Contours of the observed CO() line intensities,
(CO), integrated between -10 and +10 km/s in velocity are
presented in Fig. 1 (click here). The lowest contour lies 3 times above the
noise level. The structure of the cloud, with a sharp eastern edge, a
diffuse boundary on the west, and a comet-like extension to the south,
confirms the main features that had been seen in the low-resolution survey.
The present higher resolution does not reveal much more structure in the
interior. The few apparent clumps mainly result from velocity crowding along
the line of sight.
Figure 2: Example of composite lines recorded in the eastern cloud in
CO(), CO(
),
), and
) from more
to less intense lines
The various positions observed at 110, 220, and 230 GHz are indicated in
Fig. 1 (click here). Other positions have been sampled along the fainter edges
of the cloud, but they lacked of emission at the achieved
sensitivity. An example of bright composite lines observed in the cloud is
given in Fig. 2 (click here) for both transitions and both isotopes. The
multiple lines have been fitted by two gaussian profiles, sometimes three.
The central velocity of each component has been checked to coincide in the
four transitions. The line widths of
in CO and
in
found for the gaussians agree with those of the
single lines found elsewhere in the cloud. Each component has then been
analysed separately.
To study line ratios, the weak detections have been discarded and only lines
with an integrated intensity greater than have been retained. This
threshold is slightly lower than the customary
level since the
presence of a line at the same velocity at all four frequencies reinforces
its detection probability in a single spectrum. Hence, sets of 60 to 120
lines were prepared to compare emission at two frequencies, all at the same
angular resolution of 8.7' or 0.8 pc at 300 pc. Integrated intensities
rather than peak temperatures have been selected to study the line ratios
because of their smaller uncertainty. The observed diffuseness of the cloud
on scales larger than the angular resolution should limit the beam dilution
effects on the derived line ratios. This point is confirmed by the limited
(
) intensity variations recorded among the individual 2.3' POM-2
scans contained within one CfA beam.
The average line ratio over the cloud has not been derived, as is often
done, by taking the mean or weighted mean of the individual line ratios,
because these estimates are strongly biased - in the present case
towards low values. We studied instead the correlation between the velocity
integrated intensities measured at the and
frequencies, noted
and
,
respectively. We did not attempt linear least-square fits to the data since
the relative uncertainties on the points are comparable on both axes. Linear
fits have been obtained instead by adopting a maximum-likelihood analysis
that takes uncertainties on both axes into account. Assuming gaussian
distributions, the log-likelihood function to optimize is written as:
where (xi, yi) represent the observed point coordinates
with instrumental uncertainties and
,
and a and b characterize the linear function to be fitted together with
the parent population of points (
,
) that
is compatible with the given linear regression. The likelihood function is
therefore maximized over N+2 free parameters, N being the number of
parent or observed points. The consistency between the observed points and
their likeliest parent counterpart is used to check the relevance of the
proposed fit. Statistical errors on a, b, and
have been
derived from the information matrix (Strong 1985); quoted
errors are
.
Figure 3: Correlations between velocity-integrated CO intensities
observed in the and
transitions,
versus
, for CO
a) and
b). The fitted slopes are
and
, respectively
Figures 3 (click here)a and 3 (click here)b illustrate the tight correlation that exists
between the and
velocity-integrated
intensities for the two isotopic species of CO we have observed. The small
scatter implies that the
ratio is fairly uniform in the emitting regions and independent of the CO
brightness. Linear fits to the points give:
The intercepts are consistent with zero given the instrumental sensitivity.
The statistical error on the slopes takes into account the noise level in
each spectrum and the or
uncertainty from day-to-day
variations in the calibration of each telescope. In addition, the derived
line ratios suffer from a
systematic uncertainty in the absolute
calibration of each telescope which reduces the accuracy of the integrated
intensities significantly. Hence the transition ratios are
in CO:
in :
Because the line widths in both transitions are always nearly equal (see
Fig. 2 (click here)), the ratios of over
peak
temperatures have the same average values.
