Study of velocity centroids based on the theory of fluctuations in position-position-velocity space

We study the possibility of obtaining power spectrum of gas velocity in the turbulent interstellar medium from spatial correlation of velocity centroids (VCs) of optically thick emission lines. Combining this study with the earlier studies of centroids in Esquivel & Lazarian, we conclude that centroids are applicable for studies of subsonic/transonic turbulence for sufficiently small line-of-sight (LOS) separations at which self-absorption does not affect correlation scalings. At larger LOS separations where self-absorption becomes important, we find that there is a range of scales over which VC correlation demonstrates the universal scaling, similar to the effect found in the velocity channel analysis (VCA). In other words, for large absorptions the VCs lose their ability to reflect the spectra of turbulence. We develop analytical formalism that relates statistical properties of underlying magnetohydrodynamical (MHD) turbulence to observable scaling and anisotropy of VC correlations arising from Alfven, slow and fast modes that constitute the compressible MHD modes, and show how the VC anisotropy can be used to find the media magnetization as well as to identify and separate the contributions from these MHD modes. Our study demonstrates that VCs are complementary to the VCA. In order to study turbulent volume with insufficient resolution of single-dish telescopes, we demonstrate how the studies of anisotropy can be performed using interferometers. We also suggest that restricted VC can be constructed for absorption lines by integrating LOS velocity weighted by the optical depth. We discuss the requirements for applicability of this approach.