Annu. Rev. Astron. Astrophys. 2013. 51: 207-268 Copyright © 2013 by Annual Reviews. All rights reserved

3. A MODERN THEORETICAL PERSPECTIVE

In recent years there has been much progress in updating PDR models, and in combining hydrodynamic simulations, chemical modeling, and radiation transfer codes.

3.1. Photodissociation Regions

Since CO production mainly occurs through ion-neutral chemistry (van Dishoeck & Black 1988) the ionization structure through the PDR is important in setting the depth of the CO formation. Recombination of metal ions on PAHs can modify the abundance of free electrons. For a typical GMC with n ~ 10³ cm^-3 illuminated by a radiation field ~ 10 times the Galactic ISRF (the interstellar radiation field in the vicinity of the Sun), the _CO = 1 surface is at a depth of A_V ~ 1 when PAHs are included. Without PAHs the electron abundance stays high, and the ion-neutral chemistry is slowed due to H₃⁺ recombination. Consequently, the _CO = 1 surface is pushed deeper into the cloud (A_V ~ 2). The PAH rates are estimated by Wolfire et al. (2008) by considering the C⁰ / C⁺ ratio in diffuse lines of sight, but there is considerable uncertainty both in the rates (Liszt 2011) and in the PAH abundance and their variation with cloud depth.

Bell et al. (2006) carry out a parameter study of X_CO using PDR models with constant microturbulent line width. The calculated X_CO versus A_V plots have a characteristic shape with high X_CO at low A_V, dropping to a minimum value at A_V ~ 2-4 and then slowly rising for increasing A_V. The X_CO dependence at low A_V arises from molecular gas with low CO abundance. After the CO line intensity becomes optically thick, X_CO slowly rises again as N(H₂) increases. Increasing density up to the critical density of CO J = 1 → 0 (n(H₂)_cr,1 ~ 2200 cm^-3) enhances the CO excitation and causes X_CO to drop. In addition, the minimum moves closer to the cloud surface as does the _CO = 1 surface. An increase in X_CO by a factor of more than 100 is found when decreasing dust and metallicity to 1% of the local Galactic ISM, reflecting the larger column of H₂ at a given A_V. Bell, Viti & Williams (2007) suggest that the appropriate X_CO value to use in various extragalactic environments can be estimated from the minimum in the X_CO versus A_V plot.

Wolfire, Hollenbach & McKee (2010) use an updated version of their PDR models, including a self-consistent calculation for the median density expected from a turbulent density distribution. The models provide a theoretical basis for predicting the molecular mass fraction outside of the CO emitting region in terms of incident radiation field and gas metallicity. They assume that the median density is given by < n >_med = exp(µ), where is the volume averaged density distribution 1/r, and µ = 0.5ln(1 + 0.25 ²) (Padoan, Jones & Nordlund 1997). The sound speed that enters in the Mach number, , is calculated from the PDR model output while the turbulent velocity is given by the size-linewidth relation (Eq. 9).

3.2. Numerical Simulations

As an alternative to PDR models with simple geometries and densities, hydrodynamical models can be used to calculate the line width, and density, but with only limited spatial resolution and approximate chemistry and thermal balance. Glover & Mac Low ( 2007, 2007b) carry out simulations of the formation of molecular clouds using a modified version of the magnetohydrodynamical code ZEUS-MP. They include a time-dependent chemistry, in particular for H₂ formation and destruction, and thermal balance. Glover et al. (2010) enhance the code to account for CO chemistry in a turbulent GMC. A turbulent picture of a GMC is not one with well defined clumps surrounded by an interclump medium but one with a continuous density distribution and constant mixing between low and high density regions. The CO abundance varies within the cloud depending on the gas density and penetration of the dissociating radiation field. Calculations of X_CO are carried out by Glover & Mac Low (2011), and Shetty et al. (2011, 2011b). The latter two references include non-LTE CO excitation and line transfer in the LVG approximation. These authors carry out 3D turbulent simulations in a box of fixed size (20 pc), and turbulence generated with uniform power between wavenumbers 1 ≤ k ≤ 2, but with various initial densities and metallicities. For their standard cloud model the saturation amplitude of the 1D velocity dispersion is 2.4 km s^-1. Each line of sight has a different CO intensity, and CO and H₂ column density depending on the past and present physical conditions along it. Thus, for a given N(H₂) there is a range in X_CO values (see Figs. 5 and 6 in Shetty et al. 2011). The dispersion generally increases for clouds with lower density and lower metallicity.

Figure 7 in Shetty et al. (2011) shows the calculated mean X_CO in different A_V bins for several initial densities and metallicities. The variation in X_CO with A_V is qualitatively similar to that shown in Bell et al. (2006) for microturbulent models. Lower densities and metallicities drive X_CO to higher values due to lower CO excitation and CO/H₂ ratios respectively. At A_V 7 the models with different initial densities converge as the CO line becomes optically thick. The minimum in X_CO occurs at larger A_V in the hydrodynamic simulations compared to the one sided PDR models, likely due to the dissociating radiation incident on all sides of the box and due to its greater penetration along low density lines of sight. A range in density n = 100-1000 cm^-3 and metallicity Z = 0.1-1 Z produces a range of only X_CO,20 ~ 2-10. Thus although there can be large variations in X_CO along different lines of sight in the cloud the emission weighted X_CO is close to the typical Galactic value (Section 4). In addition, the low metallicity case has higher X_CO at low A_V but approaches the Galactic value for higher density (higher A_V) lines of sight. Although the models were run with constant box size, the dependence on A_V suggests that clouds with sufficiently small size so that CO does not become optically thick will have higher X_CO. The results seem to confirm the suggestion by Bell et al. (2006) that the mean X_CO value should be near the minimum when plotted as X_CO versus A_V.

