11institutetext: Department of Astronomy, University of Geneva, Chemin Pegasi
51, 1290 Versoix, Switzerland
11email<EMAIL_ADDRESS>22institutetext: Department of Physics and
Astronomy, University of Notre Dame, 225 Nieuwland Science Hall, Notre Dame,
IN 46556, USA 33institutetext: Astronomical Institute, Tohoku University, 6-3
Aoba, Aramaki, Aoba-ku, Sendai, Miyagi 980-8578, Japan 44institutetext:
Institut d’Astronomie et d’Astrophysique, Université Libre de Bruxelles, CP
226, 1050, Brussels, Belgium 55institutetext: Astrophysics Group, Keele
University, Keele, Staffordshire ST5 5BG, UK 66institutetext: Leibniz-Institut
für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany
77institutetext: Institute of Astronomy, University of Cambridge, Madingley
Road, Cambridge, CB3 0HA, UK 88institutetext: Department of Astronomy,
University of Texas, Austin, TX 78712, USA 99institutetext: Independent
Researcher

# Rapidly-rotating Population III stellar models as a source of primary
nitrogen

Sophie Tsiatsiou 11 Yves Sibony 11 Devesh Nandal 11 Luca Sciarini 11 Yutaka
Hirai 2233 Sylvia Ekström 11 Eoin Farrell 11 Laura Murphy 99 Arthur Choplin 44
Raphael Hirschi 55 Cristina Chiappini 66 Boyuan Liu 77 Volker Bromm 88 Jose
Groh 99 Georges Meynet 11

(Accepted XXX, Received YYY; in original form ZZZ)

###### Abstract

Context. The first stars might have been fast rotators. This would have
important consequences for their radiative, mechanical and chemical feedback.

Aims. We discuss the impact of fast initial rotation on the evolution of
massive Population III models and on their nitrogen and oxygen stellar yields.

Methods. We explore the evolution of Population III stars with initial masses
in the range of 9 $M_{\odot}$ $\leqslant M_{\rm ini}\leqslant$ 120 $M_{\odot}$
starting with an initial rotation on the Zero Age Main Sequence equal to 70%
of the critical one.

Results. We find that with the physics of rotation considered here, our
rapidly-rotating Population III stellar models do not follow a homogeneous
evolution. They lose very little mass in case mechanical winds are switched on
when the surface rotation becomes equal or larger than the critical velocity.
Impact on the ionising flux appears modest when compared to moderately-
rotating models. Fast rotation favours, in models with initial masses above
$\sim$20 $M_{\odot}$, the appearance of a very extended intermediate
convective zone around the H-burning shell during the core He-burning phase.
This shell has important consequences on the sizes of the He- and CO-cores and
thus impacts the final fate of stars. Moreover, it has a strong impact on
nucleosynthesis boosting the production of primary 14N.

Conclusions. Fast initial rotation impacts significantly the chemical feedback
of Population III stars. Observations of extremely metal-poor stars and/or
starbursting regions are essential to provide constraints on the properties of
the first stars.

###### Key Words.:

Stars: Population III – Stars: rotation – Stars: abundances – Stars: massive

##  1 Introduction

According to cosmological simulations, the first stars formed at redshifts
$z\sim 20-35$ (Abel et al., 2002; Bromm et al., 2002; Yoshida et al., 2003;
Ohkubo et al., 2009; Hirano et al., 2014; Susa, 2019) and were responsible for
the first stellar radiative, chemical, and mechanical feedback in the Universe
(see e.g. the recent review by Klessen & Glover, 2023, and references
therein). To know their evolution is therefore important for questions such as
reionisation (Bromm et al., 2002; Choudhury & Ferrara, 2005; Yoon et al.,
2012; Sibony et al., 2022), the first chemical enrichments (Meynet & Maeder,
2002b; Prantzos, 2003; Chiappini et al., 2005a; Meynet et al., 2006; Tominaga
et al., 2007; Chiappini et al., 2008; Rollinde et al., 2009; Greif et al.,
2010; Pallottini et al., 2014; Marassi et al., 2015; Sarmento et al., 2017;
Hirai et al., 2018, 2019; Corazza et al., 2022; Sanati et al., 2023), the
origin of Carbon-Enhanced Metal-Poor (CEMP) stars (Ji et al., 2015; Clarkson
et al., 2018; Liu et al., 2021b; Zepeda et al., 2023), the Population (Pop)
III progenitors of long Soft Gamma Ray Bursts (Wang et al., 2012) and of
magnetorotational supernovae (Suwa et al., 2007; Mészáros & Rees, 2010; Suwa &
Ioka, 2011; de Souza et al., 2011; Maeder & Meynet, 2012; Piro et al., 2014)
or of intermediate mass and/or merging black holes (Inayoshi et al., 2016,
2017; Regan et al., 2020; Kinugawa et al., 2020, 2021; Liu et al., 2021a;
Tanikawa et al., 2021; Santoliquido et al., 2023; Iorio et al., 2023), and for
setting the stage for the next star formation events. In addition, recent JWST
observations of high-redshift galaxies (e.g. GLASS and CEERS surveys) have
discovered a subset of N-rich galaxies (see e.g. Isobe et al., 2023). In
particular the intriguing z$\sim$11 source GN-z11 (Cameron et al., 2023), is
suggested to contain a sub-population of Pop III stars (Maiolino et al.,
2023). Such Pop III stars might be very interesting candidates for enriching
in a short timescales these high redshift galaxies in nitrogen.

The stellar feedback processes depend on characteristics of the first stellar
populations, such as the initial mass function (Bromm et al., 1999; Abel et
al., 2002; Bromm et al., 2002), the initial distribution of rotations that can
contain a large fraction of very rapidly-rotating stars (Stacy et al., 2011,
2013; Hirano & Bromm, 2018a) if magnetic braking is inefficient in the absence
of strong magnetic fields (Hirano & Bromm, 2018b; Hirano & Machida, 2022;
Kimura et al., 2023), and the degree of multiplicity (see e.g. Stacy et al.,
2010; Stacy & Bromm, 2013; Liu et al., 2021a) to cite only a few important
features. Simulations of Pop III star formation suggest that Pop III stars had
large masses, higher than 100 $M_{\odot}$ due to the lack of cooling mechanism
that prevents further fragmentation Bromm et al. (2002). However, the exact
mass range is still debated. Recent studies allow the formation of Pop III
stars with masses down to 1 $M_{\odot}$ and even lower (Yoshida et al., 2006;
Clark et al., 2011; Greif et al., 2011; Susa, 2013; Stacy et al., 2016;
Wollenberg et al., 2020; Sugimura et al., 2020; Park et al., 2021; Prole et
al., 2022; Latif et al., 2022).

Models are also needed to find observable signatures of these primordial stars
when seen as a population in high redshift star-forming regions (Schaerer,
2002, 2003; Raiter et al., 2010; Ma et al., 2017; Trussler et al., 2023) or
when they explode in energetic, very luminous transient events (Joggerst &
Whalen, 2011; Toma et al., 2011; Mesler et al., 2014; Tolstov et al., 2016;
Lee et al., 2023). Pop III stars are also privileged objects for studying some
frontiers of physics. Since they are born in dark matter mini halos they may
be impacted by the physics of the dark matter (Freese et al., 2008; Taoso et
al., 2008; Natarajan et al., 2009; Schleicher et al., 2009; Spolyar et al.,
2009; Choplin et al., 2017; Liu et al., 2019a, b, 2020). Also, some
cosmological theories predict that fundamental constants, such as the fine
structure constant, may evolve as a function of time along the cosmic history.
Pop III stellar models have been used to test the impact of different values
of the fine structure constant on the stellar nucleosynthesis in the very
early Universe (Ekström et al., 2010; Huang et al., 2019; Mori & Nomoto,
2020).

In the last two decades or so, evolutionary features of non-rotating Pop III
stellar models have been studied among others by Marigo et al. (2001); Chieffi
et al. (2001, 2002b, 2002a); Ohkubo et al. (2009); Bahena & Klapp (2010);
Heger & Woosley (2010). Works have also been done studying the impact of close
binary evolution of Pop III stars (see e.g. Chen et al., 2015; Tsai et al.,
2023). The impact of convective interactions between H- and He-burning regions
in massive Pop III stellar models has been studied by Clarkson & Herwig
(2021).

Ekström et al. (2008) explored the impact of rotation on the evolution and the
chemical feedback of Pop III stars. Murphy et al. (2021a) performed a similar
study as Ekström et al. (2008), but adopting a different setting for the
diffusion coefficients describing the mixing of the elements by rotation. More
recently, Aryan et al. (2023) using MESA computed the evolution of Pop III
rotating 25 $M_{\odot}$ models.

Rotation deeply impacts nucleosynthesis. For instance, Meynet & Maeder
(2002b); Chiappini et al. (2006); Ekström et al. (2008) have shown that the
diffusion of He-burning products into the H-burning shell induced by shear
turbulence boosts the production of isotopes such as 13C and 14N. Further
works have shown that the diffusion of nitrogen into the He-burning core
boosts the production of 19F, 22Ne, s- and p-process elements (Meynet &
Maeder, 2002a; Meynet et al., 2006; Chiappini et al., 2008; Pignatari et al.,
2008; Cescutti et al., 2013; Frischknecht et al., 2016; Limongi & Chieffi,
2018; Choplin et al., 2018, 2022) and give them a status (at least in part) of
primary elements. Let us recall that a primary element is an element whose
production does not scale with the initial metallicity but is produced from
the transformation of hydrogen and helium (see e.g. the book by Matteucci,
2001). An interesting example is the production of primary nitrogen in metal-
poor rotating stars (Meynet & Maeder, 2002a). The turbulence induced by
rotation triggers mixing between the H- and He-burning zones, allowing some
carbon and oxygen produced in the He-burning zone to diffuse into the
H-burning zone. In the H-burning shell, carbon and oxygen are transformed into
nitrogen by the action of the CNO cycle. This nitrogen is qualified as primary
because it is produced from carbon and oxygen produced by the star and not
from carbon and oxygen that were present in the star at its formation.

