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

At its core the *X*_{CO} factor represents a valiant effort
to use the bright but optically thick transition of a molecular gas
impurity to measure total molecular gas masses. How and why does
*X*_{CO} work?

Because the ^{12}CO *J* = 1 → 0 transition is
generally optically thick, its
brightness temperature is related to the temperature of the
_{CO} = 1 surface,
not the column density of the gas. *Information about
the mass of a self-gravitating entity, such as a molecular cloud, is
conveyed by its line width*, which reflects the velocity dispersion of
the emitting gas.

A simple and exact argument can be laid for virialized molecular
clouds, that is clouds where twice the internal kinetic energy equals
the potential energy. Following
Solomon et
al. (1987),
the virial mass
*M*_{vir} of a giant molecular cloud (GMC) in
M_{} is

(8) |

where *R* is the projected radius (in pc),
is
the 1D velocity dispersion (in km s^{-1};
_{3D} =
√3),
*G* is the gravitational constant (*G*
1 / 232
M_{}^{-1}
pc km^{2} s^{-2}), and *k* is the power-law index of
the spherical volume density distribution,
(*r*)
*r*^{-k}. The coefficient in front of *R*
^{2} is only
weakly dependent
on the density profile of the virialized cloud, and corresponds to
approximately 1160, 1040, and 700 for *k* = 0, 1, and 2 respectively
(MacLaren,
Richardson & Wolfendale 1988,
Bertoldi &
McKee 1992).
Unless otherwise specified, we
adopt *k* = 1 for the remainder of the discussion. This expression of
the virial mass is fairly robust if other terms in the virial theorem
(McKee &
Zweibel 1992,
Ballesteros-Paredes 2006),
such as magnetic support, can be neglected
(for a more general expression applicable to spheroidal clouds
and a general density distribution see
Bertoldi &
McKee 1992).
As long as
molecular gas is dominating the mass enclosed in the cloud radius and
the cloud is approximately virialized, *M*_{vir} will be a
good measure of the H_{2} mass.

Empirically, molecular clouds are observed to follow a size-line width relation (Larson 1981, Heyer et al. 2009) such that approximately

(9) |

with *C* 0.7 km
s^{-1} pc^{-0.5}
(Solomon et
al. 1987,
Scoville et
al. 1987,
Roman-Duval
et al. 2010).
This relation is an expression of the equilibrium supersonic
turbulence conditions in a highly compressible medium, and it is
thought to apply under very general conditions
(see Section 2.1 of
McKee &
Ostriker 2007
for a discussion). In fact, to within our current
ability to measure these two quantities such a relation is also
approximately followed by extragalactic GMCs in galaxy disks (e.g.,
Rubio, Lequeux
& Boulanger 1993,
Bolatto et
al. 2008,
Hughes et
al. 2010).

Note that insofar as the size dependence of
Eq. 9 is close to a square root, the combination
of Eqs. 8 and 9 yields that
*M*_{vir}
^{4}, and
molecular clouds that fulfill both
relations have a characteristic mean surface density,
_{GMC}, at a
value related to the coefficient of Eq. 9 so that
_{GMC} =
*M*_{vir} /
*R*^{2}
331
*C*^{2} for our chosen density profile
*r*^{-1}. We return to the question of
_{GMC} in
the Milky Way in Section 4.4.

Since the CO luminosity of a cloud, *L*_{CO}, is the
product of its area
( *R*^{2}) and
its integrated surface brightness
(*T*_{B}
√2
), then
*L*_{CO} =
√2
^{3}
*T*_{B}
*R*^{2}, where
*T*_{B} is the Rayleigh-Jeans brightness
temperature of the emission (see
Section 1.2). Using
the size-line width relation (Equation 9) to
substitute for *R* implies that *L*_{CO}
*T*_{B}
^{5}. Employing
this relation to replace
in *M*_{vir}
^{4} we
obtain a relation between *M*_{vir} and *L*_{CO},

(10) |

That is, for GMCs near virial equilibrium with approximately
constant brightness temperature, *T*_{B}, we expect an
almost linear relation
between virial mass and luminosity. The numerical coefficient in
Equation 10 is only a weak function of the density profile of the
cloud. Then using the relation between *C* and
_{GMC} we
obtain the following expression for the conversion factor

(11) |

Equations 10 and 11 rest on a
number of assumptions. We assume 1) virialized clouds with, 2) masses
dominated by H_{2} that 3) follow the size-line width relation
and 4) have approximately constant temperature. Equation 11
applies to a single, spatially resolved cloud, as
_{GMC}
is the resolved surface density.

