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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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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.
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.
We explore the impact of baryonic effects (namely stellar and AGN feedback) on the moments of pairwise velocity using the Illustris-TNG, EAGLE, cosmo-OWLS, and BAHAMAS suites of cosmological hydrodynamical simulations. The assumption that the mean pairwise velocity of the gas component follows that of the dark matter is studied here at small separations, and we find that even at pair separations of 10-20 $h^{-1}mathrm{Mpc}$ there is a 4-5% velocity bias. At smaller separations, it gets larger with strength varying depending on the subgrid prescription. By isolating different physical processes, our findings suggest that the large scale velocity bias is mainly driven by stellar rather than AGN feedback. If unaccounted for, this velocity offset could possibly bias cosmological constraints from the kinetic Sunyaev-Zeldovich effect in future cosmic microwave background (CMB) surveys. Furthermore, we examine how the first and the second moment of the pairwise velocity are affected by both the baryonic and the neutrino free-streaming effects for both the matter and gas components. For both moments, we were able to disentangle the effects of baryonic processes from those of massive neutrinos; and below pair separations of 20 $h^{-1}mathrm{Mpc}$, we find that these moments of the pairwise velocity decrease with increasing neutrino mass. Our work thus paves a way in which the pairwise velocity statistics can be utilised to constrain the summed mass of neutrinos from future CMB surveys and peculiar velocity surveys.
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