Zero and Extremely Low-metallicity Rotating Massive Stars: Evolution, Explosion, and Nucleosynthesis Up to the Heaviest Nuclei

In general, we always preferred the experimental nuclear cross sections with respect to the theoretical ones. The nuclear cross section of each process was taken from KADoNiS v0.3 (Dillmann et al. 2006) for the neutron captures and from STARLIB (Sallaska et al. 2013) in all the other cases. If a nuclear cross section was not present in any of the two databases, it was searched for in the NACRE (Angulo et al. 1999) or NACRE II (Xu et al. 2013) compilations first and eventually in the Jina Reaclib (Cyburt et al. 2010) database. The only exceptions to this procedure were the nuclear cross sections of the reactions ¹⁷O(α, γ)²¹Ne and ¹⁷O(α,n)²⁰Ne, taken from Best et al. (2011, 2013). Particular attention was paid to the computation of the nuclear cross sections of the reverse processes. As the temperature increases above roughly 2 GK, each process a(i, j)b tends to be counterbalanced by the reverse process b(j, i)a so that the net rate of each pair of processes as well as its contribution to the total nuclear energy generation rate depends on the very small difference between the forward and the reverse process. For this reason, we decided not to tabulate the nuclear cross sections of the reverse processes but to compute them at runtime for the exact value of the forward process. The advantage of this technique is that in this way the nuclear cross section of the reverse process is exactly linked to the value of the forward process, providing a more stable and precise computation of the nuclear energy generation rate. We considered as the forward process of each pair the reaction having a positive Q-value.

The databases adopted for these processes include Fuller et al. (1982), Oda et al. (1994), Pruet & Fuller (2003), Langanke & Martínez-Pinedo (2000), Suzuki et al. (2016), and Li et al. (2016). In this case, the leading databases were those of Langanke & Martínez-Pinedo (2000) and Fuller et al. (1982) because they are the ones that cover the widest ranges both in temperature and density. If a process is not present in the first nor in the second database, the most recent value among Suzuki et al. (2016), Oda et al. (1994), and Pruet & Fuller (2003) is taken. If the reaction is not present in any of the abovementioned databases, its decay rate is assumed to be the terrestrial decay. The only exception to this procedure was the decay rate of ⁵⁹Fe into ⁵⁹Co because in this case we decided to adopt the decay rate provided by Li et al. (2016). We furthermore checked that all the decay rates progressively approach the terrestrial value at low temperature and low density.

The nuclear network is an extension of the one adopted in LC18. The main upgrades are (1) a wider extension toward larger atomic masses in order to properly manage higher neutron densities and (2) a larger number of elements (in particular, we added explicitly all elements between Tc and I).

The extension (in atomic mass) of the nuclear network is mainly dictated by the neutron density because it is this parameter (together with the temperature and density) that mainly controls how far matter can move out of the stability valley, because of the neutron captures, before decaying back toward its closest stable daughter. The parameter that controls the balance between beta decay and neutron capture may be written as the ratio B between decay rate and neutron capture rate:

$$\begin{eqnarray}&&B=\displaystyle \frac{{\lambda }_{i}(T,\rho )}{{n}_{{\rm{n}}}{\left\langle \sigma v\right\rangle }_{\mathrm{in}}(T)}\end{eqnarray} \tag{ 1 }$$

where n_n is the neutron density, λ_i is the probability of decay per time unit of the nucleus i, and ${\left\langle \sigma v\right\rangle }_{\mathrm{in}}$ is the nuclear cross section of the neutron capture process on the nuclear specie i. This ratio mainly depends on the neutron density n_n and mildly on the temperature T and the density ρ. In massive stars, the s-process nucleosynthesis predominantly occurs in core and shell He burning and in shell C burning (see, e.g., LC18), and the typical physical conditions in these phases are T ∼ 2.5 × 10⁸ K and ρ ∼ 10³ g cm⁻³ and T ∼ 10⁹ K and ρ ∼ 10⁵ g cm⁻³, respectively. The maximum neutron densities that are obtained in these conditions (in standard nonrotating models) are n_n ∼ 10⁷ n cm⁻³, in core and shell He burning, and n_n ∼ 10¹¹ n cm⁻³ in shell C burning. In order to take into account an even larger neutron production, in this work, we considered a limiting neutron density up to n_n ∼ 10⁸ n cm⁻³ for He burning and n_n ∼ 10¹⁴ n cm⁻³ for C burning.

We therefore determined the maximum atomic mass of each element by requiring that B ≥ 100, i.e., that the probability of decay were at least 100 times larger than the neutron capture:

$$\begin{eqnarray}&&{\lambda }_{i}(T,\rho )\geqslant {n}_{{\rm{n}}}{\left\langle \sigma v\right\rangle }_{\mathrm{in}}(T)\times 100.\end{eqnarray} \tag{ 2 }$$

We applied this procedure to all the elements with 31 ≤ Z ≤ 65 and 80 ≤ Z ≤ 83. The nuclear network that comes out from this procedure is reported in Table 1: it includes 524 isotopes and more than 3000 reactions fully coupled to the physical evolution of the star. It is important to note that, with respect to LC18, we have now only one cluster at local equilibrium corresponding to the elements with 65 < Z < 80.

