Outflowing OH⁺ in Markarian 231: The Ionization Rate of the Molecular Gas

Potential contamination by lines of other molecular species is indicated in Figure 1. In panel (a), the OH⁺ 2₁−1₁ line is truncated at blueshifted velocities due to the proximity of the [C ii] 158 μm line; we nevertheless use the OH⁺ absorption at central velocities to compare with model predictions in Section 3. In panel (b), the OH⁺ 2₃−1₂ line is closely blended with the NH 2₂−1₁ line at ≈153.1 μm, because the NH 2₃−1₂ line at ≈153.35 μm is clearly detected. Nevertheless, the overall profile is dominated by OH⁺, as previously suggested based on observations with the Infrared Space Observatory (González-Alfonso et al. 2008). We also indicate in panel (b) the position of the very high-lying H₃O⁺ 12⁻−12⁺ transition (E_lower ≈ 1400 K), detected in Arp 220 (GA13) but absent in Mrk 231; the nearby (intrinsically weak) OH⁺ 2₁−1₂ line is also not detected. In panel (c), we detect (blueshifted) absorption in the CH ground-state doublet at 149 μm, but it is resolved from the neighboring OH⁺ 2₁−1₀ line.

The redshift of Mrk 231 places the OH⁺ 3_J−2_J' lines (E_lower ≈ 140 K, λ_rest ∼ 102 μm) redward (λ_obs ∼ 106 μm) of the short-wavelength leakage region of PACS in the 100 μm range, allowing its observation (Figure 1(d)).⁹ Since NH has a spectrum similar to OH⁺, contamination is also found in the OH⁺ 3₄−2₃ line at 101.9 μm by the NH 3₂−2₁ line at 102.0 μm, because the blend of the NH 3₃−2₂ and 3₄−2₃ lines at 102.2 μm is well detected (panel (d)). Nevertheless, the 3₃−2₂ line of OH⁺ at 101.7 μm is strongly detected with evidence for blueshifted absorption, though contamination by the NH₃ 5_K−4_K lines (with K = 0, ..., 4) should be considered (see Section 3). The strength of the intrinsically weak OH⁺ 3₂−2₁ line at 101.3 μm is doubtful owing to the proximity of the H₂O 101 μm line and the uncertain continuum level. In panel (e), the OH⁺ 4₅−3₄ and 4₄−3₃ lines are detected at systemic velocities, with hints of absorption in the 4₃−3₂ component; the OH⁺ 4_J−3_J − 1 lines are well separated from the corresponding NH lines. In panels (f) and (g), the spectra around the undetected OH⁺ 3₂−2₂ and ${5}_{J}-{4}_{J-1}$ lines are shown, and they are used in Section 3 to check for overall consistency with the models. Panel (h) shows the SPIRE spectrum around the OH⁺ ground-state 1_J–0₁ lines. The 1₁−0₁ line is contaminated on its blue side by the CO 9−8 line at 289.12 μm, the 1₂−0₁ line has potential contamination by the ground-state NH 1₂−0₁ line at ≈307.6 μm, and the emission component of the 1₀−0₁ line is most likely contaminated by NH₂ 2₀₂−1₁₁ 5/2−3/2.¹⁰ The contributions by most of these species (NH, NH₂, NH₃, and CH) are included in our model for OH⁺ below (Section 3.1.1).

The OH⁺ spectra displayed in Figure 1 show clear indications of extremely high-velocity outflowing gas, together with absorption near systemic velocities (as found for OH; GA14b; GA17). In the excited lines observed with PACS, the clearest outflow signatures are seen in the relatively strong 2₂−1₁ and 2₃−1₂ lines, obtained with high signal-to-noise ratios (S/N; Figure 1(b)). These lines have lower level energies of E_lower ≈ 50 K (see Figure 1 in GA13 for energy level diagrams for the three O-bearing molecular ions). The relatively weak OH⁺ 2₂−1₂ and 2₁−1₀ lines at ∼148 μm also show a clear indication of blueshifted absorption (Figure 1(c)). The blueshifted line wings of the uncontaminated 2₂−1₁ 152.4 μm and 2₁−1₀ 148.7 μm transitions, showing absorption out to ≈ − 1000 km s⁻¹ from the line center, closely match the line wings of the (also excited) OH84 and OH65 line profiles (Figure 2). At around systemic velocities, the OH⁺ 2₂−1₁ absorption also closely matches the OH84 line profile, including the peak absorption blueshift of ∼ −120 km s⁻¹, while the optically thinner OH⁺ 2₁−1₀ line better matches the OH65 profile. This strongly suggests that both species are formed in the same gas components. The OH⁺ 2₃−1₂ line profile also shows blueshifted absorption reaching velocities of −1200 km s⁻¹, merging with the 2₂−1₁ line (Figure 1(b)).

