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Astron. Astrophys. 331, 669-696 (1998)

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5. Discussion

CO line profiles in different transitions and isotopic variants provide valuable constraints on the spatial and velocity distribution of the emitting material. Models of the CO line emission of molecular clouds have been developed over the past twenty years. We discuss the salient properties of our data with reference to the most successful kind of models.

5.1. A short overview of existing models of radiative transfer

The intricate puzzle made up by the properties of the molecular line emission of clouds have led to increasingly sophisticated models of radiative transfer. The CO line properties are in general best reproduced by models which make the radiative transfer a local process, either in velocity space by using the Large Velocity Gradient (LVG) approximation or in space by assuming that the emitting gas is concentrated in small clumps radiatively decoupled from one another. The motivation for such models is the strong observational support for the transparency of molecular clouds at large scale, or, in other words, the fact that low CO rotational transitions do trace the mass of molecular gas, irrespective of the apparently large optical depth of the observed CO lines. This property does not hold any more for dense cores.

Homogeneous cloud models were abandoned some time ago because the densities of collisional partners ([FORMULA] or HI) inferred from CO or CS multitransition analysis turned out to be far greater than the average cloud densities derived from observed column densities and cloud sizes. On the other hand, clumpy cloud models with random velocity fields have difficulties at finding clump properties (i.e. number density, size, internal velocity dispersion and density) which would meet simultaneously the conditions of line smoothness, line shapes and large apparent opacities. Several improvements have been proposed through the years: Kwan & Sanders (1986) introduced a spectrum of clump sizes, Tauber et al. (1991) considered a large number of very small extremely dense self-gravitating clumps, Wolfire, Hollenbach & Tielens (1993) took into account the CO photodissociation in the clumps, Falgarone et al. (1991) inferred a fractal distribution of the densest emitting units from the observations. Falgarone & Phillips (1996) investigated the possibility that the local [FORMULA] density be much larger than previously estimated from multitransition analysis and concluded that it is unlikely.

An important step forward had been achieved by Baker (1976) who first recognized the paradoxical attributes of molecular lines i.e. lack of saturation signposts in spite of small [FORMULA] to [FORMULA] line ratios, comparable efficiencies of [FORMULA] and [FORMULA] at sampling molecular gas in spite of their different abundances. He proposed, following Zuckerman & Evans (1974), that macroturbulence (and more specifically the relationship between the phase-space lengthscales and the viewing geometry of the telescope beam) might alleviate the puzzle. He successfully reproduced the main attributes of the line profiles by dividing the emitting gas into comoving cells of large opacity, with a low probability of shadowing each other in the beam, thus providing the emergent lines with a low effective (or beam-averaged) optical depth, manifest as Gaussian line shapes and integrated area proportional to the total column density of molecules. Martin, Sanders and Hills (1984) (hereafter MSH) have explicitely computed the behavior of the effective optical depth of the medium, for various density distributions within the cells.

More recently, Albrecht & Kegel (1987) and Kegel, Piehler & Albrecht (1993) have computed line profiles emerging from a turbulent velocity field with a finite correlation length and shown that the larger this length compared to the size of the region over which the lines form, the better the above attributes of the CO lines are reproduced. Park & Hong (1995) have conducted 3-dimensional Monte-Carlo simulations of radiative transfer in clumpy turbulent clouds. They conclude that "clumpiness and macro-turbulence are the cloud attributes that are consistent with the observed profiles of CO lines from cold, dark, quiescent clouds". Last, turbulence has been shown to be able to reproduce the main statistical properties of molecular line profiles (Falgarone et al. 1994), and Dubinski, Narayan & Phillips (1995) have found that the most prominent variations in the line shapes of the profiles are indeed due to large scale motions in turbulence, which contain the bulk of the kinetic energy (i.e. Kolmogorov power spectrum).

In summary, all these studies agree with the idea that the CO line properties of molecular clouds are best reproduced by the macroturbulent models. In these models, a photon emitted in a phase-space location in the cloud, a cell identified either in space by a density contrast above the surrounding medium or by the local correlations of the velocity field, without any associated increase of density, or by a combination of both, is either absorbed and reemitted in the same cell or has a large probability to leave the cloud. The velocity field and/or gas distribution are such that the photons emitted by a given cell have little radiative coupling with the other cells in the cloud so that the emergent line profile is mostly determined by the averaging of the cell emission within the beam and not by radiative transfer effects intercoupling the whole cloud. The emergent line profile is therefore governed by the dilution in space and velocity of the emitting regions at each velocity. At the opposite, in the microturbulent models, a photon emerging at a given frequency may come from many phase-space locations in the cloud. The above results show that clumping in velocity space is at least as important as clumping in density to understand the characteristics of the line profiles and it is not clear what is primarily responsible for the CO spectral line properties, the turbulent velocity field and its correlations or the clumpiness of the CO emitting gas.

