Early X-Ray Flares in GRBs

Following the discovery of the gamma-ray bursts (GRBs) by the Vela satellites (Klebesadel et al. 1973) and the observations by the BATSE detectors on board the Compton Gamma-Ray Observatory (CGRO; Gehrels et al. 1993), a theoretical framework for the interpretation of GRBs was established. This materialized into the "traditional" model of GRBs developed in a large number of papers by various groups. They all agree in their general aspects: short GRBs are assumed to originate from the merging of binary neutron stars (NSs; see, e.g., Goodman 1986; Paczynski 1986; Eichler et al. 1989; Narayan et al. 1991, 1992; Mészáros & Rees 1997), and long GRBs are assumed to originate from a "collapsar" (Woosley 1993; Paczyński 1998; MacFadyen & Woosley 1999; Bromberg et al. 2013), which, in turn, originates from the collapse of the core of a single massive star to a black hole (BH) surrounded by a thick massive accretion disk (Piran 2004). In this traditional picture, the GRB dynamics follows the "fireball" model, which assumes the existence of an ultrarelativistic collimated jet (see, e.g., Shemi & Piran 1990; Meszaros et al. 1993; Piran et al. 1993; Mao & Yi 1994). The structures of long GRBs were described by either internal or external shocks (see Rees & Meszaros 1992, 1994). The emission processes were linked to the occurrence of synchrotron and/or inverse-Compton radiation coming from the jetted structure, characterized by Lorentz factors ${\rm{\Gamma }}\sim {10}^{2}\mbox{--}{10}^{3}$, in what later will become known as the "prompt emission" phase (see Section 3).

The joint X-ray, gamma-ray, and optical observations heralded by BeppoSAX and later extended by Swift discovered the X-ray "afterglow," which allowed the optical identification and the determination of the GRBs' cosmological distance. The first evidence for the coincidence of a GRB and a supernova (SN; GRB 980425/SN 1998bw) was also announced as well as the first observation of an early X-ray flare (XRT), later greatly extended in number and spectral data by the Swift satellite, the subjects of this paper. The launch of the Fermi and AGILE satellites led to the equally fundamental discovery of GeV emission both in long and short GRBs (see Section 2).

The traditional model was modified in light of these new basic information by extending the description of the "collapsar" model, adopted for the prompt emission, to both the afterglow and GeV emission. This approach, based on the gravitational collapse of a single massive star, which was initially inspired by analogies with the astrophysics of active galactic nuclei, has been adopted with the aim to identify a "standard model" for all long GRBs and vastly accepted by concordance (see, e.g., Piran 1999, 2004; Mészáros 2002, 2006; Gehrels et al. 2009; Berger 2014; Kumar & Zhang 2015). Attempts to incorporate the occurrence of an SN in the collapsar by considering nickel production in the accretion process around the BH were also proposed (MacFadyen & Woosley 1999). In 1999, a pioneering work by Fryer et al. (1999b) introduced considerations based on population synthesis computations and emphasized the possible relevance of binary progenitors in GRBs.

Since 2001, we have been developing an alternative GRB model based on the concept of induced gravitational collapse (IGC) paradigm, which involves, as progenitors, a binary system with standard components: an evolved carbon–oxygen core (${\mathrm{CO}}_{\mathrm{core}}$) and a binary companion NS. The ${\mathrm{CO}}_{\mathrm{core}}$ undergoes a traditional SN Ic explosion, which produces a new NS (νNS) and a large amount of ejecta. There is a multitude of new physical processes, occurring in selected episodes, associated with this process. The "first episode" (see Section 3) of the binary-driven hypernova (BdHN) is dominated by the hypercritical accretion process of the SN ejecta onto the companion NS. This topic has been developed in, e.g., Ruffini et al. (2001c), Rueda & Ruffini (2012), Fryer et al. (2014), and Becerra et al. (2015, 2016). These processes are not considered in the collapsar model. Our SN is a traditional Type Ic, the creation of the νNS follows standard procedure occurring in pulsar physics (see, e.g., Negreiros et al. 2012), the companion NS is a standard one regularly observed in binaries (see e.g., Rueda & Ruffini 2012; Rueda et al. 2017), and the physics of hypercritical accretion has been developed by us in a series of recent articles (see Section 3.4).

In a BdHN, the BH and a vast amount of ${e}^{+}{e}^{-}$ plasma are formed only after the accreting NS reaches the critical mass and the "second episode" starts (see Section 3.5). The main new aspect of our model addresses the interaction of the ${e}^{+}{e}^{-}$ plasma with the SN ejecta. We apply the fireshell model, which makes use of a general relativistic correct spacetime parametrization of the GRBs as well as a new set of relativistic hydrodynamics equation for the dynamics of the ${e}^{+}{e}^{-}$ plasma. Selected values of the baryon loads are adopted in correspondence with the different time-varying density distributions of the SN ejecta.

In the "third episode" (see Section 3.6), we also mention the perspectives, utilizing the experience gained from both data analysis and theory for the specific understanding of X-ray flares, to further address in forthcoming publications the more comprehensive case of gamma-ray flares, the consistent treatment of the afterglow, and finally the implication of the GeV radiation.

As the model evolved, we soon realized that the discovery of new sources was not leading to a "standard model" of long GRBs but, on the contrary, they were revealing a number of new GRB subclasses with distinct properties characterizing their light curves, spectra, and energetics (see Ruffini et al. 2016b). Moreover, these seven subclasses did not necessarily contain a BH. We soon came to the conclusion that only in the subclass of BdHNe, with an E_iso larger than 10⁵² erg, does the hypercritical accretion from the SN onto the NS lead to the creation of a newly born BH with the associated signatures in the long GRB emission (see, e.g., Becerra et al. 2015, 2016).

While our alternative model was progressing, we were supported by new astrophysical observations: the great majority of GRBs are related to SNe Ic, which have no trace of hydrogen and helium in their optical spectra and are spatially correlated with bright star-forming regions in their host galaxies (Fruchter et al. 2006; Svensson et al. 2010). Most massive stars are found in binary systems (Smith 2014) where most SNe Ic occur and which favor the deployment of hydrogen and helium from the SN progenitors (Smith et al. 2011), and the SNe associated with long GRBs are indeed of Type Ic (Della Valle 2011). In addition, these SNe associated with long bursts are broad-lined Ic SNe (hypernovae) showing the occurrence of some energy injection leading to a kinetic energy larger than that of traditional SNe Ic (Lyman et al. 2016).

The present paper addresses the fundamental role of X-ray flares as a separatrix between the two alternative GRB models and leads to the following main results, two obtained by data analysis and one obtained from the comparison of the alternative models:

• (1)  
  The discovery of precise correlations between the X-ray flares and the GRB E_iso.
• (2)  
  The radius of the occurrence of X-ray flares ($\sim {10}^{12}$ cm) and the Lorentz Gamma factor ∼2.
• (3)  
  The occurrence of a sharp break between the prompt emission phase and the flare–plateau–afterglow (FPA) phase, not envisaged in the current GRB literature. This transition is evidence of a contradiction in using the ultrarelativistic jetted emission to explain the X-ray flares, the plateau, and the afterglow.

In Section 2, we recall, following the gamma-ray observations by the Vela satellites and the CGRO, the essential role of BeppoSAX and the Swift satellite. These satellites provided X-ray observations specifically of the X-ray flares, to which our new data analysis techniques and paradigms have been applied. We also recall that the Fermi and AGILE satellites announced the existence of GeV emission, which has become essential for establishing the division of GRBs into different subclasses.

In Section 3, we update our classification of GRBs with known redshift into seven different subclasses (see Table 2). For each subclass, we indicate the progenitor "in-states" and the corresponding "out-states." We update the list of BdHNe (see Appendix A): long GRBs with ${E}_{\mathrm{iso}}\gtrsim {10}^{52}$ erg, with an associated GeV emission and with the occurrence of the FPA phase. We also recall the role of appropriate time parametrization for GRBs, which properly distinguishes the four time variables that enter into their analysis. Finally, we recall the essential theoretical background needed for the description of the dynamics of BdHNe, the role of neutrino emission in the process of hypercritical accretion of the SN ejecta onto the binary companion NS, the description of the dynamics of the ${e}^{+}{e}^{-}$–baryon plasma, and the prompt emission phase endowed with gamma-ray spikes. We then briefly address the new perspectives opened up by the present work, to be further extended to the analysis of gamma-ray flares, to the afterglow, and the essential role of each BdHN component, including the νNS. Having established the essential observational and theoretical backgrounds in Sections 2 and 3, we proceed to the data analysis of the X-ray flares.

In Section 4, we address the procedure used to compare and contrast GRBs at different redshifts, including the description in their cosmological rest frame as well as the consequent K corrections. This procedure has been ignored in the current GRB literature (see, e.g., Chincarini et al. 2010 and references therein as well as Section 11 of this paper). We then identify BdHNe as the only sources where early-time X-ray flares are identifiable. We recall that X-ray flares have neither been found in X-ray flashes nor in short GRBs. We also show that a claim of the existence of X-ray flares in short bursts has been superseded. We recall our 345 classified BdHNe (through the end of 2016). Their T₉₀,⁹ properly evaluated in the source rest frame, corresponds to the duration of their prompt emission phase, mostly shorter than 100 s. Particular attention has been given to distinguishing X-ray flares from gamma-ray flares and spikes, each characterized by distinct spectral distributions and specific Lorentz Gamma factors. The gamma-ray flares are generally more energetic and with specific spectral signatures (see, e.g., the significant example of GRB 140206A in Section 5 below). In this article we focus on the methodology of studying X-ray flares: we plan to apply this knowledge to the case of the early gamma-ray flares. Out of the 345 BdHNe, there are 211 that have complete Swift-XRT observations, and among them, there are 16 BdHNe with a well-determined early X-ray flare structure. They cover a wide range of redshifts as well as the typical range of BdHN isotropic energies ($\sim {10}^{52}\mbox{--}{10}^{54}$ erg). The sample includes all identifiable X-ray flares.

In Section 5, we give the X-ray luminosity light curves of the 16 BdHNe in our sample and, when available, the corresponding optical observations. As usual, these quantities have been K-corrected to their rest frame (see Figures 9–24 and Section 4). In order to estimate the global properties of these sources, we also examine data from the Swift, Konus-Wind, and Fermi satellites. The global results of this large statistical analysis are given in Table 3, where the cosmological redshift z, the GRB isotropic energy E_iso, the flare peak time t_p, peak luminosity L_p, duration ${\rm{\Delta }}t$, and the corresponding E_f are reproduced. This lengthy analysis has been carried out over the past years, and only the final results are summarized in Table 3.

In Section 6, we present the correlations between t_p, L_p, ${\rm{\Delta }}t$, E_f, and E_iso and give the corresponding parameters in Table 4. In this analysis, we applied the Markov Chain Monte Carlo (MCMC) method, and we also have made public the corresponding numerical codes in https://github.com/YWangScience/AstroNeuron and https://github.com/YWangScience/MCCC.

In Section 7, we discuss the correlations between the energy of the prompt emission, the energy of the FPA phase, and E_iso (see Tables 5–6 and Figures 29–31).

In Section 8, we analyze the thermal emission observed during the X-ray flares (see Table 7). We derive, in an appropriate relativistic formalism, the relations between the observed temperature and flux and the corresponding temperature and radius of the thermal emitter in its comoving frame.

In Section 9, we use the results of Section 8 to infer the expansion speed of the thermal emitter associated with the thermal components observed during the flares (see Figure 32 and Table 8). We find that the observational data imply a Lorentz factor ${\rm{\Gamma }}\lesssim 4$ and a radius of $\approx {10}^{12}$ cm for such a thermal emitter.

In Section 10, we present a theoretical treatment using a new relativistic hydrodynamical code to simulate the interaction of the ${e}^{+}{e}^{-}$–baryon plasma with the high-density regions of the SN ejecta. We first test the code in the same low-density domain of validity describing the prompt emission phase, and then we apply it in the high-density regime of the propagation of the plasma inside the SN ejecta, which we use for the theoretical interpretation of the X-ray flares. Most remarkably, the theoretical code leads to a thermal emitter with a Lorentz factor ${\rm{\Gamma }}\lesssim 4$ and a radius of $\approx {10}^{12}$ cm at transparency. The agreement between these theoretically derived values and the ones obtained from the observed thermal emission validates the model and the binary nature of the BdHN progenitors, in clear contrast with the traditional ultrarelativistic jetted models.

