X-Ray Spectroscopy in the Microcalorimeter Era. III. Line Formation under Case A, Case B, Case C, and Case D in H- and He-like Iron for a Photoionized Cloud

Microcalorimeter X-ray missions like Hitomi and the upcoming missions XRISM and Athena will provide unprecedented spectroscopic resolution. The Soft X-Ray Spectrometer (Kelley et al. 2016) on board Hitomi (Hitomi Collaboration et al. 2016) resolved the Fe xxv Kα complex in four components for the first time. A plethora of high-resolution X-ray data from these missions will be available within the next few decades. Interpreting these high-resolution spectra requires a clear understanding of the line formation processes in the X-ray emitting plasma.

Line formation in gaseous nebulae was first studied in the 1930s in a series of papers by Menzel (1937), Menzel & Baker (1937), Baker & Menzel (1938), and Baker et al. (1938) for the formation of optical H i lines. Two limiting cases were discussed: "Case A" and "Case B." Case A occurs when the nebula is optically thin and the line photons emitted by recombination escape the cloud freely. Case B occurs if the nebula is optically thick and the line photons scatter multiple times in the cloud. Higher-order Lyman lines are converted into Balmer and Lyα (or Kα) photons or into a two-photon continuum. Note that in their study, the source of the radiation was assumed to be of stellar origin. Stars in the gaseous nebulae might have strong Lyman absorption lines in the spectral energy distribution (SED), and there is almost no continuum pumping. This will be relevant to the discussion later.

A third case occurring in optically thin irradiated clouds, "Case C," was introduced by Baker et al. (1938) and later followed up by Chamberlain (1953) and Ferland (1999). In Case C, lines escape the cloud freely as in Case A. But unlike Case A, the Case C spectrum is enhanced by continuum pumping.

Some recent studies (Luridiana et al. 2009; Peimbert et al. 2016) have discussed a fourth case, "Case D," which occurs in optically thick irradiated systems. Similar to those in Case B, line photons scatter multiple times in Case D before escaping the optically thick cloud. But unlike Case B, the Case D spectrum is enhanced by continuum pumping.

Most of the previous works on Cases A, B, C, and D, both theoretical and observational, focused on the optical, ultraviolet, and infrared regimes (Menzel 1937; Menzel & Baker 1937; Baker & Menzel 1938; Baker et al. 1938; Chamberlain 1953; Soifer et al. 1981; Malkan & Sargent 1982; Hummer & Storey 1987; Keel & Windhorst 1991; Ferland 1999; Sánchez et al. 2007; Stelzer et al. 2012; Mennickent et al. 2016; Peimbert et al. 2017), with a small number of studies on X-rays—Storey & Hummer (1988, 1995) for one-electron ions and Porter & Ferland (2007) for two-electron ions. Some other previous studies on soft X-ray spectra are Paerels & Kahn (2003), Bianchi et al. (2005), Cappi et al. (2006), Guainazzi & Bianchi (2007), and Mao et al. (2018). Note that Kinkhabwala et al. (2002) have outlined many of the physical processes discussed in this paper, focusing on second- and third-row elements.

The purpose of our paper is to describe improvements to the widely disseminated code Cloudy with simultaneous radiative transfer and ionization solutions. This paper presents diagnostic diagrams making it possible to measure column densities from line intensities. Here, we discuss the line formation processes for the four Menzel and Baker cases—Cases A, B, C, and D for H-like and He-like iron in photoionized plasma. This will be essential for interpreting future high-resolution microcalorimeter observations in the presence of a photoionizing source.

This paper is the third in the series "X-Ray Spectroscopy in the Microcalorimeter Era," the first two papers of which discuss the atomic processes in a collisionally excited plasma. Chakraborty et al. (2020b) discuss line interlocking and resonant Auger destruction (Ross et al. 1978; Band et al. 1990; Ross et al. 1996; Liedahl 2005) and electron scattering escape (ESE) in the Fe xxv Kα complex. Chakraborty et al. (2020c) discuss the Case A to B transition in H- and He- like iron. The present paper explores photoionized X-ray plasma with Cloudy (Ferland et al. 2017) for a power-law SED. Note that the results shown in all three papers of this series apply in the coronal limit, although the formalism will go to equilibrium in high densities. Figures 10–12 in Ferland et al. (2017) display the coronal limit for collisionally ionized and photoionized cases. For iron (Z = 26), the coronal limit applies to electron densities lower than 10¹⁶ cm⁻³.

