INVESTIGATING THE COSMIC-RAY IONIZATION RATE IN THE GALACTIC DIFFUSE INTERSTELLAR MEDIUM THROUGH OBSERVATIONS OF H⁺₃

In diffuse molecular clouds, conditions are such that only the two lowest energy levels of H⁺₃—the (J, K) = (1, 1) para and (1, 0) ortho states—are expected to be significantly populated. As a result, observations that probe transitions arising from these two states will trace the entire content of H⁺₃ along a line of sight. We targeted three such transitions—the R(1, 1)^u, R(1, 0), and R(1, 1)^l—which are observable at 3.668083 μm, 3.668516 μm, and 3.715479 μm, respectively. As the weak H⁺₃ lines are located near various atmospheric CH₄ and HDO lines, precise removal of the telluric lines is vital for the purpose of accurately determining H⁺₃ column densities, and is facilitated by the observation of telluric standard stars. Because the R(1, 1)^u and R(1, 0) transitions are easily observed simultaneously (being separated by about 4 Å), and because together they probe the entire population of H⁺₃, they are the most frequently observed transitions.

Although observations of H⁺₃ have become more commonplace, as evidenced by the publication list in Section 1, they are still rather difficult. Given the transition dipole moments (see, e.g., Goto et al. 2002, and references therein) and a typical relative abundance of about 10⁻⁷ to 10⁻⁸ with respect to H₂, H⁺₃ absorption lines in diffuse molecular clouds are usually about 1%–2% deep, and thus require a signal-to-noise ratio (S/N) greater than about 300 on the continuum to achieve 3σ detections. As a result, background sources must have L-band (3.5 μm) magnitudes brighter than about L = 7.5 mag to be feasible targets for 8 m (or L = 5.5 mag for 4 m) class telescopes. Ideally, these background sources should also be early-type stars (OB) such that the continuum is free of stellar absorption features. A list of all background sources targeted in our study, along with select line-of-sight properties, is presented in Table 1. Additionally, to adequately sample the intrinsically narrow interstellar absorption lines requires high spectroscopic resolving power (R ≳ 20,000).

Table 1. Science Targets

		E(B − V)		Distance
Object	HD Number	(mag)	Ref.	(pc)	Ref.
κ Cas	2905	0.33	1	1370	11
HD 20041	20041	0.72	2	1400	2
HD 21483	21483	0.56	2	440	2
HD 21389	21389	0.57	2	940	2
HD 21856	21856	0.19	3	502	11
40 Per	22951	0.27	1	283	11
o Per	23180	0.31	2	280	2
BD +31 643	281159	0.85	2	240	2
ζ Per	24398	0.31	2	301	2
X Per	24534	0.59	2	590	2
Per	24760	0.09	3	196	11
ξ Per	24912	0.33	2	470	2
62 Tau	27778	0.37	2	223	2
HD 29647	29647	1.00	2	177	2
α Cam	30614	0.30	2	820	2
NGC 2024 IRS 1	...	1.69	4	450	4
χ2 Ori	41117	0.45	2	1000	2
HD 47129	47129	0.36	1	1514	12
HD 53367	53367	0.74	2	780	2
HD 73882	73882	0.70	5	1000	13
HD 110432	110432	0.51	5	300	14
o Sco	147084	0.74	4	270	11
ρ Oph D	147888	0.47	2	136	2
HD 147889	147889	1.07	2	136	2
χ Oph	148184	0.52	1	161	5
μ Nor	149038	0.38	3	1122	12
HD 149404	149404	0.68	1	417	11
ζ Oph	149757	0.32	2	140	2
HD 152236	152236	0.68	5	600	15
HD 154368	154368	0.78	5	909	16
WR 104	...	2.30	6	2600	17
HD 168607	168607	1.61	7	1100	7
HD 168625	168625	1.86	8	2800	18
BD −14 5037	...	1.57	4	1700	4
HD 169454	169454	1.12	2	930	2
W40 IRS 1A	...	2.90	9	400	9
WR 118	...	4.57	6	3900	6
WR 121	...	1.75	6	2600	6
HD 183143	183143	1.28	7	1000	7
P Cygni	193237	0.63	7	1800	7
HD 193322A	193322A	0.40	3	830	19
HD 229059	229059	1.71	2	1000	2
HD 194279	194279	1.22	7	1100	7
Cyg OB2 5	...	1.99	7	1700	7
Cyg OB2 12	...	3.35	7	1700	7
Cyg OB2 8A	...	1.60	10	1700	7
HD 204827	204827	1.11	2	600	2
HD 206267	206267	0.53	2	1000	2
19 Cep	209975	0.36	1	1300	20
λ Cep	210839	0.57	2	505	2
1 Cas	218376	0.25	2	339	2

