6 The attenuation factor

At optical and X-ray frequencies the CB is transparent and, for the spectrum of Eq. (4), the bulk of the radiation's energy is emitted around the bend frequency $\rm\nu_b$. At such relatively high frequencies, as illustrated in Fig. 1, absorption is unimportant. Thus, for optical and X-ray afterglows (DDD 2001) it suffices to know that all of the incoming electron's energy is reradiated, the spatial distribution of the radiating electrons within the CB is irrelevant. But in the radio, where absorption is important, the location of these electrons inevitably plays a role. In the next sections we argue that it is plausible that the radiating electrons be close to the surface ``illuminated'' by the ISM (Sect. 6.1), and that the values of the CB's plasma frequency (Sect. 6.2) and free-free absorption coefficient (Sect. 6.3) actually suggest that they may be relatively close to that surface. In Sect. 6.4 we deduce the final form of the attenuation factor in the CB model, characterized by a single parameter.

6.1 Electron penetration

Using numerical simulations, Achterberg al. (2001) have shown that for simple geometries the bulk of highly relativistic particles encountering a collisionless shock escape before they undergo diffusive shock acceleration. In reality, the geometry of the CB, its density distribution and its magnetic field distribution are very complicated, making the fraction of the ISM electrons that penetrate inside the CB, and their distribution there, very uncertain.

Several length scales play a role in discussing the fate of an electron that enters the CB with $\gamma_{\rm e}=\gamma(t)$. The Larmor radius is $R_{\rm L}=m_{e}\,c^2\,\gamma/\rm (e\,B)$, which is independent of $\gamma$ for B scaling as in Eq. (2). For our reference parameters, $R_{\rm L}\sim 6$ km is many orders of magnitude smaller than the CB's radius and does not play a crucial role. The length of an electron's curled-up trajectory as it radiatively loses energy is $c\,E/({\rm d}E/{\rm d}t)$ or $\sim$ $2.6\times 10^{15}$ cm for the cooling rate of Eq. (10) and an initial $\gamma=10^3$. This is only an order of magnitude larger than the reference CB's radius $R_{\rm max}$. We have no way to estimate the typical coherence size of a CB's magnetic domain $L_{\rm B}$, but the depth $D\sim (c\,\tau_\gamma\;L_{\rm B})^{1/2}$ to which an electron penetrates, even for a relatively simple magnetic mess ($L_{\rm B}$ not much smaller than $R_{\rm max}$) is smaller than the CB's radius. For $L_{\rm B}$ as small as $R_{\rm L}$, $D\sim 4\times 10^{10}$ cm, some four orders of magnitude smaller than $R_{\rm max}$. Even this concrete value is uncertain, for it depends on the surface magnetic field as B^-3/4, and the surface B-value may be different from that of Eq. (2), which is a volume average.

In addition to all of the above uncertainties, it is possible that a CB's illuminated working surface be turbulent, and harbour fast plasma motions, if only to establish local charge neutrality, which is disrupted as electrons and protons penetrate the CB to different depths. We conclude that the the fraction of ISM electrons that enter inside the CB may be small and the synchrotron-radiating electrons may be concentrated close to the CB's surface, as opposed to be acquiring a uniform distribution over the CB's volume.

6.2 The plasma frequency

The plasma frequency in a CB with an average free inner electron density $\bar n_e^{\rm free}$ is:

$$\nu_{p} = \left[{\bar n_{e}^{\rm free}\, e^2\over\pi\, m_{e}}... ...{\rm free}\over 10^7\;{\rm cm}^{-3}}\right]^{1/2}\, {\rm MHz}.$$ (19)

For a fully ionized CB $\bar n_{e}^{\rm free}=\bar n_{e}$, to whose reference value we normalized the above result (the fraction of electrons swept up from the ISM is small, relative to the total number in the CB - to whose free fraction Eq. (19) refers - but the CB is highly ionized, as shown in Appendix C).

For $\nu\!<\!\nu_{p}$the radio emission is completely damped within a typical length $\sim c/[\nu_p^2-\nu^2]^{1/2}$, much smaller than the CB's radius. At very early times, $\delta\sim 10^3$ and the Doppler-boosted value of $\nu_p$ falls in the low end of the observed range of radio signals, where a sharp cutoff is not observed. We must conclude that the (small fraction of) radiating electrons is located in a CB surface layer whose total electron density (dominated by the thermal electron constituency) is smaller than our reference average value, a one order of magnitude reduction being comfortably sufficient to move the value of $\nu_p$ to a position below the currently observed frequencies. We have explicitly checked that our fits do not improve significantly with the inclusion of $\nu_p$ as a free parameter: the minimization procedure always ``gets rid'' of the fit $\nu_p$ by choosing it somewhat below the reference value of Eq. (19), and below the lowest measured frequencies.

