© ESO, 2017

1. Introduction

Active galactic nucleus outflows may play an important role in cosmic feedback (see, e.g., Fabian 2012, for a review). For an outflow in a thin spherical shell geometry moving constantly with a radial velocity v, the mass outflow rate can be estimated via (1)where m_p is the proton mass, N_H the hydrogen column density along the line of sight, r the distance between the outflow and the central engine, Δr the radial size of the outflow, and Ω the solid angle subtended by the outflow. The kinetic power carried by the outflow is . Thus, distant and/or high-velocity outflow leads to higher mass outflow rate (Ṁ_out) and kinetic power (L_KE).

While the line-of-sight hydrogen column density (N_H) and velocity (v) of the outflow can be well constrained via spectral analysis, its solid angle (Ω) and location (r) are the main source of uncertainties in estimating the kinetic power that impacts the host galaxy. The solid angle highly depends on the exact geometry, which is not investigated here. On the other hand, the location can be immediately obtained via the definition of the ionization parameter¹ (Tarter et al. 1969; Krolik et al. 1981) if the density of the outflow is known, (2)where L is the 1–1000 Ryd (or 13.6 eV–13.6 keV) band luminosity of the ionizing source, n_H the hydrogen number density of the ionized plasma, and r the distance of the plasma with respect to the ionizing source.

However, it is not trivial to determine the density of a photoionized plasma. Three different approaches have been used to measure the density of AGN outflows. The first approach is a timing analysis where the response of the ionized outflow to changes in the ionizing continuum is monitored. A high-density plasma recombines more rapidly, thus yields a shorter recombination timescale. This approach has been used to constrain density in Mrk 509 (Kaastra et al. 2012), NGC 5548 (Ebrero et al. 2016), and NGC 4051 (Silva et al. 2016), etc. The timing analysis is in general challenging because the observations of a given object are often sparse, washing out the possible effects of variability, and lack the signal-to-noise ratio required to significantly measure the expected changes (Ebrero et al. 2016).

The second approach is a spectral analysis of density sensitive emission lines. It is well known that the ratio of intercombination to forbidden emission lines in the He-like triplets (e.g., Porquet et al. 2010) varies for plasmas with different density values. This density probe, observed in the X-ray wavelength range, has been applied to a few AGNs, e.g., NGC 4051 (Collinge et al. 2001), NGC 4593 (McKernan et al. 2003), and NGC 4151 (Schurch et al. 2004), where the upper limits of the plasma density are derived. Meanwhile, as shown in Mehdipour et al. (2015a), line absorption of He-like ion triplet lines by Li-like ions make density diagnostics complicated. In addition, for solar corona studies, metastable emission lines observed in the EUV wavelength range from Be-like (Landi & Miralles 2014), B-like (Keenan et al. 1998; Ciaravella et al. 2001; Gallagher et al. 1999; Warren & Brooks 2009), C-like (Keenan et al. 1993; Landi & Landini 1998), and N-like (Keenan et al. 2004) ions are widely used to determine the density.

The third approach is a spectral analysis where density sensitive metastable absorption lines are identified. This method has been successfully used for absorption lines observed in the UV band (e.g., Arav et al. 2015). In the X-ray band, Kaastra et al. (2004) obtain an upper limit of the outflow density in Mrk 279, with the density sensitive metastable absorption transitions 2s²–1s2s²2p in O v (Be-like) ~22.4 Å. Later, King et al. (2012) reported an upper limit of the outflow density in NGC 4051 by using the ground (11.77 Å) and metastable (11.92 Å) transitions in Fe xxii (B-like). The same transitions in Fe xxii were in fact previously used by Miller et al. (2008) in the stellar mass black hole GRO J1655–40, yielding a tight constraint on the density of a disk wind.

In this work, we carry out a systematic study of density diagnostics with ions in different isoelectronic sequences in photoionized equilibrium. A detailed calculation is made with our new photoionized plasma model PION² in the SPEX code (Kaastra et al. 1996). Given cosmic abundances (the proto-solar abundance table in Lodders et al. 2009), we consider elements including C, N, O, Ne, Mg, Si, S, Ar, Ca, and Fe. Throughout this paper we consider plasma densities in the range between 10⁶ m^-3 (or 1 cm^-3) and 10²⁰ m^-3 (or 10¹⁴ cm^-3). At n_H = 10²⁰ m^-3, metastable levels in Be-like to F-like ions can be significantly populated compared to the population of the ground level, while for Li-like, Ne-like, and Na-like ions, no metastable levels are significantly populated. Therefore, we focus on Be-like to C-like ions in Sects. 3.1 to 3.3. Due to the lack of atomic data, for N-like to F-like ions, we merely discuss Fe xx, Fe xix, and Fe xviii in Appendix A.

