ARlogo Annu. Rev. Astron. Astrophys. 2013. 51: 207-268
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The Galaxy is the only source where it is possible to determine the CO-to-H2 conversion factor in a variety of ways. It thus provides the prime laboratory to investigate the calibration and the variations of the proportionality between CO emission and molecular mass.

In the following sections we will discuss three types of XCO determinations: 1) employing virial masses, a technique that requires the ability to spatially resolve molecular clouds to measure their sizes and kinematics, 2) taking advantage of optically thin tracers of column density, such as dust or certain molecular and atomic lines, and 3) using the diffuse gamma-ray emission arising from the pion production process that takes place when cosmic rays interact with interstellar medium protons. Gamma-ray techniques are severely limited by sensitivity, and are only applicable to the Milky Way and the Magellanic Clouds. The good level of agreement between these approaches in our own galaxy is the foundation of the use of the CO-to-H2 conversion factor in other galaxies.

4.1. XCO Based on Virial Techniques

The application of the virial theorem to molecular clouds has been discussed by a number of authors, and recently reviewed by McKee & Ostriker (2007). Here we just briefly summarize the fundamental points. The virial theorem can be expressed in the Lagrangian (fixed mass) or Eulerian (fixed volume) forms, the latter particularly applicable to turbulent clouds where mass is constantly exchanged with the surrounding medium. In the somewhat simpler Lagrangian form, the virial equilibrium equation is

Equation 20 (20)

where K is the volume integral of the thermal plus kinetic energy, Ks is the surface pressure term, B is the net magnetic energy including volume and surface terms (which cancel for a completely uniform magnetic field), and W is the net gravitational energy which is determined by the self-generated gravitational potential if the acceleration due to mass external to the cloud can be neglected. In the simple case of a uniform, unmagnetized sphere virial equilibrium implies 2K + W = 0. It is useful to define the virial parameter, avir, which corresponds to the ratio of total kinetic energy to gravitational energy (Bertoldi & McKee 1992), so that avir ident 5 Rsigma2 / GM.

4.1.1. Are Clouds Virialized?

In this context, gravitationally bound objects have avir ≃ 1. Whether interstellar clouds are entities in virial equilibrium, even in a time or ensemble averaged sense (McKee 1999), is a matter of current debate. Observational evidence can be interpreted in terms of systems out of equilibrium with rapid star formation and subsequent disruption in a few Myr (e.g., Elmegreen 2000), an evolutionary progression and a typical lifetime of a few tens of Myr, long enough for clouds to become virialized (e.g., Blitz & Shu 1980b, Fukui & Kawamura 2010), or a lifetime of hundreds of Myr (e.g., Scoville & Hersh 1979). Roman-Duval et al. (2010) find a median avir approx 0.5 for clouds in the inner Galaxy, suggesting that they are bound entities where Mvir represents a reasonable measure of the molecular mass, although casting doubt on the assumption of exact virial equilibrium. Wong et al. (2011) estimate a very large scatter in avir in the Large Magellanic Cloud, but do lack an independent mass tracer so their results rest on the assumption of a fixed XCO. Observations in the outer Galaxy show another angle of the situation. Heyer, Carpenter & Snell (2001) find that clouds with Mmol > 104 Modot are self-gravitating, while small clouds with masses Mmol < 103 Modot are overpressured with respect to their self-gravity, that is, have avir ≫ 1 and are out of equilibrium. Given the observed mass function, however, such clouds represent a very small fraction of the molecular mass of the Milky Way.

In any case, observed GMC properties can be understood as a consequence of approximate energy equipartition, which observationally is very difficult to distinguish from virial equilibrium (Ballesteros-Paredes 2006). Clouds with an excess of kinetic energy, avir ≫ 1, perhaps due to ongoing star formation or SNe would be rapidly dissipated, while clouds with a dearth of kinetic energy, avir ≪ 1, would collapse at the free-fall velocity which is within 40% of the equipartition velocity dispersion and challenging to distinguish from turbulent motions in observations. Furthermore, the resulting star formation will inject energy into the cloud acting to restore the balance. Thus from the standpoint of determining cloud masses over large samples, the assumption of virial equilibrium even if not strictly correct, is unlikely to be very wrong.

4.1.2. Observational Results

The most significant study of the relation between virial mass and LCO (the mass-luminosity relation) in the Milky Way is that by Solomon et al. (1987), which encompasses 273 clouds and spans several orders of magnitude in cloud luminosity and mass. It is dominated by clouds located in the inner Galaxy, in the region of the so-called Molecular Ring, a feature in the molecular surface density of the Milky Way peaking at RGC approx 4 kpc galactocentric radius. It uses kinematic distances with an old value of the distance to the Galactic Center, Rodot = 10 kpc. We report new fits after a 0.85 scaling in all distances and sizes and 0.72 in luminosities to bring them into agreement with the modern distance scale (Rodot = 8.5 kpc). The virial mass computations assume a rho(r) propto r-1 (see Section 2.1).