In past works, it has been shown that for a given initial mass, metallicity,
and rotation, the primary nitrogen production only occurs at low metallicity,
typically at metallicities between $Z\sim 10^{-8}$ and $10^{-4}$, where $Z$ is
the mass fraction of the abundances of all the elements heavier than helium
(see e.g. Meynet et al., 2006). The reason why this mixing is efficient only
at low metallicities is mainly due to the fact that the lack of CNO element at
the beginning of the evolution makes the H-burning occur at higher
temperatures than in more metal-rich stars. As a consequence, the H- and He-
burning zones are less distant from each other in metal-poor stars, and the
entropy gradients are weaker, making any mixing either by convection or by
diffusion easier.

Previous modelling of zero metallicity stars has suggested that Pop III stars
will produce significantly less primary nitrogen than a model with a very
small initial metal content (assuming models starting with a velocity at the
Zero Age Main Sequence (ZAMS), $\upsilon_{\rm ini}$, corresponding to the same
constant fraction of the critical velocity111The critical velocity is the
velocity at the equator where the gravity is balanced by the centrifugal
force., $\upsilon_{\rm crit}$, at all metallicities). This occurs because some
degree of differential rotation is required to trigger the mixing. The
differential rotation is mainly produced by the contraction of the core at the
end of the Main-Sequence (MS) phase. Contraction in a Pop III star is not as
great as in an even slightly metal-enriched star, because as explained before,
the temperature of He-burning is not much greater than the temperature in the
H-burning core. A modest core contraction is therefore sufficient for raising
the core temperatures at the level needed for achieving radiative equilibrium.
Furthermore, the opacity in the envelope is lower facilitating the evacuation
of the energy released by contraction from the envelope. Thus, less energy is
used to expand the envelope, which also suppresses differential rotation. The
star remains in the blue part of the Hertzsprung-Russel diagram (HRD) for the
whole or at least a substantial part of the core He-burning phase222Farrell et
al. (2022) has performed very interesting numerical experiments showing how
artificially preventing the contraction actually prevents the crossing of the
Hertzsprung-Russel (HR) gap. Actually, things are more complex in the sense
that some contraction always occurs and other factors such as the distribution
of helium around the H-burning shell are important too..

In this paper, we explore Pop III stellar models rotating faster on the ZAMS
than in our previous works and how significant the production of primary
nitrogen can be in those. Therefore, instead of considering an initial
velocity of $\upsilon_{\rm ini}=0.4\,\upsilon_{\rm crit}$ as in Murphy et al.
(2021a), we shall consider $\upsilon_{\rm ini}=0.7\,\upsilon_{\rm crit}$. In
Sect. 2, we describe the ingredients of the stellar models. Some effects of
fast rotation on the evolution of primordial stars are discussed in Sect. 3. A
brief presentation of the small grid computed in this work is given in Sect.
4. We discuss the impact of fast rotation on the primary nitrogen production
in Sect. 5. Sect. 6 compares the yields obtained from models at different
metallicities, obtained by different authors. The impact of the present yields
in a simple chemical evolution model is also presented. The conclusions are
given in Sect. 7.

##  2 Ingredients of stellar models

The rapidly-rotating Pop III models have been computed with the Geneva stellar
evolution code (Genec, see a detailed description in Eggenberger et al., 2008)
with the same physical ingredients as the Pop III models published by Murphy
et al. (2021a) and models at higher metallicities by Ekström et al. (2012);
Georgy et al. (2013a); Groh et al. (2019); Eggenberger et al. (2021); Yusof et
al. (2022). The reader is invited to refer to those papers for the details of
the input physics. Only a few relevant ingredients are recalled here.

The initial composition considered is $X=0.7516$, $Y=0.2484$, and $Z=0$, with
$X$, $Y$, and $Z$ the mass fractions of hydrogen, helium, and heavy elements,
respectively. The primordial 4He abundance is coming from predicted values
from the Big Bang nucleosynthesis (Cyburt et al., 2003).

We explore the effects of fast rotation on the evolution of massive stars with
similar masses as considered in Murphy et al. (2021a). We restricted here to
masses between 9 and 120 $M_{\odot}$. More massive star models have been
recently explored using Genec in Martinet et al. (2023) and Nandal et al.
(2023a, c) although not with fast rotation.

Rotation is implemented according to the stellar-rotation theory developed by
Zahn (1992). The expressions describing the vertical and horizontal shear
diffusion coefficients are respectively from Maeder (1997) and Chaboyer & Zahn
(1992). Magnetic fields are not accounted for in the present models.

We did not account for any mass loss due to line-driven winds, as it was
discussed in Murphy et al. (2021a). Even though, there are many other
processes that they may allow Pop III stars to lose mass (see e.g. the
discussion in the introduction of Liu et al., 2021b), for example, pulsation-
driven mass loss (Volpato et al., 2023), mass loss during a mass transfer in a
close binary and mechanical mass loss, in this work only mechanical mass loss
has been taken into account. Mechanical mass loss occurs when the surface
rotation reaches the critical velocity. We estimate the mass lost mechanically
as the mass that needs to be removed from the star in order to keep its
surface rotation subcritical. A detailed account for the exact numerical
procedure is explained in Georgy et al. (2013b, Sect. 2.2).

The critical velocity is defined as the surface rotation such that the
centrifugal acceleration at the equator balances the gravity there. In this
study, we followed the expression given by Maeder & Meynet (2000) which
accounts for the fact that the critical velocity varies when the luminosity of
the star is near the Eddington luminosity (the so-called $\Omega\Gamma$ limit
where $\Omega$ is the angular velocity and $\Gamma$ the Eddington factor). The
critical rotation has two roots depending on the Eddington factor:

$\upsilon_{\rm
crit}=\begin{cases}\sqrt{\frac{2}{3}\frac{GM}{R_{\text{p,crit}}}}&,\text{$\Gamma<0.693$}\\\
\sqrt{\frac{3}{2}\upsilon_{\rm
crit,1}\frac{R_{\text{e,crit}}}{R_{\text{p,crit}}}\frac{1-\Gamma}{{V}^{\prime}(\omega))}}&,\text{$\Gamma>0.693$}\\\
\end{cases}$ (1)

where $R_{\text{p,crit}}=\frac{2}{3}R_{\text{e,crit}}$, $R_{\text{p,crit}}$
and $R_{\text{e,crit}}$ are the polar and equatorial radius at critical
velocity respectively, $V^{\prime}(\omega)$ is the ratio between the actual
volume of the deformed rotating star to a spherical volume with a radius equal
to $R_{\text{p,crit}}$, and $\omega$ is the ratio of the surface angular
velocity to the critical angular velocity (given by the first root in the
equation just above).

The limits of the convective cores have been determined using the
Schwarzschild criterion, with a step overshooting parameter of $d_{\rm
over}/H_{P}=0.1$ for the core H- and He-burning phase ($d_{\rm over}$ is the
spatial extension of the convective core above the radius given by the
Schwarzschild’s limit and $H_{P}$ is the pressure scale height at the
Schwarzschild’s boundary).

We complement the grid by Murphy et al. (2021a) for specific masses (9, 20,
60, 85, and 120 $M_{\odot}$) with models having an initial rotation rate of
$\upsilon_{\rm ini}/\upsilon_{\rm crit}=0.7$. This gives initial rotations on
the ZAMS that are between 650 and 1200 km/s (see column 2 in Table 4333Keeping
constant values for $\upsilon_{\rm ini}/\upsilon_{\rm crit}$ for all the
initial masses implies a faster initial surface velocity for more massive
stars.). At first sight, these rotations may appear quite extreme values, at
least well outside the range of surface velocities measured in the present-day
Universe. However, the perception is different if we assume that stars begin
their evolution on the ZAMS with a total angular momentum content that is kept
constant for all metallicities. Since Pop III stars of a given mass have a
much smaller momentum of inertia than solar metallicity stars, they naturally
have faster rotations. To give a numerical example, the total angular momentum
of a solar metallicity 9 $M_{\odot}$ model with an initial surface velocity
equal to 40% the critical one ($264\,{\rm km}/{\rm s}$) is $0.072\times
10^{53}\,{\rm g}\,{\rm cm}^{2}/{\rm s}$ (models by Ekström et al., 2012). The
total angular momentum of a Pop III 9 $M_{\odot}$ model with an initial
surface velocity equal to 70% the critical one ($649\,{\rm km}/{\rm s}$) is
$0.085\times 10^{53}\,{\rm g}\,{\rm cm}^{2}/{\rm s}$. The angular momenta are
not very different, but the rotation rates are.

##  3 Impact of fast rotation on primordial stars

In this section we explore the effect of fast rotation on primordial stars,
focusing on transport processes, surface He-enrichments, rotations at the
critical limit, and rotational mixing due to an effect that we call CNO shell
boost.

###  3.1 Transport processes

Figure 1: Internal profiles as a function of the radius and the Lagrangian
mass coordinate inside 9 $M_{\odot}$ models half-way through the MS
($X_{c}=0.35$) for different initial rotations: $\upsilon_{\rm
ini}/\upsilon_{\rm crit}=$ 0.2 (orange), 0.4 (red) (Murphy et al., 2021a), and
0.7 (green). The grey-shaded zone is the convective core for the moderately-
rotating model. Left panel: Radial component of the meridional circulation
velocity $U_{r}$. Middle panel: Angular velocity $\Omega$. Right panel: The
solid curve show the diffusion coefficient due to shear turbulence $D_{\rm
shear}$. The dotted curve shows the effective diffusion coefficient $D_{\rm
eff}$ (Y-axis in logarithmic scale and in units of ${\rm cm}^{2}/{\rm s}$).
For the purpose of clarity, only the curve for $D_{\rm eff}$ corresponding to
$\upsilon_{\rm ini}=0.7\,\upsilon_{\rm crit}$ is shown. In the rest of the
radiative zone $D_{\rm eff}$ is in general much smaller than $D_{\rm shear}$.

