A HERSCHEL/HIFI LEGACY SURVEY OF HF AND H₂O IN THE GALAXY: PROBING DIFFUSE MOLECULAR CLOUD CHEMISTRY^*

HIFI employs double sideband receivers and for a sideband gain ratio equal to unity, the saturated absorption of radiation at a given frequency for a transition with excitation temperature much less than ${{T}_{A}}$(cont) will reduce the measured antenna temperature (${{T}_{A}}$) to one-half the apparent continuum antenna temperature ${{T}_{A}}$(cont).

The top two panels of Figures 1–6 display the double sideband weighted average spectra for HF and p-H₂O, respectively, versus Doppler velocity in the local standard of rest frame (${{V}_{{\rm LSR}}}$) for each of the 12 sight lines probed in our survey. For most sight lines, HF is detected only in absorption, and the background continuum emission from the source itself is well modeled with a constant (red line in each panel). The flux normalized with respect to the continuum flux in a single sideband can then be expressed as,

$$\begin{eqnarray*}&&{{T}_{A}}/{{T}_{A}}({\rm cont})=(1/2)[1+{\rm exp} (-\tau )]\end{eqnarray*}$$

assuming that the sideband gain ratio is equal to unity. The latter assumption was investigated by Flagey et al. (2013) who found that the HIFI sideband gain ratio in band 5a, the band of interest here, was very stable and indeed consistent with unity.

Toward W31C, W3 IRS5, W28A, and W49N, one can see that HF emission arising from gas local to the star-forming region blended itself with HF absorption arising from foreground clouds with projected velocities similar to those of the emitting gas. Such blending also occurs for the p-H₂O line absorption toward almost all the sight lines surveyed here. In these cases, we treated the HF and p-H₂O emission as features confined to the proximity of the background continuum sources and modeled them as an additional background continuum emission that adds to the dust emission, as was done in Flagey et al. (2013). For these sight lines, we adopted a linear combination of a zeroth-order polynomial and up to three Gaussian profiles to model the shape of the observed background over the regions unaffected by blending with the foreground absorption. In Figures 1–6, we plotted our best fit models in red over the double sideband spectra of HF and p-H₂O (black lines) for the velocity regions that can be constrained by our data. Since the H₂O emission profiles were constrained using the 1113 GHz features alone, our continuum emission models are not as robust as those obtained by Flagey et al. (2013). As a result, the velocity ranges significantly affected by these continuum emission features are not discussed further here and are not displayed in the bottom two panels of Figures 1–6.

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Assuming a sideband gain ratio equal to unity, the flux normalized with respect to the continuum flux in a single sideband is then expressed as

$$\begin{eqnarray*}&&{\rm exp} (-\tau )=[{{T}_{A}}/{{T}_{A}}({\rm cont})-0.5]/[{{T}_{e}}/{{T}_{A}}({\rm cont})+0.5],\end{eqnarray*}$$

where T_e is the continuum emission arising from the HF or p-H₂O—containing gas proximate to the star-forming regions and T_A corresponds to the absorption due to the diffuse ISM foreground gas of interest for this study. The middle panels in Figures 1–6 display the resulting single sideband, continuum normalized spectrum for HF (black line) and p-H₂O (blue line) versus ${{V}_{{\rm LSR}}}$ for our sample. The horizontal black lines represent the continuum temperature normalized to unity and the zero flux level. One can see that toward all sources, the sideband gain ratios are indeed consistent with unity within our uncertainties (see Table 2).

The HF and p-H₂O transitions probed here have spontaneous radiative decay rates of 2.41 × 10⁻² s⁻¹ and 1.84 × 10⁻² s⁻¹, respectively. These large rates require high gas densities in order for the collisional de-excitation rate to equal the spontaneous radiative decay rate for both species. The foreground gas we detect through the HF and p-H₂O absorptions arises from the Galactic diffuse ISM where gas densities have been measured to be at most ∼1000 cm⁻³ (e.g., Jenkins & Tripp 2001; Sonnentrucker et al. 2007). In the absence of a significant sub-millimeter radiation field, we expect that these two species will be entirely in their ground rotational states in those foreground gas clouds (e.g., Emprechtinger et al. 2013; Flagey et al. 2013). As a result, the optical depth integrated over velocity for a gas component detected in absorption via the ground state of HF can be written as (see Neufeld et al. 2010)

$$\begin{eqnarray*}&&\int \tau dv=4.16\times {{10}^{-13}}N({\rm HF}\;{\rm c}{{{\rm m}}^{-2}})\;{\rm km}\;{{{\rm s}}^{-1}}.\end{eqnarray*}$$

Similarly, the optical depth integrated over velocity for a gas component detected in absorption via the ground state of p-H₂O is given by:

$$\begin{eqnarray*}&&\int \tau dv=4.30\times {{10}^{-13}}N({\rm p}-{{{\rm H}}_{2}}{\rm O}\;{\rm c}{{{\rm m}}^{-2}})\;{\rm km}\;{{{\rm s}}^{-1}}.\end{eqnarray*}$$

