THE CHEMISTRY OF INTERSTELLAR OH⁺, H₂O⁺, AND H₃O⁺: INFERRING THE COSMIC-RAY IONIZATION RATES FROM OBSERVATIONS OF MOLECULAR IONS

The numerical code we have developed to model the chemical and thermal structure of an opaque cloud externally illuminated by FUV flux is based on our previous PDR model described in H09. This one-dimensional code models a constant density n (the hydrogen nucleus density) slab of gas, illuminated from one side by an FUV flux of 2.7 × 10⁻³χ erg cm ⁻² s⁻¹ incident perpendicular to the slab. The unitless parameter χ is defined above in such a way that χ ∼ 1 corresponds to the average local ISRF in the FUV band (Draine 1978).¹⁰ The code calculates the steady-state chemical abundances and the gas temperature from thermal balance as a function of depth into the cloud. It incorporates 63 chemical species, ∼300 chemical reactions, and a large number of heating mechanisms and cooling processes. The chemical reactions include FUV photoionization and photodissociation; cosmic-ray ionization; neutral–neutral, ion–neutral, and electronic recombination reactions. H₂ self-shielding is included as described in H09, and CO self-shielding and the partial shielding of CO by H₂ are included as described in Visser et al. (2009). The code includes the photodissociation of molecules by "secondary" FUV photons produced (ultimately) by cosmic rays (Prasad & Tarafdar 1983). We also include reactions with charged dust grains and PAHs, and the formation of H₂, OH, H₂O, CH, CH₂, CH₃, and CH₄ on grain surfaces. The code treats the freezing of all condensable species to grain surfaces and three desorption processes: thermal desorption, photodesorption, and cosmic-ray desorption. The only significant difference in the desorption code used here versus the H09 code is the inclusion of the new (higher by factor ∼3) rate of photodesorption of CO (Öberg et al. 2009). The code does not include photodesorption by the secondary FUV photons; this process is negligible at the peaks in the OH⁺, H₂O⁺, and H₃O⁺ abundances. The code has been used to model regions which lie at hydrogen nucleus column densities N ≲ 4 × 10²² cm⁻² (or A_V ≲ 20) from the surface of a cloud. Therefore, it applies not only to the photodissociated surface region, where gas-phase hydrogen and oxygen are nearly entirely atomic and where gas-phase carbon is mostly C⁺, but also to regions deeper into the molecular cloud where hydrogen is in H₂ molecules and carbon is in CO molecules. Even in these molecular regions, the attenuated FUV field can play a significant role in photodissociating H₂O and O₂, in photodesorbing species adsorbed on grain surfaces, and in heating the gas. However, the code is now sufficiently general that it finds the steady-state solutions for abundances and temperature in any region of a molecular cloud, even where FUV is insignificant.

We emphasize that we present a steady-state model of chemical abundances as a function of depth into the cloud. The chemical timescales can be quite long, which might suggest that time-dependent models are more appropriate. For example, the timescale to convert atomic H gas to fully molecular H₂ gas is $t_{{\rm H_2}} \sim 10^9/n$ years, where n is the hydrogen nucleus density in units of cm⁻³. However, as we show below, the OH⁺ and H₂O⁺ ions peak in abundance when x(H₂) ∼ 0.03–0.1, which occur typically at A_V ∼ 0.1. (Note that abundances in this paper are defined relative to hydrogen nuclei, so that x(H₂) ≡ n(H₂)/n and n ≃ n(H) + 2n(H₂)). Therefore, the timescales for even diffuse clouds of density n ∼ 100 cm⁻³ to reach these abundances are less than ∼10⁶ years, which is shorter than the typical lifetime of a diffuse cloud (Wolfire et al. 2003). Steady-state solutions therefore generally apply, at least for computing the columns of these ions. The steady-state models may somewhat underestimate the ion columns for low-density diffuse clouds with A_V > 0.1, since the steady-state solutions can lead to lower abundances of these two ions than time-dependent solutions in the high A_V regions. This arises because the steady-state abundances of H₂ are higher than time-dependent models which start with fully atomic gas. Higher H₂ abundances lead to more destruction of OH⁺ and H₂O⁺ in the high A_V regions where H₂ and not electrons dominate the destruction of these ions. Liszt (2007) provides a detailed analysis of the time-dependent formation of H₂ and OH⁺ as a function of A_V in diffuse clouds with initial atomic conditions.

We also emphasize that our model does not include turbulent dissipation and heating of small pockets of gas along the LOS (e.g., Godard et al. 2009 and references therein). We do, however, run PDR models with small fractions of the LOS having either enhanced temperature or enhanced rates of ion–neutral drift to test the possible effects of turbulence, and find the effects on OH⁺, H₂O⁺, H₃O⁺ column densities are likely small.

For simplicity, we assume constant H nucleus density n in our models. At low A_V and low x(H₂) < 0.1, where much of the OH⁺ and H₂O⁺ columns often arise, the temperature is quite constant so that constant density implies constant thermal pressure. If thermal pressure (and not turbulence) dominates deeper into a high A_V cloud, then the density will rise as one moves from its warmer PDR surface to its CO-cooled molecular interior. In addition, self-gravity can raise the density of the interior regions at higher A_V. Still another effect is the transition of atomic hydrogen to molecular hydrogen which can raise n by a factor of two if thermal pressure is conserved. We ignore the possible rise in density, which mainly affects the second peaks of the ions deep (A_V > 4) in the cloud. If such a density enhancement occurs, then it tends to depress the abundances of OH⁺ and H₂O⁺ in the second peaks, but the abundance of H₃O⁺ at the second peak is not sensitive to hydrogen density if PAHs are present(see Section 3).¹¹ The second peak is dependent on the abundance of PAHs, very small grains (VSGs), and low-ionization potential metal ions there, which can control the electron density, n_e. The abundance of PAHs and VSGs at high A_V is uncertain due to their possible coagulation on larger grain surfaces.

One significant difference between the chemical code used in this paper and that used in H09 is the inclusion of PAHs and VSGs (radius ≲ 50 Å), which affect the ionization balance by enhancing the recombination of positive atomic ions. In the rest of this paper, we shall often use the term "PAH" to denote both PAHs and VSGs. PAHs also affect the second peaks of the ions by destroying electrons in the reaction e + PAH → PAH⁻. (The effects of PAHs on ionization balance have been treated extensively before in the literature, e.g., Lepp & Dalgarno 1988; Bakes & Tielens 1998; Weingartner & Draine 2001a; Flower & Pineau des Forêts 2003; Liszt 2003; Wolfire et al. 2003, 2008). H09 focused on the H₂O and O₂ peaks, which occur at relatively high A_V ∼ 3–7. Although the PAH abundances are not known deep in molecular clouds, H09 assumed that their abundances are low, due to coagulation of PAHs on the surfaces of larger dust grains. Here, however, we are mostly interested in the chemistry at low A_V < 1, and in particular in the chemistry of diffuse clouds. These are the regions where PAHs are observed to be present, and we use PAH parameters derived from the literature. For simplicity, we adopt a single size PAH for the PAH distribution, and assume that the standard PAH has 100 C atoms and a number abundance of x_PAH = 2 × 10⁻⁷ with respect to H nuclei (see Wolfire et al. 2008 for a discussion of the amount of carbon in PAHs; here, we adopt 100 C atoms and not the 35 C atoms that Wolfire et al. adopted because of the result of Draine & Li 2007, which suggested that the distribution in mass peaks at 100 C atoms). One of the key impacts of PAHs is in the recombination of H⁺. As shown in Figure 1 (top), cosmic-ray ionization of H can lead to OH⁺, but a competing route is the neutralization of H⁺ by an electron, PAH, or PAH⁻. PAHs therefore lower the production of OH⁺ and the columns of all the ions in the first peak. Conversely, in the second peaks, the reduction in electron abundance caused by PAHs cause fewer H⁺₃ ions to recombine with electrons, and lead to greater production rates of the ions as well as smaller destruction rates for H₃O⁺. Therefore, the columns of the ions increase with the presence of PAHs in the second peaks. We also discuss results with no PAHs or VSGs at high A_V.

Table 1, in Appendix A, lists the rate coefficients of key reactions in the pathways to the OH⁺, H₂O⁺, and H₃O⁺ ions, as well as reactions that are either new or changed since H09. Of particular note here are the photoionization of OH and H₂O, the photodissociation of OH⁺ and H₂O⁺, the treatment of PAHs—especially the photodetachment reaction rate for PAH⁻, the fine structure level population dependence in the charge exchange reaction of O with H⁺, and some minor modifications in reaction rates key to determining the abundances of the OH⁺, H₂O⁺, and H₃O⁺ ions, such as their dissociative recombination rates with electrons and their reactions rates with H₂.

