Time-dependent Photoionization Modeling of Warm Absorbers: High-resolution Spectra and Response to Flaring Light Curves

The ionization parameter is a convenient description of the ionization and temperature of a photoionized gas under the assumption of equilibrium. If so, the ionization parameter can be defined as

$$\begin{eqnarray}&&\xi =\displaystyle \frac{{L}_{\mathrm{ion}}}{{n}_{{\rm{H}}}{R}^{2}}=\displaystyle \frac{4\pi {F}_{\mathrm{ion}}}{{n}_{{\rm{H}}}},\end{eqnarray} \tag{ 1 }$$

where L_ion is the ionizing luminosity of the source, F_ion is the ionizing flux, and both of these quantities are integrated from 1 to 1000 Ry. R is the distance of gas from the ionizing source. This definition was first introduced by Tarter et al. (1969). The other way of defining the ionization parameter is $U={\int }_{{\varepsilon }_{1}}^{{\varepsilon }_{2}}{L}_{\varepsilon }/(\varepsilon 4\pi {R}^{2}{n}_{{\rm{H}}}c)d\varepsilon ,$ where L_ε is the ionizing luminosity per energy interval and c is the speed of light. U and ξ are easily convertible to each other for a given spectral energy distribution (SED) of the ionizing continuum. For the spectrum shape we adopt here, i.e., a single power law from 1 to 1000 Ry with energy index −1, the conversion is U = ξ/56.64 erg cm s⁻¹. The higher the ionization parameter, the more ionized the gas and vice versa. WA components generally exist either in the low ionization state of ξ ∼ 1 erg cm s⁻¹ or in the high ionization state of ξ ∼ 100 erg cm s⁻¹ (Laha et al. 2014).

It is important to clarify the usage of luminosity and flux. These two quantities are interconnected through Equation (1). Luminosity refers to the total ionizing luminosity emitted by the source, while flux represents the luminosity of the source divided by the area of the sphere at the specific point of interest within the cloud. The flux is a convenient description of the radiation strength and its effects on gas locally, while luminosity is a useful measure of the global energetics of the central source.

The free parameters in our models include hydrogen number density (n_H; hereafter hydrogen gas density unless otherwise stated), hydrogen column density (N_H; hereafter column density means hydrogen column density unless otherwise stated), ionization parameter (ξ), the shape of the incident radiation as described by the SED, and element abundances. We have created models for WA outflows depending mainly on the type of incident light curve. These are (1) a step-up model, (2) a flare model with multiple-zone consideration, and (3) a flare model with single-zone consideration. We refer to the step-up model as model 1, a flare model with multiple-zone consideration as model 2, and a flare model with single-zone consideration as model 3 in the subsequent sections for simplicity. The details of these models are described in the corresponding sections.

Since this work is an early step toward exploring photoionized plasma in nonequilibrium conditions, we attempt to make our results general and illustrate the general behavior of time dependence. Therefore, we have made several key approximations:

• 1.  
  We approximate the gas to be static, with no expansion of the gas and no velocity gradient. For a supersonic flow, such as in a WA, the bulk forces, whatever they are, dominate over internal dynamics in the cloud (Proga et al. 2022). If so, our models can be applied by adding a blueshift. We acknowledge that this effect can be important, but it is not included in the results presented here. We can calculate models that can be refined for specific situations, and we plan to do so in future work.
• 2.  
  We assume that the outflowing gas has constant density throughout. This assumption is partially justified by the physical thickness of the cloud, which is less than 10% with respect to the distance from the ionizing source, as shown in Paper I. If so, the imprint of any large-scale spherical flow will be small. However, we acknowledge that there may be a density gradient in the case of WAs associated with dynamical effects such as ram pressure, which will be considered in future work.
• 3.  
  We have not included emission in our radiative transfer calculation. This approximation is justified if the covering fraction of the absorber relative to the central source is small; this appears to be the case for most observed WAs. For most observed WA spectra, emission features are weak or absent (Kaspi et al. 2001; Kaastra et al. 2002). However, including emissions is on our list of tasks for future work since the emission is likely to be important in other time-dependent applications.

