The relation between_gas_and_dust_in_the_taurus_molecular_cloud

To appear in the Astrophysical Journal
Preprint typeset using L TEX style emulateapj v. 11/10/09
A

THE RELATION BETWEEN GAS AND DUST IN THE TAURUS MOLECULAR CLOUD
Jorge L. Pineda1 , Paul F. Goldsmith1 , Nicholas Chapman1 , Ronald L. Snell2 , Di Li1 , Laurent Cambr´sy3 , and
e
Chris Brunt4
1 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109-8099, USA
2 Department of Astronomy, LGRT 619, University of Massachusetts, 710 North Pleasant Street, Amherst, MA 01003, USA
3 Observatoire Astronomique de Strasbourg, 67000 Strasbourg, France
4 Astrophysics Group, School of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QL, UK

To appear in the Astrophysical Journal
arXiv:1007.5060v1 [astro-ph.GA] 28 Jul 2010

ABSTRACT
We report a study of the relation between dust and gas over a 100 deg2 area in the Taurus molecular
cloud. We compare the H2 column density derived from dust extinction with the CO column density
derived from the 12 CO and 13 CO J = 1 → 0 lines. We derive the visual extinction from reddening
determined from 2MASS data. The comparison is done at an angular size of 200′′ , corresponding to
0.14 pc at a distance of 140 pc. We find that the relation between visual extinction AV and N (CO) is
linear between AV ≃ 3 and 10 mag in the region associated with the B213–L1495 filament. In other
regions the linear relation is flattened for AV 4 mag. We find that the presence of temperature
gradients in the molecular gas affects the determination of N (CO) by ∼30–70% with the largest
difference occurring at large column densities. Adding a correction for this effect and accounting
for the observed relation between the column density of CO and CO2 ices and AV , we find a linear
relationship between the column of carbon monoxide and dust for observed visual extinctions up to
the maximum value in our data ≃ 23 mag. We have used these data to study a sample of dense cores
in Taurus. Fitting an analytical column density profile to these cores we derive an average volume
density of about 1.4 × 104 cm−3 and a CO depletion age of about 4.2 × 105 years. At visual extinctions
smaller than ∼3 mag, we find that the CO fractional abundance is reduced by up to two orders of
magnitude. The data show a large scatter suggesting a range of physical conditions of the gas. We
estimate the H2 mass of Taurus to be about 1.5 × 104 M⊙ , independently derived from the AV and
N (CO) maps. We derive a CO integrated intensity to H2 conversion factor of about 2.1×1020 cm−2 (K
km s−1 )−1 , which applies even in the region where the [CO]/[H2 ] ratio is reduced by up to two orders of
magnitude. The distribution of column densities in our Taurus maps resembles a log–normal function
but shows tails at large and low column densities. The length scale at which the high–column density
tail starts to be noticeable is about 0.4 pc.
Subject headings: ISM: molecules — ISM: structure

1. INTRODUCTION able to trace large column densities. Goldsmith et al.
Interstellar dust and gas provide the primary tools for (2008) used a 100 square degree map of 12 CO and 13 CO
tracing the structure and determining the mass of ex- in the Taurus molecular cloud to derive the distribution
tended clouds as well as more compact, dense regions of N (CO) and N (H2 ). By binning the CO data by exci-
within which new stars form. The most fundamental tation temperature, they were able to estimate the CO
measure of the amount material in molecular clouds is the column densities in individual pixels where 12 CO but not
13
number of H2 molecules along the line of sight averaged CO was detected. The pixels where neither 12 CO or
13
over an area defined by the resolution of the observa- CO were detected were binned together to estimate the
tions, the H2 column density, N (H2 ). Unfortunately, H2 average column density in this portion of the cloud.
has no transitions that can be excited under the typical Extensive work has been done to assess the reliability
conditions of molecular clouds, and therefore it cannot of CO as a tracer of the column of H2 molecules (e.g.
be directly observed in such regions. We have to rely on Frerking et al. 1982; Langer et al. 1989). It has been
indirect methods to determine N (H2 ). Two of the most found that N (CO) is not linearly correlated with N (H2 ),
common methods are observations of CO emission and as the former quantity is sensitive to chemical effects such
dust extinction. as CO depletion at high volume densities (Kramer et al.
Carbon monoxide (CO) is the second most abundant 1999; Caselli et al. 1999; Tafalla et al. 2002) and the
molecular species (after H2 ) in the Universe. Observa- competition between CO formation and destruction at
low-column densities (e.g. van Dishoeck & Black 1988;
tions of 12 CO and 13 CO together with the assumption of
Visser et al. 2009). Moreover, temperature gradients are
local thermodynamic equilibrium (LTE) and moderate
13 likely present in molecular clouds (e.g. Evans et al. 2001)
CO optical depths allow us to determine N (CO) and, affecting the correction of N (CO) for optical depth ef-
assuming an [CO]/[H2 ] abundance ratio, we can obtain fects.
N (H2 ). This method is, however, limited by the sen- The H2 column density can be independently in-
sitivity of the 13 CO observations and therefore is only ferred by measuring the optical or near–infrared light
from background stars that has been extincted by the
Jorge.Pineda@jpl.nasa.gov

2 Pineda, Goldsmith, Chapman, Snell, Li, Cambr´sy & Brunt
e

dust present in the molecular cloud (Lada et al. 1994; far-ultraviolet (FUV) photons. These effects can reduce
Cambr´sy 1999; Dobashi et al. 2005). This method is
e [CO]/[H2 ] by up to three orders of magnitude (e.g.
often regarded as one of the most reliable because it van Dishoeck & Black 1988; Liszt 2007; Visser et al.
does not depend strongly on the physical conditions of 2009). This column density regime has been studied
the dust. But this method is not without some uncer- in dozens of lines-of-sight using UV and optical ab-
tainty. Variations in the total to selective extinction and sorption (e.g. Federman et al. 1980; Sheffer et al. 2002;
dust–to–gas ratio, particularly in denser clouds like those Sonnentrucker et al. 2003; Burgh et al. 2007) as well
in Taurus, may introduce some uncertainty in the con- as in absorption toward mm-wave continuum sources
version of the infrared extinction to gas column density (Liszt & Lucas 1998). The statistical method presented
(Whittet et al. 2001). Dust emission has been also used by Goldsmith et al. (2008) allows the determination
to derive the column density of H2 (Langer et al. 1989). of CO column densities in several hundred thousand
It is, however, strongly dependent on the dust temper- positions in the periphery of the Taurus molecular cloud
ature along the line of sight, which is not always well with N (CO) ≃ 1014 − 1017 cm−3 . A comparison with
characterized and difficult to determine. Neither method the visual extinction will provide a coherent picture of
provides information about the kinematics of the gas. the relation between N (CO) and N (H2 ) from diffuse to
It is therefore of interest to compare column density dense gas in Taurus. These results can be compared
maps derived from 12 CO and 13 CO observations with with theoretical predictions that provide constraints in
dust extinction maps. This will allow us to character- physical parameters such as the strength of the FUV
ize the impact of chemistry and saturation effects in the radiation field, etc.
derivation of N (CO) and N (H2 ) while testing theoreti- Accounting for the various mechanisms affecting the
cal predictions of the physical processes that cause these [CO]/[H2 ] relative abundance allows the determination
effects. of the H2 column density that can be compared with
As mentioned before, CO is frozen onto dust grains in that derived from AV in the Taurus molecular cloud.
regions of relatively low temperature and larger volume It has also been suggested that the total molecular
densities (e.g. Kramer et al. 1999; Tafalla et al. 2002; mass can be determined using only the integrated in-
Bergin et al. 2002). In dense cores, the column densities tensity of the 12 CO J = 1 → 0 line together with the
of C17 O (Bergin et al. 2002) and C18 O (Kramer et al. empirically–derived CO-to-H2 conversion factor (XCO ≡
1999; Alves et al. 1999; Kainulainen et al. 2006) are ob- N (H2 )/ICO ). The XCO factor is thought to be depen-
served to be linearly correlated with AV up to ∼10 mag. dent on the physical conditions of the CO–emitting gas
For larger visual extinctions this relation is flattened with (Maloney & Black 1988) but it has been found to attain
the column density of these species being lower than that the canonical value for our Galaxy even in diffuse re-
expected for a constant abundance relative to H2 . These gions where the [CO]/][H2 ] ratio is strongly affected by
authors showed that the C17 O and C18 O emission is op- CO formation/destruction processes (Liszt 2007). The
tically thin even at visual extinctions larger than 10 mag large–scale maps of N (CO) and AV also allow us to assess
and therefore the flattening of the relation between their whether there is H2 gas that is not traced by CO. This
column density and AV is not due to optical depths ef- so-called “dark gas” is suggested to account for a sub-
fects but to depletion of CO onto dust grains. These stantial fraction of the total molecular gas in our Galaxy
observations suggest drops in the relative abundance of (Grenier et al. 2005).
C18 O averaged along the line-of-sight of up to a factor of Numerical simulations have shown that the proba-
∼3 for visual extinctions between 10 and 30 mag. A sim- bility density function (PDF) of volume densities in
ilar result has been obtained from direct determinations molecular clouds can be fitted by a log-normal dis-
of the column density of CO–ices based on absorption tribution (e.g. Ostriker et al. 2001; Nordlund & Padoan
studies toward embedded and field stars (Chiar et al. 1999; Li et al. 2004; Klessen 2000). The shape of
1995). At the center of dense cores, the [CO]/][H2 ] ratio the distribution is expected to be log-normal as mul-
is expected to be reduced by up to five orders of magni- tiplicative effects determine the volume density of
tude (Bergin & Langer 1997). This has been confirmed a molecular cloud (Passot & V´zquez-Semadeni 1998;
a
by the comparison between observations and radiative V´zquez-Semadeni & Garc´ 2001). A log-normal func-
a ıa
transfer calculations of dust continuum and C18 O emis- tion can also describe the distribution of column
sion in a sample of cores in Taurus (Caselli et al. 1999; densities in a molecular cloud (Ostriker et al. 2001;
Tafalla et al. 2002). The amount of depletion is not only V´zquez-Semadeni & Garc´ 2001). For some molecular
a ıa
dependent on the temperature and density of the gas, clouds the column density distribution can be well fitted
but is also dependent on the timescale. Thus, determin- by a log–normal (e.g. Wong et al. 2008; Goodman et al.
ing the amount of depletion in a large sample of cores 2009). A study by Kainulainen et al. (2009), however,
distributed in a large area is important because it al- showed that in a larger sample of molecular complexes
low us to determine the chemical age of the entire Tau- the column density distribution shows tails at low and
rus molecular cloud while establishing the existence of large column densities. The presence of tails at large col-
any systematic spatial variation that can be a result of a umn densities seems to be linked to active star–formation
large–scale dynamical process that lead to its formation. in clouds. The AV and CO maps can be used to deter-
At low column densities (AV 3 mag, mine the distribution of column densities at large scales
N (CO) 1017 cm−2 ) the relative abundance of CO while allowing us to study variations in its shape in re-
and its isotopes are affected by the relative rates gions with different star–formation activity within Tau-
of formation and destruction, carbon isotope ex- rus.
change and isotope selective photodissociation by In this paper, we compare the CO column den-
sity derived using the 12 CO and 13 CO data from

