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Issue A&A Volume 421, Number 2, July II 2004 Page(s) 387 - 398 Section Astrophysical processes DOI 10.1051/0004-6361:20047139

A&A 421, 387-398 (2004)
DOI: 10.1051/0004-6361:20047139

Velocity centroids and the structure of interstellar turbulence

I. Analytical study

F. Levrier

LERMA / LRA - UMR 8112 du CNRS, École normale supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, France

(Received 26 January 2004 / Accepted 15 March 2004 )

Abstract
We present an analytical study of the statistical properties of integrated emission and velocity centroids for a slightly compressible turbulent slab model, to retrieve the underlying statistics of three-dimensional density and velocity fluctuations. Under the assumptions that the density and velocity fields are homogeneous and isotropic, we derive the expressions of the antenna temperature for an optically thin spectral line observation, and of its successive moments with respect to the line of sight velocity component, focusing on the zeroth (intensity or integrated emission  I) and first (non-normalized velocity centroid  C) moments. The ratio of the latter to the former is the normalized centroid  C₀, whose expression can be linearized for small density fluctuations. To describe the statistics of  I, C and  C₀, we derive expansions of their autocorrelation functions in powers of density fluctuations and perform a lowest-order real-space calculation of their scaling behaviour, assuming that the density and velocity fields are fractional Brownian motions. We hence confirm, within the scope of this study, the property recently found numerically by Miville-Deschênes et al. (2003a) that the spectral index of the normalized centroid is equal to that of the full velocity field. However, it is also argued that, in order to retrieve the velocity statistics, normalization of centroids may actually not be the best way to remove the influence of density fluctuations. In this respect, we discuss the modified velocity centroids introduced by Lazarian & Esquivel (2003) as a possible alternative. In a following paper, we shall present numerical studies aimed at assessing the validity domain of these results.

Key words: ISM: structure -- methods: analytical -- turbulence

© ESO 2004

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