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The 3₃−2₂ and 3₄−2₃ fine-structure components (Figure 1(d)), with E_lower ≈ 140 K, also show indications of blue absorption, because the potential contamination by NH₃ of the blueshifted wing of the 3₃−2₂ line is expected to be weak (Section 3). No clear blueshifted absorption in the OH⁺ profiles is seen in the ${4}_{J}-{3}_{J^{\prime} }$ transitions (Figure 1(e), ${E}_{\mathrm{lower}}\approx 280$ K).

The ground-state OH⁺ 1_J–0₁ lines detected with SPIRE also show evidence for outflowing gas (Figure 1(h)), with the emission above the continuum redshifted in the three lines relative to the expected line centers, and also with clear indications of blueshifted absorption. The latter is seen in the 1₂−0₁ line profile although NH probably also contributes to the broad absorption feature (Section 3), and in the 1₁−0₁ line profile in spite of the proximity of the CO 9−8 line. Some hints of absorption are also observed in the weakest transition, 1₀−0₁.

As noted above, the strongest low-lying 2_J−1_J' lines in Figures 1(b) and (c) show the peak absorption blueshifted by ∼120 km s⁻¹, while the higher-lying lines peak at central velocities. This behavior resembles that found in the OH lines of Mrk 231 (GA14b), where the peak absorption of the OH84 transition is also blueshifted but the higher-lying lines (e.g., OH65 and OH71) peak closer to the systemic velocities (see also Figure 2). In addition, no excited OH⁺ lines show redshifted emission above the continuum. This behavior is also similar to that seen in the OH line profiles. Only the ground-state OH119 and OH79 doublet profiles show prominent redshifted emission (GA14b; GA17), just as only the ground-state lines of OH⁺ show redshifted emission features (Figure 1(h); van der Werf et al. 2010). Finally, comparison of the OH⁺ PACS spectra in Mrk 231 and Arp 220 indicates that the absorption strength (relative to the continuum) at systemic velocities is similar in both sources (J. Fischer et al. 2018, in preparation), the important difference being the high-velocity blueshifted absorption in Mrk 231 that is absent in Arp 220.

Because of the non-Gaussian shapes of the OH⁺ lines, we measured the equivalent widths and fluxes of the PACS lines in Table 2 by numerically integrating the line shapes over two velocity ranges—around systemic velocities and, when detected, in the blueshifted wings. Based on the change in the slope of the OH⁺ PACS spectra (see, e.g., Figure 2), we adopted blueshifted and systemic velocity ranges of [−1000, −300] and [−300, 200] km s⁻¹, respectively; the latter upper limit is set to avoid strong contamination by neighboring lines on the red side. The 1σ uncertainties of the equivalent widths were evaluated from the ${\sigma }_{\mathrm{rms}}^{\mathrm{norm}}$ noise of the continuum-normalized PACS spectra as ${\sigma }_{\mathrm{rms}}^{\mathrm{norm}}\times {\rm{\Delta }}V/{n}^{1/2}$, where ΔV and n are the velocity coverage and number of channels over which W_eq is measured (González-Alfonso et al. 2015). Line detection is established at ≥3σ level. The equivalent widths and fluxes of the OH⁺ ground-state lines (Table 3) as measured with SPIRE are also integrated over two velocity ranges, corresponding to the emission and absorption features. We adopted a common velocity range ([−100, 850] km s⁻¹) for the emission features, enabling comparison of the strengths of the OH⁺ 1_J−0₁ emission lines. Uncertainties for the equivalent widths of the ground-state OH⁺ lines are dominated by the remaining ripples of the apodized spectrum, rather than by random noise. A typical semi-cycle of the ripples has a width similar to that of the observed lines, so that we conservatively adopt the area of one semi-cycle (averaged over several cycles measured close to the line) as the 1σ uncertainty of the equivalent width, and adopt a detection criterion of ≥2σ.

