An Exploration of the Statistical Signatures of Stellar Feedback

All molecular clouds are observed to be turbulent, but the origin, means of sustenance, and evolution of the turbulence remain debated. One possibility is that stellar feedback injects enough energy into the cloud to drive observed motions on parsec scales. Recent numerical studies of molecular clouds have found that feedback from stars, such as protostellar outflows and winds, injects energy and impacts turbulence. We expand upon these studies by analyzing magnetohydrodynamic simulations of winds interacting with molecular clouds which vary the stellar mass-loss rates and magnetic field strength. We generate synthetic \(^{12}\)CO(1-0) maps assuming that the simulations are at the distance of the nearby Perseus molecular cloud. By comparing the outputs from different initial conditions and evolutionary times, we identify differences in the synthetic observations and characterize these using common astrostatistics. We quantify the different statistical responses using a variety of metrics proposed in the literature. We find that multiple astrostatistics, such as principle component analysis, velocity component spectrum, and dendrograms, are sensitive to changes in stellar mass-loss rates and/or magnetic field strength. This demonstrates that stellar feedback influences molecular cloud turbulence and can be identified and quantified observationally using such statistics.


Turbulence in the interstellar medium is ubiquitous and self-similar across many orders of magnitude (Brandenburg et al., 2013). Within molecular clouds, turbulence appears to play an essential role in the star formation process, regulating the efficiency at which stars form, seeding filaments and over-densities, and even potentially setting the stellar initial mass function (Padoan et al., 2014; Offner et al., 2014). While the presence of supersonic motions is readily verified and has been studied using molecular spectral lines for several decades (Larson, 1981), the origin, energy injection scale, means of sustenance, and rate of dissipation remain debated. Moreover, molecular clouds display significant variation in bulk properties, ongoing star formation, and morphology. Consequently, it seems highly likely that these differences impact the turbulent properties of the gas and leave signatures — but if so they are difficult to identify observationally.

Detailed study of the turbulence within molecular clouds is confounded by a variety of factors including observational resolution, projection effects, complex gas chemistry, and variable local conditions (e.g., Beaumont et al., 2013, and references therein). Any one molecular tracer only samples a limited set of gas densities and scales, so that reconstructing cloud kinematics reliably involves assembling a variety of tracers across different densities and scales (e.g. Gaches et al., 2015). Many studies of cloud structure instead rely on a single gas tracer like CO, which is bright and exhibits widespread emission that reflects the underlying H\(_2\) distribution (Bolatto et al., 2013; Heyer et al., 2015). Connecting such emission data to underlying turbulence and bulk cloud properties, however, is non-trivial. A variety of statistics have been proposed throughout the literature to characterize spectral data cubes and distill the complex emission information into more manageable 1D or 2D forms (e.g., Heyer et al., 1997; Rosolowsky et al., 1999; Rosolowsky et al., 2008; Burkhart et al., 2009). However, in most cases, the utility of the statistic and its interpretation is not well constrained.

Numerical simulations, which supply full 6-D information \((x,y,z, v_x, v_y, v_z)\), provide a means to study turbulence and constrain cloud properties. Prior studies have investigated how the turbulent power spectrum, inertial driving range, and fraction of compressive motions have influenced star formation (Klessen, 2001; Bate, 2009; Federrath et al., 2010). Other studies have connected simulated turbulent properties to observables such as CO emission by performing radiative post-processing (e.g., Padoan et al., 2001; Bertram et al., 2014). In some cases, this procedure is able to identify theoretical models that have good agreement with a given observation. Consequently, the most effective way to study turbulence in molecular clouds is by comparing observations with “synthetic observations", in which the emission from the simulated gas is calculated via radiative transfer post-processing (Offner et al., 2008; Offner et al., 2012). Recently, Yeremi et al. (2014) and Koch et al. (2015) performed parameter studies of magneto-hydrodynamic simulations in order to assess the sensitivity of common astrostatistics to changes in cloud velocity dispersion, virial parameter, driving scale, and magnetic field strength. They found that some statistics were responsive to changes in the temperature, virial parameter, Mach number and inertial driving range. These results suggest that certain statistics may also be sensitive to energy input and environmental variation due to ongoing star formation.