In Section 11, we present our conclusions. We first show how the traditional model, describing GRBs as a single system with ultrarelativistic jetted emission extending from the prompt emission all the way to the final phases of the afterglow and of the GeV emission, is in conflict with the X-ray flare observations. We also present three new main results that illustrate the new perspectives opened up by our alternative approach based on BdHNe.

A standard flat ${\rm{\Lambda }}$CDM cosmological model with ${{\rm{\Omega }}}_{M}=0.27$, ${{\rm{\Omega }}}_{{\rm{\Lambda }}}=0.73$, and ${H}_{0}=71$ km s⁻¹ Mpc⁻¹ is adopted throughout the paper, while Table 1 summarizes the acronyms we have used.

Table 1.  Alphabetic Ordered List of the Acronyms Used in this Work

Extended Wording	Acronym
Binary-driven hypernova	BdHN
Black hole	BH
Carbon–oxygen core	${\mathrm{CO}}_{\mathrm{core}}$
Circumburst medium	CBM
Flare–Plateau–Afterglow	FPA
Gamma-ray burst	GRB
Gamma-ray flash	GRF
Induced gravitational collapse	IGC
Massive neutron star	MNS
Neutron star	NS
New neutron star	νNS
Proper gamma-ray burst	P-GRB
Short gamma-ray burst	S-GRB
Short gamma-ray flash	S-GRF
Supernova	SN
Ultrashort gamma-ray burst	U-GRB
White dwarf	WD
X-ray flash	XRF

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The discovery of GRBs by the Vela satellites (Klebesadel et al. 1973) was presented at the AAAS meeting in February 1974 in San Francisco (Gursky & Ruffini 1975). The Vela satellites were operating in gamma-rays in the 150–750 keV energy range and only marginally in X-rays (3–12 keV; Cline et al. 1979). Soon after it was hypothesized from first principles that GRBs may originate from an ${e}^{+}{e}^{-}$ plasma in the gravitational collapse to a Kerr–Newman BH, implying an energy $\sim {10}^{54}{M}_{\mathrm{BH}}/{M}_{\odot }$ erg (Damour & Ruffini 1975; see also Ruffini 1998).

Since 1991, the BATSE detectors on the CGRO (see Gehrels et al. 1993) have been leading to the classification of GRBs on the basis of their spectral hardness and of their observed T₉₀ duration in the 50–300 keV energy band into short/hard bursts (${T}_{90}\lt 2$ s) and long/soft bursts (${T}_{90}\gt 2\,{\rm{s}}$ (Mazets et al. 1981; Dezalay et al. 1992; Klebesadel 1992; Kouveliotou et al. 1993; Tavani 1998). Such an emission was later called the GRB "prompt emission." In a first attempt, it was proposed that short GRBs originate from merging binary NSs (see, e.g., Goodman 1986; Paczynski 1986; Eichler et al. 1989; Narayan et al. 1991, 1992; Mészáros & Rees 1997) and long GRBs originate from a single source with ultrarelativistic jetted emission (Woosley 1993; Paczyński 1998; MacFadyen & Woosley 1999; Bromberg et al. 2013).

The BeppoSAX satellite, operating since 1996, joined the expertise of the X-ray and gamma-ray communities. Its gamma-ray burst monitor (GRBM) operating in the 40–700 keV energy band determined the trigger of the GRB, and two wide-field cameras operating in the 2–30 keV X-ray energy band allowed the localization of the source within an arcminute resolution. This enabled a follow-up with the narrow-field instruments (NFI) in the 2–10 keV energy band. BeppoSAX discovered the X-ray afterglow (Costa et al. 1997), characterized by an X-ray luminosity decreasing with a constant index of $\sim -1.3$ (see de Pasquale et al. 2006 as well as Pisani et al. 2016). This emission was detected after an "8 hr gap" following the prompt emission identified by BATSE. The consequent determination of the accurate positions by the NFI, transmitted to the optical (van Paradijs et al. 1997) and radio telescopes (Frail et al. 1997), allowed the determination of the GRB cosmological redshifts (Metzger et al. 1997). The derived distances of ≈5–10 Gpc confirmed their cosmological origin and their unprecedented energetics, $\approx {10}^{50}\mbox{--}{10}^{54}$ erg, thus validating our hypothesis derived from first principles (Damour & Ruffini 1975; Ruffini 1998).

To BeppoSAX goes the credit of the discovery of the temporal and spatial coincidence of GRB 980425 with SN 1998bw (Galama et al. 1998), which suggested the connection between GRBs and SNe, soon supported by many additional events (see, e.g., Woosley & Bloom 2006; Della Valle 2011; Hjorth & Bloom 2012). BeppoSAX also discovered the first "X-ray flare" in GRB 011121 closely following the prompt emission (Piro et al. 2005); see Figure 1. Our goal in this paper is to show how the X-ray flares, thanks to the observational campaign of the Swift satellite, have become the crucial test for understanding the astrophysical nature of the GRB–SN connection.

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The Swift Burst Alert Telescope (BAT), operating in the 15–150 keV energy band, can detect GRB prompt emissions and accurately determine their position in the sky within 3 arcmin. Within 90 s, Swift can re-point the narrow-field X-ray telescope (XRT), operating in the 0.3–10 keV energy range, and relay the burst position to the ground. This overcomes the "8 hr gap" in the BeppoSAX data.

Thanks to the Swift satellite, the number of detected GRBs increased rapidly to 480 sources with known redshifts. By analyzing the light curve of some long GRBs, including the data in the "8 hr gap" of BeppoSAX, Nousek et al. (2006) and Zhang et al. (2006) discovered three power-law segments in the XRT flux light curves of some long GRBs. We refer to these as the "Nousek–Zhang power laws" (see Figure 2). The nature of this feature has been the subject of a long debates, still ongoing, and is finally resolved in this article.

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We have used Swift-XRT data in differentiating two distinct subclasses of long GRBs: XRFs with ${E}_{\mathrm{iso}}\lesssim {10}^{52}$ erg and BdHNe with ${E}_{\mathrm{iso}}\gtrsim {10}^{52}$ erg (see Section 3). An additional striking difference appears between the XRT luminosities of these two subclasses when measured in their cosmological rest frames: in the case of BdHNe, the light curves follow a specific behavior that conforms to the Nousek–Zhang power law (see, e.g., Penacchioni et al. 2012, 2013; Pisani et al. 2013, 2016; Ruffini et al. 2014). None of these features are present in the case of XRFs (see Figure 3).

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Finally, the Fermi satellite (Atwood et al. 2009), launched in 2008, detects ultrahigh energy photons from 20 MeV to 300 GeV with the Large Area Telescope (LAT) and detects photons from 8 keV to 30 MeV with the Gamma-ray Burst Monitor (GBM). For the purposes of this article addressing long GRBs, the Fermi observations have been prominent in further distinguishing between XRFs and BdHNe: the Fermi-LAT GeV emission has been observed only in BdHNe and never in XRFs.

3.1. The Classification of GRBs

The very extensive set of observations carried out by the above satellites in coordination with the largest optical and radio telescopes over a period of almost 40 years has led to an impressive set of data on 480 GRBs, all characterized by spectral, luminosity, and time variability information, and each one with a well-established cosmological redshift. By classifying both the commonalities and the differences among all GRBs, it has been possible to create "equivalence relations" and divide GRBs into a number of subclasses, each one identified by a necessary and sufficient number of observables. We recall in Table 2 and Figure 4 the binary nature of all GRB progenitors and their classification into seven different subclasses (see, e.g., Ruffini et al. 2016b). In Table 2, we indicate the number of sources in each subclass, the nature of their progenitors and final outcomes of their evolution, their rest-frame T₉₀, their rest-frame spectral peak energy ${E}_{{\rm{p}},{\rm{i}}}$ and E_iso as well as the isotropic energy in X-rays ${E}_{\mathrm{iso},{\rm{X}}}$ and in GeV emission ${E}_{\mathrm{iso},\mathrm{GeV}}$, and finally their local observed number density rate. In Figure 4, we mention the ${E}_{{\rm{p}},{\rm{i}}}$–E_iso relations for these sources, including the Amati one for BdHNe and the MuRuWaZha one for the short bursts (see Ruffini et al. 2016a, 2016b), comprising short gamma-ray flashes (S-GRFs) with ${E}_{\mathrm{iso}}\lesssim {10}^{52}$ erg, authentic short GRBs (S-GRBs) with ${E}_{\mathrm{iso}}\gtrsim {10}^{52}$ erg, and gamma-ray flashes (GRFs), sources with hybrid short/long burst properties in their gamma-ray light curves, i.e., an initial spike-like harder emission followed by a prolonged softer emission observed up to ∼100 s, originating from NS–white dwarf binaries (Caito et al. 2009, 2010; Ruffini et al. 2016b). We have no evidence for an ${E}_{{\rm{p}},{\rm{i}}}$ and E_iso relation in the XRFs (see Figure 4). The Amati and the MuRuWaZha relations have not yet been theoretically understood, and as such they have no predictive power.

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Table 2.  Summary of the Seven GRB Subclasses (XRFs, BdHNe, BH–SN, Short Gamma-ray Flashes (S-GRFs), Authentic Short GRBs (S-GRBs), Ultrashort GRBs (U-GRB), and GRFs) and Their Observational Properties

	Subclass	Number	In-state	Out-state	T90	${E}_{{\rm{p}},{\rm{i}}}$	${E}_{\mathrm{iso}}$	${E}_{\mathrm{iso},{\rm{X}}}$	${E}_{\mathrm{iso},\mathrm{Gev}}$	${\rho }_{\mathrm{GRB}}$
			(Progenitor)	(Final outcome)	(s)	(MeV)	(erg)	(erg)	(erg)	(Gpc−3yr−1)
I	XRFs	82	${\mathrm{CO}}_{\mathrm{core}}$–NS	νNS–NS	∼2–103	$\lesssim 0.2$	$\sim {10}^{48}\mbox{--}{10}^{52}$	$\sim {10}^{48}\mbox{--}{10}^{51}$	⋯	${100}_{-34}^{+45}$
II	BdHNe	345	${\mathrm{CO}}_{\mathrm{core}}$–NS	νNS–BH	∼2–102	∼0.2–2	$\sim {10}^{52}\mbox{--}{10}^{54}$	$\sim {10}^{51}\mbox{--}{10}^{52}$	$\lesssim {10}^{53}$	${0.77}_{-0.08}^{+0.09}$
III	BH–SN	⋯	${\mathrm{CO}}_{\mathrm{core}}$–BH	νNS–BH	∼2–102	$\gtrsim 2$	$\gt {10}^{54}$	$\sim {10}^{51}\mbox{--}{10}^{52}$	$\gtrsim {10}^{53}$	$\lesssim {0.77}_{-0.08}^{+0.09}$
IV	S-GRFs	33	NS–NS	MNS	$\lesssim 2$	$\lesssim 2$	$\sim {10}^{49}\mbox{--}{10}^{52}$	$\sim {10}^{49}\mbox{--}{10}^{51}$	⋯	${3.6}_{-1.0}^{+1.4}$
V	S-GRBs	7	NS–NS	BH	$\lesssim 2$	$\gtrsim 2$	$\sim {10}^{52}\mbox{--}{10}^{53}$	$\lesssim {10}^{51}$	$\sim {10}^{52}\mbox{--}{10}^{53}$	$({1.9}_{-1.1}^{+1.8})\times {10}^{-3}$
VI	U-GRBs	⋯	νNS–BH	BH	$\ll 2$	$\gtrsim 2$	$\gt {10}^{52}$	⋯	⋯	$\gtrsim {0.77}_{-0.08}^{+0.09}$
VII	GRFs	13	NS–WD	MNS	∼2–102	∼0.2–2	$\sim {10}^{51}\mbox{--}{10}^{52}$	$\sim {10}^{49}\mbox{--}{10}^{50}$	⋯	${1.02}_{-0.46}^{+0.71}$

Note. In the first five columns, we indicate the GRB subclasses and their corresponding number of sources with measured z, in-states, and out-states. In the following columns, we list the ranges of T₉₀ in the rest frame, the rest-frame spectral peak energies ${E}_{{\rm{p}},{\rm{i}}}$ and ${E}_{\mathrm{iso}}$ (rest frame 1–10⁴ keV), the isotropic energy of the X-ray data ${E}_{\mathrm{iso},{\rm{X}}}$ (rest frame 0.3–10 keV), and the isotropic energy of the GeV emission ${E}_{\mathrm{iso},\mathrm{GeV}}$ (rest frame 0.1–100 GeV). In the last column, we list, for each GRB subclass, the local observed number density rate ${\rho }_{\mathrm{GRB}}$ obtained in Ruffini et al. (2016b). For details, see Ruffini et al. (2014, 2015b, 2015c), Fryer et al. (2015), Ruffini et al. (2016a, 2016b), and Becerra et al. (2016).