In CO, the mean ratio
agrees with those measured in clouds of various types. For the giant
molecular clouds in the Galactic plane, Sanders et al. (1993)
find a ratio at the solar circle of 0.9-0.95 and an average in the inner
Galaxy of
, varying little with galactocentric distance. In
nearby, dense globules such as HCL 2 in Taurus, or B157 and L1075 in Cygnus,
ratios of
and
have been found by
Cernicharo & Guélin (1987) and Robert & Pagani
(1993), respectively. Van Dishoeck et al. (1991) have
derived an average ratio of
for both high-latitude clouds
and translucent ones of visual extinction equivalent to that of Cepheus (<
2 or 3 mag). The apparent stability of these ratios has theoretical
grounds: in plane-parallel models of clouds with uniform density and
illuminated by the interstellar radiation field (Lequeux et al.
1994, and references therein), the emergent
ratio increases with cloud density from
to
, but by less than a factor of 2. The
weakness of this dependence on density results from line saturation and
decreasing gas temperature. This dependence may have been observed by
Sakamoto et al. (1994) in the Orion complex where the
ratios increase from 0.5 near the cloud edges
to
in the bright, optically thick, central ridge, around mean values
of 0.77 and 0.66 in Orion A and B, respectively. The Cepheus cloud is not
bright enough to show this effect since the
ratios in our sample do not depend on CO brightness up to
30 K km/s (see Fig. 3 (click here)a).
In , the
ratio in Cepheus agrees with those determined in the B157 and L1075 globules
(
, Robert & Pagani 1993) and in a
high-latitude clump,
of the Polaris complex (
, Boden & Heithausen 1993). The more emissive HCL 2
globule, which exhibits saturated
lines, gives line ratios
close to
(Cernicharo & Guélin 1987). In
Cepheus, the ratio peaks near unity in the denser clump of gas surrounding
the IRAS source, where the optical depth in
is close to unity.
| LTE | LVG | |||||||||||||
8 K | 10 K | 15 K | 30 K | ||||||||||||
10CO | ![]() | ![]() | ![]() | ![]() | ![]() | ||||||||||
| 116 | 116 | 116 | 116 | 116 | ||||||||||
![]() | 115 | 61 | 109 | 114 | 112 | ||||||||||
![]() | ![]() | ![]() | ![]() | ![]() | |||||||||||
![]() | 7.0 | ![]() | 1.4 | 7.9 | ![]() | 0.2 | 9.8 | ![]() | 0.5 | 13.8 | ![]() | 1.8 | 20.9 | ![]() | 7.2 |
![]() | 7.0 | ![]() | 1.4 | 7.0 | ![]() | 0.7 | 8.2 | ![]() | 1.1 | 8.8 | ![]() | 1.5 | 9.1 | ![]() | 1.7 |
![]() | ![]() | ![]() | ![]() | ![]() | |||||||||||
![]() | 1.7 | ![]() | 1.2 | 0.8 | ![]() | 0.4 | 0.7 | ![]() | 0.4 | 0.4 | ![]() | 0.2 | 0.2 | ![]() | 0.1 |
![]() | 2.3 | ![]() | 1.4 | 1.3 | ![]() | 0.7 | 1.4 | ![]() | 0.7 | 1.1 | ![]() | 0.4 | 1.0 | ![]() | 0.3 |
![]() | ![]() | ![]() | ![]() | ![]() | |||||||||||
N(CO) | 17.7 | ![]() | 12.3 | 9.9 | ![]() | 6.8 | 12.3 | ![]() | 8.5 | 9.3 | ![]() | 5.6 | 8.3 | ![]() | 4.6 |
n(![]() | 48 +96-48 | 18 +21-18 | 4.4 | ![]() | 2.2 | 1.7 | ![]() | 0.6 | |||||||
1013CO | ![]() | ![]() | ![]() | ![]() | ![]() | ||||||||||
| 60 | 46 | 46 | 46 | 46 | ||||||||||
![]() | 59 | 45 | 46 | 46 | 45 | ||||||||||
![]() | ![]() | ![]() | ![]() | ![]() | |||||||||||
![]() | 5.2 | ![]() | 1.2 | 7.5 | ![]() | 0.8 | 8.7 | ![]() | 1.5 | 10.7 | ![]() | 3.3 | 13.5 | ![]() | 8.3 |
![]() | 5.2 | ![]() | 1.2 | 5.7 | ![]() | 0.9 | 5.9 | ![]() | 1.1 | 6.1 | ![]() | 1.3 | 6.1 | ![]() | 1.3 |
![]() | ![]() | ![]() | ![]() | ![]() | |||||||||||
![]() | 0.5 | ![]() | 0.3 | 0.2 | ![]() | 0.1 | 0.1 | ![]() | 0.1 | 0.1 | ![]() | 0.1 | 0.1 | ![]() | 0.1 |
![]() | 0.5 | ![]() | 0.4 | 0.3 | ![]() | 0.2 | 0.2 | ![]() | 0.2 | 0.2 | ![]() | 0.1 | 0.2 | ![]() | 0.1 |
![]() | ![]() | ![]() | ![]() | ![]() | |||||||||||
N(13CO) | 2.3 | ![]() | 1.6 | 1.1 | ![]() | 0.9 | 1.0 | ![]() | 0.7 | 1.0 | ![]() | 0.7 | 0.9 | ![]() | 0.6 |
n(![]() | 14 +22-14 | 6 +8-6 | 2.2 | ![]() | 1.5 | 0.9 | ![]() | 0.5 | |||||||
| ![]() | ![]() | ![]() | ![]() | ![]() |
Figure 4: Number distributions of the excitation temperatures a),
optical depths in the transition b), and
molecular column-densities c), derived in the LTE approximation