5.2. Constraints on the structure of the emitting gas

In this section we explain why the line analysis of the present data set (in particular for the line-wing emission) requests the introduction of phase-space dilution. We then derive a few characteristics of the CO emitting medium, from the analytical approach of MSH.

In the LVG formalism, for instance, a constant value R(2-1/1-0) [FORMULA] over [FORMULA] (J=1-0) line temperatures ranging between 5K and 1K, characteristic of the line-wing emission in the three fields, would correspond to optically thin emission of the gas. But an optically thin solution is inconsistent with the values 3 [FORMULA] R(12/13) [FORMULA], observed in the line wings, which are smaller than the above [ [FORMULA] ]/[ [FORMULA] ] isotopic abundance ratio. Were the R(2-1/1-0) ratio slightly underestimated (see below), optically thick solutions would be a viable solution. But in that case, it would be difficult to reproduce the constancy of R(2-1/1-0) over a factor 6 in line temperature, unless a very narrow combination of density and column density variations with velocity be invoked. This latter solution would conflict with the fact that the line wing properties are the same in the three fields.

We give here a preliminary analysis of the observational properties described in Sect. 4 in the framework of macroturbulence. As will be seen, this description has its limitations too. The line intensity at velocity v in a given direction [FORMULA] on the sky is:


where the excitation temperature [FORMULA] is assumed independent of velocity and position, and [FORMULA], the effective (or beam averaged) optical depth, is defined by


[FORMULA] being the main beam pattern. [FORMULA] is the Planck function. In the formalism developed by MSH - see their Eq.(8) - the beam averaged optical depth at velocity v, in the case where the cell to cell velocity dispersion [FORMULA] is more than three times larger than the internal velocity of a cell [FORMULA], simplifies to:


where [FORMULA] is the total number of cells in the beam, irrespective of their velocity, [FORMULA] is the cell size and B the half-power beam width. [FORMULA] is the ratio of the cell effective optically thick area to [FORMULA]. It depends only on [FORMULA], the cell optical depth at its mean velocity and on the velocity law adopted for the cell internal velocity dispersion. It describes the increase of radiative coupling of a cell to the rest of the cloud as its optical depth [FORMULA] is increased. The key point here is that [FORMULA] increases slowlier than linearly with [FORMULA], so that, even for [FORMULA], [FORMULA] may remain close to a few. MSH have shown too that the departure from the linear behaviour of [FORMULA] with [FORMULA] is more pronounced for hard spheres than for cells with Gaussian or Lorentzian density distributions, because the effect of beam-averaging in space and velocity is the more pronounced the higher the contrast of opacity within the beam. Therefore, [FORMULA] may stay well below [FORMULA] even for very large values of [FORMULA]. Eq.(2) also shows that the effective optical depth depends on the surface beam filling factor of the ensemble of cells in the beam, [FORMULA], and on their dilution in velocity space, [FORMULA]. The velocity dependence of [FORMULA] is given by the cell to cell velocity distribution (here a Gaussian). These dependences are the same for different transitions. The only factor which depends on the transition is [FORMULA].

We first estimate the cell opacities, noted [FORMULA] and [FORMULA] for the [FORMULA] (J=1-0) and [FORMULA] (J=1-0) lines respectively. The R(12/13) line ratio is derived from Eq.(1). Assuming that the excitation temperatures of the [FORMULA] and [FORMULA] lines are close enough to make the ratio of [FORMULA] for the two species close to unity, we infer:


where [FORMULA] and [FORMULA] are the beam-averaged optical depths at line centroid in the [FORMULA] and [FORMULA] lines respectively. In the macroturbulent framework, the beam-averaged optical depths are smaller than or close to unity so that:


We have used the calculations of MSH for hard sphere cells with Gaussian internal velocity dispersion to derive estimates of the cell opacities. In the line wings, for isotopic abundances 40 [FORMULA] [ [FORMULA] ]/[ [FORMULA] ] [FORMULA] 70, and therefore [FORMULA], line ratios as low as [FORMULA] are obtained for [FORMULA] and [FORMULA] cell opacities [FORMULA] to 10. Cell opacities derived in the other cases considered by MSH (Gaussian or Lorentzian cell density profiles) are much larger, because in these cases the effect of beam-averaging is less pronounced.

We considered the possibility that the isotopic abundance ratio be as low as [ [FORMULA] ]/[ [FORMULA] ] [FORMULA] 20. Such a low value has been derived by Lucas & Liszt (1997) from [FORMULA] and [FORMULA] absorption measurements in molecular gas of low extinction. Indeed, it is the solution we adopt here, because, as noted by Lucas & Liszt, low abundance ratios are expected to occur in regions of weak extinction where [FORMULA] is still abundant and triggers fractionation reactions which enhance [FORMULA] relative to [FORMULA]. This is likely to be the case for the poorly shielded wing material. In this case, the [FORMULA] cell opacities are lower, [FORMULA] to 5. We rule out the optically thin solutions for the line-wing emission because the isotopic line ratios cover values of R(12/13) as low as [FORMULA] with yet the same value R(2-1/1-0) [FORMULA] 0.6.