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3.2. The Role of Time Parametrization in GRBs

Precise general relativistic rules in the spacetime parameterization of GBRs are needed (Ruffini et al. 2001a). Indeed, there are four time variables entering this discussion, which have to be properly distinguished one from another: (1) the comoving time t_com, which is the time used to compute the evolution of the thermodynamical quantities (density, pressure, temperature); (2) the laboratory time $t={\rm{\Gamma }}{t}_{\mathrm{com}}$, where as usual the Lorentz Gamma factor is ${\rm{\Gamma }}={(1-{\beta }^{2})}^{-1/2}$ and $\beta =v/c$ is the expansion velocity of the source; (3) the arrival time t_a at which each photon emitted by the source reaches an observer in the cosmological rest frame of the source, given by (see also Bianco et al. 2001; Ruffini et al. 2002; Bianco & Ruffini 2005a)

$$\begin{eqnarray}&&{t}_{a}=t-\displaystyle \frac{r(t)}{c}\cos \vartheta ,\end{eqnarray} \tag{ 1 }$$

where r(t) is the radius of the expanding source in the laboratory frame and ϑ is the displacement angle of the normal to the emission surface from the line of sight; and (4) the arrival time at the detector on the Earth, ${t}_{a}^{d}={t}_{a}(1+z)$, corrected for cosmological effects, where z is the source redshift needed in order to compare GRBs at different redshifts z. As emphasized in Ruffini et al. (2001a, p. L108), "the bookkeeping of these four different times and the corresponding space variables must be done carefully in order to keep the correct causal relation in the time sequence of the events involved." The chain of relations between these four times is given by (see e.g., Bianco et al. 2001; Ruffini et al. 2001a, 2002; Bianco & Ruffini 2005a, and see also Sections 8 and 9 for the dynamics of the flares)

$$\begin{eqnarray}{t}_{a}^{d} & = & (1+z){t}_{a}=(1+z)\left(t-\displaystyle \frac{r(t)}{c}\cos \vartheta \right)\\ & = & (1+z)\left({\rm{\Gamma }}{t}_{\mathrm{com}}-\displaystyle \frac{r({\rm{\Gamma }}{t}_{\mathrm{com}})}{c}\cos \vartheta \right).\end{eqnarray} \tag{ 2 }$$

The proper use of these four time variables is mandatory in modeling GRB sources, especially when we are dealing with a model not based on a single component but on multiple components, each characterized by a different world line and a different Lorentz Gamma factor, as is the case for BdHNe (see Sections 4 and 5).

3.3. The Role of the GRBs' Cosmological Rest Frame

3.4. Episode 1: The Hypercritical Accretion Process

3.5. Episode 2: ${e}^{+}{e}^{-}$ Pairs Colliding with the SN Ejecta

3.6. Episode 3: Ongoing Research on the Gamma-Ray Flares, Afterglow, and GeV Emission

With the increase in the number of observed GRBs, an attempt was made to analyze the X-ray flares and other processes considered to be similar in the observer reference frame, independent of the nature of the GRB type and of the value of their cosmological redshift or the absence of such a value. The goal of this attempt was to identify their "standard" properties, following a statistical analysis methodology often applied in classical astronomy (see Chincarini et al. 2007; Falcone et al. 2007; Margutti et al. 2010 as well as the review articles by Piran 1999, 2004; Mészáros 2002, 2006; Berger 2014; Kumar & Zhang 2015). We now summarize our alternative approach, having already given in the introduction and in Sections 2 and 3 the background for the observational identification and the theoretical interpretation of the X-ray flares.

As a first step, we only consider GRBs with an observed cosmological redshift. Having ourselves proposed the classification of all GRBs into seven different subclasses (see Section 3), we have given preliminary attention to verifying whether X-ray flares actually occur preferentially in some of these subclasses and if so, identifying the physical reasons determining such a correlation. We have analyzed all X-ray flares and found, a posteriori, that X-ray flares only occur in BdHNe. No X-ray flare has been identified in any other GRB subclass, either long or short. A claim of their existence in short bursts (Barthelmy et al. 2005; Fan et al. 2005; Dai et al. 2006) has been superseded: GRB 050724 with ${T}_{90}\sim 100\,{\rm{s}}$ is not a short GRB, but actually a GRF, expected to originate in the merging of an NS and a white dwarf (see Figure 4); the X-ray data for this source from XRT are sufficient to assert that there is no evidence of an X-ray flare as defined in this section. GRB 050709 is indeed a short burst. It has been classified as an S-GRF (Aimuratov et al. 2017) and has been observed by HETE with very sparse X-ray data (Butler et al. 2005), and no presence of an X-ray flare can be inferred; the Swift satellite pointed at this source too late, 38.5 hr after the HETE trigger (Morgan et al. 2005).

As a second step, since all GRBs have a different redshift z, in order to compare them we need a description of each of them in its own cosmological rest frame. The luminosities have to be estimated after doing the necessary K corrections and the time coordinate in the observer frame has to be corrected by the cosmological redshift ${t}_{a}^{d}=(1+z){t}_{a}$. This also affects the determination of the T₉₀ of each source (see, e.g., Figure 38 in Section 11 where the traditional approach by Kouveliotou et al. 1993 and Bromberg et al. 2013 has been superseded by ours).

As a third step, we recall an equally important distinction from the traditional fireball approach with a single ultrarelativistic jetted emission. Our GRB analysis envisages the existence of different episodes within each GRB, each one characterized by a different physical process and needing the definition of its own world line and corresponding Gamma factors, essential for estimating the time parametrization in the rest frame of the observer (see Section 2).

These three steps are applied in the present article, which specifically addresses the study of early X-ray flares and their fundamental role in establishing the physical and astrophysical nature of BdHNe and in distinguishing our binary model from the traditional one.

Before proceeding, let us recall the basic point of the K correction. All of the observed GRBs have a different redshift. In order to compare them, it is necessary to refer to each of them in its cosmological rest frame. This step has often been ignored in the current literature (Chincarini et al. 2007; Falcone et al. 2007; Margutti et al. 2010). Similarly, for the flux observed by the above satellites in Section 2, each instrument is characterized by its fixed energy window $[{\epsilon }_{\mathrm{obs},1};{\epsilon }_{\mathrm{obs},2}]$. The observed flux f_obs, defined as the energy per unit area and time in a fixed instrumental energy window $[{\epsilon }_{\mathrm{obs},1};{\epsilon }_{\mathrm{obs},2}]$, is expressed in terms of the observed photon number spectrum n_obs (i.e., the number of observed photons per unit energy, area, and time) as

$$\begin{eqnarray}&&{f}_{\mathrm{obs},[{\epsilon }_{\mathrm{obs},1};{\epsilon }_{\mathrm{obs},2}]}={\int }_{{\epsilon }_{\mathrm{obs},1}}^{{\epsilon }_{\mathrm{obs},2}}\epsilon \,{n}_{\mathrm{obs}}(\epsilon )d\epsilon .\end{eqnarray} \tag{ 3 }$$

It then follows that the luminosity L of the source (i.e., the total emitted energy per unit time in a given bandwidth), expressed by definition in the source cosmological rest frame, is related to f_obs through the luminosity distance D_L(z):

$$\begin{eqnarray}&&{L}_{[{\epsilon }_{\mathrm{obs},1}(1+z);{\epsilon }_{\mathrm{obs},2}(1+z)]}=4\pi {D}_{L}^{2}(z){f}_{\mathrm{obs},[{\epsilon }_{\mathrm{obs},1};{\epsilon }_{\mathrm{obs},2}]}.\end{eqnarray} \tag{ 4 }$$

The above Equation (4) gives the luminosities in different cosmological rest-frame energy bands, depending on the source redshift. To express the luminosity L in a fixed cosmological rest-frame energy band, e.g., $[{E}_{1};{E}_{2}]$, common to all sources, we can rewrite Equation (4) as

$$\begin{eqnarray}{L}_{[{E}_{1};{E}_{2}]} & = & 4\pi {D}_{L}^{2}{f}_{\mathrm{obs},\left[\tfrac{{E}_{1}}{1+z};\tfrac{{E}_{2}}{1+z}\right]}\\ & = & 4\pi {D}_{L}^{2}k[{\epsilon }_{\mathrm{obs},1};{\epsilon }_{\mathrm{obs},2};{E}_{1};{E}_{2};z]{f}_{\mathrm{obs},[{\epsilon }_{\mathrm{obs},1};{\epsilon }_{\mathrm{obs},2}]},\end{eqnarray} \tag{ 5 }$$

where we have defined the K-correction factor:

$$\begin{eqnarray}k[{\epsilon }_{\mathrm{obs},1};{\epsilon }_{\mathrm{obs},2};{E}_{1};{E}_{2};z] & = & \displaystyle \frac{{f}_{\mathrm{obs},\left[\tfrac{{E}_{1}}{1+z};\tfrac{{E}_{2}}{1+z}\right]}}{{f}_{\mathrm{obs},[{\epsilon }_{\mathrm{obs},1};{\epsilon }_{\mathrm{obs},2}]}}\\ & = & \displaystyle \frac{{\displaystyle \int }_{{E}_{1}/(1+z)}^{{E}_{2}/(1+z)}\epsilon \,{n}_{\mathrm{obs}}(\epsilon )d\epsilon }{{\displaystyle \int }_{{\epsilon }_{\mathrm{obs},1}}^{{\epsilon }_{\mathrm{obs},2}}\epsilon \,{n}_{\mathrm{obs}}(\epsilon )d\epsilon }.\end{eqnarray} \tag{ 6 }$$

If the energy range $[{\textstyle \tfrac{{E}_{1}}{1+z}};{\textstyle \tfrac{{E}_{2}}{1+z}}]$ is not fully inside the instrumental energy band $[{\epsilon }_{\mathrm{obs},1};{\epsilon }_{\mathrm{obs},2}]$, it may well happen that we will need to extrapolate n_obs within the integration boundaries $[{\textstyle \tfrac{{E}_{1}}{1+z}};{\textstyle \tfrac{{E}_{2}}{1+z}}]$.

Finally, we express each luminosity in a rest-frame energy band that coincides with the energy window of each specific instrument.

We turn now to the selection procedure for early X-ray flares. We take the soft X-ray flux light curves of each source with known redshift from the Swift-XRT repository (Evans et al. 2007, 2009). We then apply the above K correction to obtain the corresponding luminosity light curves in the rest frame 0.3–10 keV energy band. Starting from 421 Swift-XRT light curves, we found in 50 sources X-ray flare structures in the early 200 s. Remarkably, all of them are in BdHNe. We further filter our sample by applying the following criteria:

• 1.  
  We exclude GRBs with flares having a low ($\lt 20$) signal-to-noise ratio or with an incomplete data coverage of the early X-ray light curve—14 GRBs are excluded (see e.g., Figure 5).
• 2.  
  We consider only X-ray flares and do not address here the gamma-ray flares, which will be studied in a forthcoming article—eight GRBs having only gamma-ray flares are temporarily excluded (see, e.g., Figure 6). In Figure 7, we show an illustrative example of the possible co-existence of an X-ray flare and a gamma-ray flare, and a way to distinguish them.
• 3.  
  We also ignore here the late X-ray flare, including the ultralong GRB, which will be discussed in a forthcoming paper—six GRBs are consequently excluded.
• 4.  
  We ignore the GRBs for which the soft X-ray energy observed by Swift-XRT (0.3–10 keV) before the plateau phase is higher than the gamma-ray energy observed by Swift-BAT (15–150 keV) during the entire valid Swift-BAT observation. This Swift-BAT anomaly points to an incomplete coverage of the prompt emission—six GRBs are excluded (see, e.g., Figure 8).

