The edges of molecular clouds: Fractal boundaries and density structure

This paper continues a discussion of the manifestations of the highly nonlinear physical equations underlying the dynamics of the dense interstellar medium. Previously, Falgarone and Phillips confirmed that the velocity field in non-star-forming regions could be explained as a turbulent phenomenon, showing the Kolmogorov scaling of velocity dispersion with spatial extent, and proposed that the excess of large velocity deviations (line wings stronger than predicted by a Gaussian distribution) corresponds to the fundamental property of turbulent flows called intermittency. In the present work we inspect the spatial structure of the dense medium. The observations of clouds at two different distances were carried out at high angular resolution using several transitions of the carbon monoxide molecule. Cloud edge regions were selected for the study to avoid the spatial crowding of emitting components, which obscures the structure of cloud cores. The selected regions include components from the line core and line wings of the parent complexes. We find that spatial structure exists on all scales down to our best angular resolution (0.02 pc) and that the emission arises from structures small compared to the telescope beam. Maps over a large range of scale sizes are self-similar suggesting a fractal structure. This is made quantitative by the use of the area-perimeter relation. A fractional value is found for the dimension parameter, D = 1.36 (with a formal error of ±0.02), in agreement with that found for atmospheric clouds and other interstellar clouds. An unexpected result is that this dimension is the same for the largest entities (∼100 pc) which are self-gravitating and the smallest (∼0.1 pc) which are not. It is also independent of the rotational transition of CO used to map the molecular gas. In addition, the intercept of the logarithmic perimeter-area relation scales with the resolution of the observations in accord with Mandelbrot's prediction for fractals. From the study of the ¹²CO (J = 3-2) and ¹²CO (J = 2-1) lines we find that the ¹²CO (J = 3-2)/¹²CO (J = 2-1) ratio remains constant at ∼0.55 over the whole range (about a factor of 10) of line intensities available in the maps. We interpret this as a signature of uniform excitation conditions for the CO rotational transitions, throughout the entire range of column densities. It is remarkable that the gas in the line wings cannot be distinguished from that in the line cores on the basis of this ratio. The ¹³CO/¹²CO ratios show that the ¹²CO (J = 2-1) optical depth has a typical value of ∼4. We argue that the ¹²CO (J = 3-2) emission is almost thermalized and the gas seen in this transition is quite dense (10⁴ cm^-3 or more for n_H2) and cold (∼10 K). An estimate of the heating and cooling rates for these regions shows that the result is plausible. This fits with the overall framework of dense gas organized in a self-similar distribution of sizes continuing down to a threshold estimated to be smaller than 2000 AU. The dense molecular material is therefore very much non-space filling.