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arXiv:astroph/0302141v1 7 Feb 2003
International Journal of Modern Physics D
c ? World Scientific Publishing Company
On the structure of the burst and afterglow of GammaRay Bursts I:
the radial approximation
REMO RUFFINI, CARLO LUCIANO BIANCO and SHESHENG XUE
ICRA — International Center for Relativistic Astrophysics and Dipartimento di Fisica,
Universit` a di Roma “La Sapienza”, Piazzale Aldo Moro 5, I00185 Roma, Italy.
PASCAL CHARDONNET
ICRA — International Center for Relativistic Astrophysics and Universit´ e de Savoie,
LAPTH  LAPP, BP 110, F74941 AnnecyleVieux Cedex, France.
FEDERICO FRASCHETTI
ICRA — International Center for Relativistic Astrophysics and Universit` a di Trento,
Via Sommarive 14, I38050 Povo (Trento), Italy.
We have recently proposed three paradigms for the theoretical interpretation of gamma
ray bursts (GRBs). (1) The relative spacetime transformation (RSTT) paradigm em
phasizes how the knowledge of the entire worldline 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 beamtarget phase.2(3) The GRBsupernova time
sequence (GSTS) paradigm introduces the concept of induced supernova explosion in
the supernovaeGRB 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 spacetime 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 spacetime
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−
pairelectromagnetic plasma (PEM pulse), its interaction with the baryonic remnant of
the progenitor star, the further self acceleration of the e+e−pairelectromagnetic ra
diation and baryon plasma (PEMB pulse). We then study the approach of the PEMB
pulse to transparency, the emission of the proper GRB (PGRB) 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 (EAPE). The relative intensities, the time
separation and the hardness ratio of the PGRB and the EAPE 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 powerlaw 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 EAPE 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 ironlines 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, gammaray bursts
1. Introduction
1.1. The physical and astrophysical background
Gammaray 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 CGROBATSE, BeppoSAXb, Chandrac, RXTEd, XMMNewtone,
HETE2f, 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 GammaRay 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
PGRB
EAPE
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> = 1024 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 electronpositron
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 Earthbased 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
spacetime 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
spacetime 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 GammaRay 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 fourmomentum 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
spacetime 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 (g2) 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 SpaceTime 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 bookkeeping 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 GammaRay 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 spacetime 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 HeisenbergEulerSchwinger10,11
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 (PGRB) 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 EinsteinMaxwell 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 GammaRay 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 Xrays and is also quite general in
the sense that it shows relevant cosmologicaleffects. It radiates mainly in Xrays
and in γrays 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 beamtarget 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 beamtarget 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 EAPE 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 ReissnerNordstr¨ 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 GammaRay 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 ReissnerNordstr¨ 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 selforganizes 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 selfpropels 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 electronpositron pairs and electromagnetic radiation a
pairelectromagneticpulse 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 pairelectromagneticbaryon (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 PGRB) and the
acceleratedbaryonicmatter (ABM) pulse. By computing for a fixed value of the
EMBH different PEMB pulses corresponding to selected values of B in the range
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On the structure of the burst and afterglow of GammaRay 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 onedimensional (1D) hydrodynamic calculations
?10−8–10−2?, it has been possible to obtain a crucial universal diagram which is
in the PGRB 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 PGRB 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 PGRB: the hardness of the spectrum decreases
reproduced in Fig.6. In the limit of B → 10−8or smaller almost all Edyais emitted
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14R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
0
0.2
0.4
0.6
0.8
1
1e008 1e007 1e0061e005 0.00010.0010.01 0.1
(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 PGRB (the solid line) and the final
for increasing values of B, see Fig. 7.
The injector phase is concluded by the emission of the PGRB and the ABM
pulse, as the condition of transparency is reached.
The beamtarget 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 offaxis 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
spacetime 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.
Page 15
On the structure of the burst and afterglow of GammaRay Bursts I: the radial approximation15
1
10
100
1000
10000
1e0081e0071e006 1e005 0.0001 0.0010.01 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 PGRB 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
spacetime 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 offaxis emission will be studied elsewhere.58,59
We can now turn to the two eras of the beamtarget 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
powerlaw 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
Page 16
16R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
theory, we can proceed to compare and contrast the predictions of this theory with
the observational data.
1.6. The Best fit of the EMBH theory to the GRB 991216: the
global features of the solution
As expressed in section 14, we have proceeded to the identification of the only two
free parameters of the EMBH theory, Edyaand B, by fitting the observational data
from RXTE and Chandra on the decaying part of the GRB 991216 afterglow.
The afterglow appears to have three different parts: in the first part the luminosity
increases as a function of the arrival time, it then reaches a maximum and finally
monotonically decreases. In Fig. 8, we show how such a fit is actually made and
how changing the two free parameters affects the intensity and the location in time
of the peak of the afterglow. The best fit is obtained for Edya= 4.83×1053erg and
B = 3 × 10−3.
Having determined the two free parameters of the theory, we have integrated the
governing equations corresponding to these values and then obtained for the first
time the complete history of the gamma factor from the moment of gravitational
collapse to the latest phases of the afterglow observations (see Fig. 9). We have also
determined the different regimes encountered in the relation between the laboratory
time and the detector arrival time within the RSTT paradigm (see Fig. 10). We
have thus determined the entire spacetime grid of the GRB 991216 by giving
(see Tab. 1) the radial coordinate of the GRB phenomenon as a function of the
four coordinate time variables. A quick glance to Tab. 1 shows how the extreme
relativistic regimes at work lead to enormous superluminal behaviour (up to 105c!)
if the classical astrophysical concepts are adopted using the arrival time as the
independent variable. In turn this implies that any causal relation based on classical
astrophysics and the arrival time data, as often found in the current GRB literature,
is incorrect.
1.7. The explanation of the “long bursts” and the identification of
the proper gamma ray burst(PGRB)
In section 15, having determined the two free parameters of the EMBH theory, we
analyze the theoretical predictions of this theory for the general structure of GRBs.
The first striking result, illustrated in Fig. 11, shows that the peak of the afterglow
emission coincides both in intensity and in arrival time (19.87s) with the average
emission of the long burst observed by BATSE. For this we have introduced the
new concept of extended afterglow peak emission (EAPE). Once the proper space
time grid is given (see Tab. 1) it is immediately clear that the EAPE is generated
at distances of 5 × 1016cm from the EMBH. The long bursts are then identified
with the EAPEs and are not bursts at all: they have been interpreted as bursts
only because of the high threshold of the BATSE detectors (see Fig. 11). Thus the
Page 17
On the structure of the burst and afterglow of GammaRay Bursts I: the radial approximation17
1040
1042
1044
1046
1048
1050
1052
1054
102
100
102
Detector arrival time (ta
104
106
108
1016
1014
1012
1010
108
106
104
Source luminosity (ergs/(s*sterad))
Observed flux (ergs/(cm2*s))
d) (s)
a)
GRB991216 Observed afterglow data
Edya=4.49*1055 ergs
Edya=4.82*1053 ergs
Edya=5.29*1051 ergs
1040
1042
1044
1046
1048
1050
1052
1054
102
100
102
Detector arrival time (ta
104
106
108
1016
1014
1012
1010
108
106
104
Source luminosity (ergs/(s*sterad))
Observed flux (ergs/(cm2*s))
d) (s)
b)
GRB991216 Observed afterglow data
B=9.0*103
B=6.0*103
B=3.0*103
B=1.0*103
B=7.0*104
B=4.0*104
Fig. 8.
4.83 × 1053erg, Edya= 4.49 × 1055erg and B = 3 × 10−3. b) for the Edya= 4.83 × 1053, we
give the afterglow luminosities corresponding respectively to B = 9 × 10−3, 6 × 10−3, 3 × 10−3,
1 × 10−3, 7 × 10−4, 4 × 10−4.
a) Afterglow luminosity computed for an EMBH of Edya= 5.29 × 1051erg, Edya=
long standing unsolved problem of explaining the long GRBs34,33,41is radically
resolved.
Still in section 15, the search for the identification of the PGRB in the BATSE
data is described. This identification is made using the two fundamental diagrams
shown in Fig. 12 and Fig. 13. Having established the value of Edya= 4.83×1053erg
and of B = 3×10−3, it is possible from the dashed line and the solid line in Fig. 12
to evaluate the ratio of the energy EPGRB emitted in the PGRB to the energy
EBaryonsemitted in the afterglow corresponding to the determined value of B, see
the vertical line in Fig. 12. We obtain EPGRB/EBaryons = 1.58 × 10−2, which
gives EPGRB= 7.54 × 1051erg. Having so determined the theoretically expected
Page 18
18R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
1
10
100
108
1010
1012
Radial coordinate (r) (cm)
1014
1016
1018
Lorentz γ factor
•
1
•
•
3
•
•
•
♦
♦
2
4
5
6
PL
PA
I
II
III
IV
V
Fig. 9.
erg, B = 3 × 10−3is given as a function of the radial coordinate in the laboratory frame. The
corresponding values in the comoving time, laboratory time and arrival time are given in Tab. 1.
The different eras indicated by roman numerals are illustrated in the text (see sections 6,7,8,12,13),
while the points 1,2,3,4,5 mark the beginning and end of each of these eras. The points PLand
PAmark the maximum of the afterglow flux, respectively in emission time and in arrival time2
(see and sections 12,17). The point 6 is the beginning of Phase D in era V (see sections 13,17). At
point 4 the transparency condition is reached and the PGRB is emitted.
