On a GRB afterglow model consistent with hypernovae observations

On a GRB afterglow model consistent with hypernovae observations

R. Ruffini ICRA and Dipartimento di Fisica, Sapienza Università di Roma, P.le Aldo Moro 5, 00185 Rome, Italy. ICRANet, P.zza della Repubblica 10, 65122 Pescara, Italy. Université de Nice Sophia Antipolis, CEDEX 2, Grand Château Parc Valrose, Nice, France. ICRANet-Rio, Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, 22290–180 Rio de Janeiro, Brazil. M. Karlica ICRA and Dipartimento di Fisica, Sapienza Università di Roma, P.le Aldo Moro 5, 00185 Rome, Italy. ICRANet, P.zza della Repubblica 10, 65122 Pescara, Italy. Université de Nice Sophia Antipolis, CEDEX 2, Grand Château Parc Valrose, Nice, France. N. Sahakyan ICRANet-Armenia, Marshall Baghramian Avenue 24a, Yerevan 0019, Armenia. ICRANet, P.zza della Repubblica 10, 65122 Pescara, Italy. J. A. Rueda ICRA and Dipartimento di Fisica, Sapienza Università di Roma, P.le Aldo Moro 5, 00185 Rome, Italy. ICRANet, P.zza della Repubblica 10, 65122 Pescara, Italy. ICRANet-Rio, Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, 22290–180 Rio de Janeiro, Brazil. Y. Wang ICRA and Dipartimento di Fisica, Sapienza Università di Roma, P.le Aldo Moro 5, 00185 Rome, Italy. ICRANet, P.zza della Repubblica 10, 65122 Pescara, Italy. G. J. Mathews Center for Astrophysics, Department of Physics, University of Notre Dame, Notre Dame, IN, 46556, USA. ICRANet, P.zza della Repubblica 10, 65122 Pescara, Italy. C. L. Bianco ICRA and Dipartimento di Fisica, Sapienza Università di Roma, P.le Aldo Moro 5, 00185 Rome, Italy. ICRANet, P.zza della Repubblica 10, 65122 Pescara, Italy. M. Muccino ICRA and Dipartimento di Fisica, Sapienza Università di Roma, P.le Aldo Moro 5, 00185 Rome, Italy. ICRANet, P.zza della Repubblica 10, 65122 Pescara, Italy.
Abstract

We describe the afterglows of long gamma-ray-burst (GRB) within the context of a binary-driven hypernova (BdHN). In this paradigm afterglows originate from the interaction between a newly born neutron star (NS), created by an Ic supernova (SN), and a mildly relativistic ejecta of a hypernova (HN). Such a HN in turn result from the impact of the GRB on the original SN Ic. The observed power-law afterglow in the optical and X-ray bands is shown to arise from the synchrotron emission of relativistic electrons in the expanding magnetized HN ejecta. Two components contribute to the injected energy: the kinetic energy of the mildly relativistic expanding HN and the rotational energy of the fast rotating highly magnetized NS. As an example we reproduce the observed afterglow of GRB 130427A in all wavelengths from the optical ( Hz) to the X-ray band ( Hz) over times from  s to  s relative to the Fermi-GBM trigger. Initially, the emission is dominated by the loss of kinetic energy of the HN component. After  s the emission is dominated by the loss of rotational energy of the NS, for which we adopt an initial rotation period of 2 ms and a dipole/quadrupole magnetic field of  G/ G. This approach opens new views on the roles of the GRB interaction with the SN ejecta, on the mildly relativistic kinetic energy of the HN and on the pulsar-like phenomena of the NS. This scenario differs from the current ultra-relativistic treatments of the afterglow in the collapsar-fireball model and it is, instead, consistent with the current observations of the mildly relativistic regimes of X-ray flares, -ray flares and plateau emission in the BdHN.

gamma-ray burst: general — binaries: general — stars: neutron — supernovae: general — black hole physics — hydrodynamics

1 Introduction

It has been noted for almost two decades (Galama et al., 1998) that many long-duration GRBs show the presence of an associated unusually energetic supernova (SN) of type Ic (hypernova, HN) as well as of a long-lasting X-ray afterglow (Costa et al., 1997). Such HNe are unique in their spectral characteristics; they have no hydrogen and helium lines, suggesting that they are members of a binary system (Smartt, 2009). Moreover, these are broad-lined HNe suggesting the occurrence of energy injection beyond that of a normal type Ic SN (Lyman et al., 2016).

