THE DARK MOLECULAR GAS

The PDR code of Kaufman et al. (2006) is based on two main parts. The first part is derived from Kaufman et al. (1999), Wolfire et al. (1990), and Tielens & Hollenbach (1985) models and is the main program for calculating the chemistry, thermal balance, and line emission. The second part is based on the Le Petit et al. (2006) Meudon PDR code and is used for all of the molecular hydrogen processes. The Kaufman et al. (2006) code assumes a 1D UV flux normal to the cloud surface. Here, we consider the average radiation field produced by all OB associations illuminating the GMC, and the field at the cloud surface is expected to be relatively isotropic over 2π steradians. In order to account for the isotropic field, we assume a 1D flux incident at an angle of 60° to the normal. This unidirectional field has the same flux normal to the cloud surface as the isotropic field, and thus the same energy input to the cloud, and it more closely approximates the depth penetration of the field. The photo rates are modified to account for twice the path length of the incident field for a given A_V measured normal to the surface. In the Meudon portion of the code, the isotropic field is accounted for explicitly in the depth dependence of the photo rates.

We have also included several changes in the elemental abundances and chemical network that affect the carbon chemistry. We adopt the gas-phase carbon abundance ${\cal A}_{\rm C}=1.6\times 10^{-4}$ from Sofia et al. (2004), where ${\cal A}_{\rm C} = n_{\rm C}/n$ and where n_C is the gas-phase number density of elemental carbon and n is the number density of H nuclei. We also include the C⁺ radiative and dielectronic recombination rates from Badnell (2006),⁴ who have found that the rates at low temperatures can be a factor of 2–3 higher than was used in the past (e.g., Nahar & Pradhan 1997). Another change in the chemical network is to include OH production on grains from Hollenbach et al. (2009). We adopt the fastest possible rate, having each collision of O with a grain combine with H and produce OH. This limit tends to maximize CO production and minimize the dark-gas fraction. We also include the ion-dipole reaction rates for C⁺, H⁺₃, and He⁺, reacting with OH and H₂O as discussed in Hollenbach et al. (2009).

The external fluxes of FUV radiation, EUV/soft X-ray radiation, and cosmic rays are model inputs and must be specified. Associations produce bright PDRs on nearby cloud surfaces. For example, the Trapezium cluster in Orion produces the bright Orion PDR on cloud surfaces only ∼0.1 pc distant. As an association evolves, the Strömgren region expands, the embedded H ii region breaks out of the cloud, the resultant champagne flow and blister evaporation disperses the cloud and the distance between the cloud and association increases, thus allowing more of the cloud to be illuminated than the initial hot spot. The combination of cloud and association evolution, the distribution of associations around the cloud, and the continuous formation of new associations elevates the average radiation field illuminating the cloud to a field strength that is greater than the average interstellar radiation field. A detailed evaluation of this process is presented in M. Wolfire et al. (2010, in preparation). They find the average field incident on clouds is on the order of ∼20 times higher than the local Galactic interstellar field. We adopt the Draine (1978) interstellar radiation field that is equal to ∼1.7 times the Habing (1968) local interstellar field (1.6 × 10⁻³ erg cm⁻² s⁻¹ for FUV photons). We also use the notation that G₀ is the radiation field in units of the Habing field and G'₀ is the field relative to the local Galactic field found by Draine (1978), so that G'₀ = G₀/1.7. To clarify the definition of G'₀, which is consistent with the original definition given in Tielens & Hollenbach (1985), G'₀ is the ratio of the photorates at the surface (A_V = 0) of our cloud to the same rates in the Draine field in free space. Thus, G'₀ = (2πI)/(4πI_D) = 0.5I/I_D, where I is our incident isotropic intensity and I_D = 2.2 × 10⁻⁴ erg cm⁻² s⁻¹ sr⁻¹ is the Draine intensity for the local ISM.⁵ Here, we examine a range of FUV fields, 3 < G'₀ < 30, consistent with (and generously encompassing) the results of M. Wolfire et al. (2010, in preparation). Throughout this paper, we simply scale the flux by G'₀ and keep the spectral energy distribution fixed as given by the Draine (1978) field.

As discussed in Wolfire et al. (2003), the soft X-ray/EUV flux from the diffuse interstellar field can be produced by a combination of supernova remnants and stellar sources. The field shining on the molecular cloud can be enhanced above the interstellar flux due to the OB associations near the molecular cloud. We assume that we are outside any of the H ii regions associated with the OB associations, and simply scale the soft X-ray/EUV flux by the same factor that we do for the effective FUV radiation field ζ'_XR = G'₀. A corollary of this assumption is that the fraction of gas that is ionized is negligible. Wolfire et al. (2003) adopted a standard soft X-ray/EUV shielding column of neutral gas of column density N = 1 × 10¹⁹ cm⁻². The layer N < 10¹⁹ cm⁻² mainly consists of warm neutral medium (WNM) gas photoevaporated from the molecular and CNM clump surfaces and does not provide significant shielding to the FUV photodissociating radiation field. We are mainly concerned with the deeper layers and assume that the clouds are shielded by a column of this order. As we shall demonstrate in Section 5, the absorption of X-ray/EUV radiation at columns in the range 10¹⁹ cm⁻² <N ∼ 10²¹ cm⁻² leads to a drop in the thermal pressure by a factor of ∼10 in the H i layer in the cloud.