We defer the discussion of the applicability of the virial theorem to
Section 4.1.1, and the effect
of other mass components to Section 2.3. The assumption
of a size-line width relation relies on our understanding of the
properties of turbulence in the interstellar medium. The result
√*R* follows
our expectations for a highly
compressible turbulent flow, with a turbulence injection scale at
least comparable to GMC sizes. The existence of a narrow range of
proportionality coefficients, corresponding to a small interval of
GMC average surface densities, is less well understood
(for an alternative view on this point, see
Ballesteros-Paredes et al. 2011).
In fact, this narrow range could be
an artifact of the small dynamic range of the samples
(Heyer et
al. 2009).
Based on observations of the Galactic center
(Oka et al. 2001)
and starburst galaxies (e.g.,
Rosolowsky
& Blitz 2005),
_{GMC}
likely does vary with environment. Equation 11 implies that any such
systematic changes in
_{GMC} will
also lead to systematic changes in *X*_{CO}, though in
actual starburst environments the picture
is more complex than implied by Equation 11.
Section 7 reviews the case of bright, dense
starbursts in detail.

This calculation also implies a dependence of *X*_{CO} on
the physical conditions in the GMC, density and temperature. Combining
Eqs. 8 and 9 with
*M*_{vir}
*R*^{3}, where
is the gas
density, yields
^{-0.5}. Meanwhile, because CO emission is
optically thick the observed luminosity depends on the brightness
temperature, *T*_{B}, as well as the line width, so that
*L*_{CO}
*T*_{B}. Substituting
in the relationship between density and line width,

(12) |

The brightness temperature, *T*_{B}, will depend on the
excitation of the gas (Eq. 6) and the filling fraction of
emission in the telescope beam, *f*_{b}. For high density
and optical depth the excitation temperature will approach the kinetic
temperature. Under those conditions, Eq. 12 also
implies _{CO}
^{0.5}
(*f*_{b} *T*_{kin})^{-1}.

Thus, even for virialized GMCs we expect that the CO-to-H_{2}
conversion factor will depend on environmental parameters such as gas
density and temperature. To some degree, these dependencies may offset
each other. If denser clouds have higher star formation activity and
are consequently warmer, the opposite effects of
and
*T*_{B} in Eq. 12 may partially cancel yielding a conversion
factor that is closer to a constant than we might otherwise expect.

We also note that relation between mass and luminosity expressed
by Eq. 10 is not exactly linear, which is the reason
for the weak dependence of
_{CO} on
*L*_{CO} or
*M*_{mol} in Eq. 11. As a consequence,
even for GMCs that obey this simple picture,
_{CO} will
depend (weakly) on the mass of the cloud considered, varying by a factor
of ~ 4 over 3 orders of magnitude in cloud mass.

This simple picture for how the CO luminosity can be used to estimate
masses of individual virialized clouds is not immediately applicable
to entire galaxies. An argument along similar lines, however, can be
laid out to suggest that under certain conditions there should be an
approximate proportionality between the integrated CO luminosity of
entire galaxies and their molecular mass. This is known as the
"mist" model, for reasons that will become clear in a few
paragraphs. Following
Dickman, Snell
& Schloerb (1986),
the luminosity due to an
ensemble of non-overlapping CO emitting clouds is *L*_{CO}
_{i}
*a*_{i} *T*_{B}(*a*_{i})
_{i}, where
*a*_{i} is the area subtended by
cloud *i*, and *T*_{B}(*a*_{i}) and
_{i} are its
brightness temperature and
velocity dispersion, respectively. Under the assumption that the
brightness temperature is mostly independent of cloud size, and that
there is a well-defined mean, *T*_{B}, then
*T*_{B}(*a*_{i})
*T*_{B}. We can rewrite the luminosity of the cloud
ensemble as *L*_{CO}
√2
*T*_{B} *N*_{clouds} <
*R*_{i}^{2}
(*R*_{i})>,
where the brackets indicate expectation value, *N*_{clouds} is
the number of clouds within the beam, and we have used
*a*_{i} =
*R*_{i}^{2}. Similarly, the total mass of gas
inside the beam is *M*_{mol}
*N*_{clouds}
< 4 / 3
*R*_{i}^{3}
(*R*_{i})>,
where (*R*_{i}) is the volume density of a cloud
of radius *R*_{i}.
Using our definition from Eq. 2 and dropping the *i*
indices, it is then clear that

(13) |

If the individual clouds are virialized they will follow Eq. 8,
or equivalently = 0.0635
*R*√.
Substituting into Eq. 13 we find

(14) |

which is analogous to Eq. 12 (obtained for individual clouds). As Dickman, Snell & Schloerb (1986) discuss, it is possible to generalize this result if the clouds in a galaxy follow a size-line width relation and they have a known distribution of sizes. Assuming individually virialized clouds, and using the size-line width relation (Eq. 9), we can write down Eq. 13 as

(15) |

where we have introduced the explicit dependence of the coefficient of
the size-line width relation on the cloud surface density,
_{GMC}. This
equation is the analogue of Eq. 11.

