%%
%% This is file `cimsmple.tex'
%%
%%
%% IMPORTANT NOTICE:
%%
%% For the copyright see the source file.
%%
%% Any modified versions of this file must be renamed
%% with new filenames distinct from cimsmple.tex.
%%
%%
%% This generated file may be distributed as long as the
%% original source files, as listed above, are part of the
%% same distribution. (The sources need not necessarily be
%% in the same archive or directory.)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass{cimento}
%%%%%%%%%%%%%
%
%VERY IMPORTANT
%
% If you are preparing Enrico Fermi School of
% Physics report, please read the bundled file README.varenna
%
%%%%%%%%%%%%
%%%%%%%%%%%%%%%%
%
% VERY IMPORTANT
%
% In order to set a Copyright owner please use and fulfill the following command
%\setcopyright{CERN on behalf the XXXXX Collaboration}
%
%
%%%%%%%%%%%%%%%
\usepackage{graphicx} % got figures? uncomment this
\title{The gravitational wave and short gamma-ray burst \\ GW170817/SHB170817A}
%\\
%not your everyday binary neutron star merger}
\author{A. De R\'ujula}
%\from{ins:x}\ETC}
\instlist{\inst{} Instituto de F\'isica Te\'orica (UAM/CSIC), Univ. Aut\'onoma de Madrid, Spain;\\
Theory Division, CERN, CH; 1211 Geneva 23, Switzerland}
%% When only one author is present, please do not use the command \from{} near the author name.
\begin{document}
\maketitle
\begin{abstract}
This event, so far unique, beautifully confirmed the standard views on
the gravitational waves produced by a merger of two neutron stars, but
its electromagnetic multi-wavelenth observations disagreed with the
numerous initial versions of the
``standard fireball model(s)" of gamma ray bursts. Contrariwise, they provided strong
evidence in favour of the ``cannonball" model. Most uncontroversially,
a cannonball was observed at radio wavelengths, with an overwhelming
statistical significance ($>\! 17\,\sigma$), and travelling in the plane
of the sky, as expected, at an average apparent superluminal velocity $V_{app}\sim 4\, c$.
\end{abstract}
\section{Introduction}
The GW170817 event was the first binary neutron-star merger
detected with Ligo-Virgo \cite{Abbott} in
gravitational waves (GWs). It was followed by SHB170817A\footnote{SHB stands for
``Short Hard Burst", a sub-class of gamma-ray bursts (GRBs) lasting less
that $\sim 2$ s and whose photons generally have comparatively large energies.},
$1.74\!\pm\! 0.05$ s after the end
of the GW's detection. The SHB's afterglow across the electromagnetic spectrum was
used to localize its source \cite{Hjorthetal2017} to the galaxy NGC 4993, at a cosmologically
very modest redshift, $z\!=\!0.009783$.
The GW170817/SHB170817A association was the first indisputable confirmation that pairs of
neutron stars merging due to GW emission produce GRBs,
thereafter converting this suggestion \cite{Goodman2} \cite{Meszaros}
into a general consensus.
Not so well known is the fact that two
days before the discovery date, a paper appeared on arXiv
\cite{DDlargeangle}, not only reiterating the neutron star merger hypothesis,
but predicting that a SHB found in combination with a GW
would be seen far off axis. The prediction was based on
the much greater red-shift reach of GRB or SHB observations relative to the GW ones
and the fact that the $\gamma$ rays are extremely collimated:
within the volume reach of GW observations, it would be most unlikely for a
SHB to point close to the observer.
Since 1997 only two theoretical models of GRBs and their afterglows (AGs) --the standard
fireball (FB) model \cite{FBM Reviews}
and the cannonball (CB) model \cite{DD2004}-- have been extensively used
to interpret the innumerable observations. Advocates of both models have
claimed to fit the data very well. But the two models were originally
and still are quite different in their basic assumptions, despite the repeated
replacements of key assumptions of the ``standard" FB model (but not its
name) with assumptions underlying the CB model --e.g.~supernovae (SNe) of Type Ia
as progenitors of most GRBs, highly collimated ejecta made of ordinary matter,
as opposed to spherical or conical shells of an $e^+\,e^-\,\gamma$ plasma, ``jets" not necessarily
seen almost on axis. For a recent extensive discussion of the observational tests
of FB and CB models, see \cite{CTDDD}.
Significantly, and in contrast to the FB model(s), the CB model has made many successful
{\it predictions}. Among them, the large polarization of the GRB's $\gamma$ rays \cite{jet},
the precise date at which the supernova associated with GRB030329 would be
discovered {\cite{DDD030329}, the complex ``canonical" shape \cite{Dado2002}
of many GRB afterglows \cite{Vaughan,Cusumano}, the correlations
between various prompt\footnote{{\it Prompt} customarily refers to quantities measured
prior to the afterglow phase.}
observables \cite{CorrelsCB,Correlations}
%--such as the ``Amati" correlation \cite{Amati2002}--
or with AG observables \cite{Dado 2013}.
The SHB170817A event is an optimal case to tighten
the discussion of the comparisons between different models of GRBs.
The question of the apparently superluminal motion of the source
of its afterglow, discussed in chapter \ref{sec:super}, is particularly
relevant.
%\section{Description}
%This is a very short sample paper distributed with the class
%\texttt{cimento}.
%It is just a collection of examples about the syntax of commands
%which behave in a different way from the standard \LaTeX
%and/or new commands not defined in \LaTeX.
%\begin{figure}
%\includegraphics{foo} % includes figure foo.eps
%\caption{Description of the figure.}
%\end{figure}
%\subsection{Tables}
%Table~\ref{tab:pricesI}
%inserted at this point.
%
%\begin{table}
% \caption{Prices of important items.}
% \label{tab:pricesI}
% \begin{tabular}{rcl}
% \hline
% Item 1 & 1500 & EUR \\
% Item 2 & 15000 & EUR \\
% Item 3 & 1500 & dollars \\
% \hline
% Item 4 & .25 & dollars \\
% Item 5 & 1.25 & dollars \\
% Item 6 & 1 & dollars \\
% \hline
% \end{tabular}
%\end{table}
\section{The cannonball model}
The CB model is based on a direct analogy of a
phenomenon that is abundantly observed but poorly
understood: the relativistic ejecta emitted by quasars and micro-quasars.
The model \cite{jet} is illustrated in Figure \ref{fig:Figure1}.
In it, bipolar jets of highly relativistic ordinary-matter
plasmoids (a.k.a.~CBs) are assumed to be launched as a compact
stellar object is being born.
SNe of Type Ic (the broad-line stripped-envelope
ones) thus generate long-duration GRBs as the electrons in a CB raise the photons
in the SN's ``glory" by Inverse Compton Scattering (ICS) into a forward-collimated
narrow beam of $\gamma$-rays \cite{jet}.
Similarly, in the CB model, mergers of two neutron stars (NSs) or a NS and a black hole
(BH) give rise to SHBs.
In this case, the role of the glory of light is played by a Pulsar Wind Nebula (PWN)
powered by the spin-down of a newly born rapidly-rotating pulsar--
suggesting that most SHBs are produced by
NS mergers yielding a NS remnant rather than a black hole \cite{Dado2018,Dai98}.
SN-less GRBs are produced in high-mass X-ray binaries, as a NS accreting mass
from a companion suffers a phase transition to a denser object.
Finally X-ray flashes (XRFs) and some X-ray transients are simply GRBs observed from a relatively
large angle relative to the CBs' emission axis.
%\begin{figure}
%\centering
%\includegraphics[width=7.5 cm]{Glory.pdf}
%\caption{Electrons in a cannonball inverse Compton scatter photons in the glory of light
%surrounding a newly-born compact object, launching them forward as a narrow beam of
%$\gamma$ rays.}
%\label{fig:glory}
%\end{figure}
\begin{figure}[]
\centering
\includegraphics[width=11cm]{Figure1.pdf}
%\epsfig{file=epvt_shb.eps,width=9.0cm,height=9.0cm}
\caption{{\bf Left}: Electrons in a CB Compton scatter photons in the glory
of a newly-born compact object, launching them forward as a narrow beam of
$\gamma$ rays. {\bf Right}:
Comparison between the CB-model's correlation $E_p\!\propto\! 1/T_p$
% and the corresponding
and the data in GCN circulars
for resolved pulses of SGRBs.}
\label{fig:Figure1}
%\label{Fig28}
\end{figure}
%\subsection{Mathematics}
%Here is a lettered array~(\ref{e.all}), with eqs.~(\ref{e.house})
%and~(\ref{e.phi}):
%\begin{eqnletter}
% \label{e.all}
% \drm x_\sy{F} & = & 1.2\cdot10^3\un{cm}, \qquad
% \tx{where\ } \sy{F} = \tx{Fermi} \label{e.house}\\
% \phi_i & = & i\pi \label{e.phi}
%\end{eqnletter}
%\subsection{Citations}
%We're almost done, just some citations~\cite{ref:apo}
%and we will be over~\cite{ref:pul,ref:bra}.
