Magnetic Fields and Afterglows of BdHNe: Inferences from GRB 130427A, GRB 160509A, GRB 160625B, GRB 180728A, and GRB 190114C

We first briefly review the traditional afterglow models and the possible alternatives. This task has been facilitated by the publication of the comprehensive book by Zhang (2018). We focus on additional results introduced since then by the understanding of the X-ray flare (Ruffini et al. 2018e) and afterglow of GRB 130427A (Ruffini et al. 2018b) and GRB 190114C (Ruffini et al. 2019b, 2019a).

We first recall the well-known discoveries by the Beppo-SAX satellite:

• 1.  
  The discovery of the first afterglow in GRB 970228 (Costa et al. 1997);
• 2.  
  The consequent identification of the cosmological redshift of GRBs (GRB 970508; Metzger et al. 1997), which proved the cosmological nature of GRBs and their outstanding energetics; and
• 3.  
  The first clear coincidence of a long GRB with the onset of a supernova (GRB 980425/SN 1998bw, Galama et al. 1998).

Even before these discoveries, three contributors, based on first principles, formulated models for long GRBs by assuming their cosmological nature and their origination from black hole (BH) formation. At the time, these works expressed the point of view of a small minority. A parallel successful move was done by Paczynski and collaborators for short GRBs (Paczynski 1991, 1992; Narayan et al. 1992). The aforementioned three contributors are the following:

• 1.  
  Damour & Ruffini (1975) predicted that a vacuum polarization process occurring around an overcritical Kerr–Newman BH leads toward GRB energetics of up to 10⁵⁴ erg, linking their activities as well to the emergence of ultra-high-energy cosmic rays;
• 2.  
  Rees & Meszaros (1992) and Mészáros & Rees (1997) also proposed a BH as the origin of GRBs, but there, an ultrarelativistic blast wave, whose expansion follows the Blandford–McKee self-similar solution, was used to explain the prompt emission phase (Blandford & McKee 1976);
• 3.  
  Woosley (1993) linked the GRB's origin to a Kerr BH emitting an ultrarelativistic jet originating from the accretion of toroidal material onto the BH. There, the idea was presented that for long GRBs, the BH would be likely produced from the direct collapse of a massive star—a "failed" SN leading to a large BH of approximately 5M_⊙, possibly as high as 10M_⊙, a "collapsar."

1.1. Traditional Afterglow Model Originating from BH

1.2. Role of Magnetars and Spinning Neutron Stars

1.3. The Role of Binary Progenitors in GRBs

The BdHN model has been introduced to explain long-duration gamma-ray bursts (GRBs), and it is based on the induced gravitational collapse (IGC) paradigm (Rueda & Ruffini 2012) occurring in a specific binary system following a specific evolutionary path (see Figure 1 and Fryer et al. 2014; Becerra et al. 2015; Fryer et al. 2015; Rueda et al. 2019 for details).

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As Figure 1 shows, the system starts with a binary composed of two main-sequence stars, say of 15 and 12 M_⊙, respectively. At a given time, at the end of its thermonuclear evolution, the more massive star undergoes core-collapse SN and forms an NS. The system then enters the X-ray binary phase. After possibly multiple common-envelope phases and binary interactions (see Fryer et al. 2014, 2015, and references therein), the hydrogen and helium envelopes of the other main-sequence star are stripped, leaving exposed its core that is rich in carbon and oxygen. For short, we refer to it as the carbon–oxygen core (CO_core) following the literature on the subject (see e.g., Nomoto et al. 1994; Filippenko et al. 1995; Iwamoto et al. 2000; Pian et al. 2006; Yoshida & Umeda 2011). The system at this stage is a CO_core–NS binary in tight orbit (period of the order of a few minutes), which is taken as the initial configuration of the BdHN scenario in which the IGC phenomenon occurs (Fryer et al. 2014; Becerra et al. 2015, 2016, 2019).

We now proceed to describe the BdHN scenario. At the end of its thermonuclear evolution the CO_core undergoes a core-collapse SN (of type Ic in view of the hydrogen and helium absence). Matter is ejected but also a the center of the SN, a newborn NS is formed, for short referred to as νNS, to differentiate it from the accreting NS binary companion. As we shall see, this differentiation is necessary in light of the physical phenomena and corresponding observables in a BdHN associated with each of them. Owing to the short orbital period, the SN ejecta produce a hypercritical (i.e., highly super-Eddington) accretion process onto the NS companion. The material hits the NS surface developing and outward shock which creates an accretion "atmosphere" of very high density and temperature on top the NS. These conditions turn out to be appropriate for the thermal production of positron–electron (e⁺e⁻) pairs which, when annihilating, leads to a copious production of neutrino–antineutrino pairs ($\nu \bar{\nu }$), which turn out to be the most important carriers of the gravitational energy gain of the accreting matter, allowing the rapid and massive accretion to continue. We refer to Fryer et al. (2014) and Becerra et al. (2016, 2018) for details on the hypercritical accretion and the involved neutrino physics.

Depending on the specific system parameters, i.e., mass of the binary components, orbital period, SN explosion energy, etc., two possible fates for the NS are possible (see Becerra et al. 2015, 2016, 2019 for details on the relative influence of each parameter in the system). For short binary periods, i.e., ∼5 minutes, the NS reaches the critical mass for gravitational collapse and forms a BH (see, e.g., Becerra et al. 2015, 2016, 2019; Fryer et al. 2015). We have called this kind of system a BdHN type I (Wang et al. 2019b). A BdHN I emits an isotropic energy E_iso ≳ 10⁵² erg and gives rise to a new binary composed of the NS formed at the center of the SN, hereafter νNS, and the BH formed by the collapse of the NS. For longer binary periods, the hypercritical accretion onto the NS is not sufficient to bring it to the critical mass, and a more massive NS (MNS) is formed. We have called these systems BdHNe of type II (Wang et al. 2019b), and they emit energies E_iso ≲ 10⁵² erg. A BdHN II gives origin to a new binary composed of the νNS and the MNS.