The cosmic-ray flux is also associated with massive star evolution, but is not as likely to be simply proportional to G'₀ as for the EUV/soft X-ray flux. Since the temperatures of cloud interiors do not exceed T = 10–20 K, the cosmic-ray flux in cloud interiors cannot be higher than a factor of ∼2–3 times the diffuse ISM value or else the heating rate would push internal cloud temperatures higher than observed. In most of our calculations, we therefore assume that cosmic-ray rates are not enhanced by the nearby molecular cloud OB associations.

The cosmic-ray rate can also affect the formation of CO in the diffuse gas and outer molecular cloud layers through the production of OH as discussed by van Dishoeck & Black (1986, 1988). There are several observations, mainly of H⁺₃, that indicate higher cosmic-ray ionization rates by a factor of ∼10 along select lines of sight in the diffuse ISM (Indriolo et al. 2007); however, the H⁺₃ observations do not indicate higher cosmic-ray rates in the denser, molecular clouds considered here (McCall et al. 1999). We adopt our standard rate for the majority of this paper, but consider the effects of higher cosmic-ray rates in the outer portions of the cloud in Section 5.2.6.

We have applied our optically thin clump models to molecular cloud masses M(R_CO) = 1 × 10⁵ M_☉, 3 × 10⁵ M_☉, 1 × 10⁶ M_☉, and 3 × 10⁶ M_☉ for incident radiation fields G'₀ = ζ'_XR = 3, 10, and 30. The lower end of the mass range is chosen to be the lowest cloud mass that could still produce OB stars sufficient to illuminate, heat, and dissociate the molecular cloud with FUV radiation. The upper end is about 1/2 the maximum mass cloud in the Galaxy (Williams & McKee 1997). The range in radiation fields generously covers the expected elevated field from the sum of nearby stellar associations.

We show in Figure 3 (top) the calculated thermal pressures for a representative cloud of mass M(R_CO) = 1 × 10⁶ M_☉ with incident radiation fields G'₀ = ζ'_XR = 10. In this case, the two-phase thermal pressure (i.e., the thermal pressure due to radiative heating) dominates in the outer portion of the cloud, but initially drops due to absorption of EUV/soft X-rays and then due to absorption of FUV by dust. We note that the drop in two-phase pressure is greater than expected from Table 3 in Wolfire et al. (2003). They quote a decrease by a factor ∼1.8 for P^2p_th (given as P_ave in that paper) between N = 10¹⁹ cm⁻² and N = 10²⁰ cm⁻², whereas we find a drop by a factor of ∼3.7. The difference is due to the treatment of cosmic rays: here we do not scale the cosmic-ray ionization rate with G'₀ but hold it fixed at the local Galactic value. Thus, the ionization falls off faster with depth into the cloud than in the Wolfire et al. (2003) models. The turbulent thermal pressure dominates for A_V ≳ 0.001. The transition to H₂ (at $n_{{\rm H_2}}/n=0.25$) occurs at $A_V(R_{\rm H_2}) = 0.34$, the transition to optically thick CO (at τ_CO = 1) occurs at A_V(R_CO) = 1.0, and the dark-gas fraction is f_DG = 0.28.

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The calculated densities are shown in Figure 3 (bottom). We denote the clump density that would occur in two-phase equilibrium in the absence of turbulence by n_2p. For the actual clump density, we take

$$\begin{equation} n_c=\max (n_{\rm 2p},\,\langle n\rangle _{\rm med}), \end{equation} \tag{ 19 }$$

where the turbulent mass-weighted median density, 〈n〉_med, is given by Equation (14). For the case in the figure, we have n_c = n_2p for A_V ≲ 0.001 and n_c = 〈n〉_med for A_V ≳ 0.001. Both H₂ and CO form well within the turbulence-dominated region, a result that holds for all cloud masses and radiation fields considered here.⁶ Note that the volume filling factor $f_{\rm V}\equiv \bar{n}/n_c \sim 0.1$ in the outer part of the cloud where the thermal pressure dominates and two phases exist. Here, WNM fills the rest of the volume with a density of ∼10⁻²n_c. This implies the mass of the interclump medium is negligible (∼0.1) compared to the mass in clumps, as we have assumed. In the turbulence-dominated region, there is no such thing as an interclump medium, but rather a distribution of densities that fill the volume. Here, most of the mass is at densities near 〈n〉_med, as assumed.

Figure 4 shows the temperature distribution (top) and the distribution of abundances for C⁺, C⁰, CO, and OH (bottom). The CO amounts to only ∼30% of the total carbon abundance when the J = 1–0 line becomes optically thick at a CO column density of N(CO) ≈ 2 × 10¹⁶ cm⁻². Note that we do not include freeze out of H₂O on grains, which would be important at A_V ≳ 3 (Hollenbach et al. 2009), so our OH chemistry becomes somewhat unreliable at these depths. However, this happens after CO is formed and will not change our results.

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For comparison with the M(R_CO) = 1 × 10⁶ M_☉ case, we also show in Figures 5 and 6 the results for M(R_CO) = 1 × 10⁵ M_☉. Figure 5 shows the thermal pressures and densities, and Figure 6 shows the gas temperature and abundances. The lower cloud mass results in higher $\bar{n}$ (Equation (4)) and thus higher clump density (Equation (14)) and higher turbulent thermal pressure. The turbulent thermal pressure dominates the two-phase thermal pressure at all A_V. We find $A_V(R_{\rm H_2}) = 0.23$, A_V(R_CO) = 0.95, and the dark-gas fraction f_DG = 0.30.

