On the structure of the burst and afterglow of Gamma-Ray Bursts I: the radial approximation

Abstract

We have proposed three paradigms for the theoretical interpretation of gamma-ray bursts (GRBs). (1) The relative space-time transformation (RSTT) paradigm emphasizes how the knowledge of the entire world-line of the source from the moment of gravitational collapse is a necessary condition to interpret GRB data. (2) The interpretation of the burst structure (IBS) paradigm differentiates in all GRBs between an injector phase and a beam-target phase. (3) The GRB-supernova time sequence (GSTS) paradigm introduces the concept of induced supernova explosion in the supernovae-GRB association. These three paradigms are illustrated using our theory based on the vacuum polarization process occurring around an electromagnetic black hole (EMBH theory) and using GRB 991216 as a prototype. We illustrate the five fundamental eras of the EMBH theory: the self acceleration of the $e^+e^-$ pair-electromagnetic plasma (PEM pulse), its interaction with the baryonic remnant of the progenitor star (PEMB pulse). We then study the approach of the PEMB pulse to transparency, the emission of the proper GRB (P-GRB) and its relation to the ``short GRBs''. Finally the three different regimes of the afterglow are described within the fully radiative and radial approximations. The best fit of the theory leads to an unequivocal identification of the ``long GRBs'' as extended emission occurring at the afterglow peak (E-APE). The relative intensities, the time separation and the hardness ratio of the P-GRB and the E-APE are used as distinctive observational test of the EMBH theory and the excellent agreement between our theoretical predictions and the observations are documented. The afterglow power-law indexes in the EMBH theory are compared and contrasted with the ones in the literature, and no beaming process is found for GRB 991216. Comment: 96 pages, 40 figures, to appear on Int. Journ. Mod. Phys. D

Full text

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arXiv:astro-ph/0302141v1 7 Feb 2003
International Journal of Modern Physics D
c ? World Scientific Publishing Company
On the structure of the burst and afterglow of Gamma-Ray Bursts I:
the radial approximation
REMO RUFFINI, CARLO LUCIANO BIANCO and SHE-SHENG XUE
ICRA — International Center for Relativistic Astrophysics and Dipartimento di Fisica,
Universit` a di Roma “La Sapienza”, Piazzale Aldo Moro 5, I-00185 Roma, Italy.
PASCAL CHARDONNET
ICRA — International Center for Relativistic Astrophysics and Universit´ e de Savoie,
LAPTH - LAPP, BP 110, F74941 Annecy-le-Vieux Cedex, France.
FEDERICO FRASCHETTI
ICRA — International Center for Relativistic Astrophysics and Universit` a di Trento,
Via Sommarive 14, I-38050 Povo (Trento), Italy.
We have recently proposed three paradigms for the theoretical interpretation of gamma-
ray bursts (GRBs). (1) The relative space-time transformation (RSTT) paradigm em-
phasizes how the knowledge of the entire world-line of the source from the moment
of gravitational collapse is a necessary condition in order to interpret GRB data.1(2)
The interpretation of the burst structure (IBS) paradigm differentiates in all GRBs
between an injector phase and a beam-target phase.2(3) The GRB-supernova time
sequence (GSTS) paradigm introduces the concept of induced supernova explosion in
the supernovae-GRB association.3In the introduction the RSTT and IBS paradigms
are enunciated and illustrated using our theory based on the vacuum polarization pro-
cess occurring around an electromagnetic black hole (EMBH theory). The results are
summarized using figures, diagrams and a complete table with the space-time grid, the
fundamental parameters and the corresponding values of the Lorentz gamma factor for
GRB 991216 used as a prototype. In the following sections the detailed treatment of the
EMBH theory needed to understand the results of the three above letters is presented.
We start from the considerations on the dyadosphere formation. We then review the
basic hydrodynamic and rate equations, the equations leading to the relative space-time
transformations as well as the adopted numerical integration techniques. We then illus-
trate the five fundamental eras of the EMBH theory: the self acceleration of the e+e−
pair-electromagnetic plasma (PEM pulse), its interaction with the baryonic remnant of
the progenitor star, the further self acceleration of the e+e−pair-electromagnetic ra-
diation and baryon plasma (PEMB pulse). We then study the approach of the PEMB
pulse to transparency, the emission of the proper GRB (P-GRB) and its relation to the
“short GRBs”. Particular attention is given to the free parameters of the theory and to
the values of the thermodynamical quantities at transparency. Finally the three different
regimes of the afterglow are described within the fully radiative and radial approxima-
tions: the ultrarelativistic, the relativistic and the nonrelativistic regimes. The best fit
of the theory leads to an unequivocal identification of the “long GRBs” as extended
emission occurring at the afterglow peak (E-APE). The relative intensities, the time
separation and the hardness ratio of the P-GRB and the E-APE are used as distinctive
observational test of the EMBH theory and the excellent agreement between our theoret-
1

