Insights into Density and Location Diagnostics of Photoionized Outflows in X-Ray Binaries

The observed emission and absorption spectral lines in plasmas depend on the populations of the corresponding energy levels i and j (i < j). Therefore, for spectral modeling a detailed computation of their number densities n_i , n_j is needed. The total emission line intensity (ph s⁻¹ cm⁻³) at the source is

$$\begin{eqnarray}&&{I}_{{ji}}={n}_{j}{A}_{{ji}},\end{eqnarray} \tag{ 2 }$$

where A_ij (s⁻¹) is the Einstein spontaneous emission coefficient. The absorption line intensity is

$$\begin{eqnarray}&&{I}_{{ij}}={I}_{{ij},0}{e}^{-\int {n}_{i}{\sigma }_{{ij}}{dl}},\end{eqnarray} \tag{ 3 }$$

where I_ij,0 is the incident intensity around the line, σ_ij (cm²) is the PE cross section, and the integral is over the line of sight. In the general case, a proper computation of these densities requires considering both CE and PE.

In hot dense plasmas, the electron impact collisions, which depend on the electron density n_e and temperature T_e , are predominantly responsible for ionizing and exciting the ions. The CE rate coefficient (cm³ s⁻¹) from level i to j can be approximated by the van Regemorter (1962) formula:

$$\begin{eqnarray}&&{S}_{{ij}}({T}_{e})=1.7\times {10}^{-3}{f}_{{ij}}\overline{g}{E}_{{{ij}}_{[{eV}]}}^{-1}{T}_{e[K]}^{-1/2}{e}^{-\tfrac{{E}_{{ij}}}{{{kT}}_{e}}},\end{eqnarray} \tag{ 4 }$$

where E_ij = E_j − E_i is the line energy in eV, and f_ij is the oscillator strength of the transition; $\overline{g}$ is the averaged excitation Gaunt factor, which is of the order of unity. From the detailed balance argument in thermodynamic equilibrium, the rate coefficient $S{{\prime} }_{{ji}}$ of the opposite process, collisional deexcitation from j to i is related to S_ij by

$$\begin{eqnarray}&&S{{\prime} }_{{ji}}=\displaystyle \frac{{g}_{i}}{{g}_{j}}{e}^{\tfrac{{E}_{{ij}}}{{{kT}}_{e}}}{S}_{{ij}},\end{eqnarray} \tag{ 5 }$$

where g_i and g_j are the statistical weights of levels i, j, respectively.

In photoionized plasma, an external radiation source dominates the ionization and excitation processes. The PE rate (s⁻¹) from level i to j is given by

$$\begin{eqnarray}&&{R}_{{ij}}^{\mathrm{PE}}=\int {F}_{\nu }(\nu ){\sigma }_{{ij}}^{\mathrm{PE}}d\nu ,\end{eqnarray} \tag{ 6 }$$

where F_ν (ph s⁻¹ cm⁻² Hz⁻¹) is the photon flux density, and the cross section ${\sigma }_{{ij}}^{\mathrm{PE}}$ (cm²) is given by

$$\begin{eqnarray}&&{\sigma }_{{ij}}^{\mathrm{PE}}=\displaystyle \frac{\pi {e}^{2}}{{m}_{e}c}{f}_{{ij}}\phi (\nu ),\end{eqnarray} \tag{ 7 }$$

where e, m_e are the electron charge and mass, respectively, and ϕ(ν) is a normalized narrow line profile around the transition frequency. F_ν (ν) can be rewritten as

$$\begin{eqnarray}&&{F}_{\nu }(\nu )={{hF}}_{E}(E)=h\displaystyle \frac{{L}_{E}(E)}{4\pi {r}^{2}}=\displaystyle \frac{{{hLf}}_{E}(E)}{4\pi {r}^{2}E}=\displaystyle \frac{{n}_{e}\xi {{hf}}_{E}(E)}{4\pi E},\end{eqnarray} \tag{ 8 }$$