Table 1. The Nuclear Network

Element	Z	${A}_{\min }$	${A}_{\max }$	Element	Z	${A}_{\min }$	${A}_{\max }$
n	0	1	1	Br	35	79	85
H	1	1	3	Kr	36	78	89
He	2	3	4	Rb	37	84	89
Li	3	6	7	Sr	38	84	93
Be	4	7	10	Y	39	89	95
B	5	10	11	Zr	40	90	97
C	6	12	14	Nb	41	92	98
N	7	13	16	Mo	42	92	103
O	8	14	19	Tc	43	97	103
F	9	17	20	Ru	44	96	107
Ne	10	19	23	Rh	45	102	108
Na	11	21	24	Pd	46	102	114
Mg	12	23	27	Ag	47	107	114
Al	13	25	28	Cd	48	108	119
Si	14	27	32	In	49	112	119
P	15	29	34	Sn	50	112	128
S	16	31	37	Sb	51	121	131
Cl	17	33	38	Te	52	122	133
Ar	18	35	41	I	53	126	135
K	19	37	44	Xe	54	126	137
Ca	20	39	49	Cs	55	133	139
Sc	21	41	49	Ba	56	134	141
Ti	22	44	51	La	57	138	143
V	23	45	52	Ce	58	138	145
Cr	24	47	55	Pr	59	141	146
Mn	25	49	57	Nd	60	142	149
Fe	26	51	61	Pm	61	147	149
Co	27	53	62	Sm	62	147	155
Ni	28	55	65	Eu	63	151	155
Cu	29	57	66	Gd	64	152	159
Zn	30	60	71	Tb	65	159	159
Ga	31	62	72	Hg	80	202	205
Ge	32	64	77	Te	81	203	206
As	33	74	77	Pb	82	204	209
Se	34	74	83	Bi	83	208	209

Note. ${A}_{\min }$ and ${A}_{\max }$ are the minimum and maximum atomic weights of each element. Elements between Dy (Z = 66) to Au (Z = 79) are assumed to be at local equilibrium under neutron captures.

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The initial chemical composition, i.e., the composition of the gas cloud from which the star formed out, adopted in the case of zero metallicity models is not based on the theoretical computation of the Big Bang nucleosynthesis (BBN) but, on the contrary, on the abundances observed in an extremely metal-poor environment. In particular, we follow the work of Fields et al. (2020), in which they select the most reliable environments in which the primordial abundances may still be visible. In particular, the abundance of ⁴He is derived from observations of extragalactic H ii emission lines; the abundance of deuterium is estimated through the study of quasar absorption systems; the reference abundance of ⁷Li is taken from the analysis of stellar spectra of low-metallicity stars. For ³He, there are not enough reliable observations; therefore, for this isotope, we adopt the value provided by Coc et al. (2013), which is derived from the local observations of our Galaxy. The abundances of all the nuclear species having Z ≥ 4 are put to zero. The initial abundances in mass fraction adopted in this work for the zero metallicity models are shown in Table  2.

For the higher-metallicity models, we adopt a scaled solar chemical composition with the exception of C, O, Mg, Si, S, Ar, Ca, and Ti, for which we adopt an enhancement with respect to the solar value derived from the observations of low-metallicity stars, i.e., [C/Fe] = 0.18, [O/Fe] = 0.47, [Mg/Fe] = 0.27, [Si/Fe] = 0.37, [S/Fe] = 0.35, [Ar/Fe] = 0.35, [Ca/Fe] = 0.33, [Ti/Fe] = 0.23 (see, e.g., Cayrel et al. 2004; Spite et al. 2005). The adopted solar chemical composition is the one provided by Asplund et al. (2009), which corresponds to a total metallicity equal to Z = 1.345 × 10⁻². Therefore, as a result of the enhancements mentioned above, the metal fraction corresponding to [Fe/H] = −N is Z = 3.236 × 10^−(N+2). We also assume that the initial abundance of ⁴He does not significantly vary from the adopted BBN value in the range of the explored metallicity, and therefore, it is kept constant. Table 3 summarizes the metallicity range explored in this work.

3.1.1. The H Burning

The lack of nuclear species with Z ≥ 6 seriously affects the evolution of a zero metallicity star both because the very low opacity forces these stars to spend their lives as compact blue objects, and also because the lack of the energy provided by an initial abundance of CNO nuclei forces these stars to contract and raise their central temperature much more than their more metal-rich counterparts. Figure 1 shows the Hertzsprung–Russell (H-R) diagram of all models presented in this paper. The plots on the left and those on the right refer to the 15 M_⊙ and the 25 M_⊙, respectively. The two zero metallicity models are represented as black lines in the two upper panels.

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The proton–proton (PP) chain is able to produce enough energy to replace the one lost from the surface of the star just at the beginning of the central H-burning phase. Right after, the continuous increase of the mean molecular weight due to the conversion of protons in α particles leads necessarily to a progressive increase of the temperature. The low dependence of the energy generation rate of the PP chain on the temperature forces these stars to contract and heat more effectively than their more metal-rich counterparts, reaching very early a temperature high enough to partially activate the 3α nuclear reactions and therefore the synthesis of some ¹²C. The full CNO cycle activates immediately and progressively replaces the PP chain as the main energy producer: in both masses, the CNO cycle produces 50% of the nuclear energy when the central H burning is still 0.5 by mass fraction. At H_c ∼ 0.2, the CNO cycle produces more than 80% of the energy required to sustain the 15 M_⊙ and more than 90% in the 25 M_⊙. As the central temperature increases, the amount of CNO nuclei produced by the 3α increases, so that toward the end of the central H-burning phase (H_c ≃ 0.05) it reaches a mass fraction of 4 × 10⁻¹⁰ (15 M_⊙) and 6 × 10⁻¹⁰ (25 M_⊙). During the very latest phases of the central H burning, the coupling 3α + CNO cycle raises the abundance of ¹⁴N from a few times 10⁻¹⁰ to roughly 10⁻⁶ in both masses. Once the H is exhausted in the convective core, the H burning shifts in a shell where the 3α and the CNO cycle still work together, so that the level of ¹⁴N within the whole H exhausted core settles to a value of the order of 10⁻⁶ by mass fraction (again in both masses). Note that during the central H burning a convective shell forms well outside the convective core in the 25 M_⊙ (the Kippenhahn diagrams of the zero metallicity stars are shown in the two upper panels of Figure 2 while those of the other two metallicities are shown in the second and third rows).