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2R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.-S. Xue
ical predictions and the observations are documented. The afterglow power-law indexes
in the EMBH theory are compared and contrasted with the ones in the literature, and
no beaming process is found for GRB 991216. Finally, some preliminary results relating
the observed time variability of the E-APE to the inhomogeneities in the interstellar
medium are presented, as well as some general considerations on the EMBH formation.
The issue of the GSTS paradigm will be the object of a forthcoming publication and
the relevance of the iron-lines observed in GRB 991216 is shortly reviewed. The gen-
eral conclusions are then presented based on the three fundamental parameters of the
EMBH theory: the dyadosphere energy, the baryonic mass of the remnant, the interstel-
lar medium density. An in depth discussion and comparison of the EMBH theory with
alternative theories is presented as well as indications of further developments beyond
the radial approximation, which will be the subject of paper II in this series.4Future
needs for specific GRB observations are outlined.
Keywords: Afterglow, electromagnetic black hole theory, gamma-ray bursts
1. Introduction
1.1. The physical and astrophysical background
Gamma-ray bursts (GRBs) are rapidly fuelling one of the broadest scientific pursuit
in the entire field of science, both in the observational and theoretical domains.
Following the discovery of GRBs by the Vela satellites,5the observations from the
Compton satellite and BATSEahad shown the isotropic distribution of the GRBs
strongly suggesting a cosmological nature for their origin. It was still through the
data of BATSE that the existence of two families of bursts, the “short bursts” and
the “long bursts” was presented, opening an intense scientific dialogue on their
origin still active today, see e.g. Schmidt (2001)6and section 11.
An enormous momentum was gained in this field by the discovery of the after-
glow phenomena by the BeppoSAX satellite and the optical identification of GRBs
which have allowed the unequivocal identification of their sources at cosmological
distances.7It has become apparent that fluxes of 1054erg/s are reached: during
the peak emission the energy of a single GRB equals the energy emitted by all the
stars of the Universe.8
From an observational point of view, an unprecedented campaign of observations
is at work using the largest deployment of observational techniques from space with
the satellites CGRO-BATSE, Beppo-SAXb, Chandrac, R-XTEd, XMM-Newtone,
HETE-2f, as well as the HSTg, and from the ground with optical (KECKh, VLTi)
and radio (VLAj) observatories. The further possibility of examining correlations
aSee http://cossc.gsfc.nasa.gov/batse/
bSee http://www.asdc.asi.it/bepposax/
cSee http://chandra.harvard.edu/
dSee http://heasarc.gsfc.nasa.gov/docs/xte/
eSee http://xmm.vilspa.esa.es/
fSee http://space.mit.edu/HETE/
gSee http://www.stsci.edu/
hSee http://www2.keck.hawaii.edu:3636/
iSee http://www.eso.org/projects/vlt/
jSee http://www.aoc.nrao.edu/vla/html/VLAhome.shtml

