GRB-SN Association within the Binary-driven Hypernova Model

Observations of supernovae (SNe) Ic occurring after the prompt emission of long gamma-ray bursts (GRBs) are addressed within the binary-driven hypernova (BdHN) model where GRBs originate from a binary composed of a ∼10M ⊙ carbon–oxygen (CO) star and a neutron star (NS). The CO core collapse gives the trigger, leading to a hypernova with a fast-spinning newborn NS (νNS) at its center. The evolution depends strongly on the binary period, P bin. For P bin ∼ 5 min, BdHNe I occur with energies 1052–1054 erg. The accretion of SN ejecta onto the NS leads to its collapse, forming a black hole (BH) originating the MeV/GeV radiation. For P bin ∼ 10 min, BdHNe II occur with energies 1050–1052 erg and for P bin ∼ hours, BdHNe III occur with energies below 1050 erg. In BdHNe II and III, no BH is formed. The 1–1000 ms νNS originates, in all BdHNe, the X-ray-optical-radio afterglows by synchrotron emission. The hypernova follows an independent evolution, becoming an SN Ic, powered by nickel decay, observable after the GRB prompt emission. We report 24 SNe Ic associated with BdHNe. Their optical peak luminosity and time of occurrence are similar and independent of the associated GRBs. From previously identified 380 BdHN I comprising redshifts up to z = 8.2, we analyze four examples with their associated hypernovae. By multiwavelength extragalactic observations, we identify seven new episodes, theoretically explained, fortunately not yet detected in Galactic sources, opening new research areas. Refinement of population synthesis simulations is needed to map the progenitors of such short-lived binary systems inside our galaxy.


INTRODUCTION
The pioneering work of the BeppoSAX telescope, linking for the first time the success of gamma-ray astronomy with the discovery of gamma-ray bursts (GRBs) (Klebesadel et al. 1973) and the CGRO/BATSE era (Fishman et al. 1982) to the X-ray astronomy of binary X-ray sources (Giacconi & Ruffini 1978), led to the discovery of the GRB X-ray afterglow (Costa et al. 1997) and the determination of the GRB cosmological nature (Metzger et al. 1997).Following these successes, we have returned to address the fundamental issue of the observational coincidence of GRBs with Ic supernovae (SNe): 1) Our theoretical framework started with the induced gravitational collapse scenario (Rueda & Ruffini 2012) introduced to originate stellar mass black holes (BHs) powering long GRBs associated with type Ic SNe.It was soon followed by the binary-driven hypernovae (BdHNe) model (Ruffini et al. 2014a), which assumes a binary system composed of a carbon-oxygen (CO) star of 10M and a companion neutron star (NS) as the GRB progenitor.The GRB trigger occurs when the CO core collapses, originating a newborn NS (νNS) and an SN Ic.The SN ejecta accretes onto the NS companion and the νNS because of matter fallback (Becerra et al. 2019(Becerra et al. , 2022)).
2) The first evidence for such BdHN was presented by analyzing two sources: GRB 090618 at z = 0.54 (Izzo et al. 2012b,a) and GRB 090423 (Salvaterra et al. 2009;Tanvir et al. 2009;Ruffini et al. 2014b).The extraordinary result of GRB 090423 was that it was observed at z = 8.2, which was and still is the farthest GRB in our Universe with a spectroscopic confirmation.We are currently examining GRB 090429B, with a photometric redshift z = 9.4 (Cucchiara et al. 2011), within the BdHN model (Ruffini et al., in preparation).In the meantime, the existence of 380 BdHNe has been presented (Ruffini et al. 2021).Their distribution ranges from the above z = 8.2 to close extra-galactic GRBs in the local Universe.Their enormous energies range between 10 49 erg and a nearly 10 55 erg of GRB 220101A (and GRB 221009A).A crucial point is that the compact CO-NS systems of the BdHN model are the final stage of a peculiar binary evolution, short-lived and rare, as GRBs are.The probability of their occurrence in our Galaxy is extremely low.Since the progenitors are short-lived, their frequency of occurrence essentially mimics the evolution of the cosmic star formation rate with redshift, peaking at z ∼ 2-2.5 (e.g., Madau & Dickinson 2014; see also Yüksel et al. 2008;Grieco et al. 2012;Graham & Schady 2016;Graham & Fruchter 2017).Based on the low rate of long-duration GRBs in the current cosmic epoch in our Galaxy (Guetta & Della Valle 2007), which is ∼ 3 orders of magnitude lower than the observed core-collapse SN rate (Shivvers et al. 2017), the potential GRB progenitors currently ready to explode in the Milky Way are, in the most optimistic view, a handful of objects.The observed density rate of BdHN I is ∼ 1 Gpc −3 yr −1 (Ruffini et al. 2016a(Ruffini et al. , 2018b)).Therefore, it is not surprising that we can acknowledge the existence of such compact binary progenitors only through their cataclysmic fate leading to GRBs thanks to their extragalactic, cosmological nature.Interestingly, the above feature could not be fortuitous since an energetic GRB inside our Galaxy might represent a catastrophe for life on Earth (see, e.g., Chen & Ruffini 2015).
3) The crucial topic of extreme interest has been the byproducts of the GRB observations: a) the discovery of supernovae of characteristic energy of 10 49 erg associated with all different classes of BdHN, this article is dedicated to this topic; b) the discovery of seven different episodes characterizing the most general GRB and presenting new physical processes in ultrarelativistic regimes impossible to discover within our Galaxy; c) the fundamental knowledge developed in decades of observations in Earth-based accelerators pointing to vacuum polarization processes (see Ruffini et al. 2010, and references therein) are here discovered in ultrarelativistic regimes and overcritical quantum electrodynamical processes.These processes, when occurring outside our Galaxy, give the unique opportunity to extend the knowledge reached on our planet, but, at the same time, they indicate the danger of the occurrence of these events for the survival of life if they should occur in our Galaxy.An unexpected additional result has been the possibility to apply, in the comprehension of BdHNe, the still untested configuration of rapidly rotating selfgravitating systems that have attracted the attention of the greatest scientists in world history: from Isaac Newton (Principia, Book III, Propositions XVIII-XX Newton 1687) to Colin Maclaurin (MacLaurin 1742), Carl Gustav Jacob Jacobi (Jacobi 1834), George Darwin (Darwin 1886), James Hopwood Jeans (Jeans 1928), and more recently Subrahmanyan Chandrasekhar (Chandrasekhar 1969); see Section 10, for details.
A new era for relativistic astrophysics started, grounded on the classical results obtained on compact stellar X-ray sources originating from binary massive systems derived on Galactic observations (Tauris & van den Heuvel 2006), as well as on the concepts of BHs expressed by the mathematical equations of Roy Kerr (Kerr 1963) and by the mass-energy formula of Christodoulou-Ruffini (Christodoulou 1970;Christodoulou & Ruffini 1971) and Hawking (Hawking 1971) finally here reaching confirmations in extragalactic sources.It opens to the fundamental issues of understanding the role of GRBs and their intriguing possible interaction with the birth and the end of life in the Universe.
Since their discovery, the enormous energetics led to the idea that GRBs are associated with massive stars' gravitational collapse, leading to NSs or BHs.The community widely accepts the seminal proposal that mergers of NS-NS or NS-BH binaries are the progenitors of short GRBs (Goodman 1986;Paczynski 1986;Eichler, David et al. 1989;Narayan et al. 1991).For long GRBs, our sources of interest here, the traditional model is based on a collapsar, the core-collapse of a single massive star leading to a BH (or a magnetar) surrounded by an accretion disk (Woosley 1993).We refer the reader to Mészáros (2002); Piran (2004), for comprehensive reviews.
In the GRB traditional model, the prompt emission originates in the dynamics, expansion, and transparency of a fireball, an optically thick electron-positron (e − e + )photon plasma in equilibrium with baryons (Cavallo & Rees 1978;Paczynski 1986;Goodman 1986;Narayan et al. 1991Narayan et al. , 1992)).The fireball expands in a collimated relativistic jet with Lorentz factor Γ ∼ 10 2 -10 3 (Shemi & Piran 1990;Rees & Meszaros 1992;Piran et al. 1993;Meszaros et al. 1993;Mao & Yi 1994).In this picture, the interaction of internal and external shocks with the surrounding and interstellar medium is responsible for the prompt emission and the afterglow, including the very-high-energy (VHE) emission by synchrotron self-Compton radiation (Mészáros 2002;Piran 2004;MAGIC Collaboration et al. 2019;Zhang 2019).We refer to Zhang (2018) for the latest developments of the GRB traditional model.
From the energetics, dynamics, and radiation efficiency, two difficulties arise in the traditional model.1) Only a small fraction of the energy of the ultrarelativistic jet is radiated by the synchrotron emission, so much of the kinetic energy remains in the jet.2) The radiation from the jet implies the absence of afterglow in some long GRBs, while it is clear that the afterglow is present in all GRBs.
We now turn to one of this article's main topics, the GRB-SN connection.The follow-up of the optical afterglow, extended by the Neil Gehrels Swift Observatory (Barthelmy et al. 2005;Burrows et al. 2005;Roming et al. 2005), led to the discovery of the association of long GRBs with type Ic supernovae (SNe), first marked with the temporal and spatial coincidence of GRB 980425 and SN 1998bw (Galama et al. 1998).Since then, further observations have confirmed the GRB-SN connection (Woosley & Bloom 2006;Della Valle 2011;Hjorth & Bloom 2012;Cano et al. 2017).The association of GRBs with SNe Ic is possibly one of the most relevant observational clues for theoretical models.Several theoretical and observational consequences from the GRB-SN connection constrain models of GRBs and the associated SNe Ic: (i) Long GRBs and SNe have different energetics.SNe radiate energies ∼ 10 49 -10 52 erg, while GRBs show energies in the much wider range ∼ 10 49 -10 55 erg.The energy release of energetic GRBs is associated with the gravitational collapse to a BH, while SNe originate in the core-collapse of a massive star to an NS.
(ii) Most (if not all) long-duration GRBs originate from binary stars.a) In recent decades, growing evidence has shown that long-duration GRBs are associated with the explosions of massive stars.This fact has been well established both on a statistical basis, e.g., Fruchter et al. (2006); Raskin et al. (2008); Kelly et al. (2008), and from stellar evolution which, even if constraining the zero-age main-sequence (ZAMS) mass of the SN progenitor is highly model-dependent, points undoubtedly to massive stellar progenitors from the modeling of the photometric and spectroscopic follow-up of SNe-Ibc associated with GRBs, e.g., SN 1998bw, 25-40M (Woosley & Bloom 2006;Maeda et al. 2006); SN 2003dh 35-40M (Nomoto et al. 2003;Mazzali et al. 2003); SN 2003lw 25M (Mazzali et al. 2006); 2008D 30M (Tanaka et al. 2009); 2010bh 25M (Bufano et al. 2012); 2016jca 35M (Ashall et al. 2019).b) It is well known that a significant fraction of massive stars is in binaries (about 70%, e.g., Kobulnicky &Fryer 2007 andSana et al. 2012).c) In addition, although stellar evolution models predict the direct formation of a BH from the gravitational collapse for progenitor stars 25M (Heger et al. 2003), two observational facts pose serious challenges to GRB-SN models in which both a BH and SN originate from a single-star: 1) the direct gravitational collapse of a massive star to a BH should occur without an SN emission; 2) observed pre-SN progenitors  2 in Becerra et al. (2019).The binary progenitor comprises a CO star of ≈ 9M (produced by a ZAMS star of 30M ) and a 2M NS companion.The orbital period is ≈ 6 min.From left to right, each snapshot corresponds to selected increasing times where t = 0 s refers to the SN shock breakout.The upper and lower panel shows the mass density on the equatorial plane and the plane orthogonal to the latter.The reference system is rotated and translated to align the x-axis with the line joining the binary components.The origin of the reference system is located at the NS companion position.In the first snapshot at t = 40 s, particles in the NS gravitational capture region form a tail behind the NS companion.These particles then circularize around the NS, forming a thick disk visible in the second snapshot at t = 80 s.Part of the ejecta produces a fallback accretion process onto the νNS visible in the third snapshot at t = 171 s.At t = 337 s (about one orbital period), a disk structure is visible around the νNS and the NS companion.This figure has been produced with the SNsplash visualization program (Price 2011).The figure highlights the νNS is coeval with the SN explosion and remains at the SN center while the ejecta expands.It also shows that the timescale of the physical phenomena leading to the transient activity of the GRB (e.g., the hypercritical accretion and its consequences) is shorter than the timescale of changes in the orbital properties, e.g., an orbital widening or eventual binary disruption owing to mass loss (Blaauw 1961;van den Heuvel & Heise 1972;Fryer et al. 2015).
have masses 18 M (see Smartt 2009Smartt , 2015, for details).Therefore, it is unlikely that the GRB and the SN can originate from the very same single star.Indeed, it is an extreme request for the gravitational collapse of a massive star to form a collapsar, a jetted fireball, and an SN explosion.Some models attempt to supply (partial) solutions to these issues, like an efficient neutrino emission from the accretion disk (e.g.MacFadyen & Woosley 1999) or the presence of an outflow/wind where the nucleosynthesis of the nickel for the optical SN can occur (see, e.g., Kohri et al. 2005;Milosavljević et al. 2012;Lindner et al. 2012).The direct conclusion from the abovementioned points is that most long-duration GRBs occur in binaries.Indeed, Cantiello et al. (2007) tested the idea of producing rapidly rotating Wolf-Rayet (WR) stars in massive close binaries as possible progenitors of collapsars.The above facts also motivated our development of a model for long-duration GRBs that fully exploit the binary nature of progenitors.
(iii) The SNe associated with GRBs are of type Ic.The lack of hydrogen (H) and helium (He) in the spectra of type Ic SNe has the leading explanation that they originate in bare He, CO, or WR stars that lose the outermost hydrogen and helium layers dur-v ing their evolution (see, e.g., Smith et al. 2011;Teffs et al. 2020).Numerical simulations indicate that the most natural mechanism for He/CO/WR stars to get rid of their H/He envelope is from interactions with a compact-star companion (e.g., NS) through multiple mass-transfer and common-envelope phases (see, e.g., Nomoto & Hashimoto 1988;Iwamoto et al. 1994;Fryer et al. 2007;Yoon et al. 2010;Smith et al. 2011;Yoon 2015;Kim et al. 2015).
Although the above is not a complete list of possible drawbacks of the single-star scenario, it is already clear that considering alternatives is natural.In their pioneering work, Fryer et al. (1999) show that various binary stellar evolution channels can lead to diverse GRB events.This alternative binary approach has contributed, as mentioned above, in the study of short GRBs (see, e.g., Ruffini et al. 2016b;Aimuratov et al. 2017), as well as an enigmatic long-lasting GRB 060614 without SN (Della Valle et al. 2006a) interpreted as a white dwarf (WD)-NS merger (Caito et al. 2009;Rueda et al. 2018) and the weakest GRBs from WD-WD mergers (see, e.g., Rueda et al. 2019bRueda et al. , 2022b)).
We specialize in the BdHN model of long GRBs based on the IGC scenario (Rueda & Ruffini 2012).Following the evolution of stripped-envelope binaries, the BdHN model proposes as GRB progenitor a CO-NS binary at the end of the thermonuclear life of the CO star, i.e., the second core-collapse SN event in the binary lifetime.The first SN formed the NS companion of the CO star.The CO nature of the exploding star explains why the SNe associated with GRBs are type Ic.This SN explosion in the CO-NS binary triggers the physical processes that explain the seven episodes observed in the GRB (Rueda & Ruffini 2012;Izzo et al. 2012a;Fryer et al. 2014Fryer et al. , 2015;;Becerra et al. 2015Becerra et al. , 2016Becerra et al. , 2019)).Figure 1 shows an example of numerical simulation performed by Becerra et al. (2019) of the explosion of a CO star leading to a newborn NS (νNS) and the SN Ic, in the presence of an NS companion.These simulations, which include hydrodynamics, neutrino emission, and general relativistic effects, show a variety of outcomes of the system, leading to a variety of GRB events, a BdHN classification, which we discuss below.One of the most relevant results is that, among the possible fates, the NS companion can reach the point of gravitational collapse, forming a rotating, newborn Kerr BH.As recalled, the BdHN progenitors have not been simulated in population synthesis or binary stellar evolution models.Thus, in our numerical simulations, we have to use pre-SN stars resulting from the stellar evolution of single stars and assume the presence of the NS companion.Therefore, the binary evolution leading to the compact BdHN system could start with a different ZAMS mass than the one we are currently considering.Namely, single and binary evolutionary paths can lead to different ZAMS masses starting from a given pre-SN star mass.The latter scenario can lead to a less massive ZAMS progenitor than the former (see, e.g., Zapartas et al. 2019, for the case of binary progenitors of type II SNe).For the early phases of the BdHN model, we have scrutinized the simulations derived by Tauris & van den Heuvel (2006) (e.g., Figs. 16.12 and 16.15).Such simulations are based on X-ray observations of stellar evolution in our Galaxy.We generally confirm the applicability of these models up to the common-envelope phase.Following that phase, the explanation of the multiwavelength observations (from Xrays to GeV and ultrahigh energy) of long GRBs within the BdHN model predicts the existence of CO-NS binaries with orbital periods from hours to days (BdHN II and III) to minutes (BdHN I), taking into due account the relevant role of the angular momentum (see Section 2, and references therein).In view of the low occurrence rate of GRBs in a single galaxy, the necessity of forming CO-NS binaries has been evidenced only by extragalactic observations, whose comprehension has been made possible under the complementary information gained from galactic systems (e.g., Tauris & van den Heuvel 2006).
In Section 2, we recall the basics of the BdHN model and address how the interplay between the SN, νNS, and NS companion leads to the variety of long GRBs.
Section 3 recalls a relativistic formulation's framework in the source's cosmological rest frame, including the kcorrection.
In Section 4, we analyze 24 SNe associated with GRBs.We show that the SN bolometric peak luminosity and its time of occurrence in the source cosmological restframe are nearly the same for all sources (see Figs. 2  and 3).We also present the prompt gamma-ray energy (E iso ) of the associated GRB.We show that E iso spans over six orders of magnitude, while the SN bolometric peak luminosity and the time of occurrence of the peak remain relatively constant; see Figs. 4 and 5.These results constrain GRB models and will be explained within the BdHN model in the following sections.
Section 5 describes the physical phenomena in the different BdHN types and relates them to specific GRB observables, namely the seven episodes of BdHNe; see Table 2 for details.
In Section 6, after recalling the observations that made possible the identification of GRB 180720B as BdHN I (Ruffini et al. 2018c), we address the seven episodes characterizing the source as a BdHN I. vi Section 7 investigates the second BdHN I fully understood in the BdHN model: GRB 190114C (Ruffini et al. 2021;Moradi et al. 2021c).We recall the observations that identified this source as BdHN I (Ruffini et al. 2019b) and discuss its corresponding seven episodes, following an analogous presentation for GRB 180720B in Section 6.
In Section 8, we turn to the case of a BdHN II, GRB 190829A.
Since in BdHN II, the BH is not formed, the number of episodes in this GRB reduces from seven to three, which we address in detail.
In Section 9, we analyze the only example analyzed to date of a BdHN III: GRB 1711205A.Similar to BdHN II, in BdHN III, the BH is not formed.
The number of episodes in this GRB reduces from seven to two, which we present in detail.
Section 10 summarizes new physical phenomena triggered by the SN occurrence in BdHNe, not previously studied in the GRB physics literature.
Finally, we outline conclusions in Section 11.