The relation between gas and dust in the Taurus Molecular Cloud 3

is σTint =0.53 K km s−1 for 12 CO and σTint =0.23 K km s−1
∗ ∗

13
for CO. The map mean signal-to-noise ratio is 9 for
12
CO and 7.5 for 13 CO. Note that these values differ
slightly from those presented by Goldsmith et al. (2008),
as the correction for error beam pick–up produces small
changes in the noise properties of the data.
2.1. CO Column Density in Mask 2
2.1.1. The Antenna Temperature
When we observe a given direction in the sky, the an-
tenna temperature we measure is proportional to the
convolution of the brightness of the sky with the nor-
malized power pattern of the antenna. Deconvolving
the measured set of antenna temperatures is relatively
difficult, computationally expensive, and in consequence
rarely done. The simplest approximation that is made
is that the observed antenna temperature is that coming
Figure 1. Mask regions defined in the Taurus Molecular Cloud. from a source of some arbitrary size, generally that of the
Mask 2 is shown in black, Mask 1 in dark gray, and Mask 0 in light main beam, or else a larger region. It is assumed that
gray. We also show the 156 stellar members of Taurus compiled by
Luhman et al. (2006) as white circles. the measured antenna temperature can be corrected for
the complex antenna response pattern and its coupling
Narayanan et al. 2008 (see also Goldsmith et al. 2008) to the (potentially nonuniform) source by an efficiency,
with a dust extinction map of the Taurus molecular characterizing the coupling to the source. This is often
cloud. The paper is organized as follows: In Section 2 taken to be ηmb , the coupling to an uniform source of
we describe the derivation of the CO column density in size which just fills the main lobe of the antenna pat-
pixels where both 12 CO and 13 CO were detected, where tern. This was the approach used by Goldsmith et al.
12
CO but not 13 CO was detected, as well as in the region (2008). In Appendix A we discuss an improved technique
where no line was detected in each individual pixel. In which corrects for the error pattern of the telescope in
Section 3 we make pixel–by–pixel comparisons between the Fourier space. This technique introduces a “corrected
the derived N (CO) and the visual extinction for the large main–beam temperature scale”, Tmb,c . We can write the
and low column density regimes. In Section 4.1 we com- main–beam corrected temperature as
pare the total mass of Taurus derived from N (CO) and
AV . We also study how good the 12 CO luminosity to- 1 1
gether with a CO-to-H2 conversion factor can determine Tmb,c = T0 − 1 − e−τ , (1)
eT0 /Tex −1 eT0 /Tbg −1
the total mass of a molecular cloud. We study the distri-
bution of column densities in Taurus in Section 4.2. We where T0 = hν/k, Tex is the excitation temperature of the
present a summary of our results in Section 5. transition, Tbg is the background radiation temperature,
and τ is the optical depth. This equation applies to a
2. THE N(H2 ) MAP DERIVED FROM 12 CO AND 13 CO
given frequency of the spectral line, or equivalently, to a
In the following we derive the column density of given velocity, and the optical depth is that appropriate
CO using the FCRAO 14–m 12 CO and 13 CO obser- for the frequency or velocity observed.
vations presented by Narayanan et al. 2008 (see also If we assume that the excitation temperature is inde-
Goldsmith et al. 2008). In this paper we use data cor- pendent of velocity (which is equivalent to an assumption
rected for error beam pick-up using the method pre- about the uniformity of the excitation along the line–of–
sented by Bensch et al. (2001). The correction proce- sight) and integrate over velocity we obtain
dure is described in Appendix A. The correction for er- T0 C(Tex )
ror beam pick–up improves the calibration by 25–30%. Tmb,c (v)dv = (1 − e−τ (v) )dv , (2)
We also improved the determination of N (CO) compared eT0 /Tex − 1
to that presented by Goldsmith et al. (2008) by includ- where we have included explicitly the dependence of the
ing an updated value of the spontaneous decay rate and corrected main–beam temperature and the optical depth
using an exact numerical rather than approximate an- on velocity. The function C(Tex ), which is equal to unity
alytical calculation of the partition function. The val- in the limit Tbg → 0, is given by
ues of the CO column density are about ∼20% larger
than those presented by Goldsmith et al. (2008). Follow- eT0 /Tex − 1
ing Goldsmith et al. (2008), we define Mask 2 as pixels C(Tex ) = 1− . (3)
eT0 /Tbg − 1
where both 12 CO and 13 CO are detected, Mask 1 as pix-
els where 12 CO is detected but 13 CO is not, and Mask 0
as pixels where neither 12 CO nor 13 CO are detected. We 2.1.2. The Optical Depth
consider a line to be detected in a pixel when its intensity, The optical depth is determined by the difference in the
integrated over the velocity range between 0–12 km s−1 , populations of the upper and lower levels of the transition
is at least 3.5 times larger than the rms noise over the observed. If we assume that the line–of–sight is charac-
same velocity interval. We show the mask regions in Fig- terized by upper and lower level column densities, NU
ure 1. The map mean rms noise over this velocity range and NL , respectively, the optical depth is given by

e

Figure 2. Correction factors for the relation between integrated
main–beam temperature and upper level column density for a
Gaussian velocity distribution of the optical depth. The dotted
(blue) curve shows the correction factor obtained using integrals of
functions of the optical depth as given by Equation 15. The solid
(red) curve shows the correction factor employing the peak values
of the functions, given by Equation 16. Figure 3. Parameters of Mask 1 binned by 12 CO excitation tem-
perature Tex . The bottom panel shows the derived H2 density (left-
hand scale, squares) and the number of pixels in each Tex bin (right-
hand scale, triangles). The most common Tex values are between
hν0 5 − 9 K. The middle panel shows the observed (left-hand scale,
τ= φ(ν) [NL BLU − NU BUL ] , (4)
c squares) and derived (right-hand scale, triangles) 12 CO/13 CO ra-
tio. The top panel shows the derived 12 CO column density as-
where ν0 is the frequency of the transition, φ(ν) is the suming a line width of 1 Km s−1 . The H2 density, 12 CO column
line profile function, and the B’s are the Einstein B- density and derived 12 CO/13 CO ratio increase monotonically as a
coefficients. The line profile function is a function of the function of 12 CO excitation temperature.
frequency and describes the relative number of molecules
at each frequency (determined by relative Doppler veloc- gives us
ity). It is normalized such that φ(ν)dν = 1. For a
Gaussian line profile, the line profile function at line cen- c2 AUL φ(ν)NU T0 /Tex
τ (ν) = e −1 . (9)
ter is given approximately by φ(ν0 ) = 1/δνFWHM , where 8πν0 2
δνFWHM is the full width at the half maximum of the line
profile. If we integrate both sides of this equation over a range
We have assumed that the excitation temperature is of frequencies encompassing the entire spectral line of
uniform along the line of sight. Thus, we can define the interest, we find
excitation temperature in terms of the upper and lower c2 AUL NU T0 /Tex
level column densities, and we can write τ (ν)dν = e −1 . (10)
8πν0 2
NU gU −T0 /Tex
= e , (5)
NL gL
2.1.3. Upper Level Column Density
where the g’s are the statistical weights of the two levels. It is generally more convenient to describe the optical
The relationship between the B coefficients, depth in terms of the velocity offset relative to that of
gU BUL = gL BLU , (6) the nominal line center. The incremental frequency and
velocity are related through dv = (c/ν0 )dν, and hence
lets us write τ (ν)dν = (c/ν0 ) τ (v)dv. Thus we obtain
hνo BUL φ(ν)NU T0 /Tex c3 AUL NU T0 /Tex
τ (ν) = e −1 . (7) τ (v)dv = e −1 . (11)
c 8πν0 3
Substituting the relationship between the A and B coef-
ficients, We can rewrite this as
3
8πhν0 1 c3 AUL NU 1
AUL = BUL 3
, (8) = . (12)
c eT0 /Tex −1 8πν0 3 τ (v)dv


Table 1
12 CO Excitation Temperature Bins in Mask 1 and Best Estimates of Their Characteristics

Tex 12 CO/13 CO Number n(H2 ) N (CO)/δv 12 CO/13 CO

(K) Observed of Pixels (cm−3 ) (1016 cm−2 /km s−1 ) Abundance Ratio
5.5....................... 21.21 118567 250 0.56 30
6.5....................... 17.29 218220 275 0.95 30
7.5....................... 14.04 252399 275 1.6 30
8.5....................... 12.43 223632 300 2.3 32
9.5....................... 11.76 142525 300 3.6 40
10.5...................... 11.44 68091 400 4.1 45
11.5...................... 11.20 24608 500 5.3 55
12.5...................... 11.09 6852 700 6.7 69

Substituting this into Equation (2), we can write an ex- where B0 is the rotational constant of 13 CO (B0 = 5.51×
pression for the upper level column density as 1010 s−1 ) and Z is the partition function which is given
2 by
8πkν0 τ (v)dv ∞
NU = Tmb,c (v)dv. −hB0 (J+1)
hc3 AUL C(Tex ) (1 − e−τ (v) )dv Z= (2J + 1)e KTex . (18)
(13) J=0
For the calculation of the 13 CO column densities (Sec-
tion 2.1.4) we use a value for the Einstein A-coefficient The partition function can be evaluated explicitly as a
sum, but Penzias (1975) pointed out that for tempera-
of AUL =6.33×10−8 s−1 (Goorvitch 1994).
tures T ≫ hB0 /K, the partition function can be approx-
In the limit of optically thin emission for which τ (v) imated by a definite integral, which has value kT /hB0 .
≪ 1 for all v, and neglecting the background term in This form for the partition function of a rigid rotor
Equation (3)1 , the expression in square brackets is unity
molecule is almost universally employed, but it does con-
and we regain the much simpler expression tribute a small error at the relatively low temperatures of
2 dark clouds. Specifically, the integral approximation al-
8πkν0 ways yields a value of Z which is smaller than the correct
NU (thin) = Tmb,c (v)dv . (14)
hc3 AUL value. Calculating Z explicitly shows that this quantity
We will, however, use the general form of NU given in is underestimated by a factor of ∼1.1 in the range be-
Equation (13) for the determination of the CO column tween 8 K to 10K. Note that to evaluate Equation (18)
density. we assume LTE (i.e. constant excitation temperature)
We note that the factor in square brackets in Equa- which might not hold for high–J transitions. The error
tion (13) involves the integrals of functions of the optical due to this approximation is, however, very small. For
depth over velocity, not just the functions themselves. example, for Tex =10 K, only 7% of the populated states
There is a difference, which is shown in Figure 2, where is at J = 3 or higher.
we plot the two functions We can calculate the column density of 13 CO from
Equation (17) determining the excitation temperature
τ (v)dv Tex and the 13 CO optical depth from 12 CO and 13 CO
CF (integral) = , (15) observations. To estimate Tex we assume that the 12 CO
(1 − e−τ (v) )dv
line is optically thick (τ ≫ 1) in Equation (1). This
and results in
τ0
CF (peak) = , (16)
1 − e−τ0 5.53
Tex = , (19)
5.53
as a function of the peak optical depth τ0 . There is a ln 1 + 12
Tmb,c +0.83
substantial difference at high optical depth, which re-
flects the fact that the line center has the highest optical 12
where Tmb,c is the peak corrected main-beam bright-
depth so that using this value rather than the integral ness temperature of 12 CO. The excitation temperature
tends to overestimate the correction factor. in Mask 2 ranges from 4 to 19 K with a mean value of
13 13 9.7 K and standard deviation of 1.2 K.
2.1.4. Total CO column densities derived from CO and
12
CO observations.
Also from Equation (1), the optical depth as a function
of velocity of the 13 CO J = 1 → 0 line is obtained from
In LTE, the column density of the upper level (J = 1) the main-beam brightness temperature using
is related to the total 13 CO column density by
13 −1
Z hB0 J(J+1) Tmb,c (v) −1
N13 CO = NU e KTex (17) τ 13 (v) = − ln 1 − e5.29/Tex − 1 − 0.16 ,
(2J + 1) 5.29
1 This usually does not result in a significant error since in LTE
(20)
even in dark clouds Tex is close to 10 K as compared to Tbg = 2.7 13
where Tmb,c is the peak corrected main-beam brightness
K. Since Tbg is significantly less than T0 , the background term is
far from the Rayleigh–Jeans limit further reducing its magnitude temperature of 13 CO. We use this expression in Equa-
relative to that of the first term. tion (15) to determine opacity correction factor. We