Solomon et al. (1987) find a very strong correlation between Mvir and LCO, such that Mvir = 37.9 LCO0.82 with a typical dispersion of 0.11 dex for Mvir. Note the excellent agreement with the expected mass-luminosity relation in Eq. 10 using a typical CO brightness temperature TB approx 4 K (Maloney 1990). For a cloud at their approximate median luminosity, LCO approx 105 K km s-1 pc2, this yields alphaCO = 4.6 Modot (K km s-1 pc2)-1 and XCO,20 = 2.1. Because the relation is not strictly linear alphaCO will change by ~ 60% for an order of magnitude change in luminosity (Fig. 2). Therefore GMCs with lower luminosities (and masses) will have somewhat larger mass-to-light ratios and conversion factors than more luminous GMCs.

Figure 2

Figure 2. Relation between virial alphaCO and CO luminosity for GMCs in the Milky Way (Solomon et al. 1987). We have corrected the numbers in the original table to reflect the updated distance to the Galactic Center of 8.5 kpc. The dependence of alphaCO on LCO arises from the fact that the correlation between Mvir and LCO has a nonlinear slope (Mvir propto LCO0.815±0.013), following the expectations from Eq. 10 for approximately constant brightness temperature. This results in alphaCO approx 4.61 (LCO/105)-0.185, denoted by the red thick dashed line (the dispersion around this relation is ± 0.15 dex). The nominal value at LCO = 105 K km s-1 pc2 is illustrated by the thin black dashed line.

Independent analysis using the same survey by Scoville et al. (1987) yields a very similar mass-luminosity relation. After accounting for the different coefficients used for the calculation of the virial mass, the relation is Mvir = 33.5 LCO0.85. For a LCO approx 105 K km s-1 pc2 cloud this yields alphaCO = 6.0 Modot (K km s-1 pc2)-1 and XCO,20 = 2.8 (this work uses Rodot = 8.5 kpc). Interestingly, there is no substantial difference in the mass-luminosity relation for GMCs with or without HII regions (Scoville & Good 1989), although the latter tend to be smaller and lower mass, and have on average half of the velocity-integrated CO brightness of their strongly star-forming counterparts. The resulting difference in TB could have led to a displacement in the relation, according to the simple reasoning leading to Eq. 10, but it appears not to be significant.

4.1.3. Considerations and Limitations

Besides the already discussed applicability of the virial theorem, there are a number of limitations to virial studies. Some are practical, while others are fundamental to the virial technique. On the practical side, virial studies are sensitive to cloud definitions and biases induced by signal-to-noise. These will impact both the values of R and sigma used to compute the mass. In noise-free measurements isolated cloud boundaries would be defined using contours of zero emission, when in reality it is necessary to define them using a higher contour (for example, Solomon et al. 1987 use a TB ~ 4 K CO brightness contour). Scoville et al. (1987) discuss the impact of this correction, studying the "curve of growth" for R and sigma as the definition contour is changed in high signal-to-noise observations. Moreover, isolated clouds are rare and it is commonly necessary to disentangle many partially blended features along the line of sight. To measure a size clouds need to be resolved, and if appropriate the telescope beam size needs to be deconvolved to establish the intrinsic cloud size. This is a major concern in extragalactic studies, but even Galactic datasets are frequently undersampled which affects the reliability of the R and LCO determinations. Given these considerations, it is encouraging that two comprehensive studies using independent analysis of the same survey come to values of alphaCO that differ by only ~ 30% for clouds of the same luminosity.

A fundamental limitation of the virial technique is that CO needs to accurately sample the full potential and size of the cloud. For example, if because of photodissociation or other chemistry CO is either weak or absent from certain regions, its velocity dispersion may not accurately reflect the mass of the cloud. This is a particular concern for virial measurements in low metallicity regions (see Section 6), although most likely it is not a limitation in the aforementioned determinations of XCO in the inner Galaxy.

4.2. Column Density Determinations Using Dust and Optically Thin Lines

Perhaps the most direct approach to determining the H2 column density is to employ an optically thin tracer. This tracer can be a transition of a rare CO isotopologue or other chemical species (e.g., CH Magnani et al. 2003). It can also be dust, usually optically thin in emission at far-infrared wavelengths, and used in absorption through stellar extinction studies.

4.2.1. CO Isotopologues

A commonly used isotopologue is 13CO. Its abundance relative to 12CO is down by a factor approaching the 12C / 13C approx 69 isotopic ratio at the solar circle (12C/13C approx 50 at RGC approx 4 kpc, the galactocentric radius of the Molecular Ring) as long as chemical fractionation and selective photodissociation effects can be neglected (Wilson 1999). Given this abundance ratio and under the conditions in a dark molecular cloud 13CO emission may not always be optically thin, as tau1 ~ 1 requires AV ~ 5.