BDHN CLASSIFICATION
The BdHN model assumes that some long GRB progenitors are binaries composed of a CO star of mass of ∼ 10M and a companion NS of ∼ 2.0M .It also assumes that the gravitational collapse of the CO star generates an SN explosion and creates a newborn NS (νNS) at its center.The νNS with a mass of 1.5M is assumed to spin with a period of ∼ 1-100 ms.It further assumes that ∼ 7-8M are ejected during the SN explosion.The theoretical motivations and the observation constraints leading to these assumptions are given in Sections 5-10, and implications are presented in the Conclusions (Section 11).The SN ejecta drives an accretion process onto the NS companion and a fallback accretion onto the νNS.The accretion rates proceed at hypercritical rates (i.e., highly super-Eddington) due to the efficient neutrino emission (Fryer et al. 2014;Becerra et al. 2016Becerra et al. , 2018)).We differentiate three types of BdHN: I, II, and III, as a function of their overall energetics.A dependence of these energetics from the total initial angular momentum of the Co star-NS binary is evidenced.The shorter the binary period, the higher the BdHN total radiated energy.

BdHN I
We indicate by BdHN I the most energetic class of long GRBs with energies in the range of 10 52 erg E iso 10 54 erg.Their orbital period is of the order of 5 min, which implies an orbital separation of ∼ 10 10 cm, just bigger than the CO star radii (see, e.g., Fryer et al. 2014;Becerra et al. 2016Becerra et al. , 2019)).The hypercritical accretion of the SN ejecta onto the companion NS leads it to reach the critical mass, consequently forming a Kerr BH.Simulations show that the peak accretion rate onto the NS companion can reach Ṁpeak ∼ 10 −3 -10 −2 M s −1 , which implies accreting 0.5-1M in about one orbital period time (Becerra et al. 2016(Becerra et al. , 2019)).The NS gains a large angular momentum, ∆J ∼ GM NS ∆M acc /c ∼ 10 49 g cm 2 s −1 , hence it reaches the critical mass at millisecond rotation rates.The accretion energy gain when bringing the NS to the critical mass and the energy involved in the BH formation process set a lower edge of ∼ 10 52 erg of energy released in a BdHN I. Therefore, BdHNe I explain the long GRBs with energies E iso 10 52 erg (see Ruffini et al. 2018b, for details).The fallback accretion onto the νNS also proceeds at hypercritical rates, and the presence of the NS companion generates a doublepeak accretion (Becerra et al. 2019; see also Becerra et al. 2022 for recent simulations and implications).The first peak of accretion is of a few 10 −3 M s −1 and lasts for about one-tenth of the orbit (Becerra et al. 2019).The νNS reaches a high rotation period of 0.5 ms, near the mass-shedding limit (Cipolletta et al. 2015).The fast-spinning νNS gives origin to the GRB afterglow as explained in Section 5. Examples of BdHNe I are GRB 180720B (see Section 6) , GRB 190114C (Section 7), and GRB 130427A (Ruffini et al. 2019c(Ruffini et al. , 2021)).
In Rueda & Ruffini (2012); Fryer et al. (2015); Ruffini et al. (2016a), we have advanced that the CO-NS compact binaries leading to BdHN I could form in an evolution path similar to the one leading to the so-called ultra-stripped binaries (see, e.g., Tauris et al. 2015, 2017, as well as, e.g., Dewi et al. 2006;Dessart et al. 2020, for alternative stellar evolution scenarios).However, population synthesis simulations of those systems lead to binaries with orbital periods longer than the ones of the BdHN systems (see, e.g., Fig. 16.15 in Tauris & van den Heuvel 2006).We currently consider with great interest scrutinizing the possibility that the evolution following the common-envelope phases (the last evolution stages of the binary) can have a relevant role of the angular momentum of the stellar components, as suggested by the BdHN modeling of long GRBs, branching off a formation channel of BdHN systems.Certainly, BdHN progenitors can form in our own Galaxy, and likely some currently observed binary X-ray sources could, in due time, lead to a BdHN.However, it is observationally established that the probability of occurrence of a GRB in a single galaxy is extremely low, e.g., for the Milky Way, the observed GRB rate suggests one source every million years or so.The GRB detection rate in the Earth originates from extragalactic sources, which, given the GRB's enor-vii mous energetics, allow us to sample an enormous volume containing billions of galaxies, leading to nearly daily detections.We recall that the observed density rate of BdHN I is ∼ 1 Gpc −3 yr −1 (Ruffini et al. 2016a(Ruffini et al. , 2018b)), so a small subpopulation of ≈ 0.01%-0.1% of ultra-stripped binaries following such a particular evolution branch might be sufficient to explain the BdHN I population (see Fryer et al. 2015;Ruffini et al. 2016a, for details), given that ultra-stripped binaries comprise 0.1%-1% of the total SNe (Tauris et al. 2015); see also Section 11.

BdHN II
These binaries are characterized by longer orbital periods of ∼ 20-40 min, so binary separations of a few 10 10 cm.Numerical simulations show that in these binaries, the accretion rate onto the NS companion occurs at lower rates, Ṁpeak ∼ 10 −5 -10 −4 M s −1 .The NS does not reach the critical mass in these systems, so it does not form a BH.The above range of accretion rates implies that the BdHN II subclass can explain long GRBs with energies E iso ∼ 10 50 -10 52 erg (see, e.g., Ruffini et al. 2016aRuffini et al. , 2018b)).
Regarding the νNS, although the first peak of fallback accretion is similar to that of BdHN I, the second peak is considerably lower, so in the end, the fallback accretion leads the νNS to a slower rotation than its BdHN I counterpart.Still, the νNS in BdHN II reaches rotation periods of a ∼ 10 ms, sufficient to explain the afterglow by the associated synchrotron radiation; see Section 5. Examples of BdHN II are GRB 180728A (Wang et al. 2019) and GRB 190829A; see Section 8.

BdHN III
There are CO-NS binaries with orbital periods that can be even hours, corresponding to the binary separation of the order of a few 10 11 cm.The accretion rate onto the NS companion is negligible, and the SN explosion likely disrupts the binary.In these cases, the fallback accretion onto the νNS and its interaction with the SN ejecta are the only ones responsible for the long GRB emission.This BdHN III system explains lowluminous GRBs with an energy release of E iso ∼ 10 49 -10 50 erg, and the νN S reaches the period of ∼ 50-100 ms, which are sufficient to explain the afterglow by the associated synchrotron emission; see Section 5.An example of BdHN III is GRB 171205A, for which we refer the reader to the recent and detailed analysis and simulations presented in Wang et al. (2022b) and Section 9.
From all the above, all BdHNe types are endowed with an X-ray afterglow that can be explained by synchrotron radiation powered by the fast-spinning νNS.
If the binary is not disrupted by the mass loss in the SN explosion (see Fryer et al. 2015 for details), BdHNe I produce NS-BH binaries and BdHN II NS-NS binaries.In BdHN III, the SN is expected to disrupt the system.For a few minutes binary, the merger time is of the order of 10 4 yr, when they will lead to short GRBs.Given the short time to merge, the survived newborn compactobject binaries will not travel far from the long GRB site, which implies a direct link between long and short GRBs (Fryer et al. 2015;Ruffini et al. 2018b).Interestingly, the recent analysis of the population of long and short GRBs by Bianco et al. (2023) supports the above long-short GRB connection which is a unique prediction of the BdHN model.
We now turn to the observational data of 24 long GRBs and associated Ic SNe and proceed to a selected sample of two BdHN I, one BdHN II, and one BdHN III and their associated HNe.