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The optical depths $A_V(R_{\rm H_2})$ and A_V(R_CO) are presented in Figure 7 (top) for cloud masses M(R_CO) = 1 × 10⁵M_☉ and M(R_CO) = 3 × 10⁶ M_☉ and for G'₀ = ζ'_XR = 3, 10, and 30. The optical depths are measured from the cloud surface inward. Figure 7 (bottom) shows the cloud radii (measured from the cloud center) corresponding to $A_V(R_{\rm H_2}$), A_V(R_CO), and the total cloud radius. First, we see that the optical depths to both the H₂ and CO layers increase with increasing radiation fields (i.e., the transition layers are at greater column densities from the cloud surface). In addition, higher cloud masses result in (slightly) greater optical depths to the transition layers. However, Figure 7 (top) shows ΔA_V, DG to be nearly constant at ∼0.6–0.8 over our entire parameter range.

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The cloud radii are shown in Figure 7 (bottom). The radius in CO, R_CO is fixed from observations by the cloud mass and column density, (Equation (2)) and is independent of the incident radiation field. The next largest radius, $R_{\rm H_2}$, is also found to be relatively constant with radiation field, reflecting the constancy of ΔA_V, DG (Figure 7, top). The largest radius, R_tot, which encompasses the atomic surface as well as the molecular interior, is found to slightly increase with increasing radiation field, a consequence of the higher column densities required to turn the gas molecular. We also show in Figure 8 the ratio of radii R_tot/R_CO and $R_{\rm H_2}/R_{{\rm CO}}$ for cloud masses M(R_CO) = 1 × 10⁵ M_☉ and M(R_CO) = 3 × 10⁶ M_☉.

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In Figure 9, we present the calculated dark-gas fraction f_DG (see Equation (1)) for all of the cloud masses and incident radiation fields. We find that the fraction is remarkably constant with both cloud mass and radiation field at a value of f_DG ≈ 0.3. As shown in Equation (10), the constant fraction directly follows from the constant optical depth between the H₂ and CO layers, and the constant Z' and $\bar{N}$ (or $\bar{A}_V$) assumed in this case.

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The trends in $A_V(R_{\rm H_2})$, A_V(R_CO), ΔA_V, DG, and f_DG can be understood from the formation/destruction processes for H₂ and CO (Appendices B and C) and the relation between f_DG and ΔA_V, DG (Equation (10)). In Appendix B, we provide an analytic treatment of $A_V(R_{\rm H_2})$ by balancing the formation of H₂ on grains with FUV photodissociation of H₂. There we assume that the density is uniform, and since most of the mass of the cloud is in the clumps or, in the turbulent gas, at density 〈n〉_med, we use the density n_c. Strictly speaking, this density varies somewhat from R_tot to $R_{\rm H_2}$, but since the solution heavily depends on what happens at $R_{\rm H_2}$, we use the value of n_c there. The dependence on metallicity Z' ≡ Z/Z_☉, FUV field, and density n_c is given by Equation (B9):

$$\begin{equation} A_V(R_{\rm H_2}) \simeq 0.142 \ln \left[ 5.2 \times 10^3 Z^{\prime } \left({G_0^{\prime } \over {Z^{\prime } n_c}}\right)^{1.75}+1\right]. \end{equation} \tag{ 20 }$$

Here, n_c is in units of cm⁻³. We find that the column density to the transition region is a function of G'₀/n_c consistent with previous investigations (e.g., Sternberg 1988; Hollenbach & Tielens 1999; Wolfire et al. 2008; Krumholz et al. 2008; McKee & Krumholz 2010).

In Appendix C, we provide an analytic treatment of A_V(R_CO) by balancing the formation of CO by gas-phase chemistry with FUV photodissociation of CO. The dependence on G'₀, n_c, and Z' is given by Equation (C7):

$$\begin{equation} A_V(R_{{\rm CO}}) \simeq 0.102 \ln \left[ 3.3\times 10^7 \left(G_0^{\prime }\over Z^{\prime }n_c \right)^2 + 1\right], \end{equation} \tag{ 21 }$$

where we have substituted the numerical values for the constants into the equation and used T ∼ 50 K from Figure 4. Taking the difference of Equations (21) and (20), we find

$$\begin{equation} \Delta A_{V,\,\rm DG}= 0.53 - 0.045\ln \left(G_0^{\prime }\over n_c \right) - 0.097\ln \, (Z^{\prime }), \end{equation} \tag{ 22 }$$

which leads to

$$\begin{equation} \Delta A_{V,\,\rm DG}= 0.53 - 0.045\ln \left(G_0^{\prime }\over n_c \right), \end{equation} \tag{ 23 }$$

for Z' = 1. Here, we have assumed G'₀/n_c > 0.0075Z'^0.43 cm³, as is usually the case, so that one can ignore the factor of unity term inside the brackets of Equations (20) and (21). As noted in Appendices B and C, the analytic solutions for A_V(H₂) and A_V(CO) (for Z' = 1) agree with the numerical solutions to within 5% and 15%, respectively. The analytic solution for ΔA_V, DG (for Z' = 1) agrees to within 25% of the model results (Figure 7).

The principal conclusion of this analysis is that ΔA_V, DG is almost constant (Equation (23)), consistent with our numerical results in Figure 7. Equation (10) shows that the dark-gas fraction, f_DG, is a function of only ΔA_V, DG, which is nearly constant, and of $Z^{\prime }\bar{N}$ or $\bar{A}_V$. For conditions typical of the Galaxy—$\Delta A_{V,\,\rm DG}= 0.6\hbox{--}0.8$ (Figure 7), Z' = 1, and $\bar{N}=1.5\times 10^{22}$ cm⁻²(or $\bar{A}_V \simeq 8$)—we find f_DG = 0.26–0.33, in good agreement with the numerical results portrayed in Figure 9. We now discuss the reasons behind these results.

