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