The procedure consists of inverting the observed intensity of the optically thin tracer to obtain its column (or surface) density. In the case of isotopologues, this column density is converted to the density of CO using the (approximate) isotopic ratio. Inverting the observed intensity requires knowing the density and temperature structure along the line of sight, which is a difficult problem. If many rotational transitions of the same isotopologue are observed, it is possible to model the line of sight column density using a number of density and temperature components. In practice an approximation commonly used is local thermodynamic equilibrium (LTE), the assumption that a single excitation temperature describes the population distribution among the possible levels along the line of sight. It is also frequently assumed that 12CO and 13CO share the same Tex, which is particularly justifiable if collisions dominate the excitation (Tex = Tkin, the kinetic temperature of the gas). Commonly used expressions for determining N(13CO) under these assumptions can be found in, for example, Pineda et al. (2010). Note, however, that if radiative trapping plays an important role in the excitation of 12CO, Tex for 13CO will generally be lower due to its reduced optical depth (e.g., Scoville & Sanders 1987b).

Dickman (1978) characterized the CO column density in over 100 lines of sight toward 38 dark clouds, focusing on regions where the LTE assumption is unlikely to introduce large errors. The combination of LTE column densities with estimates of AV performed using star counts yields AV approx (4.0 ± 2.0) × 10-16 N(13CO) cm2 mag. Comparable results were obtained in detailed studies of Taurus by Frerking, Langer & Wilson (1982, note the nonlinearity in their expression) and Perseus by Pineda, Caselli & Goodman (2008), the latter using a sophisticated extinction determination (Lombardi & Alves 2001). Extinction can be converted into molecular column density, through the assumption of an effective gas-to-dust ratio. Bohlin, Savage & Drake (1978) determined a relation between column density and reddening (selective extinction) such that [N(HI) + 2N(H2)] / E(B - V) approx 5.8 × 1021 atoms cm-2 mag-1 in a survey of interstellar Lyalpha absorption carried out using the Copernicus satellite toward 75 lines of sight, mostly dominated by HI. For a "standard" Galactic interstellar extinction curve with RV ident AV / E(B - V) = 3.1, this results in

Equation 21 (21)

A much more recent study using Far Ultraviolet Spectroscopic Explorer observations finds essentially the same relation (Rachford et al. 2009). In high surface density molecular gas RV may be closer to 5.5 (Chapman et al. 2009), and Eq. 21 may yield a 40% overestimate (Evans et al. 2009). Using Eq. 21, the approximate relation between 13CO J = 1 → 0 and molecular column density is N(H2) approx 3.8 × 105 N(13CO). Pineda, Caselli & Goodman (2008) find a similar result in a detailed study of Perseus, with an increased scatter for AV gtapprox 5. Goldsmith et al. (2008) use these results together with an averaging method to increase the dynamic range of their 13CO and 12CO data, a physically motivated variable 12CO / 13CO ratio, and a large velocity gradient excitation analysis, to determine H2 column densities in Taurus. They find that XCO,20 approx 1.8 recovers the molecular mass over the entire region mapped, while there is a marked increase in the region of low column density, where XCO increases by a factor of 5 where N(H2) < 1021 cm-2. As a cautionary note about the blind use of 13CO LTE estimates, however, Heiderman et al. (2010) find that this relation between H2 and 13CO underestimates N(H2) by factors of 4-5 compared with extinction-based results in the Perseus and Ophiuchus molecular clouds.

4.2.2. Extinction Mapping

Extinction mapping by itself can be directly employed to determine XCO. It fundamentally relies on the assumption of spatially uniform extinction properties for the bands employed, and on the applicability of Eq. 21 to convert extinction into column density.

Frerking, Langer & Wilson (1982) determined XCO,20 approx 1.8 in the range 4 ltapprox AV ltapprox 12 in rho Oph, while the same authors found constant W(CO) for AV gtapprox 2 in Taurus. Lombardi, Alves & Lada (2006) studied the Pipe Nebula and found a best fit XCO in the range XCO,20 approx 2.9-4.2, but only for K-band extinctions AK > 0.2 (equivalent to AV > 1.8, Rieke & Lebofsky 1985). A simple fit to the data ignoring this nonlinearity yields XCO,20 ~ 2.5. The Pineda, Caselli & Goodman (2008) study of Perseus finds XCO,20 approx 0.9-3 over a number of regions. The relation between CO and H2, however, is most linear for AV ltapprox 4, becoming saturated at larger line-of-sight extinctions.

Figure 3

Figure 3. Relation between CO column density and extinction in the Taurus molecular cloud (Pineda et al. 2010). The figure shows the pixel-by-pixel relation between gas-phase CO column density (obtained from 13CO) and AV. The blue line illustrates the "average" linear relation for 3 ltapprox AV ltapprox 10, N(12CO) approx 1.01 × 1017 AV cm-2 (implying CO/H2 approx 1.1 × 10-4 for the assumed isotopic ratio). The linearity is clearly broken for AV gtapprox 10. Pineda et al. (2010) show that linearity is restored to high AV after applying a correction for CO freeze-out into dust grain mantles.