We caution that with the current spectral resolution of the apodized spectrum (∼650 km s⁻¹) the OH⁺ ground-state emission features will be contaminated by the absorption and vice versa, and that the ripples in the spectrum (Figure 1(h)) make our flux measurements somewhat uncertain (Table 3). Nevertheless, optically thin emission would imply relative fluxes for the 1₀−0₁, 1₁−0₁, and 1₂−0₁ lines of 1: 3: 5, which are the degeneracies of the upper levels.¹¹ The OH⁺ 1₀−0₁ and 1₁−0₁ emission features, however, show similar fluxes, which are a factor ∼1.4 weaker than the 1₂−0₁ emission (Table 3). This suggests both strong contamination of the 1₀−0₁ line by NH₂¹² and significant optical depth effects in the 1_J−0₁ lines.

Figures 5(a)–(g) presents our best model fit to the OH⁺ line profiles observed by PACS, and Table 4 lists the inferred OH⁺ column densities and abundances. Besides the pumping generated by the strong far-IR radiation field, we also include collisional excitation with H atoms (Lique et al. 2016) and with electrons (van der Tak et al. 2013), assuming a fractional abundance $X({e}^{-})={10}^{-3}$ (see Section 3.2.1); collisional excitation with H is found to dominate. In our model fitting, we also include predictions for NH, NH₂, NH₃, and CH. We find similar NH and OH⁺ column densities for the QC, and we adopt the same column density for NH as for OH⁺ in the outflowing components as well. Our model for NH₂ and NH₃ includes only the prediction for the QC, based on the fits to other unblended NH₂ and NH₃ lines detected across the PACS spectrum (J. Fischer et al. 2018, in preparation).

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The PACS data mainly constrain the column densities of the QC and HVC components, where OH⁺ is excited by the far-IR field. Model results for the HVC are shown in Figure 6, which compares the predicted OH84/OH⁺152 and OH65/OH⁺149 flux ratios in the blueshifted wings (the line observations that are compared in Figure 2) with the measured values, indicating OH/OH⁺ column density ratios of ∼10 in the excited outflow component. For the QC, the OH/OH⁺ ratio is ∼20 (Table 4). These column density ratios indicate very high OH⁺ abundances, ∼10⁻⁷ in the QC and even higher in the HVC for X(OH) > 10⁻⁶ (Stone et al. 2018).

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Our model for the QC, which includes an expansion velocity of 100 km s⁻¹ (GA17), reproduces the blueshift observed in the OH⁺ ${2}_{J}-{1}_{J^{\prime} }$ lines but also predicts a blueshift in higher-lying lines that is not observed. The quality of the fit is nevertheless reasonable for most lines at central and blueshifted velocities, though our spherically symmetric models predict significant redshifted reemission above the continuum in some ${2}_{J}-{1}_{J^{\prime} }$, which is not detected (see the residuals in Figure 5). In addition, an observed "bridge" of absorption joins the 2₃−1₂ and the 2₂−1₁ lines (panel (b)), and the 3₃−2₂ and the 3₂−2₁ lines (panel (d)) as well, which cannot be reproduced. These deviations of the spectral fits from the data may indicate a departure from spherical symmetry, with the outflow mainly focused along the line of sight (GA14b), or a redshifted absorption component that could represent low-velocity inflowing gas.

Figures 5(h)–(j) compare the profiles of the ground-state OH⁺ lines observed with SPIRE with the predictions of our composite model. We have not attempted to correct the spectrum for the residual ripples in the FTS apodized spectrum, and thus our results are only approximate. Two main conclusions can nevertheless be drawn from this comparison. First, the HVC and the QC together cannot account for either the absorption or the emission features observed in the OH⁺ 1_J−0₁ P-Cygni profiles, so that we require the inclusion in the model of the LEC with a high OH⁺ column density of ∼10¹⁶ cm⁻². This is a factor of only ∼5 lower than the column density derived for OH in the same LEC, and thus a very high OH⁺ abundance (>2 × 10⁻⁷) is favored. Second, the observed OH⁺ redshifted emission features, including in particular that of the uncontaminated 1₂−0₁ component (Figure 5(i)), require high volume densities (i.e., collisional excitation) at least for the QC. Otherwise, the model for the QC would predict the OH⁺ ground-state lines in absorption close to systemic velocities, which is not observed, and the emission features would be underpredicted. In the model shown in Figure 5, the density decreases from ∼10⁶ cm⁻³ in the innermost regions of the QC to ∼1.5 × 10⁵ cm⁻³ in the external parts.