%\acknowledgments
%The author acknowledge XXX, YYY.
%\begin{thebibliography}{0}
%\bibitem{ref:apo} \BY{Einstein A. \atque Fermi E.}
% \IN{Phys. Rev. A}{13}{1999}{12};
% \SAME{69}{999}{1666}.
%\bibitem{ref:pul} \BY{Newton I.}
% preprint INFN 8181.
%\bibitem{ref:bra} \BY{Bragg~B.}
% \TITLE{Complete Works}, in \TITLE{Workers Playtime}, edited by \NAME{Tizio A. \atque Caio B.} (Unexeditor, Bologna) 1997, pp.~1-10.
\section{Is SHB170817A noteworthy all by itself?}
A first question regarding SHB170817A is whether or not it is a typical SHB. The CB
model provides a strongly affirmative answer, in all respects. A first test employs the
correlations between observables predicted by this model.
Let $\gamma_0$ be the initial Lorentz factor with which a CB is launched.
Its electrons inverse-Compton-scatter the ambient photons they encounter.
This results in a $\gamma$-ray pulse of aperture $\simeq\! 1/\gamma_0\!\ll\! 1$
around the CB's direction.
Viewed by an observer at an angle $\theta$ relative to the CB's direction, the individual photons
are boosted in energy by a Doppler factor
$\delta_0\!\equiv\!\delta(t\!=\!0)\!=\!1/[\gamma_0\,(1\!-\!\beta_0\,\cos\theta)]$
or, to a good
approximation for $\gamma_0^2\!\gg\!1$ and $\theta^2\!\ll\!1$,
$\delta_0\!\simeq\!2\gamma_0/(1\!+\!\gamma_0^2\theta^2)$.
The ICS of photons of energy $\epsilon$ by a CB boosts their energy, as seen by an
observer at redshift $z$, to $E_\gamma\!=\!\gamma_0\,\delta_0\,\epsilon/(1\!+\!z)$.
Thus, the peak energy, $E_p$, of their time-integrated energy distribution satisfies
$
(1+z)\,E_p\!\approx\! \gamma_0\,\delta_0\, \epsilon_p ,
$
%\begin{equation}
%(1+z)\,E_p\!\approx\! \gamma_0\,\delta_0\, \epsilon_p ,
%\label{eq:Ep0}
%%\label{Eq2}
%\end{equation}
with $\epsilon_p$ the characteristic or peak energy of the initial photons (for the glory
of a SN $\epsilon_p\!=\!{\cal O}(1)$ eV, for a PWN $\epsilon_p\!=\!{\cal O}(1)$ keV).
The peak-time of a single $\gamma$-ray
pulse obeys $T_p\!\propto\!(1\!+\!z)/\gamma_0\, \delta_0$.
SHB170817A,
being a one-peak event, is a good case to study these observables. Indeed, one
of the simplest CB-model predictions is the $[E_p,T_p]$ correlation $E_p\!\propto\! 1/T_p$.
%\begin{equation}
%E_p\!\propto\! 1/T_p\,.
%\label{eq:EpTp}
%%\label{Eq32)
%\end{equation}
In the righthand side of Figure \ref{fig:Figure1}
this correlation is compared
with the values of $E_p$ and $T_p$ in the GCN circulars
\cite{GCN}
for resolved SGRB pulses.
%measured by the Konus-Wind and by the Fermi-GBM collaborations.
SHB170817A is where it should be.
The nearly isotropic distribution (in the CB's rest frame)
of a total number $n_\gamma$
of IC-scattered photons is beamed into an angular distribution
$dn_\gamma/d\Omega\!\approx\! (n_\gamma/4\,\pi)\,\delta^2$
in the observer's frame. Consequently, the isotropic-equivalent
total energy of the photons satisfies
$
E_{iso}\!\propto\! \gamma_0\, \delta_0^3\, \epsilon_p.
$
%\begin{equation}
%E_{iso}\!\propto\! \gamma_0\, \delta_0^3\, \epsilon_p.
%\label{eq:Eiso}
%%\label{Eq3}
%\end{equation}
Hence, both ordinary long- and short-duration GRBs, viewed
most probably from an angle $\theta\!\sim\!1/\gamma$
(for which $\delta_0\!\sim\!\gamma_0$), should satisfy \cite{CorrelsCB}
the ``Amati" correlation \cite{Amati2002},
$
(1+z)\,E_p\propto [E_{iso}]^{1/2},
$
%\begin{equation}
%(1+z)\,E_p\propto [E_{iso}]^{1/2},
%\label{eq:Corr1}
%%\label{Eq4}
%\end{equation}
shown in on the left of Figure \ref{fig:Figure2}.
Far off-axis GRBs [$\theta^2\! \gg \! 1/\gamma^2$, so that
$\delta_0\!\approx\! 2/(\gamma_0\,\theta^2)$],
have a much lower $E_{iso}$, and satisfy
$
(1+z)\,E_p\propto [E_{iso}]^{1/3}.
$
As shown on the right of Figure \ref{fig:Figure2}, SHB170817A ``is right where it should be",
since, as we shall see, it was indeed seen far off-axis.
The correlations we discussed are trivial consequences of
GRBs being narrow beams of $\gamma$ rays seen somewhat off-axis. They are not predictions
of FB models. In them, the jetted beams were generally assumed to be seen on-axis, at least
up to the observation of SHB170817A.
%\begin{figure}[]
%\centering
%\includegraphics[width=8.5 cm]{epeiso17GRBs.pdf}
%%\epsfig{file=epeiso17GRBs.eps,width=9.cm,height=9.cm}
%\caption{The $[E_p, E_{iso}]$ correlation in ordinary LGRBs viewed near axis.
%The line is the best fit, whose slope, $0.48\!\pm\! 0.02$,
%agrees with the CB model prediction of
%Equation \ref{eq:Corr1}.
%The lowest $E_{iso}$ GRB is 020903 at $z\!=\!0.25$ (HETE).}
%\label{fig:epeiso17GRBs}
%%\label{Fig3}
%\end{figure}
\begin{figure}[]
%\vspace{-2cm}
\hspace{1cm}
\includegraphics[width=11cm]{Figure2.pdf}
%\epsfig{file=epeisoallshbs.eps,width=9.cm,height=9.cm}
\caption{{\bf Left}: The $[E_p, E_{iso}]$ correlation in GRBs viewed near axis.
The line is the best fit, whose slope, $0.48\!\pm\! 0.02$,
agrees with the CB model's prediction: 1/2.
{\bf Right}: The correlation in SHBs.
The lines are the CB-model's predicted correlations.}
\label{fig:Figure2}
%\label{Fig5}
\end{figure}
A more detailed test concerning the run-of-the-mill nature of SHB170817A is
the shape of its single pulse of $\gamma$'s.
The light ``reservoir" that a CB will Compton up-scatter (a SN glory or a PWN) has a
thin thermal bremsstrahlung spectrum and a number density distribution decreasing
with distance to the CB source as $1/r^2$ \cite{ThBrem}. With these inputs it is
painless to derive a two-parameter simple expression that provides excellent
descriptions of pulse shapes \cite{Dado2009a}.
A well measured GRB pulse is shown on the left of
Figure \ref{fig:Figure3}. The pulse of SHB170817A is on the right.
Again, there is nothing atypical about it. The peak energy and peak time
of this SHB are also the expected ones \cite{CTDDD}.
\begin{figure}[]
\centering
\includegraphics[width=11.cm]{Figure3.pdf}
%\epsfig{file=pulse-shape930612.eps,width=6.cm,height=8.cm}
%\vspace{-1cm}
\caption{{\bf Left}: The pulse shape of GRB930612 measured with BATSE
aboard CGRO and
the best CB-model fit to re-bined data \cite{DDD2022}.
