X-RAY REFLECTED SPECTRA FROM ACCRETION DISK MODELS. III. A COMPLETE GRID OF IONIZED REFLECTION CALCULATIONS

As mentioned before, xillver implements the xstar routines for the calculation of the ionization structure of the gas, while making use of xstar's atomic data. The core of the xstar atomic database is described in detail in Bautista & Kallman (2001). It has been constructed using information from many sources such as CHIANTI (Landi & Phillips 2006), ADAS (Summers 2004), NIST (Ralchenko et al. 2008), TOPbase (Cunto et al. 1993), and the IRON project (Hummer et al. 1993). Over the past several years, we have dedicated a significant effort toward the investigation of the K-vacancy states in many ions. This has allowed us to systematically improve the modeling of the K lines and edges relevant for high-quality astronomical X-ray spectra.

Such atomic data sets include energy levels, wavelengths, Einstein A-coefficients, and radiative and Auger widths computed for a large number of ions using three different atomic-structure theoretical approaches, including the relativistic Hartree–Fock (Cowan 1981), the autostructure (Badnell 1986, 1997), and the multi-configuration Dirac–Fock (Berrington et al. 1987; Seaton 1987) methods. Extensive calculations of photoabsorption and photoionization cross sections have been also performed using the Breit–Pauli R-Matrix method including the effects of radiative and Auger damping by means of an optical potential (Gorczyca & Robicheaux 1999; Gorczyca & McLaughlin 2000).

In particular, calculations of the atomic data required for the spectral modeling of the K-shell photoabsorption of oxygen and nitrogen ions have been carried out in García et al. (2005) and García et al. (2009). Computations have also been carried out for the atomic structure of the isonuclear sequences of Ne, Mg, Si, S, Ar, and Ca (Palmeri et al. 2008a), nickel (Palmeri et al. 2008b), and aluminum (Palmeri et al. 2011). More recently, we have focused our attention on the iron peak and light odd-Z elements, i.e., F, Na, P, Cl, K, Sc, Ti, V, Cr, Mn, Co, Cu, and Zn producing the atomic parameters for more than three million fine-structure K lines (Palmeri et al. 2012). Photoabsorption calculations of the cross sections across the K-edge of Ne, Mg, Si, S, Ar, and Ca have been performed for ions with less than 11 electrons by Witthoeft et al. (2009), and for ions with more than 10 electrons by Witthoeft et al. (2011b). In these calculations it was shown that damping processes affect the resonances converging to the K thresholds, causing them to display symmetric profiles of constant width that smear the otherwise sharp edge at the photoionization thresholds. The Li-like to Ca-like ion stages of nickel are discussed in Witthoeft et al. (2011a). These new data sets are continually being incorporated into the xstar database in order to generate improved opacities in the K-edge regions of the ions considered.

The main differences in the current version of the atomic data with respect to the one used in García & Kallman (2010) are as follows: the inclusion of the atomic data for the n = 2 → 3 inner-shell transitions of Fe vi–xvi ions, the so-called Fe M-shell unresolved transition array (Gu et al. 2006), and the inclusion of the K-shell atomic data for the Mg, Si, S, Ar, Ca, Al, and Ni isonuclear sequences (Palmeri et al. 2008a, 2008b; Witthoeft et al. 2011b; Palmeri et al. 2011; Witthoeft et al. 2011a). The implementation of the updated radiative and dielectronic recombination rates is from Badnell (2006) and Badnell et al. (2003). Also, we have improved the resolution of the high-energy extrapolation implemented in the Fe photoionization cross sections for energies well above the K-edge. Finally, by looking at the third-row Fe ions (Fe i–viii), we found that for the densities used in our simulations (n_e ∼ 10¹⁵ cm⁻³), many metastable states are populated. Previously, we only used K-shell photoionization data for the ground state, therefore missing the contribution from these metastable states. This resulted in a weak Fe K emission line in models with low ionization parameter. This problem has been corrected in the new version of the database by including the photoionization cross sections for all these states.

Another important change in the xillver code is the definition of the radiation field that illuminates the surface of the accretion disk. In García & Kallman (2010), the incident spectrum was assumed to be a power law with a given photon index Γ for all energies in the 10⁻¹ to 2 × 10⁵ eV range. This kind of illumination is appropriate for spectra with Γ ∼ 2. However, for steeper spectra (Γ > 2) this choice of illumination creates unphysical conditions, since the incident power law will tend to have very many photons concentrated in the low-energy part of the spectrum. Thus, we have now adopted a power law that breaks at 100 eV and decays with an exponential tail for lower energies. We have also extended the energy to 1 MeV, and moved the exponential high-energy cutoff from 200 keV to 300 keV. Figure 1 shows the impact of this modification for the resulting reflected spectrum in a case where Γ = 3. The solid lines show the reflected spectra predicted by xillver using the old (black) and the new (red) definitions of the illumination. The incident power law is shown with dashed lines for each case. Both calculations have a similar ionization parameter (log ξ ∼ 2) but the integrated flux is different given the differences in the shape of the power law. Thus, the model with the new definition (broken power law) needed to be scaled down such that both incident spectra will have the same flux near 100 eV and energies above, region in which the photon index is Γ = 3. The huge impact of the illuminating spectrum on the reflected component is clear, especially at energies below 5 keV. This difference becomes more dramatic as the photon index increases. However, for harder spectra (Γ ≲ 2), the break in the incident power law has little impact on the reflected spectrum, especially in the high-energy band.