We used a linear combination of Gaussian components to fit the HF optical depth profiles over the LSR velocity ranges where the profiles showed optically thin or moderately thick HF absorption depths. For frequencies where the optical depth is large (the absorption lines are optically thick), noise fluctuations and small variations in the sideband gain ratio result in large uncertainties of the exact values of the optical depth in each frequency bin. We place a conservative limit of $\tau =3$ where (2 T_A/T_A(cont) −1) $\leqslant$ 0.05. For each sight line, the gas distribution was modeled by a linear combination of Gaussian components. The initial number of Gaussian components was determined by eye and compared to the observed spectrum using a ${{\chi }^{2}}$-minimization technique. After each run, one additional Gaussian component was added to the previous model until no significant improvement to the ${{\chi }^{2}}$ value was returned by the minimization algorithm. Given the overall similarity in component distribution between HF and p-H₂O, we adopted the best-fit model of the HF spectrum (FWHM and velocity range) as our initial guess to model the p-H₂O optical depth profiles for each sight line. We followed the same optimization procedure as for HF to derive our best fit model for the H₂O spectra. We found that for one-half of our sight lines, the number of Gaussian components required to best fit the HF and p-H₂O optical depth profiles was identical. The bottom two panels of Figures 1–6 display the HF and p-H₂O optical depth profiles (black lines). For clarity, we do not display the portion of each spectrum where blending between foreground absorption and background emission occurs. Our best fit models are overplotted as red lines only for those absorption components that are optically thin or moderately thick.

While the majority of optically thin or moderately thick absorption features from HF and p-H₂O are fitted with a single Gaussian component, some features in our sample require the use of a combination of Gaussian components. In the latter case, we sum the column densities of the individual components constituting the blended complex and report the summed column density over the gas cloud complex. To reflect these differences in sight line structures, Table 3 reports the LSR velocity range over which the optical depth absorptions were integrated rather than the individual Gaussian parameters resulting from modeling the sight lines. We only report and discuss measurements for those gas components that exhibit optically thin or moderately thick profiles ($\tau \lt 3$) for both HF and H₂O.