COSMOLOGICAL REST-FRAME TIME AND K-CORRECTION
We here introduce the conversion factor adopted in deriving a luminosity and time both in the cosmological rest-frame of the source (see Ruffini et al. 2018e).This conversion, known as k-correction, has been often neglected in the literature (Chincarini et al. 2007;Falcone et al. 2007;Margutti et al. 2010).
The observation time (t obs ) of the source is related to the time measured in the cosmological rest-frame (t rf ) on the earth by t obs = (1 + z)t rf .The observed flux f obs , namely the energy per unit area and time in a fixed detector energy bandwidth [ obs,1 ; obs,2 ], is where n obs is the photon spectrum, i.e., the number of observed photons per unit energy, area, and time.The total energy emitted in the[ obs,1 ; obs,2 ] bandwidth per unit time, which by definition is in the source cosmological rest-frame, is where D L (z) is the source luminosity distance.
To express the luminosity L in the cosmological restframe energy band, [E 1 ; E 2 ], common to all sources, we rewrite Eq.(2) as L , SN (erg.s 1 ) The plot shows the spread in data points and the lack of correlation between these two quantities.UPE stands for ultrarelativistic prompt emission, SXFs for soft X-ray flares, HXFs for hard X-ray flares, CED for classical electrodynamics, QED for quantum electrodynamics, SN for supernova, and HN for hypernova.

Physical phenomenon
BdHN type x 10 5 10 6 10 7 t p, rest (s) where the k-correction factor is defined as Throughout this article, we use a ΛCDM cosmology with H 0 = 69.6kms−1 Mpc −1 , Ω M = 0.286, Ω Λ = 0.714 for performing the k-correction related to the cosmological-rest frame of sources.
We address the observations of a sample of 24 spectroscopically well-identified SNe associated with long GRBs (GRB-SN).In Table 1, we give the name of the SN, the SN type, the cosmological redshift, our best estimate of the E iso of the associated long GRB, the peak luminosity of the SN (L p,SN ), and the time of occurrence of the peak (t p,SN ).We also give the analogous information from the literature in the following three columns.
The optical observations are performed during the long-lived multiwavelength afterglow of each GRB.As pointed out by Cano et al. (2017, and references therein), the spectroscopic analysis of the light curve close to their maxima, through the identified presence of strong absorption/emission lines (Cappellaro 2022), allows classifying the type of the SN, e.g., Ib/c or Ic-BL.The photometric observation also indicates the evidence for an emerging SN by a characteristic rise in the optical afterglow at around 7-20 days after the main GRB trigger.The rise in apparent magnitude points to the energy deposited in the expanding outflow by the decay of radioactive nickel mass synthesized during the SN explosion (see Section 5).
Since the first evidence of the GRB-SN association, GRB 980425-SN 1998bw in 2018 (Galama et al. 1998), to the end of 2019, there have been detected about 60 GRB-SN events.We collected the data from literature and catalogs (Poolakkil et al. 2021;Lien et al. 2016) . Isotropic-equivalent energy (Eγ,iso) of GRB versus the peak luminosity of the bolometric light curve of the associated SN (LP,SN).The plot shows the lack of correlation: the SN luminosities stay within an order of magnitude spread, while the GRB energy spans ∼ 6 orders of magnitude.line tables2,3 and databases.4,5Among these associations, there are 24 SNe identified spectroscopically and 26 SNe showing only a prominent "bump" in the late optical afterglow and any obtained spectra.6Interestingly, half of the sample occurred within Fermi space observatory operational era, thus extending information to the high-energy counterpart of the accompanying GRBs (Ajello et al. 2019); see Table 1.
Due to incomplete data in some of the observed GRB-SN, we cannot use the entire population.Therefore, we further focus on the 24 spectroscopically confirmed SNe associated with GRBs to the end of 2019.
The peak luminosity integrated over the optical bands is similar in all observed SNe associated with GRBs independent of their redshift; see Fig. 2. The same applies to the time of occurrence of the peak measured since the GRB trigger and is independent of the redshift of the SN; see Fig. 3.As we will point out in Section 5, the determination of the trigger time strongly depends on the luminosity of the GRB and the instrument with the indeterminacy of ∼ 10 4 s.The average peak bolometric luminosity is L p,avg = (9.45± 3.8) × 10 42 erg s −1 and the average peaking time in the rest-frame is t p,avg = (1.16± 0.24) × 10 6 s.
Quite apart from this universality, it follows from Figs. 4 and 5 that the peak luminosity of the associated SN Ic and its time of occurrence are not correlated to the E iso of the BdHN I, II, and III.
As recalled in the Introduction, we assume that the progenitors of the SN Ic associated with long GRBs are composed of a ∼ 10M CO star and a ∼ 2M companion NS.As recalled in Section 2, the same progenitors also characterize the BdHNe.In both cases, the trigger is marked by the collapse of the CO core.From the re-10 6 10 7 t p, rest (s) . Isotropic-equivalent energy (Eγ,iso) of GRB versus the peak time of luminosity of the bolometric light curve of the associated SN (tP,SN).The plot shows the lack of correlation: the SN peaking times (in the rest-frame) stay within an order of magnitude spread, while the GRB energy spans ∼6 orders of magnitude.
sults presented above, a new problem arises: how can the thermonuclear evolution of the SN Ic, characterized by a standard energy of ∼ 10 49 erg, be unaffected by the presence of BdHN I, II, and III with energies in the range of ∼ 10 49 -10 54 erg.To answer this fundamental question and the above energetic difference, we proceed in Section 5 to illustrate the physical processes in the seven fundamental episodes characterizing a most general BdHN and their spectral properties.In Sections 6-9, we provide BdHN I, II, and III examples.

BDHN EMISSION EPISODES
The advantage of introducing the BdHN model may be to bring a certain amount of clarity in a field in which a great deal of confusion exists even in interpreting the specific spectral data (see, e.g., Li 2022).
We have recalled in the Introduction the differences in addressing the fundamental question of what is considered a long GRB: in the traditional literature, the GRB is described by a single event originating from a "collapsar" and manifesting itself by an ultra-relativistic jetted emission.A much more scientifically complex and vaster picture starts from a binary progenitor.
We have also recalled how the large observational support and the equally profound theoretical comprehension following the breakthrough of the BeppoSAX promoted the unification of traditional gamma-ray astronomy to X-ray astronomy.This led to an expansion to additional multi-wavelength observations.The leading conceptual progress has emerged from explaining the spatial and temporal coincidence of two very different astrophysical events: the occurrence of SN Ic and the occurrence of long GRBs.
The BdHN model is rooted in the explanation of this coincidence, as explained in this article: we soon realized that both systems have a common origin in a progenitor composed of a CO core and a binary NS companion (see Section 1).Their evolution leads to an SN explosion which, in addition to a large amount (7-8M ) of ejecta, gives origin to a millisecond pulsar at its center.We have indicated in Section 2 the crucial role of the initial large angular momentum of the CO-NS binary systems due to the short initial binary period P bin .Three different BdHN types originate from very different energies: BdHN I with P bin of ∼ 4-5 min and energies ranging in 10 52 -10 54 erg, BdHN II with P bin ∼ 20 min and energies xiii ranging 10 50 -10 52 erg, BdHN III with P bin up to a few hours and energies below 10 50 erg.Equally remarkable is the fact that the same progenitors, as shown in Table 2 and Figs.2-5, lead to SNe Ic of a standard energy of 10 49 erg and an HN with the kinetic energy of ∼ 10 52 erg.This result points to a thermonuclear evolution of the SN Ic largely independent of the associated GRB.
The present effort is dedicated to addressing the physics and evolution of GRBs and SN Ic with quantum and classical field theories, which are currently full of conceptual holes.Within the BdHN model, we address the explanation of the above observational facts and justify the assumptions we have made.We have identified seven basic Episodes in the most general BdHN.Each Episode has been characterized through a specific new physical process, partly an extension to new extreme regimes of previously known processes or new processes introduced here for the first time.This has been made possible by observations in the extragalactic of phenomena never observed in our local Universe.Each Episode has been duly scrutinized, and the new physical laws introduced for their explanation have been validated by a time-resolved spectral analysis.The importance of these Episodes can hardly be overestimated since they offer the most reliable guide we have in classifying and interpreting the rapidly growing and already very complex observational picture.After some general considerations, we refer in the following sections to the seven specific Episodes, the related observable, and the BdHN type in which they are present.We then proceed in the following sections to specific examples; two BdHNe I in Section 6 on GRB 180720B (in Table 2 we identify the physical phenomena), on GRB 190114C in Section 7, GRB 190829A as a BdHN II in Section 8, and GRB 171205A as BdHN III in Section 9.
In the following, we refer to as"MeV" emission the radiation in the 100 keV-10 MeV energy range typical, e.g., of Fermi-GBM; "GeV" emission the radiation in the 100 MeV-10 GeV energy range, typical of Fermi-LAT, and "TeV" emission the radiation at higher energies, above 100 GeV, e.g., typical of H.E.S.S. and MAGIC.

The SN-rise
As mentioned in Section 4, the BdHN process, which includes the formation of an SN Ic and the associated GRB, is triggered by the gravitational collapse of the CO core.The early detection of this event, namely the first appearance of the SN related to the CO core collapse (SN-rise), is quite rare.It depends on various factors, including the GRB energy, the distance of the source, and especially the operation of the multi-wavelength detectors at the unpredictable moment of the occurrence of the gravitational collapse.The possible examples in BdHN I are GRB 160625B (Ruffini et al. 2021), GRB 221009A and GRB 220101A (Ruffini et al, in preparation).We are progressing in determining this episode's spectral signature, which is essential to identify the underlying physical processes originating the SN explosion.These observational features constrain SN explosion models, which still need theoretical developments to provide successful explosions in the presence of a CO core with substantial rotation and match the GRB-SN features.Although we have mentioned the difficulties in the observational identification of this Episode, we have recently identified it in a handful of GRBs (Ruffini et al., in preparation).
Subsequently to the SN-rise, the hypercritical accretion of the 7-8M onto the νNS and the NS companion shows up as Episodes of the GRB prompt emission (Becerra et al. 2016;Wang et al. 2019Wang et al. , 2022b;;Becerra et al. 2022;Wang et al. 2022b).

The νNS-rise
The prompt GRB emission starts with the transfer of energy and angular momentum due to the accretion of the SN ejecta both on a very rapidly spinning νNS and the slower rotating companion NS.The period of the νNS ranges from 1 ms in the case of a BdHN I to ∼ 100 ms periods in the case of a BdHN III.We have indicated as νNS-rise this first BdHN Episode.This process occurs in all three BdHNe types, with a characteristic CPL spectrum (see, e.g., Rueda et al. 2022a).In parallel to the νNS emission, the SN ejecta accretion that occurs on the companion NS is energetically much weaker.However, in the case of BdHN I, the hypercritical accretion onto the NS companion, a few seconds after the trigger given by the νNS-rise, leads to the formation of the BH and the new Episode of the ultra-relativistic prompt emission (UPE) occurs, with a clear CPL + thermal emission (see Section 5.2).Initially, the UPE and the νNS-rise emissions have comparable luminosities.In the case of GRB 180720B, a first νNS-rise I Episode, lasting 4.84 s, is followed by a prominent UPE I Episode lasting 1.21 s, both identifiable by their different spectral properties.Soon after, the νNS-rise II Episode starts, lasting for 3.02 s, followed by the UPE II Episode for 1.82 s; see details in Table .3. What is fascinating and identifiable is the non-interference of the emission process from the νNS-rise and the UPE.A similar behavior is present in GRB 190114C; see details in Table 4.
In both cases of GRB 180720B and GRB 190114C, the millisecond rotation of νNS has given the possibility of examining the equilibrium configurations of a triaxial Jacobi ellipsoid soon evolving into a Maclaurin spheroid xiv with possible emission of gravitational waves (Rueda et al. 2022a).Such possibility, theoretically indicated as necessary in the early evolution of the crab nebula pulsar (Ferrari & Ruffini 1969), can now be submitted to direct observations in BdHN I.
Following the νNS-rise, which again we recall exists in all BdHN types, the synchrotron radiation emitted by the rapidly spinning νNS, in the wavelengths ranging from X-rays to optical to radio gives origin to the afterglows.It is satisfactory that the afterglows are identically present in all BdHN types; see Section 5.5.
Numerical simulations show that the accretion process can be observed as a double-peak emission, where the relative time and intensity of the peaks depend on the orbital period and the angular momentum of the NS at the beginning of the accretion process (see Becerra et al. 2019Becerra et al. , 2022, for details), for details).The NS companion can reach the critical mass for BH formation before the second peak of fallback accretion onto the νNS (see Becerra et al. 2019Becerra et al. , 2022, for recent simulations).Since the accretion process and associated νNS-rise is not exclusive of binaries forming a BH, the above double-peak emission from the accretion can appear as the prompt emission in a BdHN II, as in the case of GRB 190829A (Wang et al. 2022b).The prompt emission appears without a double-peak structure in BdHN III, like in GRB 171205A (Wang et al. 2022a); see Section 9.
We refer to Section 6 for details on the νNS-rise in GRB 180720B, Section 7 for GRB 190114C, Section 8 for GRB 190829A, and Section 9 for GRB 171205A.