Figure 1 shows the part of the radial component of the meridional circulation
velocity444The radial component of the meridional circulation velocity is
given by $u_{r}(r,\theta)=U_{r}(r)P_{2}(\cos\theta)$, where $u_{r}(r,\theta)$
is the radial component of the physical velocity, $\theta$ is the co-latitude,
and $P_{2}$ the second Legendre polynomial. $U_{r}$ that depends on the radial
coordinate $r$, and the angular velocity in 9 $M_{\odot}$ stellar models half-
way through the Main Sequence (MS) phase (e.g. when the mass fraction of
hydrogen at the centre $X_{\rm c}=0.35$) for three different initial
rotations. Increasing the initial rotation significantly enhances $U_{r}$ both
in the radiative envelope (at radial coordinates equal to 0.6-0.7 $R_{\odot}$)
and just above the convective core (around 0.2 $R_{\odot}$). In the models
sketched in Fig. 1, we can identify at least three changes of sign for $U_{r}$
(when $U_{r}$ is positive the meridional currents transport the angular
momentum from the outer regions to the inner ones and the reverse when it is
negative). We see that just above the core and further out in the radiative
envelope, there are regions where the meridional currents transport the
angular momentum inwards, reinforcing there the gradient of the angular
velocity $\Omega$. This behaviour cannot be modelled by approximating the
angular momentum transport by meridional currents as a diffusive process.
Indeed a diffusive process will always tend to smooth any $\Omega-$gradient.

The interactions between the meridional currents, the effects induced by the
changes of structure, and the diffusion by shear on the angular momentum
distribution result in the variation of the angular velocity with the radius
plotted on the middle panel of Fig. 1. It shows that the gradients of $\Omega$
are steeper in the rapidly-rotating models, which implies more chemical mixing
by shear. This is consistent with the behaviour of these models in the HRD
(see Fig. 5), where indeed the rapidly-rotating models follow, once the
chemical mixing had time to be significant, a bluer and more luminous track
indicative of a more efficient chemical mixing. The steepness of the
$\Omega$-gradient is caused by the core contraction and the transport of the
angular momentum. As it has been discussed in many previous papers (see e.g.
Meynet & Maeder, 2000), in models without any magnetic fields, angular
momentum is mainly transported by meridional currents. In regions where
$U_{r}$ is positive, the angular momentum is transported from the outer layers
into the inner ones, reinforcing the gradient produced by the core contraction
and envelope expansion. As can be seen from the left panel of Fig. 1, $U_{r}$
is positive in a large part of the radiative envelope and get larger values in
the faster rotating model making thus the $\Omega$-gradient steeper. Note that
such a process cannot be reproduced when the transport of the angular momentum
by the meridional currents is described by a diffusive equation as done in
some codes.

In the 9 $M_{\odot}$ model, the mixing of the elements is mainly governed by
the vertical shear diffusion coefficient ($D_{\rm shear}$), except in a small
region above the convective core where the diffusion coefficients describing
the results of the action of both the meridional currents and of the
horizontal shear turbulence ($D_{\rm eff}$) dominates (see the right panel of
Fig. 1). In contrast, in more massive models (typically for the 60 $M_{\odot}$
model and above), the main driver for the transport of the chemical species is
$D_{\rm eff}$ (see the lower right panel of Fig. 13). The diffusion
coefficient $D_{\rm eff}$ scales with the meridional velocity (Chaboyer &
Zahn, 1992), which in turn scales as the inverse of the density. The latter
scaling implies that for a given rotation, the meridional velocities are in
general larger in the less dense more massive stars. As a numerical example,
the maximum value is nearly $8\times 10^{-5}\,{\rm cm}/{\rm s}$ in the
rapidly-rotating 9 $M_{\odot}$ model plotted in Fig. 1. In the corresponding
120 $M_{\odot}$ model, the absolute value is of the order of $1\,{\rm cm}/{\rm
s}$, hence four orders of magnitude larger (see the upper left panel of Fig.
13). Moreover, it is a negative value indicating that it transports angular
momentum from the inner layers to the outer regions. This transport appears
more efficient in rapidly-rotating stars.

###  3.2 Surface He-enrichments

Figure 2: Surface 4He enrichment at the end of the MS phase (see text) as a
function of the initial mass for all the Pop III models presented in Table 4.

In Fig. 2 we show the surface 4He enrichment at the end of the MS phase for
different initial masses and rotations. The quantity plotted is $\Delta
Y=Y_{\rm surf}-Y_{\rm ini}$, where $Y_{\rm ini}=0.2484$ is the initial
abundance of 4He. The most rapidly-rotating model (see the green curve)
achieves larger surface He enrichment than the less rapidly-rotating ones at
the end of the MS phase. Interestingly, the curves for a fixed value of
$\upsilon_{\rm ini}/\upsilon_{\rm crit}$ are non-monotonic. They present a
concave shape (see the curves for the initial rotation $0.4\,\upsilon_{\rm
crit}$ and $0.7\,\upsilon_{\rm crit}$). This shape is due to the following
effects: on the one hand, the radius of a lower initial mass star is smaller
than the one of a more massive star, implying that, for a given shear
diffusion coefficient, the mixing timescale is smaller in the case of a lower
initial mass star (the timescale of mixing due to a diffusion coefficient $D$
in a zone of spatial extent $R$ scales as $R^{2}/D$). On the other hand, the
total diffusion coefficient that is the sum of $D_{\rm shear}$ and $D_{\rm
eff}$ is in general larger in more massive stars compare the right panel of
Fig. 1 with the two lower panels of Fig. 13). This makes the mixing timescale
shorter over a given distance. These effects, together with the different
sizes of the convective cores in the different initial mass models contribute
to producing the curves shown in Fig. 2.

###  3.3 Pop III stars at the critical limit

Figure 3: Evolution of the ratio between the surface angular velocity and the
critical one as a function of the fraction of the total MS lifetime
($\tau_{\rm MS}$), for models within the mass range of 9 $M_{\odot}$
$\leqslant M_{\rm ini}\leqslant$ 120 $M_{\odot}$ with different initial
rotations. The grey dashed line indicates the critical limit. The moderately-
rotating models are from Murphy et al. (2021a).

Figure 3 shows the evolution of the ratio between the surface angular velocity
and the critical one during the MS lifetime for models with moderate and fast
rotation. The rapidly-rotating models reach the critical velocity at an
earlier stage of the evolution during the core H-burning phase (the models
with $M_{\rm ini}\geqslant 60\,{M_{\odot}}$ spend more than 50% of their MS
lifetime at the critical limit). However, the maximum mass that these model
can lose via this process is only at most 2% of their initial mass. So the
present models can be considered as evolving at nearly constant mass. We note
that it appears plausible that such stars would be surrounded by a decretion
disk as Oe-Be stars observed in the present-day Universe. They would be He-
rich stars according to the present models.

###  3.4 CNO shell boost

Figure 4: Evolutionary track for the 20 $M_{\odot}$ rapidly-rotating model
(upper panel) and Kippenhahn diagram (lower panel). The orange and purple
points overplotted on the evolutionary track in the upper panel have an
internal structure shown in the lower panel at the two times indicated by
respectively the vertical orange and purple lines. The blue-shaded zones are
the convective areas, while the white zones are the radiative ones. The blue
and green curves show the mass coordinates where hydrogen and helium are
burning, respectively. Furthermore, the solid curves correspond to the peak of
the energy generation rate and the dotted ones to the 10% of the peak energy
generation rate for each burning phase.

Ekström et al. (2008) showed that rotational mixing can drive an H-shell boost
due to the injection (by rotational mixing and/or convection) of significant
amounts of carbon and oxygen from the He-burning core. This boosts the energy
generation in the shell and may sometimes give birth to a very extended
intermediate convective zone attached to the H-burning shell. The boost
derives from the fact that the nuclear reactions of $pp$-chains are taken over
by the CNO cycle, with a very different dependence in temperature. In some
extreme cases, the H-shell may even become for a time the main source of
energy in the star, quenching the energy produced in the He-burning core. When
the mixing is gentle and progressive, the star has time to readjust and evolve
in a continuous smooth way. When the mixing is much more rapid, it may produce
a strong and abrupt change of the structure as the extreme case sketched in
Fig. 4. In the following, we shall designate such an event as a CNO shell
boost.

The upper panel of Fig. 4 shows the evolutionary track in the HRD of the
rapidly-rotating 20 $M_{\odot}$ model. The orange point along the track
indicates the end of the core H-burning phase and the beginning of the core
He-burning phase. The structure of the star at that stage is given by the
vertical orange line in the Kippenhahn diagram shown below. The core He-
burning occurs in a convective core with a mass of about 3 $M_{\odot}$, but
suddenly the convective core disappears when, due to the process described
above, carbon and oxygen are injected into the H-burning shell. This increases
the nuclear energy produced in the shell. An intermediate convective shell
appears that facilitates still more the feeding of the H-shell in carbon and
oxygen. A CNO shell boost occurs. It produces such a strong release of energy
that the intermediate convective zone extends from the mass coordinate 2 up to
12 $M_{\odot}$. The purple point along the evolutionary track shows the
position in the HRD where the CNO shell boost occurs, and the corresponding
internal structure is indicated by the vertical purple line in the Kippenhahn
diagram. From that stage onwards, the He-burning convective core disappears
for a while. The star is mainly powered by H-burning (in a very extended
shell). This produces a “new” MS track in the HRD.

