The Transition from Diffuse Molecular Gas to Molecular Cloud Material in Taurus

For C₂ excitation, the discussion in Hupe et al. (2012) forms the basis for our analysis. In particular, collisional cross sections (Lavendy et al. 1991; Robbe et al. 1992; Najar et al. 2008, 2009) and the f-value for the A–X (2, 0) band (Erman & Iwamae 1995; Kokkin et al. 2007; Schmidt & Bacskay 2007) are used. The line-of-sight column densities and their uncertainties for the gas toward HD 28975 and HD 29647 appear in Table 3. For HD 27778, the results of Sonnentrucker et al. (2007) are adopted. Results for C₂ toward HD 30122 are not available. We obtain a gas temperature, T(C₂), and total proton density, n_tot ^ex(C₂), of 10 K and greater than ∼350 cm⁻³ toward HD 29647. A remarkable outcome of this analysis is that only a value of 10 K for T(C₂) is possible, although temperatures up to 100 K were considered. For the gas toward HD 28975 and HD 27778, values of T(C₂), n_tot ^ex(C₂) are inferred to be 30 K, 250 cm⁻³ and 50 K, 150 cm⁻³, respectively. For comparison when adopting the same cross section and f-value, the values for the gas toward HD 27778 from Sonnentrucker et al. (2007) are 50 ± 10 K and 140 ± 20 cm⁻³. As discussed below, there is evidence for a weaker interstellar radiation field penetrating the TMC; adopting a value for I_IR (the strength of the adopted IR radiation field relative to the interstellar value) of 0.5 in this analysis leads to a halving of n_tot ^ex(C₂).

The effort by Goldsmith (2013), where the cross sections from Yang et al. (2010) are used, is the basis for our analysis of CO excitation. The analysis, however, uses a finer grid of gas temperatures (20, 40, 60, 80, 100 K) and adopts values from observations of H₂ or C₂ instead of generic ones. Once the gas temperature is specified, the CO excitation temperature determines the density of collision partners. As noted in Goldsmith (2013), collisions with H₂ dominate over those involving atomic hydrogen. We multiply this density, n(H₂), by a factor to convert it to total proton density from CO excitation, n_tot ^ex(CO). The adopted factor depends on properties revealed by our analyses. For diffuse molecular material like that toward HD 27778 and HD 30122, a factor of 3 is adopted, since n(H) and n(H₂) are comparable. As discussed in Section 4.1.2, the sight line toward HD 29647 is dominated by molecular gas, while that toward HD 28975 seems to represent an intermediate case. As a result, the conversion factors used are 2 and 2.5, respectively. The excitation temperatures needed for the analysis are T₁₀(CO) and, when available, T₂₁(CO) and T₃₂(CO). Since the fitting of the CO profiles provides a different set of excitation temperatures, T₀₁(CO), T₀₂(CO), and T₀₃(CO), as was done in Sheffer et al. (2008), for instance, we first transformed the latter set into the former ones through the use of Equations (2)–(4) from Goldsmith (2013).

All four sight lines have information on CO excitation. For HD 29647 and HD 28975, the IGRINS results presented here are adopted, while for HD 27778 and HD 30122, the HST results from Sheffer et al. (2008) are used. Our fitting of the CO spectrum toward HD 29647 yielded the same values for T₁₀(CO), T₂₁(CO), and T₃₂(CO) of 9.5 K considering the uncertainty in each determination of 1.0 K. Crutcher (1985) obtains CO excitation temperatures of 9.2 K (+5.1 km s⁻¹) and 7.5 K (+6.5 km s⁻¹), where the LSR velocity of the component is in parentheses, from his radio observations. For T₁₀(CO) of 9.5 K and T_k < 20 K from T(C₂), n_tot ^ex(CO) of about 1800 cm⁻³ is inferred. Even higher densities are suggested from T₂₁(CO) and T₃₂(CO). A comparable total proton density is obtained for the gas toward HD 28975 when considering T₁₀(CO) = 11.5 K and T(C₂) = 30 K. The analysis of CO toward HD 27778 and HD 30122 by Sheffer et al. (2008) yielded respective values for T₁₀(CO) of 5.3 and 3.8 K. They also derived gas temperatures from H₂ of 51 and 61 K, from which we find values for n_tot ^ex(CO) of 975 and 450 cm⁻³.

The radiation temperature, T_R , obtained from the microwave data of Crutcher (1985), as well as from the measurements by Heyer et al. (1987), van Dishoeck et al. (1991), and Liszt (2008), can be used to check for consistency in the physical conditions for the direction toward HD 29647. Crutcher (1985) used the 36-foot NRAO telescope at Kitt Peak for the J = 1 → 0 line and the 4.9 m University of Texas telescope at the Millimeter Wave Observatory for the J = 2 → 1 transition. The respective beam sizes and spectral resolutions were 60''/0.52 km s⁻¹ and 65''/0.16 km s⁻¹. Emission was seen at +5.1 and +6.5 km s⁻¹ with line widths of about 1.0 km s⁻¹ in both lines. The pairs representing these radial velocities for T_R were 5.8/4.2 K and 3.6/2.0 K for J = 1 → 0 and J = 2 → 1, respectively. The 14 m Five College Radio Observatory telescope was used by Heyer et al. (1987) for their measurements of J = 1 → 0 emission. These observations had a beam size of 45'', a spectral resolution of 0.52 km s⁻¹, and a spacing of $2^{\prime}$. The coarse spacing revealed a single component with ${T}_{R}^{* }$ of 5.9 K at +5.9 km s⁻¹ with a line width of 2.8 km s⁻¹. These results are similar to a single component at +5.7 km s⁻¹ with emission weighted by T_R found by Crutcher. It is worth noting that our measurements of CH, C₂, CN, and CO show absorption at about +6.0 km s⁻¹. The study by van Dishoeck et al. (1991) was based on data obtained with the 15 m Swedish-ESO Submillimeter Telescope for the J = 1 → 0 lines and the Caltech Submillimeter Observatory for the J = 2 → 1 and J = 3 → 2 lines. The respective beam sizes and spectral resolutions were 44''/0.13 km s⁻¹, 32''/0.043 km s⁻¹, and 20''/0.043 km s⁻¹, making these the observations with the highest spatial and spectral resolutions. Van Dishoeck et al. (1991) provided beam efficiencies so that we could convert their tabulated values for ${T}_{A}^{* }$ into ${T}_{R}^{* }$ for comparison with the earlier studies. Emission was seen at +5.2 and +7.1 km s⁻¹ and had typical line widths of about 2.0 km s⁻¹. The intensities described by ${T}_{R}^{* }$ were somewhat lower than the earlier measurements and those of Liszt (2008); they found 4.1 and 2.9 K for J = 1 → 0, 2.2 and 1.6 K for J = 2 → 1, and 2.2 and 1.1 K for J = 3 → 2. Liszt (2008) used the 12 m Arizona Radio Observatory for measurements of CO J = 1 → 0 emission; the beam size and spectral resolution were 65'' and 0.13 km s⁻¹, respectively. The emission, occurring at an average velocity of +6.14 km s⁻¹ with ${T}_{R}^{* }$ of 5.84 K, had a peak near +5.0 km s⁻¹ and a shoulder centered around +7.0 km s⁻¹; the accompanying spectrum of ¹³CO emission clearly reveals the two components.