{\bf Right}: The pulse shape of SHB170817A measured with the Fermi-GBM
\cite{20a,20c,20d,20e,20b} and its CB-model best fit, with $\chi^2/\rm {dof}\!=\!0.95$
\cite{DDD2022}. }
\label{fig:Figure3}
%\label{Fig6}
\end{figure}
%\begin{figure}[]
%\centering
%\includegraphics[width=8.5 cm]{fig02.pdf}
%%\epsfig{file=fig02.eps,width=9.cm,height=8.cm}
%\caption{The pulse shape of SHB170817A measured with the Fermi-GBM
% \cite{20a,20c,20d,20e,20b} and its CB-model best fit, with
% $\chi^2/\rm {dof}\!=\!0.95$
% \cite{DDD2022}.
%%and the best fit pulse shape given by Equations \ref{eq:pulseShape}
%%and \ref{eq:Epoft} with
%%$\Delta\!=\!0.62$ s and $\tau\!=\!0.57$ s, $\chi^2/\rm {dof}\!=\!0.95$.
%}
%\label{fig:fig02}
%%\label{Fig9}
%\end{figure}
%
%\subsection{Prompt observables of SHB170817A. Further checks}
%\label{subsec:chesks}
%
%{\it Lorentz factor.}
%In the CB model GRBs and SHBs can be understood as approximately standard
%candles viewed from different angles.
% As illustrated in Figure \ref{fig:epeisoallshbs}, low luminosity (LL)
%SHBs such as SHB170817A are ordinary (O) SHBs, but viewed far off-axis.
%Consequently, their $E_{iso}$ and $E_p$ are expected to obey the relations
%\begin{equation}
%E_{iso}{\rm (LL\,SHB)}\! \approx\! \langle\! E_{iso}{\rm (O\,SHB)}\!\rangle /
%[\gamma^2\,(1\!-\!\cos\theta)]^3 \,,
%\label{eq:LLSHBEiso}
%%\label{Eq28}
%\end{equation}
%\begin{equation}
%(1+z)\,E_p{\rm(LL\,SHB)}\!\approx\! \langle (1+z)\,E_p{\rm (O\,SHB)}\rangle /
%[\gamma^2\,(1\!-\!\cos\theta)].
%\label{eq:LLSHBEp}
%%\label{Eq29}
%\end{equation}
%
%Given the measured value $E_{iso}\!\approx\! 5.4\times 10^{46}$ erg of
%SHB170817A \cite{Goldsteinetal2017}, the mean
%value $\langle E_{iso}\rangle\!\approx\!1\times 10^{51}$ erg of
%ordinary SGRBs, and the viewing angle
%$\theta\!\approx\!28$ deg obtained --as we shall discuss in detail in section \ref{sec:superluminal}--
%from the superluminal motion of the source of
%the radio AG of SHB170817A \cite{MooleyJET}, Equation \ref{eq:LLSHBEp}
%yields $\gamma_0\!\approx\!14.7$ and $\gamma_0\,\theta\!\approx\! 7.2$.
%
%With $\gamma_0$ and $\theta$ specified,
%Equations \ref{eq:LLSHBEiso} and \ref{eq:LLSHBEp},
%result in the following additional tests of CB model predictions:
%
%{\it Peak energy.}
%Assuming that SHBs have the same redshift distribution as GRBs
%(with a mean value $\langle z\rangle \!\approx\!2$), and given the observed
%$\langle E_p\rangle\!=\!650$ keV of SHBs \cite{Goldsteinetal2017},
%one obtains $\langle(1\!+\!z)\,E_p\rangle\! \approx\! 1950$ keV.
%Consequently, Equation \ref{eq:LLSHBEp} with $\gamma_0\,\theta\!\approx\! 7.2$ and
%$z\!\approx\!1$
%yields $E_p\!\approx\!75$ keV for SHB170817A. This is to be
%compared with $E_p\!=\!82 \pm 23$ keV ($T_{90}$) reported in
%\cite{20a},
%$E_p\!=\!185 \pm 65$ keV estimated in \cite{20c},
%and $E_p\!\approx\!65\!+\!35(\!-\!14)$ keV estimated in \cite{20d},
%from the same data, with a mean value $E_p\!=\!86\!\pm\!19$ keV,
%agreeing with the expectation.
%
%{\it Peak time.}
%In the CB model the peak time $\Delta t$ after the beginning
%of a GRB or SHB pulse is roughly equal to half of its
%full width at half maximum
%(FWHM) \cite{Dado2009a}.
%Assuming again that SHBs are roughly standard candles, the
%dependence of their $\Delta t$ values on $\theta$ is
%\begin{equation}
%\Delta t{\rm (LL\,SHB)}\,\approx\! \gamma_0^2\,(1\!-\!\cos\theta)
%\langle \Delta t{\rm (O\,SHB)}\rangle,
%\label{eq:Peakt}
%%\label{Eq31)
%\end{equation}
%For $\theta\!\approx\!28$ deg,
%$\Delta t\!\approx\! 0.58$ s obtained from the prompt emission pulse
%of SHB170817A (see Figure \ref{fig:fig02}), and $\langle{\rm FWHM(SHB)}\rangle\!=55$ ms,
%Equation \ref{eq:Vapp2} results in $\gamma_0\!\approx\! 14.7 $.
% Using Equations
%\ref{eq:LLSHBEiso} and \ref{eq:LLSHBEp}, and $\gamma_0\,\theta\!\simeq \!7.2$
%one checks that this value of
%$\gamma_0$ is consistent
%with $E_{iso}\!=\!5.4\times 10^{46}$ estimated in \cite{Goldstein2017}, and
%$\langle E_p\rangle\!=\! 86\!\pm\! 19$ keV
%the mean of the estimates in \cite{Dado2018}.
%
%In the CB model the shape of resolved SHB and GRB pulses
%satisfies $2\,\Delta t\!\approx\! {\rm FWHM} \!\propto\! 1/E_p$, as illustrated in
%Figure \ref{fig:epvt_shb}. Using the observed
%$\langle {\rm FWHM(SHB)}\rangle\!\approx\!55$ ms, $\gamma_0\!\sim\!14.7$,
%and $\theta\!\approx\! 28$ deg, Equation \ref{eq:Peakt} for SHB170817A results in
%$\Delta t \!\approx\!0.63$ s, in good agreement with its observed value,
%$0.58\!\pm\! 0.06$ s.
%
%Quite obviously, the replacements we have been making of physical
%parameters by their means may not be completely reliable,
%not only because of the spread in their values, but
%also because of detection thresholds and selection effects. Yet, in checking the
%unexceptional nature of SHB170817A, they work better than one could expect.
%
%\section{SHB170817A in FB models, prompt observables}
%The correlations in Figures \ref{fig:epeisoallshbs},\ref{fig:epvt_shb} are trivial consequences of
%GRBs being narrow beams of $\gamma$ rays seen somewhat off-axis. They are not predictions
%of FB models. In them, the jetted beams were generally
%assumed to be seen on-axis, at least up to the observation
%of SHB170817A.
In FB models \cite{FBM Reviews} the GRB prompt pulses are produced by
synchrotron radiation from shock-accelerated electrons in collisions between overtaking
thin shells ejected by a central engine, or by internal shocks in the ejected conical jet.
Only for the fast decline phase of the prompt emission, and only in the limits of very
thin shells and fast cooling, falsifiable predictions have been derived
%In these limits the
%fast decline phase of a pulse was obtained from the relativistic curvature effect
\cite{Curvature,Kobayashi,NorrisHakkila}.
They result in a power law decay
$F_\nu(t)\!\propto\!(t\!-\!t_i)^{-(\beta + 2)}\nu^{-\beta}$, where $t_i$ is the beginning time
of the decay phase, and $\beta$ is the spectral index of prompt emission.
The observed decay of the SHB170817A pulse could only be reproduced by adjusting a
beginning time of the decay and replacing the constant spectral index of the FB model
by the observed time-dependent one \cite{Curvature}.
\section{The afterglow of GRBs and SHBs}
In the CB model the afterglow of GRBs is due to synchrotron radiation (SR) from the
electrons in a CB that is traveling and decelerating as it interacts with the interstellar
medium (ISM), previously fully ionized by the GRB's $\gamma$ rays. The electrons radiate
in the turbulent magnetic field generated by the merging plasmas, whose energy density
is assumed to be in equilibrium with the kinetic energy brought (in the CB's rest system)
by the ISM constituents. With these inputs, the model provides an excellent and predictive
description of the temporal and spectral dependence of GRB afterglows.
