The Fireball Model

The properties derived from the afterglow observations can be explained within the framework of the relativistic fireball models, first proposed by [Cavallo & Rees, 1978], [Goodman, 1986], [Paczynski, 1986], in which a compact source releases $\sim 10^{53}$ ergs within dozens of seconds in a $\sim 10$ Km region. Regardless of the form of energy initially released, a quasi-thermal equilibrium between radiation and matter is reached; this opaque radiation-electron-positron plasma accelerates to relativistic velocities with Lorentz factors of $\Gamma \sim 10^2-10^3$. The presence of even a small fraction of barions, around $10^{-6} M_{\odot}$, makes the plasma opaque to Thomson scattering, therefore contributing to accelerate the fireball untill a considerable fraction of the initial energy has been converted into bulk kinetic energy.

Figure: Cartoon for the Fireball Model.

The extreme characteristics of GRBs seemed to lead to a paradox called ``the compactness problem'': assuming that the initial energy is released in a volume $R < c \delta t$ ($\delta t$ is the variability timescale, $\sim$ ms), through photons with energies distributed according to a GRB spectrum, then the optical depth for pair production is extremely high: $\tau \gg 1$. On the other side, the obseved spectrum is definitely non-thermal; the solution to this paradox is given by the relativistic motion for two reasons: first, the compactness condition now becomes: $R < c \Gamma^2 \delta t$, where $\Gamma$ is the Lorentz factor; second, the rest frame energy of the photons is smaller by a factor of $\Gamma$: therefore, only a smaller fraction of them can create pairs. Thus, the optical depth is reduced below 1, provided that $\Gamma > 100$. This requirement on the Lorentz factor $\Gamma$ limits the rest mass around the burst source, thus requiring a small baryonic contamination ( $\le 10^{-5} M_{\odot}$).

Two scenarios have been proposed to convert the kinetic energy into photons: the so-called ``internal shock'' ([Narayan et al., 1992], [Rees & Mészáros, 1994]) and ``external shock'' ([Mészáros & Rees, 1993]). According to the latter, the relativistic matter runs into some external medium, either interstellar or wind earlier emitted by the progenitor, while in the former scenario the inner engine is assumed to emit many shells with different Lorentz factors and, for this, colliding into one another, therefore thermalizing a fraction of their kinetic energy.

Figure: Broad Band Synchrotron Spectrum of a GRB afterglow, according to the Fireball Model. (From [Sari, Piran & Narayan, 1998])

The most accounted picture is the following (fig. ): after the internal shocks produced the burst itself, the interactions of the expanding shells with the surrounding matter are responsible for the external shocks, which accelerate the electrons into a power law distribution $N(\gamma_e) \sim \gamma_e^{-p}$ for $\gamma_e > \gamma_m$. The lower cutoff distribution is taken to be a fixed fraction of equipartition, and a magnetic field is thought to be present behind the shock, with a certain fraction of energy of the equipartition. Therefore, the relativistic electrons emit synchrotron radiation, thus providing the spectrum observed in the afterglows (for an extensive review of the fireball model, see [Piran, 1999]). Fig.  shows the broad band sybchrotron spectrum in the cases of fast and slow cooling, respectively ([Sari, Piran & Narayan, 1998]).