from the observed CO and
lines
The radiation temperature ) of a line at frequency
,
arising from a cloud of optical depth
) filling the telescope
beam, is given for an excitation temperature
by:
Under conditions of local thermodynamic equilibrium (LTE), assuming
identical excitation temperatures for both transitions, as suggested by the
rather uniform ratios
and the low brightness temperatures recorded, the ratio of the optical
depths in the two transitions is expressed below with
representing the line width in frequency:
In order to derive the excitation temperature, optical depth and related
column-density of the emitting gas for each line, observed
velocity-integrated intensities have been preferred to peak radiation
temperatures for statistical reasons. The equations )
and
)
have been
solved numerically in
and
for
the observed intensities
and
of a line, assuming for the integration a gaussian profile
for
) with
equal to the observed width of the line.
Molecular column-densities have then been calculated in the LTE
approximation. CO and
emission have been treated
independently. Histograms of the excitation temperatures, (
)
optical depths, and column-densities found in the cloud are displayed in
Fig. 4 (click here). The mean and rms dispersion of these distributions are
given in Table 1 (click here). The data in Fig. 4 (click here)b imply optically thin
conditions in
and moderate optical depths in CO (
typically). The optical depths in the higher transition
are less than twice higher. These low optical depths and the narrow
temperature distributions do explain the uniform
ratios found in the cloud. Indeed, Eq. (3) in
the optically thin regime and values of
(in CO) and
(in
) yield excitation temperatures of (
) K and (
) K, respectively, which are totally consistent with the temperature
distributions displayed in Fig. 4 (click here)a that have a mean and rms
dispersion of (
) K in CO and (
) K in
.
The radiative transfer equation is considerably simplified for systematic
motions of the gas (Sobolev 1960; Castor 1970).
The so-called ``large velocity gradient" models provide numerical solutions
for a constant velocity gradient under the assumptions of statistical
equilibrium and a complete photon redistribution in frequency and angle. Far
from being randomly distributed, the velocity field in this cloud appears to
be highly organized and presents large gradients of amplitude (Paper III), unfortunately measured perpendicularly
to the line-of-sight. However, their presence gives support to the LVG
approximation and to the model which treats line formation at different
locations independently. We have used the model developed by Castets et al.
(1990). Its numerical simplicity allows conversion of line
intensities to volume gas density and molecular column-density, given the
local kinetic temperature,
, and two line transitions. Various
kinetic temperatures from 8 to 30 K have been tested. The choice of
collision rates (from Green & Thaddeus 1976 or Flower
& Launay 1985) and the geometry of the cloud (spherical or plane
parallel) has little influence on the inferred characteristics of the gas.
Histograms of the (
) excitation temperatures,
(
) optical depths, and column-densities found in the cloud in
CO and
are displayed in Fig. 5 (click here). The mean and rms
dispersion of these distributions are also given in Table 1 (click here). The
choice of kinetic temperature has little influence on these distributions.
On the contrary, the volume densities and the
excitation
temperatures are not constrained by the models. Kinetic temperatures below 8
K could not be tested because the adopted collision rates did not apply. At
, the LVG model still finds a solution for half of the
CO sample and all the
sample. The narrow distributions
displayed in Fig. 5 (click here) nicely corroborate the LTE findings. The
moderate
optical depths (
)
are even lower than the LTE estimates. The small (not significant)
discrepancy between the cloud average (
) temperatures found
in the LTE and LVG models may be due to our limitation in
.