Notes. References. (1) McCall et al. 2010; (2) Thorburn et al. 2003; (3) Savage et al. 1977; (4) van Dishoeck & Black 1989; (5) Rachford et al. 2009; (6) Conti & Vacca 1990; (7) McCall et al. 2002, and references therein; (8) Nota et al. 1996; (9) Shuping et al. 1999; (10) Snow et al. 2010; (11) van Leeuwen 2007; (12) Fruscione et al. 1994; (13) Nichols & Slavin 2004; (14) Chevalier & Ilovaisky 1998; (15) Ritchey et al. 2011; (16) Maíz-Apellániz et al. 2004; (17) Tuthill et al. 2008; (18) Pasquali et al. 2002; (19) Roberts et al. 2010; (20) Sheffer et al. 2008.

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Observations used in this study were taken between 1998 and 2010 with a variety of instrument/telescope combinations, including CGS4 (Mountain et al. 1990) at UKIRT, NIRSPEC (McLean et al. 1998) at Keck, Phoenix (Hinkle et al. 2003) at both KPNO and Gemini South, and CRIRES (Käufl et al. 2004) at the Very Large Telescope (VLT). A log of all H⁺₃ observations in diffuse molecular clouds utilized herein—including those previously published—can be found in Table 2.4 of Indriolo (2011). For specifics regarding instrument configurations, observing methods and the like, the reader is referred to the following sources: McCall et al. (1998, 2002, 2003); McCall (2001); Geballe et al. (1999); Indriolo et al. (2007, 2010); Indriolo (2011); Crabtree et al. (2011).

Beginning with raw data frames, our reduction procedure employed standard IRAF⁴ routines commonly used in spectroscopic data reduction. Upon extracting one-dimensional spectra, data were transferred to Igor Pro⁵ where we have macros written to complete the reduction (McCall 2001). Spectra published in Indriolo et al. (2007) were reprocessed here using a method that better accounts for bad pixels in the CGS4 array. A full description of the new reduction procedure used for CGS4 data—and all of our reduction procedures applying to H⁺₃ data in general—is presented in Indriolo (2011).

Absorption features due to H⁺₃ were fit with Gaussian functions in order to determine equivalent widths, velocity FWHM, and interstellar gas velocities. The fitting procedure uses the functional form of a Gaussian where the area (as opposed to amplitude) is a free parameter, and includes a fit to the continuum level; i.e.,

$$\begin{equation} I=I_0-\frac{A}{w\sqrt{\pi }}\exp {\left[-\left(\frac{\lambda -\lambda _0}{w}\right)^2\right]}, \end{equation} \tag{ 10 }$$

and was developed in Igor Pro. The free parameters here are the continuum level, I₀, the area of the Gaussian, A, the central wavelength of the Gaussian, λ₀, and the line width, w (note that the width used here is related to the "standard" Gaussian width, σ, by w² = 2σ²). As the continuum level has already been normalized, an area determined using this fit is by definition an equivalent width, W_λ. In the case of the R(1, 1)^u and R(1, 0) lines, both absorption features are fit simultaneously and a single best-fit continuum level is found. Uncertainties on the equivalent widths (δW_λ) and continuum level (δI)—both at the 1σ level—are found by the fitting process. To estimate the systematic uncertainties due to continuum placement, the continuum level was forced to I₀ + δI and I₀ − δI and the absorption lines re-fit. Variations in the equivalent widths due to this shift are small compared to those reported by the fitting procedure and so have been ignored (i.e., σ(W_λ) = δW_λ). Assuming optically thin absorption lines and taking transition dipole moments and wavelengths from Goto et al. (2002, and references therein), column densities are derived from equivalent widths using the standard relation:

$$\begin{equation} N(J,K)=\left(\frac{3hc}{8\pi ^3}\right)\frac{W_{\lambda }}{\lambda }\frac{1}{|\mu |^2}, \end{equation} \tag{ 11 }$$

where N(J,K) is the column density in the state from which the transition arises, h is Planck's constant, c is the speed of light, λ is the transition wavelength, and |μ|² is the square of the transition dipole moment.