Time-dependent calculations require the simultaneous solution of the equations describing the ionic level populations, temperature, and radiation field for all level populations, spatial zones, and photon energies. This corresponds to a very large number of simultaneous ordinary differential equations; the standard xstar database has ∼3 × 10⁴ energy levels. Running these full schemes is extraordinarily time-consuming and sometimes gives unstable solutions if the number of equations is too large. This necessitates a simplification of this system of equations to make it tractable. That is, we simplify the energy level structure of the ions in the gas. To do this, we have created an atomic database that describes each ion using three levels: the ground level, one excited bound level, and the continuum (ionized) level. For the one bound excited level, we adopt an ad hoc description: the level energy is 0.8 × E_th, where E_th is the ionization potential of the ion. We include electron impact collisional excitation to the level for the ions H⁰, He⁰, and He⁺ with an effective collision strength (upsilon) with a value of 1.4–2.7 depending on temperature. The excited level is a pseudolevel, and the energy of that level is chosen in such a way that the temperature versus ionization parameter (ξ) curve can agree as closely as possible with the temperature versus ionization parameter (ξ) curve obtained from the full xstar databases. This is illustrated in Figure 2 of Paper I. We include all of the ions that are in xstar in the newly created database, though the simulations in this paper include only a subset of them. Paper I gives more details about this small database.

In addition, we are limited by the number of time, position, and energy grid points that can be included in our computation while solving the discretized time-dependent equation of level population, energy balance, and radiative transfer. The energy grid we can use for such calculations is inadequate for calculating high-resolution spectra. To address this, we created a two-step technique:

• 1.  
  We first run the time-dependent photoionization code with reasonable energy, position, and energy grid numbers using the small subset of the XSTAR database, with each ion represented by two bound levels plus a continuum.
• 2.  
  The ground-level populations, which are mainly responsible for the absorption of the ionizing radiation of the source, are then used to produce high-resolution spectra using the spectral fitting model package WARMABS. ³ WARMABS uses the full XSTAR database and can handle arbitrary energy grid spacing. We then use the opacities from WARMABS to calculate the emergent absorption spectrum via the formal solution to the radiative transfer equation (including the time delays associated with light-travel time).

Model 1 has the following input parameters: gas density 10⁷ cm⁻³; column density 9.0 × 10²² cm⁻²; initial ionization parameter (ξ₁) ∼50 erg cm s⁻¹; incident ionizing spectral shape of a power law with energy index −1; initial luminosity of the source 10⁴⁴ erg s⁻¹; and it includes the elements H, He, C, O, Si, and Fe. The incident ionizing radiation flux suddenly increases by a factor of 3 at the beginning of the simulation.

In Figure 1, we present the input light curve for model 1. The initial luminosity of the source is set at 10⁴⁴ erg s⁻¹, and it is subsequently increased to 3.0 × 10⁴⁴ erg s⁻¹ at ∼20 s. We have included five vertically dotted green lines to highlight specific times for which we will display the spectra. These times are chosen such that the changed radiation field has completely emerged through the cloud. The simulation for this model is conducted for 10⁸ s.

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Figure 2 displays the evolution of transmitted spectra for model 1. Each panel illustrates the transmitted spectrum at a specific moment in time. It is important to note that the time it takes for the radiation to propagate from the source to the illuminated face of the absorber is not considered. The simulation begins with the arrival of the changed flux at the face of the cloud. The initial spectrum showcases the absorption features and characteristics before the increase in luminosity occurs. Over time, as the higher flux propagates through the absorber, the transmitted spectra in subsequent panels exhibit changes in the absorption features. These changes are a consequence of the altered ionization and excitation within the gas due to the increased luminosity of the ionizing source. The simulation is primarily on the effects of the sudden change in luminosity once it reaches the surface of the cloud, the illuminated face. Figure 2 visually represents the evolving transmitted spectra and their response to the abrupt increase in ionizing source luminosity. It provides information about the time-dependent behavior of the absorber and the impact of changes in the incident flux on its observed spectral features.