A similar figure for the 60 $M_{\odot}$ is shown in Fig. 14. Qualitatively, we
have a similar evolution as the 20 $M_{\odot}$ model. After a phase of core
He-burning (between time coordinates around 5.52 and 5.32 in logarithm of the
remaining time in years) during which no convective shell is associated with
the H-burning shell, an intermediate convective shell appears above the
H-burning region. This never succeeds in quenching the nuclear reaction in the
core but significantly reduces the size of the He-burning convective core.
Such a feature in very massive stars may prevent the star from entering the
pair-instability regime and may have importance for determining the lower
limit of the black hole mass gap (Farrell et al., 2021).

##  4 The rapidly-rotating grid

Figure 5: HRD of the Pop III models from Murphy et al. (2021a) (blue:
$\upsilon_{\rm ini}=0$, red: $\upsilon_{\rm ini}=0.4\,\upsilon_{\rm crit}$)
and the rapidly-rotating models computed for this work (green: $\upsilon_{\rm
ini}=0.7\,\upsilon_{\rm crit}$). The symbols drawn at the end of the tracks
mark the final evolutionary stage attained: a star for models that have
completed the core He-burning phase, a circle for models that are either
during the core C-burning phase or at the end of it, and a diamond for models
that are after the core C-burning phase. The models without any symbol were
stopped in the core He-burning phase. Iso-radius lines are drawn in dotted
grey. The pink arrows indicate the transition between the pp-contraction phase
and the pp-CNO-sustained phase. Table 1: Properties of Pop III models at the
end of the core H- and He-burning phases with different initial surface
velocities. Models with $\upsilon_{\rm ini}/\upsilon_{\rm crit}=$ 0, 0.2, and
0.4 are from Murphy et al. (2021a). Models that have not finished the core He-
burning phase are indicated with an horizontal dash.

|  | end MS | end He-burning
---|---|---|---
$M_{\rm ini}$ | $\upsilon_{\rm surf}$ | $\tau_{\rm H}$ | $\upsilon/\upsilon_{\rm crit}$ | $\upsilon_{\rm surf}$ |  Ysurf | $\tau_{\rm He}$ | $\upsilon/\upsilon_{\rm crit}$ | $\upsilon_{\rm surf}$ |  Ysurf
$[$$M_{\odot}$] | [km/s] | [Myrs] |  | [km/s] |  | [Myrs] |  | [km/s] |
mypink $\upsilon_{\rm ini}=0$
9 | 0 | 17.7 | 0 | 0 | 0.248 | 1.9 | 0 | 0 | 0.248
20 | 0 | 9.5 | 0 | 0 | 0.248 | 0.6 | 0 | 0 | 0.248
60 | 0 | 3.7 | 0 | 0 | 0.248 | 0.3 | 0 | 0 | 0.248
85 | 0 | 3.1 | 0 | 0 | 0.248 | 0.3 | 0 | 0 | 0.248
120 | 0 | 2.7 | 0 | 0 | 0.248 | 0.3 | 0 | 0 | 0.248
mypink $\upsilon_{\rm ini}=0.2\,\upsilon_{\rm crit}$
9 | 23.1 | 19.9 | 0.18 | 124 | 0.255 | 3.3 | 0.10 | 41 | 0.260
12 | 27.6 | 19.4 | 0.18 | 125 | 0.258 | 1.3 | 0.09 | 25 | 0.260
15 | 31.5 | 14.3 | 0.19 | 134 | 0.258 | 0.01 | 0.09 | 22 | 0.259
20 | 37.2 | 10.4 | 0.18 | 128 | 0.255 | - | - | - | -
mypink $\upsilon_{\rm ini}=0.4\,\upsilon_{\rm crit}$
9 | 372 | 20.9 | 0.38 | 274 | 0.286 | 2.5 | 0.11 | 60 | 0.286
20 | 526 | 10.7 | 0.41 | 309 | 0.277 | 1.0 | 0.44 | 192 | 0.277
60 | 613 | 4.1 | 0.87 | 657 | 0.265 | 0.4 | 0.72 | 302 | 0.269
85 | 659 | 3.3 | 0.70 | 551 | 0.265 | - | - | - | -
120 | 708 | 2.9 | 0.64 | 473 | 0.303 | - | - | - | -
mypink $\upsilon_{\rm ini}=0.7\,\upsilon_{\rm crit}$
9 | 649 | 26.6 | 0.71 | 514 | 0.370 | 2.8 | 0.17 | 19 | 0.405
20 | 833 | 11.4 | 0.87 | 628 | 0.314 | 3.0 | 0.11 | 7 | 0.552
60 | 1080 | 4.3 | 0.82 | 565 | 0.313 | 0.3 | 0.10 | 12 | 0.557
85 | 1150 | 3.6 | 1.00 | 645 | 0.349 | 0.3 | 0.03 | 3 | 0.596
120 | 1230 | 3.0 | 0.86 | 512 | 0.366 | - | - | - | -

As is well known (see e.g. Cassisi & Castellani, 1993; Marigo et al., 2001;
Ekström et al., 2008) the absence of any CNO elements in Pop III stars leaves
only $pp$-chains at the beginning for producing nuclear energy. The energy
released by this process is however not enough to compensate the losses at the
surface and the star contracts (we call here this phase the pp-contraction
phase). This occurs until temperatures are high enough to activate some
synthesis of carbon by triple $\alpha$ reactions. From this stage on, the star
reaches a radiative equilibrium in which the energy losses at the surface can
be compensated by the energy produced at the centre through nuclear reactions
involving pp and CNO reactions (pp-CNO-sustained phase).

In the evolutionary tracks presented in Fig. 5, the transition between the pp-
contraction phase and the pp-CNO-sustained phase is clearly visible in the
case of the 9 and 20 $M_{\odot}$ models. It corresponds to the points when the
effective temperature begins to decrease as the luminosity increases (pink
arrows on Fig. 5). In more massive stars, the pp-contraction phase is too
short to be visible.

In the upper mass range, the models show inflexion points (well visible for
the $\upsilon_{\rm ini}/\upsilon_{\rm crit}=0.7$ curves, less so for the
slower-rotating models) during the MS phase. These inflexion points occur when
the surface velocity reaches the critical limit. The mechanical mass loss
slows down the surface below the critical value but its secular evolution will
bring it back to the critical limit. This back-and-forth evolution produces
the oscillations seen in effective temperatures along those tracks555During
the short timescale of mechanical mass loss, the luminosity remains constant.
Since when losing mass, the star loses angular momentum and becomes less
oblate, for example, less deformed by rotation, its surface area, $S$,
decreases and hence, through the relation $L=S\sigma T_{\rm eff}^{4}$, its
effective temperature increases..

Our rapidly-rotating models are slightly shifted to the red in the first part
of the MS phase. Then in the second part, they cross the slower-rotating
track, reaching a higher luminosity. Except in the case of the 9 $M_{\odot}$,
the rapidly-rotating tracks show a turn-off at the end of the MS phase at
lower effective temperatures than their slower-rotating counterparts. These
two trends are well-known effects of rotation: the first one shows the
hydrostatic effects of rotation (see e.g. Faulkner et al., 1968; Kippenhahn et
al., 1970; Meynet & Maeder, 1997). The second one shows the mixing effects of
rotation, mainly the enrichment in He of the radiative envelope (Langer, 1992;
Meynet & Maeder, 2000, see also the discussion in Sect. 3.2) and the impact of
the mixing on the convective core mass evolution (Talon et al., 1997; Deupree,
1998; Heger & Langer, 2000).

From Fig. 5, we can see how changing the initial rotation impacts the nature
of the star at a given evolutionary stage. As an example considering the 20
$M_{\odot}$ model, the non-rotating model after the end of the core C-burning
stage is a Blue Supergiant (BSG), while the rapidly-rotating one is a Red
Supergiant (RSG). We may thus expect very different supernova (SN) light
curves from the non-rotating and rapidly-rotating models in case an explosion
occurs at the time of the final core collapse (see e.g. Yoon et al., 2012;
Aryan et al., 2023) (note that we do not expect any significant change of the
surface conditions in the few tens of years that remain after the C-burning
phase for the star to reach the final core-collapse phase).

Although the initial angular momentum of those models is high, the mixing is
not efficient enough for making them evolve homogeneously (Maeder, 1987).
Other rotating models (see e.g. Szécsi et al., 2015) show a chemical
homogeneous evolution for even slower initial rotation for masses above 26
$M_{\odot}$ at $Z=0.0002$. Here of course we have metal-free Pop III stars,
which may explain in part this difference. Besides, the physics of rotation is
different between the present models and those e.g. of Szécsi et al. (2015).
The latter include the Tayler-Spruit dynamo (Spruit, 2002; Heger et al.,
2005). We would like to remind the reader of the fact that the term “rotating
models” may actually correspond to very different physics and also to very
different outputs. The results presented here correspond to one choice (non-
ganetic models with specific expressions for the diffusion coefficients). The
impact of other choices is explored in a dedicated work (Nandal et al.,
2023b). It is still difficult to determine which kind of physics for the
transport processes due to rotation are the most favoured. At solar
metallicity, models with a more efficient angular momentum transport than
those considered in this work are favoured. Indeed, transport accounting for
the Tayler-Spruit dynamo produces much better agreement with asteroseismic
data of subgiants (Moyano et al., 2023)666Note that these models include what
has been called the Tayler-Spruit calibrated dynamo theory proposed by
Eggenberger et al. (2022). This implementation is different from the one
accounted for in the models by Szécsi et al. (2015). than non-magnetic models
as those presented here. However, we are exploring here a very special regime
of initial composition (primordial stars) where it is not guaranteed that
magnetic fields play a similar role. Also the transport of the chemical
species remains to be studied in more details. Its efficiency (and thus its
capability of driving a homogeneous evolution) depends mainly on the
expression used for the horizontal shear turbulent coefficient. Thus,
unfortunately without more dedicated study, it is still difficult to decide
which type of models are the most representative of the bulk Pop III star
populations. We just retain here that the conditions for obtaining a
homogeneous evolution heavily depends on the physics assumed for both the
transport of the chemical elements and of the angular momentum.