Even with high OH⁺ column densities in both outflowing components, Figures 5(h)–(j) show that the absorption features of the ground-state OH⁺ lines, and in particular the 1₀−0₁ line, are underestimated by the model. The three lines are optically thick in the three model components, so that an increase in column density would hardly improve the fit. The most probable reason for this discrepancy is the model underestimation of the submillimeter continuum by a factor of ≈2 at 300 μm; a moderate increase in ${T}_{\mathrm{dust}}$ or in the continuum optical depth at submillimeter wavelengths would produce deeper absorption in the 1_J−0₁ lines. On the other hand, we also note that the high densities inferred from the ground-state emission features, which are volume tracers because they probe deep into the buried nuclear regions, do not necessarily apply to the gas responsible for the absorption features in excited lines, which are surface tracers; absorption and emission features are indeed produced in different regions of the QC (see discussion in Section 3.2.1).

As noted in Section 2.2, the H₂O⁺ lines in Mrk 231 are weak and show absorption only at systemic velocities, with no evidence for blueshifted wings. Figures 7(a)–(f) show the predicted H₂O⁺ (and NH₂ in panel (a)) PACS lines of a model with H₂O⁺ column densities of 8.4 × 10¹⁶ cm⁻² and 3 × 10¹⁵ cm⁻² in the QC and the HVC, respectively. These H₂O⁺ columns are considered upper limits: while the model for the QC approximately accounts for the 2₂₁−1₁₀ 5/2−3/2 and 3₂₂−2₁₁ 5/2−3/2 lines (Figures 7(b), (c)), it overpredicts the detected H₂O⁺ 3₁₃−2₀₂ 7/2−5/2 and 5/2−3/2 lines (Figure 7(e)) and the undetected 3₂₂−2₁₁ 7/2−5/2 line (Figure 7(d)). Therefore, an H₂O⁺ column density of (4.2–8.4) × 10¹⁶ cm⁻² is favored for the QC, and $N({{\rm{H}}}_{2}{{\rm{O}}}^{+})\lesssim 3\times {10}^{15}$ cm⁻² is obtained for the HVC (Table 4).

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With these moderate columns, we emphasize the surprisingly strong emission feature associated with the ground-state H₂O⁺ 1₁₁−0₀₀ 1/2−1/2 line in the SPIRE spectrum. While the lack of apparent blueshifted absorption in this line is consistent with the low H₂O⁺ column density found for the HVC from the PACS lines, the emission feature would probably require either very high volume densities in the region of the QC where the line is generated, or formation pumping.

Unfortunately, no collisional rates between H₂O⁺ and H atoms are available, so we have attempted the following rough approach in order to check the consistency of the H₂O⁺ identification and to establish a reference density based on an (uncertain) assumption on the collisional excitation of H₂O⁺. The OH⁺–H de-excitation collisional rates reported by Lique et al. (2016) between the excited OH⁺ levels and the ground-state 0₁ state show little dependence on gas temperature and follow the approximate relationship ${C}_{u\to {0}_{1}}/{g}_{{0}_{1}}=8.5\,\times {10}^{-10}\times {\rm{\Delta }}{E}^{-0.6}$ cm³ s⁻¹, where ${g}_{{0}_{1}}=3$ is the degeneracy of the 0₁ level and ΔE is the energy of the upper level. This expression holds up to ΔE > 1000 K, i.e., up to the highest energy levels considered by the authors, N = 7. We have simply assumed that the above expression, with a (rounded-off) multiplicative constant of 10⁻⁹ (rather than 8.5 × 10⁻¹⁰) to include collisions with electrons, yields characteristic values for the de-excitation rates between the O-bearing molecular ions in collisions with H atoms, and applied it to the de-excitation rates of H₂O⁺ transitions to the ortho (0₀₀) and para (1₀₁ 3/2 and 1/2) ground levels. The excitation rates are then calculated through detailed balance with an adopted gas temperature of 500 K. For reference, the resulting collisional excitation rate from the 0₀₀ level to the 1₁₁ 1/2 level is ≈8 × 10⁻¹¹ cm³ s⁻¹. The collisional rates between excited levels are neglected.

Our overall model for H₂O⁺ also includes in Figures 7(g)–(h) calculations for H₂O and ${{\rm{H}}}_{2}{}^{18}{\rm{O}}$, the former based on the model reported in González-Alfonso et al. (2010) by also including an outflowing component; results for these species will be discussed in a future work. The main points relevant for the present study are these: (i) to approximately reproduce the strength of the H₂O⁺ 1₁₁−0₀₀ 1/2−1/2 line with our adopted collisional rates, a density of several × 10⁶ cm⁻³ in the QC is required, i.e., one order of magnitude higher than that inferred from the OH⁺ ground-state lines; (ii) the model is roughly consistent with the lack of an emission feature in the H₂O⁺ 1₁₁−0₀₀ 3/2−1/2 line (panel (h)), due to cancellation by strong H₂O 1₁₁−0₀₀ absorption in the outflow; (iii) some contribution to the absorption wing observed in panel (h) can be attributed to H₂O⁺, with the moderate column densities listed in Table 4.