The BdHNe I represent, in our binary classification of GRBs, the totality of long GRBs with energy larger than 10⁵² erg, while BdHNe II, with their energy smaller than 10⁵² erg, are far from unique, and there is a variety of long GRBs in addition to them that can have similar energetics, e.g., double white dwarf (WD–WD) mergers and NS–WD mergers (see Ruffini et al. 2016, 2018c; Wang et al. 2019b, for details).

Three-dimensional numerical SPH simulations of BdHNe have been recently presented in Becerra et al. (2019). These simulations improve and extend the previous ones by Becerra et al. (2016). A fundamental contribution of these simulations has been to provide a visualization of the morphology of the SN ejecta, which is modified from the initial spherical symmetry. A low-density cavity is carved initially by the NS companion and, once its collapses, further by the BH formation process (see also Ruffini et al. 2019a). Such an asymmetric density distribution leads to a dependence of the GRB description on the observer viewing angle—in the orbital/equatorial plane or in the plane orthogonal to it (Becerra et al. 2016; Ruffini et al. 2018e, 2018a; Becerra et al. 2019)—and on the orbital period of the binary, in the simulation of Figure 2 at about 300 s (Ruffini et al. 2018a).

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The SN transforms into a hypernova (HN) as a result of the energy and momentum transfer of the e⁺e⁻ plasma (Ruffini et al. 2018a; Becerra et al. 2019). The SN shock breakout and the hypercritical accretion can be observed as X-ray precursors (Becerra et al. 2016; Wang et al. 2019b). The e⁺e⁻ feedback also produces the gamma- and X-ray flares observed in the early afterglow (Ruffini et al. 2018e). There is then the most interesting emission episode, which is related to the νNS that originated from the SN explosion, that is, the synchrotron emission by relativistic electrons, injected from the νNS pulsar emission into the HN ejecta in the presence of the νNS magnetic field, explains the X-ray afterglow and its power-law luminosity (Ruffini et al. 2018b; Wang et al. 2019b). Finally, the HN is observed in the optical bands a few days after the GRB trigger, powered by the energy release of the nickel decay.

Figure 1 and Table 1 summarize the above correspondence between the BdHN physical process and each GRB observable, emphasizing the role of each component of the binary system. We also refer the reader to Rueda et al. (2019), and references therein, for a recent review on the physical processes at work and related observables in BdHNe I and II.

Table 1.  Summary of the GRB Observables Associated with Each BdHN I Component and Physical Phenomena

BdHN Component/Phenomena	GRB Observable
	X-Ray	Prompt	GeV–TeV	X-Ray Flares	X-Ray Plateau
	Precursor	(MeV)	Emission	Early Afterglow	and Late Afterglow
SN breakouta	$\bigotimes$
Hypercritical accretion onto the NSb	$\bigotimes$
e+e− from BH formation: transparency		$\bigotimes$
in the low baryon load regionc
Inner engine: newborn BH + B-field+SN ejectad			$\bigotimes$
e+e− from BH formation: transparency				$\bigotimes$
in the high baryon load region (SN ejecta)e
Synchrotron emission by νNS-injected					$\bigotimes$
particles on SN ejectaf
νNS pulsar-like emissionf					$\bigotimes$

Notes.

^aWang et al. (2019b). ^bFryer et al. (2014), Becerra et al. (2016), Rueda et al. (2019). ^cBianco et al. (2001). ^dRuffini et al. (2018d, 2019d, 2019a, 2019b). ^eRuffini et al. (2018e). ^fRuffini et al. (2018b), Wang et al. (2019b), and this work.

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GRB 130427A is one of the best-observed GRBs; it is located at redshift z ∼ 0.34 (Levan et al. 2013), and more than 50 observatories participated in the observation. It hits the record for brightness in gamma-ray emission, so that Fermi-GBM was saturated. It also hits the record for GeV observation, with more than 500 photons above 100 MeV received, and GeV emission observed until ∼10⁴ s (Ackermann et al. 2014).

The shape of its prompt emission consists of a ∼3 s precursor, followed by a multipeaked pulse lasting ∼10 s. At time ∼120 s, an additional flare appears, then it enters the afterglow (Maselli et al. 2014). The X-ray afterglow is observed by Swift and NuStar. Swift covers discretely from ∼150 to ∼10⁷ s (Li et al. 2015), and NuStar observes three epochs, starting approximately at 1.2, 4.8, and 5.4 days, for observational durations of 30.5, 21.2, and 12.3 ks (Kouveliotou et al. 2013). The power-law decay index of the late-time afterglow after ∼2000 s gives ∼−1.32 (Ruffini et al. 2015).

The optical spectrum reveals that 16.7 days after the GRB trigger, a typical SN Ic emerges (Xu et al. 2013; Li et al. 2018a), as predicted by Ruffini et al. (2013).

GRB 160509A, at redshift z ∼ 1.17 (Tanvir et al. 2016), is a strong source of GeV emission, including a 52 GeV photon arriving at 77 s, and a 29 GeV photon arriving ∼70 ks (Laskar et al. 2016).

GRB 160509A consists of two emission periods, 0−40 s and 280−420 s (Tam et al. 2017). The first period exhibits a single-pulse structure for sub-MeV emission, and a double-pulse structure for ∼100 MeV emission. The second period is in the sub-MeV energy range with a double-pulse structure. Swift–XRT started the observation ∼7000 s after the burst, with a shallow power-law decay of index ∼−0.6, followed by a normal decay of power-law index ∼−1.45 after 5 × 10⁴ s (Tam et al. 2017; Li et al. 2018b).

