\label{conclude}
We investigated the sensitivity of fifteen commonly applied turbulent statistics to the presence of stellar feedback. The goal of our analysis was to identify whether any of the statistics could serve as a robust indicator of feedback: a smoking gun. Our parameter study was based on magneto-hydrodynamic simulations performed by OA15 with varying magnetic field strengths and degrees of feedback from stellar winds. We first post-processed the simulations with a radiative transfer code to produce synthetic \(^{12}\)CO(1-0) emission cubes. We then computed fourteen statistical metrics using the python package turbustat (K16) and assessed the relative response of each statistic to changes in evolutionary time, magnetic field strength, and stellar mass-loss rate. Here, we focus on only those statistics found by K16 to be “good", i.e. those which responded to physical changes in parameters but were insensitive to noise fluctuations: intensity PDF, skewness, kurtosis, power spectrum, PCA, SCF, bispectrum, VCA, VCS, \(\Delta\)-variance, wavelet transform, genus, number of dendrogram features, cramer, and histogram of dendrogram feature intensities. We illustrated each statistic via a comparison between a purely turbulent output and an output with identical turbulence but with embedded stellar sources launching winds (§3).
We then computed the distance metric, as defined for each statistic by K16, for each pair of outputs (§4). This allowed us to both quantify changes and simply the comparison by reducing each pair to one characteristic number. We find that a variety of statistics exhibit sensitivity to feedback, and we present the following conclusions:
The intensity PDF, skewness and kurtosis are each sensitive to the degree of feedback, with strong wind models exhibiting very different distances than week wind models. These showed sensitivity to evolutionary time to a lesser degree but were not not strongly sensitive to magnetic field strength.
The PCA showed strong sensitivity to wind strength and weak sensitivity to magnetic field. The covariance matrix in particular exhibited strong peaks at the characteristic wind shell expansion velocity (\(v\sim 1-2\kms\)), which we predict will be visible in observational data.
SCF I think the slope is different, but need to see the angle average value.
The bispectrum shows less correlation between scales in the case with feedback, which may be the result of the shells reducing magnetic wave propagation and coupling. However, the bispectrum is also sensitive to local conditions, including the sonic and Alfvenic Mach number, which make absolute identification of feedback challenging.
VCS showed a distinct signature of feedback. The transition between the density and velocity-dominated parts of the VCS spectrum occurred at higher velocities and larger scales in the case with winds. This suggests that the breakpoint may encapsulate information about the characteristic scale of feedback. The location of this point depends upon other cloud properties, such as optical depth and the velocity dispersion, however, VCS may be used to compare cloud sub-regions.
The genus statistic, which reflects the relative number of peaks and voids, showed sensitivity to feedback at small scales: the number of voids declined when feedback was included. However, the effect was subtle and may not be useful for intercloud comparisons.
Both dendrogram statistics showed sensitivity to feedback. In the presence of feedback, the number of features followed a pure power-law rather than following off steeply as in the pure turbulent case. Prior studies find that power-law behavior is not characteristic of any cloud Mach number or magnetic field strength for purely driven turbulence. This suggests the number of features statistic may be a true scale-free metric, which could be used to identify and characterize feedback. The histogram of leaf intensities was broader in the case with feedback, which reflects the larger range of intensities associated with the increased temperatures and densities found in shells. Thus, the intensity feature histogram may be most useful for comparing cloud sub-regions.
The cramer statistic was sensitive feedback only in a binary way, since the distance metric was insensitive to the overall mass-loss rate or evolutionary time. It was, however, sensitive to magnetic field strength.
The power spectrum, VCA, wavelet transform, and \(\Delta\)-variance show little sensitivity to the presence of feedback aside from an overall offset, which would not be remarkable in comparisons of observational data.
In conclusion, our search for a smoking gun was successful. On the basis of these results, we recommend follow-up observational studies focusing on active star-forming regions utilizing PCA, SCF, VCS, genus, and dendrograms.
Although these results provide motivation for optimism, we note several caveats. The simulations neglect gravity, which should be considered in future work. We caution that many statistics presented here have two or more distinct definitions in the literature. Our conclusions hold only for the definitions stated in K16; additional studies are needed to check alternative statistical conventions. Finally, we note that the results are sensitive to the line optical depth \citep{lazarianp04,burkhart13a} and tracers with different optical depth and chemistry may yield different results \citep[e.g, ][]{swift08,gaches15}.