# Long-duration Gamma-ray Burst Progenitors and Magnetar Formation

Cui-Ying Song Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai
200240, China<EMAIL_ADDRESS>Tong Liu Department of Astronomy,
Xiamen University, Xiamen 361005, China<EMAIL_ADDRESS>
###### Abstract

Millisecond magnetars produced in the center of dying massive stars are one
prominent model to power gamma-ray bursts (GRBs). However, their detailed
nature remains a mystery. To explore the effects of the initial mass, rotation
rate, wind mass loss, and metallicity on the GRB progenitors and the newborn
magnetar properties, we evolve 227 of $10-30~{}M_{\odot}$ single star models
from the pre-main-sequence to core collapse by using the stellar evolution
code MESA. The pre-supernova properties, the compactness parameter, and
magnetar characteristics of models with different initial parameters are
presented. The compactness parameter remains a non-monotonic function of the
initial mass and initial rotation rate when the effects of vary metallicity
and “Dutch” wind scale factor are taken into account. We find that the initial
rotation rate and mass play the dominant roles in whether a star can evolve
into a GRB progenitor. The minimum rotation rate necessary to generate a
magnetar gradually reduces as the initial mass increases. The greater the
initial metallicity and “Dutch” wind scale factor, the larger the minimum
rotation rate required to produce a magnetar. In other words, massive stars
with low metallicity are more likely to harbor magnetars. Furthermore, we
present the estimated period, magnetic field strength, and masses of magnetars
in all cases. The typical rotational energy of these millisecond magnetars is
sufficient to power long-duration GRBs.

Massive stars (732) - Stellar evolutionary models (2046) - Gamma-ray bursts
(629) - Magnetars (992)

††software: MESA (Paxton et al., 2011, 2013, 2015, 2018, 2019; Jermyn et al.,
2023), MESA SDK 21.4.1 (Townsend, 2021), MatPlotLib (Hunter, 2007), NASA ADS,
py$\\_$mesa$\\_$reader, NumPy (van der Walt et al., 2011), and Pandas (Reback
et al., 2022).

##  1 Introduction

Observations of long-duration gamma-ray bursts (LGRBs) associated with core-
collapse supernovae (CCSNe, e.g., Galama et al., 1998; Bloom et al., 1999;
Hjorth et al., 2003; Stanek et al., 2003) suggest that some fraction of LGRBs
originate from the death of massive stars (see reviews by Piran, 2004; Woosley
& Bloom, 2006; Kumar & Zhang, 2015; Cano et al., 2017). After the massive star
collapses, a black hole (BH) hyperaccretion system (e.g., Woosley, 1993;
MacFadyen & Woosley, 1999; Liu et al., 2017) or a rapidly rotating neutron
star (NS) with a strong magnetic field (magnetar, e.g., Usov, 1992; Duncan &
Thompson, 1992; Wheeler et al., 2000; Zhang & Mészáros, 2001) might be formed.
In the BH hyperaccretion scenario, the annihilation of neutrinos and anti-
neutrinos escaped from neutrino-dominated accretion disks (e.g., Popham et
al., 1999; Narayan et al., 2001; Chen & Beloborodov, 2007; Gu et al., 2006;
Liu et al., 2007, 2015; Zalamea & Beloborodov, 2011; Kawanaka et al., 2013,
see a review by Liu et al. 2017) or the large-scale magnetic fields extract
the rotational energy of BHs, i.e., Blandford-Znajek mechanism, ( e.g.,
Blandford & Znajek, 1977; Lee et al., 2000; Wang et al., 2002; McKinney, 2005;
McKinney et al., 2012; Komissarov & Barkov, 2009; Tchekhovskoy et al., 2011;
Lei et al., 2013; Liu et al., 2015) to launch ultrarelativistic jets. The
release energy from the spin-down of newborn magnetars could also account for
gamma-ray bursts (GRBs, e.g., Usov, 1992; Wheeler et al., 2000; Metzger et
al., 2011, 2017; Rowlinson et al., 2013; Kaspi & Beloborodov, 2017; Hou et
al., 2021). The collimated jet production mechanism of magnetars has been
studied by some literatures (Komissarov & Barkov, 2007; Bucciantini et al.,
2008, 2009, 2012; Kiuchi et al., 2012; Siegel et al., 2014; Shankar et al.,
2021). If the ultrarelativistic jet produced by the central engine could break
out from the progenitor envelope and circumstellar medium and be along the
line of sight, the observable LGRB could be triggered.

The characteristics of LGRB progenitors are still unclear since no direct
observational evidence has been presented to reveal the association of any
progenitor and the observed LGRBs. The event rate of LGRBs is much lower than
that of CCSNe (e.g., Podsiadlowski et al., 2004; Izzard et al., 2004; Guetta
et al., 2005; Soderberg et al., 2006; Wanderman & Piran, 2010), implying that
only a small fraction of massive stars could produce LGRBs and CCSNe at the
end of their lives. Soderberg et al. (2010) estimated that the ratio of LGRBs
to Ibc supernovae (SN) is approximately 1$\%$ based on a radio survey. Georgy
et al. (2012) obtained this fraction from rotating stellar models and found
that it could exceed 25$\%$. The absence of H/He lines in the spectra of SNe
associated with LGRBs suggests that their progenitors have undergone violent
mass loss or chemical mixing of elements. Wolf-Rayet (WR) stars with striping
H/He envelopes before explosion have been proposed as LGRB progenitors (e.g.,
Woosley, 1993; Woosley & Heger, 2006; Crowther, 2007). WR stars can be
subdivided into three classes based on the exhibition of broad emission lines
in spectra. These groups include N-rich WR stars (WN) and C-rich and O-rich WR
stars (WC/WO). The origin of WR is an ongoing study. It is proposed that
rapidly rotating single massive stars could undergo quasi-chemically
homogeneous evolution (e.g., Yoon & Langer, 2005; Yoon et al., 2006; Ekström
et al., 2012; Maeder & Meynet, 2012), allowing the systems to lose their
envelopes while still retaining enough angular momentum to produce LGRBs.
Furthermore, WR stars can also be born in massive close binaries through tidal
interactions with their companion stars (e.g., Podsiadlowski et al., 1992;
Cantiello et al., 2007; Detmers et al., 2008; Yoon et al., 2010; Eldridge et
al., 2011; Langer, 2012; de Mink et al., 2013).

Several studies have investigated LGRB progenitor evolution models and
corresponding remnants (e.g., Heger et al., 2003; Hirschi et al., 2005;
Obergaulinger & Aloy, 2017; Aguilera-Dena et al., 2018; Aloy & Obergaulinger,
2021; Li et al., 2023). Yoon & Langer (2005) found that quasi-chemically
homogeneous evolving massive stars could fulfill the requirements of
collapsar. Woosley & Bloom (2006) explored the evolution of stripped-down He
cores and single star (12, 16, and 35 $M_{\odot}$) with rapid rotating
velocity, and they concluded that such stars could retain enough angular
momentum produce GRBs at core-collapse. Recently, Roy et al. (2020) showed the
effects of mass, rotation rate, and metallicity of massive stars on their
evolution into WN and subsequently into WC, which are possible Type Ic SNe/GRB
progenitors. By studying the surface helium and nitrogen mass fraction
evolution, they found that an O star with rotation rate $\Omega/\Omega_{\rm
cirt}\geq 0.4$ could evolve into the WN phase. This might denote that rapidly
rotating massive stars could satisfy the requirement of Type Ic SNe/LGRB
production. Aguilera-Dena et al. (2020) assumed that these successful
neutrino-driven explosion models would produce magnetar and power
superluminous SNe, and that failed explosion models would lead to the BH form
and power LGRBs. The subnormal distribution of the initial SN explosion energy
could naturally build the “lower mass gap” in the mass distribution of compact
objects (Liu et al., 2021). However, in addition to the collapsar, the
magnetar might also be the central engine of LGRBs. It is difficult to
determine whether the central engine of GRB is a BH or magnetar. The shallow
decay phase (Dai, 2004; Zhang et al., 2006; Dall’Osso et al., 2011; Li et al.,
2016) and “internal plateau” (Troja et al., 2007; Lyons et al., 2010; Stratta
et al., 2018) in some GRB afterglow light curves have been proposed as the
magnetar signature, but the BH hyperaccretion model cannot be ruled out,
especially for long-lasting plateaus (e.g., Yi et al., 2022). Regardless of
the type of central engine, the shallow decay phase might be related to a
precessing jet (Huang & Liu, 2021). In this paper, we focus on the progenitors
of LGRBs and the formation of magnetars.

Figure 1: (a) H-R diagram of the massive stars with different initial masses
$M/M_{\odot}=10,15$, and $30$, initial rotation rates $\Omega/\Omega_{\rm
crit}$ = 0.1 and 0.6, metallicity $Z/Z_{\odot}=0.01,0.1$, and $0.5$, and
scaling factors of the “Dutch” wind loss rate $\eta_{\rm wind}$ = 0.5, 1, and
5. (b) Evolution of the central density and temperature in the models.
Locations of hydrogen, helium, carbon, neon, oxygen, and silicon burning are
labeled. The grey dashed line indicates the $E_{F}/k_{B}T\approx 4$ electron
degeneracy curve. The endpoint of the line is marked by the pentagram,
corresponding to the time of the iron core collapse.

The physical conditions for dying massive stars to produce GRBs remain an open
question. Developments to enhance theoretical models are required to study how
various physical parameters could replicate the formation conditions and
characteristics of the GRB central engines. Conversely, to predict the final
fate of massive stars and remnants, researchers should use observational GRB
and SN data to constrain stellar evolution models.

In the framework of the 1D single star evolution scenario, we explore the
effect of initial rotation, mass, metallicity, and mass loss on the evolution
path and pre-SN properties of the massive stars. More importantly, the initial
signatures of the magnetars are discussed. This paper is structured as
follows. In Section 2, we describe the physical ingredients of our stellar
evolution models. We present our analysis of pre-SNe and the characteristics
of the protomagnetar in Section 3. Conclusions and discussion are presented in
Section 4.

##  2 Progenitor Modeling in MESA

Figure 2: The evolution of stellar mass and total angular momentum for models
with different initial parameters.

We use the Modules for Experiments in Stellar Astrophysics software package
(MESA, Paxton et al., 2011, 2013, 2015, 2018, 2019; Jermyn et al., 2023,
version 21.12.1) and MESA SDK 21.4.1 (Townsend, 2021) to simulate a large set
of 227 massive star models from the pre-main-sequence stage until the onset of
iron core collapse. The software packages are available on the website
https://docs.mesastar.org/en/release-r22.11.1/ (catalog
https://docs.mesastar.org) and on Zenodo under an open-source Creative Commons
Attribution license: https://doi.org/10.5281/zenodo.7457681 (catalog
doi:10.5281/zenodo.7457681). Here, we define collapse as the moment when the
infall velocity of any point inside the stellar model exceeds 1,000 km s-1. We
take the test suite 20M$\\_$pre$\\_$ms$\\_$to$\\_$core$\\_$collapse as our
baseline model. All inlists of the model are available on
Zenodohttps://doi.org/10.5281/zenodo.7839968 (catalog
(doi:10.5281/zenodo.7839968)). Our stellar models cover an initial mass range
between 10 and 30 $M_{\odot}$. Here we simply expect stars in that mass range
will all form NSs upon collapse. It should be noted that there is a non-
mononic pattern of NS and BH formation as a function of initial mass (e.g.,
Woosley & Heger, 2007; Schneider et al., 2021).

The initial metallicity is $Z=0.01,0.1,$ and $0.5~{}Z_{\odot}$, where $Z$ is
the mass fraction of elements heavier than helium and $Z_{\odot}$ is the solar
metallicity (e.g., Grevesse & Sauval, 1998; von Steiger & Zurbuchen, 2016).
Following Tout et al. (1996) and Pols et al. (1998), we use the initial helium
fraction $Y=0.24+2Z$ and initial hydrogen mass fraction $X=1-Y-Z$. The initial
rotational rate $\Omega/\Omega_{\rm crit}=0.1-0.8$, which corresponds to the
initial rotational velocity at the equator of the star $\sim 80-770$ km s-1.
Here, $\Omega_{\rm crit}=(GM/R_{\rm e}^{3})^{1/2}$ is the critical angular
velocity at the equator, where $M$ and $R_{\rm e}$ denote the mass of the star
and the equator radius, respectively. For stellar winds, all models are
evolved with the “Dutch” wind-loss scheme (e.g., de Jager et al., 1988;
Nieuwenhuijzen & de Jager, 1990; Nugis & Lamers, 2000; Vink et al., 2001;
Glebbeek et al., 2009), and the scaling factor $\eta_{\rm wind}=$ 0.5, 1, and
5. We treat convection using the MLT++ scheme of MESA. According to the Ledoux
criterion and standard mixing length theory (MLT) approximation (Cox & Giuli,
1968), we choose a mixing length parameter $\alpha_{\rm MLT}$ = 1.5. The
effect of semiconvection (Langer, 1991; Yoon et al., 2006) is included, and we
adopt a baseline choice of $\alpha_{\rm sc}=0.01$ except after core helium
depletion, where $\alpha_{\rm sc}=0$. Considering thermohaline mixing
(Kippenhahn et al., 1980), we set $\alpha_{\rm th}=2$ and 0 before and after
core helium depletion, respectively. The hydrodynamic mixing instabilities at
convective boundaries, called overshoot mixing, are treated as an exponential
decay process (Herwig, 2000) beyond the Schwarzschild boundary. We adopt
overshoot mixing parameters $f_{\rm ov}=0.005$ and $f_{0}=0.001$ (Paxton et
al., 2011). The uncertainty of the convective mixing and overshooting also
affect stellar evolution (Arnett et al., 2015; Farmer et al., 2016), but are
beyond the scope of this paper. The detailed definition and range of these
parameters are described and discussed by Paxton et al. (2011, 2013), Farmer
et al. (2016) and on the MESA
website111https://docs.mesastar.org/en/release-r22.05.1/.

We use the approx21$\\_$cr60$\\_$plus$\\_$co56.net network to compute nuclear
reactions in the stellar interior (Timmes, 1999; Paxton et al., 2011). This
approximate nuclear reaction network consists of 21 base isotopes plus 60Cr
and 56Co, and it is a traditional workhorse in massive star models. Nuclear
reaction rates are from JINA REACLIB (Cyburt et al., 2010), where type 2
opacity tables (Iglesias & Rogers, 1996) and the Skye (Jermyn et al., 2021)
equation of state are adopted.