Shetty et al. (2011b) investigate the results of varying the temperature, CO abundance, and turbulent line width. They find only a weak dependence on temperature with X_CO T_kin^-0.5 for 20 K < T_kin < 100 K, and thus over the range of temperatures typically found in Galactic clouds X_CO is not expected to vary significantly due to temperature. At low (fixed) CO abundance (n_CO / n_H2 ~ 10^-6) , the line does not become optically thick and thus W(CO) follows N(H₂) with constant X_CO up to at least logN(H₂) = 22.5. Varying the turbulent line width increases the CO line intensity as expected since decreasing the self-absorption allows more CO line emission to escape the cloud. They also find, however, a decreasing brightness temperature and over the range of velocity dispersion = 2-20 km s^-1, W(CO) changes as W(CO) ^0.5 instead of changing linearly. The higher turbulent velocities create dense shocks but also larger voids of low density material. The competing effects produce a dependence on that is slower than linear. Increasing the column density as well as the line width might produce a model result closer to W(CO) .

An interesting result is that X_CO is not sensitive to the internal velocity profile within the cloud but only to the total line width, however that might be generated. A cloud need not obey a power law size-linewidth relation to have the same X_CO as one with a Gaussian distribution and identical dispersion velocity. Thus, clouds need not be "virialized" nor obey a size-linewidth relation to have the same X_CO. The modeled X_CO converges for high A_V clouds and thus the X_CO factor is not expected to vary from cloud to cloud as long as the there is a large enough column density. Shetty et al. (2011b) conclude that a nearly constant X_CO is the result of the limited range in column densities, temperatures, and linewidths found in Galactic molecular clouds and that applying a constant X_CO is approximately correct to within a factor of ~ 2.

Global models of X_CO using hydrodynamic simulations to investigate the variation due to galactic environment are carried out by Feldmann, Gnedin & Kravtsov (2012) and Narayanan et al. ( 2011, 2012). A significant problem in numerical simulations is how to handle the physical conditions and/or line emission within regions smaller than the spatial grid. Feldmann, Gnedin & Kravtsov (2012) use high resolution (~ 0.1 pc) simulations from Glover & Mac Low (2011) for "sub-grid" solutions to cosmological simulations of 60 pc resolution. These models used constant T_kin = 10 K, LTE excitation for CO, with either constant CO line width or one proportional to _mol^0.5. By comparing results at their highest resolutions with those at 1 kpc, and 4 kpc, they asses the effects of spatial averaging on X_CO. The averaging tends to reduce the variation of X_CO on N(H₂) and UV radiation field intensity. At greater than kiloparsec scales, and H₂ column densities between 10²¹ cm^-2 and 10²³ cm^-2, X_CO changes by only a factor of 2. They find essentially no variation in X_CO with UV field strength between 0.1 and 100 times the Galactic ISRF. Feldmann, Gnedin & Kravtsov (2012) do find a significant variation with metallicity, with a scaling X_CO Z^-0.5 for virial line widths. Note that this relation is much shallower than found by, for example, (Genzel et al. 2012, Section 8.2). The authors suggest that low metallicity high-redshift galaxies may not obey the same gas surface-density to star formation relation observed in local disks.

The results of Narayanan et al. (2011) and Narayanan et al. (2012) will be discussed in Section 7.2. Here we note that their effective resolution of ~ 70 pc requires adopting sub-grid cloud properties. Although the surfaces of GMCs more massive than ~ 7× 10⁵ M are resolved, their internal structure is not. The resulting line and continuum transfer, and temperature and chemical structure can only be approximate. The resolution problem also enters in simulations of individual clouds. We showed in Section 1 that CO becomes optically thick within a column N(H₂) 2-3 × 10²⁰ cm^-2. Thus for sub parsec resolution ( ~ 0.1 pc) and density greater than n 10³ cm^-3, the physical conditions (temperature, density, and abundances) are averaged over the line forming region and the calculated emitted intensity can be in error. The resolution problem is much more severe for H₂ dissociation. Self-shielding of H₂ starts within a column of only N(H₂) ~ 10¹⁴ cm^-2 (de Jong, Boland & Dalgarno 1980). Thus with a resolution of ~ 0.1 pc, and density of n 10 cm^-3, the optical depth to dissociating radiation is already 10⁴ in a resolution element. A complementary approach is to apply a state-of-the-art PDR code with well resolved H₂ formation to the output of hydrodynamic simulations Levrier et al. 2012. With increasing computing power the issues of resolution will continue to improve. We suggest that high priority should be placed on creating large scale galactic simulations that are well matched to small scale simulations with resolved cloud structure. The former provide environmental conditions and cloud boundary conditions while the latter provide the chemistry and line emission in a realistic turbulent cloud. Expanding the library of GMC models with a range of column densities, line widths, and external heating, and thoroughly checking them against observations, would be most helpful.