**4. Single Stars versus Binary Stars as LGRB Progenitors**

Having discussed the connection between LGRB and Type Ic BL SNe, and the physical models that differentiate the formation of SLSNe and LGRBs, in this section, we discuss scenarios where a single and/or a binary system is a likely progenitor of LGRB. For the LGRB association with Type-Ic SN, the star needs to lose the hydrogen and helium envelopes<sup>3</sup> and have a large production of <sup>56</sup>Ni. The mass loss of a massive star is dependent on its mass, metallicity and rotation rate. The higher the rotation rate, the larger the mass loss [4,5,69]. With mass loss, the angular momentum transported from the core to the surface eventually gets lost to the interstellar medium. This loss in angular momentum reduces the massive star's rotation rate. Although the star will still have to be sufficiently rotating even after the mass loss to produce the accretion disk around the central BH, which is required for the GRB production. Therefore, the right combination of metallicity and rotation rate needs to be satisfied to create LGRB progenitors. For a detailed discussion of the effect of both rotation rate and metallicity, see Section 5.

**Single stars:** Both single and binary channels are possible for LGRB production. For counting as a single star, it might be born "single" or can be a part of a wide binary with no/minimal interaction between the binary components. In the non-rotating massive stars, heavier elements produced by nucleosynthesis in the core are quasi- chemically homogeneously mixed within the inner convective regions. There are outer layers of lighter elements with gradual decline in atomic masses in the radiative outer zones. In most non-rotating stars, the surface abundance does not evolve much from its initial abundance during the MS [4]. In rotating stars, chemical mixing due to several rotational instabilities (see Section 5 for details) dredges up the elements from the inner convective core to the surface, crossing the radiative barrier. Therefore, a rapidly rotating massive star can be quasi- chemically homogeneously mixed [4–6,57,70] without a clear chemical boundary between the inner convective core and the radiative envelope. The chemically homogeneous evolution (CHE) supplies hydrogen to the core for a longer time producing a larger He-rich core compared to the non-rotating star, leaving little or no hydrogen [57,71]. Therefore, rotating stars form relatively massive cores from modest initial masses [57]. Chemically homogeneous massive stars are observed in the Milky Way [71], even though smaller in number compared to the Magellanic clouds [72]. The major problem with CHE is that these stars lose their angular momentum while shedding off the outer H- and Helayers via line-driven wind mass loss. This channel helps to create Type Ic SNe at the expense of reduced rotation rate that might make it difficult to produce a rotating BH at the centre. Rapid rotators with *v*/*v*crit & 0.4 might satisfy the criteria for both Type-Ic SN and centrally rotating BHs that can produce an accretion disk around it. Thus, we conclude that

a sufficiently rotating massive single star can be a legitimate channel for the production of LGRB progenitor.

**Binary stars:** Binary channel is important for both short and long GRBs [71]. Binary stars are abundant both at solar and subsolar metallicities. For example, ∼60–80% MS stars in our Milky Way are in binaries [73–77]. Given the bias towards metal-poor environments for the production of GRBs, it is of particular interest to study the Large Magellanic Cloud (LMC) stellar populations. Observations of compact star-clusters NGC 1818, NGC 1805, NGC 1831 and NGC 1868 in the LMC show that almost 55–100% stars are in binaries [71]. Given the majority of the massive stars residing in binaries, it is worth investigating the physical mechanisms for gaining or retaining angular momentum in binary systems.

Similar to single stars, binary routes also transfer angular momentum to the stellar core during the MS. The star can be spun up via mass transfer from the companion star, or it can also be spun down if the star itself loses mass. It is observed that an older star in the binary gets rejuvenated by the mass transfer from its much younger companion [78,79]. In the context of the binary mass transfer, if the more massive companion loses mass, then it again becomes challenging to produce a sufficiently rotating central BH, similar to the issues faced by single stars. If the mass ratio of the primary and secondary is significantly high, then the massive component (secondary) transfers mass on to the less massive one while on the MS and core He burning phase, and in this scenario, a further little amount of mass transfer from the secondary (after its core He exhaustion) to the CO core of the primary makes the less massive primary companion to spin up and explode energetically to eject the common envelope [71,80].

