Studying velocity turbulence from doppler-broadened absorption lines: Statistics of optical depth fluctuations

We continue our work on developing techniques for studying turbulence with spectroscopic data. We show that Doppler-broadened absorption spectral lines, in particular, saturated absorption lines, can be used within the framework of the previously introduced technique termed the velocity coordinate spectrum (VCS). The VCS relates the statistics of fluctuations along the velocity coordinate to the statistics of turbulence; thus, it does not require spatial coverage by sampling directions in the plane of the sky. We consider lines with different degree of absorption and show that for lines of optical depth less than one, our earlier treatment of the VCS developed for spectral emission lines is applicable, if the optical depth is used instead of intensity. This amounts to correlating the logarithms of absorbed intensities. For larger optical depths and saturated absorption lines, we show that only wings of the line are available for the analysis. In terms of the VCS formalism, this results in introducing an additional window, whose size decreases with the increase of the optical depth. As a result, strongly saturated absorption lines only carry the information about the small-scale turbulence. Nevertheless, the contrast of the fluctuations corresponding to the small-scale turbulence increases with the increase of the optical depth, which provides advantages for studying turbulence by combining lines with different optical depths. By combining different absorption lines one can develop a tomography of the turbulence in the interstellar gas in all its complexity.