EVOLUTION OF OH AND CO-DARK MOLECULAR GAS FRACTION ACROSS A MOLECULAR CLOUD BOUNDARY IN TAURUS

The OH observations were taken using the L-band wide receiver (with frequency range 1.55–1.82 GHz) on 2013 October 28–31. We observed four OH transition lines at the rest frequencies of 1612.231, 1665.402, 1667.359, and 1720.530 MHz with the total power ON mode. Spectra were obtained with the Arecibo WAPP correlator with nine-level sampling and 4096 spectral channels for each line in each polarization. The spectral bandwidth was 3.13 MHz for a channel spacing of about 763 Hz, or 0.142 km s⁻¹. The average system temperature was about 31 K. The main beam of the antenna pattern had a FWHM beam-width of 3'. Spectra were taken at 17 × 5 positions across the Taurus boundary region (TBR), as seen in Figure 1. An integration time of 300 s per position was used resulting in an rms noise level of about 0.027 K.

H i 21 cm observations of the TBR were extracted from the results of the GALFA-H i survey (Peek et al. 2011). GALFA-H i uses ALFA, a seven-beam array of receivers mounted at the focal plane of the 305 m Arecibo telescope, to map H i emission in the Galaxy. GALFA-H i is a survey of the Galactic interstellar medium in the 21 cm line hyperfine transition of neutral hydrogen which covers a large area (13,000 deg²) with ∼4' resolution and has a high spectral resolution (0.18 km s⁻¹) with broad velocity coverage (−700 km s⁻¹ < ${{\rm{v}}}_{\mathrm{LSR}}$ < +700 km s⁻¹). Typical rms noise is 80 mK in a 1 km s⁻¹ channel.

The ¹²CO J = 1-0 and ¹³CO J = 1-0 observations were taken simultaneously between 2003 and 2005 using the 13.7 m FCRAO Telescope (Narayanan et al. 2008). The map is centered at $\alpha (2000.0)={04}^{{\rm{h}}}{32}^{{\rm{m}}}44\buildrel{\rm{s}}\over{.} 6$, $\delta (2000.0)=24^\circ 25^{\prime} 13\buildrel{\prime\prime}\over{.} 08$ with an area of $\sim 98\ {\mathrm{deg}}^{2}$. The main beam of the antenna pattern had a FWHM beam-width of $45^{\prime\prime}$ for ¹²CO and $47^{\prime\prime}$ for ¹³CO. The angular spacing (pixel size) of the resampled on-the-fly data is 20'' (Goldsmith et al. 2008), which corresponds to a physical scale of $\approx 0.014\;{\rm{pc}}$ at a distance of $D=140\ {\rm{pc}}$. The data have a mean rms antenna temperature of 0.28 and 0.125 K in channels of 0.26 and 0.27 km s⁻¹ width for ¹²CO and ¹³CO, respectively.

The locations of the positions for the telescope pointing used to study the TBR are shown in Figure 1. To examine the transition zone with higher signal-to-noise ratio, we averaged all five cuts of spectra of OH 1612 MHz, 1665 MHz, 1667 MHz, 1720 MHz, ¹²CO J = 1-0, ¹³CO J = 1-0, and H i, as shown in Figure 2. The ¹²CO J = 1-0 and ¹³CO J = 1-0 spectra were convolved to the OH beam size of 3' at each position. The emission lines of OH, ¹²CO J = 1-0, ¹³CO J = 1-0, and the absorption lines of H i are well matched in velocity. Especially, the emission lines of OH 1665 MHz at positions 10–12 all have two components and the spectral of H i at the same point have two corresponding narrow absorption components as shown in Figure 3.

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We did Gaussian fitting of the OH 1612 MHz, 1665 MHz, 1667 MHz, and 1720 MHz spectra with two Gaussian components, the fitting of ¹²CO and ¹³CO spectra with a single Gaussian component, and the fitting of H i spectra with three Gaussian components. We show the spectra and the fitted profiles in Figure 2. The line ratio between OH 1665 MHz and 1667 MHz is greater than one in the outside TBR region (TBR-O) as shown in Figure 2. Under the assumption of local thermal equilibrium (LTE) the line ratio between 1665 and 1667 MHz ranges from 0.6 to 1. The line ratio greater than one indicates a deviation from LTE for OH.

