5.3 X-rays physics

References: Shull+ 1985, Wolfire+1995, Inayoshi+2011

The ionizing effect of X-rays sources on hydrogen and helium atoms has been modeled by many authors (Shull1985,Wolfire1995,Ricotti2002,Ricotti2004,Ferrara2008,Furlanetto2010). Primary photoionization processes additionally produce energetic electrons which can subsequently cause a secondary ionization of atoms.

The total photoionization rate enhanced by secondary ionisation is

\(\zeta^{i}_{tot} = \zeta^{i}_p + \sum_{j=\rm{H, He}}\zeta^j_p\frac{n_j}{n_i}\langle\phi^i\rangle\)

where the index p stands for "primary", i runs over the atoms H and He, \(n_i\) and \(n_j\) are the species abundances, and \(\zeta^i_p\) is

\(\zeta^i_p = \frac{4\pi}{h}\int{\frac{J_X(E)}{E}e^{-\tau(E)}\sigma^i(E)dE}\)

\(h\) being the planck constant. The number of secondary ionization of H and He produced per primary electron, :math: phi^H `, and :math:phi^{He}`, are taken from Shull+ 1985 and are defined as following

\(\phi^H (E,x_e) = \left(\frac{E}{13.6 ~\mathrm{eV}} - 1\right) 0.3908(1-x_e^{0.4092})^{1.7592}\)

\(\phi^{He} (E,x_e) = \left(\frac{E}{24.6 ~\mathrm{eV}} - 1\right) 0.0554(1-x_e^{0.4614})^{1.666}\)

These fitting formulae are valid for energy $E > 100$ eV and for a gas mixture of H and He. As we are interested in the energy range between 2-10 keV we adopt the above approach. For applications which extend to lower energies different fitting formulae other than Shull+ 1985 should be used, as for instance the one discussed in the appendix of Wolfire+ 1995

The above quantities are then averaged over the X-rays spectrum as

\(\langle\phi^i\rangle = \frac{\int{J_X(E) \phi^i(E,x_e)dE}}{\int{J_X(E)}dE}\)

The flux \(J_X(E)\) is adopted from Inayoshi+ 2011 as originally introduced by Glover&Brand 2003 and expressed in terms of \(J_{21,X}\) \(erg cm^{-2} s^{-1} sr^{-1} Hz^{-1}\)

\(J_X(E) = J_{21,X} \times 10^{-21} (E/E_0)^{-1.5}\)

where we have introduced a subscript "X" to differentiate from the standard UV \(J_{21}\). Here, \(E_0\) = 1 keV and \(\tau\) is an opacity term expressed as

\(\tau = \sum_{i=\rm{H, He}}\sigma^i(E) N_i\)

with the column density defined as a function of the Jeans length

\(N_i = n_i \lambda_J\)

and

\(\lambda_J = \sqrt{\frac{\pi k_b T}{G\rho\mu m_H}}\)

where \(k_b\) is the Boltzmann constant, \(G\) the gravitational constant, \(m_H\) the proton mass, \(\rho\) the total mass density, and \(\mu\) the mean molecular weight evaluated as

\(\mu = \frac{\sum_k\rho_k}{\rho}\)

The cross sections for the photo-ionization of H and He are taken from Verner+ 1996.

The photoionization heating is given by \(\Gamma = \Gamma^{\rm H} + \Gamma^{\rm He}\)

where \(\Gamma^i\) is defined as following

\(\Gamma^i = \frac{4\pi}{h}\int{\frac{J_X(E)}{E}e^{-\tau(E)}\sigma^i(E)E^i_h(E,x_e)dE}\)

with \(E_h^i(E,x_e)\) being the fraction of primary electrons which go into heating (Shull+, 1985)

\(E^i_h(E,x_e) = (E-E^i_0)0.9971(1 - (1-x_e^{0.2663})^{1.3163})\)

with \(i = H, He\), \(E_0^H = 13.6$ eV\), and \(E_0^{He} = 24.6\) eV.

For a direct application of the above physics and its implementation in KROME we remind to http://adsabs.harvard.edu/abs/2014arXiv1408.3061L.