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Turning from Z = 0 (or, equivalently, [Fe/H] = −∞, Set Z) to [Fe/H] = −5 (Set F), the initial global abundance of the CNO nuclei (X_CNO = 3.24 × 10⁻⁷) is already high enough to fully sustain the star without the need of additional production of C in H burning. However, also at this metallicity, some C is produced toward the very end of the H burning so that once again the mass fraction of the ¹⁴N within the whole H exhausted core reaches a value of the order of 10⁻⁶ by mass fraction in both masses. The initial abundance of the CNO nuclei in Set E ([Fe/H] = −4) is higher than 10⁻⁶ so that the 3α processes at this metallicity do not even activate toward the end of the central H burning.

In general, the size of the convective core (that practically corresponds to the initial size of the He core) scales inversely with the initial metallicity because the lower the CNO abundance the larger the size of the convective core (the higher central temperature implies higher energy fluxes). But when the metallicity drops to zero, the PP chain overcomes the CNO cycle, and the lower dependence of the energy flux on the temperature inverts the trend leading to convective cores (and hence He core masses) smaller than those proper of more metal-rich stars. This explains why the He core mass increases from 3.6 M_⊙ (Set Z) to 3.9 M_⊙ (both Sets F and E) in the 15 M_⊙ and from 7.2 M_⊙ (Set Z) to 8.1 M_⊙ (both Sets F and E) in the 25 M_⊙.

The last thing worth noting is that both stars practically evolve at a constant mass in central H burning because the mass loss (that scales directly with Z) is negligible at these extreme metallicities.

3.1.2. The He Burning

3.1.3. The Advanced Burning

As already extensively discussed in literature (see, e.g., Meynet et al. 2006; CL13; LC18, and references therein), the inclusion of some angular momentum changes many evolutionary properties of the stars; among the others, it activates instabilities that drive some mixing of the matter in layers that would otherwise be in radiative equilibrium, forces a star (through the centrifugal force) to inflate its mantle much more (and much faster) than its nonrotating counterpart at the end of the central H-burning phase, and triggers episodes of (dynamical) mass loss. The red (Set Z), green (Set F), and blue symbols (Set E) in Figure 3 show the relation between the total angular momentum injected in each model and the initial rotational velocity (at the beginning of the central H-burning phase). Note that the amount of angular momentum injected in stars with the same rotational velocity scales directly with the metallicity because the higher the metallicity the more expanded the star in the MS. The diamonds refer to the 15 M_⊙ while the dots refer to the 25 M_⊙. The corresponding initial values of Ω/Ω_crit are shown in Figure 4. The same figure shows that this ratio increases in central H burning and that the fastest rotating models reach the break up velocity before the central H exhaustion. Note that the expansion of the surface that occurs in central H burning would lead, per se, to a decrease of Ω/Ω_crit in the absence of transport of the angular momentum from the interior toward the surface. It is the outward transport of angular momentum that forces Ω/Ω_crit to increase. In stars that are metal-rich enough to lose a substantial amount of mass in MS, Ω/Ω_crit is kept well below 1 by the efficient angular momentum removal operated by mass loss, while the lack of an efficient mass loss at zero and extremely low metallicity leads to the pile up of angular momentum to the surface and hence to the continuous increase of Ω/Ω_crit. The most external layers in which such a ratio reaches the value of one become unbound, and they are lost in the interstellar medium. However, the total amount of mass lost by this phenomenon is quite modest because the models reach the break up velocity only toward the end of the central H-burning phase (Figure 4). In all explored cases, the amount of mass lost by the models due to such a phenomenon never exceeds 1 M_⊙. It is worth noting that none of our models approaches the condition of quasi homogeneous mixing (see, e.g., Yoon et al. 2012; Banerjee et al. 2019), although a few of them start their evolution with a rotation velocity very close to the break up one.

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The continuous stirring of matter caused by the rotation induced instabilities leads to more extended H convective cores and also to the diffusion of the products of the central burning in the H-rich mantle. The trend of the He core mass versus the initial rotation velocity is shown in Figure 5. There is a general increase of the He core mass with the initial equatorial rotation velocity v_ini, even if with some exceptions due to the complex interplay among the various convective zones that develop during the evolution of the stars. The synthesis of fresh C by the 3α reaction in rotating stars during H burning is quite similar to that occurring in the nonrotating models: zero metallicity stars synthesize an abundance of C of the order of a few times 10⁻¹⁰ by mass fraction while no models of Sets F and E require the activation of the 3α during most of the central H burning. Similarly to what happens in the nonrotating case, toward the end of the central H burning, the models of both Sets Z and F synthesize enough C to increase the N abundance up to a value of the order of 10⁻⁶ by mass fraction in the whole He core, independently on the initial rotation velocity. Also, the rotating models of Set E behave like their nonrotating counterparts; therefore, the ¹⁴N present in the He core of these models is simply the one that descends from the initial CNO abundance.