The consistent LTE and LVG results therefore yield excitation temperatures
in the cloud in the range of in CO and
in
. These low values indicate little heating by the external UV field
- perhaps as the result of the high elevation of Cepheus above the Galactic
plane (90 pc) and the lack of massive stars in its interior or vicinity. The
small range of excitation temperatures and uniform
ratios apply to column-densities below
typically a few
molecules
(Figs. 4 (click here)c, 5 (click here)e, and 5 (click here)f). The cloud is moderately thick in
CO and optically thin in
. These rather ``transparent"
conditions justify the close excitation temperatures found in CO and
. In denser, brighter clouds such as Orion A, larger differences
have been found because the saturated lines of both isotopes probe regions
of distinct density (Castets et al. 1990). In HCL 2 for
instance (Cernicharo & Guélin 1987), CO,
,
and
are detected at extinctions above 0.5, 0.7, and 1.5
mag, respectively, which correspond to differences in temperature of several
Kelvin according to the model of Lequeux et al. (1994).
Figure 5: Number distributions of the excitation temperatures in
CO() a) and
b), of the optical depths in CO(
) c) and
d), and of the column-densities in
CO e) and
f), derived from the observed lines
and the LVG model for kinetic temperatures between 8 and 30 K
Figures 6 (click here)a and 6 (click here)b compare the velocity-integrated intensities
recorded from the CO and
isotopes. They reveal
significant fluctuations of the
) ratios
from point to point for both line transitions. While being less sensitive to
the absolute calibration of the telescopes, the
) ratios depend more on the signal-to-noise ratio of the weaker
detections and on the day-to-day variations in the calibration.
With these uncertainties, linear fits have been applied to the data in
Fig. 6 (click here) according to the method described in Sect. 3.1. They yield
averages of the
) ratios over the cloud
of
and
for the
and
transitions, respectively. The highly dispersed points are
clearly not consistent with these means.
Figure 6: Correlations between velocity-integrated intensities from the
two isotopes of CO, ) versus
),
in the
a) and
b)
transitions
Figures 7 (click here) and 8 (click here) demonstrate that the
) ratios decrease with the velocity-integrated
intensity,
), in a similar way for the
and
transitions. The high ratios recorded at low intensity
cannot be attributed to a finite instrumental sensitivity or a poor
signal-to-noise ratio in the
spectra since only firm
detections have been retained in these figures. The apparent decrease of the
ratios with
) can be explained by the rapid saturation of the
bright CO lines as shown in Figs. 7 (click here) and 8 (click here): the solid line
indicates the change in
) with
increasing
column-density as predicted by the radiative
transfer of CO lines treated under LTE assumptions. A standard isotopic
ratio
] of
(Wilson & Rood
1994) has been used to produce this curve, together with an
excitation temperature in CO and in
equal to the mean values
derived above (7.0 K and 5.2 K, respectively). To follow the increase in
optical depth and the progressive saturation of the line core, line
integrals
of the CO and
radiation temperatures have been
calculated for increasing
column-density, assuming a gaussian
velocity profile with a full-width to half-maximum (FWHM)
of
, typical of the optically thin
lines
detected in the cloud. Calculations have been conducted for both
transitions. The resulting curves in Figs. 7 (click here) and 8 (click here) nicely agree
with the envelope of the data points. The high ratios obtained at low
) arise from optically thin conditions for both isotopic species.
The asymptotic value of
measured at large
), when
line cores become optically thick, is commonly reached in
other dense globules of high optical depth:
in B157 and L1075
(Robert & Pagani 1993),
in MCLD 126.6+24.5
(Boden & Heithausen 1993),
in HCL2
(Cernicharo & Guélin 1987).
In addition to the overall decrease in
) as a function of
), a large dispersion in the
ratios about this relation can be seen in Figs. 7 (click here) and 8 (click here). The
significant fluctuations have been recorded in the two transitions and by
two different telescopes. The same behaviour has been reported in two other
clouds. In HCL 2, Cernicharo & Guélin (1987) have observed
highly dispersed
) ratios, decreasing as
a function of visual extinction AV between 1 and 7 mag, as derived from
star counts. The amplitude of the fluctuations on a 0.2 pc scale is
equivalent to that in Cepheus. Using more precise extinction measurements
from near-infrared star observations toward the dark cloud IC 5146, Lada et
al. (1994) have found that the ratios of
column-density to A V,
, present large
fluctuations at low A V and the ratios decrease with increasing A
V up to 30 mag, probably because of the saturation of the
lines. They argue that the large dispersion at low A V is not caused by
the instrument, nor by the scatter in the extinction measurements. The
fluctuations have a larger amplitude on a 0.2 pc scale than in Cepheus.