e

evaluate the integrals in Equation (15) numerically. The ties. The free parameters in the modeling are tempera-
correction factor ranges from 1 to ∼4 with a mean value ture (T ), density (n), CO column density per unit line
of 1.3 and standard deviation of 0.2. The 13 CO column width (N (CO)/δv), and the 12 CO/13 CO abundance ratio
density is transformed to 12 CO column density assuming (R). Since the excitation is determined by both density
a 12 CO/13 CO isotope ratio of 69 (Wilson 1999), which and the amount of trapping (N/δv), there is a family of
should apply for the well–shielded material in Mask 2. n − N (CO)/δv parameters that give the same excitation
temperature. The other information we have is the 13 CO
2.1.5. Correction for Temperature Gradients along the Line integrated intensity for the average spectrum in each bin.
of Sight Thus the choice of n, N (CO)/δv and R must reproduce
In the derivation of the CO column density and its the excitation temperature and the observed 12 CO/13 CO
opacity correction we made the assumption that the gas ratio. Solutions also must have an optical depth in the
12
is isothermal. But observations suggest the existence of CO J = 1 → 0 of at least 3, to be consistent with
core-to-edge temperature differences in molecular clouds the assumption that this isotopologue is optically thick.
(e.g. Evans et al. 2001) which can be found even in re- This is the same method used in Goldsmith et al. (2008),
gions of only moderate radiation field intensity. There- although this time we used the RADEX program and the
fore the presence of temperature gradients might affect updated cross-sections from LAMDA.
our opacity correction. In fact, at low excitation temperature the data can only
We used the radiative transfer code RATRAN be fit if the CO is strongly fractionated. At high excita-
(Hogerheijde & van der Tak 2000) to study the effects of tion temperature we believe that the CO is unlikely to be
temperature gradients on the determination of N (CO). fractionated, and thus, R must vary with excitation tem-
The modeling is described in the Appendix C. We found perature. We chose solutions for Mask 1 that produced
that using 12 CO to determine the excitation tempera- both a monotonically decreasing R with decreasing ex-
ture of the CO gas gives the correct temperature only citation temperature and a smoothly decreasing column
at low column densities while the temperature is over- density with decreasing excitation temperature. The so-
estimated for larger column densities. This produces an lutions are given in Table 1 and shown Figure 3. The
underestimate of the 13 CO opacity which in turn affects uncertainty resulting from the assumption of a fixed ki-
the opacity correction of N (CO). This results in an un- netic temperature and from choosing the best value for R
derestimation of N (CO). We derived a correction for is about a factor of 2 in N (CO) (Goldsmith et al. 2008).
this effect (Equation [C2]) which is applied to the data. To obtain N (CO) per unit line width for a given value
of the excitation temperature we have used a non-linear
2.2. CO Column Density in Mask 1 fit to the data, and obtained the fitted function:
The column density of CO in molecular clouds is com-
monly determined from observations of 12 CO and 13 CO N (CO) δv
−1
Tex
2.7
with the assumption of Local Thermodynamic Equilib- = 6.5 × 1013 . (21)
cm−2 km s−1 K
rium (LTE), as discussed in the previous section. The
lower limit of N (CO) that can be determined is there- We multiply by the observed FWHM line width to de-
fore set by the detection limit of the 13 CO J = 1 → 0 termine the total CO column density. The upper panel
line. For large maps, however, it is possible to determine in Figure 3 shows N (CO)/δv as a function of Tex .
N (CO) in regions where only 12 CO is detected in indi-
vidual pixels by using the statistical approach presented 2.3. CO Column Density in Mask 0
by Goldsmith et al. (2008). In the following we use this To determine the carbon monoxide column density in
approach to determine the column density of CO in Mask regions where neither 12 CO nor 13 CO were detected, we
1. average nearly 106 spectra to obtain a single 12 CO and
We compute the excitation temperature from the 12 CO 13
CO spectra. From the averaged spectra we obtain a
peak intensities for all positions in Mask 1 assuming that 12
CO/13 CO integrated intensity ratio of ≃17. We need
the emission is optically thick. The Mask 1 data is then a relatively low R to reproduce such a low observed value.
binned by excitation temperature (in 1 K bins), and the Values of R = 25 or larger cannot reproduce the observed
13
CO data for all positions within each bin averaged to- isotopic ratio and still produce 12 CO emission below the
gether. In all bins we get a very significant detection detection threshold. Choosing R = 20 and a gas kinetic
of 13 CO from the bin average. Thus, we have the exci- temperature of 15 K, we fit the observed ratio with n =
tation temperature and the observed ratio of integrated 100 cm−3 and N (CO) = 3×1015 cm−2 . This gives rise to
intensities (12 CO/13 CO) in each 1 K bin. Since positions a 12 CO intensity of 0.7 K, below the detection threshold,
in Mask 1 are distributed in the periphery of high ex- however much stronger than the Mask 0 average of only
tinction regions, it is reasonable to assume that the gas 0.18 K. Thus, much of Mask 0 must not contribute to
volume density in this region is modest, and thus LTE the CO emission. In fact, only 26% of the Mask 0 area
does not necessarily apply, as thermalization would imply can have the properties summarized above, producing
an unreasonably low gas temperature at the cloud edges. significant CO emission. Therefore, the average column
We therefore assume that 12 CO is sub-thermally excited density2 throughout Mask 0 is 7.8 × 1014 cm−2 .
and that the gas has a kinetic temperature of 15 K. We
use the RADEX program (van der Tak et al. 2007), us- 2 Note that the estimate of the CO column density in Mask 0
ing the LVG approximation, and the collision cross sec- by Goldsmith et al. (2008) did not include the ∼26% filling factor
tions from the Leiden Atomic and Molecular Database we derived here and in consequence overestimated the CO column
(LAMDA; Schïer et al. 2005), to compute line intensi-
o density in this region.


23

CO Column Density (10 17 cm −2 )
0.003

24

Visual Extinction (mag)

0

Figure 4. Maps of the CO column density (upper panel) and visual extinction (lower panel) in the Taurus Molecular cloud. The gray-scale
in the N (CO) and AV maps is expressed as the square root of the CO column density and of the visual extinction, respectively. The angular
resolution of the data in the ﬁgure is 40′′ for N (CO) and 200′′ for AV .

e

Figure 5. Histogram of the 12 CO column density distributions
in the Mask 0, 1, and 2 regions mapped in Taurus. The Mask 0
is indicated by a vertical line at N (CO) = 3 × 1015 cm−2 which
represents the column density in the CO–emitting region (26% of
the area of Mask 0; see Section 2.3). Note that we have not yet
corrected N (CO) in Mask 2 for the effect of temperature gradients Figure 6. Comparison between the visual extinction derived
in the opacity correction. from 2MASS stellar colors and the 12 CO column density derived
from 13 CO and 12 CO observations in Taurus. The dark blue line
Another option is to model the average spectra of 12 CO represents the 12 CO column density derived from AV assuming
and 13 CO matching both the ratio and intensity. Since N (H2 )/AV = 9.4 × 1020 cm−2 mag−1 (Bohlin et al. 1978) and a
now, our goal is to produce CO emission with inten- [CO]/[H2 ] abundance ratio of 1.1 × 10−4 . The gray scale repre-
sity 0.18 K, both 12 CO and 13 CO will be optically thin. sents the number of pixels of a given value in the parameter space
and is logarithmic in the number of pixels. The red contours are
Therefore we need an R that is equal to the observed 2,10,100, and 1000 pixels. Each pixel has a size of 100′′ or 0.07 pc
ratio. For R = 18, a solution with n = 100 cm−3 , at a distance of 140 pc.
δv = 1 km s−1 , and N (CO) = 7.3×1014 cm−2 fits both
the 12 CO and 13 CO average spectra for Mask 0. Note visual extinction and N (CO) are linearly correlated up
that this is very similar to the average solution (with a to about AV ≃ 10 mag. For larger visual extinctions
slightly larger R) that assumes that ∼26% of the area N (CO) is largely uncorrelated with the value of AV . In
has column density 3×1015 cm−2 and the rest 0. Thus the range 3 < AV < 10 mag, for a given value of AV ,
for a density of 100 cm−3 , the average CO column den- the mean value of N (CO) is roughly that expected for
sity must be about 7.8×1014 cm−2 in either model. Of a [CO]/[H2 ] relative abundance of ∼10−4 which is ex-
course, if we picked a different density we would get a pected for shielded regions (Solomon & Klemperer 1972;
slightly different column density. As mentioned above, Herbst & Klemperer 1973). Some pixels, however, have
the uncertainty is N (CO) is about a factor of 2. CO column densities that suggest a relative abundance
Note that the effective area of CO emission is uniformly that is reduced by up to a factor of ∼3. In the plot
spread over Mask 0. We subdivided the 12 CO data cube we show lines defining regions containing pixels with
in the Mask 0 region in an uniform grid with each bin AV > 10 mag and with 3 < AV < 10 mag and N (CO) >
containing about 104 pixels. After averaging the spectra 9 × 1017 cm−2 . In Figure 7 we show the spatial distribu-
in each bin we find significant 12 CO emission in 95% of tion of these pixels in N (CO) maps of the B213-L1457,
them. Heiles’s cloud 2, and B18-L1536 regions. White con-
3. COMPARISON BETWEEN AV AND N (12 CO) tours correspond to the pixels with AV > 10 mag and
black contours to pixels with 3 < AV < 10 mag and
In order to test our estimate of N (CO) and assess N (CO) > 9 × 1017 cm−2 . Regions with AV > 10 mag
whether it is a good tracer of N (H2 ), we compare Mask 1 are compact and they likely correspond to the center of
and 2 in our CO column density map of Taurus with a dense cores. The largest values of N (CO), however, are
dust extinction map derived from 2MASS stellar colors. not always spatially correlated with such regions. We no-
Maps of these quantities are shown in Figure 4. We also tice that large N (CO) in the AV = 3 − 10 mag range are
show in Figure 5 a histogram of the 12 CO column den- mostly located in the B213–L1457 filament. We study
sity distributions in the Mask 0, 1, and 2 regions mapped the relation between AV and N (CO) in this filament by
in Taurus. The derivation of the dust extinction map is applying a mask to isolate this region (see marked re-
described in Appendix B. The resolution of the map is gion in Figure 7). We show the relation between AV
200′′ (0.14 pc at a distance of 140 pc) with a pixel spacing and N (CO) in the B213–L1457 filament in the left hand
of 100′′ . For the comparison, we have convolved and re- panel of Figure 8. We also show this relation for the
gridded the CO column density map in order to match entire Taurus molecular cloud excluding this filament in
this resolution and pixel spacing. the right hand panel. Visual extinction and CO column
density are linearly correlated in the B213–L1457 fila-
3.1. Large N (12 CO) Column Densities ment with the exception of a few pixels that are located
We show in Figure 6 a pixel-by-pixel comparison be- in dense cores (Cores 3, 6 and 7 in Table 2). Without
tween visual extinction and 12 CO column density. The the filament the N (CO)/AV relation is linear only up to