Pineda et al. (2010) extend the aforementioned Goldsmith et al. (2008) study of Taurus by characterizing the relation between reddening (from the Two Micron All Sky Survey, 2MASS) and CO column density (derived from 13CO) to measure XCO,20 approx 2.1. They find that the relation between AV and CO flattens for AV gtapprox 10 (Fig. 3), a fact that they attribute to freeze-out of CO onto dust grains causing the formation of CO and CO2 ice mantles. Including a correction for this effect results in a linear relation to AV ltapprox 23. For AV ltapprox 3 the column density of CO falls below the linear relationship, likely due to the effects of photodissociation and chemical fractionation. Along similar lines, Heiderman et al. (2010) find that in Ophiuchus and Perseus CO can underpredict H2 with respect to AV for Sigmamol > 200 Modot pc-2 by as much as ~ 30%.

Paradis et al. (2012) recently used a high-latitude extinction map derived from 2MASS data using an extension of the NICER methodology (Dobashi et al. 2008, 2009) to derive XCO in sample of nearby clouds with |b| > 10°. They find XCO,20 approx 1.67 ± 0.08 with a somewhat higher value XCO,20 approx 2.28 ± 0.11 for the inner Galaxy region where |l| < 70°. They report an excess in extinction over the linear correlation between total gas and AV at 0.2 ltapprox AV ltapprox 1.5, suggestive of a gas phase that is not well traced by either 21 cm or CO emission. We will return to this in Section 4.2.4.

4.2.3. Dust Emission

The use of extinction mapping to study N(H2) is mostly limited to nearby Galactic clouds, since it needs a background stellar distribution, minimal foreground confusion, and the ability to resolve individual stars to determine their reddening. Most interestingly, the far-infrared emission from dust can also be employed to map the gas distribution. Indeed, dust is an extraordinarily egalitarian acceptor of UV and optical photons, indiscriminately processing them and reemitting in the far-infrared. In principle, the dust spectral energy distribution can be modeled to obtain its optical depth, taud(lambda), which should be proportional to the total gas column density under the assumption of approximately constant dust emissivity per gas nucleon, fundamentally the product of the gas-to-dust ratio and dust optical properties.

How valid is this assumption? An analysis of the correlation between taud and HI was carried out at high Galactic latitudes by Boulanger et al. (1996), who found a typical dust emissivity per H nucleon of

Equation 22 (22)

with beta = 2, in excellent accord with the recent value for high latitude gas derived using Planck observations (Planck Collaboration et al. 2011c, who prefer beta = 1.8). They also identified a break in the correlation for N(HI) gtapprox 5 × 1020 cm-2 suggestive of an increasingly important contribution from H2 to NH, in agreement with results from Copernicus (Savage et al. 1977). There is evidence that the coefficient in Eq. 22 changes in molecular gas. It may increase by factors of 2-3 at very high column densities (Schnee et al. 2008, Flagey et al. 2009, Planck Collaboration et al. 2011d), likely due to grain growth or perhaps solid state effects at low temperatures (e.g., Mény et al. 2007). Note, however, that recent work using Planck in the Galactic plane finds deltaDGR approx (0.92 ± 0.05) × 10-25 cm2 at 250 µm, with no significant variation with Galactic radius (Planck Collaboration et al. 2011b). This deltaDGR is almost identical to that observed in dust mixed with mostly atomic gas at high latitudes, suggesting that the aforementioned emissivity variations are very localized.

This excellent correlation between taud and NH is the basis for a number of studies that use dust emission to determine H2 column densities. Most notably, Dame, Hartmann & Thaddeus (2001) employed the Columbia survey of molecular gas in the Galactic plane together with the Dwingeloo-Leiden HI survey and the IRAS temperature-corrected 100 µm spectral density map by Schlegel, Finkbeiner & Davis (1998). With these data, N(H2) can be obtained using

Equation 23 (23)

which simply states that the dust optical depth (taud) is a perfect tracer of the total column density of gas when the emissivity per nucleon (deltaDGR) is known. As we just discussed, deltaDGR can be straightforwardly determined on lines of sight dominated by atomic gas, for example. The comparison of the molecular column density so derived with the observed W(CO) yields XCO,20 approx 1.8 ± 0.3, valid for |b| > 5° on large scales across the Galaxy. This study also finds evidence for a systematic increase in XCO by factors ~ 2-3 at high Galactic latitude (b > 20°), in regions with typical N(H2) ltapprox0 .5 × 1020 cm-2 on ~ 0.5° angular scales.

The excellent Planck dataset has afforded a new view on this topic (Fig. 4). Planck Collaboration et al. (2011a) produced a new taud map for the Milky Way and a new determination of XCO using Eq. 23 for lines of sight with |b| > 10°. They obtain XCO,20 = 2.54 ± 0.13, somewhat larger than the previous study. This difference is likely methodological, for example the use of different far-infrared wavelengths as well as local versus global calibrations of deltaDGR.

Figure 4

Figure 4. Planck results in the Aquila-Ophiuchus flare (Planck Collaboration et al. 2011a). The figure shows molecular gas column density with a color range N(H2) approx -1.5 × 1021 cm-2(dark blue) to 3.5 × 1021 cm-2(red). The black contours illustrate the CO emission from the Columbia survey, with values ICO approx 2, 10, 20 K km s-1. The black region lacks CO information.