In the line cores, where the bulk of the data points have [FORMULA], both the beam-averaged opacities and the cell opacities are larger, [FORMULA]. These large values are consistent with the [FORMULA] cell opacities derived from the R(13/18) ratio in the line cores (Figs. 11 to 13). In the MSH formalism, the range of observed line ratios [FORMULA] corresponds to the range of cell opacities [FORMULA] for isotopic ratios 5.5 [FORMULA] [ [FORMULA] ]/[ [FORMULA] ] [FORMULA] 7.

It is then possible to derive the other cell parameters from their optical depths, using the constraint on the excitation temperature provided by the line intensities. The [FORMULA] (J=1-0) lines span about the same intensity intervals in the three fields. Line temperatures as large as [FORMULA] K and 8K are observed in the line wings and line cores respectively. These values set lower limits to the excitation temperature equal to [FORMULA] K in the line wings and [FORMULA] K in the line cores (we note [FORMULA] the excitation temperature of the (J,J-1) pair of levels). Thermalisation of the transitions at [FORMULA] K is therefore a solution for the cells emitting in line cores, implying local densities larger than [FORMULA] [FORMULA] and [FORMULA] [FORMULA] /km s-1. Thermalisation is not achieved in cells emitting in the line wings, unless the R(2-1/1-0) line ratios are underestimated. Would the 1-0 intensities be overestimated by no more than 20%, the line intensities and line ratios would become consistent with thermalisation of the transitions at [FORMULA] K, but we ignore this possibility in what follows.

We now use the LVG formalism to determine sets of non-LTE solutions which satisfy the four constraints on the cell emission in the line wings. The two first constraints, [FORMULA] a few, and [FORMULA] K set a strict lower limit on the cell column density per unit velocity interval [FORMULA] [FORMULA] /km s-1, independent of the kinetic temperature. Then, the cell density is constrained in both directions by (i) the fact that in the macroturbulent framework the ratio R(2-1/1-0) is larger than the same ratio for individual cells, estimated in the LVG formalism. The constraint [FORMULA] sets an upper limit on the cell density for each [FORMULA] and an upper limit to [FORMULA], (ii) the line emission of invidual cells has to be brighter than the observed lines, to allow for phase-space beam dilution and this constraint sets a lower limit to the cell density at each [FORMULA].

Solutions for the cell densities exist up to large gas temperatures. They decrease from [FORMULA] to [FORMULA] [FORMULA] at [FORMULA] K to [FORMULA] to [FORMULA] [FORMULA] at [FORMULA] K, corresponding to increasing cell sizes. The additional constraint, [FORMULA], rules out the low density solutions at temperatures larger than [FORMULA] K. The corresponding cell sizes, estimated as


with [FORMULA] and [FORMULA] equal to the thermal velocity dispersion in [FORMULA], increase from [FORMULA] to 229 AU for [FORMULA] K, to [FORMULA] to 1000 AU for [FORMULA] K.

Cells emitting in the line cores may have similar sizes, although yet smaller values are possible. The cell optical depths in the line-core emission being about two orders of magnitude larger than in the wings, the local densities, for comparable cell size are as large as [FORMULA] to [FORMULA] [FORMULA]. This density range is that derived for two positions in the Polaris core from multitransition CS and C34 S observations (Gerin et al. 1997) but additional observations are required to disentangle the effects of density from those related to chemistry (Gerin et al., in preparation).

Last, we compare this cell size estimate to that derived from the condition [FORMULA] at line centroid, using Eq.(2) (line centroid here is considered as the velocity at which the line-wing emission peaks). According to Eq.(2), this condition is equivalent to:


which describes the small radiative coupling between the cells. A lower limit of the quantity [FORMULA] is in principle provided by the smoothness of the line profiles. Although the anticipated correlations of the cells in space and velocity space prevent a simple computation of the cell number in a beam from straightforward arguments on profile smoothness (Tauber et al. 1991), we give below the corresponding determination of [FORMULA] derived under the assumption of statistically independent cells (Poisson distribution). We follow a simple approach, similar to that proposed by Tauber et al. (1991) or Falgarone et al. (1992), based on a medium consisting of numerous cells, all with the same projected area, and randomly moving in the beam. The condition of smoothness of the line profile can be expressed as:


where S/N is the signal to noise ratio of the profiles. It simply expresses the fact that the channel-to-channel temperature fluctuations are only due to the statistical fluctuations of the number of cells emitting in a given channel of width [FORMULA],