There is no SN association reported; the optical signal of SNe can hardly be confirmed for GRBs with redshift >1 as the absorption is intense (Woosley & Bloom 2006).

GRB 160625B, at redshift 1.406 (Xu et al. 2016), is a bright GRB with the special quality that its polarization has been detected. Fermi-LAT has detected more than 300 photons with energy >100 MeV (Lü et al. 2017).

The gamma-ray light curve has three distinct pulses (Zhang et al. 2018; Li 2019). The first short pulse is totally thermal and lasts ∼2 s, the second bright pulse starts from ∼180 s and ends at ∼240 s, and the last weak pulse emerges at ∼330 s and lasts ∼300 s. The total isotropic energy reaches ∼3 × 10⁵⁴ erg (Alexander et al. 2017; Lü et al. 2017).

Swift–XRT starts the observation at a late time (>10⁴ s), finding a power-law behavior with decaying index ∼−1.25.

There is no SN confirmation, possibly due to the redshift being >1 (Woosley & Bloom 2006).

GRB 190114C, at redshift z ∼ 0.42 (Selsing et al. 2019), is the first GRB with TeV photon detection by MAGIC (Mirzoyan et al. 2019; MAGIC Collaboration et al. 2019). It has similar features to GRB 130427A (Wang et al. 2019a), and it has caught great attention as well.

The prompt emission of GRB 190114C starts with a multipeaked pulse, its initial ∼1.5 s is nonthermal, which is then followed by a possible thermal emission until ∼1.8 s. A confident thermal emission exists during the peak of the pulse, from 2.7 from 5.5 s. The GeV emission starts from 2.7 s, initiated by a spiky structure, then follows a power-law decay with index ∼−1.2 (Ruffini et al. 2019a). The GeV emission is very luminous; more than 200 photons with energy >100 MeV are received. The X-ray afterglow observed by Swift–XRT shows a persistent power-law decay behavior, with decaying index ∼1.35 (Wang et al. 2019a).

An continuous observational campaign lasting ∼50 days unveiled the SN emergence ∼15 days after the GRB (Melandri et al. 2019), which is consistent with the prediction of 18.8 ± 3.7 days after the GRB by Ruffini et al. (2019c).

The newborn NS at the center of the SN, i.e., the νNS, ejects high-energy particles as in traditional pulsar models. This means that these particles escape from the νNS magnetosphere through so-called "open" magnetic field lines, namely, the field lines that do not close within the light cylinder radius that determines the size of the corotating magnetosphere. These particles interact with the SN ejecta, which, by expanding in the νNS magnetic field, produce synchrotron radiation, which we discuss below. Hence, the acceleration mechanism is similar to the one occurring in traditional SN remnants but with two main differences in our case: (1) we have a ∼1 ms νNS pulsar powering the SN ejecta and (2) the SN ejecta are at a radius of ∼10¹² cm at the beginning of the afterglow, at the rest-frame time t ∼ 100 s, as the SN expands with velocity ∼0.1c.

The above distance is well beyond the light cylinder radius, so it is expected that only the toroidal component of the magnetic field, which decreases as 1/r (see Equations (4) and (12)), survives (see, e.g., Goldreich & Julian 1969 for details). Therefore, the relevant magnetic field for synchrotron radiation in the afterglow is that of the νNS, which is stronger (as shown below, at that distance it is of the order of 10⁵ G) than the one possibly produced inside the remnant by dilute plasma currents, unlike the traditional models for the emission of old (≳1 kyr) SN remnants.

In Ruffini et al. (2018b) and Wang et al. (2019b), we simulated the afterglow by the synchrotron emission of electrons from the optically thin region of the SN ejecta, which expand mildly relativistically in the νNS magnetic field. The FPA emission at times t ≳ 10² s has two origins: the emission before the plateau phase (∼5 × 10³ s) is mainly contributed by the remaining kinetic energy of the SN ejecta, and at later times, the continuous energy injection from the νNS dominates. We extend the same approach in this paper to the GRBs of Section 3.

To fully follow the temporal behavior of radiation spectra, it is necessary to solve the kinetic equation for electron distribution in the transparent region of the SN ejecta:

$$\begin{eqnarray}&&\displaystyle \frac{\partial N(\gamma ,t)}{\partial t}=\displaystyle \frac{\partial }{\partial \gamma }(\dot{\gamma }(\gamma ,t)N(\gamma ,t))+Q(\gamma ,t),\end{eqnarray} \tag{ 1 }$$

where N(γ, t) is the electron number distribution as a function of electron energy γ = E/m_ec², $\dot{\gamma }(\gamma ,t$) is the electron energy-loss rate normalized to the electron rest mass, and Q(γ, t) = Q₀(t)γ^−p is the particle injection rate, assumed to be a power law of index p, so the electrons injected are within the energy range of γ_min to γ_max. The total injection luminosity L_inj(t) is provided by the kinetic energy of the SN and the rotational energy of the νNS, here parameterized via the power-law injection power,

$$\begin{eqnarray}&&{L}_{\mathrm{inj}}(t)={\int }_{{\gamma }_{\min }}^{{\gamma }_{\max }}Q(\gamma ,t)d\gamma \simeq {L}_{0}{\left(1+\displaystyle \frac{t}{{\tau }_{0}}\right)}^{-k},\end{eqnarray} \tag{ 2 }$$

where L₀, k, and τ₀ are assessed by fitting the light-curve data. The majority of energy loss is considered to be the adiabatic energy loss and the synchrotron energy loss,

$$\begin{eqnarray}&&\dot{\gamma }(\gamma ,t)=\displaystyle \frac{\dot{R}(t)}{R(t)}\gamma +\displaystyle \frac{4}{3}\displaystyle \frac{{\sigma }_{T}}{{m}_{e}c}\displaystyle \frac{B{\left(t\right)}^{2}}{8\pi }{\gamma }^{2},\end{eqnarray} \tag{ 3 }$$