2. Methods

Unless specified otherwise, we use the spectral energy distribution (SED) of NGC 5548 in our photoionization modeling, i.e., the AGN1 SED shown in Fig. 1 of Mehdipour et al. (2016). The photoionized outflow is assumed to be optically thin with a slab geometry. The line-of-sight hydrogen column density (N_H) is 10²⁴ m^-2. The covering factor is unity. No velocity shift with respect to the central engine is assumed here, and the turbulent velocity is set to 100 km s^-1 in the calculation.

When modeling a photoionized plasma with our new PION model in the SPEX code, its thermal equilibrium, ionization balance, and level population are calculated self-consistently with detailed atomic data of relevant collisional and radiative processes, such as collisional excitation and de-excitation (FAC calculation, Gu 2008), radiative recombination (Badnell 2006; Mao & Kaastra 2016), inner shell ionization (Urdampilleta et al. 2017), etc. The thermal balance and instability curve are shown in Figs. 3–5 of Mehdipour et al. (2016). Ion concentrations are derived accordingly, and we show in Fig. 1 the ion concentrations of Si xiv (H-like) to Si v (Ne-like) as a function of the ionization parameter. The level population is also calculated simultaneously.

Fig. 1 Ion concentration of the Si isonuclear sequence (H-like to Ne-like) as a function of the ionization parameter (in units of 10-9 W m, i.e., erg s-1 cm).

3. Results

In Sect. 3.1 (for Be-like ions) to Sect. 3.3 (for C-like ions), we first show which metastable levels (Table 1) can be significantly populated in a broad density range of n_H ∈ (10⁶, 10²⁰) m^-3. We note that different ionization parameters are used to maximize the ion concentration of different ions, e.g., log ₁₀(ξ) = 1.90 for Si xi (Be-like), log ₁₀(ξ) = 2.0 for S xii (B-like), and log ₁₀(ξ) = 2.1 for Ar xiii (C-like). Second, we list the characteristic absorption lines from these metastable levels. Absorption lines from the ground level are listed together with density sensitive lines from the metastable levels, so that it is possible to tell whether the spectral resolution of a certain instrument is fine enough to distinguish these lines. If so, these lines can be used for density diagnostics.

Fig. 2 Level population ratios as a function of plasma density (in the range of 106−20 m-3 or 100−14 cm-3) at ionization parameter of maximum ion concentration in the ionization balance. The configuration of the ground () and metastable levels () are listed in Table 1.

3.1. Be-like

Figure 2 shows the ratios of metastable to ground level populations as a function of plasma density. The lifetime of the first excited level 1s²2s2p (³P₀) is rather long, so that even if the plasma density is rather low, the metastable level can still be populated up to a small percent of the ground level population. Accordingly, a plateau can be found in the level population ratio (left panel of Fig. 2). The third excited level 1s²2s2p (³P₂) can be more easily populated at lower density than the second excited level 1s²2s2p (³P₁). The rest of the excited levels in Be-like ions are not significantly populated ( < 0.01% of the ground level population) in a photoionized plasma with density n_H ≲ 10²⁰ m^-3.

Given that the metastable levels (Level 2−4) can be populated up to 20% of the ground level (Level 1) population, we list in Table 2 three sets of characteristic transitions (n_j = 2−2, 2−3, and 1−2) for each level. The corresponding wavelengths (λ) and oscillator strengths (f) of these absorption features in the X-ray (1−100 Å) and UV (100−2000 Å) wavelength ranges are listed as well.

Among the three transitions of each level in the same ion, the inner shell 1s−2p transition (denoted as n_j = 1−2) always yields the shortest wavelength, while the 2s−2p transition (n_j = 2−2) yields the longest wavelength. There are of course more transitions than we listed in Table 2. For instance, there are in total six 1s²2s2p (³P)–1s²2p² (³P) metastable transitions around λ ~ 1175 Å for C iii that have been successfully used for density diagnostics in AGN ionized outflows (e.g., Arav et al. 2015). Transitions with higher f-values are listed in Table 2 for simplicity. For the six C iii λ ~ 1175 Å lines, the three tabulated lines have f> 0.1, while the other three lines have slightly lower oscillator strength values.

