Velocity Modification of the Power Spectrum from an Absorbing Medium

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A quantitative description of the statistics of intensity fluctuations within spectral line data cubes introduced in our earlier work is extended to the absorbing media. The possibility of extracting three-dimensional velocity and density statistics from both integrated line intensity and the individual channel maps is analyzed. We find that absorption enables the velocity effects to be seen even if the spectral line is integrated over frequencies. This regime, which is frequently employed in observations, is characterized by a nontrivial relation between the spectral index of velocities and the spectral index of intensity fluctuations. For instance, when density is dominated by fluctuations at large scales, i.e., when correlations scale as r-γ, γ < 0, the intensity fluctuations exhibit a universal spectrum of fluctuations ~K-3 over a range of scales. When small-scale fluctuations of density contain most of the energy, i.e., when correlations scale as r-γ, γ > 0, the resulting spectrum of the integrated lines depends on the scaling of the underlying density and scales as K-3+γ. We show that if we take spectral line slices that are sufficiently thin, we recover our earlier results for thin-slice data without absorption. As a result, we extend the velocity channel analysis (VCA) technique to optically thick lines, enabling studies of turbulence in molecular clouds. In addition, the mathematical machinery developed enables a quantitative approach to solving other problems that involved statistical description of turbulence within emitting and absorbing gas.