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On the structure of the burst and afterglow of Gamma-Ray Bursts I: the radial approximation3
with the detection of ultra high energy cosmic rays, UHECR for short, and in
coincidence neutrinos should be reachable in the near future thanks to developments
of AUGERkand AMANDAl(see also Halzen, 20009).
Dyadosphere
Baryonic remnant
P-GRB
E-APE
r+ = 6.0*106 cm
rds = 2.4*108 cm
r = 1.2*1010 cm
r0 = 1.9*1014 cm
rPA = 5.2*1016 cm
<ρB> << <ρe+e-> ~ 105 - 1013 g/cm3
<ρB> = 1 g/cm3
<ρB> = 10-24 g/cm3
EMBH
Fig. 1.
energy density of the medium and the distances from the EMBH, in the laboratory frame and in
logarithmic scale, are given.
Selected events in the EMBH theory are represented. For each one the values of the
From a theoretical point of view, GRBs offer comparable opportunities to de-
velop entire new domains in yet untested directions of fundamental science. For the
first time within the theory based on the vacuum polarization process occurring in
an electromagnetic black hole, the EMBH theory, see Fig. 1, the opportunity exists
to theoretically approach the following fundamental issues:
(1) The extremely relativistic hydrodynamic phenomena of an electron-positron
plasma expanding with sharply varying gamma factors in the range 102to 104
and the analysis of the very high energy collision of such an expanding plasma
with baryonic matter reaching intensities 1038larger than the ones usually
obtained in Earth-based accelerators.
(2) The bulk process of vacuum polarisation created by overcritical electromagnetic
fields, in the sense of Heisenberg, Euler10and Schwinger11. This longly sought
quantum ultrarelativistic effect has not been yet unequivocally observed in
heavy ion collision on the Earth.12,13,14,15The difficulty of the heavy ion
collision experiments appears to be that the overcritical field is reached only for
kSee http://www.auger.org/
lSee http://amanda.berkeley.edu/amanda/amanda.html

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4R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.-S. Xue
time scales of the order ¯ h/mpc2, which is much shorter than the characteristic
time for the e+e−pair creation process which is of the order of ¯ h/mec2, where
mp and me are respectively the proton and the electron mass. It is therefore
very possible that the first appearance of such an effect occurs in the strong
electromagnetic fields developed in astrophysical conditions during the process
of gravitational collapse to an EMBH, where no problem of confinement exists.
(3) A novel form of energy source: the extractable energy of a black hole. The
enormous energies released almost instantly in the observed GRBs, points to the
possibility that for the first time we are witnessing the release of the extractable
energy of an EMBH, during the process of gravitational collapse itself. We can
compute and have the opportunity to study all general relativistic as well as the
associated ultrahigh energy quantum phenomena as the horizon of the EMBH
is approached and is being formed.
It is clear that in approaching such a vast new field of research, implying pre-
viously unobserved relativistic regimes, it is not possible to proceed as usual with
an uncritical comparison of observational data to theoretical models within the
classical schemes of astronomy and astrophysics. Some insight to the new approach
needed can be gained from past experience in the interpretation of relativistic effects
in high energy particle physics as well as from the explanation of some observed
relativistic effects in the astrophysical domain. Those relativistic regimes, both in
physics and astrophysics, are however much less extreme than those encountered
in GRBs.
There are three major new features in relativistic systems which have to be
properly taken into account:
(1) Practically all data on astronomical and astrophysical systems is acquired by
using photon arrival times. It was Einstein16at the very initial steps of special
relativity who cautioned about the use of such an arrival time analysis and
stated that when dealing with objects in motion proper care should be taken in
defining the time synchronization procedure in order to construct the correct
space-time coordinate grid (see Fig. 2). It is not surprising that as soon as the
first relativistic bulk motion effects were observed their interpretations within
the classical framework of astrophysics led to the concept of “superluminal”
motion. These were observations of extragalactic radio sources, with gamma
factors17∼ 10 and of microquasars in our own galaxy with gamma factor18
∼ 5. It has been recognized19that no “superluminal” motion exists if the
prescriptions indicated by Einstein are used in order to establish the correct
space-time grid for the astrophysical systems. In the present context of GRBs,
where the gamma factor can easily surpass 102, the direct application of clas-
sical concepts leads to enormous “superluminal” behaviours (see Tab. 1). An
approach based on classical arrival time considerations as sometimes done in
the current literature completely subverts the causal relation in the observed