For all three sets of transitions, the metastable to ground separation (Δλ = | λ_2,3,4−λ₁ |) increases with increasing wavelength of the lines. For the inner shell transitions, the separations are rather small with Δλ(n_j = 1−2) ≲ 0.14 Å. For the other two sets of transitions, the separations are relatively large, ranging from ~ 0.3 Å (for λ ~ 10 Å) to ~ 200 Å (for λ ~ 1000 Å), which are larger than the spectral resolution of current X-ray grating spectrometers. The metastable lines themselves, on the other hand, are closer to each other (≲ 0.2 Å for lines in the X-ray band and ≲ 3 Å for lines in the UV band). This is simply due to the fact that the three metastable levels are the fine structure splitting of the same ³P term.

3.2. B-like

For B-like ions, only the first excited level 2s²2p (²P_3/2) can be significantly populated up to a factor of two above the ground level population, as shown in Fig. 3. None of the rest of the excited levels is significantly populated ( < 0.01% of the ground level population) in a photoionized plasma with density n_H ≲ 10²⁰ m^-3. At a sufficiently high density, the ratio of metastable to ground level population follows the Boltzmann distribution (meanwhile the plasma is partially in local temperature equilibrium).

Fig. 3 Similar to Fig. 2, but for B-like ions.

Characteristic absorption lines from the ground (Level 1) and metastable level (Level 2) in B-like ions are listed in Table 3. Similar to the Be-like isoelectronic sequence (Sect. 3.1), among the three transitions for the same ion, the inner shell transition (n_j = 1−2) has the shortest wavelength and the n_j = 2−2 transition has the longest wavelength. The metastable to ground separation (Δλ) is negligible for the inner shell transition (n_j = 1−2). For the other two sets of transitions, the separations between the ground and metastable lines are Δλ(n_j = 2−3) ~ 0.15−0.30 Å and Δλ(n_j = 2−2) ~ 0.2−2.7 Å, respectively, which are larger than the spectral resolution of current X-ray grating spectrometers (see, e.g., Table 1 in Kaastra 2017).

3.3. C-like

For C-like ions, the first two excited levels 2s² 2p² (³P₁ and ³P₂) can be significantly populated up to a factor of a few of the ground level population, as shown in Fig. 4. The next two excited levels 2s² 2p² (¹D₂ and ¹S₀) can also be populated up to tens of percent of the ground level population. The rest of the excited levels are not significantly populated ( < 0.01% of the ground level population) in a photoionized plasma with density n_H ≲ 10²⁰ m^-3.

Fig. 4 Similar to Fig. 2, but for C-like ions.

Since the metastable level 2s²2p² (¹S₀) is only significantly populated at rather high density (the right panel in Fig. 4), for simplicity only transitions from the ground (Level 1) and the first three metastable levels (Level 2−4) are listed in Table 4. Again, the inner shell transition (n_j = 1−2) corresponds to the line with the shortest wavelength and the n_j = 2−2 transition corresponds to the line with the longest wavelength. For the inner shell transition, the ground to metastable separation (Δλ) is negligible. For the other two transitions (n_j = 2−2 and 2−3) that we listed here, it is easier (with Δλ ≳ 0.1 Å) to distinguish lines from the ground 2s² 2p² (³P₀) and lines from the third excited level 2s² 2p² (¹D₂). Lines from the ground and the first two excited levels are close to each other since the lower levels are the fine structure splitting of the same term ³P. Between the n_j = 2−3 and n_j = 2−2 transitions, it is more difficult to distinguish lines from the ground and the first two excited levels for the former, e.g., C i lines around λ ~ 1277 Å, N ii lines around λ ~ 533 Å.

3.4. Summary

For Be-like ions, the first three excited levels 2s2p (³P₀₋₂) can be significantly populated (≳ 1% of the ground level population) when n_H ≳ 10¹⁴ m^-3. Moreover, the third excited level 2s2p (³P₂) can be populated more easily at such a high density. For B-like ions, merely the first excited level 2s²2p (²P_3/2) can be used as a density probe. For C-like ions, the metastable levels 2s²2p² (³P₁ and ³P₂) can be more easily populated than 2s²2p² (¹D₂ and ¹S₀). On the other hand, the transition from the 2s²2p² (¹D₂) level can be more easily distinguished from the transition from the ground level 2s²2p² (³P₀).

