Constraints on the Cosmic-Ray Ionization Rate in the z ∼ 2.3 Lensed Galaxies SMM J2135–0102 and SDP 17b from Observations of OH⁺ and H₂O⁺

Over the past decade, far-infrared photometric surveys (e.g., H-ATLAS) have identified a sample of SMGs that are particularly bright due to magnification from gravitational lensing (Negrello et al. 2017). Spectroscopic redshifts have been determined for several such objects (e.g., Harris et al. 2012; Lupu et al. 2012), enabling targeted observations of transitions from various atomic and molecular species. Some of the most commonly observed species are CO, H₂O, and C, all of which are seen in emission in the star-forming regions of SMGs (e.g., Omont et al. 2013; Yang et al. 2016, 2017). Recent observations have detected CH⁺ in both emission and absorption in lensed SMGs (Falgarone et al. 2017). CH⁺ absorption generally traces cool (T ≲ 100 K), diffuse (n ≲ 100 cm⁻³) gas where hydrogen is not fully molecular, a different environment than that giving rise to the CO and H₂O emission. These are, however, conditions that favor the formation of OH⁺ and H₂O⁺, and principal component analysis of Galactic sight lines has demonstrated that CH⁺, OH⁺, and H₂O⁺ are all most commonly observed in low-density gas where hydrogen is predominantly in atomic form (Gerin et al. 2016). To study the cosmic-ray ionization rate using oxygen-bearing ions at z ∼ 2.3 we selected galaxies from Falgarone et al. (2017) that show CH⁺ absorption as potential targets. In this paper, we focus on the observations of SMM J2135−0102 and SDP 17b, and list some general properties of these galaxies in Table 1.

Table 1.  Target Galaxies

Name	IAU Name	z	z References	${{\rm{\Sigma }}}_{\mathrm{SFR}}$
				(M${}_{\odot }$ yr−1 kpc−2)	${{\rm{\Sigma }}}_{\mathrm{SFR}}$ References
SDP 17b	H-ATLAS J090302.9−014127	2.3051	L12	52 ± 36	Y17
Eyelash	SMM J2135−0102	2.3259	S10	130 ± 40	D11

References.

L12: Lupu et al. (2012); S10: Swinbank et al. (2010); Y17: Yang et al. (2017); D11: Danielson et al. (2011).

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At z ∼ 2.3, the ground state transitions of OH⁺ and H₂O⁺ that have rest-frame frequencies near 1 THz (see Table 2) are shifted down to frequencies covered by band 7 of ALMA. There are two sets of hyperfine transitions that probe the ground state of H₂O⁺ near 1115 GHz and 1139 GHz, with the former separated from the ground state transition of water at 1113.343 GHz by only about 500 km s⁻¹. Molecular emission lines in these galaxies typically have widths of a few hundred km s⁻¹, so we chose to target the 1139 GHz complex of H₂O⁺ to avoid confusion caused by potential blending of the 1115 GHz complex with water vapor. Three sets of hyperfine transitions probe the ground state of OH⁺ at 909, 971, and 1033 GHz. Observations covering the 1033 GHz transition in both target galaxies are publicly available in the ALMA archive.

Table 2.  Transition Properties

Molecule	Transition			Rest Frequency	${E}_{l}/k$	gu	gl	Aul
				(MHz)	(K)			(10−2 s−1)
	$N^{\prime} \mbox{--}N^{\prime\prime}$	$J^{\prime}$–$J^{\prime\prime}$	$F^{\prime}$–$F^{\prime\prime}$
OH+	1–0	1–1	1/2–1/2	1032998.5	0.0055	2	2	1.41
OH+	1–0	1–1	3/2–1/2	1033004.4	0.0055	4	2	0.35
OH+	1–0	1–1	1/2–3/2	1033112.9	0	2	4	0.70
OH+	1–0	1–1	3/2–3/2	1033118.6a	0	4	4	1.76
	${N}_{{K}_{a}^{{\prime} }{K}_{c}^{{\prime} }}^{{\prime} }$–$N{{\prime\prime} }_{{K}_{{a}}^{{\prime\prime} }{K}_{{c}}^{{\prime\prime} }}$	$J^{\prime}$–$J^{\prime\prime}$	$F^{\prime}$–$F^{\prime\prime}$
o-H2O+	111–000	1/2–1/2	1/2–1/2	1139536.6	0.0053	2	2	0.37
o-H2O+	111–000	1/2–1/2	3/2–1/2	1139555.8	0.0053	4	2	1.48
o-H2O+	111–000	1/2–1/2	1/2–3/2	1139648.7	0	2	4	2.93
o-H2O+	111–000	1/2–1/2	3/2–3/2	1139667.9a	0	4	4	1.83