The theoretically computed gamma factor for the parameter values Edya= 4.83 × 1053
intensity of the PGRB, a second fundamental observable parameter, which is also
a function of Edya and B, is the arrival time delay between the PGRB and the
peak EAPE, determined in Fig. 13. From Tab. 1, we have that the detector arrival
time of the PGRB occurs at 8.41×10−2s, corresponding to a radial coordinate of
1.94×1014cm, a comoving time of 21.57s, a laboratory time of 6.48×103s and an
arrival time of 4.21×10−2s. At this point, the gamma factor is 310.1. The peak of
the EAPE occurs at a detector arrival time of 19.87s, corresponding to a radial
coordinate of 5.18 × 1016cm, a comoving time of 5.85 × 103s, a laboratory time
of 1.73 × 106s and an arrival time of 9.93s (see Tab. 1). The delay between the
PGRB and the peak of the EAPE is therefore 19.78s, see Fig. 13. The theoretical
prediction on the intensity and the arrival time uniquely identifies the PGRB with
the “precursor” in the GRB 991216 (see Fig. 3). Moreover, the hardness of the
PGRB spectra is also evaluated in this section. As pointed out in the conclusions,
the fact that both the absolute and relative intensities of the PGRB and EAPE
have been predicted within a few percent accuracy as well as the fact that their
Page 19
On the structure of the burst and afterglow of GammaRay Bursts I: the radial approximation19
104
105
106
107
108
100
101
102
103
104
105
106
107
108
Laboratory time (t) (s)
Detector arrival time (ta
d) (s)
A
B
C
D
Fig. 10.
(t) measured at the GRB source. The solid curve is computed using the exact formula given in
Eq.(37). The dasheddotted curve is computed using the approximate formula given in Eq.(39)
and often used in the current literature. We distinguish four different phases. Phase A: There is
a linear relation between t and td
a, given by Eq.(134) in the text (dashed line). Phase B: There
is an “effective” powerlaw relation between t and td
C: No analytic formula holds and the relation between t and td
the integration of the complete equations of energy and momentum conservation (Eqs.(104,105)).
Phase D: As the gamma factor approaches γ = 1, the relation between t and td
goes to t = td
Relation between the arrival time (td
a) measured at the detector and the laboratory time
a, given by Eq.(139) (dotted line). Phase
ahas to be directly computed by
aasymptotically
a(light gray line).1
arrival time has been computed with the precision of a few tenths of milliseconds,
see Tab. 1 and Fig. 14, can be considered one of the major successes of the EMBH
theory.
1.8. On the powerlaws, beaming and temporal structures in the
afterglow of GRB 991216.
In section 17 a piecewise description of the afterglow by the expansion of the funda
mental hydrodynamical equations given by Taub (1948)60and Landau & Lifshitz61
have allowed the determination of a powerlaw index for the dependence of the af
terglow luminosity on the photon arrival time at the detector. It is evident that
the determination of the powerlaw index is very sensitive to the basic assumptions
made for the description of the afterglow, as well as to the relations between the
different temporal coordinates which have been clarified by the RSTT paradigm.1
Page 20
20R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
Table 1.
1,2,3,4,5,6,PL,PAare the same reported in Fig. 9, while the point F is the endpoint of the sim
ulation. It is particularly important to read the last column, where the apparent motion in the
radial coordinate, evaluated in the arrival time at the detector, leads to an enormous “superlumi
nal” behaviour, up to 9.55 × 104c. This illustrates well the impossibility of using such a classical
estimate in regimes with gamma factors up to 310.1.
Gamma factors for selected events and their spacetime coordinates. The points marked
Point
r (cm)
τ(s)
t(s)
ta(s)
td
a(s)
γ
“Superluminal”
v ≡
r
td
a
The Injector Phase
12.354 × 108
1.871 × 109
4.486 × 109
7.080 × 109
9.533 × 109
1.162 × 1010
0.0
1.550 × 10−2
2.141 × 10−2
2.485 × 10−2
2.715 × 10−2
2.868 × 10−2
0.0
5.886 × 10−2
1.463 × 10−1
2.329 × 10−1
3.148 × 10−1
3.845 × 10−1
0.0
4.312 × 10−3
4.523 × 10−4
4.594 × 10−3
4.627 × 10−3
4.644 × 10−3
0.0
8.625 × 10−3
9.046 × 10−3
9.187 × 10−3
9.253 × 10−3
9.288 × 10−3
1.000
10.08
20.26
30.46
40.74
49.70
0
7.23c
16.5c
25.7c
34.4c
41.7c
21.162 × 1010
1.186 × 1010
1.234 × 1010
1.335 × 1010
1.389 × 1010
2.868 × 10−2
2.889 × 10−2
2.949 × 10−2
3.144 × 10−2
3.279 × 10−2
3.845 × 10−1
3.923 × 10−1
4.083 × 10−1
4.423 × 10−1
4.603 × 10−1
4.644 × 10−3
4.646 × 10−3
4.655 × 10−3
4.706 × 10−3
4.753 × 10−3
9.288 × 10−3
9.292 × 10−3
9.311 × 10−3
9.413 × 10−3
9.506 × 10−3
49.70
38.06
24.21
15.14
12.94
41.7c
42.6c
44.2c
47.3c
48.7c
31.389 × 1010
2.326 × 1010
6.913 × 1010
1.861 × 1011
9.629 × 1011
3.205 × 1013
1.943 × 1014
3.279 × 10−2
5.208 × 10−2
9.694 × 10−2
1.486 × 10−1
3.112 × 10−1
3.958
21.57
4.603 × 10−1
7.733 × 10−1
2.304
6.206
32.12
1.069 × 103
6.481 × 103
4.753 × 10−3
5.369 × 10−3
6.086 × 10−3
6.446 × 10−3
6.978 × 10−3
1.343 × 10−2
4.206 × 10−2
9.506 × 10−3
1.074 × 10−2
1.217 × 10−2
1.289 × 10−2
1.396 × 10−2
2.685 × 10−2
8.413 × 10−2
12.94
20.09
50.66
100.1
200.3
300.1
310.1
48.7c
72.2c
1.89 × 102c
4.82 × 102c
2.30 × 103c
3.98 × 104c
7.70 × 104c
The BeamTarget Phase
41.943 × 1014
6.663 × 1015
2.863 × 1016
4.692 × 1016
5.177 × 1016
5.878 × 1016
6.580 × 1016
7.025 × 1016
7.262 × 1016
9.058 × 1016
1.136 × 1017
1.539 × 1017
2.801 × 1017
3.624 × 1017
4.454 × 1017
21.57
7.982 × 102
3.114 × 103
5.241 × 103
5.853 × 103
6.791 × 103
7.811 × 103
8.506 × 103
8.895 × 103
1.236 × 104
1.866 × 104
3.819 × 104
2.622 × 105
6.702 × 105
1.433 × 106
6.481 × 103
6.481 × 103
9.549 × 105
1.565 × 106
1.727 × 106
1.961 × 106
2.195 × 106
2.343 × 106
2.422 × 106
3.021 × 106
3.788 × 106
5.134 × 106
9.351 × 106
1.213 × 107
1.500 × 107
4.206 × 10−2
1.164
5.057
8.775
9.933
11.82
14.03
15.66
16.61
26.66
52.84
2.000 × 102
7.278 × 103
3.860 × 104
1.439 × 105
8.413 × 10−2
2.328
10.11
17.55
19.87
23.63
28.06
31.32
33.23
53.32
1.057 × 102
4.000 × 102
1.455 × 104
7.719 × 104
2.877 × 105
310.1
310.0
300.0
270.0
258.5
240.0
220.0
207.0
200.0
150.0
100.0
50.02
10.00
5.001
2.998
7.70 × 104c
9.55 × 104c
9.45 × 104c
8.92 × 104c
8.69 × 104c
8.30 × 104c
7.82 × 104c
7.48 × 104c
7.29 × 104c
5.67 × 104c
3.58 × 104c
1.28 × 104c
6.42 × 102c
1.57 × 102c
51.6c
PA
PL
54.454 × 1017
4.830 × 1017
5.390 × 1017
6.422 × 1017
1.034 × 1018
1.433 × 106
1.928 × 106
2.873 × 106
5.387 × 106
2.903 × 107
1.500 × 107
1.635 × 107
1.844 × 107
2.271 × 107
5.002 × 107
1.439 × 105
2.381 × 105
4.643 × 105
1.291 × 106
1.552 × 107
2.877 × 105
4.762 × 105
9.285 × 105
2.581 × 106
3.103 × 107
2.998
2.500
2.000
1.500
1.054
51.6c
33.8c
19.4c
8.30c
1.11c
61.034 × 1018
1.202 × 1018
2.903 × 107
4.979 × 107
5.002 × 107
7.150 × 107
1.552 × 107
3.140 × 107
3.103 × 107
6.280 × 107
1.054
1.025
1.11c
6.38 × 10−1c
F
1.248 × 10185.706 × 1077.894 × 1073.731 × 1077.461 × 1071.0005.58 × 10−1c
The different powerlaw indices obtained are compared and contrasted with the
ones in the current literature (see Tab. 2). As a byproduct of this analysis, see also
the conclusions, there is a perfect agreement between the observational data and
Page 21
On the structure of the burst and afterglow of GammaRay Bursts I: the radial approximation 21
1042
1044
1046
1048
1050
1052
101
100
101
102
Detector arrival time (ta
103
104
105
106
107
108
1014
1012
1010
108
106
Source luminosity (ergs/(s*sterad))
Observed flux (ergs/(cm2*s))
d) (s)
Theoretical curve
GRB991216 Observed afterglow data
GRB991216 Observed peak value
BATSE noise level
Fig. 11.
the BATSE data on the major burst, by a unique afterglow curve leading to the parameter values
Edya= 4.83 × 1053erg,B = 3 × 10−3. The horizontal dotted line indicates the BATSE noise
threshold. On the left axis the luminosity is given in units of the energy emitted at the source,
while the right axis gives the flux as received by the detectors.