where R(t) is the size of the emitter, σ_T is the Thomson cross section, and B(t) is the magnetic field strength expected to have a toroidal configuration given by

$$\begin{eqnarray}&&B(t)={B}_{0}{\left(\displaystyle \frac{R(t)}{{R}_{0}}\right)}^{-1},\end{eqnarray} \tag{ 4 }$$

where B₀ is the magnetic field strength at the distance R₀. The final bolometric synchrotron luminosity from this system gives

$$\begin{eqnarray}&&{L}_{\mathrm{syn}}(\nu ,t)={\int }_{1}^{{\gamma }_{\max }}N(\gamma ,t){P}_{\mathrm{syn}}(\nu ,\gamma ,B(t))d\gamma .\end{eqnarray} \tag{ 5 }$$

As we have introduced in Section 1, the thermal emission during the FPA phase indicates a mildly relativistic velocity, ∼0.5−0.9c, at time ∼100 s (Ruffini et al. 2015, 2018e, 2019d; Wang et al. 2019b). We adopt this value as the initial velocity and radius of the transparent part of the SN ejecta.

For later stages at around 10⁶ s, when a sizable front shell of SN ejecta becomes transparent, we adopt the velocity of ∼0.1c obtained through observations of Fe II emission lines (see, e.g., Xu et al. 2013). We make the simplest assumption of a uniformly decelerating expansion during the time interval 10² ≲ t ≲ 10⁶ s. The SN ejecta remain in the coasting phase for hundreds of years (see e.g., Sturner et al. 1997); therefore, we adopt a constant velocity from 10⁶ s until 10⁷ s.

Following the above discussion and our data analysis, we describe the expansion velocity as

$$\begin{eqnarray}\dot{R}(t)=\left\{\begin{array}{ll}{v}_{0}-{a}_{0}\,t & {10}^{2}\lt t\lt {10}^{6}{\rm{s}}\\ {v}_{f} & {10}^{7}\gt t\gt {10}^{6}{\rm{s}}\end{array}\right.\end{eqnarray} \tag{ 6 }$$

with typical value v₀ = 2.4 × 10¹⁰ cm s⁻¹, a₀ = 2.1 × 10⁴ cm s⁻², and v_f = 3 × 10⁹ cm s⁻¹.

It is appropriate to clarify how the model parameters presented in this table are obtained: R₀ and τ₀ are fixed by the observed thermal component at around 10² s, from which we obtain the radius and expansion velocity of the SN front. The minimum and maximum energies of the injected electrons, γ_min and γ_max, are fixed once B is given. L₀ is fixed by a normalization of the observed source luminosity. The power-law index of the energy injection rate, p, is fixed to the value p = 3/2. The parameter k is fixed to produce the power-law decay of the late-time X-ray data. Therefore, the "free parameter" to be obtained is B₀.

In Ruffini et al. (2018b), we have given detailed fitting parameters and figures of GRB 130427A. In this article, we additionally fit GRB 160625B and confirm that the mildly relativistic model is capable of producing the GRB afterglow. As shown in Table 2 and Figure 3, our model fits very well the optical and the X-ray spectrum but not the GeV data. This is in agreement with the BdHN paradigm because the GeV emission is expected to be explained by the newborn BH activity and not by the νNS one (Ruffini et al. 2019d). On the other hand, radio data show a lack of expected flux, which comes from synchrotron self-absorption processes that are rather complicated to model in the current numerical framework but can be thoroughly ignored at frequencies above 10¹⁴ Hz.

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Comparing their fitting parameters, GRB 130427A and GRB 160625B are similar except for the constant of injection power L₀ (see Equation (2)). Such similarities can be extended to others. It can be seen from Figure 4 that taking everything else as similar, from the magnetic field strength and structure to expansion evolution, simulated light curve of GRB 190114C at the relevant times can be obtained from that of GRB 130427A by scaling L₀ by a factor of 1/5.

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The injection power index k ∼ 1.5 from the fitting suggests that the quadrupole emission from a pulsar dominates the late-time afterglow. As we will see below, the complementary analysis allows the initial rotation period of the νNS as well as an independent estimate of its magnetic field structure to be inferred.

Being just born, the νNS must be rapidly rotating, and as such it contains abundant rotational energy:

$$\begin{eqnarray}&&E=\displaystyle \frac{1}{2}I{{\rm{\Omega }}}^{2},\end{eqnarray} \tag{ 7 }$$

where I is the moment of inertia and Ω = 2π/P_νNS is the angular velocity. For a millisecond νNS and I ∼ 10⁴⁵ g cm², the total rotational energy E ∼ 2 × 10⁵² erg. Assuming that the rotational energy loss is driven by magnetic dipole and quadrupole radiation, we have

$$\begin{eqnarray}\begin{array}{rcl}{L}_{\mathrm{NS}}(t) & = & \displaystyle \frac{{dE}}{{dt}}=-I{\rm{\Omega }}\dot{{\rm{\Omega }}}\\ & = & -\displaystyle \frac{2}{3{c}^{3}}{{\rm{\Omega }}}^{4}{B}_{\mathrm{dip}}^{2}{R}_{\nu \mathrm{NS}}^{6}{\sin }^{2}{\chi }_{1}\left(1+{\eta }^{2}\displaystyle \frac{16}{45}\displaystyle \frac{{R}_{\nu \mathrm{NS}}^{2}{{\rm{\Omega }}}^{2}}{{c}^{2}}\right),\end{array}\end{eqnarray} \tag{ 8 }$$

where

$$\begin{eqnarray}&&{\eta }^{2}=({\cos }^{2}{\chi }_{2}+10{\sin }^{2}{\chi }_{2})\displaystyle \frac{{B}_{\mathrm{quad}}^{2}}{{B}_{\mathrm{dip}}^{2}},\end{eqnarray} \tag{ 9 }$$

with χ₁ and χ₂ the inclination angles of the magnetic moment, and B_dip and B_quad are the dipole and quadrupole magnetic fields, respectively. The parameter η measures the quadrupole to dipole magnetic field strength ratio.