4.2.4. CO-Faint Molecular Gas and Diffuse Lines of Sight

Most interestingly, Planck Collaboration et al. (2011a) observe a tight linear correlation between taud and N(HI) + 2XCO W(CO) for AV ltapprox 0.4 and 2.5 ltapprox AV ltapprox 10, with an excess in taud in the intermediate range (Fig. 5). This excess can be understood in terms of a component of H2 (or possibly a combination of H2 and cold, opaque HI) that emits weakly in CO and is prevalent at 0.4 ltapprox AV ltapprox 2.5 (the explanation is not unique, since the methodology cannot distinguish it from a change in dust emissivity in that narrow AV regime, but such possibility appears unlikely). This molecular component arises from the region in cloud surfaces where gas is predominantly H2 but most carbon is not in CO molecules because the extinction is too low, essentially the PDR surface (see Section 6.1 and Fig. 1). This component is frequently referred to as "CO-dark molecular gas" or sometimes simply "dark gas" (Grenier, Casandjian & Terrier 2005, Wolfire, Hollenbach & McKee 2010). In this review we will refer to it as "CO-faint," which is a more accurately descriptive name. The existence of molecular gas with low CO abundance has been noted previously in theoretical models (e.g., van Dishoeck & Black 1988), in observations of diffuse gas and high-latitude clouds (e.g., Lada & Blitz 1988), and in observations of irregular galaxies (e.g., Madden et al. 1997). In this context, the results by Planck Collaboration et al. 2011a are in qualitative agreement with already discussed observations that show an increase in XCO at low molecular column densities (e.g., Goldsmith et al. 2008, Paradis et al. 2012).

Figure 5

Figure 5. Correlation between taud at 350 µm and total hydrogen column density NH for XCO,20 = 2.3 (for |b| > 10°, Planck Collaboration et al. 2011a). The color scale represents the logarithm of the number of lines-of-sight, and the blue dots the result of NH binning. The red dashed lines indicate AV = 0.37 and AV = 2.5. The red solid line represents the best linear fit for low NH. Note that it is also a good fit to the AV gtapprox 2.5 points. The excess in the binned correlation over the red line for 0.37 ltapprox AV ltapprox 2.5 is either an indication of "CO-faint" molecular gas, or possibly a combination of high optical-depth "opaque" HI with "CO-faint" H2, or a change in the dust emissivity over that AV regime.

The Planck observations are also in qualitative agreement with the analysis by Grenier, Casandjian & Terrier (2005), who correlated the diffuse gamma-ray emission over the entire sky with templates derived from the HI, CO, and dust, finding a component of gas not traced by CO evident in local clouds at high latitudes. Their analysis finds that this component is as important, by mass, as the "CO-bright" H2 component in several of these clouds, and increasingly more important for smaller cloud masses. The recent analyses based on Fermi data by Abdo et al. (2010d) and Ackermann et al. (2012) are also qualitatively compatible with these results, finding that the "CO-faint" component amounts to 40%-400% of the "CO-bright" mass in the Cepheus, Polaris, Chamaleon, R Cr A, and Cassiopeia clouds (small local molecular clouds).

The ionized carbon far-infrared fine structure emission provides an additional probe of molecular gas at low AV. Large scale [CII] observations of the (2P3/22P1/2) fine-structure transition in the Milky Way and external galaxies suggest it is due to a combination of emission from the Cold Neutral Medium (CNM) and from PDRs located in the surfaces of GMCs (Stacey et al. 1991, Shibai et al. 1991, Bennett et al. 1994). The contribution from [CII] in the diffuse ionized gas, however, could also be important (Heiles 1994, Madden et al. 1993), particularly along certain lines-of-sight (Velusamy et al. 2012).

In regions where most of the emission arises in PDRs [CII] has the potential to trace the "CO-faint" molecular regime at low AV. Langer et al. (2010) analyze 16 lines-of-sight in the plane of the Galaxy and find that in about half of them the observed [CII] intensity can be entirely explained as due to carbon in atomic gas in the CNM. The other half, however, exhibits [CII] / N(HI) ratios that are too large to be due to atomic gas and may have molecular to atomic ratios as large as N(H2) / N(HI) ~ 6. While the very brightest [CII] components investigated arise from dense (n > 105 cm-3) PDRs exposed to intense radiation fields, most of the [CII] emission in these molecular lines-of-sight can be explained as originating in the surfaces of modestly dense GMCs (n ~ [3-300] × 103 cm-3) exposed to at most a few times the local interstellar radiation field at the solar circle (Pineda et al. 2010b). Following the reasoning in Velusamy et al. (2010) and Langer et al. (2010), a very approximate relation between [CII] emission and "CO-faint" H2 is N(H2) ~ 1.46 × 1020 W([CII]) - 0.35 N(HI) cm-2, for W([CII]) in K km s-1 (see Eq. 7 to convert between Jy and K). We caution that the coefficients correspond to the Milky Way carbon abundance and are very dependent on the assumed physical conditions, particularly the densities (we use n(HI) ~ 200 cm-3, n(H2) ~ 300 cm-3, and THI ~ TH2 ~ 100 K).