We have also included in Table 1 rates that are important in producing H⁺ by chemical means rather than by cosmic-ray ionization of H in PDRs. There are two main chemical routes. The first is initiated by the FUV photoionization of C to C⁺. The C⁺ reacts with OH to form CO⁺. The CO⁺ charge exchanges with H to form H⁺. The second is also initiated by FUV photoionization of C to C⁺. The C⁺ reacts with H₂ to form CH⁺. The CH⁺ is photodissociated by FUV photons to produce H⁺. Both these reaction chains are very much enhanced by high (≳ 300 K) temperatures in gas with appreciable H₂. The first chain is enhanced because high gas temperatures lead to significant amounts of OH being formed by the reaction H₂ + O → OH + H. This reaction has an activation barrier of ΔE/k = 3160 K and so is insignificant for low gas temperatures. The second chain similarly is enhanced because the reaction C⁺ + H₂ → CH⁺ + H has an activation barrier of ΔE/k = 4640 K (see Table 1).¹² In regions where either of these two chains dominate the production of H⁺ over cosmic-ray ionization, the ions OH⁺, H₂O⁺, and H₃O⁺ only provide upper limits to the cosmic-ray ionization rates.

Other alternate routes to the production of OH⁺ and H₂O⁺ ions include three routes which produce OH or H₂O without the ions: the production of water on grain surfaces followed by photodesorption of the water to produce gas-phase OH and H₂O, the radiative association of O with H to form OH, and the reaction of FUV-pumped H₂ in excited vibrational states with O to form OH. The gas-phase OH and H₂O can be photoionized by FUV photons to produce OH⁺ and H₂O⁺ (see Table 1). Unlike the routes described in the preceeding paragraphs, these routes are never dominant in producing the ion column densities.

An analogous route to the formation of H₂O on grain surfaces followed by photodesorption, but one not treated in this work, is the time-dependent evaporation of water ice which occurs around newly formed stars. The rapid rise in embedded luminosity heats the dust grains above about 100 K, and the icy mantles on grains are then thermally evaporated. This sudden injection of high abundances of water vapor into the gas is followed by reaction of the gas-phase water with HCO⁺ and H⁺₃ to form H₃O⁺. Eventually, the system relaxes to the steady-state chemistry described in this paper, but for a short time, there might be a large enhancement in H₃O⁺ (Millar et al. 1991, PvDK92). If this release of H₂O from the icy grain surface to the gas occurs in regions with elevated FUV fields, then the photoionization of H₂O could also result in enhanced H₂O⁺ abundances (Gupta et al. 2010)

Table 2, in Appendix A, lists the gas-phase elemental abundances, the PAH properties, and the grain surface area per H nucleus adopted in our code.

As noted above, our opaque molecular cloud models invoke a constant density n slab, illuminated from one side by a one-dimensional normal FUV flux χ. We solve for the gas temperature and the gas phase and ice abundances of the various species as a function of depth in the slab. Note that depth is synonymous with hydrogen nucleus column N [=N(H)+2N(H₂)] into the cloud or A_V into the cloud. We take N = 2 × 10²¹A_V cm⁻². We follow the chemistry to A_V ∼ 20, beyond which there is little contribution to the columns of OH⁺, H₂O⁺, and H₃O⁺ ions.

We use the primary cosmic-ray ionization rate per H atom as an input parameter, since our code calculates the secondary ionizations caused by cosmic rays and these depend on the H₂ and electron abundances. In order to probe the range of cosmic-ray ionization rates suggested in the literature, we include cases with primary ionization rates of ζ_crp =2 × 10⁻¹⁷ s⁻¹ and 2 × 10⁻¹⁶ s⁻¹ per H atom, which correspond to total rates (including secondary ionizations) of about ζ_crt ∼ 3 − 5 × 10⁻¹⁷ s⁻¹ and 3–5 × 10⁻¹⁶ s⁻¹, respectively. Note that the primary rate per H₂ molecule is 2ζ_crp. We also adopt ζ_crp as the primary rate for He atoms; however, He has insignificant secondary ionizations. Although there is evidence that the cosmic-ray ionization rate may decrease with depth (e.g., Rimmer & Herbst 2011; Rimmer et al. 2012; Indriolo & McCall 2012, and discussion later in this paper), for simplicity we assume that the primary cosmic-ray ionization rate does not vary with depth into a cloud in a given model. The main effect of this assumption is that we may have overestimated the abundances and columns of the OH⁺ and H₂O⁺ ions in the second (deeper) peak relative to the first peak. However, since we vary ζ_crp in our parameter study, the reader can use lower ζ_crp for the columns in the second peak if so desired. The abundance of H₃O⁺ in the second peak is not sensitive to ζ_crp if PAHs are present, as we will show below.

The main parameters that we explore for our one-sided, opaque molecular cloud models are the gas density n, the incident FUV flux χ, and the primary cosmic-ray ionization rate ζ_crp. We study the parameter space 10 cm⁻³ <n < 10⁷ cm⁻³, 1 <χ < 10⁶, and 2 × 10⁻¹⁷ s⁻¹ < ζ_crp <2 × 10⁻¹⁶ s⁻¹. We explore the sensitivity of the columns of OH⁺, H₂O⁺, and H₃O⁺ ions to assumptions about the PAH chemistry, the elemental gas-phase abundances of low-ionization potential metals, and the rate coefficient for the formation of H₂ on grain surfaces.

The diffuse cloud models treat a constant density slab of total thickness A_Vt illuminated on both sides by a one-dimensional normal FUV flux χ/2. We explore the parameter space 10^−17.5 s⁻¹ < ζ_crp/n₂ < 10^−14.5 s⁻¹, 0.01 < A_Vt < 3, 30 cm⁻³ <n < 300 cm⁻³, 1 <χ < 10, and 1 ⩽ χ/n₂ ⩽ 10 where n₂ = n/100 cm⁻³. The models of Wolfire et al. (2003) provide a good estimate of χ/n₂ in the Galaxy by estimating χ from the star formation rate and the dust opacity as a function of galactocentric radius R, and setting the thermal pressure to provide two phases, a cold diffuse cloud phase and a warm intercloud medium that fills most of the volume. Assuming R = 8.5 kpc as the solar location, Wolfire et al. find χ = 1.0, n = 33 cm⁻³, and χ/n₂ = 3.0 for diffuse clouds at R = 8.5 kpc; χ = 2.35, n = 49 cm⁻³, and χ/n₂ = 4.8 at R = 5 kpc; χ = 3.0, n = 54 cm⁻³, and χ/n₂ = 5.5 at R = 4 kpc; and χ = 3.8, n = 60 cm⁻³, χ/n₂ = 6.3 at R = 3 kpc. Therefore, our prime region for study is χ/n₂ = 3–6. We are especially interested in how the ratio of the columns N(OH⁺)/N(H₂O⁺) and N(OH⁺)/N(H) vary as functions of A_Vt, χ/n and ζ_crp/n.

We note that the columns our models predict are columns perpendicular to the one-sided (molecular cloud) slab or two-sided (diffuse cloud) slab. If the slabs are viewed at an angle θ with respect to the normal, then the observed columns will increase by (cos θ)⁻¹. Another effect that will obviously raise the columns is if there are more than one diffuse cloud (in a given velocity range) along the LOS or if the molecular cloud is clumpy and FUV scattering then introduces several "surfaces" along the LOS. However, as we will show, ζ_crp/n in diffuse clouds can be estimated from our models from the ratios N(OH⁺)/N(H₂O⁺) and N(OH⁺)/N(H), and the ratios are independent of the geometric effects. Nevertheless, there is some degeneracy in the solution for ζ_crp/n, depending on the combination of the A_Vt of a single cloud, χ/n, and the enhancement in columns created by the geometrical effects.

3.1.1. The Chemical and Thermal Structure of Individual Clouds

In order to understand the columns of OH⁺, H₂O⁺, and H₃O⁺ ions produced as a function of n, χ, and ζ_crp, we first study the detailed chemical and thermal structure of a few specific (standard) cases. Figure 2 shows the chemical abundances as a function of depth A_V into the cloud for the case n = 10² cm⁻³, χ = 1, and ζ_crp = 2 × 10⁻¹⁷ s⁻¹. This case is chosen not only because it may be appropriate for the ambient ISRF incident on a relatively low-density giant molecular cloud (GMC), but also because the surface (A_V ≲ 3) structure (i.e., T and chemical abundances as a function of depth or A_V) is illustrative of the depth structure of a diffuse or translucent cloud. In addition, the cosmic-ray rate may be appropriate to the interior of molecular clouds.

