$ for models more massive than 20 M⊙.

Table 1: Properties of the cores and convective zones at different
evolutionary stages, and the mass at the final stage.

| Core properties | ICZ | OCZ | M${}_{\text{fin}}$
---|---|---|---|---
Crit. | M${}_{\text{ini}}$ | ${\frac{\upsilon_{\text{ini}}}{\upsilon_{\text{crit}}}}$ | He | CO |  | pre-He-b | He-b | post He-b |
| $M_{\odot}$ |  | $M_{\odot}$ | $M_{\odot}$ | $M_{\odot}$ | $M_{\odot}$
Led. | $120$ | $0.4$ | $55.6$ | $25.3$ | – | – | – | – | $30.8$
Sch. | $120$ | $0.4$ | $33.7$ | $14.9$ | – | – | – | – | $19.0$
Led. | $120$ | – | $54.4$ | $23.5$ | $50.8-58.8$ | – | – | – | $28.6$
Sch. | $120$ | – | $55.2$ | $25.5$ | $50.8-59.8$ | – | – | – | $30.9$
Led. | $85$ | $0.4$ | $42.7$ | $18.1$ | – | – | – | – | $22.7$
Sch. | $85$ | $0.4$ | $42.8$ | $21.4$ | – | – | – | – | $26.4$
Led. | $85$ | – | $37.0$ | $14.8$ | $34.9-44.6$ | – | – | – | $19.0$
Sch. | $85$ | – | $37.4$ | $14.7$ | $35.0-44.0$ | – | – | – | $18.6$
Led. | $60$ | $0.4$ | $30.6$ | $12.9$ | $27.4-34.4$ | – | – | – | $16.8$
Sch. | $60$ | $0.4$ | $31.4$ | $14.0$ | $27.6-34.1$ | – | - | – | $18.0$
Led. | $60$ | – | $24.5$ | $10.4$ | $25.2-33.6$ | – | – | – | $13.9$
Sch. | $60$ | – | $24.9$ | $9.14$ | $23.2-33.5$ | – | – | – | $12.5$
Led. | $40$ | $0.4$ | $18.5$ | $11.6$ | $17.0-27.7$ | $30.7$ | $30.1$ | – | $15.4$
Sch. | $40$ | $0.4$ | $18.7$ | $8.98$ | $18.0-29.6$ | $30.4$ | $29.3$ | – | $12.3$
Led. | $40$ | – | $14.6$ | $9.41$ | $14.0-22.2$ | – | $23.8$ | – | $12.9$
Sch. | $40$ | – | $14.8$ | $10.4$ | $13.7-25.3$ | – | $31.8$ | – | $13.7$
Led. | $32$ | $0.4$ | $13.5$ | $10.5$ | $13.3-22.9$ | – | – | – | $14.1$
Sch. | $32$ | $0.4$ | $13.5$ | $7.03$ | $12.4-25.8$ | – | – | – | $10.1$
Led. | $32$ | – | $10.8$ | $8.29$ | $17.2-21.6$ | – | $16.0$ | – | $11.3$
Sch. | $32$ | – | $11.0$ | $7.70$ | $10.6-23.3$ | – | $25.5$ | – | $10.9$
Led. | $25$ | $0.4$ | $9.1$ | $6.91$ | $7.0-16.8$ | – | $10.7$ | – | $10.4$
Sch. | $25$ | $0.4$ | $9.1$ | $6.53$ | $8.6-17.5$ | – | $19.7$ | – | $9.7$
Led. | $25$ | – | $7.6$ | $6.23$ | $12.0-15.6$ | – | $9.8$ | – | $9.3$
Sch. | $25$ | – | $7.7$ | $5.29$ | $7.5-17.2$ | – | $18.7$ | – | $8.3$
Led. | $20$ | $0.4$ | $6.2$ | $4.44$ | $7.4-12.4$ | – | $7.2$ | $7.6$ | $7.3$
Sch. | $20$ | $0.4$ | $6.3$ | $4.36$ | $5.6-12.5$ | – | $14.6$ | – | $7.2$
Led. | $20$ | – | $5.3$ | $4.18$ | $6.1-12.3$ | – | $8.2$ | – | $7.0$
Sch. | $20$ | – | $5.4$ | $3.65$ | $5.5-12.6$ | – | $13.4$ | $6.4$ | $8.6$
Led. | $15$ | $0.4$ | $3.8$ | $2.66$ | – | $5.8$ | $5.1$ | $5.2$ | $12.4$
Sch. | $15$ | $0.4$ | $3.8$ | $2.68$ | $3.9-7.5$ | – | $5.9$ | $5.2$ | $11.1$
Led. | $15$ | – | $3.3$ | $2.28$ | $6.2-8.4$ | $5.0$ | $4.8$ | $4.6$ | $13.0$
Sch. | $15$ | – | $3.3$ | $2.11$ | $3.5-7.6$ | – | $5.8$ | $4.3$ | $13.2$
Led. | $12$ | $0.4$ | $2.5$ | $1.69$ | – | $3.6$ | $3.6$ | $3.8$ | $11.1$
Sch. | $12$ | $0.4$ | $2.5$ | $1.82$ | $2.8-5.1$ | $4.9$ | $4.9$ | $3.9$ | $10.2$
Led. | $12$ | – | $2.3$ | $1.43$ | $4.4-5.5$ | $3.9$ | $3.5$ | $3.3$ | $11.1$
Sch. | $12$ | – | $2.3$ | $1.26$ | $2.6-5.0$ | $2.6$ | $3.2$ | $3.0$ | $11.3$
Led. | $9$ | $0.4$ | $1.5$ | $0.93$ | – | $2.4$ | $2.3$ | $2.5$ | $8.7$
Sch. | $9$ | $0.4$ | $1.5$ | $1.18$ | – | $2.4$ | $2.3$ | $3.1$ | $8.5$
Led. | $9$ | – | $1.4$ | $0.82$ | – | $2.4$ | $2.4$ | $2.3$ | $8.6$
Sch. | $9$ | – | $1.4$ | $0.83$ | – | $2.3$ | $2.2$ | $1.2$ | $8.8$
Led. | $7$ | $0.4$ | $1.0$ | $0.60$ | – | $1.7$ | $1.7$ | $1.2$ | $6.9$
Sch. | $7$ | $0.4$ | $1.0$ | $0.61$ | – | $1.7$ | $1.7$ | $1.3$ | $6.9$
Led. | $7$ | – | $0.9$ | $0.54$ | – | $1.7$ | $1.7$ | $1.7$ | $6.9$
Sch. | $7$ | – | $0.9$ | $0.57$ | – | $1.6$ | $1.6$ | $1.0$ | $6.9$

##  3 Convective criterion and internal structure of the models

Figure 1: Relative mass of the convective core as a function of central He
(from 1 to 0) during the He-burning phase for the non-rotating Ledoux (red)
and Schwarzschild (blue) models between 7 and 15 M⊙.

Changing the convective criterion impacts the sizes of the convective zones
and thus the chemical structure of a star. Before discussing the impacts on
the observable outputs of the stellar models using different criteria, we
discuss here the direct impact on various internal properties of our stellar
models. In Table 1, we present for each model the masses of the He- and CO
cores at the end of the central H- and He-burning phases, respectively, the
maximum extent (in Lagrangian mass coordinates) of the intermediate convective
zone (ICZ) before the beginning of core He-burning, and the deepest reach of
the outer convective zone (OCZ) at different evolutionary stages.

We define the cores in the following way: the He core is the region inside the
mass coordinate where at the first time the stellar layers are scanned from
the surface towards the interior the mass fraction of helium $Y>0.9$. The
boundary of the CO core is where for a similar scanning, it crosses
$Y<10^{-2}$.

The masses of the He cores at the end of the MS phase obtained with the two
different criteria show very small differences. For non-rotating models, the
differences are always smaller than 3%111We deduce this number by computing
the Schwarzschild mass minus the Ledoux mass divided by the Ledoux mass.. The
same difference is obtained by comparing rotating models. These differences
are blurred by other uncertainties pertaining to the models and are consistent
with no effect due to the change in the convective criterion. This is
consistent with the fact that in phases during which the convective core size
decreases in mass, no differences are expected when the Schwarzschild or the
Ledoux criterion are applied because there is no mean molecular weight
gradient just above the core.

The only exception is the case of the rotating 120 M⊙ model. The core mass
obtained with the Schwarzschild criterion (33 M⊙) is much lower than the core
mass obtained with the Ledoux model (55.6 M⊙). The structure evolution of the
two rotating 120 M⊙ models shows intermediate convective shells that appear
just above the core in the Schwarzschild models and quickly merge with the
core. This grants more fuel to the core, prolongs the main sequence by
$\sim$100 kyr, and causes greater mass-loss of the star.

Rotation increases the mass of the He core at the end of the MS phase. This
increase is very similar for the Schwarzschild and Ledoux criteria. It is 5-6%
for the 9 M⊙, reaches a maximum for a mass about 40 M⊙ where it amounts to
26-27%, and then decreases because mass loss by stellar winds becomes
important and thus blurs the effects of rotation. The rise in the He-core mass
with initial mass is linked to the higher efficiency of rotational mixing when
the initial mass increases (see the discussion in Sect. 3.1 of Maeder &
Meynet, 2000).

Figure 1 shows how the mass of the convective core evolves during the core He-
burning phase in the non-rotating stellar models for initial masses between 9
and 15 M⊙. The rotating cases (not shown here) exhibit very similar
qualitative behaviours as those shown in Fig. 1: Their core masses increase
during helium burning, and there are frequent breathing pulses222These
breathing pulses are commonly agreed to be numerical artefacts and not of a
physical nature. that inject helium from the H-burning shell into the core.
Interestingly, the convective cores are more massive in the 12 and 15 M⊙
models when the Ledoux criterion is applied. At first sight, this appears to
be counter-intuitive. A stricter criterion for instability (Ledoux) should
produce smaller convective cores than a less strict criterion (Schwarzschild).
However, changing the criterion also affects the sizes of other convective
zones in the models, and the changes in these other convective zones in turn
affect the size of the convective core (see below).

The CO-core masses (see Table 1) present modest differences for masses below
and including 15 M⊙. The largest differences in the higher-mass star range are
due to differences in the mass lost by stellar winds.

As expected, when the Schwarzschild criterion is used, larger intermediate
convective zones associated with the H-burning shell appear. This has
important consequences for the evolution of the He core, the occurrence of a
blue loop (Maeder & Meynet, 2001), and for the surface abundances of stars,
which evolve back to the blue after having lost a large amount of mass during
the red supergiant stage (Saio et al., 2013b; Georgy et al., 2014b, 2021). The
ICZ tends to increase the contribution of the H-burning shell to the total
luminosity (as it transports energy more efficiently), and as a result, the
star reacts to this by decreasing the energy generated in the He-burning core.
The Schwarzschild core has a lower central temperature than the Ledoux core
and is therefore less dense. The Schwarzschild core has the same extent in
radius as the Ledoux core, but its mass is lower, as shown in the bottom
panels of Fig 1.

A convective envelope does not appear in the high-mass range (above 60 M⊙) as
a result of mass loss, which prevents the stars from evolving in the red part
of the Hertzsprung-Russell diagram (HRD). The outer convective zones for the
40 M⊙ models are very small. The 32 M⊙ initial mass is a transition case,
without a convective envelope for the rotating models and with a significant
envelope, reaching down to 25.5 M⊙ for Schwarzschild and 16 M⊙ for Ledoux, in
the non-rotating case. For masses below and including 25 M⊙, all the models
have significant convective envelopes during the core He-burning phase. Models
between 12 and 25 M⊙ computed with the Ledoux criterion have deeper outer
convective zones, but this is not the case for the 7 and 9 M⊙ (see Sect. 4.2).

It might appear surprising that adopting a criterion that hinders convection
can increase the extent of the outer convective zone. This extent also
strongly depends on the position of the star in the HRD during the core He-
burning phase, however: A hotter position during that phase decreases the
extent of the outer convective zone. The position in the HRD at which core He-
burning occurs is very sensitive to the variation in the abundances above the
core as well as to the presence or absence of an intermediate convective zone.
The intermediate convective zone typically tends to make the star more compact
(i.e. blue), which then reduces the size of the convective envelope. In this
manner, a criterion favouring convection (Schwarzschild) can produce a larger
ICZ and consequently a less extended OCZ.

