2 The forward shock

The synchrotron spectrum from relativistic electrons that are continuously accelerated into a power law energy distribution is always given by four power law segments, separated by three critical frequencies: $\nu_{\rm sa}$ the self absorption frequency, $\nu_{\rm c}$ the cooling frequency and $\nu_{\rm m}$ the characteristic synchrotron frequency.

Using the relativistic shock jump condition and assuming that the electrons and the magnetic field acquire fractions $\epsilon_{\rm e}$and $\epsilon_{\rm B}$ of the equipartition energy, we obtain:  

$$\nu_{\rm m}=1.1\ 10^{19}~{\rm Hz}\left( \frac{\epsilon_{\rm ... ...\rm B}}{0.1}\right) ^{1/2}(\frac{\gamma}{300} )^{4}n_{1}^{1/2}.$$ (1)

$$\nu_{\rm c}=1.1\ 10^{17}~{\rm Hz}\left( \frac{\epsilon_{\rm ... ...ft( \frac{\gamma }{300}\right) ^{-4}n_{1}^{-3/2}t_{\rm s}^{-2},$$ (2)

$$F_{\nu ,\max }=220~{\rm \mu Jy}\ {\rm D}_{28}^{-2}\left( \fr... ...}\left( \frac{\gamma }{300}\right) ^{8}n_{1}^{3/2}t_{\rm s}^{3}$$ (3)

 

$$\nu _{\rm sa}=220~{\rm GHz}\left( \frac{\epsilon_{\rm B}}{0.... ...( \frac{\gamma }{300}\right) ^{28/5}n_{1}^{9/5}t_{\rm s}^{8/5}.$$ (4)

These scalings generalize the adiabatic scalings obtained by Sari et al. (1998) to an arbitrary hydrodynamic evolution of $\gamma(t)$.

For typical parameters, during the early afterglow $\nu_{\rm c} < \nu_{\rm m}$, so fast cooling occurs. The spectrum of fast cooling electrons is described by four power laws: (i) For $\nu <\nu_{\rm sa}$ self absorption is important and $F_{\nu }\propto \nu ^{2}$. (ii) For $\nu _{\rm sa}<\nu <\nu _{\rm c}$ we have the synchrotron low energy tail $F_{\nu }\propto \nu ^{{1/3}}$. (iii) For $\nu _{\rm c}<\nu <\nu _{m}$ we have the electron cooling slope $F_{\nu }\propto \nu ^{-1/2}$. (iv) For $\nu \gt\nu_{\rm m}$ $F_{\nu }\propto \nu ^{-p/2}$, where p is the index of the electron power law distribution.

In the early afterglow, the Lorentz factor is initially constant. After that the evolution can be of two types (Sari 1997). Thick shells, which corresponds to long bursts, begin to decelerate with $\gamma(t)\sim t^{-1/4}$. Only later there is a transition to deceleration with $\gamma(t)\sim t^{-3/8}$. The light curves for such bursts can be obtained by substituting these scalings in Eqs. (1-4). However, the complex internal shocks GRB signal would overlap, for these long bursts, the smooth external shock afterglow signal. The separation of the observations to GRB and early afterglow would be rather difficult.

For thin shells, that correspond to short bursts, there is no intermediate stage of $\gamma(t)\sim t^{-1/4}$. There is a single transition, at the time $t_\gamma=( 3E/32\pi \gamma _{0}^{8}nm_{\rm p}c^{5}) ^{1/3}$, from a constant velocity to self-similar deceleration with $\gamma(t)\sim t^{-3/8}$. The possible light curves are illustrated in Fig. 1. As the intial afterglow peaks several dozen seconds after the GRB there should be no difficulty to detect it.

The detection of a delayed emission which fits the light curves of Fig. 1, would enable us to determine $t_\gamma$. Using $t_\gamma$ we could proceed to estimate the initial Lorentz factor:  

$$\gamma _{0}= 240E_{52}^{1/8}n_{1}^{1/8}\left( t_\gamma/10{\rm s}\right) ^{-3/8}.$$ (5)

If the second peak of GRB 970228, delayed by 35 s, is indeed the afterglow rise, then $\gamma_0 \sim 150$ for this burst.

  

Figure 1: Light curves of the forward (solid) and reverse (dashed) shocks in three energy bands