where h is the Planck constant, F_E is given in units of ph s⁻¹ cm⁻² keV⁻¹, L_E (ph s⁻¹ keV⁻¹) is the photon luminosity density, and f_E (E) is the normalized spectral energy distribution (SED), usually taken from 1–1000 Ry. In Equation (8) we used the expression for ξ of Equation (1) with the energy luminosity L = ∫EL_E dE. By substituting Equations (7) and (8) into Equation (6) one gets

$$\begin{eqnarray}&&{R}_{{ij}}^{\mathrm{PE}}={n}_{e}\displaystyle \frac{{{he}}^{2}\xi {f}_{{ij}}{f}_{E}({E}_{{ij}})}{4{m}_{e}{{cE}}_{{ij}}}.\end{eqnarray} \tag{ 9 }$$

In this section, we compare the relative roles of CE and PE in populating level j from level i, for different ionizing sources and absorbing plasma conditions. Before solving the full collisional radiative model with PE, which takes into account all populating and depopulating processes of all levels, we focus on a two-level system. The ratio of the CE and PE rates (Equations (4) and (9)) for a specific i to j transition is

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{{ij}}^{\mathrm{PE}}}{{n}_{e}{S}_{{ij}}}=\displaystyle \frac{{{he}}^{2}\xi {f}_{E}({E}_{{ij}}){f}_{{ij}}}{4{S}_{{ij}}{E}_{{ij}}{m}_{e}c}=8.7\displaystyle \frac{\xi }{{10}^{3}}{\left(\displaystyle \frac{{T}_{e}}{{10}^{6}K}\right)}^{1/2}{f}_{E}({{kT}}_{e}),\end{eqnarray} \tag{ 10 }$$

where we assumed E_ij = kT_e and $\overline{{\bf{g}}}={\bf{1}}$. The above ratio is computed here for two types of SEDs, a power law (PL) with photon index 1.5 < Γ < 2.5, typical of AGNs, and a multiple-T blackbody disk spectrum with maximal source temperature 0.3 keV < kT_s < 3 keV, typical of XRBs. We compute this ratio for two different T_e values and for ξ = 10³, which is the ionization parameter of peak formation of the Fe⁺²¹ ion (Kallman et al. 2009; Section 3 therein), but the scaling with ξ is trivial. The ratios computed from Equation (10) are presented in Table 1. The disk spectra are computed with the diskbb model (Mitsuda et al. 1984) in Xspec (Arnaud 1996). Note that since f_E ∝ E^−Γ+1, the special case of Γ = 1.5 eliminates the ${T}_{e}^{1/2}$ dependence. The tabulated ratios show that the PE/CE rate ratios range between 0.3 and 20. This demonstrates that both CE and PE can be significant and need to be considered in most cases. Specifically, the GRO J1655-40 outburst, which is discussed in Section 3, has a diskbb spectrum with kT_s = 1.3 keV (Miller et al. 2008), where the ratio of Equation (10) is of the order of unity (bold text in Table 1).

Another process that can populate excited levels is radiative recombination (RR), in which an electron is captured into a bound state and a photon is emitted. The rate coefficient for RR to the ground level α^RR (cm³ s⁻¹) is given by Cillié (1932):

$$\begin{eqnarray}&&{\alpha }^{{RR}}=3.2\times {10}^{-6}\displaystyle \frac{{e}^{{E}_{\mathrm{ion}}/{{kT}}_{e}}}{{{\bf{n}}}^{3}{T}_{e}^{3/2}}{E}_{1}\left(\displaystyle \frac{{E}_{\mathrm{ion}}}{{{kT}}_{e}}\right),\end{eqnarray} \tag{ 11 }$$

where n is the principal quantum number, E_ion is the ionization energy, and E₁ is the exponential integral:

$$\begin{eqnarray}&&{E}_{1}(x)={\int }_{x}^{\infty }\displaystyle \frac{{e}^{-t}}{t}{dt}.\end{eqnarray} \tag{ 12 }$$

The ratio between the PE and RR rates is (Equations (9), (11))

