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\subsubsection{Velocity Channel Analysis and Velocity Coordinate Spectrum} \label{VCA}
%SSRO Note power spectra - lower case; to cite text add label as \label{VCA} and then add (e.g., \ref{VCA}) in the text
Velocity Channel Analysis (VCA) and Velocity Coordinate Spectrum (VCS) are techniques that isolate how fluctuations in velocity contribute to differences between spectral cubes. VCA produces a 1D power spectrum as a function of spatial frequency, while VCS yields a 1D power spectrum as a function of velocity-channel frequency (frequency equivalent of velocity). For
runs outputs W1T2t0.2 and
W2T2t0, T2t0, we
first compute the three-dimensional power
spectrum (of their Fourier transforms). spectrum. To obtain the VCA, we calculate a one-dimensional power spectrum by integrating the 3D power spectrum over the velocity channels and then radially averaging over the two-dimensional spatial frequencies. A portion (look back when you get a chance to see what this portion physically is) of the
now resultant 1D power spectrum is then fit to a power law.
We derive the VCS from different integration techniques. We For VCS, we reduce each 3D power spectrum to one dimension by averaging over the spatial frequencies.
The VCS This yields two distinct power
laws that correspond to certain (different?) simulation (cloud?) parameters driving gas motion at different scales. laws, which we fit individually using the segmented linear model described in K15. The fit at larger scales describes bulk gas velocity-dominated motion; the fit at smaller scales describes gas density-dominated motions (Chepurnov and Lazarian 2009).
To %To quantity the VCS, we fit the 1D power spectrum to the segmented linear model used Koch et al.
(2015) (or cite the paper describing the model?). (2015).
Figures 5 and 6 show the VCA and VCS power law
fits fits, respectively, for
runs outputs W1T2t0.2 and
W2T2t0. T2t0. VCA
generates produces similarly sloped power laws for both runs, but there is a constant horizontal offset. This implies that at all spatial
scales, run scales output W1T2t0.2, our run with feedback, has more energy than that of
run W2T2t0, our run without feedback. The output T2t0. However, since the curves are otherwise indistinguishable, we conclude VCA is not useful for characterizing feedback properties or comparing the turbulent properties of different clouds.
In Figure 6, VCS also shows a horizontal offset between the two
curves, confirming curves.
%confirming that there is more power in velocities at all scales in the case with
feedback. We feedback
However, we also note a difference in both VCS power-law fits, and, more importantly, the break point between the two fits. {\bf the breakpoint: what it means and how it is determined needs to be described. The specific position and how this corresponds to a spatial or velocity scale needs to be noted: e.g. box size ~ 1 = 5pc, so k=0.1 = 0.5 pc; velocity range = +/-20 km/s, so k=0.1 is ~ 2 km/s. }
The
run output without feedback
(our %Can use both w/wo feedback and pure turbulence.
%(our purely turbulent run? Go back and denotes everything like this?)
appears to have a larger range over which it is dominated by velocity fluctuations ($k\simeq 1-0.05$). The velocity-dominated regime is smaller for the run with feedback ($k\simeq 1-0.1$), such that changes in gas density affect a greater portion of the structure in the cloud
emission.% emission. Thus, variation in the breakpoint location, could provide useful insight into the underlying turbulent driving and the importance of feedback.
% (here than in the other run...trying to be grammatically correct with comparisons while including sentence variation--not sure if this sentance is grammatically correct). %SSRO Good job. made some small edits to address content.