The excitation calculations for CO emission were carried out using the RADEX code (van der Tak et al. 2007). As with the analysis noted above, the rate coefficients for collisions with H₂ were from Yang et al. (2010). We assumed equal abundances of ortho- and para-H₂, but the dependence on the spin state of H₂ is small for diffuse cloud temperatures, typically ±<10% from the average at 10 K, and much less at higher temperatures. For the J = 1 → 0 CO transition, the rate coefficients for para-H₂ collisions on CO at 10 (100) K are 3.3 × 10⁻¹¹ (3.5 × 10⁻¹¹) cm³ s⁻¹, and those for ortho-H₂ on CO are 3.8 × 10⁻¹¹ (3.5 × 10⁻¹¹) cm³ s⁻¹. An expanding spherical cloud with v proportional to radius or a plane-parallel slab is invoked to handle radiative transfer. The models are based on the large velocity gradient approximation. Optical depths approach values of 100 for J = 1 → 0 and J = 2 → 1 when considering column densities of 10¹⁸ cm⁻², as seen toward HD 29647.

We considered cases with T_k equal to 10 and 20 K and line widths of 2.0 km s⁻¹ and sought agreement with the excitation temperatures found in our CO measurements and with the values of T_R from CO emission (Crutcher 1985; Heyer et al. 1987; van Dishoeck et al. 1991). Separate models were run with densities differing by 0.25 in the log. These densities of n(H₂) were multiplied by 2 to yield total proton densities, as noted above for HD 29647. For a given density, the results from the plane-parallel slab yielded slightly higher excitation and radiation temperatures. The calculations with T_k of 20 K produced excitation temperatures larger than observed, for values of n_tot ^ex(CO) greater than 200 cm⁻³ as found in our other analyses. Moreover, the fact that T_k needs to be 10 K confirms our results from C₂ excitation. Total proton densities of about 350–600 cm⁻³ best match the observations, within a factor of 2 or so of our other determinations.

We also performed calculations to predict line intensities for CO emission toward HD 28975. We considered T_k of 20 and 30 K from our C₂ analysis and line widths of 2.0 and 3.0 km s⁻¹, seeking agreement with the excitation temperatures found from the analysis of the IR spectra. A value of 1.78 × 10¹⁷ cm⁻² was adopted for N_tot(CO), close to the value found from IGRINS spectra. The most consistent results using RADEX were n_tot ^ex(CO) between 180 and 320 cm⁻³ with values of T_R of about 10 (7) K for the J = 1 → 0 (2 → 1) line. The plane-parallel model produced satisfactory results for T_k of 20 K. We multiply n(H₂) by 2.5 to infer a total proton density of about 450–800 cm⁻³ for n_tot ^ex(CO), as suggested by the discussion in Section 4.1.2. It is not surprising that the radiation temperatures are higher in this case in light of the higher kinetic and excitation temperatures but comparable total density. This range in n_tot ^ex(CO) is consistent with the chemical results discussed below and within a factor of 3 or so of those from C₂ excitation.

The degree of excitation observed in CN in diffuse molecular gas differs from the situation for C₂ and CO, the other molecules examined here. When local sources of excitation are present, the CN excitation temperature that corresponds to transitions between the N = 0 and N = 1 levels, T₀₁(CN), may be somewhat higher than the temperature of the CMB, and only measurements with high S/N are able to discern the effect of local sources. The difficulty arises because CN has a much larger dipole moment than CO; the homonuclear C₂ molecule has no dipole moment. Excitation of the N = 2 level from T₁₂(CN) is even harder to detect in diffuse gas. For typical values of the ionization fraction, electron impacts will always dominate over collisions with neutral species in populating the upper rotational levels of CN. The results on column density for rotational levels in CN that are used for the analysis of its excitation appear in Table 4; fits to the profiles are provided in Figure 6.

In our analysis of CN excitation toward stars in the Taurus region, we consider local excitation by collisions with electrons with rate coefficients from Harrison et al. (2013) and, for completeness, the effects of collisions from ortho- and para-H₂ molecules (Kalugina & Lique 2015) and neutral He atoms (Lique et al. 2010). The analysis yields the electron density that corresponds to the measured excitation temperature in cases where T₀₁(CN) is statistically greater than T_CMB. From our McDonald observations of CN absorption toward HD 29647, we find an excitation temperature of T₀₁(CN) = 2.816 ± 0.127 K. If the kinetic temperature of the gas in this direction T_k equals T(C₂) and T(CO) of ≈10 K, as indicated by the analyses of C₂ and CO excitation, then the small excess in the CN rotational temperature would indicate that n(e) is about 0.19 cm⁻³. For n_tot(H) of about 1500 cm⁻³, as suggested by the analyses of CO excitation above and CN chemistry below, the electron fraction would be x(e) ∼ 1.3 × 10⁻⁴, similar to the value expected for low-density diffuse gas. These calculations assume that the H₂ ortho-to-para ratio is that determined by the kinetic temperature, which for the gas toward HD 29647 is assumed to be 10 K. However, even if the ortho-to-para ratio were larger than this, due to turbulent dissipation, for example, the results on n(e) and x(e) would not be significantly different. Our analysis of the UVES data on CN toward HD 29647 yielded T₀₁(CN) of 2.722 ± 0.102 K, which is consistent with no additional excitation over that from the CMB. However, at the 2σ level, this measurement is consistent with T₀₁(CN) < 2.926 K, which would indicate that n(e) < 0.45 cm⁻³, in agreement with the McDonald result. The weighted average of the two measurements yields 2.759 ± 0.080 K for a conservative 2σ limit of 2.919 K and a limit on x(e) of 2.9 × 10⁻⁴. Similarly, the CN excitation temperature that we find toward HD 28975, T₀₁(CN) = 2.617 ± 0.134 K, is consistent with T_CMB. At the 2σ level, the upper limit of T₀₁(CN) is 2.885 K for this sight line and yields n(e) less than 0.25 cm⁻³ for T_k of 30 K from C₂ excitation. The values of n_tot(H) inferred from CO excitation (1900 cm⁻³) and CN chemistry (1200 cm⁻³) suggest that x(e) < 2.1 × 10⁻⁴.