5.2.1. Insensitivity to CO Cloud Mass, M(R_CO)

5.2.2. Insensitivity to Ambient Radiation Field, G'₀

5.2.3. Dependence on Cloud Column Density, $\bar{N}$

Equation (10) shows that the dark-gas fraction is sensitive to the mean column density of the cloud. Heyer et al. (2009) have argued that for Galactic molecular clouds, the mean column density within R_CO is about half the value we adopted from Solomon et al. (1987), or $\bar{N}_{22} = 1.5/2 = 0.75$. As Heyer et al. (2009) point out, this value is quite approximate, since it depends on an uncertain correction for non-LTE effects in the excitation of ¹³CO. Bolatto et al. (2008) has found the same values for the mean column density ($\bar{N}_{22} = 0.75$) and turbulent velocity in the SMC as Heyer adopts for the Galactic clouds.

To investigate the effects of variations in the mean column density, we have also run $\bar{N}_{22} = 0.75$ models, including the $\bar{N}^{1/2}$ scaling for σ(r) suggested by Heyer et al. (2009; Equation (18)). Figure 10 (bottom) shows the low column density ($\bar{N}_{22} = 0.75$) results for M(R_CO) = 1 × 10⁶ M_☉ and Z' = 0.5, 1, and 1.9. We find higher dark-gas fractions in the $\bar{N}_{22} = 0.75$ models compared to the $\bar{N}_{22} = 1.5$ models; in particular, for Z' = 1, $\bar{N}_{22} = 0.75$, and G'₀ = 10, the dark-gas fraction f_DG ∼ 0.5, or $M(R_{\rm H_2})/M(R_{{\rm CO}})\sim 2$. The change in f_DG is almost entirely due to the change in mean column density; the change in σ(r) accounts for only a few percent of the increase. There is currently no evidence for such high values of dark-gas mass in the local Galaxy for the high mass clouds that we model here and we favor the higher, $\bar{N}_{22} = 1.5$, value. We note that Grenier et al. (2005) find higher dark mass fractions in local clouds with very low masses (M(R_CO) < 3 × 10⁴ M_☉). However, these clouds when observed with high spatial resolution CO observations reveal quite small mean column densities, $\bar{N}_{22} \lesssim 0.2$ (Mizuno et al. 2001; Yamamoto et al. 2006), where we expect the dark-gas fraction to be higher than f_DG ∼ 0.3.

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5.2.4. Dependence on Metallicity, Z'

5.2.5. Dependence on Visual Extinction Through the Cloud, $\bar{A}_V$

Equation (10) shows that f_DG depends only on $\Delta A_{V,\,\rm DG}/ \bar{A}_V$, where $\bar{A}_V\equiv 5.26 Z^{\prime } \bar{N}_{22}$ is the mean visual extinction through the cloud. As discussed above, ΔA_V, DG is only weakly dependent on Z', $\bar{N}_{22}$, G₀, and n_c. Therefore, the main parameter that controls the dark-gas fraction is the visual extinction through the cloud, $\bar{A}_V$. Figure 11 clearly shows this dependence; it plots f_DG against $\bar{A}_V$ for two different values of Z' = 0.5 and 1.9, holding G'₀ fixed at 10 and adjusting the gas density (see Equations (4), (14), (15), and (18)) as $\bar{N}$ changes. Although, at fixed $\bar{A}_V$, the Z' = 0.5 case does have a slightly higher value of f_DG than the Z' = 1.9 case,⁷ the main parameter controlling f_DG is $\bar{A}_V$. If we fix $\bar{N}$, lowering Z' lowers the $\bar{A}_V$ through the cloud, and this is the cause of the large change in f_DG seen in Figure 10. Essentially, lowering $\bar{A}_V$ raises the mass of the dark gas located in the surface region relative to the mass of the interior CO gas, which is fixed in our model. For $\bar{A}_V=2$, there is considerable shielded H₂ gas out to large radius around the fixed mass CO "core." For $\bar{A}_V=30$, the CO "core" takes up much of the cloud, and the dark gas is but a thin shell on the surface.

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We have fixed the mass in the CO region in our models because that is what is generally observed. However, to better understand the dependence of f_DG on $\bar{A}_V$, it is easier to consider the alternate case of a cloud of fixed total mass (i.e., H i + H₂ + CO mass) and variable Z'. As Z' is lowered, the $\bar{A}_V$ through the cloud is lowered. The radius of the CO region, R_CO, shrinks because of the reduced shielding in the cloud. The mass of the CO region drops as R²_CO, since we have assumed that $\bar{n} \propto r^{-1}$. However, the ΔA_V, DG is relatively constant through the dark-gas shell around the CO, so the column through the shell rises as Z'⁻¹. The mass in the H₂ dark gas, $M(R_{\rm H_2}) - M(R_{{\rm CO}})$, changes as R²_eff times the column in the shell, where $R_{\rm CO} < R_{\rm eff}< R_{\rm H_2}$. Taking R_eff = R_CO to obtain the minimum mass in the dark gas, we see that the ratio of dark-gas mass to mass in the CO region goes as Z'⁻¹ in this case. Therefore, f_DG increases as $\bar{A}_V$ decreases (Z' decreases) in this example.