##  4 Evolutionary tracks and lifetimes

Figure 2: Evolutionary tracks of our stellar models in the HRD, colour-coded
by the central helium mass fraction $Y_{c}$. The grey shaded area represents
the Cepheid instability strip (Tammann et al., 2003).

We show the evolutionary tracks for all models in Fig. 2. Main-sequence tracks
are as expected to be similar in the Schwarzschild and Ledoux models. The main
difference occurs in the rotating 120 M⊙ models, as mentioned in Sect. 3.

After the MS phase, the largest differences between the tracks (whether
rotating or not) computed with the Schwarzschild and Ledoux criteria occur for
the mass range between 7 and 40 M⊙. In the upper mass range (between 60 and
120 M⊙), stellar winds are the main factor governing the evolution of the
stars. In the next paragraphs, we discuss in more detail how changing the
convection criterion affects the first crossing of the Hertzsprung gap (Sect.
4.1), the blue loops (Sect. 4.2), the properties of the Cepheids (Sect. 4.3),
the properties of stars ending their evolution as red supergiants (Sect. 4.4),
and the properties of stars ending their evolution as yellow (YSG), blue
supergiants (BSG), or as Wolf-Rayet stars (Sect. 4.5).

###  4.1 First crossing of the Hertzsprung gap

Figure 3: Effective temperature towards the end of the main sequence ($Y_{\rm
c}>0.8$) and core He-burning for non-rotating (solid curves) and rotating
(dashed curves) Ledoux (red) and Schwarzschild (blue) models. Upper row:
Masses between 7 and 15 M⊙. Lower row: Masses between 20 and 40 M⊙. In each
row, the vertical axes in all panels share the same limits, which are
indicated in the leftmost panels. Figure 4: Kippenhahn diagrams showing the
post-MS phases for the four models computed with an initial mass of 15 M⊙. The
vertical axis shows the mass coordinate, and the numbered lines show isoradius
levels. Left column: Schwarzschild models. Right column: Ledoux models. Upper
row: Non-rotating models. Lower row: Rotating models. The intermediate
convective zone above the H-burning shell is clear in the Schwarzschild models
and is associated with the slower expansion of the envelope.

Figure 3 shows the evolution of the effective temperature as a function of the
mass fraction of helium at the centre $Y_{c}$ for all the models with initial
masses between 7 and 40 M⊙.

Models with masses below and including 12 M⊙ all begin their core He-burning
phase when the star is a red supergiant, with $\log{(T_{\rm
eff}\leavevmode\nobreak\ \rm{[K]})}\sim 3.6$. For the 15-25 M⊙ mass range, the
Ledoux criterion favours a quick crossing of the Hertzsprung gap, whether the
models are computed with or without rotation. This is likely to be related to
the fact that the use of the Ledoux criterion restrains the apparition and/or
the extension of the intermediate convective zone associated with the
H-burning shell, as shown in Fig. 4, which shows Kippenhahn diagrams for the
post-MS phases of the four 15 M⊙ models. This tends to produce steeper
gradients of H and He in that zone and favours helium ignition in the core in
the red supergiant stage (Farrell et al., 2022). The models of 15 and 20 M⊙
with the Schwarzschild criterion (as well as the non-rotating 25 M⊙ model)
show a different behaviour. These models ignite helium in their cores when the
star still has a high effective temperature ($\log{T_{\rm
eff}\leavevmode\nobreak\ \rm{[K]}}>4.0$), whether it rotates or not. This is
linked to the larger intermediate convective zone in these models when the
Schwarzschild criterion is used. A tentative explanation is that convection is
more effective at transporting energy than radiation, and therefore, the
presence of this ICZ allows the increased luminosity generated by the
contraction of the core to reach the surface more easily. In the Ledoux
models, however, energy is transported by radiation, and a larger fraction of
it is deposited into the envelope itself, causing its rapid expansion and a
drop in its luminosity. In both cases, the core contraction timescales are
similar; the ICZ only affects the timescale of the envelope expansion. This
means that at the onset of core helium burning, the Schwarzschild star remains
in a blue part of the HRD.
The cases of the more massive models (equal to or higher than 25 M⊙) result
from intricate interactions between convection, rotation, and mass loss. In
general, helium ignition in the core of these models always occurs at an
effective temperature below $\log{T_{\rm eff}\leavevmode\nobreak\
\rm{[K]}}=4.0$. The only exception is the non-rotating 25 M⊙ Schwarzschild
model.

The evolution of the luminosity during the first crossing varies depending on
the convective criterion. In general, the Ledoux models for masses between 15
and 20 M⊙ experience a stronger decrease in luminosity before going up the
Hayashi line than Schwarzschild models because these models are in a stronger
radiative disequilibrium than the Schwarzschild models. Because they expand
more rapidly, a larger amount of energy per unit time is tapped from the
gravitational contraction of the core to expand the envelope. This tends to
decrease the amount of energy that is radiated away. This decrease is to some
extent a consequence of the time taken to cross the Hertzsprung gap. If it
occurs on a sufficiently long timescale, the decrease in luminosity is modest.
When it occurs on a rapid timescale, the luminosity drop is marked. To
summarise, the Schwarzschild criterion favours the apparition of an ICZ above
the H-burning shell after the main sequence, and for the reasons already
described just above, this favours He ignition in the core when the star is
still in the blue part of the HRD. The crossing of the Hertzsprung gap occurs
on a longer timescale (nuclear, about 500 kyr) than with the Ledoux criterion
(thermal, about 30 kyr). With the Schwarzschild criterion, the envelope has
more time to radiate the excess energy produced by the contraction of the core
away, and the Hertzsprung gap crossing occurs at roughly constant luminosity.
The Ledoux criterion suppresses the ICZ, implying a shorter timescale for the
Hertzsprung gap crossing. A large part of the energy radiated by the core
contraction is absorbed by the envelope. As a result, the luminosity at the
surface decreases significantly during the crossing.

###  4.2 Blue loops

Figure 5: 4He profiles at different times before and during core He-burning
for the two non-rotating 7 M⊙ models.

Fig. 3 shows that for initial masses above 20 M⊙ the effective temperature can
increase during core He-burning. These phases are not considered as blue loops
for two reasons, however. First, these stars end their lifetimes at high
effective temperatures as BSG or WR stars. Second, this evolution is mainly
driven by mass loss and not by some internal evolution due to changes in the
hydrostatic structure. In the rest of this section, we thus focus on the mass
domain between 7 and 15 M⊙.

For the lower-mass models, blue loops, characterised by an increase in
effective temperature (blueward evolution) around $Y_{c}\sim 0.4-0.3$, appear
and are followed by a decrease (redward evolution) at the end of core He-
burning. The physics of the blue loops was discussed in detail in previous
papers (see e.g. Lauterborn et al., 1971; Maeder & Meynet, 2001; Anderson et
al., 2014). Many aspects of the stellar models influence the blue loops: for
instance the rate of the ${}^{14}N(p,\gamma)^{15}O$ reaction (Halabi et al.,
2012), of the ${}^{12}C(\alpha,\gamma)^{16}O$ reaction (Brunish & Becker,
1990), the undershooting below the outer convective zone (Lai & Li, 2011), the
helium and metal content (Bono et al., 2000), and rotation (Zhao et al.,
2023). Recently, the occurrence of blue loops was used to constrain the
magnetic moment of massive neutrinos (Mori et al., 2020) or the impact of
axions (Friedland et al., 2013). Stellar models accounting for these effects
undergo an additional loss of energy, and this tends to suppress the blue
loops. This demonstrates that if the evacuation of energy from the core is
facilitated, then this favours the suppression of the blue loops. Here, we
focus on only one aspect, namely the impact of the convective criterion.

It has been shown that the blue loops may be linked to subtle differences in
the abundance distributions near the H-burning shell, and that excess helium
above the shell will suppress the blue loop (see e.g. Walmswell et al., 2015,
who propose that the increase in mean molecular weight caused by having helium
instead of hydrogen above the shell is the main culprit in suppressing the
blue loop). Lauterborn et al. (1971) have proposed a criterion based on the
gravitational potential of the core in order to decide whether a model
produces a blue loop. Stars whose core has a gravitational potential higher
than a given critical value remain along the Hayashi line, while those whose
core has a gravitational potential lower than this critical value develop a
blue loop. Interestingly, the quantity to be compared to the critical value is
the core potential only if there is a steep chemical gradient just above the
He core or near the H-burning-shell. Otherwise, the gravitational potential of
the core has to be multiplied by a factor that is larger for a milder
gradient, thus inducing the model to stay along the Hayashi line.

The non-rotating 7 and 9 M⊙ models computed with the Ledoux criterion show no
blue loops, while the 7 and 9 M⊙ models with the Schwarzschild criterion have
well-developed loops. Because these two masses behave very similarly in the
features of their overall evolution, we focus on the 7 M⊙ models, and the
results qualitatively apply to the 9 M⊙ models. Figure 5 shows the 4He
profiles at different times before and during core He-burning for the two non-
rotating 7 M⊙ models. In the Schwarzschild model, the outer convective zone
has equalised the abundances from the surface down to a mass near 1.6 M⊙,
reducing the extent of the zone with a chemical composition gradient (see the
line corresponding to 4Hec where the gradient is between the mass coordinates
1.3 and 1.6 M⊙). This occurs at the very beginning of the core He-burning
phase. In the Ledoux model, the base of the outer convective zone always
remains above the mass coordinate of 2.1 M⊙ and the zone of the chemical
composition gradient is larger (from 1.3 to 2.1 M⊙), thus milder, than in the
Schwarzschild model. This acts to prevent the formation of a blue loop. The
Ledoux model also has an excess of helium above the hydrogen-burning shell (at
1.7-2.1 M⊙), which suppresses the blue loop (see again Walmswell et al.,
2015). The Ledoux models have this excess helium but the Schwarzschild models
lack it because the mean molecular weight gradient above the hydrogen-burning
shell stabilises this region against convection when the Ledoux criterion is
used. As a result, the convective envelope does not reach down far enough to
equalise the helium abundance to the surface value, leaving a helium excess
above the shell.

Models of 7 and 9 M⊙ with rotation present less marked differences when the
convective criterion is changed from the Schwarzschild to the Ledoux
criterion. Rotational mixing blurs the differences that arise from different
choices of the criterion for convection, and chemical composition gradients as
well as helium abundances above the H-burning shell are similar for those
models.

For the non-rotating 12 M⊙, we do not observe any blue loop or any difference
between the Schwarzschild and the Ledoux models. The only 12 M⊙ model showing
a blue loop is the rotating Ledoux model. This model would thus predict the
most luminous Cepheids of those discussed in this paper (excluding the 15 M⊙
models that cross the Cepheid instability strip very briefly).

###  4.3 Cepheid properties

Figure 6: Bar plot showing the durations of the different Cepheid phases for
masses at which at least one of the four (Ledoux, Schwarzschild, rotating,
non-rotating) models has a Cepheid phase. We distinguish between the first
crossing (redward crossing of the Hertzsprung gap after the MS), the second
crossing (blueward during the blue loop), and the third crossing (redward
during the blue loop). These three moments are indicated in the small top
right inset by coloured lines and dots with corresponding colours. Left panel:
Non-rotating model. Right panel: Rotating model.

Figure 7: Bar plot showing the surface abundances of ${}^{14}\rm{N}$ before
(cyan bar) and during (yellow bar) the blue loop for stars between 7 and 15
M⊙. The dotted line indicates the initial surface abundance. Left panel: Non-
rotating model. Right panel: Rotating model.

Figure 8: Bar plot showing the surface velocities of rotating stars during the
first crossing of the Hertzsprung gap and at three different times of the blue
loop for stars between 7 and 15 M⊙. These four moments are indicated in the
small top right inset by coloured lines and dots with corresponding colours.
Left panel: Equatorial velocity $\upsilon_{\rm surf}$. Right panel: Ratio of
the equatorial to the critical velocity $\upsilon_{\rm surf}/\upsilon_{\rm
crit}$.

Here, we discuss results concerning stars that cross the Cepheid instability
strip (shown as the grey shaded region in Fig. 2 and in the small panels of
Figs. 6 and 8). The initial masses of these stars are between 7 and 15 M⊙.