The UPE phase
The UPE phase is the first new process that has made possible the extrapolation of the well-known quantum electrodynamics (QED) process of vacuum polarization, which for a long time approached in earth-bound experiments without reaching observational support, and now observing the new regime of overcritical fields in extragalactic astrophysics sources (see Ruffini et al. 2010, and references therein).
These processes were pioneered by decades of theoretical works in the 1930s by Paul Dirac (Dirac 1930), Gregory Breit and John Archibald Wheeler (Breit & Wheeler 1934) and by Fritz Sauter (Sauter 1931a,b), Werner Heisenberg and Hans Euler (Euler 1936;Heisenberg & Euler 1936), and later in the 1940s by Julian Schwinger (Schwinger 1948(Schwinger , 1949a,b),b), and Richard Feynmann (Feynman 1948(Feynman , 1949a,b),b); see e. g.Cherubini et al. (2009); Ruffini et al. (2010).Despite many efforts, the inverse of the Breit-Wheeler process, namely pair creation by two photons, was never observed in Earth-bound experiments neither in the past at DESY and SLAC, nor in the present in Brookhaven and Darmstadt, nor at ELI https://eli-laser.eu/ or XFEL https: //www.xfel.eu.It is today clear that these processes are routinely observed in GRBs on the vastest possible energy scales up to 10 54 erg/s, on the shortest time intervals up to 10 −9 s, and highest energies up to ∼ 10 18 eV.
A novel hierarchical (self-similar ) structure has been evidenced in the UPE spectra of GRB 190114C and GRB 180720B, composed of a black body (BB) plus a cutoff power-law (CPL) model; see sections 6 and 7. Namely, the spectra of the UPE, rebinned in time intervals up to a fraction of a second, are all fitted by analogous BB+CPL models.This feature implies a microscopic phenomenon at work on ever shorter timescales.The explanation of the UPE phase of these BdHN I require the interplay of general relativity, QED, and plasma physics in an overcritical regime, which has been observed for the first time.
In BdHN I, ionized matter and the magnetic field inherited from the collapsed NS surround the newborn Kerr BH.These three components comprise the inner engine that drives the GRB radiation above MeV energies, i.e., the prompt and the GeV emission (Ruffini et al. 2019c;Rueda & Ruffini 2020;Moradi et al. 2021b;Ruffini et al. 2021;Moradi et al. 2021c).
The QED process at work in the UPE originates in the vacuum polarization of the BH vicinity by the electric field, E, induced by the gravitomagnetic interaction of the Kerr BH and the magnetic field, B 0 .At the BH horizon, r = r H = (1 + √ 1 − α 2 )GM/c 2 , the electric field is approximately given by (see, e.g., Ruffini et al. 2019c;Rueda & Ruffini 2020) where M , J, α = cJ/(GM 2 ), and Ω H = c α/(2 r H ) are, respectively, the BH mass, angular momentum, dimensionless spin parameter, and angular velocity.The last expression introduces the effective charge (the BH has zero net charge), defined by Q eff = (G/c 3 )2B 0 J (see Ruffini et al. 2019c;Rueda & Ruffini 2020;Moradi et al. 2021b, for details).
For a magnetic field strength B 0 > 2B c /α 0 , or conversely, for an initial BH spin parameter α 0 ≥ 2B c /B 0 , the induced electric field is initially overcritical, i.e., E(r H ) ≥ E c = m 2 e c 3 /(e ) ≈ 1.32×10 16 V cm −1 .Therefore, in a short time-scale of the order of the Compton time, ∼ /(m e c 2 ) ≈ 10 −21 s, the approximate vacuum around the BH is rapidly filled with electron-positron pairs (e + e − ), forming an optically thick plasma.The e + e − pairs self-accelerate and engulf baryons from the xv low-density medium around the BH.The plasma reaches transparency at large distances from the BH (e.g., R tr ∼ 10 9 cm), with large Lorentz factor (e.g., Γ ∼ 10 2 ; see Moradi et al. 2021c).There is no single transparency event but a train of transparencies that continues when the electric field reaches the critical value.This occurs when the spin parameter has been reduced from its initial value, α 0 , to α ∼ 2B c /B 0 .
The e + e − plasma energy comes from the electric energy stored in the electric field induced by the interaction of the external magnetic field and the gravitomagnetic field of the Kerr BH.Thus, the ultimate energy reservoir is the BH extractable energy, E ext = (M − M irr )c 2 , where M irr is the BH irreducible mass.The latter is related to the other BH parameters by the massenergy formula (Christodoulou 1970;Christodoulou & Ruffini 1971;Hawking 1971) As shown in (Moradi et al. 2021c;Rastegarnia et al. 2022), each transparency process reduces the BH angular momentum by a small fractional amount ∆J/J ∼ 10 −9 , leading to a slightly smaller angular momentum J * = J − ∆J.The BH mass changes by ∆M ≈ Ω H ∆J/c 2 (keeping the BH irreducible mass approximately constant in the process), so ∆M/M ∼ ∆J/J.Therefore, the system starts a new process with the same magnetic field B 0 , kept constant, and a new effective charge of Q * eff = Q eff −∆Q eff , with ∆Q eff /Q eff = ∆J/J.We refer to Section 6 (and Rastegarnia et al. 2022) for details on the UPE phase in GRB 180720B, and Section 7 (and Moradi et al. 2021c) for GRB 190114C.
The UPE structure has been found as well in GRB 160625B (z = 1.406), extending from t rf = 77.72 s to t rf = 87.70 s, and GBR 160509A (z = 1.17), spanning from t rf = 4.84 s to t rf = 8.53 s (see Li et al. 2023, for more details).There, the detailed time-resolved spectral analysis of the UPE phase of GRB 160625B has been given in Table 2, Fig. 4, as well as the luminosity and the temperature of the thermal components as a function of the rest-frame time in Fig. 5.The same analysis has been carried out for the UPE phase of GRB 160509A, presented in Table 4, Fig. 8, as well as Fig. 9 of Li et al. (2023).Although the UPE has been successfully analyzed in both sources, we are verifying the remaining six Episodes.
Thus, the UPE is expected to be present only in the prompt emission of BdHN I.The nuNS-rise instead dominates the prompt emission of BdHN II.We advance the possibility that a UPE-like emission could also occur under some conditions around a highly magnetic, fast-rotating NS, and the differences between the two cases could be checked through the prompt emission of BdHNe I and II.

High-energy jetted (GeV) emission
The UPE ends when the strength of the induced electric field becomes lower than the critical field's.Hence, the vacuum polarization's QED process is no longer active.Yet, the induced electric field is sufficiently large to power the GeV emission by the following classical electrodynamics (CED) process.The electric field accelerates charged particles that move along and spiral around the magnetic field lines given the magnetic dominance, i.e., B 2 − E 2 > 0, leading to radiation by acceleration, e.g., synchrotron emission.In particular, for a magnetic field aligned and parallel to the BH spin, electrons move outward in the polar region around the BH rotation axis (θ = 0) comprised at angles −60 • θ 60 • in the northern hemisphere, and the analogous region in the southern hemisphere because of the reflection symmetry of the Kerr BH spacetime.For the involved pitch angles (see, e.g., Moradi et al. 2021b, for details), those electrons emit most of the synchrotron radiation at GeV energies with a luminosity that explains the observed GeV radiation in (some, see below) long GRBs (Ruffini et al. 2019c;Rueda & Ruffini 2020;Moradi et al. 2021b).We refer the reader to Rueda et al. (2022b) for a fully general relativistic treatment of the above process.As for the UPE phase, the BH extractable energy powers the GeV emission, which decreases with time following a powerlaw with an index of α GeV = −1.19± 0.04.Thus, the mass and angular momentum of the BH keeps decreasing with time.In this case, each process of emission extracts a fraction of the BH mass-energy ∆M/M ∼ 10 −18 and angular momentum ∆J/J ∼ 10 −16 (see, e.g., Moradi et al. 2021b;Rueda et al. 2022b).
Unlike the isotropic afterglow emission, which originates from the νNS and is present in all types of BdHN, the GeV radiation occurs only in BdHN I since the Kerr BH power it and is anisotropic, occurring in a doublecone of semi-aperture angle ≈ 60 • , centered on the BH rotation axis.Therefore, it is not observable in every BdHN I, which explains the absence of observed GeV emission in a fraction of them (see Ruffini et al. 2021, for details).
We refer to Section 6 for details on the GeV emission in GRB 180720B (see also Ruffini et al. 2019c), and Section 7 for GRB 190114C (see also Rueda & Ruffini 2020;Moradi et al. 2021b).

The BH echoes
The hypercritical accretion onto the NS companion and the consequent BH formation in BdHN I decrease xvi the matter density around the BH (Becerra et al. 2019).Numerical simulations show that the expanding e + e − plasma causes a further decrease of the density from 10 −7 g cm −3 to a value as low as 10 −14 g cm −3 .The collision and partial reflection of the expanding e + e − plasma with the cavity walls generates emission, known as cavity, characterized by a spectrum similar to a Comptonized blackbody with a peak energy of a few hundreds of keV (Ruffini et al. 2019a).
The density of the matter surrounding the newborn BH site is highly asymmetric (see Fig. 1).Consequently, the number of baryons that the e + e − plasma loads during its expansion have an angular dependence.The transparency of the plasma in regions with B 10 −2 explains the radiation of the UPE phase, being B the baryon load parameter.The transparency in regions with B ∼ 50 and Lorentz factors of Γ 5 explain the SXFs and HXFs (see Ruffini et al. 2018e, for numerical simulations).The emission is visible at intermediate angles between the binary plane and the rotation axis (see, e.g., Ruffini et al. 2021).We notice that low Lorentz factors Γ 5 are indeed inferred from the time-resolved analysis of the X-ray data, which rule out any ultrarelativistic bulk motion (e.g., massive jets) of the emitter (see Ruffini et al. 2018e, for details).
We expect SXFs and/or HXFs to appear only in BdHN I since they are related to the transparency in the high-density regions of the e + e − plasma, originated in the formation of the newborn Kerr BH (explained above in the UPE).However, the emission is not observable in every BdHN I because of the angular dependence of the emission, which becomes visible only for lines-of-sight close to the binary plane (Ruffini et al. 2018e).

Multiwavelength (X, optical, radio) afterglow
In the BdHN scenario, the synchrotron radiation generated by relativistic electrons in the ejecta expanding in the magnetized medium provided by the νNS magnetic field, and powered by the νNS rotational energy, explains the afterglow emission in the X-rays, optical, and radio wavelengths (Ruffini et al. 2018a;Wang et al. 2019;Rueda et al. 2020Rueda et al. , 2022a)).
Because the afterglow emission depends only on the existence of the νNS, the SN ejecta, and the synchrotron radiation from an isotropic distribution of pitch angles is isotropic, the afterglow synchrotron emission must be present in all BdHNe.Indeed, the X-ray afterglow is observed in all the 380 BdHN I identified in Ruffini et al. (2021), and in all observed BdHN II and III, as shown in this article, which proves that the afterglow emission is spherically symmetric with excellent approximation.A further implication comes from the nature of the BdHN progenitor.Every gravitational collapse of a CO star with a sufficient short orbital period must necessarily lead to a νNS (see Conclusions).
A semi-analytic theoretical treatment of the above synchrotron emission in BdHN can be found in Rueda et al. (2022a); Wang et al. (2022b).The synchrotron luminosity follows a power-law behavior with the same power-law index in all energy bands.The fit of the multiwavelength afterglow data with the above model gives information on the SN ejecta expansion velocity, the νNS magnetic field, the energy and distribution of electrons in the ejecta, and the power injected by the νNS into the SN ejecta.This description of the GRB afterglow within the BdHN scenario differs from traditional GRB models, which consider that an ultra-relativistic jet with Lorentz factor > 100 produces the prompt emission and then continues to expand, leading to the afterglow by the synchrotron emission from the accelerated electrons swept in.
In general, the X-ray emission has the contribution of the synchrotron emission and the νNS pulsar.The νNS pulsar luminosity is characterized by a plateau, followed by a power-law decay at times longer than the characteristic spin-down timescale.Thus, in the X-rays, the sum of the synchrotron and the pulsar emission can result in a power-law luminosity that is shallower than the power-law luminosity of pure synchrotron radiation.Therefore, from the energetics of the afterglow, and the fit of the X-ray light curve, it is possible to infer the evolution of the νNS rotation period and magnetic field strength (see, e.g.Ruffini et al. 2018a;Wang et al. 2019;Rueda et al. 2020;Ruffini et al. 2021;Rueda et al. 2022a;Wang et al. 2022b).