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On the structure of the burst and afterglow of Gamma-Ray Bursts I: the radial approximation5
t
t0
t0
∆t
+
∆ta
R
R0
r
Fig. 2.
and the arrival time interval ∆ta for a pulse moving with velocity v in the laboratory time (solid
line). We have indicated here the case where the motion of the source has a nonzero acceleration.
The arrival time is measured using light signals emitted by the pulse (dotted lines). R0 is the
distance of the observer from the EMBH, t0 is the laboratory time corresponding to the onset of
the gravitational collapse, and r is the radius of the expanding pulse at a time t = t0+ ∆t.1
This qualitative diagram illustrates the relation between the laboratory time interval ∆t
astrophysical phenomenon.
(2) One of the clear successes of relativistic field theories has been the understand-
ing of the role of four-momentum conservation laws in multiparticle collisions
and decays such as in the reaction: n → p+e−+¯ νe. From the works of Pauli and
Fermi it became clear how in such a process, contrary to the case of classical
mechanics, it is impossible to analyze a single term of the decay, the electron
or the proton or the neutrino or the neutron, out of the context of the global
point of view of the relativistic conservation of the total four momentum of
the system. This in turn involves the knowledge of the system during the entire
decay process. These rules are routinely used by workers in high energy particle

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6R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.-S. Xue
physics and have become part of their cultural background. If we apply these
same rules to the case of the relativistic system of a GRB it is clear that it is just
impossible to consider a part of the system, e.g. the afterglow, without taking
into account the general conservation laws and whole relativistic history of the
entire system. The description of the afterglow alone, as has been given at times
in the literature, indeed possible within the framework of classical astronomy
and astrophysics, is not viable in a relativistic astrophysics context where the
space-time grid necessary for the description of the afterglow depends on the
entire previous relativistic part of the worldline of the system (see also section
14).
(3) The lifetime of a process has not an absolute meaning as special and general
relativity have shown. It depends both on the inertial reference frame of the lab-
oratory and of the observer and on their relative motion. Such a phenomenon,
generally expressed in the “twin paradox”, has been extensively checked and
confirmed to extremely high accuracy as a byproduct of the elementary particle
physics (g-2) experiment.20This situation is much more extreme in GRBs due
to the very large (in the range 102–104) and time varying (on time scales rang-
ing from fractions of seconds to months) gamma factors between the comoving
frame and the far away observer (see Fig. 9). Moreover in the GRB context
such an observer is also affected by the cosmological recession velocities of its
local Lorentz frame.
1.2. The Relative Space-Time Transformations: the RSTT
paradigm and current scientific literature
Here are some of the reasons why we have recently presented a basic relative space-
time transformation (RSTT) paradigm1to be applied prior to the interpretation
of GRB data.
The first step is the establishment of the governing equations relating:
a) The comoving time of the pulse (τ)
b) The laboratory time (t)
c) The arrival time at the detector (ta)
d) The arrival time at the detector corrected for cosmological expansion (td
The book-keeping of the four different times and corresponding space variables must
be done carefully in order to keep the correct causal relation in the time sequence
of the events involved.
As formulated the RSTT paradigm contains two parts: the first one is a nec-
essary condition, the second one a sufficient condition. The first part reads: “the
necessary condition in order to interpret the GRB data, given in terms of the arrival
time at the detector, is the knowledge of the entire worldline of the source from the
gravitational collapse”.
Clearly such an approach is in contrast with articles in the current literature
which emphasize either some qualitative description of the sources or some quanti-
a)