We start by running models at the pre-main-sequence stage and activating
rotation when near the zero-age-main-sequence. The implementation of rotation
in MESA closely follows Heger et al. (2000) and Heger et al. (2005). Six
different rotationally induced mixing processes are included: dynamical shear
instability $D_{\rm DSI}$, Solberg-Høiland instability $D_{\rm SHI}$, secular
shear instability $D_{\rm SSI}$, Eddington-Sweet circulation $D_{\rm ES}$,
Goldreich-Schubert-Fricke $D_{\rm GSF}$, and Spruit-Tayler dynamo $D_{\rm
ST}$, which could lead to angular momentum redistribution and chemical mixing.
It is assumed that dynamical shear instability is weak and thus is set to 0.
It should be noted that the Spruit-Tayler dynamo only operates in radiative
regions of the star and produces poloidal magnetic fields. The toroidal
magnetic fields generated by differential winding are orders of magnitude
larger than the poloidal magnetic fields (Heger et al., 2005). The turbulent
viscosity $\nu=D_{\rm conv}+D_{\rm sem}+D_{\rm DSI}+D_{\rm SHI}+D_{\rm
SSI}+D_{\rm ES}+D_{\rm GSF}+D_{\rm ST}$, which is determined as the sum of the
diffusion coefficients for convection $D_{\rm conv}$, semiconvection $D_{\rm
sem}$, and rotationally induced instabilities. The factor $f_{\rm c}$ denotes
the contribution of the rotationally induced instabilities to the diffusion
coefficient $D=D_{\rm conv}+D_{\rm sem}+f_{\rm c}(D_{\rm DSI}+D_{\rm
SHI}+D_{\rm SSI}+D_{\rm ES}+D_{\rm GSF}+D_{\rm ST})$. We assume that $f_{\rm
c}=1/30$ in this paper (Heger et al., 2000).

Figure 3: Specific angular momentum (a), angular velocity (b), toroidal
magnetic field strength (c), and density (d) variations as a function of the
radius at the onset of core collapse.

Figure 4: Specific angular momentum (a), angular velocity (b), toroidal
magnetic field strength (c), and density (d) variations as a function of the
mass coordinate at the onset of core collapse.

Figure 5: Final stellar mass $M_{\rm f}$, He core mass $M_{\rm He~{}core}$, CO
core mass $M_{\rm CO~{}core}$ and iron core mass $M_{\rm Fe~{}core}$ at the
beginning of core collapse as a function of the initial rotational rate
$\Omega/\Omega_{\rm crit}$.

Figure 6: Final stellar mass $M_{\rm f}$, He core mass $M_{\rm He~{}core}$, CO
core mass $M_{\rm CO~{}core}$ and iron core mass $M_{\rm Fe~{}core}$ at the
beginning of core collapse as a function of initial stellar mass when
$\Omega/\Omega_{\rm crit}=0.6$.

Figure 7: Hydrogen and helium mass as a function of initial rotation rate
(upper panels) and initial stellar mass (bottom panels) for pre-SN models.
Different initial masses, metallicities, and wind loss scaling factors are
indicated by different colors.

Hertzsprung-Russell (H-R) diagram and evolution of the central density and
temperature for some examples of our models are shown in Figure 1. The
different color lines correspond to different initial stellar masses
$M=10,15,$ and $30M_{\odot}$, initial rotation rates $\Omega/\Omega_{\rm
crit}=0.1$ and $0.6$, initial metallicity $Z=0.01,0.1$, and $0.5~{}Z_{\odot}$,
and the “Dutch” wind scale factor $\eta_{\rm wind}=0.5,1$, and $5$. The
endpoint of lines is marked by a pentagram, corresponding to the time of the
iron core collapse. In Figure 1(b), locations of hydrogen, helium, carbon,
neon, oxygen, and silicon burning are labeled. The grey dashed line indicates
the $E_{F}/k_{B}T\approx 4$ electron degeneracy curve. The results show that
the faster rotating model evolves toward the blue part of the H-R diagram and
ends its life with a WR star. The slower rotating model evolves toward the red
part of the H-R diagram and becomes red supergiants (RSG). At the same initial
rotation rate, the larger the initial mass, the higher the luminosity and
effective temperature of the model evolving to the end. The luminosity
generally increases with increasing initial metallicity when the initial mass
is the same. Fast-rotating stars have higher central temperatures and
densities than low-rotating stars. At the same initial rotation rate, the more
massive the stars are, the higher the central density and temperature.

In Figure 2, we present the evolution of stellar mass and total angular
momentum. Similar to Figure 1, different colored lines indicate vary initial
parameters. Comparing the different colored lines in the diagram, we can see
that the initial mass plays a decisive role in the lifetime of the star. More
rapid rotation, lower metallicity, and a smaller scaling factor of wind can
slightly extend the star lifetime. The final mass significantly decreases with
increasing initial rotation rate. Stars with higher metallicity and larger
“Dutch” wind scale factors have smaller stellar masses at the beginning of
core-collapse for models with the same initial mass and fast rotation. This
may be due to the fact that while larger stellar winds and high metallicity
increase the mass loss on the one hand, they also reduce the stellar lifetime.
For models with the same initial rotation rate, the larger the initial mass,
the larger the total initial angular momentum. High rotation rate
significantly increases angular momentum loss in stellar life.

##  3 Results

###  3.1 Pre-SN Properties

We summarize the properties of all models at the end of their evolution in
Table 1 of the Appendix, listing the initial mass, metallicity, rotational
rate, and corresponding velocity at the equator, the “Dutch” wind scale
factor, the initial total angular momentum, final mass, ages, radius,
effective temperature and luminosity, final angular momentum and masses of the
He/CO/Fe core at the end of the calculation. The compactness parameter, mass,
average magnetic field strength, and rotation period of the protomagnetar are
also presented in the table. To show the final structure of pre-SN models, we
present the specific angular momentum, angular velocity, toroidal magnetic
field strength, and density variations as a function of the radius (Figure 3)
and the mass coordinate (Figure 4). It is clear to see that the radius of an
RSG that evolved from a star with a lower initial rotation rate is two orders
of magnitude higher than the radius of a WR star that evolved from a star with
a larger initial rotation rate. The fast rotation causes the star to lose its
envelope, which affects its surface parameters. The specific angular momentum,
angular velocity, toroidal magnetic field strength, and density profiles are
flatter in the interior of faster-rotating stellar models. Convection
transports angular momentum from the inside of the convection zone to the
outside of the same zone, which produces a sawtooth shape of special angular
momentum. During the evolution, $\Omega$ increases significantly in the
central region when the core of the star contracts, and then the $\Omega$
curve flattens under the action of convection. In the pre-SN stage, the
angular velocity $\Omega$ reaches values of the order of 0.1 s-1 in the
center. The expansion of the outer layers of RSG leads to the value of
$\Omega$ decreasing sharply at the surface. Fast-rotating stars have higher
magnetic field strengths in the inner regions and higher surface densities.

In Figure 5, we display the final stellar masses $M_{\rm f}$, helium core
masses $M_{\rm He~{}core}$, carbon-oxygen core masses $M_{\rm CO~{}core}$, and
iron core mass $M_{\rm Fe~{}core}$ of massive stars at the beginning of iron
core collapse as a function of the initial rotational rate
$\Omega$/$\Omega_{\rm crit}$. The He and CO core masses are measured using the
mass coordinate, where the mass fraction of hydrogen is less than 0.1 for
$M_{\rm He~{}core}$ and the mass fraction of helium decreases below 0.1 for
$M_{\rm CO~{}core}$. Here, we define the iron core mass boundary as the
outermost location where the Si mass fraction is less than 0.1. It should be
noted that this definition is a bit arbitrary, as massive cores often have
very shallow chemical gradients. The radius and composition of the stellar
core might change with different core mass definitions (e.g., Sukhbold &
Woosley, 2014; Laplace et al., 2021). It is difficult to precisely define the
stellar core boundary because there is a composition and density gradient at
the edge. Moreover, the stellar core mass and boundary are also affected by
semiconvection and overshoot (Schootemeijer et al., 2019).

It is obvious from Figures 5(a) and (b) that $M_{\rm He~{}core}$ values
approximate $M_{\rm f}$ values when $\Omega$/$\Omega_{\rm crit}\geq$ 0.4. For
the more massive and lower metallicity stars, the corresponding values
$\Omega$/$\Omega_{\rm crit}$ $\geq$ 0.3. In other words, stars almost entirely
lose their hydrogen envelopes and become naked He cores. Note that $M_{\rm
f}$, $M_{\rm He~{}core}$, $M_{\rm CO~{}core}$ and $M_{\rm Fe~{}core}$ show
little change when $\Omega/\Omega_{\rm crit}\geq 0.4$. The values of $M_{\rm
f}$, $M_{\rm He~{}core}$, and $M_{\rm CO~{}core}$ are positively correlated
with the initial mass and negatively correlated with the initial metallicity
$Z$ and a scaling factor of wind loss $\eta_{\rm wind}$ when the initial
rotational rate is low. This is similar to models without rotation. Small
fluctuations in the values of $M_{\rm f}$, $M_{\rm He~{}core}$, and $M_{\rm
CO~{}core}$ when $\Omega$/$\Omega_{\rm crit}\geq$ 0.4. For high initial
rotation rate, rotation plays an important role in enhanced stellar mass loss
(Paxton et al., 2013) according to the prescription, i.e., $\dot{M}\propto
1/(1-\Omega/\Omega_{\rm crit})^{0.43}$.

Similar to Figure 5, $M_{\rm f}$, $M_{\rm He~{}core}$, $M_{\rm CO~{}core}$,
and $M_{\rm Fe~{}core}$ of massive stars at the beginning of iron core
collapse as a function of the initial stellar mass are presented in Figure 6.
Here, we set $\Omega$/$\Omega_{\rm crit}=0.6$. For models with lower
metallicity and smaller “Dutch” wind scale factor, $M_{\rm f}$, $M_{\rm
He~{}core}$, and $M_{\rm CO~{}core}$ increase as the initial mass increases,
except for some fluctuations. For models with higher metallicity and larger
“Dutch” wind scale factor, the values of $M_{\rm f}$, $M_{\rm He~{}core}$, and
$M_{\rm CO~{}core}$ climb up and then decline. $M_{\rm Fe~{}core}$ of massive
stars with different initial $Z$ and $\eta_{\rm wind}$ fluctuates with
$\Omega$ and stellar masses, and there is no obvious correlation. Iron core
masses in our models range from 1.29 to 1.92 $M_{\odot}$, and most of them are
concentrated between $1.4-1.6~{}M_{\odot}$. According to some previous studies
(e.g. Sukhbold & Woosley, 2014; Schneider et al., 2021), $M_{\rm Fe~{}core}$
appears to peak in the range of $20-25~{}M_{\odot}$, and BH are produced at
the corresponding peak. The position of the peak is influenced by the
metallicity (Woosley & Heger, 2007). There is a peak of $M_{\rm Fe~{}core}$ in
the mass range of $20-25~{}M_{\odot}$ for zero metallicity star. For stars
with solar metallicity, this peak corresponds to masses around 40 $M_{\odot}$.
However, there is no peak in our model and no BH formation, probably because
of the large initial rotation rate $\Omega/\Omega_{\rm crit}$ of the model in
Figure 6. All models in this diagram have gone through WR phase, and lost
large amount of mass. This is similar to the fact that exceptionally massive
stars ($\sim 100~{}M_{\odot}$) can still produce NSs, probably both as a
consequence of mass loss. Furthermore, the mapping from $M_{\rm CO~{}core}$
and $M_{\rm Fe~{}core}$ to initial mass could differ greatly in single stars
compared to stripped stars (Schneider et al., 2021).

Figure 7 demonstrates the variation of hydrogen and helium masses with initial
rotation rate and initial stellar mass for the pre-SN model. As shown in the
upper panel of Figure 7, when $\Omega$/$\Omega_{\rm crit}$ $\geq$ 0.4, most of
pre-SN models have hydrogen and helium mass below 0.001 and 0.5 $M_{\odot}$,
respectively. Only a small fraction of models with larger “Dutch” wind scale
factor have helium mass below 0.2$M_{\odot}$. It should be noted that the
hydrogen and helium mass show slight changes when $\Omega$/$\Omega_{\rm crit}$
$\geq$ 0.4. In lower panel of Figure 7, we can find that the hydrogen masses
fluctuate between 10-5 and 10-3 $M_{\odot}$ for the vast majority of models as
the initial mass increases. For models with higher metallicity and larger
“Dutch” wind scale factor, the helium mass decreases significantly with the
growth of initial mass. Among them, a small percentage of models with initial
masses greater than 20 $M_{\odot}$ have helium masses less than 0.2
$M_{\odot}$.

Massive stars without hydrogen envelopes could core-collapse and give rise to
Type Ib or Ic SNe. Type Ib or Ic SNe are distinguished based on the presence
or absence of helium lines in the spectrum. Sometimes, it is difficult to
distinguish them from limited observational data. Most progenitor models still
retain some helium after the core collapse. The different characteristics of
the progenitor stars which lead to the difference between Type Ib and Ic SNe
remain unknown (Hachinger et al., 2012; Dessart et al., 2011, 2012, 2022).
Therefore, these stars are usually collectively referred to as Type Ibc SNe.
The maximum H/He mass contained in progenitor that would give rise to a Type
Ib/Ic SN appearance is uncertain. It is proposed that the hydrogen mass of
progenitors for Type Ib SN should less than 0.001 $M_{\odot}$ (Yoon et al.,
2010; Dessart et al., 2011). Gilkis & Arcavi (2022) estimate this threshold
mass is 0.033 $M_{\odot}$. Estimates of the limit helium mass included in the
progenitors of SNe Ic from previous work range from 0.06 to 0.6 $M_{\odot}$
(Georgy et al., 2009; Yoon et al., 2010; Hachinger et al., 2012). If the
progenitor of Type Ib SN should contain hydrogen masses below 0.001
$M_{\odot}$ and the progenitors of Type Ic SN contain helium masses below 0.2
$M_{\odot}$, most of our models can produce Type Ib SNe and only a small
fraction produce Type Ic SNe. If the upper limit on the mass of helium hidden
in the progenitor of type Ic SNe are relaxed a bit, more than half of the
models can produce Type Ic SNe. Moreover, Eldridge & Tout (2004) and Eldridge
(2005) used surface helium abundance $Y_{\rm surface}$ to estimate the SNe
type produced by the progenitors. If $Y_{\rm surface}$ $\textless$ 0.3, a Type
Ic SN occurs; if 0.3 $\textless$ $Y_{\rm surface}$ $\textless$ 0.7, they label
the star as Type Ibc due to uncertain, and if $Y_{\rm surface}$ $\textgreater$
0.7, a Type Ib SN results. The vast majority of the SN type created by our
progenitor models can be designated as Type Ibc, with only a small fraction
being SN Ic or Ib, based on the $Y_{\rm surface}$ value. The rates of SN Ib
and Ic estimated by the rotating single star models is consistent with
observations (e.g. Georgy et al., 2009; Groh et al., 2013). Only a small
fraction of Type Ic SN are observed in association with LGRBs (Woosley &
Bloom, 2006; Cano et al., 2017). It is uncertain what proportion of single and
binary channels (e.g. Podsiadlowski et al., 1992; Eldridge et al., 2011)
contribute to the production of Type Ic SNe.

The explodability of model is explored according to compactness, which was
defined as (O’Connor & Ott, 2011)

$\xi_{\rm M_{r}}=\frac{M_{r}/M_{\odot}}{R(M_{\rm bary}=M_{r})/1000\rm~{}km},$
(1)

where $M_{r}=2.5~{}M_{\odot}$ was chosen to evaluate the compactness parameter
(e.g., Ugliano et al., 2012; Sukhbold & Woosley, 2014), which corresponds to
the point where the collapse speed first reaches $1,000~{}\rm km~{}s^{-1}$.
This has been pointed out as a rough indicator of whether the collapse of a
nonrotating stellar core could lead to a successful neutrino-driven explosion
($\xi_{2.5}<0.45$) or form a BH ($\xi_{2.5}>0.45$). While this criterion is
not sufficient to accurately predict the fate of rapidly spinning stars, it
can still reveal structural features of the stellar core (e.g., Ertl et al.,
2016; Müller et al., 2016; Aguilera-Dena et al., 2020).