Table 3.  Column Density Measurements

${{V}_{{\rm LSR}}}$ Range	N(HF)	N(p-H2O)	N(H2O)$_{{\rm tot}}$	$\frac{N{{({{{\rm H}}_{2}}{\rm O})}_{{\rm tot}}}}{N({\rm HF})}$
(km s−1)	(1012 cm−2)	(1012 cm−2)	(1012 cm−2)
W49N
+30, +34b	12.74 ± 0.22	4.82 ± 0.16	19.27 ± 0.66	1.51 ± 0.06
+43, +48	2.91 ± 0.25	1.47 ± 0.27	5.87 ± 1.07	2.02 ± 0.41
+49, +53b	5.73 ± 0.27	1.80 ± 0.18	7.19 ± 0.73	1.25 ± 0.14
+53, +55	6.37 ± 0.24	3.04 ± 0.25	12.18 ± 1.01	1.91 ± 0.17
+55, +58	5.63 ± 0.22	3.80 ± 0.28	15.22 ± 1.11	2.70 ± 0.23
+64, +71	8.36 ± 0.89	2.82 ± 0.88	11.27 ± 3.51	1.35 ± 0.44
W51
+4, +6	4.88 ± 0.17	0.88 ± 0.16	3.53 ± 0.62	0.72 ± 0.13
+6, +10	11.39 ± 1.07	5.37 ± 0.11	21.49 ± 4.54	1.89 ± 0.44
+11, +15	1.46 ± 0.83	0.22 ± 0.08	0.88 ± 0.32	0.60 ± 0.40
+19, +21	0.38 ± 0.18	<0.12	<0.48	<1.25
+21, +24	0.37 ± 0.19	<0.11	<0.44	<1.20
+24, +27	0.74 ± 0.18	0.16 ± 0.05	0.64 ± 0.20	0.87 ± 0.34
+40, +46	7.77 ± 0.25	4.71 ± 0.54	18.82 ± 2.15	2.42 ± 0.29
+47, +50	5.99 ± 0.20	1.99 ± 0.20	7.95 ± 0.79	1.33 ± 0.14
G29.96-0.02
+3, +5	1.55 ± 0.59	0.53 ± 0.17	2.12 ± 0.70	1.37 ± 0.69
+7, +11	20.36 ± 6.13	16.31 ± 0.47	65.22 ± 1.86	3.20 ± 0.97
+11, +15	6.06 ± 1.85	2.22 ± 0.23	8.88 ± 0.92	1.47 ± 0.32
+16, +18	1.43 ± 0.67	0.37 ± 0.14	1.48 ± 0.56	1.03 ± 0.60
+51, +56	3.79 ± 0.92	0.56 ± 0.13	2.22 ± 0.52	0.59 ± 0.20
+57, +62	8.18 ± 2.30	2.35 ± 0.35	9.41 ± 1.40	1.15 ± 0.37
+65, +73	22.49 ± 2.44	9.17 ± 0.59	36.70 ± 2.35	1.63 ± 0.21
+74, +77	1.80 ± 0.90	0.22 ± 0.08	0.90 ± 0.30	0.50 ± 0.30
+89, +93	2.75 ± 0.84	1.67 ± 0.55	6.67 ± 2.18	2.42 ± 1.08
G34.3 + 0.1
+8, +10	0.57 ± 0.19	<0.25	<1.00	<1.74
+10, +13	15.19 ± 5.17	9.66 ± 0.46	38.62 ± 1.85	2.54 ± 0.87
+13, +16	3.13 ± 0.16	1.11 ± 0.16	4.43 ± 0.62	1.42 ± 0.21
+17, +26	2.36 ± 0.53	0.43 ± 0.32	1.71 ± 1.28	0.73 ± 0.57
+27, +30b	12.63 ± 0.39	7.79 ± 1.22	31.17 ± 4.87	2.47 ± 0.39
+40, +44	2.00 ± 0.19	2.34 ± 0.47	9.36 ± 1.89	4.68 ± 1.04
+44, +47	6.54 ± 0.35	4.33 ± 0.64	17.32 ± 2.56	2.65 ± 0.42
+48, +50	13.26 ± 0.33	4.05 ± 0.52	16.20 ± 2.08	1.22 ± 0.16
W33A
+20, +23	0.51 ± 0.09	0.29 ± 0.18	1.15 ± 0.71	2.27 ± 1.46
+24, +27	7.05 ± 0.38	3.61 ± 0.51	14.45 ± 2.05	2.05 ± 0.31
+43, +46	1.24 ± 0.14	0.92 ± 0.37	3.69 ± 1.49	2.97 ± 1.25
W31C
+41, +47	>23.4	18.58 ± 0.35	74.32 ± 1.39	<3.2
DR21(OH)
−11,−9	0.52 ± 0.12	<0.12	<0.47	<0.90
+11, +13b	4.36 ± 0.17	0.90 ± 0.11	3.59 ± 0.43	0.83 ± 0.10
+13, +16	3.43 ± 0.37	1.03 ± 0.72	4.11 ± 2.89	1.20 ± 0.85
W3(OH)
−22, −18	1.70 ± 0.14	0.19 ± 0.10	0.76 ± 0.40	0.45 ± 0.24
−13, −8	6.61 ± 0.32	4.90 ± 0.29	19.61 ± 1.14	2.97 ± 0.25
−6, −4	1.34 ± 0.22	0.47 ± 0.10	1.89 ± 0.38	1.40 ± 0.37
−4, −2	1.61 ± 0.14	0.77 ± 0.12	3.08 ± 0.47	1.91 ± 0.34
−2, +0	2.58 ± 0.16	0.56 ± 0.23	2.25 ± 0.93	0.87 ± 0.37
+0, +2	8.19 ± 0.11	6.78 ± 0.08	27.13 ± 0.31	3.31 ± 0.06
W3 IRS5
−22, −17	7.05 ± 0.63	3.20 ± 0.24	12.82 ± 0.94	1.82 ± 0.21
−6, −3	1.56 ± 0.34	0.51 ± 0.23	2.06 ± 0.92	1.32 ± 0.66
−2, +2	3.61 ± 0.45	1.04 ± 0.38	4.17 ± 1.50	1.15 ± 0.44
G-0.02-0.07
-46,-37	17.34 ± 1.78	12.34 ± 9.89	49.37 ± 39.57	2.85 ± 2.30
+13, +18	5.17 ± 0.66	2.14 ± 1.44	8.55 ± 5.74	1.65 ± 1.13
G-0.13-0.08
−45, −41	6.01 ± 1.21	1.52 ± 1.15	6.09 ± 4.58	1.01 ± 0.86
+20, +25	12.26 ± 1.89	6.47 ± 2.68	25.88 ± 10.74	2.11 ± 0.93
+26, +30	4.36 ± 1.56	1.82 ± 0.90	7.26 ± 3.62	1.66 ± 1.02

^aN(H₂O)$_{{\rm tot}}$ was derived from our N(p-H₂O) measurements assuming an ortho–para ratio of 3 for water (Flagey et al. 2013).

^bOur N(p-H₂O) value is higher by up to a factor of three than that reported by Flagey et al. (2013) toward W49N and G34.3+0.1; our N(p-H₂O) value is lower by ∼50% than that reported by Flagey et al. (2013) toward DR21(OH). Details can be found in Section 6.2.

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We have used a modified version of the diffuse cloud models presented in Neufeld et al. (2005) and Neufeld & Wolfire (2009). The modifications include ice freeze out and grain-surface chemistry as in Hollenbach et al. (2009), the gas-phase reactions and radiation field attenuation as in Wolfire et al. (2010), and the oxygen chemistry as in Hollenbach et al. (2012). The model consists of a slab of gas of constant hydrogen nucleus density ${{n}_{{\rm H}}}$, and total width ${{A}_{V,{\rm tot}}}$, which is illuminated by the interstellar radiation field, χ, measured in units of the Draine (1978) field. The gas temperature and the abundances of atomic and molecular species are calculated as a function of A_V through the cloud under the assumptions of thermal balance and chemical equilibrium. We assume the slab is embedded in an isotropic interstellar radiation field of value χ in free-space and thus $\chi /2$ is incident on opposite sides of the slab. We use the single ray approximation given in Wolfire et al. (2010) for attenuation of the isotropic field.