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{R}_{{ij}}^{\mathrm{PE}}}{{n}_{e}{\alpha }^{{RR}}} & = & 1.7\times {10}^{-6}\displaystyle \frac{\xi {f}_{E}({E}_{{ij}}){f}_{{ij}}{T}_{e}^{3/2}}{{E}_{{ij}}{E}_{1}\left(\tfrac{{E}_{\mathrm{ion}}}{{{kT}}_{e}}\right)}{e}^{-{E}_{\mathrm{ion}}/{{kT}}_{e}}\\ & = & 3.2\times {10}^{6}\displaystyle \frac{\xi }{{10}^{3}}{\left(\displaystyle \frac{{T}_{e}}{{10}^{6}K}\right)}^{1/2}{f}_{E}({{kT}}_{e}).\end{array}\end{eqnarray} \tag{ 13 }$$

In the last equality we assumed E_ij = E_ion = kT_e and f_ij = 0.1. The value of a few millions on the right-hand side indicates that recombination is likely negligible for all source conditions. The ratio, computed for two different spectral shapes and two temperatures (similar to Table 1) is presented in Table 2.

Both PE/CE and PE/RR ratios decrease with increasing temperature for a PL spectrum, and increase with temperature for a diskbb spectrum. This is due to the energy dependence of f_E , which can decrease faster or slower with E, compared to ${T}_{e}^{1/2}$ (Equations (10) and (13)). This two-level model indicates that in all cases where kT_e ∼ E_ion, including GRO J1655-40 (bold text in Table 2), the PE/RR ratio is ∼10⁵ − 10⁶. Hence, RR can be safely neglected, while PE must be considered.

We can also consider a lower temperature of T_e = 10⁵ K, for which ξ will also be lower ξ ≈ 10^1.5 (see Figure 6 in Kallman et al. 2004). For a PL spectrum, the PE/CE rate ratios are between 0.1 and 4 and the PE/RR ratios are (0.05–1.4) × 10⁶. This demonstrates that at lower temperatures as well, CE and PE are both important, while RR is not. T_e = 10⁵ K is not considered here for diskbb spectra, as typical X-ray binaries have much higher radiation temperatures kT_s > 0.3 keV.

PI transitions can also (populate or) depopulate the excited levels of the ion. The PI rate from level i is given in a similar fashion to that of the PE rate (Equation (6)):

$$\begin{eqnarray}&&{R}_{i}^{{PI}}=\int {F}_{\nu }(\nu ){\sigma }_{i}^{{PI}}d\nu .\end{eqnarray} \tag{ 14 }$$

${\sigma }_{i}^{{PI}}$(cm²) is the PI cross section, which is given by Karzas & Latter (1961):

$$\begin{eqnarray}&&{\sigma }_{i}^{{PI}}=\left(\displaystyle \frac{64\pi {ng}}{3\sqrt{3}}\right)\alpha {a}_{0}^{2}{\left(\displaystyle \frac{{E}_{\mathrm{ion}}}{E}\right)}^{3},\end{eqnarray} \tag{ 15 }$$

where g is the Gaunt factor, which is of the order of unity, a₀ = ℏ²/m_e e² is the Bohr radius, and α = e²/ℏ c is the fine structure constant. To demonstrate the difference in orders of magnitude between PE and PI, we evaluate the rates at E = E_ion = 1 keV (Equations (6), (7), (8), (14), (15)):

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{R}_{{ij}}^{\mathrm{PE}}}{{R}_{i}^{{PI}}}(E={E}_{\mathrm{ion}}=1\,\mathrm{keV})=\displaystyle \frac{0.002\times {f}_{E}(1\,{keV})}{8\times {10}^{-18}}\\ \,=\,2.5\times {10}^{13}.\end{array}\end{eqnarray} \tag{ 16 }$$

This result shows that the PI population of excited levels is insignificant compared to PE. This is not surprising given that PE is a resonant process. The PI rates are computed fully in our numerical analysis in Section 3, but there too they are negligible.