We can perform the same analyses on the UVES data for HD 27778 shown in Table 4 and compare them to the results by Roth & Meyer (1995). We obtained column densities for the N = 0 and 1 levels of (9.46 ± 0.47) × 10¹² and (4.44 ± 0.08) × 10¹² cm⁻², while Roth & Meyer (1995) found values of $({10.87}_{-0.69}^{+0.74})\times {10}^{12}$ and $({4.57}_{-0.086}^{+0.083})\times {10}^{12}$ cm⁻², respectively. The two sets of results agree within about 1.2σ considering the mutual uncertainties. The corresponding values for T₀₁(CN) of 2.934 ± 0.084 K (present) and ${2.747}_{-0.101}^{+0.096}$ K (Roth & Meyer) are consistent at the combined 1.0σ level; our determination and their 2σ limit are indistinguishable. Our CN excitation temperature suggests an electron density of 0.34 cm⁻³ when adopting a value for T_k of 50 K from H₂ and C₂ observations. With n_tot(H) of about 700 cm⁻³ from our analyses of CO excitation and CN chemistry, an ionization fraction of 4.9 × 10⁻⁴ is inferred. The weighted average of the two excitation temperatures is 2.856 ± 0.064 K, indicating values for n(e) and x(e) of 0.21 cm⁻³ and 3.0 × 10⁻⁴, respectively.

These values for x(e) can be compared with the results for observed C⁺ abundances in diffuse clouds of about 1.4 × 10⁻⁴ (Sofia et al. 1997) and the upper limit for the gas toward HD 27778 of 1.1 × 10⁻⁴ (Sofia et al. 2004); C⁺ is expected to be the dominant source of electrons in this material. Further discussion with respect to the carbon budget appears in Section 4.1.2. The results in Table 4 for HD 30122, where the CN column density is much less but the relative uncertainties in N_CN(0) and N_CN(1) are greater, yield an excitation temperature that is not distinguishable from T_CMB.

Crutcher (1985) also detected the (N, J, F) = (1, 3/2, 5/2)–(0, 1/2, 3/2) and (1, 3/2, 3/2)–(0, 1/2, 1/2) lines of CN toward HD 29647 with the NRAO 36-foot telescope with a beamwidth of about 1'. Each line has a velocity component at v_LSR of +5.1 km s⁻¹, and the stronger F = 5/2–3/2 transition shows weak emission extending to about +7 km s⁻¹. Thus, the component structure is similar to the ¹²CO emission lines. According to Crutcher, the two CN lines have T_R of 0.059 and 0.042 K and line widths of 1.0 km s⁻¹. We also modeled the emission from these lines with RADEX, using the collisional data from Lique et al. (2010). These results for He were scaled by 1.37 to approximate collisions with H₂. Like the models of ¹²CO emission discussed above, we considered gas temperatures of 10 and 20 K, where the scaling is most likely to apply. Because CN has a larger dipole moment than CO and as a consequence a larger critical density, we only chose a line width of 1 km s⁻¹, consistent with the observations for CN (Crutcher 1985) and CO (van Dishoeck et al. 1991). In light of the RADEX results for CO, we only considered geometries representing an expanding sphere. The models most consistent with the measurements by Crutcher (1985) suggest H₂ densities of 1000 cm⁻³ for 10 K and about 600 cm⁻³ for 20 K, corresponding to total proton densities of 2000 and 1200 cm⁻³, respectively. We favor the 10 K results because this temperature was required to reproduce the results for C₂ and CO excitation above. This leads to comparable total proton densities for CO and CN excitation. For completeness, we note that the optical depths for the two CN emission lines are about 5 and 10 (i.e., weaker vs. stronger emission) and that T_ex(CN) is about 2.8 K, like we found from the observations at visible wavelengths.

The environments of HD 27778 and HD 30122 seem to be dominated by the diffuse interstellar medium (ISM). No bright filament is visible in the far-IR, and the mid-IR does not reveal any dark cloud either. The dust temperature map is quite homogeneous in the area surrounding both stars, with values between 14.5 and 15 K. The dust temperature maps from Flagey et al. (2009) indicate values of 14.7 K (averaged within a 3'-radius region centered on HD 27778) and 14.9 K (averaged within a 3'-radius region centered on HD 30122). Figures 7 and 8 show the mid-IR and far-IR emission in the environment of the two stars. The distances to HD 27778 and HD 30122 are 224 ± 2 pc and 256 ± 4 pc, respectively, according to the Gaia DR2 (see Table 1), locating them both well beyond the TMC, which is about 150 pc away as discussed in more detail below.

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The environment of HD 28795 is dominated by the presence of a dark cloud about 10' to the SE. A filamentary structure visible in the far-IR emission extends from the dark cloud south of HD 28975, though the emission levels remain low directly toward the star (see Figure 9). The presence of a nearby dark cloud might be the cause of the high T_ex(CO) found from IGRINS spectra. The dust temperature averaged within a 3'-radius region centered on HD 28975 is 14.0 K, which indicates that the line of sight might be dominated by somewhat colder gas than toward HD 27778 and HD 30122. The distance to HD 28795 is 194 ± 2 pc according to the Gaia DR2 (Table 1), indicating that this star is also well beyond the TMC.

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The environment of HD 29647 is the most intriguing of all background stars in this paper. This star is located within the Heiles Cloud-2 (HCL-2) and was previously associated with an IRAS source (IRAS 04380+2553) by Whittet et al. (2004). Its distance was initially measured by Hipparcos at about 180 pc, which would place the star a few tens of parsecs behind the TMC. Whittet et al. (2004) used extinction curves toward this star and other nearby stars, as well as IRAS images, to conclude that (1) HD 29647 is "embedded in or in very close proximity to dust that is being warmed, at least in part, by radiation from the star itself" and (2) it is located behind TMC-1 and halfway within a diffuse screen that contributes about 3.6 mag of visual extinction. At higher angular resolution, an IR nebula centered on HD 29647 and 5' in radius is clearly detected in the mid- to far-IR (3.6–160 μm). The nebula is IRAS 04380+2553, and it is not detected at shorter wavelengths because visual extinction along the line of sight is so large.

In the mid-IR, two dark filaments are located to the ENE and S of HD 29647. In the far-IR (160 μm and beyond), these filaments appear in emission, as they trace cold and large dust grains. The dust temperature map of Flagey et al. (2009) tells a similar story with cold filaments surrounding a "warm spot" at the location of HD 29647. The temperature they derive toward the star is 15.4 K (averaged within a 3'-radius region centered on the peak in dust temperature, 42'' away from HD 29647), while it is about 13.5 K in the filaments and about 14.5 K in the surrounding diffuse medium.

Figure 10 shows a three-color mid-IR image of the HD 29647 surroundings (4.5, 8.0, and 24 μm) with far-IR contours (250 μm). The IR nebula seems to be located exactly in a "hole" delineated by the filamentary structure observed in the far-IR, which, at least visually, could be an indication that the two are related, though a fortuitous alignment cannot be ruled out. We also note that the distance to HD 29647 has been revised by Gaia DR2 measurements and now is 155 ± 2 pc, which would put the star somewhat closer to TMC-1.

