Appendix D: Synchrotron and Bremsstrahlung self absorption

We argue that the self-attenuation in the CB of the observed radio waves is dominated by free-free absorption.

The density of energetic electrons

In the rest system of a CB, the ISM electrons arrive at a rate ${\rm d}\,N_{\rm e}/{\rm d}t\simeq n_p\,\gamma\, \pi\, R_{\rm max}^2\, c.$A fraction $\eta$ of their energy, $E_{\rm b}=m_{e}\, c^2\, \gamma,$ is synchrotron re-radiated. Let $n_{\rm eff}$ be the density of the emitting electrons, so that:

$$\eta\, E_{\rm b}\, {{\rm d}N_{\rm e}\over {\rm d}t}=\int F_{_... ...{\rm eff}\over {\rm d}E_{\rm e}}\,{1\over \tau(E_{\rm e})}\; ,$$ (46)

with $\tau(E_{\rm e})=E_{\rm e}/({\rm d}E_{\rm e}/{\rm d}t)$ the cooling time for electrons of energy E_e, determined from Eqs. (2) and (10) to be:

$$\tau(E_{\rm e})=4\,{m_{e}\over m_{p}}\,{1\over n_{p}\,\sigma_{_{\rm T}}\,c\,\gamma_{\rm e}^3} \cdot$$ (47)

For the electron spectrum of Eq. (13), the second integral in Eq. (46) is dominated by energies $E_{\rm e}\sim E_{\rm b}$( $\rm\gamma_e\sim \gamma_b=\gamma$). Thus, for a uniform distribution of electrons in the CB's volume V, we obtain:

 

$$n_{\rm eff} \simeq -\,{\eta\over V}\,{{\rm d}N_{\rm e}\over {\rm d}t}\,\tau(E_{\rm b})$$

$${m_{e}\over m_{p}}\,{3\,\eta\over R_{\rm max}\,\sigma_{_T}\,\gamma^2} \simeq (11\,\eta\, {\rm cm}^{-3})\left[10^3\over\gamma\right]^2 ,$$ = (48)

where we have used our reference $R_{\rm max}$. This density could be somewhat higher if the emitting electrons are concentrated on the CB's front surface. Note that this density increases with time like $[\gamma(t)]^{-2}$.

Synchrotron self absorption

For a power-law distribution, ${\rm d}n_{e}/{\rm d}\gamma_{\rm e}= (p-1)\,n_{\rm eff}\, \gamma_{\rm e}^{-p}$, the correct attenuation coefficient for synchrotron self absorption at frequency $\nu$ is (e.g., Shu 1991, Eqs. (19.37), (38)):

$$\chi_\nu= K_0\, (p-1)\,n_{\rm eff}\, {B_\perp}^{(p+2)/2}\, \nu^{-(p+4)/2}\, ,$$ (49)

where:

$$K_0={c\, r_{\rm e}\over 4\, \sqrt{3}}\, \left ({3\, e\over 2... ...{3\, p+2\over 12}\right) \Gamma\left({3p+22\over 12}\right),$$ (50)

with $r_{\rm e}= {\rm e}^2/m_{e}\, c^2.$ An observed frequency $\rm\nu_{obs}$ was emitted at $\nu(t)=(1+z)\, \nu_{ob}/\delta(t)$ in the CB's rest frame. As an example, $\nu_{\rm ob}=5$ GHz from a decelerating CB with $\delta(t)\simeq \gamma_0/2 \simeq 500$, if emitted at a typical z=1, corresponds to $\nu=20$ MHz in the CB rest frame. For our reference parameters, the synchrotron self absorption coefficient in the CB is $\chi_\nu \simeq 2.4\times 10^{-10}\,\eta\, \rm cm^{-1}$. Since $n_{\rm eff}\propto \gamma^{-2}$, $B\propto \gamma$, and, after a few observer's days, $\delta\sim 2\gamma$ and $\gamma(t)\sim t^{-1/3}$, $\chi_\nu$ decreases with time like $\gamma^{(p+1)}\sim t^{-(p+1)/3}\sim t^{-1.1}$.

Bremsstrahlung self absorption

The X-ray AG is dominated first by bremsstrahlung from plasma electrons and later by synchrotron radiation from the swept up high energy electrons (DDD 2001). The observed X-ray flux, or the theoretical UV flux in the CB rest frame, can be used to show that the CB is partially ionized during the radio AG observations and that the ionized fraction of the CB plasma is proportional to $\gamma(t)$ (see Appendix C). The logarithmic dependence of the plasma temperature in the Saha equation on the fractional ionization, keeps the CB's temperature nearly constant during its  AG phase. Consequently, in the CB rest frame, the free-free attenuation at a fixed frequency is proportional to $[\gamma(t)]^2$ and the free-free (bremsstrahlung) absorption coefficient is that of Eq. (20).

The temperature of the partially ionized CB is of ${\cal{O}}(1)$ eV and almost constant during the observed AG. For $\sim$20 MHz emission from a thermal plasma at such temperature, $g\sim 10$and for one tenth of the typical bulk CB density, $\bar n_{e}\sim 10^6~ \rm cm^{-3}$, one obtains from Eq. (20) $\chi_\nu\simeq 4\times 10^{-10}\,\rm cm^{-1}$, which is $\sim$$1.7/\eta$ larger than the synchrotron absorption coefficient of the energetic electrons in the CB (the values of $\eta$ are listed in Table 3). At a fixed observer frequency, $\nu =(1+z)\,\nu_{\rm ob}/\delta$, the free-free opacity of the CB decreases roughly like $\sim$ $\gamma^2\sim t^{-2/3}$ compared with the $\sim$t^-1.1 decline of the synchrotron self opacity.

The conclusion is that free-free absorption is dominant for as long as the ionization of the CB is considerable.