The organization of this paper is as follows. Section 2 discusses the theoretical framework of Cases A, B, C, and D. Section 3 lists the simulation parameters used for our calculations. Section 4 describes the results. Section 5 discusses the total emitted spectrum within the energy range 0.1–10 keV. Section 6 describes the effects of background continuum opacities like electron scattering opacity. Section 7 discusses our results. We refer to the transitions going from n = 2, 3, 4 to n = 1 in H-like iron as Lyα, Lyβ, and Lyγ and those in He-like iron as Kα, Kβ, and Kγ. Transitions going from n = 3 to n = 2 are called Hα in H-like iron and Lα in He-like iron. This nomenclature is inspired by the Siegbahn notation (Siegbahn 1916) as implemented in, for instance, Gabriel (1972), Fukumura & Tsuruta (2004), and Koyama et al. (2007).

Lines are formed under Case A, Case B, Case C, or Case D conditions. Case A and Case B occur in collisionally ionized clouds. Case A, Case B, Case C, and Case D occur in photoionized clouds. Figure 1 shows the four cases for the two ionizing conditions. Line formation processes in collisionally ionized clouds have been discussed in the first two papers of this series. This paper solely focuses on photoionized clouds.

Zoom In Zoom Out Reset image size

A schematic representation of all four cases is shown in Figure 2 for a simplified three-level system. Low-column-density (optically thin) regions can be described by Case A or Case C depending on the ionizing radiation. In both cases, line photons escape the cloud without any scattering. Case A occurs when the continuum radiation hitting the cloud has strong absorption features in the Lyman lines. There is no continuum pumping in the Lyman lines emitted by the cloud. Lines in the Case A limit are solely formed by radiative recombination and cascades from higher levels. ³ Case C occurs if the continuum source striking the optically thin cloud does not contain Lyman absorption lines and the emitted Lyman lines are enhanced by continuum pumping. As a result, the Case C spectrum is always brighter than Case A.

Zoom In Zoom Out Reset image size

The ratio of the Case C to the Case A line intensities can be calculated from the ratio of the continuum pumping rate to the rate of recombination (r). In equilibrium, r is equal to the rate of photoionization (Γ_n ):

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{n}={\int }_{{\nu }_{o}}^{\infty }\displaystyle \frac{4\pi \,{J}_{\nu }}{h\nu }\ {\alpha }_{\nu }\ d\nu \quad ({{\rm{s}}}^{-1})\end{eqnarray} \tag{ 1 }$$

where J_ν (erg cm⁻² s⁻¹ sr⁻¹ Hz⁻¹) is the mean intensity per unit frequency per unit solid angle of the incident radiation and α_ν is the photoionization cross section (cm²) for the atom/ion by photons of energy h ν.

The rate of continuum pumping of a line from level l to level u is given by

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{l}={B}_{{lu}}{J}_{l}={f}_{{lu}}\displaystyle \frac{\pi {e}^{2}}{{mc}}\left(\displaystyle \frac{4\pi \,{J}_{l}}{h\nu }\right)\quad ({{\rm{s}}}^{-1})\end{eqnarray} \tag{ 2 }$$

where B_lu is the Einstein coefficient, f_lu is the oscillator strength, and the other symbols have their usual meanings.

For a power-law (f_ν ∝ ν⁻¹) model in H-like iron for the Lyα transition in a simple two-level system,

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Gamma }}}_{l}}{r}\sim 16.\end{eqnarray} \tag{ 3 }$$

This implies that Lyα line intensities are ∼16 times enhanced in Case C as compared to Case A. The calculated ratio is approximate, as the real calculation will have many pumping lines and many different branching ratios. This ratio is approximately in agreement with the line intensities listed in Table 1 obtained from our Cloudy simulations, which shows that the Case C Lyα in H-like iron is ∼10 times enhanced as compared to Case A.