Figure 5 shows the bolometric light curves (∼5 times brighter than the Swift–XRT light curves inferred from the fitting) of GRBs 160625B, 160509A, 130427A, 190114C, and 180728A, respectively. We show that the νNS luminosities L_NS(t) fit the light curves. We report the fitting νNS parameters—the dipole (B_dip) and quadrupole (B_quad) magnetic field components, the initial rotation period (P_{ν NS})—and assume a νNS of mass and radius of 1.4M_⊙ and 10⁶ cm, respectively. The results are also summarized in Table 3. It also becomes clear from this analysis that the νNS emission alone is not able to explain the emission of the FPA phase at early times 10²–10³ s. As we have shown, that emission is mainly powered by the mildly relativistic SN kinetic energy.

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Table 3.  Observational Properties of the GRB and Inferred Physical Quantities of the νNS of the Corresponding BdHN Model that Fits the GRB Data

GRB	Type	Redshift	Eiso	PνNS	Erot	Bdip	Bquad
			(erg)	(ms)	(erg)	(G)	(G)
130427A	BdHN I	0.34	$1.40\times {10}^{54}$	0.95	3.50 × 1052	6.0 × 1012	2.0 × 1013 ∼ 6.0 × 1014
160509A	BdHN I	1.17	1.06 × 1054	0.75	5.61 × 1052	4.0 × 1012	1.3 × 1014 ∼ 4.0 × 1014
160625B	BdHN I	1.406	3.00 × 1054	0.5	1.26 × 1053	1.5 × 1012	5.0 × 1013 ∼ 1.6 × 1014
190114C	BdHN I	0.42	2.47 × 1053	2.1	7.16 × 1051	5.0 × 1012	1.5 × 1015 ∼ 5.0 × 1015
180728A	BdHN II	0.117	2.73 × 1051	3.5	2.58 × 1051	1.0 × 1013	3.5 × 1015 ∼ 1.1 × 1016

Note. Column 1: GRB name; column 2: identified BdHN type; column 3: the isotropic energy released (E_iso) in gamma-rays; column 4: cosmological redshift (z); column 5: νNS rotation period (P_νNS); column 6: νNS rotational energy (E_rot); columns 7 and 8: strength of the dipole (B_dip) and quadrupole (B_quad) magnetic field components of the νNS. The quadrupole magnetic field component is given in the range where the upper limit is three times the lower limit; this is brought about by the freedom of the inclination angles of the magnetic moment. During the fitting, we consistently assume the NS mass of 1.4M_⊙ and the NS radius of 10⁶ cm for all three cases. The fitted light curves are shown in Figure 5; the parameters of GRB 1340427A and 180728A are taken from Wang et al. (2019b).

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Having estimated the magnetic field structure and the rotation period of the νNS from the fit of the data of the FPA phase at times 10²–10⁷ s, we can now assess their self-consistency with the expected values within the BdHN scenario.

First, let us adopt the binary as tidally locked, i.e., the rotation period of the binary components is synchronized with the orbital period. This implies that the rotation period of the CO_core is P_CO = P, where P denotes the orbital period. From the Kepler law, the value of P is connected to the orbital separation a_orb and to the binary mass as

$$\begin{eqnarray}&&{P}_{\mathrm{CO}}=P=2\pi \sqrt{\displaystyle \frac{{a}_{\mathrm{orb}}^{3}}{{{GM}}_{\mathrm{tot}}}},\end{eqnarray} \tag{ 10 }$$

where G is the gravitational constant and M_tot = M_CO + M_NS is the total mass of the binary, where M_CO and M_NS are the masses of the CO_core and the NS companion, respectively. Thus, M_CO = M_Fe + M_ej, with M_Fe and M_ej the masses of the iron core (which collapses and forms the νNS) and the ejected mass in the SN event, respectively.

The mass of the νNS is ${M}_{\nu \mathrm{NS}}\approx {M}_{\mathrm{Fe}}$. The rotation period, ${P}_{\nu \mathrm{NS}}$, is estimated from that of the iron core, P_Fe, by applying angular momentum conservation in the collapse process, i.e.,

$$\begin{eqnarray}&&{P}_{\nu \mathrm{NS}}={\left(\displaystyle \frac{{R}_{\nu \mathrm{NS}}}{{R}_{\mathrm{Fe}}}\right)}^{2}P,\end{eqnarray} \tag{ 11 }$$

where R_νNS and R_Fe are the radius of the νNS and of the iron core, respectively, and we have assumed that the pre-SN star has uniform rotation; so, ${P}_{\mathrm{Fe}}={P}_{\mathrm{CO}}=P$.

Without loss of generality, in our estimates we can adopt a νNS order-of-magnitude radius of 10⁶ cm. As we shall see below, a more careful estimate is that for the CO_core progenitor (which tells us the radius of the iron core) and the orbital period/binary separation, which affect additional observables of a BdHN.

It is instructive to appreciate the above statement with specific examples; for these we use the results of Wang et al. (2019b) for two BdHN archetypes: GRB 130427A for BdHN I and GRB 180827A for BdHN II. Table 3 shows, for the above GRBs, as well as for GRB 190114C, GRB 160625B, and GRB 160509A, some observational quantities (the isotropic energy released E_iso and the cosmological redshift), the inferred BdHN type, and the properties of the νNS (rotation period P_{ν NS}, rotational energy, and the strength of the dipole and quadrupole magnetic field components).