5.2.6. Variation with Cosmic Rays

First, we note that the dark-gas fraction f_DG = 0.28 for our standard model is in good agreement with the observations of Grenier et al. (2005), who found f_DG ≈ 0.3, for the four local clouds in their sample that are more massive than M(R_CO)>3 × 10⁴ M_☉.⁸ We note that there is considerable scatter in their dark mass fraction for these four clouds, which may indicate scatter in the average extinctions through the clouds, but that the average value of the dark mass fraction is close to our theoretical prediction. In addition, Abdo et al. (2010) from observations of nearby resolved clouds find an average value of f_DG ≈ 0.34 for their two massive clouds. Grenier et al. (2005) and Abdo et al. (2010) find f_DG higher than 0.3 in very low mass clouds (M(R_CO) < 3 × 10⁴ M_☉) where the mean extinction is observed to be much less than our standard value and a higher f_DG is consistent with our prediction.

Next, we compare our numerical results with the observations of H i halos around molecular clouds in the Galaxy. Wannier et al. (1983) observed H i halos around eight molecular clouds and concluded they extend several parsecs beyond the CO and have a thickness of a few parsecs. In addition, the halos are "warm," having temperatures of at least 50–200 K in order to be seen in emission over the background. We note that the use of "warm" in their title has been interpreted by others to mean gas at temperatures of ∼8000 K, but in fact they cite temperatures of order a few 100 K.

Andersson et al. (1991) and Andersson & Wannier (1993) included additional data and analysis and report H i integrated intensities between 700 K km s⁻¹ and 4300 K km s⁻¹ and characteristic depths of about 4.7 pc. Based on model fits, they estimate H i densities $n_{{\rm H}\,\hbox{\scriptsize\sc i}} \sim 25\hbox{--}125$ cm⁻³ and temperatures T ∼ 50–200 K. We note that they suggested that either the formation rate of H₂ was lower than the standard rate or the FUV field was about 10 times the interstellar value in order to simultaneously match the CO and H i observations. At that time they favored the lower H₂ formation rate, since an increased FUV field would produce too little OH compared to observation. However, subsequent work by Hollenbach et al. (2009) showed that OH is formed at greater abundances than previously thought when grain surface chemistry is included. Thus, their suggestion that the FUV field is enhanced near GMCs is consistent with a similar result by M. Wolfire et al. (2010, in preparation) in modeling the average FUV field illuminating a star-forming molecular cloud.

For cloud masses in the range M(R_CO) = (1–10) × 10⁵ M_☉ and for G'₀ = 10–30, we find H i integrated intensities from 965 K km s⁻¹ to 4100 K km s⁻¹ and a range in H i halo thickness from 1 pc to 10 pc. These are in good agreement with the observations. At $A_V(R_{\rm H_2})$, our temperature range T = 70–80 K and density range $\bar{n} = 45\hbox{--}147$ cm⁻³ agree with that of Andersson & Wannier (1993). Therefore, we consider the model to be in very good agreement with these observations.

We note that CO-to-H₂ conversion factors have been inferred from gamma-ray and far-infrared observations, and also from virial mass estimates. The gamma-ray (Strong & Mattox 1996) and far-infrared (Dame et al. 2001) estimates include the "dark gas" and are X = 1.9 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ and X = 1.8 × 10²⁰ (K km s⁻¹)⁻¹ from gamma-ray observations and far-infrared observations, respectively. However, these values are Galactic averages and observations of nearby resolved clouds show a variable X increasing toward the outer Galaxy (Abdo et al. 2010). The Abdo et al. observations separate the "dark gas" and CO components and generally show lower X ratios (X ∼ 0.87 ± 0.05 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹). Depending on whether the dark gas is mixed with other components, the X value could be lower if mixed with H i or raised if mixed with CO. The Solomon et al. (1987) conversion factor was based on a virial cloud estimate and for an idealized cloud would not include the "dark gas." McKee & Ostriker (2007) re-evaluated their conversion, accounting for an 8.5 kpc distance to the Galactic center and including the He mass, and found a revised Solomon et al. value of X = 1.9 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹. In general, the X ratios derived from virial mass estimates exceed those based on gamma-ray observations from resolved clouds by factors of 1.5–3.0 (Abdo et al. 2010). However, turbulent mixing of the bright CO and dark gas could easily cause the virial estimate to include some of the dark gas. Therefore, one should note that CO to H₂ conversion factors are sometimes normalized to implicitly include the dark mass, even though it lies outside the CO gas.

One of the important approximations in our work is that the density fluctuations in the molecular cloud—the clumps—are optically thin, or A_V < 1 through each clump. This approximation can now be justified in light of our results: the fact that the dark-gas fraction primarily depends on $\bar{A}_V$, the mean visual extinction through the entire cloud, but only weakly on the density and radiation field, implies that it is insensitive to the distribution of matter within the cloud. Consider an extreme example of a cloud with very opaque clumps, namely a cloud with clumps so opaque that their mean extinction is equal to that of the entire cloud, $\bar{A}_V\sim 8$. We assume that the cloud with opaque clumps has the same mean $\bar{A}_V$, and clump density, n_c, as the cloud with transparent clumps. For simplicity, we assume that there is no radial variation in the properties of the clumps. Let a be the clump radius. Then the number of clumps, ${\cal N}$, is determined by the condition that the clumps occupy a fraction f_V of the volume:

$$\begin{equation} {\cal N}a^3=f_{\rm V}R_{\rm tot}^3. \end{equation} \tag{ 24 }$$

