Physical X-ray reflection models

The physical situation with a real disk atmosphere is much more complex. We can consider the atmosphere as a layer of gas that is heated from below by the main body of the accretion disk (which, in turn, is heated by the ``viscous'' dissipation of gravitational potential energy) and irradiated from above by high-energy radiation from the corona. To determine the structure of the disk atmosphere, we must model the radiative transfer, and solve for the thermal and ionization balance of each fluid element. Furthermore, the atmosphere almost certainly responds dynamically to the heating/irradiation, thereby requiring us to solve the equations of hydrodynamics (or, more generally MHD) at the same time. Finally, we must acknowledge the fact that radiation pressure may well be important in many disks of interest.

Figure: Theoretical ionized reflection spectra for different ionization parameters, as computed using the constant density models of David Ballantyne and collaborators [181]. From bottom to top, the curves show the reflected X-ray spectrum for ionization parameters of $\log\xi=1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5$ and $5.0$. The detailed behavior is discussed in the main text.

This is clearly a very complex problem, and progress to date has been made only by considering rather special cases. The first step towards realistic disk reflection models maintained the assumption of a stationary, time-independent, uniform density structure, and assumed that the atmosphere was in thermal and ionization equilibrium. By solving the radiative transfer problem in this fixed background, the thermal and ionization structure as a function of depth in the atmosphere can be obtained [182,183,184,185]. Hence, the resulting X-ray reflection spectrum can be determined. A useful quantity in the discussion of photoionized atmospheres is the ionization parameter,

$$\xi(r)=\frac{4\pi F_{\rm x}(r)}{n(r)},$$ 29

where $F_{\rm x}(r)$ is the X-ray flux received per unit area of the disk at a radius $r$, and $n(r)$ is the comoving electron number density: it measures the ratio of the photoionization rate (which is proportional to $n$) to the recombination rate (proportional to $n^2$). The constant density ionization models suggest that the value of $\xi$ delineates four regimes of behavior (see Fig. 8).

1. $\xi <100$ ergs cm s$^{-1}$: This is the ``neutral reflection'' regime since the X-ray reflection resembles that from cold gas containing neutral metals (as in Fig 7). There is a cold iron line at 6.4 keV, and the Compton backscattered continuum only weakly contributes to the observed spectrum at this energy. There is a weak iron K=shell edge at $7.1{\rm\thinspace keV}$.
2. 100 ergs cm s$^{-1}<\xi <500$ ergs cm s$^{-1}$: In this ``intermediate ionization'' regime, the iron is in the form of FeXVII-FeXXIII and there is a vacancy is the L-shell ($n=2$) of the ion. These ions can resonantly absorb the corresponding K$\alpha$ line photons. Following each absorption, there can be another fluorescent emission (with the consequence that the photon has effectively been scattered) or de-excitation via the Auger effect. A given iron K$\alpha$ photon can be trapped in the slab via the resonant scattering until it is terminated by a Auger event. Thus, only a few line photons can escape the disk leading to a very weak iron line. The reduced opacity below the iron edge due to ionization of the lower-$Z$ elements leads to a moderate iron absorption edge.
3. 500 ergs cm s$^{-1}<\xi <5000$ ergs cm s$^{-1}$: In this ``high ionization'' regime, the ions are too highly ionized to permit the Auger effect (which requires at least 2 L-shell electrons prior to photoionization). While the line photons are still subject to resonant scattering, the lack of a destruction mechanism ensures that they can escape the disk. The result is K$\alpha$ iron emission of FeXXV and FeXXVI at 6.67keV and 6.97keV respectively. Due to the high levels of ionization and subsequent lack of photoelectric opacity characterizing the low-$Z$ elements, the Compton backscattered continuum is a significant contributor to the observed emission at $6{\rm\thinspace keV}$. Thus, there is a large iron absorption edge in the observed spectrum.
4. $\xi >5000$ ergs cm s$^{-1}$: In this ``fully ionized'' regime, the disk is too highly ionized to produce any atomic signatures. There is no iron emission line or edge.

More recent variants of these fixed density models have improved the treatment of Compton scattering in the upper atmosphere [181], and have assumed a Gaussian density profile $\rho(z) = \rho(0) \exp[-(z/z_0)^2]$ [158]. The qualitative behavior is, however, unchanged.

Yet more realistic models require us to relax the assumption of a fixed density structure. The next simplest, yet physical, assumption is to suppose that the disk atmosphere is in hydrostatic (or, more generally, hydromagnetic) equilibrium. The density, temperature and pressure must then adjust so as to achieve hydrostatic equilibrium. When such hydrostatic, irradiated disk-atmosphere models are constructed, a qualitatively new phenomenon occurs -- the thermal ionization instability (TII) [186].

The origin of the TII is as follows. The equations defining local thermal and ionization equilibrium for a specified density admit a unique solution. On the other hand, if the pressure is specified, there are certain values of pressure for which there exist more than one equilibrium thermal and ionization state (with the intermediate temperature equilibria often proving to be thermally unstable). This is the thermal ionization instability. The primary consequence of the TII is that, over certain ranges of pressure, a multiphase medium can form with comparatively cool regions co-existing in pressure equilibrium with very hot regions.

Now consider moving to greater and greater depths in a hydrostatic disk atmosphere. As the pressure increases monotonically, one will encounter layers of the atmosphere in which the material is subject to the TII and, hence, multiple temperature and ionization structures are allowed. The exact solution actually realized by a particular fluid element depends upon the history of that element, but there are general arguments suggesting that it will usually exhibit the highest temperature solution[186,159]. As one transits from regions (i.e., pressures) where the TII operates to regions where it does not, there will be abrupt changes in the temperature, density and ionization state of the material. In practice, a highly-ionized and well-defined hot ``skin'' will develop, overlying the colder parts of the disk that are more relevant for producing interesting X-ray reflection features. If this ionized skin has a low optical depth, it will be largely irrelevant to the observed reflection spectrum. However, if the electron scattering optical depth of the skin approaches unity, it will cause a smearing of any sharp features in the reflection spectrum (due to Compton recoil) as well as an apparent dilution of the observed reflection features (since it will Compton scatter X-ray continuum photons back towards the observer without any appreciable spectral change). For strongly irradiated disks, the hot skin will be optically-thick and act as a ``Compton mirror'', simply reflecting back the irradiating spectrum (with slight modifications caused by electron recoil which only become important for very hard X-rays; see for example [187]). If the irradiating spectrum is sufficiently hard ($\Gamma < 2$), the Compton mirror can form before the disk is capable of producing observable H- or He- like K$\alpha$ iron line emission.

When faced with real data, how is one to make progress given the complexities involved in modeling the reflection spectrum? A major simplification, pointed out by David Ballantyne and collaborators [181], is that self-consistent reflection spectra can be approximately mapped to more manageable constant-density models, albeit with different inferred reflection fractions (due to dilution by an ionized skin) and ionization parameters. Thus, for all practical purposes, X-ray observers to date have used the constant density models when attempting to fit real data.