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The very high temperature at which the H burning occurs in models of Sets Z and F leads to the He ignition and to the formation of a convective core immediately after the central H exhaustion when the models are still on the blue side of the H-R diagram. The rotation induced instabilities, which in H burning led to both an increase of the H convective core and to the diffusion of the products of the nuclear burning into the H-rich mantle, in He burning continuously stir matter between the He convective core and the H-burning shell. The first consequence of such a stirring is that, similarly to what happens in H burning, the final CO cores are more massive than in the nonrotating case (Figure 6). The second one is that the continuous ingestion of fresh He in the convective core leads to a lower final abundance of ¹²C in the He exhausted core. Figure 7 shows the final ¹²C abundance as a function of the initial rotation velocity for both masses. The third one is that, as already found by Meynet & Maeder (2002), Chieffi & Limongi (2013), a continuous slow exchange of matter between the He convective core and the H-burning shell leads to a large enhancement of the products of the CNO (chiefly ¹⁴N and ¹³C) in both the He convective core and the whole radiative zone that separates the border of the convective core from the H-burning shell. We call such a strong continuous reprocessing of matter between the He- and the H-burning entanglement. The consequences of this entanglement will be discussed in Section 3.3.

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If the initial rotation is fast enough, a significant mixing of C might drive the formation of H convective shell, which grows progressively in mass producing a shape that in the Kippenhahn diagram looks like an eagle's beak. Such a peculiar H convective shell forms in the 025z700, 025z000, 015f450, 015f800, 015e600, and 015e700 models. Although there is a clear general trend that shows that such a convective shell forms in the fastest rotating stars, one should not expect a strict correlation because rotation leads simultaneously to an expansion of the H mantle (lower densities and temperature imply lower H-burning rates), to the increase of the C abundance in the H-rich mantle (which implies higher H-burning rates), and to larger He core masses (that would speed up the evolution of the He core, reducing the time available for the secular instabilities to bring freshly synthesized C in the H-rich mantle). The Kippenhahn diagrams of all the rotating models are shown in Figures 8–13.

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The path of all rotating models in the H-R diagram reflects the complex interplay occurring among the various convective zones that form during the evolution of these stars and is shown in the various panels of Figure 1. Also the rotating models ignite He as a blue supergiant (BSG), but, at variance with their nonrotating counterparts that spend all their central He-burning lifetime as BSG, they are able (in many cases) to almost reach their Hayashi track on a nuclear timescale. The rotating 15 M_⊙ of Set Z, for example, reaches their Hayashi track toward the end of the central He burning, consuming up to 20% of the He as a red supergiant (RSG). By the way, this means that extremely metal-poor stars may populate the Hertzsprung gap in central He burning. All models able to approach their Hayashi track, however, overcome their Eddington luminosity when their surface temperature drops below ∼8000 K, lose most of their H-rich mantle on a dynamical timescale, and turn back to a BSG configuration on a thermal timescale. Note that the super-Eddington regime is always reached in the atmosphere of the star, within 10⁻⁴ M_⊙ from the surface, well before the formation of an extended convective envelope, in layers where the superadiabatic gradient practically coincides with the radiative one. This is a quite new result since the extremely metal-poor stars (including the Population III) have never been expected to experience mass-loss events, but, on the contrary, they were believed to evolve at constant mass unless their rotational velocity reaches the break up velocity. The model with v_ini = 150 km s⁻¹ is the only one whose luminosity never overcomes the Eddington one and hence remains an RSG up to the final collapse. The rotating 25 M_⊙ of Set Z, on the contrary, remains to the blue side of the H-R diagram expanding only moderately during the central He burning. The reason is the presence of an H convective shell that slows down or even prevents the expansion of the models toward their Hayashi track. Only the models with an initial rotation velocity equal to 450 and 600 km s⁻¹ reach their Hayashi track because of the smaller mass size of the H convective shell, and only the one started with v_ini = 600 km s⁻¹ exceeds its Eddington luminosity, and loses most of its envelope becoming again a BSG. The path in the H-R diagram of the rotating models of both Sets F and E is quite similar to that of the models of Set Z. All of them ignite He as BSG and move on a nuclear timescale toward their Hayashi track. Once again, only the models that do not form an extended H convective shell become RSG; the others remain BSG all along their central He-burning phase (see Figure 1). All rotating models that are able to reach their Hayashi track before the end of the central He burning overcome their Eddington luminosity when their surface temperature drops to 8000 K, and start losing dynamically a large fraction of their H-rich mantle becoming again BSG.

We have already discussed in Section 3.1.2 that the physical evolution of any model beyond the central He burning is basically independent on the initial mass and metallicity since the two main drivers of the evolution in the advanced burning phases are the CO core mass and the amount of ¹²C left by the He burning. In addition to this, the advanced burning phases are almost independent also on the initial rotation velocity because (a) the CO core is so compact that the centrifugal force is not able to distort significantly the shape of the star with respect to a spherical configuration, and (b) the rotation induced instabilities do not have time to grow so that neither the angular momentum nor the chemicals are transported any more by the rotation driven instabilities. The only exception is the transport of the angular momentum in the convective regions: since we assume instantaneous mixing of the angular momentum in all the convective regions, we force Ω to become flat in each convective region even in the advance burning (as discussed in CL13).