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The extension of the spectrum to higher energies also has an important influence on the reflection calculations. This is particularly important in models on the other extreme of the photon index parameter space. In the case of a very hard illumination spectrum (Γ < 2), models with similar ionization parameters will differ in the number of photons that are concentrated in the high-energy part of the spectrum. Figure 2 shows the resulting xillver models for an incident power law with Γ = 1.4, log ξ = 2.8, and solar abundances. For the black and red curves, the energy range extends upward to 200 keV and 1 MeV, respectively. In panel (a), the dashed lines show the incident power-law spectra, while solid lines are the reflected spectra for each case. The emission lines in the model with the high-energy coverage (red) are considerably weaker than those in the original model (black). This is most evident for the Fe K complex around 6–7 keV, and for the O vii Lα near 650 eV. The reason for this difference is that in models with the high-energy extension, the region of the atmosphere near the surface is more strongly heated by the high-energy photons, and thus its temperature is higher than in the original model. This can be seen in panel (b) of Figure 2, where the temperature profiles in the vertical direction of the disk are plotted as function of the Thomson optical depth. Clearly, the two models converge at large depths (τ_T ∼ few), but the one in which the illumination extends to higher energies results in a considerably hotter atmosphere near the surface. Although we expect the Fe features to be produced well inside the disk, line photons that escape through a hotter gas will tend to suffer a larger energy shift from Compton scattering, with similar probabilities for up- or downscattering. This makes the line profile broader and more symmetric.

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A new library of reflection spectra has been produced using the new version of the code xillver. This set of models covers a wide range of parameter values that are relevant for fitting the spectra of accreting sources. Each model is characterized by the photon index Γ of the incident power-law radiation, the ionization parameter ξ, and the iron abundance A_Fe with respect to the solar value (Grevesse & Sauval 1998). In order to have a library of models suitable for both AGNs and GBHs, we have produced models covering photon indices in the Γ = 1.2–3.4 range, in steps of 0.2. Observations typically show that the reflection signatures from accretion disks in AGNs are produced in material at a lower ionization stage than those observed from GBHs (García et al. 2011), consistent with the picture that the accretion disk in the latter are much hotter than in the former. Consequently, we vary the ionization parameter over a wide range to cover both classes of sources. We have produced models with ξ = 1, 2, 5, 10, 20, 50, ..., 10⁴, 2 × 10⁴, 5 × 10⁴ erg cm s⁻¹. Finally, we also allowed the iron abundance to be treated as a free parameter, given the importance of the Fe K emission profile in most accreting sources. For simplicity, the abundance of all the other elements considered, namely, H, He, C, N, O, Ne, Mg, Si, S, Ar, and Ca, is kept fixed to the solar values of Grevesse & Sauval (1998). Thus, models with A_Fe = 0.5, 1, 5, and 10 were calculated, taking into account both sub- and super-solar Fe abundances. Here, A_Fe = 1 corresponds to an iron abundance of 2.5 × 10⁻⁵ with respect to hydrogen. These choices of parameter values resulted in a library of 720 synthetic reflection spectra that can be used for the modeling of the reprocessed component in X-ray observations. The present set of models is provided as a single file⁸ in Flexible Image Transport System (FITS) format, which can be loaded into the fitting package xspec⁹ via the atable model.

Figure 3 shows a sub-group of the resulting reflected spectra for three values of the photon index Γ and a range of ionization parameters. The Fe abundance is set to the solar value (A_Fe = 1) for all the models shown here. Panels (a), (b), and (c) correspond to Γ = 1.4, 2, and 2.6, respectively. Each panel shows the reflected spectra calculated for a different value of the ionization parameter. From bottom to top, each curve corresponds to ξ = 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000, 2000, 5000 erg cm s⁻¹, respectively. Because the gas density is held fixed at n_e = 1.2 × 10¹⁵ cm⁻³ for all these calculations, increasing the ionization parameter is equivalent to increasing the flux F_x of the illuminating source, where F_x = ξn_e/4π. The spectra are plotted in units of EF_E (equivalent to νF_ν if plotted in frequency), so that a power law of Γ = 2 is shown as a horizontal line. Note that no rescaling or shift is applied; instead, each curve is color-coded according to its corresponding value of log ξ, to improve clarity. This figure provides a general overview of the effect that the ionization parameter has on the spectrum reflected from an optically thick plane-parallel slab. The original power-law shape of the illuminating continuum (for the lowest value of ξ), shown in black dashed lines, suffers drastic modifications due to both absorption and emission. The spectra are shown in the entire energy range included in the calculations (0.1–10⁶ eV). In the low-energy part of the spectrum (0.1–10 eV), the continuum is dominated by bremsstrahlung emissivity, despite the fact that this is where the illumination flux is decreased due to the cutoff imposed at 100 eV. Emission lines and absorption edges from H, He, and C are clearly visible for most of the models with low and intermediate ionization. In the 10–10⁴ eV energy range, the photoelectric opacity dominates over the electron scattering opacity. Therefore, this region is where most of the absorption occurs, yielding large departures from the original power-law continuum. Bremsstrahlung emissivity decreases rapidly for energies above ∼100 eV; however, many emission lines from all the ions included in these calculations remain visible over the entire energy range. At higher energies (>10⁴ eV), electron scattering is the dominant source of opacity since the cross section for photoelectric absorption decays as ∼E⁻³, while the Klein–Nishina cross section for electron scattering remains fairly constant. Thus, in this spectral region Compton scattering is the only relevant process and the reflected spectrum depends on the shape of the original illuminating field.