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The spectra presented in Figure 2 constitute many absorption features. The absorption features spanning the energy range ≃0.2–0.7 keV primarily arise from the absorption of carbon (C v and C vi), silicon, and iron L-shell ions. The second group of absorption features, located at ≃0.75 keV, predominantly results from iron ions. The absorption edge at ≃1 keV is primarily influenced by iron and oxygen. The interaction between these ions and incident radiation gives rise to the distinctive absorption edge observed in this energy range. Throughout the spectrum, absorption lines arise from a wide range of ions associated with various elements. In each panel, three different spectra are presented: the red curve represents the high flux equilibrium spectrum, the orange spectrum corresponds to the low flux equilibrium spectrum, and the blue curve represents the time-dependent spectrum. The green curves are obtained by multiplying the low flux equilibrium spectrum by three. These reference spectra help to visualize how the ionized absorber changes its opacity over time owing to the sudden change in the luminosity of the source from low to high. These spectra allow for a comparative analysis of the evolving absorption features and their variations in response to changing flux conditions. By examining the figure and the different spectra presented, it is possible to gain a deeper understanding of the time-dependent behavior of the absorption features, as well as the contributions of different ions and elements to the overall spectrum.

The first panel of Figure 2 corresponds to the case before the changed radiation field arrives at the back of the cloud. The time it takes light to cross this cloud for this model is ≃3 × 10⁵ s. This is why the orange and blue spectra overlap until this time. When the altered radiation arrives at the back of the cloud, the blue spectrum changes toward the red spectrum and overlaps with the green spectrum because the gas has not yet had a chance to change its ionization state, as seen in the second panel. Since the opacity of the gas cloud decreases over time owing to a high ionizing radiation field, the gas becomes more transparent, and the spectrum starts to move closer to the red curve and farther from the green curves. This model takes ∼5.8 × 10⁶ s for the gas to reach an equilibrium state corresponding to the high flux. This is the time including the effect of photoionization and thermal equilibrium. The timescale of thermal equilibrium is much longer than the photoionization equilibrium (Paper I).

We have generated a ratio plot, depicted in Figure 3, to illustrate the dynamic evolution of absorber opacity. To achieve this, we created two distinct ratios against energy at various time points. The blue curves represent the ratios of spectra resulting from time-dependent photoionization equilibrium calculations (blue spectra in Figure 2) to those of the high flux equilibrium spectra (red spectra in Figure 2). On the other hand, the green curves denote the ratios of three times the low flux equilibrium spectrum (green spectra in Figure 2) to the ultimate high flux equilibrium spectra.

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The green line within Figure 3 serves as a reference line for comparing alterations in transmitted absorption spectra at various times during the simulation. The blue curve in the first panel corresponds to the time before the altered radiation field reached the back of the cloud. This is why the blue and green curves differ in absorption features and continuum. In the second panel, both of these curves overlap each other. The altered radiation field has just arrived at the back of the cloud by this time. In the rest of the panel, the green curve starts to depart at absorption features because the gas starts to be ionized and becomes transparent as time passes. The opacity of the gas changes until the blue line becomes nearly horizontal, and the ratio becomes 1. This corresponds to the final flux ionization state of the gas.

Model 2 has gas density of 10⁷ cm⁻³, initial luminosity of the source of 10⁴⁴ erg s⁻¹, initial ionization parameter (ξ₁) of ∼81 erg cm s⁻¹, column density of 10²² cm⁻², and the incident ionizing spectral shape of the power law with energy index −1, and it includes elements H, He, C, N, O, Ne, Mg, Si, S, and Fe. The incident ionizing radiation flux suddenly increases by a factor of 10. The cloud in this model is divided into 60 spatial zones. The simulation is run for 10⁵ s.

