Appendix C: Photoionization and recombination in the CB

We argue that the synchrotron radiation in a CB is intense enough to maintain its plasma partially ionized with ion and free electron densities proportional to $\gamma(t)$.

The bound-free cross section for photoionization of atomic hydrogen in its nth excited state by photons with frequency above the ionization threshold, $\nu_n=3.29\times 10^{15}/n^2\, \rm Hz$, is given by $\sigma_\nu(n)=n\,\sigma_1\,\bar{g}_n (\nu/\nu_n)^{-3}$, with $\sigma_1= {64\, \alpha\, \pi\, a_0^2/(3\, \sqrt{3}}) \simeq 7.91\times 10^{-18}$ cm² ( $a_0=0.53\times 10^{-8}$ cm is the Bohr radius and $\bar{g}_n$ is the Gaunt factor for photoabsorption by hydrogen). For the surface flux of photons of Eq. (6), we obtain an ionization rate of the nth level of atomic hydrogen:

$$R_{\rm i}(n)= {\eta\, n_{e}\, m_{e}\, c^3 \gamma(t)^2\, n\, \... ...\, \nu_n}\, \left({\nu_{\rm b}\over\nu_n}\right)^{(p-2)/ 2}.$$ (44)

For the reference values of n_e, p and $\gamma=\gamma_0$, the ionization rate of the ground state is $R_{\rm i}(1)\sim 1.1\times 10^{-2}\,\eta\,\rm s^{-1}$.

The recombination rate per unit volume of hydrogen in an hydrogenic CB is (Osterbrock 1989):

$$R_{\rm rec}\simeq 1.0\times 10^{-12}\,\bar n_{e}\, \left[{T\over 10^4\,{\rm K}}\right]^{-0.7}\,{\rm s}^{-1} .$$ (45)

For the reference value of the CB parameters, and $T\,=\,10\,000$ K the recombination rate of hydrogen is $R_{\rm rec}\sim 1.1\times 10^{-5}\, \rm s^{-1}$. In quasi equilibrium, the ionization rate per unit volume, which is proportional to the number density of recombined hydrogen atoms, must be equal to the recombination rate per unit volume, which is proportional to the product of the ion and free electron densities in the CB. Thus, initially the CB is highly ionized. But, for small values of $\eta$ and later times when $\gamma(t)$ becomes sufficiently small, equilibrium between the ionization and recombination rates in the partially ionized hydrogenic plasma results in $\bar n_{i}=\bar n_{e}\propto \gamma(t)\, .$