In the context of these calculations, CO works as a molecular mass tracer in galaxies because its intensity is proportional to the number of clouds in the beam, and because through virial equilibrium the contribution from each cloud to the total luminosity is approximately proportional to its mass, as discussed for individual GMCs. This is the essence of the "mist" model: although each particle (cloud) is optically thick, the ensemble acts optically thin as long as the number density of particles is low enough to avoid shadowing each other in spatial-spectral space.

Besides the critical assumption of non-overlapping clouds, which could
be violated in environments of very high density leading to "optical
depth" problems that may render CO underluminous, the other key
assumption in this model is the virialization of individual clouds,
already discussed for GMCs in the Milky Way. The applicability of a
uniform value of
_{CO} across
galaxies relies on three
assumptions that should be evident in Eq. 15: a similar value for
_{GMC},
similar brightness temperatures for the CO emitting gas, and a
similar distribution of GMC sizes that determines the ratio of the
expectation values <*R*^{2}> /
<*R*^{2.5}>. Very little is known
currently on the distribution of GMC sizes outside the Local Group
(Blitz et
al. 2007,
Fukui &
Kawamura 2010),
and although this is a potential source of
uncertainty in practical terms this ratio is unlikely to be
the dominant source of galaxy-to-galaxy variation in
_{CO}.

**2.3. Other Sources of Velocity Dispersion**

Because CO is optically thick, a crucial determinant of its luminosity
is the velocity dispersion of the gas,
. In our discussion for
individual GMCs and ensembles of GMCs in galaxies we have assumed that
is ultimately
determined by the sizes (through the size-line
width relation) and virial masses of the clouds. It is especially
interesting to explore what happens when the velocity dispersion of
the CO emission is related to an underlying mass distribution that
includes other components besides molecular gas. Following the
reasoning by
Downes, Solomon
& Radford (1993)
(see also
Maloney &
Black 1988,
Downes &
Solomon 1998)
and the discussion in Section 2.1, we can write a
cloud luminosity *L*_{CO}
for a fixed *T*_{B} as *L*_{CO}^{*} =
*L*_{CO}
^{*} /
,
where the asterisk indicates quantities where the
velocity dispersion of the gas is increased by other mass components,
such as stars. Assuming that both the molecular gas and the total
velocity dispersion follow the virial velocity dispersion due to a
uniform distribution of mass,
=
√*GM* / 5*R*,
then
*L*_{CO}^{*} = *L*_{CO}
√*M*^{*} /
*M*_{mol},
where *M*^{*} represents the total mass within
radius *R*. Substituting *L*_{CO} =
*M*_{mol} /
_{CO} yields the
result _{CO}
*L*_{CO}^{*} =
√*M*_{mol}
*M*^{*}.

Therefore, the straightforward application of
_{CO} to the
observed luminosity *L*_{CO}^{*} will yield an
overestimate of the molecular gas
mass, which in this simple reasoning will be the harmonic mean of the
real molecular mass and the total enclosed mass. If the observed
velocity dispersion is more closely related to the circular velocity,
as may be in the center of a galaxy, then
^{*}
√*GM*^{*} /
*R* and the result of applying
_{CO} to
*L*_{CO}^{*} will be an even larger
overestimate of *M*_{mol}. The appropriate value of the
CO-to-H_{2}
conversion factor to apply under these circumstances in order to
correctly estimate the molecular mass is

(16) |

where is a geometrical
correction factor accounting for the
differences in the distributions of the gas and the total mass, so
that
(^{*} /
)^{2}. In the
extreme case of a uniform distribution of gas responding to the
potential of a rotating disk of stars in a galaxy center,
~ 5. Everything else
being equal, in a case where *M*^{*} ~ 10
*M*_{mol} the straight application of a standard
_{CO} in a
galaxy center may lead to
overestimating *M*_{mol} by a factor of ~ 7.