We show the change of peak intensity of two components in OH 1665 MHz and a single component in ¹³CO across the TBR in Figure 4. Component 1 of OH 1665 MHz spectrum at 5.3 km s⁻¹ appears after position 4 and gets stronger across the TBR. Component 2 at 6.8 km s⁻¹ appears in the TBR-O and gets faint in the inside TBR region (TBR-I) and disappears after position 13. The central velocities of ¹³CO shift from 6.3 to 5.7 km s⁻¹. We assume that each component of the OH emission indicates a gas stream. When the intensity of OH component 2 is stronger than that of component 1 in TBR-O, the central velocity of ¹³CO is located at 6.3 km s⁻¹. When the intensity of OH component 1 is stronger than that of component 2 in TBR-I, the central velocity of ¹³CO is located at 5.7 km s⁻¹. The central velocity of ¹³CO shifts in the same way as the central velocity of stronger intensity OH component shifts. In Figure 3 we see that component 2 of OH 1665 gradually becomes fainter and disappears at position 13. At the same time the central velocity of component 1 gradually shifts from 5.3 km s⁻¹ at position 9 to 5.8 km s⁻¹ at position 13, which indicates that the collision of two streams results in the final central velocity being located between the velocities of the two components. The central velocity of the final combined stream is located closer to component 1, which has a stronger emission line in TBR-I. This is consistent with the assumption of different amounts of ¹³CO emission at different velocities. Not only OH 1665 MHz but also ¹³CO shows the central velocity shifting after the streams collide from position 10 to position 13 in Figure 3.

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We show the change of line width along the cut direction in Figure 5. We can clearly see that the line widths of ¹²CO and ¹³CO peak at the TLB. This is mainly due to the two gas streams that contribute to the line widths of ¹²CO and ¹³CO. The line width of OH 1667 MHz is much wider than that of other lines in TBR-O. This is mainly caused by the weak emission lines of OH 1667 MHz in TBR-O. Since the height of OH 1667 MHz is small in TBR-O, the fitted Gaussian profile tends to be flatter than that at other positions. So the line width of OH 1667 MHz is two times wider than that of the other three lines in TBR-O. Component 2 of OH 1667 MHz whose central velocity is at 6.8 km s⁻¹ is very weak in some positions in TBR-I, which leads to a wider line width of component 1 as shown in Figure 2. The width of component 2 in TBR-I is wider than that of other three lines in TBR-I, which leads to the average line width of OH 1667 behaving in an erratic way. The line width of OH 1665 MHz is nearly constant across the TBR.

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The line width of ¹²CO is significantly greater than that of ¹³CO. Opacity may be one of the reasons. Owing to the different abundance of ¹²CO and ¹³CO, ${\tau }_{12}$ is almost 70 times larger than ${\tau }_{13}$. When ${\tau }_{12}$ is much larger than unity, the term 1-exp[$-{\tau }_{12}\phi ({\rm{\Delta }}{\boldsymbol{v}}){\rm{]}}$ in the line profile of ¹²CO is much wider than that of ¹³CO, where $\phi ({\rm{\Delta }}{\boldsymbol{v}})$ is a Gaussian profile of the velocity offset from line center ${\rm{\Delta }}{\boldsymbol{v}}$. We made an estimate of the line width ratio between ¹²CO and ¹³CO only considering the opacity. In TBR-O the optical depth range is from 0.05 to 0.4. We took position 8 having ${\tau }_{13\mathrm{CO}}=0.2$ as an example. The line width ratio between ¹²CO and ¹³CO is 1.4, which is much less than the observed ratio 2.3. Other broadening mechanisms must occur in the TBR. Park & Hong (1995) took different broadening mechanisms, such as micro-turbulence and macro-turbulence, into consideration and found that the line width ratio between ¹²CO and ¹³CO can range from 1 to 3 depending on the physical condition of the gas, which just corresponds to the line width ratio between ¹²CO and ¹³CO in our work. The wider line width of ¹²CO may be the result of larger turbulence due to the more extended distribution of ¹²CO. According to Larson's law (Larson 1981), a larger scale of a molecular cloud leads to a larger velocity dispersion. Considering current geometrical model of photodissociation region (Tielens 2005), the larger self-shielding threshold of ¹³CO makes it more constrained to the inner region of the boundary, which yields a narrower line width than that of ¹²CO.

We can also see the central velocities of component 1 in OH 1665 MHz shifting from 5.3 to 6.0 km s⁻¹ at positions 10–12 in Figure 3. This velocity gradient may be caused by colliding streams or gas flow at the TLB. The description of the OH spectra and components across TBR can be found in Table 1.