An important update from the Neufeld & Wolfire (2009) paper is a new measurement for the rate of the reaction

$$\begin{eqnarray*}&&{\rm F}\;+\;{{{\rm H}}_{2}}\to {\rm HF}\;+\;{\rm H}\end{eqnarray*}$$

(Tizniti et al. 2014). This reaction is the dominant formation route for HF in regions with either high or low molecular fractions. The previously used rate was based on a fit to the calculations of Zhu et al. (2002) for an assumed ${{{\rm H}}_{2}}$ population in local thermodynamic equilibrium (LTE). The new rate is based on experimental measurements between 11 and 295 K using a supersonic flow technique. Tizniti et al. (2014) provide a fit to the rate coefficient versus temperature assuming ${{{\rm H}}_{2}}$ in LTE. Figure 8 compares the new and old rate using the fitted functions. Over temperatures expected in diffuse gas $T\sim 50-100$ K, the new rate is a factor of ∼2.5 times lower than the old rate and thus at low ${{{\rm H}}_{2}}$ abundance, the new rate results in a lower HF abundance by a factor of ∼2.5. The HF abundance increases with ${{{\rm H}}_{2}}$ abundance until all of the gas phase F is locked in HF, at which point the HF abundance is insensitive to the ${{{\rm H}}_{2}}\;+\;{\rm F}$ reaction rate.

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Additional updates to the previous codes include fine-structure collision rates for O i and C i with H⁰ from Abrahamsson et al. (2007) and revised photo rates with polycyclic aromatic hydrocarbons (PAHs) using the Draine (1978) radiation field with linear yield functions for the ionization of PAH⁰ and electron detachment of PAH⁻. Finally, an important parameter for the gas-phase production of water is the cosmic-ray ionization rate, ζ. We use the definition that ζ is the primary rate per hydrogen nucleus. The ionization rate is often quoted as the total rate per ${{{\rm H}}_{2}}$ which is a factor of ∼2.3 times larger in molecular gas.

We have run a series of models varying the total extinction through the cloud ${{A}_{V,{\rm tot}}}$, the density ${{n}_{{\rm H}}}$, the cosmic-ray ionization rate ζ, and the incident radiation field χ. The parameters are intended to cover a range of values expected in diffuse molecular clouds (e.g., Sonnentrucker et al. 2007; Goldsmith 2013; Indriolo et al. 2013; Phillips et al. 2013, p. 283). The cloud extinction varies between ${{A}_{V,{\rm tot}}}\sim 0.05$ mag (depending on other parameters) and ${{A}_{V,{\rm tot}}}=4$ mag. The density varies between ${{n}_{{\rm H}}}=50\;{\rm c}{{{\rm m}}^{-3}}$ and ${{n}_{{\rm H}}}=900\;{\rm c}{{{\rm m}}^{-3}}$. As a standard model we use a radiation field of $\chi =1$, but examine the case $\chi =3$ as expected for the field in the inner Galaxy (Wolfire et al. 2003). We adopt a standard cosmic-ray ionization rate of $\zeta =2\times {{10}^{-16}}$ s⁻¹. This rate is close to the average rate of $\sim 1.5\times {{10}^{-16}}\;{{{\rm s}}^{-1}}$ estimated from ${\rm H}_{3}^{+}$ column density measurements over 50 lines of sight by Indriolo & McCall (2012b). It is also close to the rate of $\sim 2.1\times {{10}^{-16}}\;{{{\rm s}}^{-1}}$ estimated by Indriolo et al. (2012a) in the sight line toward W51. Previous diffuse ISM studies adopted cosmic-ray ionization rates that are lower than the standard value we adopted by a factor of ∼10. We investigate the effects of lower rates on our predictions as well. For the gas-phase abundances of carbon and oxygen we adopt the values of 1.6 $\times \;{{10}^{-4}}$ (Sofia et al. 2004) and 3.9 $\times \;{{10}^{-4}}$ (Cartledge et al. 2004), respectively. For fluorine, we use a gas-phase abundance value of 1.8 $\times \;{{10}^{-8}}$ from Neufeld et al. (2005) which amounts to ∼60% of the solar abundance with the remainder depleted onto grains. Our results for the HF abundance will scale linearly with the adopted gas-phase abundance of fluorine.

Figure 9 shows our model predictions for the HF column density relative to the H₂ column density as a function of total visual extinction in the cloud ${{A}_{V,{\rm tot}}}$ (panel a), as a function of the total ${{{\rm H}}_{2}}$ column in the cloud $N({{{\rm H}}_{2}})$ (panel b), and as a function of the ${{{\rm H}}_{2}}$ fraction in the cloud $f({{{\rm H}}_{2}})=2N({{{\rm H}}_{2}})/[N({{{\rm H}}^{0}})+2N({{{\rm H}}_{2}})]$ (panel c). For the models shown in Figure 9, χ is fixed at $\chi \;=$ 1 in units of the Draine (1978) field. The $N({\rm HF})/N({{{\rm H}}_{2}})$ ratio is predicted to rise monotonically with increasing ${{A}_{V,{\rm tot}}}$, $N({{{\rm H}}_{2}})$, and $f({{{\rm H}}_{2}})$ and varies by at most a factor of ∼3 while ζ and ${{n}_{{\rm H}}}$ vary by at least a factor of 10 over the entire parameter range considered here.

