Simple X-ray reflection models

Initially, we explore a model for the disk atmosphere that is clearly oversimplified but helps elucidate the physics of X-ray reflection. We suppose that the surface of the accretion disk can be modeled as a semi-infinite slab of uniform density gas, irradiated from above by a continuum X-ray spectrum produced in the disk corona via thermal Comptonization. Furthermore, we assume that the hydrogen and helium are fully ionized, but all other elements (collectively referred to as ``metals'' by astronomers) are neutral. This is a crude approximation to the situation found in a ``cold'' AGN accretion disk. Now we consider the possible fates of an incident X-ray photon. Firstly, the photon can be Compton scattered by either the free-electrons associated with the ionized hydrogen and helium, or the outer electrons of the other elements. Secondly, the photon can be photoelectrically absorbed by one of the neutral atoms. For this to happen, the photon must possess an energy above the threshold energy for the particular photoelectric transition. The transitions with the largest cross-sections are those associated with the photo-ejection of a K-shell (i.e. $n=1$ shell) electron. Following K-shell photoionization, the resulting ion commonly de-excites in one of two ways, both of which start with an L-shell ($n=2$) electron dropping into the K-shell. In the first case, the excess energy is radiated as a $K\alpha$ line photon⁹ (i.e., fluorescence). In the second case, the extra energy is carried away via the ejection of a second L-shell electron (i.e., autoionization or the Auger effect). The fluorescent yield of a species gives the probability that the excited ion will de-excite via fluorescence rather than autoionization.

Figure 7: Results of a simple Monte Carlo simulation demonstrating the ``reflection'' of an incident power-law X-ray spectrum (shown as a dashed line) by a cold and semi-infinite slab of gas with cosmic abundances. In the accretion disk setting, one would observe the sum of the direct power-law continuum and the reflection spectrum -- the principal observables are then the cold iron K$\alpha$ fluorescent line at 6.40keV and a ``Compton reflection hump'' peaking at $\sim 30$keV. Figure from reference [178].

Figure 7 shows the results of a Monte-Carlo simulation modeling these processes when a power-law X-ray continuum with photon index $\Gamma=2$ is incident on a gaseous slab [24,178], assuming the slab has cosmic abundances [179]. At soft X-ray energies, the albedo of the slab is very small due to the photoabsorption by the metals in the slab. However, at hard X-ray energies, this photoabsorption becomes unimportant (i.e., the photoelectric cross section falls to small values) and most of the X-rays incident on the slab are Compton scattered back out of the slab. Associated with the photoionization of metals in the slab, there is a spectrum of fluorescent emission lines. Due to the combination of high fluorescent yield and large cosmic abundance, the most prominent such fluorescence is the K$\alpha$ line of iron at 6.40keV. It is interesting to note in Fig. 7 the weak ``shoulder'' on the low-energy side of the iron-K$\alpha$ line. This feature corresponds to line photons that have Compton scattered, and hence lost energy due to electron recoil, before escaping the disk. It is often referred to as the ``Compton shoulder''.

It is customary to measure the relative strengths of astrophysical emission lines via the use of equivalent widths. The equivalent width of an emission line is the energy (or wavelength) range over which the continuum radiation contains a flux equal to that contained in the emission line. For the case presented in Fig. 7 the equivalent width of the iron line when viewed in the combined direct$+$reflected spectrum is approximately 180eV. The other lines are much weaker, with equivalent widths at least an order of magnitude less [180].