Table 1. The Change of OH Spectral Lines Across the Boundary in Taurus

Position	Offset	OH Spectral Lines
ID	(')	1612 MHz		1665 MHz		1667 MHz		1720 MHz
		${\mathfrak{C}}$1 a	${\mathfrak{C}}$2 b	${\mathfrak{C}}1$	${\mathfrak{C}}2$	${\mathfrak{C}}1$	${\mathfrak{C}}2$	${\mathfrak{C}}1$	${\mathfrak{C}}2$
1 (outer)	−24.0	$w{\mathfrak{e}}$ c	${\mathfrak{a}}$ d	${\mathfrak{n}}$	${\mathfrak{e}}$	${\mathfrak{n}}$	$w{\mathfrak{e}}$	${\mathfrak{n}}$	$w{\mathfrak{e}}$
7 (${{\rm{I}}}_{{{\rm{H}}}_{2}}\;{peak}$)	−6.0	$w{\mathfrak{e}}$	${\mathfrak{a}}$	${\mathfrak{n}}$	${\mathfrak{e}}$	$w{\mathfrak{e}}$	$w{\mathfrak{e}}$	${\mathfrak{n}}$	${\mathfrak{e}}$
9 (boundary)	0.0	${\mathfrak{e}}$ e	$w{\mathfrak{a}}$ f	${\mathfrak{e}}$	${\mathfrak{e}}$	${\mathfrak{e}}$	$w{\mathfrak{e}}$	${\mathfrak{a}}$	${\mathfrak{e}}$
11 (${{\rm{N}}}_{{{\rm{H}}}_{2}}$ peak)	6.0	${\mathfrak{e}}$ ($\to$ g )	${\mathfrak{n}}$ h	${\mathfrak{e}}(\to )$	${\mathfrak{e}}$	${\mathfrak{e}}(\to )$	$w{\mathfrak{e}}$	${\mathfrak{a}}(\to )$	$w{\mathfrak{e}}$
12 (NCO peak)	9.0	${\mathfrak{e}}(\to )$	${\mathfrak{n}}$	${\mathfrak{e}}(\to )$	$w{\mathfrak{e}}$	${\mathfrak{e}}(\to )$	${\mathfrak{n}}$	${\mathfrak{a}}(\to )$	${\mathfrak{n}}$
17 (inner)	24.0	${\mathfrak{e}}(\to )$	${\mathfrak{n}}$	${\mathfrak{e}}(\to )$	${\mathfrak{n}}$	${\mathfrak{e}}(\to )$	${\mathfrak{n}}$	${\mathfrak{a}}(\to )$	${\mathfrak{n}}$

Notes.

^a ${\mathfrak{C}}1$ means the component at 5.3 km s⁻¹. ^b ${\mathfrak{C}}2$ means the component at 6.8 km s⁻¹. ^c $w{\mathfrak{e}}$ means weak emission. ^d ${\mathfrak{a}}$ means absorption. ^e ${\mathfrak{e}}$ means emission. ^f $w{\mathfrak{a}}$ means weak absorption. ^g $\to$ means the shifting of central velocity of component 1. ^h ${\mathfrak{n}}$ means neither emission nor absorption.

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From the fitting result of ¹²CO J = 1-0 and ¹³CO J = 1-0 above, we can calculate the column density of ¹³CO. The column density of ¹³CO in the upper level (J = 1) can be written as

$$\begin{eqnarray}&&{N}_{u{,}^{13}\mathrm{CO}}=\displaystyle \frac{8\pi k{\nu }^{2}}{{{hc}}^{3}{A}_{\mathrm{ul}}}\int {T}_{b}(V){dV},\end{eqnarray} \tag{ 1 }$$

where k is Boltzmann's constant, h is Planck's constant, c is the speed of light, A_ul is the spontaneous decay rate from the upper level to the lower level, and T_b is the brightness temperature. A convenient form of this equation is

$$\begin{eqnarray}&&\left(\displaystyle \frac{{N}_{u{,}^{13}\mathrm{CO}}}{{\mathrm{cm}}^{-2}}\right)=3.6\times {10}^{14}\int \left(\displaystyle \frac{{T}_{b}}{{\rm{K}}}\right)d\left(\displaystyle \frac{V}{\mathrm{km}\;{{\rm{s}}}^{-1}}\right).\end{eqnarray} \tag{ 2 }$$