In cases where H⁺₃ absorption lines are not detected, upper limits to the equivalent width are computed as

$$\begin{equation} W_{\lambda }<\sigma \lambda _{\rm pix}\sqrt{{\cal N}_{\rm pix}}, \end{equation} \tag{ 12 }$$

where σ is the standard deviation on the continuum level near the expected H⁺₃ lines, λ_pix is the wavelength coverage per pixel, and ${\cal N}_{\rm pix}$ is the number of pixels expected in an absorption feature. Upper limits to the column density are then determined via Equation (11). Column densities, equivalent widths, velocity FWHM, and interstellar gas velocities for all of the sight lines presented in Figures 1–13 are reported in Table 2.

As discussed in Section 2, the chemistry associated with H⁺₃ in diffuse molecular clouds is rather simple. Assuming steady-state conditions, the formation and destruction rates of H⁺₃ can be equated. In the simplified version of diffuse cloud chemistry—where destruction occurs only via dissociative recombination with electrons—the result is

$$\begin{equation} \zeta _{2}n({\rm H}_2)=k_en({\rm H}_3^+)n_e, \end{equation} \tag{ 13 }$$

where ζ₂ is the ionization rate of H₂, n's are number densities, and k_e is the H⁺₃–electron recombination rate coefficient (see Table 3). Substituting the electron fraction (defined as x_e ≡ n_e/n_H, where n_H ≡ n(H) + 2n(H₂)) into Equation (13) and solving for the ionization rate gives

$$\begin{equation} \zeta _2=k_{e}x_{e}n_{\rm H}\frac{n({\rm H}_3^+)}{n({\rm H}_2)}. \end{equation} \tag{ 14 }$$

Observations cannot measure changes in these parameters along a line of sight, so we assume a uniform cloud with path length L and constant x_e, k_e, n_H, and n(H⁺₃)/n(H₂). In this case, the H⁺₃ and H₂ densities can by definition be replaced with N(H⁺₃)/L and N(H₂)/L, respectively, such that

$$\begin{equation} \zeta _2=k_{e}x_{e}n_{\rm H}\frac{N({\rm H}_3^+)}{N({\rm H}_2)} \end{equation} \tag{ 15 }$$

gives the cosmic-ray ionization rate in a diffuse molecular cloud.

While Equation (15) only considers one formation and destruction mechanism for H⁺₃, comparison to a more complete chemical reaction network shows that it is a robust approximation given diffuse cloud conditions. Assuming steady state for H⁺₂ where destruction by electron recombination and charge transfer to protons (reactions (3) and (4)) are considered, and steady state for H⁺₃ where destruction by proton transfer to CO and O (reactions (6), (7), and (8)) is accounted for, gives the equations

$$\begin{equation} \zeta _2n({\rm H}_2)=k_{3}n_en({\rm H}_2^+)+k_{4}n({\rm H})n({\rm H}_2^+)+k_{2}n({\rm H}_2)n({\rm H}_2^+), \end{equation} \tag{ 16 }$$

$$\begin{equation} k_{2}n({\rm H}_2)n({\rm H}_2^+)=k_en_en({\rm H}_3^+)+k_{\rm CO}n({\rm CO})n({\rm H}_3^+)+k_{\rm O}n({\rm O})n({\rm H}_3^+). \end{equation} \tag{ 17 }$$

Solving for the ionization rate and making similar substitutions as before results in

$$\begin{eqnarray} \zeta _2&=&\frac{N({\rm H}_3^+)}{N({\rm H}_2)}n_{\rm H}\left[k_{e}x_e+k_{\rm CO}x({\rm CO})+k_{\rm O}x({\rm O})\right]\nonumber\\ &&\times\, \left[1+\frac{2k_{3}x_e}{k_{2}f_{{\rm H}_2}}+\frac{2k_{4}}{k_{2}}\left(\frac{1}{f_{{\rm H}_2}}-1\right)\right], \end{eqnarray} \tag{ 18 }$$

where relevant rate coefficients are given in Table 3, k_CO = k₆ + k₇, and $f_{{\rm H}_2}\equiv 2n({\rm H}_2)/n_{\rm H}$ is the fraction of hydrogen nuclei in molecular form.

The cosmic-ray ionization rate as a function of electron fraction is plotted in Figure 14 for both Equations (15) and (18). In all four panels, the thick solid line shows the linear relationship given by Equation (15), while the various other curves show the ionization rate determined using Equation (18) and different values of x(CO), x(O), and $f_{{\rm H}_2}$. In the first panel x(CO) is set to 10⁻⁶, 10⁻⁵, and 10⁻⁴ with x(O) = 10⁻⁸ and $f_{{\rm H}_2}=1$. In the second panel x(O) is set to 10⁻⁶, 10⁻⁵, 10⁻⁴, and 3 × 10⁻⁴ with x(CO) = 10⁻⁸ and $f_{{\rm H}_2}=1$. In the third panel $f_{{\rm H}_2}$ is set to 0.67, 0.5, and 0.1 with x(CO) = x(O) = 10⁻⁸.