Note that for this correction to apply the emission has to be
optically thick throughout the medium. Otherwise any increase in line
width is compensated by a decrease in brightness, keeping the
luminosity constant. Thus this effect is only likely to manifest
itself in regions that are already rich in molecular gas.
Furthermore, it is possible to show that an ensemble of virialized
clouds that experience cloud-cloud shadowing cannot explain a lower
*X*_{CO}, simply because there is a maximum attainable
luminosity. Therefore we expect *X*_{CO} to drop in regions
where the CO emission is extended throughout the medium, and not confined to
collections of individual self-gravitating molecular clouds. This
situation is likely present in ultra-luminous infrared galaxies
(ULIRGs), where average gas volume densities are higher than the
typical density of a GMC in the Milky Way, suggesting a pervading
molecular ISM (e.g.,
Scoville, Yun
& Bryant 1997).
Indeed, the reduction of *X*_{CO} in
mergers and galaxy centers has been modeled in detail by
Shetty et
al. (2011b)
and
Narayanan et al.
(2011,
2012),
and directly observed (Section 5.2,
Section 5.3,
Section 7.1,
Section 7.2)

Although commonly the emission from ^{12}CO *J* = 1
→ 0 transition is
optically thick, under conditions such as highly turbulent gas motions
or otherwise large velocity dispersions (for example stellar outflows
and perhaps also galaxy winds) then emission may turn optically thin.
Thus it is valuable to consider the optically thin limit on the
value of the CO-to-H_{2} conversion factor. Using Eq. 5,
the definition of optically thin emission (*I*_{J}
= _{J}
[*B*_{J}(*T*_{ex}) -
*B*_{J}(*T*_{cmb})], where
*B*_{J} is the Planck function
at the frequency _{J}
of the *J* → *J* - 1 transition,
*T*_{ex} is the excitation temperature, and
*T*_{cmb} is the
temperature of the Cosmic Microwave Background), and the definition of
antenna temperature *T*_{J}, *I*_{J}
= (2 *k*
_{J}^{2}
/ *c*^{2}) *T*_{J}, the
integrated intensity of the *J*
→ *J* - 1 transition can be written as

(17) |

The factor *f*_{cmb} accounts for the effect of the Cosmic
Microwave Background on the measured intensity,
*f*_{cmb} = 1 - (*e*^{h
J
/ k Tex} - 1) /
(*e*^{h J /
k Tcmb}
- 1). Note that *f*_{cmb} ~ 1 for *T*_{ex}
≫ *T*_{cmb}.

The column density of H_{2} associated with this integrated
intensity
is simply N(H_{2}) = 1 / *Z*_{CO}
_{J =
0}^{∞} *N*_{J},
where *Z*_{CO} is the CO abundance relative to molecular
hydrogen, *Z*_{CO} = CO / H_{2}.
For a Milky Way gas phase carbon abundance, and assuming all gas-phase
carbon is locked in CO molecules, *Z*_{CO}
3.2 ×
10^{-4}
(Sofia et
al. 2004).
Note, however, that what matters is the
integrated *Z*_{CO} along a line-of-sight, and CO may become
optically thick well before this abundance is reached (for example,
Fig. 1). Indeed,
Sheffer et
al. (2008)
analyze *Z*_{CO}
in Milky Way lines-of-sight, finding a steep *Z*_{CO}
4.7 ×
10^{-6} (N(H_{2})
/ 10^{21} cm^{-2})^{2.07} for
N(H_{2}) > 2.5 × 10^{20} cm^{-2},
with an order of magnitude scatter (see also
Sonnentrucker
et al. 2007).

When observations in only a couple of transitions
are available, it is useful to assume local thermodynamic equilibrium
applies (LTE) and the system is described by a Boltzmann distribution
with a single temperature. In that case the column density will be
*N*(CO) = *Q*(*T*_{ex})
*e*^{E1 / k Tex}
*N*_{1} / *g*_{1},
where *E*_{1} is the energy of the *J* = 1 state
(*E*_{1} / *k*
5.53 K for CO), and
*Q*(*T*_{ex}) =
_{J =
0}^{∞}
*g*_{J}
*e*^{-EJ / kTex}
corresponds to the partition function at temperature
*T*_{ex} which can be approximated as
*Q*(*T*_{ex}) ~ 2 *k* *T*_{ex} /
*E*_{1} for rotational transitions when
*T*_{ex}
≫ 5.5 K
(Penzias 1975
note this is accurate to ~ 10% even down
to *T*_{ex} ~ 8 K). Using Eq. 17 we can then write

(18) |

Consequently, adopting *Z*_{CO} = 10^{-4} and using
a representative *T*_{ex} = 30 K, we obtain

(19) |

or _{CO}
0.34
M_{} (K km
s^{-1} pc^{2})^{-1}.
These are an order of magnitude smaller than the typical values of
*X*_{CO} and
_{CO} in the
Milky Way disk, as we will discuss in
Section 4. Note that they are approximately
linearly dependent on the assumed *Z*_{CO} and
*T*_{ex} (for
*T*_{ex} ≫ 5.53 K). For a similar calculation that
also includes an expression for non-LTE, see
Papadopoulos
et al. (2012).