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On the structure of the burst and afterglow of Gamma-Ray Bursts I: the radial approximation7
tative description of the afterglow era by itself.
In the current literature several attempts have addressed the issue of the sources
of GRBs. They include scenarios of binary neutron stars mergers,21,22,23,24black
hole / white dwarf25and black hole / neutron star binaries,26,27hypernovae,28
failed supernovae or collapsars,29,30supranovae.31,32Only those based on binary
neutron stars have reached the stage of a definite model and detailed quantitative
estimates have been made. In this case, however, various problems have surfaced: in
the general energetics which cannot be greater than ∼ 3×1052erg, in the explana-
tion of “long bursts”,33,34and in the observed location of the GRB sources in star
forming regions.35In the remaining cases attention was directed to a qualitative
analysis of the sources without addressing the overall problem from the source to
the observations. The necessary details to formulate the equations of the dynamical
evolution of the system are generally missing.
Other models in the literature have addressed the problem of only fitting the
data of the afterglow observations by a phenomenological analysis. They are sepa-
rated into two major classes:
The “internal shock model”, first introduced by Rees & M´ esz´ aros (1994),36
by far the most popular one, has been developed in many different aspects, e.g. by
Paczy´ nski & Xu (1994),37Sari & Piran (1997),38Fenimore (1999)39and Fenimore
et al. (1999)40. The underlying assumption is that all the variabilities of GRBs in
the range ∆t ∼ 1ms up to the overall duration T of the order of 50s are determined
by a yet undetermined “inner engine”. The difficulties of explaining the long time
scale bursts by a single explosive model has evolved into a subclass of approaches
assuming an “inner engine” with extended activity (see e.g. Piran, 2001,41and
references therein).
The “external shock model”, also introduced by M´ esz´ aros & Rees (1993),42
is less popular today. It relates the GRB light curves and time variabilities to
interactions of a single thin blast wave with clouds in the external medium. The
interesting possibility has been recognized within this model, that GRB light curves
“are tomographic images of the density distribution of the medium surrounding the
sources of GRBs” (Dermer & Mitman, 199943) see also Dermer, Chiang & B¨ ottcher
(1999),44Dermer (2002)45and references therein. In this case, the structure of the
burst is assumed not to depend directly on the “inner engine” (see e.g. Piran,
2001,41and references therein).
All these works encounter the above mentioned difficulty: they present either
a purely qualitative or phenomenological or a piecewise description of the GRB
phenomenon. By neglecting the earlier phases, their space-time grid is undefined
and as we will explicitly show in the following, results are reached at variance from
the ones obtained in a complete and unified description of the GRB phenomenon.
We show in the following how such a unified description naturally leads to new
characteristic features both in the burst and afterglow of GRBs.

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8R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.-S. Xue
1.3. The EMBH Theory
In a series of papers, we have developed the EMBH theory46which has the ad-
vantage, despite its simplicity, that all eras following the process of gravitational
collapse are described by precise field equations which can then be numerically
integrated.
Starting from the vacuum polarization process ` a la Heisenberg-Euler-Schwinger10,11
in the overcritical field of an EMBH first computed in Damour & Ruffini (1975),47
we have developed the dyadosphere concept.48
The dynamics of the e+e−-pairs and electromagnetic radiation of the plasma
generated in the dyadosphere propagating away from the EMBH in a sharp pulse
(PEM pulse) has been studied by the Rome group and validated by the numerical
codes developed at Livermore Lab.49
The collision of the still optically thick e+e−-pairs and electromagnetic radiation
plasma with the baryonic matter of the remnant of the progenitor star has been
again studied by the Rome group and validated by the Livermore Lab codes.50The
further evolution of the sharp pulse of pairs, electromagnetic radiation and baryons
(PEMB pulse) has been followed for increasing values of the gamma factor until
the condition of transparency is reached.51
As this PEMB pulse reaches transparency the proper GRB (P-GRB) is emitted2
and a pulse of accelerated baryonic matter (the ABM pulse) is injected into the
interstellar medium (ISM) giving rise to the afterglow.
1.4. The GRB 991216 as a prototypical source
Until this stage, the EMBH theory has been done from first principles based on the
exact solutions of the Einstein-Maxwell equations implied by the EMBH uniqueness
theorem as well as on the quantum description of the vacuum polarization process
in overcritical electromagnetic fields. Turning now to the afterglow, the variety of
physical situations that can possibly be encountered are very large and far from
unique: the description from first principles is just impossible. We have therefore
proceeded to properly identify what we consider a prototypical GRB source and
to develop a theoretical framework in close correspondence with the observational
data.
We present the criteria which have guided us in the selection of the GRB source
to be used as a prototype before proceeding to an uncritical comparison with
the theory. It is now clear, since the observations of GRB 980425, GRB 991216,
GRB 970514 and GRB 980326 that the afterglow phenomena can present, espe-
cially in the optical and radio wavelengths, features originating from phenomena
spatially and causally distinct from the GRB phenomena. There is the distinct
possibility that phenomena related to a supernova can be erroneously attributed
to a GRB. This problem has been clearly addressed by the GRB supernova time
sequence (GSTS) paradigm in which the time sequence of the events in the GRB
supernova phenomena has been outlined.3This has led to the novel concept of an

