Velocity centroids and the structure of interstellar turbulence^*

I. Analytical study

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

Corresponding author: levrier@lra.ens.fr

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 [CITE] 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 [CITE] as a possible alternative. In a following paper, we shall present numerical studies aimed at assessing the validity domain of these results.