One fundamental puzzle in star formation is why the efficiency at which dense gas forms stars is only a few percent per free fall time (Krumholz, 2014). Early three-dimensional hydrodynamic simulations discovered that supersonic turbulence decays rapidly and predicted that without additional energy input turbulence should decay significantly within a dynamical time (Stone et al., 1998; Mac Low, 1999). This implies that gravity should be able to efficiently form stars after a dynamical time. However, turbulence observed within molecular clouds does not appear to weaken and star formation efficiencies are small after several dynamical times (Krumholz et al., 2007) One explanation for the longevity of observed turbulence is that motions are driven internally via feedback from forming or evolved stars (Krumholz et al., 2014, and references therein). In principle this should introduce a characteristic energy input scale (Carroll et al., 2009; Hansen et al., 2012; Offner et al., 2015), which should impact turbulent statistics. However, from an observational prospective, stellar feedback is messy and identifying clear feedback signatures is complex for the reasons mentioned above. Disentangling feedback signatures from the turbulent background and assessing their role is challenging since any low-velocity motions excited by feedback are lost in the general cloud turbulence (Swift et al., 2008; Arce et al., 2010; Arce et al., 2011).

Few prior numerical or observational studies have examined the response of turbulent statistics to stellar feedback. Several studies of the most commonly computed turbulent statistic, the velocity power spectrum, find that it may be sensitive to feedback. In numerical simulations, turbulence shaped by both isolated and clustered outflows exhibits a steepened velocity power spectrum (Nakamura et al., 2007; Cunningham et al., 2009; Carroll et al., 2009). In observations of NGC1333, Swift et al. (2008) identified a break in the power spectrum of the \(^{13}\)CO intensity moment map, which they attribute to a characteristic scale associated with the embedded protostellar outflows (the break is absent in the \(^{12}\)CO data). Brunt et al. (2009) and (Padoan et al., 2009) reexamine the NGC1333 spectral cubes with principle component analysis and velocity coordinate spectrum method, respectively, but find no evidence of outflow driving and conclude that the turbulence is instead predominantly driven on large scales. Numerical simulations of point-source (supernovae) driving also discovered changes in the spectral slope but found no obvious critical injection scale (Joung et al., 2006).

Probability distribution functions (PDFs) of densities, intensities or velocities are also commonly computed (e.g., Nordlund et al., 1999; Lombardi et al., 2006; Federrath et al., 2008). Both observations and simulations suggest that gravity shapes the distribution at high densities (citation not found: kainulainen09) Collins et al., 2012), but the impact of feedback on PDFs is less clear. Beaumont et al. (2013) showed that observed CO velocity distributions extend to higher velocities than synthetic observations of simulations containing pure large-scale turbulence and gravity; they attribute this difference to expanding shells associated with stellar winds. Offner et al. (2015) confirm that when winds are included in simulations a high-velocity tail appears. In contrast, the column density probability distribution does not appear sensitive to the inclusion of stellar feedback (Beaumont et al., 2013).

The impact of feedback on higher order statistics, such as principle component analysis (PCA), the spectral correlation function (SCF), dendrograms, the bispectrum and many others, is even less well explored (Rosolowsky et al., 1999; Heyer et al., 1997; Rosolowsky et al., 2008; Burkhart et al., 2009). Burkhart et al. (2010), in analyzing HI maps of the Small Magellenic Cloud, noted the possible signature of supernovae on the bispectrum, which appears as break around \(\sim\)160 pc. It seems likely that stellar feedback influences other higher order statistics as well.