We remark that the density inferred for the QC from the ground-state H₂O⁺ lines, several × 10⁶ cm⁻³, should only be taken as a qualitative confirmation of the results obtained for OH⁺, but the absorption lines seen by PACS and the emission lines seen by SPIRE do not sample the same regions (Section 3.2.1). Our subsequent analysis (Section 3.2) is based on the column densities inferred from the absorption lines, where the densities may be significantly lower.

As seen in Section 2.3, little evidence is found in the PACS spectrum of Mrk 231 for H₃O⁺, and an upper limit for the H₃O⁺ column density in the QC is estimated from the model shown in Figure 8. With $N({{\rm{H}}}_{3}{{\rm{O}}}^{+})=8.4\times {10}^{16}$ cm⁻², i.e., just the same column density as used for H₂O⁺ in Figure 7, the model accounts for some hints of absorption in Figures 8(a), (b), and is also compatible with the undetected lines shown in Figures 8(c) and (d). No significant upper limits on the H₃O⁺ column density in the outflowing components could be estimated from the absence of line wings.

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The predicted chemical model abundances of OH⁺, OH, H₂O⁺, and H₃O⁺ and their ratios are plotted in Figure 9 as a function of ζ/n_H and compared with the ranges allowed by the model fits to the observations of the QC. The most reliable observational values are the abundance ratios shown in the left panels, for which an uncertainty of a factor 1.5 is adopted. Because H₃O⁺ is not clearly detected in Mrk 231, we adopt a large range of 4–40 for the ${\mathrm{OH}}^{+}/{{\rm{H}}}_{3}{{\rm{O}}}^{+}$ ratio. The absolute abundances are shown in the right panels, for which a higher uncertainty, of a factor of 2, is adopted. Chemical models are shown for T_gas = 150 and 550 K, and for two fiducial values of the molecular fraction, ${f}_{{{\rm{H}}}_{2}}=0.25$ and 0.50.

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Inspection of Figure 9 highlights the unique role of OH⁺ in probing the ionization rate of the molecular gas, because it is the only species for which the abundance increases monotonically with ζ/n_H up to the most extreme values (for fixed T_gas and ${f}_{{{\rm{H}}}_{2}}$). By contrast, the abundances of all other considered species decline with increasing ζ/n_H above $\mathrm{several}\,\times {10}^{-17}$ cm³ s⁻¹. In addition, Figure 9 indicates that results for both the absolute abundances and abundance ratios are sensitive to both ${f}_{{{\rm{H}}}_{2}}$ and ${T}_{\mathrm{gas}}$. Once OH⁺ is formed, it can be destroyed either by generating H₂O⁺ or through the competing dissociative recombination that breaks the pathway to generate H₂O⁺ and H₃O⁺. With increasing ${T}_{\mathrm{gas}}$, the recombination rate decreases and the channel that produces H₂O⁺ and H₃O⁺ (i.e., ${\mathrm{OH}}^{+}+{{\rm{H}}}_{2}\to \ {{\rm{H}}}_{2}{{\rm{O}}}^{+}+{\rm{H}}$ and ${{\rm{H}}}_{2}{{\rm{O}}}^{+}+{{\rm{H}}}_{2}\to \ {{\rm{H}}}_{3}{{\rm{O}}}^{+}+{\rm{H}}$) is reinforced, lowering the ${\mathrm{OH}}^{+}/{{\rm{H}}}_{2}{{\rm{O}}}^{+}$ and ${\mathrm{OH}}^{+}/{{\rm{H}}}_{3}{{\rm{O}}}^{+}$ ratios. In addition, the OH abundance is boosted at high T_gas mainly because the activation barrier of ${\rm{O}}+{{\rm{H}}}_{2}\to \ \mathrm{OH}+{\rm{H}}$ is overcome (e.g., Elitzur & de Jong 1978), thus increasing the $\mathrm{OH}/{\mathrm{OH}}^{+}$ ratio. On the other hand, the abundance of OH⁺ relative to the other species also decreases with increasing ${f}_{{{\rm{H}}}_{2}}$, because the above reactions that generate H₂O⁺ and H₃O⁺ proceed more efficiently. Therefore, to explain the inferred abundance ratios (shaded regions in Figure 9), higher values of $\zeta /{n}_{{\rm{H}}}$ are required if higher values for ${f}_{{{\rm{H}}}_{2}}$ and T_gas are adopted.