The total ¹³CO column density ${N}_{{\rm{tot}}}$ is related to the upper-level column density N_u through (Li 2002)

$$\begin{eqnarray}&&{N}_{\mathrm{tot}{,}^{13}\mathrm{CO}}={f}_{u}{f}_{\tau }{f}_{b}{N}_{u{,}^{13}\mathrm{CO}}.\end{eqnarray} \tag{ 3 }$$

In the equation above, the level correction factor F_u can be calculated analytically under the assumption of LTE as

$$\begin{eqnarray}&&{f}_{u}=\displaystyle \frac{Q({T}_{{\rm{ex}}})}{{g}_{u}\mathrm{exp}\left(-\displaystyle \frac{h\nu }{{{kT}}_{{\rm{ex}}}}\right)},\end{eqnarray} \tag{ 4 }$$

where g_u is the statistical weight of the upper level. ${T}_{{\rm{ex}}}$ is the excitation temperature and $Q({T}_{{\rm{ex}}})={{kT}}_{{\rm{ex}}}/{{hB}}_{0}$ is the LTE partition function, where B₀ is the rotational constant (Tennyson 2005). A convenient form of the LTE partition function is $Q({T}_{{\rm{ex}}})\approx {T}_{{\rm{ex}}}/2.76\;{\rm{K}}$. The correction factor for opacity is defined as

$$\begin{eqnarray}&&{f}_{\tau }=\displaystyle \frac{\int {\tau }_{13}{dv}}{\int (1-{e}^{-{\tau }_{13}}){dv}},\end{eqnarray} \tag{ 5 }$$

and the correction for the background

$$\begin{eqnarray}&&{f}_{b}={\left[1-\displaystyle \frac{{e}^{\frac{h\nu }{{{kT}}_{{\rm{ex}}}}}-1}{{e}^{\frac{h\nu }{{{kT}}_{{\rm{bg}}}}}-1}\right]}^{-1},\end{eqnarray} \tag{ 6 }$$

where ${\tau }_{13}$ is the opacity of the ¹³CO transition and ${T}_{{\rm{bg}}}$ is the background temperature, assumed to be 2.7 K.

The ¹³CO opacity is estimated as follows. Assuming equal excitation temperatures for the two isotopologues, the ratio of the brightness temperature of ¹²CO to that of ¹³CO can be written as

$$\begin{eqnarray}&&\displaystyle \frac{{T}_{b,12}}{{T}_{b,13}}=\displaystyle \frac{1-{e}^{-{\tau }_{12}}}{1-{e}^{-{\tau }_{13}}}.\end{eqnarray} \tag{ 7 }$$

Assuming ${\tau }_{12}\gg 1$, the opacity of ¹³CO can be written as

$$\begin{eqnarray}&&{\tau }_{13}=-\mathrm{ln}\left(1-\displaystyle \frac{{T}_{b,13}}{{T}_{b,12}}\right).\end{eqnarray} \tag{ 8 }$$

The excitation temperature ${T}_{{\rm{ex}}}$ is obtained from the ¹²CO intensity. First, the maximum intensity in the spectrum of each pixel is found. This quantity is denoted by ${T}_{{\rm{max}}}$. The excitation temperature is calculated by solving the following equation:

$$\begin{eqnarray}&&{T}_{{\rm{max}}}=\displaystyle \frac{h\nu }{k}\left[\displaystyle \frac{1}{{e}^{\frac{h\nu }{{{kT}}_{{\rm{ex}}}}}-1}-\displaystyle \frac{1}{{e}^{\frac{h\nu }{{{kT}}_{{\rm{bg}}}}}-1}\right],\end{eqnarray} \tag{ 9 }$$

where h, k, and ν are Planck's constant, Boltzmann's constant, and the central frequency of ¹²CO $J=1\to 0$ line (115.27 GHz), respectively.

To examine the LTE assumption when calculating the physical parameters of CO, we used a spherical 1D non-LTE spectral analysis radiative transfer model, RADEX (van der Tak et al. 2007) to derive the excitation temperature of ¹²CO and ¹³CO. We took position 10 as an example. We assumed the number density of H₂ to be 400 cm⁻³, which is given by Orr et al. (2014). We also assumed the kinetic temperature to be 15 K, which is widely applied in the relatively diffuse region in the Taurus molecular cloud (e.g., Goldsmith et al. 2008). The resulting excitation temperature of ¹²CO and ¹³CO are ${T}_{\mathrm{ex}{,}^{12}\mathrm{co}}$ = 7.7 K and ${T}_{\mathrm{ex}{,}^{13}\mathrm{co}}$ = 6.8 K, with a difference of about 10%. The assumption that the excitation temperatures of ¹²CO and ¹³CO are equal seems reasonable. Owing to the difference of excitation temperature between ¹²CO and ¹³CO, the derived column density of ¹³CO also has an error about 10%.