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Finally, we have found 16 BdHNe satisfying all of the criteria to be included in our sample. Among them, seven BdHNe show a single flare. The other nine BdHNe contain two flares: generally, we exclude the first one, which appears to be a component from the gamma-ray spike or gamma-ray flare, and therefore select the second one for analysis (see, e.g., Figure 7).

These 16 selected BdHNe cover a wide range of redshifts. The closest one is GRB 070318 with redshift z = 0.84, and the farthest one is GRB 090516A with redshift z = 4.11. Their isotropic energy is also distributed over a large range: five GRBs have energies of the order of 10⁵² erg, nine GRBs of the order of 10⁵³ erg, and two GRBs have extremely high isotropic energies ${E}_{\mathrm{iso}}\gt {10}^{54}$ erg. Therefore, this sample is well-constructed although the total number is limited.

We now turn to the light curves of each of these 16 GRBs composing our sample (see Figures 9–24). The blue curves represent the X-rays observed by Swift-XRT, and the green curves are the corresponding optical observations when available. All of the values are in the rest frame and the X-ray luminosities have been K-corrected. The red vertical lines indicate the peak time of the X-ray flares. The rest-frame luminosity light curves of some GRBs show different flare structures compared to the observed count flux light curves. An obvious example is GRB 090516A, which follows from comparing Figure 18 in this paper with Figure 1 in Troja et al. (2015). The details of the FPA, as well as their correlations or the absence of correlation with E_iso, are given in the next section.

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We then conclude that in our sample, there are Swift data for all GRBs: Konus-Wind observed GRBs 080607, 080810, 090516A, 131030A, 140419A, 141221A, and 151027A, while Fermi detected GRBs 090516A, 140206, 141221A, and 151027A. The energy coverage of the available satellites is limited, as mentioned in Section 2: Fermi detects the widest photon energy band, from 8 keV to 300 GeV, Konus-Wind observes from 20 keV to 15 MeV, and Swift-BAT has a narrow coverage from 15 keV to 150 keV. No GeV photons were observed, though GRB 090516A and 151027 were in the Fermi-LAT field of view. This contrasts with the observations of S-GRBs for which, in all of the sources so far identified and within the Fermi-LAT field of view, GeV photons were always observed (Ruffini et al. 2016a, 2016b) and can always freely reach a distant observer. These observational facts suggest that NS–NS (or NS–BH) mergers leading to the formation of a BH leave the surrounding environment poorly contaminated with the material ejected in the merging process (≲10⁻²–${10}^{-3}\,{M}_{\odot }$) and therefore the GeV emission, originating from the accretion on the BH formed in the merger process (Ruffini et al. 2016a) can be observed. On the other hand, BdHNe originate in CO_core–NS binaries in which the material ejected from the CO_core explosion ($\approx {M}_{\odot }$) greatly pollutes the environment where the GeV emission has to propagate to reach the observer (see Section 3). This, together with the asymmetries of the SN ejecta (see Section 3 and Becerra et al. 2016), lead to the possibility that the GeV emission in BdHNe can be "obscured" by the material of the SN ejecta, explaining the absence of GeV photons in the above cases of GRBs 090516A and 151027.

We derive the isotropic energy E_iso by assuming the prompt emission to be isotropic and by integrating the prompt photons in the rest-frame energy range from 1 keV to 10 MeV (Bloom et al. 2001). None of the satellites is able to cover the entire energy band of E_iso, so we need to fit the spectrum and find the best-fit function, then extrapolate the integration of energy by using this function. This method is relatively safe for GRBs observed by Fermi and Konus-Wind, but six GRBs in our sample have been observed only by Swift, so we uniformly fit and extrapolate these six GRBs by power laws and cutoff power laws; we then take the average value as E_iso. In general, our priority in computing E_iso is Fermi, Konus-Wind, then Swift. In order to take into account the expansion of the universe, all of our computations consider the K correction. The formula of K correction for E_iso varies depending on the best-fit function. The energy in the X-ray afterglow is computed in the cosmological rest-frame energy band from 0.3 to 10 keV. We smoothly fit the luminosity light curve using an algorithm named locally weighted regression (Cleveland & Devlin 1988), which provides a sequence of power-law functions. The corresponding energy in a fixed time interval is obtained by summing up all of the integrals of the power laws within it. This method is applied to estimate the energy of the flare E_f as well as the energy of the FPA phase up to 10⁹ s, E_FPA. An interesting alternative procedure was used in Swenson & Roming (2014) to fit the light curve and determine the flaring structure with a Bayesian Information method. On this specific aspect, the two treatments are equally valid and give compatible results.

Table 3 contains the relevant energy and time information of the 16 BdHNe of the sample: the cosmological redshift z, E_iso, the flare peak time t_p, the corresponding peak luminosity L_p, the flare duration ${\rm{\Delta }}t$, and the energy of the flare E_f. To determine t_p, we apply a locally weighted regression, which results in a smoothed light curve composed of power-law functions: the flare peak is localized where the power-law index is zero. Therefore, t_p is defined as the time interval between the flare peak and the trigger time of Swift-BAT.¹⁰ Correspondingly, we find the peak luminosity L_p at t_p and its duration ${\rm{\Delta }}t$, which is defined as the time interval between a start time and an end time where the luminosity is half of L_p. We have made public the entire details including the codes online.¹¹

Table 3.  GRB Sample Properties of the Prompt and Flare Phases

GRB	z	T90 (s)	Eiso (erg)	tp (s)	Lp (erg s−1)	${\rm{\Delta }}t$ (s)	Ef (erg)	${\alpha }_{f}$
060204B	2.3393	40.12	$2.93(\pm 0.60)\times {10}^{53}$	100.72 ± 6.31	$7.35(\pm 2.05)\times {10}^{49}$	17.34 ± 6.83	$8.56(\pm 0.82)\times {10}^{50}$	2.73
060607A	3.082	24.49	$2.14(\pm 1.19)\times {10}^{53}$	66.04 ± 4.98	$2.28(\pm 0.48)\times {10}^{50}$	18.91 ± 3.84	$3.33(\pm 0.32)\times {10}^{51}$	1.72
070318	0.84	28.80	$3.41(\pm 2.14)\times {10}^{52}$	154.7 ± 12.80	$6.28(\pm 1.30)\times {10}^{48}$	63.80 ± 19.82	$3.17(\pm 0.37)\times {10}^{50}$	1.84
080607	3.04	21.04	$1.87(\pm 0.11)\times {10}^{54}$	37.48 ± 3.60	$1.14(\pm 0.27)\times {10}^{51}$	15.63 ± 4.32	$1.54(\pm 0.24)\times {10}^{52}$	2.08
080805	1.51	31.08	$7.16(\pm 1.90)\times {10}^{52}$	48.41 ± 5.46	$4.66(\pm 0.59)\times {10}^{49}$	27.56 ± 9.33	$9.68(\pm 1.24)\times {10}^{50}$	1.25
080810	3.35	18.25	$5.00(\pm 0.44)\times {10}^{53}$	51.03 ± 6.49	$1.85(\pm 0.53)\times {10}^{50}$	12.38 ± 4.00	$1.80(\pm 0.17)\times {10}^{51}$	2.37
081008	1.967	62.52	$1.35(\pm 0.66)\times {10}^{53}$	102.24 ± 5.66	$1.36(\pm 0.33)\times {10}^{50}$	18.24 ± 3.63	$1.93(\pm 0.16)\times {10}^{51}$	2.46
081210	2.0631	47.66	$1.56(\pm 0.54)\times {10}^{53}$	127.59 ± 13.68	$2.23(\pm 0.21)\times {10}^{49}$	49.05 ± 6.49	$8.86(\pm 0.54)\times {10}^{50}$	2.28
090516A	4.109	68.51	$9.96(\pm 1.67)\times {10}^{53}$	80.75 ± 2.20	$9.10(\pm 2.26)\times {10}^{50}$	10.43 ± 2.44	$7.74(\pm 0.63)\times {10}^{51}$	3.66
090812	2.452	18.77	$4.40(\pm 0.65)\times {10}^{53}$	77.43 ± 16.6	$3.13(\pm 1.38)\times {10}^{50}$	17.98 ± 4.51	$5.18(\pm 0.61)\times {10}^{51}$	2.20
131030A	1.293	12.21	$3.00(\pm 0.20)\times {10}^{53}$	49.55 ± 7.88	$6.63(\pm 1.12)\times {10}^{50}$	33.73 ± 6.55	$3.15(\pm 0.57)\times {10}^{52}$	2.22
140206A	2.73	7.24	$3.58(\pm 0.79)\times {10}^{53}$	62.11 ± 12.26	$4.62(\pm 0.99)\times {10}^{50}$	26.54 ± 4.31	$1.04(\pm 0.59)\times {10}^{51}$	1.73
140301A	1.416	12.83	$9.50(\pm 1.75)\times {10}^{51}$	276.56 ± 15.50	$5.14(\pm 1.84)\times {10}^{48}$	64.52 ± 10.94	$3.08(\pm 0.22)\times {10}^{50}$	2.30
140419A	3.956	16.14	$1.85(\pm 0.77)\times {10}^{54}$	41.00 ± 4.68	$6.23(\pm 1.45)\times {10}^{50}$	14.03 ± 5.74	$7.22(\pm 0.88)\times {10}^{51}$	2.32
141221A	1.47	9.64	$6.99(\pm 1.98)\times {10}^{52}$	140.38 ± 5.64	$2.60(\pm 0.64)\times {10}^{49}$	38.34 ± 9.26	$7.70(\pm 0.78)\times {10}^{50}$	1.79
151027A	0.81	68.51	$3.94(\pm 1.33)\times {10}^{52}$	183.79 ± 16.43	$7.10(\pm 1.75)\times {10}^{48}$	163.5 ± 30.39	$4.39(\pm 2.91)\times {10}^{51}$	2.26

Note. This table contains the redshift z, the T₉₀ in the rest frame, the isotropic energy E_iso, the flare peak time t_p in the rest frame, the flare peak luminosity L_p, the flare duration where the starting and ending time correspond to half of the peak luminosity ${\rm{\Delta }}t$, the flare energy E_f within the time interval ${\rm{\Delta }}t$, and ${\alpha }_{f}$ the power-law index from the fitting of the flare's spectrum.

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We then establish correlations between the above quantities characterizing each luminosity light curve of the sample with the E_iso of the corresponding BdHN. We have relied heavily on the MCMC method and iterated 10⁵ times to obtain the best fit of the power law and their correlation coefficient. The main results are summarized in Figures 25–28. All of the codes are publicly available online.¹² We conclude that the peak time and the duration of the flare, as well as the peak luminosity and the total energy of flare, are highly correlated with E_iso, with correlation coefficients larger than 0.6 (or smaller than −0.6). The average values and the 1σ uncertainties are shown in Table 4.

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Table 4.  Power-law Correlations Among the Quantities in Table 3

Correlation	Power-law Index	Coefficient
${E}_{\mathrm{iso}}-{t}_{p}$	$-0.290(\pm 0.010)$	$-0.764(\pm 0.123)$
${E}_{\mathrm{iso}}-{\rm{\Delta }}t$	$-0.461(\pm 0.042)$	$-0.760(\pm 0.138)$
${E}_{\mathrm{iso}}-{L}_{p}$	$1.186(\pm 0.037)$	$0.883(\pm 0.070)$
${E}_{\mathrm{iso}}-{E}_{f}$	$0.631(\pm 0.117)$	$0.699(\pm 0.145)$

Note. The values and uncertainties (at the 1σ confidence level) of the power-law index and of the correlation coefficient are obtained from 10⁵ MCMC iterations. All relations are highly correlated.