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On the structure of the burst and afterglow of Gamma-Ray Bursts I: the radial approximation9
induced supernova.3This problem will be addressed in a forthcoming paper.52
Fig. 3.
Rapid Burst Response53); b) The afterglow emission of GRB 991216 as seen by XTE and Chandra
(reproduced from Halpern et al., 200054)
a) The peak emission of GRB 991216 as seen by BATSE (Reproduced from BATSE
In view of these considerations we have selected GRB 991216 as a prototypical
case (see Fig. 3) for the following reasons:

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10R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.-S. Xue
(1) GRB 991216 is one of the strongest GRBs in X-rays and is also quite general in
the sense that it shows relevant cosmologicaleffects. It radiates mainly in X-rays
and in γ-rays and less than 3% is emitted in the optical and radio bands.54
(2) The excellent data obtained by BATSE on the burst53is complemented by the
data on the afterglow acquired by Chandra55and RXTE.56Also superb data
have been obtained from spectroscopy of the iron lines.55
(3) A value for the slope of the energy emission during the afterglow as a function
of time has been obtained: n = −1.6457and n = −1.616± 0.067.54
1.5. The interpretation of the burst structure: the IBS paradigm
and the different eras of the EMBH theory
The comparison of the EMBH theory with the data of the GRB 991216 and its
afterglow has naturally led to a new paradigm for the interpretation of the burst
structures (IBS paradigm)) of GRBs.2The IBS paradigm reads: “In GRBs we
can distinguish an injector phase and a beam-target phase. The injector phase in-
cludes the process of gravitational collapse, the formation of the dyadosphere, as
well as Era I (the PEM pulse), Era II (the engulfment of the baryonic matter of the
remnant) and Era III (the PEMB pulse). The injector phase terminates with the P-
GRB emission. The beam-target phase addresses the interaction of the ABM pulse,
namely the beam generated during the injection phase, with the ISM as the target.
It gives rise to the E-APE and the decaying part of the afterglow”. The detailed
presentations of these results are the main topic of this article.
We recall that the injector phase starts from the moment of gravitational
collapse and encompasses the following eras:
The Zeroth Era: the formation of the dyadosphere. In section 2 we review the
basic scientific results which lie at the basis of the EMBH theory: the black hole
uniqueness theorem, the mass formula of an EMBH, the process of vacuum polar-
ization in the field of an EMBH. We also point out how after the discovery of the
GRB afterglow the reexamination of these results has led to the novel concept of
the dyadosphere of an EMBH. We have investigated this concept in the simplest
possible case of an EMBH depending only on two parameters: the mass and charge,
corresponding to the Reissner-Nordstr¨ om spacetime. We recall the definition of the
energy Edyaof the dyadosphere as well as the spatial distribution and energetics of
the e+e−pairs. See Fig. 4.
In order to analyse the time evolution of the dyadosphere we give in the three
following sections the theoretical background for the needed equations.
In section 3 we give the general relativistic equations governing the hydrody-
namics and the rate equations for the plasma of e+e−-pairs.
In section 4 we give the governing equations relating the comoving time τ to the
laboratory time t corresponding to an inertial reference frame in which the EMBH
is at rest and finally to the time measured at the detector tawhich, to finally get
td
a, must be corrected to take into account the cosmological expansion.