A range of physical conditions, and in particular in density, T_gas, ${f}_{{{\rm{H}}}_{2}}$, and $\zeta /{n}_{{\rm{H}}}$, will be naturally present in the QC, though densities of several × 10⁴ cm⁻³ are most likely prevalent (Mashian et al. 2015). At densities ≳5 × 10⁴ cm⁻³, the gas cools efficiently and thermal balance (including only CR heating) indicates that T_gas will be moderate even for high ζ ∼ 5 × 10⁻¹³ s⁻¹ (<200 K; Papadopoulos et al. 2011, 2014). Observationally, the CO lines from J_up = 5 to 11 are fitted in Mrk 231 with T_gas ≈ 300 K (Mashian et al. 2015). We thus expect ${T}_{\mathrm{gas}}$ in the range between the 150 and 550 K cases shown in Figure 9, but probably closer to the low-${T}_{\mathrm{gas}}$ solution.

To account for the inferred abundances and ratios, the required value of ζ/n_H increases by a factor of ∼5 if ${f}_{{{\rm{H}}}_{2}}$ is increased from 0.25 to 0.50 (Figure 9). Nevertheless, the combination of very warm T_gas = 550 K and ${f}_{{{\rm{H}}}_{2}}\gtrsim 0.50$ is ruled out, because it grossly overpredicts the abundances of OH, H₂O⁺, and H₃O⁺ unless an extremely high ζ/n_H > 10⁻¹⁶ cm³ s⁻¹ is assumed, but this would hardly be compatible with a high molecular fraction (GA13). Indeed, the case of ${f}_{{{\rm{H}}}_{2}}=0.50$ is only compatible with T_gas = 150 K, because it gives $\zeta /{n}_{{\rm{H}}}\gtrsim 5\times {10}^{-17}$ cm³ s⁻¹, which is still consistent with a maximum ${f}_{{{\rm{H}}}_{2}}^{\max }\sim 0.5$ (GA13). On the other side, all molecular abundances start to decline strongly at ${f}_{{{\rm{H}}}_{2}}\lt 15$% (not shown), and the molecular chemistry is simply suppressed. Our favored solution is therefore T_gas = 150–500 K and ${f}_{{{\rm{H}}}_{2}}=0.15-0.5$, with decreasing T_gas for increasing ${f}_{{{\rm{H}}}_{2}}$, for which we infer ζ/n_H ∼ (3 ± 2) × 10⁻¹⁷ cm³ s⁻¹. The electron abundance is ∼10⁻³.

The quoted solution gives results compatible with the observational constraints for the $\mathrm{OH}/{\mathrm{OH}}^{+}$, ${\mathrm{OH}}^{+}/{{\rm{H}}}_{2}{{\rm{O}}}^{+}$, and ${\mathrm{OH}}^{+}/{{\rm{H}}}_{3}{{\rm{O}}}^{+}$ ratios, and also nicely brackets the absolute abundances estimated for OH⁺, OH, H₂O⁺, and H₃O⁺. In particular, the predicted range for the abundance of OH ((0.8–5) × 10⁻⁶) is consistent with the value inferred in ULIRGs from the OH 35 μm transition (Stone et al. 2018). As will be shown in a future work, however, these physical conditions underestimate the H₂O abundance. A range of physical values will naturally be present in the QC, and it is quite possible that H₂O preferentially forms and survives in warm, high-density regions (i.e., ζ/n_H lower than average). The strong submillimeter line of H₂O⁺ could also partially arise there.

We can now evaluate the role of formation pumping, relative to collisional pumping, in the emission of strong ground-state lines of OH⁺. The number of photons generated per unit time and volume in the three ground-state 1_J−0₁ lines via formation pumping is ${\gamma }_{\mathrm{pump}}^{{1}_{J}-{0}_{1}}\sim {n}_{{\rm{H}}}\,\zeta \,{\epsilon }_{{\mathrm{OH}}^{+}}$ (GA13), where ${\epsilon }_{{\mathrm{OH}}^{+}}$ is the efficiency with which ionizations of hydrogen are transferred to OH⁺ production (Neufeld et al. 2010). We adopt a value ${\epsilon }_{{\mathrm{OH}}^{+}}=0.5$, which is an upper limit because neutralization of H⁺ on small grains or PAHs is expected to decrease the efficiency of OH⁺ formation (Hollenbach et al. 2012; Indriolo et al. 2012). Ignoring excitation by the radiation field, the corresponding value generated through collisional excitation is ${\gamma }_{\mathrm{col}}^{{1}_{J}-{0}_{1}}\sim {f}_{{0}_{1}}\,{n}_{{\rm{H}}}^{2}\,X({\mathrm{OH}}^{+}){\sum }_{u}{C}_{{0}_{1}\to u}$, where ${f}_{{0}_{1}}$ is the fractional population in the ground ${0}_{1}$ level, and the sum extends to all collisional rates from the 0₁ level to any other excited one (yielding 1.55 × 10⁻⁹ cm³ s⁻¹ at 300 K; Lique et al. 2016). The ratio of these quantities is