The metastable levels in Be-like ions are less populated than B-like and C-like ions at the same density. For all three isoelectronic sequences, it is rather difficult to distinguish inner shell transitions from the ground and metastable levels.

The wavelengths and oscillator strengths can be obtained from the SPEX atomic code and table (SPEXACT) v3.04. We cross-checked the wavelengths of the lines in Tables 2 to 4 with those from NIST v5.3 (Kramida et al. 2015) when available.

4. Discussion

4.1. Ionization parameter dependence

In Sect. 3, for each ion, the level population is calculated given a certain ionization parameter. Here we discuss the level population dependence on both ionization parameter and density. Since the majority of the n_j = 2−2 and n_j = 2−3 transitions from ions in the same isoelectronic sequence share the same lower and upper levels (Tables 2 to 4), we take these transitions from Be-like Si xi to C-like Si ix for this exercise. The ionization dependence can be applied to the rest of the ions in the same sequence. We note that inner shell transitions (n_j = 1−2) are excluded since it is difficult to distinguish lines from ground and metastable levels in any case (Sect. 3.4). We choose three different ionization parameters for each ion, corresponding to the maximum ion concentration and ~ 90% of the maximum ion concentration.

As shown in Fig. 5, the optical depth at the line center (τ₀ ∝ N_if) varies with ionization parameter and density. Similarly, the metastable to ground equivalent width ratios (EW_meta/EW_ground) involving the 2s2p (³P₂) metastable level in Be-like ions and the 2s²2p² (¹D₂) metastable level in C-like ions are sensitive to both ionization parameter and density of the plasma. Nonetheless, the ratios of metastable to ground equivalent width involving the 2s²2p (²P_3/2) metastable level in the B-like sequence and the 2s²2p² (³P₁ and ³P₂) metastable levels in the C-like sequence are only sensitive to the density of the plasma, which makes them ideal density probes.

We caution that adopting a different ionizing SED yields a different ionization balance for the plasma. That is to say, the exact value of the ionization parameters where the ion concentration reaches a maximum or 90% of the maximum in Fig. 6 differs for different SEDs (Table 5). The SED adopted here is representative of a typical Seyfert 1 galaxy. Of course, the exact values of τ₀ and EW also depend on the hydrogen column density (N_H) and turbulence velocity (v_b) of the ionized outflow.

4.2. Domain of density and ionization parameter diagnostics

Ions from different isoelectronic sequences cover an extensive area in the n_H – ξ two-dimensional parameter space. In Fig. 7, each box corresponds to a certain ion. Within the box, the density of the plasma can be well constrained with lines from the metastable levels. Above (below) the box, only a lower (upper) limit can be obtained.

The box width gives the range of ionization parameters where lines from the ground and metastable levels are expected to be detected. The lower and upper boundaries correspond to 1 /e of the maximum ion concentration using the AGN1 SED. Again we caution that a different ionizing SEDs yield different ionization parameters³.

The box height corresponds to the density of the plasma. The lower and upper boundaries of the box corresponds to 10% and 99% of the maximum ratio of metastable to ground level population. Only the 2s²2p (²P_3/2) metastable level in B-like ions and the 2s²2p² (³P₁ and ³P₂) metastable levels in C-like ions are used for the calculation since their corresponding metastable-to-ground-EW ratios barely depend on the ionization parameter (Fig. 6).

Be-like ions are not included in Fig. 7 because the level population depends on both the ionization parameter (thus SED dependent) and density, and because the metastable levels are less populated than B-like and C-like ions at the same density (Sect. 3.4).

In addition, we also show the distance of the plasma inferred from the definition of ionization parameter (Eq. (2)). Within the same isoelectronic sequence, high-Z ions probe higher density, higher ionization parameters and smaller distances, while low-Z ions measure lower density, lower ionization parameters, and larger distances. For ions in the same isonuclear sequence (i.e., the same Z), less ionized ions can probe lower density, thus larger distances.