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The dependence of the N(HF)/N(H₂) ratio on the total cloud extinction ${{A}_{V,{\rm tot}}}$ can be understood by considering the dominant formation and destruction processes for HF at the cloud center. The detailed ${\rm HF}$ chemistry is presented in Neufeld et al. (2005) and Neufeld & Wolfire (2009). Briefly, the production of HF proceeds by the reaction of ${{{\rm H}}_{2}}$ with F, while the destruction occurs through reactions with ${{{\rm C}}^{+}}$, ${\rm H}_{3}^{+}$, He⁺, and Si⁺ plus photodissociation. For the low density models, ${{{\rm C}}^{+}}$ dominates the destruction of HF, to ${{A}_{V,{\rm tot}}}\sim 1.5$ mag, at which point electrons recombining with ${{{\rm C}}^{+}}$ lead to a lower ${{{\rm C}}^{+}}$ abundance and higher HF abundance. At high density (${{n}_{{\rm H}}}\geqslant 300\;{\rm c}{{{\rm m}}^{-3}}$), the recombination of ${{{\rm C}}^{+}}$ is drawn to the cloud surface leading to higher HF columns even for clouds of low ${{A}_{V,{\rm tot}}}$. Increasing cloud column densities results in a drop in the C⁺ abundance and reactions with less abundant ions dominate the destruction of HF. Thus the HF destruction rates fall and the N(HF)/N(H₂) ratio rises.

Since both the HF column density and ${{{\rm H}}_{2}}$ column density depend on the ${{{\rm H}}_{2}}$ abundance, the ratio does not depend directly on the ${{{\rm H}}_{2}}$ abundance. In addition, since both the formation of HF and destruction of HF have the same dependence on density ($\propto \ {{n}^{2}}$), the HF fractional abundance is not directly density dependent. As a result, we only see a weak density dependence in Figure 9.

The HF abundance predictions are also mostly insensitive to variations in ζ. Note that a slight increase of the HF abundance of at most 10% is predicted at ${{A}_{V,{\rm tot}}}$ = 4 mag for ${{n}_{{\rm H}}}\;\leqslant$ 100 cm⁻³ when increasing ζ by a factor of 10. Cosmic rays play a role in the destruction of both HF and H₂, as they produce ${\rm H}_{3}^{+}$ and He⁺ which then react with HF and H₂. The decrease in the H₂ abundance is greater than that in the HF abundance which, in turn, results in the ∼10% increase in the N(HF)/N(H₂) ratio seen in Figure 9.

The grid of model calculations presented in the previous section predicts that the column density of HF with respect to H₂ varies between $N({\rm HF})/N({{{\rm H}}_{2}})\;\sim$ 0.9 $\times {{10}^{-8}}$ at the cloud surface (${{A}_{V,{\rm tot}}}\;\sim$ 0.08 mag) in the low density regime to $N({\rm HF})/N({{{\rm H}}_{2}})\sim 3.3\times {{10}^{-8}}$ at the cloud center (${{A}_{V,{\rm tot}}}\sim$ 4 mag) in the high density regime considered here. As a result, HF is predicted to be a very sensitive probe of H₂, as it can trace H₂ both in mostly diffuse atomic gas (f (H₂) $\leqslant$ 0.1) and in mostly diffuse molecular gas (0.1 $\lt \;f$(H₂) $\leqslant$ 0.9) in the ISM. The HF abundance in our models never reaches the value of 3.6 × 10⁻⁸ expected if all gas-phase F is locked into gas-phase HF; a value derived from FUV measurements of the abundance of gas-phase atomic fluorine (e.g., Snow et al. 2007). The leveling off around a value of $3.3\;\times \;{{10}^{-8}}$ seen in Figure 9 for ${{n}_{{\rm H}}}$ = 900 cm⁻³, is due to the incomplete conversion of F into HF in the outer regions of the cloud. If the gas becomes fully molecular slightly before all F is locked in HF, then the total column density ratio N(HF)/N(H₂) is less than the F abundance.

Following the first Herschel/HIFI detections of the ground rotational transition of HF toward sight lines comprised mostly of diffuse atomic/molecular clouds (Neufeld et al. 2010; Sonnentrucker et al. 2010), numerous additional HF detections were reported both with the high resolution HIFI data and the lower resolution SPIRE and PACS data (e.g., Kirk et al. 2010). At this point in time, HF has been detected in the Galactic Center (e.g., Goicoechea et al. 2013; Sonnentrucker et al. 2013), in hot cores around massive proto-stars (e.g., Phillips et al. 2010; Emprechtinger et al. 2012), in the ISM of extra-galactic sources up to z = 2.6 (e.g., Monje et al. 2011, 2014) and in the Galactic disk (Flagey et al. 2013; this work). These studies clearly established the ubiquitous presence of the HF molecule in the ISM, as predicted by our models and as expected from a tracer of H₂.