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We estimate the distance between HD 29647 and the IR nebula, adopting the method used by Federman et al. (1991) based on the prescription of Draine & Salpeter (1979). In particular, the distance r to a point source whose luminosity is L and with dust temperature T_d of the cloud is given by

$$\begin{eqnarray}&&r={\left({{LQ}}_{\star }/16\pi \sigma {T}_{d}^{4}{Q}_{\mathrm{ir}}\right)}^{1/2},\end{eqnarray} \tag{ 2 }$$

where Q_⋆ is the effective absorption efficiency for the source spectrum, Q_ir is the Planck-averaged emissivity at T_d , and σ is the Stefan–Boltzmann constant. We determined Q_⋆ from optical properties for silicates (Draine 1985) and Q_ir from Draine & Lee (1984). The grain radius was taken to be 0.1 μm. Use of a spectral type of B9III Hg–Mn (Mooley et al. 2013) and T_⋆ of 10,550 K (Cox 2000) resulted in a value of 0.330 for Q_⋆. Selecting a T_⋆ of 11,500 K (Mooley et al. 2013) with or without an estimate for the effect of line blanketing in this metal-rich star led to Q_⋆ values differing by only 15%. The adopted values for R_⋆ and Q_ir were 3.9 × 10¹¹ cm (5.6 R_☉; Cox 2000) and 2.51 × 10⁻⁴. With T_d of 15.4 K, the estimate for the distance between HD 29647 and the TMC is about 1 pc.

The final step in this analysis is to determine the likelihood that HD 29647 lies in the diffuse gas on the far side of the TMC, as suggested by Whittet et al. (2004). We infer the extent of the diffuse gas surrounding the TMC by deriving the distance between HD 30122 and the nearby dark cloud, L1538, based on the map (Figure 4) of Galli et al. (2019). The Galactic coordinates (l, b) for HD 30122 and the cloud are (17662, −1403) and (17550, −1305), yielding 15 on the sky or 4 pc for a distance of 150 pc. If the total extent of the diffuse envelope surrounding the TMC is about twice this distance (10 pc), a gas density of about 100 cm⁻³ is obtained when the value of N_tot(H) from N(K i) in the direction of HD 29647 is adopted (see Section 4.1.1). Such a value is very typical of diffuse gas. Therefore, it appears that HD 29647 lies within the diffuse material behind the TMC.

Three groups used Gaia DR2 parallaxes to study the distance to the Taurus star-forming region and associated molecular material. Yan et al. (2019) traced the cloud in Planck 857 GHz emission (Planck Collaboration et al. 2014), deriving a distance of ${145}_{-16}^{+12}$ pc, though they see evidence for two components. Building on the work of Schlafly et al. (2014), Zucker et al. (2019) used Gaia DR2 results when available and allowed R_V to vary. Their average distance was 141 ± 2 ± 7 pc, where the first number gives the statistical uncertainty and the second the systematic uncertainty. Zucker et al. (2019) also subdivided the clouds in their sample, finding a bimodel distribution for Taurus—132 and 152 pc. Using the interactive software given by Zucker et al. (2019), ¹¹ we obtain distances to the clouds in front of our stars of $\sim {162}_{-4}^{+5}$ pc (HD 27778), ${160}_{-8}^{+6}$ pc (HD 28975), ${123}_{-8}^{+11}$ pc (HD 29647), and $\sim {162}_{-4}^{+5}$ pc (HD 30122). The estimates for HD 27778 and HD 30122 lie somewhat beyond the region considered in this study. Last, Galli et al. (2019) used Gaia DR2 and VLBI techniques to obtain distances to young stellar objects in Taurus. They divided the TMC into clusters for different dark clouds and analyzed the results through Bayesian statistics. For HD 29647 (represented by clusters called HCL-2), distances of 140.2 (139.9) pc, both with uncertainties of 1.3 pc, were inferred. The directions toward HD 27778 and HD 28975 (represented by a single cluster containing L1524 and L1529) had a distance of 129.0 ± 0.8 pc. It is not clear that the same material was sampled in the three studies for the directions that are our focus, and there is the possibility that small-scale structure is present in the chosen regions since our analysis is based on absorption along an infinitesmal pencil beam. Still, we can infer two things of importance to our work: the TMC is about 150 pc away so that HD 29647 lies within the diffuse material on the far side of the cloud, and the other three stars are much beyond the cloud.

The total number of protons, N_tot(H) = (N(H i) + 2N(H₂)), along a line of sight is directly obtained from UV observations of H i and H₂ absorption. Of the four sight lines in our study, only that toward HD 27778 has such a determination. From the numerous measurements on diffuse molecular clouds, two other methods yield values of N_tot(H). The first uses E(B − V) from Bohlin et al. (1978), 〈N_tot(H)/E(B − V)〉 = 5.8 × 10²¹ cm⁻² mag⁻¹. The second method is based on the column density of neutral potassium, N(K i), from Welty & Hobbs (2001). Welty & Hobbs found linear relationships between log(N(K i)) and log(N_tot(H)); we adopted the one that included all high-resolution measurements and the most reliable medium-resolution ones—log(N(K i)) = −27.21 ± 2.79 + (1.84 ± 0.13)log(N_tot(H)). Toward HD 27778, the measured value for N_tot(H) is 2.3 × 10²¹ cm⁻² (Ritchey et al. 2018; see also Cartledge et al. 2001, 2004). Using E(B − V) from Table 1 and the total N(K i) from our analysis of the UVES spectrum, we obtain values for N_tot(H) of 2.1 × 10²¹ cm⁻² and 1.9 × 10²¹ cm⁻², respectively, within 20% of the UV results.

Therefore, we apply the two methods to the data for HD 30122, HD 28975, and HD 29647. For the gas toward HD 30122, we obtain 1.3 × 10²¹/1.6 × 10²¹ cm⁻² from E(B − V)/N(K i), while for HD 28975 we find values of 3.5 × 10²¹/3.7 × 10²¹ cm⁻², respectively. The consistency found for the material toward HD 27778, HD 30122, and HD 28975 is not evident when applying the methods for the line of sight toward HD 29647, 6.3 × 10²¹/3.6 × 10²¹ cm⁻². Although there are numerous stellar features near the weak K i doublet at 4044 and 4047 Å in the UVES spectrum of HD 29647, we were able to measure N(K i) from the optically thin line at 4047 Å, obtaining a value of (2.6 ± 0.6) × 10¹² cm⁻². This value agrees with the one obtained from our fit of the line at 7698 Å, and so it appears that the amount of depletion onto grains is enhanced by about a factor of 2 for potassium toward this star.

We can also estimate the extent of the molecular material along the four sight lines. For the very molecule-rich directions toward HD 28975 and HD 29647 we consider the total proton column densities discussed here, while for the diffuse molecular gas toward HD 27778 and HD 30122 we adopt the molecular hydrogen column densities from Rachford et al. (2002) and Sheffer et al. (2008), respectively. When dividing these column densities by the gas densities found from CO excitation and CN chemistry, we find that the molecular portion of the TMC in these directions is approximately 1 pc, varying by only 50%.