In contrast, Case B occurs in the high-column-density limit (N_H ≥ 10^21.5 cm⁻²). In this limit, Lyman line optical depths are large enough for photons to undergo multiple scatterings and so are converted into Hα (or Lα) and Lyα (or Kα) photons or into a two-photon continuum. The cloud becomes self-shielding, stopping the continuum pumping despite the presence of a continuum radiation source in the cloud. Typically, Case B describes the line formation in most observed optically thick nebulae (Osterbrock & Ferland 2006).

Case D occurs if Lyman line optical depths are large but the cloud does not entirely become self-shielding to the external radiation. Luridiana et al. (2009) argued that a real nebula in the optically thick limit would be better represented by Case D than by Case B. Their study reported a significant contribution of continuum pumping to Balmer emissivity in the H ii region. As far as we know, Case D has not been studied in the X-ray to date. Our calculations for the X-ray regime in the optically thick limit show substantial enhancement in the Lyman and Balmer line intensities in H- and He-like iron. This will be further elaborated in Section 4.4.

This section discusses the simulation parameters used in Cloudy. We aim to establish a standard mathematical framework of line formation through Case A, Case B, Case C, and Case D. The simulation parameters have been chosen accordingly. All the simulations are done using the development version of Cloudy with a hydrogen density of 1 cm⁻³. To make the simplest case, the shape of the incident radiation field is assumed to be a power-law SED:

$$\begin{eqnarray}&&{f}_{\nu }\propto {\nu }^{-\alpha }\end{eqnarray} \tag{ 4 }$$

with α = 1.

The intensity of the radiation field is characterized by the ionization parameter (U), defined with the following ratio:

$$\begin{eqnarray}&&U=\displaystyle \frac{{{\rm{\Phi }}}_{{\rm{H}}}}{{n}_{{\rm{H}}}\,c}\end{eqnarray} \tag{ 5 }$$

where Φ_H is the flux of hydrogen-ionizing photons, n_H is the hydrogen density, and c is the speed of light.

Much of the X-ray literature uses the ξ ionization parameter defined by Kallman & Bautista (2001). For our α = 1 SED, ξ = 1 corresponds to a U of 0.01767.

Figure 3 shows the variation of different ionization stages in iron with the log of U. The top x-axis shows the log of ξ. The left panel of the figure shows a linear plot, and the right panel shows a log plot. We choose log U = 2 (log ξ = 3.75) to maximize the quantity of H- and He-like iron in the cloud. This also minimizes the overlap between He- and Li-like iron ions, as Li-like iron selectively changes the He-like spectra by line interlocking (Chakraborty et al. 2020b). The linear figure also shows our choice of ionization parameter with a black vertical line. Note that our choice of ionization parameter is in agreement with the range of ξ for highly charged iron mentioned in Kallman et al. (2004; log ξ ≥ 2), who also discussed K lines in iron in a photoionized cloud for a power-law SED.

Zoom In Zoom Out Reset image size

The cloud temperature (T) is obtained from the energy equilibrium equation from a heating–cooling balance to replicate the actual physical temperature of a photoionized cloud. The computed temperature is T ∼ 6 × 10⁶ K at log U = 2. The variation of T with log U is shown in Figure 4. Note that photoionized clouds are a lot cooler than collisionally ionized clouds. In fact, this equilibrium temperature is about 8 times smaller than that of the collisionally ionized plasma considered in the first two papers of this series.

Zoom In Zoom Out Reset image size

This section explores different conditions for lines to form in the presence of a photoionizing source emitting in X-rays. Table 1 shows a comparison between selected line intensities (including the Kα complex) for Case A, Case B, Case C, and Case D conditions for H- and He-like iron. The quantity listed in the table is the line intensity (I) per unit thickness (d), I/d. As the line intensities increase as the cloud's size/thickness increases, I/d tracks the scaled change in the line intensities for all four cases or the transition between them. Although I/d has the same units as the emissivity of a line, 4π j, it does not have the same physical interpretation. All the I/d listed in Table 1 are observed in nature except for Case B_classic, which will be further discussed in Section 4.2. The line wavelengths listed in the table are taken from the National Institute of Standards and Technology ⁴ (version 5.8; Kramida et al. 2018).