The compactness parameter $\xi_{2.5}$ is calculated according to Equation (1).
As shown in Figure 8, $\xi_{2.5}$ varies with the initial rotation rate
$\Omega/\Omega_{\rm crit}$ and initial mass $M_{\rm ini}$ of progenitors. The
$\xi_{2.5}$ = 0.45 line separates models that could explode, which are driven
by neutrino (gray) and disallowed (white) regions. It is easy to find that
most of our progenitor models could explode successfully. The models with
$Z=0.01~{}Z_{\odot}$ and $\eta_{\rm wind}$ = 1 appear to have rather low
compactness. The values of the compactness parameter are less affected by the
initial velocity of stars, especially for $\eta_{\rm wind}$ = 1. The
compactness parameter is positively correlated with the mass of the NSs in all
cases. The models with the same initial rotation rate, the compactness
parameter varies with the initial mass in a non-monotonic way. The non-
monotonic behavior of the compactness for nonrotating massive stars has been
investigated by some literatures. Sukhbold & Woosley (2014) found that the
timing and location of several carbon and oxygen convective shells play a
major role in generating this non-monotonic structure. The final compactness
can be quite complicated because of the multiple episodes of shell burning. It
is also sensitive to the initial mass, metallicity, convection way, and
nuclear physics in code. The BHs are predicted to be produced by stars in the
compactness peaks. When compared to a single star, the compactness in a
stripped star shifts to a higher CO core mass (Woosley, 2019; Ertl et al.,
2020; Schneider et al., 2021).

It is worth noting that the compactness parameter is really only meaningful
for neutrino-driven explosions because these involve a threshold process.
Neutrino heating mechanism could account for SNe with energy approximately
1051 ergs (see a review by Janka, 2012). For some more energetic SNe,
magnetohydrodynamic mechanisms are required to drive explosions (e.g., Akiyama
et al., 2003; Burrows et al., 2007; Shankar et al., 2021). The characteristics
and mass distribution of the remnants are also influenced by the supernova
explosion mechanism (e.g., Fryer et al., 2012, 2022a).

###  3.2 Protomagnetar signatures

Figure 8: Compactness parameters $\xi_{2.5}$ of progenitor models as a
function of initial rotation rate $\Omega/\Omega_{\rm crit}$ and stellar mass.
The $\xi_{2.5}=0.45$ line separates models that could explode, which are
driven by neutrino (gray) and disallowed (white) regions.

Figure 9: The mass $M_{\rm NS}$, periods $P_{\rm NS}$ and the average magnetic
field strength $B_{\rm NS}$ of the protomagnetar as a function of initial
stellar mass for models with different initial metallicity. All models
involved in the diagram have the same “Dutch” wind scale factor $\eta_{\rm
wind}=1$, and they eventually evolved into WR stars.

Figure 10: The mass $M_{\rm NS}$, periods $P_{\rm NS}$ and the average
magnetic field strength $B_{\rm NS}$ of the protomagnetar as a function of
initial stellar mass for models with different “Dutch” wind scale factor. All
models involved in the diagram have the same initial metallicity
$Z=0.01Z_{\odot}$ , and they eventually evolved into WR stars.

Figure 11: The total rotation energy $E_{\rm rot}$ as a function of periods
$P_{\rm NS}$. All models involved in the diagram eventually evolved into WR
stars.

Figure 12: Ranges of masses and initial rotation rates required to produce
magnetars from stellar models with different initial metallicity and “Dutch”
wind scale factors. Only models with parameters for blue regions above the
gray online can evolve into WR stars, which might eventually undergo SN
explosions or GRBs when the core collapses, producing magnetars.

After a massive star collapses, a proto-NS might be born in the center. In
this paper, we take the iron core masses as the NS baryonic masses $M_{\rm
b}$. The binding energy will be released by emitting neutrinos during the iron
core collapse to the NS. Following Lattimer & Prakash (2001), the NS
gravitational mass $M_{\rm NS}$ can be calculated by

$\frac{M_{\rm b}-M_{\rm NS}}{M_{\rm NS}}\approx\frac{0.6~{}GM_{\rm NS}/R_{\rm
NS}c^{2}}{1-0.5~{}GM_{\rm NS}/R_{\rm NS}c^{2}}.$ (2)

The change in NS mass due to material fallback in accretion during the
explosion is negligible, especially for stars with zero-age-main-sequence
masses less than 30 $M_{\odot}$. The results of accurate calculations,
including fallback, exhibit good agreement with this simple approach (e.g.,
Sukhbold et al., 2016; Ertl et al., 2020).

The Spruit-Tayler dynamo operates in radiative regions and taps into
differential rotation to produce magnetic fields (e.g., Spruit, 2002; Petrovic
et al., 2005; Heger et al., 2005). The corresponding toroidal and radial
magnetic field strength are (Heger et al., 2005)

$B_{\rm\phi}=2\pi^{1/2}\rho^{1/2}\nu_{\rm e}^{1/4}q^{1/2}\omega^{3/4}r^{1/2},$
(3) $B_{\rm r}=2\pi^{1/2}\rho^{1/2}\nu_{\rm
e}^{3/4}q^{1/2}\omega^{1/4}r^{-1/2},$ (4)

where $\nu_{\rm e}$ is the effective viscosity, $q=d$ ln $\omega/d$ ln $r$ is
the shear. It is assumed that the initial magnetic field is sufficiently weak.
The radial component $B_{\rm r}$ is wound up by differential rotation, which
leads to the dominance of toroidal magnetic field $B_{\rm\phi}\gg B_{\rm r}$.
A detailed description of the dynamo mechanism has been presented in Spruit
(2002). It should be noted that MESA doesn’t calculate any Spruit-Tayler
magnetic fields in convective regions (Paxton et al., 2013). The surface
magnetic field produced by a dynamo (e.g., Brandenburg & Subramanian, 2005) in
the convective zone is also not included in star evolution.

The average strength of dynamo-generated magnetic fields inside the iron core
is given by

$<B_{\phi}>=\frac{\int_{0}^{M_{\rm Fe}}B_{\phi}(m)dm}{\int_{0}^{M_{\rm
Fe}}dm}.$ (5)

Assuming the conservation of magnetic flux and iron core contracted
homogeneously to the NS with radius $R_{\rm NS}=12$ km, we obtain the mean
magnetic field strength of NSs.

The natal spin rates of NSs can be determined by angular momentum within the
iron core:

$P_{\rm NS}=\frac{2\pi I_{\rm NS}}{J_{\rm Fe}}.$ (6)

Here, the moment of inertia of the NSs is set as $I_{\rm NS}$ = 0.35$M_{\rm
NS}R_{\rm NS}^{2}$ (e.g., Lattimer & Prakash, 2001).

In Figure 9, the mass $M_{\rm NS}$, periods $P_{\rm NS}$ and the mean magnetic
field strength of the protomagnetar as a function of initial stellar mass for
models with different initial metallicity. All models involved in the diagram
have the same “Dutch” wind scale factor $\eta_{\rm wind}=1$. The upward
triangles, downward triangles, and circles represent different metallicity
$Z=0.01,0.1,$ and $0.5~{}Z_{\odot}$, respectively. The initial rotation rate
$\Omega/\Omega_{\rm crit}$ is indicated by the color bar, which varies from
$0.3-0.8$. As mentioned before, it is only the models with initial rotation
rates greater than 0.3 or 0.4 that can eventually evolve into WR stars, which
could produce SNe or/and GRBs at core collapse. It is easy to see that the
mass of most proto-NSs born in higher initial metallicity
($Z=0.5~{}Z_{\odot}$) progenitors are lower than those of progenitors produced
by lower initial metallicity ($Z=0.01$ and $0.1~{}Z_{\odot}$). Almost all
$P_{\rm NS}$ of proto-NSs born in progenitors with $Z=0.1~{}Z_{\odot}$ are
shorter than those of progenitors with $Z=0.01$ and $0.5~{}Z_{\odot}$. The
higher the initial mass, the smaller the $P_{\rm NS}$, and this phenomenon is
more evident in models with $Z=0.1$ and $0.5~{}Z_{\odot}$. The rotation rate
of NSs does not increase monotonically with the initial rotation rate of the
progenitor, which may be related to the fact that the angular momentum loss
during star evolution is influenced by several factors. The mean magnetic
fields $B_{\rm NS}$ of most proto-NSs born from progenitors with
$Z=0.1~{}Z_{\odot}$ are larger than those of progenitors with $Z=0.01$ and
$0.5~{}Z_{\odot}$ when $\eta_{\rm wind}$ = 1.

Similarly, Figures 10(a)-(c) correspond to $P_{\rm NS}$, $M_{\rm NS}$, and
$B_{\rm NS}$ of proto-NSs as a function of initial stellar mass for models
with different “Dutch” wind scale factor, respectively. All models involved in
the diagram have the same initial metallicity $Z$ = 0.01 $Z_{\odot}$, and they
eventually evolved into WR stars. The upward triangles, downward triangles,
and circles represent the “Dutch” wind scale factor $\eta_{\rm wind}$ = 0.5,
1, and 5, respectively. We can find that the larger the “Dutch” wind scale
factor, the smaller the mass of the final produced NSs for models with the
same initial metallicity. Models with $\eta_{\rm wind}$ = 0.5 produce proto-
NSs with larger masses, shorter periods, and stronger mean magnetic fields
than those with $\eta_{\rm wind}$ = 1 and 5. The masses $M_{\rm NS}$ span from
1.29 to 1.61 $M_{\odot}$, with a median value of 1.4 $M_{\odot}$. The $P_{\rm
NS}$ range from $1.0-2.6$ ms. Even after taking into account the effects of
initial rotation rate, stellar wind loss, and metallicity, progenitor stars
with larger initial masses are still more likely to form fast-spinning NSs.
The more massive NSs are more likely to have faster rotation rates and
stronger magnetic field strengths. Regardless, these NSs are classified as
millisecond magnetars.

Here we take the iron core mass as the baryonic mass of NS, then convert it to
NS gravitational mass through an analytical, radius-dependent approach
(Lattimer & Prakash, 2001) and do not consider detailed SNe explosion physics.
The material fallback process during an SN explosion can affect the final
properties of compact remnants (Zhang et al., 2008; Fryer et al., 2012;
Ugliano et al., 2012; Janka, 2013; Müller et al., 2019; Ertl et al., 2020,
e.g.,). First, the mass of NS could grow by the accretion of fallback
material. However, if a large amount of matter could fall on the newly born
NS, a BH might form. Second, recent three-dimensional simulations of
nonrotating progenitors indicate that the NS can be spun up to millisecond
periods by fallback (Chan et al., 2020). Müller et al. (2016) obtained a
multimodal NS mass distribution with peaks around 1.15, 1.45, and possibly
$1.9~{}M_{\odot}$. According to the simulation results of Sukhbold et al.
(2016), there is some bimodality in the distribution, with peaks around 1.25
and 1.4 $M_{\odot}$. Their maximum NS is 1.68 $M_{\odot}$, and the minimum is
1.23 $M_{\odot}$. Our result exhibits good agreement with these previous SNe
explosion simulations.

To explore the ability of newborn magnetars to power GRBs, the total rotation
energy $E_{\rm rot}$ and $P_{\rm NS}$ of magnetars born from different values
of initial mass, metallicity, “Dutch” wind scale factor, and initial rotation
rate are shown in Figure 11. All of the progenitor stars in the diagram are WR
stars that have lost their hydrogen envelope. $E_{\rm rot}$ ranges from $\sim
5\times 10^{51}$ to $\sim 2\times 10^{52}$ ergs. We conclude that stars with
larger initial masses tend to produce rapidly rotating magnetars with more
rotational energy, even when different initial rotation rates and stellar wind
losses are taken into account. This phenomenon is especially evident for stars
with $Z=0.01~{}Z_{\odot}$. Considering that the envelopes are easily disrupted
by jets in most cases, the above results indicate that our models could meet
the energy requirements of most of the observed LGRBs, including some
collapsar-origin SGRBs (e.g. Zhang et al., 2009; Levesque et al., 2010; Thöne
et al., 2011; Xin et al., 2011). In other words, we present theoretical
evidence on magnetar-origin GRBs by simulating the evolution of fast-rotating,
low-metallicity, and massive stars.

In Figure 12, we present how the ranges of masses and initial rotation rates
required to produce magnetars from stellar models vary with different initial
metallicity and “Dutch” wind scale factors. The model of a star with initial
mass and rotation rate in the red region eventually evolved into RSG to
possibly produce Type II SNe. The model with parameters within the blue region
evolved into a WR star, which might eventually undergo SN explosions or GRBs
when the core collapses, producing magnetars. It can be found that the greater
the metallicity and “Dutch” wind scale factor, the greater the minimum
rotation rate required to produce a magnetar. For a star with a mass around 10
$M_{\odot}$, when $(Z/Z_{\odot},\eta_{\rm wind})=(0.5,1)$, the initial
rotation rate at least 0.5 to evolve into a WR star. When
$(Z/Z_{\odot},\eta_{\rm wind})=(0.01,0.5),(0.01,1)$, this value of 0.4 is
sufficient. As the initial mass increases, the minimum rotation rate required
to form a magnetar gradually decreases.

##  4 Conclusions and discussion

We use MESA code to simulate a grid of stellar models with different initial
rotation rates ($\Omega/\Omega_{\rm crit}=0.1-0.8$), masses
($10-30~{}M_{\odot}$), metallicity ($Z=0.01,0.1$, and $0.5~{}Z_{\odot}$),
“Dutch” wind scaling factors ($\eta_{\rm wind}=0.5,1$, and $5$). All models
evolve from the pre-main-sequence stage until core collapse. The effects of
the initial rotation, mass, metallicity, and mass loss on the formation and
characteristics of the GRB progenitor and protomagnetar are explored. We find
that the initial rotation rate and mass play a dominant role in whether a star
can evolve into a GRB progenitor. As the initial mass increases, the minimum
rotation rate required to form a magnetar gradually decreases. The initial
metallicity and the “Dutch” wind scaling factors affect the minimum initial
rotation rate required to produce a magnetar. The greater the metallicity and
“Dutch” wind scale factor, the greater the minimum rotation rate. In other
words, massive stars with low metallicity are more likely to harbor magnetars.

The final stellar, helium, and carbon-oxygen core masses roughly increase with
increasing initial mass and decrease moderately with increasing initial
rotation rate for most cases. We also discuss the progenitors of the different
types of supernovae. Most of our progenitor models would give rise to Type Ib
SNe, and a small fraction of them will produce Type Ic. Furthermore, we find
that the compactness parameter remains a non-monotonic function of the initial
mass and initial rotation rate when the effects of different metallicity and
wind mass loss are considered.

The period of a protomagnetar ranges from $1.0-3.2$ ms with a median value of
1.6 ms, and the mass ranges from $1.18-1.68~{}M_{\odot}$ with a median value
of 1.4 $M_{\odot}$. The average magnetic field strength is concentrated on the
order of $\sim 10^{14}$ G. The $E_{\rm rot}$ can be up to about $2\times
10^{52}$ ergs. The rotational energy, magnetic field strength, and period of
the protomagnetar are satisfied to power the typical LGRBs.