Zoom In Zoom Out Reset image size

The main chemical result is that the OH⁺, H₂O⁺, and H₃O⁺ ions all have a peak at 0.03 < A_V < 0.3, and then have a second peak at A_V ∼ 6 for this combination of n and χ. We first discuss the surface peaks at low A_V. OH⁺ peaks at A_V ∼ 0.03, where the molecular hydrogen abundance is x(H₂) ∼ 0.01. The H₂O⁺ peaks slightly deeper, at A_V ∼ 0.1, where x(H₂) ∼ 0.1. Finally, H₃O⁺ peaks at A_V ∼ 0.3, where x(H₂) ∼ 0.25. Appendix B provides an approximate analytic solution to the chemistry that explains the OH⁺ first peak, and its relation to x(H₂). From A_V ∼ 0.01 to either A_V ∼ 0.03 (OH⁺) or A_V ∼ 0.1 (H₂O⁺) or A_V ∼ 0.3 (H₃O⁺), the three ions rise in abundance with increasing A_V because of the rise in the H₂ abundance, which drives the cosmic-ray-produced O⁺ to the molecular ions. At larger A_V but for x(H₂) significantly less than its maximum value of 0.5, the abundance of OH⁺ and H₂O⁺ plateau at the peak value because here both their formation and destruction rates are proportional to x(H₂). H₂O⁺ tends to peak at somewhat higher A_V than OH⁺ because, even at the peak of OH⁺ abundance, not all cosmic-ray ionizations lead to H₂O⁺ and so its abundance continues to rise as the H₂ abundance rises with increasing A_V. Finally, as x(H₂) approaches 0.5, two effects lead to a drop in the OH⁺ and H₂O⁺ abundances with increasing A_V. One is that the formation rates of these ions saturate as nearly every cosmic-ray ionization leads to their production, whereas the destruction rates still scale as x(H₂), which increases with increasing A_V. The other dominant effect is that the H abundance drops so that the OH⁺ formation rate via the top chain of reactions in Figure 1 drops. The bottom chain at relatively low A_V is not as efficient at producing OH⁺, as the electrons are relatively abundant (x_e ∼ 10⁻⁴) in these surface regions, and H⁺₃ recombines with electrons rather than forming OH⁺.¹³ The H₃O⁺ is destroyed not by H₂, but by electrons, whose abundance stays quite constant (supplied mostly by C⁺ but with possible contribution by H+ at high ζ_crp/n) at the cloud surface. The H₃O⁺ starts to drop in abundance once the gas becomes predominantly H₂, due to the second effect described above.

We next discuss the second deeper peak in the OH⁺, H₂O⁺, and H₃O⁺ ions. As one moves deeper into the cloud (typically, A_V ≳ 2), the electron abundance starts to drop. As this happens, a greater fraction of the cosmic-ray ionizations of H₂ leads to the production of the three ions, and their formation rates rise. The destruction of OH⁺ and H₂O⁺ is by H₂, which now has constant abundance (the gas is fully H₂), so the destruction rates hold constant. Therefore, these two ions rise in abundance. Finally, they peak and fall in abundance for A_V > 6 because gas-phase oxygen freezes out as water ice, and again the oxygen reaction with H⁺₃ cannot compete with H⁺₃ electronic recombination or its reaction with CO. Therefore, the formation rates of all three ions drop. H₃O⁺ behaves somewhat more dramatically, because its destruction is mainly by dissociative recombination with electrons. Thus, as the electron abundance drops, not only is the formation rate of H₃O⁺ enhanced, but the destruction rate is suppressed. Therefore, H₃O⁺ rises to much higher abundances than OH⁺ and H₂O⁺ in the second peak. It also drops at very high A_V because gas-phase elemental oxygen from which H₃O⁺ forms freezes out as water ice.

In this particular case, there is a column of N(OH⁺) = 2 × 10¹¹ cm⁻² in the first surface peak and 4 × 10¹¹ cm⁻² in the deeper peak, for H₂O⁺ the columns are 1.4 × 10¹¹ cm⁻² and 2 × 10¹¹ cm⁻², and for H₃O⁺ the columns are 1.1 × 10¹¹ cm⁻² and 9.4 × 10¹³ cm⁻². The columns of OH⁺ and H₂O⁺ are not detectable. Typically, columns of ≳ 10¹² cm⁻² are needed for detection via absorption spectroscopy. In particular, observations of OH⁺ often imply columns ≳ 10¹³ cm⁻², which suggest that higher cosmic-ray rates are required. Therefore, for the rest of our standard cases, we use ζ_crp =2 × 10⁻¹⁶ s⁻¹. Most of the H₃O⁺ column is produced in the second deeper peak, and in our model the predicted columns are detectable. For an absorption measurement, a background submillimeter source behind or in a cloud with A_V > 6 from observer to submillimeter source is required.

Figure 3 shows the case n = 10² cm⁻³, χ = 1, and ζ_crp = 2 × 10⁻¹⁶ s⁻¹; in other words, the same as Figure 2 but with 10 times the cosmic-ray ionization rate. The abundances of all three ions in the first peak rise in proportion to ζ_crp. The OH⁺ and H₂O⁺ ion abundances in the second peak also scale roughly with ζ_crp. The column of N(OH⁺) = 2.2 × 10¹² cm⁻² in the first peak and 3.2 × 10¹² cm⁻² in the deeper peak; for H₂O⁺ the columns are 1.5 × 10¹² cm⁻² and 6.9 × 10¹² cm⁻²; and for H₃O⁺ the columns are 9.2 × 10¹¹ cm⁻² and 7.3 × 10¹³ cm⁻². Because the electron abundance is low in the second peak, a significant fraction (0.1–0.3) of cosmic-ray ionizations of H₂ lead to OH⁺ and H₂O⁺, and their destruction is by H₂, which does not change in abundance with varying ζ_crp. Thus, the scaling of OH⁺ and H₂O⁺ abundances and columns with ζ_crp.¹⁴ However, the H₃O⁺ abundance in the second peak does not rise linearly with ζ_crp, but stays fairly constant, because although the formation rate scales with ζ_crp, the destruction rate also increases as ζ_crp is raised, due to the higher electron abundances produced by the enhanced cosmic-ray flux. Unlike OH⁺ and H₂O⁺, which are destroyed by H₂, H₃O⁺ is destroyed by dissociative recombinations with electrons. In fact, to first order, we would expect the abundance of H₃O⁺ to scale as ζ_crp/n_e = ζ_crp/(x_en) at the second peak.

Zoom In Zoom Out Reset image size

The abundance of electrons x_e deep in the cloud depends on the uncertain PAH abundance deep in the cloud. If the PAH abundance remains as high as is indicated in diffuse clouds and cloud surfaces (as we assume in our standard models), then we obtain the following result. Electrons are formed by cosmic-ray ionization of H₂. Electrons are mainly destroyed by collisional attachment to neutral PAHs, and the PAHs are mostly neutral. Therefore, the abundance of electrons x_e∝ ζ_crp/n. As a result, if PAHs are abundant in this deep peak, then we predict that x(H₃O⁺) ∝ ζ_crp/(x_en) will be independent of both n and ζ_crp! Note that Figure 3 compared to Figure 2 shows that the electron abundance in the second peak does scale roughly as ζ_crp, and that the abundance of H₃O⁺ in the second peak does not change significantly as we increase the cosmic-ray ionization rate by 10. Appendix B presents an analytic solution for x(H₃O⁺) near the second peak if PAHs are present. Since cosmic-ray ionization of H₂ is similar to X-ray ionization of H₂, this result implies that, if PAHs are present, regions of enhanced X-ray ionization will not show enhanced H₃O⁺ columns. We discuss in Section 3.1.3 the case of no PAHs at high A_V.