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On the structure of the burst and afterglow of Gamma-Ray Bursts I: the radial approximation11
+Q-Q
cme
ℏ
-
-
-
-
-
+
+
+
+
+
-
-
-
-
-
+
+
+
+
+
-
-
-
-
-
+
+
+
+
+
plasma (e+e– γ)
+
−=∆
rrr
ds
Fig. 4.
by a concentric set of shells of capacitors, each one of thickness ¯ h/mec and producing a number
of e+e−pairs of the order of ∼ Q/e on a time scale of 10−21s, where Q is the EMBH charge.
The shells extend in a region ∆r, from the horizon r+ to the dyadosphere outer radius rds(see
text). The system evolves to a thermalised plasma configuration.
The dyadosphere of a Reissner-Nordstr¨ om black hole can be represented as constituted
In section 5 we describe the numerical integration of the hydrodynamical equa-
tions and the rate equation developed by the Rome and Livermore groups. This
entire research program could never have materialized without the fortunate inter-
action between the complementary computational techniques developed by these
two groups. The validation of the results of the Rome group by the fully general
relativistic Livermore codes has been essential both from the point of view of the
validity of the numerical results and the interpretation of the scientific content of
the results.
The Era I: the PEM pulse. In section 3 by the direct comparison of the inte-
grations performed with the Rome and Livermore codes we show that among all

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12R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.-S. Xue
possible geometries the e+e−plasma moves outward from the EMBH reaching a
very unique relativistic configuration: the plasma self-organizes in a sharp pulse
which expands in the comoving frame exactly by the amount which compensates
for the Lorentz contraction in the laboratory frame. The sharp pulse remains of
constant thickness in the laboratory frame and self-propels outwards reaching ul-
trarelativistic regimes, with gamma factors larger than 102, in a few dyadosphere
crossing times. We recall that, in analogy with the electromagnetic (EM) pulse
observed in a thermonuclear explosion on the Earth, we have defined this more
energetic pulse formed of electron-positron pairs and electromagnetic radiation a
pair-electromagnetic-pulse or PEM pulse.
The Era II: We describe the interaction of the PEM pulse with the baryonic
remnant of mass MB left over from the gravitational collapse of the progenitor
star. We give the details of the decrease of the gamma factor and the corresponding
increase in the internal energy during the collision. The dimensionless parameter
B = MBc2/Edyawhich measures the baryonic mass of the remnant in units of the
Edyais introduced. This is the second fundamental free parameter of the EMBH
theory.
The Era III: We describe in section 8 the further expansion of the e+e−plasma,
after the engulfment of the baryonic remnant of the progenitor star. By direct
comparison of the results of integration obtained with the Rome and the Livermore
codes it is shown how the pair-electromagnetic-baryon (PEMB) plasma further
expands and self organizes in a sharp pulse of constant length in the laboratory
frame (see Fig. 5). We have examined the formation of this PEMB pulse in a wide
range of values 10−8< B < 10−2of the parameter B, the upper limit corresponding
to the limit of validity of the theoretical framework developed.
In section 9 it is shown how the effect of baryonic matter of the remnant,
expressed by the parameter B, is to smear out all the detailed information on
the EMBH parameters. The evolution of the PEMB pulse is shown to depend only
on Edyaand B: the PEMB pulse is degenerate in the mass and charge parameters
of the EMBH and rather independent of the exact location of the baryonic matter
of the remnant.
In section 10 the relevant thermodynamical quantities of the PEMB pulse, the
temperature in the different frames and the e+e−pair densities, are given and the
approach to the transparency condition is examined. Particular attention is given
to the gradual transfer of the energy of the dyadosphere Edyato the kinetic energy
of the baryons EBaryonsduring the optically thick part of the PEMB pulse.
In section 11, as the condition of transparency is reached, the injector phase
is concluded with the emission of a sharp burst of electromagnetic radiation and
an accelerated beam of highly relativistic baryons. We recall that we have respec-
tively defined the radiation burst (the proper GRB or for short P-GRB) and the
accelerated-baryonic-matter (ABM) pulse. By computing for a fixed value of the
EMBH different PEMB pulses corresponding to selected values of B in the range

