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2 Summary

In the CB model a GRB jet consists of $n_{_{\rm CB}}$ cannonballs, typically a few, each of them generating a prominent pulse in the $\gamma$-ray signal, as they reach the transparent outskirts of the shell and/or prior ``wind" of their associated supernova (SN). We assume CBs to be made of ordinary matter, mainly hydrogen, and to enclose a magnetic field maze, as is the case for the observed ejections from quasars and microquasars. The interstellar medium (ISM) the CBs traverse in the AG phase has been previously partially ionized by the GRB radiation and is fully ionized by Coulomb interactions as it enters the CB. In analogy to processes occurring in quasar and microquasar ejections, the ionized ISM particles are multiple scattered, in a ``collisionless'' way, by the CBs' turbulent magnetic fields. In the rest system of the CB the ISM swept-up nuclei are isotropically re-emitted, exerting an inwards force on the CB's surface. This allows one to compute explicitly the CB's radius as a function of time (DDD 2001). The radius, for typical parameters, and in minutes of observer's time, reaches a constant $R_{\rm max}$ of a few times 1014 cm.

The ISM nuclei (mainly protons) that a CB scatters also decelerate its flight: its Lorentz factor, $\gamma(t)$, is calculable. Travelling at a large $\gamma$ and viewed at a small angle $\theta$, the CB's emissions are strongly relativistically aberrant: in minutes of observer's time, the CBs are parsecs away from their source. For a constant CB radius and an approximately constant ISM density, $\gamma(t)$ has an explicit analytical expression, as discussed in Appendix A. Typically $\rm\gamma=\gamma(0)/2$ at a distance of order 1 kpc from the source, and $\gamma(0)\sim 10^3$. Due to a limited observational sensitivity, GRBs have been detected only up to angles $\theta$ of a few times $1/\gamma(0)$.

The ISM electrons entering a CB are caught up and bounce off its enclosed magnetic domains acquiring a predictable power-law energy spectrum, as we argue in Sect. 3. In the CB's rest system ${\rm d}n_{e}/{\rm d}E\propto E^{-2}$ below an energy $E_{\rm b}(t)\simeq \gamma(t)\, m_{e}\, c^2$, steepening to ${\rm d}n_{e}/{\rm d}E\propto E^{-(p+1)}$, with $p\simeq 2.2$, above this energy[*]. The energy $E_{\rm b}$ does not correspond to the conventional synchrotron ``cooling break'' but to the injection bend at the energy at which electrons enter the CB with a Lorentz factor $\gamma(t)$. In Sect. 4 we discuss the observational support of the existence of the injection bend, which is strong. Given the very large magnetic and radiation energy densities in the CB, the usual cooling break (at the energy at which the energy-loss rate due to synchrotron emission and inverse Compton scattering equals that due to bremsstrahlung, adiabatic losses and escape) happens only at subrelativistic energies, as discussed in Appendix B. The magnetic energy-density in a CB (DDD 2001) is:

 \begin{displaymath}U_B= {B^2\over8\, \pi}\sim {1\over 4}\,\gamma^2\,n_p\, m_p\, c^2,
\end{displaymath} (1)

with np the ISM baryon density (seen as $\gamma\, n_p$ by the CB in its rest system). Thus, the magnetic field is:

 \begin{displaymath}B(t)\sim 3 \;\left[{n_p\over 10^{-3}\,{\rm cm}^3}\right]^{1/2}
\left[{\gamma(t)\over 10^3}\right]\; {\rm Gauss}.
\end{displaymath} (2)

The pitch-angle averaged characteristic synchrotron-radiation frequency of electrons of energy $E=E_{\rm b}$ is (Rybicki & Lightman 1979):

 \begin{displaymath}\nu_{\rm b}(t)\sim 0.29\,{3\over 4}\; \gamma(t)^2\,\nu_{\rm L}
\end{displaymath} (3)

where $\nu_{\rm L}=e\,B/(2\,\pi\,m_{e}\,c)$ is the Larmor frequency in the CB enclosed magnetic field B. To a good approximation, in the CB rest system and prior to cumulation, absorption and limb-darkening corrections, the synchrotron radiation has a spectral shape:

\begin{displaymath}\nu\,{{\rm d}n_\gamma\over {\rm d}\,\nu} \propto
f_{\rm sync...
...{\rm b}(t)]^{-1/2}\over
\sqrt{1+[\nu/\nu_{\rm b}(t)]^{(p-1)}}}
\end{displaymath}