This has led to our suggestion (e.g. Ruffini et al., 2001c; Izzo et al., 2012) of a model for long GRBs associated with SNe Ic. In this paradigm, the progenitor is a carbon-oxygen star (CO) in a tight binary system with a neutron star (NS). As the CO explodes in a type Ic SN it produces a new NS (hereafter NS) and ejects a remnant of few solar masses, some of which is accreted onto the companion NS (Rueda & Ruffini, 2012). The accretion onto the companion NS is hypercritical, i.e. highly super-Eddington, reaching accretion rates of up to a tenth of solar mass per second, for the most compact binaries with orbital periods of few minutes (Fryer et al., 2014). The NS gains mass rapidly, reaching the critical mass, within few seconds. The NS then collapses to a black hole (BH) with the consequent emission of the GRB (Fryer et al., 2015). In this picture the BH formation and the associated GRB occurs some seconds after the initiation of the SN. The high temperature and density reached during the hypercritical accretion and the NS collapse lead to a copious emission of pairs which form an pair plasma that drives the GRB (see e.g. Becerra et al., 2015, 2016; Ruffini et al., 2016). The expanding SN remnant is reheated and shocked by the injection of the pair plasma from the GRB explosion (Ruffini et al., 2017b).

The shocked-heated SN, originally expanding at c, is transformed into an HN reaching expansion velocities up to c (Ruffini et al., 2014, 2015). A vast number of totally new physical processes are introduced that must be treated within a correct classical and quantum general relativistic approach (see e.g. Ruffini et al., 2017b, and references therein). The ensemble of these processes, addressing causally disconnected phenomena, each characterized by specific world lines, ultimately leads to a specific Lorentz factor. This ensemble comprises the binary-driven hypernova (BdHN) paradigm (Ruffini et al., 2016).

In this article we extend this novel approach to the analysis of the BdHN afterglows. The existence of regularities in the X-ray luminosity of BdHNe, expressed in the observer cosmological rest-frame, has been previously noted leading to the Muccino-Pisani power-law behavior (Pisani et al., 2013; Ruffini et al., 2014). The aim of this article is to now explain the origin of these power-law relations and to understand their physical origin and their energy sources.

The kinetic energy of the mildly relativistic expanding HN at c following the -ray flares and the X-ray flares, as well as the overall plateau phase, appears to have a crucial role (Ruffini et al., 2014). Equally crucial appears to be the contribution of the rotational energy electromagnetically radiated by the NS. As we show in this article, the power-law luminosity in the X-rays and in the optical wavelengths, expressed as a function of time in the GRB source rest-frame, could not be explained without their fundamental contribution. We here indeed assume that the afterglow originates from the synchrotron emission of relativistic electrons injected in the magnetized plasma of the HN, using both the kinetic energy of expansion and the electromagnetic energy powered by the rotational energy loss of the NS.

As an example, we apply this new approach to the afterglow of GRB 130427A associated to the SN 2013cq, in view of the excellent data available in X-rays, optical and radio wavelengths. We fit the spectral evolution of the GRB from 604 to  s and over the observed frequency bands from 10 Hz to 10 Hz. We present our simulations of the afterglow of GRB 130427A suggesting that a total energy of order  erg, composed of the kinetic energy of the HN and the rotational energy of the NS with a dipole/quadrupole magnetic field  G/ G and rotation period 2 ms, has been injected into the electrons confined within the expanding magnetized HN.