Using absorption from the C ii intercombination line at 2325 Å, Sofia et al. (1997) determined that the gas-phase carbon abundance in diffuse clouds is 1.4 × 10⁻⁴. Since C⁺ is the main source for the electron fraction in diffuse material, our limits and values for x(e) from CN excitation (Section 3.2.3) suggest that higher-precision CN measurements are needed to place constraints on C⁺ for the carbon budget. However, with the CO column density ranging from 10¹⁵ to 10¹⁸ cm⁻² for the four directions in the present study, we can compare the CO contribution to the carbon budget. We adopt the values of N_tot(H) discussed in the previous section—2.3 × 10²¹ cm⁻² (HD 27778) from observations, as well as 1.3 × 10²¹ (HD 30122), 3.5 × 10²¹ (HD 28975), and 6.3 × 10²¹ (HD 29647) cm⁻² from E(B − V). Besides singly ionized carbon and CO, another significant contribution to the carbon budget comes from neutral carbon.

For HD 27778, Sofia et al. (2004) obtained an upper limit of 1.1 × 10⁻⁴ for the fractional abundance of C⁺, x(C⁺) = n(C⁺)/n_tot(H). Though cruder, we can look at the estimate used in our chemical model to track the conversion of C⁺ into CO (Federman et al. 1994), α = [1 + 14 × (τ_UV − 2)/5], where α is the percentage of C⁺ remaining. For this direction, α is 0.27, or x(C⁺) is 3.8 × 10⁻⁵, comfortably below the upper limit of Sofia et al. (2004). Moreover, Burgh et al. (2010) quote log(N(C i)) of 15.06, or x(C) of at least 5.0 × 10⁻⁷, and Sheffer et al. (2008) give log(N(CO)) = 16.09, or x(CO) of 5.4 × 10⁻⁶. For this sight line only about 4% of the gas-phase carbon in diffuse gas is in CO; it is not clear where the missing carbon resides.

Because Jenkins & Tripp (2011) were not able to measure the amount of C i toward HD 30122, a result of severe blending from stellar features, the other three directions in our sample only have information on CO (and the crude measure for C⁺ from α). We start with the gas toward HD 30122, where τ_UV indicates no conversion of C⁺. Using the results of Sheffer et al. (2008), N(CO) of 7.04 × 10¹⁴ cm⁻², we infer a CO abundance of 5.4 × 10⁻⁷ relative to N_tot(H). Here, too, essentially all of the carbon is in C⁺. Turning to the sight line toward HD 28975, N(CO) is 1.45 × 10¹⁷ cm⁻² and x(CO) is 4.1 × 10⁻⁵, or about 30% of the available carbon. With a value for α of 0.17, about 50% of the carbon appears to be in neutral carbon. For the gas toward HD 29647, N(CO) is 9.56 × 10¹⁷ cm⁻², or x(CO) = 1.5 × 10⁻⁴; all of the interstellar carbon appears to be in the form of CO. Because τ_UV is so large, α is only 0.07 and little of the carbon could be in C⁺. This is consistent with our conclusions reached through CO fractionation that C⁺ is only present in the outer portion of the cloud, where the species CH⁺, CH, and C₂ reside. Moreover, Whittet et al. (1989) found an upper limit for the CO column of 5 × 10¹⁶ cm⁻² in solid form, or 5% of the gas-phase abundance. Therefore, it seems that the line of sight toward HD 29647 probes the dark molecular TMC, with a gas temperature cold enough (10 K) to form solid CO, but has not yet shown any evidence for it. With a temperature of 30 K, the material toward HD 28975 is too warm for CO depletion onto grains.

Sheffer et al. (2008) sought relationships among observed quantities in their CO survey. Of particular interest for the present study is the correspondence between column densities of CO and CH for N(CO) > 10¹³ cm⁻², which they represented as

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{\rm{log}}(N({\rm{CO}})) & = & -(22.3\pm 11.3)\\ & & +(2.80\pm 0.85){\rm{log}}(N({\rm{CH}})).\end{array}\end{array}\end{eqnarray} \tag{ 3 }$$

Adopting our results for N(CH), the predicted values for N(CO) toward HD 27778, HD 30122, HD 28975, and HD 29647 are 2.6 × 10¹⁵ cm⁻², 4.4 × 10¹⁴ cm⁻², 1.14 × 10¹⁶ cm⁻², and 1.79 × 10¹⁶ cm⁻², respectively. The observed CO column densities are 1.23 × 10¹⁶ cm⁻² and 7.04 × 10¹⁴ cm⁻² for the material toward HD 27778 (Sheffer et al. 2007) and HD 30122 (Sheffer et al. 2008); Table 3 provides the column densities for the other two sight lines. We first focus on the sight lines toward HD 27778 and HD 30122. The observed values for N(CO) are 4.7 and 1.6 times larger, respectively, than the values inferred from the relationship above. Looking at Figure 9 from Sheffer et al. (2008), a plot of log(N(CO)) versus log(N(CH)), the data point for HD 27778 lies near the upper left boundary of the sample, while the data point for HD 30122 is in the middle of the sample. This indicates that the material toward these two stars is representative of diffuse molecular gas and that the gas density is higher toward HD 27778 (as inferred from Figure 17 in Sheffer et al. 2008). This is borne out by our analyses of molecular excitation and CN chemistry.

The differences between observed and predicted N(CO) are greater for the gas toward HD 28975 and HD 29647, with ratios of 13 and 53. The data points for both directions lie beyond the upper boundary shown in Figure 9 of Sheffer et al. (2008). The data point for HD 28975 would appear approximately near the point for HD 200775, the illuminating source for the reflection nebula NGC 7023. That for HD 29647 would appear about a dex higher, with a CO column density similar to that for a dark cloud (see Figure 6(b) in Sheffer et al. (2008)). We note that point (N(CO), N(H₂)) for HD 200775 in Figure 6(b) of Sheffer et al. (2008) is within the boundary represented by dark clouds. However, the cause for these outliers differs. In NGC 7023, the enhanced flux of UV radiation from HD 200775 preferentially destroys CH relative to CO (and H₂), a consequence of the protection arising from CO (and H₂) self-shielding. Since there is no evidence for an enhancement in UV flux in its sight line, the extreme case of HD 29647 is likely the result of converting CH molecules into other molecules, including CO. In a study of CH emission from molecular clouds, Mattila (1986) found the CH column density leveled off in a plot N(CH) versus A_B , consistent with the chemical model of Boland & de Jong (1984) that suggested a value of about 2 × 10¹³ cm⁻². In their study of HCL-2 at 38 resolution, Sakai et al. (2012) found CH column densities falling below the relationship between CH and H₂ from Sheffer et al. (2008). The position of TMC-1(NH₃), which is closest to the sight line toward HD 29647, has a line width of about 1.4 km s⁻¹ (or b-value of 0.8 km s⁻¹, similar to our observations) and N(CH) of 1.79 × 10¹⁴ cm⁻². This column density is three times larger than what we infer, but their measurements sample the full extent of the cloud. Moreover, subsequent work by Magnani and colleagues (e.g., Magnani & Onello 1995; Magnani et al. 1998, 2003) presented evidence that CH emission was a reliable tracer of H₂ in the diffuse envelopes of dark clouds. In light of our results for HD 29647, we believe that this star probes the outer predominantly molecular portion of the TMC, although it is embedded in the diffuse molecular material on the far side of the cloud (Whittet et al. 2004, Section 3.5). This is described in more detail in the next section. As for the portion of the TMC in front of HD 28975, we consider it a diffuse molecular cloud because only about 30% of the available carbon is in the form of CO.