Figure 6 shows the durations of the different Cepheid phases. As is already
well known, the first crossing is very short in all cases: It is much shorter
than the blue loop when one occurs. As a result, most observed Cepheids should
be undergoing a blue loop rather than crossing the Hertzsprung gap for the
first time. For non-rotating models, the Schwarzschild criterion in general
favours a longer duration of the Cepheid phase. The 7 and 9 M⊙ Schwarzschild
models undergo blue loops and the Ledoux models do not. Moreover, the 15 M⊙
Schwarzschild models start core helium-burning at an effective temperature of
$\log{(T_{\rm eff}[K])}\sim 4$ and thus subsequently evolve on a helium
nuclear timescale, whereas the Ledoux stars begin core helium-burning as red
supergiants and thus cross on a much shorter Kelvin-Helmholtz timescale.
For rotating models, the situation is less different between the Schwarzschild
and Ledoux models, although the case of the 9 M⊙ shows very significant
differences: the Schwarzschild model spends 110 kyr as a Cepheid, while the
Ledoux model spends only 20 kyr. We find no obvious correlation between the
overall duration of the Cepheid phase and the stellar mass.

Figure 7 shows the surface abundances of ${}^{14}\rm{N}$ before and during the
blue loop. In non-rotating models, there is no nitrogen surface enrichment
during the MS phase, and as a result, no model shows nitrogen enrichment
during the first crossing. Non-rotating models show an increase in surface
nitrogen when they go through the RSG stage, where some nitrogen is dredged up
to the surface through convection. This dredge-up occurs before the blue loop
and is thus visible during the blue loops for the 7 and 9 M⊙ models. For
rotating models, the impact of rotational mixing during the MS is apparent in
the surface ${}^{14}\rm{N}$ during the first crossing as it is larger than the
initial value for all models. Interestingly, in the nitrogen surface abundance
of the 9 M⊙ Schwarzschild models, the blue-loop value obtained for the non-
rotating model is equivalent to that given by the rotating model before the
blue loop. This shows that the surface nitrogen abundance by itself is a poor
indicator of the stellar phase (before or after the blue loop), unless its
surface velocity is measured (see more on this point below). The nitrogen
surface abundance increases with the initial mass, and it is slightly higher
for Schwarzschild than for Ledoux models. It is higher in rotating models than
in non-rotating models during the blue loop.

Figure 8 shows the surface velocities of rotating stars at the three moments
described above, as well as at the hottest point of the blue loop. In the 9 M⊙
Schwarzschild model, the surface rotation speed along the blue loop strongly
increases during blueward evolution (from 10 km/s up to nearly 100 km/s or
nearly 70% of the critical velocity). Interestingly, it decreases very fast
during the redward evolution after reaching the hottest point of the blue
loop. The strong surface acceleration arises because during the RSG phase, an
extended convective envelope is present. This convective zone rotates as a
solid body, which means that the angular momentum is highest at the outer
border of the convective zone. When the star contracts, the angular momentum
accumulated by convection in the outer layers produces a rapid acceleration.
The star loses angular momentum by winds during the blue loop, and it has a
larger radius after than before (the effective temperature is similar, but the
luminosity has increased). As a result, when it evolves back to the red, its
situation is not the same as before the blue loop: Its surface velocity is
lower. The Ledoux model shows a qualitatively similar behaviour, but the
different evolution of the outer convective zone makes the evolution of the
surface rotation slightly different.

###  4.4 Red supergiants

In this and the next section, we examine the impact of the convective
criterion on different subclasses of post-MS stars. These subclasses are
defined in Table 2.

Table 2: Definitions of stellar subtypes in populations.

Subtype | Definition
---|---
MS | $X_{c}{\textsuperscript{$\dagger$}}\leavevmode\nobreak\ \geq 10^{-5}$
Post-MS | $X_{c}<10^{-5}$
SG | Post-MS & $\log{(L\leavevmode\nobreak\ [L_{\odot}])}>4$ & $X_{s}{\textsuperscript{$\dagger$}}\leavevmode\nobreak\ >0.35$
BSG | SG & $\log{(T_{\rm eff}\leavevmode\nobreak\ [\rm{K}])}\geq 3.9$
YSG | SG & $3.66<\log{(T_{\rm eff}\leavevmode\nobreak\ [\rm{K}])}<3.9$
RSG | SG & $\log{(T_{\rm eff}\leavevmode\nobreak\ [\rm{K}])}\leq 3.66$
WR |  $\log{(T_{\rm eff}\leavevmode\nobreak\ [\rm{K}])}>4$ & $X_{s}\leq 0.35$
WNL | WR & $X_{s}\geq 10^{-5}$
WNE | WR & $X_{s}<10^{-5}$ & $C_{s}{\textsuperscript{$\dagger$}}\leavevmode\nobreak\ \leq$$N_{s}$$\dagger$
WNC | WR & $X_{s}<10^{-5}$ & $0.1\leq$ $C_{s}/N_{s}\leq 10$
WC | WR & $X_{s}<10^{-5}$ & $C_{s}$ \+ $O_{s}{\textsuperscript{$\dagger$}}\leavevmode\nobreak\ \leq$ $Y_{\rm s}{\textsuperscript{$\dagger$}}\leavevmode\nobreak\ $ (numb. frac.)
WO | WR & $X_{s}<10^{-5}$ & $C_{s}$ \+ $O_{s}$ ¿ $Y_{\rm s}$ (numb. frac.)

  * $\dagger$

$X$, $Y$, $C$, $N$, and $O$ refer to the mass fractions (unless ‘numb. frac.’
is specified, in which case they are the number fractions) of 1H, 4He, 12C,
14N, and 16O, respectively. The subscripts $c$ (centre) and $s$ (surface)
refer to where in a star these abundances are taken.

Here, we discuss whether the convective criterion affects the maximum
luminosity of red supergiants. Fig. 2 shows that the maximum luminosity
reached for red supergiants is not much affected by the convective criterion,
but it is affected by rotation. Rotating models produce an upper limit for the
RSG luminosity of about $\log{(L\leavevmode\nobreak\ [L_{\odot}])}=5.4$, while
the non-rotating models extend this upper limit to values of about
$\log{(L\leavevmode\nobreak\ [L_{\odot}])}=5.7$.

When we now compare the upper luminosity of red supergiants that remain red
until the end of their evolution and thus will be progenitors of a type II SN,
then the upper limits are lower than those in the previous paragraph because
mass loss by stellar winds induces a blueward evolution and thus produces
yellow or blue progenitors in the higher-luminosity range (see Sect. 4.5).
This upper limit in luminosity for red supergiant core-collapse progenitors is
about $\log{(L\leavevmode\nobreak\ [L_{\odot}])}=5.2$ for the non-rotating
Schwarzschild models, $\log{(L\leavevmode\nobreak\ [L_{\odot}])}=4.95$ for the
non-rotating Ledoux models, $\log{(L\leavevmode\nobreak\ [L_{\odot}])}=5.1$
for rotating Schwarzschild models, and $\log{(L\leavevmode\nobreak\
[L_{\odot}])}=5.05$ for the rotating Ledoux models. We note that the models
that spend a larger fraction of their core He-burning phase in the red
supergiant phase have slightly lower upper limits. This is expected because
their mass is reduced by strong RSG winds for a longer duration. This tends to
lower the minimum mass above which stars later evolve blueward. For instance,
in the non-rotating case (with the largest difference in maximum RSG
luminosity), the most luminous Schwarzschild RSG has an initial mass of 20 M⊙
while for the Ledoux RSG, the initial mass is 15 M⊙ because the 20 M⊙ model
becomes blue after its mass-loss episode.

###  4.5 Yellow and blue supergiants, luminous blue variables, and Wolf-Rayet
stars

Table 3: Properties of the stellar models at the end of the H-, He-, and C-
burning phases.