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The energy of the prompt emission is proportional to E_iso if and only if spherical symmetry is assumed: this clearly follows from the prompt emission time-integrated luminosity. We are now confronted with a new situation: the total energy of the FPA emission up to 10⁹ s (E_FPA) is also proportional to E_iso, following the correlation given in Tables 5 and 6, and Figure 29. What is clear is that there are two very different components where the energy of the dyadosphere ${E}_{{e}^{+}{e}^{-}}$ is utilized: the energy E_prompt of the prompt emission and the energy E_FPA of the FPA, i.e., ${E}_{{e}^{+}{e}^{-}}={E}_{\mathrm{iso}}={E}_{\mathrm{prompt}}+{E}_{\mathrm{FPA}}$. Figures 30 and 31 show the distribution of ${E}_{{e}^{+}{e}^{-}}={E}_{\mathrm{iso}}$ between these two components.

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Table 5.  GRB Sample Properties of the Prompt and FPA Phases

GRB	z	Eiso (erg)	EFPA (erg)
060204B	2.3393	$2.93(\pm 0.60)\times {10}^{53}$	$6.02(\pm 0.20)\times {10}^{51}$
060607A	3.082	$2.14(\pm 1.19)\times {10}^{53}$	$2.39(\pm 0.12)\times {10}^{52}$
070318	0.84	$3.41(\pm 2.14)\times {10}^{52}$	$4.76(\pm 0.21)\times {10}^{51}$
080607	3.04	$1.87(\pm 0.11)\times {10}^{54}$	$4.32(\pm 0.96)\times {10}^{52}$
080805	1.51	$7.16(\pm 1.90)\times {10}^{52}$	$6.65(\pm 0.42)\times {10}^{51}$
080810	3.35	$5.00(\pm 0.44)\times {10}^{53}$	$1.67(\pm 0.14)\times {10}^{52}$
081008	1.967	$1.35(\pm 0.66)\times {10}^{53}$	$6.56(\pm 0.60)\times {10}^{51}$
081210	2.0631	$1.56(\pm 0.54)\times {10}^{53}$	$6.59(\pm 0.60)\times {10}^{51}$
090516A	4.109	$9.96(\pm 1.67)\times {10}^{53}$	$3.34(\pm 0.22)\times {10}^{52}$
090812	2.452	$4.40(\pm 0.65)\times {10}^{53}$	$3.19(\pm 0.36)\times {10}^{52}$
131030A	1.293	$3.00(\pm 0.20)\times {10}^{53}$	$4.12(\pm 0.23)\times {10}^{52}$
140206A	2.73	$3.58(\pm 0.79)\times {10}^{53}$	$5.98(\pm 0.69)\times {10}^{52}$
140301A	1.416	$9.50(\pm 1.75)\times {10}^{51}$	$1.42(\pm 0.14)\times {10}^{50}$
140419A	3.956	$1.85(\pm 0.77)\times {10}^{54}$	$6.84(\pm 0.82)\times {10}^{52}$
141221A	1.47	$6.99(\pm 1.98)\times {10}^{52}$	$5.31(\pm 1.21)\times {10}^{51}$
151027A	0.81	$3.94(\pm 1.33)\times {10}^{52}$	$1.19(\pm 0.18)\times {10}^{52}$

Note. This table lists z, E_iso, and the FPA energy E_FPA from the flare until 10⁹ s.

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Table 6.  Power-law Correlations Among the Quantities in Table 5

Correlation	Power-law Index	Coefficient
Eiso–EFPA	$0.613(\pm 0.041)$	$0.791(\pm 0.103)$
Eiso–${E}_{\mathrm{FPA}}/{E}_{\mathrm{iso}}$	$-0.005(\pm 0.002)$	$0.572(\pm 0.178)$

Note.  The statistical considerations of Table 4 are valid here as well.

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As a consequence of the above, in view of the presence of the companion SN remnant ejecta (see Becerra et al. 2016 for more details), we assume here that the spherical symmetry of the prompt emission is broken. Part of the energy due to the impact of the ${e}^{+}{e}^{-}$ plasma on the SN is captured by the SN ejecta, and gives rise to the FPA emission as originally proposed by Ruffini (2015). We shall return to the study of the impact between the plasma and the SN ejecta in Section 10 after studying the motion of the matter composing the FPA in the next few sections.

It can also be seen that the relative partition between E_prompt and E_FPA strongly depends on the value of ${E}_{{e}^{+}{e}^{-}}$: the lower the GRB energy, the higher the FPA energy percentage, and consequently the lower the prompt energy percentage (see Figure 31).

In Becerra et al. (2016), we indicate that both the value of ${E}_{{e}^{+}{e}^{-}}$ and the relative ratio of the above two components can in principle be explained in terms of the geometry of the binary nature of the system: the smaller the distance is between the CO_core and the companion NS, the shorter the binary period of the system, and the larger the value of ${E}_{{e}^{+}{e}^{-}}$ .

We discuss now the profound difference between the prompt emission, which we recall is emitted at distances of the order of 10¹⁶ cm away from the newly born BH with ${\rm{\Gamma }}\approx {10}^{2}\mbox{--}{10}^{3}$, and the FPA phase. We focus on a further fundamental set of data, which originates from a thermal emission associated with the flares.¹³ Only in some cases is this emission so clear and prominent that it allows the estimation of the flare expansion speed and the determination of its mildly relativistic Lorentz factor ${\rm{\Gamma }}\lesssim 4$, which creates a drastic separatrix both in the energy and in the Gamma factor between the astrophysical nature of the prompt emission and of the flares.

Following the standard data reduction procedure of Swift-XRT (Romano et al. 2006; Evans et al. 2007, 2009), X-ray data within the duration of flare are retrieved from the United Kingdom Swift Science Data Centre (UKSSDC)¹⁴ and analyzed by Heasoft.¹⁵ Table 7 shows the fit of the spectrum within the duration ${\rm{\Delta }}t$ of the flare for each BdHN of the sample. As a first approximation, in computing the radius, we have assumed a constant expansion velocity of $0.8c$ indicated for some BdHNe, such as GRB 090618 (Ruffini et al. 2014) and GRB 130427A (Ruffini et al. 2015c). Out of 16 sources, seven BdHNe have highly confident thermal components (significance $\gt 0.95$; see boldfaced entries in Table 7), which means that the addition of a blackbody spectrum improves a single power-law fit (which is, conversely, excluded at the 2σ confidence level). These blackbodies have fluxes in a range from 1% to 30% of the total flux and share a similar order of magnitude radii, i.e., ∼10¹¹–10¹² cm. In order to have a highly significant thermal component, the blackbody radiation itself should be prominent as well as its ratio to the nonthermal part. Another critical reason is that the observable temperature must be compatible with the satellite bandpass. For example, Swift-XRT observes in the 0.3–10 keV photon energy band, but the hydrogen absorption affects the lower energy part (∼0.5 keV), and data are not always adequate beyond 5 keV, due to the low effective area of satellite for high-energy photons. The reliable temperature only ranges from 0.15 keV to 1.5 keV (since the peak photon energy is equal to the temperature times 2.82), so the remaining nine GRBs may contain a thermal component in the flare but outside the satellite bandpass.

Table 7.  Radii and Temperatures of the Thermal Components Detected Within the Flare Duration ${\rm{\Delta }}t$

GRB	Radius (cm)	${{kT}}_{\mathrm{obs}}$ (keV)	Significance
060204B	${\bf{1.80}}(\pm {\bf{1.11}})\times {{\bf{10}}}^{11}$	${\bf{0.60}}(\pm {\bf{0.15}})$	${\bf{0.986}}$
060607A	${\bf{1.67}}(\pm {\bf{1.01}})\times {{\bf{10}}}^{11}$	${\bf{0.92}}(\pm {\bf{0.24}})$	${\bf{0.991}}$
070318	unconstrained	$1.79(\pm 1.14)$	0.651
080607	${\bf{1.52}}(\pm {\bf{0.72}})\times {{\bf{10}}}^{12}$	${\bf{0.49}}(\pm {\bf{0.10}})$	${\bf{0.998}}$
080805	$1.12(\pm 1.34)\times {10}^{11}$	$1.31(\pm 0.59)$	0.809
080810	${\bf{2.34}}(\pm {\bf{4.84}})\times {{\bf{10}}}^{11}$	${\bf{0.61}}(\pm {\bf{0.57}})$	${\bf{0.999}}$
081008	${\bf{1.84}}(\pm {\bf{0.68}})\times {{\bf{10}}}^{12}$	${\bf{0.32}}(\pm {\bf{0.03}})$	${\bf{0.999}}$
081210	unconstrained	$0.80(\pm 0.51)$	0.295
090516A	unconstrained	$1.30(\pm 1.30)$	0.663
090812	$1.66(\pm 1.84)\times {10}^{12}$	$0.24(\pm 0.12)$	0.503
131030A	${\bf{3.67}}(\pm {\bf{1.02}})\times {{\bf{10}}}^{12}$	${\bf{0.55}}(\pm {\bf{0.06}})$	${\bf{0.999}}$
140206A	${\bf{9.02}}(\pm {\bf{2.84}})\times {{\bf{10}}}^{11}$	${\bf{0.54}}(\pm {\bf{0.07}})$	${\bf{0.999}}$
140301A	unconstrained	unconstrained	0.00
140419A	$1.85(\pm 1.17)\times {10}^{12}$	$0.23(\pm 0.05)$	0.88
141221A	$1.34(\pm 2.82)\times {10}^{12}$	$0.24(\pm 0.24)$	0.141
151027A	$1.18(\pm 0.67)\times {10}^{12}$	$0.29(\pm 0.06)$	0.941

Note. The observed temperatures ${{kT}}_{\mathrm{obs}}$ are inferred from fitting with a power-law plus blackbody spectral model. The significance of a blackbody is computed by the maximum likelihood ratio for comparing nested models and its addition improves a fit when the significance is $\gt 0.95$. The radii are calculated assuming mildly relativistic motion ($\beta =0.8$) and isotropic radiation. The GRBs listed in boldface have prominent blackbodies, with radii of the order of ∼10¹¹–10¹² cm. Uncertainties are given at the 1σ confidence level.

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We now attempt to perform a more refined analysis to infer the value of β from the observations. We assume that during the flare, the blackbody emitter has spherical symmetry and expands with a constant Lorentz Gamma factor. Therefore, the expansion velocity β is also constant during the flare. The relations between the comoving time t_com, the laboratory time t, the arrival time t_a, and the arrival time t_a^d at the detector, given in Equation (2), in this case become

$$\begin{eqnarray}{t}_{a}^{d} & = & {t}_{a}(1+z)=t(1-\beta \cos \vartheta )(1+z)\\ & = & {\rm{\Gamma }}{t}_{\mathrm{com}}(1-\beta \cos \vartheta )(1+z).\end{eqnarray} \tag{ 7 }$$

We can infer an effective radius R of the blackbody emitter from (1) the observed blackbody temperature T_obs, which comes from the spectral fit of the data during the flare; (2) the observed bolometric blackbody flux ${F}_{\mathrm{bb},\mathrm{obs}}$, computed from T_obs and the normalization of the blackbody spectral fit; and (3) the cosmological redshift z of the source (see also Izzo et al. 2012). We recall that ${F}_{\mathrm{bb},\mathrm{obs}}$ by definition is given by

$$\begin{eqnarray}&&{F}_{\mathrm{bb},\mathrm{obs}}=\displaystyle \frac{L}{4\pi {D}_{L}{(z)}^{2}},\end{eqnarray} \tag{ 8 }$$

where D_L(z) is the luminosity distance of the source, which in turn is a function of the cosmological redshift z, and L is the source bolometric luminosity (i.e., the total emitted energy per unit time). L is Lorentz invariant, so we can compute it in the comoving frame of the emitter using the usual blackbody expression,

$$\begin{eqnarray}&&L=4\pi {{R}_{\mathrm{com}}}^{2}\sigma {{T}_{\mathrm{com}}}^{4},\end{eqnarray} \tag{ 9 }$$

where R_com and T_com are the comoving radius and the comoving temperature of the emitter, respectively, and σ is the Stefan–Boltzmann constant. We recall that T_com is constant over the entire shell due to our assumption of spherical symmetry. From Equations (8) and (9), we then have

$$\begin{eqnarray}&&{F}_{\mathrm{bb},\mathrm{obs}}=\displaystyle \frac{{{R}_{\mathrm{com}}}^{2}\sigma {{T}_{\mathrm{com}}}^{4}}{{D}_{L}{(z)}^{2}}.\end{eqnarray} \tag{ 10 }$$