Best fit of the afterglow data of Chandra, RXTE as well as of the range of variability of
the theoretical predictions, implying that the assumptions adopted for the descrip
tion of the afterglow are valid and therefore that there is no evidence for a beamed
emission in GRB 991216.
In section 19 the role of the inhomogeneities in the interstellar matter has been
analysed in order to explain the observed temporal substructures in the BATSE
data on GRB 991216. From the data of Tab. 1 and the highly “superluminal”
behaviour of the source in the region of the EAPE, it is concluded that the observed
time variability in the intensity of the emission
inhomogeneities in the interstellar matter: (∆nism/nism) ∼ 5. The typical size of
the scattering region is estimated to be 5 × 1016cm, and these are the typical
sizes and density contrasts found in interstellar clouds. Since the emission of the
EAPE occurs at typical dimensions of the order of 5 × 1016cm, the observed
inhomogeneities are probing the structure of the interstellar medium, and have
nothing to do with the “inner engine” of the source. These conclusions, reached in
the radial approximation of the afterglow adopted in this article, have been proved
to hold in the more general case when offradial emission is taken into account.58,59
?∆I/I?
∼ 5 can be traced to
Page 22
22 R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
0
108
0.2
0.4
0.6
0.8
1
107
106
105
104
103
102
E/Edya
B
Fig. 12.
by the EMBH theory corresponding to the values of the parameters determined in Fig. 11, as
a function of B. Details are given in section 15. The vertical line corresponds to the value B =
3 × 10−3.
Relative intensities of the EAPE (dashed line) and the PGRB (solid line), as predicted
1.9. The observation of the iron lines in GRB 991216: on a
possible GRBsupernova time sequence
In section 20 the program of using GRBs to further explore the region surround
ing the newly formed EMBH is carried one step further by using the observations
of the emitted iron lines.55This gives us the opportunity to introduce the GRB
supernova time sequence (GSTS) paradigm and to introduce as well the novel con
cept of an induced supernova explosion. The GSTS paradigm reads: A massive
GRBprogenitor star P1of mass M1undergoes gravitational collapse to an EMBH.
During this process a dyadosphere is formed and subsequently the PGRB and the
EAPE are generated in sequence. They propagate and impact, with their photon
and neutrino components, on a second supernovaprogenitor star P2 of mass M2.
Assuming that both stars were generated approximately at the same time, we expect
to have M2< M1. Under some special conditions of the thermonuclear evolution of
the supernovaprogenitor star P2, the collision of the PGRB and the EAPE with
the star P2can induce its supernova explosion.
Using the result presented in Tab. 1 and in all preceding sections, the GSTS
paradigm is illustrated in the case of GRB 991216. Some general considerations on
the nature of the supernova progenitor star are also advanced.
Page 23
On the structure of the burst and afterglow of GammaRay Bursts I: the radial approximation 23
0.0001
0.001
0.01
0.1
1
10
100
1000
0.00010.0010.01
Arrival time (ta
d) delay between PGRB and EAPE (s)
B
Edya=4.49*1055 erg
Edya=4.83*1053 erg
Edya=5.29*1051 erg
Fig. 13.
function of the B parameter for three selected values of Edya.
The arrival time delay between the PGRB and the peak of the EAPE is plotted as a
Some general considerations on the EMBH formation are presented in section 21.
The general conclusions are presented in section 22.
The understanding of all these points has led to the formulation of the second
part, namely the sufficient condition of the RSTT paradigm which 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. In order to meet this condition, given a proper theoretical
description and the correct governing equations, it is sufficient to know the energy
of the dyadosphere and the mass of the remnant of the progenitor star”.
2. The zeroth era: the process of gravitational collapse and the
formation of the dyadosphere
We first recall the three theoretical results which lie at the basis of the EMBH
theory.
In 1971 in the article “Introducing the Black Hole”,62the theorem was advanced
that the most general black hole is characterized uniquely by three independent
parameters: the massenergyM, the angular momentum L and the charge Q making
it an EMBH. Such an ansatz, which came to be known as the “uniqueness theorem”
has turned out to be one of the most difficult theorems to be proven in all of physics
Page 24
24R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
Fig. 14.
EAPE.
A qualitative diagram showing the full picture of the model, with both PGRB and
and mathematics. The progressin the proof has been authoritatively summarized by
Carter (1997).63The situation can be considered satisfactory from the point of view
of the physical and astrophysical considerations. Nevertheless some fundamental
mathematical and physical issues concerning the most general perturbation analysis
of an EMBH are still the topic of active scientific discussion.64
In 1971 it was shown that the energy extractable from an EMBH is governed
by the massenergy formula,65
E2
BH= M2c4=
?
Mirc2+Q2
2ρ+
?2
+L2c2
ρ2
+
, (1)
with
1
ρ4
+
?G2
c8
??Q4+ 4L2c2?≤ 1, (2)
where
S = 4πρ2
+= 4π(r2
++
L2
c2M2) = 16π
?G2
c4
?
M2
ir,(3)
is the horizon surface area, Mir is the irreducible mass, r+ is the horizon radius
and ρ+ is the quasispheroidal cylindrical coordinate of the horizon evaluated at
the equatorial plane. Extreme EMBHs satisfy the equality in Eq.(2). Up to 50% of
Page 25
On the structure of the burst and afterglow of GammaRay Bursts I: the radial approximation 25
the massenergy of an extreme EMBH can in principle be extracted by a special
set of transformations: the reversible transformations.65
In 1975, generalizing some previous results of Zaumen (1975)66and Gibbons
(1975),67Damour & Ruffini (1975)47showed that the vacuum polarization process
` a la HeisenbergEulerSchwinger10,11created by an electric field of strength larger
than
Ec=m2
ec3
¯ he
(4)
can indeed occur in the field of a KerrNewmann EMBH. Here meand e are respec
tively the mass and charge of the electron. There Damour and Ruffini considered an
axially symmetric EMBH, due to the presence of rotation, and limited themselves
to EMBH masses larger then the upper limit of a neutron star for astrophysical
applications. They purposely avoided all complications of black holes with mass
smaller then the dual electron mass of the electron
may lead to quantum evaporation processes.68They pointed out that:
?
m⋆
e=
c¯ h
Gme=
m2
Planck
me
?
which
(1) The vacuum polarization process can occur for an EMBH mass larger than the
maximum critical mass for neutron stars all the way up to 7.2 × 106M⊙.
(2) The process of pair creation occurs on very short time scales, typically
is an almost perfect reversible process, in the sense defined by Christodoulou
Ruffini, leading to a very efficient mechanism of extracting energy from an
EMBH.
(3) The energy generated by the energy extraction process of an EMBH was found
to be of the order of 1054erg, released almost instantaneously. They concluded
at the time “this work naturally leads to a most simple model for the explanation
of the recently discovered γray bursts”.
¯ h
mec2, and
After the discovery of the afterglow of GRBs and the determination of the cos
mological distance of their sources we noticed the coincidence between the theoret
ically predicted energetics and the observed ones in Damour & Ruffini (1975):47we
returned to our theoretical results developing some new basic theoretical concepts,46,69,48,49,50
which have led to the EMBH theory.
As a first simplifying assumption we have developed our considerations in the
absence of rotation with spherically symmetric distributions. The spacetime is then
described by the ReissnerNordstr¨ om geometry, whose spherically symmetric metric
is given by
d2s = gtt(r)d2t + grr(r)d2r + r2d2θ + r2sin2θd2φ , (5)
where gtt(r) = −
The first new result we obtained is that the pair creation process does not occur
at the horizon of the EMBH: it extends over the entire region outside the horizon in
which the electric field exceeds the critical value given by Eq. 4. Since the electric
?
1 −2GM
c2r+Q2G
c4r2
?
≡ −α2(r) and grr(r) = α−2(r).
Page 26
26R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
field in the ReissnerNordstr¨ om geometry has only a radial component given by70
E (r) =Q
r2, (6)
this region extends from the horizon radius
r+= 1.47 · 105µ(1 +
?
1 − ξ2) cm (7)
out to an outer radius46
r⋆=
?¯ h
mc
?1
2?GM
c2
?1
2?mp
m
?1
2?e
qp
?1
2?
Q
√GM
?1
2
= 1.12 · 108?
µξ cm, (8)
where we have introduced the dimensionless mass and charge parameters µ =
Q
(M√G)≤ 1, see Fig. 4.
The second new result has been to realize that the local number density of
electron and positron pairs created in this region as a function of radius is given by
M
M⊙,
ξ =
ne+e−(r) =
Q
4πr2?¯ h
mc
?e
?
1 −
?r
r⋆
?2?
,(9)
and consequently the total number of electron and positron pairs in this region is
N◦
e+e− ≃Q − Qc
e
?
1 +(r⋆− r+)
¯ h
mc
?
,(10)
where Qc= Ecr2
The total number of pairs is larger by an enormous factor r⋆/(¯ h/mc) > 1018
than the value Q/e which a naive estimate of the discharge of the EMBH would
have predicted. Due to this enormous amplification factor in the number of pairs
created, the region between the horizon and r⋆is dominated by an essentially high
density neutral plasma of electronpositron pairs. We have defined this region as the
dyadosphere of the EMBH from the Greek duas, duadsos for pairs. Consequently we
have called r⋆the dyadosphere radius r⋆≡ rds.46,69,48The vacuum polarization
process occurs as if the entire dyadosphere are subdivided into a concentric set of
shells of capacitors each of thickness ¯ h/mec and each producing a number of e+e−
pairs on the order of ∼ Q/e (see Fig. 4). The energy density of the electronpositron
pairs is given by
+.