The low-column-density (optically thin) limit represents Cases A and C, and the high-column-density (optically thick) limit represents Cases B and D. Therefore, the I/d listed in the table for Cases A and C are computed at N_H = 10¹⁹ cm⁻² and those for Cases B and D at N_H = 10²⁴ cm⁻². The continuous variation of I/d with hydrogen column density and the transition from Case A to B and Case C to D are shown in Figure 5.

Zoom In Zoom Out Reset image size

4.1. Case A

4.2. Case B

The top-right panel of Figure 2 shows the Case B condition in a cloud. Continuum pumping is disabled as in Section 4.1. As shown in Figure 5, column densities higher than N_H ∼ 10^21.5 cm⁻² begin to make a transition to the Case B limit. Two types of Case B are shown—Case B_classic and Case B. The dashed and solid lines in the top panel of Figure 5 represent Case B_classic and Case B, respectively. What we refer to as Case B throughout the text considers all the physical processes observed in nature in the X-ray limit, including electron scattering and line overlap. The concept of electron scattering is described later in this section and in Section 6. Case B_classic is the Menzel–Baker Case B values studied in the 1930s, which do not include electron scattering or line overlap. In Cloudy, electron scattering can be disabled with the following command:

no electron scattering.

We find that for X-ray emission from H- and He-like iron, Case B_classic remains a pedagogical scenario.

Table 1 shows the I/d values for Case B_classic and Case B (observed in nature) at N_H = 10²⁴ cm⁻². In the typical Menzel–Baker Case B_classic, as a consequence of the conversion of higher-n Lyman lines into Lyα (or Kα) and Balmer lines, Lyα (or Kα) intensity increases, Lyβ (or Kβ) and higher-n Lyman line intensities decrease, and Hα (or Lα) and higher-order Balmer line intensities increase as compared to the Case A limit. The Case B_classic column in the table exactly reflects this behavior for both H- and He-like iron. For instance in H-like iron, I/d in Lyα increases by ∼32%, Lyβ decreases by ∼80%, and Hα increases by ∼80% at N_H = 10²⁴ cm⁻². In He-like iron, I/d in z increases by ∼40%, x, y, and w increase very slightly (≤4%), Kβ decreases by ∼3–7 times, and Lα increases by ∼3–13 times as compared to Case A.

However, the observed Case B values are quite different from the Case B_classic values. In fact, in H-like iron, all Lyman I/d values decrease as compared to the Case A limit, including Lyα. H-like Lyα decreases by ∼50%. In He-like iron, selected Kα lines (x, y, and w) show a decrease up to ∼24%.

Such a decrease in the line intensities in the observed Case B mainly occurs due to electron scattering, as described in Section 6. When line photons scatter off high-speed electrons, they are largely Doppler-shifted from their line-center. Lines with the largest optical depths are more likely to exhibit a reduction in their line intensities as they are more likely to scatter. Figure 6 shows the optical depth of certain Lyman and Balmer lines in H- and He-like iron. In H-like iron, Lyα, Lyβ, and Lyγ intensities are reduced due to electron scattering because of their large optical depths. In He-like iron, w, y, Kβ, and Kγ intensities are reduced. The optical depth in z is negligible at N_H = 10²⁴ cm⁻², and thus I/d for z is not affected by electron scattering. The reduction in I/d for x is due not to electron scattering but to a series of processes described in the appendix of Chakraborty et al. (2020b). This explains the observed Case B behavior in both Table 1 and Figure 5.