We show the change of excitation temperature of ¹³CO and the change of column density of ¹³CO along the cut direction in Table 2 and Figure 6. The excitation temperature ${T}_{{\rm{ex}}}$ of ¹³CO increases crossing the boundary. The column density of ¹³CO shows a peak at position 13 and 14 inside the boundary.

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Table 2. Parameters along the Boundary

Position	${T}_{\mathrm{ex}{,}^{13}\mathrm{co}}$	${N}_{{}^{{\rm{13}}}\mathrm{co}}$
	(K)	(${10}^{14}\;{\mathrm{cm}}^{-2}$)
1	6.5	2.5
2	6.3	2.3
3	5.9	2.8
4	5.8	3.1
5	5.6	3.2
6	5.6	4.0
7	6.3	5.6
8	6.3	5.6
9	6.9	13
10	7.6	39
11	7.8	33
12	7.7	42
13	7.6	59
14	8.5	58
15	8.5	34
16	8.0	20
17	8.5	23

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We modeled the linear boundary as a cylinder with a radius-dependent H₂ volume density structure of the following form (King 1962),

$$\begin{eqnarray}{n}_{{{\rm{H}}}_{2}}(r)=\left\{\begin{array}{cc}{n}_{c}{a}^{2}/({r}^{2}+{a}^{2}) & :r\leqslant R\\ 0 & :r\gt R\end{array}.\right.\end{eqnarray} \tag{ 10 }$$

This functional form was used in analysis of the Taurus region by Pineda et al. (2010).

This profile has a flat high-density center which transitions to a region of power-law decay and finally at a truncating radius, the density is set to zero. It assumes only three parameters: the central core density n_c , a parameter characterizing the width of the central core a, and the truncating radius R. Orr et al. (2014) fitted the profile with visual extinction data for this region. The fitted values for the density profile parameters are ${n}_{c}=626$ cm⁻³, $a=0.457$ pc, and $R=1.80$ pc. The axis of the cylindrical distribution is centered $11\buildrel{\,\prime}\over{.} 5$ to the southwest of the boundary position between position 12 and position 13.

The H₂ volume density of 17 observed points along the boundary is obtained from Orr et al. (2014).

RADEX takes the following inputs: kinetic temperature (${T}_{{\rm{k}}}$), density of H₂ (${n}_{{{\rm{H}}}_{2}}$), H₂ ortho-to-para ratio (OPR), background temperature (${T}_{{\rm{bg}}}$), column density of OH (${N}_{{\rm{OH}}}$), and line width (${\rm{\Delta }}{v}_{{\rm{OH}}}$). Rates for collisional excitation of OH are taken from Offer et al. (1994).

The ${n}_{{{\rm{H}}}_{2}}$ is an input parameter estimated based on the cylindrical model in Orr et al. (2014). The Galactic background emission is estimated to be about 0.8 K by extrapolating the standard interstellar radiation field (ISRF) to the L band (Winnberg et al. 1980). ${T}_{{\rm{bg}}}$ = 3.5 K is thus used in the simulation.

We vary the ${T}_{{\rm{k}}}$, H₂, OPR, and ${N}_{{\rm{OH}}}$ to find the optimum model by minimizing ${\chi }^{2}$ for the four OH lines, defined as

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}\displaystyle \frac{{({I}_{{\mathrm{model}}_{i}}-{I}_{{\mathrm{obs}}_{i}})}^{2}}{{\sigma }_{{\mathrm{obs}}_{i}}^{2}},\end{eqnarray} \tag{ 11 }$$

where I_obs is the four observed by OH lines' instensities, I_model is the model lines generated by RADEX, and ${\sigma }_{\mathrm{obs}}^{2}$ is the rms of the four observed OH lines.