ǫ(r) =
Q2
8πr4
?
1 −
?r
rds
?4?
, (11)
(see Figs. 2–3 of Preparata, Ruffini & Xue, 1998a69). The total energy of pairs
converted from the static electric energy and deposited within the dyadosphere is
then
Edya=1
2
Q2
r+(1 −r+
rds)
?
1 −
?r+
rds
?2?
. (12)
Page 27
On the structure of the burst and afterglow of GammaRay Bursts I: the radial approximation27
1e+49
1e+50
1e+51
1e+52
1e+53
1e+54
1e+55
1e+56
1e+57
1e+58
1e+59
1e+60
1e+61
Ergs
.1e2.1e3.1e4.1e5 1e+05
M/Solar mass
Fig. 15.
as a function of the mass M in solar mass units for selected values of the charge parameter
ξ = 1,0.1,0.01 (from top to bottom) for an EMBH, the case ξ = 1 reachable only as a limiting
process. For comparison we have also plotted the maximum energy extractable from an EMBH
(dotted lines) given by eq. (1). Details in Preparata, Ruffini & Xue (2001).71
The energy extracted by the process of vacuum polarization is plotted (solid lines)
As we will see in the following this is one of the two fundamental parameters of
the EMBH theory (see Fig. 16). In the limit
which coincides with the energy extractable from EMBHs by reversible processes
r+
rds→ 0, Eq.(12) leads to Edya→1
2
Q2
r+,
Page 28
28R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
(Mir = const.), namely EBH− Mir=
pair density given by Eq.(9) and to the sizes of the crosssections for the process
e+e−↔ γ + γ, the system is expected to thermalize to a plasma configuration for
which
1
2
Q2
r+,65see Fig. 15. Due to the very large
ne+ = ne− ∼ nγ∼ n◦
e+e−,(13)
where n◦
The third new result which we have introduced for simplicity is that for a given
Edya we have assumed either a constant average energy density over the entire
dyadosphere volume, or a more compact configuration with energy density equal
to the peak value. These are the two possible initial conditions for the evolution of
the dyadosphere (see Fig. 17).
e+e−is the total number density of e+e−pairs created in the dyadosphere.69,48
1058
1058
1057
1057
1056
1056
1055
1055
1054
1054
1053
1053
1052
1052
000.20.20.40.40.60.60.80.811
ξξ
101
101
102
102
103
103
104
104
105
105
µµ
Fig. 16.
two parameters µ ξ, only the solutions below the continuous heavy line are physically relevant.The
configurations above the continuous heavy lines correspond to unphysical solutions with rds< r+
Selected lines corresponding to fixed values of the Edyaare given as a function of the
These three old and three new theoretical results permit a good estimate of the
general energetics processes originating in the dyadosphere, assuming an already
formed EMBH. In reality, if the data become accurate enough, the full dynamical
description of the dyadosphere formation mentioned above will be needed in order
to follow all the general relativistic effects and characteristic time scales of the
approach to the EMBH horizon72,73,74,75, see also section 21.
Page 29
On the structure of the burst and afterglow of GammaRay Bursts I: the radial approximation 29
1e+024
1e+025
1e+026
1e+027
1e+028
1e+029
1e+030
1e+031
1e+0071e+008
energy density (erg/cm3)
Laboratory radius (cm)
Fig. 17.
first one (dashed line) fixes the energy density equal to its peak value, and computes an “effective”
dyadosphere radius accordingly. The second one (dotted line) fixes the dyadosphere radius to its
correct value, and assumes an uniform energy density over the dyadosphere volume. The total
energy in the dyadosphere is of course the same in both cases. The solid curve represents the real
energy density profile.
Two different approximations for the energy density profile inside the dyadosphere. The
Below we shall concentrate on the dynamical evolution of the electronpositron
plasma created in the dyadosphere. We shall first examine in the next three sections
the governing equations necessary to approach such a dynamical description.
3. The hydrodynamics and the rate equations for the plasma of
e+e−pairs
The evolution of the e+e−pair plasma generated in the dyadosphere has been
treated in two papers.49,50We recall here the basic governing equations in the
most general case in which the plasma fluid is composed of e+e−pairs, photons
and baryonic matter. The plasma is described by the stressenergy tensor
Tµν= pgµν+ (p + ρ)UµUν,(14)
where ρ and p are respectively the total proper energy density and pressure in the
comoving frame of the plasma fluid and Uµis its fourvelocity, satisfying
gtt(Ut)2+ grr(Ur)2= −1 , (15)
Page 30
30R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
where Urand Utare the radial and temporal contravariant components of the
4velocity.
The conservation law for baryon number can be expressed in terms of the proper
baryon number density nB
(nBUµ);µ= g−1
= (nBUt),t+1
2(g
1
2nBUν),ν
r2(r2nBUr),r= 0 .(16)
The radial component of the energymomentum conservation law of the plasma
fluid reduces to
∂p
∂r+∂
∂t
−1
?(p + ρ)UtUr
?∂gtt
?+1
r2
∂
∂r
?r2(p + ρ)UrUr
∂r(Ur)2
?
2(p + ρ)
∂r(Ut)2+∂grr
?
= 0 .(17)
The component of the energymomentum conservation law of the plasma fluid equa
tion along a flow line is
Uµ(Tµν);ν= −(ρUν);ν− p(Uν);ν,
= −g−1
= (ρUt),t+1
2(g
1
2ρUν),ν− pg−1
r2(r2ρUr),r
(Ut),t+1
r2(r2Ur),r
2(g
1
2Uν),ν
+ p
?
?
= 0 . (18)
Defining the total proper internal energy density ǫ and the baryonic mass density
ρBin the comoving frame of the plasma fluid,
ǫ ≡ ρ − ρB,ρB≡ nBmc2,(19)
and using the law (16) of baryonnumber conservation, from Eq. (18) we have
(ǫUν);ν+ p(Uν);ν= 0 . (20)
Recalling thatdV
time for the plasma fluid, we have along each flow line
dτ= V (Uµ);µ, where V is the comoving volume and τ is the proper
d(V ǫ)
dτ
+ pdV
dτ
=dE
dτ
+ pdV
dτ
= 0 ,(21)
where E = V ǫ is the total proper internal energy of the plasma fluid. We express
the equation of state by introducing a thermal index Γ(ρ,T)
Γ = 1 +p
ǫ.
(22)
We now turn to the second set of governing equations describing the evolution
of the e+e−pairs. Letting ne− and ne+ be the proper number densities of electrons
and positrons associated with pairs and nb
electrons, we clearly have
e− the proper number densities of ionized
ne− = ne+ = npair,nb
e− =¯ZnB,(23)
Page 31
On the structure of the burst and afterglow of GammaRay Bursts I: the radial approximation31
where npair is the number of e+e−pairs and¯Z the average atomic number
¯Z < 1 (¯Z = 1 for hydrogen atom and¯Z =1
equation for electrons and positrons gives,
1
2<
2for general baryonic matter). The rate
(ne+Uµ);µ= (ne+Ut),t+1
r2(r2ne+Ur),r
e−(T))ne+(T)
e−)ne+?,
r2(r2ne−Ur),r
= σv?(ne−(T) + nb
− (ne− + nb
(24)
(ne−Uµ);µ= (ne−Ut),t+1
= σv [ne−(T)ne+(T) − ne−ne+],
e−Uµ);µ= (nb
(25)
(nb
e−Ut),t+1
r2(r2nb
e−Ur),r
= σv?nb
e−(T)ne+(T) − nb
e−ne+?, (26)
where σv is the mean of the product of the annihilation crosssection and the
thermal velocity of the electrons and positrons, ne±(T) are the proper number
densities of electrons and positrons associated with the pairs, given by appropriate
Fermi integrals with zero chemical potential, and nb
density of ionized electrons, given by appropriate Fermi integrals with nonzero
chemical potential µe at an appropriate equilibrium temperature T. These rate
equations can be reduced to
e−(T) is the proper number
(ne±Uµ);µ= (ne±Ut),t+1
r2(r2ne±Ur),r
= σv?ne−(T)ne+(T) − ne−ne+?,
e−Uµ);µ= (nb
(27)
(nb
e−Ut),t+1
r2(r2nb
e−(T)
nb
e−
e−Ur),r= 0,(28)
Frac ≡
ne±
ne±(T)=nb
. (29)
Equation (28) is just the baryonnumber conservation law (16) and (29) is a rela
tionship satisfied by ne±,ne±(T) and nb
The equilibrium temperature T is determined by the thermalization processes
occurring in the expanding plasma fluid with a total proper energy density ρ gov
erned by the hydrodynamical equations (16,17,18). We have
e−,nb
e−(T).
ρ = ργ+ ρe+ + ρe− + ρb
e− + ρB,(30)
where ργ is the photon energy density, ρB ≃ mBc2nB is the baryonic mass den
sity which is considered to be nonrelativistic in the range of temperature T under
consideration, and ρe± is the proper energy density of electrons and positrons pairs
given by
ρe± =
ne±
ne±(T)ρe±(T),(31)
Page 32
32R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
where ne± is obtained by integration of Eq.(27) and ρe±(T) is the proper energy
density of electrons(positrons) obtained from zero chemical potential Fermi integrals
at the equilibrium temperature T. On the other hand ρb
the ionized electrons coming from the ionization of baryonic matter
e− is the energy density of
ρb
e− =
nb
e−(T)ρb
e−
nb
e−(T),(32)
where nb
density of ionized electrons obtained from an appropriate Fermi integral of nonzero
chemical potential µeat the equilibrium temperature T.