Some features of the Pop III stellar models at the end of the core H- and He-
burning phase are presented in Table 4. Columns $1-2$ give the initial mass
and the initial rotation on the ZAMS. Columns $3-6$ show respectively the
duration of the MS lifetime, and the values at the end of the core H-burning
phase of the ratios between the surface to the critical velocity, the linear
surface velocities at the equator, and the mass fractions of helium at the
surface. The same quantities for the core He-burning phase are indicated in
columns $7-10$.

From Table 4, looking at column 10, we see that the non-rotating models show
no increase in the surface He abundance between the end of the core H-burning
phase and that of the core He-burning phase. This comes from the fact that the
non-rotating models show no outer convective envelope during the core He-
burning phase. They burn their helium in the core when the star is on the blue
side of the HR diagram and thus no important outer convective zone is present.
Intermediate and outer convective envelopes appear (if any) only after the
core He-burning phase. In contrast, rapidly-rotating models show surface He
enrichment that is significantly increased with respect to the value reached
at the end of the core H-burning phase. This is a consequence of the outer and
intermediate convective zones that are already appearing during the core He-
burning phase in those models.

Comparisons of the lifetimes of the core H- and He-burning phase for different
initial rotations are given in columns 3 and 7 of Table 4. The MS phase of the
rapidly-rotating 9 $M_{\odot}$ model is longer by about 50% than the duration
for the non-rotating one. Much smaller effects are obtained in higher masses.
The increase is only 10% for the 120 $M_{\odot}$ model. Since the differences
between the moderately-rotating models (see Murphy et al., 2021a) and the
present rapidly-rotating ones are modest during the MS phase for both the
duration and the evolution in the HRD, we expect that the results on
ionisation fluxes based on the moderate rotation rate models (Murphy et al.,
2021b) are not changed much if rapidly-rotating models are used instead.

##  5 Primary nitrogen production in rotating models

Figure 6: The chemical structure as a function of the Lagrangian mass
coordinate in the interior of the Pop III grids for $9M_{\odot}\leqslant
M_{\rm ini}\leqslant$ 60 $M_{\odot}$. The figures are at the half-way through
the core He-burning phase ($Y_{c}=0.5$) and at the last model that has been
computed in different evolutionary stages, HeC is between the core He- and
C-burning phase, CNe is the stage between the core C- and the Ne-burning
phase, and O is during the core O-burning phase. The first column of plots
corresponds to the moderately-rotating models with 40% of the critical
rotation, and the middle and last columns to the rapidly-rotating models with
70% of the critical rotation. Each row refers to a different initial mass, the
first row corresponds to 9 $M_{\odot}$, the second to 20 $M_{\odot}$, and the
third to 60 $M_{\odot}$. Each coloured curve represents a different element,
as it is noted on the upper left plot. The light brown zones are the
convective areas of each model. The moderately-rotating models have been taken
from Murphy et al. (2021a). The solid curves correspond to the most common
isotope of an element, while the dashed curves to the second most common
isotope. Figure 7: The abundance of 14N as a function of the Lagrangian mass
coordinate during different stages of the core He-burning phase for the 20
$M_{\odot}$ model with different initial rotations, zero (dotted curves),
moderate (dashed curves), and fast (solid curves). The blue curves left
correspond to the beginning of the core He-burning phase ($Y_{c}=0.9$), the
green curves (middle) to the half-way through it ($Y_{c}=0.5$), and the red
curves (right) to the end of it ($Y_{c}=0$). The non-rotating and moderately-
rotating models were taken from Murphy et al. (2021a). Figure 8: The evolution
of the total mass of 14N inside the rapidly-rotating stellar models during
their evolution. For the 20 $M_{\odot}$ model (orange), the non-rotating and
moderately-rotating models are also indicated with the dotted and dashed
curves. On the x-axis the fraction of burning phases is given, where the core
H-burning phase is for 0$-$1, the core He-burning phase is for 1$-$2, and the
advanced burning phases are for 2$-$3\. The non-rotating and moderately-
rotating models were taken from Murphy et al. (2021a).

Figure 6 shows the evolution of the chemical structure in different rotating
models at different stages of their evolution. The light grey areas are the
convective zones of the stars, while the white ones are the radiative zones.

The first column shows the structure at the middle of the core He-burning
phase in models starting with a rotation equal to 40% of the critical limit.
We see that in those models some carbon and oxygen are brought by rotational
mixing into the H-burning shell inducing a small production of nitrogen (see
e.g. the green curve bump at a mass around 4 $M_{\odot}$ in the 20 $M_{\odot}$
model). Some of this nitrogen can diffuse into the He-burning core and be
transformed into 22Ne (see the dashed purple curve into the He-burning core,
e.g. below the mass coordinate 2.5 $M_{\odot}$).

The panels in the middle column show models similar to those of the panels of
the first column but now considering an initial rotation of 70% of the
critical one. Here we see that the bumps in nitrogen are much more prominent.
More specifically, comparing the rapidly-rotating models, we see that nitrogen
dominates in the outer regions as we move towards to the surface in the 20
$M_{\odot}$ model. This exceptional fraction of nitrogen can be attributed to
the presence of the convective shell in this model, a feature absent in the 9
model and occurring a later evolutionary phase in the 60 $M_{\odot}$. We can
also see that 13C (see the red dashed curve) is produced in the H-burning
shell (let us recall that 12C is transformed partially into 13C by the CNO
cycle). Interestingly also in the 20 $M_{\odot}$ model, a significant quantity
of 19F appears in the He-burning core (this point will be discussed in a
dedicated paper, Tsiatsiou et al. (in prep.)). Finally, 22Ne also reaches a
high abundance in the He-core (see the purple dashed curve) as a result of the
diffusion/mixing of nitrogen into the He-burning core.

The panels on the right show the same models as those of the middle panel but
at a more advanced evolutionary stage, indicated in the caption of Fig. 6. We
see that in general, the abundance of nitrogen has increased in the outer
layers of the star. This is because of the continuing build-up of this element
in the H-burning shell due to the diffusion of carbon and oxygen into it. The
appearance of convective zones can redistribute nitrogen over larger regions
in the envelope. We can note the very high level of nitrogen in the outer
layers of the 20 and 60 $M_{\odot}$ models.

Figure 7 allows us to see more clearly the impact of the initial rotation.
These three panels compare the distribution of nitrogen as a function of the
Lagrangian mass coordinate in the 20 $M_{\odot}$ model with three different
initial rotations at three different evolutionary stages of the core He-
burning phase. We can note that the nitrogen is mostly built in the early
phases of the core He-burning. Besides, in the rapidly-rotating models
($\upsilon_{\rm ini}/\upsilon_{\rm crit}=0.7$), the increase is impressive. We
see that even the central regions are enriched in nitrogen. This is a
consequence of the very large intermediate convective zone that actually
extends nearly to the centre (as can be seen in the lower panel of Fig. 4) and
thus brings nitrogen in very central regions. When He-burning restarts in the
core, a new convective core appears that maintains for a while at least a high
level of nitrogen abundance.

In Fig. 8, the solid curves show how the integrated mass of nitrogen inside a
model varies when the star evolves for different rapidly-rotating models. The
integrated mass at a given time $t$ is given by
$M_{{}^{14}\text{N}}=\int_{0}^{M_{tot}(t)}{}^{14}X(M_{r},t)\,{\rm d}M_{r}$,
where $M_{tot}(t)$ is the actual total mass of the star at the time $t$,
${}^{14}X(M_{r},t)$ the mass fraction of 14N at the Lagrangian coordinate
$M_{r}$ at time $t$.

We see that in the 9 $M_{\odot}$ model, the increase of $M_{{}^{14}\text{N}}$
occurs progressively during the whole core He-burning phase. For the higher
initial mass models, the increase is more step-like. As just seen above, it
occurs at the very beginning of the core He-burning phase in the 20
$M_{\odot}$ model and at later and later core He-burning stage in more and
more massive stars. In these higher initial mass models, convection plays an
important role in connecting regions rich in both H- and He-burning products.

We have also plotted for the 20 $M_{\odot}$ model the evolution of
$M_{{}^{14}\text{N}}$ for the non-rotating and moderately-rotating models
(respectively the orange dotted and dashed curves). Interestingly, even the
non-rotating model shows some step-like increase of $M_{{}^{14}\text{N}}$
during the core C-burning phase. The moderately-rotating model does not show
such a feature but we cannot discard the fact that it may present one in a
more advanced stage. We can note that in non-rotating models such connection
between the He- and H-burning regions leading to significant primary nitrogen
production seems to be restricted to a narrow mass domain (Ekström et al.,
2010; Limongi & Chieffi, 2012), while rotational mixing not only boosts the
primary nitrogen production in this mass domain but also extends the mass
domain over which this process occurs across the whole mass range of stars
considered here (see Fig. 9).

