Velocity modification of HI power spectrum

The distribution of atomic hydrogen in the Galactic plane is usually mapped using the Doppler shift of 21 cm emission line, and this causes the modification of the observed emission spectrum. We calculate the emission spectrum in velocity slices of data (channel maps) and derive its dependence on the statistics of velocity and density fields. We find that, (1) if the density spectrum is steep, i.e., n < -3, the large k asymptotics of the emissivity spectrum are dominated by the velocity fluctuations; and (2) the velocity fluctuations make the emission spectra shallower, provided that the data slices are sufficiently thin. In other words, turbulent velocity creates small-scale structure that can erroneously be identified as clouds. The effect of thermal velocity is very similar to the change of the effective slice thickness, but the difference is that, while an increase of the slice thickness increases the amplitude of the signal, the increase of the turbulent velocity leaves the measured intensities intact while washing out fluctuations. The contribution of fluctuations in warm H I is suppressed relative to those in the cold component when the velocity channels used are narrower than the warm H I thermal velocity and small angular scale fluctuations are measured. We calculate how the spectra vary with the change of velocity slice thickness and show that the observational 21 cm data is consistent with the explanation that the intensity fluctuations within individual channel maps are generated by turbulent velocity fields. As the thickness of velocity slices increases, density fluctuations begin to dominate emissivity. This allows us to disentangle velocity and density statistics. The application of our technique to Galactic and SMC data reveals spectra of density and velocity with power law indexes close to -11/3. This is a Kolmogorov index, but the explanation of the spectrum as due to the Kolmogorov-type cascade faces substantial difficulties. We generalize our treatment for the case of a statistical study of turbulence inside individual clouds. The mathematical machinery developed is applicable to other emission lines.