∼4 magnitudes of extinction. In Section 3.1.1 we will
see that the deviation from a linear N (CO)/AV relation
is mostly due to depletion of CO molecules onto dust
grains. Depletion starts to be noticeable for AV ≥ 4 mag.
Therefore, pixels on the B213–L1457 filament appear to
show no signatures of depletion. This can be due either
to the filament being chemically young in contrast with
the rest of Taurus, or to the volume densities being low
enough that desorption processes dominate over those of
adsorption. If the latter case applies, and assuming a
volume density of n(H2 ) = 103 cm−3 (low enough to not
show significant CO depletion but still larger than the
critical density of the 13 CO J = 1 → 0 line), this fila-
ment would need to be extended along the line-of-sight
by 0.9–3 pc for 3 < AV < 10 mag. This length is much
larger than the projected thickness of the B213–L1495 fil-
ament of ∼0.2 pc but comparable to its length of ∼7 pc.
We will study the nature of this filament in a separate
paper.
Considering only regions with AV < 10 mag and
N (CO) > 1017 cm−2 (see Section 3.2) we fit a straight
line to the data in Figure 6 to derive the [CO]/[H2 ]
relative abundance in Mask 2. A least squares fit re-
sults in N (CO)/cm−2 = (1.01 ± 0.008) × 1017 AV /mag.
Assuming that all hydrogen is in molecular form
we can write the ratio between H2 column density
and color excess observed by Bohlin et al. (1978) as
N (H2 )/EB−V =2.9×1021 cm−2 mag−1 . We combine this
relation with the ratio of total to selective extinction
RV = AV /EB−V ≃ 3.1 (e.g. Whittet 2003) to ob-
tain N (H2 )/AV = 9.4 × 1020 22cm−2 mag−1 . Combin-
ing the N (H2 )/AV relation with our fit to the data,
we obtain a [CO]/[H2 ] relative abundance of 1.1×10−4.
Note that, as discussed in Appendix B, grain growth
would increase the value of RV up to ∼4.5 in dense re-
gions (Whittet et al. 2001). Due to this effect, we esti-
mate that the derived AV would increase up to 20% for
AV ≤10 mag. This would reduce the N (H2 )/AV con-
version but also increase the AV /N (CO) ratio. Thus the
derived [CO]/[H2 ] abundance is not significantly affected.
3.1.1. CO depletion
The flattening of the AV –N (CO) relation for AV >
10 mag could be due to CO depletion onto dust grains.
This is supported by observations of the pre-stellar core
B68 by Bergin et al. (2002) which show a linear increase
in the optically thin C18 O and C17 O intensity as a func-
tion of AV up to ∼7 mag, after which the there is a
turnover in the intensity of these molecules. This is simi-
lar to what we see in Figure 6. Note, however, AV alone is
not the sole parameter determining CO freeze-out, since
this process also depends on density and timescale (e.g.
Bergin & Langer 1997).
Following Whittet et al. (2010), we test the possibility
Figure 7. N (CO) maps of the B213–L1495 (top), Heiles’s cloud that effects of CO depletion are present in our observa-
2 (middle), and B18–L1536 (bottom) regions. The white contours tions of the Taurus molecular cloud by accounting for
denote regions with AV > 10 mag, while the black contours denote
regions with AV < 10 mag and N (CO) > 9 × 1017 cm−2 (see the column of CO observed to be in the form of ice on
Figure 6). The blue contour outlines approximately the B213– the dust grains. Whittet et al. (2007) measured the col-
L1457 filament. umn density of CO and CO2 ices3 toward a sample of
stars located behind the Taurus molecular cloud. They
3 It is predicted that oxidation reactions involving the CO
molecules depleted from the gas–phase can produce substantial
amounts of CO2 in the surface of dust grains (Tielens & Hagen

e

Figure 8. Pixel–by–pixel comparison between AV and N (CO) in the B213–L1457 filament (left) and the entire Taurus molecular cloud
without this filament (right).

grains is given by

N (CO)total = N (CO)ice + N (CO2 )ice .
ice (24)
Thus, for a given AV the total CO column density is
given by

N (CO)total = N (CO)gas−phase + N (CO)total .
ice (25)
We can combine our determination of the column den-
sity of gas-phase CO with that of CO ices to plot the
total N (CO) as a function of AV . The result is shown
in Figure 9. The visual extinction and N (CO)total are
linearly correlated over the entire range covered by our
data, extending up to AV = 23 mag. This result confirms
that depletion is the origin of the deficit of gas-phase CO
seen in Figure 6.
In Figure 10 we show the ratio of N (CO)total to
N (CO)gas−phase as a function of AV , for AV greater than
Figure 9. The same as Figure 6 but including the estimated 10. The drop in the relative abundance of gas-phase CO
column density of CO and CO2 ices. For comparison we show from our observations is at most a factor of ∼2. This is in
the relation between visual extinction and N (CO) derived from
observations of rare isotopic species by Frerking et al. (1982) (see agreement with previous determinations of the depletion
Appendix C) which also include the contribution for CO and CO2 along the line of sight in molecular clouds (Kramer et al.
ices. 1999; Chiar et al. 1995).
find that the column densities are related to the visual
extinction as 3.1.2. CO Depletion Age
In this Section we estimate the CO depletion age (i.e.
N (CO)ice the time needed for CO molecules to deplete onto dust
= 0.4(AV − 6.7), AV > 6.7 mag, (22)
1017 [cm−2 ] grains to the observed levels) in dense regions in the Tau-
and rus Molecular Cloud. We selected a sample of 13 cores
that have peak visual extinction larger than 10 mag and
N (CO2 )ice that AV at the edges drops below ∼0.9 mag (3 times the
= 0.252(AV − 4.0), AV > 4.0 mag. (23)
1017 [cm−2 ] uncertainty in the determination of AV ). The cores are
located in the L1495 and B18–L1536 regions (Figure 7).
We assume that the total column of CO frozen onto dust Unfortunately, we were not able to identify individual
1982; Ruffle & Herbst 2001; Roser et al. 2001). Since the timescale
cores in Heiles’s Cloud 2 due to blending.
of these his reactions are short compared with the cloud’s lifetime, We first determine the H2 volume density structure of
we need to include CO2 in order to account for the amount of CO our selected cores. Dapp & Basu (2009) proposed using
frozen into dust grains along the line–of–sight. the King (1962) density profile,


Figure 10. Ratio of N (CO)total to N (CO)gas−phase plotted as
a function of AV for the high extinction portion of the Taurus
molecular cloud. The line represents our fit to the data.

nc a2 /(r2 + a2 ) r≤R
n(r) = (26)
0 r > R,
which is characterized by the central volume density nc ,
a truncation radius R, and by a central region of size a
with approximately constant density.
The column density N (x) at an offset from the core
center x can be derived by integrating the volume density
along a line of sight through the sphere. Defining Nc ≡
2anc arctan(c) and c = R/a, the column density can be
written

Nc
N (x) =
1 + (x/a)2
c2 − (x/a)2
× arctan( )/ arctan(c) . (27)
1 + (x/a)2
This column density profile can be fitted to the data.
The three parameters to fit are (1) the outer radius R,
(2) the central column density Nc (which in our case is
AV,c ), and (3) the size of the uniform density region a.
We obtain a column density profile for each core by
fitting an elliptical Gaussian to the data to obtain its
central coordinates, position angle, and major and mi-
nor axes. With this information we average the data in
concentric elliptical bins. Typical column density pro-
files and fits to the data are shown in Figure 11. We
give the derived parameters of the 13 cores we have
analyzed in Table 2. We convert the visual extinction
at the core center AV,c to H2 column density assuming
N (H2 )/AV = 9.4 × 1020 cm−2 mag−1 . We use then the
definition of column density at the core center (see above)
to determine the central volume density nc (H2 ) from the Figure 11. Typical radial distributions of the visual extinction
fitted parameters. in the selected sample of cores. The solid lines represent the cor-
With the H2 volume density structure, we can derive responding fit.
the CO depletion age for each core. The time needed for

e

Table 2
Core Parameters

Core ID α(J2000) δ(J2000) AV,c a Radius nc (H2 ) Mass Depletion Age
[mag] [pc] [pc] [104 cm−3 ] [M⊙ ] [105 years]
1 04:13:51.63 28:13:18.6 22.4±0.5 0.10±0.004 2.01±0.30 2.2±0.11 307±102 6.3±0.3
2 04:17:13.52 28:20:03.8 10.7±0.3 0.19±0.021 0.54±0.12 0.7±0.09 56±43 3.4±1.5
3 04:18:05.13 27:34:01.6 12.3±1.2 0.16±0.054 0.32±0.18 1.0±0.40 29±63 1.3±3.1
4 04:18:27.84 28:27:16.3 24.2±0.4 0.13±0.005 1.27±0.08 1.9±0.07 258±52 3.8±0.2
5 04:18:45.66 25:18:0.4 9.4±0.2 0.09±0.005 2.00±0.93 1.1±0.07 110±81 10.9±0.8
6 04:19:14.99 27:14:36.4 14.3±0.6 0.12±0.010 0.89±0.15 1.3±0.12 93±47 3.1±0.6
7 04:21:08.46 27:02:03.2 15.2±0.3 0.08±0.003 1.12±0.08 1.9±0.07 90±19 2.9±0.2
8 04:23:33.84 25:03:01.6 14.4±0.3 0.11±0.004 0.93±0.07 1.4±0.06 94±23 5.1±0.3
9 04:26:39.29 24:37:07.9 15.6±0.5 0.09±0.006 1.48±0.23 1.7±0.12 143±58 2.3±0.3
10 04:29:20.71 24:32:35.6 17.2±0.4 0.13±0.006 2.48±0.39 1.3±0.07 371±127 4.5±0.3
11 04:32:09.32 24:28:39.0 16.0±0.5 0.09±0.006 3.50±2.41 1.7±0.13 347±347 3.2±0.4
12 04:33:16.62 22:42:59.6 12.2±0.5 0.08±0.007 1.66±0.64 1.5±0.14 110±82 6.3±0.7
13 04:35:34.29 24:06:18.2 12.5±0.3 0.11±0.006 1.64±0.35 1.2±0.07 145±65 2.0±0.4