The article is organized as follows. In Sec. 2 we summarize how the BdHN treatment compares and contrasts with the traditional collapsar-fireball model of the GRB afterglow which is based on a single ultra-relativistic jet. In Sec. 3 we present the data reduction of GRB 130427A. In Sec. 4 we examine the basic parameters of the NS relevant for this analysis such as the rotation period, the mass, the rotational energy, and the magnetic field structure. We introduce in Sec. 5 the main ingredients and equations relevant for the computation of the synchrotron emission of the relativistic electrons injected in the magnetized HN. In Sec. 6 we set up the initial/boundary conditions to solve the model equations of Sec. 5. In Sec. 7 we compare and contrast the results of the numerical solution of our synchrotron model, the theoretical spectrum and light-curve, with the afterglow data of GRB 130427A at early times  s. We also show the role of the NS in powering the late,  s, X-ray afterglow. Finally, we outline our conclusions in Sec. 8 outlining some possible further observational predictions of our model.

2 On BdHNe versus traditional collapsar-fireball approach

That seven different GRB subclasses, all with binary systems as progenitors composed of various combinations of white dwarfs, CO, NSs and BHs and that only in three of these subclasses BHs are formed, was clearly established in (Ruffini et al., 2016).Far from being just a morphological classification, the identification of these systems and their properties has been made possible by the unprecedented quality and extents of the data ranging from X-ray, to the -ray, to the GeV emission as well as in the optical and in the radio. A comparable effort has been progressing in the theoretical field by introducing new paradigms and developing consistently the theoretical framework.

The main lesson gained from BdHN paradigm, one of the most numerous of the above seven subclasses, has been the successful identification, guided by the observational evidence, of a vast number of independent processes within a GRB. These processes have been successfully interpreted by their theoretical comprehension. For each process it has been possible to formulate the corresponding field equations, to integrate them to obtain the corresponding Lorentz factor as well as their space-time evolution. This is precisely what has been done in the recent publications for the X-ray flares (Ruffini et al., 2017b), for the prompt radiation and the -ray flares and for the extended thermal X-ray emission signing the transformation of a SN into an HN (Ruffini et al., 2017a).

Here we extend the BdHN model to the study of the afterglow utilizing the fundamental concept of the NS which has not been identified in its fundamental role in the previous publications. In this paper we are not going to address the topic of the GeV GRB radiation which is addressed separately.

We note that the BdHN paradigm as well as the model for the afterglow introduced here are in sharp contrast to the traditional model for long-duration GRBs and their associated afterglow (see e.g. reviews in Piran, 1999; Mészáros, 2002, 2006; Kumar & Zhang, 2015). The traditional fireball model consists of a single highly relativistic jet () originating from an already existing BH (collapsar). There is a large body of literature on this scenario (Sari et al., 1996; Waxman, 1997; Sari et al., 1998; Chevalier & Li, 2000; Kumar, 2000; Waxman, 2004; Zhang et al., 2006; Fan & Piran, 2006; Granot et al., 2006; Grupe et al., 2007; Duffell & MacFadyen, 2015; Kumar & Zhang, 2015). In all of these papers it is tacitly assumed that a single ultra-relativistic regime extends all the way from the prompt radiation, to the plateau phase (Nousek et al., 2006), all the way to the GeV emission and to the latest power-law of the afterglow. The equations of motion of this ultra-relativistic regime have been adopted by mutual consensus and not confronted nor directly derived from the observations. They have been uncritically assumed, and often have lead to discrepancies between the observations and the theoretical description (Troja et al., 2015, 2017). This approach based on a single process clearly contrast with the many different causally unrelated physical processes occurring in the BdHN.