| End of H-burning | End of He-burning | End of C-burning
---|---|---|---
Crit. | M${}_{\text{ini}}$ | $\upsilon_{\text{ini}}$ | $\upsilon_{\text{ini}}/\upsilon_{\text{crit}}$ | $\bar{v}_{\text{MS}}$ | $\tau_{\text{H}}$ | M | $\upsilon_{\text{surf}}$ | $Y_{\text{surf}}$ | $\text{N}/\text{C}$ | $\text{N}/\text{O}$ | $\tau_{\text{He}}$ | M | $\upsilon_{\text{surf}}$ | $Y_{\text{surf}}$ | $\text{N}/\text{C}$ | $\text{N}/\text{O}$ | $\tau_{\text{C}}$ | M | $\upsilon_{\text{surf}}$ | $Y_{\text{surf}}$ | $\text{N}/\text{C}$ | $\text{N}/\text{O}$
| $M_{\odot}$ |  | km s-1 | Myr | $M_{\odot}$ | km s-1 | mass fract. | Myr | $M_{\odot}$ | km s-1 | mass fract. | kyr | $M_{\odot}$ | km s-1 | mass fract.
Led. | $120$ | $0.4$ | $397$ | $120$ | $3.140$ | $64$ | $3.5$ | $0.93$ | $78.11$ | $86.75$ | $0.305$ | $30.9$ | $10.1$ | $0.24$ | 0 | 0 | $0.010$ | $30.8$ | $21.7$ | $0.24$ | 0 | 0
Sch. |  | $0.4$ | $389$ | $110$ | $3.178$ | $34.7$ | $0.6$ | $0.97$ | $70.21$ | $92.04$ | $0.350$ | $19.1$ | $2$ | $0.27$ | 0 | 0 | $0.030$ | $19$ | 0 | $0.26$ | 0 | 0
Led. |  | – | – | – | $2.664$ | $63.3$ | – | $0.78$ | $95.88$ | $77.06$ | $0.311$ | $28.7$ | – | $0.23$ | 0 | 0 | $0.012$ | $28.6$ | – | $0.23$ | 0 | 0
Sch. |  | – | – | – | $2.672$ | $63.7$ | – | $0.78$ | $94.93$ | $77.28$ | $0.307$ | $31$ | – | $0.24$ | 0 | 0 | $0.008$ | $30.9$ | – | $0.24$ | 0 | 0
Led. | $85$ | $0.4$ | $372$ | $133$ | $3.698$ | $43.4$ | $6.1$ | $0.97$ | $68.74$ | $92.14$ | $0.330$ | $22.8$ | $12.4$ | $0.26$ | 0 | 0 | $0.028$ | $22.7$ | $24.9$ | $0.25$ | 0 | 0
Sch. |  | $0.4$ | $368$ | $124$ | $3.715$ | $49.4$ | $2.8$ | $0.93$ | $81.25$ | $84.61$ | $0.319$ | $26.5$ | $10.6$ | $0.27$ | 0 | 0 | $0.013$ | $26.4$ | $22.7$ | $0.26$ | 0 | 0
Led. |  | – | – | – | $3.014$ | $49.6$ | – | $0.61$ | $114.15$ | $69.54$ | $0.345$ | $19.1$ | – | $0.25$ | 0 | 0 | $0.036$ | $19$ | – | $0.25$ | 0 | 0
Sch. |  | – | – | – | $3.025$ | $49.2$ | – | $0.62$ | $113.02$ | $69.92$ | $0.348$ | $18.7$ | – | $0.25$ | 0 | 0 | $0.031$ | $18.6$ | – | $0.25$ | 0 | 0
Led. | $60$ | $0.4$ | $346$ | $149$ | $4.393$ | $39.1$ | $6.7$ | $0.78$ | $69.84$ | $37.89$ | $0.361$ | $16.9$ | $30.4$ | $0.27$ | 0 | 0 | $0.047$ | $16.8$ | $52.1$ | $0.26$ | 0 | 0
Sch. |  | $0.4$ | $346$ | $138$ | $4.466$ | $38.5$ | $4.3$ | $0.82$ | $78.76$ | $49.34$ | $0.355$ | $18.1$ | $31.2$ | $0.28$ | 0 | 0 | $0.037$ | $18$ | $24.3$ | $0.28$ | 0 | 0
Led. |  | – | – | – | $3.521$ | $36.4$ | – | $0.48$ | $147.38$ | $59.45$ | $0.383$ | $14$ | – | $0.27$ | 0 | 0 | $0.083$ | $13.9$ | – | $0.27$ | 0 | 0
Sch. |  | – | – | – | $3.531$ | $36.3$ | – | $0.49$ | $145.23$ | $60.67$ | $0.396$ | $12.6$ | – | $0.28$ | 0 | 0 | $0.114$ | $12.5$ | – | $0.27$ | 0 | 0
Led. | $40$ | $0.4$ | $314$ | $168$ | $5.769$ | $31.8$ | $4.3$ | $0.56$ | $13.40$ | $4.61$ | $0.417$ | $15.5$ | $62.9$ | $0.23$ | 0 | $0.001$ | $0.061$ | $15.4$ | 0 | $0.22$ | 0 | 0
Sch. |  | $0.4$ | $314$ | $166$ | $5.699$ | $32$ | $19.4$ | $0.55$ | $12.73$ | $4.31$ | $0.423$ | $12.4$ | $50.5$ | $0.28$ | 0 | 0 | $0.123$ | $12.3$ | $56.8$ | $0.28$ | 0 | 0
Led. |  | – | – | – | $4.433$ | $36.5$ | – | $0.27$ | $0.29$ | $0.12$ | $0.472$ | $13$ | – | $0.16$ | 0 | 0 | $0.087$ | $12.9$ | – | $0.16$ | 0 | 0
Sch. |  | – | – | – | $4.441$ | $36.5$ | – | $0.27$ | $0.29$ | $0.12$ | $0.472$ | $13.9$ | – | $0.932$ | $140.20$ | $70.80$ | $0.069$ | $13.7$ | – | $0.99$ | $85.60$ | $88.30$
Led. | $32$ | $0.4$ | $305$ | $188$ | $6.701$ | $28.1$ | $15.9$ | $0.45$ | $6.07$ | $1.89$ | $0.484$ | $14.3$ | 0 | $0.70$ | $159.09$ | $16.49$ | $0.117$ | $14$ | 0 | $0.94$ | $97.23$ | $62.10$
Sch. |  | $0.4$ | $306$ | $187$ | $6.640$ | $28.2$ | $12.7$ | $0.44$ | $5.84$ | $1.81$ | $0.521$ | $10.2$ | $52.8$ | $0.29$ | $0.01$ | $0.01$ | $0.213$ | $10.1$ | 0 | $0.28$ | $0.01$ | $0.01$
Led. |  | – | – | – | $5.202$ | $30.1$ | – | $0.27$ | $0.29$ | $0.12$ | $0.536$ | $11.5$ | – | $0.99$ | $71.25$ | $90.27$ | $0.135$ | $11.3$ | – | $0.99$ | $51.56$ | $90.04$
Sch. |  | – | – | – | $5.209$ | $30.1$ | – | $0.27$ | $0.29$ | $0.12$ | $0.542$ | $11.2$ | – | $0.63$ | $145.69$ | $32.95$ | $0.186$ | $10.9$ | – | $0.99$ | $56.82$ | $77.97$
Led. | $25$ | $0.4$ | $291$ | $208$ | $7.895$ | $23.6$ | $80.9$ | $0.34$ | $3.14$ | $0.86$ | $0.60$ | $10.4$ | $2$ | $0.64$ | $168.65$ | $5.19$ |  |  |  |  |  |
Sch. |  | $0.4$ | $295$ | $209$ | $7.903$ | $23.6$ | $82.2$ | $0.34$ | $3.13$ | $0.86$ | $0.618$ | $9.9$ | $0.9$ | $0.82$ | $138.02$ | $14.18$ | $0.269$ | $9.7$ | 0 | $0.93$ | $100.50$ | $25.12$
Led. |  | – | – | – | $6.324$ | $24.2$ | – | $0.27$ | $0.29$ | $0.12$ | $0.624$ | $9.5$ | – | $0.70$ | $143.82$ | $59.48$ | 0.296 | $9.3$ | – | $0.84$ | $102.67$ | $68.93$
Sch. |  | – | – | – | $6.312$ | $24.2$ | – | $0.27$ | $0.29$ | $0.12$ | $0.692$ | $8.8$ | – | $0.54$ | $195.70$ | $6.69$ | $0.433$ | $8.3$ | – | $0.83$ | $119.92$ | $72.30$
Led. | $20$ | $0.4$ | $279$ | $216$ | $9.536$ | $19.5$ | $173$ | $0.30$ | $2.35$ | $0.59$ | $0.780$ | $7.8$ | $0$ | $0.41$ | $143.10$ | $1.45$ | $0.927$ | $7.3$ | 0 | $0.49$ | $151.00$ | $6.60$
Sch. |  | $0.4$ | $274$ | $217$ | $9.507$ | $19.5$ | $175$ | $0.30$ | $2.42$ | $0.60$ | $0.85$ | $7.6$ | $0$ | $0.60$ | $106.34$ | $3.73$ | $0.866$ | $7.2$ | 0 | $0.75$ | $127.30$ | $15.88$
Led. |  | – | – | – | $7.822$ | $19.7$ | – | $0.27$ | $0.29$ | $0.12$ | $0.854$ | $7.3$ | – | $0.63$ | $151.76$ | $53.99$ | $0.911$ | $7$ | – | $0.71$ | $128.11$ | $64.66$
Sch. |  | – | – | – | $7.743$ | $19.7$ | – | $0.27$ | $0.29$ | $0.12$ | $0.873$ | $9$ | – | $0.50$ | $57.86$ | $3.89$ | $1.219$ | $8.6$ | – | $0.51$ | $81.52$ | $4.39$
Led. | $15$ | $0.4$ | $259$ | $202$ | $13.610$ | $14.7$ | $145$ | $0.29$ | $2.37$ | $0.54$ | $1.134$ | $12.5$ | $0.3$ | $0.36$ | $6.11$ | $0.99$ | $2.944$ | $12.4$ | $0.2$ | $0.36$ | $6.36$ | $1.00$
Sch. |  | $0.4$ | $271$ | $201$ | $13.449$ | $14.7$ | $160$ | $0.29$ | $2.34$ | $0.53$ | $1.450$ | $11.2$ | $0.1$ | $0.41$ | $8.31$ | $1.24$ | $1.374$ | $11.1$ | $0$ | $0.43$ | $9.52$ | $1.35$
Led. |  | – | – | – | $11.153$ | $14.8$ | – | $0.27$ | $0.29$ | $0.12$ | $1.221$ | $13.1$ | – | $0.28$ | $1.68$ | $0.45$ | $3.684$ | $13$ | – | $0.29$ | $1.84$ | $0.49$
Sch. |  | – | – | – | $11.021$ | $14.8$ | – | $0.27$ | $0.29$ | $0.12$ | $1.314$ | $13.3$ | – | $0.30$ | $2.04$ | $0.51$ | $4.79$ | $13.2$ | – | $0.34$ | $3.07$ | $0.74$
Led. | $12$ | $0.4$ | $248$ | $196$ | $18.711$ | $11.9$ | $173$ | $0.28$ | $1.93$ | $0.45$ | $1.723$ | $11.2$ | $0.5$ | $0.34$ | $6.00$ | $0.89$ | $6.434$ | $11.1$ | $0.3$ | $0.34$ | $6.41$ | $0.91$
Sch. |  | $0.4$ | $262$ | $198$ | $18.373$ | $11.9$ | $193$ | $0.28$ | $1.89$ | $0.45$ | $2.097$ | $10.3$ | $0.4$ | $0.33$ | $5.33$ | $0.83$ | $3.566$ | $10.2$ | $0.3$ | $0.35$ | $6.27$ | $0.92$
Led. |  | – | – | – | $15.395$ | $11.9$ | – | $0.27$ | $0.29$ | $0.12$ | $1.862$ | $11.1$ | – | $0.28$ | $1.58$ | $0.42$ | $7.60$ | $11.1$ | – | $0.29$ | $1.74$ | $0.46$
Sch. |  | – | – | – | $15.336$ | $11.9$ | – | $0.27$ | $0.29$ | $0.12$ | $2.123$ | $11.4$ | – | $0.30$ | $1.84$ | $0.49$ | $7.927$ | $11.3$ | – | $0.31$ | $1.98$ | $0.53$
Led. | $9$ | $0.4$ | $230$ | $185$ | $31.638$ | $9$ | $192$ | $0.27$ | $1.34$ | $0.36$ | $3.369$ | $8.8$ | $1.6$ | $0.33$ | $5.44$ | $0.82$ | $1.794$ | $8.7$ | $0.9$ | $0.33$ | $5.73$ | $0.83$
Sch. |  | $0.4$ | $248$ | $188$ | $31.216$ | $9$ | $198$ | $0.27$ | $1.40$ | $0.37$ | $3.760$ | $8.6$ | $0.6$ | $0.33$ | $5.72$ | $0.83$ | $3.143$ | $8.5$ | $0.3$ | $0.34$ | $6.10$ | $0.86$
Led. |  | – | – | – | $26.268$ | $9$ | – | $0.27$ | $0.29$ | $0.12$ | $3.404$ | $8.7$ | – | $0.27$ | $1.03$ | $0.27$ |  |  |  |  |  |
Sch. |  | – | – | – | $26.274$ | $9$ | – | $0.27$ | $0.29$ | $0.12$ | $3.490$ | $8.8$ | – | $0.28$ | $1.61$ | $0.42$ |  |  |  |  |  |
Led. | $7$ | $0.4$ | $219$ | $175$ | $51.514$ | $7$ | $184$ | $0.27$ | $0.94$ | $0.28$ | $6.505$ | $6.9$ | $2.1$ | $0.32$ | $4.57$ | $0.75$ |  |  |  |  |  |
Sch. |  | $0.4$ | $235$ | $178$ | $51.001$ | $7$ | $188$ | $0.27$ | $0.97$ | $0.29$ | $6.901$ | $6.9$ | $2$ | $0.32$ | $4.68$ | $0.76$ |  |  |  |  |  |
Led. |  | – | – | – | $43.078$ | $7$ | – | $0.27$ | $0.29$ | $0.12$ | $6.506$ | $6.9$ | – | $0.27$ | $1.05$ | $0.28$ |  |  |  |  |  |
Sch. |  | – | – | – | $41.745$ | $7$ | – | $0.27$ | $0.29$ | $0.12$ | $6.914$ | $6.9$ | – | $0.28$ | $1.52$ | $0.40$ |  |  |  |  |  |

The mass domain between 20 and 40 M⊙ is a transition domain between the stars
that evolve into the red supergiant stage and remain a red supergiant until
the end of their lifetimes and the stars, for instance the 60 M⊙ models, that
never evolve into a RSG, but become a luminous blue variable before evolving
into a WR phase. In this transition mass domain, the star can end its
evolution as a red, yellow, or blue supergiant, or become a Wolf-Rayet star
(in the latter case after having been a red supergiant for a while). The
tracks show a complex behaviour resulting from intricate effects involving
convection, rotational mixing, and mass loss.

Below an initial mass M${}_{\rm RSG}\sim 15\,{\rm M}_{\odot}$, stars end their
lives as red supergiants (see Sect. 4.4). Stars initially between MRSG and
M${}_{\rm RSG-WR}\sim 20-25\,{\rm M}_{\odot}$ end their lives as a yellow or
blue supergiants. For initial masses between MRSG-WR and M${}_{\rm WR}\sim
40\,{\rm M}_{\odot}$, stars end their lives as Wolf-Rayet stars after a red
supergiant stage. We note that this mass range of stars that experience both
an RSG and a WR phase is expected to be narrow because at the moment, apart
from Westerlund-1 (see e.g. Clark et al., 2005; Crowther et al., 2006;
Negueruela et al., 2010; Beasor et al., 2021, we present isochrones of our
models compared with observed data of massive evolved stars in Westerlund-1 in
Sect. 7.2), there is no single-age stellar population in which both red
supergiants and Wolf-Rayet stars are observed at the same time. Finally, stars
with initial masses above MWR enter the WR phase without previously becoming a
RSG.

These limits appear to be more sensitive to rotation than to the convective
criterion (at least for the initial rotation speed considered here). In
general, rotation tends to decrease the limits MRSG-WR and MWR (see more
below). Because MWR is shifted to lower values than MRSG-WR in rotating
models, this implies that rotation suppresses or at least disfavours the
production of WR stars with RSG progenitors (at least for a single-star
evolution).

