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Universität zu Köln
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Mathematisch-Naturwissenschaftliche Fakultät
Fachgruppe Physik

I. Physikalisches Institut

Extension of the Delta-variance analysis

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The Delta-variance analysis was introduced by Stutzki et al. (1998) and systematically studied and applied to observational data by Bensch et al. (2001). It measures the relative amount of structure on a given scale by filtering an observed map by a radially symmetric wavelet with a characteristic length scale l and computing the total variance in the convolved map. The spectrum of these Delta-variance values as a function of the filter size l provides a measure for the scaling behaviour of the different components contributing to an observed structure. It can easily separate observational artifacts in the map structures resulting from a finite signal-to-noise of the measurements and a finite telescope beam size from the inherent structure of the cloud studied.

This first implementation had, however, several shortcomings. It was not able to take into account a possible variation of the signal-to-noise ratio across different points in the map, the computation by convolution in the spatial coordinates was very time-consuming, and the selection of the filter shape was somewhat arbitrary and did not provide an exact value for the scales traced. In the last SFB period we have considerably improved the Delta-variance analysis method solving the first two problems by an appropriate treatment of the filtering in Fourier space (Ossenkopf et al. 2005a). It makes use of a supplementary significance function by which each data point is weighted. This allows to distinguish the influence of variable noise from actual small scale structure in the maps and it helps to deal with the boundary problem in non-periodic and/or irregularly bounded maps. By enabling a computation in Fourier space the analysis is considerably accelerated.

The new method thus can be performed efficiently for irregular maps or observations with a variable noise across the map. The use of a significance function is also essential to analyze derived quantities like centroid velocities. Here, two thresholds given by the ntegrated line intensities have to be used: a lower threshold below which all data have to be ignored and an upper threshold above which the significance of the data is not further improved by decreasing the noise.

Systematically studying the influence of the filter shape on the detection of characteristic structures and spectral indices we determined the actual relation between the detection scale and the characteristic filter size l and found that the optimum Delta-variance filter as a reasonable compromise between slope stability and scale detection is given by a Mexican hat filter with a ratio between the diameters of the core and the annulus of 1.5.

pic-4a pic-4b
1.3mm continuum map rho Oph taken by Motte et al. (1998) in a linear intensity scale together with the map of weights given by the square root of the number of integrations at each point

 

pic-4c Delta-variance spectrum of the rho Oph map with the full weighting function plotted with systematic error bars indicating the statistical uncertainty of the measurement. The connecting line represents the power-law fit to the self-similar part of the spectrum including the beam convolution effect.

Applying the new method to the dust emission map of rho Oph by Motte et al. (1998) shows that the spatial scaling behaviour there can be described perfectly by a power law interconnecting the range of small clumps and more massive cores. The method reproduces the size of the dominant cores but we find no difference in the behaviour between cores and clumps identified by Motte et al. (1998) in the data. The region represents an intermediate stage in the molecular cloud evolution showing both signatures of the typical molecular cloud scaling behaviour and the formation of condensed cores. When analysing the velocity structure of the Polaris Flare we find that a universal power law connects scales from 0.03 pc to 3 pc. However, a plateau in the Delta-variance spectrum around 5 pc indicates that the visible large scale velocity gradient is not converted directly into a turbulent cascade.

Contact: Volker Ossenkopf