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The first and second panels show that Equation (18) only differs significantly from Equation (15) when x_e ≲ 10⁻⁵, and for high fractional abundances of CO and O (≳ 10⁻⁴). These deviations occur when proton transfer to CO and O comes to dominate over electron recombination as the primary destruction pathways for H⁺₃. The third panel shows that $f_{{\rm H}_2}$ affects the ionization rate for all values of x_e, but that rather low molecular hydrogen fractions are necessary to significantly alter the inferred value of ζ₂. With $f_{{\rm H}_2}=0.5$ (i.e., twice as many H atoms as H₂ molecules) the value given by Equation (18) is about 1.6 times that from Equation (15), while for $f_{{\rm H}_2}=0.67$ (i.e., equal number of H and H₂) the value given by Equation (18) is about 1.3 times that from Equation (15). This deviation is caused by the larger relative abundance of atomic hydrogen destroying H⁺₂ before it can form H⁺₃, and is represented by the final term in Equation (18). The destruction of H⁺₂ by electron recombination does not play a major role in the chemical network used here, and its influence can only be seen in the slight deviation between Equations (15) and (18) when x_e ∼ 10⁻².

Although some of the sight lines studied herein have molecular hydrogen fractions of $f_{{\rm H}_2}=0.2$ as determined from observations of H and H₂ (Savage et al. 1977; Rachford et al. 2002, 2009), it must be remembered that these are line-of-sight fractions, while Equation (18) requires local fractions. As it is expected that there will be atomic gas along a line of sight that is not associated with a cloud containing H₂, the line-of-sight value of $f_{{\rm H}_2}$ always underestimates the value of $f_{{\rm H}_2}$ within a diffuse molecular cloud. It is generally assumed that the interior of a diffuse molecular cloud has conditions where $0.67<f_{{\rm H}_2}<1$ (i.e., somewhere between half and all of the hydrogen is in molecular form), such that reaction (4) will not be very important.

Using average diffuse molecular cloud conditions (x(O) = 3 × 10⁻⁴, Cartledge et al. 2004 and Jensen et al. 2005; x(CO) ∼ 10⁻⁶, Sonnentrucker et al. 2007; $f_{{\rm H}_2}\gtrsim 0.67$) in Equation (18) produces the dashed curve in the last panel of Figure 14, and at x_e = x(C⁺) = 1.5 × 10⁻⁴ (Cardelli et al. 1996; Sofia et al. 2004) the resulting ionization rate is only 1.33 times larger than that inferred from Equation (15). This demonstrates that Equation (15) is a good approximation to Equation (18) in the regions we study, and so we adopt Equation (15) in computing ζ₂.

With Equation (15), the cosmic-ray ionization rate (ζ₂) can be determined from the H⁺₃ column density (N(H⁺₃)), H₂ column density (N(H₂)), total hydrogen density (n_H), electron fraction (x_e), and H⁺₃–electron recombination rate coefficient (k_e). The electron fraction can be approximated by the fractional abundance of C⁺ assuming that nearly all electrons are the result of singly photoionized carbon. In sight lines where C⁺ has been observed, the measured value of x(C⁺) is used for x_e. For all other sight lines, the average fractional abundance measured in various diffuse clouds, x(C⁺) ≈ 1.5 × 10⁻⁴ (Cardelli et al. 1996; Sofia et al. 2004), is adopted for x_e. Uncertainties in x_e are assumed to be ±20%, i.e., ±3 × 10⁻⁵. The H⁺₃–electron recombination rate coefficient has now been measured in multiple laboratory experiments (e.g., McCall et al. 2004; Kreckel et al. 2005, 2010) with consistent results and is presented in Table 3. When available the spin temperature of H₂, T₀₁, and its uncertainty are used in calculating k_e; otherwise, an average value of 70 ± 10 K is adopted. The total hydrogen number density is difficult to determine, but various studies have estimated n_H using a rotational excitation analysis of observed C₂ lines (Sonnentrucker et al. 2007), an analysis of H and the J = 4 level of H₂ (Jura 1975), or a thermal pressure analysis of fine structure lines of C i (Jenkins et al. 1983). Number densities from these studies are presented in Table 4 when available. For sight lines without estimated densities, the rough average value of n_H = 200 cm⁻³ is adopted. In all cases, uncertainties in n_H are assumed to be ±50% of the reported values. Absorption lines from electronic transitions of H₂ have been observed in the UV along many of the sight lines in this study (from which the aforementioned values of T₀₁ are derived; Savage et al. 1977; Rachford et al. 2002, 2009). In sight lines where H₂ has not been observed, two other methods were used to estimate N(H₂). The preferred method uses column densities of CH determined from observations in combination with the relation N(CH)/N(H₂) = 3.5^+2.1_−1.4 × 10⁻⁸ from Sheffer et al. (2008). In sight lines where neither H₂ nor CH has been observed, N(H₂) is estimated from the color excess, E(B − V), using the relation N_H ≈ E(B − V)5.8 × 10²¹ cm⁻² mag⁻¹ from Bohlin et al. (1978), and assuming $f_{{\rm H}_2}\approx 2N({\rm H}_2)/N_{\rm H}=0.67$. Molecular hydrogen column densities determined both from observations and estimates are presented in Table 4.

