Velocity coordinate spectrum: geometrical aspects of observations

TL;DR

This paper formalizes the geometric factors affecting the Velocity Coordinate Spectrum (VCS) technique for turbulence analysis in spectral line data, providing asymptotic descriptions for different observational configurations.

Contribution

It introduces a unified geometric framework for VCS, enabling analysis of various observational setups and their asymptotic behaviors, especially for diffuse interstellar medium studies.

Findings

Derived explicit formulas for geometric terms in VCS.

Provided asymptotic behavior for parallel and crossing lines of sight.

Validated results through direct calculations and numerical simulations.

Abstract

We analyze a technique of obtaining turbulence power spectrum using spectral line data along the velocity coordinate, which we refer to as Velocity Coordinate Spectrum (VCS). We formalize geometrical aspects of observation through a single factor, "geometric term". We find that all variety of particular observational configurations can be described using correspondent variants of this term, which we explicitly calculate. This allows us to obtain asymptotics for both parallel lines of sight and crossing lines of sight. The latter case is especially important for studies of turbulence within diffuse ISM in Milky Way. For verification of our results, we use direct calculation of VCS spectra, while the numerical simulations are presented in a companion paper.