Zoom In Zoom Out Reset image size

4.3. Case C

4.4. Case D

4.5. Case A/Case C to Case B/Case D Transition

What drives the Case A to Case B or Case C to Case D transition in a real astronomical scenario is the variation in column density from the low-column-density (optically thin) to the high-column-density (optically thick) limit. Figure 5 shows these transitions for H-like and He-like iron in a photoionized cloud.

When Cases A and B were first discussed in the 1930s, the source of the radiation was assumed to be stellar with strong Lyman absorption lines and no continuum pumping. Thus, galactic nebulae with strong absorption features show the Case A to Case B transition under the variation in column density. The Case A to Case B transition also occurs in any collisionally ionized cloud, as discussed in the first two papers of this series (Chakraborty et al. 2020b, 2020c). Chakraborty et al. (2020c) showed that the Fe xxv Kα line ratios calculated with Cloudy are in excellent agreement with the line ratios observed by Hitomi (Hitomi Collaboration et al. 2016) for the outer region of the Perseus core (see Figure 14 in their paper). At the best-fit hydrogen column density of the hot gas at the Perseus core (N_H,hot = 1.88 × 10²¹ cm⁻²) reported by Hitomi Collaboration et al. (2018a), line formation processes can be best described by Case A.

In extragalactic environments, such as a cloud photoionized by an active galactic nucleus (AGN) SED with no Lyman absorption lines, the line formation in the low-column-density limit will be described by Case C in the optically thin limit and by Case D in the optically thick limit until the cloud becomes very optically thick to stop continuum pumping.

Figure 7 shows the variation of I/d in H-like iron with the hydrogen column density for the most complex system observed in nature (the Case C to D transition) to the simplest possible system (the Case A to B_classic transition). The Case C to B transition shows the observed I/d values in an irradiated cloud, which includes continuum pumping and electron scattering. The Case A to B transition shows the observed I/d with no continuum pumping. Case A to B_classic shows the classic Menzel–Baker transition with no continuum pumping and no electron scattering.

Zoom In Zoom Out Reset image size

Figure 8 shows the total observed X-ray spectrum coming from a photoionized cloud for Cases A, B, C, and D within the energy range 0.1–10 keV. The spectrum is generated at the resolving power of XRISM (R ∼ 1200) at 6 keV set to Cloudy. Similar to that in Section 4, the y-axis of the figure has been scaled to show the total emission (ν F_ν ) per unit thickness (d) of the cloud, ν F_ν /d. ν F_ν is the sum of the total continuum emission and discrete line intensity (I) multiplied by R:

$$\begin{eqnarray}&&\nu {F}_{\nu }=\nu {F}_{\nu }^{\mathrm{continuum}}+{RI}.\end{eqnarray} \tag{ 6 }$$

Zoom In Zoom Out Reset image size

In Cloudy, νF_ν can be stored with the command

save emitted continuum

added to the input script.

The top and bottom rows in Figure 8 overplot the total emission spectrum for Case A with Case C and that for Case B with Case D. The column densities set to the cloud for calculating the spectra are the same as those in Section 4. The left and right panels show the same plots in linear and log scale, respectively. Clearly, Case C is enhanced as compared to the Case A spectrum due to continuum pumping. Case D is also brighter than Case B due to partial continuum pumping.

Figure 9 shows a simplified plot for a hydrogen-and-iron-only model under Case D conditions. This is not what is observed in nature. The purpose of this figure is to present the components of the spectra coming from H- and He-like iron in a less complicated form. The Lyman and Balmer lines are marked black, and the ionization edges of H-like and He-like iron at ∼9.3 keV and ∼8.8 keV are marked blue.

Zoom In Zoom Out Reset image size

The background continuum opacity of the photoionized cloud consists of two types of opacities—absorption opacity and scattering opacity. Figure 10 shows these two types of opacities. Absorption opacity mostly comes from the photoelectric absorption/photoionization opacity. ⁵ Near the Kα complex, absorption opacity is many orders of magnitude smaller than the scattering opacity, making it unimportant without affecting the line spectrum. Thus we only discuss the effects of scattering opacity.