By examining the BdHN models simulated in Becerra et al. (2019) (see, e.g., Table 2 there), we have shown in Wang et al. (2019b) that the Model "25m1p08e" fits the observational requirements of GRB 130427A, and the Model "25m3p1e" those of GRB 180827A. These models have the same binary progenitor components: the ≈6.8 M_⊙ CO_core (R_Fe ∼ 2 × 10⁸ cm) developed by a 25M_⊙ zero-age main-sequence star (see Table 1 in Becerra et al. 2019) and a 2 M_⊙ NS companion. For GRB 130427A, the orbital period is P = 4.8 minutes (binary separation a_orb ≈ 1.3 × 10¹⁰ cm), resulting in P_νNS ≈ 1.0 ms while, for GRB 180827A, the orbital period is P = 11.8 minutes (a_orb ≈ 2.6 × 10¹⁰ cm), so a less compact binary, which leads to P_νNS ≈ 2.5 ms.

We turn now to perform a further self-consistency check of our picture. Namely, we make a cross-check of the estimated νNS parameters obtained first from the early afterglow via synchrotron emission, and then from the late X-ray afterglow via the pulsar luminosity, with respect to expectations from NS theory.

Up to factors of order unity, the surface dipole B_s and the toroidal component B_t at a distance r from the surface are approximately related as (see, e.g., Goldreich & Julian 1969)

$$\begin{eqnarray}&&{B}_{t}\approx {\left(\displaystyle \frac{2\pi {R}_{\nu \mathrm{NS}}}{{{cP}}_{\nu \mathrm{NS}}}\right)}^{2}\left(\displaystyle \frac{{R}_{\nu \mathrm{NS}}}{r}\right){B}_{s}.\end{eqnarray} \tag{ 12 }$$

Let us analyze the case of GRB 130427A. By equating Equations (4) and (12), and using the values of B₀ = 5 × 10⁵ G and R₀ = 2.4 × 10¹² cm obtained from synchrotron analysis by Ruffini et al. (2018b), and P_νNS = P₀ ≈ 1 ms from the pulsar activity in the late-afterglow analysis, we obtain B_s ≈ 2 × 10¹³ G. This value has to be compared with that obtained from the constraint that the pulsar luminosity power the late afterglow, B_dip = 6 × 10¹² G (see Table 3). If we use the parameters B₀ = 1.0 × 10⁶ G and R₀ = 1.2 × 10¹¹ cm from Table 2 for GRB 160625B, and the corresponding P_νNS = P₀ ≈ 0.5 ms, we obtain B_s ≈ 6.8 × 10¹¹ G, compared with B_dip ≈ 10¹² G (see Table 3). An even better agreement can be obtained by using a more accurate value of the νNS radius, which is surely bigger than the fiducial value R_νNS = 10⁶ cm we have used in these estimates.

We attribute the spin-down energy of the νNS to the energy injection of the late-time afterglow. By fitting the observed emission through the synchrotron model, the spin period and the magnetic field of the νNS can be inferred. In Wang et al. (2019b), we have applied this approach to GRB 130427A and GRB 180728A; here we apply the same method to the recent GRB 190114C and to other two, GRB 160509A and GRB 160625B, for comparison. In Figure 5, we plot the energy injection from the dipole and quadrupole emission of νNS; the fitting results indicate 190114C leaves a νNS of spin period 2.1 ms, with dipole magnetic field ${B}_{\mathrm{dip}}=5\times {10}^{12}$ G, and a quadrupole magnetic field >10¹⁵ G. The fitting parameters of all the GRBs are listed in table 3. Generally, the NS in the BdHN I system spins faster, of period ≲2 ms, and contains more rotational energy ≳10⁵² erg. We notice that GRB 160625B has the shortest initial spin period of P = 0.5 ms, which is exactly on the margin of the rotational period of an NS at the Keplerian sequence. For an NS of mass 1.4 M_⊙ and radius 12 km, its Keplerian frequency f_K ≃ 1900 (Lattimer & Prakash 2004; Riahi et al. 2019), corresponding to the spin period of P ≃ 0.5 ms.

From Equations (10) and (11), the orbital separation of the binary system relates to the spin of νNS as ${a}_{\mathrm{orb}}\propto {P}_{\nu \mathrm{NS}}^{2/3}$. Therefore, with the knowledge of the binary separation of GRB 130427A, ∼1.35 × 10¹⁰ cm, the spin period of ∼1 ms, and the newly inferred spin of GRB 190114C of ∼1.2 ms, assuming these two systems have the same mass and radius as the CO_core and the νNS, we obtain the orbital separation of GRB 190114C as ∼1.52 × 10¹⁰ cm.

The self-consistent value obtained for the orbital period/separation gives a strong support to our basic assumptions: (1) owing to the system compactness, the binary components are tidally locked, and (2) the angular momentum is conserved in the core-collapse SN process.

We would like to recall that it has been shown that purely poloidal field configurations are unstable against adiabatic perturbations; for nonrotating stars, it has been first demonstrated by Wright (1973), Markey & Tayler (1973; see also Flowers & Ruderman 1977). For rotating stars, similar results have been obtained, e.g., by Pitts & Tayler (1985). In addition, Tayler (1973) has shown that purely toroidal configurations are also unstable. We refer the reader to Spruit (1999) for a review of the different possible instabilities that may be active in magnetic stars. In this vein, the dipole–quadrupole magnetic field configuration found in our analyses with a quadrupole component dominating in the early life of the the νNS is particularly relevant. They also give support to theoretical expectations pointing to the possible stability of poloidal–toroidal magnetic field configurations on timescales longer than the collapsing time of the pre-SN star; for details, see, e.g., Tayler (1980) and Mestel (1984).