3 GRB 130427A data

GRB 130427A is well-known for its high isotropic energy  erg, SN association and multi-wavelength observations (Ruffini et al., 2015). It triggered Fermi-GBM at 07:47:06.42 UT on April 27 2013 (von Kienlin, 2013), when it was within the field of view of Fermi-LAT. A a long-lasting ( s) burst of ultra-high energy ( MeV– GeV) radiation was observed (Ackermann et al., 2014). Swift started to follow from 07:47:57.51 UT,  s after the GBM trigger, observing a soft X-ray ( keV) afterglow for more than  days (Maselli et al., 2014). NuStar joined the observation during three epochs, approximately , and  days after the Fermi-GBM trigger, providing rare hard X-ray ( keV) afterglow observations (Kouveliotou et al., 2013). Ultraviolet, optical, infrared, radio observations were also performed by more than satellites and ground-based telescopes, within which Gemini-North, NOT, William Herschel, and VLT confirmed the redshift of (Levan et al., 2013; Xu et al., 2013a; Wiersema et al., 2013; Flores et al., 2013), and NOT found the associated supernova SN 2013cq (Xu et al., 2013b). We adopt the radio, optical and the GeV data from various published articles and GCNs (Perley et al., 2014; Maselli et al., 2014; von Kienlin, 2013; Sonbas et al., 2013; Xu et al., 2013b; Ruffini et al., 2015). The soft and hard X-rays, which are one the main subjects of this paper, are analyzed from the original data downloaded from Swift repository 111http://www.swift.ac.uk and NuStar archive 222https://heasarc.gsfc.nasa.gov/docs/nustar/nustar_archive.html. We follow the standard data reduction procedure using Heasoft 6.22 with relevant calibration files333http://heasarc.gsfc.nasa.gov/lheasoft/, and generate the spectra by XSPEC 12.9 (Evans et al., 2007, 2009). During the data reduction, the pile-up effect in the Swift-XRT are corrected for the first time bins (see Fig. 4) before  s (Romano et al., 2006). The NuStar spectrum at  s is inferred from the closest first  s of the NuStar third epoch at  days, by assuming the spectra at these two times have the same cutoff power-law shape but different amplitudes. The amplitude at  s is computed by fitting the NuStar light-curve. A K-correction is implemented for transferring observational data to the cosmological rest frame (Bloom et al., 2001).

4 Model for energetics and properties of the new fast-rotating NS

The angular momentum conservation implies that the NS should be rapidly rotating. For instance, we expect that the gravitational collapse of an iron core of radius  cm of a carbon-oxygen star leading to a Ic SN, rotating with a period  min, implies an initial NS rotation period  ms. Thus we should expect that the NS has a large amount of rotational energy available to power the SN remnant. In order to evaluate such a rotational energy we need to know the structure properties of fast rotating NSs, which we adopt from Cipolletta et al. (2015).

The structure of NSs in uniform rotation is obtained by numerical integration of the Einstein equations in axial symmetry and the stability sequences are described by two parameters, e.g.: the baryonic mass (or the gravitational mass/central density) and the angular momentum (or the angular velocity/polar to equatorial radius ratio). The stability of the star is bounded by (at least) two limiting conditions (see e.g. Stergioulas, 2003, for a review). The first is the mass-shedding or Keplerian limit: for a given mass (or central density) there is a configuration whose angular velocity equals the one of a test particle in circular orbit at the stellar equator. Thus, the matter at the stellar surface is marginally bound to it and any small perturbation causes mass loss/mass shedding to bring the star back to stability or otherwise to bring it to a point of dynamical instability. The second is the secular axisymmetric instability: in this limit the star becomes unstable against axially symmetric perturbations and is expected to evolve first quasi-stationarily to then find a dynamical instability point where gravitational collapse takes place. This instability sequence thus leads to the NS critical mass and it can be obtained via the turning-point method by Friedman et al. (1988). In Cipolletta et al. (2015) the values of the critical mass were obtained for the NL3, GM1 and TM1 EOS and the following fitting formula was found to describe them with a maximum error of 0.45%:

(1)

where is a dimensionless angular momentum parameter, is the NS angular momentum, and are parameters that depend on the nuclear EOS, and is the critical mass in the non-rotating case (see Table 1).

EOS    (km)    (km)  (ms)
NL3 17.35
GM1 16.12
TM1 15.98

Note. – In the last column we report the rotation period of the fastest possible configuration which corresponds to the one of the critical mass configuration (i.e. secularly unstable) that intersects the Keplerian mass-shedding sequence.

Table 1: Critical mass (and corresponding radius) obtained in Cipolletta et al. (2015) for selected parameterizations of nuclear EOS.

The configurations lying along the Keplerian sequence are also the maximally rotating ones (given a mass or central density). The fastest rotating NS is the configuration at the crossing point between the Keplerian and the secular axisymmetric instability sequences. Fig. 1 shows the minimum rotation period and the rotational energy as a function of the NS gravitational mass for the NL3 EOS.