Zoom In Zoom Out Reset image size

Scattering opacity mostly comes from the scattering of line photons by high-speed thermal electrons that leads to a process called ESE. The concept of ESE has been elaborated in Section 7 of the first paper of this series (Chakraborty et al. 2020b).

As a result of scattering off high-speed electrons, a fraction of the line photons are largely Doppler-shifted from their line-center. These Doppler-shifted photons create super-broad Gaussian profiles. The observed spectrum will include these broad Gaussian profiles as well as the actual sharp line profiles for the fraction of photons that were not scattered (Miller et al. 2002; Hanke et al. 2009).

In Cloudy, these broad Gaussian profiles can be excluded with the following Cloudy command: ⁶

no scattering intensity,

reporting only the intensities of the sharp line profiles.

Figure 11 shows the total scaled emission near the Fe xxv Kα complex. For simplicity, the widths of the sharp line profiles are assumed to be coming from the thermal velocity of the iron ions only. The presence of turbulence will change the widths of the sharp components, but the physics of electron scattering will be the same. At T = 6 × 10⁶ K, the temperature of our simulated cloud, the FWHMs of these sharp line profiles at E ∼ 6.7 keV are Δ${E}_{\mathrm{FWHM}}^{\mathrm{sharp}}=2\sqrt{\mathrm{ln}2}$ $\tfrac{{u}_{\mathrm{Dop}}}{c}$ E ∼ 1.6 eV, where ${u}_{\mathrm{Dop}}=\sqrt{\tfrac{2{kT}}{{m}_{\mathrm{Fe}}}}$ ∼ 43 km s⁻¹. At the same temperature, the FWHMs of the broad line profiles are Δ${E}_{\mathrm{FWHM}}^{\mathrm{broad}}$ ∼ 0.5 keV, where u_Dop = $\sqrt{\tfrac{2{kT}}{{m}_{{\rm{e}}}}}$ ∼ 13,500 km s⁻¹. This implies that the height of broad Gaussians will be orders of magnitude smaller than the sharp components and will be difficult to detect by telescope (Torrejón et al. 2010).

Zoom In Zoom Out Reset image size

The left panel in Figure 11 shows the changes in the total scaled emission in log scale for the hydrogen column densities N_H = 10²⁰, 10²², and 10²⁴ cm⁻² in the presence of continuum pumping. The broad Gaussians shown in the figure are solely from the electron scattering of the photons in the Fe xxv Kα complex. We do not show the broad Gaussians from the other lines to keep the figure simple.

The right panel shows a zoomed-in version of the sharp line profiles for all three column densities on a linear scale. As continuum pumping is present, ν F_ν /d at N_H = 10²⁰ cm⁻² represents the Case C limit, and N_H = 10²⁴ cm⁻² represents the Case D limit.

It can be seen from the figure that the w line intensity reduces significantly with the increase in optical depth/column density. There are two factors responsible for this reduction. First, the continuum pumping begins to become partially blocked with the increase in optical depth. Second, w has the largest line optical depth among all He-like transitions and is more likely to suffer electron scattering. Of course, when observed by future high-resolution telescopes like XRISM and Athena, the electron-scattered broad Gaussian component of w will be much fainter than the sharp component. But the reduction in the sharp w line intensity (or the line intensity of any resonance line) with increasing optical depth can serve as a powerful optical depth/column density diagnostic.

Note that Gilfanov et al. (1987) have discussed the effects of resonance scattering and shown distortion in the radial surface brightness profile due to migration of resonance X-ray line photons from the cluster center to the outer region. This will lead to a suppression of line fluxes in the central region of a cluster. For example, in Perseus, this factor was reported to be ∼1.3 for w (τ ∼ 1) near the cluster center by Hitomi Collaboration et al. (2018b). Of course, the suppression factor will be different in other systems depending on the geometry and optical depth.

The calculations presented in this paper represent a general study for a photoionized system irradiated with a power-law SED. From our Cloudy calculation, the suppression in w line intensity at τ = 1 is ∼1.43 due to the joint contribution of partial continuum pumping and electron scattering. Therefore it is safe to say that the change in the resonance line intensities due to these two factors can be as important as the resonance scattering effects. Our current model predicts the total emission from a symmetric geometry, so scattering has no effect on the emergent intensity. In addition to what we report in this paper, the Gilfanov et al. (1987) resonance scattering geometric correction has to be applied to match the observed spectra.