The CB-model's description of the early AGs of SHBs and ``Supernova-less" GRBs differs from
the one of ordinary (SN-generated) GRBs. It is simply the isotropic radiation
from a pulsar wind nebula (PWN), powered by a newly born rapidly-rotating pulsar, and has an
expected luminosity \cite{Dado5} satisfying
$
L(t,t_b)/L(t=0)\!=\!(1\!+\!t/t_b)^{-2},
$
%\begin{equation}
%L(t,t_b)/L(t=0)\!=\!(1\!+\!t/t_b)^{-2},
%\label{eq:PWN}
%\end{equation}
with $t_b\!= \! P(0)/2\, \dot P(0)$, where $P(0)$ and $\dot P(0)$ are
the pulsar's initial period and its time derivative.
%This is shown in Figure \ref{fig:XOAG990510MSP} for GRB 990510.
This {\it universal behaviour} \cite{Dado2017} describes well
the AG of all the SN-less GRBs and SHBs with a well
sampled AG during the first few days after burst. This is demonstrated
on the left of Figure \ref{fig:Figure4}
%%Figure \ref{fig:XAG10GRBMSP}
%for
%the X-ray AG of twelve SN-less GRBs, and in Figure \ref{fig:XAGS12SHBMSP}
for the twelve SHBs \cite{Dado2018}
from the Swift XRT light curve repository \cite{Swift}
that were well sampled in the mentioned period.
The bolometric light curve of SHB170817A \cite{Drout2017}
is shown on the right of Figure \ref{fig:Figure4}. The two-parameter
[$L(0)$ and $t_b$] CB-model fit is excellent. SHB170817A, once more time,
is not deviant.
\begin{figure}[]
\centering
\includegraphics[width=11 cm]{Figure4.pdf}
%\epsfig{file=XAGS12SHBMSP.eps,width=9.0cm,height=9.0cm}
\caption{{\bf Left}: Comparison between the normalized light curve
of the X-ray AG of 11 SHBs
with a well sampled AG measured with Swift's XRT \cite{Swift}
during the first couple of days after burst
and the predicted universal behavior of \cite{Dado2017}.
{\bf Right}: Comparison between the observed \cite{Drout2017} bolometric light curve
of SHB170817A and the prediction of \cite{Dado2017},
assuming the presence of a milli-second pulsar
with $L(0)\!=\!2.27\times 10^{42}$ erg/s and
$t_b\!=\!1.15$ d. The fit has $\chi^2/{\rm dof}\!=\!1.04$.}
\label{fig:Figure4}
%\label{Fig21}
\end{figure}
%\begin{figure}[]
%\centering
%\includegraphics[width=8.5 cm]{fig04a.pdf}
%%\epsfig{file=fig04a.eps,width=9.0cm,height=9.0cm}
%\caption{Comparison between the observed \cite{Drout2017} bolometric light curve
%of SHB170817A and the universal light curve of Equation \ref{eq:PWN},
%assuming the presence of a milli-second pulsar
% with $L(0)\!=\!2.27\times 10^{42}$ erg/s and
%$t_b\!=\!1.15$ d. The fit has $\chi^2/{\rm dof}\!=\!1.04$.}
%\label{fig:fig04a}
%%\label{Fig29}
%\end{figure}
\subsection{The late-time afterglow of SHB170817A}
The PWN-powered early AG decreases with time extremely fast and
is eventually overtaken, in the CB model, by the synchrotron radiation from CBs.
In the case of SHB170817A, the AG was very well observed up to extremely
late times: almost three years \cite{Makha}. To discuss this subject
within the page-number limit I have to skip details contained in \cite{CTDDD}.
For a CB travelling in a constant-density ISM, the spectral energy density of its AG
is $F_\nu(t,\nu)\!\propto\! t^{0.72 \pm 0.03}\nu^{-0.56\!\pm\!0.06}$, where I used
the observed \cite{MooleySL} spectral index $0.56\!\pm\!0.06$,
which extends from the radio (R) band, through the optical (O) band, to
the X-ray band. If the CB moves out into a wind-like ISM density
distribution (proportional to $r^{-2}$)
$F_{\nu}(t,\nu) \propto t^{-2.12\!\pm\!0.06}\nu^{-0.56\!\pm\!0.06}$.
These results agree with the observations
\cite{Makha},
%$0.86\!\pm\!0.04$ and $-1.92\!\pm\!0.12$,
as shown in the upper part of Figure \ref{fig:RadioAG}.
%The observed spectral energy density (SED) flux of the {\it unabsorbed} SR,
%%X-rays,
%$F_\nu(t)\!=\!\nu\,dN_\nu/d\nu$, has the form (see, e.g.~Eqs.~(28)-(30) in \cite{Dado2002}),
%\begin{equation}
%F_{\nu} \propto n(t)^{(\beta_x+1)/2}\,[\gamma(t)]^{3\,\beta_x-1}\,
%[\delta(t)]^{\beta_x+3}\, \nu^{-\beta_x}\, ,
%\label{eq:Fnu}
%%\label{Eq8became7}
%\end{equation}
%where $n$ is the baryon density of the external medium encountered
%by the CB at a time $t$ and $\beta_x$ is the spectral index
%of the emitted X-rays, $E\,dn_x/dE\propto E^{-\beta_x}$.
%The CBs are decelerated by the swept-in ionized material. Energy-momentum conservation
%for such a plastic collision\footnote{The original assumption in the CB model was that the
%interactions between a CB and the ISM were elastic. It was later realized, in view of the
%shape of AGs at late times, that a plastic collision was a better approximation
%in the AG phase.}
%--between a CB of baryon number $N_B$, approximately constant radius $R$ \cite{DDCRs},
%and initial Lorentz factor $\gamma_0\!\gg\! 1$, propagating in an
%approximately constant-density ISM-- implies that the CB
%decelerates according to \cite{Dado2009a}:
%\begin{equation}
%\gamma(t) \simeq {\gamma_0\over \left[\sqrt{(1+\theta^2\,\gamma_0^2)^2 +t/t_d}
% - \theta^2\,\gamma_0^2\right]^{1/2}}\,,
%\label{eq:decelerate}
%%\label{Eq9became8}
%\end{equation}
%where $t$ is the time in the observer frame since the beginning of the AG emission
%by a CB, and $t_d$ is its deceleration time-scale:
%\begin{equation}
%t_d\!\simeq\!{(1\!+\!z)\, N_{_B}/ 8\,c\, n\,\pi\, R^2\,\gamma_0^3}.
%\label{eq:td}
%%\label{Eq10}
%\end{equation}
%
%As long as the Lorentz factor of a decelerating CB is such that $\gamma^2\!\gg\! 1$,
% $\gamma\,\delta\!\approx\!1/(1\!-\!\cos\theta)$ and
%the spectral energy density of its
% synchrotron AG --Equation \ref{eq:Fnu}-- can be rewritten as
%\begin{equation}
%F_{\nu}(t,\nu)\propto n(t)^{\beta_\nu+1/2}\,[\gamma(t)]^{2 \beta_\nu -4}\,\nu^{-\beta_\nu}.
%\label{eq:Fnu2}
%%\label{Eq33}
%\end{equation}
\begin{figure}[]
\centering
%\includegraphics[width=11.5 cm]{RadioAG.001.jpeg}
\includegraphics[width=9 cm]{RadioSHB170817Abis.pdf}
\includegraphics[width=11 cm]{Figure5.pdf}
%\epsfig{file=FB_AG170817A4.eps,width=8.cm,height=8.cm}
\caption{{\bf Up}: Radio, optical and X-ray observations of the AG of SHB170817A,
figure 2 of \cite{Makha}.
% The horizontal axis is the time after merger, in days.
The radio light curve
%measured until 940 days post-merger,
is scaled to 3 GHz using the spectral index $-0.584$.
The early-time trend expected in the CB model is the rising
black-dashed line. The late-time trend, also black-dashed, is for an assumed
$1/r^2$ ISM density decline encountered by the CB after day $\sim\! 150$.
{\bf Down Left}: Best fit light curves
of an off-axis structured jet model
\cite{Lazzati1} before December 2017
(first version of \cite{Lazzati1} arXiv dated 171217).
%December 8th, 2017).