Notes. All data were obtained from the Cologne Database for Molecular Spectroscopy (CDMS; Müller et al. 2005) on 2017 October 27. OH⁺ data are from Bekooy et al. (1985). H₂O⁺ data are from Mürtz et al. (1998), with minor frequency shifts due to the findings of Neufeld et al. (2010) and Muller et al. (2016).

^aIndicates the strongest of the hyperfine transitions for a specific ΔJ, which was used to set the velocity scale during our analysis.

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Two of the observations presented here were made as part of program ADS/JAO.ALMA#2016.1.00285.S: H₂O⁺ at 1139GHz in SMM J2135−0102 and SDP 17b. Three basebands, each with a single spectral window, were placed in the same sideband to provide continuous spectral coverage across the widest possible frequency range (about 3000 km s⁻¹), with the middle window centered on the strongest hyperfine component of the targeted transition. A frequency resolution of 31.25 MHz provides a velocity resolution of about 30 km s⁻¹. SDP 17b was observed for 1481 s on 2016 November 26 targeting H₂O⁺, with J0854+2006 as the bandpass calibrator, J0909+0121 as the phase calibrator, and J1058+0133 as the flux calibrator. SMM J2135−0102 was observed for 1451 s on 2017 April 14 targeting H₂O⁺, with J2134−0153 as the phase calibrator, J2232+1143 as the bandpass calibrator, and Titan as the flux calibrator.

The observations targeting OH⁺ at 1033 GHz toward SMM J2135−0102 and SDP 17b were made as parts of programs ADS/JAO.ALMA#2012.1.00175.S and ADS/JAO.ALMA#2013.1.00569.S, respectively. Both placed two basebands with a single spectral window each in the upper sideband to provide near-continuous coverage over 2500–3000 km s⁻¹ at 31.25 MHz spectral resolution. SMMJ2135−0102 was observed for 3568 s on 2014 August 19 targeting OH⁺ with J2134−0153 as the phase calibrator and J2148+0657 as the bandpass, amplitude, and flux calibrator. SDP 17b was observed for 1965 s on 2015 June 15 targeting OH⁺, with J0909+0121 as the phase calibrator and J1058+0133 as the bandpass, amplitude, and flux calibrator.

Figures 2 and 3 show OH⁺ and H₂O⁺ spectra toward SMM J2135−0102. The NW and SE images of the galaxy are reflected about the z = 2.3 critical curve (Swinbank et al. 2010, 2011), and the eight subregions sample roughly the same location along the major axis in the two images (see Figure 1). While the continuum level flux density is about 1.5 times higher in the NW image than in the SE image, the OH⁺ and H₂O⁺ absorption line profiles for the mirrored regions agree with each other within the noise levels. It is clear that the absorption profiles of OH⁺ and H₂O⁺ change with position along the major axis of the galaxy, shifting to larger velocities with increasing distance from the critical curve. This is visible in the first moment maps of OH⁺ and H₂O⁺ absorption shown in Figure 1, panels (c) and (f), respectively, as a velocity gradient along the major axis of the galaxy. Position–velocity maps of CH⁺ absorption (Falgarone et al. 2017) and of CO J = 6–5 and 1–0 emission (Swinbank et al. 2011) show a similar behavior. Together, these observed velocity gradients indicate large-scale rotation of both the warm gas and dust where CO 6–5 and rest-frame 1 THz continuum emission arises, as well as the cool foreground gas giving rise to OH⁺, H₂O⁺, and CH⁺ absorption. Correlation between the strength of continuum emission and foreground absorption further suggests a relationship between the two components.