The classic SN emission powered by nickel decay
Finally, the emission is observed in the optical band powered by the energy release of nickel decay (into cobalt) in the SN ejecta.We refer to Rueda et al. (2021); Rueda (2021); Rueda et al. (2019a) for recent reviews on the BdHN scenario of long GRBs and the related physical phenomena.
The nuclear energy released by the decay of nickel into cobalt within the SN ejecta powers the observed energy of the SN Ic emission.The SNe associated with GRBs are similar to each other irrespectively on the GRB energetics (see, e.g., Cano et al. 2017 and this article).The GRB-SN connection is one of the most relevant observational properties constraining GRB models.We introduce in this article additional observational features of the GRB-associated SNe and discuss how they constrain GRB models.xvii Therefore, within the BdHN model, the SN optical emission is always present and observable for z <1 with current telescopes or z >1 for future missions.Using the BdHN model, we have successfully predicted the time of occurrence and luminosity of the SN optical emission for the BdHN I, GRB 130427A (Ruffini et al. 2013), GRB 190114C (Ruffini et al. 2019b), GRB 211023A (Aimuratov et al. 2021), and GRB 221009A (Aimuratov et al. 2022a); for the BdHN II, GRB 180728A (Ruffini et al. 2018d), and GRB 190829A (Wang et al. 2022b); for the BdHN III, GRB 171205A (Wang et al. 2022b).
Having given the details of the physical origin of each episode and the information about the time-resolved spectral analysis, we now turn to specific examples of two BdHNe I; GRB 180720B in Section 6, and GRB 190114C in Section 7, one BdHN II; GRB 190829A in Section 8, and one BdHN III; GRB 171205A in Section 9. ).In the X-rays, the Swift-XRT started to observe the GRB afterglow from 91 s after the Fermi-GBM trigger (Siegel et al. 2018), MAXI/GSC at 296 s (Negoro et al. 2018) and NuStar from 243 ks to 318 ks (Bellm & Cenko 2018).In the optical and nearinfrared, the 1.5-m Kanata telescope observed the source at 78 s from the GRB trigger time (Sasada et al. 2018).Complementary observations in the optical, infrared, and radio telescopes are also reported in Martone et al. (2018) 2019).With the redshift, z = 0.654, identified by the Fe II and Ni II lines in the optical observations by the VLT/X-shooter telescope (Vreeswijk et al. 2018), the GRB 180720B isotropic energy is E iso = 5.92 × 10 53 erg (Ruffini et al. 2018f;Abdalla et al. 2019;Fraija et al. 2019).
We summarize in Table 3 the name of each episode, their physical event, the duration, the spectrum, E iso , and the physical phenomena originating each event.Similarly, in Fig. 6, we represent the luminosity in wavelengths ranging from radio to TeV and show the spectra corresponding to each physical process.
The νNS-rise I.The radiation originating from the fallback of the SN ejecta onto the νNS (Becerra et al. 2019(Becerra et al. , 2022)).The first evidence of this episode in GRB 180720B, referred to as the νNS-rise, extends from t rf = 0 s to t rf = 4.84 s time interval, with isotropic energy of E iso = (1.53 ± 0.09) × 10 53 erg.A Band model best fits its spectrum with E p = 1064 keV, α = −0.99,and β = −2.00.
The UPE I.This episode pinpoints the first emission originating from the BH (BH-rise).The UPE I of GRB 180720B occurs from t rf = 4.84 s to t rf = 6.05 s.Its measured isotropic energy is E MeV UPEI = (6.37 ± 0.48) × 10 52 erg, and its spectrum is best fitted by a CPL+BB model (index α = −1.13,cutoff energy E c = 2220.569keV, and blackbody (BB) temperature kT = 50.31keV in the observer frame).
The νNS-rise II.It spans from t rf = 6.05 s to t rf = 9.07 s.The isotropic energy of this phase is E MeV νNS = (1.13 ± 0.04) × 10 53 erg, and its spectrum is best fitted by a CPL model (α = −0.98,and E c = 737 keV, in the observer frame).
The UPE II.It is evidenced by the first significant observed GeV photon at t rf = 7.06 s.The UPE phase is also continued during this phase (UPE II), which lasts from t rf = 9.07 s to t rf = 10.89 s, with isotropic energy of E MeV UPEII = (1.6 ± 0.95) × 10 53 erg.A CPL+BB model with the following model parameters of α = −1.06+0.01 −0.01 , E c = 1502.5+88.6  −87.5 keV and kT = 39.8 +1.6 −1.6 keV best fits the spectrum.
xix Table 3.The episodes and afterglows of GRB 180720B.This table reports the name, the underlying astrophysical process, the duration (s), the best-fit spectrum, and the isotropic energy (erg) for each event in GRB 180720B.GRB 180720B has a redshift z = 0.654 and T total 90 = 29.56s (corrected in the rest frame).The NS-rise in GRB 180720B is not observable because of the formation of the BH.The GeV emission.The 0.1-10 GeV emission of GRB 180720B observed by Fermi-LAT starts at t rf = 7.01 s.The highest photon energy corresponding to this GRB is 4.9 GeV, which was detected 137 seconds after the Fermi-GBM trigger (Ronchi et al. 2020).The luminosity rises up to t rf ∼ 40 s.After t rf ∼ 40 s the GeV luminosity follows a temporal decaying luminosity of L GeV = 4.6 × 10 53 t −1.94±0.0.13 erg s −1 .It has a total isotropic energy of E iso,GeV = (2.2±0.2)×10 52erg.

Episode
The radio, optical, and X-ray afterglows.The X-ray afterglow luminosity observed by Swift-XRT starts at t rf = 52 s with a time decaying luminosity of L X = 2.5 × 10 53 t −1.44±0.01erg s −1 and its isotropic energy is E iso,X = 2.61 × 10 52 erg.The X-ray afterglow is accompanied by the radio, optical, and TeV afterglows with isotropic energies of E iso,radio = 2.21 × 10 46 erg, E iso,opt = 6.1 × 10 50 erg, and E iso,TeV = 2.4 × 10 50 erg, respectively.
In Rueda et al. (2022a), the above afterglows of GRB 180720B have been explained within the synchrotron scenario described in Section 5.5.The X-ray afterglow of GRB 180720B exhibits two distinct power-laws, the first at times 10 2 -10 3 s and the second at times > 10 4 s (there is a data gap at 10 3 -10 4 s).The X-ray luminosity in the time interval 10 2 -10 3 s exhibits a shallower power-law than the pure synchrotron luminosity, as evidenced by comparing it with the power-laws of the optical and radio synchrotron at times > 10 4 s.The above is explained by the contribution of the νNS magneticbraking radiation (see Section 5.5).Around 10 2 s, the critical synchrotron radiation energy falls below the keV range, so the X-rays synchrotron luminosity decays exponentially afterward.At lower energies, the power-law behavior remains.The subsequent dominance of the pulsar emission in the observed X-ray emission has allowed us to infer the strength of the magnetic field dipole and quadrupole and the rotation period of the νNS.We refer to Rueda et al. (2022a) for more details.There is a technical difficulty in detecting the early (from the GRB trigger up to a few tens of seconds) X-ray afterglow by Swift-XRT.Only recently, thanks to the cosmological time dilation effect, it has been possible to pinpoint this νNS emission in its early phase using high-z sources (Bianco et al., submitted).The extrapolation of the X-ray afterglow power-law behavior, backward in time from 10 4 s, indicates our theoretical prediction at early times, confirmed in the few cases where observations have allowed us to do it.
The optical SN.As a BdHN I source, GRB 180720B was expected to have an associated SN emission, with an optical peak at 21.8 ± 4.3 day after the trigger (Ruffini et al. 2018f).Unfortunately, no telescope observed the source at those times to confirm the SN appearance.
In conclusion, the total energy released by the GRB 180720B is E tot = 6.5 × 10 53 erg of which 3.57 × 10 53 erg is due to the BH with mass with a lower limit of M = 2.4M and initial spin with an upper limit of α = 0.6.The remaining 2.93 × 10 53 erg is due to the accreting νNS with the period of 1 ms.

GRB 190114C AS AN EXAMPLE OF BDHN I
GRB 190114C was first detected by the Fermi -GBM (Hamburg et al. 2019), and the Neil Gehrels Swift Burst Alert Telescope (BAT) (Gropp et al. 2019).The highestenergy GeV photon detected by Fermi -LAT (with a boresight angle of 68 degrees) is a 22.9 GeV event which is observed 15 s after the GBM trigger (Kocevski et al. 2019).Nordic Optical Telescope (NOT) announced the redshift of z = 0.424 (Selsing et al. 2019) which leads the isotropic energy of E iso = (2.48 ± 0.22) × 10 53 erg.The late-time 0.3-10 keV light curve observed by Swift X-ray Telescope (XRT) revealed a temporal power-law decay (D'Elia et al. 2019).Given the above observations, at time 15:29:54 GMT on January 15, 2019, we identified by Ruffini et al. (2019b) this GRB as a BdHN I and predicted that an optical SN should appear in the same location of the GRB within 18.8 ± 3.7 days, which indeed was confirmed by Melandri et al. (2019).This successful prediction and the following detection of TeV radiation by MAGIC (Mirzoyan et al. 2019) have made GRB 190114C a prototype in which all the BdHN phases have been observed (Ruffini et al. 2019d).
The GRB 190114C reveals different episodes of specific astrophysical processes identified in the timeresolved spectral analysis; see Table 4 and Fig. 7.
We summarize in Table 4 the name of each episode, their physical event, the duration, the spectrum, E iso , and the physical phenomena originating each event.Similarly, in Fig. 7, we represent the luminosity in wavelengths ranging from radio to TeV and show the spectra corresponding to each physical process.
The νNS-rise I.With an isotropic energy of E iso = (3.52 ± 0.15) × 10 52 erg, it extends from t rf = 0 s to t rf = 0.79 s time interval.Its spectrum is best fitted by a CPL model with E c = 710 +21.3  −26.1 .The UPE I.It starts from t rf = 0.79 s and ends at t rf = 1.18 s.Its spectrum is best fitted by a cutoff power-law plus blackbody (CPL+BB) with the parameters of power-law index α = −0.62 +0.03 −0.03 , cut-off energy E c = 524.7 +20.1 −20.1 , temperature, kT = 18.4 +0.5 −0.5 keV, with isotropic energy of E iso = (1.00 ± 0.11) × 10 53 erg.
The νNS-rise II.With an isotropic energy of E iso = (3.75 ± 0.11) × 10 52 erg, it spans from t rf = 1.18 s to LGeV = (4.6 ± 2.9) × 10 53 t −1.94±0.04erg.The prediction of the associated SN by (Ruffini et al. 2019b) has been successfully observed by (Melandri et al. 2019) and has made GRB 190114C as a prototype of BdHN I (Moradi et al. 2021c) to study the properties of GRB-SN sources.The rest-frame visual absolute magnitude of the SN associated with GRB 190114C is ∼ −18 mag Melandri et al. (2019), which is ∼ 1 mag less than famous SN 1998bw (Patat et al. 2001).This fainter brightness can be due to the extinction of this event (Kann et al. 2019).The energetic of the Episodes are given in section 7 and Table 4. xxii Figure 8. Time-resolved spectral analysis of UPE II phase of GRB 190114C from t = 2.7 s (t rf = 1.9 s) to t = 5.5 s (t rf = 3.9 s).The self-similar spectral structure is present when (a) the time interval is divided into two parts, (b) four parts, (c) eight parts, and (d) sixteen parts, respectively.The plot is adapted from Ruffini et al. (2019d) with the authors' permission.
t rf = 1.9 s time interval.Its spectrum is best fitted by a CPL model with E c = 770 +22.4  −21.8 .The UPE II.It is signed by a CPL+BB spectrum with power-law index α = −0.71+0.02 −0.02 , cut-off energy E c = 717.6 +25.4  −25.4 , temperature, kT = 111.64+2.5 −2.5 keV, and a self-similar structure deduced from an appropriate timeresolved analysis (Moradi et al. 2021c); see Fig. 8.With an isotropic energy of E iso = (1.47 ± 0.20) × 10 53 erg, it starts from t rf = 1.9 s, and ends at t rf = 3.99 s.The following mass and spin parameter of the newborn BH have been inferred, M = 4.5M , and α = 0.54, respectively (see, e.g.Moradi et al. 2021c, for details).
The Cavity.It extends from t rf = 11 s to t rf = 17 s.Its spectrum is best fitted by a CPL model with the photon index α = −1.67 and the cutoff energy E c = 251 keV.The enclosure of the companion NS and the accreted material (i.e., ≈ 10 57 baryons) inside the BH horizon creates a large cavity of ≈ 10 11 cm around it.The density distribution around the newborn BH has been inferred in Ruffini et al. (2019a), and the data have confirmed the spatial extension of the cavity (see Table 4 and Fig. 7).
The HXF and SXF.It is demonstrated in Ruffini et al. (2018g) and Ruffini et al. (2021) that the HXF and SXF are observable when the BdHNe viewing angle is closed to the equatorial plane of the binary progenitors.GRB 190114C is a BdHNI observed with a viewing angle orthogonal to the orbital plane of the GRB binary (Ruffini et al. 2021).Therefore, the HXF and SXF of GRB 190114C are not observable from the polar axis.
The GeV emission.The onset of the GeV radiation is also signed by the first GeV photon in the range 0.1-100 GeV observed by Fermi -LAT.The total energy emitted by this source in the above GeV range is E GeV = (1.8 ± 0.9) × 10 53 erg (Ruffini et al. 2021), comparable to the energy observed by the GBM.
The radio, optical, and X-ray afterglows.The X-ray afterglow luminosity observed by Swift-XRT starts at t rf = 52 s with a temporal decaying luminosity of L X = 5.14×10 52 t −1.37±0.05erg s −1 and its equivalent isotropic energy is E iso,X = 3.2 × 10 52 erg.The X-ray afterglow of GRB 190114C is accompanied by the radio, optical, and TeV afterglows with isotropic energies of E iso,radio = 3.31 × 10 46 erg, E iso,opt = 7.1 × 10 50 erg, and E iso,TeV = 4.0 × 10 51 erg, respectively.These afterglows originated from synchrotron radiation powered by the interaction of the νNS, with an initial period of P 0 = 1 ms, and the SN ejecta Rueda et al. (2022a) (see Table 4 and Fig. 7).
The optical SN.The optical signal of SN 2019jrj, a typical GRB-associated SN Ic (see Figs. 5 and 2), peaks at ∼ 10 6 s (see also Fig. 7).Deducing certain physical properties of SN 2019jrj is difficult due to the relatively  2019).The νNS is located at the center of the dark blue spot accumulating material around it.And at the center of the green spot, the NS companion is also accreting SN ejecta.Also, we notice that a portion of the SN ejecta is flowing back towards νNS due to the distortion of SN ejecta caused by the companion NS.
In conclusion, the total energy released by the GRB 180720 is E tot = 3.8 × 10 53 erg of which 1) 2.7 × 10 53 erg is due to the BH with mass with a lower limit of M = 4.53M and initial spin with an upper limit of α = 0.54, 2) 1.1 × 10 53 erg is due to the accreting νNS with the period of 1 ms, and 3) 3 × 10 49 erg is due to the optical SN emission corresponding to the HN ejecta with the kinetic energy of 2.5 × 10 52 erg.