The condition that the clumps have the same $\bar{A}_V$ as the cloud as a whole gives

$$\begin{equation} n_c a=\bar{n} R_{\rm tot}=f_{\rm V}n_cR_{\rm tot}, \end{equation} \tag{ 25 }$$

so that a = f_VR_tot. When viewed from the outside, the fraction of the cloud that is covered by clumps (the projected covering factor) is then

$$\begin{equation} C=\frac{{\cal N}\pi a^2}{\pi R_{\rm tot}^2}=1, \end{equation} \tag{ 26 }$$

as expected. From the perspective of a clump inside the cloud, this means that about half the sky is covered by clumps, since, on average, the path length from a clump to the surface of the cloud is half the path length through the entire cloud. The average radiation field incident on a clump is therefore reduced by about a factor of 2 compared to the radiation field incident on the cloud as a whole.

We can now estimate the dark-gas fraction of a cloud of these opaque clumps. Each clump will have a dark-gas fraction that is almost the same as that of the entire cloud, modified only by the reduction in G'₀ by about a factor of 2. Since the extinction of the dark-gas layer depends only weakly on the intensity of the radiation field, $\Delta A_{V,\,\rm DG}\propto 0.045\ln G_0^{\prime }$, this modification is very slight for typical values of $\bar{A}_V\sim 8$ (see Equation (10)). This argument does not prove that clumps with $\bar{A}_V$ intermediate between the large value ∼8 assumed here and the small values $\bar{A}_V < 1$ assumed in the rest of this work will also have the same dark-gas fraction. However, because this range $1 < \bar{A}_V < 8$ is small, it strongly suggests that this is the case. We conclude that inclusion of finite optical depth of clumps in a cloud is unlikely to significantly alter our conclusions.

Our models solve for the steady state abundances of the atomic and molecular species. In this subsection, we examine the assumption of steady state chemistry in the turbulent molecular cloud surfaces by comparing the chemical and dynamical times at the cloud depth $A(R_{\rm H_2})$. The chemical timescale, t_chem, is the time for atomic gas to become completely molecular ($n_{\rm H_2} = 0.5 \langle n\rangle _{\rm med}$), thus $t_{\rm chem}= 0.5/({\cal R} \langle n\rangle _{\rm med})$ where ${\cal R}=3\times 10^{-17}Z^{\prime }$ cm³ s⁻¹ is the rate coefficient for H₂ formation on dust grains (see Appendix B). To compute the dynamical timescale, t_dyn, we require the characteristic distance for the turbulence to bring molecular gas from the interior to the surface, and to bring atomic gas from the surface to the interior, where the interior is A_V greater than $A_V(R_{\rm H_2})$. This characteristic distance is $d_{\rm dyn} = 1.9\times 10^{21} A_V(R_{\rm H_2})/\bar{n}$. The dynamical timescale is this distance divided by the turbulent velocity for lengthscale d_dyn, t_dyn = d_dyn/σ(d_dyn) where σ is the 1D turbulent velocity dispersion (Equation (18)).

For a cloud mass M(R_CO) = 10⁵ M_☉ and Z' = 1, we find $\bar{n}\approx 140\;$cm⁻³ and 〈n〉 ≈ 600 cm⁻³ at $A_V(R_{\rm H_2}) = 0.22$ and thus t_chem ∼ 2.8 × 10¹³ s, d_dyn ∼ 1.0 pc, and t_dyn ∼ 4.3 × 10¹³ s. Comparing chemical and dynamical timescales, we find t_chem/t_dyn = 0.65. Similarly, for M(R_CO) = 10⁶ M_☉, we find t_chem/t_dyn = 0.48. For t_chem/t_dyn ≲ 1, a steady state approximation is valid since gas has time to reach steady state before significant turbulent mixing. We find ratios less than one, but only marginally so, and thus we expect some affects from turbulence. We note however that turbulence brings atomic gas inward past $A_V(R_{\rm H_2})$ into the molecular gas, and at the same time it brings molecular gas to the surface. Thus, the first-order effect is to spread out the rise of $x_{\rm H_2}$, but not to move $A_V(R_{\rm H_2})$. In addition, we have demonstrated in Section 5.3 that the steady state model is in good agreement with observations, and thus the effects of turbulence are modest.

We have constructed models of molecular clouds to investigate the fraction of gas that is mainly H₂ and contains little CO. These conditions exist where the CO is photodissociated into C and C⁺ but the gas is molecular H₂ due to either H₂ self-shielding or dust-shielding. Such conditions can exist either on the surfaces of molecular clouds or the surfaces of clumps contained within such clouds. Observations indicate that the mass in this "dark gas" can be as high as 30% of the total molecular mass in the local Galaxy. Previous theoretical plane-parallel models of individual PDRs have indicated that such a layer should exist (e.g., van Dishoeck & Black 1988), but here we construct models directed toward molecular clouds as a whole while including observational constraints on cloud mass, radius, average density, and linewidth and theoretical considerations of the likely strengths of the UV fields impinging on GMCs.

We assume that the surface of each cloud is isotropically illuminated over 2π steradians by a soft X-ray/EUV field and an FUV radiation field. We use the standard cosmic-ray ionization rate of 1.8 × 10⁻¹⁷ s⁻¹ per hydrogen nucleus everywhere in the cloud for all cases but one test case. There is evidence from observations of H⁺₃ that the cosmic-ray ionization rate is a factor of 10 higher in some portions of the diffuse ISM (Indriolo et al. 2007), but there is no indication that such rates apply in molecular cloud interiors (McCall et al. 1999). In preliminary work of M. Wolfire et al. (2010, in preparation), we find that the average FUV field on clouds is ∼20 times the local Galactic interstellar field. This elevated field arises from the distribution of OB associations around the cloud.