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The broad component of CH⁺ emission seen in SMM J2135−0102 is not definitively detected in OH⁺. There is a hint of broad OH⁺ emission in some of the spectra (see, e.g., NW2, NW3, and NW4 in Figure 2) between about −500 and 1000 km s⁻¹, but it is difficult to determine if this feature is real without wider frequency coverage. It is also possible that the small synthesized beam used for the OH⁺ observations (018 × 014) resolves out the emission seen in CH⁺ observations. There does, however, appear to be a narrow (FWHM ∼ 100 km s⁻¹) component of OH⁺ emission centered at about −250 km s⁻¹ toward some of the regions closer to the z = 2.3 critical curve (e.g., NW2, NW3, SE2, SE3). The final emission signature seen in the OH⁺ spectra of SMM J2135−0102 is at the extreme blue end (−700 km s⁻¹ to −900 km s⁻¹) toward regions NW5, NW6, NW7, SE5, and SE6. This emission is due to the CO J = 9–8 transition at 1036.9124 GHz and was previously reported by Danielson et al. (2011). There is no indication of H₂O⁺ emission in SMM J2135−0102.

Spectra of OH⁺ and H₂O⁺ toward SDP 17b are shown in the left and right portions of Figure 4, respectively. The gap between about 800 and 1100 km s⁻¹ in the OH⁺ spectra is due to noncontinuous frequency coverage between the adjacent spectral windows. The OH⁺ line profile in the spectrum covering the entire source resembles that of CH⁺ (Falgarone et al. 2017), with strong absorption from about −800 to 200 km s⁻¹, and what appears to be part of a broad emission component from about 200 to 800 km s⁻¹. A quantitative analysis of this emission component is severely impacted by the aforementioned gap in coverage and the lack of data beyond −800 km s⁻¹, so we do not attempt such an analysis. The OH⁺ absorption profile changes between the different subregions, most notably reaching 75% absorption in the south region—a region that accounts for less than 10% of the total continuum flux density—while only reaching about 50% absorption elsewhere. It is unlikely that there is a contribution from CO 9–8 in these spectra as observations covering both the CO 7–6 and 8–7 transitions resulted in nondetections (Yang et al. 2017). The H₂O⁺ absorption profile in SDP 17b is both weaker and narrower than that of OH⁺, although its full extent in velocity is difficult to determine given the relatively low signal-to-noise (S/N) level of the continuum. Like OH⁺, H₂O⁺ absorption is strongest in the south region. First moment maps of OH⁺ and H₂O⁺ absorption (Figure 1 panels (i) and (l), respectively) do not show any rotation signature, and the extreme velocities at the edges of these maps are the result of low S/N in the spectra extracted from those pixels.

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In order to determine column densities of OH⁺ and H₂O⁺, the spectra presented in Figures 2 through 4 are first normalized to a scale where unity represents complete transmission and zero is complete absorption. This is done by dividing the spectra by a moving average that was interpolated across velocities showing absorption or narrow emission. Normalized transmission spectra are then converted to optical depth spectra using the standard relation $\tau =-\mathrm{ln}(\mathrm{transmission})$. Velocity ranges were defined to fully encompass absorption features and are reported in Table 3 along with the integrated optical depths ($\int \tau {dv}$) measured over those velocity intervals. Uncertainties in the integrated optical depths are estimated from the standard deviation of the continuum level of transmission spectra. The standard deviation of the normalized continuum is determined over a line-free region, and this is assumed to be the error in each channel. This error is divided by the transmission level to compute the 1σ uncertainty in optical depth in each channel, and the uncertainty in the integrated optical depth is taken to be the product of Δv (the channel width in km s⁻¹) and the square root of the sum of variances in optical depth for each channel that was included in the integration.