We now need the relation between T_com and the observed blackbody temperature T_obs. Considering both the cosmological redshift and the Doppler effect due to the velocity of the emitting surface, we have

$$\begin{eqnarray}{T}_{\mathrm{obs}}({T}_{\mathrm{com}},z,{\rm{\Gamma }},\cos \vartheta ) & = & \displaystyle \frac{{T}_{\mathrm{com}}}{(1+z){\rm{\Gamma }}(1-\beta \cos \vartheta )}\\ & = & \displaystyle \frac{{T}_{\mathrm{com}}{ \mathcal D }(\cos \vartheta )}{1+z},\end{eqnarray} \tag{ 11 }$$

where we have defined the Doppler factor ${ \mathcal D }(\cos \vartheta )$ as

$$\begin{eqnarray}&&{ \mathcal D }(\cos \vartheta )\equiv \displaystyle \frac{1}{{\rm{\Gamma }}(1-\beta \cos \vartheta )}.\end{eqnarray} \tag{ 12 }$$

Equation (11) gives us the observed blackbody temperature of the radiation coming from different points of the emitter surface, corresponding to different values of $\cos \vartheta$. However, since the emitter is at a cosmological distance, we are not able to resolve spatially the source with our detectors. Therefore, the temperature that we actually observe corresponds to an average of Equation (11) computed over the emitter surface:¹⁶

$$\begin{eqnarray}{T}_{\mathrm{obs}}({T}_{\mathrm{com}},z,{\rm{\Gamma }}) & = & \,\displaystyle \frac{1}{1+z}\displaystyle \frac{{\displaystyle \int }_{\beta }^{1}{ \mathcal D }(\cos \vartheta ){T}_{\mathrm{com}}\cos \vartheta d\cos \vartheta }{{\displaystyle \int }_{\beta }^{1}\cos \vartheta d\cos \vartheta }\\ & = & \,\displaystyle \frac{2}{1+z}\displaystyle \frac{\beta (\beta -1)+\mathrm{ln}(1+\beta )}{{\rm{\Gamma }}{\beta }^{2}(1-{\beta }^{2})}{T}_{\mathrm{com}}\\ & = & \,{\rm{\Theta }}(\beta )\displaystyle \frac{{\rm{\Gamma }}}{1+z}{T}_{\mathrm{com}},\end{eqnarray} \tag{ 13 }$$

where we defined

$$\begin{eqnarray}&&{\rm{\Theta }}(\beta )\equiv 2\,\displaystyle \frac{\beta (\beta -1)+\mathrm{ln}(1+\beta )}{{\beta }^{2}}.\end{eqnarray} \tag{ 14 }$$

We have used the fact that due to relativistic beaming, we observe only a portion of the surface of the emitter defined by

$$\begin{eqnarray}&&\beta \leqslant \cos \vartheta \leqslant 1,\end{eqnarray} \tag{ 15 }$$

and we used the definition of Γ given in Section 3. Therefore, inverting Equation (13), the comoving blackbody temperature T_com can be computed from the observed blackbody temperature T_obs, the source cosmological redshift z. and the emitter Lorentz Gamma factor as follows:

$$\begin{eqnarray}&&{T}_{\mathrm{com}}({T}_{\mathrm{obs}},z,{\rm{\Gamma }})=\displaystyle \frac{1+z}{{\rm{\Theta }}(\beta ){\rm{\Gamma }}}{T}_{\mathrm{obs}}.\end{eqnarray} \tag{ 16 }$$

We can now insert Equation (16) into Equation (10) to obtain

$$\begin{eqnarray}&&{F}_{\mathrm{bb},\mathrm{obs}}=\displaystyle \frac{{{R}_{\mathrm{com}}}^{2}}{{D}_{L}{(z)}^{2}}\sigma {T}_{\mathrm{com}}^{4}=\displaystyle \frac{{{R}_{\mathrm{com}}}^{2}}{{D}_{L}{(z)}^{2}}\sigma {\left[\displaystyle \frac{1+z}{{\rm{\Theta }}(\beta ){\rm{\Gamma }}}{T}_{\mathrm{obs}}\right]}^{4}.\end{eqnarray} \tag{ 17 }$$

Since the radius R_lab of the emitter in the laboratory frame is related to R_com by

$$\begin{eqnarray}&&{R}_{\mathrm{com}}={\rm{\Gamma }}{R}_{\mathrm{lab}},\end{eqnarray} \tag{ 18 }$$

we can insert Equation (18) into Equation (17) and obtain

$$\begin{eqnarray}&&{F}_{\mathrm{bb},\mathrm{obs}}=\displaystyle \frac{{(1+z)}^{4}}{{{\rm{\Gamma }}}^{2}}{\left(\displaystyle \frac{{R}_{\mathrm{lab}}}{{D}_{L}(z)}\right)}^{2}\sigma {\left[\displaystyle \frac{{T}_{\mathrm{obs}}}{{\rm{\Theta }}(\beta )}\right]}^{4}.\end{eqnarray} \tag{ 19 }$$

Solving Equation (19) for R_lab, we finally obtain the thermal emitter's effective radius in the laboratory frame:

$$\begin{eqnarray}&&{R}_{\mathrm{lab}}={\rm{\Theta }}{(\beta )}^{2}{\rm{\Gamma }}\displaystyle \frac{{D}_{L}(z)}{{(1+z)}^{2}}\sqrt{\displaystyle \frac{{F}_{\mathrm{bb},\mathrm{obs}}}{\sigma {T}_{\mathrm{obs}}^{4}}}={\rm{\Theta }}{(\beta )}^{2}{\rm{\Gamma }}{\phi }_{0},\end{eqnarray} \tag{ 20 }$$

where we have defined ${\phi }_{0}$,

$$\begin{eqnarray}&&{\phi }_{0}\equiv \displaystyle \frac{{D}_{L}(z)}{{(1+z)}^{2}}\sqrt{\displaystyle \frac{{F}_{\mathrm{bb},\mathrm{obs}}}{\sigma {T}_{\mathrm{obs}}^{4}}}.\end{eqnarray} \tag{ 21 }$$

In astronomy, the quantity ${\phi }_{0}$ is usually identified with the radius of the emitter. However, in relativistic astrophysics, this identity cannot be straightforwardly applied, because the estimate of the effective emitter radius R_lab in Equation (20) crucially depends on the knowledge of its expansion velocity β (and, correspondingly, of Γ).

It must be noted that Equation (20) above gives the correct value of R_lab for all values of $0\leqslant \beta \leqslant 1$ by taking all of the relativistic transformations properly into account. In the non-relativistic limit ($\beta \to 0$, ${\rm{\Gamma }}\to 1$), we have, respectively:

$$\begin{eqnarray}&&\mathop{{\rm{\Theta }}\longrightarrow 1}\limits_{\beta \to 0},\quad \quad \mathop{{{\rm{\Theta }}}^{2}\longrightarrow 1}\limits_{\beta \to 0},\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&\mathop{{T}_{\mathrm{com}}\longrightarrow {T}_{\mathrm{obs}}(1+z)}\limits_{\beta \to 0},\quad \quad \mathop{{R}_{\mathrm{lab}}\longrightarrow {\phi }_{0}}\limits_{\beta \to 0},\end{eqnarray} \tag{ 23 }$$

as expected.

An estimate of the expansion velocity β can be deduced from the ratio between the variation of the emitter effective radius ${\rm{\Delta }}{R}_{\mathrm{lab}}$ and the emission duration in laboratory frame ${\rm{\Delta }}t$, i.e.,

$$\begin{eqnarray}&&\beta =\displaystyle \frac{{\rm{\Delta }}{R}_{\mathrm{lab}}}{c{\rm{\Delta }}t}={\rm{\Theta }}{(\beta )}^{2}{\rm{\Gamma }}(1-\beta \cos \vartheta )(1+z)\displaystyle \frac{{\rm{\Delta }}{\phi }_{0}}{c{\rm{\Delta }}{t}_{a}^{d}},\end{eqnarray} \tag{ 24 }$$

where we have used Equation (20) and the relation between ${\rm{\Delta }}t$ and ${\rm{\Delta }}{t}_{a}^{d}$ given in Equation (7). We then have

$$\begin{eqnarray}&&\beta ={\rm{\Theta }}{(\beta )}^{2}\displaystyle \frac{1-\beta \cos \vartheta }{\sqrt{1-{\beta }^{2}}}(1+z)\displaystyle \frac{{\rm{\Delta }}{\phi }_{0}}{c{\rm{\Delta }}{t}_{a}^{d}},\end{eqnarray} \tag{ 25 }$$

where we used the definition of Γ given in Section 3.

For example, in GRB 081008, we observe a temperature of ${T}_{\mathrm{obs}}=(0.44\pm 0.12)$ keV between ${t}_{a}^{d}=280\,{\rm{s}}$ and ${t}_{a}^{d}=300\,{\rm{s}}$ (i.e., 20 s before the flare peak time), and a temperature of ${T}_{\mathrm{obs}}=(0.31\pm 0.05)$ keV between ${t}_{a}^{d}=300\,{\rm{s}}$ and ${t}_{a}^{d}=320\,{\rm{s}}$ (i.e., 20 s after the flare peak time, see the corresponding spectra in Figure 32). In these two time intervals, we can infer ${\phi }_{0}$, and by solving Equation (25) and taking the errors of the parameters properly into account, get the value of $\langle \beta \rangle$ corresponding to the average expansion speed of the emitter from the beginning of its expansion up to the upper bound of the time interval considered. The results so obtained are listed in Table 8. Moreover, we can also compute the value of $\langle \beta \rangle$ between the two time intervals considered above. For $\cos \vartheta =1$, namely along the line of sight, we obtain $\langle \beta \rangle ={0.90}_{-0.31}^{+0.06}$ and $\langle {\rm{\Gamma }}\rangle ={2.34}_{-1.10}^{+1.29}$. In conclusion, no matter what the details of the approximation adopted, the Lorentz Gamma factor is always moderate, i.e., ${\rm{\Gamma }}\lesssim 4$.

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Table 8.  List of the Physical Quantities Inferred from the Thermal Components Observed During the Flare of GRB 081008

Time Interval	$280\,{\rm{s}}\leqslant {t}_{a}^{d}\leqslant 300\,{\rm{s}}$	$300\,{\rm{s}}\leqslant {t}_{a}^{d}\leqslant 320\,{\rm{s}}$
Tobs (keV)	0.44 ± 0.12	0.31 ± 0.05
${\phi }_{0}$ (cm)	$(5.6\pm 3.2)\times {10}^{11}$	$(1.44\pm 0.48)\times {10}^{12}$
$\langle \beta {\rangle }_{(\cos \vartheta =1)}$	${0.19}_{-0.11}^{+0.10}$	${0.42}_{-0.12}^{+0.10}$
$\langle {\rm{\Gamma }}\rangle$	${1.02}_{-0.02}^{+0.03}$	${1.10}_{-0.05}^{+0.07}$
Rlab (cm)	$(7.1\pm 4.1)\times {10}^{11}$	$(2.34\pm 0.78)\times {10}^{12}$

Note. For each time interval, we summarize the observed temperature T_obs, ${\phi }_{0}$, the average expansion speed $\langle \beta \rangle$ computed from the beginning up to the upper bound of the considered time interval, and the corresponding average Lorentz factor $\langle {\rm{\Gamma }}\rangle$ and laboratory radius R_lab.

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10.1. Necessity for a New Hydrodynamic Code for $10\leqslant B\leqslant {10}^{2}$

As stated above, there are many different components of BdHNe: following episode 1 of the hypercritical accretion of the SN ejecta onto the NS, the prompt emission occurs with ${\rm{\Gamma }}\approx {10}^{2}\mbox{--}{10}^{3}$, which represents the most energetic component accelerated by the ${e}^{+}{e}^{-}$ plasma; a third component, which encompasses the X-ray flare with ${\rm{\Gamma }}\lesssim 4$ and represents only a fraction of ${E}_{{e}^{+}{e}^{-}}$ ranging from 2% to 20% (see Figure 31); finally, there are in addition the gamma-ray flare and the late X-ray flares, which will be addressed in a forthcoming publication, as well as the late afterglow phases, which have been already addressed in Pisani et al. (2013, 2016) but whose dynamics will be discussed elsewhere. As already mentioned, for definiteness, we address here the case of X-ray flares.