 \begin{displaymath}K(p) \equiv
{\sqrt{\pi}\over \Gamma\left[{2\, p-1\over 2(p-1...
...ft[{2\, p-3\over 2(p-1)}\right]}
\simeq{p-2\over 2\,(p-1)}\; ,
\end{displaymath} (4)

where we have normalized $f_{\rm sync}(\nu)$ to a unit integral over all frequencies and the approximation is good to better than 8% precision in the range $2<p\leq 2.6$. Note that, for $\nu\gg\nu_{\rm b}$, $f_{\rm syn}\propto \nu_{\rm b}^{(p-2)/2}$ i.e., it is independent of $\nu_{\rm b}$ for p=2. For $p\sim 2.2,$ the extremely weak dependence of $f_{\rm syn}$ on $\rm\nu_b$ in the optical and X-ray bands was neglected in DDD 2001.

For the first $\sim $103 s of observer's time, a CB is still cooling fast and emitting via thermal bremsstrahlung (DDD 2001), but after that the CB emissivity integrated over frequency is equal to the energy deposition rate of the ISM electrons in the CB[*]:

 \begin{displaymath}{{\rm d}E\over {\rm d}t}
\simeq \eta\, \pi\, R_{\rm max}^2\, n_{e}\, m_{e}\, c^3\, \gamma(t)^2,
\end{displaymath} (5)

where $n_{e}\gamma$ is the ISM electron density in the CB rest system and $\eta$ is the fraction of ISM electrons that enter the CB and radiate there the bulk of their incident energy. In the early afterglow Eq. (5) must be modified to account for the fact that the bulk of the radio emission by the incoming ISM electrons is delayed by the time it takes them to cool down to energies much lower than their initial one. This implies that Eq. (5) must be modified by a multiplicative ``cumulation factor'' ${\rm C}(\nu,t)$, which is $\approx$1 at optical and X-ray wavelengths, as discussed in detail in Sect. 5. Two other factors, discussed in Sects. 6 and 7, distinguish radio waves from higher-frequency emissions: attenuation by self-absorption and limb darkening; they introduce two extra factors $A_{_{\rm CB}}[\nu]$and $L_{_{\rm CB}}(\nu,\theta_{_{\rm CB}})$, with $\theta_{_{\rm CB}}$ a direction of emission relative to the CB's velocity vector. Normalized as in Eq. (5) and corrected by all these factors, the afterglow energy flux density of a CB, in its rest system, is:
 
$\displaystyle F_{_{\rm CB}}[\nu,t,\theta_{_{\rm CB}}] \simeq
\eta\, \pi\,R_{\rm...
...nu,t)
C(\nu,t)\, A_{_{\rm CB}}[\nu]\, L_{_{\rm CB}}(\nu,\theta_{_{\rm CB}})\, ,$     (6)

to be summed over $n_{_{\rm CB}}$ for a jet with that number of cannonballs. This expression, for $\rm\nu \gg \nu _b$ and the second row set to unity, reproduces the optical and X-ray AG result discussed in DDD 2001.

An observer in the GRB progenitor's rest system, viewing a CB at an angle $\theta$ (corresponding to $\rm\theta_{_{CB}}$ in the CB's proper frame), sees its radiation Doppler-boosted by a factor $\delta$:

\begin{displaymath}\delta(t)\equiv
{1\over\gamma(t)\,(1-\beta(t)\cos\theta)}
\simeq {2\,\gamma(t)\over 1+\theta^2\gamma(t)^2},
\end{displaymath}


 
$\displaystyle \cos\theta_{_{\rm CB}}=
{\cos\theta-\beta(t)\over 1-\beta(t)\,\cos\theta}
\simeq{1-\theta^2\gamma(t)^2\over1+\theta^2\gamma(t)^2}$     (7)

where the approximations are valid in the domain of interest for GRBs: large $\gamma$ and small $\theta$. Since the CB is catching-up[*] with the radiation it emits, $\delta$ is also the relative time aberration: ${\rm d}t_{\rm obs}={\rm d}t_{_{\rm CB}}/\delta$. The observed spectral energy density is modulated by a factor $\delta^3$, two powers of $\delta$ reflecting the relativistic forward collimation of the radiation emitted in the CB's rest system. The AG spectral energy density $F_{\rm obs}$seen by a cosmological observer at a redshift z (Dar & De Rújula 2000a), is:

 \begin{displaymath}F_{\rm obs}[\nu,t]\simeq
{A_{\rm Gals}\, (1+z)\,\delta^3
...
...[{(1+z)\,\nu\over\delta(t)},{\delta(t)\,t\over 1+z}
\right] ,
\end{displaymath} (8)

where $A_{\rm Gals}$ represents the absorption in the host galaxy and the Milky Way, $F_{_{\rm CB}}$is as in Eq. (6), and $D_{\rm L}$ is the luminosity distance (we use throughout a cosmology with $\Omega=1$ and $\Omega_\Lambda=0.7$). In the CB model, the extinction in the host galaxy may be time dependent: in a day or so, CBs typically move to kiloparsec distances from their birthplace, where the extinction should have drastically diminished.