Because Welty & Hobbs (2001) concluded that there is a close correspondence between K i and CH, we look at this for our lines of sight. In particular, Welty & Hobbs (2001) find an N(K i)/N(CH) ratio of 0.043 for their sample. The ratios for gas toward HD 27778, HD 28975, HD 29647, and HD 30122 are 0.027, 0.052, 0.050, and 0.040, respectively. Though the ratio toward HD 27778 is somewhat lower than the others, all four ratios are within about 20% of the mean value given by Welty & Hobbs (2001). In light of our findings for K i and CH toward HD 29647, we examine why the ratio does not differ, although evidence exists for smaller relative amounts of these two species, by considering the highest-resolution spectra for K i from our previously unpublished data acquired with the coudé 6-foot camera on HJST. There is a hint of an asymmetry in the profile (see Figure 11). Fitting it with two velocity components yields velocities of +5.6 and +7.6 km s⁻¹, b-values of 0.72 and 0.48 km s⁻¹, and relative fractions of 0.88 and 0.12 (yielding a total column of 2.58 × 10¹² cm⁻², much like the column from our most recent spectrum from HJST). The component velocities are similar to those found for CO emission (+5.1 and +6.5 km s⁻¹, Crutcher 1985; +5.2 and +7.1 km s⁻¹, van Dishoeck et al. 1991; +5.4 and +6.7 km s⁻¹, our estimate from the ¹³CO spectrum in Liszt 2008). The FHWMs of the emission lines are about 1 km s⁻¹ (Crutcher 1985), equivalent to b-values of about 0.6 km s⁻¹. The redder component in the ¹³CO spectra of Crutcher (1985) and Liszt (2008) is narrower, much like we infer from fitting the spectrum in Figure 11. Since Crutcher only detected HCO⁺, HCN, and CN emission from the bluer component (with weaker wings in HCO⁺ and CN), Messinger et al. (1997) inferred that a dense clump with v_LSR of+5.1 km s⁻¹ was present toward this star. We suggest that the presence of the dense clump results in lower column densities for both K i and CH in this direction, and the relationship between these species in diffuse gas found by Welty & Hobbs (2001) maintains their relative abundances.

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We now examine how our study fits into the global picture developed from measurements across the TMC (Goldsmith et al. 2008; Narayanan et al. 2008; Flagey et al. 2009; Pineda et al. 2010), as well as other observations of interstellar gas in Taurus (Lee et al. 2006; Paradis et al. 2012). While we already considered the results on dust emission from Flagey et al. (2009) for our targets in Section 3.5, some of their other findings are worth discussing. The average dust temperature across the TMC is 14.5 ± 1.0 K, with colder filaments seen in the far-IR emission maps in the middle of warmer material. One such filament lies in the vicinity of the sight line to HD 28975, and multiple filaments surround the sight line to HD 29647. Emission from polycyclic aromatic hydrocarbons (PAHs) and very small grains (VSGs) are correlated, but there is no relationship with emission from large grains. Abundance variations on subparsec scales of a factor of a few are seen in the PAH and VSG emission, including the more diffuse portions of the TMC with A_V up to a few mag. The far-IR dust opacity was shown to correlate very well with the visual extinction derived from near-IR images and indicated an increase in dust emissivity relative to the diffuse ISM. Their models of the large-scale dust emission revealed an overall reduction in the interstellar radiation field, which was incorporated into our analyses above; the strength of the field also showed significant variations in different regions of the TMC.

Goldsmith et al. (2008) created a 100 deg² map of the TMC in emission from the J = 1 → 0 lines of CO and ¹³CO, sampled on a 20'' grid. The map was divided into three regions: one without molecular emission, one with only CO emission, and one showing emission from both isotopologues. Intensities were integrated over velocity from +2 to +9 km s⁻¹. As seen in our Figure 1, the molecular emission is very filamentary. Peak emission occurs between +5 and +8 km s⁻¹, the interval where we observe strong molecular absorption. Goldsmith et al. (2008) note, however, that emission at about +10 km s⁻¹ (where we see absorption toward HD 26751 in Appendix C) might not be associated with the TMC. The pixels with no emission suggested an upper limit of N(CO) of 7.5 × 10¹⁵ cm⁻², indicating that the direction toward HD 30122 probed this region of the map. The region showing only CO emission contained about half the gas, with N(H₂) less than 2.1 × 10²¹ cm⁻² and an average value for N(CO) of about 3.6 × 10¹⁶ cm⁻². The sight line toward HD 27778 may probe this type of material. For the region containing both CO and ¹³CO emission, the average N(CO) spanned a range from 1.3 × 10¹⁷ cm⁻² to approximately 1 × 10¹⁸ cm⁻², consistent with our interpretation for the kind of material probed by HD 28975 and HD 29647. Goldsmith et al. (2008) also used the models by van Dishoeck & Black (1988) to place the results into context, finding that (1) the region with no CO emission had H₂ columns less than 10²¹ cm⁻², (2) the region with only CO emission had values between 10²¹ cm⁻² and about 3 × 10²¹ cm⁻², and (3) the region with emission from both CO and ¹³CO ranged from (1.5 to 10) × 10²¹ cm⁻². For the material analyzed in this paper, we have a measurement (toward HD 27778) and estimates (toward HD 29647 and HD 30122) for N(H₂) in units of 10²⁰cm⁻², 6.2 (Rachford et al. 2002), ∼30 (see Sections 4.1.1, 4.1.3), and 4.4 (from our N(CH) and an N(CH)/N(H₂) ratio of 3.5 × 10⁻⁸ from Sheffer et al. 2008). Clearly, the sight line toward HD 30122 is similar to regions without CO emission, and HD 29647 samples a region with both isotopologues. It appears that HD 27778 probes gas that is somewhere between regions with and without CO emission based on the CO and H₂ results for this sight line. Narayanan et al. (2008) described details of the observations used in creating the maps and showed the emission in 1 km s⁻¹ intervals. For the four lines of sight discussed here, the emission peaked at velocities between 5 and 7 km s⁻¹, where we see the strongest molecular absorption.