5. Density diagnostics for the ionized outflow in NGC 5548

NGC 5548 is the archetypal Seyfert 1 galaxy. As such, it exhibits all the typical spectral features seen in type 1 Seyfert galaxies. The broadband (from optical to hard X-ray) spectral energy distribution (SED) and various properties from the multiphase ionized outflow are well studied (e.g., Kaastra et al. 2014; Mehdipour et al. 2015b; Arav et al. 2015; Ebrero et al. 2016). NGC 5548 was observed with Chandra in January 2002 with both high- and low-energy transmission grating spectrometers (HETGS and LETGS), which allow us to study the X-ray absorption features in a wide wavelength range (~2−60 Å). Moreover, the 2002 spectra have the best signal-to-noise ratio of all the high-resolution grating spectra of NGC 5548.

Here we reanalyze these spectra to search for density diagnostic lines. The HETGS (ObsID: 3046, with ~ 150 ks exposure) and LETGS (ObsID: 3383 and 3045, with ~ 340 ks total exposure) spectra are optimally binned (Kaastra & Bleeker 2016) and fitted simultaneously. Fits to LEG, MEG, and HEG spectra are restricted to the 11−60 Å, 4−19 Å, and 1.8−10 Å wavelength range, respectively.

Six PION components are used to account for the six components in the ionized outflow (Kaastra et al. 2014; Ebrero et al. 2016). Assuming a low plasma density (n_H = 10⁶ m^-3 or 1 cm^-3, kept frozen) for all six PION components, the best-fit C-statistics are 5146.0 (denoted as the baseline C-statistics in the following) with a degree of freedom of 4843. The hydrogen column density (N_H), ionization parameter (ξ), turbulent velocity (v_b), and outflow velocity (v_out) for the PION components (Table 6) are consistent with values found in previous studies by Kaastra et al. (2014) and Ebrero et al. (2016).

For each PION component, we then vary its density from n_H = 10⁶ m^-3 to 10²⁰ m^-3, with one step per decade. All the other parameters are kept frozen. The deviation of C-statistics (ΔC) from the baseline fit are demonstrated in Fig. 8.

For the least ionized component A, the X-ray spectra with λ ≲ 60 Å are insensitive to the density. The density sensitive lines that can be distinguished from the ground absorption lines of O iv (B-like) and Ne v (C-like) are at longer wavelength range (λ ≳ 100 Å, Tables 3 and 4). That is to say, high-resolution UV spectra are required to determine the density for this component.

Meanwhile, we find a lower limit (at the confidence level of 3σ) of n_H ≳ 10¹³ m^-3 for component B. Figure 9 shows the LETGS spectrum in the neighborhood of Si ix absorption lines (~ 56 Å), where the C-statistics are improved at high density. When n_H ≳ 10¹³ m^-3, the population of the ground level 2s²2p² (³P₀) decreases, while the population of the metastable levels 2s²2p² (³P₁ and ³P₂) increases (Fig. 4). Accordingly, the ground absorption line at 56.15 Å (in the observed frame) is shallower, while the metastable absorption lines at 56.20 Å and 56.25 Å are deeper. The 3σ upper limit of the distance of component B is accordingly 0.23 pc. The inferred density and distance disagree with the results reported in Ebrero et al. (2016), where a timing analysis (Sect. 1) is used. For the photoionized absorber component B, Ebrero et al. (2016) report n_H ∈ (2.9, 7.1) × 10¹⁰ m^-3 and d ∈ (13, 20) pc, both at the confidence level of 1σ. The upper limits of density is based on the non-detection of variability on smaller timescales. The authors also note that there is a marginal hint of recombination timescale of ~ 4 and ~ 60 days, which would indicate a density lower limit (1σ) of ~ 10¹⁰⁻¹¹ m^-3, which would agree more with our results.

Fig. 5 Optical depth at the line center (τ0) for characteristic lines from ground and metastable levels in Be-like Si xi to C-like Si ix. The assumptions (SED, geometry, column density, turbulence, etc.) of the calculation are described in Sect. 2.

Fig. 6 Equivalent width (EW) ratios for characteristic lines from ground and metastable levels in Be-like Si xi to C-like Si ix. The assumptions (SED, geometry, column density, turbulence, etc.) of the calculation are described in Sect. 2. For those lines with non-negligible optical depth (τ0 ≳ 1), the exact EWs depend on the line broadening profiles.

