Quantification of molecular cloud structure using the Δ-variance

I. Physikalisches Institut der Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany

Corresponding author: F. Bensch, bensch@ph1.uni-koeln.de

Received: 4 September 2000
Accepted: 7 November 2000

Abstract

We present a detailed study of the Δ-variance as a method to quantify molecular cloud structure. The Δ-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 Δ-variance allows to determine the power spectral index β. We present algorithms to determine the Δ-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 , edge effects can bias the structure parameters when determined by means of a Fourier transform analysis. In contrast, the Δ-variance provides a reliable estimate for the spectral index β, if determined in the spatial domain. The effects of noise and beam smoothing are analytically represented in a leading order approximation. This allows to use the Δ-variance of observed maps even at scales where the influence of both effects becomes significant, allowing to derive the spectral index β over a wider range and thus more reliably than possible otherwise. The Δ-variance is applied to velocity integrated spectral line maps of several clouds observed in rotational transitions of ¹²CO and ¹³CO. 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 ∼0.5 pc the spectral index is remarkably uniform for the different clouds and transitions observed (). Significantly larger values () 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).