There are two additional distinctive features caused by rotation. The first one is that rotation inhibits the penetration of the He convective shell in the H-rich mantle (a feature that in nonrotating models leads to the synthesis of a large amount of primary ¹⁴N; Heger & Woosley 2010; Limongi & Chieffi 2012, and reference therein). Basically, all authors who published over the years with models of zero or extremely low metallicity found that in some mass intervals the He convective shell penetrates the H-rich mantle whose main effect is to produce a large amount of primary ¹⁴N. In the nonrotating models presented in this work, such a merging occurs in both the 25 M_⊙ of Sets Z and F (see Figure 2). Vice versa, none of the corresponding rotating models shows such a merging (with one exception). The main reason is that the stirring of the matter driven by the rotation leaves a quite smooth He profile at the central He exhaustion (instead of the essentially vertical profile typical of the nonrotating models, see also CL13; and LC18). As a consequence, the He convective shell forms in a region of variable He; and therefore, its abundance is much lower than in the nonrotating case (where it is of the order of 1); such a lower abundance reduces significantly the energy produced by the 3α and therefore the capability of the outer border of the He convective shell to penetrate into the H-rich layers. It must also be considered that the distance, in mass, between the He core and the CO core masses increases with the initial rotation velocity, and therefore, the base of the H-rich mantle is farther away from the outer border of the He convective shell than in the case of nonrotating models. The only exception to this general trend is the 25 M_⊙ of Set Z, rotating at 300 km s⁻¹. The reason is easily understood by looking at the upper right panel in Figure 9. In this case, in fact, the H convective shell is particularly wide and extends down to the region of variable He since the beginning of the central He burning. This means that (1) the He convective shell forms much closer to the H-rich mantle, and (2) the H profile is so steep that any ingestion (even modest) of H immediately leads to a H-burning runaway event because of its high concentration (almost the initial one).

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The second distinctive feature of the rotating models is the penetration of the O convective shell well within the C-rich zone soon after the central Si exhaustion in many rotating models. In particular, this phenomenon occurs in all the rotating 15 M_⊙ models of the three metallicities (except for the models 015z150 and 015z800), while it does not occur at all in the 15 M_⊙ nonrotating models nor in any of the 25 M_⊙, rotating or not. The Kippenhahn diagrams of the 15 M_⊙ models (Figures 8, 10, and 12) show clearly such an occurrence. The reason that leads most of the 15 M_⊙ rotating models to experience the penetration of the O convective shell within the Ne- and C-rich layers is quite difficult to determine robustly because the rotating ones end the central He burning with a much lower C abundance (and hence, in turn, Ne abundance) with respect to their nonrotating counterparts, and this occurrence affects all the advanced evolutionary phases. However, we note that both rotating and nonrotating models develop an O convective shell that closely approaches the Ne-rich layers; but while in the first case, the merging occurs; in the second one, it does not. A closer look at the models at the time of the merging (i.e., soon after the disappearance of the Si convective shell) shows that the models with a lower Ne abundance (the rotating ones) show a smaller entropy jump at the base of the Ne burning shell with respect to their respective nonrotating models. The role of the Ne abundance in determining the entropy jump may be understood by considering that such a jump would vanish as the Ne abundance would turn to zero because in this limiting case any difference, in the nuclear energy generation and in the chemical composition, would disappear. Figure 11 shows, as a typical example, a comparison between the entropy profiles of the 015e000 (nonrotating, with no merging) and the 015e300 (mildly rotating, with an extended merging). The figure clearly shows the presence of a larger entropy barrier in the nonrotating model with respect to that present in the rotating one at the base of the ²⁰Ne-rich zone (at $\mathrm{log}\ P\sim 22.7$). To be absolutely clear, we are not saying that the smaller entropy barrier present in the rotating models determines, per se, the penetration of the O convective shell in the Ne-rich region but, simply, that a lower entropy barrier certainly favors (i.e., represents a smaller obstacle) the growth of the O convective shell. An additional point worth being stressed here is that also the criterion adopted to determine the outer (actually any) border of the O convective shell plays a pivotal role. The adoption of the Ledoux criterion instead of the Schwarzschild one, e.g., would obviously disfavor the penetration of the outer border of the convective shell across the chemical discontinuity. Note, however, that also in this respect a lower Ne abundance would in any case imply a lower gradient of molecular weight. The possible occurrence of a merging of the O convective shell in the C-rich zone is of great interest because it would reduce the yields of the products of the C burning and increase those of the O burning. The reason is that the merging spreads the products of the C burning down to the base of the O convective shell (where they are easily destroyed by the passage of the shock wave), and, vice versa, it spreads out the products of the O burning where they can survive to the passage of the shock wave (see Ritter et al. 2018; Roberti et al. 2023; and Section 3.3). This problem certainly deserves a deeper analysis, and we plan to address this problem in the next future. The 25 M_⊙ models do not show such a behavior of the O convective; because, in this case, even in the models with a low ¹²C mass fraction (and therefore, in turn, a low ²⁰Ne mass fraction), the outer border of the O convective shell always remains quite far from the O/Ne discontinuity.