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By looking at the overall shape of the reflected continuum shown in Figure 3, it is possible to distinguish between two main regimes: (1) the high-ionization case, where the resulting spectra mostly shows very narrow emission features, while the continuum still resembles the original shape of the illuminating power law; and (2) the low-ionization case, where the emerging spectra are a combination of a very rich and complex set of emission-line profiles superimposed on a strongly absorbed and modified continuum, which significantly departs from the original power law. The specific value of the ionization parameter that separates these two regimes depends, to a certain degree, on the photon index Γ. In fact, for the Γ = 1.4 case shown in panel (a) this transition is quite obvious, as a drastic change in the reflected spectra can be seen between models with log ξ = 2.3 and 2.7. These changes are the result of large differences in the ionization balance solutions, as can be seen in Figure 4, which shows the corresponding temperature profiles for each one of the models shown in Figure 3, color-coded in the same way. Because larger ionization implies a larger illumination flux, models with high ξ values are systematically hotter than those with low ξ values. The illuminating radiation, incident at the surface of the slab (at τ_T = 10⁻⁴ for our proposes), penetrates the first layers heating the gas. The amount of heating not only depends on the illuminating flux, but also on the particular shape of the radiation field; e.g., note that the models for small values of Γ are consistently hotter than those with large values. We shall discuss these effects further in the next section. As photons are scattered or absorbed and re-emitted at different energies, the shape of the continuum is modified reducing the net heating in the gas. At some point H and He recombine which increases the cooling very rapidly. The gas then suffers a rather sudden transition to a lower temperature, as can be seen for most of these models. This transition occurs at deeper regions in the slab for larger ionization parameters and for softer input spectra (large Γ). Regardless of the ionization parameter or the shape of the ionizing radiation, in all cases the field thermalizes and the gas eventually reaches the same lower temperature around (0.5–1) × 10⁵ K. This is the limit in which the illuminating radiation no longer contributes to the ionization of the material and its temperature is simply set by the density of the gas.

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To illustrate in more detail the effects of the ionization parameter on the reflected spectra, we will concentrate on models with one particular value of the photon index. Further, we will examine the low- and high-ionization models separately. Thus, Figure 5 shows some of the reflected spectra resulting from models with Γ = 2 in the 10² to 5 × 10⁵ eV energy range, which is the spectral band typically covered in X-ray observations. Panel (a) shows four models with low ionization, specifically ξ = 1, 5, 20, and 100, multiplied by factors of 1, 10², 10⁴, and 10⁶, respectively. Their corresponding temperature profiles are curves 1, 3, 5, and 7, from left to right, in panel (b) of Figure 4. The dashed line represents the illuminating power-law spectrum for the model with the lowest ionization parameter (ξ = 1). As mentioned before, these models display a strong decrease of the continuum flux due to photoelectric absorption, in particular for energies below ∼20 keV. Nevertheless, a very rich and complex set of fluorescent emission lines due to K-, L-, and M-shell transitions from many ions is also present, superimposed on a highly absorbed continuum. The general spectral shape of the lowest ionization models (ξ = 1–20) is very similar, where only small changes are seen. The temperature in these models is relatively low (T ≲ 10⁶ K), and the gas settles to a low-temperature regime at a small optical depth (τ_T ≲ 0.1), meaning that most of the slab remains neutral. The emission due to the Kα and Kβ transitions in Fe is distinctive at ∼6.4 and ∼7.1 keV, respectively. The case for ξ = 100 is where the changes in the reflected spectra become more evident, in both the continuum absorption and in the emission features. The higher temperature in this model allows for the excitation of elements such as Mg, Al, Si, S, Ar, and Ca, which produces more fluorescence lines in the 1–10 keV region. The Fe Kα emission becomes slightly broader and the Kβ is less intense. Some RRC can also be seen near ∼0.5–1 keV.

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Panel (b) of Figure 5 shows the reflected spectra for the next five values of the ionization parameter considered here (ξ = 200, 500, 1 × 10³, 2 × 10³, and 5 × 10³). Each curve has been rescaled by a constant factor to improve clarity. These factors are 1, 10, 10², 10³, and 10⁵ from low to high ionization, respectively. As before, the dashed line shows the illuminating power-law spectrum corresponding to the model with the lower ionization (ξ = 200). These are the models considered as high ionization, since the continuum is not highly affected by the photoelectric opacity. This can be seen by comparing, for example, the reflected flux at 10² eV in the spectrum for ξ = 200 and in the one for ξ = 1, shown in panel (a). As mentioned before, increasing the ionization raises the temperature of the illuminated region of the slab. This region also extends to deeper zones as the radiation is able to ionize the gas at larger optical depths, as can be seen in the corresponding temperature profiles (see panel (b) of Figure 4). The increase in the temperature has two main effects. On the one hand, it affects the ionization of the gas. The ions from low-Z elements are completely stripped from all their electrons, while the ions of the heavier elements are partially ionized. This changes the emission lines produced inside the gas, and thus the emerging spectra lacks emission from low-Z ions, progressively showing more emission from highly ionized O, Ne, Ar, Ca, and Fe as the illumination increases. On the other hand, the high temperature affects the way line photons are scattered by electrons, given the dependence of Equations (6) and (7) on the kinetic energy of the electrons. When 4 kT ∼ E, the probability for a photon to either gain or lose energy after each scattering becomes comparable, which effectively produces a broadened and more symmetric line profile. This effect is particularly evident in the Kα emission lines from O and Fe, observed at ∼0.65 and ∼6.9 keV, respectively. We shall return to a more detailed discussion of the spectral features in Section 3.5.

It is clear from the discussion in the previous section that both the net flux and the spectral shape of the ionizing radiation incident on the surface of the slab have a great impact on the ionization balance of the gas, and thus on the reflected spectrum that will emerge at the surface. For this particular library of models we have adopted a power-law shape for the illumination spectrum, thus the general shape is controlled by changing the value of the photon index Γ. Figure 6 shows a sub-group of the reflected spectra calculated for various conditions. Panels (a), (b), (c), and (d) show the resulting models for a given ionization parameter, i.e., ξ = 10, 10², 10³, and 10⁴, respectively. Each panel contains the spectra of models for all the values of the photon index Γ considered in our calculations (Γ = 1.2–3.4), color-coded accordingly. Since the goal of this figure is to show the general trends in the spectra introduced by changing Γ, all the curves are plotted with the same normalization (thus, no rescaling was applied). It is quite obvious how the changes in the ionizing continuum affect the resulting continuum of the emergent radiation, as expected. However, the ionization structure of the gas seems to follow different trends on Γ for different values of ξ.