6.3 Free-free attenuation

At the MHz frequencies in the CB system corresponding to the observed radio frequencies, the synchrotron emission is strongly attenuated by free-free absorption (inverse bremsstrahlung) in the CB; free-free absorption dominates over self-synchrotron absorption, as shown in Appendix D. For a hydrogenic plasma, the free-free absorption coefficient at radio frequencies is:

$$\chi_\nu \simeq 0.018\, g_{\rm ff}\,\bar n_{e}\, \bar n_{i}\, T^{-3/2}\, \nu^{-2}\, {\rm cm}^{-1}\, ,$$ (20)

where $\bar n_{i}\simeq \bar n_{e}$ is the free ion density in the CB, in units of cm^-3, T is the plasma temperature in degrees Kelvin, $\nu$ is in Hertz and  $g_{\rm ff}$ is a Gaunt factor for free-free emission, of $\cal {O}$(10) at the relevant frequencies.

The opacity $\tau_\nu$ of a surface layer of depth D is:

$$\tau_\nu(D,t) =\int_{R_{\rm max}-D}^{R_{\rm max}}\chi_\nu \, {\rm d}r\equiv \bar \chi_\nu[t]\,D.$$ (21)

Equilibrium between photoionization of atomic hydrogen in the CB by synchrotron radiation and recombination of free electrons and protons to hydrogen keeps the CB plasma partially ionized during the observed AG. The Coulomb relaxation rate in the CB is very fast because of its high density. Consequently, the CB plasma is approximately in quasi thermal equilibrium. Because of the exponential dependence of the Saha equation on temperature, and the high ionization rate, the CB's temperature is kept practically constant around a few eV, and the ion density and free electron density become proportional to $\gamma(t)$ (Appendix C). Using T₀= 10⁵K and the reference average densities $\bar n_{e}=\bar n_i\simeq 10^7\, \rm cm^{-3}$ in Eqs. (20) and (21), we obtain:

$$\tau_\nu(D,t)\sim 1.4\times 10^2 \left[{D\over R_{\rm max}}\r... ...~GHz}\over \nu}\right]^2 \, \left[{T\over 10^5}\right]^{-3/2},$$ (22)

which is very large for $D\sim {\cal{O}}(R_{\rm max})$. A reduction in surface or average CB density of one order of magnitude or more - which, as we have seen, renders unobservable the unobserved plasma-frequency cutoff - reduces $\tau_\nu$ by two orders of magnitude, or more. For $D\sim R_{\rm max}/10$, this would make $\tau(D)$, as required, of order unity at the peak frequency $\sim$10² GHz of the early-time spectrum of Fig. 1 (at which time the observed and CB frequencies differ by a factor $\delta/(1+z)\sim 10^3$).

The conclusion is that a reasonable deviation of the properties of the CB from their reference bulk average values (a reduction of the total number-density of free electrons in a synchrotron-emitting surface layer) implies, not only that the plasma-frequency break is not observable in the current data, but also that the magnitude of the free-free attenuation is the required one. Our ignorance of the depth, temperature and density of ions and electrons in the radio-emitting surface of a CB can be absorbed into a single parameter: a characteristic absorption frequency, $\rm\nu_a$, in the opacity of Eqs. (20)-(22):

$$\tau_\nu\equiv\left[{\nu\over\nu_a}\right]^{-2}\,\left[{\gamma(t)\over \gamma_0}\right]^{2}\cdot$$ (23)

The frequency dependence of the free-free attenuation, $\chi_\nu\propto\nu^{-2}$, is fairly well supported by the observed radio spectra at their lowest frequencies, as our comparisons with observations in Sects. 8 and 9 demonstrate.

6.4 Attenuation in slabs and spheres

We do not know a priori the geometry of the working surface from which a CB's synchrotron radiation is emitted. In the case of optical AGs this is immaterial, for the CB is transparent to radiation at the corresponding CB-system wavelengths: the bulk of the radiation energy is emitted at these frequencies. For the case of radio AGs, attenuation is important and the shape of the emitting surface layers plays a role: the expression for attenuation as a function of opacity is geometry-dependent.

For a planar-slab geometry, the familiar expression for the attenuation is:

$$\rm {\it A}[\nu]={1-e^{-\tau_\nu}\over \tau_\nu}\cdot$$ (24)

For the emission from a sphere of constant properties, we obtain:

$$\rm {\it A}[\nu]= {3\, \tau_\nu^2-6\, \tau_\nu+6\, [1-e^{-\tau_\nu}]\over \tau_\nu^3}\, ,$$ (25)

while for the emission from a thin spherical surface, the result of Eq. (24) is recovered.

For the sake of definiteness, we adhere to CBs that are spherical in their rest system. This means that, as the frequencies increase and the CB evolves from being opaque to being transparent, we should use an attenuation evolving from Eq. (25) to Eq. (24). Rather than doing that, we have checked explicitly that our results are insensitive to the use of one or the other form, and used the simpler one.