Figure 4 shows the same case as Figure 3, but plots a parameter , first discussed by N10. Here, the parameter is defined as the rate per unit volume of formation of OH⁺ divided by the total (not primary) rate per unit volume of cosmic-ray ionization of H and H₂. In effect, is an efficiency parameter in determining the formation of OH⁺ from cosmic rays. If PAH, PAH⁻ and electron abundances are relatively low, and H₂ and gas-phase O abundances are high, then is near unity. Essentially, one needs O⁺ to react with H₂ before H⁺ reacts with e, PAH, or PAH⁻. Although this qualitative limit is clear from the chemical pathways shown in Figure 1, we derive in Appendix B an analytic formula for which makes this statement more quantitative. Roughly, the condition for to be of the order of unity is

$$\begin{eqnarray} &&\left({ x({\rm H_2})\over 0.5}\right)\left({x({\rm O})\over 10^{-4}}\right) \gtrsim 0.028e^{230/T} \left[\left({x_e \over 3\times 10^{-4}}\right)\right.\nonumber\\ &&\quad\left. +\,4.4\left({x({\rm PAH^-})\over 1.5\times 10^{-8}}\right) + 2.7\left({x({\rm PAH})\over 1.85\times 10^{-7}}\right) \right]. \end{eqnarray} \tag{ 1 }$$

However, if the reverse is true, then H⁺ can recombine with PAH, PAH⁻, or electrons and disrupt the chain of reactions that lead to OH⁺, leading to low (<1) values of . Figure 4 also plots the temperature and repeats the plots of the abundances of H₂ and electrons, since they help determine the value of , as well as the abundance of OH⁺ (see Appendix B). Finally, we add the abundance of H⁺₃ to Figure 4 since it is also used to estimate cosmic-ray rates (e.g., Indriolo et al. 2007, Indriolo & McCall 2012). Because H⁺₃ is formed by the reaction of H₂ with H⁺₂ and often destroyed by electrons (see Figure 1), we see the H⁺₃ abundance rise with A_V as the H₂ abundance rises and the electron abundance falls. In general, the H⁺₃ probes the cosmic-ray ionization rates at higher A_V than the first peak in OH⁺.

Zoom In Zoom Out Reset image size

Figures 5 and 6 show the case n = 10⁴ cm⁻³, χ = 100, and ζ_crp = 2 × 10⁻¹⁶ s⁻¹. This case is nearly identical to the standard case in H09, and is representative of a GMC surface illuminated by an FUV field somewhat higher than the ISRF because of the presence of nearby O and B stars. The total cosmic-ray ionization rate (∼3–5 × 10⁻¹⁶ s⁻¹ including secondary ionizations) may represent values on the surfaces of GMCs, but may be somewhat high for the interior. We see from Figures 4 and 6 that the gas temperatures of the molecular interiors of these clouds at A_V > 5 are of the order of 30 K for n = 10⁴ cm⁻³ and 70 K for n = 100 cm⁻³, due to cosmic-ray heating when ζ_crp = 2 × 10⁻¹⁶ s⁻¹. Typically, temperatures in molecular cloud interiors are observed to be ≲ 30 K, suggesting that such high cosmic-ray ionization rates may not be appropriate for molecular cloud interiors. However, as we shall see, such high cosmic-ray rates are required to explain observations of diffuse clouds, which should have the same cosmic-ray rates as the surfaces (A_V ≲ 2) of GMCs. This suggests that ζ_crp may be higher on the surface of a molecular cloud than deep in its interior.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

If chemistry is driven by FUV photoreactions and particle–particle reactions, such as H/H₂ chemistry, then the chemical abundances mainly depend on the ratio χ/n. Therefore, one expects and sees that the H₂ abundance of Figure 5 closely matches that of Figures 2 and 3, which have the same χ/n ratio. The H₂ abundance in the higher χ case is a bit lower because of FUV photodissociation of H₂ in FUV-pumped excited vibrational states of H₂.

However, the molecular ion abundances in the first peak depend to first order on the ratio ζ_crp/n (see Appendix B) and are not too sensitive to χ. Thus, in this n = 10⁴ cm⁻³ case, their abundances in the first peak drop by a factor of nearly 10–30 compared to Figure 3 (which has the same value of ζ_crp) as the density rises by 100 from the n = 10² cm⁻³ assumed in Figure 3. Slight differences in electron abundances, H₂ abundances, and T explain the divergence from the expected n⁻¹ dependence.

The second peaks of OH⁺ and H₂O⁺ nearly scale as n⁻¹, as expected. However, the second peak of the H₃O⁺ abundance (∼10⁻⁸) is independent of n, as predicted above by the scalings of the electron density with n and ζ_crp if PAHs are present.

Figures 7 shows the case n = 10⁶ cm⁻³, χ = 10⁵, and ζ_crp = 2 × 10⁻¹⁶ s⁻¹. This case may represent strongly illuminated PDRs such as may occur around embedded compact or ultracompact H ii regions, or possibly embedded protostars illuminating the opaque walls of outflow cones. This high density and high FUV flux case was chosen because most of the column of all three ions is produced not by cosmic-ray ionization, but by other chemical reactions. In Figure 7, we see an enormous enhancement of OH⁺ abundance at A_V ∼ 1.6. Here, T ∼ 1000 K and at the same time the abundance of H₂ is moderately high, ∼10⁻². At these elevated temperatures, as discussed in Section 2.1, the H₂ can react rapidly with O to form OH or with C⁺ to form CH⁺, leading to reaction chains that make OH⁺, H₂O⁺, and H₃O⁺. One of the key heating mechanisms providing this high T is the FUV pumping and the H₂ formation pumping of excited vibrational levels of H₂, followed by collisional de-excitation of these levels which leads to gas heating (e.g., Tielens & Hollenbach 1985). The essential point is that the OH⁺, H₂O⁺, and H₃O⁺ columns are not provided by cosmic-ray ionization, and therefore cannot diagnose the cosmic-ray ionization rate, except to give an upper limit.

Zoom In Zoom Out Reset image size

3.1.2. Contour Plots of Integrated Columns of Ions

Figure 8 shows the contours of the integrated (from A_V = 0 to A_V = 20) columns of OH⁺ ions for a primary cosmic-ray ionization rate of ζ_crp = 2 × 10⁻¹⁶ s⁻¹ per H atom, and plotted as functions of n and χ. Note that the OH⁺ columns include both the first and second peaks. We emphasize that these are columns perpendicular to the face of the PDR slab. If clouds are observed obliquely, proportionately more column will be in the LOS. Similarly, if we are observing a slab illuminated on both sides, then the columns will be raised by a factor of two if the slab is quite optically thick (A_V ≫ 1) so that first and second peaks occur on both sides. The upper right hand portion of the figure is blacked out because radiation pressure, photoelectric emission, and photodesorption forces on dust grains, when χ/n₂ ≳ 300, drives dust rapidly through the PDR (Weingartner & Draine 2001b). Such high ratios of χ/n rarely occur in nature, and require a much more detailed PDR code.

Zoom In Zoom Out Reset image size

Figure 8 shows two main features. First, at low χ ≲ 1000 or low n ≲ 10⁴ cm⁻³, the columns of OH⁺ are roughly proportional to n⁻¹ and almost independent of χ, as discussed above. There is a weak dependence on χ for χ ≲ 1000 because higher χ leads to higher gas T, and the O⁺ abundance rises with exp(−230 K/T). From this figure, it is clear that since OH⁺ columns of at least 10¹² cm⁻² are needed to make absorption observations feasible, low-density (n ≲ 300 cm⁻³) clouds should be targeted if ζ_crp = 2 × 10⁻¹⁶ s⁻¹. It should also be noted, as is obvious in Figures 2, 3, and 5, that both the first and second peaks contribute to the OH⁺ column so that there is substantial column at both A_V < 1 and A_V > 1. The OH⁺ in this case is quite cold, T ≲ 200 K. Second, these relations completely break down in the upper right portion (n ≳ 10⁴ cm⁻³ and χ ≳ 10³) of this contour plot. Here, as discussed above, the high χ and n (high density brings the H/H₂ interface closer to the surface, where the gas is warm) lead to very warm (T ∼ 1000 K) H₂ near the surface (A_V ≲ 2) of the cloud. This warm H₂ drives reactions that produce OH⁺ without the need for cosmic-ray ionization. In this way, observable columns of OH⁺ can be formed at these elevated values of χ and n, and the columns occur at moderate A_V ∼ 2. The OH⁺ in this case is quite warm, T ≳ 300 K, and its column is independent of ζ_crp, but depends on n and χ.

