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Register a new userVelocity centroids and the structure of interstellar turbulence: I. Analytical study
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F. Levrier
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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 C_0, whose expression can be linearized for small density fluctuations. To describe the statistics of I, C and C_0, 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-Deschenes, Levrier and Falgarone (2003, ApJ, 593, 831) 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 and Esquivel (2003, ApJL, 592, 37) as a possible alternative. In a following paper, we shall present numerical studies aimed at assessing the validity domain of these results.
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Velocity statistics is a direct probe of the dynamics of interstellar turbulence. Its observational measurements are very challenging due to the convolution between density and velocity and projection effects. We introduce the projected velocity structure function, which can be generally applied to statistical studies of both sub- and super-sonic turbulence in different interstellar phases. It recovers the turbulent velocity spectrum from the projected velocity field in different regimes, and when the thickness of a cloud is less than the driving scale of turbulence, it can also be used to determine the cloud thickness and the turbulence driving scale. By applying it to the existing core velocity dispersion measurements of the Taurus cloud, we find a transition from the Kolmogorov to the Burgers scaling of turbulent velocities with decreasing length scales, corresponding to the large-scale solenoidal motions and small-scale compressive motions, respectively. The latter occupy a small fraction of the volume and can be selectively sampled by clusters of cores with the typical cluster size indicated by the transition scale.
The interstellar turbulence is magnetized and thus anisotropic. The anisotropy of turbulent magnetic fields and velocities is imprinted in the related observables, rotation measures (RMs), and velocity centroids (VCs). This anisotropy provides valuable information on both the direction and strength of the magnetic field. However, its measurement is difficult especially in highly supersonic turbulence in cold interstellar phases due to the distortions by isotropic density fluctuations. By using 3D simulations of supersonic and sub-Alfvenic magnetohydrodynamic(MHD) turbulence, we find that the problem can be alleviated when we selectively sample the volume-filling low-density regions in supersonic MHD turbulence. Our results show that in these low-density regions, the anisotropy of RM and VC fluctuations depends on the Alfvenic Mach number as $rm M_A^{-4/3}$. This anisotropy-$rm M_A$ relation is theoretically expected for sub-Alfv enic MHD turbulence and confirmed by our synthetic observations of $^{12}$CO emission. It provides a new method for measuring the plane-of-the-sky magnetic fields in cold interstellar phases.
Guided by the duality of turbulence (random versus coherent we seek coherent structures in the turbulent velocity field of molecular clouds, anticipating their importance in cloud evolution. We analyse a large map (40 by 20) obtained with the HERA multibeam receiver (IRAM-30m telescope) in a high latitude cloud of the Polaris Flare at an unprecedented spatial (11) and spectral (0.05 km/s) resolutions in the 12CO(2-1) line. We find that two parsec-scale components of velocities differing by ~2 km/s, share a narrow interface ($<0.15$ pc) that appears as an elongated structure of intense velocity-shear, ~15 to 30 km/s/pc. The locus of the extrema of line--centroid-velocity increments (E-CVI) in that field follows this intense-shear structure as well as that of the 12CO(2-1) high-velocity line wings. The tiny spatial overlap in projection of the two parsec-scale components implies that they are sheets of CO emission and that discontinuities in the gas properties (CO enrichment and/or increase of gas density) occur at the position of the intense velocity shear. These results disclose spatial and kinematic coherence between scales as small as 0.03 pc and parsec scales. They confirm that the departure from Gaussianity of the probability density functions of E-CVIs is a powerful statistical tracer of the intermittency of turbulence. They disclose a link between large scale turbulence, its intermittent dissipation rate and low-mass dense core formation.
We review the use of velocity centroids statistics to recover information of interstellar turbulence from observations. Velocity centroids have been used for a long time now to retrieve information about the scaling properties of the turbulent velocity field in the interstellar medium. We show that, while they are useful to study subsonic turbulence, they do not trace the statistics of velocity in supersonic turbulence, because they are highly influenced by fluctuations of density. We show also that for sub-Alfvenic turbulence (both supersonic and subsonic) two-point statistics (e.g. correlation functions or power-spectra) are anisotropic. This anisotropy can be used to determine the direction of the mean magnetic field projected in the plane of the sky.
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