We varied ${T}_{{\rm{k}}}$, OPR, and ${N}_{{\rm{OH}}}$ to obtain the best fit to the observation at position 1 shown in Figure 7. The best-fitting ${T}_{{\rm{k}}}$, OPR, and ${N}_{{\rm{OH}}}$ are 31 K, 0.2, and $3.7\times {10}^{14}\;{\mathrm{cm}}^{-2}$, respectively. We also calculated the column density of OH at position 1 (assuming it to be optically thin with no background emission) from the integrated intensity (in K km s⁻¹) of the 1667 MHz line through (e.g., Turner & Heiles 1971; Knapp & Kerr 1973):

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{\mathrm{OH}}}{{\mathrm{cm}}^{-2}}=2.22\times {10}^{14}\displaystyle \frac{\int {T}_{b}(V){dV}}{{\rm{K}}\;\mathrm{km}\;{{\rm{s}}}^{-1}}.\end{eqnarray} \tag{ 12 }$$

With these assumptions ${N}_{{\rm{OH}}}$ is $0.5\times {10}^{14}$ cm⁻². When the excitation temperature of OH is so low that the background cannot be neglected, we have a correction factor F_bg (e.g., Harju et al. 2000; Suutarinen et al. 2011)

$$\begin{eqnarray}&&{f}_{\mathrm{bg}}=\displaystyle \frac{1}{1-{T}_{{\rm{bg}}}/{T}_{{\rm{ex}}}}.\end{eqnarray} \tag{ 13 }$$

When ${T}_{{\rm{bg}}}$ $\simeq$ 3.5, K and ${T}_{{\rm{ex}}}$ $\simeq$ 4 K, F_bg $\simeq$ 8, which yields ${N}_{{\rm{OH}}}$ $\simeq$ $4\times {10}^{14}\quad {\mathrm{cm}}^{-2}$, which is almost the same as the fitted ${N}_{{\rm{OH}}}$ from RADEX (denoted ${N}_{{\rm{OH}}}^{\mathrm{RADEX}}$). The problem with using the "LTE method" is that we do not know the excitation temperature of OH. Instead we have to assume an excitation temperature for OH, which has a major effect on the correction factor f_bg. Only with a statistical equilibrium calculation (e.g., RADEX) can we know the excitation temperature of OH exactly. We should be cautious about the low excitation temperature of OH, otherwise we may underestimate the column density of OH by a factor of eight or more in many conditions where the LTE method is used.

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Compared with ${N}_{{\rm{OH}}}^{\mathrm{RADEX}}$, the calculated ${N}_{{\rm{OH}}}$ through Equations (12) and (13) (noted as ${N}_{{\rm{OH}}}^{\mathrm{LTE}}$ representing the "LTE method") is almost the same as ${N}_{{\rm{OH}}}^{\mathrm{RADEX}}$. The fitted opacity (Table 3) based on the model is relatively smaller ($\tau \;\sim$ 0.04–0.3), which is consistent with the assumption of optical thin in the "LTE method."

Table 3. OH Physical Parameters at Position 1

n ${}_{{{\rm{H}}}_{2}}$ = 62.6 cm−3 OPR = 0.20 ${T}_{{\rm{k}}}=31.0$ K
${N}_{{\rm{OH}}}=3.7\times {10}^{14}\;{\mathrm{cm}}^{-2}$
Line	${T}_{{\rm{ex}}}$	τ	${T}_{{\rm{A}}}$
	(K)		(K)
1612	3.1	0.045	−0.025
1665	4.1	0.163	0.098
1667	3.8	0.331	0.062
1720	5.4	0.024	0.055

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Equation (12) is used to calculate ${N}_{{\rm{OH}}}$ and assumes LTE, but the observed line ratio between OH 1665 and 1667 MHz is obviously greater than one, indicating that the OH at position 1 is far from LTE. The critical density of OH HFS lines at 10–50 K is 1–20 cm⁻³, indicating that OH excitation is dominated by a collision in TBR with n = 70–600 cm⁻³. Compared with OH, low-J ¹²CO lines have a larger critical density of about 2000 cm⁻³. In most cases when collision dominate the excitation, the LTE assumption is reasonable. Surprisingly, OH HFS lines are far from LTE, which has a low critical density. ¹²CO is consistent with LTE as discussed in Section 3.2 with RADEX analysis. Considering for the complex energy level of OH, some pumping mechanism must be operative to yield the non-LTE of OH. We will have a detailed discussion of this in Section 5. A most possible mechanism is C-shock. The line ratio between 1665 and 1667 MHz and the integrated intensity of 1667 MHz across the boundary are listed in Table 4. In TBR-O the line ratio is greater than one. In TBR-I the line ratio is between ≃0.5 and ≃0.8, which is within the range allowed by LTE.