{\bf Down, Right}: Best fit light curves up to April 2018,
obtained from a structured jet model \cite{Lazzati2}. Reported in
version 4 of \cite{Lazzati1}, arXiv dated 180511.}
%May 11th 2018. }
\label{fig:RadioAG}
%\label{Fig30}
\end{figure}
%\begin{equation}
%F_\nu(t,\nu)\!\propto\! t^{0.72 \pm 0.03}\nu^{-0.56\!\pm\!0.06},
%\label{eq:Fnu3}
%%\label{Eq34}
%\end{equation}
%where we used the observed \cite{MooleySL} $\beta_\nu\!=\!0.56\!\pm\!0.06$,
%which extends from the radio (R) band, through the optical (O) band, to
%the X-ray band.
%These CB-model approximate rise and fall power-law
% time dependences of the light curves of the ROX afterglow
%of SHB170817, with temporal indices $0.72\!\pm\!0.03$
%and $-2.12\!\pm\!0.06$, respectively, are in good agreement with
%the power-law indices extracted in \cite{MooleyKP8},
%$0.78\!\pm\!0.05$ and $-2.41+0.26/-0.42$, respectively, in \cite{MooleyKP8}.
%\begin{figure}[]
%\centering
%\includegraphics[width=6.52 cm]{FB_AG170817A5.pdf}
%\includegraphics[width=7.5 cm]{FB_AG170817A7.pdf}
%%\includegraphics[width=6.52 cm]{FB_AG170817A5.pdf}
%%\includegraphics[width=7.5 cm]{FB_AG170817A7.pdf}
%%\epsfig{file=FB_AG170817A5.eps,width=9.cm,height=9.cm}
%\caption{{\bf Above:} Best fit light curves
%of an off-axis structured jet model
%\cite{Lazzati1} to the ROX
%AGs of SHB170817A measured before December 2017
%(Figure from the first version of \cite{Lazzati1} posted in the arXiv on December 8th, 2017).
%{\bf Below:} Best fit light curves to
%the ROX AG of SHB170817A up to April 2018,
%obtained from a structured jet model \cite{Lazzati2}. Reported in
%version 4 of \cite{Lazzati1} posted in the arXiv on May 11th 2018.}
%\label{fig:FB_AG170817A5}
%%\label{Fig34}
%\end{figure}
%Summarizing: The data in Figure \ref{fig:RadioAG} extend
%from radio to X-ray frequencies and, when corrected with the observed
%spectral index, satisfactorily lie close to a single curve
%(the time and frequency dependences of Equations
%\ref{eq:Fnu3} and \ref{eq:Fnu4}
%factorize). The dashed rising
%trend is the CB-model's prediction for an assumed constant
%density of the ISM encountered by the CB, which is generally an excellent
%approximation. The subsequent decline follows
%from the assumption that, at an observer's time $\sim\,150$ days,
%the ISM density began to decrease as $1/r^2$.
%SHB170817A was only exceptional in that the observations lasted long
%enough for this ISM-density transition to be very clearly observable.
\subsection{FB-model interpretations of SHB170817A}
Soon after the discovery of the late-time AG
of SHB170817A, many FB model best fits to the initially
rising light curves were published. They involved completely different models and multiple
best-fit parameters (e.g.~\cite{MooleyKP7} and references therein). As new observations were made,
the proponents of FB model(s) put to use their large flexibility, see e.g.~the arXiv versions 1-4 of
\cite{Lazzati1}.
All these models --with a chocked jet cocoon, conical or structured jets at
$\theta\!=\!0$, or not--
failed to correctly predict the subsequent data.
This is demonstrated in the lower left part of Figure \ref{fig:RadioAG}.
When the AG break around day 150 and its subsequent fast
decline were observed, the
structured jet model with its dozen or so adjustable parameters
had no problem to accommodate this behavior, see
the lower right part of Figure \ref{fig:RadioAG}.
% HASTA AQUI
%
%In November 2017 the authors of \cite{MooleyKP7} concluded that
%{\it The off axis jet scenario as a viable explanation of
%the radio afterglow of SHB170817A is ruled out} and that a {\it chocked
%jet cocoon} is most likely the origin of the gamma rays and rising
%AG of SHB170817A.
%Their claim was based on a best fit (Figure 2 in \cite{MooleyKP7})
%to the 3 GHz radio observations obtained with ATCA and VLA before November 2017.
%In April 2018, the observers among the authors of \cite{Dobie2018} reported
%that their 2-9 GHz radio observations of GW170817 covering the
%period 125-200 days post-merger, taken with the Australia
%Telescope Compact Array and the Karl G.~Jansky Very Large Array,
%unexpectedly peaked at day $149\!\pm\!2$ post merger and
%declined thereafter. RXO observations of
%the AG were continued until two years
%after burst. They are shown in Figure \ref{fig:RadioAG}
%%\ref{fig:FB_AG170817A4}
%(Figure 2 of \cite{MooleyKP8}). The parametrization is
% a smoothly broken power-law \cite{Beuermann99} with a
%temporal index $\alpha\!=\!0.84\!\pm\!0.05$ on the rise, peak
%time $149\!\pm\! 2$ day, and a temporal index $1.6\!\pm\!0.2$ on
%the decay.
%In October 2018 the authors of \cite{MooleyKP8} reached conclusions opposite
% to their earlier ones \cite{MooleyKP7,Dobie2018} and to their previous
%arXiv versions. To wit, in \cite{MooleyKP8} they reported a {\it strong
%jet signature in the late-time light-curve of GW170817}, and
%They justified their new conclusion by the fact that the post break
%flux density parametrized as $F_\nu(t)\!\propto\!
%t^{\alpha}\nu^{\beta}$ yields
% $\beta\!=\!-0.54\!\pm\! 0.06$ and $\alpha\!=\!\!-2.17$,
%consistent with the FB model prediction
% $\alpha\!=\!-p$ post break with $p$ the power-law
% index of the energy distribution of the radiating electrons \cite{Sari98}.
%concluded that {\it while the early-time radio emission
%was powered by a wider-angle outflow (cocoon), the late-time emission was most
%likely dominated by an energetic and narrowly-collimated jet, with an opening angle of
%$<\! 5$ degrees, and observed from a viewing angle of about 20 degrees.}
%The cited FB model interpretations are not self-consistent for various reasons: \\
%(a) The
%relation $\alpha\!=\! -p$ is only valid for a
%conical jet with a fast lateral expansion ($V_\perp\!\approx\!c$)
%that has stopped propagating after the jet break
%time \cite{Sari98}. The fast spreading and the stopped
%propagation of the jet are not supported by the VLBI
%observations of the radio AG of SHB170817A \cite{MooleyJET}, which
%show a superluminal compact source (a CB), rather than
%the cited features.\\
%(b) The relation $\alpha\!=\!-p$ post break
%is seldom satisfied by GRB AGs and often yields $p\!<\! 2$. \\
%(c) Due to rather large measurement errors, it is not yet clear that the
%temporal behavior of the AG of SHB170817A after break can be well
%parametrized by a broken power-law satisfying
%$\alpha\!=\! - p$.\\
%(d)
%All types of FB models --with conical or structured jets-- used to fit the multi-band afterglow
%of SHB170817A failed to correctly predict the subsequent data.
%This is demonstrated, for example, by the arXiv versions 1-4 of
%\cite{Lazzati1} where the evolution of the AG was
%first incorrectly predicted by a structured jet with a
%relativistic, energetic core surrounded by slower and less energetic
%wings, propagating in a low density ISM, as shown in
%the upper part of Figure \ref{fig:FB_AG170817A5}.
%When the AG break around day 150 and its subsequent fast
%decline were observed, the
%structured jet model with its dozen or so adjustable parameters
%had no problem to accommodate this behavior, see
%the lower part of Figure \ref{fig:FB_AG170817A5}.
Neither could the CB model foretell the change of slope in Figure \ref{fig:RadioAG},
but its explanation is simple. It only required changing
the ISM density from a constant to a $1/r^2$ behavior, a one-parameter change,
consistent with the CB starting to exit the galaxy.
\section{The CB's superluminal velocity in GRBs and SHBs}
\label{sec:super}
The first observation of an apparent superluminal velocity of a source in the plane of
the sky was reported \cite{Kapteyn}
in 1902, and since 1977 in many observations of
jets launched by quasars, blazars, and micro-quasars. The interpretation
of this kind of observations within the framework of special relativity was provided by Paul Courderc
%in his beautiful article
%{\it Les Aur\'eoles Lumineuses des Novae}
in 1939 \cite{Courdec39Rees1996}.