• 1.  
  Line formation processes were broadly categorized into two cases in the 1930s—Case A and Case B (Menzel 1937; Menzel & Baker 1937; Baker & Menzel 1938; Baker et al. 1938). At that time, the SED ionizing the cloud was assumed to have strong Lyman absorption lines. There was no continuum pumping/fluorescence to enhance the spectra. Some examples are O-stars in starburst galaxies, planetary nebulae, etc. (Osterbrock & Ferland 2006). But in extragalactic environments such as a cloud photoionized by an AGN SED or galaxies with quasars with no Lyman absorption lines, continuum pumping will significantly enhance the spectra. This led to the discovery of a third case in the late 1930s—Case C (Baker et al. 1938), which describes optically thin clouds in the presence of continuum pumping. A fourth case, Case D, was discovered recently (Luridiana et al. 2009), which describes spectra from an optically thick cloud in the presence of continuum pumping.
• 2.  
  Figure 3 shows a schematic representation of all four cases.

  • (a)  
    Under Case A conditions, lines are formed by radiative recombination and cascades from higher levels. Lyman lines escape the optically thin cloud without scattering/absorption.
  • (b)  
    Case B occurs when higher-n Lyman lines are converted to Balmer lines and a Lyα (or Kα) or two-photon continuum due to multiple scatterings in an optically thick cloud.
  • (c)  
    Lines are formed under Case C when Lyman lines are enhanced by continuum pumping and freely escape the optically thin cloud.
  • (d)  
    Case D occurs when multiple scattering and continuum pumping in Lyman lines occur together in an optically thick cloud so that the lines are partially enhanced.
• 3.  
  This paper is dedicated to understanding line formation processes through Case A, Case B, Case C, and Case D in X-ray-emitting photoionized plasma with Cloudy. We study H- and He-like iron emitting in the X-ray with improved Cloudy energies in excellent agreement with future microcalorimeter observations (Chakraborty et al. 2020a). In our simulations, we use a power-law SED to illuminate the cloud and an equilibrium temperature of 6 × 10⁶ K computed from the heating–cooling balance. Refer to Section 3 for details on the simulation parameters. As the absolute line intensities (I) increase with cloud thickness (d), we compare the line intensities per unit thickness of the cloud (I/d) for estimating the scaled differences between all four cases.
• 4.  
  Table 1 lists the line intensity (I) per unit thickness (d) of the cloud, I/d, for Case A, Case B, Case C, and Case D conditions observed in nature. To generate optically thin and optically thick conditions in the cloud, the I/d for Cases A and C are computed at N_H = 10¹⁹ cm⁻², and those for Cases B and D are computed at N_H = 10²⁴ cm⁻². The Menzel–Baker Case B values are listed under the Case B_classic column in the table. Lyα in H-like iron and Kα in He-like iron in Case B_classic are enhanced as compared to the corresponding Case A values due to the conversion of higher-n Lyman lines into Lyα (or Kα) plus Balmer lines. But in real astronomical sources, the presence of electron scattering reduces the observed Case B values. In H-like iron, I/d for Lyα decreases by ∼50%. In He-like iron, x, y, and w exhibit a decrease up to ∼24%. The Case C values are the brightest of all the four cases due to the free escape of Lyman photons following continuum pumping. The Lyα and Kα transitions in H- and He-like iron are up to ∼10 and ∼27 times enhanced as compared to the corresponding Case A values. The Case D values are smaller than the Case C values but bigger than the Case B values, as they are partially enhanced by continuum pumping. For H-like iron, the Case D I/d for Lyα is ∼36% enhanced as compared to the corresponding Case B value. Lyβ is ∼71% enhanced, and Hα is ∼36% enhanced. In He-like iron, Kα is enhanced up to ∼109%, Kβ is enhanced up to ∼208%, and Lα is enhanced up to ∼80%.
• 5.  