In order to solve the populations of all levels of a given ion, we refer to a collisional radiative model with PE. The level i population n_i is solved from a set of linear rate equations:

$$\begin{eqnarray}&&\displaystyle \begin{array}{c}\frac{{{dn}}_{i}}{{dt}}=0=\displaystyle \sum _{j\lt i}\left({n}_{j}{n}_{e}{S}_{{ji}}+{n}_{j}{R}_{{ji}}^{\mathrm{PE}}\right)+\displaystyle \sum _{k\gt i}\left({n}_{k}{A}_{{ki}}+{n}_{k}{n}_{e}S{{\prime} }_{{ki}}\right)\\ -\,\displaystyle \sum _{j\lt i}\left({n}_{i}{A}_{{ij}}+{n}_{i}{n}_{e}S{{\prime} }_{{ij}}\right)-\displaystyle \sum _{k\gt i}\left({n}_{i}{n}_{e}{S}_{{ik}}+{n}_{i}{R}_{{ik}}^{\mathrm{PE}}\right)-{n}_{i}{R}_{i}^{PI}\end{array}\,.\end{eqnarray} \tag{ 17 }$$

The first and second terms represent the populating processes of level i from lower and higher levels. The third and fourth terms represent the depopulating processes of level i to lower and higher levels. The last term is the depopulating PI. In a steady state, i.e., ${{dn}}_{i}/{dt}=0$, Equation (17) reduces to a set of linear algebraic equations, which is solved to get the set of population densities {n_i }. These results are presented in Section 3.

The n₂/n₁ population ratio computed with CE, but not PE, is shown in Figure 1 (left panel), assuming T = 10⁶ K. It shows a critical density of n_e,c = 6 × 10¹² cm⁻³ in which n₂/n₁ starts to increase nonlinearly. At much higher n_e > 10²¹ cm⁻³, n₂/n₁ tends to the Boltzmann population ratio, ${n}_{2}/{n}_{1}\,=2\,\exp (-{E}_{21}/{kT})=1.722$, where the gas is in local thermodynamic equilibrium (LTE). The right-hand side panel of Figure 1 shows the total rates of the dominant transitions to level 2. At low densities n_e ≪ n_e,c , n₂ is populated predominantly by the weak collisional excitation from level 1 to 2 (2s²2p_1/2 − 2s²2p_3/2), which then decays back to level 1. As n_e approaches n_e,c , the $2s2{p}_{3/2}^{2}$ levels (7, 9, and 10) get increasingly populated, and their total transition rates to level 2 surpass that of 1 to 2, thus dramatically increasing the n₂/n₁ ratio. The crossing point of the 1 to 2 and 10 to 2 rates can be used to define the critical density of n_e,c = 6 × 10¹² cm⁻³. The increase in the n₂/n₁ ratio in the left-hand side of Figure 1 is similar to that found in Miller et al. (2008), while the results of (Mao et al. 2017, Figure 3 therein) indicate slightly lower densities to reach n₂/n₁ = 1.

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The next case we consider is a model with only the PE terms, and no CE. We used the measured diskbb spectrum of GRO J1655-40 with kT_s = 1.3 keV, and a luminosity of L_X = 5 × 10³⁷ erg s⁻¹ between 0.65 and 10 keV (Miller et al. 2008), which implies a bolometric luminosity of L = 5.7 × 10³⁷ erg s⁻¹. The n₂/n₁ population ratio is shown in Figure 2 as a function of distance from the ionizing source.