Observation and theory demonstrated that GRB rates and properties vary with
redshift due to the different distributions of metallicity, star formation
rate, and initial mass distributions of massive stars at different redshifts
(e.g., Yonetoku et al., 2004; Hirschi et al., 2005; Langer & Norman, 2006;
Madau & Dickinson, 2014; Taggart & Perley, 2021). The evolution of LGRB
progenitors is also influenced by redshift (Yoon et al., 2006; Fryer et al.,
2022b). However, the analysis of the metallicity of GRB host galaxies reveals
that GRBs favor subsolar metallicity, even considering the redshift effect
(Levan et al., 2016). Therefore, we only consider three subsolar metallicity
values $Z=0.01,0.1,$ and $0.5~{}Z_{\odot}$ in our stellar models.

It is unclear how much the evolutionary pathways of single and binary stars
contributed to the production of Type Ibc SN and GRB progenitors (e.g.,
Langer, 2012). Here, we do not consider the interaction of binary stars and
focus on exploring the effects of different parameters on single star
evolution and the formation of LGRB central engines. The evolution of binary
stars is more complicated than that of a single star. In addition to some
usual uncertainties in single star evolution, as we mentioned above, the
evolution path and outcome of a binary star system also depend on the initial
mass ratio, orbital eccentricity, material, and angular momentum exchange.
Binary channels might also be important for the formation of GRB progenitors
(e.g., Fryer et al., 1999; Fryer & Heger, 2005; Eldridge et al., 2011). In a
close binary system, the mass gainer can be spun up to break up the rotation
velocity through mass and angular momentum transport (e.g., Cantiello et al.,
2007). Recently, Hu et al. (2023) simulated binary evolution models with
efficient angular momentum transport mechanisms and found that fast-spinning
magnetars could form in these systems, powering the stripped-envelope SNe.
Therefore, the binary progenitors for LGRBs do not require a fast initial
rotation velocity to strip the stellar hydrogen envelope and retain sufficient
angular momentum to produce a GRB jet (e.g., Podsiadlowski et al., 2010;
Chrimes et al., 2020). Moreover, Laplace et al. (2021) studied the systematic
differences in the core density and composition structure of solar metallicity
nonrotating single and donor stars in binary systems with an initial mass
range of $11-21~{}M_{\odot}$. They concluded that single stars systematically
possessed more massive He cores than binary-striped stars with the same
initial mass. Lloyd-Ronning (2022) proposed that radio-loud GRBs are produced
by interacting binary systems, while radio-quiet GRBs originate from the
collapse of single stars.

There is no firm conclusion regarding the upper and lower limits of NS mass
born in nature (Özel & Freire, 2016). It is difficult to form NSs with masses
below 1.2 $M_{\odot}$ through iron-core collapse SN explosions. Some low-mass
NSs candidates have been observed. Rawls et al. (2011) found NS mass of
$1.0\pm 0.1~{}M_{\odot}$ for 4U 1538-52, and $1.04\pm 0.09~{}M_{\odot}$ for
SMC X-1. Özel et al. (2012) reanalyzed these systems and estimate the mass are
$1.18\pm 0.25~{}M_{\odot}$ and $0.93\pm 0.12~{}M_{\odot}$, respectively.
Recently, Doroshenko et al. (2022) reported a strangely light NS with
$0.77^{+0.20}_{-0.17}M_{\odot}$ in the central of SN remnant HESS J1731-347.
Low-mass NSs may be born in an electron-capture SNe explosion with 8$-$10
$M_{\odot}$ progenitors (Lattimer, 2012). The well-measured massive pulsar
masses, such as J1614+2230 (1.97$\pm$0.04 $M_{\odot}$), J0348+0432
(2.01$\pm$0.04 $M_{\odot}$), J2215+5135 (2.27$\pm$0.17 $M_{\odot}$), and
J0740+6620 (2.14$\pm$0.1 $M_{\odot}$), indicate that the maximum NS mass can
be at least 2 $M_{\odot}$ (Demorest et al., 2010; Antoniadis et al., 2013;
Linares et al., 2018; Cromartie et al., 2020). The observation of a compact
object in a binary merger GW190814 (Abbott et al., 2020) could be an
indication of the most massive NS mass reaching $2.6-2.9~{}M_{\odot}$
(Godzieba et al., 2021). The NS mass distributions in binary systems have been
investigated by many authors (Thorsett & Chakrabarty, 1999; Fryer et al.,
2012). Woosley (2019), Ertl et al. (2020), and Woosley et al. (2020)
calculated the evolution and explosion of a grid of nonrotating helium stars
and studied the remnant mass distributions. They simplify binary star
evolution and assume that the entire hydrogen envelope is promptly removed by
binary interaction at the moment of helium core ignition. Their results show
that the median NS masses in binary systems are in the range of
$1.32-1.37~{}M_{\odot}$ and vary slightly with mass loss and metallicity
(Woosley et al., 2020). They set the lightest and heaviest NS gravitational
mass at 1.24 and 2.3 $M_{\odot}$, respectively. Previous observations have
shown overwhelming evidence that the mass distribution of NSs is bimodal
(e.g., Özel et al., 2012; Antoniadis et al., 2016; Alsing et al., 2018), with
two peaks at 1.3 $M_{\odot}$ and 1.5$-$1.7 $M_{\odot}$.

The distribution of the compactness parameter for nonrotating stars has been
studied by some authors (O’Connor & Ott, 2011; Ugliano et al., 2012; Sukhbold
& Woosley, 2014; Müller et al., 2016). The approximate critical value between
black hole and NS formation varies in different studies. Müller et al. (2016)
proposed a semianalytic method to determine explodability and found that
$\xi_{2.5}=0.278$ is the best value for discriminating successful or failed
explosions. Aguilera-Dena et al. (2020) calculated the compactness of 42
rotating massive star models with initial equatorial rotation velocities of
$600~{}\rm km~{}s^{-1}$ and $Z=0.02~{}Z_{\odot}$. They show that semianalytic
exploding models are compatible with the BH and NS thresholds
$\eta_{2.5}=0.45$. They suggested that the compactness is a nonmonotonic
function of the initial mass for rapidly rotating stars. In this paper, we
explore the effects of the initial rotation rate, initial mass, metallicity
and mass loss on compactness. The results show that the compactness parameter
remains a nonmonotonic function of the initial mass and initial velocity when
the effects of different metallicity and wind mass loss are considered. In the
future, we will test the accuracy of the compactness parameter as a criterion
for rotating massive star explosions by simulating the process of SN
explosion. We will also investigate the effects of the initial mass,
metallicity and rotational rate on the SN explosion energy, ejecta mass and
remnant characteristics.

We appreciate Prof. Alexander Heger and Prof. Bernhard Müller for their
constructive suggestions and comments, and thank Prof. Dong Lai, Dr. Tuan Yi,
Dr. Zhenyu Zhu, and Shuai Zha for helpful discussion. The computations in this
paper were run on the $\pi$2.0 cluster supported by the Center for High
Performance Computing at Shanghai Jiao Tong University. This work was
supported by the National Natural Science Foundation of China under grants
12103033, 12173031, and 12221003.

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The properties of the star models at the end of their evolution are shown in
Table 1.

Table 1: Properties of the models at the end of their evolution.