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By comparing the two extreme values of ξ (panels (a) and (d)), a completely opposite behavior is seen. In the low-ionization case (ξ = 10), models with low Γ values are the ones with the most absorption, suggesting a much colder gas than those for high Γ values. This is consistent with the temperature profiles obtained for these models, which are shown in panel (a) of Figure 7. Therefore, in the low-ionization regime, the softer the input spectrum (large Γ value), the hotter and more ionized the gas becomes. Looking at the high-ionization case in panel (d) (ξ = 10⁴), the opposite occurs: the models with harder illumination spectra (lowest Γ) are completely ionized, owing to the large gas temperature. Thus, in the high-ionization regime, the harder the input spectra, the hotter and more ionized the gas becomes. The intermediate regimes plotted in panels (b) and (c) of Figures 6 and 7 show the transition between these two regimes. For ξ = 10², the gas temperature is low for the harder spectrum (Γ = 1.2), and increases with the photon index up to Γ ∼ 2.4, where it starts decreasing again. For the case with ξ = 10³ (panel (c)), the transition occurs at Γ ∼ 1.8.

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The reason for this change of behavior is related to the processes that contribute to the heating and cooling of the gas. In general terms, there are two main competing mechanisms: photoionization heating plus recombination cooling, and Compton heating and cooling due to electron scattering. The low-ionization models are those for which the illumination is relatively low. In this case, photoionization is the dominant process that heats the illuminated layers of the slab. Because the heating rate due to photoionization is essentially given by the radiation field flux times the photoelectric opacity of the gas, and since the latter is dominant at energies below ∼10 keV, a very hard input spectrum will produce much less heating than a soft spectrum, owing to the lack of low-energy photons. The contrary is true in the high-ionization regime where the dominant process is Compton heating and cooling. As discussed in Section 3.1 of García & Kallman (2010), in this limit the gas temperature approaches an asymptotic value, the Compton temperature, given by

$$\begin{equation} T_C = \frac{\langle E\rangle }{4k}, \end{equation} \tag{ 11 }$$

where

$$\begin{equation} \langle E\rangle = \frac{\int F(E){EdE}}{\int F(E){dE}} \end{equation} \tag{ 12 }$$

is the mean photon energy, which is a quantity that only depends on the spectral shape. In Figure 8, we show the resulting Compton temperature as a function of the photon index Γ in the range of our calculations, which agrees very well with the temperature of the hot layer in the high-ionization models shown in panel (d) of Figure 7.¹⁰ Physically this makes sense, since for the input spectra with low Γ values most of the photons are concentrated in the high-energy part of the spectrum, where Compton scattering becomes very important. Therefore, in the high-ionization regime, hard spectra are more efficient in heating the illuminated layers of the slab.

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The elemental abundances considered in a photoionization calculation can potentially affect the ionization balance, temperature structure, and ultimately the observable spectral features in the reprocessed radiation. The total amount of a particular element changes the continuum opacity, which in turn affects the photoionization heating rate. At the same time, the abundance of a particular element influences the strength of the emission and absorption features due to bound–bound and bound–free transitions. Given the relevance of the Fe emission in the analysis of the X-ray spectra from accreting sources, we have carried out calculations in which the Fe abundance, normalized to its solar value, is varied between sub-solar, solar, and super-solar values. All the other elements considered in these calculations are set to their solar values. Figure 9 shows a comparison of the reflection calculations for different values of the iron abundance A_Fe. Left panels show the temperature profiles, while right panels show the corresponding reflected spectrum in the 10–10⁵ eV energy range. In all the panels, each curve corresponds to one particular value of A_Fe = 0.5, 1, 5, and 10, where A_Fe = 1 corresponds to 2.5 × 10⁻⁵ of Fe with respect to H (Grevesse et al. 1996). In each one of the right panels, the plotted spectra have been rescaled for clarity. The scaling factors are, from bottom to top, 10⁻², 1, 10², and 10⁴. Top, medium, and bottom panels correspond to ionization parameters ξ = 10, 10², and 10³, respectively. The photon index is set to Γ = 2 in all these models. The general tendency is the same in all these simulations. The increase of the Fe abundance induces more heating in the illuminated layers due to the increase in the opacity (and thus a larger photoionization rate), which raises the gas temperature. However, because continuum absorption is also increased, the radiation field thermalizes at a smaller depth for the high-abundance models, as can be seen from the temperature curves. Note that the increase in temperature is more subtle in the high-ionization case (top panel), since in this regime the Compton heating and cooling is the dominant process that controls the gas temperature.

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The effects of the Fe abundance are evident in the reflected spectra as well, both in the emission and the absorption features. The 10²–10⁴ eV energy range clearly shows a substantial reduction of the flux due to the increase in the continuum opacity; meanwhile, the Fe K-edge near 8 keV grows deeper as A_Fe becomes larger. At the same time, all the Fe emission features are affected as well, which is mostly evident in the Fe K emission complex in the 6–8 keV region. The strength of the whole emission profile increases when the abundance is high, as expected. In the high-ionization models (top panel), there is a distinctive RRC profile right before the Fe K-edge, which becomes substantially more prominent for A_Fe = 10. Also, there are enhanced emission lines near 100 eV and just above 1 keV, which correspond to M- and L-shell transitions in iron, respectively. However, these emission profiles are noticeable only for the ξ = 100 model (middle panel), in particular the L-shell. The reason for this is that these transitions occur in a rather narrow range of ionization stages. If the gas is very neutral, the photoelectric absorption reduces considerably the number of ionizing photons at those energies, reducing the number of excitations from the L-shell. On the contrary, if the gas is hot and ionized, the fraction of Fe ions with L-shell electrons is very low (i.e., Fe i–xvi), as most of them are stripped already. This is of great relevance, since this feature can be used to constrain both the ionization of the gas as well as the iron abundance. One particular example is the Seyfert 1 galaxy 1H 0707-495, which X-ray spectrum shows evidence of a very intense emission in both the Fe K- and L-shell regions. Fits using reflection models require a high Fe abundance at an ionization parameter consistent with the present analysis (Fabian et al. 2012; Dauser et al. 2012).