The distribution of temperature, density, and abundances within the H i, H₂, and CO layers are calculated using the PDR code of Kaufman et al. (2006). In constructing model clouds from the PDR output, we impose the constraint from Solomon et al. (1987) that the typical column density through the cloud is $\bar{N}_{22}= 1.5$, independent of cloud mass and independent of radius inside a given cloud (since $\bar{n} \propto r^{-1}$). The locally averaged density $\bar{n}$ and radius of the CO cloud, R_CO, as functions of the CO cloud mass, M(R_CO) follow from Equations (2) and (4). In our notation, M(R_CO) is the molecular mass contained within the τ_CO = 1 surface (of the CO J = 1–0 transition) including the mass of H₂ and He, and R_CO is the radius of this CO photosphere. We impose a volume-averaged density distribution $\bar{n}(r)$ that behaves as $\bar{n}\propto 1/r$ throughout all layers of the cloud. Note that in regions where the thermal pressure P_th lies between P_min and P_max both warm (T ≳ 7000 K) and cold (T ≲ 500 K) gas solutions are possible. In this regime, we use the cold solution from our model results with n_c, the density of the cold clumps, and $\bar{n}$, the average over cold and warm gas components. The warm gas fills the volume but contains little of the mass.

We test two extreme limits for clumps within the cloud, one in which all clumps are optically thin to the incident FUV field, and one in which all clumps are optically thick to the incident FUV field. In the optically thin approximation, the PDR model output as a function of A_V directly gives the distribution in A_V throughout the cloud. In the optically thin limit, we test two different models for the cloud density distribution. First, we use constant thermal pressure models; and second, we include two sources of thermal pressure, radiative heating (which by itself would lead to a two-phase equilibrium), and supersonic turbulence. The distribution of two-phase thermal pressure is calculated as in Wolfire et al. (2003) and stored in a look-up table as a function of total column density and molecular fraction (see Figure 2). This pressure drops as one moves into the cloud due to the absorption of the radiation responsible for heating the gas. Supersonic turbulence is characterized by a mass-weighted median density $\langle n \rangle _{\rm med}=\bar{n}\exp (\mu)$, with $\mu = 0.5\ln (1\,{+}\,0.25{\cal M}^2)$ (Padoan et al. 1997). When the two-phase pressure drops below P^turb_th ≡ x_t〈n〉_medkT, where x_t is the sum over the abundances of all species relative to hydrogen nuclei, we assume that the turbulence maintains the gas at a thermal pressure P^turb_th. The sound speed that enters in the Mach number is calculated from the PDR model output, while the turbulent velocity is given by the linewidth–size relation (Equation (18)).

In the limit of very optically thick clumps (with optical depths at least as large as the average optical depth through the cloud), we find that the average radiation field on a clump ranges between about G'₀/2 and G'₀. In this limit, the dark mass fraction of a clump is the same as the dark mass fraction of the entire cloud. Since the dark-gas mass fraction of a spherical clump or a spherical cloud is insensitive to the incident FUV field (see Equations (10) and (22)), the dark mass fraction of the cloud does not significantly change compared with the case of a cloud made up of optically thin clumps.

We have tested the steady state assumption for the chemistry by comparing the time to form molecular hydrogen, t_chem, with the dynamical time, t_dyn, for turbulence to bring molecular gas from the interior to the surface and to bring atomic gas to the interior. We find t_chem/t_dyn ∼ 0.7 for a cloud mass of M(R_CO) = 1 × 10⁵ M_☉ and t_chem/t_dyn ∼ 0.5 for a cloud mass of M(R_CO) = 1 × 10⁶ M_☉. Thus, we expect modest affects due to turbulence; mainly to spread out the transition from atomic to molecular hydrogen but not to move $A_V(R_{\rm H_2})$. We also note that steady state models agree well with observations.

For our standard cloud mass, M(R_CO) = 1 × 10⁶ M_☉, and incident radiation field, G'₀ = ζ'_XR = 10, we find dark-gas mass fractions of f_DG = 0.28 and f_DG = 0.31 for constant thermal pressures of P_th = 10⁵ K cm⁻³ and P_th = 10⁶ K cm⁻³, respectively. These correspond to a total molecular mass $M(R_{\rm H_2}) \approx 1.4 M(R_{{\rm CO}})$. For models that include both thermal and turbulent pressures, we find essentially the same results as for the cases with constant thermal pressure. The variation in f_DG ranges from 0.25 to 0.33 over a range in G'₀ from 3 to 30 and a range in cloud mass from 10⁵ M_☉ to 3 × 10⁶ M_☉ (Figure 9). Figure 3 shows the distribution in thermal pressures and densities; and Figure 4 shows the distribution in temperatures and chemical abundances for the standard model.

The constant value of f_DG for fixed $\bar{A}_V$ can be understood from the analytic solutions for $A_V(R_{\rm H_2})$ and A_V(R_CO) in Appendices B and C and the expression for f_DG in Equation (10). In the limit of G'₀/n > 0.0075Z'^0.43 cm³, both $A_V(R_{\rm H_2})$ and A_V(R_CO) increase as ln(G'₀/n). Thus, the optical depth through the H₂ layer ΔA_V, DG is a weak function of G'₀/n and Z' (Equation (22)) and is nearly constant over our parameter space. Furthermore, we find that f_DG is a function of only ΔA_V, DG and the mean extinction through the cloud $\bar{A}_V \equiv 5.26 Z^{\prime } \bar{N}_{22}$ (Equation (10)); thus, for a given $\bar{A}_V$, the dark-gas fraction is constant. However, f_DG significantly increases if $\bar{A}_V$ decreases.