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The main physical properties of the stars at the onset of the core collapse are shown in Table 6 of Appendix B. The final fate of a massive star, i.e., if a successful explosion will occur or not, depends on the interplay between the amount of energy stored in the shock wave (that would push matter outward) and the binding energy (that opposes to the ejection of the mantle). As already discussed in Section 3.1.2, the binding energy at the onset of the collapse is the result of the overlap of the various nuclear burning phases that followed each other in the hydrostatic evolution of the star. Since rotation modifies somewhat the various burning stages, it is interesting to see how rotation modifies the binding energy of a stellar model, which directly reflects, in turn, the final mass–radius relation. We already introduced the compactness ξ_2.5. Another interesting point in which it is interesting to evaluate the compactness is at the CO core, i.e., ξ_CO, since it reflects the behavior of the C convective shells in the advanced burning stages. Figure 15 shows the trend of the compactness, evaluated in these two points, as a function of the initial rotation velocity and of the metallicity. ξ_2.5 shows a modest increase with the initial rotation velocity, while ξ_CO shows a significantly larger dependence on it. This suggests that, generally speaking, rotating models would not be more difficult to explode. However, since the compactness parameter can give at most a preliminary idea of the final fate of a star, we plan to address in the future such a problem with the aid of codes able to follow the collapse and rebounce of the collapsing core (see, e.g., Boccioli et al. 2023).

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In this section, we discuss basically how rotation affects the yields of the various nuclear species at these extreme metallicities. The key differences induced by rotation on the chemical evolution of a star are essentially two: the first one is that rotation leads to larger CO core masses and lower C abundances at the end of the central He burning, and this implies that rotating stars tend to behave as stars of moderately larger mass, and the second one is the entanglement between the central He burning and the shell H burning. The consequence of this phenomenon is the formation of a robust concentration of ¹⁴N (and ¹³C) within the He core in all models. The ¹⁴N engulfed in the He convective core is rapidly converted in ²²Ne while ¹³C is fully destroyed by the (α, n) nuclear reaction becoming therefore a primary neutron source. The ¹⁴N and the ¹³C that instead remain locked between the border of the convective core and the base of the H shell are left substantially unaltered because of the relatively low temperature of this region. We call this region, rich in ¹⁴N, ¹³C, and in general products of the CNO cycle, CNO pocket. Toward the end of the central He burning, the ²²Ne(α,n)²⁵Mg nuclear reaction activates in the convective core, providing therefore a quite robust primary neutron source. This primary neutron source depends of course on the efficiency of the entanglement between the He and H burning, which, in turn, is strictly connected to the initial rotation velocity. A useful tool able to quantify the efficiency of the chemical mixing between the H- and the He-burning regions in the rotating models is the χ parameter, defined in LC18 as follows:

$$\begin{eqnarray}\begin{array}{rcl}\chi ({\rm{N}},\mathrm{Mg}) & = & \displaystyle \frac{{\rm{X}}{(}^{14}{\rm{N}})}{14}+\displaystyle \frac{{\rm{X}}{(}^{18}{\rm{F}})}{18}+\displaystyle \frac{{\rm{X}}{(}^{18}{\rm{O}})}{18}+\displaystyle \frac{{\rm{X}}{(}^{22}\mathrm{Ne})}{22}\\ & & +\,\displaystyle \frac{{\rm{X}}{(}^{25}\mathrm{Mg})}{25}+\displaystyle \frac{{\rm{X}}{(}^{26}\mathrm{Mg})}{26},\end{array}\end{eqnarray} \tag{ 3 }$$

and it is simply the sum by number fraction of all the nuclei between ¹⁴N and ²⁶Mg involved in the conversion of N into Mg. This parameter remains constant if the He- and the H-burning regions do not interact; because, in this case, the ¹⁴N burning may populate only nuclei of the chain that goes from the ¹⁴N to the ²⁵Mg/²⁶Mg. Conversely, in rotating stars, the amount of ¹⁴N increases continuously because of the progressive conversion of freshly made ¹²C into ¹⁴N. Figure 16 shows the strong correlation between the variation of the χ parameter and the initial rotation velocity. Since χ is basically a proxy of the amount of ²²Ne present in the core, the neutron density in He burning increases as the initial rotation velocity increases, turning from the negligible values of 10²–10⁴ n cm⁻³ (typical of the nonrotating models at these metallicities) up to values of the order of ∼10⁷ n cm⁻³, i.e., very close to that of solar metallicity models.

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At zero metallicity, this quite large neutron density does not allow in any case the synthesis of nuclei heavier than Zn, not even the s-process weak component because of the lack of seed nuclei. The high neutron density, in this case, leads to the synthesis of several neutron-rich nuclei like, e.g., ²⁵⁻²⁶Mg, ²³Na, and, in progressively faster rotating stars, of ²⁷Al, ^28−29−30Si, ³¹P, and ^32−33−34S.

At higher metallicity (Sets F and E), the presence of seed nuclei, coupled to the high neutron-to-seed ratio, allows the synthesis of heavy nuclei up to, in some cases, the strong component (i.e., up to Pb).

In the following, for simplicity, we will mainly discuss Sr, Ba, and Pb to describe the climbing of the matter along the chart of the nuclides as a function of the initial metallicity and rotation velocity.

Figure 17 shows a snapshot of the internal structure of four representative models at the end of the central He burning for Sets F and E. All four models show a consistent production of the weak component (represented by ⁸⁸Sr), and the 15 M_⊙ shows an increase of the elements of the main component (represented by ¹³⁸Ba). The nucleosynthesis extends up to ²⁰⁸Pb in both 15 M_⊙ models. Note that also ¹⁹F shows a pronounced peak in both 15 M_⊙, while the 25 M_⊙ does not. It is worth noting that the core He-burning phase produces the bulk of the final amount of the s-process elements.