Table 3.  Absorption Line Properties and Inferred Parameters

Region	Vel. Range	OH${}^{+}\int \tau {dv}$	H2O${}^{+}\int \tau {dv}$	$N({\mathrm{OH}}^{+})$	$N({{\rm{H}}}_{2}{{\rm{O}}}^{+})$	$N({\mathrm{OH}}^{+})$/$N({{\rm{H}}}_{2}{{\rm{O}}}^{+})$
	(km s−1)	(km s−1)	(km s−1)	(1014 cm−2)	(1014 cm−2)
SMM J2135–0102
Full	[−220, 610]	264.6 ± 3.5	69.3 ± 1.8	12.88 ± 0.17	3.86 ± 0.10	3.33 ± 0.10
NW	[−220, 610]	259.6 ± 5.1	69.0 ± 1.7	12.63 ± 0.25	3.85 ± 0.09	3.28 ± 0.10
NW1	[−220, 280]	⋯	43.6 ± 8.4	⋯	2.43 ± 0.47	⋯
NW2	[−150, 280]	231.1 ± 103.1	50.5 ± 6.6	11.25 ± 5.02	2.82 ± 0.37	4.00 ± 1.86
NW3	[−150, 280]	341.8 ± 29.3	62.8 ± 2.7	16.64 ± 1.43	3.50 ± 0.15	4.75 ± 0.46
NW4	[−220, 340]	373.9 ± 20.1	98.2 ± 2.8	18.20 ± 0.98	5.47 ± 0.16	3.32 ± 0.20
NW5	[−150, 450]	361.5 ± 11.5	103.9 ± 1.6	17.59 ± 0.56	5.79 ± 0.09	3.04 ± 0.11
NW6	[−130, 610]	398.4 ± 10.1	86.1 ± 2.6	19.39 ± 0.49	4.80 ± 0.14	4.04 ± 0.16
NW7	[−130, 610]	255.1 ± 20.6	67.4 ± 3.0	12.42 ± 1.00	3.76 ± 0.17	3.30 ± 0.30
NW8	⋯	⋯	⋯	⋯	⋯	⋯
SE	[−220, 610]	247.2 ± 4.4	66.4 ± 3.0	12.03 ± 0.21	3.70 ± 0.17	3.25 ± 0.16
SE1	[−220, 280]	⋯	33.8 ± 8.7	⋯	1.88 ± 0.48	⋯
SE2	[−150, 280]	166.4 ± 46.1	21.1 ± 6.7	8.10 ± 2.24	1.18 ± 0.37	6.89 ± 2.90
SE3	[−150, 280]	234.8 ± 25.4	46.4 ± 5.4	11.43 ± 1.24	2.59 ± 0.30	4.42 ± 0.70
SE4	[−220, 340]	420.3 ± 26.3	88.3 ± 3.1	20.46 ± 1.28	4.92 ± 0.17	4.16 ± 0.30
SE5	[−150, 450]	393.7 ± 27.2	89.1 ± 4.0	19.16 ± 1.32	4.97 ± 0.22	3.86 ± 0.32
SE6	[−130, 610]	336.3 ± 13.7	70.8 ± 4.6	16.37 ± 0.67	3.95 ± 0.26	4.15 ± 0.32
SE7	[−130, 610]	164.1 ± 14.9	63.8 ± 7.3	7.99 ± 0.73	3.56 ± 0.41	2.25 ± 0.33
SE8	⋯	⋯	⋯	⋯	⋯	⋯
SDP 17b
Full	[−755, 235]	339.6 ± 3.8	22.8 ± 2.6	16.53 ± 0.18	1.27 ± 0.14	13.00 ± 1.49
North	[−755, 235]	275.2 ± 7.5	33.0 ± 4.7	13.39 ± 0.37	1.84 ± 0.26	7.28 ± 1.06
East	[−755, 235]	323.0 ± 10.6	23.6 ± 2.6	15.72 ± 0.52	1.32 ± 0.14	11.95 ± 1.37
South	[−755, 235]	626.6 ± 16.4	57.0 ± 4.6	30.50 ± 0.80	3.18 ± 0.26	9.60 ± 0.81
West	[−755, 235]	374.4 ± 4.6	30.0 ± 3.6	18.22 ± 0.22	1.67 ± 0.20	10.90 ± 1.31

Note. Integrated optical depths are determined over the tabulated velocity ranges from the spectra in Figures 2 through 4. Column densities are calculated from Equation (5) as described in Section 5.1. Measured quantities in corresponding mirrored regions toward SMM J2135–0102 are in agreement within 3σ uncertainties.