Fig. 7 Domain of density and ionization parameter where metastable absorption line from B-like (solid) and C-like (dotted) ions can be used for diagnostics in the case of photoionized equilibrium (see Sect. 4.2 for a detailed description). Dashed lines indicate the distance of the photoionized plasma with respect to the central engine. L37 is the 1 to 1000 Ryd luminosity in units of 1037 W.

Fig. 8 Deviation of C-statistics (ΔC) from the baseline fit (C-stat. = 5146.0 and d.o.f. = 4843) with varying plasma density (nH) for each photoionized absorber component (A–F). Component A refers to the least ionized photoionized absorber and F refers to the most ionized photoionized absorber (Table 6).

Fig. 9 PION modeling of the LEG spectrum of NGC 5548 with nH = 1010 m-3 (upper panel) and 1014 m-3 (lower panel). The spectral bin size is 0.025 Å. The overall C statistics is 5146.0 (d.o.f. = 4843) at low density and 5124.7 (d.o.f. = 4842) at high density. The two density sensitive metastable absorption lines of Si ix (C-like), from component B, at 55.20 Å and 56.25 Å (in the observed frame) are labeled with ∗.

Fig. 10 PION modeling of the spectra of NGC 5548 with nH = 1010 m-3 (left panels) and 1019 m-3 (right panels). The spectral bin sizes are 0.025 Å (for LEG) and 0.01 Å (for MEG), respectively. The overall C statistics is 5146.0 (d.o.f. = 4843) at low density and 5158.9 (d.o.f. = 4842) at high density. The density sensitive metastable absorption lines of Fe xxii (B-like) and Fe xxi (C-like), from component F, are labeled with ∗.

For components C–F, upper limits are obtained with n_H ≲ 10¹⁹ m^-3 (above 3σ for components C, E, and F) and ≲ 10²⁰ m^-3 (2.6σ for component D). We show in Fig. 10 the spectra in the neighborhood of Fe xxii (B-like) and Fe xxi (C-like) with the density of component F set to n_H = 10¹⁰ m^-3 (left panels) and 10¹⁹ m^-3 (right panels). At high density the metastable absorption lines at 12.08 Å (Fe xxii), at 12.50 Å (Fe xxi), and at 12.60 Å (Fe xxi) are deeper. This overestimation of absorption lines contradicts the data and leads to poorer C-statistics. Of course, the spectra are crowded in this wavelength range, so that density diagnostics are challenging. The 3σ lower limits of the distance of components C−F are a few light-hours (~ 10¹² m). The obtained density and distance for components C−F do not contradict the results (n_H ≳ 10¹⁰ m^-3 and d ≲ 1 pc) reported in Ebrero et al. (2016).

Due to the narrow wavelength coverage and rather limited effective area of current grating instruments, and the lack of atomic data for N-like to F-like ions, density diagnostics using absorption lines from the metastable levels are not very effective. Once the N-like to F-like atomic data are included, a significant portion of the n_H – ξ parameter space can be covered. This, combined with the next generation of spectrometers on board Arcus (Smith et al. 2016) and Athena (Nandra et al. 2013), will allow us to identify the presence/absence of these density-sensitive absorption lines (Kaastra et al. 2017), thus tightly constraining the location and the kinetic power of AGN outflows. We refer the readers to Fig. 6 of Kaastra et al. (2017) for a simulated Arcus spectrum of NGC 5548, compared with the observed 2002 LETGS spectrum.

Acknowledgments

We thank the referee for the constructive comments and suggestions. SRON is supported financially by NWO, the Netherlands Organization for Scientific Research.

References

Appendix A: N-like Fe xx to F-like Fe xviii

Currently, in SPEXACT v3.04, the atomic data for the N-like to F-like isoelectronic sequences are lacking, except for Fe xx, Fe xix, and Fe xviii. Here we repeat the same analyses described above, but for N-like to F-like Fe. For both N-like Fe xx and O-like Fe xix, the first two excited levels can be populated up to 50%, compared to the ground level population, while only the first excited level of F-like Fe xviii can be populated significantly (Fig. A.1). Characteristic transitions from these ground and metastable levels are listed in Table A.2.