Although somewhat extreme, 1H 0707-495 is not the only case where Fe is found to be overabundant based on predictions from reflection models. In fact, at least for AGNs, it is commonly the case that super-solar iron abundance is required to fit the observed X-ray spectra (Fabian 2006). A possible explanation for the apparent extreme Fe abundance in some AGNs has been proposed by Reynolds et al. (2012), on the grounds of radiative levitation in the accretion disk. If the radiative force exerted on an Fe ion exceeds the vertical gravity, this could cause iron to diffuse toward the photosphere of the disk, enhancing its abundance. But, in general, there is no particular reason why the other elements should be considered at their solar values, except the simplicity inherent in such an approximation. Given the capability of xillver to treat any particular choice of elemental abundances, it can be used to produce smaller set of models custom made for any specific situation. This could be of great use in peculiar systems such as ultra-compact X-ray binaries, where a prominent O vii Kα emission line observed in the X-ray spectra is thought to originate from reflection in an accretion disk overabundant in oxygen (Madej et al. 2010; Madej & Jonker 2011). Also, emission lines from H-like S, Ar, and Ca ions detected in some low-mass X-ray binaries suggest reprocessing material with compositions different from solar (D'Aì et al. 2009; di Salvo et al. 2009; Egron et al. 2012).

Figure 3 shows the great complexity of the reprocessed spectrum emerging from an illuminated, optically thick slab. Meanwhile, the models presented here constitute a high-resolution representation of only one of the components observed in the X-ray spectrum of accreting sources, namely, the reflected component. In reality, one observes a composite spectrum that includes the original power-law (presumably coronal) component plus possibly a thermal blackbody-like component. Additionally, if the reflection occurs within a few gravitational radii from the compact object, Doppler and gravitational redshifts will smear the spectral profiles. Finally, absorption due to intervening gas such as warm absorbers and outflows can also be present. We shall discuss some of the most prominent and representative of these features that one expects to observe given the capabilities of current detectors.

3.5.1. The Fe K-shell Emission

Undoubtedly, the emission complex near 6–7 keV, which is due to transitions from the inner K-shell of Fe ions, is the most prominent atomic feature in the X-ray spectrum of accreting sources. It is this feature that provides the clearest evidence for reflection of high-energy photons in the relatively cold, optically thick material of an accretion disk. The ubiquity of this feature has been established observationally for a large number of sources (e.g., Gottwald et al. 1995; Winter et al. 2009; Ng et al. 2010; Fukazawa et al. 2011) for the following two reasons. First, the fluorescence yield, i.e., the probability of emission of a photon rather than an Auger electron, is proportional to Z⁴, where Z is the nuclear charge. Second, the Fe K-shell lines are emitted in a clean region of the X-ray spectrum (6–8 keV), where few other ions emit or absorb radiation. Furthermore, in this energy range galactic absorption is negligible and most detectors operate quite effectively.

Figure 10 shows a compilation of all the radiative transitions from Fe ions in the 6–10 keV energy range that are included in our database¹¹ (cf. Figure 3 in Kallman et al. 2004). There are a total of 2735 lines, all of which correspond to K-shell transitions. The open circles show the line energy plotted against the ionization stage of each ion, as determined by the relation Z − N_e + 1, where Z is the nuclear charge and N_e is the number of electrons in the ion. Thus, an ionization stage of 2 corresponds to Fe ii (single ionized), 3 to Fe iii (double ionized), and so forth. Filled circles show the most intense transitions, i.e., those with large transition probabilities (here we have chosen A_r > 10¹³ s⁻¹). Ionization increases upward: the H- and He-like ions are near the top and the neutrals near the bottom. The big group of points to the left corresponds to the Kα transitions (n = 1 → 2, with n being the principal quantum number), and the smaller group to the right (for which most energies are above 7 keV) corresponds to the Kβ transitions (n = 1 → 3). Note that what is commonly referred to as the neutral Fe K line at 6.4 keV is in fact a combination of several transitions that span the energy range 6.39–6.43 keV; this feature can be produced by many ions ranging from Fe i up to Fe xvii. For Fe xviii and more ionized ions, the transition energy spread is larger and the average energy moves monotonically toward higher energies as the ionization stage increases. For the He- and H-like ions (Fe xxv–xxvi), there are only a few lines with energies around 6.9 keV. Notice also that the Kβ transitions are only produced up to Fe xvii, since for higher ionization stages the n = 3 shell is empty. The line energy for the Kβ transitions varies between ∼7.04 keV for Fe ii up to ∼7.19 keV for Fe xvii.