GRB 190829A AS AN EXAMPLE OF BDHN II
GRB 190829A triggered the Fermi -GBM at 19:55:53 UT on 2019 August 29 (Fermi GBM Team 2019).Swift-BAT was triggered 51 s later.The Swift-XRT started observing 148.3 s later after the Fermi trigger (Dichiara et al. 2019).Swift-UVOT (Dichiara et al. 2019), Half Meter Telescope (HMT) (Xu et al. 2019), Nordic Optical Telescope (NOT) (Heintz et al. 2019) andGTC Hu et al. (2021) detected the redshift of z = 0.0785 ± 0.005,  The SN-rise is not observed for this source.Two pulses are observed in the Fermi-GBM and The Neil Gehrels Swift-BAT light curves (Wang et al. 2022b).The initial pulse rises at time −0.70 s, peaks at 1.02 s, and declines at time 7.46 s.After a time delay of 35.65 s, the second, more luminous pulse begins at 43.11 s, peaks at 47.89 s, and declines at 59.34 s.All the times are indicated in the rest frame.A cutoff power-law function best fits the first pulse.It has isotropic energy 4.25 ± 1.02 × 10 49 erg s −1 and averaged luminosity 4.84 ± 1.16 × 10 48 erg s −1 .The second pulse shows a Band function spectrum.It is nearly one order of magnitude more energetic than the first pulse, with isotropic energy 3.56 ± 0.50 × 10 50 erg and an averaged luminosity is (2.05±0.29)×10 49erg s −1 ; see Table 5 for the summary of the GRB 190829A episodes.
The accretion onto the NS companion and the enhanced fallback accretion onto the νNS are responsible for the above two pulses.The comparison of GRB 190829A, especially the time separation between two pulses (∼ 50 s), with several CO-NS binaries simulated in Becerra et al. (2019) and Becerra et al. (2022), indicates as the possible progenitor of this GRB a binary composed of a CO star and an NS with an orbital period in the range 20-40 min.
Figure 9 shows the visualization of the threedimensional numerical simulation that shows the νNS and the NS companion surrounded by high-density material and undergoing their corresponding accretion processes.
The first peak corresponds to the SN ejecta accretion onto the companion NS; see details in Wang et al. (2022b).A part of the ejecta is altered by the companion NS and flows back to the νNS, leading to a second fallback accretion episode onto the νNS, leading to the second peak.
At the time > 100 s, the afterglow started and was observed by Swift-XRT for the soft X-ray band, GTC for the optical band, and AMI-LA for the radio band, as shown in figure 10.The X-ray afterglow from ∼ 1000 s follows a power-law decay with an index of ∼ −1.1.A single power-law function best fits its spectrum with a photon index ∼ −2.15.The optical and radio light curves share similar power-law behavior.The total energy released till 10 7 s is ∼ 4 × 10 50 erg.We attribute this energy to the rotational energy of the νNS, which leads to an initial period of 8 ms (Wang et al. 2022b).
In addition, the ejected mass by the CO core collapse, M ej = 5.67 ± 0.72M , contributes in three different ways: 1) in spinning up the νNS, which then releases dipole and/or multipole radiation; 2) in the accretion on the NS; and 3) in the kinetic energy, E K = (1.35± 0.51) × 10 52 erg, of the remaining SN ejecta mov- ing with mildly relativistic velocities.All these three components contribute to the overall energetics, which reaches its peak emission within the first 100 s.
The optical emission of the SN Ic 2019oyw, due to a nickel mass of M Ni = (0.5 ± 0.1) M , commonly occurs around ∼ 10 6 s with the emission of 10 49 erg; see Figs. 2-5, (see Cano et al. 2017;Hu et al. 2021, for an in-depth discussion of the SN observation and calculations).
The remaining kinetic energy of expansion of the ejecta leads to establishing the HN associated with GRB 190829A with the total energy of 1.35 × 10 52 erg kinetic energy+all the radiation energy).
In addition to being a very close GRB at z = 0.0785, which has allowed a specially significant data analysis of GRB 190829A, one of the remarkable peculiarities of this source, has been the discovery of the TeV emission very similar to the case of GRB 180720B and GRB 190114C.In all these systems, the TeV emission behavior closely follows the ∼ 10% level of the X-ray afterglow power-law emission.This is the most significant since being a BdHN II, no BH is present in this source, which suggest linking the TeV radiation to the νNS activity.However, the explanation of the TeV emission within the BdHN model needs still further research that we are currently pursuing (see Section 10.4).We can now conclude that the total energy of BdHN 190829A, observed in the keV, sub-MeV, TeV, optical, and radio bands, is E tot > 8.5610 50 erg.
The prompt emission maintains its luminosity of 10 46 − 10 47 erg s −1 for ∼ 100 s then drops following a power-law, see figure 11.Its spectrum is best fitted by a cutoff power-law function with peak energy 148.55 ± 121.97 keV and low-energy power-law index −1.10 ± 0.35.The total isotropic energy within the T 90 of BAT gives E iso = (1.71± 0.35) × 10 49 erg, see Wang et al. (2022b) and table 6 for details.
As we discussed in the previous section for BdHN II, there are three episodes of accretion, and the last two are unique features of BdHNe.In the case of GRB 171205A, the progenitor system is a single CO star or a CO-NS binary with negligible interaction between the binary components because of a large orbital separation.Hence, only the first fallback accretion onto the νNS is expected.We also discussed that a large part ) GRB 171205A, z=0.0368E iso = 5.7 × 10 49 erg x =1.12 Swift-XRT (0.3-10 keV) Swift-BAT (15-50 keV) Swift-UVOT (B) Supernova bump-B 1.4GHz Figure 11.BdHN III: GRB 171205A.Luminosity light-curves obtained from Swift-BAT in 15-50 keV, Swift-XRT in 3-10 keV and Swift-UVOT in V and B band.After t rf ∼ 10 5 s it follows a decaying power-law with index αX = 1.12 ± 0.08 and amplitude of AX = (1.1 ± 0.8) × 10 48 erg s −1 .The optical and radio data were taken from D'Elia et al. ( 2018) and Maity & Chandra (2021), and the X-ray data were retrieved from the Swift-XRT repository.The blue color indicates an SN bump.
of the energy from the accretion propagates inside the SN ejecta and accelerates its outermost layer, which has a steep density gradient, to a mild-relativistic speed of Lorentz factor < 10.The fast-moving material produces the luminosity of < 10 47 erg s −1 for some minutes, which is often missed by Fermi -GBM or Swift-BAT.But for GRB 171205A, one of the nearest GRBs at redshift z = 0.0368, this weak signal is resolvable and detected by Swift-BAT, shown as the initial hundreds of seconds of prompt emission.This physical picture is similar to the hot cocoon, which is produced by a narrow jet passing through the shells of the progenitor (see, e.g.Mészáros & Rees 2001;Ramirez-Ruiz et al. 2002;Zhang et al. 2004;Nakar & Piran 2017).The difference comes from the outflow in our picture having a clear accretion origin onto the νNS, which emits radiation at a wider opening angle.The heated SN ejecta emits thermal emissions, a temperature of ∼ 80 eV is observed by Swift-XRT in the initial ∼ 400 s (see Fig. 2 in Wang et al. 2022b), then cools to optical bands observed by Swift-UVOT, VLT/X-shooter, and GTC/OSIRIS.
Different from more luminous GRBs, the emission from the accelerated fast-moving material has an obvious impact on the observation of the weak GRB 171205A.Before the transparency time ∼ 10 5 s of the fast-moving material of mass ∼ 10 −2 M , the X-ray and optical light-curves form a long plateau phase (see Fig. 11).The growing transparent part of the fast-moving material dominates the X-ray flux through the synchrotron mechanism, and the thermal radiation from the rest opaque part dominates the optical flux.After ∼ 10 5 s, the X-ray light curve decays as a typical powerlaw of power-law index ∼ −1, and optical emission starts to be taken over by the emission from the radioactive decay of SN ejecta.The 1000 days radio observation by uGMRT (Maity & Chandra 2021) shows the radio flux rises till ∼ 10 7 then decays as a power-law, and no jet break signature was observed, indicating the outflow has a wide opening angle.Like GRB 190829A, the same synchrotron simulation for the fast-moving material was applied on GRB 171205A (Wang et al. 2022b).To fit the power-law decay behavior of the X-ray and radio afterglow, an νNS with an initial magnetic field of ∼ 3 × 10 13 G and a spin period of 58 ms is required (see Fig. 5 in Wang et al. 2022b).
Table 6.The episodes of GRB 171205A.The first episode of prompt emission contains energy from the fallback accretion onto the νNS and the emission from the heated SN ejecta; the latter contributes the most energy.The optical afterglow emission is dominated by the cooling of fast-moving ejecta and the supernova nickel radioactive decay.The synchrotron emission mainly contributes to the X-ray and radio bands.Nickel decay Mej = 4.9 ± 0.9 M MNi = 0.18 ± 0.01 M EK = (2.4 ± 0.9) × 10 52 erg 10.

NEW PHYSICS REGIMES IN HYPERNOVAE AND GRBS PHYSICS
The above description of the richness of physical phenomena triggered by the SN in the BdHN brings us to new physics in the explanation of long GRBs and which deserves to be highlighted.Below, we summarize new physics regimes made possible by understanding long GRBs and HNe.

Evidence from triaxiality in the νNS early evolution
The νNS-rise and the afterglow emission are powered by the rotational energy of the νNS.The recent analysis of the νNS parameters and energetics in GRB 180720B and GRB 190114C (Rueda et al. 2022a), has shown that the νNS at the beginning of the νNS-rise, is characterized by a rotation period at the verge of the bifurcation point of the Maclaurin sequence of equilibrium spheroids into the Jacobi ellipsoidal sequence.The presence of the highly spinning νNS deserves deeper attention in the core collapse of the CO star.
Therefore, the νNS might have evolved from a triaxial Jacobi-like ellipsoid into the axially symmetric Maclaurin spheroid by emission of gravitational waves, as anticipated in early models of pulsars (Ostriker & Gunn 1969;Ferrari & Ruffini 1969;Ruffini & Wheeler 1971), and theoretically verified by Chandrasekhar (1970);Miller (1974).The triaxial configuration lives for a short time, i.e., about less than a second, due to the copious emission of gravitational waves, before the GRB emission.
The gravitational-wave emission could be, in principle, detected for sources located at distances closer than 100 Mpc (see Rueda et al. 2022a, for details).This appears to be the only emission of gravitational waves associated with the long GRB in the BdHN scenario: the corecollapse leading to the νNS radiates poor gravitational waves (∼ 10 −7 M c 2 ∼ 10 47 erg; see Dimmelmeier et al. 2002;Fryer & New 2011).In addition, given the stringent limits on the ultrarelativistic jetted emission, both in the GeV radiation and in the X-ray afterglow, previous gravitational waves estimates (e.g.Leiderschneider & Piran 2021) do not apply (Rueda et al. 2022b).

QED radiation process in the UPE
At every expansion and transparency of the e + e − plasma, the energy radiated by the plasma is paid by the Kerr BH that reduces its mass and angular momentum by amounts ∆M and ∆J, respectively (see Section 5 for details).The lower value of the BH spin leads to a lower value of the induced electric field, which implies that a new self-expansion and transparency can occur with a lower e + e − plasma energy (Moradi et al. 2021c;Rastegarnia et al. 2022).The QED process and the approach to transparency are analogous.Still, the plasma parameters are different, which explains the hierarchical structure and similarity of the spectra in the time-resolved analysis of the UPE.

Classic electrodynamics radiation in the GeV emission xxix
At the end of the UPE phase, the induced electric field is still sufficiently high to power the GeV emission of the GRB, which is emitted in the polar regions above and below the BH within an angle ≈ 60 • from the polar axis.The radiation power, timescale, and the energy stored in the electric field to accelerate the electrons confabulate to power luminosities of the order of 10 51 erg s −1 in the GeV domain for magnetic fields B 0 ∼ 10 11 G (Ruffini et al. 2019c;Moradi et al. 2021b;Rueda et al. 2022b).The acceleration and radiation process occurs thanks to the magnetic dominance, B 2 − E 2 > 0, and the existence of regions where the component of the electric field parallel to the magnetic field is non-zero, i.e., E • B = 0.As for the UPE, the rotational energy of the BH, the reservoir, powers this radiation process.The extension of this approach to AGNs (e.g., M87*; see Moradi et al. 2021b).

Additional knowledge from the Physics frontier: the TeV emission
As we have shown in the above sections, the SN has triggered not only the path to the new physical processes and understandings of phenomena in the BdHN, but there is also a focus on the part of GRB radiation that is not yet theoretically understood and has only recently begun in earth band experiments: TeV radiation.
In particular, what is most impressive is the presence of the TeV radiation in the prompt phase of BdHN I GRB 190114C (Ruffini et al. 2021) as well as in the afterglow of a BdHN I, GRB 180720B (Rueda et al. 2022a;Rastegarnia et al. 2022), and in the afterglow of a BdHN II, GRB 190829A (Wang et al. 2022b).
The first crucial information possibly contributing to the understanding of these processes is the fact that the energy flux of the TeV is 10%-60% of the energy flux of the afterglow.The second essential information is that TeV emission has been observed in the case of the BdHN II, GRB 190829A, hence, without a BH (Wang et al. 2022b).These two observations lead to privilege an energy emission of the TeV radiation linked to the rapidly spinning νNS emission.All the above has driven us to predict the TeV luminosity of GRB 221009A (Aimuratov et al. 2022b).
Finally, new perspectives emerge from the knowledge on the seven Episodes of BdHNe presented in this article for long GRBs, for the analysis of short GRBs previously studied, e.g., GRB 140619B (Ruffini et al. 2015), GRB 090510 (Ruffini et al. 2016b), GRB 081024B and GRB 140402A (Aimuratov et al. 2017).