Our study considers the incident light curve, as depicted in Figure 4. The red curve represents the incident light curve at the illuminated face of the cloud, whereas the green curve is at the back of the cloud. This incident light curve exhibits three distinct regions, each with its own characteristics. The first region corresponds to a linear increase in source luminosity until 10⁴ s. During this phase, the flare is in its initial rising state, gradually increasing in intensity. The second region of the light curve represents a high luminosity state that remains constant for an additional 10⁴ s. This phase corresponds to the peak of the flare, where the source maintains a steady high luminosity. After the high state, the third region shows a linear decrease in flux over a duration of 10⁴ s. This marks the declining phase of the flare, where the luminosity gradually decreases with time until it reaches a low luminosity state. Overall, the entire variability time of the flare is 3 × 10⁴ s, which encompasses the rising, peak, and declining phases.

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In our multiple-zone modeling approach, we consider the propagation of the flare through the gas cloud. As the flare progresses deeper into the cloud, its temporal behavior changes, resulting in a smoother variation compared to the initial light curve. This effect can be observed by examining the green line in Figure 4, which represents the light curve obtained at the back of the cloud.

The time for the flare to travel through the cloud is determined by the sum of three components: the variability time of 3 × 10⁴ s, the light-travel time, and the propagation time. The light-travel time is the time it takes for the flare to travel from the front of the cloud to the back in the absence of any absorption. In our model, the light-travel time is estimated to be ∼3.3 × 10⁴ s. Therefore, the total time for the flare to traverse through the entire cloud is ∼7.0 × 10⁴ s, as shown in Figure 4. The propagation time primarily comes from the variable absorption process of ionizing radiation within the cloud. It depends on the ratio between the number of atomic species undergoing ionization within a specific spherical shell and the rate of ionizing photons emitted by the source in case of increasing source luminosity. The propagation time, therefore, depends on the ionization and recombination time of the ions. For a more comprehensive and quantitative description of the propagation time, please refer to Paper I. For reference, the photoionization and recombination times of some important ions are presented in Table 1. These times are calculated at the illuminated face of the cloud, using equilibrium calculation for the low luminosity value of the source. These times are taken from the simulation and hence include some errors. Most of the error appears as a result of the coarse energy grids and the simplified database we have used.

This study examines the response of a gas cloud to changes in the radiation field. It is important to note that this response is not instantaneous and simultaneous throughout the cloud. Instead, there is a time delay in the response of the gas at different locations within the cloud. This delay is primarily influenced by the time it takes for the radiation to propagate through the cloud and the absorption processes that occur.

To illustrate this, Figure 5 presents the ion fraction profile of oxygen (O viii) at different times during the cloud's evolution. The ion fraction profile represents the relative abundance of O viii ions within the cloud at various spatial locations. The value of the O viii ion fraction is ≃0.13 in our simulation. This differs from the value at this same ionization parameter from Kallman & McCray (1982), which was ≃0.5. The ion abundance curve for O viii is shifted when these two calculations are compared. The ion fraction versus ξ curves peaks at ξ ≃ 15 erg cm s⁻¹ for the calculations shown here, while it peaks at ξ ≃ 50 erg cm s⁻¹ for the 1982 calculations. For the standard XSTAR code plus database, the peak occurs at ξ ≃ 25 erg cm s⁻¹. Reasons for these differences include differing energy grids, the simplified version of the atomic database used in our current simulations, and other numerical factors.

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At the initial time, when the radiation field starts to impact the cloud, the ion fraction profile of O viii exhibits its initial distribution. As time progresses, the radiation field propagates through the cloud, reaching deeper into its interior. This leads to changes in the ionization state of the gas.

As depicted in Figure 5, the ion fraction profile of O viii at later times shows variations compared to the initial profile. The changes in the ion fraction occur gradually as the radiation field penetrates further into the cloud, resulting in a modified ionization state of oxygen. The time is shown in the box for each panel. The first left panel corresponds to the initial flux equilibrium ion fraction profile in the gas. This is before the gas gets flares. In the successive panels of the left column, the ion fractions are shown for increasing time. When the gas receives the flare, the ion fraction starts to change at the face of the cloud first. The ion fraction at the face of the cloud started to change at ∼3 × 10³ s, gradually went down as the flare went up, and reached a minimum at ∼2.1 × 10⁴ s (the last panel of the left column) and started to increase as the flare decreased and reached the initial value again at time ∼3.9 × 10⁴ s (sixth panel of the right column).