The N(HF)/N(H₂) values currently available in the literature range from 1.0 × 10⁻⁸ to 2.5 × 10⁻⁸ (e.g., Phillips et al. 2010; Sonnentrucker et al. 2010; Monje et al. 2011; Emprechtinger et al. 2012) and are fully consistent with our model predictions. With the exception of the HF/H₂ ratio estimated by Indriolo et al. (2013) and discussed below, those ratios were estimated indirectly from simultaneous measurements of HF and either CH or ¹³CO along these sight lines. The CH column densities were converted into H₂ columns using the most recent estimate of the CH–H₂ relationship N(CH)/N(H₂) = 3.6 × 10⁻⁸ (Sheffer et al. 2008). As mentioned earlier, the CH–H₂ relationship exhibits a standard deviation of about 0.2 dex that is real and thought to be related to the chemical pathways involved in the formation of CH in the diffuse ISM. The $^{13}$CO column densities where converted to H₂ columns for a given carbon Galactic isotopic ratio and for a given CO-to-H₂ ratio; the latter two quantities also vary by factors of a few in the Galactic disk (Langer & Penzias 1990; Sonnentrucker et al. 2007). With these caveats in mind, it is interesting to note that the range in the N(HF)/N(H₂) ratio measured to-date with Herschel is fully consistent with our model calculations, thus validating our understanding of both the physical conditions and the new reaction rates involved in the chemistry of interstellar fluorine.

A few direct measurements of N(HF)/N(H₂) were recently obtained by Indriolo et al. (2013) for a handful of targets sampling diffuse molecular clouds where N(H₂) and N(HF) were derived from FUV and near-IR observations, respectively. For all sight lines where measurements of both quantities were obtained, Indriolo et al. (2013) derived N(HF)/N(H₂) ratios between 0.5 and 1.4 × 10⁻⁸. Of particular interest are the results obtained toward the translucent sight line HD154368. This sight line was best modeled to be comprised of diffuse molecular material with ${{n}_{{\rm H}}}$ = 325 cm⁻³, χ = 3 and ζ = 0.5 × 10⁻¹⁶ s⁻¹ by Spaans et al. (1998) while FUV measurements based on C₂ observations yielded ${{n}_{{\rm H}}}$ = 240 cm⁻³ (Sonnentrucker et al. 2007). Our model calculations for the HF abundance using the parameters from Spaans et al. (1998) and an average density of ${{n}_{{\rm H}}}$ = 300 cm⁻³ yield a ratio of N(HF)/N(H₂) = 2.14 × 10⁻⁸ toward this sight line. Our model predictions are consistent with the direct measurements of Indriolo et al. (2013) of N(HF)/N(H₂) = 1.15 ± 0.41 within 3σ. In summary, all measurements obtained to date—whether direct or indirect—corroborate the prediction that HF can be used as a surrogate tracer of H₂ in the diffuse ISM down to very low molecular fractions where CH is typically below our detection level. Since direct H₂ measurements are not available for the entire set of HF and H₂O absorption features we report in this survey, we will use the HF column densities we derive as proxies for H₂ in the remainder of the paper.

Figure 10 shows the predictions of the PDR model described in the previous section for the ${{{\rm H}}_{2}}{\rm O}$ column density relative to the H₂ column density as a function of total visual extinction in the cloud (${{A}_{V,{\rm tot}}}$, panel a), as a function of the total ${{{\rm H}}_{2}}$ column density in the cloud ($N({{{\rm H}}_{2}})$, panel b), and as a function of the molecular fraction in the cloud ($f({{{\rm H}}_{2}})$, panel c). The UV field is given in units of the Draine (1978) field and is fixed to χ = 1.

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The ${{{\rm H}}_{2}}{\rm O}$ chemistry was discussed in detail by van Dishoeck & Black (1986), Hollenbach et al. (2009, 2012), and van Dishoeck et al. (2013). As in the case of HF, the variations in the H₂O abundance reflect the competition between the formation and destruction processes that contribute over the parameter range explored here. In brief, the destruction of ${{{\rm H}}_{2}}{\rm O}$ is dominated by photodissociation and, at low A_V, has some contribution by reaction with ${{{\rm C}}^{+}}$. The production of ${{{\rm H}}_{2}}{\rm O}$ proceeds by a combination of surface chemistry on grains and ion-neutral chemistry in the gas phase. When the molecular fraction is low the ion-neutral chemistry in the gas phase is initiated by cosmic-ray ionization of atomic H, while when the molecular fraction is high, the ion-neutral chemistry is initiated by cosmic-ray ionization of ${{{\rm H}}_{2}}$. In both the atomic and molecular branches the rate of production of ${{{\rm H}}_{2}}{\rm O}$ increases with the ${{{\rm H}}_{2}}$ abundance since both branches require reactions with ${{{\rm H}}_{2}}$ to proceed in the gas phase. When both the ${{{\rm H}}_{2}}$ and electron abundances are high, the production of water can be reduced due to electron recombinations with ${\rm H}_{3}^{+}$. When the ${{{\rm H}}_{2}}$ abundance is low, but the electron abundance remains high, the water production can be reduced by electron recombination with ${{{\rm H}}^{+}}$ or by neutralization of ${{{\rm H}}^{+}}$ by ${\rm PA}{{{\rm H}}^{-}}$ or by PAH. Finally, when the H₂ abundance is high, but the electron abundance is low, the gas-phase production of H₂O proceeds efficiently through ion-molecule reactions.