Page 13

On the structure of the burst and afterglow of Gamma-Ray Bursts I: the radial approximation 13
Fig. 5.
(Livermore code) and slab calculations (Rome code) as a function of the radial coordinate (in units
of dyadosphere radius) in the laboratory frame. The calculations show an excellent agreement.
Comparison of gamma factor for the one-dimensional (1-D) hydrodynamic calculations
?10−8–10−2?, it has been possible to obtain a crucial universal diagram which is
in the P-GRB and a negligible fraction is emitted in the kinetic energy EBaryons
of the baryonic matter and therefore in the afterglow. On the other hand in the
limit B → 10−2which is also the limit of validity of our theoretical framework,
almost all Edyais transferred to EBaryonsand gives origin to the afterglow and the
intensity of the P-GRB correspondingly decreases. We have identified the limiting
case of negligible values of B with the process of emission of the so called “short
bursts”. A complementary result reinforcing such an identification comes from the
thermodynamical properties of the P-GRB: the hardness of the spectrum decreases
reproduced in Fig.6. In the limit of B → 10−8or smaller almost all Edyais emitted

Page 14

14R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.-S. Xue
0
0.2
0.4
0.6
0.8
1
1e-0081e-0071e-006 1e-0050.0001 0.0010.010.1
(Energy)/(Edya)
B
Fig. 6.
kinetic energy EBaryonsof baryonic matter (the dashed line) in units of the total energy of the
dyadosphere (Edya) are plotted as functions of the B parameter.
At the transparent point, the energy radiated in the P-GRB (the solid line) and the final
for increasing values of B, see Fig. 7.
The injector phase is concluded by the emission of the P-GRB and the ABM
pulse, as the condition of transparency is reached.
The beam-target phase, in which the accelerated baryonic matter (ABM)
generated in the injector phase collides with the ISM, gives origin to the afterglow.
Again for simplicity we have adopted a minimum set of assumptions:
(1) The ABM pulse is assumed to collide with a constant homogeneous interstellar
medium of number density nism∼ 1cm−3. The energy emitted in the collision
is assumed to be instantaneously radiated away (fully radiative condition). The
description of the collision and emission process is done using spherical sym-
metry, taking only the radial approximation neglecting all the delayed emission
due to off-axis scattered radiation.
(2) Special attention is given to numerically compute the power of the afterglow
as a function of the arrival time using the correct governing equations for the
space-time transformations in line with the RSTT paradigm.
(3) Finally some approximate solutions are adopted in order to determine the power
law exponents of the afterglow flux and compare and contrast them with the
observational results as well as with the alternative results in the literature.

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On the structure of the burst and afterglow of Gamma-Ray Bursts I: the radial approximation 15
1
10
100
1000
10000
1e-0081e-0071e-0061e-0050.00010.0010.01 0.1
Energy Peak (Ep) (KeV)
B
Fig. 7.
measured in the laboratory frame is plotted as function of the B parameter.
The energy corresponding to the peak of the photon number spectrum in the P-GRB as
In this paper we only consider the above mentioned radial approximation and a
spherically symmetric distribution in order to concentrate on the role of the correct
space-time transformations in the RSTT paradigm and illustrate their impact on
the determination of the power law index of the afterglow. This topic has been
seriously neglected in the literature. Details of the role of beaming and on the
diffusion due to off-axis emission will be studied elsewhere.58,59
We can now turn to the two eras of the beam-target phase:
The Era IV: the ultrarelativistic and relativistic regimes in the afterglow. In
section 12 the hydrodynamic relativistic equations governing the collision of the
ABM pulse with the interstellar matter are given in the form of a set of finite dif-
ference equations to be numerically integrated. Expressions for the internal energy
developed in the collision as well as for the gamma factor are given as a function
of the mass of the swept up interstellar material and of the initial conditions. In
section 17 the infinitesimal limit of these equations is given as well as analytic
power-law expansions in selected regimes.
The Era V: the approach to the nonrelativistic regimes in the afterglow. In
section 13 it is stressed that this last era often discussed in the current literature
can be described by the same equations used for era IV.
Having established all the governing equations for all the eras of the EMBH