In DDD 2001 we fit, in the CB model, the R-band AG light curves of GRBs. The fit involves five parameters per GRB: the overall normalization; $\theta$: the viewing angle; $\gamma_0$: the t=0 value of the Lorentz factor; $x_\infty$: the ``deceleration'' parameter of the CBs in the ISM; and the spectral index p. The value of p, obtained from the temporal shape of the afterglow, is in every case very close to the expectation p=2.2, and - within the often large uncertainties induced by absorption - with the observed spectra from optical frequencies to X-rays (DDD 2001).

In this paper we complete our previous work by making broad-band fits to the data at all available radio and optical frequencies. In so doing, we need to introduce just one new ``radio'' parameter: an ``absorption frequency'' $\rm\nu_a$, corresponding to unit CB opacity at a reference frequency. We set p=2.2 so that the extension to a broad-band analysis does not involve an increase in the total number of parameters. We have to refer very often to the values of the parameters that our previous experience with the CB model made us choose as reference values. For convenience, these are listed in Table 1.

 
Table 1: Reference parametersa.
\begin{table}\begin{displaymath}
\begin{array}{ p{0.2\linewidth}l p{0.7\line...
...e}$ ). \lq\lq Ambient''
numbers refer to the ISM, not the CBs.
\end{list} \end{table}

The predictions of the CB model, for typical parameters, are summarized in Fig. 1. The energy density spectra at radio to optical frequencies are shown, at various times after the GRB, in the upper panel. The spectral slopes before and soon after the peak frequency are $\rm 3/2$ and -(p-1)/2, as indicated. The spectra peak at a frequency at which self-attenuation in the CBs results in an opacity of $\cal {O}$(1). At frequencies well above the frequency $\rm\nu_b$ characterizing the injection bend, the spectrum steepens to a slope -p/2. In the figure's lower panel we show light curves at various radio frequencies. At large times and for $\rm\nu \gg \nu _b$ - which is the case at all frequencies in the example of Fig. 1, whose parameters are close to those of GRB 000301c - they tend to $t^{-2\,(p+1)/3}$, this behaviour being reached at earlier times, the higher the frequency. For $\rm\nu\ll \nu_b$, the corresponding limiting behaviour is $\approx$t-4/3, observable at low frequencies in the cases of GRBs 991216, 991208 and 000418. All of the above predictions are robust: they do not depend on the detailed form of the attenuation, cumulation and limb-darkening factors. The early rise of the light curves does depend on such details, on which we shall have to invest a disproportionate effort in Sects. 5 to 7.

  \begin{figure}
\includegraphics[width=8.8cm,clip]{MS2654f1a.eps}\vspace*{3mm}
\includegraphics[width=8.8cm,clip]{MS2654f1b.eps}\end{figure} Figure 1: Typical predictions for the CB model's radio afterglow. Upper panel: spectra at different times, from 1 to 300 days. The peak frequencies correspond to CB self-opacities of $\cal {O}$(1). The black dots are the location of the synchrotron frequency corresponding to the injection bend. Lower panel: Light curves at different radio frequencies, from 350 to 1.43 GHz. The asymptotic curve is $t^{-2\,(p+1)/3}$(for $\rm\nu \gg \nu _b$, as is the case at all frequencies shown in this figure).

The CB model provides an excellent description of the data, as discussed in Sects. 4, 8 and 9. In the case of GRB 980425, for which the optical AG is dominated by SN1998bw, we used the parameters that fit its X-ray afterglow (DDD 2001) and the GRB's fluence (Dar & De Rújula 2000a) to argue that they are not exceptional. The CB-model's description of the radio data for this GRB/SN pair is excellent: there is nothing peculiar about this GRB, nor about its associated supernova, as we discuss in detail in Sect. 9, along with the question of the angular separation in the sky of the SN and the associated CBs, which may have been, or may still be, observable.

The apparent sky velocities of cosmological CBs are extremely superluminal and their angular velocities happen to be of the same order of magnitude as those of galactic pulsars. This implies that CB velocities can possibly be extracted from their observed radio scintillations, as discussed in Sect. 12.


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