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Column densities in the rotational states from which the observed transitions arise—in these cases, the ground rotational states of OH⁺ and H₂O⁺—are calculated from integrated optical depths as

$$\begin{eqnarray}&&{N}_{l}=\displaystyle \frac{8\pi }{{c}^{3}}\displaystyle \sum _{S}{g}_{l}\displaystyle \sum _{T}{A}_{{ul}}^{-1}{g}_{u}^{-1}{\nu }^{3}\int \tau {dv},\end{eqnarray} \tag{ 5 }$$

where c is the speed of light, g_l and g_u are the lower and upper state statistical weights, respectively, A_ul is the spontaneous emission coefficient, ν is the transition frequency, the first sum is over g_l in all of the substates that make up the ground rotational state, and the second sum is over ${A}_{{ul}}^{-1}{g}_{u}^{-1}{\nu }^{3}$ for all of the transitions that are contributing to absorption (see Table 2 for transition properties). As the energy difference between the substates is negligible, their relative populations can be approximated solely by their statistical weights. This equation also makes the simplifying assumption that the absorption from all hyperfine components is at the same frequency. Given the width of the absorption features and the instrumental spectral resolution, ignoring the shift of order 30 km s⁻¹ between the different hyperfine transitions is a reasonable approximation. To compute column density in units of cm⁻² when $\int \tau {dv}$ has units of km s⁻¹, the constants outside of the integral in Equation (5) are 4.87 × 10¹² and 4.18 × 10¹² for the 1033 GHz and 1139 GHz transitions, respectively.

Determining total column densities for a species requires accounting for the population in excited states in addition to the ground state. OH⁺ and H₂O⁺ absorption primarily arises in low-density (n_H < 100 cm⁻³) gas (Indriolo et al. 2015) where collisional excitation is slow compared to spontaneous emission. The cosmic microwave background (CMB) at z = 2.3 has a blackbody temperature of T_CMB ≈ 9 K, and thus does not significantly populate the first rotationally excited state of OH⁺ (∼45 K). As such, the column density in the ground rotational state is a good approximation of the total OH⁺ column density. The case for H₂O⁺ is more complicated because the two identical protons lead to both ortho and para nuclear spin modifications that do not communicate via radiative transitions or nonreactive collisions, meaning there are effectively two separate ground states. The first excited state of o-H₂O⁺ is about 55 K above the ground ortho state (the state probed by the 1139 GHz transition) such that it is not significantly populated by CMB radiation. The ortho-to-para ratio expected based solely on statistical weights is 3:1, and we assume this value when computing the total H₂O⁺ column density from our observations of ortho-H₂O⁺. Total column densities of OH⁺ and H₂O⁺ determined from all of our spectra are reported in Table 3, with uncertainties computed from the uncertainties in integrated optical depth discussed above. However, it is possible that we have underestimated column densities if saturated absorption is being masked by other effects. These might include emission from the upper state in the observed transitions filling in part of the line, narrow (in velocity), spectrally unresolved absorption components, or nonuniform and/or incomplete screening of the background continuum by the absorbing gas. Indeed, we likely see the last effect toward SDP 17b (compare the absorption depth of OH⁺ in the full and south spectra in the top left and bottom left panels of Figure 4, respectively), and it may operate on smaller spatial scales within the defined subregions as well. The column densities reported in Table 3 then, are really the averages of all column densities in front of infinitesimally small solid angles (dΩ) of the background continuum region, each weighted by the continuum flux within dΩ.⁷ Still, it is possible that these are only lower limits if saturated absorption is masked by the above effects.

The formation of OH⁺ and H₂O⁺ begins with the ionization of H, and the abundances of our target species can be used in constraining the ionization rate. While analytical expressions derived from steady-state chemistry have frequently been used for this purpose (e.g., Indriolo et al. 2015), detailed chemical models have shown that those expressions are only valid over a limited range of physical conditions (Neufeld & Wolfire 2017). In particular, the expressions are invalid at high ionization rates (ζ_H ≳ 10⁻¹⁵ s⁻¹), where the ionization of H becomes the primary source of free electrons, and the OH⁺ abundance no longer scales linearly with ζ_H (Indriolo et al. 2015). Additionally, those expressions require estimates of the atomic hydrogen column density (N(H)), hydrogen nucleon density (${n}_{{\rm{H}}}\equiv n({\rm{H}})+2n({{\rm{H}}}_{2})$), electron fraction (${x}_{e}\equiv {n}_{e}/{n}_{{\rm{H}}}$), and OH⁺ production efficiency (), all of which are either poorly constrained or unconstrained in our target galaxies. We thus choose to employ the chemical models presented in Neufeld & Wolfire (2017), removing the need to adopt unconstrained values of x_e and .