Figure 1: Rotational energy and period of NSs along the Keplerian sequence for the NL3 EOS.

We turn now to the magnetosphere properties. Within the traditional model of pulsars (Goldreich & Julian, 1969), in a rotating, highly magnetized NS, a corotating magnetosphere is enforced up to a maximum distance , where is the speed of light and is the angular velocity of the star. This defines the so-called light cylinder since corotation at larger distances imply superluminal velocities of the magnetospheric particles. The last -field line closing within the corotating magnetosphere is located at an angle from the star’s pole. The -field lines that originate in the region between and (referred to as magnetic polar caps) cross the light cylinder and are called “open” field lines. Charged particles leave the star moving along the open field lines and escaping from the magnetosphere passing through the light cylinder.

At large distances from the light cylinder the magnetic field lines becomes radial and thus the magnetic field geometry is dominated by the toroidal component which decreases with the inverse of the distance. For typical pulsar magnetospheres it is expected to be related with the poloidal component of the field at the surface, , as (see Goldreich & Julian, 1969, for details)

(2)

up to a factor of order unity. Thus, as the SN remnant expands it finds a magnetized medium with a different value of the -field. We adopt a magnetic field of the form

(3)

with . We then seek the value of which fits best the data (see Secs. 57).

According to the previous agreement we have found between our model and GRB data (see e.g. Becerra et al., 2016; Ruffini et al., 2017b), we shall adopt values for and the expansion velocity (see below Secs. 57) and leave the parameter to be set by the fit of the afterglow data. We then compare and contrast the results with that expected from the NS theory.

5 Model for the Optical and X-ray Spectrum of the Afterglow

The origin of the observed afterglow emission is interpreted as due to the synchrotron emission of electrons accelerated in an expanding magnetic HN ejecta444We note that synchrotron emission of electrons in fast cooling regime has been previously applied in GRBs but to explain the prompt emission (see e.g. Uhm & Zhang, 2014).. A fraction of the kinetic energy of the ejecta is converted, through a shockwave, to accelerated particles (electrons) above GeV and TeV energies — enough to emit photons up to the X-ray band by synchrotron emission. Depending on the shock speed, number density, magnetic field, etc., different initial energy spectra of particles can be formed. In the most common cases, the accelerated particle distribution function can be described by a power law in the form of

(4)

where is the electron Lorentz factor, and are the minimum and maximum Lorenz factors, respectively. is the number of injected particles per second per energy, originating from the remnant impacted by the pair plasma of the GRB.

After the electrons are injected with the spectrum given by Eq. (4), the evolution of the particle distribution at a given time can be determined from the solution of the kinetic equation that describes the time-evolution of the electrons taking into account the particle energy losses (Kardashev, 1962)

(5)

where is the characteristic escape time and is the cooling rate. In the present case, for electrons, the escape time is much longer than the characteristic time of the cooling rate (fast cooling regime). The term includes various electron energy loss processes, such as synchrotron and inverse-Compton cooling as well as adiabatic losses due to the expansion of the emitting region. For the magnetic field considered here, the dominant cooling process for higher energy electrons is synchrotron emission (the electron cooling timescale due to inverse-Compton scattering is significantly longer) while adiabatic cooling can dominate for the low energy electrons at later phases. By introducing the expansion velocity of the remnant and its radius , the energy loss rate of electrons can be written as

(6)

where is the Thomson cross section and is the magnetic field strength. From the early X-ray data we find that the initial expansion velocity of GRB 130427A at times  s is (Ruffini et al., 2015), which then decelerates to at  s, as inferred from the SN optical data (Xu et al., 2013b). In our model we assume that the ejecta initially linearly decelerates until  s, and then it expands with a constant velocity of . In that case radius and expansion velocity of ejecta are given by

(7)
(8)

where  cm s,  cm s, and  cm s.

Due to the above decelerating expansion of the emitting region, the magnetic field decreases. Therefore we adopt a magnetic field that scales as with . We shall show below (see Sec. 7) that the data best fit with . This corresponds to conservation of magnetic flux for the longitudinal component.