%\begin{equation}
%V_{app}\!=\!{\beta\,c \sin\theta \over (1\!+\!z)(1\!-\!\beta \cos\theta)}\,\approx\,
%{\beta\,c\,\gamma\,\delta \sin\theta \over (1\!+\!z) }\, \cdot
%\label{eq:Vapp}
%\end{equation}
%For $\gamma\!\gg\!1$, $V_{app}$ has a maximum value
%$2\,\gamma\,c/(1\!+\!z)$ at $\sin\theta\!=\!1/\gamma$.
%The predicted superluminal velocity of the jetted CBs cannot be verified during
%the prompt emission phase, because of its short duration and the large cosmological
%distances of GRBs. But the superluminal velocity of the jet in far off-axis, i.e.~nearby
%low-luminosity SHBs and GRBs, can be obtained from high resolution follow-up
% measurements of their AGs \cite{Dar2000b}. Below, two cases are treated in detail:
% SHB170817A and GRB030329.
\subsection{The superluminally moving source of SHB170817A}
\label{sec:superluminal}
The VLBI/VLBA observations of the
radio AG \cite{MooleyJET} of SHB170817A provided images
of an AG source escaping from the GRB location with
superluminal celerity. Such a behavior in GRBs was predicted within the CB
model \cite{DD2004} two decades ago \cite{MooleyJET}.
Figure \ref{fig:MooleyRadio},
borrowed from \cite{MooleyJET}, shows the displacement with time of a compact radio source
%--as seen before in micro-quasars and blazars \cite{DD2004}--, rather than
%an unresolved image of a GRB and its AG expanding with a
%superluminal speed, as claimed before in the case of GRB030329
% \cite{Taylor2004}.
%The figure displays
%the angular locations of the radio source
%moving
%away in the plane of the
%sky from the SHB location by
by $2.68\!\pm\!0.3$ mas between day 75 and day 230.
In \cite{MooleyJET} this image is called ``a jet". It is in fact a time-lapse capture of
the moving CB emitted (approximately) towards us by the
fusion of the neutron stars.
\begin{figure}[]
\centering
\includegraphics[width=10.5 cm]{Figure6.pdf}
%\epsfig{file=Mooley_SL.eps,width=9.0cm,height=9.0cm}
\caption{Proper motion of the radio counterpart of GW170817.
Its authors \cite{MooleyJET} explain: {\it The
centroid offset positions (shown by $1\,\sigma$ error bars) and
$3\,\sigma$-$12\,\sigma$ contours of the radio source detected 75 d
(black) and 230 d (red) post-merger with VLBI at 4.5 GHz.
The radio source is consistent with being unresolved at both epochs.
The shapes of the synthesized beam
for the images from both epochs are shown as dotted ellipses in the
lower right corner. The proper motion vector of the radio source has
a magnitude of $2.7\!\pm\!0.3$ mas}. The $1\,\sigma$
domains have been colored
not to deemphasize the effectively point-like (unresolved) nature of the source (a CB). }
\label{fig:MooleyRadio}
%\label{Fig27}
\end{figure}
%We have estimated that, for the CB responsible for SHB170817A,
%$\gamma\!\sim\! 14.7$, that is $\beta\!\sim\!0.998$.
A source with a velocity $\beta\,c$ at redshift $z$, viewed from an angle $\theta$ relative to its
direction of motion and recorded by the local arrival times of its emitted photons has an
apparent velocity in the plane of the sky:
$V_{app}\!=\!{\beta\,c\,\gamma\,\delta \sin\theta /(1\!+\!z) }$.
In the excellent $\beta\!=\!1$ approximation one may write
\begin{equation}
V_{app}\!\approx\!
{c\,\sin\theta\over (1\!+\!z)\,(1\!-\!\cos\theta)}\!\approx\!
{D_A\,\Delta \theta_s\over (1\!+\!z)\Delta t}\,,
\label{eq:Vapp2}
%\label{Eq28)
\end{equation}
which we have also expressed
in terms of observables:
$\Delta\theta_s$ is the angle by which the source is seen to have moved in a time
$\Delta t$; $D_A\!=\!39.6$ Mpc is the angular distance (for the local value
$H_0\!=\!73.4 \pm 1.62\,{\rm km/s\, Mpc}$ obtained from Type Ia SNe \cite{Riess 2016})
to SHB170817A and its host galaxy NGC 4993,
at $z\!=\! 0.009783$ \cite{Hjorthetal2017}.
The location of the VLBI-observed source
--which moved $\Delta\theta_s\!=\! 2.70\!\pm\!0.03$ mas in a time
$\Delta t\!=\!155$ d (between days 75 and 230)-- implies
$V_{app} \!\approx\! (4.0\pm 0.4)\,c$, which, solving for the
viewing angle $\theta$
in Equation \ref{eq:Vapp2}, results in $\theta\!\approx\! 27.8 \pm 2.9$ deg.
This value agrees with $\theta_{\rm GW}\!=\!25\pm 8$ deg,
the angle between the direction to the source and
the rotational axis of the binary system,
obtained from --only--
gravitational wave observations \cite{Mandeletc2017},
for the same $H_0$ \cite{Riess 2016}.
%was ejected along the rotation axis of the binary system.
More strikingly, one can invert the order of the previous concordance.
If the value of $\theta_{\rm GW}$ implied by the GW observations is input in
Equation \ref{eq:Vapp2}, the result is a correct prediction of the magnitude
of the observed superluminal velocity. So simple!, this is a ``multi-messenger"
collaboration working at its best.
\subsection{The two superluminally moving sources of GRB030329}
This GRB was the subject of a unique open
%--though not always accepted for publication--
controversy between advocates and critics of FB and CB models.
As mentioned in the Introduction, the CB model was used to fit the
early AG of GRB030329 and to predict the discovery date of its associated
SN, SN2003dh. This being a two-pulse GRB, the fits to its $\gamma$ rays and to its
two-shoulder AG curve consequently involved two cannonballs. The prediction of the amount of
their superluminal motions, based on the approximation of a constant ISM density,
turned out to be wrong \cite{Taylor2003}. Subsequent observations of the AG showed a series of
very clear re-brightenings, interpreted in the CB model as encounters of the CBs with ISM
over-densities \cite{ManyBumpAG}. Corrected by the consequent faster slow-down
of the CBs' motion, the new CB-model results were not a prediction, but
were not wrong (see \cite{SL030329} and its Figure 2 for details not mentioned here).
The authors of \cite{Taylor2003}
analized their data in terms of a single radio source, in spite of the fact that, with a
significance of $20\,\sigma$,
they saw two: {\it Much less easy
to explain is the single observation 52 days after the burst of an additional radio
component 0.28 mas northeast of the main afterglow.}
Whether there was one or two sources of the AG is the crux of the (factual)
clash between
FB- and CB-model interpretations \cite{SL030329}.
%Another critique by G.~B.~Taylor et al.~\cite{Taylor2003} was:
%{\it
% A more general problem for the cannonball model is the absence of rapid fluctuations
%in the radio light curves of GRB 030329 \cite{Berger}}. That would be true for a sufficiently
%point-like source, but for CBs it is not correct. For the sake of definiteness, we discuss
%the issue for this particular GRB:
%
%Initially \cite{DDCRs}, the radius of a CB in its rest frame is assumed to
%increase at the speed of sound in a relativistic plasma, $c_s\!=\!c/\sqrt{3}$.
%At an early observer's time $T$ its radius, $R$, has increased to:
%\begin{equation}
%R(T,\theta)\!\approx\! {c_s\over (1+z)}\int_0^T\delta(t,\theta)\,dt,
%\label{eq:CBradius}
%\end{equation}
%where use has been
%made of the relation between $T$ and the time in the CB's rest system.
%When the first radio observations started as early as $T\!=\!2.7$ days after burst,
%the result of Equation \ref{eq:CBradius}
%for this GRB --for the parameters of the CB-model description of its AG and the
%deceleration law of Equation \ref{eq:decelerate}-- is $R(T)\!\approx\!5.7\!\times\! 10^{17}$ cm.
%This is more than an order of magnitude larger than the largest source size
%that could still have resulted in diffractive scintillations. Case closed.