Hence, data from HCL 2, IC 5146, and Cepheus suggest that large intrinsic
fluctuations in the
abundance or its excitation conditions
occur in the outer layers of some molecular clouds. Similar conclusions have
been reached by Langer et al. (1989) studying the
ratio in Barnard 5. The fluctuations
appear at A V below
in Cepheus,
in
HCL 2 and
3 mag in IC 5146.
Figure 7: Observed ) ratios as a
function of
) in the
transition. The
LTE model curve (thick line) is given for a line width of 2.1 km/s and an
excitation temperature of 7.0 K in CO. The thin lines correspond to
variations of
in temperature a) and
in line width b)
Figure 8: Observed ) ratios as a
function of
) in the
transition. The
LTE model curve (thick line) is given for a line width of 2.1 km/s and an
excitation temperature of 7.0 K in CO. The thin lines correspond to
variations of
in temperature
The height of the LTE model curve in Figs. 7 (click here) and 8 (click here) is determined
by the excitation temperature in CO, the FWHM
of the assumed line profile, and the isotopic abundance ratio
; it is insensitive to the excitation temperature in
. The regions spanned by the model curve for plausible variations of
and
in Cepheus are shown in
Figs. 7 (click here)a and 7 (click here)b, respectively. The selected range of
in
corresponds to the rms dispersion of the CO temperature
distribution found in the cloud (see Fig. 4 (click here)a and Table 1 (click here)).
The interval of
in
represents the measured
velocity dispersion from line to line. Hence, it seems unlikely that the
large scatter observed in the
) ratios
be caused by variations in the lines width or in CO excitation temperature.
In particular, unrealistically cold temperatures or narrow lines would be
required to account for the lowest ratios recorded at low or medium
). These points have been checked individually to be inconsistent
with the modelled
) ratio expected from
their particular temperature and line width. Only one point could be
reconciled with the model. In addition, these points have ``normal"
excitation temperatures in
, close to the mean.
The ) ratios in the 2-1 transition
have also been studied at the full resolution of the POM-2 telescope, i.e.
2.3' or 0.2 pc locally, to check for possible beam dilution effects. The
measured ratios and intensities at 2.3' are totally consistent with those
presented in Fig. 8 (click here) at the reduced resolution of 8.7', i.e. 0.8
pc in the cloud. Beam dilution in
cannot explain the lowest
) ratios which, on the contrary, require
enhanced
emission. One could, however, imagine clumps of gas,
unresolved both in CO and
, and dense enough to exhibit low
) ratios (hardly sensitive to beam
dilution). In Figs. 7 (click here) or 8 (click here), these points would appear as simply
``shifted" to abnormally low
) intensity. Yet the consistency
between the 8.7' and 2.3' data rules out this possibility.
In conclusion, the observed fluctuations in the ) ratio are likely to be due to
abundance variations
along the line of sight. The isotopic ratio
must be
lowered to values between 4 and 10 for the model curve to reach these
points. The lowest
) ratios recorded
may be explained by isotopic fractionation in a low density and cold
(
) ambient gas. Fractionation would be particularly active in
the extended envelope of Cepheus where the temperature is low and visual
extinction amounts only to
(Lebrun 1986),
implying that much of the carbon must be in the form of
.
Observations of
emission at
would be highly
desirable to check this effect.
These competing effects introduce a large uncertainty in the ``mean"
) value obtained for a cloud and may be
responsible for the differences observed between Cepheus and other clouds.
Among the translucent and high-latitude clouds the ratios range from 3 to 29
around a mean value of about 10 (van Dishoeck et al. 1991).
Variations by a factor of 4 occur in the Orion A data of Castets et
al. (1990). Observing the fluctuations and dependence of the
) ratio with
) or extinction in a
variety of other clouds would give valuable constraints on the theoretical
modelling of the photodissociation and fractionation processes in the outer
layers of clouds. As a drawback, the intrinsic scatter introduces a large
uncertainty in the
mass derivation from
observations.