  The total emission spectrum for Case A, B, C, and D conditions within the energy range 0.1–10 keV is shown in Figure 8. The spectrum includes the continuum emission as well as the line emission described in the previous paragraphs. The resolving power (R) for our Cloudy simulations is set at R ∼ 1200, which is the resolving power of XRISM at ∼6 keV. The figure shows Case A overplotted with Case C and Case B overplotted with Case D in linear and log scale. Clearly, the line emissions in the Case C spectrum are brighter than those in Case A, and the line emissions in the Case D spectrum are brighter than those in Case B due to continuum pumping and partial continuum pumping, respectively.
• 6.  
  Electron scattering opacity can play an important role in deciding the line intensities in optically thick clouds. Line intensities for Case B and Case D can be reduced because of this. The top panel of Figure 5 and Table 1 show the deviation of the observed Case B values, which includes the effect of electron scattering, from Case B_classic, which does not include electron scattering. The I/d values shown in Table 1 and the bottom panel of Figure 5 for Case D also include electron scattering.
• 7.  
  Due to the electron scattering opacity, the line photons are scattered off high-speed electrons and are Doppler-shifted from their line-center. These scattered photons form Gaussians with super-broad bases. Line photons that are not scattered have a sharp base equivalent to their thermal width (and turbulent width if turbulence is present). Figure 11 shows these sharp and broad components coming from the Fe xxv Kα complex in the presence of continuum pumping for hydrogen column densities N_H = 10²⁰ cm⁻², 10²² cm⁻², and 10²⁴ cm⁻². N_H = 10²⁰ cm⁻² is the Case C limit, and N_H = 10²⁴ cm⁻² is the Case D limit. The broad components will be much fainter than the sharp components when detected by high-resolution telescopes. The observed sharp components for the resonance lines will exhibit significant changes in their line fluxes with variation in column density (and optical depth). It can be seen in Figure 11 that the w line intensity decreases significantly with an increase in column density. Such reduction in w is due to the following two factors: (a) Continuum pumping becomes partially blocked with the increase in optical depth. (b) The large optical depth of w makes it more likely to be scattered by electrons. A combination of (a) and (b) will reduce the line intensity of w or any other resonance line significantly with an increase in column density, which can serve as a powerful diagnostic in measuring the column density/optical depth of the cloud. From our Cloudy simulation, we get a suppression in w line intensity of ∼1.43 due to the above two factors at τ = 1, which is as important as the resonance scattering effects described by Gilfanov et al. (1987). As our current Cloudy model assumes a symmetric geometry, the effects of resonance scattering are not included in our calculation. A real observed spectrum will correspond to a resonance-scattering-corrected Cloudy-generated spectrum as shown in this paper.
• 8.  
  After the discovery of Cases A and B, these two cases have been widely discussed in the literature for the optical, ultraviolet, and infrared regimes, with limited studies on X-rays. As far as we know, there has been no discussion on X-ray spectra under Case C and Case D conditions. Case D is the least discussed of all four cases, as ideally, at very high column densities, Case D should be no different from Case B values, as mentioned in Section 4.4. But Table 1 and the bottom panel of Figure 8 show that even at a column density as high as N_H = 10²⁴ cm⁻² in a cloud illuminated with a power-law SED, Case D deviates considerably from Case B for X-ray emission from H- and He-like iron. This deviation will certainly be detected by future high-resolution telescopes with microcalorimeter technology. We emphasize the fact that Case C and Case D deserve far more attention than they have been given to date, especially because they could be the best representation of the emission spectra from irradiated extragalactic sources with a broad range of column densities.

We thank the referee of this paper for his/her very helpful comments. We thank Stefano Bianchi and Anna Ogorzalek for their valuable comments. We acknowledge support from the National Science Foundation (1816537, 1910687), NASA (17-ATP17-0141, 19-ATP19-0188), and STScI (HST-AR-15018). M.C. also acknowledges support from STScI (HST-AR-14556.001-A).