Liszt, Pety & Lucas (2010) measure XCO in diffuse gas by first estimating the total hydrogen column from dust continuum emission (using the map by Schlegel, Finkbeiner & Davis 1998), and subtracting the observed HI column density (c.f., Eq. 23). They select lines-of-sight with HCO+ absorption spectra against bright extragalactic continuum sources. The CO emission, together with the measured H2 column provides a measure of XCO. Surprisingly, Liszt, Pety & Lucas (2010) and Liszt & Pety (2012) find mean values in diffuse gas similar to those in GMCs. There are large variations, however, about the mean with low XCO (bright CO) produced in warm T ~ 100 K diffuse gas and high XCO (faint CO) produced at low N(H2) column densities. The authors argue that these variations mainly reflect the CO chemistry and its dependence on ultraviolet radiation field, density, and total column density, rather than the H2 column density. We note that the observed CO column densities cannot be produced in steady-state PDR models (.e.g., Sonnentrucker et al. 2007). Enhanced CO production might occur through the "CH+ channel" driven by non-thermal ion-neutral reactions (Federman et al. 1996, Visser et al. 2009) or by pockets of warm gas and ion-neutral reactions in turbulent dissipation regions (Godard, Falgarone & Pineau des Forets 2009). Density fluctuations in a turbulent median might also increase the CO production (Levrier et al. 2012). Thus, although the mean XCO diffuse cloud value is similar to GMCs, the CO emission from diffuse gas cannot be easily interpreted as a measure of the molecular column except perhaps in a statistical sense.

4.3. XCO Based on Gamma-Ray Observations

Diffuse gamma-ray emission in the Galaxy is chiefly due to three processes: neutral pion production and subsequent decay in collisions between cosmic-rays and interstellar matter, bremsstrahlung emission due to scattering of cosmic-ray electrons by interstellar matter, and inverse Compton scattering of low energy photons by cosmic-ray electrons. The first of these processes is the dominant production channel for diffuse gamma-rays with energies above 200 MeV, although at high Galactic latitude there will be an increasingly important inverse Compton component (Bloemen 1989). Interestingly, the fact that interactions between cosmic rays and nucleons give rise to diffuse gamma-ray emission can be used to count nucleons in the ISM, and indeed the use of XCO to represent the ratio N(H2) / W(CO) was introduced for the first time in gamma-ray work using observations from the COS B satellite (Lebrun et al. 1983).

Accounting for the pion decay and bremsstrahlung processes, and neglecting the contribution from ionized gas, the basic idea behind modeling the emission is to use a relation Igamma= Sigma єgamma,HI(Ri)[N(HI)i + 2 YCO W(CO)i] (Bloemen 1989). Here Igamma is the diffuse gamma-ray emission along a line-of-sight, єgamma is the HI gas emissivity (a function of Galactocentric radius R), and YCO is a parameter that takes into account that emissivity in molecular clouds may be different than in atomic gas, due to cosmic-ray exclusion or concentration, XCO = YCO єgamma,HI / єgamma,H2 (Gabici, Aharonian & Blasi 2007, Padovani, Galli & Glassgold 2009). This situation is analogous to that presented in dust emission techniques, where emissivity changes in the molecular and atomic components will be subsumed in the resulting value of XCO.

Recent analyses use the approach Igamma approx qHI N(HI) + qCOW(CO) + qEBV E(B - V)res + …, where the first three terms account for emission that is proportional to the HI, the "CO-bright" H2, and a "CO-faint" H2 component that is traced by dust reddening or emission residuals (the additional terms not included account for an isotropic gamma-ray background and the contribution from point sources, e.g., Abdo et al. 2010d). The dust residual template consists of a dust map (e.g., Schlegel, Finkbeiner & Davis 1998) with a linear combination of N(HI) and W(CO) fitted and removed.

4.3.1. Observational Results

A discussion of the results of older analyses can be found in Bloemen (1989), here we will refer to a few of the more recent results using the Compton Gamma Ray Observatory and Fermi satellites.

Strong & Mattox (1996) analyzed the EGRET all-sky survey obtaining XCO,20 approx 1.9 ± 0.2. Similar results were obtained by Hunter et al. (1997) for the inner Galaxy, and by Grenier, Casandjian & Terrier (2005) for clouds in the solar neighborhood (see also Section 4.2.4). Strong et al. (2004) introduce in the analysis a Galactic gradient in XCO, in an attempt to explain the discrepancy between the derived єgamma(R) and the distributions of supernova remnants and pulsars, which trace the likely source of cosmic rays in supernovae shocks. Matching the emissivity to the pulsar distribution requires a significant change in XCO between the inner and outer Galaxy (XCO,20 approx 0.4 for R ~ 2.5 kpc to XCO,20 approx 10 for R > 10 kpc).