Propagation of the flare caused the ion fraction to decrease, reach the minimum, remain at a minimum for some time, and increase until it acquired the final low flux equilibrium value at all points in the cloud. The equilibration time at the back of the cloud is the sum of light-travel (∼3.3 × 10⁴ s), flare (∼3.0 × 10⁴ s), and response time (∼6.0 × 10³ s), which comes to be ∼6.9 × 10⁴ s (the first panel of the right column). Therefore, it took ∼6.9 × 10⁴ s for the whole gas to return to equilibrium. We note that the times in the panels are increasing going down in the first column and increasing going up in the second column.

The time evolution of the different ions at different spatial points in the cloud can be seen in Figure 6. Each panel corresponds to a different ion except the first (top left) panel. The top left panel shows the temperature evolution in the cloud at different locations. The time-dependent calculation predicts a temperature that evolved very little during the simulation. The temperature changed only by ∼10⁴ K for a flare, which increased in flux by a factor of 10. This temperature change is small compared to the change found if the gas were in equilibrium at the corresponding flux. The temperature changes by an order of magnitude in equilibrium approximation because the ionizing luminosity also changes by the same amount. The small increase in temperature in the time-dependent calculation is attributed to the longer thermal timescale of low-density plasma. The ionizing source stays at a high luminosity state for 10⁴ s. This is much shorter than when it takes plasma to be thermalized. The thermal timescale of the plasma is about ∼10⁵ s for this model (Paper I). In the rest of the panels, we have shown the evolution of representative ions such as C vi, O vii, O viii, Si xii, Si xiv, Fe xv, and Fe xvi. The blue curves come from the time-dependent calculation, whereas the orange lines come from the equilibrium approximation. The line that evolves earlier corresponds to the ion fraction at the illuminated face of the cloud, and the one that evolves slowest is at the back of the cloud; the rest are inside the cloud. In this plot, we have shown ion fractions at seven different points in the cloud, including the illuminated face and back of the cloud. The inner spatial points are at ∼1.7 × 10¹⁴ cm, ∼3.4 × 10¹⁴ cm, ∼5.0 × 10¹⁴ cm, ∼6.8 × 10¹⁴ cm, and ∼8.5 × 10¹⁴ cm from the illuminated face. Note that the physical size of the cloud for this model is 10¹⁵ cm.

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For this model, C vi, O vii, and O viii ions come to the final low flux equilibrium ion fraction value relatively quickly compared to the other ions, such as Si xii, Si xiv, Fe xv, and Fe xvi. The time it takes for an ion to come to equilibrium corresponds to the time it takes the absorption spectrum to come to equilibrium at the relevant energies. The spectral evolution will be discussed in more detail in Section 4.4.

The red dashed vertical line in all the panels indicates the time of the flare, which is 3 × 10⁴ s as seen in the light curve. The black dashed vertical line represents the time it takes for the flare to emerge completely from the cloud. This time is the sum of the flare time (3 × 10⁴ s) plus the light-travel time (∼3.3 × 10⁴ s) of the cloud. The green dashed horizontal line is the low flux equilibrium value. All the panels show that, even if the flare has already passed, the ion fraction is still changing. However, the timescale to come to equilibrium is different for different ions. Ions such as C vi, O vii, O viii, and Fe xv come to the equilibrium value by the time the simulation ends. However, ions such as Si xiii, Si xiv, and Fe xvi do not come all the way to the equilibrium values by the end of the simulation. They are close to the equilibrium when the simulation is ended; we estimate that it would take an additional ∼5 × 10⁴ s to reach equilibrium for all these ions.

