5 The cumulation factor

Three factors that are irrelevant in the optical and X-ray domains play a role in the description of the longer radio wavelengths and the early radio AG. In this section we discuss the first one of them.

Electrons that enter a CB with an injection Lorentz factor $\gamma(t)$are rapidly Fermi accelerated to a distribution that we have argued to be roughly that of Eq. (9). On a longer time scale, they lose energy by synchrotron radiation, and their energy distribution evolves as in Eq. (11). Electrons with a large $\gamma\sim {\cal{O}}\,[\gamma(t)]$ emit synchrotron radiation, with no significant time-delay, at the observer's optical and X-ray wavelengths. But the emission of radio is delayed by the time it takes the electrons to ``descend'' to an energy at which their characteristic emission is in the observer's radio band. At the start of the afterglow, when equilibrium conditions have not yet been reached, this implies a dearth of radio emission relative to the higher-frequency bands. This introduces a ``cumulation factor'' ${\rm C}(\nu,t)$ in Eq. (6).

Consider a fixed observed radio frequency $\nu_{\rm obs}$. It corresponds to a time changing frequency $\nu=(1+z)\nu_{\rm obs}/\delta(t)$ in the CB system. The CB electrons preferentially emitting at this frequency (over an unconstrained range of pitch angles) are those whose Lorentz factor $\rm\gamma_e$ satisfies the relation $\rm\nu \sim 0.22\,\gamma_e^2\,\nu_L$, in analogy to Eq. (3). To estimate the time $\rm\Delta t$ it takes an electron to decelerate from $\gamma\sim \gamma(t)$ to $\rm\gamma=\gamma_e$, substitute the magnetic energy density of Eq. (1) into the electron energy loss of Eq. (10) and integrate, to obtain

 

$$\Delta t {3\,m_{e}\over n_{p}\,m_{p}\,\sigma_T\,c}\,{1\over\gamma^2}\, \left({1\over \gamma_{\rm e}}-{1\over \gamma}\right)$$ =

$$[8.27\times 10^7\,{\rm s}]\left[{{10^{-3}\,{\rm cm}^{-3}}\over{n_... ...10^6\over \gamma^2}\, \left({1\over \gamma_{\rm e}}-{1\over \gamma}\right)\cdot$$ = (15)

The function $\gamma$ is given by Eq. (37) of Appendix A, which we may rewrite as:

$${1\over \gamma^2} = {1\over \gamma_0^2}\left[1+{t\over t_0}\right]$$

$$t_0\equiv [5.14\times 10^7\, {\rm s}] \left[{x_\infty\over 1\;{\rm Mpc}}\right] \left[{10^3\over \gamma_0}\right]^2.$$ (16)

The electrons emitting the observed radio frequencies have $\rm\gamma_e\!\sim\! {\cal{O}}(1)$, so that the proper CB times $\Delta t$ and t₀ are of ${\cal{O}}(1)$ year, corresponding to observer's times - foreshortened by a factor $(1+z)/\delta$ - of ${\cal{O}}(1)$ day. For optical and radio observations $\gamma_{\rm e}\!\sim\! {\cal{O}}(\gamma)$there is no significant delay in their emission. Moreover, the electron accumulation rate ( $\eta \pi\,R_{\rm max}^2\,n_{e}\,c\,\gamma$ in the CB system) is orders of magnitude larger than the characteristic synchrotron cooling time $E/({\rm d}E/{\rm d}t)$ of Eq. (10), even for $\gamma\sim 10^3$. Thus, the optical and X-ray AG emission starts as soon as the CB is transparent to its enclosed radiation: for each CB, a few observer's seconds after the corresponding $\gamma$-ray pulse (DDD 2001). The radio signal, on the other hand, must await a time $\Delta t$ for the cumulated electrons to cool down.

The simple way to parametrize the frequency-dependent ``cumulation effect'' is to use the expression for the total number of electrons N(t) incorporated by the CB up to time t (Eq. (40) of Appendix A) and to posit:

$$C(\nu,t)={N(t-\Delta t)\over N(t)}\,\Theta(t-\Delta t),$$ (17)

where the frequency dependence is via $\Delta t=\Delta t(\nu,t)$, and the sharp start at $t>\Delta t$ is an artifact of our simplifications. For optical and X-ray frequencies, $\Delta t=0$ and $C(\nu,t)=1$. In practice we find that, except for GRB 980425 whose viewing angle is exceptionally large, one may also use within the radio band an approximation to Eq. (17):

$$C(\nu,t)\sim C(t)=\left[1-{\gamma(t)\over\gamma_0}\right]^{1/2},$$ (18)

which is also frequency independent.