Moreover, in Fig. 10 we plot the integrated abundance of nitrogen (green) and
oxygen (blue) as a function of the Lagrangian mass coordinate for the 20
$M_{\odot}$ rapidly-rotating model at its last evolutionary stage (during the
core O-burning phase). We show in the plot where the remnant mass ($M_{\rm
rem}$ in grey dashed line) is located for the model, as well as the typical
mass of a neutron star (NS) ($M_{\rm NS}$ in light grey dashed line). It is
clear that the selection of the remnant mass can have a strong impact on the
oxygen yields. Typically we see in Fig. 10 that the integrated mass of
nitrogen, starting the integration in mass from the surface increases when
considering still deeper layers until the lagrangian mass around 11
$M_{\odot}$ is reached. Below that mass, the nitrogen mass does not longer
increase because below this mass, nitrogen is completely destroyed. Oxygen
remains abundant in much deeper layers since O-burning only occurs in the very
central parts of the star. As a consequence, the integrated mass of oxygen
continues to increase when deeper layers are added to the mass ejected.

Table 2: Stellar yields of nitrogen and oxygen for different Pop III and very
metal-poor stellar models. The asterisk after the initial mass indicates that
the model is still during the core He-burning phase. The Pop III models with
$\upsilon_{\rm ini}/\upsilon_{\rm crit}=$ 0, 0.2, and 0.4 are from Murphy et
al. (2021a) and the models with $Z=10^{-5}$ are from Sibony et al.
(submitted).

$M_{\rm ini}$ | $M_{\rm CO}$ | $M_{\rm rem}$ | Final |  14N |  16O
---|---|---|---|---|---
$[{M_{\odot}}]$ | $[{M_{\odot}}]$ | $[{M_{\odot}}]$ | fate | $[{M_{\odot}}]$ | $[{M_{\odot}}]$
operamauve Z=0
mypink $\upsilon_{\rm ini}=0$
9 | 0.718 | 0.718 | WD | 2.52E-08 | 0.109
20 | 4.047 | 1.517 | NS | 1.39E-03 | 1.719
60 | 22.575 | 26.575 | BH | 1.01E-04 | 4.42E-07
85 | 31.768 | 35.768 | BH | 2.26E-05 | 0.008
120 | 52.887 | 34.640 | PPISN | 9.02E-06 | 15.969
mypink $\upsilon_{\rm ini}=0.2\,\upsilon_{\rm crit}$
9 | 0.940 | 0.940 | WD | 2.89E-05 | 0.173
12 | 1.385 | 1.385 | WD | 1.27E-04 | 0.173
15 | 2.180 | 1.400 | NS | 2.88E-04 | 0.814
20* | 19.99 | 19.99 | BH | 0.0 | 0.0
mypink $\upsilon_{\rm ini}=0.4\,\upsilon_{\rm crit}$
9 | 0.895 | 0.895 | WD | 0.0011 | 0.304
20 | 3.981 | 1.785 | NS | 2.26E-07 | 2.006
60* | 19.373 | 23.373 | BH | 8.54E-07 | 1.06E-08
85* | 30.465 | 34.465 | BH | 7.15E-06 | 2.07E-06
120* | 56.397 | 30.550 | PPISN | 0.0007 | 16.540
mypink $\upsilon_{\rm ini}=0.7\,\upsilon_{\rm crit}$
9 | 1.389 | 1.389 | WD | 0.004 | 0.551
20 | 8.679 | 12.679 | BH | 0.207 | 0.213
60 | 14.442 | 18.442 | BH | 3.760 | 0.914
85 | 33.07 | 37.070 | BH | 0.706 | 3.068
120* | 61.645 | 0.0 | PISN | 0.318 | 26.707
operamauve $Z=10^{-5}$
mypink $\upsilon_{\rm ini}=0$
9 | 1.009 | 1.009 | WD | 1.91E-05 | 0.123
20 | 3.748 | 7.748 | BH | 3.79E-05 | -8.67E-05
60 | 20.925 | 24.925 | BH | 1.70E-04 | -9.96E-05
85 | 31.622 | 35.622 | BH | 2.51E-04 | 0.001
120 | 47.977 | 41.620 | PPISN | 2.66E-04 | 6.292
mypink $\upsilon_{\rm ini}=0.4\,\upsilon_{\rm crit}$
9 | 1.119 | 1.119 | WD | 0.008 | 0.529
20 | 2.980 | 6.980 | BH | 0.171 | 0.216
60 | 13.419 | 17.419 | BH | 2.902 | 1.750
85 | 37.358 | 41.358 | BH | 0.051 | 0.652
120 | 59.437 | 0.0 | PISN | 0.012 | 53.17

We use here the classical definition of the stellar yield of a given isotope.
It is the quantity of that isotope newly synthesised and ejected by the star.
Since in the present models, the mass lost in the course of the evolution is
negligible, only the mass ejected at the time of the core collapse has to be
accounted for. Thus, we only need to know the remnant mass to calculate the
stellar yields. To obtain the final fate and remnant mass we use the results
of Farmer et al. (2019); Patton & Sukhbold (2020), following the method
outlined in Sibony et al. (2023) and in Nandal, Sibony & Tsiatsiou (2024) for
different CO core masses. The stellar yields are obtained by integrating the
abundances of an element above the remnant mass and by subtracting the initial
quantity of the element, as indicated in Maeder (1992). Of course for Pop III
stellar models, the mass of a metal isotope ejected is the stellar yield of
that isotope.

The results are presented in Table LABEL:table:yields_Patton for different
metallicities and different initial rotations. The initial mass is given in
column 1. In columns 2 and 3, the CO core and the remnant masses are given,
respectively. The CO core mass corresponds to the mass inside the Lagrangian
mass coordinate where the mass fraction of helium drops below $10^{-2}$ at the
end of the core He-burning phase. We also provide the possible final fate of
the stellar models in column 4. Lastly, the stellar yields of nitrogen and
oxygen are given in columns 5 and 6, respectively.

The models have been computed until the end of the core He-burning phase (some
have been pushed further), so the yields that we obtain may still be affected
by the more advanced stages of the evolution. However, the changes for the two
isotopes considered here remain modest and thus the yields obtained here are a
good approximation of the yields that would be obtained from pre-SN models.
Also, the bulk of nitrogen and oxygen are located sufficiently away from the
central conditions so that they will be affected very little by explosive
nucleosynthesis.

Figure 9 shows the stellar yields of nitrogen and oxygen. The upper panels
show the yields for Pop III models with different initial rotations. The non-
rotating and moderately-rotating models have been taken from Murphy et al.
(2021a). We have to note that the stellar yields from Murphy et al. (2021a)
are re-calculated for this work, since we have a different method of
calculating the remnant mass. In Murphy et al. (2021a) the remnant mass is
calculated according to Maeder (1992). The middle panels show the yields from
the rapidly-rotating Pop III models together with the moderately-rotating
metal-poor models ($Z=10^{-5}$) from Sibony et al. (submitted). The two lower
panels show the yields for Pop III models with different initial rotations and
from different authors. The non-rotating models have been taken from Limongi &
Chieffi (2012) and Murphy et al. (2021a), and the moderately-rotating from
Ekström et al. (2008). We shall discuss these two lower panels in Sect. 6.3.

The production of 14N is boosted in the rapidly-rotating models as can be seen
by comparing the solid green curve with the dashed blue (no rotation) and
orange (moderate rotation) curves. The moderately-rotating models show smaller
yields of nitrogen than the non-rotating ones in the mass range between about
10 and 87 $M_{\odot}$. As explained in Sect. A.3, this results from very
different behavior of the H-burning shell in the two models. Actually in the
non-rotating models, the structure show that the H- and the He-burning shell
are much nearer from each other than in the rotating model. This favours
mixing between these shells in the non-rotating model after the core He-
burning phase.

Regarding the comparison between the Pop III models and those with
$Z=10^{-5}$, we see that the rapidly-rotating Pop III stars may contribute
even more than the moderately-rotating $Z=10^{-5}$ models (see middle left
panel). The $Z=10^{-5}$ models in the high mass range lose significant amounts
of mass by stellar winds. As a numerical example the 85 $M_{\odot}$ model at
$Z=10^{-5}$ loses about 30 $M_{\odot}$. This has a strong impact on the
structure, notably making the H-shell to be less active, more distant from the
He-burning region and thus making primary nitrogen production less efficient.
In Fig. 10, we show how the yield of nitrogen would vary when considering
different remnant masses for the 20 $M_{\odot}$ model. For instance, for the
remnant mass given in Table LABEL:table:yields_Patton, the yield is given by
the ordinate of the intersection between the nitrogen curve and the vertical
dashed line labelled $M_{\rm rem}$ in Fig. 10. Considering lower values for
the remnant mass would not change the nitrogen yield much. Considering larger
values would on the contrary significantly impact the yield, even suppressing
it completely if the whole final mass of the star were to remain locked into
the stellar remnant black hole.

The yields for the main isotope of oxygen show significant differences between
the rapidly-rotating, non-rotating and moderately-rotating models. Most of
these differences reflect a complex interaction between the effects of
rotation on the size of the CO core and the process of computing the remnant
mass. Changing the remnant mass may strongly impact the yields of oxygen.
Looking at Fig. 10, we see that in case the rapidly-rotating 20 $M_{\odot}$
model produces a 12.7 $M_{\odot}$ black hole or a 1.6 $M_{\odot}$ neutron
star, the yield of oxygen varies from a value of 0.2 $M_{\odot}$ up to a value
of nearly 6 $M_{\odot}$. This high sensitivity of the oxygen yields on the
remnant mass explains the very large difference between the yields of the non-
rotating/moderately-rotating 60 $M_{\odot}$ models and the rapidly-rotating
one. In the rapidly-rotating model, a smaller CO core mass is produced (14.4
$M_{\odot}$ instead of 22.6 $M_{\odot}$ in the non-rotating model). This is a
result, as already explained above, of the extended intermediate convective
zone attached to the H-burning shell when a CNO shell boost occurs. This has
the consequence of reducing the He- and then the CO core mass and finally the
remnant mass in the rapidly-rotating model. It has a dramatic impact on the
oxygen yields passing from a value equal to $4.4\times 10^{-7}$ $M_{\odot}$ in
the non-rotating model to a value equal to 0.91 $M_{\odot}$ in the rapidly-
rotating one. This illustrates the great sensitivity of the oxygen yield on
the final fate of a star, a point that was already discussed in detail by
Maeder (1992). Given this and since the final fate of massive stars is still a
topic associated with great uncertainties, we provide in our
webpage777https://www.unige.ch/sciences/astro/evolution/en/database/ the
structure of our last computed models, so that other remnant masses can be
considered to obtain the yields.