In this paper, we aim to extend the study by Koch et al. (2015) by applying a suite of turbulent statistics to simulations with feedback from stellar winds. The simulated stellar winds produce parsec scale features and excite motions of several \(\kms\) as a result of their expansion (Offner et al., 2015). While protostellar outflows may also leave imprints in the turbulent distribution, winds appear to inject more energy on larger scales which leaves a more distinct imprint on the gas velocity distribution (Arce et al., 2011). By performing the analysis on synthetic CO spectral cubes, we aim to identify discriminating statistical diagnostics to apply to observed clouds that can pinpoint and constrain feedback: the “smoking gun".

In, §\ref{methods} we describe the numerical simulations, production of synthetic CO data cubes, and astrostatistical toolkit. We examine the response of each statistic to the presence of stellar winds in §\ref{comparisons}. In §\ref{distance}, we compare changes in the statistics between all pairs of outputs and assess the sensitivity to mass-loss rate, evolutionary time, and magnetic field strength. We summarize our conclusions in §\ref{conclude}.



Numerical Simulations

In this paper, we analyze the magneto-hydrodynamic simulations performed by Offner et al. (2015) of a small group of wind-launching stars, which are embedded in a turbulent molecular cloud. We refer the reader to that paper for full numerical details. Table \ref{simprop} summarizes the simulation properties for the specific evolutionary times we analyze here. In brief, the calculations are performed using the orion adaptive mesh refinement code (e.g., Li et al., 2012). They include supersonic turbulence, magnetic fields, and five star particles endowed with a prescription for launching isotropic stellar winds. The domain size for all runs is 5 pc and the molecular gas is initially 10 K with a velocity dispersion of \(2.0\kms\). The turbulent realization, magnetic field strength and wind properties vary between runs as stated in Table \ref{simprop}.

CO Emission Modeling

As in Offner et al. (2015), we post-process each output with the radiative transfer code radmc-3d1 in order to compute the \(^{12}\)CO (1-0) emission. We solve the equations of radiative statistical equilibrium using the Large Velocity Gradient (LVG) approach (Shetty et al., 2011). We perform the radiative transfer using the densities, temperatures, and velocities of the simulations flattened to a uniform \(256^3\) grid. We convert to CO number density by defining \(n_{\rm H_2} = \rho/(2.8 m_p)\) and adopting a CO abundance of [\(^{12}\)CO/H\(_2\)] =\(10^{-4}\) (Frerking et al., 1982). Gas above 800 K or with \(n_{\rm H_2} < 10\) cm\(^{-3}\) is set to a CO abundance of zero. This effectively means that gas inside the wind bubbles is CO-dark. The CO abundance in regions with densities \(n_{\rm H_2} > 2 \times 10^4\) cm\(^{-3}\) is also set to zero, since CO freezes-out onto dust grains at higher densities (Tafalla et al., 2004). In the radiative transfer calculation, we include sub-grid turbulent line broadening by setting a constant micro-turbulence of 0.25\(\kms\). The data cubes have a velocity range of \(\pm 20\kms\) and a spectral resolution of \(\Delta v = 0.156~ \kms\). To mimic the effects of observational noise, we add Gaussian noise with a standard deviation of \(\sigma_{\rm rms} = 0.1\)K. This is comparable to the noise in the FCRAO \(^{12}\)CO COMPLETE survey of local star forming regions (Ridge et al., 2006).

We produce synthetic observations of the nearby Perseus molecular cloud by setting the spectral cubes at a distance of 250 pc. The emission units are converted to temperature (K) using the Rayeigh-Jeans approximation. Noise is considered in some of the statistics as described below.

Statistical Analysis

We perform the statistical analysis using TurbuStat 2, a Python package developed by K15 that contains 16 turbulent statistics culled from the literature. Table ??? summarizes the statistics contained in this astrostatistical toolkit. K15 provide a detailed description of each turbulent statistic, and so we give only a brief overview here. TurbuStat measures differences between spectral cubes by computing a pseudo-distance metric as proposed in Yeremi et al. (2014).

Below we group the statistics into three categories based on their method of analysis: intensity statistics quantify emission distributions, Fourier statistics analyze N-dimensional power spectra obtained through spatial integration techniques, and morphology statistics characterize structure of the emission. We focus on those statistics deemed by K15 to be “good", i.e., those which exhibit a response to changes in underlying physical parameters rather than to statistical fluctuations in the data. (And then other part in sec 4).