An alternative channel to obtain the necessary angular momentum is the merger of two companions in a binary. The proposed mechanisms for this channel are either the merger of two He cores or a NS/BH with a He core. The latter channel is thought to be the possible cause of the Christmas-day burst, GRB 101225A [81]. In a merger event, the orbital angular momentum of the two components of the binary is combined in a single merged object. In the case of He core-He core merging, there is a little time lapse between the merging event and SN. In the other scenario of BH-He core merger, BH, in principle, can immediately produce the LGRB via mass accretion from the He-core [71].

Although there are several channels to produce LGRBs via binary evolution, it is yet not certain if they can produce LGRBs at the necessary rate to match the observed LGRB populations. In each of these binary channels, obtaining the required angular momentum is a major issue, and there is a limited parameter space in initial mass, mass ratio and separation between the two components that can produce the necessary angular momentum (as discussed above) in order to create LGRBs, and therefore, each of these routes contributes to only moderate LGRB-production rates. Hence, even though the majority of the massive stars are in binaries, the binary route of LGRB production is less favoured compared to CHE of single stars to match the observed LGRB production rate, especially at sub-solar metallicity [71].

#### **5. Properties of WR Stars Required for LGRB Candidates**

Following the discussion in Sections 2 and 3, it is clear that we need sufficiently rotating massive stars for simultaneous production of LGRBs and Type-Ic SNe. The star should rotate at a speed that helps the stellar core retain enough angular momentum to create an accretion disk when collapsing into a BH, and also produce H- and He- depleted SN. Wind mass loss plays a vital role in determining the end phase angular momentum. Metallicity, being the significant determinant of wind mass loss, along with rotation, decides the final fate of a massive star. Therefore, in this section, we discuss the importance of rotation and metallicity in evolving a massive star into a WN star that eventually might become WC star later in its evolution. The eventual transition to the WC phase is essential because WC stars are believed to be the progenitors of Type-Ic SNe [7,82] from the theoretical perspective. In this section, we do not directly study the properties of LGRB progenitors and their dependence on mass, rotation rate, and metallicity. We rather follow an indirect

approach—we study the mass, rotation rate, and metallicity dependence of massive stars that evolve into WN, and subsequently to WC stars which are SN Type-Ic progenitors. We follow this indirect method because Type-Ic SNe are proxies for LGRB association given their observed correlation.

**Rotation rate:** In this section, we investigate the minimum rotation rate required for an O star to evolve into WN phase. To do that, we study the evolution of surface helium and nitrogen mass fractions for varying mass, metallicity and rotation rate because a certain surface enhancement in He and N makes the transition from O to WN stars. Requirements of He- and N- surface enrichments that determine the WN Late-type (L) phase is given in Table 1.

**Table 1.** Taken from [5]. Criteria to classify stars as O, WNL, WNE, WC, and WO based on the surface enhancement of various elements and core burning status. Surface state and core state refer to conditions at the stellar surface and the centre of the star, respectively; *X*<sup>Q</sup> is the mass fraction of element Q. See [5] for detail.


Figure 1 shows the surface He mass fraction contours as a function of time and *v*/*v*crit for a range of mass and metallicity. We notice that all masses show WNL features with surface <sup>4</sup>He mass fraction &40% quite early on the MS, *<sup>t</sup>* . 0.5 <sup>×</sup> *<sup>t</sup>*MS, if they are moderately or rapidly rotating with *v*/*v*crit & 0.4 irrespective of metallicity. This value of *v*/*v*crit is also supported by the theory of massive star formation [83]. Therefore, we use *v*/*v*crit = 0.4 as our fiducial case. At solar metallicity, stars show surface enhancement of He when they have lost .50% of their initial masses on the MS irrespective of rotation rate. However, at low metallicities, [Fe/H] . −1.0, we see the surface He enrichment only for *v*/*v*crit & 0.4. In these metal-poor moderately or rapidly rotating stars, the rotational mixing of chemical elements dredges up the heavy nucleosynthetic by-products from the core to the surface, even though weak mass loss does not strip off much of their outer layers.