Having intrinsically defined the equilibrium temperature T in Eq.(30), we can
also analogously evaluate the total pressure
e− is obtained by integration of Eq.(28) and ρe−(T) is the proper energy
p = pγ+ pe+ + pe− + pb
e− + pB,(33)
where pγis the photon pressure, pe± and pb
e− are given by
pe± =
ne±
ne±(T)pe±(T),
nb
e−
nb
(34)
pb
e− =
e−(T)pb
e−(T), (35)
the pressures pe±(T) are determined by zero chemical potential Fermi integrals,
and pb
e−(T) is the pressure of the ionized electrons, evaluated by an appropriate
Fermi integral of nonzero chemical potential µe at the equilibrium temperature
T. In Eq.(33), the ion pressure pB is negligible by comparison with the pressures
pγ,e±,e−(T), since baryons and ions are expected to be nonrelativistic in the range
of temperature T under consideration. Finally using Eqs.(30,33) we compute the
thermal factor Γ of the equation of state (22).
It is clear that the entire set of equations considered above, namely Eqs.(16,17,18)
with equation of state given by Eq.(22) and the rate equation (27), have to be in
tegrated satisfying the total energy conservation for the system. The boundary
conditions adopted here are simply purely ingoing conditions at the horizon and
purely outgoing conditions at radial infinity. The calculation is initiated by deposit
ing a proper energy density (11) between the ReissnerNordstr¨ om horizon radius
r+and the dyadosphere radius rds, following the approximation presented in Fig.15
The total energy deposited is given by Eq.(12).
4. The equations leading to the relative spacetime
transformations
In order to relate the above hydrodynamic and pair equations with the observations
we need the governing equations relating the comoving time to the laboratory time
corresponding to an inertial reference frame in which the EMBH is at rest and
finally to the time measured at the detector, which must also include the effect
Page 33
On the structure of the burst and afterglow of GammaRay Bursts I: the radial approximation33
of the cosmological expansion. These transformations have been the object of the
relative spacetime transformations (RSTT) Paradigm.1
For signals emitted by a pulse moving with velocity v in the laboratory frame,1
we have the following relation between the interval of arrival time ∆ta and the
corresponding interval of laboratory time ∆t (see Fig. 2):
∆ta=
?
t0+ ∆t +R0− r
c
?
−
?
t0+R0
c
?
= ∆t −r
c.
(36)
For simplicity in what follows we indicate by tathe interval of arrival time mea
sured from the reception of a light signal emitted at the onset of the gravitational
collapse. Analogously, t indicates the laboratory time interval measured from the
time of the gravitational collapse. In this case, Eq.(36) can be written simply as:
ta= t −r
c= t −
?t
0v (t′)dt′+ rds
c
, (37)
where the dyadosphere radius rdsis the value of r at t = 0. We consider here only
the photons emitted along the line of sight from the external surface of the pulse.
The arrival time spreading due to the angular dependence and that due to the
thickness of the pulse will be considered elsewhere.58,59The solution of Eq.(37)
has the expansion:
ta= t −a1
ct −1
2
a2
ct2− ..., (38)
so the relation between taand t is in general highly nonlinear.
If and only if the expansion of the pulse is such that r(t) = vt with v ≃ c, Eq.(37)
can be written, neglecting rds, in the following simplified form (see Fig. 10):
ta≃ t
?
1 −v
c
?
= t
?1 −v
c
??1 +v
c
c
?
?1 +v
?
≃
t
2γ2. (39)
This formula has been uncritically and widely applied in all articles dealing with
GRBs. It is clear, however, that the knowledge of ta, which is indeed essential for
any physical interpretation of GRB data, depends on the definite integral given
in Eq.(37) whose integration limits in the laboratory time extend from the onset
of the gravitational collapse to the time t relevant for the observations. Such an
integral is not generally expressible as a simple linear relation or even by any explicit
analytic relation since we are dealing with processes with variable gamma factor
unprecedented in the entire realm of physics (see Figs. 9 and Fig. 10). Any linear
approximation of the kind given in Eq.(39) with γ constant or changing with time76
misses a crucial feature of the GRB process and is therefore erroneous in this
context.
To relate the time in the laboratory frame to the time in the detector frame
we have to do one additional step: the two frames are related by a transformation
Page 34
34R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
which is a function of the cosmological expansion. We recall that the geometry of
the spacetime of the universe is described by the RobertsonWalker metric:
ds2= dt2− R2(t)
?
dr2
1 − kr2+ r2dϑ2+ r2sinϑ2dϕ2
?
,(40)
where R(t) is the cosmic scale factor and k is a constant related to the curvature
of the threedimensional space (k = 0,+1,−1 corresponds to flat, close and open
space respectively). The wavelength of an electromagnetic wave travelling from the
point P1(t1,r1,ϑ1,ϕ1) to the point P◦(t◦,r◦,ϑ◦,ϕ◦) where the observer is located
is related to the redshift parameter z by
z =λ◦− λ1
λ1
,(41)
where λ◦is the wavelength of the radiation for the observer and λ1for the emitter.
We have the following general relation:
1 + z = (1 + zu)(1 + zo)(1 + zs),(42)
where z is the total redshift due to the motion of the source zs, the motion of
the observer zoand the cosmological redshift zu. In the following we will assume
zo<< 1 and zs<< 1 so z = zu. In terms of the scale factor R(t) the relation (41)
gives
λ◦
λ1
=R(to)
R(t1)= 1 + z =ω1
ω0
(43)
where ω1 and ω0 are the frequencies associated to λ1 and λ0 respectively. This
frequency ratio then relates the time elapsing at the source with the time elapsing
at the detector due to the cosmological expansion.
We can now define the corrected arrival time td
is related to taby
ameasured at the detector, which
td
a= ta(1 + z), (44)
where z is the cosmological redshift of the GRB source. In the case of GRB 991216
we have z ≃ 1.00.
The observed flux is the flux which crosses the surface 4π(R(to)r)2but this
flux is lower by a factor 1 + z due to the redshift energy of the photons and by
another factor 1+z due to the fact that the number of photons at reception is less
than the number at emission. Thus we can define a luminosity distance by:
d2
L= R2
or2(1 + z)2. (45)
Then the observed flux is related to the absolute luminosity of the GRB by the
following relation:
l =
L
4πd2
L
, (46)
Page 35
On the structure of the burst and afterglow of GammaRay Bursts I: the radial approximation35
where the luminosity distance dLis simply related to the proper distance dp= Ror
by dL= dp(1 + z). The observed total fluence f is related to the total energy E of
the GRB by the following relation:
f =E(1 + z)
4πd2
L
(47)
Then the cosmological effect is taken into account by the definition of the proper
distance Ror which depends on the cosmological parameters: the Hubble constant
H◦=˙R(t◦)/R(t◦) at time t◦and the matter density ρ◦or ΩM= ρ◦/ρcrit, where
ρcrit=
The computation of the proper distance is then simply given by the relation :
3H2
8πG.
◦
dp=
c
Ho
?z
0
dz
F(z),(48)
where F(z) =
In the case of the Friedman flat universe, ΩM= 1 and we have:
?ΩM(1 + z)3.
dp(z) =2c
Ho
?
1 −
1
√1 + z
?
.(49)
So the measurement of the redshift gives us the luminosity distance via a cos
mological scenario. With the measurement of the flux we can deduce the proper
luminosity of the burst and from the measurement of the total fluence the total
energy so we are then able to find the Edya.
5. The numerical integration of the hydrodynamics and the rate
equations
5.1. The Livermore code
A computer code77,78has been used to evolve the spherically symmetric general
relativistic hydrodynamic equations starting from the dyadosphere.49
We define the generalized gamma factor γ and the radial 3velocity in the lab
oratory frame Vr
γ ≡
?
1 + UrUr,Vr≡Ur
Ut. (50)
From Eqs.(5, 15), we then have
(Ut)2= −1
gtt(1 + grr(Ur)2) =
1
α2γ2.(51)
Following Eq.(19), we also define
E ≡ ǫγ,D ≡ ρBγ,and˜ ρ ≡ ργ(52)
so that the conservation law of baryon number (16) can then be written as
∂D
∂t
= −α
r2
∂
∂r(r2
αDVr). (53)
Page 36
36R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
Eq.(18) then takes the form,
∂E
∂t
= −α
r2
∂
∂r(r2
αEVr) − p
?∂γ
∂t+α
r2
∂
∂r(r2
αγVr)
?
.(54)
Defining the radial momentum density in the laboratory frame
Sr≡ α(p + ρ)UtUr= (D + ΓE)Ur,(55)
we can express the radial component of the energymomentum conservation law
given in Eq.(17) by
∂r(r2
−α
= −α
r2
?M
In order to determine the numberdensity of e+e−pairs, we turn to Eq.(27).
Defining the e+e−pair density in the laboratory frame Ne± ≡ γne± and Ne±(T) ≡
γne±(T), where the equilibrium temperature T has been obtained from Eqs.(30)
and (31), and using Eq.(51), we rewrite the rate equation given by Eq.(27) in the
form
∂Ne±
∂tr2
These equations are integrated starting from the dyadosphere distributions given
in Fig. 17 and assuming as usual ingoing boundary conditions on the horizon of the
EMBH.
∂Sr
∂t
= −α
r2
∂
αSrVr) − α∂p
?∂gtt
∂r(r2
r2−Q2
r3
∂r
2(p + ρ)
∂r(Ut)2+∂grr
αSrVr) − α∂p
??D + ΓE
∂r(Ur)2
?
∂
∂r
− α
γ
???γ
α
?2
+(Ur)2
α4
?
