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The observed prompt emission must be generated by energetic particles that have been accelerated within the collisionless shocks. The most likely process is synchrotron emission, even though there is some evidence that a simple synchrotron spectra does not fit all bursts [321] (but see however, [16] who finds consistency with the synchrotron model). I consider here, the different physical ingredient that determine the emission process: particle acceleration, magnetic field amplification, synchrotron emission and inverse Compton emission that could be relevant in some cases.

A. Relativistic Shocks

Shocks involve sharp jumps in the physical conditions. Conservation of mass, energy and momentum determine the Hugoniot shock jump conditions across the relativistic shocks for the case when the upstream matter is cold (see e.g. Blandford and McKee [30]):

Equation 10 (10)

where n1, 2,e1, 2 are the number density and the energy density (measured in the local rest frame) of the matter upstream (region 1) and downstream (region 2). I have assumed that the energy density in region 1 is very small compared to the rest mass density. Gamma is the Lorentz factor of the fluid just behind the shock and Gammash is the Lorentz factor of the shock front (both measured in the rest frame of the upstream fluid). The matter is compressed by a factor Gamma across a relativistic shock. The pressure, or the internal energy density behind the shock is of order Gamma2 n1 mp c2. Thus, in the shock's rest frame the relative "thermal" energy per particle (downstream) is of the same order of the kinetic energy per particle (ahead of the shock) upstream. Put differently the shock converts the `ordered' kinetic energy to a comparable random kinetic energy. In an ultra-relativistic shock the downstream random velocities are ultra-relativistic.

Similar jump conditions can be derived for the Magnetic fields across the shock. The parallel magnetic field (parallel to the shock front) B|| is compressed and amplified:

Equation 11 (11)

The perpendicular magnetic field Bperp remains unchanged.

The energy distribution of the (relativistic) electrons and the magnetic field behind the shock are needed to calculate the Synchrotron spectrum. In principle these parameters should be determined from the microscopic physical processes that take place in the shocks. However, it is difficult to estimate them from first principles. Instead I define two dimensionless parameters, epsilonB and epsilone, that incorporate our ignorance and uncertainties [288, 305, 367]. It is commonly assumed that these energies are a constant fraction of the internal energy behind the shock (see however, Daigne and Mochkovitch [68]). I denote by epsilone and by epsilonB the ratio between these energies and the total internal energy:

Equation 12 (12)

One usually assumes that these factors, epsilone,B, are constant through out the burst evolution. One may even expect that they should be constant from one burst to another (as they reflect similar underlying physical processes). However, it seems that a simple model that assumes that these parameters are constant during the prompt burst cannot reproduce the observed spectrum [68]. This leads to explorations of models in which the equipartition parameters epsilone,B depend on the physical conditions within the matter.

In GRBs, as well as in SNRs the shocks are collisionless. The densities are so small so that mean free path of the particles for collisions is larger than the typical size of the system. However, one expects that ordered or random magnetic fields or alternatively plasma waves will replace in these shocks the role of particle collisions. One can generally use in these cases the Larmour radius as a typical shock width. A remarkable feature of the above shock jump conditions is that as they arise from general conservation laws they are independent of the detailed conditions within the shocks and hence are expected to hold within collisionless shocks as well. See however [263] for a discussion of the conditions for collisionless shocks in GRBs.

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