It remains the question of how, during the process of gravitational collapse, the magnetic field increases its strength to the observed NS values. This is still one of the most relevant open questions in astrophysics, which at this stage is out of the scope of the present work. We shall mention here only one important case, which is the traditional explanation for the NS magnetic field strength based on the amplification of the field by magnetic flux conservation. The flux conservation implies ${{\rm{\Phi }}}_{i}=\pi {B}_{i}{R}_{i}^{2}={{\rm{\Phi }}}_{f}=\pi {B}_{f}{R}_{f}^{2}$, where i and f stand for the initial and final configurations and R_i,f the corresponding radii. The radius of the collapsing iron core is of the order of 10⁸–10⁹ cm, while the radius of the νNS is of the order of 10⁶ cm; therefore, the magnetic flux conservation implies an amplification of 10⁴–10⁶ times the initial field during the νNS formation. Therefore, a seed magnetic field of 10⁷–10⁹ G is necessary to be present in the iron core of the pre-SN star to explain a νNS magnetic field of 10¹³ G. The highest magnetic fields observed in main-sequence stars leading to the pre-SN stars of interest are of the order of 10⁴ G (Spruit 2009). If the magnetic field is uniform inside the star, then the value of the magnetic field observed in these stars poses a serious issue to the magnetic flux conservation hypothesis for the NS magnetic field genesis. A summary of the theoretical efforts to understand the possible sources of the magnetic field of an NS can be found in Spruit (2009).

The BH in a BdHN I is formed from the gravitational collapse of the NS companion of the CO_core, which reaches critical mass by the hypercritical accretion of the ejecta of the SN explosion of the CO_core. Hence, the magnetic field surrounding the BH derived in the previous section to explain the GeV emission should originate from the collapsed NS. In fact, the magnetic field of the νNS evaluated at the BH position is too low to be relevant in this discussion. As we shall see, the magnetic field inherited from the collapsed NS can easily reach values of the order of 10¹⁴ G. Instead, the magnetic field of the νNS at the BH site is ${B}_{\mathrm{dip}}{\left({R}_{\nu \mathrm{NS}}/{a}_{\mathrm{orb}}\right)}^{3}=10$ G, adopting fiducial parameters according to the results of Table 3: a dipole magnetic field at the νNS surface B_dip = 10¹³ G, a binary separation of a_orb = 10¹⁰ cm, and a νNS radius of ${R}_{\nu \mathrm{NS}}={10}^{6}\,\mathrm{cm}$.

Having clarified this issue, we proceed now to discuss the nature of the field. Both the νNS and the NS follow an analogous formation channel, namely, they are born from core-collapse SNe. In fact, to reach the BdHN stage, the massive binary has to survive two SN events: the first SN which forms the NS and the second one which forms the νNS (core collapse of the CO_core). Figure 1 shows the evolutionary path of a massive binary leading to a BdHN I. It is then clear that the NS companion of the CO_core will have magnetic field properties analogous to those of the νNS, which were discussed in the previous section. Therefore, we can conclude that the BH forms from the collapse of a magnetized and fast-rotating NS.

In this scenario, the magnetic field of the collapsing NS companion should then be responsible for the magnetic field surrounding the BH. Only a modest amplification of the initial field from the NS, which is ∼10¹³ G, is needed to reach the value of 10¹⁴ G around the newborn BH. Then, even the single action of magnetic flux conservation can suffice to explain the magnetic field amplification. The BH horizon is r₊ ∼ GM/c², where M can be assumed to be equal to the NS critical mass, say, 3 M_⊙, so r₊ ≈ 4.4 km. The NS at the collapse point, owing to high rotation, will have a radius in excess of the typically adopted 10 km (Cipolletta et al. 2015); let us assume a conservative range of 12–15 km. These conditions suggest that magnetic flux conservation magnifies the magnetic field in the BH formation by a factor of 7–12. Therefore, a seed field of 10¹³ G present in the collapsing NS is enough to explain the magnetic field of 10¹⁴ G near the newborn BH.

It is worth clarifying a crucial point: the magnetic field has to remain anchored to some NS material, which guarantees its existence. It is therefore expected that some part of the NS does not take part in the BH formation. Assuming that magnetic flux is conserved during the collapse, then the magnetic energy is a constant fraction of the gravitational energy during the entire process, so only high rotation (see, e.g., Becerra et al. 2016) and some degree of differential rotation (see, e.g., Shibata et al. 2006) of the NS at the critical mass point are responsible for some fraction of the NS matter to avoid remaining outside with sufficient angular momentum to orbit the newborn BH (see, e.g., Figure 6).

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The three-dimensional simulations of BdHNe presented in Becerra et al. (2019) show that the part of the SN ejecta surrounding the BH forms a torus-like structure around it. The aforementioned matter from the NS with high angular momentum will add to this orbiting matter around the BH. In the off-equatorial directions, the density is much smaller (Ruffini et al. 2018b; Becerra et al. 2019; see also Ruffini et al. 2019a). This implies that on the equatorial plane, the field is compressed, while in the axial direction, the matter accretion flows in along the field lines.

Our inner engine, the BH + magnetic field configuration powering the high-energy emission in a BdHN I, finds additional support in numerical simulations of magnetic and rotational collapse into a BH. The first numerical computer treatment of the gravitational collapse to a BH in the presence of magnetic fields starts with the pioneering two-dimensional simulations by Wilson (1975; see Figure 6(a), reproduced from Wilson 1978). These works already showed the amplification of the magnetic field in the gravitational collapse process. Rotating magnetized gravitational collapse into a BH has been more recently treated in greater detail by three-dimensional simulations, which have confirmed this picture and the crucial role of the combined presence of magnetic field and rotation (Dionysopoulou et al. 2013; Nathanail et al. 2017; Most et al. 2018).