Figure 9 shows the contours of the integrated (from A_V = 0 to A_V = 20) columns of H₂O⁺ ions for a primary cosmic-ray ionization rate of ζ_crp = 2 × 10⁻¹⁶ s⁻¹ per H atom. This figure can be compared with Figure 8, which treats the same cosmic-ray case for OH⁺. We focus on regions where these ions might be detectable, that is, for columns >10¹² cm⁻², and in the cosmic-ray-dominated zones of n and χ. In these low-density regions, comparison of Figure 9 with 8 reveals that the column ratios are of order unity, as might be expected. The formation rates of these two ions are nearly the same (stemming from the cosmic-ray ionization rate) and the destruction rates of these two molecules are nearly the same (via H₂ in the regions where most of the columns are generated). Note that in Figures 2, 3, and 5, the local OH⁺ abundance peaks at lower values of A_V than the H₂O⁺ abundance, and that Figures 8 and 9 present integrated columns through a high A_V slab. Therefore, we can obtain higher column ratios of these two molecular ions if we truncate our slabs to small diffuse clouds with A_V < 1 (see below Section 3.2.2).

Zoom In Zoom Out Reset image size

We do not provide a contour plot similar to Figures 8 and 9 but with different ζ_crp because the results are simple to describe. For χ ≲ 1000, the OH⁺ and H₂O⁺ columns scale as ζ_crp. For χ ≳ 1000, the columns are independent of ζ_crp because cosmic rays do not produce the OH⁺ and H₂O⁺. Forcing a fit with the expected (to first order) ζ_crp/n dependence in the cosmic-ray-dominated region χ ≲ 100, we find roughly (factor of two) the simple expressions

$$\begin{equation} N({\rm OH^+}) \sim 5\times 10^{12} \chi ^{1/3} \left({{100\ {\rm cm^{-3}}}\over n}\right)\left({\zeta _{{\rm crp}} \over {2\times 10^{-16} \ {\rm s^{-1}}}}\right) \ {\rm cm^{-2}} \end{equation} \tag{ 2 }$$

and

$$\begin{equation} N({\rm H_2O^+}) \sim 10^{13}\chi ^{1/4} \left({{100\ {\rm cm}^{-3}}\over n}\right) \left({\zeta _{{\rm crp}} \over { 2\times 10^{-16} \ {\rm s^{-1}}}}\right) \ {\rm cm^{-2}}. \end{equation} \tag{ 3 }$$

We emphasize that these columns are the total OH⁺ and H₂O⁺ columns in both the first and second peaks.

We do not present integrated (from A_V = 0 to A_V = 20) columns of H₃O⁺ ions because, as discussed above, most of their columns arise at high A_V ∼ 6 and this second peak column is quite constant as a function of n, χ, and ζ_crp if PAHs are present at the same abundances as in diffuse clouds. With PAHs present in the second peak, we find that the H₃O⁺ column is slightly dependent on the gas density, varying from 10¹⁴ cm⁻² at n ∼ 100 cm⁻³ to 10^13.5 cm⁻² at n ∼ 10⁶ cm⁻³. At densities of n = 100 cm⁻³ and with χ = 1 and ζ_crp = 2 × 10⁻¹⁶ s⁻¹, we remind the reader (see discussion of Figure 3) that N(H₃O⁺) =9.2 × 10¹¹ cm⁻² in the first peak; like OH⁺ and H₂O⁺ the column of H₃O⁺ in the first peak scales with ζ_crp/n so that even lower densities or higher ζ_crp are needed to make H₃O⁺ detectable in the first peak (e.g., in a diffuse foreground cloud).

3.1.3. Sensitivity to PAH Abundance and Other Parameters

Diffuse clouds have relatively small columns, A_V ≲ 1, whereas translucent clouds have columns intermediate between diffuse clouds and GMCs, 1 ≲ A_V ≲ 5. Although GMCs have sufficient column to incorporate both peaks of the OH⁺, H₂O⁺, and H₃O⁺ abundances, diffuse and translucent clouds typically (with the possible exception of the highest A_V translucent clouds) only contain the surface peak. In fact, a diffuse cloud may have such small A_V that the cloud may truncate the full peak of one or all of the ions. Therefore, the parameter A_Vt, the total A_V through the cloud, becomes an important new parameter in determining, for example, ratios of the columns of the ions.

Both diffuse and translucent clouds have insufficient column to be gravitationally bound, so generally they are in thermal pressure equilibrium with the ISM. The local typical value of the thermal pressure is nT ∼ 3 − 4 × 10³ cm⁻³ K, and this pressure may rise by factors of 2–3 for clouds in the molecular ring of the Galaxy (Wolfire et al. 2003). As discussed above (Section 2.2), depending on the galactocentric radius R, they have densities 30 cm⁻³ < n < 100 cm⁻³, temperatures T ∼ 50–100 K, and incident FUV fluxes characterized by 1<χ < 4. The ratio χ/n₂, critical to the photochemistry, varies typically from about 3–6, although we explore a somewhat larger range 3–10 here.

In the low-density regime, n < 10³ cm⁻³, appropriate for diffuse and translucent clouds, there is a scaling of the results of thermochemical models. The parameters which control the thermochemical structure of a cloud are just χ/n, ζ_crp/n, and A_Vt. We note that this scaling does not apply to the denser PDR models discussed in the previous section. At high density, n > 10³ cm⁻³, the cooling of the gas is suppressed by collisional de-excitation of the fine structure states of C⁺ and O. Therefore, higher density gas will typically give higher gas temperatures, even when χ/n is held constant. However, given that this scaling applies for diffuse and translucent clouds, we generally in this section plot our results as functions of χ/n, ζ_crp/n, and A_Vt. We note that this implies that the observations may give us a measure of ζ_crp/n, but to get an estimate of ζ_crp we will then have to estimate the gas density n in the region observed.

3.2.1. The Total Column of OH⁺ through a Diffuse Cloud of Size A_Vt

Figure 10 plots the column of OH⁺, N(OH⁺), as a function of A_Vt for four values of ζ_crp/n₂ (recall, n₂ ≡ n/100 cm⁻³) and for χ/n₂ = 3.16. The dotted lines plot x_c(H₂), the abundance of H₂ at cloud center, and these values appear on the right of the figure. Recall that A_Vt is the total A_V through the diffuse slab, measured perpendicular to the slab and the ion columns plotted are perpendicular to the slab.

Zoom In Zoom Out Reset image size

Figure 10 shows that N(OH⁺) increases with ζ_crp/n and with A_Vt, although one sees a saturation of the column with A_Vt at A_Vt ≳ 0.3 for the lower three cases of ζ_crp/n. This saturation is due to the peaking of the local OH⁺ abundance at lower values of A_V. Figure 10 shows that N(OH⁺) correlates with x_c(H₂), and that moderate molecular hydrogen abundances x_c(H₂) ≳ 0.03 are required to obtain substantial columns of OH⁺. Observed OH⁺ columns in broad (Δv ∼ 20 km s⁻¹, see Section 4.1) velocity components in the ISM are of the order of 3 × 10¹³–3 × 10¹⁴ cm⁻². With this value of χ/n₂ = 3.16, we see that it requires very high cosmic-ray ionization rates, ζ_crp/n₂ ≳ 3 × 10⁻¹⁶ s⁻¹ and high A_Vt ≳ 0.3 for a single cloud, seen face on, to produce these columns. Therefore, it appears that geometrical effects such as seeing several clouds along the LOS, or viewing the cloud obliquely, may be required to match observation. In fact, as we discuss in Section 4.1, many clouds may to required to cover the broad (Δv ∼ 20 km s⁻¹) absorption components, since single clouds would have much narrower velocity widths. Because of geometrical effects and variation in A_Vt and χ/n, the observation of the column of OH⁺ is not sufficient to tightly restrict ζ_crp/n.

Figure 11 plots N(OH⁺) as a function of A_Vt for four values of ζ_crp/n₂ and for χ/n₂ = 10; in other words, the same as Figure 10 but with a ratio of χ/n that is 3.16 times higher. The main effect of raising χ/n is to push the H/H₂ transition deeper into the cloud to higher A_V. This is seen in the plot of x_c(H₂), which rises to 0.01 at A_Vt ∼ 0.3 in this case, compared to A_Vt ∼ 0.1 in the previous case of χ/n₂ = 3.16. Since H₂ is required to make OH⁺, this moves the peak of OH⁺ to higher values of A_V, and thus pushes the rise of N(OH⁺) to higher values of A_Vt than in Figure 10. However, this first peak has more column of OH⁺ compared with the case χ/n₂ = 3.16. Therefore, to produce the observed columns of OH⁺ now requires only ζ_crp/n₂ ≳ 3 × 10⁻¹⁷ s⁻¹ and high A_Vt ≳ 1 for a single cloud, seen face on.