CO molecules to deplete to a specified degree onto dust
grains is given by (e.g. Bergin & Tafalla 2007),

−1
5 × 109 n(H2 )
tdepletion = ln(n0 /ngas ), (28)
yr cm−3
where n0 is the total gas–phase density of CO before
depletion started and ngas the gas-phase CO density at
time tdepletion. Here we assumed a sticking coefficient4
of unity (Bisschop et al. 2006) and that at the H2 vol-
ume densities of interest adsorption mechanisms domi-
nate over those of desorption (we therefore assume that
the desorption rate is zero).
To estimate n0 /ngas we assume that CO depletion oc-
curs only in the flat density region of a core, as for larger
radii the volume density drops rapidly. Then the total
column density of CO (gas–phase+ices) in this region is
given by N (CO)flat ≃ 2anc (1.1 × 10−4 ). The gas-phase
CO column density in the flat density region of a core
is given by Ngas−phase (CO) = N (CO)flat − N (CO)total ,
flat
ice
where N (CO)total can be derived from Equation (25).
ice
Assuming that the decrease in the [CO]/[H2 ] relative
abundance in the flat region is fast and stays constant
toward the center of the core (models from Tafalla et al.
(2002) suggest an exponential decrease), then n0 /ngas ≃
N (CO)flat /Ngas−phase (CO). The derived CO depletion
flat

ages are listed in Table 2. Note that the fitted cores
might not be fully resolved at the resolution of our AV
map (200′′ or 0.14 pc at a distance of 140 pc). Although
n0 /ngas is not very sensitive to resolution, due to mass
conservation, we might be underestimating the density
at the core center. Therefore, our estimates of the CO
depletion age might be considered as upper limits.
In Figure 12 we show the central density and the cor-
responding depletion age of the fitted cores as a function
of AV . The central volume density is well correlated
with AV but varies only over a small range: its mean
value and standard deviation are (1.4 ± 0.4)×104 cm−3 .
Still, the moderate increase of n(H2 ) with AV com-
pensates for the increase of N (CO)total /N (CO)gas−phase
with AV to produce an almost constant depletion age.
Figure 12. (upper panel) The central H2 volume density as a
The mean value and standard deviation of tdepletion are function of the peak AV for a sample of 13 cores in the Taurus
molecular cloud. The line represent a fit to the data. (lower panel)
4 The sticking coefficient is defined as how often a species will
CO depletion age as a function of AV for the sample of cores.
remain on the grain upon impact (Bergin & Tafalla 2007).


(4.2 ± 2.4) × 105 years. This suggests that dense cores at- Sheffer et al. (2008) is much smaller than that shown in
tained their current central densities at a similar moment Figure 13. This indicates that we are tracing a wider
in the history of the Taurus molecular cloud. range of physical conditions of the gas. The excitation
temperature of the gas observed by Sheffer et al. (2008)
3.2. Low N (12 CO) column densities does not show a large variation from Tex =5 K while we
In the following we compare the lowest values of the observe values between 4 and 15 K.
CO column density in our Taurus survey with the visual In Figure 13 we see that some regions can have large
extinction derived from 2MASS stellar colors. In Fig- [12 CO]/[H2 ] abundance ratios but still have very small
ure 13 we show a comparison between N (CO) and AV column densities (AV = 0.1 − 0.5 mag). This can be
for values lower than 5 magnitudes of visual extinction. understood in terms of a medium which is made of an
The figure includes CO column densities for pixels lo- ensemble of spatially unresolved dense clumps embedded
cated in Mask 1 and 2. We do not include pixels in Mask in a low density interclump medium (Bensch 2006). In
0 because its single value does not trace variations with this scenario, the contribution to the total column den-
AV . Instead, we include a horizontal line indicating the sity from dense clumps dominates over that from the
derived average CO column density in this Mask region. tenuous inter–clump medium. Therefore the total col-
We show a straight line (blue) that indicates N (CO) ex- umn density is proportional to the number of clumps
along the line–of–sight. A low number of clumps along a
pected from a abundance ratio [12 CO]/[H2 ]=1.1×10−4 line–of–sight would give low column densities while in the
(Section 3.1). The points indicate the average AV in a interior of these dense clumps CO is well shielded against
N (CO) bin. We present a fit to this relation in Figure 14. FUV photons and therefore it can reach the asymptotic
The data are better described by a varying [12 CO]/[H2 ] value of the [CO]/[H2 ] ratio characteristic of dark clouds.
abundance ratio than a fixed one. This might be caused
by photodissociation and fractionation of CO which can 4. DISCUSSION
produce strong variations in the CO abundances between
UV-exposed and shielded regions (van Dishoeck & Black 4.1. The mass of the Taurus Molecular Cloud
1988; Visser et al. 2009). To test this possibility we in- In this section we estimate the mass of the Taurus
clude in the figure several models of these effects pro- Molecular Cloud using the N (CO) and AV maps. The
vided by Ruud Visser (see Visser et al. 2009 for details). masses derived for Mask 0, 1, and 2 are listed in Table 3.
They show the relation between AV and N (H2 ) for dif- To derive the H2 mass from N (CO) we need to apply an
ferent values of the FUV radiation field starting from appropriate [CO]/[H2 ] relative abundance for each mask.
χ = 1.0 to 0.1 (in units of the mean interstellar radia- The simplest case is Mask 2 where we used the asymp-
tion field derived by Draine 1978). All models have a totic 12 CO abundance of 1.1×10−4 (see Section 3.1). We
kinetic temperature of 15 K and a total H volume den- corrected for saturation including temperature gradients
sity of 800 cm−3 which corresponds to n(H2 ) ≃ 395 cm−3 and for depletion in the mass calculation from N (CO).
assuming n(H i)= 10 cm−3 . (This value of n(H2 ) is close These corrections amount to ∼319 M⊙ (4 M⊙ from the
to the average in Mask 1 of 375 cm−3 .) The observed re- saturation correction and 315 M⊙ from the addition of
lation between AV and N (CO) cannot be reproduced by the column density of CO–ices). For Mask 1 and 0, we
a model with a single value of χ. This suggests that the use the fit to the relation between N (H2 ) and N (CO)
gas have a range of physical conditions. Considering the shown in Figure 14. As we can see in Table 3, the masses
average value of AV within each bin covering a range in derived from AV and N (CO) are very similar. This con-
N (CO) of 0.25 dex, we see that for an increasing value firms that N (CO) is a good tracer of the bulk of the
of the visual extinction, the FUV radiation field is more molecular gas mass if variations of the [CO]/[H2 ] abun-
and more attenuated so that we have a value of N (CO) dance ratio are considered.
that is predicted by a model with reduced χ. Most of the mass derived from AV in Taurus is in Mask
We also include in Figure 13 the fit to the observa- 2 (∼49%). But a significant fraction of the total mass lies
tions from Sheffer et al. (2008) toward diffuse molecular in Mask 1 (∼28%) and Mask 0 (∼23%). This implies that
Galactic lines–of–sight for log(N (H2 )) ≥ 20.4. The fit mass estimates that only consider regions where 13 CO is
seems to agree with the portion our data points that detected underestimate the total mass of the molecular
agree fairly well with the model having χ = 1.0. Since gas by a factor of ∼2.
Sheffer et al. (2008) observed diffuse lines-of-sight, this We also estimate the masses of high–column density
suggests that a large fraction of the material in the Tau- regions considered by Goldsmith et al. (2008) that were
rus molecular cloud is shielded against the effect of the previously defined by Onishi et al. (1996). In Table 4 we
FUV illumination. This is supported by infrared obser- list the masses derived from the visual extinction as well
vations in Taurus by Flagey et al. (2009) that suggest as from N (CO). Again, both methods give very similar
that the strength of the FUV radiation field is between masses. These regions together represent 43% of the total
χ = 0.3 and 0.8. mass in our map of Taurus, 32% of the area, and 46%
Sheffer et al. (2008, see also Federman et al. 1980) of the 12 CO luminosity. This suggest that the mass and
showed empirical and theoretical evidence that the scat- 12
CO luminosity are uniformly spread over the area of
ter in the AV − N (CO) relation is due to variations of our Taurus map.
the ratio between the total H volume density (ntotal =
H
A commonly used method to derive the mass of molec-
nH + 2nH2 ) and the strength of the FUV radiation field. ular clouds when only 12 CO is available is the use
The larger the volume density or the weaker the strength of the empirically derived CO–to–H2 conversion factor
of the FUV field the larger the abundance of CO relative (XCO ≡ N (H2 )/ICO ≃ MH2 /LCO ). Observations of γ-
to H2 . Note that the scatter in the observations from rays indicate that this factor is 1.74×1020 cm−2 (K km

e

Figure 13. Comparison between the visual extinction derived from 2MASS stellar colors and the 12 CO column density derived from
13 CO and 12 CO observations in Taurus for AV < 5 mag. The blue line represents the 12 CO column density derived from AV assuming
N (H2 )/AV = 9.4 × 1020 cm−2 mag−1 (Bohlin et al. 1978) and a [CO]/[H2 ] abundance ratio of 1.1 × 10−4 . The gray scale represents the
number of pixels of a given value in the parameter space and is logarithmic in the number of pixels. The red contours are 2,10,100, and
1000 pixels. The black lines represent several models of selective CO photodissociation and fractionation provided by Ruud Visser (see
text). The light blue line represents the fit to the observations from Sheffer et al. (2008) toward diffuse molecular Galactic lines-of-sight
for log(N (H)2 ) ≥ 20.4. The horizontal line represents the average N (CO) derived in Mask 0. Each pixel has a size of 100′′ or 0.07 pc at a
distance of 140 pc.

s−1 pc−2 )−1 or M (M⊙ )=3.7LCO (K Km s−1 pc2 ) in our
Galaxy (Grenier et al. 2005). To estimate XCO in Mask
2, 1, and 0 we calculate the 12 CO luminosity (LCO ) in
these regions and compare them with the mass derived
from AV . We also calculate XCO from the average ratio
of N (H2 ) (derived from AV ) to the CO integrated inten-
sity ICO for all pixels in Mask 1 and 2. For Mask 0, we
used the ratio of the average N (H2 ) (derived from AV
) to the average CO integrated intensity obtaining after
combining all pixels in this mask region. The resulting
values are shown in Table 3. The table shows that the
difference in XCO between Mask 2 and Mask 1 is small
considering that the [CO]/[H2 ] relative abundance be-
tween these regions can differ by up to two orders of
magnitude. The derived values are close to that found in
our Galaxy using γ-ray observations. For Mask 0, how-
ever, XCO is about an order of magnitude larger than in
Mask 1 and 2.
Finally we derive the surface density of Taurus by com-
paring the total H2 mass derived from AV (15015 M⊙ )
and the total area of the cloud (388 pc2 ). Again, we
Figure 14. The average N (H2 ) and AV as a function of N (CO) assumed that in Mask 0 the CO–emitting region occu-
in Mask1. N (H2 ) is estimated from AV assuming N (H2 )/AV =
9.4 × 1020 cm−2 mag−1 (Bohlin et al. 1978). pies 26% of the area. The resulting surface density is
∼39 M⊙ pc−2 which is very similar to the median value
of 42 M⊙ pc−2 derived from a large sample of galactic


Figure 15. Probability density function of the visual extinc-
tion in the Taurus molecular cloud. The solid line corresponds
to a Gaussian fit to the distribution of the natural logarithms of
AV / AV . This fit considers only visual extinctions that are lower
than 4.4 mag, as the distribution deviates clearly from a Gaussian
for larger visual extinctions. (see text).
molecular clouds by Heyer et al. (2009).