It is important to note that the equilibrium ion fraction values are different from those from the time-dependent calculation. Ions such as C vi, O vii, and O viii have the same equilibrium ion fraction pattern as they do for time-dependent models. They differ only in magnitude by some fraction. However, for ions such as Si xiii, Si xiv, Fe xv, and Fe xvi, the equilibrium ion fraction is qualitatively different from those from a time-dependent calculation. The equilibrium ion fraction curves have an "M shape" structure owing to the flare. That is, at low flux early in the flare and late in the flare, the ion fractions increase with increasing flux and decrease with decreasing flux. In the middle of the flare, the ion fractions decrease with increasing flux and increase with decreasing flux as the parent element is ionized to higher ion stages. This behavior makes the time-dependent modeling qualitatively different from equilibrium calculations.

The most significant feature of the behavior of some ion fractions versus time is asymmetry. There is a tail-like structure in ions such as Si xiii, Si xiv, Fe xv, and Fe xvi. This implies that when the flux is in the increasing phase of the flare, the ion fraction values change more rapidly than when the flare is in the declining phase. It is important to keep in mind that the time-dependent ion fraction value at any time during evolution does depend on the condition at the previous times together with input parameters such as ionization parameter, time of the flare, density, etc. Once the gas experiences the flare, and if the variability time of the flare is comparable to the photoionization and/or recombination timescale, a delay in the response occurs. This is just because the gas did not have enough time to respond to the change in the radiation field. The reason behind this is the behavior of the dominant microscopic timescale behind the increasing and declining phases of the flare. During the increasing phase, the gas starts to change its ionization state to a higher value, which is determined by photoionization, whereas recombination time plays a dominant role during the declining phase of the flare. Furthermore, the recombination is dominated by dielectronic recombination for these ions, and this process is very temperature sensitive. As shown in Paper I, the temperature is determined by the thermal timescale, i.e., the cooling time, and for this model the cooling time is longer than the recombination time. Thus, the gas remains stuck at a higher temperature late into the simulation, and at this temperature the dielectronic recombination rate is less than it was at the initial equilibrium. This results in the asymmetric ion fraction versus time distribution for these ions. Successive ionization and recombination of the ions with the delayed response result in the complicated shape of ion evolution for these ions.

Figure 7 showcases the individual elemental contributions to the absorption spectrum. The model includes several elements such as H, He, C, N, O, Ne, Mg, Si, S, and Fe to simulate the response of a cloud to a flaring incident light curve. The figure highlights the relative importance of each element in contributing to the overall absorption spectrum. Among the elements considered, O (oxygen), Si (silicon), S (sulfur), and Fe (iron) dominate the absorption features in the spectrum. While the focus is on these key elements, it is important to note that the full spectrum encompasses absorption lines from all the included elements. The absorption features arise from a combination of edges and lines associated with the various ions. The energy range shown in the figure is up to 2 keV, as most ions primarily absorb radiation in the soft X-ray band. This range allows for a comprehensive representation of the absorption features resulting from the elemental contributions. The final model spectrum has three deep continuum absorption features at ∼0.3, ∼0.75, and ∼1 keV. These are denoted by first, second, and third absorption features in the spectrum described in the following section. In subsequent sections, the figure will be further explored to elucidate the specific absorption features, including edges and lines, present in the spectrum. This analysis will provide valuable insights into the ionization states, physical properties, and dynamics of the gas cloud under investigation.

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Figure 8 illustrates the temporal evolution of a modeled observed spectrum in response to a flare-like variation in the ionizing source. The figure consists of eight panels, each representing a specific time interval and displaying the corresponding absorption spectrum. The panels are divided into two sets of four. The first four panels depict the absorption spectrum during the passage of the flare through the cloud, while the last four panels represent the absorption spectrum after the flare has passed. The orange curve in each panel corresponds to an equilibrium spectrum generated using the instantaneous flux. This spectrum represents the expected absorption features without any temporal variations. The blue curves in the figure represent time-dependent photoionization spectra at different points in time during the evolution. These spectra showcase the changes in the absorption features as the flux varies over time at the face of the cloud. By observing the figure, one can see how the absorption spectrum evolves over time in response to the changing flux from the ionizing source. The differences between the yellow and blue curves highlight the impact of temporal variations on the absorption features of the spectrum.