Zoom In Zoom Out Reset image size

3.2.2. N(OH⁺)/N(H₂O⁺) and N(OH⁺)/N(H) through a Diffuse Cloud of Size A_Vt

The results on the total column of OH⁺ suggest that other observations must be brought into play to further restrict the cosmic-ray ionization rate. Since H₂O⁺ is often observed as well, we first consider the ratio N(OH⁺)/N(H₂O⁺). As seen in Figures 2, 3, and 5, the H₂O⁺ abundance peaks at higher values of A_V than the OH⁺ abundance. Therefore, the ratio of the columns of these two ions may give us some measure of A_Vt.

Figure 12 plots the ratio N(OH⁺)/N(H₂O⁺) as a function of A_Vt for four values of ζ_crp/n₂ and for χ/n₂ = 3.16. As expected, at low values of A_Vt ≲ 0.1, the ratio N(OH⁺)/N(H₂O⁺) is high, ∼10–30, as we have not yet reached high enough values of A_V to include the H₂O⁺ peak. However, as A_Vt increases, the ratio of the columns continues to drop until at A_Vt ≳ 1, the ratio is approximately 2–3, except in the case of the highest cosmic-ray rate ζ_crp/n₂ = 3.16 × 10⁻¹⁵ s⁻¹.¹⁵ Once we have incorporated both the OH⁺ and the H₂O⁺ abundance peaks, the column ratios are near unity, as noted in Section 3.1.

Zoom In Zoom Out Reset image size

Figure 13 plots the ratio N(OH⁺)/N(H₂O⁺) as a function of A_Vt for four values of ζ_crp/n₂ and for χ/n₂ = 10; in other words, the same cases as Figure 12 except the ratio χ/n is raised by a factor of 3.16. The rise in χ/n pushes the H/H₂ transition and the peaks of OH⁺ and H₂O⁺ to higher values of A_V. Therefore, compared to Figure 12, we see the drop in the N(OH⁺)/N(H₂O⁺) ratio at higher values of A_Vt. The observed ratios of N(OH⁺)/N(H₂O⁺) range from 3 to 15. If χ/n₂ = 3.16, and if we assume that ζ_crp/n₂ ≲ 3 × 10⁻¹⁶ s⁻¹, then Figure 12 suggests that A_Vt ∼ 0.1–0.3 to obtain these ratios. However, if this is true, then Figure 10 implies that even with as high a value of ζ_crp/n₂ = 3 × 10⁻¹⁶ s⁻¹, we will need a "geometrical factor" of ∼10 in order to obtain N(OH⁺) ∼10¹⁴ cm⁻². This geometrical factor is the combination of many clouds along the LOS, along with the enhancement in column due to viewing angle of the cloud. The situation changes with χ/n₂ = 10, as seen in Figure 13. Here, again assuming that ζ_crp/n₂ ≲ 3 × 10⁻¹⁶ s⁻¹, the observed ratios can be obtained with A_Vt ∼ 1–3. In this case, using Figure 11, the geometrical factor needs to be only ∼1–3 to produce OH⁺ columns of 10¹⁴ cm⁻² when ζ_crp/n₂ ∼ 3 × 10⁻¹⁶ s⁻¹.

Zoom In Zoom Out Reset image size

Observations of H i 21 cm are often available along the same LOS as the OH⁺ and H₂O⁺ measurements. In this case, the column of HI, N(H), along the LOS associated with the velocity feature of the ions can be estimated. To be precise, the observations directly give N(H)/T, where T is an average temperature along the LOS. In Figures 14 and 15, we plot the ratio N(OH⁺)/[N(H)/T₂] on the vertical axis (T₂ = T/100 K) and the ratio N(OH⁺)/N(H₂O⁺) on the horizontal axis for two fixed values of χ/n₂ = 3.16 and 10. The results of our two-sided diffuse cloud models are shown as contours on this figure. As noted above, higher average abundances of OH⁺ (or N(OH⁺)/N(H)) requires higher ζ_crp/n. On the other hand, higher N(OH⁺)/N(H₂O⁺) requires either low A_Vt or high χ/n. In the next section, we apply Figures 10, 11, 14, and 15 to the observational data to constrain ζ_crp/n along sight lines to W49N.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

3.2.3. The Possible Effects of Turbulence

As discussed in Section 1, recent Herschel observations of bright Galactic continuum sources have revealed OH⁺ and H₂O⁺ absorption features arising in multiple foreground clouds along the targeted sight lines (Gerin et al. 2010; N10). The three crosses in Figure 14 denote the N(OH⁺)/N(H₂O⁺) and N(OH⁺)/[N(H)/T₂] ratios for the two such sources for which results have been reported to date: G10.6–0.4 (a.k.a. W31C) (the lower limits) and W49N (2 points to the right). Those given for G10.6–0.4 apply to all the material along the sight line, while those given for W49N apply to two separate velocity intervals (30–50 km s⁻¹ and 50–78 km s⁻¹). The OH⁺ absorption in G10.6–0.4 is largely saturated, and thus the plotted values are lower limits. For W49N, the columns of H derived assumed T = 100 K or T₂ = 1; the quoted values of N(H)/T₂ are 6.95 × 10²¹ cm⁻² for the 30–50 km s⁻¹ feature and 7.23 × 10²¹ cm⁻² for the 50–78 km s⁻¹ feature. The total columns N(OH⁺) observed in these two velocity intervals are 3.6 × 10¹⁴ and 2.1 × 10¹⁴ cm⁻², respectively.

The N(OH⁺)/N(H₂O⁺) ratios measured toward W49N (∼10) and the wide velocity range of the absorbing clouds suggest that the absorbing material is comprised of multiple clouds of small extinction. The LOS to W49N is approximately 11 kpc long, and passes as close as R = 5 kpc to the Galactic center. Assuming χ/n₂ = 3.16, models with A_Vt = 0.32 and 0.25 best account for all the data (Figure 14), with cosmic-ray ionization rates of ∼6 and 4 × 10⁻¹⁶ n₂ s⁻¹, respectively, for the 30–50 km s⁻¹ and 50–78 km s⁻¹ velocity intervals. As seen in Figure 10 for χ/n₂ = 3.16, the column of OH⁺ in a single cloud of size A_Vt = 0.32 is N(OH⁺) ≃ 2 × 10¹³ cm⁻² and for A_Vt = 0.25 is N(OH⁺) ≃ 1.5 × 10¹³ cm⁻². Therefore, we either need ∼15 clouds along the LOS or a smaller number but with geometrical enhancement effects. Note that a single cloud might have an OH⁺ absorption line width of only ∼1–3 km s⁻¹. Therefore, to cover the broad absorption features we are modeling (Δv = 20 or 28 km s⁻¹), we need 10–30 clouds spread out along the LOS so that their galactic rotational velocities coupled with their individual line widths cover this velocity range. Therefore, this χ/n₂ = 3.16 model has barely enough clouds to produce the relatively smooth absorption feature observed. Assuming χ/n₂ = 10, models with A_Vt ≃ 0.75 and A_Vt ≃ 0.63 and with ζ_crp/n₂ ∼ 2 × 10⁻¹⁶ and 1.2 × 10⁻¹⁶ s⁻¹, respectively, best account for the data (see Figure 15). With these values of A_Vt and ζ_crp/n₂, inspection of Figure 11 shows that we require ∼12 clouds along the LOS. This value of χ/n₂ has even fewer clouds to cover the broad absorption line observed. Moreover, the temperatures in clouds with χ/n₂ = 10 are higher (∼200 K) than is typically observed near the solar neighborhood (Heiles & Troland 2003). And finally, the Wolfire et al. (2003) results suggest that along the LOS to W49N, the average χ/n₂ ∼ 4. A similar modeling of the case χ/n₂ = 1, not shown in the figures, gives ζ_crp/n₂ = 2.5 × 10⁻¹⁵ and 2.0 × 10⁻¹⁵ s⁻¹ with A_Vt = 0.16 and 0.08, respectively. Again, because of the high cosmic-ray rates, the gas temperatures are high in these clouds (∼150 K). In addition, these cosmic-ray rates seem improbably high. This model is driven to high cosmic-ray rates in part because otherwise (with low χ/n₂) the gas is cold, which reduces the rate that H⁺ can charge exchange with O. Therefore, we finally conclude that our models suggest typical A_Vt ∼ 0.3, χ/n₂ ∼ 3, and ζ_crp/n₂ ∼ 6 × 10⁻¹⁶ s⁻¹ for the 30–50 km s⁻¹ component, and ζ_crp/n₂ ∼ 4 × 10⁻¹⁶ s⁻¹ for the 50–78 km s⁻¹ component.