Table 4. Cosmic-ray Ionization Rates

	N(H+3)	σ[N(H+3)]	H+3	N(H2)	σ[N(H2)]	H2	nH	Density	ζ2	σ(ζ2)
Object	(1013 cm−2)	(1013 cm−2)	Reference	(1020 cm−2)	(1020 cm−2)	Reference	(cm−3)	Reference	(10−16 s−1)	(10−16 s−1)
HD 20041	23.7	5.13	1,2	11.4	6.24	5	200		9.48	7.59
HD 21389	8.67	0.92	1	5.57	3.06	5	200		7.11	5.54
ζ Per	6.26	0.52	1	4.75	0.95	6	215	10	5.55	3.18
X Per	7.34	0.92	1	8.38	0.89	7	325	10	5.85	3.54
HD 29647	11.9	1.11	1	22.9	12.5	5	200		2.38	1.85
HD 73882	9.02	0.50	3	12.9	2.39	7	520	10	9.71	5.57
HD 110432	5.22	0.17	3	4.37	0.29	7	140	10	3.86	2.10
HD 154368	9.37	1.32	1	14.4	3.99	7	240	10	4.19	2.62
WR 104	23.2	1.54	2	44.7	22.3	8	200		2.37	1.76
HD 168607	6.60	0.63	1	17.4	8.47	5	200		1.73	1.28
HD 168625	10.6	0.92	1	16.3	7.95	5	200		2.96	2.17
HD 169454	5.93	0.34	1	16.6	8.37	5	300	10	2.45	1.83
W40 IRS 1A	26.9	3.54	1	56.3	28.2	8	300	11	3.27	2.45
WR 118 (5)	29.4	7.07	2	43.3	21.7	8	200		3.10	2.41
WR 118 (47)	30.8	5.73	2	45.5	22.7	8	200		3.10	2.36
WR 121	22.0	2.60	2	34.0	17.0	8	200		2.96	2.21
HD 183143 (7)	11.3	2.98	2	4.86	2.38	5	200		10.6	8.23
HD 183143 (24)	13.5	2.81	2	7.91	3.85	5	200		7.82	5.93
HD 229059	37.4	1.20	1	65.7	51.2	5	200		2.60	2.47
Cyg OB2 5	24.0	3.29	2	15.2	7.39	5	225	10	8.13	6.03
Cyg OB2 12	34.3	5.89	2,4	80.0	69.1	5	300	10	2.93	3.04
HD 204827	19.0	2.54	1	20.9	10.2	5	450	10	9.32	6.92
λ Cep	7.58	1.17	1	6.88	0.48	7	115	10	2.84	1.61
κ Cas	...	0.91	1	1.88	0.20	6	200		...	1.78
HD 21483	...	2.30	1	11.9	5.91	5	200		...	0.88
HD 21856	...	2.25	1	1.10	0.12	6	200		...	8.42
40 Per	...	0.87	1	2.92	0.53	6	80	12	...	0.57
o Per	...	0.47	1	4.09	0.79	6	265	10	...	0.85
BD +31 643	...	4.63	1	12.4	4.39	7	200		...	1.67
Per	...	0.38	1	0.33	0.07	6	15	13	...	0.36
ξ Per	...	0.51	1	3.42	0.53	6	300	13	...	1.98
62 Tau	...	1.79	1	6.23	1.12	7	280	10	...	2.06
α Cam	...	1.48	1	2.17	0.26	6	150	13	...	2.17
χ2 Ori	...	2.75	2	4.90	1.05	9	200		...	2.79
HD 47129	...	2.91	1	3.51	0.58	6	200		...	3.81
HD 53367	...	0.44	1	11.1	1.44	9	200		...	0.20
o Sco	...	0.41	1	19.1	9.81	5	225	10	...	0.11
HD 147888	...	2.25	1	2.97	0.61	9	215	10	...	9.25
HD 147889	...	0.87	1	28.8	14.0	5	525	10	...	0.36
χ Oph	...	0.52	1	4.26	1.02	6	185	10	...	0.64
μ Nor	...	0.58	1	2.77	0.49	6	200		...	1.09
HD 149404	...	1.10	1	6.17	0.53	9	200		...	0.87
ζ Oph	...	0.37	1,2	4.49	0.42	6	215	10	...	0.39
HD 152236	...	0.69	1	5.34	1.32	9	200		...	0.63
BD −14 5037	...	1.11	1	38.6	23.6	5	200		...	0.13
P Cygni	...	1.84	2	3.14	1.61	5	200		...	2.67
HD 193322A	...	1.82	1	1.21	0.18	6	200		...	6.52
HD 194279	...	4.59	2	23.7	11.9	8	200		...	0.88
Cyg OB2 8A	...	4.77	1	9.26	4.58	5	200		...	2.35
HD 206267	...	1.47	1	7.18	0.59	7	200		...	0.97
19 Cep	...	1.28	1	1.21	0.18	6	200		...	4.60
1 Cas	...	1.41	1	1.40	0.19	6	200		...	4.18
NGC 2024 IRS 1	...	1.27	1	49.0	24.5	8	10000		...	0.14

