Quantification of molecular cloud structure using the Δ-variance

We present a detailed study of the Delta -variance as a method to qunatify molecular cloud structure. The Delta -variance was introduced by Stutzki et al. (1998) to analyze the drift behaviour of scalar functions and is used to characterize the spatial structure of observed molecular cloud images. For fractional Brownian motion structures (fBm-fractals), characterized by a power law power spectrum and random phases, the Delta -variance allows to determine the power spectral index beta. We present algorithms: to determine the a-variance for discretely sampled maps and study the influence of white noise, beam smoothing and the finite spatial extent of the maps. We find that for images with beta > 3, edge effects can bias the structure parameters when determined by means of a Fourier transform analysis. In contrast, the Delta -variance provides a reliable estimate for the spectral index beta, if detcrnlined in the spatial domain. The effects of noise and beam smoothing are analytically represented in a lending order approximation. This allows to use the Delta -variance of observed maps even at scales where the influence of both effects becomes significant, allowing to derive the spectral index beta over a wider range and thus moro reliably than possible otherwise. The Delta -variance is applied to velocity integrated spectral line maps of several clouds observed in rotational transitions of (12)CO and (13)CO. We find that the spatial structure of the emission is well characterized by a power law power spectrum in all cases. For linear scales larger than similar to0.5 pc the spectral index is remarkably uniform for the different clouds and transitions observed (2.5 less than or equal to beta less than or equal to 2.8). Significantly larger values (beta greater than or similar to 3) are found for observations made with higher linear resolution toward the molecular cloud MCLD 123.5+24.9 in the Polaris Flare, indicating a smoother spatial structure of the emission at small scales (<0.5 pc).