In Section 3.5, we showed that our model successfully explains the entire prompt emission as originating from the transparency of an initially optically thick ${e}^{+}{e}^{-}$ plasma with a baryon load $B\lt {10}^{-2}$ reaching ${\rm{\Gamma }}\approx {10}^{2}$–10³ and the accelerated baryons interacting with the clouds of the CBM. The fundamental equations describing the dynamics of the optically thick plasma, its self-acceleration to ultrarelativistic velocities, and its interaction with the baryon load have been described in Ruffini et al. (1999, 2000). A semi-analytic approximate numerical code was developed, which assumed that the plasma expanded as a shell with a constant thickness in the laboratory frame (the so-called "slab" approximation; see Ruffini et al. 1999). This semi-analytic approximate code was validated by comparing its results with the ones obtained by numerically integrating the complete system of equations for selected values of the initial conditions. It turns out that the semi-analytic code is an excellent approximation to the complete system of equations for $B\lt {10}^{-2}$, which is the relevant regime for the prompt emission, but this approximation is not valid beyond this limit (see Ruffini et al. 1999, 2000 for details).

We examine here the possibility that the energy of the X-ray flare component also originates from a fraction of the ${e}^{+}{e}^{-}$ plasma energy (see Figure 31) interacting with the much denser medium of the SN ejecta with $10\lesssim B\lesssim {10}^{2}$. The above-mentioned semi-analytic approximate code cannot be used for this purpose, since it is valid only for $B\lt {10}^{-2}$, and therefore, thanks to the more powerful computers we have at present, we move on here to a new numerical code to integrate the complete system of equations.

We investigate if indeed the dynamics to be expected from an initially pure ${e}^{+}{e}^{-}$ plasma with a negligible baryon load relativistically expanding in the fireshell model, with an initial Lorentz factor ${\rm{\Gamma }}\sim 100$, and then impacting such an SN ejecta can lead, reaching transparency, to the Lorentz factor ${\rm{\Gamma }}\lesssim 4$ inferred from the thermal emission observed in the flares (see Tables 7 and 8, and Figure 32).

We have performed hydrodynamical simulations of such a process using the one-dimensional relativistic hydrodynamical (RHD) module included in the freely available PLUTO¹⁷ code (Mignone et al. 2011). In the spherically symmetric case considered here, only the radial coordinate is used and the code integrates partial differential equations with two variables: radius and time. This permits the study of the evolution of the plasma along one selected radial direction at a time. The code integrates the equations of an ideal relativistic fluid in the absence of gravity, which can be written as follows:

$$\begin{eqnarray}&&\displaystyle \frac{\partial (\rho {\rm{\Gamma }})}{\partial t}+{\rm{\nabla }}.(\rho {\rm{\Gamma }}{\boldsymbol{v}})=0,\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {m}_{r}}{\partial t}+{\rm{\nabla }}.({m}_{r}{\boldsymbol{v}})+\displaystyle \frac{\partial p}{\partial r}=0,\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial { \mathcal E }}{\partial t}+{\rm{\nabla }}.({\boldsymbol{m}}-\rho {\rm{\Gamma }}{\boldsymbol{v}})=0,\end{eqnarray} \tag{ 28 }$$

where ρ and p are, respectively, the comoving fluid density and pressure, ${\boldsymbol{v}}$ is the coordinate velocity in natural units (c = 1), ${\rm{\Gamma }}={(1-{{\boldsymbol{v}}}^{2})}^{-\tfrac{1}{2}}$ is the Lorentz Gamma factor, ${\boldsymbol{m}}=h{{\rm{\Gamma }}}^{2}{\boldsymbol{v}}$ is the fluid momentum, m_r its radial component, ${ \mathcal E }$ is the internal energy density, and h is the comoving enthalpy density, which is defined by $h=\rho +\epsilon +p$. In this last definition, is equal to ${ \mathcal E }$ measured in the comoving frame. We define ${ \mathcal E }$ as follows:

$$\begin{eqnarray}&&{ \mathcal E }=h{{\rm{\Gamma }}}^{2}-p-\rho {\rm{\Gamma }}.\end{eqnarray} \tag{ 29 }$$

The first two terms on the right-hand side of this equation coincide with the T⁰⁰ component of the fluid energy–momentum tensor ${T}^{\mu \nu }$, and the last one is the mass density in the laboratory frame.

Under the conditions discussed in Appendix B, the plasma satisfies the equation of state of an ideal relativistic gas, which can be expressed in terms of its enthalpy as

$$\begin{eqnarray}&&h=\rho +\displaystyle \frac{\gamma p}{\gamma -1},\end{eqnarray} \tag{ 30 }$$

with $\gamma =4/3$. Fixing this equation of state completely defines the system, leaving the choice of the boundary conditions as the only remaining freedom. To compute the evolution of these quantities in the chosen setup, the code uses the Harten–Lax–van Leer-contact Riemann solver. Time integration is performed by means of a second-order Runge–Kutta algorithm, and a second-order total variation diminishing scheme is used for spatial reconstruction (Mignone et al. 2011). Before each integration step, the grid is updated according to an adaptive mesh refinement algorithm, provided by the CHOMBO library (Colella et al. 2003).

It must be emphasized that the above equations are equivalent (although written in a different form) to the complete system of equations used in Ruffini et al. (1999, 2000). To validate this new numerical code, we compare its results with the ones obtained with the old semi-analytic "slab" approximate code in the domain of its validity (i.e., for $B\lt {10}^{-2}$), finding excellent agreement. As an example, in Figure 33 we show the comparison between the Lorentz Gamma factors computed with the two codes for one particular value of ${E}_{{e}^{+}{e}^{-}}$ and B.

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We can then conclude that for $B\lt {10}^{-2}$, the new RHD code is consistent with the old semi-analytic "slab" approximate one, which in turn is consistent with the treatment done in Ruffini et al. (1999, 2000). This is not surprising, since we already stated that the above system of equations is equivalent to the one considered in Ruffini et al. (1999, 2000).

Having validated the new RHD code in the region of parameter space where the old semi-analytic one can also be used, we now explore the region of $B\gt {10}^{-2}$, which is relevant for the interaction of the plasma with the SN ejecta.

10.2. Inference from the IGC Scenario for the Ejecta Mass Profile

10.3. The Density Profile of the Ejecta and the Reaching of Transparency

We now use the results of a simulation with the following binary parameters: the NS has an initial mass of $2.0\,{M}_{\odot };$ the ${\mathrm{CO}}_{\mathrm{core}}$ obtained from a progenitor with a zero-age main-sequence mass ${M}_{\mathrm{ZAMS}}=30\,{M}_{\odot }$ leads to a total ejecta mass of $7.94\,{M}_{\odot }$ and follows an approximate power-law profile ${\rho }_{\mathrm{ej}}^{0}\approx 3.1\times {10}^{8}{(8.3\times {10}^{7}/r)}^{2.8}$ g cm⁻³. The orbital period is $P\approx 5\,\mathrm{minutes}$, i.e., a binary separation $a\approx 1.5\times {10}^{10}$ cm. For these parameters, the NS reaches the critical mass and collapses to form a BH.

Figure 34 shows the SN ejecta mass that is enclosed within a cone of 5° of the semi-aperture angle, whose vertex is at the position of the BH at the moment of its formation (see the lower-left panel of Figure 6 in Becerra et al. 2016), and whose axis is along various directions measured counterclockwise with respect to the line of sight. Figure 35 shows instead the cumulative radial mass profiles within a selected number of the aforementioned cones. We can see from these plots how the ${e}^{+}{e}^{-}$ plasma engulfs different amounts of baryonic mass along different directions due to the asymmetry of the SN ejecta created by the presence of the NS binary companion and the accretion process onto it (see Becerra et al. 2016).

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In these calculations, we have chosen initial conditions consistent with those of the BdHNe. At the initial time, the ${e}^{+}{e}^{-}$ plasma has ${E}_{{e}^{+}{e}^{-}}=3.16\times {10}^{53}$ erg, a negligible baryon load, and is distributed homogeneously within a region of radii on the order of 10⁸–10⁹ cm. The surrounding SN ejecta, whose pressure has been assumed to be negligible, has a mass density radial profile given by

$$\begin{eqnarray}&&\rho \propto {({R}_{0}-r)}^{\alpha },\end{eqnarray} \tag{ 31 }$$

where the parameters R₀ and α, with $2\lt \alpha \lt 3$, as well as the normalization constant, are chosen to fit the profiles obtained in Becerra et al. (2016) and represented in Figure 35. The initial radial velocity is taken to be ${v}_{r}\propto r$ in order to reproduce the homologous expansion of the SN ejecta before its interaction with the plasma. Every choice of these parameters corresponds to studying the evolution along a single given direction.

The evolution from these initial conditions leads to the formation of a shock and to its subsequent expansion until reaching the outermost part of the SN. In Figure 36, we show the radial distribution profiles of the velocity and mass density ${\rho }_{\mathrm{lab}}$ in the laboratory frame inside the SN ejecta as a function of r for B = 200 at two selected values of the laboratory time. The velocity distribution peaks at the shock front (with a Lorentz Gamma factor ${\rm{\Gamma }}\lesssim 4$), and behind the front it forms a broad tail of accelerated material with $0.1\lesssim \beta \lesssim 1$.

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Figure 37 shows the Lorentz Γ factor at the transparency radius R_ph, namely the radius at which the optical depth τ, calculated from the observer's line of sight, is equal to 1. If we assume a constant cross-section, τ becomes Lorentz invariant, and therefore we can compute it in laboratory coordinates in the following way:

$$\begin{eqnarray}&&\tau ={\int }_{{R}_{\mathrm{ph}}}^{\infty }{dr}\,{\sigma }_{T}\,{n}_{{e}^{-}}(r),\end{eqnarray} \tag{ 32 }$$

where ${\sigma }_{T}=6.65\times {10}^{-25}$ cm² is the Thomson cross-section, and the electron density is related to the baryon mass density by means of the formula ${n}_{{e}^{-}}=\rho \,{\rm{\Gamma }}/{m}_{P}$, where m_P is the proton mass, the mass of the electrons and positrons is considered to be negligible with respect to that of the baryons, and we have assumed one electron per nucleon on average. The values of Γ at $r={R}_{\mathrm{ph}}$ computed in this way are shown in Figure 37, as a function of the time measured in the laboratory frame, for several values of $B\gt {10}^{-2}$ corresponding to the expansion of the ${e}^{+}{e}^{-}$ plasma along several different directions inside the SN ejecta (see Figures 34 and 35).

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We conclude that the relativistic expansion of an initially pure ${e}^{+}{e}^{-}$ plasma (see Figure 33), interacting with an SN ejecta with the above-described induced asymmetries (see Figures 39–40), leads to the formation of a shock that reaches the outermost part of the ejecta with Lorentz Gamma factors at the transparency radius ${\rm{\Gamma }}({R}_{\mathrm{ph}})\lesssim 4$. This is in striking agreement with the one inferred from the thermal component observed in the flares (see Section 9). The spacetime diagram of the global scenario is represented in Figure 39. Clearly in this approach neither ultrarelativistic jetted emission nor synchrotron or inverse-Compton processes play any role.

11.1. Summary

11.2. Conclusions

11.3. Perspectives

Far from representing solely a criticism of the traditional approach, in this paper, (1) we exemplify new procedures in data analysis—see Sections 4 to 7, (2) we open up the topic to an alternative style of conceptual analysis which adopts procedures well-tested in high-energy physics and not yet appreciated in the astrophysical community—see Sections 8–10, and (3) we introduce new tools for simulation techniques affordable with present-day large computer facilities—see figures in Section 11, which, if properly guided by a correct theoretical understanding, can be particularly helpful in the visualization of these phenomena.