CONCLUSIONS
A new era in physics and astrophysics started in 1996 when the Beppo SAX satellite promoted the extension of the observational techniques from the gamma-rays, the domain where GRBs were initially discovered, to the X-rays, optical, and radio observations.Further extensions to GeV, TeV, and VHE emissions observations were soon implemented.Three main discoveries were made possible at the time: a) the presence in long GRBs of an afterglow with long-lasting X-ray emission (Costa et al. 1997).As we here show, these afterglows have contributed significantly to the long GRB understanding; b) the cosmological nature of the GRBs, implying energies up to 10 54 erg (Metzger et al. 1997); and c) the outstanding spatial and temporal coincidence between the Type Ic SN 1998bw, with optical emission of 10 49 erg (Galama et al. 1998), and the long GRB 980425 of 10 48 erg (Ruffini et al. 2007).This article is rooted in explaining this outstanding coincidence and illustrates, as well, the exponential growth of knowledge in physics and astrophysics made possible by an equally impressive growth of new technologies.
We have recalled in the Introduction the earlier description of long GRBs as originating from a single BH and an ultrarelativistic jet, the "collapsar" model.The lengthy and gradual evolution to a binary progenitor follows the pioneering work of Fryer et al. (1999).A further change of perspective happened with the introduction of the concept of induced gravitational collapse (IGC Rueda & Ruffini 2012).There the idea was advanced that BHs in long GRBs were not primordial but could be created by reaching the critical mass of an already existing accreting NS during the evolution of the binary progenitor.We have also recalled how motivated by a multiyear inquiry of long GRBs, and we finally proposed the BdHN model with a ∼ 10M CO core and ∼ 2M NS binary companion as progenitors for long GRBs.The CO core collapse triggers the GRB event.
We have also recalled how the BdHN approach has gained relevance because of the observed spatial and temporal coincidences of long GRBs with type Ic SNe.Most SN Ic progenitors assume ultra-stripped binaries based on a multi-year effort evolution analysis.This fact has been a guiding factor in further developing our BdHN model, which naturally leads to comprehending the occurrence of the SN Ic in coincidence with a family of long GRBs, presented in this article.
Section 2 recalls that the BdHN model assumes that the gravitational collapse of the CO core necessarily leads to an SN with 7-8M ejecta and a millisecond spinning νNS of 1.5M , at its center.Both theoretical arguments and observational evidence for these assumptions are later justified in the article.Still, in Section 2, we recall that ultra-stripped binaries comprise 0.1-1% of the total SNe; so the BdHN I population could be ex-xxx plained by a small subpopulation of 0.01-0.1% of them (see, e.g., Fryer et al. 2015).It is interesting to explore if that branch could only occur under specific conditions in the last evolution stages of the binary evolution after the common-envelope phase.The description of the multiwavelength phenomenology of long GRBs with the BdHN model predicts the formation of CO-NS binaries with orbital periods from hours to days (BdHN II and III) to minutes (BdHN I), with a crucial role of the angular momentum.These binaries could be eventually observed in the Galaxy or nearby galaxies by sensitive facilities, e.g., like the James Webb Space Telescope (JWST).In addition, thanks to the cosmological time dilation, we have identified in BdHN at high redshift (e.g., GRB 220101A at z = 4.2, GRB 090423 at z = 8.2, GRB 090429B at z = 9.4) crucial information of the νNS-rise emission in Swift-XRT data (Bianco et al. 2023).The JWST is also gaining information on the galaxies hosting high redshift GRBs like the above ones.We have advanced that such νNS-rise emission identified in high redshift sources could be observed coincident with GWs in nearby sources by a new satellite overcoming the 43 s gap between the GRB trigger and the Swift-XRT observations (see Bianco et al. 2023, for details).
This article addresses the identification of the separatrix properties of the CO core's gravitational collapse occurring in CO-NS binaries and leading, alternatively, to a single SN Ic or a similar SN Ic and a variety of long GRBs.It is shown that the most general BdHN, in addition to a standard Ic SN, leads to 1) an HN 10 3 times more energetic than a typical SN Ic, 2) to long GRBs, much more energetic than the SN Ic, in the range of 10 49 -10 54 erg, 3) these long GRBs being subdivided in BdHN I, BdHN II, and BdHN III.
From observations and theoretical analysis, we illustrate in Section 2 the BdHN I with energies between 10 52 erg and 10 54 erg, the only BdHNe where the IGC process forms a BH, BdHN II with energies between 10 50 erg and 10 52 erg, and BdHN III with energies below 10 50 erg.For each BdHN type, we have identified the typical CO-NS orbital period and the νNS spin: the former ranges from ∼ 4-5 min in BdHN I, ∼ 20 min in BdHN II, and to a few hours in BdHN III.The νNS spin ranges between 1 and 100 ms.A long-lasting X-ray afterglow is associated with each GRB and is present in all BdHN types.Specific examples are given in Section 6-9.
Already on these results, the important conclusion can be inferred that BdHNe are intrinsically dominated by a large amount of rotational energy; see text and references therein: 1.The νNS spin inferred from the energetics of the Xray afterglows has an initial dimensionless angular momentum a/M = cJ/(GM 2 ), being J and M the νNS angular momentum and mass, of ∼ 0.5 for BdHN I down to ∼ 10 −3 in BdHN III.We have given an example of how the fast spinning νNS in GRB 180720B initially follows a Jacobi ellipsoid sequence (Rueda et al. 2022a), an absolute first in relativistic astrophysics.
2. The BH is formed only in BdHN I by the IGC process due to the accretion of SN ejecta onto the companion NS.Also, in this case, an initial dimensionless parameter ∼ 0.5 of the BH has been inferred from the two BHs in BdHNI, GRB 180720B (see Section 6) and GRB 19014C (see Section 7).
3. As recalled above, the CO core gravitational collapse originates the entire energetics of the BdHN.
Traditionally, the initial rotational energy of the CO core is assumed to be zero.Possibly the largest change of paradigm introduced by the BdHN model has been to point out that the zero angular momentum traditionally assumed in the description of the collapse of the CO core is untenable.In the BdHN model, the CO core has to be close to corotation with the binary NS companion: this implies, for a binary companion NS of ∼ 4 min orbital period, a CO core with a/m ∼ 1, assuming a radius ∼ 10 10 cm and a mass ∼ 10M .All efforts should be directed at gaining observational evidence for this corotation and developing an SN explosion model consistent with this assumption.
In Section 3, we have recalled relativistic transformations to evaluate the time measurement and the bolometric luminosities in the rest frame of the source.In Section 4, we present a selected sample of 24 spectroscopically confirmed SN Ic and their associated long GRBs (see Table 1).The main outcome is that all observed SNe Ic have peak luminosities around an average value of 9.45 × 10 42 erg s −1 independently of the source redshift (see Fig. 2).The time of occurrence of the peak optical luminosity, measured from the GRB trigger, peaks at an average value of 1.16 × 10 6 s (see Fig. 3), again independently of the redshift of the source.
The properties of the associated GRBs for the selected three BdHNe classes are correspondingly summarized: 1) Figure 4 shows that the luminosity of the SN Ic has roughly the same value, BdHNe E iso ranges from 10 48 erg to 10 54 erg.2) Figure 5 shows that the time of occurrence of the peak luminosity of the SN Ic is also independent of the energetics of the associated BdHN.
3) The HN energy is 10 3 times larger than the common xxxi SN Ic.This decoupling between the GRBs and the Ic SN was highlighted in a pioneering work of Zeh et al. (2004) where this problem was announced, which we quantify and explain.
In Section 5, we indicate the BdHN approach in addressing using quantum and classical field theories, the conceptual description of a selected number of Episodes are then subjected to observational scrutiny via a timeresolved spectral analysis in the rest frame of the BdHN.The case is presented for the necessity of introducing and verifying new physical laws: either in extrapolating well-known physical laws already studied on Earthbound experiments (see, e. g., Ruffini et al. 2010), now extended to new more extreme regimes encountered for the first time in extragalactic sources.This is the case of the classical electrodynamics processes extended to overcritical fields.Equally important has been to review the introduction of new physical laws in the quantum electrodynamics regimes to probe the process of rotational energy extraction from non-stationary and nonasymptotically flat Kerr solution as explaining the high energy GeV emission of GRBs.Particularly important has been the observational verification of the energy extraction process from a Kerr BH embedded in a fully ionized low-density plasma with a non-flat asymptotic solution given by a magnetic field aligned with the rotation axis of the Kerr solution.These new approaches, previously published in specific cases, are here directly applied in interpreting all seven Episodes of the most general BdHN, which are here briefly recalled, and details are presented in Section 5.
Section 5.0: the SN-rise.We introduce, in this Episode (0), the first appearance of the CO core collapse and the SN explosion.This episode has been possibly observed in three BdHNe, i.e., GRB 160625B (Ruffini et al. 2021), GRB 221009A and GRB 220101A (Ruffini et al., in preparation), and needs further examples to verify its spectrum unambiguously.What makes this Episode's observation particularly difficult is its intrinsically low luminosity, with total energy ∼ 10 52 erg, that in all three above BdHN I precedes, by a time interval between 1 s and 100 s, the νNS-rise and the UPE, the two first Episodes of the prompt radiation of energy 10 53 -10 54 erg (see also Ruffini et al. 2021).
Section 5.1: the νNS-rise.This Episode is identifiable by CPL spectra and its time of occurrence, manifesting the early presence of rapidly spinning νNS.Their periods range from ∼ 1 ms in BdHN I to ∼ 100 ms in BdHN III.The νNS-rise occurs in all BdHN types.It is followed by the synchrotron emission emitted by the νNS interacting with the SN ejecta and leading to the three components afterglow: in the X-ray, in the optical and the radio, further examined in Section 5.5.One of the main results reached in the analysis of the νNS-rise in the two BdHN I, GRB 180720B and GRB 190114C, has been the first observations of an initial triaxial Jacobi ellipsoid evolving in a Maclaurin spheroid, with possible emission of gravitational waves.It is interesting that the presence of afterglows in all GRBs (observed in 380 BdHN I and all BdHN II and III) necessarily also implies the presence of νNS in all GRBs.
Section 5.2: the UPE phase.The SN accretion onto the binary NS companion, soon after the first observation of the νNS-rise, leads to the formation by the IGC process of a rapidly spinning Kerr BH whose presence is highlighted by the emission of the Ultra-relativistic Prompt Emission (UPE).In this section, we present an extended introduction of the theoretical works developed to extend to overcritical fields, i.e., E ≥ E c = m 2 e c 3 /(e ) ≈ 1.32 × 10 16 V cm −1 , to the multiyear theoretical works on vacuum polarization.This treatment is now finally reaching its observational verification in the GRBs.The overcritical field is due to an effective charge given by Q eff = 2B 0 JG/c 3 , being B 0 the magnetic field and J the angular momentum of the Kerr BH.These verifications on two selected BdHNe I, GRB 190114C, and GRB 180720B, have allowed explaining the existence of detailed new spectral features with the presence of self-similarities and structures on ever-decreasing time intervals to 10 −9 s.The UPE phase has allowed to test observationally and verify the validity of the Christodoulou-Ruffini (Christodoulou 1970;Christodoulou & Ruffini 1971)-Hawking (Hawking 1971) mass-energy formula.This has allowed us to estimate the initial mass-energy of the Kerr BH and the associated magnetic field, B 0 , in both BdHNe I examined.
Section 5.3: High-energy jetted (GeV) emission.In this section, we study the high energy GeV emission originating from the classical electrodynamics process that transitions from the overcritical field, characterizing the UPE phase, to an undercritical field.The theoretical analysis of the emission originated from a Kerr BH in the presence of a magnetic field of ∼ 10 10 -10 11 G has allowed inferring an emission of the GeV radiation within a cone of half-opening of ≈60 • (Rueda et al. 2022b).This has been confirmed by the statistical analysis of the 54 BdHNe observed by Fermi-LAT.Only 25 emit the GeV radiation, and the remaining 29, confirming not observable given the beamed radiation (Ruffini et al. 2021).Equally important has been the specific temporal power-law behavior of the GeV luminosity, well evidenced in Section 6 dedicated to GRB 180720B and Section 7 dedicated to GRB 190114C.xxxii Section 5.4: The BH echoes.The cavity radiation, explained by the collision and partial reflection of the expanding e + e − with the cavity's wall, originated from the BH formation (Ruffini et al. 2019a), is evidenced for GRB 180720B in Section 6 and GRB 190114C in Section 7. The HXFs and SXFs, previously explained by the interaction of the expanding e + e − with the surrounding accretion matter, are observable in sources with observation angle in the equatorial plane of the BdHN (Ruffini et al. 2018g).These processes are identified in Section 6 for GRB 180720B.Neither HXF nor SXF is present in GRB 190114C, given the viewing angle orthogonal to the plane of orbit.
Section 5.5: Multiwavelength (X, optical, radio) afterglow.In this section, the afterglow's multi-wavelength X-ray, optical, and radio emissions are recalled with references to their theoretical treatments.We here recall that the afterglows are observed in all BdHN types, implying a large angle emission perfectly explained in terms of the synchrotron radiation emission process originating in a millisecond period of spinning νNS as described in the following Sections 6 and 7.The afterglow is observed in all BdHNe, implying that all these GRBs originate from a CO-NS binary.
Section 5.6: The classic SN emission powered by nickel decay.In this section, we address the optical SN emission due to the nickel decay well expressed by the theoretical work of Nadyozhin (see, e.g., Nadyozhin's lectures, Nadyozhin 2011a,b) and Arnett (Arnett 1982).The crucial point is to recall that SN Ic is present in all BdHN types and observable with current telescopes for z 1.New telescopes, e.g., the James Webb Space Telescope, should probe the presence of an SN, which is predicted to exist also for higher z values, following the BdHN model.We refer to Table 1 for a summary.
We turn then to the two examples of BdHNe I.In Section 6, we have summarized the results of GRB 180720B, and in Section 7 of GRB 190114C.In Section 8, we give the example of a BdHN II, GRB 190829A, and finally, in Section 9, the case of a BdHN III, GRB 171205A.
For each source, we have given: 1) the complete references to the observational papers we have utilized to perform the theoretical and the time-resolved spectral analysis; 2) a Figure summarising the luminosities for each Episode as a function of the rest-frame time and concerning the specific instruments and bandwidths.The same figure shows the specific examples of the spectra of each Episode; and 3) again, for each source, we present a Table summarizing the names of the observed Episodes: for each, we give the name of the event, the duration, the spectrum the corresponding E iso and the underlying physical phenomena.A specific time-resolved spectral analysis of the UPE phase is exemplified in the case of GRB 190114C.In addition to the complete material for the description of two BdHN I, one BdHN II, and one BdHN III, we would like to mention that preliminary results have already been obtained for the UPE phase of two additional BdHN I, namely GRB 160626B and GRB 160509A (Li et al. 2023).There, one can find the detailed UPE analysis for GRB 160625B in Table 2, Fig. 4, as well as Fig. 5, and for GRB 160509 in Table 4, Fig. 8,and Fig. 9.We are currently working on the identification of the other six Episodes present in both sources.
Following the above, we identify the primary energy source of all BdHN, independently of their type.The most remarkable property which has allowed us to understand the nature of GRBs has been the possibility to retrace back from the extraordinary observed spectral data, the specific energy sources, and their fundamental new physics.This has been made possible by the guidance of the BdHN model.We refer to the Figures and the Tables in the text and the references to the data acquisitions we have performed.
In Section 10, we briefly highlight the three main topics in which the analysis of the BdHN has promoted new research perspectives with the discovery of new physical laws and the verification of existing laws in new regimes made possible by the unique GRBs and HNe observations.The study of rotating figures of equilibrium represents one of the topics of research in which the best intellectuals have addressed their attention for over two centuries: from the self-gravitating Maclaurin spheroids to the discovery of the triaxial Jacobi ellipsoids.Now, for the first time, we have given evidence that triaxial ellipsoids can play a fundamental role in relativistic astrophysics and be the most prominent source of gravitational waves (Rueda et al. 2022a).Furthermore, far from being a conclusion, this is just the beginning of a new era in relativistic astrophysics leading to a new understanding of the physics of gravitational collapse of the creation of new physical systems by gravitational fission, to a new physics of SN explosion based on quantum and classical electrodynamics process coupled to the rotational energy extraction.
Similarly, we have indicated the perspectives of classic and quantum electrodynamics energy extraction processes from rotating NS and Kerr BHs.The examples of the UPE phase and the GeV emission are here recalled just as interesting prototypes to be further extended.But far from being self-exhaustive, the GRB observations still present new challenges by the observations of vast amounts of TeV radiation up to luminosities of 10 52 erg s −1 .At the same time, these emissions have recently xxxiii been observed, in very low fluxes, in Earth-based accelerators, e.g., at CERN.Possibly, the most exciting new perspective is that there is evidence that this most energetic emission does not originate from a rotating BH, as already shown in this article.
It looks equally promising the interpretation of previous results on short GRBs using the knowledge acquired from the BdHN seven Episodes, e.g., in GRB 140619B (Ruffini et al. 2015), GRB 090510 (Ruffini et al. 2016b), GRB 081024B and GRB 140402A (Aimuratov et al. 2017).
Having said all the above, we can return to explain the enormous energetic difference between an SN Ic and the associated HN and long GRB through the occurrence of seven specific episodes in the most general BdHN leading to these concluding remarks: 1.The associated SN Ic bolometric energy of 10 49 erg originates from the nuclear physics process leading to the decay of a common amount of a fraction of 0.2 M to 0.5 M of nickel (see e. g.Nadyozhin 2011a,b;Arnett 1982), remarkably similar in all BdHN sources.The same explanation regarding nuclear physics applies to explain the common time of occurrence of the SN peak emission, identified as soon as the relativistic corrections are implemented.
2. The HNe in BdHN have kinetic energies of 10 52 erg originating from the kinetic energy of 7-8 M ejecta, expanding mildly-relativistically, observed in all BdHN types.
3. Both the above kinetic energy and the formation of a highly spinning millisecond νNS at the SN center should find an explanation in a CO core-collapse, duly considering the contribution of the rotational energy, again observed in all BdHN types.
Turning now to the GRBs: 1.The X-ray, optical, and radio emission of the afterglow, present in all BdHN types, ranging from a few 10 52 erg in BdHN I (GRB 190114C) down to 10 49 erg in BdHN III (GRB 171205A), are powered by the synchrotron emission originating from the rotational energy of the νNS interacting with the SN ejecta.The νNS initial rotation period is 1-100 ms.
2. The MeV and GeV emissions observed in the prompt radiation phase, present only in BdHN I, ranging 10 52 -10 54 erg, are powered by quantum and classical electrodynamics process originating from the extractable rotational energy of a Kerr BH, embedded in a fully ionized lowdensity plasma.The Kerr solution is neither stationary nor asymptotically flat, but in the presence of a magnetic field, B 0 , aligned with its rotation axis and fulfilling the Christodoulou-Ruffini (Christodoulou 1970;Christodoulou & Ruffini 1971)-Hawking (Hawking 1971) massenergy formula.For the latest developments, see Rueda and Ruffini (submitted).
3. Only the MeV emission in the prompt radiation of BdHN II, of ∼ 10 52 erg (see Table 5), originates from the accretion of the SN ejecta into the slowing rotating binary NS companion.
We can then conclude, generally, that SNe Ic associated with long GRBs originate from CO-NS binary progenitors.
We advance the hypothesis that most CO-NS binaries, with a binary period longer than a few hours, lead only to SN Ic, without any associated pulsar, GRB, or HN.This point can be easily tested observationally.A CO core, with an initial a/M ∼ 1, endowed with an initial magnetic field of ∼ 10 3 G, and density of ∼ 10 4 g cm −3 , can indeed lead, in the process of gravitational collapse, to a triaxial ellipsoid.The consequent fission, Roche lobe bifurcation, can lead to a fast spinning νNS and a most powerful explosion.Like in the UPE phase, this process is expected to be driven by a quantum electrodynamical process originating from an overcritical "effective charge".This overcritical field can complete the comprehension of the GRB-SN connection and lead to a new understanding of some of the current open issues.This will undoubtedly sign a good starting point for approaching the yet unsolved problem of the SN explosion, mainly examined in the absence of rotation.But this brings us to a different topic: the multi-century works on the rotating equilibrium configurations, as recalled above, developed by Elie Cartan, Bernhard Riemann, James Hopwood Jeans and summarized in a series of articles by Subrahmanyan Chandrasekhar (Chandrasekhar 1969), also in collaboration with Enrico Fermi (Chandrasekhar & Fermi 1953).These works are finally reaching the test of astrophysical observations in relativistic astrophysics, which is only partly the article's topic today.
It has been a pleasure to collaborate with a knowledgeable anonymous referee who has promoted an in-depth and constructive dialogue with us.We are developing the BdHN model in a series of papers and are in continuous contact with many scientists and collaborators.Particularly important have been the discussions with xxxiv Roy Patrick Kerr and with editors and referees of the journals: they have constructively contributed to finalizing the presentation of our research.To all of them goes our heartfelt thanks.L.M.B. is supported by the Vicerrectoría de Investigación y Extensión -Universidad Industrial de Santander Postdoctoral Fellowship Program No. 2023000107.