=20mm

Progenitor Parameters | Stellar Properties at Core Collapse | Protomagnetar
---|---|---
_$M$_ | _Z_ | _$\Omega$_ | _$V_{\rm equ}$_ | _$\eta_{\rm wind}$_ | _$J_{\rm ini}$_ | _M f_ | _Ages_ | _R f_ | log _T eff_ | log _L_ | _J ${}_{\rm He~{}core}$_ | _M ${}_{\rm He~{}core}$_ | _J ${}_{\rm CO~{}core}$_ | _M ${}_{\rm CO~{}core}$_ | _J ${}_{\rm Fe~{}core}$_ | _M ${}_{\rm Fe~{}core}$_ | _$\xi_{2.5}$_ | _M NS_ | _B NS_ | _P NS_
(_M ⊙)_ | (_Z ⊙)_ | ($\Omega_{\rm crit}$) | (km s-1) |  | (1052ergs s) | (_M ⊙)_ | (Myr) | (_R ⊙)_ | (K) | (_L ⊙)_ | ( 1050 ergs s) | (_M ⊙)_ | (1049 ergs s) | (_M ⊙)_ | (1048 ergs s) | (_M ⊙)_ |  | (_M ⊙)_ | (1014 G) | (ms)
10 | 0.01 | 0.1 | 89 | 0.5 | 0.31 | 9.97 | 28.31 | 558.77 | 3.59 | 4.81 | 0.61 | 3.51 | 1.15 | 1.98 | 4.19 | 1.47 | 0.01 | 1.33 | 1.69 | 2.00
10 | 0.01 | 0.3 | 265 | 1 | 0.91 | 9.66 | 39.34 | 25.73 | 4.35 | 5.17 | 1.90 | 6.20 | 6.91 | 4.12 | 7.03 | 1.65 | 0.25 | 1.48 | 2.84 | 1.32
10 | 0.01 | 0.3 | 265 | 0.5 | 0.91 | 9.97 | 38.49 | 29.79 | 4.28 | 5.04 | 1.41 | 5.38 | 4.54 | 3.43 | 5.15 | 1.47 | 0.11 | 1.33 | 2.04 | 1.62
10 | 0.01 | 0.4 | 348 | 5 | 1.17 | 7.37 | 41.50 | 6.58 | 4.66 | 5.24 | 1.73 | 6.56 | 7.51 | 4.58 | 5.57 | 1.64 | 0.27 | 1.47 | 0.79 | 1.66
10 | 0.01 | 0.5 | 428 | 0.5 | 1.40 | 5.87 | 43.95 | 0.92 | 5.12 | 5.36 | 3.17 | 5.87 | 3.62 | 3.44 | 4.96 | 1.61 | 0.23 | 1.44 | 1.50 | 1.83
10 | 0.01 | 0.6 | 502 | 1 | 1.58 | 4.28 | 45.02 | 0.95 | 5.04 | 5.05 | 0.76 | 4.28 | 2.48 | 2.87 | 4.05 | 1.50 | 0.10 | 1.35 | 2.46 | 2.10
10 | 0.01 | 0.6 | 502 | 5 | 1.58 | 6.33 | 44.51 | 2.05 | 4.91 | 5.20 | 1.31 | 6.11 | 5.81 | 4.30 | 4.07 | 1.54 | 0.19 | 1.39 | 0.96 | 2.15
10 | 0.01 | 0.6 | 502 | 0.5 | 1.58 | 5.98 | 44.93 | 0.68 | 5.15 | 5.24 | 3.66 | 5.98 | 8.64 | 4.18 | 5.63 | 1.51 | 0.20 | 1.36 | 1.68 | 1.52
10 | 0.01 | 0.7 | 571 | 0.5 | 1.71 | 5.93 | 45.77 | 0.68 | 5.15 | 5.23 | 3.61 | 5.93 | 8.43 | 4.13 | 5.18 | 1.59 | 0.21 | 1.42 | 2.07 | 1.73
10 | 0.01 | 0.8 | 593 | 0.5 | 1.56 | 5.82 | 44.95 | 0.70 | 5.16 | 5.27 | 3.13 | 5.82 | 6.20 | 4.00 | 4.42 | 1.59 | 0.24 | 1.43 | 1.67 | 2.03
10 | 0.1 | 0.1 | 82 | 0.5 | 0.31 | 9.73 | 28.48 | 657.59 | 3.55 | 4.81 | 0.27 | 3.47 | 0.82 | 2.02 | 2.99 | 1.47 | 0.02 | 1.33 | 0.80 | 2.80
10 | 0.1 | 0.6 | 461 | 1 | 1.58 | 7.35 | 44.70 | 1.73 | 5.03 | 5.55 | 1.87 | 7.20 | 8.51 | 5.21 | 4.52 | 1.63 | 0.26 | 1.46 | 1.53 | 2.03
10 | 0.1 | 0.6 | 461 | 0.5 | 1.58 | 5.85 | 45.62 | 0.78 | 5.11 | 5.17 | 2.87 | 5.85 | 5.55 | 4.12 | 5.73 | 1.53 | 0.21 | 1.37 | 2.95 | 1.51
11 | 0.01 | 0.4 | 354 | 1 | 1.40 | 5.21 | 36.60 | 0.67 | 5.21 | 5.44 | 1.08 | 5.21 | 4.96 | 3.76 | 4.40 | 1.62 | 0.20 | 1.45 | 1.35 | 2.08
11 | 0.01 | 0.6 | 510 | 1 | 1.90 | 4.50 | 38.43 | 0.81 | 5.09 | 5.11 | 0.81 | 4.50 | 2.93 | 3.07 | 4.18 | 1.54 | 0.18 | 1.39 | 2.20 | 2.09
11 | 0.01 | 0.6 | 510 | 0.5 | 1.90 | 6.35 | 38.36 | 0.68 | 5.25 | 5.61 | 3.74 | 6.35 | 8.90 | 4.48 | 5.77 | 1.64 | 0.26 | 1.46 | 2.00 | 1.60
11 | 0.1 | 0.6 | 469 | 1 | 1.90 | 7.91 | 38.02 | 1.51 | 5.01 | 5.34 | 2.33 | 7.81 | 11.00 | 5.70 | 4.80 | 1.55 | 0.16 | 1.39 | 1.72 | 1.83
11 | 0.1 | 0.6 | 469 | 0.5 | 1.90 | 6.28 | 38.65 | 0.81 | 5.11 | 5.21 | 2.57 | 6.28 | 8.06 | 4.42 | 4.97 | 1.53 | 0.13 | 1.37 | 1.80 | 1.74
11 | 0.5 | 0.4 | 272 | 1 | 1.48 | 9.08 | 33.67 | 757.09 | 3.59 | 5.08 | 1.16 | 5.31 | 4.08 | 3.43 | 5.28 | 1.52 | 0.15 | 1.37 | 1.79 | 1.64
11 | 0.5 | 0.5 | 334 | 1 | 1.76 | 6.69 | 37.09 | 1.11 | 5.08 | 5.35 | 1.39 | 6.69 | 6.15 | 4.70 | 2.98 | 1.55 | 0.19 | 1.39 | 0.38 | 2.95
11 | 0.5 | 0.6 | 392 | 1 | 1.99 | 6.52 | 37.90 | 1.11 | 5.08 | 5.36 | 1.04 | 6.52 | 4.70 | 4.60 | 2.68 | 1.51 | 0.17 | 1.36 | 1.41 | 3.20
12 | 0.01 | 0.6 | 518 | 1 | 2.24 | 4.86 | 33.26 | 0.62 | 5.17 | 5.21 | 0.94 | 4.86 | 3.87 | 3.44 | 3.82 | 1.59 | 0.24 | 1.43 | 1.42 | 2.35
12 | 0.01 | 0.6 | 518 | 5 | 2.24 | 7.24 | 32.79 | 1.25 | 5.10 | 5.57 | 2.09 | 7.23 | 9.38 | 5.18 | 4.69 | 1.58 | 0.18 | 1.42 | 1.84 | 1.91
12 | 0.1 | 0.1 | 85 | 0.5 | 0.45 | 11.72 | 20.79 | 688.92 | 3.57 | 4.92 | 0.74 | 4.10 | 1.96 | 2.43 | 3.43 | 1.43 | 0.04 | 1.30 | 1.04 | 2.38
12 | 0.1 | 0.6 | 476 | 1 | 2.26 | 8.19 | 33.00 | 0.80 | 5.15 | 5.37 | 2.56 | 8.19 | 13.30 | 6.07 | 5.00 | 1.53 | 0.24 | 1.37 | 4.31 | 1.73
12 | 0.1 | 0.6 | 476 | 0.5 | 2.26 | 8.07 | 33.12 | 0.70 | 5.18 | 5.35 | 4.22 | 8.07 | 18.10 | 5.95 | 3.38 | 1.51 | 0.16 | 1.36 | 1.05 | 2.53
12 | 0.5 | 0.6 | 398 | 1 | 2.38 | 6.71 | 32.89 | 0.97 | 5.09 | 5.28 | 1.41 | 6.71 | 7.08 | 4.81 | 4.46 | 1.53 | 0.18 | 1.38 | 2.09 | 1.95
13 | 0.01 | 0.6 | 525 | 1 | 2.61 | 5.02 | 29.34 | 0.87 | 5.17 | 5.52 | 0.96 | 5.02 | 4.23 | 3.58 | 4.39 | 1.57 | 0.19 | 1.41 | 1.45 | 2.03
13 | 0.01 | 0.6 | 525 | 5 | 2.61 | 7.83 | 28.70 | 1.06 | 5.11 | 5.44 | 2.39 | 7.83 | 11.70 | 5.74 | 5.11 | 1.64 | 0.26 | 1.46 | 1.53 | 1.81
13 | 0.01 | 0.6 | 525 | 0.5 | 2.61 | 7.16 | 29.20 | 0.67 | 5.18 | 5.30 | 4.03 | 7.16 | 13.50 | 5.18 | 5.69 | 1.47 | 0.17 | 1.33 | 3.62 | 1.47
13 | 0.1 | 0.6 | 483 | 1 | 2.64 | 8.68 | 29.06 | 0.91 | 5.13 | 5.38 | 2.74 | 8.68 | 14.90 | 6.56 | 4.86 | 1.51 | 0.13 | 1.36 | 3.61 | 1.76
13 | 0.1 | 0.6 | 483 | 0.5 | 2.64 | 7.41 | 29.33 | 0.70 | 5.25 | 5.65 | 3.39 | 7.41 | 13.80 | 5.46 | 5.73 | 1.50 | 0.14 | 1.36 | 1.97 | 1.49
14 | 0.01 | 0.3 | 280 | 1 | 1.71 | 13.91 | 21.79 | 67.45 | 4.16 | 5.25 | 2.34 | 7.25 | 9.31 | 5.01 | 6.95 | 1.76 | 0.48 | 1.56 | 2.10 | 1.41
14 | 0.01 | 0.6 | 531 | 1 | 3.00 | 5.32 | 26.14 | 0.61 | 5.18 | 5.25 | 1.05 | 5.32 | 5.00 | 3.86 | 4.03 | 1.59 | 0.21 | 1.42 | 1.80 | 2.23
14 | 0.01 | 0.6 | 531 | 5 | 3.00 | 7.91 | 25.81 | 0.93 | 5.11 | 5.34 | 2.42 | 7.91 | 12.10 | 5.88 | 4.62 | 1.50 | 0.14 | 1.35 | 2.29 | 1.84
14 | 0.1 | 0.6 | 490 | 1 | 3.05 | 9.28 | 25.90 | 1.03 | 5.12 | 5.45 | 3.59 | 9.28 | 18.70 | 7.02 | 4.08 | 1.48 | 0.11 | 1.34 | 1.14 | 2.06
14 | 0.1 | 0.6 | 490 | 0.5 | 3.05 | 7.87 | 26.12 | 0.61 | 5.21 | 5.36 | 3.60 | 7.87 | 15.40 | 5.77 | 6.43 | 1.61 | 0.23 | 1.44 | 2.50 | 1.41
14 | 0.5 | 0.6 | 412 | 1 | 3.22 | 7.58 | 25.32 | 0.91 | 5.11 | 5.31 | 1.87 | 7.58 | 10.30 | 5.64 | 4.21 | 1.50 | 0.15 | 1.36 | 1.24 | 2.03
15 | 0.01 | 0.2 | 190 | 1 | 1.33 | 14.95 | 17.20 | 570.08 | 3.68 | 5.18 | 1.82 | 6.06 | 6.65 | 4.07 | 6.31 | 1.64 | 0.26 | 1.46 | 2.41 | 1.46
15 | 0.01 | 0.2 | 190 | 5 | 1.33 | 14.84 | 16.68 | 109.82 | 3.98 | 4.97 | 1.30 | 5.53 | 4.45 | 3.59 | 5.57 | 1.57 | 0.14 | 1.41 | 1.75 | 1.59
15 | 0.01 | 0.4 | 372 | 1 | 2.52 | 5.68 | 22.57 | 0.59 | 5.18 | 5.22 | 1.13 | 5.68 | 5.26 | 4.18 | 4.47 | 1.63 | 0.22 | 1.46 | 1.41 | 2.05
15 | 0.01 | 0.4 | 372 | 5 | 2.52 | 8.61 | 22.33 | 0.99 | 5.11 | 5.37 | 2.90 | 8.61 | 16.00 | 6.55 | 4.96 | 1.49 | 0.11 | 1.34 | 1.82 | 1.71
15 | 0.01 | 0.4 | 372 | 0.5 | 2.52 | 8.09 | 22.48 | 0.58 | 5.22 | 5.37 | 4.23 | 8.09 | 17.20 | 6.00 | 5.70 | 1.51 | 0.20 | 1.36 | 2.39 | 1.51
15 | 0.01 | 0.5 | 458 | 1 | 3.02 | 5.55 | 23.07 | 0.60 | 5.17 | 5.18 | 1.10 | 5.55 | 4.50 | 4.05 | 4.38 | 1.54 | 0.19 | 1.38 | 1.52 | 1.99
15 | 0.01 | 0.6 | 537 | 1 | 3.42 | 5.50 | 23.56 | 0.64 | 5.15 | 5.18 | 1.08 | 5.50 | 4.97 | 4.02 | 3.76 | 1.58 | 0.19 | 1.41 | 0.81 | 2.37
15 | 0.01 | 0.6 | 537 | 5 | 3.42 | 8.18 | 23.34 | 0.79 | 5.16 | 5.37 | 2.52 | 8.18 | 13.50 | 6.19 | 4.55 | 1.56 | 0.19 | 1.40 | 1.49 | 1.94
15 | 0.01 | 0.6 | 537 | 0.5 | 3.42 | 7.79 | 23.48 | 0.58 | 5.22 | 5.37 | 4.00 | 7.79 | 15.50 | 5.75 | 5.01 | 1.55 | 0.20 | 1.39 | 2.21 | 1.75
15 | 0.01 | 0.7 | 612 | 1 | 3.72 | 5.34 | 23.94 | 0.64 | 5.16 | 5.19 | 1.01 | 5.34 | 4.84 | 3.85 | 4.19 | 1.58 | 0.18 | 1.42 | 0.99 | 2.13
15 | 0.01 | 0.7 | 612 | 0.5 | 3.72 | 7.75 | 23.88 | 0.54 | 5.24 | 5.38 | 3.98 | 7.75 | 16.10 | 5.69 | 5.99 | 1.58 | 0.23 | 1.41 | 2.40 | 1.49
15 | 0.01 | 0.8 | 658 | 1 | 3.69 | 5.44 | 23.91 | 0.58 | 5.19 | 5.25 | 1.07 | 5.44 | 4.78 | 3.98 | 4.48 | 1.63 | 0.24 | 1.46 | 1.41 | 2.05
15 | 0.01 | 0.8 | 658 | 5 | 3.69 | 8.15 | 23.68 | 0.83 | 5.14 | 5.36 | 2.55 | 8.15 | 13.40 | 6.14 | 5.24 | 1.57 | 0.19 | 1.41 | 2.03 | 1.69
15 | 0.01 | 0.8 | 658 | 0.5 | 3.69 | 7.95 | 23.83 | 0.60 | 5.21 | 5.36 | 4.52 | 7.95 | 16.90 | 5.80 | 6.64 | 1.61 | 0.22 | 1.44 | 1.96 | 1.37
15 | 0.1 | 0.1 | 88 | 1 | 0.69 | 12.70 | 14.51 | 719.67 | 3.59 | 5.04 | 1.19 | 4.89 | 3.93 | 3.09 | 4.13 | 1.53 | 0.15 | 1.37 | 2.70 | 2.09
15 | 0.1 | 0.1 | 88 | 0.5 | 0.69 | 13.52 | 15.55 | 748.97 | 3.61 | 5.13 | 1.46 | 5.66 | 5.27 | 3.77 | 4.59 | 1.53 | 0.19 | 1.38 | 1.56 | 1.89
15 | 0.1 | 0.2 | 175 | 1 | 1.36 | 14.63 | 16.44 | 697.36 | 3.61 | 5.10 | 0.98 | 5.39 | 3.38 | 3.49 | 4.51 | 1.57 | 0.16 | 1.41 | 1.83 | 1.97
15 | 0.1 | 0.2 | 175 | 0.5 | 1.36 | 14.49 | 16.71 | 750.76 | 3.62 | 5.17 | 1.81 | 5.94 | 6.18 | 3.99 | 5.48 | 1.59 | 0.22 | 1.43 | 1.73 | 1.64
15 | 0.1 | 0.3 | 261 | 0.5 | 1.99 | 14.76 | 19.35 | 670.84 | 3.66 | 5.24 | 1.55 | 6.94 | 5.85 | 4.74 | 3.44 | 1.29 | 0.38 | 1.18 | 0.04 | 2.16
15 | 0.1 | 0.4 | 343 | 1 | 2.57 | 10.19 | 22.24 | 0.70 | 5.22 | 5.51 | 3.34 | 10.19 | 19.40 | 7.99 | 5.70 | 1.79 | 0.47 | 1.59 | 1.95 | 1.75
15 | 0.1 | 0.4 | 343 | 0.5 | 2.57 | 12.01 | 22.24 | 1.15 | 5.16 | 5.71 | 8.89 | 12.01 | 42.40 | 9.31 | 7.74 | 1.69 | 0.35 | 1.50 | 2.88 | 1.22
15 | 0.1 | 0.5 | 422 | 0.5 | 3.08 | 7.78 | 23.11 | 0.61 | 5.21 | 5.35 | 3.02 | 7.78 | 14.00 | 5.73 | 6.03 | 1.55 | 0.21 | 1.40 | 3.10 | 1.46
15 | 0.1 | 0.6 | 495 | 1 | 3.48 | 9.78 | 23.33 | 0.70 | 5.21 | 5.47 | 3.94 | 9.78 | 21.70 | 7.55 | 6.49 | 1.64 | 0.26 | 1.47 | 1.50 | 1.42
15 | 0.1 | 0.6 | 496 | 0.5 | 3.48 | 7.88 | 23.55 | 0.67 | 5.19 | 5.35 | 3.71 | 7.88 | 14.70 | 5.74 | 5.56 | 1.58 | 0.20 | 1.42 | 2.08 | 1.61
15 | 0.1 | 0.7 | 565 | 1 | 3.79 | 9.21 | 23.69 | 0.81 | 5.16 | 5.39 | 3.12 | 9.21 | 18.30 | 7.15 | 5.24 | 1.55 | 0.15 | 1.40 | 2.22 | 1.68
15 | 0.1 | 0.8 | 622 | 1 | 3.90 | 9.43 | 23.79 | 0.79 | 5.17 | 5.45 | 3.37 | 9.43 | 19.40 | 7.30 | 6.66 | 1.56 | 0.23 | 1.40 | 3.37 | 1.32
15 | 0.1 | 0.8 | 623 | 0.5 | 3.90 | 7.49 | 24.06 | 0.66 | 5.18 | 5.33 | 3.32 | 7.49 | 13.50 | 5.52 | 5.07 | 1.57 | 0.15 | 1.41 | 3.91 | 1.76
15 | 0.5 | 0.1 | 74 | 1 | 0.73 | 12.09 | 14.79 | 901.88 | 3.55 | 5.07 | 0.97 | 5.09 | 3.47 | 3.32 | 3.93 | 1.51 | 0.18 | 1.36 | 0.79 | 2.17
15 | 0.5 | 0.2 | 148 | 1 | 1.44 | 12.72 | 16.57 | 938.26 | 3.56 | 5.16 | 1.62 | 5.86 | 5.84 | 3.93 | 4.52 | 1.51 | 0.21 | 1.36 | 2.09 | 1.90
15 | 0.5 | 0.3 | 219 | 1 | 2.12 | 12.11 | 18.44 | 877.22 | 3.59 | 5.19 | 0.95 | 6.17 | 3.62 | 4.17 | 3.81 | 1.65 | 0.27 | 1.47 | 0.78 | 2.43
15 | 0.5 | 0.4 | 291 | 1 | 2.70 | 9.55 | 21.29 | 3.49 | 4.86 | 5.47 | 2.96 | 9.30 | 14.70 | 7.02 | 5.37 | 1.56 | 0.27 | 1.40 | 1.06 | 1.64
15 | 0.5 | 0.6 | 421 | 1 | 3.66 | 7.84 | 22.85 | 0.77 | 5.16 | 5.35 | 1.63 | 7.84 | 9.37 | 5.91 | 4.09 | 1.56 | 0.26 | 1.40 | 1.00 | 2.15
15 | 0.5 | 0.7 | 481 | 1 | 3.99 | 7.74 | 23.16 | 0.82 | 5.14 | 5.34 | 1.70 | 7.74 | 9.88 | 5.84 | 3.98 | 1.60 | 0.23 | 1.44 | 2.46 | 2.27
15 | 0.5 | 0.8 | 536 | 1 | 4.18 | 7.72 | 23.39 | 0.79 | 5.15 | 5.34 | 1.63 | 7.72 | 9.50 | 5.82 | 4.32 | 1.58 | 0.23 | 1.41 | 1.03 | 2.06
16 | 0.01 | 0.6 | 543 | 1 | 3.86 | 5.80 | 21.45 | 0.58 | 5.19 | 5.24 | 1.19 | 5.80 | 5.10 | 4.23 | 4.32 | 1.62 | 0.23 | 1.45 | 1.01 | 2.12