**2.5. Insights from Cloud Models**

A key ingredient in further understanding *X*_{CO} in
molecular clouds is
the structure of molecular clouds themselves, which plays an important
role in the radiative transfer. This is important both for the
photodissociating and heating ultraviolet radiation, and for the
emergent intensity of the optically thick CO lines.

The CO *J* = 1 → 0 transition arises well within the
photodissociation
region (PDR) in clouds associated with massive star formation, or
even illuminated by the general diffuse interstellar radiation field
(Maloney &
Black 1988,
Wolfire,
Hollenbach & Tielens 1993).
At those depths gas heating is dominated by the grain photoelectric
effect whereby stellar far-ultraviolet photons are absorbed by dust
grains and eject a hot electron into the gas. The main parameter
governing grain photoelectric heating is the ratio
*T*_{kin}^{0.5} / *n*_{e}, where
is a measure of the
far-ultraviolet field
strength, and *n*_{e} is the electron density. This process
will produce hotter gas and higher excitation in starburst galaxies.
At the high densities of extreme starbursts, the gas temperature and
CO excitation may also be enhanced by collisional coupling between
gas and warm dust grains.

Early efforts to model the CO excitation and luminosity in molecular clouds
using a large velocity gradient model were carried out by
Goldreich
& Kwan (1974).
The CO luminosity-gas mass relation was investigated by
Kutner &
Leung (1985)
using microturbulent models, and by
Wolfire,
Hollenbach & Tielens (1993)
using both microturbulent and macroturbulent
models. In microturbulent models the gas has a
(supersonic) isotropic turbulent velocity field with scales
smaller than the photon mean free path.
In the macroturbulent case the scale size of the turbulence is much
larger than the photon mean free path, and the emission arises from
separate Doppler shifted emitting elements. Microturbulent models
produce a wide range of CO *J* = 1 → 0 profile shapes,
including centrally
peaked, flat topped, and severely centrally self-reversed, while most
observed line profiles are centrally peaked. Macroturbulent models,
on the other hand, only produce centrally peaked profiles if
there are a sufficient number of "clumps" within the beam with
densities *n*
10^{3} cm^{-3} in order to
provide the peak brightness temperature.
Falgarone et
al. (1994)
demonstrated that a turbulent velocity field can
produce both peaked and smooth line profiles, much closer to
observations than macroturbulent models.

Wolfire,
Hollenbach & Tielens (1993)
use PDR models in which the chemistry and thermal balance was calculated
self-consistently as a function of depth into the cloud. The
microturbulent PDR models are very successful in matching and
predicting the intensity of low-J CO lines and the emission from many
other atomic and molecular species
(Hollenbach
& Tielens 1997,
Hollenbach
& Tielens 1999).
For example, the nearly constant ratio of [CII] / CO
*J* = 1 → 0 observed in both Galactic and extragalactic sources
(Crawford et
al. 1985,
Stacey et
al. 1991)
was first explained by PDR models as
arising from high density (*n*
10^{3}
*cm*^{-3}) and high UV
field (
10^{3})
sources in which both the [CII] and CO are
emitted from the same PDR regions in molecular cloud surfaces. These
models show that the dependence on CO luminosity with incident
radiation field is weak. This is because as the field
increases the _{CO}
= 1 surface is driven deeper into the cloud where the dominant heating
process, grain photoelectric heating, is weaker. Thus the dissociation
of CO in higher fields regulates the temperature where
_{CO} = 1.

PDR models have the advantage that they calculate the thermal and
chemical structure in great detail, so that the gas temperature is
determined where the CO line becomes optically thick. We note however,
that although model gas temperatures for the CO *J* = 1 → 0
line are
consistent with observations the model temperatures are typically too
cool to match the observed high-J CO line emission (e.g.,
Habart et
al. 2010).
The model density structure is generally
simple (constant density or constant pressure), and the velocity is
generally considered to be a constant based on a single microturbulent
velocity. More recent dynamical models have started to combine
chemical and thermal calculations with full hydrodynamic simulations.