In the middle right panel of Fig. 9, the oxygen yields for the $Z=10^{-5}$
models with $\upsilon_{\rm ini}/\upsilon_{\rm crit}=0.4$ are compared with the
Pop III oxygen yield from models with $\upsilon_{\rm ini}/\upsilon_{\rm
crit}=0.7$. Some masses show very similar values, and the others significant
differences, without any systematic effects. Again here the behaviour is very
dependent on the way the remnant masses are computed.

Figure 9: The stellar yields of nitrogen (left) and oxygen (right) for Pop III
and very low-metallicity ($Z=10^{-5}$) stars, for different initial rotations
and from different authors (Ekström et al., 2012; Limongi & Chieffi, 2012;
Murphy et al., 2021a, Sibony et al. submitted). The ranges of initial
equatorial velocities corresponding to the different coloured curves are
indicated. Figure 10: Variation of the mass of nitrogen and oxygen in the
outer layers of the star as a function of the lagrangian mass taken as the
lower limit of the integration. This lower limit can be called the mass of the
remnant. The case of the rapidly-rotating 20 $M_{\odot}$ model is shown. The
vertical grey dashed line denotes the remnant mass as adopted in the present
paper, while the horizontal grey dashed line denotes the nitrogen yield. The
lighter grey dashed line denotes the typical mass of a neutron star.

##  6 Discussion

In this section, we shall discuss three points: How the primary nitrogen
production depends on the metallicity considering the whole range from $Z=0$
to $Z=0.020$; How the present yields compare with other works; How our yields
impact the variation of N/O as a function of O/H when incorporated into a
simple chemical evolution model.

###  6.1 The metallicity dependence

Figure 11: The stellar yields of nitrogen for the 9 (blue) and 20 $M_{\odot}$
(orange) models at different metallicities ($0\leqslant Z\leqslant 0.020$).
The circles connected by the solid curve correspond to models with an initial
rotation of $\upsilon_{\rm ini}=0.4\,\upsilon_{\rm crit}$ and the diamonds of
$\upsilon_{\rm ini}=0.7\,\upsilon_{\rm crit}$. The moderately-rotating models
are from Murphy et al. (2021a, $Z=0$), Sibony et al. (submitted, $Z=10^{-5}$),
Groh et al. (2019, $Z=0.0004$), Georgy et al. (2013a, $Z=0.002$), Ekström et
al. (2012, $Z=0.014$), Eggenberger et al. (2021, $Z=0.006$), and Yusof et al.
(2022, $Z=0.020$). Figure 12: Evolution of N/O as a function of O/H ratios for
stellar models in the metallicity range of $0\leqslant Z\leqslant 0.020$
(coloured curves) and observed abundances of the Milky Way’s stars (grey
points). Yields from models with different initial rotations are used (see
Sect. 6.3). The observed data have been taken from the SAGA database (Suda et
al., 2008, 2011; Yamada et al., 2013). The yellow star represent the Sun’s
abundances from Asplund et al. (2009).

Figure 11 shows how the stellar yields vary as a function of the initial
metallicity for the 9 (orange) and 20 $M_{\odot}$ (blue) models. The solid
curves show the results when the initial rotation is $\upsilon_{\rm
ini}/\upsilon_{\rm crit}=0.4$ at every metallicity, while the orange diamonds
indicate the results when fast rotation is considered for $Z=0$.

Let us first discuss the case of the 20 $M_{\odot}$ models. Schematically, we
see that for the metallicity range between about 0.0004 and 0.020 with
$\upsilon_{\rm ini}/\upsilon_{\rm crit}=0.4$, the yields in nitrogen remain
modest. They are even negative in the upper metallicity range. For those
metallicities, nitrogen is mainly produced by the transformation of the
initial carbon and oxygen abundances and nearly no nitrogen is produced by the
diffusion of carbon and oxygen produced in the He-core into the H-burning
shell. Yields are negative at high $Z$ because the mass of the ejecta where
nitrogen is destroyed (typically matter processed by He-burning) is larger
than the mass of the ejecta where nitrogen is produced.

It is only in the metallicity range between 0 and 0.0004 that primary nitrogen
is produced thanks to rotational mixing. Starting from $Z=0$, the increase of
the yield with $Z$ is mainly due to rotational mixing that is favoured in
slightly metal-rich models. As explained before, the differential rotation
that drives the mixing increases when the metallicity increases. Beyond the
peak, the decrease of the yield when the metallicity increases further is due
to the larger and larger distance between the He- and H-burning zones. The 20
$M_{\odot}$ rapidly-rotating model is available only for $Z=0$. For that
model, the primary nitrogen production reaches a level that is larger than the
peak value shown at a metallicity $Z=10^{-5}$ with a moderate initial
rotation.

The 9 $M_{\odot}$ Pop III models with $\upsilon_{\rm ini}/\upsilon_{\rm
crit}=0.4$ do not show any strong peak between $Z=0$ and 0.0004. Primary
nitrogen production in this metallicity range depends sensitively on the
initial mass. It is interesting to note here that the central temperatures
during the H-burning phase are smaller in less massive stars. Thus, decreasing
the mass, everything else being kept constant, also tends to increase the
change of entropy between the He- and H-burning zones making the mixing more
difficult between the two zones.

###  6.2 Comparisons with other works

Comparing the nitrogen yields in Fig. 9 for the non-rotating models computed
with Genec by Murphy et al. (2021a) (dashed blue curve) with those of Limongi
& Chieffi (2012) (blue dot-dashed curve), one can note the following points.
First, our yields for the 9 $M_{\odot}$ model are in line with those of
Limongi & Chieffi (2012). Qualitatively we also obtain a bump that begins
around 20 $M_{\odot}$. Since our grid does not consider the same initial
masses, it is difficult to say more at the moment. We can just add that our
yield for the 85 $M_{\odot}$ is also in line with the yield given by Limongi &
Chieffi (2012).

If we compare the nitrogen yields by Ekström et al. (2010) for rotating models
computed with Genec with those presented in this work, we obtain significant
differences. This is in part due to the faster initial rotation considered
here but also due to the fact that we used in the present models different
diffusion coefficients for rotational mixing.

The differences between the yields in oxygen are mainly due to the different
methods used for estimating the remnant mass. It is interesting to note that
yields of Ekström et al. (2008) for rotating models and those of Limongi &
Chieffi (2012) for non-rotating Pop III stars are very similar. Actually, in
these two papers, the authors used remnant masses that are in general much
smaller than those in this work. This gives larger oxygen yields that do not
differ much between the rotating and non-rotating models.

These yields differ significantly from those obtained with the rapidly-
rotating models computed here and even more from the non-rotating models of
Murphy et al. (2021a). As explained already, this comes from the use of the
method by Patton & Sukhbold (2020) for estimating the remnant mass.

###  6.3 Impact on the early chemical evolution of galaxies

Figure 12 shows the impact of the yields with different initial rotations in a
simple galactic chemical evolution (GCE) model. Here, we computed the GCE with
the one-zone closed-box model compiled in the chemical evolution library
(celib, Saitoh, 2017; Hirai et al., 2021). This model adopts the initial mass
function of Kroupa (2001) for 0.1$-$120 $M_{\odot}$ and stellar lifetimes in
Portinari et al. (1998). We assume stars are constantly formed for 13 Gyr from
a gas cloud of $10^{11}$ $M_{\odot}$ with a star formation efficiency of 0.08
Gyr-1. Note that rapidly-rotating models ($\upsilon_{\rm ini}/\upsilon_{\rm
crit}=0.7$) have been computed only for Pop III models. Therefore, we adopt
the rapidly-rotating model for $Z$ = 0 and the rotating model with
$\upsilon_{\rm ini}/\upsilon_{\rm crit}=0.4$ from Ekström et al. (2012);
Georgy et al. (2013a); Groh et al. (2019); Eggenberger et al. (2021); Yusof et
al. (2022); Sibony et al. (submitted) for metallicities above zero in the
green curve. The red curve corresponds to the models above and to the models
from Murphy et al. (2021a) with $\upsilon_{\rm ini}/\upsilon_{\rm crit}=0.4$
for the metallicity $Z=0$. Lastly, the blue curve corresponds to non-rotating
models from all the works above. Observed data of the Milky Way’s stars were
taken from the SAGA database (Suda et al., 2008, 2011; Yamada et al., 2013).
We excluded CEMP stars with [C/Fe] $>+$0.7, which might be affected by binary
mass transfer.

We see that depending on the initial rotation, very different values of the
initial plateau for $\log({\rm O/H})+12<5$ are obtained, covering more than
two orders of magnitude. Because of the increased nitrogen production in the
model with $\upsilon_{\rm ini}/\upsilon_{\rm crit}=0.7$, the $\log({\rm N/O})$
value at $\log({\rm O/H})+12=5$ is predicted to be $-$1.3, while the non-
rotating model predicts $\log({\rm N/O})=-3.3$. The peaks of the $\log({\rm
N/O})$ seen around $\log({\rm O/H})+12=6.3$ in the rotating models are caused
by the maximum yields of 14N at $Z=10^{-5}$ shown in Fig. 11. Most stars with
$\log({\rm O/H})+12<7$ are within the N/O ratios predicted in models with
$\upsilon_{\rm ini}/\upsilon_{\rm crit}=0,\,0.4$ and 0.7. The increasing trend
seen in the observations for $\log({\rm O/H})+12>7$ could be explained by the
contribution of asymptotic giant branch (AGB) stars, which are not included in
this model (e.g. Chiappini et al., 2006). This example is just to illustrate
that indeed the increase of primary nitrogen production obtained in the
present models can have a significant impact at very low metallicities. In a
complementary paper, we shall publish the yields obtained from our stellar
grids of models covering the whole metallicity domain and further works
including these new yields will allow us to predict the evolution of other
elemental ratios.