We give three specific examples of our new approach and indicate as well, when necessary, some disagreements with current approaches:

• (A)  
  The first step in any research on GRBs is to represent the histogram of T₉₀ for the GRB subclasses. We report in Figure 38 the T₉₀ values for all of the GRB subclasses we have introduced (see Ruffini et al. 2016b). The values reported are both in the observer frame (left panel; see, e.g., Kouveliotou et al. 1993; Bromberg et al. 2013) and properly converted to the cosmological rest frame of the sources (right panel). The large majority of papers on GRBs have been neglecting the cosmological corrections and subdivision in the subclasses, making impossible the comparison of T₉₀ among different GRBs (see, e.g., Falcone et al. 2007; Chincarini et al. 2010).
• (B)  
  For the first time, we present a simplified spacetime diagram of BdHNe (see Figure 39). This spacetime diagram emphasizes the many different emission episodes, each one with distinct corresponding Lorentz Gamma factors and consequently leading through Equation (2) to a specific value of their distinct times of occurrence in the cosmological rest frame of the GRB (see Figure 39). In all episodes we analyzed for the X-ray flares, and more generally for the entire FPA phase, there is no need for collapsar-related concepts. Nevertheless, in view of the richness of the new scenario in Figure 39, we have been examining the possibility that such concepts can play a role in additional episodes, either in BdHNe or in any of the additional six GRB subclasses, e.g., in S-GRBs. These results are being submitted for publication. The use of spacetime diagrams in the description of GRBs is indeed essential in order to illustrate the causal relation between the source in each episode, the place of occurrence, and the time at detection. Those procedures have been introduced long ago in the study of high-energy particle physics processes and codified in textbooks. Our group, since the basic papers (Ruffini et al. 2001a, 2001b, 2001c), has widely shared these spacetime formulations (see, e.g., in Taylor & Wheeler 1992) and also extended the concept of the quantum S-Matrix (Wheeler 1937; Heisenberg 1943) to the classic astrophysical regime of the many components of a BdHN, introducing the concept of the cosmic matrix (Ruffini et al. 2015c). The majority of astrophysicists today make wide use of the results of nuclear physics in the study of stellar evolution (Bethe 1991) and also of Fermi statistics in general relativity (Oppenheimer & Volkoff 1939). They have not yet been ready, however, to approach these additional concepts more typical of relativistic astrophysics and relativistic field theories, which are necessary for the study of GRBs and active galactic nuclei.
• (C)  
  The visual representation of our result (see Figure 40) has been made possible thanks to the simulations of SN explosions with the core-collapse SN code developed at Los Alamos (see, e.g., Fryer et al. 1999a, 2014; Frey et al. 2013), the smoothed-particle-hydrodynamics-like simulations of the evolution of the SN ejecta accounting for the presence of an NS companion (Ruffini et al. 2016b), and the possibility of varying the parameters of the NS, of the SN, and of the distance between the two to explore all possibilities (Becerra et al. 2015; Ruffini et al. 2016b). We recall that these signals occur in each galaxy every ∼hundred million years, but with their luminosity of $\sim {10}^{54}$ erg, they can be detected in all 10⁹ galaxies. The product of these two factors gives the "once per day" rate. They are not visualizable in any other way, but analyzing the spectra and time of arrival of the photons now, and simulating these data on the computer, we see that they indeed already occurred billions of years ago in our past light cone, and they are revived by scientific procedures today.

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We are happy to acknowledge fruitful discussions with Fulvio Melia, Tsvi Piran, and Bing Zhang in our attempt to make these new results clearer for a broader audience, and with Roy Kerr on the implication of the GRB GeV emission for the Kerr–Newman solution. We also thank the second referee for important suggestions. We acknowledge the continuous support of the MAECI. This work made use of data supplied by the UK Swift Science Data Center at the University of Leicester. M.Ka. and Y.A. are supported by the Erasmus Mundus Joint Doctorate Program Grant N. 2014–0707 from EACEA of the European Commission. M.M. and J.A.R. acknowledge the partial support of the project N 3101/GF4 IPC-11 and the target program F.0679 0073-6/PTsF of the Ministry of Education and Science of the Republic of Kazakhstan.

We present here in Table 9 the complete list of the 345 BdHNe observed through the end of 2016, which includes the 161 BdHNe already presented in Pisani et al. (2016).

Table 9.  List of the BdHNe Considered in This Work

GRB	z	${E}_{\mathrm{iso}}$ a	LXb	Early Flarec	ULd	${T}_{90}$ e	Instrumentf	Referenceg
970228	0.695	1.65 ± 0.16				80	B-SAX	(1)
970828	0.958	30.4 ± 3.6				90	BATSE	(2)
971214	3.42	22.1 ± 2.7				40	BATSE	IAUC 6789
980329	3.5	267 ± 53				54	B-SAX	(3)
980703	0.966	7.42 ± 0.74				400	BATSE	GCN 143
990123	1.6	241 ± 39				63.3	BATSE	GCN 224
990506	1.3	98.1 ± 9.9				131.33	BATSE	GCN 306
990510	1.619	18.1 ± 2.7				75	BATSE	GCN 322
990705	0.842	18.7 ± 2.7				42	B-SAX	(4)
991208	0.706	23.0 ± 2.3				68	Ulysses	(5)

Notes. It is composed of 345 sources spanning 12 years of Swift/XRT observation activity. In the table, we report important observational features: the redshift z, the isotropic energy E_iso, the observing instrument in the gamma-ray band, and the corresponding reference from which we take the gamma-ray spectral parameters in order to estimate E_iso.

^aIn units of 10⁵² erg. ^b"LX" marks the sources with Swift/XRT data observed up to times larger than 10⁴ s in the rest frame after the initial explosion. ^c"C" and "E" mark the sources showing an early flare in Swift/XRT, and they stand for "confirmed" and "excluded," respectively. The 16 "C" sources compose the sample considered in the present paper. ^d"UL" stands for ultralong, indicating sources with ${T}_{90}\gtrsim 1000\,{\rm{s}}$. ^eObserved T₉₀ (s). ^f"B-SAX" stands for BeppoSAX/GRBM; "BATSE" stands for Compton-GRO/BATSE; "Ulysses" stands for Ulysses/GRB; "KW" stands for Konus/WIND; "HETE" stands for HETE-2/FREGATE; "Swift" standss for Swift/BAT; "Fermi" stands for Fermi/GBM. ^g(1) Frontera et al. (1998), (2) Ruffini et al. (2015a), (3) in 't Zand et al. (1998), (4) Amati et al. (2000), (5) Hurley et al. (2000), (6) in't Zand et al. (2001), (7) Barraud et al. (2003), (8) Shirasaki et al. (2008), (9) Cenko et al. (2006).

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We give here details concerning the determination of the value of the index γ and verify the accuracy of our assumption $\gamma =4/3$ adopted in the equation of state of the plasma (30). This index is defined as

$$\begin{eqnarray}&&\gamma \equiv 1+\displaystyle \frac{p}{\epsilon }.\end{eqnarray} \tag{ 33 }$$

The total internal energy density and pressure are computed as

$$\begin{eqnarray}&&\epsilon ={\epsilon }_{{e}^{-}}+{\epsilon }_{{e}^{+}}+{\epsilon }_{\gamma }+{\epsilon }_{B}\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&p={p}_{{e}^{-}}+{p}_{{e}^{+}}+{p}_{\gamma }+{p}_{B},\end{eqnarray} \tag{ 35 }$$

where the subscript B indicates the contributions of the baryons in the fluid. The number and energy densities, as well as the pressure of the different particles, can be computed in natural units ($c={\hslash }={k}_{B}=1$) using the following expressions (see, e.g., Landau & Lifshitz 1980):

$$\begin{eqnarray}&&{n}_{{e}^{-}}=A\,{T}^{3}{\int }_{0}^{\infty }f(z,T,{m}_{e},{\mu }_{{e}^{-}}){z}^{2}\,{dz}\end{eqnarray} \tag{ 36 }$$

$$\begin{eqnarray}&&{n}_{{e}^{+}}=A\,{T}^{3}{\int }_{0}^{\infty }f(z,T,{m}_{e},{\mu }_{{e}^{+}}){z}^{2}\,{dz}\end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray}{\epsilon }_{{e}^{-}} & = & A\,{T}^{4}{\displaystyle \int }_{0}^{\infty }f(z,T,{m}_{e},{\mu }_{{e}^{-}})\\ & & \times \sqrt{{z}^{2}+{({m}_{e}/T)}^{2}}\,{z}^{2}\,{dz}\,-\,{m}_{e}\,{n}_{{e}^{-}}\end{eqnarray} \tag{ 38 }$$

$$\begin{eqnarray}{\epsilon }_{{e}^{+}} & = & A\,{T}^{4}{\displaystyle \int }_{0}^{\infty }f(z,T,{m}_{e},{\mu }_{{e}^{+}})\\ & & \times \sqrt{{z}^{2}+{({m}_{e}/T)}^{2}}\,{z}^{2}\,{dz}\,-\,{m}_{e}\,{n}_{{e}^{+}}\end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray}&&{p}_{{e}^{-}}=A\,\displaystyle \frac{{T}^{4}}{3}{\int }_{0}^{\infty }f(z,T,{m}_{e},{\mu }_{{e}^{-}})\displaystyle \frac{{z}^{4}}{\sqrt{{z}^{2}+{({m}_{e}/T)}^{2}}}\,{dz}\end{eqnarray} \tag{ 40 }$$

$$\begin{eqnarray}&&{p}_{{e}^{+}}=A\,\displaystyle \frac{{T}^{4}}{3}{\int }_{0}^{\infty }f(z,T,{m}_{e},{\mu }_{{e}^{+}})\displaystyle \frac{{z}^{4}}{\sqrt{{z}^{2}+{({m}_{e}/T)}^{2}}}\,{dz}\end{eqnarray} \tag{ 41 }$$

$$\begin{eqnarray}&&{\epsilon }_{\gamma }=a\,{T}^{4}\end{eqnarray} \tag{ 42 }$$

$$\begin{eqnarray}&&{p}_{\gamma }=\displaystyle \frac{a\,{T}^{4}}{3}\end{eqnarray} \tag{ 43 }$$

$$\begin{eqnarray}&&{\epsilon }_{B}=\displaystyle \frac{3}{2}{n}_{N}\,T\end{eqnarray} \tag{ 44 }$$

$$\begin{eqnarray}&&{p}_{B}={n}_{N}\,T,\end{eqnarray} \tag{ 45 }$$

where

$$\begin{eqnarray}&&f(z,T,m,\mu )=\displaystyle \frac{1}{{e}^{\sqrt{{z}^{2}+{(m/T)}^{2}}-\mu /T}+1}\end{eqnarray} \tag{ 46 }$$

is the Fermi–Dirac distribution, m_e is the electron mass, n_N the nuclei number density, $a=8{\pi }^{5}{k}_{B}^{4}/15{h}^{3}{c}^{3}=7.5657\,\times {10}^{-15}$ erg cm⁻³ K⁻⁴ the radiation constant, and $A=15a/{\pi }^{4}$. If the pair annihilation rate is zero, i.e., if the reaction ${e}^{-}+{e}^{+}\leftrightarrows 2\gamma$ is in equilibrium, then the equality ${\mu }_{{e}^{-}}=-{\mu }_{{e}^{+}}\equiv \mu$ holds, since the equilibrium photons have zero chemical potential. Besides, charge neutrality implies that the difference in the number of electrons and positrons is equal to the number of protons in the baryonic matter, which can be expressed as

$$\begin{eqnarray}&&{n}_{{e}^{-}}(\mu ,T)-{n}_{{e}^{+}}(\mu ,T)=Z\,{n}_{B},\end{eqnarray} \tag{ 47 }$$

where n_B is the baryon number density and $1/2\lt Z\lt 1$ is the average number of electrons per nucleon. The number density n_B is related to the other quantities as

$$\begin{eqnarray}&&\rho ={m}_{p}\,{n}_{B}+{m}_{e}({n}_{{e}^{-}}+{n}_{{e}^{+}}),\end{eqnarray} \tag{ 48 }$$

where m_p is the proton mass. If the baryons are only protons, then Z = 1 and n_N = n_B. Together with Equation (47), this completely defines the mass density as a function of (μ, T). The equation of state that relates the pressure with the mass and internal energy densities is thus defined implicitly as the parametric surface

$$\begin{eqnarray}&&\{(\rho (\mu ,T),\epsilon (\mu ,T),p(\mu ,T)):\,T\gt 0,\mu \geqslant 0\}\end{eqnarray} \tag{ 49 }$$

that satisfies all of the above relations.

In the cases relevant for the simulations performed in Section 10, we indeed have that the index γ in the equation of state of the plasma, Equation (30), satisfies $\gamma =4/3$ with a maximum error of 0.2%.