3 The electron spectrum

3.1 Numerical simulations

The acceleration of charged particles by a moving CB is not substantially different from some of the cases already studied in the literature, the acceleration of cosmic rays and electrons having attracted an enormous amount of attention since Fermi's first analysis in 1949 (for an excellent introduction, see Longair 1994). The most efficient and thus promising mechanism is the ``first-order'' acceleration of particles by fast-moving shocks, extensively studied analytically and numerically since the pioneering works of Axford et al. (1977), Krymsky (1977), Bell (1978) and Blandford & Ostriker (1978). The analysis closest to the case at hand is that of Ballard & Heavens (1992), who studied acceleration by relativistic shocks, with the charged particles deflected by highly disordered magnetic fields, rather than, as it is generally assumed, by small irregularities in an otherwise uniform field. The ``relativistic'' and ``chaotic'' inputs are what make this work particularly relevant to the case of particle acceleration by and within CBs.

Ballard and Heavens study numerically, for various values of a moving discontinuity's Lorentz factor ranging up to $\rm\gamma_s=5$, the result of its collision with an isotropic ensemble of particles with $\rm\gamma_p=100$. They find that, for $\rm\gamma_s=5$, the resulting particle energy distribution has a break (in this reference system) at $\rm\gamma\sim 10\, \gamma_p$, at which point it steepens. The particles below the break have a dominantly very forward motion: they are the ones which have been upscattered just once. Given this hint, it is easy to reproduce the numerical results in an analytic approximation. In the shock's rest system, the energy of the particles that have been scattered only once is equal to their incoming energy: the break in the spectrum seen in the simulations is a kinematical break occurring roughly at the injection energy. ``Observed'' in the system in which the shock is travelling at $\rm\gamma_s=5$, this injection bend is very reminiscent of the familiar synchrotron-cooling ``break'', but it has little to do with it; indeed, in the simulations of Ballard & Heavens (1992) cooling was entirely neglected.

3.2 A simple analysis

Consider the CB in its rest system and temporarily postpone the discussion of cooling. The ISM electrons impinge on the CB in a fixed direction with a Lorentz factor equal to that of the CB in the GRB progenitor's rest system, $\gamma_{\rm e}=\gamma(t)$. The electrons not having ``bounced back'' off the CB's strong magnetic field, or having done it only once, retain the incoming energy, $E_{\rm b}(t)=\gamma(t)\,m_{e}\, c^2$, so that their energy distribution is: ${\rm d}n_{e}/{\rm d}E\propto \delta[E-E_{\rm b}(t)]$. A very robust (i.e. detail independent) feature of the studies of acceleration by relativistic shocks is that the particles having bounced more than once acquire a spectrum ${\rm d}n_{e}/{\rm d}E\propto E^{-p}$, with p=2 in analytical approximations and $p\sim 2.2$in numerical simulations. A few bounces are sufficient to attain such a spectrum. The CB is a system of finite transverse dimensions and the magnetic field contrast between its interior and its exterior is very large. Thus, we do not expect the same electron to bounce many times off the CB, as the latter catches up with it. The acceleration should occur mainly within the CB as charged particles bounce off its chaotically moving magnetic domains, and it should be very fast and efficient, since the injection is highly relativistic and there is no distinction between ``first and second order Fermi'' processes. The overall ``source'' spectrum of relativistic electrons is:

 

$${{\rm d}n_e^{\rm s}\over {\rm d}{E}}\sim A_1(t)\,E_{\rm b}(t)\,\d... ... A_2(t)\,\Theta[E-E_{\rm b}(t)] \left[{E\over E_{\rm b}(t)}\right]^{-p}\!\!\! ,$$ (9)

with A₁ and A₂ of comparable magnitude and a time dependence which is that of the rate, $\eta\,\pi\,R_{\rm max}^2\,c\,n_{e}\,\gamma(t)$, at which electrons enter the CB.

3.3 The spectrum of cooled electrons

The electron energy loss by synchrotron radiation is:

$$-{{\rm d}E\over {\rm d}t}= A_{\rm S}\,\beta^2\,E^2 ,$$

$$A_{\rm S} \equiv {B^2\over 6\,\pi}\,{\sigma_{\rm T}\,c\over (m_{e}\,c^2)^2}\, ,$$ (10)

with $\beta\approx 1$ for the relativistic energies of interest and $\sigma_{\rm T}=0.665$ barn the Thomson cross-section. Let the rate at which fresh electrons are supplied by the ISM be called R. The electron source distribution of Eq. (9) ``ages'' by cooling so that:

$${\partial\over\partial t}\left[{{\rm d}n_{e}\over {\rm d}E}\r... ...{\rm d}E}\right]+ R\,{{\rm d}n_{e}^{\rm s}\over {\rm d}E}\cdot$$ (11)

At times longer than the synchrotron cooling time, the electron distribution tends to a time-independent ${{\rm d}n_{e}/ {\rm d}E}$, obtained by equating to zero the l.h.s. of Eq. (11) and integrating it with the source function of Eq. (9):

 

$${{\rm d}n_{e}\over {\rm d}E}\sim A_1(t)\,\Theta[E_{\rm b}(t)-E]\,... ...er p-1}\,\Theta[E-E_{\rm b}(t)]\,\left[{E\over E_{\rm b}(t)}-1\right]^{-(p+1)}.$$ (12)

Admittedly, the process of acceleration that we have discussed is not well understood, our derivation is heuristic and Eq. (12) is not even a continuous function (the step function in Eq. (9) should not be so abrupt, the magnetic energy in Eq. (10) should not have a fixed value). All we want to conclude from this exercise is that, when the probability of an electron to have been ``kicked'' only once is not negligible (A₁ comparable to A₂), the electron spectrum has an injection bend at $E\sim E_{\rm b}(t)$, around which its spectral index changes by $\sim$1 from $\sim$2 to $\sim$p+1. We choose to characterize this behaviour by the function:

$${{\rm d}n_{e}\over {\rm d}E} \propto {E^{-2}\over \sqrt{1 + [E/E_{\rm b}(t)]^{2\,(p-1)}}}\cdot$$ (13)

Note how similar the injection bend is to a cooling break (also a spectral steepening by roughly one unit) even though their origins are so different. The observational evidence for an injection bend at the injection energy turns out to be strong, as we proceed to show.