4.2. Column density probability density function
Numerical simulations have shown that the probabil-
ity density function (PDF) of volume densities in molec-
ular clouds can be fitted by a log-normal distribution.
This distribution is found in simulations with or with-
out magnetic fields when self-gravity is not important
(Ostriker et al. 2001; Nordlund & Padoan 1999; Li et al.
2004; Klessen 2000). A log-normal distribution arises as
the gas is subject to a succession of independent compres-
sions or rarefactions that produce multiplicative varia-
tions of the volume density (Passot & V´zquez-Semadeni
a
1998; V´zquez-Semadeni & Garc´ 2001). This effect
a ıa
is therefore additive for the logarithm of the vol-
ume density. A log-normal function can also de- Figure 16. Probability density function of the visual extinction
scribe the distribution of column densities in a molec- for Mask 1 (upper panel) and Mask 2 (lower panel) in the Taurus
molecular cloud. The solid line corresponds to a Gaussian fit to
ular cloud if compressions or rarefactions along the the distribution of the natural logarithm of AV / AV . The fit
line of sight are independent (Ostriker et al. 2001; for Mask1 considers only visual extinctions that are larger than
V´zquez-Semadeni & Garc´ 2001).
a ıa Note that log– 0.24 mag, while the fit for Mask 2 includes only visual extinctions
normal distributions are not an exclusive result of su- that are less than 4.4 mag (see text).
personic turbulence as they are also seen in simulations
with the presence of self–gravity and/or strong magnetic function of the form
fields but without strong turbulence (Tassis et al. 2010).
Deviations from a log-normal in the form of (ln(x) − µ)2
f (lnx) = Npixels exp − , (29)
tails at high or low densities are expected if the 2σ 2
equation of state deviates from being isothermal
(Passot & V´zquez-Semadeni 1998; Scalo et al. 1998).
a where µ and σ 2 are the mean and variance of ln(x). The
This, however, also occurs in simulations with an isother- mean of the logarithm of the normalized column den-
mal equation of state due to the effects of self-gravity sity is related to the dispersion σ by µ = −σ 2 /2. In all
√
(Tassis et al. 2010). Gaussian fits, we consider N counting errors in each
In Figure 15 we show the histogram of the natural log- bin.
arithm of AV in the Taurus molecular cloud normalized The distribution of column densities derived from the
by its mean value (1.9 mag). Defining x ≡ N/ N , where visual extinction shows tails at large and small AV . The
N is the column density (either AV or N (H2 ) ), we fit a large–AV tail starts to be noticeable at visual extinc-

e

Table 3
Properties of Different Mask Regions in Taurus

Region # of Pixelsa Mass from Mass from Area [CO]/[H2 ] LCO b
XCO = N (H2 )/ICO c
XCO = M/LCO
13 CO and 12 CO AV
[M⊙ ] [M⊙ ] [pc2 ] [K km s−1 pc2 ] [cm−2 /(K km s−1 )] [M⊙ /(K km s−1 pc2 )]
Mask 0 52338 3267 3454 63d 1.2×10−6 130 1.2×1021 26
Mask 1 40101 3942 4237 185 variable 1369 1.6×1020 3.1
Mask 2 30410 7964 7412 140 1.1×10−4 1746 2.0×1020 4.2
Total 122849 15073 15103 388 3245 2.3×1020 4.6

a At the 200′′ resolution of the AV map.
b Calculated from the average ratio of N (H2 ), derived from AV , to CO integrated intensity for each pixel.
c Total mass per unit of CO luminosity.
d Effective area of CO emission based in the discussion about Mask 0 in Section 2.3.

Table 4
Mass of Different High Column Density Regions in Taurus

Region # of Pixels Mass from Mass from Area LCO
13 CO and 12 CO AV
[M⊙ ] [M⊙ ] [pc2 ] [K km s−1 pc2 ]
L1495 7523 1836 1545 35 461
B213 2880 723 640 13 155
L1521 4026 1084 1013 19 236
HCL2 3633 1303 1333 17 221
L1498 1050 213 170 5 39
L1506 1478 262 278 7 68
B18 3097 828 854 14 195
L1536 3230 474 579 15 134
Total 26917 6723 6412 125 1509

possible to determine whether it has a physical origin
or it is an effect of noise. The distribution is well fit-
ted by a log–normal for AV smaller than 4.4 mag. We
searched in our extinction map for isolated regions with
peak AV 4.4 mag. We find 57 regions that satisfy this
requirement. For each region, we counted the number
of pixels that have AV 4.4 mag and from that calcu-
lated their area, A. We then determined their size using
L = 2 (A/π). The average value for all such regions
is 0.41 pc. This value is similar to the Jeans length,
which for Tkin =10 K and n(H2 ) = 103 cm−3 is about
0.4 pc. This agreement suggests that the high–AV tail
might be a result of self–gravity acting in dense regions.
Kainulainen et al. (2009) studied the column density dis-
tribution of 23 molecular cloud complexes (including the
Taurus molecular cloud) finding tails at both large and
small visual extinctions.
Kainulainen et al. (2009) found that high–AV tails are
only present in active star–forming molecular clouds
while quiescent clouds are well fitted by a log–normal.
We test whether this result applies to regions within
Taurus in Figure 16 where we show the visual extinc-
tion PDF for the Mask 1 and 2 regions. Mask 1 includes
Figure 17. Probability density function of the H2 column den- lines–of–sights that are likely of lower volume density
sity derived from N (CO) in Mask 2 with an angular resolution of than regions in Mask 2, and in which there is little star
47′′ (0.03 pc at the distance of Taurus, 140 pc). The solid line corre- formation. This is illustrated in Figure 1 where we show
sponds to a Gaussian fit to the distribution of the natural logarithm the distribution of the Mask regions defined in our map
of N (H2 )/ N (H2 ) . The fit considers H2 column densities that are
lower than 4×1021 cm−2 (or ∼4 mag).
overlaid by the compilation of stellar members of Taurus
by Luhman et al. (2006). Most of the embedded sources
tions larger than ∼4.4 mag. The low–AV tail starts to be in Taurus are located in Mask 2. Note that the normal-
noticeable at visual extinctions smaller than ∼0.26 mag, ization of AV is different in the two mask regions. The
which is similar to the uncertainty in the determination average value of AV in Mask 1 is 0.32 mag and in Mask 2
of visual extinction (0.29 mag), and therefore it is not is 2.1 mag. In Mask 1 we see a tail for low–AV starting


at about 0.2 mag. Again, this visual extinction is close to rized as follows,
the uncertainty in the determination of AV . For larger
visual extinctions the PDF appears to be well fitted by a • We have improved the derivation of the CO
log–normal distribution. In case of the visual extinction column density compared to that derived by
PDF in Mask 2, we again see the tail at large AV starting Goldsmith et al. (2008) by using an updated value
at about 4.4 mag. For lower values of AV the distribution of the spontaneous decay rate and using exact nu-
is well represented by a log–normal. merical rather than approximate analytical calcu-
We can use our CO map of Taurus at its original reso- lation of the partition function. We also have used
lution (47′′ which corresponds to 0.03 pc at the distance data that has been corrected for error beam pick–
of Taurus, 140 pc) to study the column density PDF at up using the method presented by Bensch et al.
higher resolution than the 200′′ AV map (Figure 17). We (2001).
estimate N (H2 ) from our CO column density map in
Mask 2 by applying a constant [CO]/[H2 ] abundance ra- • We find that in the Taurus molecular cloud the col-
tio of 1.1×10−4 (Section 3.1). The average H2 column umn density and visual extinction are linearly cor-
density in Mask 2 is 3 × 1021 cm−2 . We do not con- related for AV up to 10 mag in the region associated
sider Mask 1 because of the large scatter found in the with the B213–L1495 filament. In the rest of Tau-
[CO]/[H2 ] abundance ratio (Section 3.2). In the figure rus, this linear relation is flattened for AV 4 mag.
we see that the distribution is not well fitted by a log– A linear fit to data points for AV < 10 mag and
normal. As for AV , the PDF also shows a tail for large N (CO) > 1017 cm−2 results in an abundance of
column densities that starts to be noticeable at about CO relative to H2 equal to 1.1×10−4.
4 × 1021 cm−2 (or AV ≃ 4 mag). Therefore, the high–
column density excess seems to be independent of the • For visual extinctions larger than ∼4 mag the CO
spatial scale at which column densities are sampled. We column density is affected by saturation effects and
repeated the procedure described above to search for iso- freezeout of CO molecules onto dust grains. We
lated cores in our map with N (H2 ) > 4 × 1021 cm−2 and find that the former effect is enhanced due to the
obtained an average size for cores of 0.5 pc, which is con- presence of edge–to–center temperature gradients
sistent to that obtained in our AV map . Note that at this in molecular clouds. We used the RATRAN ra-
resolution we are not able to account for effects of tem- diative transfer code to derive a correction for this
perature gradients and of CO depletion along the line of effect.
sight, as this requires knowledge of AV at the same reso- • We combined the column density of CO in ice form
lution. We therefore underestimate the number of pixels derived from observations towards embedded and
in the H2 column density PDF for N (H2 ) 1×1022 cm−2 field stars in Taurus by Whittet et al. (2007) with
while we overestimate them for N (H2 ) 1 × 1022 cm−2 . the saturation–corrected gas–phase N (CO) to de-
But the number of pixels (∼7000) affected by those ef- rive the total CO column density (gas–phase+ices).
fects represent only 9% of the number of pixels (∼81000) This quantity is linearly correlated with AV up to
that are in excess relative to the log–normal fit between the maximum extinction in our data ∼23 mag.
3 × 1021 and 1 × 1022 cm−2 , and therefore the presence
of a tail at large–N (H2) is not affected. Note that this • We find that the gas–phase CO column density is
also affected our ability to identify isolated regions in reduced by up to a factor of ∼2 in high–extinction
the N (H2 ) map. We were able to identify only 40 cores regions due to depletion in the Taurus molecular
compared with the 57 found in the AV map. cloud.
In summary, we find that the distribution of column
densities in Taurus can be fitted by a log–normal distri- • We fit an analytical column density profile to
bution but shows tails at low and high–column densities. 13 cores in Taurus. The mean value and stan-
The tail at low–column density may be due to noise and dard deviation of the central volume density are
thus needs to be confirmed with more sensitive maps. (1.4 ± 0.4)×104 cm−3 . We use the derived volume
We find that the tail at large column densities is only density profile and the amount of depletion ob-
present in the region where most of the star formation served in each core to derive an upper limit to the
is taking place in Taurus (Mask 2) and is absent in more CO depletion age with a mean value and standard
quiescent regions (Mask 1). The same trend has been deviation of (4.2 ± 2.4) × 105 years. We find lit-
found in a larger sample of clouds by Kainulainen et al. tle variation of this age among the different regions
(2009). Here we suggest that the distinction between within Taurus.
star-forming and non star-forming regions can be found
even within a single molecular cloud complex. The pres- • For visual extinctions lower than 3 mag we find that
ence of tails in the PDF in Taurus appears to be inde- N (CO) is reduced by up to two orders of magni-
pendent of angular resolution and is noticeable for length tude due to the competition between CO formation
scales smaller than 0.41 pc. and destruction processes. There is a large scatter
5. CONCLUSIONS
in the AV –N (CO) relation that is suggestive of dif-
ferent FUV radiation fields characterizing the gas
In this paper we have compared column densities de- along different lines–of–sight.
rived from the large scale 12 CO and 13 CO maps of the
Taurus molecular cloud presented by Narayanan et al. • The mass of the Taurus molecular cloud is about
2008 (see also Goldsmith et al. 2008) with a dust extinc- 1.5×104 M⊙ . Of this, ∼49% is contained in pixels
tion map of the same region. This work can be summa- where both 12 CO and 13 CO are detected (Mask 2),