Mass loss by stellar winds (which increases when the luminosity increases and
the effective temperature decreases) begins to become a dominant feature in
this mass domain (it becomes an even more important feature for higher initial
masses). As a numerical example, the 20 M⊙ models lose significant amounts of
mass during the post-MS phase (see Table 3), ending core He-burning with a
total mass of about 7-9 M⊙ (for comparison, the 15 M⊙ models reach the end of
He-burning at around 11-13 M⊙). The masses of the 25 M⊙ models at the end of
He-burning are about 9-10 M⊙ (this is higher than for the 20 M⊙ models, but
more mass has been lost by the 25 M⊙ models). The 32 and 40 M⊙ models
experience even stronger mass loss (losing up to 22 and 28 M⊙ for the 32 and
40 M⊙ models, respectively). They all end core He-burning at high
($\log{T_{\rm eff}\leavevmode\nobreak\ \rm{[K]}}>4.4$) effective temperatures,
and some of them lose so much mass that they become Wolf-Rayet stars.

While all the stars including and above 20 M⊙ lose large amounts of mass, the
mass of the core remains an increasing function of the initial mass. In other
words, the 20 M⊙ models reach the end of He-burning with a lower total mass
than the 15 M⊙ models, but their CO cores are more massive (3.65-4.44 M⊙
compared with 2.11-2.66 M⊙). Furthermore, the models between 20 and 40 M⊙
experience much stronger surface nitrogen enrichment than those below 20 M⊙.

Figure 9: Bar plot showing the durations of the different Wolf-Rayet phases
for masses at which at least one of the four (Ledoux, Schwarzschild, rotating,
and non-rotating) models has a Wolf-Rayet phase. Left panel: Non-rotating
models. Right panel: Rotating models.

The mass domain above this transition mass range is dominated by mass loss.
Not many differences are therefore visible between Schwarzschild and Ledoux
models. The most noticeable difference occurs for the 120 $M_{\odot}$ models,
as discussed in Sect. 3.

Figure 9 shows the durations of the different Wolf-Rayet phases (stacked in
chronological order) for the masses where at least one of the four models
reaches a WR phase. A first striking difference appears between the rotating
and non-rotating models. Rotation produces longer WR phases. This is in line
with previous studies (Meynet & Maeder, 2003; Georgy et al., 2012). The time
spent as WR stars is longer for rotating than for non-rotating models because
rotation drives additional mass loss and mixing. More mixing implies that less
mass loss is required to uncover hydrogen-poor layers. Rotation also favours
the WNL phase, decreases WNE and WC durations, and prevents the WO phase.
Interestingly, rotation increases the WNC-phase duration.

Above and including 60 M⊙ the Schwarzschild and Ledoux models spend similar
times (within 6%) as Wolf-Rayet stars because mass loss is the dominating
effect here. This agrees with the fact that the differences between Ledoux and
Schwarzschild models are mostly due to the ICZ. They do not appear in the most
massive stars due to the strong mass loss. As a result, the choice of
criterion for convection intervenes as a second-order effect.

For the mass range between 25 and 40 M⊙, the differences induced by changing
from the Ledoux to Schwarzschild criterion are more marked. Without rotation,
the Ledoux criterion produces Wolf-Rayet stars for lower initial masses than
the Schwarzschild criterion. These Ledoux models spend a longer time of their
core He-burning phase as an RSG than the Schwarzschild models. As a result,
mass loss is stronger for the former, and they reach the WR phase earlier (as
a result, the WR phase lasts longer).

##  5 Final properties of the models

Table 4: Properties of all the stars at the last computed model, remnant types
and masses, and averaged mass fractions of different elements above the
remnant mass.

Crit. | M${}_{\text{ini}}$ | ${\frac{\upsilon_{\text{ini}}}{\upsilon_{\text{crit}}}}$ | M${}_{\text{fin}}$ | $\log{T_{\text{eff, fin}}}$ | $\log{L_{\text{fin}}}$ | Remnant |  1H |  4He |  12C |  16O |  $\mathcal{Z}$$\dagger$
---|---|---|---|---|---|---|---|---|---|---|---
| $M_{\odot}$ |  | M⊙ | K | $L_{\odot}$ | Type | M⊙ | mass frac.
Led. | 120 | 0.4 | 30.8 | 5.4 | 6.2 | BH | 30.8 | 0.00 | 0.11 | 0.343 | 0.494 | 0.049
Sch. | 120 | 0.4 | 19.0 | 5.4 | 5.9 | BH | 19.0 | 0.00 | 0.12 | 0.395 | 0.418 | 0.064
Led. | 120 | – | 28.6 | 5.3 | 6.2 | BH | 28.6 | 0.00 | 0.12 | 0.358 | 0.469 | 0.051
Sch. | 120 | – | 30.9 | 5.3 | 6.2 | BH | 30.9 | 0.00 | 0.12 | 0.346 | 0.469 | 0.061
Led. | 85 | 0.4 | 22.7 | 5.4 | 6.0 | BH | 22.7 | 0.00 | 0.11 | 0.374 | 0.430 | 0.082
Sch. | 85 | 0.4 | 26.4 | 5.4 | 6.1 | BH | 26.4 | 0.00 | 0.12 | 0.367 | 0.451 | 0.058
Led. | 85 | – | 19.0 | 5.4 | 6.0 | BH | 19.0 | 0.00 | 0.12 | 0.389 | 0.424 | 0.066
Sch. | 85 | – | 18.6 | 5.4 | 6.0 | BH | 18.6 | 0.00 | 0.12 | 0.391 | 0.426 | 0.065
Led. | 60 | 0.4 | 16.8 | 5.4 | 5.9 | BH | 16.8 | 0.00 | 0.12 | 0.394 | 0.428 | 0.060
Sch. | 60 | 0.4 | 18.0 | 5.4 | 5.9 | BH | 18.0 | 0.00 | 0.12 | 0.396 | 0.418 | 0.064
Led. | 60 | – | 13.9 | 5.4 | 5.8 | BH | 13.9 | 0.00 | 0.13 | 0.411 | 0.406 | 0.057
Sch. | 60 | – | 12.5 | 5.3 | 5.7 | BH | 12.5 | 0.00 | 0.12 | 0.413 | 0.401 | 0.061
Led. | 40 | 0.4 | 15.4 | 5.4 | 5.8 | BH | 15.4 | 0.00 | 0.10 | 0.389 | 0.458 | 0.051
Sch. | 40 | 0.4 | 12.3 | 5.3 | 5.6 | BH | 12.3 | 0.00 | 0.12 | 0.411 | 0.420 | 0.045
Led. | 40 | – | 12.9 | 5.3 | 5.7 | BH | 12.9 | 0.00 | 0.09 | 0.370 | 0.478 | 0.061
Sch. | 40 | – | 13.7 | 4.6 | 5.7 | BH | 13.7 | 0.00 | 0.56 | 0.165 | 0.204 | 0.073
Led. | 32 | 0.4 | 14.0 | 4.5 | 5.7 | BH | 14.0 | 0.00 | 0.53 | 0.207 | 0.225 | 0.031
Sch. | 32 | 0.4 | 10.1 | 5.3 | 5.5 | BH | 10.1 | 0.00 | 0.12 | 0.403 | 0.441 | 0.032
Led. | 32 | – | 11.3 | 4.6 | 5.6 | BH | 11.3 | 0.00 | 0.58 | 0.150 | 0.210 | 0.058
Sch. | 32 | – | 10.9 | 4.6 | 5.6 | BH | 10.9 | 0.00 | 0.71 | 0.112 | 0.119 | 0.058
Led. | 25 | 0.4 | 10.4 | 4.3 | 5.4 | BH | 10.4 | 0.02 | 0.58 | 0.183 | 0.188 | 0.030
Sch. | 25 | 0.4 | 9.7 | 4.4 | 5.5 | NS | 1.57 | 0.00 | 0.20 | 0.209 | 0.525 | 0.070
Led. | 25 | – | 9.3 | 4.4 | 5.5 | BH | 9.3 | 0.01 | 0.74 | 0.106 | 0.105 | 0.042
Sch. | 25 | – | 8.3 | 4.4 | 5.4 | NS | 1.55 | 0.00 | 0.33 | 0.171 | 0.417 | 0.081
Led. | 20 | 0.4 | 7.3 | 3.8 | 5.3 | NS | 1.65 | 0.00 | 0.30 | 0.219 | 0.454 | 0.028
Sch. | 20 | 0.4 | 7.2 | 4.3 | 5.3 | NS | 1.65 | 0.00 | 0.29 | 0.165 | 0.427 | 0.113
Led. | 20 | – | 7.0 | 4.0 | 5.2 | NS | 1.51 | 0.01 | 0.37 | 0.134 | 0.351 | 0.141
Sch. | 20 | – | 8.6 | 3.6 | 5.2 | NS | 1.77 | 0.17 | 0.48 | 0.072 | 0.174 | 0.105
Led. | 15 | 0.4 | 12.4 | 3.6 | 5.1 | NS | 1.55 | 0.42 | 0.40 | 0.047 | 0.100 | 0.036
Sch. | 15 | 0.4 | 11.1 | 3.6 | 5.1 | BH | 11.1 | 0.39 | 0.45 | 0.045 | 0.094 | 0.019
Led. | 15 | – | 13.0 | 3.6 | 4.9 | NS | 1.4 | 0.51 | 0.36 | 0.032 | 0.065 | 0.037
Sch. | 15 | – | 13.2 | 3.6 | 4.9 | NS | 1.4 | 0.49 | 0.41 | 0.025 | 0.045 | 0.033
Led. | 12 | 0.4 | 11.1 | 3.6 | 4.8 | NS | 1.4 | 0.49 | 0.42 | 0.025 | 0.051 | 0.021
Sch. | 12 | 0.4 | 10.2 | 3.6 | 4.9 | NS | 1.4 | 0.46 | 0.38 | 0.041 | 0.102 | 0.026
Led. | 12 | – | 11.1 | 3.5 | 4.7 | NS | 1.4 | 0.56 | 0.37 | 0.018 | 0.030 | 0.017
Sch. | 12 | – | 11.3 | 3.6 | 4.7 | NS | 1.4 | 0.56 | 0.37 | 0.016 | 0.031 | 0.019
Led. | 9 | 0.4 | 8.7 | 3.6 | 4.3 | WD | 0.93 | 0.52 | 0.39 | 0.016 | 0.039 | 0.031
Sch. | 9 | 0.4 | 8.5 | 3.5 | 4.6 | WD | 1.18 | 0.48 | 0.41 | 0.024 | 0.050 | 0.03
Led. | 9 | – | 8.7 | 3.6 | 4.1 | WD | 0.82 | 0.58 | 0.36 | 0.022 | 0.026 | 0.011
Sch. | 9 | – | 8.8 | 3.6 | 3.9 | WD | 0.83 | 0.58 | 0.36 | 0.007 | 0.020 | 0.034
Led. | 7 | 0.4 | 6.9 | 3.6 | 3.7 | WD | 0.60 | 0.54 | 0.38 | 0.026 | 0.048 | 0.013
Sch. | 7 | 0.4 | 6.9 | 3.6 | 3.8 | WD | 0.61 | 0.54 | 0.36 | 0.029 | 0.058 | 0.013
Led. | 7 | – | 6.9 | 3.6 | 3.8 | WD | 0.54 | 0.59 | 0.35 | 0.027 | 0.029 | 0.011
Sch. | 7 | – | 6.9 | 3.6 | 3.5 | WD | 0.57 | 0.59 | 0.33 | 0.028 | 0.035 | 0.014

  * •

Notes: We took the stellar properties at the end of core C-burning for the
rotating 9 M⊙ models and all models 12 M⊙ and above, at the end of core He-
burning for the 7 M⊙ and non-rotating 9 M⊙ models. In the case of direct
collapse to a black hole, we give the averaged mass fractions of elements
above the mass of the CO core.

  * $\dagger$

Usually, Z refers to metallicity: $Z=1-X-Y$, where $X$ and $Y$ are the mass
fractions of 1H and 4He, respectively. Because we refer to metallicity
excluding 12C and 16O, we used the calligraphic
$\mathcal{Z}=1-X-Y-X(^{12}\text{C})-X(^{16}\text{O})$.

Figure 10: Mass of the ${}^{4}\text{He}$ core (left panel) and mass ratio of
the He and CO cores (right panel) at the last computed stage as a function of
the initial mass for the different models. Stars (blue: Schwarzschild, and
red: Ledoux) represent non-rotating models, and crosses (green: Schwarzschild,
and orange: Ledoux) show rotating models.