A better approach is to ask, "What distribution of cloud A_Vt or hydrogen nucleus column N through the cloud will we encounter in traversing the ∼11 kpc to W49N?" H i 21 cm observational data suggest that dn_cl/dN∝N⁻², where n_cl is the number of clouds for N ≳ 2.6 × 10²⁰ cm⁻², and proportional to N⁻¹ for N ≲ 2.6 × 10²⁰ cm⁻² (Heiles & Troland 2005).¹⁷ We have crudely integrated a distribution of clouds with this dependence, with the constant of proportionality derived by requiring the integral to obtain a total column of N(H) =7 × 10²¹ cm⁻², the observed atomic H column in each velocity component of W49N. We use N = 6 × 10²¹ cm⁻² as an upper limit for the integration; above this, we enter small number statistics since the total column is of this order. The lower limit does not enter the integration significantly, as long as it is ≪2.6 × 10²⁰ cm⁻², since these clouds contain very little of the total column of any species. The integration removes A_Vt as a variable, and leaves only the parameters ζ_crp/n₂ and χ/n₂ as free parameters to match the observed OH⁺/H₂O⁺ ratio and the total column of OH⁺. We find that with χ/n₂ = 3.16 and ζ_crp/n₂ = 3.7 × 10⁻¹⁶ s⁻¹, we obtain N(OH⁺) ≃ 2.1 × 10¹⁴ cm⁻² and N(H₂O⁺) ≃ 2.4 × 10¹³ cm⁻². These values match the velocity interval 50–78 km s⁻¹ extremely well. Increasing the cosmic-ray rate to ζ_crp/N₂ = 6.3 × 10⁻¹⁶ s⁻¹, we obtain N(OH⁺) ≃ 3.6 × 10¹⁴ cm⁻² and N(H₂O⁺) ≃ 4.0 × 10¹³ cm⁻², a very good fit to the lower velocity component. Considering the small number statistics, especially of the higher A_Vt clouds, this is in very good agreement. Interestingly, the main contribution to N(OH⁺) comes from near A_Vt = 0.2–0.6. This is because of the dependence of N(OH⁺) on A_Vt (see Figure 10). Note that N(OH⁺) rises sharply with A_Vt until about A_Vt = 0.3, and then it levels off. Therefore, the integral is weighted to that size cloud in the range of the turnover (A_Vt ∼ 0.3–0.6), which has the peak OH⁺ abundance (averaged through the cloud). In addition, the cloud distribution steepens when A_Vt > 0.13, thereby decreasing the contribution from higher A_Vt clouds. The total number of clouds in the range A_Vt ≃ 0.1–1 is approximately 20. We note that the H₂O⁺ columns have greater contribution from higher A_Vt clouds, which are less numerous, and therefore we predict the H₂O⁺ absorption feature to have greater fluctuations.

We next examine the cases χ/n₂ = 1 and 10. Assuming χ/n₂ = 10, we find that we need cosmic-ray rates of 1.1 × 10⁻¹⁵ s⁻¹ and 7 × 10⁻¹⁶ s⁻¹ to match the OH⁺ columns in the two velocity components, respectively. However, the OH⁺/H₂O⁺ column ratio is 12.1, somewhat higher than observed, and again, the clouds are too warm (∼200 K). The high field pushes the peaks of OH⁺ and H₂O⁺ deeper into the cloud, and our clouds tend to truncate the H₂O⁺ column, leading to the high OH⁺/H₂O⁺ ratio. We need much higher cosmic-ray rates than our solution above for χ/n₂ = 10 with clouds of single A_Vt ∼ 0.7 because the cloud distribution leads to a significant column of H from clouds of low A_Vt which have very little OH⁺. Assuming χ/n₂ = 1, we find that we need cosmic-ray rates of 2 × 10⁻¹⁵ s⁻¹ and 1.2 × 10⁻¹⁵ s⁻¹ to match the OH⁺ columns in the two velocity components, respectively. However, the OH⁺/H₂O⁺column ratio is 5.6, lower than observed. With low χ/n₂, the clouds tend to contain both OH⁺ and H₂O⁺ peaks, driving the ratio down. We conclude that the best fit is with χ/n₂ = 3.16, giving ζ_crp/n₂ ∼ 6 × 10⁻¹⁶ s⁻¹ and 4 × 10⁻¹⁶ s⁻¹ for the two velocity components, in good agreement with our analysis above that did not use the cloud distribution. However, the cloud distribution gives added weight to the conclusion that χ/n₂ ∼ 3.

The cosmic-ray rates derived from our models are roughly two to four times larger than those inferred by N10, who used a simple analytic treatment to infer cosmic-ray ionization rates of 0.6 and 1.2 × 10⁻¹⁶ (n₂/) s⁻¹, where is the fraction of cosmic-ray ionizations that lead to OH⁺. Using pure gas-phase model results from the Meudon PDR code (Le Petit et al. 2006), N10 found that lay in the range 0.5–1.0 for a wide variety of cloud conditions; this would imply cosmic-ray ionization rates in the range ∼0.6–2.4 × 10⁻¹⁶ n₂ s⁻¹ for the material along the sight line. The main source of the difference between our models and those of N10 is our inclusion of PAH⁻ – H⁺ recombination and the charge exchange of H⁺ with neutral PAH in the present work, processes—previously omitted—that reduce the fraction of cosmic-ray ionizations that lead to OH⁺ production (in our models ∼ 0.1–0.3 in the regions where most of the OH⁺ lies).

Figures 16 and 17 repeat Figures 14 and 15, except that the rates of PAH and PAH⁻ neutralization of H⁺ have been reduced by >4, so that they are negligible compared to the radiative recombination of H⁺ with electrons (see Appendix B). Note that this reduction could be achieved by either lower PAH abundances or lower PAH rate coefficients with H⁺ or a combination of both. Another possibility is increased photoionization rates of PAH⁻ and PAH. Figure 16 shows that the best fit to the data for χ/n = 3.16 is A_Vt ∼ 0.316 and ζ_crp/n₂ ∼ 1–2 × 10⁻¹⁶ s⁻¹, in complete agreement with N10. Therefore, the PAH chemistry is extremely important in deriving cosmic-ray rates from the OH⁺ and H₂O⁺ ions. We consider the rates from Figures 16 and 17 lower limits to the cosmic-ray rates, whereas the rates from Figures 14 and 15 could be considered upper limits, although they represent our best estimates of PAH chemistry at this time.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The cosmic-ray ionization rates suggested by the predictions shown in Figure 14 are also a factor ∼5–10 larger than those typically inferred from observations of H⁺₃ in the Galactic disk (Indriolo et al. 2007; Indriolo & McCall 2012). Indriolo & McCall conclude, however, that cosmic-ray rates vary by more than an order of magnitude over various sight lines, from ζ_crp ∼0.7 × 10⁻¹⁶ s⁻¹ to 4.6 × 10⁻¹⁶ s⁻¹, with a mean of ∼1.5 × 10⁻¹⁶ s⁻¹ (note that we have converted their total ionization rate per H₂ to our primary ionization rate per H nucleus by dividing their values by 2.3; we also note that their typical density was n ∼ 200 cm⁻³, so their results imply ζ_crp/n₂ that are two times smaller than these values). Their highest values of ζ_crp/n₂ are still a factor of ∼2 below what we find for W49N, so there appears to be a small discrepancy. We point out that the cosmic-ray rate derived from H⁺₃ observations is directly proportional to the assumed electron abundance. Indriolo & McCall assumed that the electrons were supplied by C⁺, with an abundance of 1.5 × 10⁻⁴. However, in our models of diffuse clouds, where typically the gas density is ∼100 cm⁻³, the cosmic-ray ionization of H can produce comparable or greater abundances of H⁺ than C⁺. Therefore, we find for ζ_crp/n₂ = 2 × 10⁻¹⁶ s⁻¹, for example, that x_e ≃ 3 × 10⁻⁴ when x(H₂) ≃ 0.1 and x_e ≃ 2 × 10⁻⁴ when x(H₂) ≃ 0.4. This implies that the Indriolo et al. rates may need to be increased, by factors as much as ∼2, depending on ζ_crp/n₂. Given the small number of sources toward which N(OH⁺)/N(H₂O⁺) and N(OH⁺)/N(H) ratios have so far been reported, the proven variation in cosmic-ray rates along different lines of sight, the lack of a common sight line to compare the cosmic-ray rates derived by H⁺₃ versus by OH⁺and H₂O⁺, the possibility of higher electron abundances than assumed by Indriolo and coworkers, and the lack of certainty in PAH chemistry, it is not yet possible to draw firm conclusions about whether our results differ significantly from that of Indriolo et al. What is clear is that cosmic-ray rates in the diffuse ISM are higher than was previously thought.