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Once He is exhausted in the convective core, He burning shifts in shell where an extended convective shell soon develops. Although the neutron density reached at the base of the convective shell reaches values in the range 10^8–10 n cm⁻³, the overall contribution of the He convective shell to the synthesis of the heavy elements is quite modest. Figure 18 shows a snapshot of the four structures already shown in Figure 17 once the He convective shell is fully developed.

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Although the He convective shell does not play an important role in the synthesis of the trans-Fe elements, it is actually the nursery of F. ¹⁹F, in fact, is synthesized by the sequence of reactions first identified by Forestini et al. (1992): ¹⁴N(α, γ)¹⁸F(β⁺)¹⁸O(p,α)¹⁵N(α, γ)¹⁹F, where the protons necessary to feed the ¹⁸O come from the ¹⁴N(n,p)¹⁴C, while the neutrons are produced by the ¹³C(α,n). It is not easy to find an environment in which this chain of reactions may work because of the simultaneously need of both neutrons and ¹⁴N. The He-burning shell in rotating massive stars perfectly embodies these requirements because of the large buffer of ¹⁴N and ¹³C present in the CNO pocket between the CO core and the He core. When the He convective shell develops and extends within the He core, it engulfs a large amount of ¹⁴N and ¹³C that are quickly brought to the He-burning temperature where the above quoted reactions may easily occur (see, e.g., CL13; LC18). ¹⁹F may reach a concentration as high as 10⁻⁴ in mass fraction in the fastest rotating models.

The last neutron production site occurs in C burning, more specifically in the convective C shell burning where the ²²Ne survived the central He burning reacting with the α particles provided by the ¹²C(¹²C,α)γ nuclear reaction, pours a substantial amount of neutrons. Although the contribution of the C burning to the synthesis of the elements heavier than Zn is modest, it is worth noting that, even if the C convective shell does not increases the yields of the heavy nuclei, it does not even destroy them significantly (Figure 19). Typical temperatures and densities at the base of the C convective shell are T = 1.2–1.5 GK, and ρ = 1–2 × 10⁵ g cm⁻³, and the neutron densities span the range 10¹¹–10¹³ n cm⁻³.

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Beyond the C burning, no s-process nuclei production is possible because the temperature is high enough that they are progressively and completely photodisintegrated.

There is however another phenomenon that occurs in the 15 M_⊙ rotating models. In fact, as discussed in Section 3.2, during the very late stages of their evolution, the O convective shell merges with the C shell and ingests a sizable amount of fresh ¹²C. The result of this phenomenon is the formation of a large O convective shell in which part of the products of the C shell burning is reprocessed. In particular, the most abundant nuclei that emerge from this interaction are the typical products of the O burning (see discussion in Section 6). This extended convective region preserves these intermediate and Fe-peak nuclei from being reprocessed by the explosive nucleosynthesis. In the case of the element above Fe, the effect of such an ingestion is the destruction of the heaviest isotopes and an increase of the abundances of the elements of the weak component in the extended convective shell. Figure 20 shows the result of such a mixing in two representative cases.

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As discussed in the previous sections (Sections 3.1.2 and 3.2), in several cases, the He convective shell penetrates the H-rich region, and the extra energy produced by the proton ingestion in the He shell leads to the consequent formation of a large convective region that rapidly covers a large part of the H-rich envelope. This phenomenon has been associated to the synthesis of primary ¹⁴N at early times. In both nonrotating 25 M_⊙ stars, the abundance of ¹⁴N in the extended convective shell reaches a concentration of ∼3 × 10⁻³ in mass fraction. But this merging leads also to the production of a strong neutron flux due to α captures on ¹³C. Very high neutron densities may be obtained, typical of the so-called intermediate neutron capture process (i-process), i.e., n_n ∼ 10¹⁴ n cm⁻³. In the 025z000 model, similarly to the He-burning phase in the rotating case, these neutrons are absorbed by lighter nuclei, populating the light and intermediate nuclei up to Ti. The most abundant isotopes in the newly formed H convective shell are ¹⁴N, ¹³C, ²²Ne, and ²³Na. In the 025f000 model, the most abundant nuclei are ⁸⁹Y, ¹⁰⁵Pd, ¹¹¹Cd, and ¹²⁵Sb; therefore, the neutron capture nucleosynthesis favors in this case the synthesis of some nuclei belonging mostly to the weak component. Note that ¹²⁵Sb is an unstable nucleus; therefore, it will contribute to the abundance of ¹²⁵Te. Figure 21 shows the internal structure before (left panels) and after (right panels) the merging of the H and He convective shell in the nonrotating 25 M_⊙ stars at[Fe/H] = −∞ (upper panels) and –5 (lower panels).