Comparing the density-sensitive metastable levels for Be-like Fe xxiii to F-like Fe xviii (Fig. A.1), we find that, in general, the metastable levels in C-like Fe xxi can be more easily populated when n_H ≳ 10¹⁶ m^-3. In addition, when n_H ≲ 10¹⁶ m^-3, the first excited level 2s²2p⁴ (³P₀) of O-like Fe xix can be populated up to 10% of the ground level population,

Fig. A.1 Ratio of metastable to ground level population as a function of density of the Fe isonuclear sequence (from Be-like to F-like) at the temperature of maximum ion concentration in ionization equilibria. The configuration and notation of the ground (Level 1) and metastable levels (Level 2 and 3) are listed in Table 1.

which has been pointed out in Kaspi et al. (2001). Nonetheless, the density-sensitive line (13.420 Å) of O-like Fe xix is weaker than the ground level absorption lines (13.521 Å and 13.492 Å). Meanwhile, the density-sensitive line (13.420 Å) of O-like Fe xix is sensitive to both ionization parameter and density (Fig. A.2). Additionally, from the observational point of view, we caution that the n_j = 2−3 transitions of Fe xviii at 14.203 Å (ground) and 14.121 Å (metastable) might be blended with the O viii absorption edge at 14.228 Å.

Fig. A.2 Equivalent width (EW) ratios for characteristic absorption lines of O-like Fe xix. The two lines with λ =13.521 Å and 13.492 Å are from the ground level. The line with λ =13.420 Å is from the first excited level. Dashed (ξ = 2.3), solid (ξ = 2.4, where ion concentration reaches its maximum), and dotted (ξ = 2.5) lines indicate different ionization parameters of the photoionized plasma.

Appendix B: Comparison of the level population calculation with CHIANTI

CHIANTI (Del Zanna et al. 2015) is a widely used atomic code for analyzing emission line spectra from astrophysical sources. Similar to SPEX, CHIANTI also provides detailed calculations (e.g., level populations). Therefore, here we compare the level population calculations for collisional ionized equilibrium (CIE) plasmas, using the latest version of CHIANTI v8.0.8 (with atomic database v8.0.6⁴) and SPEX v3.04 (with SPEXACT v3.04) to show the effects of different atomic codes.

For simplicity, we only compare the population of the first five levels (i.e., the ground level and the first four excited levels) of the Fe isonuclear sequence from He-like Fe xxv to Ne-like Fe xvii. The level populations are calculated in both codes using the same abundance table (Lodders et al. 2009), the same ionization balance (Bryans et al. 2009), and the same plasma temperature (where the ion concentration reaches its maximum). Figure B.1 shows the level population ratio with respect to the ground as a function of plasma density ranging from 10⁶ m^-3 to 10²⁰ m^-3 (or 10¹ cm^-3 to 10¹⁴ cm^-3).

Fig. B.1 Level population ratio for the first four excited levels (level indices 2–5) of the Fe isonuclear sequence from He-like Fe xxv to Ne-like Fe xvii for collisional ionized equilibrium (CIE) plasma, using SPEXACT v3.04 (solid line) and CHIANTI v8.0.8 (dashed line). The CHIANTI atomic database v8.0.6 is used.

For He-like Fe xxv and Li-like Fe xxiv, the level population ratios provided by the two codes are consistent with each other. For Be-like Fe  xxiii to Ne-like Fe xvii, the level population ratios share the same increasing and/or flattening trend, whereas the two codes yield slightly different level population ratios (a factor of few). This is not totally unexpected, given the fact that the two codes do not use the same atomic data for individual atomic processes, and the two codes use different means of implementing the atomic data for the calculation (interpolation or parameterization). For instance, the CHIANTI atomic database includes various atomic data with n ≤ 5 for almost all ions with Z ≤ 30 in the He-like to Ne-like isoelectronic sequences, while the SPEXACT includes atomic data with n ≤ 8 for He-like to C-like ions with Z ≤ 30, and only Fe ions in the N-like to Ne-like isoelectronic sequences. The atomic data collected in both codes are state-of-the-art, yet still incomplete and not 100% accurate. Furthermore, the SPEX code parameterizes almost all the atomic data (e.g., Mao & Kaastra 2016; Urdampilleta et al. 2017), while CHIANTI uses interpolation instead.