These CO and ¹³CO maps formed the basis for additional analyses. In particular, Pineda et al. (2010) used them for a comparison of H₂ columns derived from dust extinction with the Two Micron All Sky Survey (2MASS) on scales of 200''. Other improvements to the analyses included recent molecular data, adoption of RADEX for radiative transfer, and comparisons with the updated photochemical code of Visser et al. (2009). The TMC again was divided into three regions based on CO and ¹³CO emission. Emphasis was placed on results for the CO-to-H₂ conversion factor, x(CO) = N(H₂)/I_CO, where I_CO is the integrated intensity of line emission. A typical value for the conversion factor throughout the cloud was 2.1 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹. Using our estimate for N(H₂) from the previous paragraph and I_CO from Liszt (2008), we find a very similar value for x(CO) toward HD 29647, reinforcing our conclusion that this star probes dark cloud material. For A_V less than 3 mag, the conversion factor could be two orders of magnitude smaller. Pineda et al. (2010) averaged nearly 10⁶ spectra for the region showing no emission and obtained an average value for N(CO) of 7.8 × 10¹⁴ cm⁻², much like the value toward HD 30122. This reveals the effort needed to obtain CO column densities typically found in diffuse molecular clouds (e.g., Sheffer et al. 2008). Pineda et al. (2010) found a linear correlation between visual extinction and N(CO) for A_V from 3 to 10 mag with a CO/H₂ ratio of ∼10⁻⁴, as we found for the gas toward HD 29647. Their modeling results based on the photochemistry described by Visser et al. (2009) reveal other similarities to our analyses. For A_V less than 5 mag, their data are best described by a varying CO/H₂ ratio, and for T_k of 15 K, the models yield an n_tot(H) of about 800 cm⁻³; our analyses suggest that such densities apply to the material toward HD 28975 and HD 29647 (and possibly toward HD 27778). The models also indicate a reduced radiation field and that half the mass of the cloud is in the region without detectable amounts of ¹³CO.

There are two large-scale surveys including the TMC that provide useful comparisons. First, Paradis et al. (2012) follow the approach taken by Planck Collaboration et al. (2011), except they use extinction instead of far-IR emission to study the presence of CO-dark gas and the CO-to-H₂ conversion factor. The visual extinction comes from Dobashi (2011) and is based on 2MASS data. The more diffuse regions give the optimal value for A_V /N_tot(H); for the solar neighborhood the value is 6.53 × 10⁻²² mag cm², which agrees very well with the UV result of Bohlin et al. (1978) for R_V of 3.1. The global conversion factor is 1.67 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹, with significant variations, and it is 2.27 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ for gas in Taurus, comparable to what Pineda et al. (2010) found. Paradis et al. (2012) obtained average values for the locations of the H-to-H₂ transition (A_V of ∼0.2 mag) and the H₂-to-CO transition (A_V of ∼1.5 mag). The location of the H-to-H₂ transition agrees with the results from UV absorption (Savage et al. 1977), and that of the H₂-to-CO transition is consistent with our conclusions that the gas probed by HD 28975 and HD 29647 is predominantly molecular. These authors also find a significant fraction of the molecular content in Taurus in the form of CO-dark gas. Second, Lee et al. (2006) describe diffuse far-UV observations of the Taurus region acquired with SPEAR/FIMS. Because the spatial resolution is coarse (pixel size of 02 by 02 smoothed by 3 pixels), detailed comparisons with our results are not possible. The measurements can distinguish cloud cores from halos surrounding the cores. The map includes directions toward HD 27778, HD 28975, HD 29647, HD 30122, and HD 26571. The sight lines toward HD 28975 and HD 29647 sample edges of cores, while the other sight lines lie within halos. Cloud cores, with A_V > 1.5 mag, obscure background far-UV radiation, while the flux seen from halos comes from scattered foreground light. Molecular hydrogen fluorescence, which is only seen from halos, is examined with the model of Black & van Dishoeck (1987). The low values for gas density (∼50 cm⁻³) and N(H₂) (0.8 × 10²⁰) are likely caused by the large pixels. Of note is that again a reduced radiation field is required to reproduce the observations.

A number of efforts examined chemical species associated with diffuse molecular gas in specific dark clouds or regions of the TMC. Since these species are seen along the lines of sight to our stellar background sources, we now describe how our results fit into the perspective of these findings for molecular cloud envelopes. The works by Sakai et al. (2012) and Ebisawa et al. (2015) studied CH emission from HCL-2 and OH emission from material to the east of the cloud, respectively. We already noted that the CH column density lies below the extension of the relationship for diffuse clouds (Sakai et al. 2012); here we focus on the kinematics. For two of the four regions examined, both a narrow component (∼0.3 km s⁻¹) and a broad component (∼1.5 km s⁻¹) are seen. Where a narrow component is observed, it contributes about 20% to the total CH column density. Only the narrow component is present in the dense core, while the broad component is more extended and represents the diffuse envelope. The transition from narrow to broad component is probably sharp, but their measurements with a $3\buildrel{\,\prime}\over{.} 2$ beam could not resolve it. The presence of the two components might be caused by dissipation of turbulence. Only the broad component is observed in the region closest to HD 29647, TMC-1(NH₃).

All four hyperfine transitions in OH were measured by Ebisawa et al. (2015). The material to the east of HCL-2 is more diffuse and lies about 4.5 pc from the direction of HD 29647. Intensity anomalies caused by non-LTE effects were found, allowing them to infer gas temperatures over a wide range in density (10²–10⁷ cm⁻³). Toward the center of their strip map acquired with an $8\buildrel{\,\prime}\over{.} 2$ beam, T_k is 53 ± 1 K and increases to about 60 K at the boundaries. The averaged spectrum yields 60 ± 3 K and N(OH) of (4.4 ± 0.3) × 10¹⁴ cm⁻². The lines only reveal the broader component (∼1.3 km s⁻¹). The emission appears to arise from CO-dark gas. The line of sight toward HD 27778 probes similar material and has a comparable amount of OH (Felenbok & Roueff 1996), (1.02 ± 0.04) × 10¹⁴ cm⁻². The somewhat smaller value toward the star could be a consequence of the much larger beam used in the radio measurements.