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It is these line energies and intensities that we use to analyze the emission in the Fe K region of the reflected spectra. Figure 11 shows, in the 5–8 keV band, the reflected spectra for Γ = 2 and A_Fe = 1 for different values of the ionization parameter in the range ξ = 5–5 × 10³. Ionization increases downward in the figure. Starting at the lowest level of ionization (ξ = 5–50), the first few models show similar groups of Fe lines, with two distinctive emission lines centered at ∼6.4 keV and ∼7.1 keV, which correspond respectively to the Kα and Kβ lines. The line energies indicate that the emission is dominated by Fe ions with ionization stages lower than Fe xvii. The smaller feature below 6.4 keV, which is produced by Compton downscattering of line photons, is usually referred to as the Compton shoulder. For models with ξ ≳ 200, the Kβ line is no longer visible, which suggests that the emission is dominated by ions at higher ionization stages. In particular, models with ξ = 200 and 500 show a rich complex of emission lines at energies between 6.4 to 6.7 keV. At ξ = 10³, the emission is centered at ∼6.7 keV and ∼6.9 keV, which implies that the gas is highly ionized and most of the Fe ions are hydrogenic or He-like. In this state the temperature is high, and the overall line emission profile is thermally broadened into a symmetric profile via Compton scattering. Kallman et al. (2004) present a more detailed but similar analysis of the Fe K emission complex based on xstar simulations, which includes atomic data for Fe; however, they do not consider energy redistribution due to Compton scattering.

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3.5.2. Lower-Z Elements

In addition to the important Fe K emission complex in the reflected spectrum, our code includes many emission lines of astrophysically relevant ions at lower energies, which are due to such elements as C, N, O, Ne, Mg, Si, S, Ar, and Ca. The intensities of these lines tend to increase as the illuminating spectrum softens (i.e., as Γ increases) and as the flux correspondingly increases at lower energies where photoelectric absorption dominates. Figures 12–14 show the reflected spectra for Γ = 3 and A_Fe = 1 and for all the values of the ionization parameter considered in our library. The spectra cover the 0.3–4.5 keV energy range, which is the band that contains most of the inner-shell transitions from low-Z ions. At the top of each figure is indicated the energy of the strongest Kα emission line for each ion in the isonuclear sequence of each of the elements contained in our atomic database (except for the neutral and single-ionized cases). While these Kα lines are not at all the only lines considered in our models, they nevertheless give an indication of the energy range one expects most of the features of a given element to appear.

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Figure 12 shows spectra for the lowest ionization models, namely, ξ = 1, 2, 5, 10, and 20 (with ionization increasing from bottom to top). At the lowest energies (0.3–1 keV), the lines are indistinguishably blended together. At higher energies, emission lines from low ionization states of Mg, Si, and S are evident, and relatively weak lines due to Ar and Ca Kα are also present. The RRC due to H-like Ne at ∼1.4 keV becomes visible in models with ξ ≳ 5. Such features are produced by recombining electrons with energies that exceed the ion binding energy. The excess energy is radiated as a photon. The energy of a typical RRC photon is E = E_IP + kT_e, where E_IP is the ionization potential of the ion and T_e is the electron temperature.

For higher ionization parameters, the reflected spectra are dominated by RRC features, as shown in Figure 13, which displays spectra ranging from ξ = 50 at the bottom to ξ = 10³ at the top. Also visible are higher-excitation emission lines of Mg, Si, S, and Ar. The H-like oxygen RRC at ∼0.87 keV is quite strong for ξ = 50 (bottom curve), and it weakens as ξ increases. Strong emission features due to Kα transitions of Mg, Si, and S appear at ∼1.4, 1.8, and 2.45 keV, respectively. The latter feature corresponds to a blend of the S xv Kα line and the Si xiii RRC (García et al. 2011).

Figure 14 shows the last five models of the series with the ionization parameter increasing from ξ = 2 × 10³ (bottom) to ξ = 5 × 10⁴ (top). For these large values of ξ, fewer emission lines are present. However, they are quite intense because the soft illuminating spectrum is rich in low-energy photons. The strongest emission lines are produced by H-like ions of O, Mg, Si, and S with energies of ∼0.654, 1.47, 2.01, and 2.63 keV, respectively. The RRC from H-like Si and S are clearly seen at ∼2.67 and 3.5 keV, respectively. Generally, Kα fluorescence from the low ionization states of O, Mg, Si, S, Ar, and Ca ions is most important at low column depths because of the higher K-shell opacity of these elements relative to Fe. Conversely, Fe Kα is more important at relatively high columns, i.e., τ_T ∼ 1.

Considering the several reflection models currently available, reflionx (Ross & Fabian 2005) is the one that is most similar to the models presented in this paper. It has been widely used by the scientific community in analyzing spectral data for many observations of various black hole and neutron star sources. We therefore benchmark our results by making a detailed comparison of xillver and reflionx, while describing the advantages of our model.

In reflionx, the radiation field is separated into the direct component of the illuminating radiation, which is unaffected by either scattering or absorption (assumed to be $\propto e^{-\tau _{\nu }}$), and the diffuse component, which results from Compton scattering and emission within the slab. The Compton processes are treated by solving the Fokker–Planck diffusion equation, which includes a modified Kompaneets operator (Ross et al. 1978; Ross 1979). Incident photons that are Compton scattered in the slab contribute to the emissivity of the diffuse field, and it is assumed that their distribution in energy is described by a Gaussian with central energy and energy dispersion given by Equations (6) and (7). Due to the deficiency of the Fokker–Planck equation in handling steep gradients in the energy spectrum, resonance lines are treated using the escape probabilities technique. In addition to fully ionized species, the Ross & Fabian (2005) calculations include C iii–iv, N iii–vii, O iii–viii, Ne iii–x, Mg iii–xii, Si iv–xiv, S iv–xvi, and Fe iv–xxvi. However, none of the neutral or near-neutral ions are included in their models. The cross sections for photoionization are calculated from the fits of Verner & Yakovlev (1995); in the case of Fe, transition probabilities and Auger decay rates are taken from Kaastra & Mewe (1993). The items we have just highlighted are among the most important differences between the atomic data employed in the two models (see Section 2.1). It is also important to mention that the Fe Kα lines treated in reflionx are the recombination lines of Fe xxvi and Fe xxv, and the fluorescence lines of Fe vi–xvi, while the Kα fluorescence of Fe xvii–xxii is assumed to be suppressed by autoionization. All Kβ and higher n resonances are also neglected.