16 | 0.01 | 0.6 | 543 | 5 | 3.87 | 8.76 | 21.26 | 0.97 | 5.11 | 5.38 | 2.95 | 8.76 | 16.60 | 6.72 | 4.59 | 1.44 | 0.10 | 1.31 | 4.83 | 1.80
16 | 0.1 | 0.6 | 501 | 1 | 3.94 | 10.22 | 21.17 | 0.68 | 5.22 | 5.50 | 3.84 | 10.22 | 22.50 | 8.01 | 6.34 | 1.83 | 0.60 | 1.62 | 2.53 | 1.61
16 | 0.1 | 0.6 | 501 | 0.5 | 3.94 | 8.37 | 21.42 | 0.64 | 5.21 | 5.40 | 4.14 | 8.37 | 18.40 | 6.13 | 5.31 | 1.54 | 0.17 | 1.38 | 2.73 | 1.64
16 | 0.5 | 0.6 | 422 | 1 | 4.20 | 8.01 | 20.83 | 0.76 | 5.17 | 5.38 | 2.11 | 8.01 | 12.90 | 6.13 | 4.40 | 1.52 | 0.16 | 1.37 | 1.50 | 1.96
17 | 0.01 | 0.6 | 549 | 1 | 4.32 | 5.86 | 19.73 | 0.65 | 5.19 | 5.32 | 1.16 | 5.86 | 5.44 | 4.28 | 4.35 | 1.56 | 0.16 | 1.40 | 1.44 | 2.03
17 | 0.1 | 0.6 | 507 | 1 | 4.42 | 10.34 | 19.44 | 0.78 | 5.18 | 5.45 | 3.78 | 10.34 | 22.70 | 8.03 | 6.38 | 1.63 | 0.24 | 1.46 | 2.36 | 1.44
17 | 0.1 | 0.6 | 507 | 0.5 | 4.42 | 8.80 | 19.61 | 0.73 | 5.17 | 5.37 | 4.12 | 8.80 | 17.80 | 6.53 | 5.27 | 1.54 | 0.14 | 1.39 | 4.83 | 1.66
18 | 0.01 | 0.6 | 554 | 1 | 4.81 | 6.03 | 18.22 | 0.56 | 5.20 | 5.26 | 1.18 | 6.03 | 5.90 | 4.45 | 4.17 | 1.61 | 0.23 | 1.44 | 1.11 | 2.18
18 | 0.01 | 0.6 | 554 | 5 | 4.81 | 8.48 | 18.06 | 0.92 | 5.11 | 5.31 | 2.56 | 8.48 | 14.70 | 6.43 | 5.53 | 1.46 | 0.09 | 1.32 | 1.78 | 1.50
18 | 0.01 | 0.6 | 554 | 0.5 | 4.81 | 9.10 | 18.14 | 0.61 | 5.22 | 5.41 | 4.92 | 9.10 | 21.90 | 6.71 | 7.87 | 1.66 | 0.27 | 1.48 | 3.13 | 1.19
18 | 0.1 | 0.6 | 512 | 0.5 | 4.93 | 9.46 | 18.14 | 0.78 | 5.17 | 5.40 | 4.83 | 9.46 | 22.00 | 7.08 | 5.67 | 1.45 | 0.10 | 1.31 | 3.25 | 1.46
18 | 0.5 | 0.6 | 431 | 1 | 5.28 | 8.59 | 17.56 | 0.80 | 5.21 | 5.61 | 2.44 | 8.59 | 14.80 | 6.56 | 4.23 | 1.51 | 0.12 | 1.36 | 2.48 | 2.03
19 | 0.01 | 0.6 | 559 | 1 | 5.32 | 6.45 | 16.93 | 0.59 | 5.19 | 5.25 | 1.38 | 6.45 | 7.04 | 4.77 | 4.43 | 1.48 | 0.16 | 1.34 | 2.16 | 1.90
19 | 0.01 | 0.6 | 559 | 5 | 5.32 | 7.46 | 16.84 | 0.50 | 5.22 | 5.22 | 1.98 | 7.46 | 19.40 | 7.43 | 3.69 | 1.50 | 0.11 | 1.35 | 1.26 | 2.31
19 | 0.01 | 0.6 | 559 | 0.5 | 5.32 | 9.59 | 16.87 | 0.59 | 5.24 | 5.46 | 5.31 | 9.59 | 23.90 | 7.13 | 3.97 | 1.50 | 0.18 | 1.35 | 1.32 | 2.14
19 | 0.1 | 0.6 | 516 | 1 | 5.46 | 10.21 | 16.71 | 0.64 | 5.23 | 5.49 | 3.64 | 10.21 | 19.30 | 7.55 | 5.70 | 1.69 | 0.35 | 1.50 | 1.86 | 1.66
19 | 0.1 | 0.6 | 516 | 0.5 | 5.46 | 9.34 | 16.87 | 0.75 | 5.20 | 5.51 | 4.33 | 9.34 | 21.00 | 7.03 | 5.50 | 1.46 | 0.10 | 1.32 | 2.51 | 1.51
19 | 0.5 | 0.6 | 436 | 1 | 5.86 | 8.74 | 16.36 | 0.79 | 5.15 | 5.36 | 2.55 | 8.74 | 16.00 | 6.75 | 3.73 | 1.47 | 0.11 | 1.33 | 1.40 | 2.24
20 | 0.01 | 0.1 | 100 | 1 | 1.14 | 19.63 | 10.95 | 657.71 | 3.70 | 5.41 | 3.70 | 8.71 | 18.00 | 6.43 | 6.71 | 1.64 | 0.28 | 1.46 | 1.92 | 1.37
20 | 0.01 | 0.1 | 100 | 5 | 1.14 | 17.76 | 11.31 | 644.03 | 3.71 | 5.42 | 4.09 | 9.01 | 19.60 | 6.70 | 7.24 | 1.61 | 0.24 | 1.44 | 2.70 | 1.25
20 | 0.01 | 0.2 | 199 | 1 | 2.26 | 19.88 | 12.02 | 429.99 | 3.78 | 5.33 | 3.54 | 8.86 | 15.90 | 6.44 | 7.41 | 1.84 | 0.59 | 1.62 | 2.49 | 1.38
20 | 0.01 | 0.2 | 199 | 5 | 2.26 | 18.93 | 12.93 | 516.81 | 3.79 | 5.53 | 3.53 | 10.39 | 21.40 | 7.65 | 8.07 | 1.85 | 0.58 | 1.63 | 2.50 | 1.27
20 | 0.01 | 0.3 | 296 | 1 | 3.32 | 9.63 | 14.74 | 0.64 | 5.21 | 5.41 | 3.11 | 9.63 | 16.40 | 7.20 | 4.96 | 1.52 | 0.12 | 1.37 | 1.83 | 1.74
20 | 0.01 | 0.3 | 296 | 5 | 3.32 | 9.13 | 14.54 | 0.55 | 5.23 | 5.35 | 2.97 | 9.13 | 29.70 | 9.13 | 6.01 | 1.56 | 0.18 | 1.40 | 1.79 | 1.46
20 | 0.01 | 0.3 | 296 | 0.5 | 3.32 | 18.01 | 14.58 | 2.24 | 5.06 | 5.89 | 27.30 | 18.01 | 42.20 | 8.93 | 10.40 | 1.86 | 0.61 | 1.64 | 5.26 | 1.00
20 | 0.01 | 0.4 | 390 | 1 | 4.29 | 6.74 | 15.21 | 0.57 | 5.20 | 5.28 | 1.41 | 6.74 | 7.40 | 5.01 | 4.11 | 1.50 | 0.18 | 1.35 | 1.59 | 2.07
20 | 0.01 | 0.4 | 390 | 5 | 4.30 | 8.13 | 15.07 | 0.53 | 5.22 | 5.28 | 2.42 | 8.13 | 24.20 | 8.13 | 4.80 | 1.51 | 0.11 | 1.36 | 1.62 | 1.78
20 | 0.01 | 0.4 | 390 | 0.5 | 4.29 | 10.27 | 15.15 | 0.67 | 5.21 | 5.44 | 5.79 | 10.27 | 27.90 | 7.77 | 6.08 | 1.51 | 0.13 | 1.36 | 3.04 | 1.41
20 | 0.01 | 0.5 | 480 | 1 | 5.15 | 6.58 | 15.56 | 0.58 | 5.20 | 5.27 | 1.40 | 6.58 | 7.45 | 4.88 | 4.87 | 1.53 | 0.17 | 1.37 | 1.49 | 1.78
20 | 0.01 | 0.5 | 479 | 5 | 5.16 | 7.94 | 15.47 | 0.40 | 5.29 | 5.33 | 2.31 | 7.94 | 23.10 | 7.94 | 4.74 | 1.50 | 0.18 | 1.35 | 1.41 | 1.79
20 | 0.01 | 0.5 | 480 | 0.5 | 5.15 | 9.72 | 15.48 | 0.74 | 5.18 | 5.43 | 4.84 | 9.72 | 22.80 | 7.22 | 4.87 | 1.50 | 0.11 | 1.35 | 1.46 | 1.75
20 | 0.01 | 0.6 | 564 | 1 | 5.84 | 6.65 | 15.83 | 0.55 | 5.21 | 5.28 | 1.47 | 6.65 | 7.17 | 4.91 | 4.82 | 1.57 | 0.20 | 1.41 | 2.65 | 1.84
20 | 0.01 | 0.6 | 563 | 5 | 5.86 | 7.50 | 15.76 | 0.49 | 5.28 | 5.47 | 1.91 | 7.50 | 18.80 | 7.47 | 5.12 | 1.63 | 0.20 | 1.46 | 1.48 | 1.79
20 | 0.01 | 0.6 | 564 | 0.5 | 5.84 | 9.91 | 15.76 | 0.67 | 5.21 | 5.45 | 5.45 | 9.91 | 25.40 | 7.38 | 5.45 | 1.55 | 0.14 | 1.39 | 2.49 | 1.61
20 | 0.01 | 0.7 | 644 | 1 | 6.36 | 6.48 | 16.07 | 0.53 | 5.21 | 5.27 | 1.38 | 6.48 | 7.12 | 4.84 | 5.11 | 1.57 | 0.21 | 1.41 | 1.87 | 1.73
20 | 0.01 | 0.7 | 643 | 5 | 6.38 | 7.65 | 15.97 | 0.41 | 5.28 | 5.30 | 2.07 | 7.65 | 20.70 | 7.65 | 4.98 | 1.56 | 0.18 | 1.41 | 2.09 | 1.78
20 | 0.01 | 0.7 | 644 | 0.5 | 6.36 | 9.82 | 16.00 | 0.55 | 5.26 | 5.47 | 5.39 | 9.82 | 25.00 | 7.35 | 7.15 | 1.61 | 0.22 | 1.45 | 3.25 | 1.27
20 | 0.01 | 0.8 | 716 | 1 | 6.67 | 6.34 | 16.22 | 0.55 | 5.23 | 5.35 | 1.29 | 6.34 | 6.59 | 4.71 | 4.24 | 1.61 | 0.23 | 1.44 | 1.60 | 2.15
20 | 0.01 | 0.8 | 715 | 5 | 6.68 | 7.50 | 16.12 | 0.45 | 5.29 | 5.40 | 1.93 | 7.50 | 18.80 | 7.46 | 5.16 | 1.60 | 0.18 | 1.43 | 1.51 | 1.75
20 | 0.01 | 0.8 | 716 | 0.5 | 6.67 | 9.76 | 16.14 | 0.50 | 5.28 | 5.46 | 5.24 | 9.76 | 25.60 | 7.49 | 6.55 | 1.61 | 0.23 | 1.44 | 2.94 | 1.39
20 | 0.1 | 0.1 | 92 | 1 | 1.18 | 15.00 | 10.45 | 854.89 | 3.62 | 5.31 | 2.77 | 7.45 | 11.60 | 5.32 | 5.95 | 1.63 | 0.26 | 1.46 | 2.51 | 1.54
20 | 0.1 | 0.3 | 264 | 1 | 3.41 | 18.73 | 13.36 | 934.99 | 3.67 | 5.56 | 3.21 | 10.37 | 14.50 | 7.72 | 4.92 | 1.47 | 0.16 | 1.33 | 2.29 | 1.70
20 | 0.1 | 0.3 | 273 | 0.5 | 3.41 | 16.40 | 14.47 | 16.90 | 4.62 | 5.87 | 9.18 | 15.61 | 53.60 | 12.55 | 4.73 | 1.85 | 0.60 | 1.63 | 0.53 | 2.17
20 | 0.1 | 0.4 | 360 | 1 | 4.42 | 12.62 | 14.87 | 0.83 | 5.28 | 5.92 | 4.75 | 12.62 | 30.20 | 10.04 | 6.59 | 1.58 | 0.22 | 1.41 | 3.52 | 1.35
20 | 0.1 | 0.4 | 360 | 0.5 | 4.42 | 11.66 | 15.09 | 0.45 | 5.34 | 5.64 | 6.45 | 11.66 | 32.30 | 8.81 | 7.46 | 1.60 | 0.26 | 1.44 | 2.08 | 1.21
20 | 0.1 | 0.5 | 443 | 1 | 5.29 | 11.91 | 15.28 | 0.54 | 5.29 | 5.58 | 4.65 | 11.91 | 27.60 | 9.25 | 5.42 | 1.58 | 0.27 | 1.41 | 3.61 | 1.64
20 | 0.1 | 0.5 | 443 | 0.5 | 5.30 | 9.79 | 15.47 | 0.75 | 5.18 | 5.41 | 4.19 | 9.79 | 20.70 | 7.32 | 5.20 | 1.49 | 0.11 | 1.34 | 4.68 | 1.63
20 | 0.1 | 0.6 | 521 | 1 | 6.01 | 11.60 | 15.58 | 0.59 | 5.27 | 5.56 | 4.62 | 11.60 | 29.00 | 9.18 | 5.98 | 1.65 | 0.30 | 1.48 | 2.74 | 1.56
20 | 0.1 | 0.6 | 521 | 0.5 | 6.01 | 9.42 | 15.76 | 0.73 | 5.18 | 5.40 | 3.79 | 9.42 | 19.40 | 7.11 | 4.76 | 1.49 | 0.11 | 1.34 | 2.60 | 1.78
20 | 0.1 | 0.7 | 596 | 0.5 | 6.55 | 9.33 | 16.00 | 0.77 | 5.17 | 5.39 | 4.20 | 9.33 | 20.40 | 7.05 | 5.35 | 1.47 | 0.09 | 1.33 | 2.66 | 1.56
20 | 0.1 | 0.8 | 664 | 1 | 6.89 | 11.76 | 15.96 | 0.51 | 5.32 | 5.65 | 5.41 | 11.76 | 33.50 | 9.52 | 6.65 | 1.76 | 0.44 | 1.56 | 4.44 | 1.48
20 | 0.1 | 0.8 | 665 | 0.5 | 6.88 | 9.09 | 16.14 | 0.50 | 5.27 | 5.44 | 4.28 | 9.09 | 20.40 | 6.86 | 6.32 | 1.68 | 0.30 | 1.49 | 2.50 | 1.49
20 | 0.5 | 0.4 | 305 | 1 | 4.72 | 9.60 | 14.50 | 1.01 | 5.11 | 5.41 | 2.77 | 9.60 | 16.10 | 7.25 | 4.14 | 1.53 | 0.11 | 1.37 | 0.82 | 2.09
20 | 0.5 | 0.6 | 440 | 1 | 6.47 | 9.02 | 15.28 | 0.87 | 5.15 | 5.41 | 2.68 | 9.02 | 15.30 | 6.69 | 4.97 | 1.46 | 0.10 | 1.32 | 2.64 | 1.68
21 | 0.01 | 0.6 | 568 | 1 | 6.40 | 6.56 | 14.90 | 0.54 | 5.22 | 5.28 | 1.35 | 6.56 | 7.33 | 4.89 | 4.21 | 1.54 | 0.21 | 1.39 | 1.29 | 2.07
21 | 0.01 | 0.6 | 568 | 5 | 6.39 | 8.08 | 14.84 | 0.55 | 5.21 | 5.29 | 2.32 | 8.08 | 23.20 | 8.08 | 5.64 | 1.49 | 0.12 | 1.35 | 2.33 | 1.51
21 | 0.01 | 0.6 | 568 | 0.5 | 6.40 | 10.31 | 14.82 | 0.76 | 5.18 | 5.45 | 5.73 | 10.31 | 26.30 | 7.68 | 5.77 | 1.44 | 0.10 | 1.31 | 3.69 | 1.43
21 | 0.1 | 0.6 | 525 | 1 | 6.58 | 11.82 | 14.67 | 0.59 | 5.27 | 5.57 | 4.68 | 11.82 | 28.70 | 9.30 | 6.63 | 1.55 | 0.23 | 1.39 | 3.54 | 1.32
21 | 0.1 | 0.6 | 525 | 0.5 | 6.58 | 9.70 | 14.81 | 0.71 | 5.19 | 5.42 | 3.83 | 9.70 | 19.90 | 7.29 | 5.15 | 1.53 | 0.13 | 1.38 | 3.02 | 1.68
21 | 0.5 | 0.6 | 443 | 1 | 7.11 | 9.02 | 14.37 | 0.99 | 5.15 | 5.53 | 2.69 | 9.02 | 15.50 | 6.77 | 5.30 | 1.60 | 0.16 | 1.43 | 1.66 | 1.70
22 | 0.01 | 0.6 | 572 | 1 | 6.97 | 6.93 | 14.05 | 0.59 | 5.20 | 5.29 | 1.49 | 6.93 | 7.83 | 5.15 | 4.73 | 1.56 | 0.17 | 1.40 | 1.37 | 1.87
22 | 0.01 | 0.6 | 572 | 5 | 6.98 | 7.97 | 13.98 | 0.46 | 5.27 | 5.35 | 1.95 | 7.97 | 19.00 | 7.92 | 4.88 | 1.51 | 0.21 | 1.36 | 1.65 | 1.75
22 | 0.01 | 0.6 | 572 | 0.5 | 6.97 | 10.43 | 13.99 | 0.66 | 5.22 | 5.48 | 5.31 | 10.43 | 25.50 | 7.72 | 7.62 | 1.66 | 0.19 | 1.48 | 3.02 | 1.22
22 | 0.1 | 0.6 | 530 | 0.5 | 7.18 | 9.88 | 13.99 | 0.67 | 5.21 | 5.43 | 3.84 | 9.88 | 19.60 | 7.41 | 5.96 | 1.51 | 0.16 | 1.36 | 2.60 | 1.44
22 | 0.5 | 0.6 | 448 | 1 | 7.75 | 9.08 | 13.57 | 1.08 | 5.09 | 5.36 | 2.74 | 9.08 | 16.20 | 6.88 | 5.20 | 1.42 | 0.08 | 1.29 | 1.64 | 1.56
23 | 0.01 | 0.6 | 577 | 1 | 7.56 | 6.98 | 13.31 | 0.52 | 5.23 | 5.32 | 1.48 | 6.98 | 8.08 | 5.21 | 4.69 | 1.57 | 0.21 | 1.41 | 2.76 | 1.89
23 | 0.01 | 0.6 | 576 | 5 | 7.58 | 6.35 | 13.32 | 0.48 | 5.23 | 5.23 | 1.04 | 6.35 | 6.06 | 4.80 | 3.50 | 1.50 | 0.16 | 1.35 | 2.26 | 2.43
23 | 0.01 | 0.6 | 577 | 0.5 | 7.56 | 11.02 | 13.24 | 0.57 | 5.26 | 5.53 | 5.92 | 11.02 | 28.30 | 8.19 | 8.15 | 1.82 | 0.50 | 1.60 | 2.28 | 1.24
23 | 0.1 | 0.6 | 533 | 1 | 7.80 | 12.45 | 13.13 | 0.54 | 5.30 | 5.61 | 5.27 | 12.45 | 30.40 | 9.57 | 6.74 | 1.60 | 0.27 | 1.43 | 4.88 | 1.34
23 | 0.1 | 0.6 | 533 | 0.5 | 7.80 | 10.03 | 13.25 | 0.63 | 5.22 | 5.44 | 4.12 | 10.03 | 21.50 | 7.59 | 6.08 | 1.57 | 0.20 | 1.41 | 2.23 | 1.46
23 | 0.5 | 0.6 | 451 | 1 | 8.44 | 9.16 | 12.86 | 0.74 | 5.18 | 5.43 | 2.83 | 9.16 | 16.60 | 6.95 | 4.11 | 1.56 | 0.19 | 1.41 | 3.37 | 2.15
24 | 0.01 | 0.6 | 580 | 5 | 8.18 | 6.45 | 12.65 | 0.44 | 5.25 | 5.25 | 1.03 | 6.45 | 6.00 | 4.89 | 3.83 | 1.59 | 0.20 | 1.42 | 1.86 | 2.34
24 | 0.01 | 0.6 | 581 | 0.5 | 8.16 | 11.41 | 12.57 | 0.50 | 5.30 | 5.56 | 6.18 | 11.41 | 29.40 | 8.48 | 7.58 | 1.70 | 0.36 | 1.51 | 2.86 | 1.26
24 | 0.1 | 0.6 | 537 | 1 | 8.44 | 12.59 | 12.48 | 0.75 | 5.23 | 5.61 | 5.78 | 12.59 | 31.20 | 9.49 | 4.87 | 1.50 | 0.14 | 1.36 | 1.66 | 1.75
24 | 0.1 | 0.6 | 537 | 0.5 | 8.44 | 10.92 | 12.57 | 0.59 | 5.24 | 5.48 | 4.82 | 10.92 | 24.80 | 8.23 | 7.28 | 1.65 | 0.29 | 1.47 | 6.77 | 1.27
24 | 0.5 | 0.6 | 453 | 1 | 9.17 | 9.17 | 12.24 | 1.00 | 5.11 | 5.39 | 2.93 | 9.17 | 16.70 | 6.96 | 5.36 | 1.52 | 0.09 | 1.37 | 2.28 | 1.61
25 | 0.01 | 0.1 | 103 | 0.5 | 1.70 | 24.93 | 8.65 | 110.83 | 4.11 | 5.48 | 3.71 | 9.73 | 17.40 | 7.08 | 7.03 | 1.92 | 0.66 | 1.69 | 2.26 | 1.51