###  5.1 Core masses

Figure 10 shows the CO-core masses and the mass ratios of the He and CO core
as functions of the initial mass for the four sets of models at the end of
their evolution. The masses of the cores are defined as in Table 1 (here, we
discuss their values at the last computed stage, however). The ratio of the
mass of the He core and that of the CO core decreases from 2 to roughly 1.3
between 7 and 32 M⊙ and then remains constant around 1.2-1.3, indicating that
the He-burning shell sits relatively farther out from the C-burning core in
lower-mass models than in higher-mass models.

The differences among models of the same mass in the left panel of Fig. 10
show that rotating models have a 10% larger core on average (except for the
rotating 32, 40, and 120 M⊙ Schwarzschild models) than non-rotating models.
There is no clear trend in the differences between Schwarzschild and Ledoux
models, however. The impact of rotation is stronger than that of changing the
convective criterion.

The strongest impact of changing from the Ledoux to Schwarzschild criterion
appears for the 7 and 9 M⊙ stellar model. The ratio of He- to CO-core mass can
vary from 1 to 2, depending on the criterion. This mass domain covers the
transitions between stars that would produce white dwarfs and those that would
produce neutron stars at the end of their lifetimes. This transition has been
studied for instance by Siess (2006, 2007). This final fate depends on whether
ignition of carbon occurs in degenerate, mildly degenerate, or non-degenerate
conditions. A star whose central region is for a large part in mildly
degenerate conditions can still succeed in igniting (off-centre) carbon, but
this requires a more massive CO core than in non-degenerate conditions. It is
therefore expected that a larger part of the central regions for models with
the lowest ratio of the He- to the CO-core mass lies in the degenerate domain.
Only models that ignite carbon in non-degenerate or very mildly degenerate
conditions (i.e. the rotating 9 M⊙ models) show high ratios of the He to CO
core.

###  5.2 Properties of the compact object progenitors

We list integrated chemical abundances (in mass fractions) of 1H, 4He, 12C,
and 16O in the models at the end of core C-burning (core He-burning for stars
that do not reach the end of C-burning) in Table 4. This moment is close
enough to the end of evolution that the abundances of the isotopes shown in
the ejecta will not change dramatically (see e.g. the Appendix Table in
Limongi & Chieffi, 2018). Furthermore, explosive nucleosynthesis during the
supernova is not expected to significantly affect the yields of these four
elements. We computed the remnant types and masses using the models from
Patton & Sukhbold (2020), whose predictions concerning the remnant type
(neutron star or black hole) were based on the ‘Ertl criterion’ (Ertl et al.,
2016, 2020). For each star, we took the mass of the CO core $M_{\rm CO}$
(defined, like in Table 1, as the region within which the mass fraction of
helium $Y<10^{-2}$) and the mass fraction of 12C in the core $X_{\rm{}^{12}C}$
at the end of helium-burning. When $M_{\rm CO}<1.4\,M_{\odot}$, then we
considered the remnant to be a white dwarf with a mass of $M_{\rm WD}=M_{\rm
CO}$. When $1.4\,M_{\odot}<M_{\rm CO}<2.5\,M_{\odot}$, the remnant is a
neutron star with a mass of $M_{\rm NS}=1.4\,M_{\odot}$. When
$2.5\,M_{\odot}<M_{\rm CO}<10\,M_{\odot}$, we consulted the top panel of Fig.
3 of Patton & Sukhbold (2020) to determine whether the star will explode or
implode (without ejecting anything). If it explodes, then we estimated the
baryonic333We do not provide the gravitational masses of the neutron stars.
One can compute them using the equation of state of their choice. One example
for a relation between baryonic and gravitational mass can be found in Eq.
(39) of Griffiths et al. (2022). Typically the gravitational mass of a neutron
star will be 10-20% smaller than its baryonic mass. mass of the resulting
neutron star to be $M_{\rm NS}=M_{4}$, where $M_{4}$ is defined in Patton &
Sukhbold (2020, we use their Table 1 which provides the values of $M_{4}$ for
a range of $(M_{\rm CO},X_{\rm{}^{12}C})$ at the end of core helium-burning).
If it implodes, then the mass of the resulting black hole is $M_{\rm
BH}=M_{\rm fin}$. Finally, if $10\,M_{\odot}<M_{\rm CO}<30\,M_{\odot}$, the
outcome is a black hole, with $M_{\rm BH}=M_{\rm fin}$. This equality between
the black hole mass and the final mass of the star may be an overestimation,
but the underlying assumption is that the entire star collapses into the black
hole and that no matter is ejected. Any value between $M_{\rm CO}$ and $M_{\rm
fin}$ would be a reasonable estimate for $M_{\rm BH}$. The case $M_{\rm
CO}>30\,M_{\odot}$ could yield a pulsational or regular pair-instability
supernova, but this applies to none of our stars. We then integrated the
abundances of 1H, 4He, 12C, and 16O above the remnant (between its mass
coordinate and the surface). If the remnant is a black hole, we list the
integrated chemical abundances above the mass coordinate of the CO core. We
grouped all other elements into the quantity $\mathcal{Z}$ (usually, the
metallicity $Z$ includes 12C and 16O, and therefore, we used the calligraphic
$\mathcal{Z}=1-X-Y-X(^{12}\text{C})-X(^{16}\text{O})$). We also provide the
mass, effective temperature, and luminosity of the final models.

Using the criteria defined above on $M_{\rm CO}$, as well as the grid of CO
cores with varying 12C core mass fractions computed by Patton & Sukhbold
(2020), we can predict the type of compact object that would remain after the
end of stellar evolution. All of the 7 and 9 M⊙ models would yield white
dwarfs (WD in Table 4). Of the 12, 15, and 20 M⊙ models, the rotating 15 M⊙
Schwarzschild model would become a black hole, while all the others would
become neutron stars. In the 25 M⊙ stars, the Ledoux models become black holes
and the Schwarzschild models become neutron stars. All stars 32 M⊙ and above
become black holes.

The composition of the envelopes shows that, as expected for stars that reach
the WR phase, the models above 20 M⊙ do not have any 1H at the end of their
evolution. More generally, the integrated abundances of both 1H and 4He
decrease with increasing initial mass.

Conversely, the quantity of metals (including 12C and 16O) in the envelope is
an increasing function of the initial mass. We find no striking consistent
difference between the Schwarzschild and Ledoux models. The differences mostly
concern the integrated 12C and 16O abundances: in some cases, one model
produces more 12C and less 16O than its counterpart, but there is no clear-cut
effect.

The case of the 25 M⊙ models is interesting because the two Ledoux models are
predicted to become black holes and the Schwarzschild models to become neutron
stars. There is a striking difference in the compositions of their envelopes:
the former are much richer in 4He ($Y_{\rm env}$ of 0.58 and 0.74 against 0.20
and 0.33 for the Schwarzschild models). The latter, conversely, are much
richer in 12C and 16O (with the most marked differences being in the 16O
abundance: $X(^{16}\rm{O})_{\rm env}$ of 0.525 and 0.417 against 0.188 and
0.105 for the Ledoux models).

In the 32 and 40 M⊙ models, the rotating 32 M⊙ and non-rotating 40 M⊙
Schwarzschild models present very different 12C and 16O abundances from the
other models of the same mass. The former is much richer in 12C and 16O (and
thus poorer in 4He) than the other 32 M⊙ models; the latter is much poorer in
12C and 16O (and thus richer in 4He) than the other 40 M⊙ models.

From the 1H and 4He content of the models at the last computed stage, we can
infer the type of core-collapse supernova that is expected to occur for all
the exploding models. All the models of 12 and 15 M⊙ contain significant
amounts of 1H (about 50% of the envelope mass), and therefore, we expect type
II SNe from them (except for the rotating Schwarzschild 15 M⊙ model, which we
expect to directly collapse into a black hole). Of the 20 M⊙ stars, only the
non-rotating Schwarzschild model is expected to lead to a type II SN because
it still contains a non-negligible amount of 1H. We expect the other 20 M⊙
models to die in type Ib SNe because they have no 1H left, but still contain
4He. The same holds for the 25 M⊙ Schwarzschild models, for which we expect an
explosion.

Finally, we theoretically predict the remnant masses. The most massive neutron
star comes from the non-rotating 20 M⊙ Schwarzschild model, with a baryonic
mass $M_{\rm NS}=1.77M_{\odot}$. The least massive black hole results from the
non-rotating 25 M⊙ Ledoux model, with $M_{\rm BH}=9.3{\rm M}_{\odot}$.
Acknowledging that taking $M_{\rm BH}=M_{\rm fin}$ may be an overestimation of
the black hole mass, we can estimate the absolute minimum predicted black hole
mass from that of the CO cores (taking $M_{\rm BH}=M_{\rm CO}$). In this case,
the 15 M⊙ rotating Schwarzschild model yields the lowest-mass black hole, with
$M_{\rm BH}=2.68{\rm M}_{\odot}$.

##  6 Stellar population synthesis

###  6.1 The population synthesis models

The population synthesis results we present in this work were computed using
the Syclist code (Georgy et al., 2014a), using the stellar models from Ekström
et al. (2012, computed with the Schwarzschild criterion), and from Saio et al.
(2013a) and Georgy et al. (2014b, 20 and 25 M⊙ models computed with the Ledoux
criterion) as input. We added all the other Ledoux models that we computed for
the current paper in order to cover masses from 7 to 120 M⊙. Syclist generates
a stellar population with initial masses sampled according to the Salpeter
IMF, and it saves the state of the population at each requested time step. To
mimic continuous star formation rates, we summed the populations at each time
step, and we then counted stars and classified them into subtypes according to
some of their properties. The subtypes are defined in Table 2.

We generated two types of clusters. One type consisted of a burst of star
formation at $t=0$, and the other type had continuous star formation for 60
Myr. The second type allowed us to simulate a stationary state and derive
equilibrium relative abundances of stellar subtypes. We also generated
isochrones of our populations to compare them to the observed populations of
evolved massive stars in Westerlund-1.

###  6.2 Case of a burst of star formation

Figure 11: Number of stars of certain types alive at each time, normalised by
the total initial number of stars generated in a synthetic cluster for the
case of a burst of star formation at t=0. The solid lines represent stars
computed with the Schwarzschild criterion, and the dashed lines show the same
for the Ledoux criterion. Top panels: Blue and red supergiants. The vertical
axis is logarithmic, and the time axis extends to 60 Myr. Bottom panels: Wolf-
Rayet stars. The vertical axis is linear and in units of $10^{-3}$, and the
time axis extends to 12 Myr (there are no WR stars later on). Left panels:
Non-rotating models. Right panels: Rotating models.

Figure 11 shows the number of blue and red supergiants (top panels) and Wolf-
Rayet stars (bottom panels) alive at each time in the case of a burst of star
formation, normalised by the initial number of stars. For non-rotating stars,
the Ledoux clusters produce far fewer blue and many more red supergiants than
the Schwarzschild clusters. This is attributable to the different crossing
durations of the Hertzsprung gap above 15 M⊙ (the Schwarzschild models start
core He-burning as BSGs, the Ledoux models start as RSGs), and to a larger
extent (because of the IMF slope) to the blue loop behaviour, which is present
in Schwarzschild stars between 7 and 12 M⊙ but is absent in Ledoux models.

This is not so much the case for rotating stars because, as mentioned
previously, rotation tends to mitigate the differences between the two sets of
models, and the blue loop behaviour is much more similar in rotating models.
The difference in these models occurs just before 20 Myr, which is when the
Ledoux 12 M⊙ model undergoes its blue loop while the Schwarzschild model does
not.

Finally, we note that the non-rotating Wolf-Rayet populations (especially the
WNL stars) are quite different, with Ledoux clusters clearly hosting more WR
stars than Schwarzschild clusters. This is expcected because the WR stage is
reached for non-rotating Ledoux stars of lower masses (down to 25 M⊙), and it
lasts longer for these lower masses (see the left panel of Fig. 9) than in the
Schwarzschild models. WNE, WNC, and WO type stars are very rare in both types
of clusters, and the differences in WC number densities are not significant.

###  6.3 Stationary regime populations

Figure 12: Density maps in the HRD of the Schwarzschild and Ledoux non-
rotating populations in the case of continuous star formation (stationary
regime). We add the tracks of the 7, 15, 25, and 60 M⊙ models. Left:
Schwarzschild model. Right: Ledoux model. Figure 13: Density maps in the HRD
of the Schwarzschild and Ledoux rotating populations in the case of continuous
star formation (stationary regime). We add the tracks of the 7, 15, 25, and 60
M⊙ models. Left: Schwarzschild model. Right: Ledoux model. Table 5: Number
ratios of different classes of stars (stationary regime population with
continuous star formation for 60 Myr).