The surveys by Goldsmith et al. (2008), Narayanan et al. (2008), and Pineda et al. (2010) were used in more focused efforts. Of most relevance to our results, we discuss measurements of emission from H₂ (Goldsmith et al. 2010), from C i and C ii (Orr et al. 2014), and from CH and OH (Xu & Li 2016; Xu et al. 2016). Goldsmith et al. (2010) obtained data on S(0) through S(3) transitions in H₂ for three boundary regions (see Figure 1), a Linear Edge (R. A. (J2000) = 4^h 38^m 00^s and $\mathrm{decl}.({\rm{J}}2000)=+27^\circ \,00^{\prime} \,00^{\prime\prime}$ at the center of the nominal slit position), the Filament (R. A. (J2000) = 4^h 50^m 30^s and $\mathrm{decl}.({\rm{J}}2000)=+25^\circ \,17^{\prime} \,30^{\prime\prime}$), and the Globule (R. A. (J2000) = 4^h 26^m 45^s and $\mathrm{decl}.({\rm{J}}2000)=+25^\circ \,39^{\prime} \,00^{\prime\prime}$). The Linear Edge and the Globule regions lie near the center of Figure 1, while the Filament is near the eastern edge of the TMC. The study emphasized the Linear Edge because it was less complex. The other studies acquired their data on the Linear Edge as well. The emission and relative populations were analyzed with the Meudon PDR code (Le Petit et al. 2006), but the high-excitation temperature (210 K) suggested the presence of other processes such as the dissipation of turbulence. The observed ortho-to-para ratio revealed the effects of turbulent diffusion, bringing colder interior gas to the surface. The peak intensity occurred beyond the edge in ¹³CO emission for the Linear Edge, where the column densities for J = 2 through 5 were 1.3 × 10¹⁸ cm⁻², 1.6 × 10¹⁷ cm⁻², ∼ 1.1 × 10¹⁶ cm⁻², and ∼ 8.5 × 10¹⁵ cm⁻², respectively. These column densities can be compared with the results for the sight line toward HD 27778 (Jensen et al. 2010), which samples similar material: 3.0 × 10¹⁸ cm⁻², 3.5 × 10¹⁷ cm⁻², 4.4 × 10¹⁵ cm⁻², and 2.1 × 10¹⁴ cm⁻², respectively. The UV data for HD 27778 show more excitation in the lower rotational levels and less in the highest ones. The line of sight toward HD 210839 (λ Cep) in the survey by Jensen et al. (2010) has column densities for these levels that are most similar to the Spitzer measurements (Goldsmith et al. 2010). Somewhat lower column densities were derived for the Filament.

Orr et al. (2014) also utilized the Meudon PDR code in an attempt to reproduce their observations. The most consistent model indicated the need for a weak radiation field, a low ambient ¹²C/¹³C ratio of 43, a primary cosmic-ray ionization rate of about 5 × 10⁻¹⁷ s⁻¹, and significant sulfur depletion. The emission from neutral carbon and the CO isotopologues appears to come from dark cloud material. While no C ii emission was seen, the modeling results reveal a rapid decline in C⁺ abundance with depth into the cloud. Orr et al. (2014) suggested that ionized carbon was associated with the diffuse component of the cloud. They provided integrated intensities for CO (≈13 K km s⁻¹) and ¹³CO (3–4 K km s⁻¹). For two of the directions in our survey, HD 27778 and HD 29647, Liszt (2008) quoted respective values of 7.17 (0.43) and 16.35 (5.84) K km s⁻¹, where the numbers in parentheses are for ¹³CO. Again these values imply that the absorption toward HD 27778 samples diffuse material just beyond the edge, while that toward HD 29647 is more like a dark cloud. The predictions for a rapid decline in C⁺ abundance are consistent with the small amount of C⁺ we infer for the gas toward HD 29647.

Observations of CH and OH emission from the Linear Edge were described by Xu et al. (2016) and Xu & Li (2016). Two velocity components are present: one changes from +5.3 to +6.0 km s⁻¹, while the other appears at +6.8 km s⁻¹. The bluer emission comes from the region where ¹³CO emission is present; the redder component is mainly confined to the gas beyond the edge associated with only detectable amounts of CO. The OH column density is about 50% greater in the more diffuse portions of the strip map, with values of about 4 × 10¹⁴ cm⁻². In other words, the column densities along the strip intersecting the Linear Edge are about 2−4 times greater than what is measured in the diffuse molecular gas toward HD 27778. Column densities for CH were more uncertain because limited information on excitation temperature and optical depth was available. The authors interpret the shift in velocity, the apparent excess in CH at A_V less than about 1 mag, and line anomalies in the OH satellite line at 1712 MHz as evidence for a continuous or C-shock caused by colliding streams or gas flow.

Our CH⁺ results can provide further insight into this phenomenon because C-shocks would be a site of significant production for this molecular ion (e.g., Flower et al. 1985; Draine & Katz 1986). Subsequent reactions can transform CH⁺ into CH. This route proceeds in low-density gas (<100 cm⁻³) because CH⁺ is destroyed rapidly through reactions with H, H₂, and electrons at higher densities. Another route at higher densities synthesizes CH (as well as CN and most of the CO) and is initiated by a reaction involving radiative association, C⁺ + H₂ → CH₂ ⁺ + photon. According to Pan et al. (2005), there is a linear relationship between CH associated with CH⁺ and CH⁺ in terms of column densities. In other words, when CH is made in the network involving CH⁺, the molecules have comparable column densities. According to Table 2, each of the velocity components containing absorption from both CH and CH⁺ has a much smaller column of CH⁺. The same applies for the results on the sight line toward HD 26571 in Appendix C. Thus, these five directions that sample diffuse and dark molecular clouds, the same type of material seen in CO and ¹³CO in the TMC (Goldsmith et al. 2008; Pineda et al. 2010), contain little CH⁺. While turbulence is clearly present in the form of nonthermal line widths, evidence for significant amounts of shocked gas appears less convincing.

Rybarczyk et al. (2020) measured small-scale features traced by H i absorption toward radio sources having component structure separated by less than a few arcminutes, or linear scales from 10³ to 10⁵ au. Resolved velocity components were treated separately in the analysis. Both changes in peak optical depth and line width contribute to the observed variation. These appear to be a transient feature of the cold neutral medium. Differences in optical depth of about 0.05 are common, but reaching factors of 10 larger in some instances. The median change in optical depth corresponds to changes in H i column density of about 3.5 × 10¹⁹ cm⁻², though for a sight line in Taurus (3C 111) the variation in column density is ∼2 × 10²⁰ cm⁻². For gas in the TMC this corresponds to density variations of 1000 cm⁻³. Only limits are available for another source in Taurus (3C 123), which lies closer to the directions of the stars in our sample. It is important to note that if the material is sheet-like, the density estimate could be a factor of a few smaller (Heiles 1997). The inferred densities are similar to what we find for the molecular gas probed by HD 27778, HD 28975, HD 29647, and HD 30122. Considering absorption from the same set of species as in the present paper, Pan et al. (2001) detected spatial variations in molecular column densities across members of HD 206267 A/C/D and HD 217035 A/B with separations (10³–10⁴ au) like those used in Rybarczyk et al. (2020). Such a study is not possible for most of the material in the TMC because similar systems tend to be too faint or are in front of the TMC. One interesting exception is the companion to HD 27778 (BD +23°683), a star with B and V of 8.40 and 8.2, respectively, about 30'' (4600 au) from the primary (see Figure 7).