Table 4. OH Parameters Along the Boundary

Position	Ratio(1665/1667)	Intensity(1667)
		(${\rm{K}}\quad \mathrm{km}\quad {{\rm{s}}}^{-1}$)
1	1.6 ±  0.18	0.21 ±  0.02
2	1.2 ±  0.14	0.20 ±  0.02
3	1.3 ±  0.13	0.22 ±  0.02
4	1.2 ±  0.12	0.24 ±  0.02
5	1.5 ±  0.14	0.29 ±  0.02
6	1.6 ±  0.13	0.26 ±  0.02
7	1.3 ±  0.10	0.35 ±  0.02
8	1.2 ±  0.08	0.31 ±  0.02
9	1.1 ±  0.05	0.42 ±  0.02
10	0.51 ±  0.02	0.49 ±  0.01
11	0.54 ±  0.02	0.53 ±  0.02
12	0.52 ±  0.02	0.50 ±  0.01
13	0.61 ±  0.02	0.49 ±  0.01
14	0.69 ±  0.02	0.48 ±  0.01
15	0.72 ±  0.02	0.42 ±  0.01
16	0.77 ±  0.03	0.40 ±  0.01
17	0.83 ±  0.04	0.37 ±  0.01

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The change of physical parameters across the boundary is partially listed in Table 5.

Table 5. Change of Physical Parameters Across the Boundary in Taurus

Position	Offset	Physical Parameters
ID	(')	${T}_{k}$ (K)		${N}_{\mathrm{OH}}$(1014 cm−2)		Av (mag)	${N}_{\mathrm{OH}}/{N}_{{{\rm{H}}}_{2}}$ $({10}^{-7})$	DGF a
		${\mathfrak{C}}1$	${\mathfrak{C}}2$	${\mathfrak{C}}1$	${\mathfrak{C}}2$
1 (outer)	−24.0	⋯	37	⋯	2.9	0.4	${7.2}_{-2}^{+3}$	${0.75}_{-0.1}^{+0.1}$
7 (${{\rm{I}}}_{{{\rm{H}}}_{2}}\;\mathrm{peak}$)	−6.0	25	35	2.9	1.3	1.3	${3.5}_{-0.3}^{+0.7}$	${0.70}_{-0.1}^{+0.1}$
9 (boundary)	0.0	26	32	1.5	4.9	1.8	${3.7}_{-0.2}^{+0.4}$	${0.13}_{-0.3}^{+0.2}$
11 (${{\rm{N}}}_{{{\rm{H}}}_{2}}$ peak)	6.0	10	27	1.5	4.5	2.4	${2.6}_{-0.3}^{+0.5}$	${0.03}_{-0.4}^{+0.2}$
12 (${{\rm{N}}}_{\mathrm{CO}}$ peak)	9.0	23	26	2.1	2.5	2.6	${1.9}_{-0.2}^{+0.4}$	${0.07}_{-0.4}^{+0.2}$
17 (inner)	24.0	24	⋯	3.3	⋯	1.5	${2.3}_{-0.3}^{+0.5}$	${0.12}_{-0.3}^{+0.2}$

Note.

^a DGF means dark gas fraction.

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Owing to the different cross sections between OH and two spin symmetries of H₂, i.e., ortho-H₂ and para-H₂ (Offer et al. 1994), the OPR plays a significant role in the excitation of OH. The exact value of the OPR is important for producing the observed OH 1665/1667 intensity ratio in certain positions, such as for position 9, as shown in Figure 8. Overall the derived column density is insensitive to the numerical value of the OPR, such as for position 10 as shown in Figure 8. We will discuss this issue in a separate paper and focus on the dark gas and OH abundance content in the present work.

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We define the CO-dark molecular gas (DMG) fraction or dark gas fraction (DGF) as

$$\begin{eqnarray}&&\mathrm{DGF}=1-\displaystyle \frac{{N}_{{\rm{CO}}}\times {10}^{4}}{{N}_{{{\rm{H}}}_{2}}},\end{eqnarray} \tag{ 14 }$$

