3. Results and discussion

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.

3.1. Line-ratio determination

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 (x_i, y_i) 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

3.2. ratios

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

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

48 ⁺⁹⁶_-48 18 ⁺²¹_-18 4.4 2.2 1.7 0.6

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

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

3.3. Excitation temperatures, opacities, and column-densities

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

3.4. W(¹²CO)/W(¹³CO) ratios

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 A_V 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.