Ratio | $\upsilon_{\rm ini}/\upsilon_{\rm crit}$ | Schwarzschild | Ledoux
---|---|---|---
Post-MS $/$ MS | $0$ | $0.152$ | $0.144$
| $0.4$ | $0.134$ | $0.123$
BSG $/$ RSG | $0$ | $0.61$ | $0.13$
| $0.4$ | $0.42$ | $0.31$
WR $/$ RSG | $0$ | $0.045$ | $0.070$
| $0.4$ | $0.106$ | $0.096$
WNE $/$ WNL | $0$ | $6.1\times 10^{-3}$ | $2.8\times 10^{-3}$
| $0.4$ | $8.5\times 10^{-5}$ | 0
WO $/$ WC | $0$ | $0$ | $0.082$
| $0.4$ | $0$ | $0.014$
WNC $/$ WR | $0$ | $0.024$ | $0.013$
| $0.4$ | $0.022$ | $0.017$
WN $/$ WR | $0$ | $0.60$ | $0.79$
| $0.4$ | $0.81$ | $0.81$
WC $/$ WN | $0$ | $0.677$ | $0.243$
| $0.4$ | $0.235$ | $0.232$

We now discuss populations in the stationary regime. Table 5 shows the number
ratios of different classes of stars in this equilibrium state, where we
simulated populations with a constant continuous star formation episode
lasting 60 Myr. The ratios of post-MS to MS stars are very similar in
Schwarzschild and Ledoux models: we find differences of 5% (non-rotating) and
9% (rotating), and the Schwarzschild post-MS to MS ratio is higher than the
Ledoux ratio because Ledoux models have a slightly longer main sequence and
shorter helium-burning phases. The ratios of BSGs to RSGs are higher for the
Schwarzschild populations regardless of the rotation scheme, although the
difference is more marked for populations of non-rotating stars (the BSG to
RSG ratio is almost five times higher for Schwarzschild than for Ledoux
populations); this echoes what we mentioned in Sect. 6.2, and we attribute
this difference to the different blue loop behaviour and to the durations of
the Hertzsprung gap crossing.

The WR to RSG ratio is quite interesting in the non-rotating case. Ledoux
populations produce more RSGs, but also more WR stars than Schwarzschild
populations. The overproduction is higher for the WR stars, as
$\rm{WR}/\rm{RSG}_{\rm Ledoux}>\rm{WR}/\rm{RSG}_{\rm Schwarzschild}$. Finally,
while the populations produce similar numbers of WC stars (see the lower
panels of Fig. 11), the ratio of WC to WN stars is three times higher for the
non-rotating Schwarzschild population because the Ledoux population contains
more WN(L) stars.

Figures 12 (non-rotating) and 13 (rotating) show density maps in the HRD of
populations generated with continuous star formation. The densities are higher
in the lower parts of the diagram because more lower-mass stars are generated
due to the Salpeter IMF we used. The comparison of populations both with and
without rotation shows that the MS regions are the same for the Schwarzschild
and Ledoux models.

We note for the non-rotating populations that they contain a region of higher
density that corresponds to the blue loops of the Schwarzschild models, but
this region is absent from the Ledoux models. We also see differences in the
supergiant stars above $\log{(L\leavevmode\nobreak\ [L_{\odot}])}\sim 4.5$
because the 20 to 40 M⊙ models experience a blue and/or redward evolution in
diverse parts of the HRD.

For the rotating populations, the blue loop stars are present in both sets of
models, but their high-density regions extend to lower surface temperatures
for the Ledoux models ($\log{T_{\rm eff}\leavevmode\nobreak\ \rm{[K]}}\sim
3.85$) than for the Schwarzschild models ($\log{T_{\rm
eff}\leavevmode\nobreak\ \rm{[K]}}\sim 3.95$). This would not be noticeable
from the HRD tracks alone (the loops even extend farther bluewards for the
rotating Ledoux stars), which means that the Ledoux models spend less time at
the blue edge of the loop than the Schwarzschild models do. Fig. 13 shows that
the higher-density region of the blue loops corresponds to the hottest point
along the loop of 7 M⊙ Schwarzschild stars, but the equivalent point of the
Ledoux-star loop extends to a higher temperature. The high-luminosity part of
the blue-loop region reaches higher luminosities for the Ledoux models
($\log{(L\leavevmode\nobreak\ [L_{\odot}])}\sim 4.6$) than for the
Schwarzschild models ($\log{(L\leavevmode\nobreak\ [L_{\odot}])}\sim 4.3$)
because the threshold for blue loops is between 12 and 15 M⊙ for the former
and between 9 and 12 M⊙ for the latter. We also see differences in the higher
luminosity supergiants, which we explain in a similar way to those of the non-
rotating models.

##  7 Comparison with previous works and observations

###  7.1 Comparison with previous theoretical works

While this paper presents the first grid that spans such a wide mass range in
the comparison of Schwarzschild and Ledoux models at solar metallicity, with
and without rotation, it is not the first to study the impact of various
convective parameters on stellar evolution. In this section, we compare our
conclusions with a few such previous theoretical works.

Schootemeijer et al. (2019) used MESA (Paxton et al., 2011, 2013, 2015, 2018)
to compute models between 9 and 100 M⊙, with and without rotation, at SMC
metallicity ($Z=0.002$). They used the Ledoux criterion and varied parameters
related to the strength of overshooting and efficiency of semiconvective
mixing. The models relevant to our comparison are those with overshooting
parameter $\alpha_{\rm ov}=0.11$, and the most extreme values of the
semiconvection parameter $\alpha_{\rm sc}$ = 0.01 and 300 (or 100 because they
discussed more results related to that value). They found that very efficient
semiconvection (large $\alpha_{\rm sc}$, Schwarzschild in the current paper)
leads to more time being spent by stars as blue supergiants. For instance,
these models start to burn helium in the core at a higher effective
temperature than those with inefficient semiconvection. Fig. 2 (top panel for
$\alpha_{\rm ov}=0.11$) of Schootemeijer et al. (2019) shows that blue loops
occur in the lower-mass range for efficient semiconvection, but not when it is
inefficient. All of these observations lead to a higher ratio of blue to red
supergiants when semiconvective mixing is efficient. Qualitatively, this
agrees very well with what we find. The specific values for the mass range of
the blue loops, the effective temperatures at the beginning of core helium
burning, and the ratio of blue to red supergiants differ from what we predict,
but that is to be expected because their models were computed at SMC
metallicity (which is almost an order of magnitude lower than ours).

Kaiser et al. (2020) also used MESA to compute non-rotating models of 15, 20,
and 25 M⊙ at solar metallicity. They computed models with both the
Schwarzschild and Ledoux criteria and used a free parameter $f_{\rm CBM}$,
where CBM stands for convective boundary mixing, to vary the amount of
additional mixing at the convective boundary (larger $f_{\rm CBM}$ corresponds
to more mixing). This parameter intervenes in a diffusive model rather than
the penetrative overshoot that we used in this study. They found that the
main-sequence evolution is not affected by the choice of criterion for
convective stability as long as CBM is included (in our case, as long as
overshooting is included). They also discussed the intermediate convective
zones and found that ICZs that are larger and remain for longer mean that
stars continue to be blue until the ICZ recedes. Their stars computed with the
Schwarzschild criterion have stronger (meaning larger and longer-living) ICZs
than those computed with the Ledoux criterion. As a result, their
Schwarzschild stars start core helium-burning as blue supergiants and the
Ledoux stars start as red supergiants. Their 15 M⊙ Ledoux models also cross
the HRD much faster after the main sequence, and this is correlated with a
drop in surface luminosity. Overall however, their results are more affected
by the value of $f_{\rm CBM}$ than by the choice of Schwarzschild and Ledoux
criterion.

Anders et al. (2022) performed a 3D hydrodynamical simulation of a convective
zone adjacent to a semiconvective (Ledoux-stable but Schwarzschild-unstable)
region. They found that overshooting mixes the semiconvective region,
increasing the size of the convective zone by a process they called
entrainment. After a few thousand convective-overturn times, the Ledoux and
Schwarzschild criteria predict the same convective boundary. The consequence
is that as the title of Anders et al. (2022) states, ‘Schwarzschild and Ledoux
are equivalent on evolutionary timescales’. This is the case during the main
sequence (when the evolutionary timescale is much longer than the convective-
overturn timescale). However, when the two timescales are of similar order,
Ledoux predicts the instantaneous location of the boundary, whereas
Schwarzschild provides its location in a stationary state. This agrees with
what we find: that the largest differences are introduced during the post-MS
expansion, which occurs on a rapid timescale.

###  7.2 Comparison with observations

Figure 14: HRD showing isochrones at $\log{(\rm{Age\leavevmode\nobreak\
[yr]})}=6.7,\,6.8,\,6.9,\,{\rm and}\leavevmode\nobreak\ 7.0$ (corresponding to
ages of 5, 6.3, 7.9, and 10 Myr) for our four sets of grids, compared with
observations of evolved massive stars in the Westerlund-1 cluster. We use the
data from Yusof et al. (2022) for the observed stars. Left column:
Schwarzschild models. Right column: Ledoux models. Upper row: Non-rotating
models. Lower row: Rotating models.

We briefly present here a comparison of our four sets of models with the
evolved massive stars in the Westerlund-1 (Wd1) cluster. Wd1, at a distance of
$\sim$4 kpc (Beasor et al., 2021), is the best-studied young cluster in the
galactic disk, and it contains both RSG, YSG, and WR stars. Its metallicity is
unknown because the gas associated with its formation has been dispersed.
Based on its age, however, we may expect that its metallicity could be higher
than that in our study (which is why Yusof et al. (2022) used it as a
comparison for their super-solar metallicity ($Z=0.020$) models). We
nevertheless chose to also compare our models to Wd1 because it is the best-
studied cluster containing both cool supergiants and Wolf-Rayet stars. We
again used SYCLIST (Georgy et al., 2014a), this time, to generate isochrones
for our four grids of models. We used the data from Yusof et al. (2022) for
the observed evolved stars in Wd1.

Figure 14 shows isochrones for our four sets of grids and the observed data of
evolved massive stars in Wd1. While no isochrone perfectly fits the observed
distribution of stars, we find that the non-rotating Ledoux isochrone at an
age of 6.3 Myr best incorporates the presence of RSG and WR stars, as well as
the spread in luminosity of the RSGs. The 10 Myr Ledoux isochrones (both
rotating and non-rotating) fit the observed YSGs. We recall the possibility
that a cluster formation event lasted a few million years, typically between
10 and 6 Myr ago, and that real stellar populations have varied rotation
rates. Moreover, multiple star evolution and cluster dynamics will impact the
observable properties of stars and induce a scatter in their luminosities, and
this can explain some of the observed data.

##  8 Conclusions and discussion

In the above sections, we discussed the possible effect of changing the
convective criterion on stellar model outputs with and without rotation. In
many aspects, the effects appear to be relatively minor compared to those of
rotation or mass-loss. We summarise the main results from this work below.

  * •

The main-sequence phase is not much affected by the choice of the convection
criterion.

  * •

The Schwarzschild criterion does not necessarily lead to more extended
convective zones. It does facilitate the formation of an extended intermediate
convective zone at the end of the main sequence (especially for stars with
initial masses between 15 and 32 M⊙), however, which gives rise to significant
differences in the post-MS evolution of these stars.

  * •

One such difference occurs during the first crossing of the HRD, where stars
computed with the Schwarzschild criterion can start core He-burning as BSG,
whereas the Ledoux stars start as RSG. This has important implications in the
subsequent mass loss of the models. This in turn affects the ratio of blue to
red supergiants as well as the way in which case B mass transfer occurs in
close binary systems. Because the Schwarzschild criterion causes longer
crossing times, it predicts more nitrogen-poor Cepheids (those undergoing
their first crossing of the HRD) than the Ledoux criterion. Observations of
nitrogen-poor Cepheids would definitely tip the scale in favour of the
Schwarzschild criterion (or efficient semiconvection).