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Far from the source (r ≫ r_c = 10¹¹ cm) the exciting flux is low and the effect of PE is weak, thus n₂ ≪ n₁. As r decreases, the flux (∝r⁻²) increases, PE of excited levels increases, and radiative cascades increasingly populate level 2. Around the critical distance r_c = 10¹¹ cm, n₂ approaches n₁. At closer distances, PE from level 2 becomes significant, and balances the cascade effect. This is seen as the plateau on the low-r side of Figure 2. The right-hand side panel of Figure 2 shows the total transition rates of the dominant levels populating n₂ around r_c = 10¹¹ cm. Farther from the source (r > r_c ) level 16 (2s²3d_3/2), which is photoexcited from level 1 by X-rays (the 11.77 Å transition) dominates the population of level 2. Below the critical distance, n₂ increases on the account of n₁, thus PE from level 1 to level 16 diminishes, while the rate for PE from level 2 to level 17 (2s²3d_5/2, which is the X-ray 11.92 Å transition), and the decay back to level 2, dominate. Level 10 has a strong radiative transition to level 2, therefore it populates level 2 at all distances. The crossing point of the 16 to 2 (fed by 1–16) and 17–2 rates can be used to define the critical distance of r_c = 10¹¹ cm, where level 2 is significantly populated by PE. The plotted distances in Figure 2 actually represent fluxes, derived from the assumed L. If L is in fact higher than the assumed 5.7 × 10³⁷ erg s⁻¹, for example the central source is obscured to us but visible to the outflow, the derived distances would also be higher ($r\propto \sqrt{L}$).

Our final results for n₂/n₁ when including both CE and PE are presented in Figure 3, and compared to only-CE and only-PE models at two temperatures of T = 10⁶, 10⁷ K. The difference between T = 10⁶ K and T = 10⁷ K is statistically insignificant. The figure shows that neglecting PE results in an n_e value that is higher by a factor of 4, and thus a distance smaller by a (square root) factor of 2. The CE+PE model for Fe⁺²¹ assumes ξ = 10³; a higher value will move the CE+PE curve to lower densities since the PE rates will be higher (see Equation (9)).

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We remeasured the equivalent widths (EWs) of the 11.77 and 11.92 Å lines by fitting Voigt profiles (Figure 4, left panel) and obtained values of (8.4 ± 0.5) mÅ and (6.0 ± 0.5) mÅ, respectively, providing a ratio of 0.71 ± 0.07 (1σ). n₁ and n₂ were obtained by comparison of the EWs to a theoretical curve of growth (Figure 4, right panel). This results in n₂/n₁ = 0.76 ± 0.08, which is consistent with ${n}_{2}/{n}_{1}={0.71}_{-0.09}^{+0.05}$ in Miller et al. (2008), but not with n₂/n₁ ≅ 0.2 ± 0.1 in Tomaru et al. (2023), who presumably obtained the EWs of the lines from a global fit. Plotting the n₂/n₁ confidence region on the theoretical density curve in Figure 3, indicates n_e = (2.6 ± 0.5) × 10¹³ cm⁻³.

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Table 3 summarizes the n_e and r values for the three current models: only CE, only PE, and CE+PE; r was calculated using ξ. The present best estimate for density and distance are n_e = (2.6 ± 0.5) × 10¹³ cm⁻³ and r = (4.4 ± 0.4) × 10¹⁰ cm. Note how the eventual differences between models are a factor of a few, proving again the comparable roles of CE and PE, as inferred from the simplified the two-level hydrogenic approximation in Section 2.2. Miller et al. (2008) report a density value of n_e = (5 ± 1) × 10¹³ cm⁻³, when PE was neglected, and with a different collisional radiative model than ours. Tomaru et al. (2023) included both CE and PE, but obtained a density of n_e = (0.28 ± 0.09) × 10¹³ cm⁻³, because of their low n₂/n₁ value (see Figure 3). We stress that the different distances obtained by different authors depend appreciably also on their assumed ξ value, as r ∝ ξ^−1/2.

In the numerical models, when computing the case of only CE and only PE separately, the results are independent of ξ. Only when needing to scale CE versus PE does the model require an assumption for ξ. On the other hand, one can estimate a critical ξ_c from n_e,c and r_c (Figures 1 and 2). In the above case of Fe⁺²¹ we obtain ${\xi }_{c}=L/({n}_{e,c}{r}_{c}^{2})=950$, which is almost exactly the value of 10³ obtained from photoionization balance (Kallman et al. 2009), which we used above. This provides yet another consistency test for the comparable roles of CE and PE, once absorption from excited levels are identified, as also demonstrated in Equation (10) and in Table 1.