.(56)
= −α∂
∂r(r2
αNe±Vr) + σv(N2
e±(T) − N2
e±)/γ2,(57)
5.2. The Rome code
In the following we recall a zeroth order approximation of the fully relativistic equa
tions of the previous section:49
(i) Since we are mainly interested in the expansion of the e+e−plasma away from
the EMBH, we neglect the gravitational interaction.
(ii) We describe the expanding plasma by a special relativistic set of equations.
(iii) In contrast with the previous treatment where the evolution of the density pro
files given in Fig. 17 are followed in their temporal evolution leading to a pulselike
structure, selected geometries of the pulse are a priori adopted and the correct one
validated by the complete integration of the equations given by the Livermore codes.
In analogy to Eq.(21), from Eq.(16) we have along each flow line in the general
case in which baryonic matter is present
d(nBV )
dτ
= 0. (58)
Page 37
On the structure of the burst and afterglow of GammaRay Bursts I: the radial approximation37
For the expansion of a shell from its initial volume ∆V◦ to the volume ∆V , we
obtain
n◦
B
nB
∆V◦
where ∆V is the volume of the shell in the laboratory frame, related to the proper
volume ∆V in the comoving frame by ∆V = γ(r)∆V, where γ(r) defined in Eq.(50)
is the gamma factor of the shell at the radius r.
Similarly from Eq.(21), using the equation of state (22), along the flow lines we
obtain
=∆V
=
∆Vγ(r)
∆V◦γ◦(r),(59)
dlnǫ + ΓdlnV = 0.(60)
Correspondingly we obtain for the internal energy density ǫ along the flow lines
ǫ◦
ǫ
=
?∆V
∆V◦
?Γ
=
?∆V
∆V◦
?Γ?γ(r)
γ◦(r)
?Γ
,(61)
where the thermal index Γ given by (22) is a slowlyvarying function with values
around 4/3. It can be computed for each value of ǫ,p as a function of ∆V .
The overall energy conservation requires that the change of the internal proper
energy of a shell is compensated by a change in its bulk kinetic energy. We then
have49
dK = [γ(r) − 1](dE + ρBdV ). (62)
In order to model the relativistic expansion of the plasma fluid, we assume that
E and D as defined by Eq.(52) are constant in space over the volume ∆V . As a
consequence the total energy conservation for the shell implies49
(ǫ◦+ ρ◦
B)γ2
◦(r)∆V◦= (ǫ + ρB)γ2(r)∆V,(63)
which leads the solution
γ(r) = γ◦(r)
?
(ǫ◦+ ρ◦
(ǫ + ρB)∆V
B)∆V◦
. (64)
Corresponding to Eq.(57) we obtain the equation for the evolution of the e±
numberdensity as seen by an observer in the laboratory frame
∂
∂t(Ne±) = −Ne±
1
∆V
∂∆V
∂t
+ σv
1
γ2(r)(N2
e±(T) − N2
e±) . (65)
Eqs.(59), (61), (64) and (65) are a complete set of equations describing the rel
ativistic expansion of the shell. If we now turn from a single shell to a finite
distribution of shells, we can introduce the average values of the proper internal
energy, baryonmass,baryonnumber and pairnumber densities (¯ ǫ, ¯ ρB, ¯ nB, ¯ ne±) and
¯E ≡ ¯ γ¯ ǫ,¯D ≡ ¯ γ¯ ρB,¯ Ne± ≡ ¯ γ(r)¯ ne± for the PEMpulse, where the average ¯ γfactor
is defined by
¯ γ =1
V
?
V
γ(r)dV, (66)
Page 38
38R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
and V is the total volume of the shell in the laboratory frame. The corresponding
equations are given in Ruffini, Salmonson, Wilson & Xue (1999).49Having defined
all its governing equations we can now return to the description of the different eras
of the GRB phenomena.
6. The era I: the PEM pulse
We have assumed that, following the gravitational collapse process, a region of very
low baryonic contamination exists in the dyadosphere all the way to the remnant
of the progenitor star.
Recalling Eq.(9) the limit on such baryonic contamination, where ρBcis the
massenergy density of baryons, is given by
ρBc≪ mpne+e−(r) = 3.2 · 108?rds
Near the horizon r ≃ r+, this gives
r
?2?
1 −
?
r
rds
?2?
(g/cm3).(67)
ρBc≪ mpne+e−(r) = 1.86 · 1014
?ξ
µ
?
(g/cm3),(68)
and near the radius of the dyadosphere rds:
ρBc≪ mpne+e−(r) = 3.2 · 108
?
1 −
?
r
rds
?2?
r→rds
(g/cm3).(69)
Such conditions can be easily satisfied in the collapse to an EMBH, but not neces
sarily in a collapse to a neutron star.
Consequently we have solved the equations governing a plasma composed solely
of e+e−pairs and electromagnetic radiation, starting at time zero from the dyado
sphere configurations corresponding to constant density in Fig. 17. The Livermore
code49has shown very clearly the self organization of the expanding plasma in a
very sharp pulse which we have defined as the pairelectromagnetic pulse (PEM
pulse), in analogy with the EM pulse observed in nuclear explosions. In order to
further examine the structure of the PEM pulse with the simpler procedures of
the Rome codes we have assumed49three alternative patterns of expansion of the
PEM pulse on which to try the simplified special relativistic treatment and then
compared the results with the fully general relativistic hydrodynamical results:
• Spherical model: we assume the radial component of the fourvelocity Ur(r) =
Ur
R, where U is the radial component of the fourvelocity at the moving outer
surface r = R(t) of the PEM pulse and the ¯ γfactor and the velocity Vrare
3
8U3
?
¯ γ =
?
2U(1 + U2)
3
2− U(1 + U2)
?
1
2
− ln(U +1 + U2),Vr=Ur
¯ γ
; (70)
Page 39
On the structure of the burst and afterglow of GammaRay Bursts I: the radial approximation39
this distribution expands keeping an uniform density profile which decreases
with time similar to a portion of a Friedmann Universe.
• Slab 1: we assume U(r) = Ur = const., the constant width of the expanding
slab D = R◦in the laboratory frame of the PEM pulse, while ¯ γ and Vrare
?
this distribution does not need any averaging process.
• Slab 2: we assume a constant width R2−R1= R◦of the expanding slab in the
comoving frame of the PEM pulse, while ¯ γ and Vrare
¯ γ =
1 + U2
r,Vr=Ur
¯ γ
;(71)
¯ γ =
?
1 + U2
r(˜ r),Vr=Ur
¯ γ,
(72)
This distribution needs an averaging procedure and R1< ˜ r < R2, i.e. ˜ r is an
intermediate radius in the slab.
These different assumptions lead to three different distinct slopes for the mono
tonically increasing ¯ γfactor as a function of the radius (or time) in the laboratory
frame, having assumed for the energy of dyadosphere Edya= 3.1 × 1054erg (see
Fig. 18). In principle, we could have an infinite number of models by defining ar
bitrarily the geometry of the expanding fluid in the special relativistic treatment
given above. To find out which expanding pattern of PEM pulses is the physically
realistic one, we need to compare and contrast the results of our simplified mod
els (performed in Rome) with the numerical results based on the hydrodynamic
Eqs.(53,54,56) (obtained at Livermore).49Details of the iterative method used to
solve the special relativistic equation can be found in Ruffini, Salmonson, Wilson
& Xue (1999).49
It is manifest from the results (see Fig. 18) that the slab 1 approximation (con
stant thickness in the laboratory frame) is in excellent agreement with the Livermore
results (open squares).
The remarkable validation of the special relativistic treatment of the PEM
pulse,49allows us to easily estimate the related quantities of physical and astro
physical interest in the model, like the e+e−pair densities as a function of the
laboratory time, the temperature of the plasma in the comoving and laboratory
frames, the reheating ratio as a function of the e+e−pair annihilation for a variety
of initial conditions.49
7. The era II: the interaction of the PEM pulse with the remnant
of the progenitor star
The PEM pulse expands initially in a region of very low baryonic contamination
created by the process of gravitational collapse. As it moves further out the baryonic
remnant (see Fig. 1) of the progenitor star is encountered. As discussed in section
21 below, the existence of such a remnant is necessary in order to guarantee the
overall charge neutrality of the system: the collapsing core has the opposite charge
Page 40
40R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
Fig. 18.
PEMpulse are compared with the results of the one dimensional hydrodynamic code for an energy
of dyadosphere Edya= 3.1 × 1054erg. The 1D code has an expansion pattern that strongly
resembles that of a shell with constant thickness in the laboratory frame.
Gamma factor as a function of radius. Three models for the expansion pattern of the
of the remnant and the system as a whole is clearly neutral. The number of extra
charges in the baryonic remnant negligibly affects the overall charge neutrality of
the PEM pulse.79,75
The baryonic matter remnant is assumed to be distributed well outside the
dyadosphere in a shell of thickness ∆ between an inner radius rin and an outer
radius rout = rin+ ∆ at a distance from the EMBH at which the original PEM
pulse expanding in vacuum has not yet reached transparency. For the sake of an
example we choose
rin= 100rds,∆ = 10rds.(73)
Page 41
On the structure of the burst and afterglow of GammaRay Bursts I: the radial approximation41
The total baryonic mass MB = NBmp is assumed to be a fraction of the dyado
sphere initial total energy (Edya). The total baryonnumber NB is then expressed
as a function of the dimensionless parameter B given by
B =NBmpc2
Edya
,(74)
where B is a parameter in the range 10−8− 10−2and mp is the proton mass.