e

∼28% where 12 CO is detected but 13 CO is not present in regions associated with star formation,
(Mask 1), and ∼23% where neither 12 CO nor 13 CO while the more quiescent positions in Taurus do
are detected (Mask 0). not show this feature. This tail is independent of
the resolution of the observations.
• We find that the masses derived from CO and AV
are in good agreement. For Mask 2 and Mask 0 we
used a [CO]/[H2 ] relative abundance of 1.1×10−4
and 1.2×10−6, respectively. For Mask 1, we used a
variable [CO]/[H2 ] relative abundance taken from a
fit to the average relation between AV and N (CO)
We would like to thank Douglas Whittet for the idea to
in this region, with −6.7 < log([CO]/[H2 ]) ≤ −3.9.
add the column density of CO–ices to the gas–phase CO
• We also compared the mass derived from AV with column densities, Jonathan Foster for providing sample
the 12 CO J = 1 → 0 luminosity for the regions de- extinction maps that were used to test the implementa-
rived above. For Mask 1 and 2 these two quantities tion of the NICER algorithm used here, John Black, Ed-
are related with a CO–to–H2 conversion factor of wine van Dishoeck, and Ruud Visser for helpful discus-
about 2.1×1020cm−2 (K km s−1 )−1 . The derived sions about the formation/destruction processes affect-
CO–to–H2 conversion factor is in agreement with ing CO at low column densities, specially Ruud Visser
that found in our Galaxy using γ–ray observations. for providing results of his recent calculations, Kostas
In Mask 0, however, we find a larger the conversion Tassis for discussions about the nature of column den-
factor of 1.2×1021cm−2 (K km s−1 )−1 . sity distributions in molecular clouds, and Marko Krˇo c
for sharing his H i map of Taurus. J.L.P was supported
• We studied the distribution of column densities in by an appointment to the NASA Postdoctoral Program
Taurus. We find that the distribution resembles at the Jet Propulsion Laboratory, California Institute of
a log–normal but shows tails at large and low Technology, administered by Oak Ridge Associated Uni-
column densities. The length scale at which the versities through a contract with NASA. This research
high–column density tail starts to be noticeable was carried out at the Jet Propulsion Laboratory, Cal-
is about 0.4 pc, which is similar to the Jeans ifornia Institute of Technology and was supported by a
length for a T =10 K and nH2 = 103 cm−3 gas, grant from the National Science Foundation. This re-
suggesting that self–gravity is responsible for its search has made use of NASA’s Astrophysics Data Sys-
presence. The high–column density tail is only tem Abstract Service.

APPENDIX
ERROR BEAM CORRECTION
The FCRAO 14m telescope is sensitive not only to emission that couples to the main beam (with efficiency ηmb = 0.45
at 115 GHz and ηmb = 0.48 at 110 GHz) but also to emission distributed on scales comparable to the error beam
(30′ ). For emission extended over such large-scale, the coupling factor (including the main beam contribution) is the
forward spillover and scattering efficiency, ηfss = 0.7, at both frequencies. The error beam pickup, also known as “stray
radiation”, can complicate the accurate calibration of the measured intensities: a straightforward scaling of the data
by 1/ηmb can significantly overestimate the true intensity in regions where emission is present on large angular scales.
Given the wide range of angular sizes of the structures in Taurus, it is clear in general that neither ηmb nor ηfss will
give optimum results.
To accurately scale the FCRAO data, it is essential to remove the error beam component before scaling the intensities
to the main beam scale. Methods for correcting millimeter-wave data for error beam pickup have been discussed
by Bensch et al. (2001), who introduce the “corrected main beam temperature scale” (Tmb,c ) with which optimum
calibration accuracy is achieved by scaling the data by 1/ηmb after removal of radiation detected by the error beam.
To remove the error beam component we use the second of the methods described in Bensch et al. (2001). The error
beam component is removed in Fourier space directly from the FCRAO data, and the intensities are converted to the
Tmb,c scale, by the following method:
˜∗ ∗
• The Fourier transform of the antenna temperature is taken: TA = F T (TA )
• The following correction is applied to each velocity slice in the cube:
−π 2 (θeb − θmb )(kx + ky ) −1
2 2 2 2
˜ ˜∗
Tmb,c = TA (ηmb + ηeb exp( ))
4 ln(2)
where θeb is the FWHM of the error beam, θmb is the FWHM of the main beam, ηeb = ηfss − ηmb , and kx , ky
are the wavenumbers along the x, y (RA, decl.) directions respectively.
• The inverse Fourier transform is performed, with only the real part of the result being retained:
˜
Tmb,c = Re (IF T (Tmb,c )). The imaginary part is consistent with round–off errors.
At low spatial frequencies, the correction factor is ∼ 1/ηfss while at high spatial frequencies, the correction factor is
∼ 1/ηmb . The effective correction factor at any point is determined therefore by the spatial structure of the emission


Figure 18. (left) Color (H − Ks ) versus (J − H) for stars observed in the control field. (right) Intrinsic J − H and H − Ks colors of
Main Sequence, Giant, and Supergiant stars (taken from Koornneef 1983). These stars are indicated by circles in the left panel.

in the vicinity of that point. More detailed information and quantitative analysis of the above procedure can be
found in Brunt et al (2010, in prep) and Mottram & Brunt (2010, in prep). For typical applications, a naive scaling
by 1/ηmb overestimates the true intensities, as inferred from comparison to CfA survey data (Dame et al. 2001), by
around 25–30%. For reference, an overestimation of ∼ 50% would be applicable if ηfss were appropriate everywhere.
The spatially variable correction factor afforded by the method used here therefore offers a higher fidelity calibration
of the data.
THE EXTINCTION MAP
We have used the 2MASS point source catalog to create an near–infrared extinction map of Taurus. This was done
using an implementation of NICER (Lombardi & Alves 2001) from Chapman (2007). The 2MASS catalog we used has
1039735 (∼1 million) stars over an area between RA=04:03:51.6 and 05:05:56.6 and decl.=+19:24:14.4 and +30:50:24
(J2000). We use the compilation by Luhman et al. (2006) to remove 156 stars that are known to be members of
Taurus. The map generated has an angular resolution of 200′′ and is Nyquist sampled with a pixel spacing of 100′′ ,
corresponding to 0.07 pc at a distance of 140 pc. The resolution of the map was determined by that of the H i map
used to correct the data for the contribution of H i to the total extinction (see below). The final extinction map is
shown in Figure 4.
We constructed extinction maps in nearby regions around Taurus with the goal of finding a field that does not show
significant extinction, so it can be used as a control field to estimate the intrinsic (J − H) and (H − Ks ) stellar colors.
We selected a region corresponding to a 2◦ × 2◦ box centered at RA=03:50:44.7 and decl.=+27:46:54.1 (J2000). The
mean (± weighted standard deviation) values for stars in this box are 0.454±0.157 mag for (J − H) and 0.114±0.074
mag for (H − Ks ). We also computed the covariance matrix for the (J − H) and (H − Ks ) colors. The on-axis elements
2 2
of this matrix are σJ−H and σH−Ks , the dispersions of the (J − H) and (H − Ks ) colors in the control field, while
the two off–axis elements are identical to each other, with a value of 0.006. In Figure 18 we show the color (H − Ks )
versus (J − H) of stars in the control field. We also show the intrinsic color of Main Sequence, Giant, and Supergiant
stars. Apart from the scatter due to photometric errors, there is large scatter in the intrinsic colors due to different
stellar types in the control field. The mean values, weighted standard deviation and off–axis covariance matrix are
input to the NICER routine and with them we correct for the different sources of scatter of the intrinsic colors in the
control field. Note that Padoan et al. (2002) used an intrinsic (H − Ks ) color of 0.13 mag in their extinction map of
Taurus. The difference relative to that in our control field is thus seen to be small.
We transformed the (J − H) and (H − Ks ) colors to AV using an extinction curve from Weingartner & Draine
(2001) with a ratio of selective to total extinction RV =3.1. Note that this RV is derived towards diffuse regions (AV ≤
1.5 mag). At larger volume densities the value of RV is expected to increase up to 4.5 at the center of dense cores
due to grain growth by accretion and coagulation (Whittet et al. 2001). Considering that a given line–of–sight might
intersect both dense and diffuse regions, Whittet et al. (2001) estimated an effective RV that increases up to ∼ 4.0 for
AV ≃ 10 mag. For such a value of AV we expect that for a given total hydrogen column density, the visual extinction
increases by about 20% due to enhanced scattering as the grain sizes increases (Draine 2003).
Figure 19 shows the relation between the estimated visual extinction in the Taurus molecular cloud and the formal
error per pixel (i.e. error propagation from the error in the estimation of AV for each star) and the number of stars
per pixel. The errors in AV range from ∼0.2 mag at low extinctions to ∼1.3 mag at large visual extinctions. The

e

Figure 19. The formal error per pixel (left) and the number of stars per pixel (right) as a function of the visual extinction estimated in
the Taurus molecular cloud.