The initial injection rate of particles, , depends on the energy budget of ejecta and on the efficiency of converting from kinetic to non-thermal energy. This can be defined as

(9)

where it is assumed that varies in time, based on the recent analyses of BdHNe which show that the X-ray light curve of GRB 130724A decays in time following a power-law of index (Ruffini et al., 2015; see Fig. 2). In our interpretation, the emission in the optical and X-ray bands is produced from synchrotron emission of electrons: if one assumes the electrons are constantly injected (), this will produce constant synchrotron flux. Thus we assume that the luminosity of electrons change from an initial value as follows:

(10)

where the index will be set in order to obtain a synchrotron luminosity (see Eq. 11) that fits the afterglow light-curve (see details below in Secs. 6 and 7).

The kinetic equation given in Eq. (5) has been solved numerically. The discretized electron continuity equation (5) is re-written in the form of a tridiagonal matrix which is solved using the implementation of the “tridiag” routine in Press et al. (1992). We have carefully tested our code by comparing the numerical results with the analytic solutions given in Kardashev (1962).

The synchrotron luminosity temporal evolution is calculated using with

(11)

where is the synchrotron spectra for a single electron which is calculated using the parameterization of the emissivity function of synchrotron radiation presented in Aharonian et al. (2010).

6 Initial Conditions for GRB 130724A

In Ruffini et al. (2017b) an analysis was completed for seven subclasses of GRBs including 345 identified BdHNe candidates, one of which is GRB 130724A that was seen in the Swift-XRT data and analyzed in detail in Ruffini et al. (2015). From the host-galaxy identification it is known that this burst occurred at a redshift . After transforming to the cosmological rest-frame of the burst and properly correcting for effects of the cosmological redshift and Lorentz time dilation, one can infer a time duration  s for 90% of the GRB emission. The isotropic energy emission in the range of –10 keV in the cosmological rest-frame of the burst is also deduced to be  erg and the total emission in the power-law afterglow can be inferred (Ruffini et al., 2015).

Fig. 2 shows the slope of the light-curve, defined by the logarithmic time derivative of the luminosity: slope = . This slope is obtained by fitting the luminosity light-curve in the cosmological rest-frame, using a machine learning, locally weighted regression (LWR) algorithm. We have made publicly available the corresponding technical details and codes to perform this calculation at: https://github.com/YWangScience/AstroNeuron. The green line is the slope of the soft X-ray, in the  keV range, and the blue line corresponds to the optical R-band, centered at  nm. The solid line covers the time when the data are well observed, while the dashed line, corresponds to an epoch in which observational data are missing. The rapid change of the slope implies variations of the energy injection, different emission mechanisms or different emission phases. The slope of the soft X-ray varies sharply at early times when various complicated GRB components (prompt emission, gamma-ray flare, X-ray flare) are occurring. Hence, we do not attempt to explain this early part with the synchrotron emission model defined above. We only consider times later than  s. Also we note that, at times later than  s, the slopes of the X-ray and R bands reach a common value of , indicated as a red line.

Figure 2: The slope of the afterglow light-curve of BdHN 130427A, defined by the logarithmic time derivative of the luminosity: slope = . This slope is obtained by fitting the luminosity light-curve in the cosmological rest-frame, using a machine learning, locally weighted regression (LWR) algorithm. For the corresponding technical details and codes we refer the reader to: https://github.com/YWangScience/AstroNeuron. The green line is the slope of the soft X-ray emission, in the  keV range, and the blue line corresponds to the optical R-band, centered at  nm.

Furthermore, we are not interested in explaining the GeV emission observed in most of BdHNe (when LAT data are available) with the synchrotron radiation model proposed here. Such emission has been explained in Ruffini et al. (2015) as originating from the further accretion of matter onto the newly-formed BH. This explanation is further reinforced by the fact that a similar GeV emission, following the same power-law decay with time, is also observed in the authentic short GRBs (S-GRBs; short bursts with  erg; see Ruffini et al., 2016) which are expected to be produced in NS-NS mergers leading to BH formation (Ruffini et al., 2016; Aimuratov et al., in preparation).