%
%To summarize the FB advocates' views on this GRB:
Quite forcefully Bloom and collaborators \cite{Bloom2003}
stated: {\it
%Owing to the proximity and bright radio emission, high-resolution ($\sim\!1$ pc)
Very Long Baseline Array
imaging of the compact afterglow was used by Frail (2003) \cite{Taylor2003}
to unequivocally disprove the
cannonball model for the origin of GRBs.} However,
referring to their ``second source" the authors of \cite{Taylor2003} admitted:
{\it This component requires a high average velocity of 19c and cannot be readily
explained by any of the standard models. Since it is only seen at a
single frequency, it is remotely possible that this image is an artifact of the calibration.}
As for this second source, will the dictum {\it seeing is believing} be rejected,
with the image of a moving CB in Figure \ref{fig:MooleyRadio} somehow serving
to disprove the cannonball model?
%\section{Conclusion}
%
%The cannonball model provides a successful and very simple and consistent
% interpretation of SHB170817A.
%A serious limitation of the model is that the emission of relativistic
%blobs of matter in core-collapse supernovae and binary-neutron-star mergers is not
%theoretically well understood. Neither is, in detail, the fate of a CB traveling in the interstellar
%medium. This is not surprising, for general-relativistic catastrophic
%magneto-hydrodynamics is not a simple discipline. Hopefully the renewed
%interest on neutron stars and their mergers provoked by GW170817 will lead
%to reinvigorated efforts on these subjects.
\acknowledgments
This project has received funding/support from the European Union's
Horizon 2020 research and innovation programme under the Marie
Sklodowska-Curie grant agreement No 860881-HIDDeN.
\begin{thebibliography}{0}
\bibitem{Abbott}%[1]
B. P. Abbott, et al. (Ligo-Virgo Collaboration), Nature, {\bf 551}, 85 (2017)
[arXive:1710.05835].\\
B. P. Abbott, et al. (Ligo-Virgo Collaboration) ApJ, {\bf 848}, L13 (2017)
[arXiv:1710.05834].\\
B. P. Abbott, et al. (Ligo-Virgo Collaboration) PRL, {\bf 119}, 161101 (2017)
[arXiv:1710.05832].\\
B. P. Abbott, et al. (Ligo-Virgo Collaboration) ApJ, {\bf L12}, 848 (2017)
[arXiv:1710.05833].
\bibitem{Hjorthetal2017}%[2]
J. Hjorth, et al., ApJ, {\bf 848} L31 (2017), [arXiv:1710.05856].
\bibitem{Goodman2}% [3]
J. Goodman, A. Dar, S. Nussinov, ApJ, {\bf 314}, L7 (1987).
\bibitem{Meszaros}% [4]
M. J. Rees, P. Meszaros, MNRAS, {\bf 25}, 29 (1992).
\bibitem{DDlargeangle}%[5]
S. Dado and A. Dar, arXiv:1708.04603.
\bibitem{FBM Reviews}%[6]
A partial list of reviews include:\\
T. Piran, Phys. Rep. {\bf 314}, 575 (1999) [arXiv:astro-ph/9810256].\\
T. Piran, Phys. Rep., {\bf 333}, 529 (2000) [arXiv:astro-ph/9907392].\\
P. Meszaros, ARA\&A, {\bf 40}, 137 (2002) [arXiv:astro-ph/0111170].\\
T. Piran Rev. Mod. Phys. {\bf 76}, 1143 (2004) [arXiv:astro-ph/0405503].\\
B. Zhang, P. Meszaros, Int. J. Mod. Phys. A, {\bf 19}, 2385 (2004) [astro-ph/0311321].\\
P. Meszaros, Rep. Prog. Phys. {\bf 69}, 2259 (2006) [arXiv:astro-ph/0605208].\\
B. Zhang, Chin. J. Astron. Astrophys. {\bf 7}, 1 (2007) [arXiv:astro-ph/0701520].\\
E. Nakar, Phys. Rep. {\bf 442}, 166 (2007) [astro-ph/0701748].\\
E. Berger, ARA\&A, {\bf 52}, 45 (2014) [arXiv:1311.2603].\\
P. Meszaros, M. J. Rees, (2014) eprint [arXiv:1401.3012].\\
A. Pe'er, AdAst, {\bf Vol. 2015}, 22 (2015) [arXiv:1504.02626].\\
P. Kumar, B. Zhang, Phys. Rep. {\bf 561}, 1 (2015) [arXiv:1410.0679].\\
Z. Dai, E. Daigne, P. Meszaros, SSRv, {\bf 212}, 409 (2017).
\bibitem{DD2004}%[7]
A. Dar, A. De R\'ujula, Phys. Rep. {\bf 405}, 203 (2004) [arXiv:astro-ph/0308248].\\
Kee ideas underlying the CB model were adopted from:\\
N. J. Shaviv, A. Dar, ApJ, {\bf 447}, 863 (1995) [arXiv:astro-ph/9407039].\\
A. Dar, ApJ, {\bf 500}, L93 (1998) [arXiv:astro-ph/9709231].\\
A. Dar A\&AS, {\bf 138}, 505 (1999) [arXiv:astro-ph/9902017].\\
A. Dar, R. Plaga, A\&A {\bf 349}, 259 (1999) [arXiv:astro-ph/9902138].\\
A. Dar, A. De R\'ujula, 2000 e-print [arXiv:astro-ph/0012227].\\
S. Dado, A. Dar, A. De R\'ujula, A\&A, {\bf 388}, 1079D (2002) [arXiv:astro-ph/0107367].
\bibitem{CTDDD}%[8]
S. Dado and A. Dar, [arXiv:1810.03514].\\
See also S. Dado, A. Dar, A. De R\'ujula, [arXiv:2204.04128].
\bibitem{jet}%[8]
N. J. Shaviv, A. Dar, ApJ, {\bf 447}, 863 (1995) [arXiv:astro-ph/9407039].\\
A. Dar, A. De R\'ujula, Phys. Rept. {\bf 405}, 203 (2004) [arXiv:astro-ph/0308248].
\bibitem{DDD030329}%[9]
S. Dado, A. Dar and A. De R\'ujula, Astrophys.J.Lett. {\bf 594}, L89 (2003)
[astro-ph/0304106].
\bibitem{Dado2002}% [10]
S. Dado, A. Dar, A. De R\'ujula, A\&A, {\bf 388}, 1079 (2002) [arXiv:astro-ph/0107367].\\
S. Dado, A. Dar, A. De R\'ujula, ApJ, {\bf 646}, L21 (2006) [arXiv:astro-ph/0512196].
\bibitem{Vaughan}% [11]
S. Vaughan, et al., ApJ, {\bf 638}, 920 (2006) [arXiv:astro-ph/0510677].
\bibitem{Cusumano}% [12]
G. Cusumano, et al., ApJ, {\bf 639}, 316 (2006) [arXiv, astro-ph/0509689].
\bibitem{CorrelsCB}
A. Dar and A. De R\'ujula, [arXiv:astro-ph/0012227].
\bibitem{Correlations}%[13]
S. Dado, A. Dar and A. De R\'ujula, Astrophys. J. {\bf 663}, 400 (2007).
\bibitem{Dado 2013}%[14]
S. Dado, A. Dar, A\&A, {\bf 558}, A115 (2013) [arXiv:1303.2872].
\bibitem{Dado2018}%[15]
S. Dado, A. Dar, PRD. {\bf 99}, 123031 (2019) [arXiv:1807.08726].
For earlier ideas see \cite{Dai98}.