$$\begin{eqnarray}&&\displaystyle \frac{{\gamma }_{\mathrm{pump}}^{{1}_{J}-{0}_{1}}}{{\gamma }_{\mathrm{col}}^{{1}_{J}-{0}_{1}}}\sim \displaystyle \frac{(\zeta /{n}_{{\rm{H}}})\,{\epsilon }_{{\mathrm{OH}}^{+}}}{{f}_{{0}_{1}}\,X({\mathrm{OH}}^{+}){\displaystyle \sum }_{u}{C}_{{0}_{1}\to u}}\sim \displaystyle \frac{0.1}{{f}_{{0}_{1}}}\times \displaystyle \frac{\zeta /{n}_{{\rm{H}}}}{3\times {10}^{-17}\,{\mathrm{cm}}^{3}\,{{\rm{s}}}^{-1}},\end{eqnarray} \tag{ 1 }$$

where $X({\mathrm{OH}}^{+})={10}^{-7}$ is adopted (Figure 9). In our best-fit models, ${f}_{{0}_{1}}$ lies in the range ∼0.2–0.6, and thus collisional excitation appears to dominate.¹⁴

In summary, the combination of Herschel observations, radiative transfer models, and chemical models favors $\zeta /{n}_{{\rm{H}}}\sim (3\pm 2)\times {10}^{-17}$ cm³ s⁻¹ in the torus of Klöckner et al. This is similar to, though likely somewhat higher than, the value inferred in the nuclear region of Arp 220 (GA13), and indeed the continuum-normalized absorption at systemic velocities in the excited OH⁺ and H₂O⁺ lines is similar in both sources. One obvious difference, however, is the high densities inferred in Mrk 231 from the ground-state lines of OH⁺, n_H ∼ 2 × 10⁵ cm⁻³ (Section 3.1.1), pointing to a very high value of ζ in the nuclear region. The caveat, however, is that the column densities and abundances are primarily derived from the (excited) absorption lines that are formed in front of the continuum source, while the above density is inferred from the (ground-state) emission lines, which do not have the above requirement and are primarily formed in a different region of the torus. In fact, while the peak absorption of the excited OH⁺ ${2}_{J}-{1}_{J^{\prime} }$ lines is blueshifted, the peak emission of the ground-state OH⁺ ${1}_{J}-{0}_{J^{\prime} }$ lines is redshifted (Figure 5), unambiguously indicating distinct formation regions for the absorption and emission lines. Observations of CO 2−1 and 3−2 show that the emission of the redshifted line wing is stronger than that of the blueshifted wing (Feruglio et al. 2015). Therefore, the density in the environments where the absorption lines are generated may be different, and probably lower than estimated from the emission lines. Nevertheless, the warm CO spectral line energy distribution observed in the source indicates that the overall density in the nuclear region of Mrk 231 is at least several × 10⁴ cm⁻³ (van der Werf et al. 2010; Mashian et al. 2015), from which we conservatively estimate ζ ∼ 10⁻¹² s⁻¹ with an uncertainty of up to a factor ∼3. This value is still >10³ times the mean value found in the Milky Way (Indriolo et al. 2015), >10 times the value estimated in a spiral galaxy at z = 0.89 (Muller et al. 2016), and ≳10 times the maximum values found in the Galactic center by Indriolo et al. (2015).

Because of the limited observational constraints for the outflowing gas, we model the HVC and the LEC with a single chemical model and consider only the abundances of OH, OH⁺, and H₂O⁺ (Figure 10). In the outflow, a very low abundance ratio of $\mathrm{OH}/{\mathrm{OH}}^{+}\sim 10$ is inferred (Table 4), which points to values of ζ/n_H even higher than in the QC.