Additional support can be also found in the context of binary NS mergers. Numerical simulations have indeed shown that the collapse of the unstable massive NS formed in the merger into a BH leads to a configuration composed of a BH surrounded by a nearly collimated magnetic field and an accretion disk (see Duez et al. 2006a, 2006b; Shibata et al. 2006; Stephens et al. 2007, 2008 for details). Three-dimensional numerical simulations have also been performed and confirm this scenario (Rezzolla et al. 2011). In particular, it is appropriate to underline the strong analogy between Figure 6(a) taken from Wilson (1978) with Figure 6(b) reproduced in this paper from Rezzolla et al. (2011). It is also interesting that the value of the magnetic field close to the BH estimated in Rezzolla et al. (2011) along the BH spin axis, 8 × 10¹⁴ G , is similar to the value of 3 × 10¹⁴ G needed for the operation of the "inner engine" of GRB 130427A (Ruffini et al. 2018d). What is also conceptually important is that the uniform magnetic field assumed by the Wald solution should be expected to reach a poloidal configuration already relatively close to the BH. This already occurs in the original Wilson (1978) solution confirmed by the recent and most detailed calculation by Rezzolla et al. (2011); see Figures 6(a) and (b).

Although the above simulations refer to the remnant configuration of a binary NS merger, the post-merger configuration is analogous to the one developed for BdHNe I related to the newborn BH, which we have applied in our recent works (see, e.g., Ruffini et al. 2018b, 2018d, 2019d, 2019a; Wang et al. 2019b, and references therein), and which is supported by the recently presented three-dimensional simulations of BdHNe (see Becerra et al. 2019 for details).

Before closing, let us indicate the difference between the NS merger and the BdHN. In the case of the BdHN, the gravitational collapse leading to the BH with the formation of a horizon creates a very-low-density cavity of 10⁻¹⁴ g cm⁻³ with a radius of ∼10¹¹ cm in the SN ejecta; see Figure 1 and Figure 7, reproduced from Ruffini et al. (2019b). The presence of such a low-density environment is indeed essential for the successful operation of the "inner engine."

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Reaching a poloidal configuration already close to the BH in the Wald solution and the existence of the cavity are crucial factors in the analysis of the propagation of the photons produced by synchrotron radiation and in reaching the transparency condition of the "inner engine" of the BdHNe (Ruffini et al. 2019d).

Our general conclusions were reached by comparing and contrasting the observations of GRB 130427A, GRB 160509A, GRB 160625B, GRB 180728A, and GRB 190114C:

• 1.  
  From the analysis of GRB 130427A (Ruffini et al. 2018a) and GRB 190114C presented here (see Figures 4 and 5), we conclude that the early (t ∼ 10²–10⁴ s) X-ray emission during the FPA phase is explained by the injection of ultrarelativistic electrons from the νNS into the magnetized expanding ejecta, producing synchrotron radiation. The magnetic field inferred in this part of the analysis is found to be consistent with the toroidal/longitudinal magnetic field component of the νNS. The dominance of this component is expected at distances much larger (∼10¹² cm) than the light cylinder radius in which this synchrotron emission occurs. No data on the other GRBs considered in this paper are available in this time interval.
• 2.  
  Using the data of all present GRBs, we concluded that at times t ≳ 10³–10⁴ s of the FPA phase, the power-law decaying luminosity is dominated by the pulsar magnetic-braking radiation. We have inferred a dipole+quadrupole structure of the νNS magnetic field, with the quadrupole component initially dominant. The strength of the dipole component is about 10¹²–10¹³ G while that of the quadrupole can be of order 10¹⁵ G (see Figure 4 and Table 3). As clearly shown in Figures 4 and 5, νNS solely with the dipole + quadrupole magnetic field structure cannot explain the emission of the early FPA phase, which is dominated by SN emission.
• 3.  
  We have checked that the magnetic field of the νNS, inferred independently in the two above regimes of the FPA phase, give values in very good agreement. The νNS magnetic fields obtained from the explanation of the FPA phase, at times 10²–10³ s, by synchrotron radiation, and at times t≳10⁴ s by pulsar magnetic braking, are in close agreement (see Section 4, Table 3 and Figure 5).
• 4.  
  In Section 5, we have shown the consistency of the inferred νNS parameters with the expectations in the BdHN scenario. In particular, we have used the rotation period of the νNS inferred from the FPA phase at times t ≳ 10³–10⁴ s , we have inferred the orbital period/separation assuming tidal synchronization of the binary and angular momentum conservation in the gravitational collapse of the iron core leading to the νNS. This inferred binary separation is shown to be in excellent agreement with the numerical simulations of the binary progenitor in Wang et al. (2019b).

Before concluding, in view of the recent understanding gained on the "inner engine" of the high-energy emission of the GRB (Ruffini et al. 2019d), we can also conclude the following:

• 1.  
  The magnetic field along the rotational axis of the BH is rooted in the magnetosphere left by the binary companion NS prior to the collapse.
• 2.  
  While in the equatorial plane the field is magnified by magnetic flux conservation, in the axial direction, the matter accretion flows in along the field lines; see Figure 2 and Becerra et al. (2019). Indeed, three-dimensional numerical simulations of the gravitational collapse into a BH in the presence of rotation and a magnetic field confirm our picture; see Figure 6 and Rezzolla et al. (2011), Dionysopoulou et al. (2013), Nathanail et al. (2017), and Most et al. (2018).
• 3.  
  The clarification reached regarding the role of SN accretion both in the NS and in the νNS, the stringent limits imposed on the Lorentz factor of the FPA phase, and the energetic requirement of the "inner engine" inferred from recent publications clearly point to an electrodynamical nature of the "inner engine" of the GRB, occurring close to the BH horizon, as opposed to the traditional, gravitational massive blast-wave model.

We acknowledge the public data from the Swift and Fermi satellites. We appreciate the discussion with Prof. She-sheng Xue and the suggestions from the referee.