Here, the velocity distribution of all the cells is assumed to be a square distribution, but a more realistic Gaussian distribution would not significantly change the conclusions (see Tauber et al. 1991). The smoothness of the line profiles is illustrated by the set of [FORMULA] (J=2-1) line profiles obtained at the (0,0) positions (Fig. 3b). The rms noise level measured on the baseline of these spectra is [FORMULA] mK for a velocity resolution [FORMULA] 0.1 km s-1. The line intensities in the wings being [FORMULA] 2K, a signal to noise ratio of [FORMULA] 30 is characteristic of our best [FORMULA] (J=2-1) profiles. We assume that, if all spectra were observed with comparable signal-to-noise ratio, these numbers would feature the rule, not the exception. Substituting the corresponding upper limit of [FORMULA] into Eq.(4) thus provides an upper limit to the cell size,


[FORMULA] pc being the IRAM beam at 230GHz, at an adopted average distance of 150 pc for the sources. Although the hypothesis made on the cell statistics are very crude, this upper limit is not inconsistent with the estimates based on the line ratios. Better determinations require appropriate statistics for the cells, especially in phase space, and a detailed modelling of the radiative transfer.

5.3. The variations of the peak [FORMULA] line temperatures with linewidths

To illustrate the trend, visible in Fig. 8, that in Polaris, the more intense the [FORMULA] lines, the narrower they are, we have performed Gaussian fits in the [FORMULA] (J=1-0), [FORMULA] (J=2-1) and [FORMULA] (J=1-0) lines. Fig. 14 displays the half-power widths of the fitted Gaussians versus their peak values for the [FORMULA] (J=1-0) lines, in the Polaris and L1512 fields. The results are the same with the other lines. In Polaris (small diamonds), there is a clear decline of the line widths as the line peak increases over a large range of linewidths (from [FORMULA] 0.9 km s-1 to 0.2 km s-1) while in L1512 (large diamonds), the decline is still visible but over a much smaller range of line widths, from [FORMULA] 0.3 to 0.2 km s-1. The power law dependence of the decrease (eye-fitted) is approximately [FORMULA] for the main family of points belonging to both sources.

[FIGURE] Fig. 14. The half-power width versus peak line intensity of the Gaussians fitted to the [FORMULA] (J=1-0) lines in Polaris (small diamonds) and L1512 (large diamonds).

It is interesting to note that in Polaris, the line width decrease is associated with a systematic variation of the Gaussian centroid velocity (Fig. 15). This variation is due to the gradient along the filament visible in the channel maps of Fig. 5b. In other words, the decrease in line width is monotonous along the [FORMULA] filament, with the smallest line widths in the North-East, and the largest in the South-West.

[FIGURE] Fig. 15. The centroid velocity versus half-power width of the Gaussians fitted to the [FORMULA] (J=1-0) lines in a Polaris and b L1512.

One simple interpretation of this relation between line width and line peak is the following. The line width being a good indicator of the total velocity dispersion of the cells, [FORMULA], its decrease tends to reduce the dilution in velocity space of the emission of the cells (assuming that their other properties remain the same), and therefore tends to increase the emergent intensity. The decrease of [FORMULA] might also be associated with an increased surface filling factor of the cells, which would explain why the line width dependence with the line intensity is not strictly an inverse power law. The concomitant increase of the R(2-1/1-0) ratio with the line intensity suggests an increase of the excitation temperature, either due to an increase of the radiative coupling between cells or to an increase of density, up to thermalisation of the transitions. This issue requires further observations of high dipole moment molecules.

We may speculate that the line-wing material has a large suprathermal velocity dispersion and line-wing emission is weak mostly because originating in coherent cells highly diluted in space and velocity. In regions where dissipation of the non-thermal support has occurred, the space-velocity filling factor of the cells increases and the line emission bears the signatures of the increased radiative coupling among the cells: these are the regions corresponding to the line-core emission. Within these regions, the relation between the peak line intensity and the line width suggests a similar link between the line intensity and the decreasing velocity dilution of the cells in a medium where turbulence is dissipating. In the Polaris field, the line width decrease may be interpreted as due to a gradual loss within [FORMULA] 0.2pc (the length of the bright [FORMULA] filament in Figs. 5b) of the suprathermal support of the cell ensemble and a correlated increase of their radiative coupling. In L1512, the spatial transition between the line-wing and line-core material is very sharp as shown by the maps of Fig. 4. The line-core emission properties seem to be more uniform through the map, possibly because the turbulence dissipation is already completed there, as suggested by the extremely narrow lines (Fig. 14). A further evolutionary step is that of L134A in which self-reversed profiles indicate that radiative coupling among all the parts of the cloud is important.

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© European Southern Observatory (ESO) 1998

Online publication: February 16, 1998