The solar abundances of the elements in reflionx are taken from Morrison & McCammon (1983), while in xstar, and thus in xillver, we adopt the values of Grevesse & Sauval (1998). Table 1 shows the values reported from both sources for each element, and the last column shows the ratio of the two values. Apart from N and Ar, all the abundances used in xillver are lower than those used in reflionx. In particular, O is lower by ∼10% and Fe by ∼30%, while, most notably, Ne is ∼80% lower. The choice of abundance model is relatively unimportant, as illustrated in Figure 15, which shows reflected spectra computed using xillver for ξ = 10 (left panel) and ξ = 10³ (right panel). Results obtained using the model of Grevesse & Sauval (1998) are shown in black, and those obtained using the model of Morrison & McCammon (1983) in red. The differences in each panel are small. For the higher Morrison & McCammon (1983) abundances, the continuum is slightly depressed and some of the strongest lines are somewhat weaker. These small differences are unlikely to be important in analyzing real data.

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Although the solar abundances assumed for both reflionx and xillver are fixed and different, both models allow the Fe abundance to be varied. This allows us to compensate for the abundance differences in making direct comparisons of the two models. One such set of comparisons is shown in Figure 16, where the emergent spectra for different values of both the ionization parameter and the photon index are plotted for xillver (in black), and reflionx (in red). Top, middle, and bottom panels are for Γ = 1.4, 2, and 2.6, respectively. Each panel shows three pairs of curves, corresponding to ξ = 10, 10², and 10³, from bottom to top, respectively. The spectra are from the FITS tables that are accessible via the atable model in XSPEC. To ensure that the energy resolution is the same for all the spectra, in all cases we used a logarithmic dummy response of 1000 energy points over the energy range 0.1–1000 keV. Furthermore, appropriate grid point were chosen in the parameter space to avoid problems that might be introduced by the interpolation procedure. The normalization for both models is the same because we use consistent definitions for the ionization parameter and the power-law spectrum. Thus, all the parameters were set to be identical for the two models, with the exception of the iron abundance: it was set to A_Fe = 1.32 in xillver in order to compensate for the difference in the solar values assumed in each model, as discussed above.

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In general, the two models are in better agreement for low ionization and soft spectra, as in the models for Γ = 2.6 shown in the bottom panel of Figure 16. However, even in these cases there are important differences: for the bottom spectra with ξ = 10, the xillver model is more absorbed at low energies than the reflionx model. While the energies and intensities of the strongest lines are in good agreement, some weaker features at ∼2 keV and ∼3 keV are only present in the xillver spectrum, probably due to the differences in the atomic data. These particular features are much stronger for the harder spectra shown in the middle and top panels. The differences in the atomic data also affect the Fe K emission profile near 6–8 keV. For example, Fe Kα emission at ∼6.4 keV is somewhat more intense in our model and Kβ is conspicuously absent in the reflionx spectra. Concerning the Fe continuum, one might expect significant differences because the models use very different photoionization cross sections. However, the models agree very well on the depth of the Fe K-edge near 7.2 keV, especially at low ionization. As mentioned above, the discrepancies are most pronounced for high ionization and for the hard spectra shown in the top panel (Γ = 1.4). The difference for the pair of curves for ξ = 10³ is remarkable: the xillver continuum below ∼1 keV is at least an order of magnitude fainter than the reflionx continuum; the extremely intense O Lyα emission line at ∼0.65 keV in our spectrum is scarcely present in the reflionx spectrum; and the Fe K line in the latter spectrum is much broader than in the former.

These large discrepancies for high ionization and hard spectra are unexpected and there is no ready explanation. The lack of emission lines from low-Z elements suggests that (for the same input parameters) the slab is hotter and more ionized for reflionx than for xillver. We test this idea by comparing the performance of the two models for different values of the ionization parameter: in Figure 17 we compare the two Γ = 1.4 models where for reflionx we have doubled the normalization and reduced the ionization parameter to ξ = 500 (half the value for xillver), in order to match the fluxes at the higher energies. The shape and intensity of the Fe K line is now in better agreement. However, the reflionx continuum is still considerably higher at low energies; as a consequence, the reflected spectrum is even softer than the incident power-law spectrum.

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In order to test whether our code is reliably modeling this low-energy continuum, we compared our results with those computed using the apec model (Foster et al. 2012). For this comparison, with xillver we computed the reflection spectrum from a thin slab (τ_T = 10⁻²) for solar abundances at very low ionization with the temperature held fixed at 10⁶ K. These parameters closely match the case of a collisionally ionized gas whose spectrum is dominated by bremsstrahlung emission. Figure 18 shows the comparison. Setting aside the differences in the emission lines (a result of the differences in the atomic data sets), the spectra are in good agreement, especially the level of the continuum. This comparison gives us confidence in xillver's implementation of the free–free emissivity.

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An important quantity to compare these models is the EW of the Fe K emission complex. Following García et al. (2011), we use the well-known formula,

$$\begin{equation} {\rm EW}= \int _{5.5\, \mathrm{keV}}^{7.2 \,\mathrm{keV}}\frac{F(E) - F_c(E)}{F_c(E)}dE, \end{equation} \tag{ 13 }$$

where F(E) is the flux of the reflected spectrum and F_c(E) is the flux in the continuum. The 5.5–7.2 keV range of integration covers all Fe emission features, including the Kβ lines. The continuum is approximated by a straight line that passes through the endpoints of the energy band. Clearly, this straight-line continuum and this particular choice of integration limits are somewhat arbitrary and by no means constitutes an accurate determination of a physical quantity, but it suffices to compare the two models. Figure 19 shows plots of the EWs versus the ionization parameter for both xillver and reflionx, with Γ increasing from top to bottom. For both models, the EWs tend to decrease as the ionization increases. At low ionization, the EWs are similar, although the values for xillver are consistently higher, which is expected because the reflionx model lacks the Fe Kβ lines. Meanwhile, at high ionization (ξ ≳ 10³), where the Fe K emission is dominated by the H- and He-like ions, the agreement in the EWs is also good, but values for the reflionx model are in this case somewhat greater for large Γ (bottom panel).