The analysis of Neufeld & Wolfire (2017), first presented by Hollenbach et al. (2012), utilizes a chemical network designed specifically to focus on processes important to oxygen chemistry to predict the abundances of OH⁺ and H₂O⁺ in a model interstellar cloud. We provide a brief description of the chemical model here, and for more details the interested reader is referred to Hollenbach et al. (2012) and Neufeld & Wolfire (2017), and references therein. The model cloud is a plane–parallel slab with uniform gas density that is illuminated on both sides by a UV radiation field. Certain parameters, including the cosmic-ray ionization rate, metallicity (Z), hydrogen nucleon density, total visual extinction through the cloud (A_V), and ultraviolet radiation field (χ_UV, in multiples of the mean interstellar radiation field from Draine 1978), are specified as input. Equilibrium temperature and steady-state chemical abundances are computed as a function of depth into the cloud, and integration through the cloud provides predicted column densities for a given set of model input parameters. By running the chemical model using several different combinations of input parameters we can investigate clouds with a range of physical conditions.

The range of input parameters that we chose to explore is as follows. Metallicities in submillimeter galaxies at z ∼ 2 have been estimated to be roughly solar (Bothwell et al. 2016), so we use the chemical models with Z = Z_☉ in our analysis. CH⁺ absorption in the Galactic disk primarily arises in gas with 30 cm${}^{-3}\leqslant {n}_{{\rm{H}}}\leqslant 50$ cm⁻³, and the same is likely true of the absorption presented by Falgarone et al. (2017) so we adopt models with n_H = 50 cm⁻³. The UV radiation fields in SMGs are estimated to be 10²–10⁴ times stronger than in the Milky Way (Wardlow et al. 2017) due to prolific star formation. However, those estimates come from dense, molecular gas that is in close proximity to the regions where stars are forming. Falgarone et al. (2017) concluded that the CH⁺ absorption they observe arises from massive, low-density gas halos that extend to the order of 10 kpc around these compact (∼1 kpc diameter) galaxies. The intensity of the radiation field will decrease with increasing distance from the sites of star formation, and nearly all of the UV flux will be attenuated outside of the dusty starburst regions. Because it is not clear exactly where the gas giving rise to OH⁺ and H₂O⁺ absorption is located in these galaxies, we run the suite of chemical models using χ_UV = 0.1, 1, 10, 100, and 1000 to investigate a range of potential UV radiation fields. Finally, we sample the parameter space bounded by 0.0003 mag ≤ A_V ≤ 8 mag and 9 × 10⁻¹⁹ s⁻¹≤ ζ_H ≤ 9 × 10⁻¹⁵ s⁻¹ (specific values are listed in Neufeld & Wolfire 2017) to explore a range of cloud thicknesses and cosmic-ray ionization rates.