Notes. The sight lines toward WR 118 and HD 183143 have two distinct velocity components observed in H⁺₃ by McCall et al. (2002), and each of these components is labeled with a number in parentheses that corresponds to its average LSR velocity. When no reference for n_H was available the value was set to 200 cm⁻³, an average density for diffuse molecular clouds. Upper limits to the cosmic-ray ionization rate should be taken as three times σ(ζ₂). H⁺₃ References. (1) This work; (2) McCall et al. 2002; (3) Crabtree et al. 2011; (4) McCall et al. 1998. H₂ References. (5) Calculated from CH column densities using the relation N(CH)/N(H₂) = 3.5^+2.1_−1.4 × 10⁻⁸ from Sheffer et al. (2008); (6) Savage et al. 1977; (7) Rachford et al. 2002; (8) calculated from E(B − V) (see Table 1) using the relation N_H ≈ E(B − V)5.8 × 10²¹ cm⁻² mag⁻¹ from Bohlin et al. (1978), and assuming $2N({\rm H}_2)/N_{\rm H}=f_{{\rm H}_2}=0.67$; (9) Rachford et al. 2009. $\mathbf {n_{\rm H}}$ References. (10) Sonnentrucker et al. 2007; (11) Shuping et al. 1999; (12) Jenkins et al. 1983; (13) Jura 1975.

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While Table 2 gives individual column densities for the lowest lying ortho and para levels of H⁺₃, values of N(H⁺₃) in Table 4 are equal to the sum of the column densities in the (1, 1) and (1, 0) states (i.e., N(H⁺₃) = N(1, 1) + N(1, 0)). In cases where observations of both the R(1, 1)^u and R(1, 1)^l lines produce independent values of N(1, 1), a variance weighted average,

$$\begin{equation} N=\sum _{i=1}^{n} \left(N_i/\sigma _i^2\right)/\sum _{i=1}^{n} \left(1/\sigma _i^2\right), \end{equation} \tag{ 19 }$$

is used to determine N(1, 1). Because there are only two independent measurements of N(1, 1), the uncertainty of the weighted average, S[N(1, 1)], is determined using an unbiased estimator of a weighted population variance for small samples. This is given by

$$\begin{equation} S^2=\frac{V_1}{V_1^2-V_2}\sum _{i=1}^{n}w_i(N_i-\mu ^*)^2, \end{equation} \tag{ 20 }$$

where

$$\begin{equation*} w_i=\frac{1}{\sigma _i^2}, V_1=\sum _{i=1}^{n}w_i, V_2=\sum _{i=1}^{n}w_i^2, \end{equation*}$$

and μ* is the weighted average determined from Equation (19). The uncertainty in the total H⁺₃ column density is then computed as usual by adding σ[N(1, 1)] and σ[N(1, 0)] in quadrature. Upper limits to the H⁺₃ column density should be taken as 3σ[N(H⁺₃)].