Regarding the model parameters, the initial velocity of the expanding ejecta is expected to be  cm s (Ruffini et al., 2015) from the thermal black body emission. Similarly, the radius at the beginning of the X-ray afterglow should be  cm. This corresponds to an expansion timescale of  s. These values are consistent with our previous theoretical simulations of BdHNe (Becerra et al., 2016). For our simulation of this burst we include all expected energy losses (synchrotron and adiabatic energy losses). However, the escape timescale was assumed to be large so that its effect could be neglected.

7 Results

Our modeling for the broadband spectral energy distribution (SED) of GRB 130724A for different periods is shown in Fig. 3. The corresponding parameters are given in Table 2. The radio emission is due to low-energy electrons that accumulate for longer periods. That is why the radio data are not included in the model. Only the optical and X-ray emissions are interpreted as due to synchrotron emission of electrons. Such emission, for instance at 604 s, is produced in a region with a radius of  cm and a magnetic field of  G. For this field strength synchrotron self-absorption can be significant as estimated following Rybicki & Lightman (1979). At the initial phases, when the system is compact and the magnetic field is large, synchrotron-self absorption can be neglected for the photons with frequencies above  Hz. Otherwise it is important. Thus, it is effective in reducing the radio flux predicted by the model, but not the optical and X-ray emission.

The optical and X-ray data can be well fit by a single power-law injection of electrons with and with initial minimum and maximum energies of ( GeV) and ( GeV), respectively. Due to the fast synchrotron cooling, the electrons are cooled rapidly forming a spectrum of for and for . The slope of the synchrotron emission () below the frequency defined by (e.g., ) is . This explains well both the optical and X-ray data.

For frequencies above , the slope is which continues up to . Since and depend on the magnetic field, they decrease with time, e.g. at  s,  Hz and  Hz. Due to the changes in the initial particle injection rate and magnetic field, the synchrotron luminosity also decreases. This is evident from Fig. 4, where the observed optical and X-ray light-curves of GRB 130427A are compared with the theoretical synchrotron emission light-curve obtained from Eq. (11). In this figure we also show the electron injection power given by Eq. (10). Here, it can be seen how the synchrotron luminosity fits the observed decay of the afterglow luminosity with the correct power-law index (see also Fig. 2).

Parameter Value
Table 2: Parameters used for simulation of GRB 130724A.
Figure 3: Model evolution of synchrotron spectral luminosity at various times compared with measurements in various spectral bands for GRB 130724A.
Figure 4: Optical and X-ray light-curves for GRB 130427A compared with the theoretical synchrotron light-curve from Eq. (11). We also show the electron injection power given by Eq. (10).

The SN ejecta is expected to become transparent to the NS radiation at around  s. Thus, we now discuss the pulsar emission that might power the late ( s) X-ray afterglow light-curve.

The late X-ray afterglow also shows a power-law decay of index which, as we show below, if powered by the pulsar implies the presence of a quadrupole magnetic field in addition to the traditional dipole one.

Thus, we adopt a dipole+quadrupole magnetic field model (see Pétri, 2015, for details). The luminosity from a pure dipole () is

(12)

where = 0 degrees gives the axisymmetric mode alone whereas = 90 degrees gives the mode alone. The braking index, following the traditional definition , is in this case .

On the other hand, the luminosity from a pure quadrupole field () is

(13)

where the different modes are easily separated by taking = 0 and any value of for , (, ) = (90, 0) degrees for and (, ) = (90, 90) degrees for . The braking index in this case is .

Thus, the quadrupole to dipole luminosity ratio is:

(14)

where

(15)

It can be seen that for the mode, and for the mode. For a  ms period NS, if , the quadrupole emission is about of the dipole emission, if , the quadrupole emission increases to times the dipole emission; and for a  ms pulsar, the quadrupole emission is negligible when , or only of the dipole emission even when . From this result one infers that the quadrupole emission dominates in the early fast rotation phase, then the NS spins down and the quadrupole emission drops faster than the dipole emission and, after tens of years, the dipole emission becomes the dominating component.