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Figure 8 comprises multiple panels, each representing a specific time corresponding to different vertical lines in the associated light curve. In the first panel, the time corresponds to the period before the flare reaches the back of the cloud. At this point, both the equilibrium and time-dependent spectrum overlap significantly. The absence of significant divergence indicates that the spectra are similar, as the increased flux from the flare has not yet reached the cloud's back. This time interval aligns with the first vertical line in the light curve.

The second panel corresponds to a time of ∼3.7 × 10⁴ s (the sum of the light-crossing time plus the time it took for the flare to reach this luminosity level); the luminosity of the source reaches ∼4 × 10⁴⁴ erg s⁻¹. In this state, the equilibrium spectrum and the time-dependent spectrum begin to diverge. The equilibrium spectrum exhibits relatively less absorption compared to the time-dependent spectrum. The first and third absorption features, observed in the time-dependent spectrum, are nearly absent in the equilibrium spectrum. This difference arises from the higher ionization level of the gas due to the increased flux value.

The third panel represents the high luminosity state with source luminosity ∼10⁴⁵ erg s⁻¹, crucially showcasing the absence of absorption features in the equilibrium spectrum. In this state, the equilibrium spectrum shows minimal absorption, with most features being eliminated, except for a few weak absorption lines. In contrast, the time-dependent spectrum continues to exhibit absorption features. The time corresponding to this particular state is ∼4.6 × 10⁴ s.

Finally, the fourth panel represents the point at which the source's luminosity begins to decrease, which corresponds to the fourth vertical dashed line from the left in the light curve in Figure 4.

The remaining four panels in Figure 8 depict the time evolution of the emergent spectra after the flare has completely emerged from the cloud. These spectra are compared to the instantaneous equilibrium spectra. In these panels, even after the flare has passed, the time-dependent spectra continue to evolve, gradually approaching a low-luminosity equilibrium absorption spectrum. The ongoing evolution is particularly noticeable in the edges and some lines in the spectra. The absorption lines below 0.25 keV, around 1.3 keV, and at approximately 1.8 keV exhibit clear changes and evolution over time. The first absorption edge shows an apparent evolution pattern. This is because of the effect of longer equilibrium timescale of ions such as Si xiii, Si xiv, Fe xv, and Fe xvi as shown in Figure 6. As the gas within the cloud settles into a low-luminosity ionization equilibrium state, these lines continue to shift and evolve. The evolving nature of the spectra is indicative of the ongoing processes and adjustments occurring within the gas cloud. It highlights the time it takes for the ionization and excitation states of the gas to reach a stable equilibrium configuration after the passage of the flare.

According to the model, the cloud takes ∼9.1 × 10⁴ s to reach this low-luminosity ionization equilibrium state. During this period, the time-dependent spectra gradually converge toward the equilibrium absorption spectrum. By studying the time evolution of the emergent spectra in these panels, one can gain insights into the relaxation and settling processes of the gas cloud following the flare event. The changes in the absorption features, particularly in the edges and lines, provide valuable information about the ionization and the timescales involved in reaching a stable equilibrium state.

The absorption features change rapidly during the evolution. This is primarily due to the ionization of Fe from M-shell ions such as Fe xv, Fe xvi, and Fe xvii, whose simple atomic structure does not have the rich inner shell spectra of the M-shell ions but contributes most to the absorption. It is also worth pointing out that the effects of saturation can be to make absorption features vary less with time than the abundances of their ions. Thus, strong lines appear to be less variable than the ions. Turbulent broadening plays a role in determining the importance of saturation. In AGN outflows, typical turbulent velocities are ∼10³ km s⁻¹, which will reduce the optical depths for many resonance lines. These effects are included in our simulation.