We shall see below the paramount importance of B in the determination of the
features of the GRBs. We will see in section 9 the sense in which B and Edyacan
be considered to be the only two free parameters of the EMBH theory for the entire
GRB family, the so called “long bursts”. We shall see in section 11 that for the so
called “short bursts” the EMBH theory depends on the two other parameters µ,
ξ, since in that case B = 0. The baryon number density n◦
constant
Bis assumed to be a
¯ n◦
B=NB
VB,¯ ρ◦
B= mp¯ n◦
Bc2.(75)
As the PEM pulse reaches the region rin < r < rout, it interacts with the
baryonic matter which is assumed to be at rest. In our simplified quasianalytic
model we make the following assumptions to describe this interaction:
• the PEM pulse does not change its geometry during the interaction;
• the collision between the PEM pulse and the baryonic matter is assumed to be
inelastic,
• the baryonic matter reaches thermal equilibrium with the photons and pairs of
the PEM pulse.
These assumptions are valid if: (i) the total energy of the PEM pulse is much
larger than the total massenergy of baryonic matter MB, 10−8< B < 10−2, (ii)
the ratio of the comoving number density of pairs and baryons at the moment of
collision ne+e−/n◦
pulse has a large value of the gamma factor (100 < ¯ γ).
In the collision between the PEM pulse and the baryonic matter at rout> r > rin
, we impose total conservation of energy and momentum. We consider the collision
process between two radii r2,r1satisfying rout> r2> r1> rinand r2− r1≪ ∆.
The amount of baryonic mass acquired by the PEM pulse is
Bis very high (e.g., 106< ne+e−/n◦
B< 1012) and (iii) the PEM
∆M =MB
VB
4π
3(r3
2− r3
1),(76)
where MB/VBis the meandensity of baryonic matter at rest. The conservation of
total energy leads to the estimate of the corresponding quantities before (with “◦”)
and after such a collision
(Γ¯ ǫ◦+ ¯ ρ◦
B)¯ γ2
◦V◦+ ∆M = (Γ¯ ǫ + ¯ ρB+∆M
V
+ Γ∆¯ ǫ)¯ γ2V, (77)
Page 42
42R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
where ∆¯ ǫ is the corresponding increase of internal energy due to the collision.
Similarly the momentumconservation gives
(Γ¯ ǫ◦+ ¯ ρ◦
B)¯ γ◦U◦
rV◦= (Γ¯ ǫ + ¯ ρB+∆M
V
+ Γ∆¯ ǫ)¯ γUrV, (78)
where the radial component of the fourvelocity of the PEM pulse is U◦
and Γ is the thermal index. We then find
r=
?¯ γ2
◦− 1
∆¯ ǫ =1
Γ
?
(Γ¯ ǫ◦+ ¯ ρ◦
a
√a2− 1,
B)¯ γ◦U◦
¯ γUrV
a ≡¯ γ◦
rV◦
− (Γ¯ ǫ + ¯ ρB+∆M
∆M
(Γ¯ ǫ◦+ ¯ ρ◦
V
)
?
, (79)
¯ γ =
U◦
r
+
B)¯ γ◦U◦
rV◦.(80)
These equations determine the gamma factor ¯ γ and the internal energy density
¯ ǫ = ¯ ǫ◦+ ∆¯ ǫ in the capture process of baryonic matter by the PEM pulse.
The effect of the collision of the PEM pulse with the remnant leads to the
following results50as a function of the B parameter defined in Eq.(74):
1) an abrupt decrease of the gamma factor given by
γcoll= γ◦
1 + B
?γ◦2(2B + B2) + 1,(81)
where γ◦is the gamma factor of the PEM pulse prior to the collision and B is given
by Eq.(74),
2) an increase of the internal energy in the comoving frame Ecolldeveloped in the
collision given by
Ecoll
Edya
=
?γ◦2(2B + B2) + 1
γ◦
−
?1
γ◦
+ B
?
,(82)
3) a corresponding reheating of the plasma in the comoving frame but not in the
laboratory frame, an increase of the number of e+e−pairs and correspondingly an
overall increase of the opacity of the pulse. See details in section 10.
8. The era III: the PEMB pulse
After the engulfment of the baryonic matter of the remnant the plasma formed
of e+e−pairs, electromagnetic radiation and baryonic matter expands again as a
sharp pulse, namely the PEMB pulse. The calculation is continued as the plasma
fluid expands, cools and the e+e−pairs recombine until it becomes optically thin:
?
R
dr(ne± +¯ZnB)σT ≃ O(1), (83)
where σT = 0.665 · 10−24cm2is the Thomson crosssection and the integration
is over the radial interval of the PEMB pulse in the comoving frame. We have
first explored the general problem of the PEMB pulse evolution by integrating
the general relativistic hydrodynamical equations with the Livermore codes, for a
Page 43
On the structure of the burst and afterglow of GammaRay Bursts I: the radial approximation43
total energy in the dyadosphere of 3.1 × 1054erg and a baryonic shell of thickness
∆ = 10rdsat rest at a radius of 100rdsand B ≃ 1.3 · 10−4.
In total analogy with the special relativistic treatment for the PEM pulse, pre
sented in section 6 (see also Ruffini, Salmonson, Wilson & Xue, 199949), we obtain
for the adiabatic expansion of the PEMB pulse in the constantslab approximation
described by the Rome codes the following hydrodynamical equations with ρB?= 0
¯ n◦
¯ nB
B
=V
V◦
?V
=
V¯ γ
V◦¯ γ◦,
?Γ
(Γ¯ ǫ◦+ ¯ ρ◦
(Γ¯ ǫ + ¯ ρB)V
∂V
(84)
¯ ǫ◦
¯ ǫ
=
V◦
?
=
?V
V◦
B)V◦
?Γ?¯ γ
¯ γ◦
?Γ
, (85)
¯ γ = ¯ γ◦
, (86)
∂
∂t(Ne±) = −Ne±1
V
∂t+ σv1
¯ γ2(N2
e±(T) − N2
e±).(87)
In these equations (r > rout) the comoving baryonic mass and number densities
are ¯ ρB= MB/V and ¯ nB= NB/V , where V is the comoving volume of the PEMB
pulse.
We compare and contrast (see Fig. 5) the bulk gamma factor as computed from
the Rome and Livermore codes, where excellent agreement has been found. This
validates the constantthickness approximation in the case of the PEMB pulse as
well. On this basis we easily estimate a variety of physical quantities for an entire
range of values of B.
For the same EMBH we have considered five different cases: a shell of baryonic
mass with (1) B ≃ 1.3 · 10−4; (2) B ≃ 3.8 · 10−4; (3) B ≃ 1.3 · 10−3; (4) B ≃
3.8 · 10−3; (5) B ≃ 1.3 · 10−3). The results of the integration given in detail in
Ruffini, Salmonson, Wilson & Xue (2000)50show that for the first parameter range
the PEMB pulse propagates as a sharp pulse of constant thickness in the laboratory
frame, but already for B ≃ 1.3 · 10−2the expansion of the PEMB pulse becomes
much more complex and the constantthickness approximation ceases to be valid;
see Ruffini, Salmonson, Wilson & Xue (2000)50for details.
It is particularly interesting to evaluate the final value of the gamma factor of
the PEMB pulse when the transparency condition given by Eq.(83) is reached as
a function of B, see Fig. 19. For a given EMBH, there is a maximum value of the
gamma factor at transparency. By further increasing the value of B the entire Edya
is transferred into the kinetic energy of the baryons; see also section 11. Details are
given in Ruffini, Salmonson, Wilson & Xue (2000).50
In Fig. 20 we plot the gamma factor of the PEMB pulse versus the radius for
different amounts of baryonic matter. The diagram extends to values of the radial
coordinate at which the transparency condition given by Eq.(83) is reached. The
Page 44
44R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
100
1000
10000
100000
1e0081e0071e0061e0050.00010.0010.01
Lorentz γ factor at the transparency point
B
Fig. 19.
the B parameter. The asymptotic value (the dashed line) Edya/(MBc2) is also plotted.
The gamma factor (the solid line) at the transparent point is plotted as a function of
“asymptotic” gamma factor
¯ γasym≡
Edya
MBc2
(88)
is also shown for each curve. The closer the gamma value approaches the “asymp
totic” value (88) at transparency, the smaller the intensity of the radiation emitted
in the burst and the larger the amount of kinetic energy left in the baryonic matter.
9. The identification of the free parameters of the EMBH theory
Within the approximation presented in section 2 the EMBH is characterized by two
parameters: µ and ξ. The energy of the dyadosphere is expressed in terms of these
two parameters by Eq.(12).
There is an entire family of EMBH solutions with different values of µ and ξ
corresponding to the same value of Edya(see Fig. 16). These solutions are physically
different with respect to the density of electronpositron pair distributions given by
Eq.(9), as well as to their energy density given by Eq.(11). A clear example of
such a degeneracy is given in Fig. 21 where the two limiting energy density profiles
approximating the dyadosphere as introduced in Fig. 17 are given for three different
Page 45
On the structure of the burst and afterglow of GammaRay Bursts I: the radial approximation45
Fig. 20.
radius for selected values of B for the typical case Edya= 3.1 × 1054erg. The asymptotic values
γasym = Edya/(MBc2) = 104,103,102are also plotted. The collision of the PEM pulse with the
baryonic remnant occurs at r/rds= 100 where the jump occurs and the PEMB pulse starts.
The gamma factors are given as functions of the radius in units of the dyadosphere
EMBH configurations corresponding to the same value of Edya= 3.1 × 1054erg.