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To constrain ${f}_{{{\rm{H}}}_{2}}$ and the density ${n}_{{\rm{H}}}$, we follow Richings & Faucher-Giguère (2017, 2018) to characterize the physical conditions in the high-velocity (partially) molecular outflows driven by luminous AGNs. In their thermochemical 3D models, the ISM gas is swept out in a forward shock driven by a hot shocked bubble, attaining high T_gas ∼ 10⁷ K where all molecules are destroyed. The molecular gas re-forms downstream, and one key result of these models is that there is a range in the post-shock density (n_H, hence also in the pre-shock density n_H0) that allows for high velocities of the outflowing molecular gas: if n_H is too low, the post-shock gas cannot cool in time to form molecules, while if n_H0 is too high, the post-shock gas does not approach the observed high velocities. Values of n_H0 around 10 cm⁻³ and post-shock gas densities around a few × 10³ to 10⁴ cm⁻³ appear to be an appropriate compromise between quick enough H₂ formation and high outflowing velocities (Richings & Faucher-Giguère 2017), and these are indeed the typical densities we found in our best-fit model for the HVC (GA17). In addition, the conversion to molecular gas is far from complete, with ${f}_{{{\rm{H}}}_{2}}\sim 0.25$ being favored for high-velocity outflows (Richings & Faucher-Giguère 2018).

Chemical models for the abundances and ratios of OH, OH⁺, and H₂O⁺ in Figure 10 therefore use ${f}_{{{\rm{H}}}_{2}}\sim 0.25$, and models for T_gas = 1000 K are also considered because the gas is very warm at the "knee" of the H₂ formation. Results for the abundance ratios are then consistent with ζ/n_H ∼ (0.5−2) × 10⁻¹⁶ cm³ s⁻¹. Papadopoulos et al. (2011) show that for an ionization rate of 5 × 10⁻¹³ s⁻¹ (due to CRs, see Section 4.2) and gas densities of 5 × 10³ cm⁻³, ${T}_{\mathrm{gas}}$ is well above 200 K even at high visual extinctions, and this is a lower limit because of additional heating sources in the post-shock gas. Therefore, the solutions with high T_gas = 550–1000 K are favored for the outflowing gas, and yield ζ/n_H ∼ (1–2) × 10⁻¹⁶ cm³ s⁻¹ and ζ ∼ 5 × 10⁻¹³ s⁻¹. This solution yields absolute abundances of OH⁺ and H₂O⁺ above the observationally determined ranges, probably meaning that only a fraction of the outflowing column of gas is subject to the above extreme conditions. For the HVC, where $N({{\rm{OH}}}^{+})\sim 3\times {10}^{16}$ cm⁻² (Table 4), the corresponding hydrogen column density is 4.3 × 10²² cm⁻² for $X({{\rm{OH}}}^{+})\,\sim 7\times {10}^{-7}$, i.e., less than half the total column density of the HVC component. As the molecular gas cools and its density increases (with the concomitant decrease of ζ/n_H), the OH⁺ abundance is expected to decrease (Figure 10), and thus we expect that OH⁺ primarily probes the most extreme, partially molecular environments associated with high T_gas and ζ/n_H—including the molecular formation region of the post-shock gas.

One caveat to the favored solution concerns the shallow H i 21 cm absorption found by Morganti et al. (2016) in the outflow of Mrk 231. For ${f}_{{{\rm{H}}}_{2}}=0.25$, we expect a column density N(H i$)\sim 3.2\times {10}^{22}$ cm⁻² in the HVC, and about half of this column is at blueshifted velocities $| v| \gt 500$ km s⁻¹. While this is still consistent with the observed H i 21 cm absorption at blueshifted velocities if T_spin ≳ 1000 K, the model would overestimate the observed absorption if T_spin were significantly lower. On the other hand, at the expected densities >10³ cm⁻³ the H i 21 cm line is thermalized and T_spin = T_gas (Morganti et al. 2016). Therefore, the OH⁺ and H i 21 cm absorption at high blueshifted velocities are expected to arise in the same outflowing component as long as ${T}_{\mathrm{gas}}$ is high. Nevertheless, the shallow H i absorption may be better explained if a significant fraction of the synchrotron 21 cm continuum is generated in the forward shock (Faucher-Giguère & Quataert 2012; Nims et al. 2015; Liu et al. 2017, see also Section 4.3), i.e., ahead of the neutral and molecular outflowing gas, in such a way that only a fraction of the radio continuum is background relative to H i (see Figures 1 in Faucher-Giguère & Quataert 2012; Zubovas & King 2014; Richings & Faucher-Giguère 2017).