From all of the different model runs, we construct a set of predicted OH⁺ and H₂O⁺ column densities for clouds with known A_V, χ_UV, and ζ_H. Using these results, we plot contours of constant ζ_H/n₅₀ in the parameter space ${\mathrm{log}}_{10}[N({\mathrm{OH}}^{+})/N({\rm{H}})]$ versus $N({\mathrm{OH}}^{+})/N({{\rm{H}}}_{2}{{\rm{O}}}^{+})$ as shown in Figure 5, where each panel corresponds to a different value of ${\chi }_{\mathrm{UV}}/{n}_{50}$ (the use of ${n}_{50}\equiv {n}_{{\rm{H}}}/[50$ cm⁻³] here clearly highlights the effects of varying density). With $N({\mathrm{OH}}^{+})$ and $N({{\rm{H}}}_{2}{{\rm{O}}}^{+})$ determined from our observations, only $N({\rm{H}})$ is needed to infer ionization rates from Figure 5. There are no observations of hydrogen in our target galaxies, so it is necessary to estimate N(H) from observations of another species. The recent observations of CH⁺ absorption in these galaxies are likely the best tracers of the same gas component giving rise to OH⁺ and H₂O⁺ absorption (Gerin et al. 2016). Surveys of CH⁺ in the Galaxy have shown that the average relative abundance between CH⁺ and hydrogen ($X({\mathrm{CH}}^{+})\,=N({\mathrm{CH}}^{+})/{N}_{{\rm{H}}}$, where ${N}_{{\rm{H}}}\equiv N({\rm{H}})+2N({{\rm{H}}}_{2})$) in the solar neighborhood is about $X({\mathrm{CH}}^{+})=7.6\times {10}^{-9}$ (Godard et al. 2014). There is large scatter between individual measurements though, with values between about 2 × 10⁻⁹ and 2 × 10⁻⁸, meaning any estimate of the hydrogen column density from CH⁺ will be highly uncertain. Additionally, the molecular hydrogen fraction (${f}_{{{\rm{H}}}_{2}}\equiv 2n({{\rm{H}}}_{2})/{n}_{{\rm{H}}}$) is required to determine the column density of atomic hydrogen, N(H), given the column density of hydrogen nuclei, N_H. We adopt the mean value of ${f}_{{{\rm{H}}}_{2}}=0.053\pm 0.026$ reported by Indriolo et al. (2015) in gas containing OH⁺ and H₂O⁺ for use in calculating N(H) via

$$\begin{eqnarray}&&N({\rm{H}})=(1-{f}_{{{\rm{H}}}_{2}})\displaystyle \frac{N({\mathrm{CH}}^{+})}{X({\mathrm{CH}}^{+})}.\end{eqnarray} \tag{ 6 }$$

Average column densities of CH⁺ toward SMM J2135−0102 and SDP 17b (i.e., determined from spectra extracted from the solid angle encompassing all of the continuum emission) are reported as $N({\mathrm{CH}}^{+})=(8.3\pm 1.4)\times {10}^{14}$ cm⁻² and $N({\mathrm{CH}}^{+})=(4.7\,\pm 1.9)\times {10}^{14}$ cm⁻², respectively (Falgarone et al. 2017), but for the same reasons discussed in Section 5.1 these may be lower limits. With this caveat, we estimate $N({\rm{H}})=({1.0}_{-0.7}^{+3.6})\times {10}^{23}$ cm⁻² and $N({\rm{H}})=({5.9}_{-4.5}^{+25})\times {10}^{22}$ cm⁻² for SMM J2135−0102 and SDP 17b, respectively. Uncertainties are dominated by the scatter in $X({\mathrm{CH}}^{+})$ as we adopt the limiting values of 2 × 10⁻⁹ and 2 × 10⁻⁸ in our analysis. Additional uncertainties are introduced if ${f}_{{{\rm{H}}}_{2}}$ is larger than the adopted value. The maximum value of ${f}_{{{\rm{H}}}_{2}}$ in portions of the model clouds where most of the OH⁺ and H₂O⁺ reside is about 0.3, so this would cause about a 30% overestimate in N(H). Note that these values of N(H) exceed the expected total column densities of our thickest model clouds (N_H = 1.4 × 10²²cm⁻², assuming N_H/A_V = 1.8 × 10²¹ cm⁻² mag⁻¹; Predehl & Schmitt 1995), necessitating absorption from multiple clouds along the line of sight.

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While we have measured $N({\mathrm{OH}}^{+})$ and $N({{\rm{H}}}_{2}{{\rm{O}}}^{+})$ in each subregion and have reported their ratio in Table 3, the synthesized beams from observations of each molecule in the same galaxy do not match, as seen in Figure 1. As a result, the spatial extent probed by our defined regions is slightly different for OH⁺ and H₂O⁺ and should be considered a further source of uncertainty in our analysis. The mismatch is more extreme for OH⁺ and CH⁺ as the latter only has a single average column density reported for each galaxy. Keeping in mind these caveats, our estimates of ${\mathrm{log}}_{10}[N({\mathrm{OH}}^{+})/N({\rm{H}})]$ and $N({\mathrm{OH}}^{+})/N({{\rm{H}}}_{2}{{\rm{O}}}^{+})$ from observations are plotted in the various panels of Figure 5.