Using Equation (15), and taking the values described above and in Table 4, cosmic-ray ionization rates are inferred for all diffuse molecular clouds where H⁺₃ observations have been made. These values of ζ₂ are presented in Column 10 of Table 4, with uncertainties in Column 11. As for the H⁺₃ column densities, upper limits to the cosmic-ray ionization rate should be taken as 3σ(ζ₂).

For one sight line however, that toward NGC 2024 IRS 1, a different analysis is used because recent observations suggest that the interstellar material is more likely dense than diffuse (T. Snow 2011, private communication). In this case, values appropriate for dense clouds (x_e = 10⁻⁷, $f_{{\rm H}_2}=1$) are adopted, effectively simplifying Equation (18) to

$$\begin{equation} \zeta _2=\frac{N({\rm H}_3^+)}{N({\rm H}_2)}n_{\rm H}\left[k_{\rm CO}x({\rm CO})+k_{\rm O}x({\rm O})\right]. \end{equation} \tag{ 21 }$$

The density and temperature are also set to average dense cloud values (n_H = 10⁴ cm⁻³, T = 30 K). The large CO column density (N(CO) = 1.26 × 10¹⁸ cm⁻²; T. Snow 2011, private communication) results in x(CO) = 1.28 × 10⁻⁴, demonstrating that most of the carbon is in molecular form, and thus validating the low value of x_e assumed above. Oxygen abundances are typically about two times carbon abundances (Lodders 2003), such that half of all O is expected to be in the form of CO. We assume the remainder to be in atomic form, and use x(O) = x(CO) in Equation (21). The resulting upper limit on the cosmic-ray ionization rate toward NGC 2024 IRS 1 is ζ₂ < 4.2 × 10⁻¹⁷ s⁻¹. This lower value is expected in dense cloud conditions, as will be discussed below, and is excluded from our analysis of the distribution of ionization rates in diffuse clouds.

With ionization rates and upper limits inferred for 50 diffuse cloud sight lines, it is interesting to study the distribution of our entire sample. Cosmic-ray ionization rates with 1σ uncertainties and 3σ upper limits are plotted in Figure 15. In order to find the probability density function for the cosmic-ray ionization rate in our sample, all of these data points are combined. Each sight line with an H⁺₃ detection is treated as a Gaussian function with mean and standard deviation equal to the ionization rate and uncertainty reported in Columns 10 and 11 of Table 4. Each sight line with a non-detection of H⁺₃ is treated as a Gaussian function with mean equal to zero and standard deviation equal to the uncertainty in Column 11 of Table 4. The cosmic-ray ionization rate cannot be negative, and it is reasonable to assume that in diffuse clouds there is some "floor" value below which ζ₂ does not fall caused by the continuous diffusion of particles throughout the Galactic disk (Webber 1998). We set this floor value to 10⁻¹⁷ s⁻¹—a rough average of ionization rates computed from proposed local interstellar proton spectra (e.g., Spitzer & Tomasko 1968; Webber 1998)—and truncate all of the Gaussian functions at this floor.⁶ Each function is then re-normalized so that the area under the curve above the floor ionization rate is equal to 1. The truncated Gaussians are added together, and the final distribution is again normalized to have an area equal to 1.

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The resulting probability density function, f(ζ₂), is plotted in the top panel of Figure 16. Because of the logarithmic x-axis, it is difficult to tell "by eye" which ionization rates are most probable. However, this is easily seen by scaling f(ζ₂) by ζ₂, as shown in the middle panel of Figure 16. By integrating ζ₂f(ζ₂), we find the mean value of the cosmic-ray ionization rate to be 3.5 × 10⁻¹⁶ s⁻¹, marked by the vertical dashed line. The minimum range of ionization rates (in log space) that contains 68.3% (Gaussian 1σ equivalent) of the area under f(ζ₂) is bounded by 5.0 × 10⁻¹⁷ s⁻¹ on the left and 8.8 × 10⁻¹⁶ s⁻¹ on the right, and is shown by the shaded region. The bottom panel of Figure 16 shows the cumulative distribution function, F(ζ₂), which gives the probability that the ionization rate is below a particular value.