Figure 1 .
Figure 1.SPH simulation of a BdHN I: model "30m1p1eb" of Table2inBecerra et al. (2019).The binary progenitor comprises a CO star of ≈ 9M (produced by a ZAMS star of 30M ) and a 2M NS companion.The orbital period is ≈ 6 min.From left to right, each snapshot corresponds to selected increasing times where t = 0 s refers to the SN shock breakout.The upper and lower panel shows the mass density on the equatorial plane and the plane orthogonal to the latter.The reference system is rotated and translated to align the x-axis with the line joining the binary components.The origin of the reference system is located at the NS companion position.In the first snapshot at t = 40 s, particles in the NS gravitational capture region form a tail behind the NS companion.These particles then circularize around the NS, forming a thick disk visible in the second snapshot at t = 80 s.Part of the ejecta produces a fallback accretion process onto the νNS visible in the third snapshot at t = 171 s.At t = 337 s (about one orbital period), a disk structure is visible around the νNS and the NS companion.This figure has been produced with the SNsplash visualization program(Price 2011).The figure highlights the νNS is coeval with the SN explosion and remains at the SN center while the ejecta expands.It also shows that the timescale of the physical phenomena leading to the transient activity of the GRB (e.g., the hypercritical accretion and its consequences) is shorter than the timescale of changes in the orbital properties, e.g., an orbital widening or eventual binary disruption owing to mass loss(Blaauw 1961;van den Heuvel & Heise 1972;Fryer et al. 2015).

Figure 2 .
Figure 2. GRB redshifts (z) versus the values of peak luminosity of the bolometric light curve of the associated SN (LP,SN).The plot shows the spread in data points and the lack of correlation between these two quantities.

Figure 3 .
Figure 3. GRB redshifts (z) versus the peak time of luminosity of the bolometric light curve of the associated SN (tP,SN).The plot shows the lack of correlation between these two quantities.

Figure 6 .
Figure 6.Luminosity light-curve of GRB 180720B and spectra related to the different Episodes identified in GRB 180720B.The energetics of the Episodes are given in Section 6 and Table3.See alsoRastegarnia et al. (2022) for the analysis of the UPE phase.

Figure 7 .
Figure 7. BdHNe I: GRB 190114C.Luminosity light-curves obtained from Fermi-GBM, in 10 keV-10 MeV, Fermi-LAT in 0.1 GeV-10 GeV, Swift-BAT in 15 keV-50 keV, Swift-XRT in 3 keV-10 keV and optical R-band.The late X-ray afterglow luminosity of BdHN I GRB 190114C observed by Swift-XRT is best fit by a temporal decaying power law of LX = (2.5 ± 0.4) × 10 53 t 1.44±0.01erg s −1 .The light curve of Fermi-LAT in is fitted by temporal decaying power law of LGeV = (4.6 ± 2.9) × 10 53 t −1.94±0.04erg.The prediction of the associated SN by (Ruffini et al. 2019b) has been successfully observed by (Melandri et al. 2019) and has made GRB 190114C as a prototype of BdHN I (Moradi et al. 2021c) to study the properties of GRB-SN sources.The rest-frame visual absolute magnitude of the SN associated with GRB 190114C is ∼ −18 mag Melandri et al. (2019), which is ∼ 1 mag less than famous SN 1998bw(Patat et al. 2001).This fainter brightness can be due to the extinction of this event(Kann et al. 2019).The energetic of the Episodes are given in section 7 and Table4.

Figure 9 .
Figure 9. Ongoing accretion process of SN ejecta onto the νNS and the NS companion, simulated in Becerra et al. (2019).The νNS is located at the center of the dark blue spot accumulating material around it.And at the center of the green spot, the NS companion is also accreting SN ejecta.Also, we notice that a portion of the SN ejecta is flowing back towards νNS due to the distortion of SN ejecta caused by the companion NS.

HFigure 10 .
Figure10.BdHN II: GRB 190829A.Luminosity light-curves obtained from H.E.S.S. in 200 GeV -4 TeV, Fermi-GBM in 10 keV-10 MeV, Swift-BAT in 15-50 keV, Swift-XRT in 3-10 keV and i band and radio band.An SN component at ∼ 10 6 is indicated as the blue color.The power-law fitting of the X-ray, shown as a green dotted line, gives a power-law index of −1.1.The T0 is taken from the trigger of Fermi-GBM to which the initial time of other telescopes is aligned.

Table 2 .
Physical phenomena occurring in BdHN I, II, and III, and their associated observations in the GRB data.References in the table: a , Gamma-ray Coordinates Network (GCN), 1 on-, = 9.45 × 10 42 erg.s 1 L

Table 4 .
The episodes and afterglows of GRB 190114C.This table reports the name, the underlying astrophysical process, the duration (s), the best-fit spectrum, and the isotropic energy (erg) for each event in GRB 190114C.GRB 190114C has a redshift z = 0.424 and T total 90 = 81.4s (corrected in the rest frame).

Table 5 .
The episodes of GRB 190829A.The episodes of accretion onto the companion star and the νNS are triggered by SN explosion.According to the BdHN terminology, they can be classified as sub-episodes of SN-rise.Times are measured in the source rest frame.