##  7 Conclusion

We have explored the impact of fast rotation on the evolution of Pop III stars
with some discussion on their chemical feedback. The main conclusions are
briefly synthesised below:

  1. 1.

We obtain that for initial masses between 9 and 60 $M_{\odot}$, primary
nitrogen production in Pop III stellar models beginning their evolution with
70% of the critical velocity on the ZAMS is similar to the primary nitrogen
production in models beginning their evolution with 40% of the critical
velocity on the ZAMS at a metallicity of $Z=10^{-5}$.

  2. 2.

The appearance of the extended intermediate convective zone induced by the
injection of carbon and oxygen produced in the He-burning region into the
H-burning shell by diffusion/convection may have a dramatic impact on the
structure and the evolution of the star. This intermediate convective zone may
quench the He-burning in some model in the core for a while or at least
significantly reduce the He-burning core (a process called CNO shell boost, as
in Ekström et al., 2008). This has an impact on the nature of the stellar
remnant and on nucleosynthesis.

  3. 3.

Models with a mass equal to or larger than 60 $M_{\odot}$ spend more than half
of their MS lifetime at the critical velocity limit. However, the mass lost
mechanically remains very modest, the most massive model (120 $M_{\odot}$)
loses 2% of its initial mass at the end of its evolution.

  4. 4.

With the physics used here, models starting with 70% of the critical velocity
on the ZAMS show no evidence of following a chemically homogeneous evolution
trend. Actually, their tracks during the MS phase follow the classical redward
evolution with the models rotating faster showing larger extension towards
lower effective temperatures than slower rotators. The impact of fast rotation
on the ionisation outputs appears modest in the present models.

  5. 5.

Depending on the initial rotation at the ZAMS, a Pop III star can end its
stellar lifetime as a BSG or a RSG. The rapidly-rotating models will be RSGs,
as opposed to the BSGs from the slower-rotating models (for $M_{ini}>20$
$M_{\odot}$).

This work illustrates the fact that the physics of rotational mixing is a key
point to resolve in order to predict stellar yields, especially for the first
generations of stars. The physics of rotation is in our view as important as
the impact of close binary evolution. Indeed, if mass transfer in close
binaries may strongly impact the evolution of stars, primary nitrogen
production is essentially a question related to the internal mixing. Rotation
appears as an excellent candidate for driving this mixing (note that close
binary evolution and rotation are tightly intertwined through the exchange of
angular momentum between orbits and spins of stars). An interesting feature of
the rotational mixing is that it shows a dependence on the metallicity quite
in line with the fact that important sources of primary nitrogen are needed
only at low metallicities (see e.g. the discussion in Chiappini et al.,
2005b).

This work also underlines the importance of the initial rotation of Pop III
stars. Moreover, even if we had a good knowledge of the rotational velocity
distribution in the first Pop III stars, significant uncertainties persist
regarding the physics of rotation. Similar work as discussed here should be
done with other approaches for describing the transport of angular momentum
and of chemical species by rotation and also by other types of instabilities
(improvement in the physics of convection, internal waves, magnetic
instabilities). There is some hope that testing stellar physics on stars that
we can observe in the present-day Universe will allow us to constrain these
mechanisms in a still tighter way. Then one can assume that this physics will
also apply to Pop III stellar models. At the moment and likely for a still
very long period there will be no way of directly measuring the rotation
periods of Pop III stars from observations. Thus the only way to get
information about that is to observe the consequences of stellar models
computed with various initial rotations, ideally with different physics for
the transport mechanisms, on the early chemical evolution of galaxies (as the
origin of the chemical composition of CEMP stars and of the bulk of the
galactic halo stars). A possible exception could be provided by cases of
extreme flux amplification due to strong gravitational lensing (e.g.
Zackrisson et al., 2015). Recently, such extreme lensing has been invoked to
explain the Earendel stellar source (Welch et al., 2022), which has a non-
negligible probability of being Pop III, although a metal-poor, massive Pop II
star is more likely (Schauer et al., 2022). Other channels for getting
information on the evolution of the first massive star generations is through
the predictions of the rate of various types of supernovae (see the reviews by
Frebel & Norris, 2015; Meynet & Maeder, 2017, and references therein).
Recently nitrogen-rich high redshift galaxies have been observed (Oesch et
al., 2016). Implications of the present rapidly-rotating models in this
context are discussed in another paper (Nandal et al. in prep.).

###### Acknowledgements.

ST, YS, DN, SE and GM 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). LS and SE have received
support from the SNF project No 212143. EF has received support from the SNF
project number 200020_212124. YH has received JSPS KAKENHI Grant Numbers
JP22KJ0157, JP20K14532, JP21H04499, JP21K03614, JP22H01259. BL is supported by
the Royal Society University Research Fellowship.

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##  Appendix A Additional figures

###  A.1 Case of the rapidly-rotating 120 $M_{\odot}$ model

Figure 13 compares the variation of $U_{\rm r}$, $\Omega$, $D_{\rm shear}$ and
$D_{\rm eff}$ as a function of the Lagrangian mass during the core H-burning
phase when the mass fraction of hydrogen $X_{\rm c}$ is equal to 0.35 for the
120 $M_{\odot}$ models with a moderate (red curves) and a fast (green curves)
initial rotation. The rapidly-rotating model shows stronger meridional
currents (see the green curve in the upper left panel), which transport
angular momentum from inside to outside. The value of $D_{\rm eff}$ is larger
by many orders of magnitudes than $D_{\rm shear}$ in the radiative envelope
and governs the mixing of the elements in that region.

Figure 13: Variation of $U_{\rm r}$ (upper left panel), $\Omega$ (upper right
panel), $D_{\rm shear}$ (solid curves) and $D_{\rm eff}$ (dashed curves)
(lower panel) as a function of the Lagrangian mass coordinate in the Pop III
120 $M_{\odot}$ models with different initial rotations. The moderately-
rotating model was taken from Murphy et al. (2021a).

###  A.2 CNO shell boost in a Pop III 60 $M_{\odot}$ model

Figure 14: Comparing stages for the 60 $M_{\odot}$ rapidly-rotating model as
in Fig. 4.

The upper panel of Fig. 14 shows the HRD of the rapidly-rotating Pop III 60
$M_{\odot}$ model. Coloured points are overplotted. The orange point indicates
when the core He-burning phase begins, while the purple point indicated the
stage when an extended intermediate convective zone appears as a result of
injections of carbon and oxygen from the He-burning core to the H-burning
shell. Meanwhile, the convective He-burning core is reduced. Those points
correspond to the vertical coloured lines in the lower Kippenhahn plot,
respectively.

###  A.3 Reason why rotating models may produce less nitrogen than non-
rotating ones.

Figure 15: Kippenhahn diagrams for the 20 $M_{\odot}$ Pop III models with zero
(left panel) and moderate (right panel) rotation.

In Fig. 15, we have an example of 20 $M_{\odot}$ Pop III models with zero and
moderate rotation. We notice that the H- and He-burning shells are very close
to each other in the non-rotating model. More precisely, in the non-rotating
model, the mass separating the H- to the He-burning shell is a fraction of 1
$M_{\odot}$ during the core C-burning phase. While, in the moderately-rotating
model, during the same phase, the mass separating the H- and He-burning shell
is more than 2 $M_{\odot}$. This is in our view an indirect consequence of the
fact that in the moderately-rotating model (right panel), a small convective
zone occurs above the H-shell at the very beginning of the core He-burning
phase (due to the rotational mixing some C and O migrate into the H-shell and
boost its energy production). This has for an effect to make the mass of the
convective He-core smaller during the first part of the He-burning phase. If
the H-shell is replenished in hydrogen by the convective shell, it stays
longer in deeper regions of the star, so it migrates slowly towards outer
regions. In the non-rotating model, the H-burning shell at the beginning of
the core He-burning phase migrates much faster outwards than in the rotating
model (the shell is not replenished by the intermediate convective zone and
thus progresses more rapidly outwards). This makes the He-core to also
increases more rapidly in mass. As a result of this behaviour the two shells,
just after the end of the core He-burning phase are separated by less mass in
the non-rotating model than in the rotating one. Compare, for instance, in the
two Kippenhahn diagram (see Fig. 15) the mass between the blue and green solid
lines at a time 4. Due to the greater proximity in the non-rotating model, a
CNO shell boost occurs. As a result, we see that the H-shell shifts to a
position at a smaller lagrangian mass coordinate. So we see how complex can be
this behaviour. In the rotating model a modest early CNO boost occurs that
create conditions that disfavour any further CNO shell boost. In the rotating
model the early CNO shell boost does not occur in absence of any rotational
mixing, but it creates conditions more favourable for a CNO boost occurring at
the end of the core He-burning phase.

Moreover, at the end of core He-burning phase, the moderately-rotating mode
has higher nitrogen yield (approximately the moderately-rotating has
${}^{14}{\rm N}=3.2\times 10^{-7}$, while the non-rotating has ${}^{14}{\rm
N}=1.2\times 10^{-7}$). Therefore, we see that those yields on the Fig. 9 are
a result after the core He-burning phase.