The evolution of the NS rotation and luminosity are given by

(16)

where is the moment of inertia. The solution is

(17)

where

(18)

and

(19)

The first and the second derivative of the angular velocity are

(20)
(21)

Therefore the braking index is

(22)

that in the present case ranges from to . From Eqs. (1720) we can compute the evolution of total pulsar luminosity as

(23)

Figure 5 shows the luminosity obtained from the above model for a pulsar with a radius of  cm,  G, an initial rotation period  ms, and for selected values of the parameter . This figure shows that the theoretical luminosity of pulsar is close to the soft X-ray luminosity observed in GRB 130427A when is around . This means, if choosing the harmonic mode , the quadrupole magnetic field is about times stronger than the dipole magnetic field. The luminosity of the pulsar before  s is mainly powered by the quadrupole emission, which is tens of times higher than the dipole emission. At about years the dipole emission starts to surpass the quadrupole emission and continues to dominate thereafter.

Figure 5: The observed luminosity of GRB 130427A in the  keV (grey points), and the theoretical luminosity from a pulsar for selected quadrupole to dipole magnetic field ratio and quadrupole angles in color lines. Other parameters of the pulsar are fixed: initial spin period  ms, dipole magnetic field  G, inclination angle , mass , radius  cm.

It is important to check the self-consistency of the estimated NS parameters obtained first from the early afterglow via synchrotron emission and then from the late X-ray afterglow via the pulsar luminosity. We can obtain from Eqs. (3) and (2), via the values of and from Table 2 and for  ms, an estimate of the dipole field at the NS surface from the synchrotron emission powering the early X-ray afterglow,  G. This value is to be compared with the one we have obtained from the pulsar luminosity powering the late afterglow,  G. The self-consistency of the two estimates is remarkable. In addition, the initial rotation period  ms for the NS is consistent with our estimate in Sec. 4 based upon angular momentum conservation during the gravitational collapse of the iron core leading to the NS. It can also be checked from Fig. 1 that is longer than the minimum period of a NS, which guarantees the gravitational and rotational stability of the NS.

8 Conclusions

We have constructed a model for a broad frequency range of the observed spectrum in the afterglow of BdHNe. We have made a specific fit to the BdHN 130427A as a representative example. We find that the parameters of the fit are consistent with the BdHN interpretation for this class of GRBs.

We have shown that the optical and X-ray emission of the early ( s s) afterglow is explained by the synchrotron emission by electrons expanding in the HN threading the magnetic field of the NS. At later times the HN becomes transparent and the electromagnetic radiation from the NS dominates the X-ray emission. We have inferred that the NS possesses an initial rotation period of 2 ms and a dipole magnetic field of (5–7) G. It is worth mentioning that we have derived the strength of the magnetic dipole independently by the synchrotron emission model at early times ( s) and by the magnetic braking model powering the late ( s) X-ray afterglow and show that they are in full agreement.

In this paper we proposed a direct connection between the afterglow of a BdHN and the physics of a newly born fast-rotating NS. This establishes a new self-enhancing understanding both of GRBs and young SNe which could be of fundamental relevance for the understanding of ultra-energetic cosmic rays and neutrinos as well as new ultra high energy phenomena.

It appears to be now essential to extend our comprehension in three different directions: 1) understanding of the latest phase of the afterglow; 2) the possible connection with historical supernovae; as well as 3) to extend observations from space of the GRB afterglow in the GeV and TeV energy bands. These last observations are clearly additional to the current observations of GRBs and GRB GeV radiation, originating from a Kerr-Newman BH and totally unrelated to the physics and astrophysics of afterglows.

One of the major verifications of our model can come from observing, in still active afterglows of historical GRBs, the pulsar-like emission from the NS we here predict, and the possible direct relation of the Crab Nebula to a BdHN is now open to further examination.

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.K., is supported by the Erasmus Mundus Joint Doctorate Program Grant N.2014–0707 from EACEA of the European Commission. J.A.R. acknowledges 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. Work at the University of Notre Dame (G.J.M.) is supported by the U.S. Department of Energy under Nuclear Theory Grant DE-FG02-95-ER40934. N.S. acknowledges the support of the RA MES State Committee of Science, in the frames of the research project No 15T-1C375.

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