We show the evolution of surface nitrogen mass fraction in Figure 2, similar to Figure 1. The surface N enrichment is consistent with the values inferred for WNL stars. For metal rich stars with [Fe/H] = 0.0, the surface enrichment is ∼17 times the initial N abundance (6.73 <sup>×</sup> <sup>10</sup>−<sup>3</sup> ). This results in surface N mass fraction of 0.011, a factor of 2 less than that is observed in WNL stars in Arches cluster [84]. For metal-poor stars, the surface enrichment is a factor to 28–30 compared to their initial abundances of 6.97 <sup>×</sup> <sup>10</sup>−<sup>4</sup> , and 7.024 <sup>×</sup> <sup>10</sup>−<sup>5</sup> for [Fe/H] = −1.0 and −2.0, respectively.

**Metallicity:** Having discussed the required rotation rate that is favourable for WN and eventual WC production, in this section, we discuss the metallicity range that might be optimum for the production of Type-Ic SN. To study this, we run our models for a mass grid of 10 M to 150 M with a mass resolution ∆*M* = 5 M until the end of core <sup>12</sup>C exhaustion (*t*C) for our fiducial rotation rate, *v*/*v*crit = 0.4, for three metallicities, [Fe/H] = 0, −1.0, −2.0, similar to Figure 1 of [5]. We use the 1-D stellar evolution code MESA [85–87] Isochrone Stellar Tracks-II (MIST-II [4,5], Dotter et al., 2021, in prep.). For details of mass-grid and the simulation setup, see [4,5].

**Figure 1.** <sup>4</sup>He surface mass fraction, as denoted by the colorbar, as a function of time and rotation rate, *v*/*v*crit for three metallicities, [Fe/H] = 0.0 (leftmost panels), −1.0 (middle panels), and −2.0 (rightmost panels), and for three masses, 80 M (top panels), 100 M (middle panels), 150 M (bottom panels). We normalize the time by the main-sequence (MS) lifetime, *t*MS. The orange dotted lines show the points where 20% and 50% of the initial mass are lost. For details, see the discussion of Figure 1 of [4].

**Figure 2.** Same as Figure 1, except for the N color contour denoting the <sup>14</sup>N mass-fraction compared to the initial <sup>14</sup>N (14Nini) abundance. <sup>14</sup>Nini are 6.73 <sup>×</sup> <sup>10</sup>−<sup>3</sup> , 6.97 <sup>×</sup> <sup>10</sup>−<sup>4</sup> , and 7.024 <sup>×</sup> <sup>10</sup>−<sup>5</sup> , respectively, for [Fe/H] = 0.0, −1.0, −2.0, respectively. For details, see the discussion of Figure 2 of [4].

Definitions of massive star classifications, such as O, WNL, WN Early-type (E), WC, WO, based on surface elemental mass fraction and central burning state, are given in Table 1. For a detailed discussion of our classifications, see [5]. Here, we briefly summarize them:


Having discussed definitions of different classifications of massive stars, we show the fraction of time a star of a given initial mass spends in these individual phases in Figure 3. We find that stars <45 M spend their entire lives as O stars, and beyond this mass, they spend a significant fraction of their lifetimes in the WNL phase for solar metallicity. They enter WNL phase for even lower mass stars ∼20 M–25 M for subsolar metallicities, [Fe/H] . −1.0, independent of metallicity, as can be seen also in Figures 1 and 2. Nonetheless, there is a peculiar metallicity evolution of the stars that show WC and WO features. At solar metallicity, WC and WO features appear for masses &55 M. At subsolar metallicities; however, the minimum mass (*M*min) for showing up WC and WO features takes an anomalous turn. It shifts to a larger mass &115 M for [Fe/H] = −1.0, and then again comes down to a lower mass ∼50 M–55 M for [Fe/H] = −2.0. Metal-rich stars undergo strong mass loss, and the mass loss rate decreases with decreasing metallicities. Along with the mass-loss, stars lose angular momentum and therefore slow down, causing less rotational dredge-up at later epochs of their evolutions. Therefore, on the one hand, metal-rich stars, such as solar metallicity stars in our model grid, show heavy metals, such as C and O, on their surfaces when significant mass-loss exposes the metal-rich inner cores [4] for *M*min ∼ 55 M. On the other hand, significantly metal-poor stars, for example [Fe/H] = −2.0 in our models, can retain sufficient angular momentum due to weaker mass loss, and therefore might have surfaces enriched with C and O for *M*min ∼ 50 M because of rotational dredge-up, even though they do not expose their metal-rich inner cores. At intermediate metallicity, however, mass loss rates are too weak to expose the metal-rich inner cores but strong enough to lose the angular momentum and therefore to inhibit the chemical mixing driven by rotational dredge-up. This might be the reason why only the most massive stars &115 M show WC- WO- features at intermediate metallicity [Fe/H] = −1.0. This, however, needs to be speculated in detail, and we plan to address this issue in our future paper Roy et al., 2021, in prep. Note that this result may depend strongly on the adopted mass-loss schemes and their metallicity dependance, and we will explore that in our follow-up paper Roy et al., 2021, in prep.

**Figure 3.** The fraction of time each star with a specific intial mass (horizontal-axis) spends in a particular phase classified as O, WNL, WNE, WC, WO, until we stop our simulations at the end of core <sup>12</sup>C exhaustion (*t*C), similar to Figure 1 of [5]. These different phases of massive stars are shown by different colors indicated in the figure legend. The definitions for these claasifications of different phases are given in Table 1. Models shown here are for our fiducial rotation rate, *v*/*v*crit = 0.4, and the three panels refer to three metallicities, as indicated.

Having discussed the metallicity evolution of *M*min, we expect to have a larger number of WC and WO stars at solar metallicity, and the number to decrease at [Fe/H] = −1.0 and to increase again at [Fe/H] = −2.0. Theoretically, WC and WO stars should be progenitors of Type-Ic SN as they lose their H and He envelopes [7,82]. Therefore, we expect to see a larger number of Type-Ic SNe at solar metallicity decreasing at intermediate subsolar metallicity and increasing again at significantly lower metallicity of [Fe/H]∼ −2.0. In addition, there is a one-to-one correlation between LGRB and Type-Ic SNe. We conclude that LGRB numbers starting from solar metallicity might reduce initially with decreasing metallicity and then increase again for significantly metal-poor stars at [Fe/H] = −2.0. In metal-poor environments, the LGRB production rate is also favourable because at low metallicities, stellar cores can retain enough angular momentum to form an accretion disk around the collapsing core that creates the central BH. Observations of LGRB also agree with this theoretical prediction that the LGRB rate increases with decreasing metallicities, however, not at excessively low metallicity (e.g., [88–90]). In fact, most observations show that there is a rapid drop-off in the LGRB rate somewhere between solar [90,91] and 1/3 solar [92], i.e., between the Milky Way (almost solar) and the Small Magellanic Cloud (≈half solar). However, to draw a tight constraint on the metallicity evolution of LGRB rate from both theoretical and observational perspectives, one needs to have observations at metallicities much lower than SMC, and also models with finer metallicity resolutions at these low metallicities. We plan to study this in detail in a follow-up paper Roy et al., 2021, in prep.

Even though most LGRBs are observed to be associated with SN Type-Ic, there are a few exceptions as discussed in Section 2. Therefore, for a conclusive study of LGRB progenitors and their possible connection to SN Type-Ic, we need to investigate the properties of stellar cores and their explodability criteria as given in Equations (1)–(5), and also their dependence on mass, rotation rate, and metallicity in detail. We plan to study all these detailed theoretical aspects in a follow-up paper Roy et al., 2021, in prep.