The sensitivity of Fermi has been a boon for studies of diffuse gamma-ray emission in our Galaxy. Abdo et al. (2010d) and Ackermann et al. (2011, 2012b, 2012d) analyze the emission in the solar neighboorhood and the outer Galaxy. Taken together, they find a similar XCO for the Local arm and the interarm region extending out to R ~ 12.5 kpc, XCO,20 ~ 1.6-2.1 depending on the assumed HI spin temperature. The very local high-latitude clouds in the Gould Belt have a lower XCO,20 approx 0.9, and appear not to represent the average properties at the solar circle (Ackermann et al. 2012b, 2012c).

It is important to note, however, that the Fermi studies follow the convention XCO ident qCO / 2qHI with the parameters defined in the previous section. Thus, their definition of XCO does not include the "CO-faint" envelope that is traced by the dust residual template: it only accounts for the H2 that is emitting in CO. This is different from the convention adopted in the dust studies discussed in the previous section, where all the H2 along a line-of-sight is associated with the corresponding CO. Ackermann et al. (2012b) report XCO,20 approx 0.96, 0.99, 0.63 for Chameleon, R CrA, and the Cepheus/Polaris Flare region, respectively using the Fermi convention. The ratio of masses associated with the CO and the dust residual template (reported in their Table 4) suggests that we should correct these numbers by factors of approximately 5, 2, and 1.4 respectively to compare them with the dust modeling. This correction results in XCO,20 approx 4.8, 2, 0.9, bringing these clouds in much better (albeit not complete) agreement with the results of dust modeling and the expectation that they be underluminous — not overluminous — in CO due to the presence of a large "CO-faint" molecular component.

4.3.2. Considerations and Limitations

We have already mentioned a limitation of gamma-ray studies of XCO, the degree to which they may be affected by the rejection (or generation) of cosmic rays in molecular clouds. A major limitation of gamma-ray determinations is also the poor angular resolution of the observations. Another source of uncertainty has become increasingly apparent with Fermi, which resolves the Magellanic Clouds in gamma-ray emission (Abdo et al. 2010, 2010b). In the Clouds, the distribution of emission does not follow the distribution of gas. Indeed, the emission is dominated by regions that are not peaks in the gas distribution, but may correspond to sites of cosmic ray injection (e.g., Murphy et al. 2012), suggesting that better knowledge of the cosmic ray source distribution and diffusion will have an important impact on the results of gamma-ray studies.

Ackermann et al. (2012) carry out a thorough study of the impact of systematics on the global gamma-ray analyses that include a cosmic-ray generation and propagation model, frequently used to infer Galactic XCO gradients. They find that the XCO determination can be very sensitive to assumptions such as the cosmic ray source distribution and the HI spin temperature, as well as the selection cuts in the templates (in particular, the dust template). Indeed, the value in the outer Galaxy in their analysis is extremely sensitive to the model cosmic ray source distribution. The magnitude of a Galactic XCO gradient turns out to also be very sensitive to the underlying assumptions (see Fig. 25 in Ackermann et al. 2012). What appears a robust result is a uniformly low value of XCO near the Galactic center (R ~ 0-1.5 kpc). This was already pointed out by much earlier gamma-ray studies (Blitz et al. 1985). The authors also conclude that including the dust information leads to an improvement in the agreements between models and gamma-ray data, even on the Galactic plane, suggesting that "CO-faint" gas is an ubiquitous phenomenon.

Table 1. Representative XCO values in the Milky Way disk

Method XCO / 1020 References
  cm-2(K km s-1)-1  

Virial 2.1 Solomon et al. (1987)
  2.8 Scoville et al. (1987)
Isotopologues 1.8 Goldsmith et al. (2008)
Extinction 1.8 Frerking, Langer & Wilson (1982)
  2.9-4.2 Lombardi, Alves & Lada (2006)
  0.9-3.0 Pineda, Caselli & Goodman (2008)
  2.1 Pineda et al. (2010)
  1.7-2.3 Paradis et al. (2012)
Dust Emission 1.8 Dame, Hartmann & Thaddeus (2001)
  2.5 Planck Collaboration et al. (2011a)
gamma-rays 1.9 Strong & Mattox (1996)
  1.7 Grenier, Casandjian & Terrier (2005)
  0.9-1.9 * Abdo et al. (2010d)
  1.9-2.1 * Ackermann et al. (2011 , 2012d)
  0.7-1.0 * Ackermann et al. (2012b, 2012c)

* Note difference in XCO convention (Section 4.3).

4.4. Synthesis: Value and Systematic Variations of XCO in the Milky Way

There is an assuring degree of uniformity among the values of XCO obtained through the variety of methodologies available in the Milky Way. Representative results from analyses using virial masses, CO isotopologues, dust extinction, dust emission, and diffuse gamma-ray radiation hover around a "typical" value for the disk of the Milky Way XCO approx 2 × 1020 cm-2(K km s-1)-1 (Table 1). This fact, combined with the simple theoretical arguments outlined in Section 2 as to the physics behind the H2-to-CO conversion factor, as well as the results from elaborate numerical simulations discussed in Section 3.2, strongly suggests that we know the mass-to-light calibration for GMCs in the disk of the Milky Way to within ± 0.3 dex certainly, and probably with an accuracy closer to ±0.1 dex (30%). This is an average number, valid over large scales. Individual GMCs will scatter around this value by a certain amount, and individual lines-of-sight will vary even more.