Herschel has also detected absorption by H₃O⁺ in foreground material along the sight line to G10.6 – 0.4 (Gerin et al. 2010). The average H₃O⁺/H₂O⁺ ratio along the sight line is 0.7, but the distribution of H₃O⁺ is clearly different from that of H₂O⁺ and OH⁺, with the H₃O⁺ concentrated within a single narrow velocity component where the H₂ fraction is presumably largest. This cloud is unassociated with the source, but is probably the foreground cloud with the largest A_V along the LOS. It appears in the HF spectrum (Neufeld et al. 2010) and CO spectrum (C. Vastel 2012, private communication; see also Corbel & Eikenberry 2004), suggesting a high molecular fraction and therefore substantial A_V. The H₃O⁺ column in this narrow component is ∼1–2 × 10¹³ cm⁻², consistent with our predictions for the second peak in a high A_V cloud.

PvDK92 searched for the 396, 364, and 307 GHz inversion-rotation lines of H₃O⁺ in 20 Galactic targets—including GMCs, star-forming regions, Galactic center sources, and a few evolved stars—and reported definitive detections of H₃O⁺ emission from the G34.3+0.15 hot core, two positions in Sgr B2 (a Galactic center GMC), and the W3 IRS5, W3(OH), W51M, and Orion KL high-mass star-forming regions. In the case of Orion and Sgr B2, H₃O⁺ emissions had been previously discovered by Wootten et al. (1991). The H₃O⁺ column densities derived by PvDK92 for the high-mass cores W3 IRS5, W3(OH), W51M are fairly similar, ranging from a few × 10¹³ to a few × 10¹⁴ cm⁻², and argue for a similar environment and formation mechanism. Given the large critical densities for the observed transitions, PvDK92 inferred relatively high densities (n > 10⁶ cm⁻³) and temperatures (T > 50 K) for the H₃O⁺emitting gas. The H₃O⁺ column densities observed in these high-mass cores are in good agreement with the predictions of our models for either the case in which PAHs and/or VSGs are assumed to be present or the case where PAHs and low-ionization potential metals are highly depleted¹⁸; the PAH case predicts H₃O⁺ column densities of 3 × 10¹³–10¹⁴ cm⁻², regardless of density n or ζ_crp.

Herschel's HIFI instrument has allowed the detection of far-infrared and submillimeter H₃O⁺ lines that are not accessible to ground-based observatories. Several H₃O⁺ emission lines in the THz domain have been reported toward high-mass YSOs in the W3 IRS5 region (Benz et al. 2010). A rotation diagram combining HIFI data with lower frequency data from PvDK92 indicates a rotational temperature T_rot ∼ 239 K, suggesting that the observed emission may arise either from hot and dense gas or is radiatively excited by continuum photons from hot dust grains. Benz et al. (2010) inferred N(H₃O⁺) = 8.5 ± 2 × 10¹³ cm⁻² and interpreted the H₃O⁺ emission as arising from the outflow walls heated and irradiated by the FUV radiation field from massive YSOs. Their inferred H₃O⁺ column densities and rotational temperatures are consistent with our models of dense gas illuminated by a relatively strong FUV field (where large kinetic temperatures are achieved at relatively low A_V ∼ 1–2).

For sources where a strong infrared radiation field is present, submillimeter H₃O⁺ line emission can be accompanied by far-infrared absorption within an extended envelope. The H₃O⁺ J_K = 2⁻₁ − 1₁⁺ and 2⁻₀ − 1₀⁺ inversion-rotation and 1⁻₁ − 1₁⁺ pure-inversion FIR absorption lines at 100.577, 100.869, and 181.054 μm were detected by ISO toward the optically thick FIR continuum emission of Sgr B2 (Goicoechea & Cernicharo 2001). An H₃O⁺ column density of ∼1.6 × 10¹⁴ cm⁻² was inferred in the warm and low-density extended envelope of Sgr B2 (n = 10³–10⁴ cm⁻³, χ = 10³–10⁴ from [O i] and [C ii] observations; Goicoechea et al. 2004). However, the p-H₃O⁺ 3⁺₂ − 2₂⁻ 364 GHz emission line was subsequently mapped around the Sgr B2(M) core at 18'' angular resolution with APEX by van der Tak et al. (2006), who inferred column densities ∼10¹⁵–10¹⁶ cm⁻² on the basis of a detailed excitation model. This value is much larger than the estimates obtained from FIR absorption lines, presumably because the latter only probe the outer envelope of the source whereas the submillimeter observations probe material to a much larger depth. The H₃O⁺ column densities inferred by van der Tak et al. (2006) are much larger than our model predictions for externally illuminated clouds; this discrepancy likely reflects the presence of a strong internal luminosity source in Sgr B2 that heats the dust grains within the interior and prevents the freezeout of oxygen nuclei (see Appendix C).

Thanks to the much improved sensitivity of space- and ground-based receivers and detectors, molecular ions can now be detected outside the Milky Way. Extragalactic H₃O⁺ was first detected through James Clerk Maxwell telescope observations of the p-H₃O⁺ 364 GHz emission line toward the prototypical ultraluminous infrared galaxy Arp 220, and toward M82, an evolved starburst (van der Tak et al. 2008). H₃O⁺ was subsequently detected toward IC342, NGC 253, NGC 1068, NGC 4418, and NGC 6240, and upper limits obtained toward IRAS 15250 and Arp 299 galaxies (Aalto et al. 2011). Except for IC342 and M82, the typical H₃O⁺ column densities (∼10¹⁵–10¹⁶ cm⁻²) derived from extragalactic H₃O⁺ detections are much larger than the predictions of our models for single clouds. These observations are of H₃O⁺ in emission and the authors assume T_ex ∼ T_rad to obtain these columns with error bars of a factor of two. As in the case of Sgr B2, the discrepancy between our models and the observations may reflect the effect of enhanced dust temperatures that prevent the freezeout of elemental oxygen. Alternatively, there may be a large number of clouds along the LOS.

For the starburst galaxies M82 and IC342, however, where H₃O⁺ column densities (∼10¹⁴ cm⁻²) are inferred from the observations, enhanced dust temperatures or ionization rates are not apparently required. For example, van der Tak et al. (2008) inferred N(H₃O⁺) ≃ 1.1 × 10¹⁴ cm⁻² in M82 from an emission line that contained a combination of a broad (Δv ∼ 260 km s⁻¹) and a narrow (Δv ∼ 40 km s⁻¹) component. In addition, a recent Herschel/HIFI detection (Weiß et al. 2010) of the o-H₂O⁺ 1₁₁ − 0₀₀ ground-state line in absorption toward M82 revealed N(H₂O⁺) ≃ 2.9 × 10¹⁴ cm⁻², arising from a Δv ≃ 77 km s⁻¹ feature. The large width of the lines suggest numerous clouds in the beam. Overall, these numbers are consistent with our models with the H₂O⁺ arising from the first (and possibly second) peak of numerous relatively low-density clouds, whereas the H₃O⁺ arises from the second peaks of many molecular clouds in the beam.

Table 1 presents some of the key reactions and/or some of the reaction rates that have changed since H09. We emphasize that this is not a complete listing of the ∼300 chemical reactions in the PDR code described in H09. The PAH rates are scaled by the factor Φ_PAH, a scaling factor introduced in Wolfire et al. (2008) that incorporates the uncertainties in PAH reaction rates, PAH sizes, and PAH abundances. Wolfire et al. (2008) found from a semi-empirical fit of our PDR models to C, C⁺, H, and H₂ column densities in diffuse clouds that Φ_PAH ∼ 0.5, and we have adopted that value in all of our PDR models.

Table 2 presents the standard gas-phase abundances and grain properties that we have adopted in most of our PDR models. As discussed in the text, we have also run models with these values changed to test the sensitivity of the results to the standard values. In particular, we have run models with x(PAH) = 0; with x(Si), x(Fe), x(S), and x(Mg) all set to 10⁻⁸ or 10⁻⁵; and with x(O) = 4.5 × 10⁻⁴.