To further explain the physical regimes where the various processes dominate, we start with a typical diffuse cloud visual extinction (${{A}_{V,{\rm tot}}}\;=$ 0.5 mag), density (${{n}_{{\rm H}}}\;=$ 100 ${\rm c}{{{\rm m}}^{-3}}$), and cosmic-ray ionization rate ($\zeta \;=$ 2 × 10⁻¹⁶ s⁻¹), and then discuss the variation in the H₂O production, while varying these parameters. For the ${{A}_{V,{\rm tot}}}\sim 0.5$ mag and ${{n}_{{\rm H}}}\leqslant 100\;{\rm c}{{{\rm m}}^{-3}}$ models, the production of ${{{\rm H}}_{2}}{\rm O}$ is dominated by ion-neutral chemistry through the cosmic-ray ionization of atomic hydrogen. To first order the ${{{\rm H}}_{2}}{\rm O}$ abundance scales as $\zeta /{{n}_{{\rm H}}}$ and thus increasing density results in a lower $N({{{\rm H}}_{2}}{\rm O})/N({{{\rm H}}_{2}})$ ratio for a given cosmic-ray rate.

For models ${{n}_{{\rm H}}}\geqslant 300\;{\rm c}{{{\rm m}}^{-3}}$, the molecular fraction rises and the atomic hydrogen abundance falls sufficiently so that the ion-molecule chemistry proceeds mainly through cosmic-ray ionization of ${{{\rm H}}_{2}}$. However, for these models, the electron abundance remains high and the gas-phase production of ${{{\rm H}}_{2}}{\rm O}$ is reduced due to the recombination of ${\rm H}_{3}^{+}$, an intermediary in this particular H₂O production route. As a result, ${{{\rm H}}_{2}}{\rm O}$ is produced mainly by reactions on grains. In the limit of ${{{\rm H}}_{2}}{\rm O}$ produced by grain chemistry alone and destroyed by photodissociation, the $N({{{\rm H}}_{2}}{\rm O})/N({{{\rm H}}_{2}})$ ratio is a function of ${{n}_{{\rm H}}}/\chi$ (Hollenbach et al. 2009).

For the ${{A}_{V,{\rm tot}}}\sim$ 0.5 mag, ${{n}_{{\rm H}}}\;=$ 300 ${\rm c}{{{\rm m}}^{-3}}$, and $\zeta \;=$ 0.7 × 10⁻¹⁶ s⁻¹ model, only ∼15% of the water is predicted to be produced through gas-phase reactions. For the lowest cosmic-ray rate we tested ($\zeta =0.2\times {{10}^{-16}}\;{{{\rm s}}^{-1}}$), the models with ${{n}_{{\rm H}}}\geqslant 100\;{\rm c}{{{\rm m}}^{-3}}$ and $\chi =1$ are dominated by grain surface chemistry and increasing the density ${{n}_{{\rm H}}}$ produces more water and, hence, higher $N({{{\rm H}}_{2}}{\rm O})/N({{{\rm H}}_{2}})$.

At lower ${{A}_{V,{\rm tot}}}$ ($\lt 0.5$ mag), curves for which grain surface chemistry dominates are seen to rise. Since the grain surface chemistry reactions do not depend on the ${{{\rm H}}_{2}}$ abundance, the curves rise as the ${{{\rm H}}_{2}}$ abundance drops. At higher ${{A}_{V,{\rm tot}}}\;\geqslant$ 2 mag, for all models, the electron abundance drops sufficiently due to absorption of FUV photons, so that water can be produced efficiently by ion-molecule chemistry.

We note that Dulieu et al. (2013) have presented experimental evidence to suggest that OH and H₂O that formed on bare silicate grain surfaces can be chemically desorbed upon formation. We consider the effects of chemical desorption by assuming that 30% of the OH is immediately desorbed and 70% of the H₂O is immediately desorbed (see their Figure 4). We find at most a ∼60% increase in the H₂O column density over our model parameter space. The effect is largest where surface chemistry dominates the production of H₂O and for the smallest cloud column densities. The increase drops to ∼40% for a total cloud column density of ${{A}_{V,{\rm tot}}}$ = 2.0, and drops to ∼10% for a total cloud column density of ${{A}_{V,{\rm tot}}}$ = 4.0. We also note that Minissale et al. (2013, 2014) have found that O-atom diffusion can be quite rapid on grain surfaces leading to the production of O₂ and O₃. This process is important deep in molecular clouds where the incidence of oxygen atoms exceeds that of H. These conditions are, however, not applicable to the models presented here.

Since H₂ is not observable directly in the sub-millimeter efforts are made to find surrogate tracers of H₂ in this wavelength domain in order to obtain abundance measurements relevant to test our knowledge of diffuse cloud chemistry. In Section 5, we demonstrated that HF can be used as a surrogate tracer of H₂, down to much lower molecular fractions than the more typical tracers such as CH. Since we do not have H₂ or CH column density measurements for all 47 features presented here, we use HF as a tracer of H₂ and discuss the H₂O abundance relative to HF—N(H₂O)/N(HF)—in the remainder of this paper.

As described in Section 4, we detected absorption from H₂O toward all 12 sight lines included in our survey and we measured its column density in a total of 47 cloud components (see Table 3). We compared our H₂O column density measurements against those reported by Flagey et al. (2013) for the 17 gas components in common to both studies; these are detected in the foreground to W49N, W51, G34.3+0.1, W33A and DR21(OH). We find that the two sets of measurements are consistent within 2 σ for 13 components. For four components, the H₂O column densities we measured are either lower by 50% (DR21(OH)) or higher by up to a factor of three (G34.3+0.1 and W49N) than those reported by Flagey et al. (2013). In all four cases, the background continua local to these gas components are somewhat affected by the presence of water emission local to the sources; the discrepancies we noted above are readily explained by the different methods we adopted to account for the background emission toward each source and give a measure of the modeling uncertainty in such cases. The H₂O column density measurements we report for these four velocity components (marked with an asterisk in Table 3) should be used with caution.