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Significant discrepancies appear when one compares the two models at higher levels of ionization, 10² ≲ ξ ≲ 10³: the EWs for the reflionx models decrease drastically, especially for the softer spectra (bottom panel). This behavior may occur because reflionx does not include the Fe Kα lines for most of the second-row ions (Fe xvii–xxii) under the assumption that these lines are suppressed by Auger resonant destruction (Ross et al. 1996). xillver, on the other hand, predicts strong Kα emission at the energies at which many of these ions are expected to emit (e.g., see Figures 10 and 11). xstar, and consequently xillver, automatically takes into account the effects of Auger decay by accurately calculating the branching ratios for Auger versus fluorescence emission; however, it does not include resonant scattering of the lines. This omission seems reasonable because it is plausible that velocity gradients or turbulence in an accretion disk will remove photons from the line core, thereby reducing resonant scattering. Nevertheless, because we disregard this mechanism we may be overpredicting the line intensity for some models. As one additional complication, Liedahl (2005) has shown that Auger resonance destruction is a selective process that highly attenuates only a limited subset of Fe Kα lines. We conclude that further analysis is required in order to accurately quantify the emission from L-shell Fe ions.

We have also compared our results to those obtained using the reflection models pexrav and pexriv (Magdziarz & Zdziarski 1995), which compute the reflected spectrum from a completely neutral disk and an ionized disk, respectively. These models are an implementation of a semi-analytic Green's function (Lightman & White 1988) that models Compton reflection of X-rays by cold electrons while including bound–free continuum opacity. Fluorescence emission line are however not included. The left panel of Figure 20 compares the reflected spectrum predicted by xillver with that predicted by pexrav for an incident power-law spectrum with Γ = 2 and an exponential cutoff at 300 keV. Because pexrav models a completely neutral gas, we use the lowest ionization parameter considered in xillver, namely, ξ = 1. The two models agree fairly well in the level of the reflected continuum, in particular near the Fe K line region. The edge energy and EW of the feature at ∼7.2 keV are similar. Our model shows more absorption than pexrav in the low-energy part of the spectrum. However, the xillver model cannot be considered completely neutral; there is some heating near the surface (see the leftmost curve in the middle panel of Figure 4).

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The right panel in Figure 20 compares a reflected spectrum from an ionized disk (ξ = 10³) computed using xillver with one computed using pexriv. At the outset, we note that pexriv, unlike both xillver and reflionx, does not solve for the ionization balance of the slab. Instead, pexriv assumes an isothermal medium with a maximum temperature of T = 10⁶ K. Both the ionization parameter and the gas temperature are allowed to vary. This approach results in the large discrepancies that are evident in both the level of the continuum and the edge features. The level of the continuum is similar only for energies below 1 keV. For higher energies, the pexriv spectrum is highly absorbed and its Fe K-edge is much deeper. This comparison demonstrates for the case of an ionized reflector the importance of accurately modeling the ionization balance. pexriv's assumption of an isothermal gas is a poor one for large values of the ionization parameter. Another limitation of pexriv that was pointed out by Fabian & Ross (2010) is the lack of blurring effects of Compton scattering, which smears the K-edges.

We now discuss in turn the differences at high energies (≳ 20 keV) between our results and those delivered by reflionx and pexrav/pexriv. At these energies, photoelectric absorption is negligible, and the only source of opacity is Compton scattering. Furthermore, the gas temperature has only a very small effect on the Comptonized spectrum. Therefore, any differences in the model spectra can be attributed to how each model treats scattering of photons by cold electrons. xillver implements a Gaussian convolution for all energies and at all depths (Section 2). reflionx uses this approximation for the lines only, while using a modified Kompaneets operator to treat the diffuse radiation.

We tested the accuracy of our approximation by comparing with results obtained using an MC code, which we treat as the gold standard. This code, which extends the work of Matt (2002) and Magdziarz & Zdziarski (1995), simulates Compton scattering and radiative transfer through neutral gas for a semi-infinite slab (Eikmann et al. 2012). Included in this code are photoabsorption for all elements with Z ⩽ 30 and fluorescent line emission using the fluorescent yields from Kaastra & Mewe (1993). Compton scattering is modeled using the proper relativistic scattering dynamics and the differential Klein–Nishina cross section.

Figure 21 compares reflected spectra computed using xillver and the MC code for Γ = 2, ξ = 1 and solar abundances. The result for reflionx is also shown for these same parameters. Overall, there is a good agreement between the three models. At energies ≲ 10 keV, xillver and the MC simulation predict very nearly the same level of the continuum and the same spectrum of intense emission lines. For example, the ∼6.4 keV Fe Kα line has a very similar intensity and shape, although the MC code predicts a ∼7.2 keV Kβ line that is somewhat more intense. The main differences occur above the ∼7.2 keV Fe K-edge, in the continuum feature referred to as the Compton hump. The MC simulated spectrum has a somewhat shallower Fe K-edge than the other two spectra, and its ∼20–60 keV is somewhat fainter. Above ∼100 keV, the reflionx spectrum is the better match to the MC spectrum.

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As a bottom line, the differences between the MC simulation and either xillver or reflionx are not larger than 30%, which is reasonable given the simplistic approximation used in those models. However, this level of performance may not be adequate for the analysis of such current and future X-ray missions as NuSTAR (Harrison et al. 2010) and eROSITA (Predehl et al. 2011). Furthermore, the Compton-hump spectrum can provide important constraints on disk inclination. Therefore, our goal is to improve this aspect of the code in future versions of xillver.