The three configurations correspond respectively to the three different pairs (µ,ξ):
(10,0.76),?102,0.27?,?103,0.10?.
tion 6 and Ruffini, Salmonson, Wilson & Xue (1999)49is clearly different in the
three cases. It is remarkable that when the collision with the remnant of the pro
genitor star is considered all these differences disappear. As usual (see section 7) we
describe the baryonic content of the remnant by the parameter B. The PEMB pulse
generated after the collision with the baryonic matter depends uniquely on the two
parameters Edyaand B. In Fig. 22 the temperature in the laboratory frame is given
The corresponding dynamical evolution of the PEM pulse introduced in sec
Page 46
46R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
1e+024
1e+026
1e+028
1e+030
1e+032
1e+034
4e+008 6e+008 8e+008 1e+009
energy density (erg/cm3)
Laboratory radius (cm)
µ=103, ξ=0.10
1e+024
1e+026
1e+028
1e+030
1e+032
1e+034
1e+0081e+009
energy density (erg/cm3)
Laboratory radius (cm)
µ=102, ξ=0.27
1e+024
1e+026
1e+028
1e+030
1e+032
1e+034
1e+0071e+0081e+009
energy density (erg/cm3)
Laboratory radius (cm)
µ=10, ξ=0.76
Fig. 21.
and with different values of the two parameters µ and ξ are given. The three different configurations
are markedly different in their spatial extent as well as in their energydensity distribution.
Three different dyadospheres corresponding to the same value of Edya= 3.1 × 1054erg
for the PEM pulse and the PEMB pulse corresponding to the three configurations
of Fig.21 and B = 4 × 10−3. It is clear that while for the PEM pulse era the three
configurations are markedly different, they do converge to a common behaviour in
the PEMB pulse era.
If we turn now to the effect of the distance between the EMBH and the baryonic
remnant, we see that this degeneracy is further extended: while the three PEM
Page 47
On the structure of the burst and afterglow of GammaRay Bursts I: the radial approximation47
0.001
0.01
0.1
1
10
100
0.00010.0010.010.11 10100100010000100000
Temperature in the laboratory frame (MeV)
Laboratory time (t) (s)
µ=10, ξ=0.76
µ=102, ξ=0.27
µ=103, ξ=0.10
Fig. 22.
in the laboratory frame, corresponding to the three configurations presented in Fig. 21 is given
as a function of the laboratory time. The three different curves converge to a common one in the
PEMB pulse era, which is therefore only a function of the Edyaand B. The difference among
the three curves in the early part of the PEMB pulse follows from having located the baryonic
matter at a distance of 50(rds−r+), which is different in the three cases. Such difference become
negligible at large distances in the later phases of the evolution.
The temperature of the plasma during the PEM pulse and PEMB pulse eras, measured
pulse eras are quite different, the common PEMB pulse era is largely insensitive
to the location of the baryonic remnant, see Fig. 23. We have plotted the three
gamma factors in the PEM pulse era corresponding to the different configurations
of Fig. 21 and B = 10−2, in the two cases the baryonic remnant is positioned at
different distances from the EMBH.
If the PEM pulse has reached extreme relativistic regimes, the common value
γcollto which the three gamma factors drop in the collision with the baryonic matter
of the remnant can be simply expressed by the large gamma limit of Eq.(81)
γcoll=
B + 1
√B2+ 2B,
(89)
while the internal energy Ecoll developed in that collision is simply given by the
corresponding limit of Eq.(82)
Ecoll
Edya
= −B +
?
B2+ 2B .(90)
This approximation applies when the final gamma factor at the end of the PEM
Page 48
48R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
pulse era is larger than γcoll, upper panel in Fig. 23.
1
10
100
1000
1e+0081e+0091e+0101e+0111e+0121e+013 1e+014
Lorentz γ factor
Laboratory radius (cm)
µ=10, ξ=0.76
µ=102, ξ=0.27
µ=103, ξ=0.10
1
10
100
1000
1e+0081e+009 1e+0101e+0111e+012 1e+0131e+014
Lorentz γ factor
Laboratory radius (cm)
µ=10, ξ=0.76
µ=102, ξ=0.27
µ=103, ξ=0.10
Fig. 23.
function of the radial coordinate in the laboratory frame. The two figures correspond to a baryonic
remnant positioned respectively at rin= 50(rds− r+) (above) and at rin= 5(rds− r+). Again
the convergence to a common behaviour, uniquely a function of Edyaand B for the late stages of
the PEMB pulse, is manifest.
The gamma factors for the three configurations considered in Fig. 21 are given as a
Turning from these general considerations to the GRB data, this degeneracy in
the PEMB pulse eras and their dependence on only two parameters Edyaand B has
far reaching astrophysical implications for the identification of the source of GRBs.
As we will see in the conclusions all the information obtainable from GRBs with
a large value of the parameter B will lead to the determination of the above two
parameters. An entire family of degenerate astrophysical solutions in the range of
charges and masses given in Fig. 16 are possible. The direct knowledge of the mass
Page 49
On the structure of the burst and afterglow of GammaRay Bursts I: the radial approximation49
and charge of the EMBH can only be gained from the PEM pulse or from GRBs
with very small values of B — the so called “short bursts”, see section 11 and the
conclusions.
10. The approach to transparency: the thermodynamical
quantities
As the condition of transparency expressed by Eq.(83) is reached the injector phase
terminates. The electromagnetic energy of the PEMB pulse is released in the form
of freestreaming photons — the proper GRB. The remaining energy of the PEMB
pulse is released as an acceleratedbaryonicmatter (ABM) pulse.
We now proceed to the analysis of the approach to the transparency condition.
It is then necessary to turn from the pure dynamical description of the PEMB
pulse described in the previous sections to the relevant thermodynamic parameters.
Also such a description at the time of transparency needs the knowledge of the
thermodynamical parameters in all previous eras of the GRB.
As above we shall consider as a typical case an EMBH of Edya = 3.1 × 1054
erg and B = 10−2. The considerations will refer to a dyadosphere configuration
described by the two limiting approximations shown in Fig. 17.
One of the key thermodynamical parameters is represented by the temperature
of the PEM and PEMB pulses. It is given as a function of the radius both in the
comoving and in the laboratory frames in Fig. 24. Before the collision the PEM
pulse expands keeping its temperature in the laboratory frame constant while its
temperature in the comoving frame falls.49. In fact Eqs.(63,64) are equivalent to
d(ǫγ2V)
dt
= 0, (91)
where the baryon massdensity is ρB= 0 and the thermal energydensity of photons
and e+e−pairs is ǫ = σBT4(1 + fe+e−), σB is the Boltzmann constant and fe+e−
is the Fermiintegral for e+and e−. This leads to
ǫγ2V = Edya,T4γ2V = const.(92)
Since e+and e−in the PEM pulse are extremely relativistic, we have the equation
of state p ≃ ǫ/3 and the thermal index (22) Γ ≃ 4/3 in the evolution of PEM pulse.
Eq.(92) is thus equivalent to
T3¯ γV ≃ const.(93)
These two equations (91) and (93) result in the constancy of the laboratory tem
perature T¯ γ in the evolution of the PEM pulse.
It is interesting to note that Eqs.(92) and (93) hold as well in the crossover
region where T ∼ mec2and e+e−annihilation takes place. In fact from the conser
vation of entropy it follows that asymptotically we have
(V T3)T<mec2
(V T3)T>mec2=11
4
, (94)
Page 50
50R. Ruffini, C.L. Bianco, P. Chardonnet, F. Fraschetti, S.S. Xue
exactly for the same reasons and physics scenario discussed in the cosmological
framework by Weinberg, see e.g. Eq. (15.6.37) of Weinberg (1972). The same con
siderations when repeated for the conservation of the total energy ǫγV = ǫγ2V
following from Eq. (91) then lead to
(V T4γ)T<mec2
(V T4γ)T>mec2=11
4
.(95)
The ratio of these last two quantities gives asymptotically
T◦= (Tγ)T>mec2 = (Tγ)T<mec2, (96)
where T◦is the initial average temperature of the dyadosphere at rest.
During the collision of the PEM pulse with the remnant we have an increase in
the number density of e+e−pairs (see Fig. 25). This transition corresponds to an
increase of the temperature in the comoving frame and a decrease of the temperature
in the laboratory frame as a direct effect of the dropping of the gamma factor (see
Fig. 20).
After the collision we have the further acceleration of the PEMB pulse (see
Fig. 20). The temperature now decreases both in the laboratory and the comoving
frame (see Fig. 24). Before the collision the total energy of the e+e−pairs and the
photons is constant and equal to Edya. After the collision
Edya= EBaryons+ Ee+e− + Ephotons, (97)
which includes both the total energy Ee+e− + Ephotonsof the nonbaryonic compo
nents and the kinetic energy EBaryonsof the baryonic matter
EBaryons= ¯ ρBV (¯ γ − 1). (98)
In Fig. 26 we plot both the total energy Ee+e− + Ephotons of the nonbaryonic
components and the kinetic energy EBaryonsof the baryonic matter as functions of
the radius for the typical case Edya= 3.1×1054erg and B = 10−2. Further details
are given in Ruffini, Salmonson, Wilson & Xue (2000)50.
11. The PGRBs and the “short bursts”. The end of the injector
phase.
We now analyze the approach to the transparency condition given by Eq.(83). For
selected values of B we give the energy EPGRBof the PGRB, and EBaryonsof the
ABM pulse. We clearly have
Edya= EPGRB+ EBaryons. (99)
Taking into account the results shown in Figs. 24–26, we can repeat all the
considerations for selected values of B. We shall examine values of B ranging from
B = 10−8only up to B = 10−2: for larger values of B our constant slab approxi
mation breaks down. We will see in the following that this range does indeed cover
the most relevant observational features of the GRBs.
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