  * •

The occurrence of blue loops is affected by the choice of a Schwarzschild and
Ledoux criterion: non-rotating 7-9 M⊙ Ledoux stars do not have them, but
Schwarzschild stars do. This is probably due to different chemical profiles
and helium abundances near and above the H-burning shell, and it also affects
the ratio of blue to red supergiants.

  * •

The duration of and the surface velocities (for rotating stars) during the
Cepheid phase are influenced by the choice of the convection criterion. The
Schwarzschild criterion predicts longer-lasting and faster-rotating Cepheids
than the Ledoux criterion. The Ledoux criterion predicts the most massive
Cepheids.

  * •

Red supergiants computed with the Ledoux criterion reach a lower maximum
luminosity than those computed with the Schwarzschild criterion.

  * •

The impact of the criterion for convection on stars with initial masses
between 20 and 40 M⊙ is complex because of the interplay with rotation and
mass loss. For instance, the Ledoux criterion produces non-rotating WR stars
for lower initial masses than the Schwarzschild criterion, but the inverse
occurs for rotating stars.

  * •

The impact of changing the convective criterion on stars whose initial masses
are above 60 M⊙ is modest because their evolution is dominated by mass loss.

  * •

The main differences relative to the endpoint of stellar evolution occur for
stars with initial masses of 25 to 40 M⊙. For the 25 M⊙ the Ledoux criterion
predicts the final compact objects to be black holes, while the Schwarzschild
criterion predicts neutron stars. If these stars still undergo a supernova,
the Ledoux criterion predicts much stronger helium signatures than the
Schwarzschild criterion because the amount of helium in their envelope is
larger.

  * •

Our population synthesis models imply different number ratios of evolved stars
depending on the convection criterion, especially for non-rotating stars: The
Schwarzschild criterion predicts much higher ratios of BSGs to RSGs and WCs to
WNs than the Ledoux criterion. Conversely, the Ledoux criterion predicts a
higher ratio of WRs to RSGs.

We recall, however, that the comparison presented in this work was made for a
given choice of mass-loss rates, step overshoot, physics of rotation, and
metallicity, and only for single-star evolution. Without performing detailed
evolutionary calculations, it is not possible to know how the picture we
presented in this work would change when one, a few, or all of the physical
ingredients would change.

###### Acknowledgements.

The authors would like to thank the anonymous referee for their very
insightful comments which greatly improved the quality of this manuscript. The
authors have received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement No 833925, project STAREX).

## References

  * Anders et al. (2022) Anders, E. H., Jermyn, A. S., Lecoanet, D., et al. 2022, ApJ, 928, L10
  * Anderson et al. (2014) Anderson, R. I., Ekström, S., Georgy, C., et al. 2014, A&A, 564, A100
  * Arnett & Meakin (2016) Arnett, W. D. & Meakin, C. 2016, Reports on Progress in Physics, 79, 102901
  * Baraffe et al. (2023) Baraffe, I., Clarke, J., Morison, A., et al. 2023, MNRAS, 519, 5333
  * Beasor et al. (2021) Beasor, E. R., Davies, B., Smith, N., Gehrz, R. D., & Figer, D. F. 2021, The Astrophysical Journal, 912, 16
  * Böhm-Vitense (1958) Böhm-Vitense, E. 1958, ZAp, 46, 108
  * Bono et al. (2000) Bono, G., Caputo, F., Cassisi, S., et al. 2000, ApJ, 543, 955
  * Brunish & Becker (1990) Brunish, W. M. & Becker, S. A. 1990, ApJ, 351, 258
  * Chun et al. (2018) Chun, S.-H., Yoon, S.-C., Jung, M.-K., Kim, D. U., & Kim, J. 2018, ApJ, 853, 79
  * Clark et al. (2005) Clark, J., Negueruela, I., Crowther, P., & Goodwin, S. 2005, Astronomy & Astrophysics, 434, 949
  * Cristini et al. (2017) Cristini, A., Meakin, C., Hirschi, R., et al. 2017, MNRAS, 471, 279
  * Crowther (2001) Crowther, P. A. 2001, in Astrophysics and Space Science Library, Vol. 264, The Influence of Binaries on Stellar Population Studies, ed. D. Vanbeveren (Dordrecht: Kluwer Academic Publishers), 215
  * Crowther et al. (2006) Crowther, P. A., Hadfield, L., Clark, J., Negueruela, I., & Vacca, W. 2006, Monthly Notices of the Royal Astronomical Society, 372, 1407
  * de Jager et al. (1988) de Jager, C., Nieuwenhuijzen, H., & van der Hucht, K. A. 1988, A&AS, 72, 259
  * Eggenberger et al. (2008) Eggenberger, P., Meynet, G., Maeder, A., et al. 2008, Ap&SS, 316, 43
  * Ekström et al. (2012) Ekström, S., Georgy, C., Eggenberger, P., et al. 2012, A&A, 537, A146
  * Ertl et al. (2016) Ertl, T., Janka, H. T., Woosley, S. E., Sukhbold, T., & Ugliano, M. 2016, ApJ, 818, 124
  * Ertl et al. (2020) Ertl, T., Woosley, S. E., Sukhbold, T., & Janka, H. T. 2020, ApJ, 890, 51
  * Farrell et al. (2022) Farrell, E., Groh, J. H., Meynet, G., & Eldridge, J. J. 2022, MNRAS, 512, 4116
  * Freytag et al. (1996) Freytag, B., Ludwig, H.-G., & Steffen, M. 1996, A&A, 313, 497
  * Friedland et al. (2013) Friedland, A., Giannotti, M., & Wise, M. 2013, Phys. Rev. Lett., 110, 061101
  * Georgy et al. (2012) Georgy, C., Ekström, S., Meynet, G., et al. 2012, A&A, 542, A29
  * Georgy et al. (2014a) Georgy, C., Granada, A., Ekström, S., et al. 2014a, A&A, 566, A21
  * Georgy et al. (2011) Georgy, C., Meynet, G., & Maeder, A. 2011, A&A, 527, A52
  * Georgy et al. (2014b) Georgy, C., Saio, H., & Meynet, G. 2014b, MNRAS, 439, L6
  * Georgy et al. (2021) Georgy, C., Saio, H., & Meynet, G. 2021, A&A, 650, A128
  * Gräfener & Hamann (2008) Gräfener, G. & Hamann, W.-R. 2008, A&A, 482, 945
  * Griffiths et al. (2022) Griffiths, A., Eggenberger, P., Meynet, G., Moyano, F., & Aloy, M.-Á. 2022, A&A, 665, A147
  * Halabi et al. (2012) Halabi, G. M., El Eid, M. F., & Champagne, A. 2012, ApJ, 761, 10
  * Herwig et al. (2006) Herwig, F., Freytag, B., Hueckstaedt, R. M., & Timmes, F. X. 2006, ApJ, 642, 1057
  * Horst et al. (2021) Horst, L., Hirschi, R., Edelmann, P. V. F., Andrássy, R., & Röpke, F. K. 2021, A&A, 653, A55
  * Jones et al. (2017) Jones, S., Andrassy, R., Sandalski, S., et al. 2017, MNRAS, 465, 2991
  * Kaiser et al. (2020) Kaiser, E. A., Hirschi, R., Arnett, W. D., et al. 2020, MNRAS, 496, 1967
  * Kato (1966) Kato, S. 1966, PASJ, 18, 374
  * Lai & Li (2011) Lai, X.-J. & Li, Y. 2011, Research in Astronomy and Astrophysics, 11, 1351
  * Langer et al. (1983) Langer, N., Fricke, K. J., & Sugimoto, D. 1983, A&A, 126, 207
  * Lauterborn et al. (1971) Lauterborn, D., Refsdal, S., & Roth, M. L. 1971, A&A, 13, 119
  * Lawlor et al. (2015) Lawlor, T. M., Young, T. R., Teffs, J., & MacDonald, J. 2015, MNRAS, 450, 1618
  * Limongi & Chieffi (2018) Limongi, M. & Chieffi, A. 2018, ApJS, 237, 13
  * Maeder (1987) Maeder, A. 1987, A&A, 173, 247
  * Maeder (1997) Maeder, A. 1997, A&A, 321, 134
  * Maeder & Mermilliod (1981) Maeder, A. & Mermilliod, J. C. 1981, A&A, 93, 136
  * Maeder & Meynet (1989) Maeder, A. & Meynet, G. 1989, A&A, 210, 155
  * Maeder & Meynet (2000) Maeder, A. & Meynet, G. 2000, ARA&A, 38, 143
  * Maeder & Meynet (2001) Maeder, A. & Meynet, G. 2001, A&A, 373, 555
  * Martinet et al. (2021) Martinet, S., Meynet, G., Ekström, S., et al. 2021, A&A, 648, A126
  * Meakin & Arnett (2007) Meakin, C. A. & Arnett, D. 2007, ApJ, 667, 448
  * Meynet & Maeder (2003) Meynet, G. & Maeder, A. 2003, A&A, 404, 975
  * Mori et al. (2020) Mori, K., Balantekin, A. B., Kajino, T., & Famiano, M. A. 2020, ApJ, 901, 115
  * Negueruela et al. (2010) Negueruela, I., Clark, J. S., & Ritchie, B. W. 2010, A&A, 516, A78
  * Nugis & Lamers (2000) Nugis, T. & Lamers, H. J. G. L. M. 2000, A&A, 360, 227
  * Patton & Sukhbold (2020) Patton, R. A. & Sukhbold, T. 2020, MNRAS, 499, 2803
  * Paxton et al. (2011) Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3
  * Paxton et al. (2013) Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208, 4
  * Paxton et al. (2015) Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS, 220, 15
  * Paxton et al. (2018) Paxton, B., Schwab, J., Bauer, E. B., et al. 2018, ApJS, 234, 34
  * Reimers (1977) Reimers, D. 1977, A&A, 61, 217
  * Saio et al. (2013a) Saio, H., Georgy, C., & Meynet, G. 2013a, MNRAS, 433, 1246
  * Saio et al. (2013b) Saio, H., Georgy, C., & Meynet, G. 2013b, in Astronomical Society of the Pacific Conference Series, Vol. 479, Progress in Physics of the Sun and Stars: A New Era in Helio- and Asteroseismology, ed. H. Shibahashi & A. E. Lynas-Gray, 47
  * Schootemeijer et al. (2019) Schootemeijer, A., Langer, N., Grin, N. J., & Wang, C. 2019, A&A, 625, A132
  * Scott et al. (2021) Scott, L. J. A., Hirschi, R., Georgy, C., et al. 2021, MNRAS, 503, 4208
  * Siess (2006) Siess, L. 2006, A&A, 448, 717
  * Siess (2007) Siess, L. 2007, A&A, 476, 893
  * Stothers & Chin (1973) Stothers, R. & Chin, C.-W. 1973, ApJ, 179, 555
  * Stothers & Chin (1975) Stothers, R. & Chin, C. W. 1975, ApJ, 198, 407
  * Stothers & Chin (1976) Stothers, R. & Chin, C. W. 1976, ApJ, 204, 472
  * Sylvester et al. (1998) Sylvester, R. J., Skinner, C. J., & Barlow, M. J. 1998, MNRAS, 301, 1083
  * Tammann et al. (2003) Tammann, G. A., Sandage, A., & Reindl, B. 2003, A&A, 404, 423
  * van Loon et al. (1999) van Loon, J. T., Groenewegen, M. A. T., de Koter, A., et al. 1999, A&A, 351, 559
  * Viallet et al. (2015) Viallet, M., Meakin, C., Prat, V., & Arnett, D. 2015, A&A, 580, A61
  * Vink et al. (2000) Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. 2000, A&A, 362, 295
  * Vink et al. (2001) Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. 2001, A&A, 369, 574
  * Walmswell et al. (2015) Walmswell, J. J., Tout, C. A., & Eldridge, J. J. 2015, MNRAS, 447, 2951
  * Woodward et al. (2015) Woodward, P. R., Herwig, F., & Lin, P.-H. 2015, ApJ, 798, 49
  * Yusof et al. (2022) Yusof, N., Hirschi, R., Eggenberger, P., et al. 2022, MNRAS, 511, 2814
  * Zahn (1991) Zahn, J.-P. 1991, A&A, 252, 179
  * Zahn (1992) Zahn, J.-P. 1992, A&A, 265, 115
  * Zhao et al. (2023) Zhao, L., Song, H., Meynet, G., et al. 2023, A&A, 674, A92