Figure 20. Histogram of the visual extinction in Taurus associated with H i for the full range of 21 cm velocities (light gray) and for the
velocity range between 0 and 20 km s−1 (dark gray). The histogram has been determined for the region where both 12 CO and 13 CO are
detected.

average error is 0.29 mag while the average number of stars per pixel is 28. As expected, the number of stars per pixel
decreases as the extinction increases.
There is a large filamentary Hi structure, extending away from the Galactic plane, which coincides with the eastern
part of Taurus. Based on distances of molecular clouds at the end of the filament, we assume that this filament lies
between the Taurus background stars and the Earth. Dust in the filament will thus contribute to the total extinction
measured. We estimate the contribution to the visual extinction from dust associated with H i using the Arecibo map
from Marco Krˇo (PhD Thesis, Cornell University, in preparation). In Figure 20 we show a histogram of the visual
c
extinction associated with positions in the H i map where both 12 CO and 13 CO are detected in our Taurus map. We
show the extinction for the full range of velocities and for the range between 0 to 20 km s−1 (similar to the velocity
range at which CO emission is observed). We correct the AV map by extinction associated with neutral hydrogen in
the latter range (see below). The average correction is ∼0.3 mag.
In order to see whether some H i velocity components are foreground to the 2MASS stars, we examine a field with
complex H i velocity structure. We choose a region northwest of the Taurus molecular cloud that shows small visual
extinction (RA = 04:57:25.472 and decl. = 29:07:0.81). Figure 21a shows a histogram of the visual extinction without
correction, corrected for H i over the entire velocity range, and corrected for H i over the 0 to 20 km s−1 range. We
also show in Figure 21b the average H i spectrum in the selected field. The negative velocity components produce


Figure 21. (left) Histogram of the visual extinction calculated in a field with complex velocity structure (see text) with and without
correction for extinction associated with H i. (right) The H i spectrum averaged over the this field.

significant excess reddening associated with H i that is inconsistent with the extinction determined from 2MASS stars.
We therefore conclude that H i components with negative velocities are background to the 2MASS stars. This confirms
the correctness of excluding negative velocities for determining the H i-associated extinction correction for Taurus.
The exact velocity range used is a source of uncertainty of a few tenths of a magnitude in the extinction.
We finally note that the widespread H i emission is also present in the control field. The control field is contaminated
by ∼0.12 mag of visual extinction associated with H i. This contribution produces a small overestimation of the
intrinsic colors in the control field. Therefore, since we determine visual extinctions based on the difference between
the observed stellar colors in Taurus and those averaged over the control field, we have added 0.12 mag to our final
AV map of Taurus.
CORRECTION FOR TEMPERATURE GRADIENTS ALONG THE LINE–OF–SIGHT
In order to assess the impact of core-to-edge temperature gradients in the estimation of N (CO), we use the radiative
transfer code RATRAN (Hogerheijde & van der Tak 2000). With RATRAN we calculate 12 CO and 13 CO line profiles
and integrated intensities from a model cloud and use them to estimate N (CO), following our analytic procedure
(Section 2), which we then compare with that of the original model cloud.
The model is a spherical cloud with a truncated power-law density profile. We adopt a density profile
n(r)=ns (r/rc )−α for 0.3rc ≤ r ≤ rc , and constant density, n(r) = ns (0.3)−α in the central portion of the cloud
(r < 0.3rc ). Here, rc is the cloud radius and ns the density at the cloud surface. In our models we use a power–law
exponent of α = 1.96 for the density profile, a density at the cloud surface of 2×104 cm−3 , and a cloud radius of
5×1016 cm (0.015 pc at a distance of 140 pc). The values of the density at the cloud surface and the power–law
exponent are taken from the fit to the extinction profile of the prestellar core B68 (Alves et al. 2001). We adopt a
FWHM line-width in the model of ∆v=1.33 km s−1 , in order to have line widths that are consistent with those in
pixels having AV > 10 mag (Figure 22). We use the 13 CO and 12 CO emission resulting from the radiative transfer
calculations to derive, using Equation (17), the CO column density (N (CO)emission ) that will be compared with that
of the model cloud (N (CO)model ). We trace CO column densities between 3×1016 cm−2 and 1×1018 cm−2 .
We consider the case of isothermal clouds and of clouds with temperature gradients. Temperature gradients can
be produced, for example, when clouds are externally illuminated by the interstellar radiation field (e.g. Evans et al.
2001). We adopt a temperature profile given by
2
r
T (r) = (Ts − Tc ) + Tc , (C1)
rc
where Tc and Ts are the temperature at the cloud center and surface, respectively.
We run isothermal cloud models with kinetic temperatures of 8, 9, 10,12, and 15 K. In the case of clouds with
temperature gradients we consider the same range of temperatures for the cloud surface and Tc = 8 K for the cloud
center. We choose this value because the balance between the dominant heating and cooling mechanisms in dense and
shielded regions, namely cosmic-ray heating and cooling by gas-grain collisions, typically results in this range of tem-

e

peratures (Goldsmith 2001). The selected range of temperatures match the observed range of excitation temperatures
for AV > 10 mag (Figure 22).

Figure 22. Pixel-by-pixel comparison between the visual extinction (AV ) and the 12 CO-derived excitation temperature (left ) and 13 CO
FWHM line–width (right).

In the upper rows of Figure 23 we show the excitation temperature derived from the model 12 CO emission as a
function of N (CO)model. The left panel corresponds to isothermal clouds and the right panel to clouds with temperature
gradients. We also show the line-of-sight (LOS) averaged kinetic temperature as a function of N (CO)model . In clouds
with temperature gradients, low column densities are on average warmer than larger column densities. For isothermal
clouds Tex and the model kinetic temperature are almost identical for all values of N (CO)model . In the case of clouds
with temperature gradients we see that, although the average LOS kinetic temperature decreases for large CO column
densities, the derived excitation temperature shows little variation with N (CO)model , tracing only the temperature at
the cloud surface. This is a result of 12 CO becoming optically thick close to the cloud surface and therefore the Tex
determined in this manner applies only to this region. In clouds with temperature gradients, using 12 CO to calculate
the excitation temperature overestimates its value for regions with larger column densities, where most of the 13 CO
emission is produced.
We show the 13 CO opacity (Equation [20]) for both isothermal clouds and clouds with temperature gradients as a
function of N (CO)model in the middle panels of Figure 23. We also show opacities calculated from the model cloud
13
CO column densities assuming LTE (τLTE ). When the cloud kinetic temperature is constant, both opacities show
good agreement for all sampled values of N (CO)model . In contrast, for clouds with temperature gradients, opacities
derived from the model line emission are lower than τLTE by up to a factor of ∼3. The differences arise due to the
overestimation of the excitation temperature in regions with large N (CO), as Equation [20] assumes a constant value
of Tex .
The relation between N (CO)emission and N (CO)model is shown in the lower panels of Figure 23. Isothermal clouds
show almost a one-to-one relation between these two quantities whereas clouds with temperature gradients show that
the relation deviates from linear for large N (CO)model. This is produced by the underestimation of opacities that
affect the correction for this quantity (Equation [15]). The difference between N (CO)emission and the expected CO
column density is about 20%.
In the following we use the relation between N (CO)model and N (CO)emission to derive a correction to the observed
CO column densities. We notice that the difference between these quantities does not show a strong dependence in
the cloud surface temperature. This is because all models have the same temperature at the cloud center. Since the
observed excitation temperatures lie between ∼ 9 − 15 K for AV > 10 mag (Figure 23), and the excitation temperature
derived from 12 CO is similar to the kinetic temperature at the cloud surface, we average all models from Ts = 9 K to
15 K in steps of 1 K to derive a correction function. In Figure 24 we show the difference between the model and derived
CO column density (N (CO)diff = N (CO)model − N (CO)emission ) as a function of the model CO column density. To
this relation we fit a polynomial function given by
N (CO)diff
= 0.05N (CO)1.9 − 0.25N (CO)model + 0.17
model (C2)
1017 cm−2


Figure 23. (upper row) Excitation temperature derived from the 12 CO emission as function of N (CO)model for isothermal cloud models
(left ) and models with temperature gradients (right). In both panels we also show the model cloud kinetic temperature averaged along
the line of sight, Tkin los , as a function of N (CO)model . (middle row) The 13 CO opacity versus N (CO)model for the models shown in the
upper row. The opacity was derived using Equation (20). We also show the opacity calculated from the model cloud N(13 CO) with the
assumption of LTE. (lower row) Column density of CO calculated from the 12 CO and 13 CO emission using Equation (17) versus the model
cloud N (CO). The straight line corresponds to an one–to–one relation.

e

Figure 24. Difference between the expected N (CO)model and the derived N (CO)emission as a function of the model CO column density.
The red line represents a second-order polynomial fit.
To apply this correction we made a rough estimate of the gas–phase CO column density as a function of visual
extinction. We use the observations by Frerking et al. (1982) of C18 O and the rarer isotopic species C17 O and 13 C18 O
in the direction of field stars located behind Taurus. We convert the observed column densities into N (CO) assuming
[CO]/[C18 O]=557, [C18 O]/[C17 O]=3.6, and [C18 O]/[13 C18 O]=69 (Wilson 1999). Due to their low abundances, these
species are likely not affected by saturation. Note that, however, they are still sensitive to the determination of
the excitation temperature. Frerking et al. (1982) presented column densities as lower limits when 12 CO is used to
determine Tex (12 CO) (average ∼10 K) and as upper limits when they used Tex (12 CO)/2 (i.e. ∼5 K) as the excitation
temperature. The kinetic temperature in dense regions is likely to be in between 5 and 10 K (Goldsmith 2001) and
therefore, assuming that the isotopologues are thermalized, the excitation temperature should also have a value in
this range. Thus, we use the average value between upper and lower limits of the CO column density to determine its
relation with AV . For the visual extinction at the positions observed by Frerking et al. (1982), we use updated values
derived by Shenoy et al. (2008) from infrared observations5 . We note that the visual extinction correspond to a single
star while the Frerking et al. (1982) observations are averaged over a 96′′ beam. We constructed an extinction map
of Taurus with 96′′ resolution and an extinction curve that matches that adopted by Shenoy et al. (2008) in order to
compare with their determination of AV . We found that the visual extinctions always agree within ±0.4 mag.
In Figure 25 we show the relation between N (CO) and AV with and without the correction for the effects of temper-
ature gradients along the line–of–sight. For reference we include the values of N (CO) derived from the observations
by Frerking et al. (1982). The error bars denote the upper and lower limits to the CO column density mentioned
above. Although our determination of the gas–phase N (CO)/AV relation is necessarily approximate, the validity of
the correction for the effects of temperature gradients along the line–of–sight is confirmed by the good agreement
between AV and N (CO) up to AV ≃23 mag after the addition of the column density of CO–ices (Section 3.1.1).

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