which represents the fraction of H₂ that cannot be traced by CO emission. ${N}_{{\rm{CO}}}$ is obtained from the LTE calculation in Section 3.2 with the assumption that the abundance ratio of ¹²CO to ¹³CO is 65 (Langer & Penzias 1993; Liszt 2007). ${N}_{{{\rm{H}}}_{2}}$ is obtained from integration of the density profile in Section 4.1 along the line of sight and the visual extinction is obtained from the relation N(H${}_{2})/{A}_{{\rm{V}}}=9.4\times {10}^{20}$ cm⁻² mag⁻¹, assuming that the hydrogen is predominately in molecular form and standard grain properties are appropriate for the diffuse ISM (Orr et al. 2014). We averaged the column density of H₂ within the OH beam size (∼3') at each position to make all the calculations refer to the same beam size. The variation of ${T}_{{\rm{k}}}$, ${N}_{{\rm{OH}}}$, ${N}_{{\rm{OH}}}/{N}_{{{\rm{H}}}_{2}}$, and DGF across the boundary as a function of extinction A_v is shown in Figure 9. When the extinction A_v increases from 0.4 to 2.6 mag, the kinetic temperature, the abundance of OH, and the DGF all decrease. Especially, the DGF decreases from 80% to 20%, which means the amount of molecular gas that cannot be traced by CO is three times larger than that of molecular gas traced by CO when the extinction is below 1.4 mag. Empirically, CO intensities have been used as an indicator of the total molecular mass in the Milky way and in galaxies through the so-called "X-factor" with numerous well-known caveats. In other words, we may seriously underestimate the amount of molecular gas through "X-factor" in low-extinction clouds or regions of galaxies.

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We parametrize the trend of ${N}_{{\rm{OH}}}/{N}_{{{\rm{H}}}_{2}}$ and DGF in an exponential law and a Gaussian profile, respectively, as shown in Figure 10. The trend of ${N}_{{\rm{OH}}}/{N}_{{{\rm{H}}}_{2}}$ can be fitted as

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{{\rm{OH}}}}{{N}_{{{\rm{H}}}_{2}}}=1.5\times {10}^{-7}+0.9\times {10}^{-7}\times \mathrm{exp}\left(-\displaystyle \frac{{A}_{v}}{0.81}\right).\end{eqnarray} \tag{ 15 }$$

When the visual extinction is much larger than 0.48 mag, ${N}_{{\rm{OH}}}/{N}_{{{\rm{H}}}_{2}}$ remains roughly a constant $1.5\times {10}^{-7}$. This constant indicates the abundance of OH at large visual extinctions within molecular clouds. A calculation using the Meudon PDR Code (Le Petit et al. 2006) with conditions appropriate for the TBR yields reasonably good agreement with OH abundance [OH]/[H₂] for moderate extinction (A_v  ∼ 2). The prediction of [OH]/[H₂] at extinctions at or below 1 mag is far lower (by a factor of 80) than we derive, suggesting that there may be an additional channel of OH production active, possibly due to the shock (e.g., Draine & Katz 1986a) produced by the colliding streams. When shock waves propagate through the molecular ISM the ambient gas is compressed, heated, and accelerated. When temperature is above 300 K, the neutral–neutral reactions become important, which yield the overabundance of OH (Neufeld et al. 2002).

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The trend of the DGF can be fitted as

$$\begin{eqnarray}&&{\rm{DGF}}=0.90\times \mathrm{exp}\left[-{\left(\displaystyle \frac{{A}_{v}-0.79}{0.71}\right)}^{2}\right].\end{eqnarray} \tag{ 16 }$$

The peak of the DGF is located at A_v = 0.79 mag and the FWHM is 1.4 mag. When 0.79 < ${A}_{v}\lt 2.5$ mag, the DGF decreases sharply with increasing A_v . This behavior of the DGF is expected theoretically, e.g., Planck & Fermi Collaborations (2015) found that CO-dark molecular dominates the molecular columns at A_v  $\leqslant$ 1.5 mag and Paradis et al. (2012) found an obvious deviation of the amount of gas traced by H i and ¹²CO from that of the extinction data in the extinction range between 0.3 and 2 mag, with neither the low extinction (≤0.3 mag) nor the high extinction (≥2 mag) showing significant deviation. In regions with sufficiently low visual extinction, H₂ cannot survive due to the destruction by UV photons. On the other hand, when the visual extinction is high enough, ¹²CO can be formed and survive. At this point the molecular gas can be well traced by ¹²CO so that the DGF also drops at a high visual extinction. When the visual extinction is between 0.4 and 1.3 mag, the abundance of H₂ is already a layer fraction of the total hydrogen abundance, but ¹²CO has not formed completely so the DGF peaks under these conditions.

It is important to emphasize that the parameterized trends above can only be applied to the molecular gas with visual extinction between 0.4 and 1.3 mag for which we have OH data and have modeled the lines successfully.