2 Set up and calculation

The emitting region consists of a conical shell of opening angle $2\Theta_{{\rm \scriptscriptstyle{J}}}$,the base of which is at $r_{\rm b}=r_{\rm o}+\beta c t$ (where $\beta=\sqrt{1 -1/\Gamma^2}, \Gamma\gg 1$, and $r_{\rm o} =\Gamma^2 c t_{\rm var}$ is the dissipation radius) and its front is moving at $\beta_{\rm sh,f}$($\sim {\cal{O}}(10^{-1}))$ in the flow's frame. The emitting region is assumed homogeneous at any given moment. This is valid as long as the region is not expanding faster than the injected electrons can fill it up (i.e., $\beta_{\rm sh,f} \ll 1$). ${\rm e}^-$s are injected at a constant rate per unit volume with the same spectral shape, i.e., a relativistic Maxwellian peaking at $\gamma_{\rm m,o}$ and a power law tail of slope -3 extending up to $\gamma_{\rm M,o}$ (determined by the size of the emitting region). I solve the continuity equation along its characteristics. I take into account continuous particle injection (over $t_{\rm inj}$), cooling through sy ($\vert{\rm d}\gamma/{\rm d}t\vert _{sy} = 1.29 \ 10^{-9} B^2 (\gamma^2 -1$)), IC ($\vert{\rm d}\gamma/{\rm d}t\vert _{{\rm \scriptscriptstyle{IC}}}= 1.36\ 10^{-17}(\gamma^2-1) \int_0^{m_{\rm e} c^2/h \gamma}{\cal{I}}^{'}_{sy,\nu'} {\rm d}\nu'$ and the sy intensity integral is evaluated iteratively so that IC cooling is taken into account self-consistently) and adiabatic expansion of the shell $\vert{\rm d}\gamma/{\rm d}t\vert _{\rm ad}= \gamma({\rm d}n_{\rm p}/n_{\rm p} +\beta/(r_{\rm o} +\beta t))$. The observer lies at an angle $\Theta_{\rm o}$ from the symmetry axis of the jet. Thus, the frequency of a photon emitted at a angle $\theta$ with respect to the jet axis is boosted by a Doppler factor ${\cal{D}}= \left\{\Gamma \left[1- \beta \cos(\theta+\Theta_{\rm o})\right]\right\}^{-1}$. The observed spectrum is calculated by integrating the contributions from each volume element over the visible area and the duration of the emission.

For typical values of the physical parameters, the BATSE range spectrum is entirely optically thin. The different scenarios examined include the following: Continuous injection in a thin shell (curvature determines the lightcurve); Continuous injection in a thick shell (thickness determines lightcurve); Magnetic field decay with the regions expansion (conserving either flux or total energy); Smooth time dependence of injected ${\rm e}^-$ number (conserving either total number or number density); The jet is viewed off the axis of symmetry.