In short, considering only the systematic uncertainties introduced by the different atomic data used in SPEX and CHIANTI, with the same density diagnostics in Be-like to Ne-like ions, the two codes are expected to estimate the plasma density to a similar order of magnitude. However, it is not clear which part(s) of the atomic data and/or code contribute to what percentage of the total “error” budget (i.e., the discrepancy between the codes) for the level population calculation of a specific level in a specific ion.

All Tables

All Figures

	Fig. 1 Ion concentration of the Si isonuclear sequence (H-like to Ne-like) as a function of the ionization parameter (in units of 10-9 W m, i.e., erg s-1 cm).
In the text

	Fig. 2 Level population ratios as a function of plasma density (in the range of 106−20 m-3 or 100−14 cm-3) at ionization parameter of maximum ion concentration in the ionization balance. The configuration of the ground () and metastable levels () are listed in Table 1.
In the text

	Fig. 3 Similar to Fig. 2, but for B-like ions.
In the text

	Fig. 4 Similar to Fig. 2, but for C-like ions.
In the text

	Fig. 5 Optical depth at the line center (τ0) for characteristic lines from ground and metastable levels in Be-like Si xi to C-like Si ix. The assumptions (SED, geometry, column density, turbulence, etc.) of the calculation are described in Sect. 2.
In the text

	Fig. 6 Equivalent width (EW) ratios for characteristic lines from ground and metastable levels in Be-like Si xi to C-like Si ix. The assumptions (SED, geometry, column density, turbulence, etc.) of the calculation are described in Sect. 2. For those lines with non-negligible optical depth (τ0 ≳ 1), the exact EWs depend on the line broadening profiles.
In the text

	Fig. 7 Domain of density and ionization parameter where metastable absorption line from B-like (solid) and C-like (dotted) ions can be used for diagnostics in the case of photoionized equilibrium (see Sect. 4.2 for a detailed description). Dashed lines indicate the distance of the photoionized plasma with respect to the central engine. L37 is the 1 to 1000 Ryd luminosity in units of 1037 W.
In the text

	Fig. 8 Deviation of C-statistics (ΔC) from the baseline fit (C-stat. = 5146.0 and d.o.f. = 4843) with varying plasma density (nH) for each photoionized absorber component (A–F). Component A refers to the least ionized photoionized absorber and F refers to the most ionized photoionized absorber (Table 6).
In the text

Fig. 9 PION modeling of the LEG spectrum of NGC 5548 with nH = 1010 m-3 (upper panel) and 1014 m-3 (lower panel). The spectral bin size is 0.025 Å. The overall C statistics is 5146.0 (d.o.f. = 4843) at low density and 5124.7 (d.o.f. = 4842) at high density. The two density sensitive metastable absorption lines of Si ix (C-like), from component B, at 55.20 Å and 56.25 Å (in the observed frame) are labeled with ∗.

In the text

Fig. 10 PION modeling of the spectra of NGC 5548 with nH = 1010 m-3 (left panels) and 1019 m-3 (right panels). The spectral bin sizes are 0.025 Å (for LEG) and 0.01 Å (for MEG), respectively. The overall C statistics is 5146.0 (d.o.f. = 4843) at low density and 5158.9 (d.o.f. = 4842) at high density. The density sensitive metastable absorption lines of Fe xxii (B-like) and Fe xxi (C-like), from component F, are labeled with ∗.

In the text

	Fig. A.1 Ratio of metastable to ground level population as a function of density of the Fe isonuclear sequence (from Be-like to F-like) at the temperature of maximum ion concentration in ionization equilibria. The configuration and notation of the ground (Level 1) and metastable levels (Level 2 and 3) are listed in Table 1.
In the text

	Fig. A.2 Equivalent width (EW) ratios for characteristic absorption lines of O-like Fe xix. The two lines with λ =13.521 Å and 13.492 Å are from the ground level. The line with λ =13.420 Å is from the first excited level. Dashed (ξ = 2.3), solid (ξ = 2.4, where ion concentration reaches its maximum), and dotted (ξ = 2.5) lines indicate different ionization parameters of the photoionized plasma.
In the text

	Fig. B.1 Level population ratio for the first four excited levels (level indices 2–5) of the Fe isonuclear sequence from He-like Fe xxv to Ne-like Fe xvii for collisional ionized equilibrium (CIE) plasma, using SPEXACT v3.04 (solid line) and CHIANTI v8.0.8 (dashed line). The CHIANTI atomic database v8.0.6 is used.
In the text