# 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.

Updated