There are, however, some systematic departures from this value in particular regimes, as suggested by the simple theoretical arguments. As pointed out by several studies, the Galactic center region appears to have an XCO value 3-10 times lower than the disk. In addition to the aforementioned gamma-ray results (Blitz et al. 1985, Strong et al. 2004, Ackermann et al. 2012), a low XCO in the center of the Milky Way was obtained by analysis of the dust emission (Sodroski et al. 1995), and the virial mass of its clouds (Oka et al. 1998, Oka et al. 2001). We will see in Section 5 that this is not uncommonly observed in the centers of other galaxies. It is likely due to a combination of enhanced excitation (clouds are hotter) and the dynamical effects discussed in Section 2.3.

Departures may occur in some nearby, high-latitude clouds, for example the Cepheus/Polaris Flare region. A critical step to understand the magnitude or even existence of such discrepancies will be to derive XCO for diffuse gamma ray observations using conventions matched to other fields. We have attempted some estimate of the correction in the Chameleon, R CrA, and the Cepheus/Polaris Flare. In Cepheus/Polaris even after trying to account for the difference in XCO convention, the gamma-ray studies find a low XCO value suggesting that those local clouds are overluminous in CO by a factor ~ 2, or maybe that their dust emissivity is different. Low values are not consistent with results from dust emission techniques (Dame, Hartmann & Thaddeus 2001, Planck Collaboration et al. 2011a). On the other hand, they are known to be turbulent to very small ( ~ 0.01 pc) scale (Miville-Deschênes et al. 2010), and turbulent dissipation may play a role in exciting CO (Ingalls et al. 2011).

Finally, observations of XCO in the outer galaxy are still sparse, although some of them suggest high values of XCO (for example, Brand & Wouterloot 1995, Heyer, Carpenter & Snell 2001). This can be understood in terms of an increasing dominance of the "CO-faint" molecular gas associated probably with decreasing metallicities in the outer disk (see, Section 6). In nearby, resolved molecular clouds there is compelling evidence that a regime of high XCO exists at low extinctions and column densities (e.g., Goldsmith et al. 2008, Pineda et al. 2010), where most of the carbon in the gas phase is not locked in CO molecules.

It seems clear that CO becomes a poor tracer of H2 at low column densities. Most studies agree that this phenomenon occurs for AV ltapprox 2 (Goldsmith et al. 2008, Planck Collaboration et al. 2011a). How much molecular gas exists in this regime is still a subject of study, but it seems to represent a substantial, perhaps even dominant fraction of the H2 near the solar circle (Grenier, Casandjian & Terrier 2005, Wolfire, Hollenbach & McKee 2010, Planck Collaboration et al. 2011a). It is, however, unlikely to be a large fraction of the total molecular mass of the Galaxy, which is dominated by clouds in the inner galaxy. Nonetheless, we will see that this "CO-faint" phase almost surely constitutes most of the molecular gas in low metallicity systems.

Using these conclusions, what is the characteristic surface density of a GMC, SigmaGMC, in the Milky Way? The precise value, or even whether it is a well-defined quantity, is a matter of study. Using the survey by Solomon et al. (1987) updated to the new distance scale and assuming XCO,20 = 2, we find a distribution of surface densities with SigmaGMC approx 150-70+95 Modot pc-2 (± 1 sigma interval). Using instead the distribution of 331 sigma2 / R to evaluate surface density (see discussion after Eq. 9), we obtain SigmaGMC approx 200-80+130 Modot pc-2. Heyer et al. (2009) use 13CO observations of these clouds, finding SigmaGMC approx 40 Modot pc-2 over the same cloud areas but concluding that it is likely an underestimate of at least a factor of 2 due to non-LTE and optical depth effects (SigmaGMC approx 144 Modot pc-2 using a more restrictive 13CO contour instead of the original 12CO to define surface area, Roman-Duval et al. 2010). Real clouds have a range of surface densities. Heiderman et al. (2010) analyze extinction-based measurements for 20 nearby clouds, calculating their distribution of Sigmamol. It is clear that many of these clouds do not reach the characteristic SigmaGMC of clouds in the inner Galaxy; for the most part the clouds with low mass tend to have low surface densities. Nonetheless, several of the most massive clouds do have SigmaGMC gtapprox 100 Modot pc-2. The mass-weighted (area-weighted) SigmaGMC are 160, 150, 140, and 110 (140, 140, 100, and 90) Modot pc-2 for Serpens-Aquila, Serpens, Ophiuchus, and Perseus respectively according to their Table 2. By comparison, SigmaGMC approx 85 Modot pc-2 for a sample of nearby galaxies, many of them dwarf (Bolatto et al. 2008). Note that SigmaGMC is likely to be a function of the environment. In the Galactic Center, for example, Oka et al. (1998, 2001) report a size-line width relation where the coefficient C is five times larger than that observed in the Milky Way disk, suggesting a SigmaGMC that is 25 times larger (see also Rosolowsky & Blitz 2005).

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