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Given that the H⁺₃ formation process starts with H₂, one might expect that abundances of the two species should be correlated. Figure 17 shows N(H⁺₃) versus N(H₂) for the entire sample of sight lines studied herein. Also included are H⁺₃ column densities in sight lines near the supernova remnant IC 443 (Indriolo et al. 2010), as well as in dense clouds (McCall et al. 1999; Kulesa 2002; Brittain et al. 2004; Gibb et al. 2010). It is clear that for diffuse cloud sight lines where H⁺₃ has been detected, relative abundances tend to cluster about 10⁻⁷. A re-arrangement of Equation (15),

$$\begin{equation*} \frac{N({\rm H}_3^+)}{N({\rm H}_2)}=\frac{\zeta _2}{k_{e}x_{e}n_{\rm H}}, \end{equation*}$$

suggests that scatter about a central value of N(H⁺₃)/N(H₂) is likely due to variations in ζ₂ and electron density (x_en_H) between sight lines.

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The much lower value of N(H⁺₃)/N(H₂) in dense clouds (10⁻⁸ to 10⁻⁹) is the result of a higher density, different destruction partner, and lower ionization rate. Replacing electrons with CO as the dominant destruction partner leaves N(H⁺₃)/N(H₂) inversely proportional to k_COx(CO)n_H. The rate coefficient for destruction of H⁺₃ by CO is about 2 orders of magnitude slower than by electrons (see Table 3), the density of dense clouds is about 2–3 orders of magnitude larger than diffuse clouds (Snow & McCall 2006), and the fractional abundance of electrons in diffuse clouds should be about equal to that of CO in dense clouds. This means that ζ₂ must be about 0–2 orders of magnitude lower in dense clouds than in diffuse clouds to produce the observed results, and several studies of ionized species in dense clouds support such a trend (e.g., van der Tak & van Dishoeck 2000; Kulesa 2002; Hezareh et al. 2008).

Beyond studying the cosmic-ray ionization rate in and of itself, it is insightful to search for correlations between ζ₂ and various other sight line parameters. One of the most fundamental relationships to study is that between ζ₂ and location within the Galaxy. The cosmic-ray ionization rate is plotted against Galactic longitude in the bottom panel of Figure 18. The top panel is a map in Galactic coordinates and the middle panel gives the distance to background sources such that the reader can easily trace the ionization rates in the bottom panel to an actual on-sky position. There are no apparent gradients in Figure 18, and comparisons of ζ₂ with both heliocentric distance and Galactocentric radius of background sources do not show any trends, thus suggesting that the mechanism responsible for variations in the cosmic-ray ionization rate does not operate on large scales. To study variations of ζ₂ on small spatial scales, we investigate some of the closest sight line pairings from our data.

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6.2.1. HD 168607 and HD 168625

6.2.2. Cyg OB2 Association

6.2.3. Per OB2 Association

6.2.4. Ophiuchus–Scorpius Region

Given that there is no evidence for large-scale gradients in the cosmic-ray ionization rate, it is possible that properties of the observed clouds themselves are responsible for variations in ζ₂, and will show some correlation with the cosmic-ray ionization rate. A parameter with which ζ₂ has been predicted to vary is the hydrogen column density, N_H (Padovani et al. 2009). This is because the energy spectrum of cosmic rays is expected to change with depth into a cloud. Lower-energy particles—those most efficient at ionization—will lose all of their energy to ionization interactions in the outer regions of a cloud, leaving only higher-energy particles to ionize the cloud interior. As such, the ionization rate in a cloud interior should be lower than in a cloud exterior. Because lower-energy particles can operate through a larger portion of clouds with lower column densities (Cravens & Dalgarno 1978), it is expected that the inferred ζ₂ will decrease with increasing N_H. A plot of ζ₂ versus N_H (and equivalent E(B − V)) is shown in Figure 19. In sight lines with multiple distinct velocity components, the total line-of-sight N_H has been divided by the number of components to better approximate the size of each individual cloud. Included in Figure 19 are data from four dense cloud sight lines observed in H⁺₃ by Kulesa (2002), five dense cloud sight lines observed in H¹³CO⁺ by van der Tak & van Dishoeck (2000), and one dense cloud sight line observed in HCNH⁺ by Hezareh et al. (2008) for the purpose of extending the relationship to much higher N_H. There does not appear to be a correlation between ζ₂ and N_H when only the diffuse molecular cloud sight lines are considered. However, when the dense cloud ionization rates (most of which are of order a few times 10⁻¹⁷ s⁻¹) are included, it seems that ζ₂ does decrease in sight lines with higher column densities.

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