The population of an atomic level obeys the kinetic equation involving the atomic rates into and out of the level,

$$\begin{eqnarray}&&\displaystyle \frac{{{dn}}_{i,{\rm{X}}}}{{dt}}=\displaystyle \sum _{j=1}^{p}{n}_{j,{\rm{X}}}{R}_{{ji}}-\displaystyle \sum _{k=1}^{p}{n}_{i,{\rm{X}}}{R}_{{ik}},\end{eqnarray} \tag{ A1 }$$

and the equation of number conservation,

$$\begin{eqnarray}&&\displaystyle \sum _{i=1}^{p}{n}_{i,{\rm{X}}}={{xn}}_{{\rm{H}}},\end{eqnarray} \tag{ A2 }$$

where n_i,X are the level populations in ith energy level of element X in units of cm⁻³, which is equal to the product of the fractional elemental abundance and total hydrogen number gas density, n_H. Parameter x is the fractional elemental abundance of species X relative to hydrogen. R_ji is the transition rate from the jth to ith energy level contributed by atomic processes, including photoexcitation, photoionization, collisional excitation, collisional ionization, recombination, charge transfer, and radiative decay. The values of rates for R_ij are taken from XSTAR. Parameter p is the total number of energy levels considered for the particular element. As discussed in Section 3, we adopt a simplified level structure so that, for an element with atomic number Z, p = 2Z + 1. Here and in what follows, we adopt the values of the atomic constants (e.g., photoionization cross sections, atomic energy levels, collision rate coefficients) from the xstar database (Bautista & Kallman 2001).

The temperature variation in the gas is given by the following differential equation:

$$\begin{eqnarray}&&\displaystyle \frac{{dT}}{{dt}}=\displaystyle \frac{2}{3{{kn}}_{t}}\left({{\rm{\Gamma }}}^{\mathrm{heat}}-{{\rm{\Lambda }}}^{\mathrm{cool}}-\displaystyle \frac{3}{2}{kT}\displaystyle \frac{{{dn}}_{t}}{{dt}}\right),\end{eqnarray} \tag{ A3 }$$

where T is the temperature, Γ^heat and Λ^cool are total heating and cooling rates in units of erg s⁻¹ cm⁻³, k is the Boltzmann constant, and n_t is the total number density of particles, i.e., the sum of all electrons, ions, and neutral atoms. The last term on the right-hand side of this equation is the rate of change in the internal energy density associated with the rate of change of the total number of free particles in the gas. This is negligible when the gas is highly ionized; otherwise, it explicitly couples the level populations with the temperature equation.

The equation describing the time evolution of the radiation field in spherical symmetry is (Hatchett et al. 1976; García et al. 2013)

$$\begin{eqnarray}&&\displaystyle \frac{1}{c}\displaystyle \frac{\partial {L}_{\varepsilon }(R,t)}{\partial t}+\displaystyle \frac{\partial {L}_{\varepsilon }(R,t)}{\partial R}=4\pi {R}^{2}{j}_{\varepsilon }-{\kappa }_{\varepsilon }(R,t){L}_{\varepsilon }(R,t),\end{eqnarray} \tag{ A4 }$$

where R is the distance in the cloud from the ionizing source, L_ε (R, t) is the specific luminosity of the ionizing source in erg s⁻¹ erg⁻¹, j_ε is the local emissivity in erg s⁻¹ cm⁻³ erg⁻¹, and κ_ε (R, t) is the total extinction coefficient in cm⁻¹. In our calculations in this paper, the extinction comes from only absorption and we do not include emission, so the equation becomes

$$\begin{eqnarray}&&\displaystyle \frac{1}{c}\displaystyle \frac{\partial {L}_{\varepsilon }(R,t)}{\partial t}+\displaystyle \frac{\partial {L}_{\varepsilon }(R,t)}{\partial R}=-{\kappa }_{\varepsilon }(R,t){L}_{\varepsilon }(R,t).\end{eqnarray} \tag{ A5 }$$

The rest of the equations for different timescales and their comparison are mentioned in detail in our previous work (Paper I). We have generated a mini version of the full XSTAR database, including only three levels; one ground, one excited, and one ionized level. This newly generated database is then tested with multiple methods and verified to produce adequate accuracy in our time-dependent computation. The readers are encouraged to review our previous paper (Paper I). The recombination, photoionization, and light-travel time play an important role in the evolution and equilibration of the gas when it experiences variable ionizing radiation. The recombination time is the most important out of them, which essentially depends on the density of the gas.