Our numerical results compare well with observations of the local Galactic dark-gas fraction, which Grenier et al. (2005) find to be f_DG ∼ 0.3, when averaged over their four most massive clouds with masses between 3 × 10⁴and3 × 10⁵ M_☉. Lower mass clouds observed by Grenier et al. (2005) are observed to have low $\bar{A}_V$ and thus high dark-gas fractions consistent with our prediction. Our H i integrated intensities of 965 K km s⁻¹ to 4100 s⁻¹, H i cloud-halo thickness of 1–10 pc, average cloud densities of $\bar{n} \sim 45\hbox{--}150$ cm⁻³, and average H i temperatures of ∼70–80 K are in good agreement with the observations of H i cloud halos observed by Wannier et al. (1983), Andersson et al. (1991), and Andersson & Wannier (1993).

We have carried out several additional tests to assess the dependence of our results on the cosmic-ray ionization rate, the clump optical depth, the mean cloud column density, the metallicity, and the mean visual extinction through the cloud. We ran our standard model with the cosmic-ray ionization rate a factor of 10 higher at A_V < 2, as suggested by H⁺₃ observations in some regions of the diffuse ISM, and found that this enhanced cosmic-ray rate only slightly increases the dark-gas fraction. In the limit of very optically thick clumps, we find no significant change in f_DG.

Heyer et al. (2009) have suggested that the mean column density (and therefore the mean visual extinction $\bar{A}_V$) in Galactic CO clouds is about half the value found by Solomon et al. (1987), i.e., $\bar{N}_{22} \simeq 0.75$ or $\bar{A}_V \simeq 4$. The results of changing the mean column density, the metallicity, and the mean visual extinction through the cloud are illustrated in Figures 10 and 11. We find that the dark-gas fraction increases with lower extinction since the dark gas occupies a larger fraction of the cloud (see Equation (10)). There is also a weak dependence of ΔA_V, DG on the mean column density that tends to slightly mitigate the dominant effect. Lower mean columns lead to lower mean densities (Equation (4)) and lower n_c (Equations (14) and (19)), and thus lower ΔA_V, DG (Equation (22)). We examine metallicities appropriate for the LMC (Z' = 0.5), for the local Galaxy (Z' = 1), and for the molecular ring at R = 4.5 kpc (Z' = 1.9). In general, f_DG increases as the metallicity drops for fixed columns because the mean extinction through the cloud decreases, which raises the ratio of the surface dark gas to the interior CO gas (again, see Equation (10)). There is also a weak dependence of ΔA_V, DG on Z' that also slightly increases ΔA_V, DG (or f_DG) with decreasing Z' (Equation (22)) even if the column is changed so that $\bar{A}_V$ remains fixed. We note that in the case of varying $\bar{N}$ and Z', but at constant $\bar{N}Z^{\prime }$ or $\bar{A}_V$ (see Figure 11), the change in f_DG is entirely due to the weak dependencies of ΔA_V, DG on $\bar{N}$ and Z' as noted above and shown in Equation (22). We also examine the case for dust scaling as Z'² (=0.25), while gas-phase metals scale as Z' (=0.5). We find f_DG is larger than when both metals and gas scale together.

In Appendices B and C, we derive analytic solutions for $A_V(R_{\rm H_2})$ and A_V(R_CO) as functions of density n, FUV field G'₀, and metallicity Z'. For the case of H₂, we find an expression for the abundance of H₂ by balancing formation and destruction processes and then solving for the position where $n_{{\rm H_2}} = 0.25 n$. Similar studies have been carried out by, for example, Sternberg (1988) and McKee & Krumholz (2010). We find our fits are good to ±5% for all models presented in this paper. The fits to CO are found by integrating the expression for the abundance of CO to a column density of N(CO) = 2 × 10¹⁶ cm⁻², where τ_CO = 1 (the optical depth of the CO J = 1–0 transition). The fits are generally good to within ±15% except for the lowest Z' and G'₀ model, where the fit is good to ±25%. We have also derived analytic expressions for ΔA_V, DG and f_DG in the main text (Equations (22) and (10), respectively).

The overall result of this paper is a theoretical derivation that f_DG ≃ 0.3 ± 0.08 for GMCs with $\bar{A}_V \simeq 8$ in our Galaxy. Therefore, a significant fraction of the molecular gas in our Galaxy lies outside the CO gas. As discussed in Section 2.4.2, some calibrations of the CO line intensities to H₂ mass take into account this "dark" H₂ gas. However, it is important to be aware of this component since it significantly contributes to the gamma ray, infrared/submillimeter continuum, and [C ii] 158 μm emission from clouds in galaxies. Its contribution to the star formation in a galaxy is as yet undetermined. The importance of this component increases as the metallicity decreases, such as in the outer regions of galaxies or in low-metallicity galaxies.

Although the gas in the C⁺/H₂ layer is termed "dark gas," it emits [C ii] 158 μm line emission and the dust in the dark layer emits infrared continuum. In a subsequent paper, we will estimate the emission from the "dark gas." In addition, a future paper will model in detail the lower metallicity clouds found, for example, in the SMC and early universe.