\bibitem{Dai98}%[17]
E. G. Blackman, I, Yi, ApJ, {\bf 498}, L31 (1998) [arXiv:astro-ph/9802017].\\
Z. G. Dai, T. Lu, PRL, {\bf 81}, 4301 (1998) [arXiv:astro-ph/9810332].\\
Z. G. Dai, T. Lu, A\&A, {\bf 333}, L87 (1998) [arXiv:astro-ph/9810402].\\
B. Zhang, P. Meszaros, ApJ, {\bf 552}, L35 (2001) [arXiv:astro-ph/0011133].\\
Z. G. Dai et al., Science, {\bf 311}, 1127 (2006), [arXiv:astro-ph/0602525].\\
B. D. Metzger, E. Quataert, T.A. Thompson, MNRAS, {\bf 385}, 1455 (2008) [arXiv:0712.1233].\\
B. P. Gompertz, P.T. OBrien, G.A. Wynn, MNRAS, {\bf 438}, 240 (2014) [arXiv:1311.1505].\\
A. Rowlinson, et al., MNRAS {\bf 409}, 531 (2010) [arXiv:1007.2185].\\
B. D. Metzger, A. L. Piro, MNRAS, {\bf 439}, 3916 (2014) [arXiv:1311.1519].\\
H. Lu, et al., MNRAS, ApJ, {\bf 805}, 89 (2015) [arXiv:1501.02589].\\
S. Gibson, et al., MNRAS, {\bf 470} 4925 (2017) [arXiv:1706.04802].
\bibitem{GCN}
The Gamma Ray Coordinates Network,
https://gcn.gsfc.nasa.gov/
\bibitem{Amati2002}% [18]
L. Amati, F. Frontera, M. Tavani, et al., A\&A, {\bf 390}, 81 (2002) [arXiv:astro-ph/0205230].
\bibitem{ThBrem}
See, e.g.~W. C. Straka and C. J. Lada,
ApJ., {195}, 563 (1975) and references therein.
\bibitem{Dado2009a}% [42]
For LGRBs, see S. Dado, A. Dar, A. De R\'ujula, ApJ, {\bf 696}, 994 (2009)
[arXiv:0809.4776].\\
For SGRBs, see S. Dado, A. Dar, 2018 e-print [arXiv:1808.08912].
%aqui \bibitem{Goldsteinetal2017}%[19]
A. Goldstein, et al., ApJ, {\bf 848}, L14 (2017) [arXiv:1710.05446].
\bibitem{DDD2022}
A. Dar, S. Dado and A. De R\'ujula, to be published.
\bibitem{MooleyJET}%[20]
K. P. Mooley, et al., Nature, {\bf 561}, 355 (2018) [arXiv:1806.09693].
\bibitem{20a}%[21]
A. von Kienlin, et al., 2017, GCN Circular 21520.
\bibitem{20c}%[22]
A. Goldstein, et al., ApJ, {\bf 848}, L14 (2017) [arXiv:1710.05446].
\bibitem{20d}%[23]
A. S. Pozanenko, et al., ApJ, {\bf 852}, 30 (2018) [arXiv:1710.05448].
\bibitem{20e}
V. Savchenko, et al., ApJ, {\bf 848}, L15 (2018) [arXiv:1710.05449].
\bibitem{20b}%
Goldstein, A., et al. 2017, GCN Circular 21528.
\bibitem{Goldstein2017}%[81]
A. Goldstein, et al., ApJ, {\bf 848}, L14 (2017) [arXiv:1710.05446].\bibitem{Curvature}% [39]
E. E. Fenimore, C. D. Madras, S. Nayakshin, AJ, {\bf 473}, 998 (1996) [arXiv:astro-ph/9607163].\\
P. Kumar, A. Panaitescu, 2000, ApJ, {\bf 541}, L51 (2000), [arXiv:astro-ph/0006317].\\
F. Ryde, V. Petrosian, AJ, {\bf 578}, 290 (2002), [arXiv:astro-ph/0206204].\\
D. Kocevski, F. Ryde, E. Liang, ApJ, {\bf 596}, 389 (2003) [arXiv:astro-ph/0303556].\\
C. D. Dermer, ApJ, {\bf 614}, 284 (2004) [arXiv:astro-ph/0403508].\\
E. W. Liang, et al., ApJ, {\bf 646}, 351 (2006) [arXiv:astro-ph/0602142].\\
A. Panaitescu, NC, {\bf B121}, 1099 (2006) [arXiv:astro-ph/0607396].
\bibitem{Kobayashi}% [44]
S. Kobayashi, T. Piran, R. Sari, ApJ, {\bf 490}, 92 (1997) [arXiv:astro-ph/9705013].
\bibitem{NorrisHakkila}% [45]
J. P. Norris, et al., ApJ, {\bf 627}, 324 (2005) [arXiv:astro-ph/0503383].\\
J. Hakkila1, et al., ApJ, {\bf 815}, 134 (2015) [arXiv:1511.03695].
%\bibitem{Norris}%[3]
%J. P. Norris, et al., Nature, {\bf 308}, 434 (1984). \\
%C. Kouveliotou, et al., ApJ, {\bf 413}, L101 (1993).
\bibitem{Dado5}%[108]
S. Dado, A. Dar, Phys. Rev. {\bf D101}, 063008 (2019), arXiv:1907.10523.
\bibitem{Dado2017}%[26]
S. Dado, A. Dar, ApJ, {\bf 855}, 88 (2018) [arXiv:1710.02456]
and references therein.
\bibitem{Swift}%[51]
Swift-XRT GRB light curve repository, UK Swift Science Data Centre, Univ. of Leicester:\\
P. A. Evans, et al., A\&A, {bf 469}, 379 (2007) [arXiv:0704.0128].\\
P. A. Evans, et al., MNRAS, {\bf 397}, 1177 (2009) [arXiv:0812.3662].
\bibitem{Drout2017}%[71]
M. R. Drout, et al., Science, {\bf 358}, 1570 (2017) [arXiv:1710.054431].
\bibitem{Makha}
S. Makhathini et al. 2011, arXiv:2006.02382v3.
% aqui \bibitem{DDCRs}
A. Dar and A. De R\'ujula, Phys.Rept. {\bf 466,} 179 (2008).
\bibitem{MooleySL}%[83]
K. P. Mooley, et al., ApJ, {\bf 868}, L11 (2018) [arXiv:1810.12927].
%aqui \bibitem{MooleyKP8}%[102]
K. P. Mooley, et al., ApJ, {\bf 868}, L11 (2018), [arXiv:1810.12927].
\bibitem{MooleyKP7}%[100]
K. P. Mooley, et al., Nature, {\bf 554}, 207 (2018) [arXiv:1711.11573V1].
% aqui \bibitem{Dobie2018}%[101]
D. Dobbie et al. ApJ {\bf 858}, L15 (2018) [arXiv:1803.06853V3].
\bibitem{Lazzati1}%[105]
D. Lazzati, et al., e-print arXiv:1712.03237v1 (2017).
\bibitem{Lazzati2}%[106]
D. Lazzati, et al., PRL, {\bf 120} 241103 (2018) [arXiv:1712.03237v4].
\bibitem{Kapteyn}%[84]
J.C. Kapteyn, 1902, Astron. Nachr., {\bf 157} 201 (1902).
\bibitem{Courdec39Rees1996}%[85]
P. Courdec, Annales d'Astrophysique {\bf 2}, 271 (1939)\\
M. J. Rees, Nature, {\bf 211}, 468 (1966).
%aqui \bibitem{Dar2000b}%[86]
A. Dar, A. De R\'ujula, 2000, e-print arXiv:astro-ph/0008474.\\
S. Dado, A. Dar, A. De R\'ujula, 2016, e-print [arXiv:1610.01985].
\bibitem{Riess 2016}%[94]
A. G. Riess, et al., ApJ, {\bf 826} 56 (2016) [arXive:1604.01424].\\
A. G. Riess, et al. ApJ, {\bf 861}, 126 (2018) [arXiv:1804.10655].
\bibitem{Mandeletc2017}%[95]
I. Mandel, ApJ, {\bf 853}, L12 (2018), [arXiv:1712.03958]. See Table 1.\\
S. Nissanke, et al., ApJ, {\bf 725}, 496 (2010) [arXive:0904.1017].
\bibitem{Taylor2003}
G. B. Taylor, et al., ApJ, {\bf 609}, L1 (2004) [arXiv:astro-ph/0405300v1].
\bibitem{ManyBumpAG}
S. Dado, A. Dar and A. De R\'ujula, astro-ph/0402374
\bibitem{SL030329}
S. Dado, A. Dar and A. De R\'ujula, astro-ph/0406325.
\bibitem{Berger}
Berger E. et al., Nature, {\bf 426}, 154 2003).
\bibitem{Bloom2003}
J.S. Bloom et al.
The Astronomical Journal {\bf 127}, 252-263 (2003), [astro-ph/0308034].
\end{thebibliography}
% \begin{figure}
%\centering
% \includegraphics[width=.4\linewidth]{Glory.pdf}
% \includegraphics[width=.45\linewidth]{epeiso17GRBs.pdf}
%\caption{A figure with two subfigures}
%\label{fig:test}
%\end{figure}
\end{document}
%%
%% End of file `cimsmple.tex'.