25 | 0.01 | 0.2 | 205 | 1 | 3.38 | 24.72 | 9.92 | 109.61 | 4.15 | 5.64 | 7.79 | 13.45 | 43.00 | 10.49 | 10.00 | 1.80 | 0.43 | 1.60 | 5.70 | 1.00
25 | 0.01 | 0.3 | 306 | 1 | 4.96 | 9.38 | 11.32 | 0.64 | 5.21 | 5.39 | 2.46 | 9.38 | 13.70 | 7.06 | 5.33 | 1.51 | 0.12 | 1.36 | 2.27 | 1.60
25 | 0.01 | 0.4 | 404 | 1 | 6.43 | 7.63 | 11.62 | 0.51 | 5.25 | 5.35 | 1.66 | 7.63 | 9.37 | 5.72 | 4.09 | 1.57 | 0.21 | 1.41 | 1.21 | 2.17
25 | 0.01 | 0.4 | 404 | 0.5 | 6.43 | 11.74 | 11.56 | 0.48 | 5.32 | 5.58 | 5.81 | 11.74 | 29.00 | 8.80 | 6.72 | 1.73 | 0.38 | 1.54 | 2.01 | 1.44
25 | 0.01 | 0.5 | 497 | 1 | 7.72 | 7.67 | 11.85 | 0.53 | 5.24 | 5.36 | 1.73 | 7.67 | 9.59 | 5.75 | 4.39 | 1.52 | 0.20 | 1.37 | 2.12 | 1.97
25 | 0.01 | 0.5 | 497 | 0.5 | 7.72 | 11.83 | 11.79 | 0.50 | 5.31 | 5.61 | 6.28 | 11.83 | 29.80 | 8.75 | 7.02 | 1.68 | 0.33 | 1.50 | 2.09 | 1.34
25 | 0.01 | 0.6 | 585 | 1 | 8.79 | 7.46 | 12.07 | 0.49 | 5.26 | 5.39 | 1.65 | 7.46 | 9.09 | 5.58 | 4.80 | 1.55 | 0.23 | 1.40 | 1.27 | 1.83
25 | 0.01 | 0.6 | 584 | 5 | 8.82 | 6.56 | 12.07 | 0.42 | 5.27 | 5.27 | 1.15 | 6.56 | 6.64 | 4.96 | 4.40 | 1.63 | 0.23 | 1.45 | 1.21 | 2.08
25 | 0.01 | 0.6 | 585 | 0.5 | 8.79 | 11.45 | 11.99 | 0.51 | 5.30 | 5.57 | 5.52 | 11.45 | 27.10 | 8.53 | 7.09 | 1.76 | 0.45 | 1.56 | 2.41 | 1.39
25 | 0.01 | 0.7 | 669 | 1 | 9.60 | 7.42 | 12.24 | 0.50 | 5.25 | 5.35 | 1.64 | 7.42 | 9.19 | 5.57 | 5.24 | 1.58 | 0.23 | 1.42 | 2.19 | 1.70
25 | 0.01 | 0.7 | 669 | 0.5 | 9.60 | 11.86 | 12.17 | 0.49 | 5.31 | 5.59 | 6.91 | 11.86 | 33.50 | 8.86 | 7.37 | 1.66 | 0.28 | 1.48 | 3.03 | 1.27
25 | 0.01 | 0.8 | 747 | 1 | 10.10 | 7.52 | 12.34 | 0.51 | 5.24 | 5.34 | 1.70 | 7.52 | 9.40 | 5.63 | 4.50 | 1.58 | 0.21 | 1.41 | 1.45 | 1.98
25 | 0.01 | 0.8 | 747 | 0.5 | 10.10 | 11.36 | 12.29 | 0.60 | 5.26 | 5.54 | 5.70 | 11.36 | 27.00 | 8.37 | 7.27 | 1.79 | 0.54 | 1.58 | 2.23 | 1.37
25 | 0.1 | 0.1 | 95 | 1 | 1.76 | 13.86 | 8.96 | 1271.00 | 3.61 | 5.62 | 6.40 | 11.93 | 34.00 | 9.29 | 8.47 | 1.72 | 0.36 | 1.53 | 4.09 | 1.14
25 | 0.1 | 0.2 | 190 | 1 | 3.49 | 23.86 | 9.46 | 955.17 | 3.70 | 5.71 | 4.36 | 11.00 | 21.50 | 8.28 | 5.63 | 1.82 | 0.57 | 1.61 | 0.46 | 1.80
25 | 0.1 | 0.2 | 190 | 0.5 | 3.49 | 24.36 | 9.58 | 1085.14 | 3.65 | 5.62 | 5.08 | 12.42 | 26.90 | 9.68 | 6.87 | 1.79 | 0.45 | 1.58 | 2.48 | 1.45
25 | 0.1 | 0.3 | 283 | 1 | 5.13 | 19.98 | 10.77 | 1501.66 | 3.62 | 5.81 | 7.04 | 16.04 | 32.40 | 12.02 | 6.66 | 1.76 | 0.39 | 1.56 | 3.08 | 1.47
25 | 0.1 | 0.4 | 372 | 1 | 6.65 | 12.87 | 11.39 | 0.55 | 5.30 | 5.63 | 5.54 | 12.87 | 53.70 | 12.80 | 4.58 | 1.53 | 0.16 | 1.38 | 3.13 | 1.90
25 | 0.1 | 0.4 | 372 | 0.5 | 6.65 | 13.20 | 11.54 | 0.63 | 5.27 | 5.65 | 7.34 | 13.20 | 39.10 | 10.19 | 5.43 | 1.53 | 0.14 | 1.38 | 2.11 | 1.60
25 | 0.1 | 0.5 | 458 | 1 | 7.99 | 12.69 | 11.68 | 0.71 | 5.24 | 5.61 | 5.75 | 12.69 | 31.70 | 9.63 | 5.20 | 1.53 | 0.13 | 1.38 | 2.75 | 1.67
25 | 0.1 | 0.6 | 540 | 1 | 9.10 | 12.51 | 11.91 | 0.71 | 5.23 | 5.60 | 5.81 | 12.51 | 31.60 | 9.47 | 6.83 | 1.57 | 0.13 | 1.41 | 6.02 | 1.30
25 | 0.1 | 0.7 | 620 | 1 | 9.95 | 11.94 | 12.06 | 0.65 | 5.28 | 5.70 | 5.16 | 11.94 | 27.20 | 9.04 | 7.14 | 1.63 | 0.20 | 1.46 | 4.28 | 1.29
25 | 0.1 | 0.7 | 619 | 0.5 | 9.95 | 11.21 | 12.15 | 0.51 | 5.32 | 5.66 | 5.71 | 11.21 | 27.20 | 8.30 | 6.95 | 1.77 | 0.50 | 1.57 | 2.08 | 1.42
25 | 0.1 | 0.8 | 693 | 0.5 | 10.50 | 10.79 | 12.26 | 0.65 | 5.23 | 5.50 | 5.36 | 10.79 | 24.70 | 7.88 | 7.60 | 1.81 | 0.57 | 1.60 | 2.56 | 1.33
25 | 0.5 | 0.4 | 315 | 1 | 7.20 | 9.32 | 11.16 | 0.63 | 5.20 | 5.37 | 3.11 | 9.32 | 16.70 | 7.04 | 5.02 | 1.53 | 0.15 | 1.38 | 2.69 | 1.73
25 | 0.5 | 0.5 | 389 | 1 | 8.62 | 8.54 | 11.44 | 0.50 | 5.26 | 5.39 | 2.57 | 8.54 | 25.10 | 8.51 | 5.77 | 1.72 | 0.35 | 1.53 | 1.35 | 1.67
25 | 0.5 | 0.6 | 458 | 1 | 9.87 | 9.11 | 11.70 | 0.91 | 5.13 | 5.38 | 2.90 | 9.11 | 16.20 | 6.90 | 5.03 | 1.48 | 0.08 | 1.34 | 2.05 | 1.67
25 | 0.5 | 0.6 | 458 | 1 | 9.88 | 9.04 | 11.65 | 0.72 | 5.17 | 5.34 | 2.98 | 9.04 | 15.70 | 6.83 | 4.49 | 1.52 | 0.09 | 1.37 | 3.77 | 1.92
26 | 0.01 | 0.6 | 588 | 1 | 9.44 | 7.61 | 11.54 | 0.55 | 5.23 | 5.33 | 1.66 | 7.61 | 9.36 | 5.72 | 3.73 | 1.56 | 0.18 | 1.40 | 1.56 | 2.37
26 | 0.01 | 0.6 | 588 | 5 | 9.44 | 5.79 | 11.62 | 0.43 | 5.25 | 5.21 | 0.82 | 5.79 | 4.82 | 4.38 | 3.83 | 1.62 | 0.23 | 1.45 | 1.17 | 2.38
26 | 0.01 | 0.6 | 588 | 0.5 | 9.44 | 11.66 | 11.48 | 0.50 | 5.30 | 5.57 | 5.46 | 11.66 | 26.70 | 8.66 | 6.68 | 1.68 | 0.34 | 1.50 | 1.90 | 1.41
26 | 0.1 | 0.6 | 544 | 1 | 9.77 | 12.34 | 11.37 | 0.60 | 5.27 | 5.58 | 5.47 | 12.34 | 52.90 | 12.27 | 5.83 | 1.48 | 0.13 | 1.34 | 3.71 | 1.45
26 | 0.5 | 0.6 | 460 | 1 | 10.60 | 8.79 | 11.22 | 0.71 | 5.17 | 5.32 | 2.81 | 8.79 | 15.00 | 6.67 | 4.88 | 1.45 | 0.09 | 1.32 | 2.92 | 1.70
27 | 0.01 | 0.6 | 592 | 1 | 10.10 | 7.82 | 11.07 | 0.52 | 5.25 | 5.37 | 1.73 | 7.82 | 9.71 | 5.87 | 4.08 | 1.55 | 0.19 | 1.39 | 1.25 | 2.15
27 | 0.1 | 0.6 | 547 | 1 | 10.50 | 12.41 | 10.91 | 0.63 | 5.25 | 5.57 | 5.20 | 12.41 | 51.00 | 12.38 | 5.00 | 1.47 | 0.14 | 1.33 | 2.84 | 1.68
27 | 0.1 | 0.6 | 548 | 0.5 | 10.50 | 11.82 | 10.97 | 0.52 | 5.30 | 5.58 | 5.27 | 11.82 | 27.10 | 8.88 | 6.37 | 1.58 | 0.23 | 1.42 | 2.53 | 1.40
27 | 0.5 | 0.6 | 463 | 1 | 11.40 | 8.89 | 10.76 | 0.68 | 5.18 | 5.33 | 2.87 | 8.89 | 15.50 | 6.73 | 5.17 | 1.51 | 0.10 | 1.36 | 3.87 | 1.65
28 | 0.01 | 0.6 | 595 | 1 | 10.80 | 8.22 | 10.64 | 0.53 | 5.24 | 5.38 | 1.93 | 8.22 | 10.90 | 6.18 | 4.48 | 1.55 | 0.19 | 1.40 | 1.96 | 1.96
28 | 0.01 | 0.6 | 594 | 5 | 10.80 | 5.64 | 10.73 | 0.52 | 5.20 | 5.16 | 0.77 | 5.64 | 4.49 | 4.27 | 3.62 | 1.54 | 0.14 | 1.38 | 2.10 | 2.41
28 | 0.01 | 0.6 | 595 | 0.5 | 10.80 | 12.95 | 10.57 | 0.53 | 5.31 | 5.64 | 7.26 | 12.95 | 69.80 | 12.88 | 4.92 | 1.55 | 0.22 | 1.39 | 3.88 | 1.78
28 | 0.1 | 0.6 | 550 | 1 | 11.20 | 12.28 | 10.48 | 0.59 | 5.27 | 5.58 | 5.29 | 12.28 | 51.70 | 12.24 | 4.20 | 1.51 | 0.15 | 1.36 | 1.62 | 2.04
28 | 0.1 | 0.6 | 550 | 0.5 | 11.20 | 11.69 | 10.55 | 0.50 | 5.30 | 5.57 | 4.96 | 11.69 | 26.40 | 8.86 | 6.86 | 1.57 | 0.23 | 1.41 | 4.52 | 1.29
28 | 0.5 | 0.6 | 466 | 1 | 12.20 | 8.76 | 10.36 | 0.64 | 5.19 | 5.34 | 2.73 | 8.76 | 25.70 | 8.66 | 4.45 | 1.46 | 0.10 | 1.32 | 2.49 | 1.86
29 | 0.01 | 0.6 | 599 | 1 | 11.50 | 8.33 | 10.24 | 0.52 | 5.25 | 5.38 | 1.93 | 8.33 | 10.90 | 6.27 | 4.38 | 1.53 | 0.19 | 1.37 | 1.09 | 1.97
29 | 0.1 | 0.3 | 288 | 1 | 6.69 | 21.15 | 9.32 | 61.34 | 4.37 | 6.02 | 12.20 | 19.47 | 65.20 | 15.32 | 8.99 | 1.82 | 0.49 | 1.60 | 4.70 | 1.12
29 | 0.1 | 0.6 | 554 | 1 | 11.90 | 12.27 | 10.13 | 0.53 | 5.35 | 5.78 | 5.35 | 12.27 | 52.50 | 12.23 | 6.45 | 1.61 | 0.24 | 1.44 | 3.95 | 1.40
29 | 0.1 | 0.6 | 554 | 0.5 | 11.90 | 11.97 | 10.17 | 0.48 | 5.32 | 5.60 | 4.93 | 11.97 | 27.20 | 9.19 | 5.74 | 1.57 | 0.24 | 1.41 | 2.32 | 1.55
29 | 0.5 | 0.6 | 469 | 1 | 13.00 | 9.02 | 9.96 | 0.41 | 5.31 | 5.43 | 2.79 | 9.02 | 26.80 | 8.96 | 5.68 | 1.57 | 0.25 | 1.41 | 2.06 | 1.56
30 | 0.01 | 0.1 | 106 | 1 | 2.35 | 29.79 | 7.16 | 63.99 | 4.26 | 5.62 | 3.74 | 11.07 | 19.40 | 8.28 | 6.05 | 1.80 | 0.54 | 1.59 | 1.24 | 1.66
30 | 0.01 | 0.2 | 211 | 0.5 | 4.66 | 29.70 | 8.61 | 117.05 | 4.23 | 6.03 | 11.70 | 18.32 | 59.80 | 14.10 | 9.06 | 1.82 | 0.56 | 1.61 | 3.79 | 1.12
30 | 0.01 | 0.3 | 314 | 1 | 6.86 | 13.59 | 9.27 | 0.67 | 5.26 | 5.64 | 5.15 | 13.59 | 50.20 | 13.53 | 5.54 | 1.51 | 0.14 | 1.36 | 3.93 | 1.55
30 | 0.01 | 0.3 | 314 | 0.5 | 6.86 | 14.20 | 9.33 | 0.62 | 5.29 | 5.69 | 7.23 | 14.20 | 38.40 | 10.86 | 5.94 | 1.51 | 0.14 | 1.36 | 4.28 | 1.44
30 | 0.01 | 0.4 | 415 | 1 | 8.90 | 8.94 | 9.53 | 0.48 | 5.28 | 5.44 | 2.15 | 8.94 | 12.20 | 6.77 | 4.85 | 1.64 | 0.24 | 1.46 | 1.43 | 1.90
30 | 0.01 | 0.4 | 415 | 5 | 8.90 | 6.12 | 9.57 | 0.42 | 5.26 | 5.24 | 0.92 | 6.12 | 5.35 | 4.63 | 3.94 | 1.60 | 0.22 | 1.43 | 1.52 | 2.29
30 | 0.01 | 0.4 | 415 | 0.5 | 8.90 | 13.92 | 9.48 | 0.70 | 5.25 | 5.66 | 7.46 | 13.92 | 71.80 | 13.85 | 5.56 | 1.59 | 0.14 | 1.43 | 2.12 | 1.62
30 | 0.01 | 0.5 | 511 | 1 | 10.70 | 8.57 | 9.72 | 0.63 | 5.20 | 5.36 | 1.92 | 8.57 | 10.70 | 6.42 | 4.11 | 1.48 | 0.14 | 1.33 | 1.56 | 2.04
30 | 0.01 | 0.5 | 511 | 0.5 | 10.70 | 13.76 | 9.65 | 0.64 | 5.28 | 5.67 | 7.72 | 13.76 | 38.50 | 10.40 | 6.12 | 1.56 | 0.15 | 1.40 | 3.97 | 1.44
30 | 0.01 | 0.6 | 602 | 1 | 12.20 | 8.50 | 9.88 | 0.60 | 5.21 | 5.36 | 2.02 | 8.50 | 11.40 | 6.38 | 5.17 | 1.53 | 0.17 | 1.38 | 1.31 | 1.68
30 | 0.01 | 0.6 | 602 | 5 | 12.20 | 5.06 | 10.01 | 0.49 | 5.20 | 5.14 | 0.63 | 5.06 | 3.67 | 3.81 | 3.60 | 1.48 | 0.16 | 1.34 | 1.35 | 2.34
30 | 0.01 | 0.6 | 602 | 0.5 | 12.20 | 13.72 | 9.82 | 0.69 | 5.25 | 5.65 | 7.76 | 13.72 | 74.40 | 13.64 | 6.28 | 1.53 | 0.14 | 1.38 | 5.70 | 1.38
30 | 0.01 | 0.7 | 690 | 1 | 13.40 | 8.59 | 10.02 | 0.63 | 5.20 | 5.36 | 2.07 | 8.59 | 11.80 | 6.48 | 5.19 | 1.55 | 0.15 | 1.39 | 1.59 | 1.69
30 | 0.01 | 0.7 | 690 | 0.5 | 13.40 | 13.55 | 9.96 | 0.70 | 5.25 | 5.64 | 7.52 | 13.55 | 71.70 | 13.47 | 4.88 | 1.49 | 0.12 | 1.35 | 5.03 | 1.74
30 | 0.01 | 0.8 | 773 | 1 | 14.10 | 8.37 | 10.11 | 0.61 | 5.21 | 5.35 | 1.97 | 8.37 | 11.00 | 6.29 | 3.99 | 1.50 | 0.15 | 1.35 | 6.30 | 2.13
30 | 0.01 | 0.8 | 773 | 0.5 | 14.10 | 13.55 | 10.05 | 0.70 | 5.25 | 5.64 | 7.50 | 13.55 | 72.30 | 13.48 | 5.49 | 1.49 | 0.13 | 1.35 | 2.56 | 1.55
30 | 0.1 | 0.1 | 98 | 1 | 2.44 | 27.05 | 7.46 | 1310.05 | 3.63 | 5.72 | 8.38 | 14.26 | 46.60 | 11.34 | 7.26 | 1.77 | 0.52 | 1.57 | 3.91 | 1.36
30 | 0.1 | 0.2 | 194 | 0.5 | 4.82 | 29.15 | 7.99 | 1028.65 | 3.67 | 5.67 | 5.93 | 13.64 | 32.70 | 10.56 | 5.85 | 1.82 | 0.57 | 1.61 | 0.59 | 1.73
30 | 0.1 | 0.5 | 471 | 1 | 11.10 | 12.67 | 9.60 | 0.62 | 5.26 | 5.58 | 5.22 | 12.67 | 51.70 | 12.65 | 5.45 | 1.52 | 0.22 | 1.37 | 4.03 | 1.58
30 | 0.1 | 0.5 | 471 | 0.5 | 11.10 | 12.83 | 9.65 | 0.62 | 5.27 | 5.61 | 5.44 | 12.83 | 30.80 | 9.93 | 5.34 | 1.53 | 0.14 | 1.38 | 3.92 | 1.63
30 | 0.1 | 0.6 | 556 | 1 | 12.70 | 12.91 | 9.76 | 0.63 | 5.26 | 5.60 | 5.74 | 12.91 | 56.40 | 12.88 | 6.40 | 1.61 | 0.15 | 1.44 | 2.92 | 1.42
30 | 0.1 | 0.6 | 557 | 0.5 | 12.70 | 12.35 | 9.81 | 0.51 | 5.31 | 5.61 | 5.32 | 12.35 | 28.30 | 9.36 | 6.43 | 1.62 | 0.25 | 1.45 | 3.01 | 1.42
30 | 0.1 | 0.7 | 639 | 0.5 | 13.90 | 12.39 | 9.95 | 0.54 | 5.30 | 5.62 | 6.22 | 12.39 | 30.30 | 9.16 | 5.27 | 1.56 | 0.22 | 1.40 | 1.82 | 1.67
30 | 0.5 | 0.3 | 244 | 1 | 7.77 | 14.54 | 8.65 | 0.58 | 5.34 | 5.82 | 7.33 | 14.54 | 69.30 | 14.39 | 6.89 | 1.71 | 0.32 | 1.52 | 3.65 | 1.39
30 | 0.5 | 0.4 | 322 | 1 | 10.10 | 9.53 | 9.23 | 0.61 | 5.27 | 5.63 | 3.00 | 9.53 | 28.90 | 9.46 | 4.52 | 1.54 | 0.16 | 1.38 | 2.84 | 1.92
30 | 0.5 | 0.5 | 398 | 1 | 12.10 | 9.07 | 9.42 | 0.49 | 5.27 | 5.42 | 2.67 | 9.07 | 14.40 | 6.86 | 3.88 | 1.52 | 0.20 | 1.37 | 1.43 | 2.22
30 | 0.5 | 0.6 | 471 | 1 | 13.90 | 9.14 | 9.63 | 0.63 | 5.20 | 5.36 | 2.89 | 9.14 | 27.70 | 9.07 | 5.46 | 1.48 | 0.11 | 1.34 | 2.42 | 1.54
30 | 0.5 | 0.7 | 543 | 1 | 15.30 | 9.53 | 9.76 | 0.69 | 5.19 | 5.40 | 3.21 | 9.53 | 17.40 | 7.20 | 5.45 | 1.49 | 0.13 | 1.34 | 1.52 | 1.55
30 | 0.5 | 0.8 | 610 | 1 | 16.20 | 9.42 | 9.50 | 0.62 | 5.21 | 5.39 | 3.04 | 9.42 | 29.20 | 9.35 | 4.71 | 1.53 | 0.14 | 1.37 | 2.02 | 1.84

