deletions | additions       diff --git a/subsection_Fourier_Statistics_In_this__.tex b/subsection_Fourier_Statistics_In_this__.tex  index a2f74d2..8008254 100644  --- a/subsection_Fourier_Statistics_In_this__.tex  +++ b/subsection_Fourier_Statistics_In_this__.tex 
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In Figure 6, VCS also shows a horizontal offset between the two curves.   %confirming that there is more power in velocities at all scales in the case with feedback  However, we also note a difference in both VCS power-law fits, and, more importantly, the break point between the two fits. Physically, this transition point indicates the scale at which the dispersion of the density fluctuations is equal to the mean density (e.g, Lararian \& Pogosyan 2006). Lazarian \& Pogosyan (2008) define this break as $k_{cr} = \Delta V_{r_0}^{-1} \simeq \sigma(L)^{-1} (r_0/L)^{-1/4}$, where $\sigma(L) $\sigma(L)$%  = D_z(L)^{1/2}$ $D_z(L)$ is the variance  is the velocity dispersion on the cloud scale $L$,$D_z(L)$ is the variance,  and the scaling exponent assumes supersonic turbulence. For T2t0 $L=5$pc $L=5$ pc  and the velocity dispersion is $\sigma =1.4$\kms \footnote{This is computed as the  second moment of the velocity dispersion is spectral cube:  $\sigma =1.4$\kms, = \Sigma_i I(v_i)(v_i - \bar v)^2 dv/\Sigma I(v_i) dv}$, where $I$ is the intensity.},  which gives $V_{r_0}= 1.0$ \kms and $r_0\sim 1.2$ pc. % The synthetic obs value value is slightly higher at 1.4 and 2.1, respectively. Note that I've assumed kv to be in units of dv=40km/s/256. The rms dispersion is the sqrt of the variance. 1D sim dispersion is 1.15, 1.25, respectively. For W2T2t0.2, $k_{\rm cr} \simeq 0.1$, so $V_{r_0}= 1.6$ \kms. The velocity dispersion is slightly higher ($\sigma_{\rm 1D}=1.25$\kms), which gives $r_0=1.5$pc.  %{\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. }