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Nitrogen is synthesized by the CNO cycle in the hydrostatic H-burning stage, and it is therefore a typical secondary element in nonrotating stars. The only exception occurs at zero metallicity where the He convective shell engulfs part of the H-rich mantle in some models. The abrupt ingestion of protons at temperatures of the He burning in a C-rich environment leads to a burst of synthesis of nitrogen (primary). We obtain such an ingestion in the two nonrotating models 025z000 and 025f000. As already discussed in Sections 3.2 and 3.3, rotation plays a critical role in the production of N. Figure 23 shows the N yield as a function of the initial rotation velocity for all the computed models. Both masses and all three metallicities show a steep increase of the N yield up to v_ini = 300 km s⁻¹ and a rough plateau for larger initial velocities. It is worth noting that rotation may lead to N yields as high as those provided by the nonrotating massive stars where the penetration of the He convective shell in the H-rich mantle occurs. Therefore, the rather high and somewhat unexpected N abundance plateau obtained from observations in low-metallicity halo stars of the Milky Way (Chiappini et al. 2005, 2006; Grisoni et al. 2021) could be explained in terms of the nucleosynthesis of these very low-metallicity massive stars (rotating and not). By the way, the temporal evolution of N in the context of the chemical evolution of the Milky Way and its connection with the average rotation velocity of the stellar generations of different metallicities has been addressed in detail by Prantzos et al. (2018) and later on by, e.g., Grisoni et al. (2020, 2021) and Franco et al. (2021). Of course, a conclusive analysis on this subject would require additional models as well as a quantitative estimate of the contribution of these stars and of the initial distribution of their rotation velocities to the enrichment of N in the early stages of evolution of the Galaxy and of the Universe in general.

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The MS of processes that leads to the synthesis of fluorine is the one identified by Forestini et al. (1992) and described in Section 3.3. Prantzos et al. (2018) have shown that rapidly rotating massive stars can be responsible of the synthesis of fluorine at low metallicity (see also Grisoni et al. 2020; Franco et al. 2021). Here we confirm such a result also at extremely low metallicities. Figure 24 shows the dependence of the ¹⁹F yields on the initial rotation velocity for both masses and the three initial rotation velocities. The plot shows a good trend with the velocity, while it is almost independent of the initial metallicity, resembling that of N. In particular, it increases with the initial rotation velocity and then bends toward a rough plateau for v_ini > 300 km s⁻¹, reaching values as high as 10⁻⁴–10⁻³ M_⊙. Note, however, that the nonrotating models in which the H–He shell merging occurs do not contribute significantly to the synthesis of F.

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Let us consider first the sum of the yields of all the elements heavier than Zn. Figure 25 shows this quantity as a function of the initial rotation velocity for stars of Sets F and E. Note that zero metallicity stars do not synthesize any appreciable amount of elements beyond Zn (Chieffi & Limongi 2004) and hence do not appear in this plot. The quantity plotted in the figure is the net total yield, defined as the difference between the final and the initial sum of the yields, of all the elements with Z > 30 up to Z = 83 (hereinafter S3083). An overall look at the figure shows a definite increase of S3083 with the initial rotation velocity. A closer look shows a few other features worth being mentioned. First of all, while three out of the four nonrotating models (015f000, 015f000, and 025f000) barely reach a global abundance of the order of 10⁻¹⁰–10⁻⁹ M_⊙, the model 025f000 shows a much larger increase of S3083 (of the order of 10⁻⁷ M_⊙, green dot). Such a high abundance is the consequence of the merging of the He convective shell with the H-rich mantle. The abrupt ingestion of protons down to the He-burning layers triggers a short burst of neutrons due to the activation of the ¹³C(α,n)¹⁶O nuclear reaction rate that leads to a nonnegligible neutron capture nucleosynthesis. The second thing worth noting is that, for each mass, S3083 increases up to a rotation velocity of the order of 300 km s⁻¹ and then bends considerably for large initial rotation velocities. In the 15 M_⊙ of Set F, the rotation increases S3083 up to a factor of the order of 50 times the initial value while the increase may become 10 times larger (i.e., 500 times) in the 15 M_⊙ of Set E. In the 25 M_⊙, S3083 may increase up to a factor 1000 with respect to the nonrotating model while it remains roughly flat in the 25 M_⊙ of Set E and close to the value obtained in the nonrotating case.

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The global picture shown by Figure 25 does not say anything about the distribution of the elements between Ga and Bi but only how much matter flows above Zn as a function of the mass and initial rotation velocity. In order to understand how matter populates different parts of the chart of the nuclides, we will take advantage of the existence of the three magic neutron numbers 50, 82, and 126. Nuclei having these numbers of neutrons have in fact a minimum of the neutron capture nuclear cross section and therefore constitute the main barriers to the flux of the matter toward heavier nuclei. Depending on the neutron-to-seed ratio and the neutron exposure, matter may stop at the first, second, or third neutron closure shell.

As proxies of the nuclei that have a magic neutron number, we consider the following three key nuclei: ⁸⁸Sr(N = 50), ¹³⁸Ba(N = 82), and ²⁰⁸Pb(N = 126). Figure 26 shows in the left column the dependence of the yields of these three nuclei on the initial rotation velocity for both Sets F and E, while the trend with respect to the Δχ (which basically quantify the amount of primary ¹⁴N produced by the entanglement between the He and H burning; see Section 3.3) is shown in the right column. The yields of all three nuclei scale directly with the initial rotation velocity (and Δχ), a clear indication of the increase of the neutron flux with the initial rotation velocity. The main increase in the yield of ⁸⁸Sr occurs for initial rotation velocities in the range 150–300 km s⁻¹; the dependence becomes much weaker above 300 km s⁻¹. This behavior may be easily understood by looking at Figure 16 that shows that the largest increase in the Δχ occurs between 150 and 300 km s⁻¹. Set E, in particular, shows a saturation of the Sr yield for both masses. The left panels in Figure 26 that refer to Ba and Pb also show an increase of the yields with the initial rotation velocity even if with some scatter while the corresponding right panels show a much tighter scaling. The reason is obviously that the yields directly depend on the Δχ (because it is directly responsible of the neutron flux) and only indirectly on the initial rotation velocity (because it determines the synthesis of primary ¹⁴N in central He burning).

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