Notes. The columns are the initial mass $M$ (Column 1), the initial
metallicity $Z$ (Column 2), the initial rotation rate $\Omega$ (Column 3) and
corresponding velocity at the equator $V_{\rm equ}$ (Column 4), the “Dutch”
wind scale factor $\eta_{\rm wind}$ (Column 5), the initial total angular
momentum $J_{\rm ini}$ (Column 6), final mass $M_{\rm f}$ (Column 7), final
star ages (Column 8), final radius $R_{\rm f}$ (Column 9), final effective
temperature $\log T_{\rm eff}$ (Column 10) and luminosity $\log L$ (Column 11)
in logarithmic form, final angular momentum in He core $J_{\rm He~{}core}$
(Column 12), He core mass at the end of calculation $M_{\rm He~{}core}$
(Column 13), final angular momentum in CO core $J_{\rm CO~{}core}$ (Column
14), CO core mass at the end of calculation $M_{\rm CO~{}core}$ (Column 15),
final angular momentum in Fe core $J_{\rm Fe~{}core}$ (Column 16), Fe core
mass at the end calculation $M_{\rm Fe~{}core}$ (Column 17), the compactness
parameter $\xi_{2.5}$ (Column 18), the NS mass $M_{\rm NS}$ (Column 19), the
average surface magnetic field strength $B_{\rm NS}$ (Column 20), and NS
rotation period $P_{\rm NS}$ (Column 21). He, CO and Fe core masses have been
measured using $10\%$ mass fraction point X(H) $\leq 0.1$ and X(He) $\geq 0.1$
for $M_{\rm He~{}core}$, X(He) $\leq 0.1$ and X(C+O) $\geq 0.1$ for $M_{\rm
CO~{}core}$, X(Si)$\leq 0.1$ and X(iron) $\geq 0.1$ for $M_{\rm Fe~{}core}$.
For all cases, “iron” includes any species with A > 46.