Flagey et al. (2013) reported no dependence of the H₂O abundance relative to H₂ on Galactocentric distance for the six sources they considered. For those sources located in the inner Galaxy in our extended survey, we computed a Galactocentric distance for each cloud we detected in HF and H₂O based on the source longitude and ${{V}_{{\rm LSR}}}$ to see whether the variations in the N(H₂O)/N(HF) we measure show a dependence on position in the Galaxy. We find no dependence of the N(H₂O)/N(HF) ratio with Galactic radius either, a conclusion consistent with that of the smaller Flagey et al. (2013) survey. Our results therefore add weight to the conclusion that the variations in the N(H₂O)/N(HF) ratio we measure can be considered representative throughout the Galactic disk. With a median value of N(H₂O)/N(HF) = 1.51 in diffuse clouds, H₂O also appears as an alternative tracer of H₂ (within a factor of 2.5) in the absence of HF or CH spectra, as already suggested in Flagey et al. (2013).

The comparison of our model predictions displayed in Figures 9 and 10 shows that, in the diffuse molecular cloud regime (${{f}_{{{{\rm H}}_{2}}}}\leqslant 0.5$), the column density ratio of H₂O-to-HF varies from 0.5 $\leqslant \;N$(H₂O)/N(HF) $\leqslant$ 3, all models considered. Our models predict that the water abundance relative to HF varies by up to a factor of six, depending on the local gas physical conditions. The model comparisons further show that the variations in the $N({{{\rm H}}_{2}}{\rm O})/N({{{\rm H}}_{2}})$ ratio are much larger (a factor of 2–4) than the variations in the $N({\rm HF})/N({{{\rm H}}_{2}})$ ratio (at most a factor of two,) both as a function of cloud column density for a given set of parameters and between models with different input parameters. Our models further predict that the variations in the N(H₂O)/N(HF) are mostly driven by variations in the H₂O column density, rather than in the HF column densities for a given set of model parameters.

In Figure 11 we display the N(H₂O)/N(HF) ratio versus N(HF) for all 47 foreground clouds with optically thin or moderately thick absorptions. The measurements toward each sight line are represented by a particular symbol and color that are listed at the top of the panel. We overplot our model predictions for the N(H₂O)/N(HF) ratio onto our measurements using the same symbol and color coding as in Figures 9 and 10. The range of model parameters we use is listed at the top left of the panel. We use a log–log scale for both axes for clarity purposes. We show 1σ uncertainties for all measurements and 3σ uncertainties are plotted as downward arrows for upper limits in the water abundance. This figure shows that the N(H₂O)/N(HF) ratio varies by about a factor of five overall; our observational results are therefore fully consistent with the model predictions presented here and known to best bracket the physical conditions in diffuse clouds.

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Our sight line analyses (Section 2) indicated that HF and H₂O are co-located in the gas components we detect throughout the Galactic disk, meaning that both molecules are subject to the same physical conditions within each gas clump along the sight lines. As a result, the variations we measure are not caused by differences in the spatial distributions of HF or H₂O along a given sight line (geometric effects) but are caused by true variations in the molecular content for a given gas component.

In particular, Figure 11 displays a subtle break in the water abundance relative to HF once N(HF) = 5 × 10¹² cm⁻². For N(HF) $\leqslant$ 5 × 10¹² cm⁻², our measurements indicate that the N(H₂O)/N(HF) ratio varies by about a factor of six to nine from ∼0.5 to 4.6. The lowest water abundances relative to HF are best reproduced by models with low cosmic-ray ionization rates ($\zeta \leqslant 0.7\;\times$ 10⁻¹⁶ s⁻¹) over the density range known to be representative of the diffuse ISM while the highest water abundances relative to HF are best represented by $\zeta \sim 2\;\times$ 10⁻¹⁶ s⁻¹ and ${{n}_{{\rm H}}}\;\leqslant$ 100 cm⁻³. The prevalence of mixtures of physical conditions within the galactic disk diffuse ISM and along any given sightline has long been recognized and is clearly evidenced here again, as multiple model runs can accommodate our measurements for this particular HF column density range.

For N(HF) >5 × 10¹² cm⁻², however, our measurements are not consistent with models using $\zeta \leqslant 0.7\;\times$ 10⁻¹⁶ s⁻¹. As a matter of fact, over 80% of our measurements in this specific HF column density range show water abundances relative to HF consistent with model predictions for single clouds with $\zeta =2\;\times$ 10⁻¹⁶ s⁻¹, $\chi =1$ and ${{{\rm n}}_{{\rm H}}}\;\leqslant$ 100 cm⁻³. The variations in the measured N(H₂O)/N(HF) in this HF regime amount to at most a factor of three, thus, pointing toward a population of low gas density clouds with properties intrinsically very similar. Considering that the HF abundance relative to H₂ currently measured varies by at most a factor of ∼2 (see Section 5), we conclude that the variations we measured in the N(H₂O)/N(HF) ratio are mostly driven by variations in the H₂O/H₂ ratio within the gas clouds we probe.