Our analysis extends the K15 study by examining simulations including feedback from stellar winds and considering higher turbulent resolution. However, our simulation suite does not utilize experimental design to set the parameter values. As discussed in Yeremi et al. (2014), comparisons between outputs in one-factor-at-a-time approaches may give a misleading signal since the statistical effects are not fully calibrated.

  1. dullemond/software/radmc-3d/

  2. link to package

[Ryan - when we get out of authorea I’ll take the table in Word you made and put it in latex format]

Test table.\label{simprop}

lccccc W1_T1_t1 & 13.5 & 0.1 & 0.1 & 41.7 & 200 &
W1_T2 & 13.5 & 0.1 & 0.2 & 41.7 & 200 &
W2_T2 & 13.5 & 0.1 & 0.2 & 4.5 & 200 &
W2_T3 &5.6 & 0.6 & 0.1 & 4.5 & 200 &
W2_T4 &30.1 & 0.02 & 0.1 & 4.5 & 200 &

Statistical Comparisons


We calculate the statistics and distance between each pair of spectral cubes. However, since few prior statistical studies have investigated the impact of feedback, we first investigate the statistical response to feedback specifically by comparing two fiducial outputs: W1T2t0.2 and T2t0. T2t0 is a simulated turbulent molecular cloud (turbulent realization T2) prior to wind launching. W1T2t0.2 begins with the same turbulence as T2t0 (T2), but we follow its evolution for another 0.2 Myr (t0.2) with wind launching model W1. Thus, the two runs begin with the same turbulent seed, but the turbulence in one run is shaped by feedback, while in the other the turbulence is “pristine". For each statistic in Table ???, we compare the results produced with runs W1T2t0.2 and T2t0 and identify qualitative differences.

Intensity Statistics

In this section we discuss statistics that quantify the intensity distribution: the probability distribution function (PDF), skewness, kurtosis, principle component analysis (PCA), and the spectral correlation function (SCF). K15 find that the Tsallis statistic does not reliably discriminate between different input physics, so we do not include it here. We also compute the Cramer statistic, which we discuss in \(\S4\).

Probability Distribution Function

We calculate the probability distribution function (PDF) of the normalized integrated intensity maps, weighted by their respective errors. Figure 1 shows our two PDFs. Both runs exhibit log-normal behaviors, although the output with winds is more peaked around the mean. For integrated intensities larger than 1 (unit), the PDF of run T2t0 falls off at a faster rate than the PDF of W1T2t0.2 can’t see this now due to plot range. These differences arise because the winds create shells with CO-brightened rims that exhibit intensities higher than those created by the strongest shocks in the case of pure turbulence.

The width of the density and column density PDFs increase with Mach number (e.g., Nordlund et al., 1999; Ostriker et al., 2001). In the strong wind case, the effective Mach number is about 10% higher (OA15), however, this is not sufficient to explain the difference in Figure 1. The intensity distribution of the case with winds is broadened by the combination of increased densities and temperatures (the shells are warmer than the ambient turbulent gas), which enhance the CO excitation.

to be clarified: (1) as plotted this is not a log-normal. (2) What are the “weighted errors"-some assumed noise? and is the minimum around 5d-2 set by this added noise?

Fig 1—Probability Distribution functions (PDFs) of the integrated intensity moment maps for runs W1T2t0.2 (blue) and T2t0 (green). The integrated intensities are normalized to have a mean of 0 and standard deviation of 1.

Skewness and Kurtosis

We follow the analysis of Koch et al. (2015) to compute the skewness and kurtosis, which are the third- and fourth- order statistical moments of the PDF, respectively. Skewness is a measure of the symmetry of the data distribution. Data that is symmetric around the center point have low skewness. If there is an excess of high values, the skewness will be positive, while an excess of low values produces negative skewness. Kurtosis quantifies the “peakiness" of the distribution. Normally distributed data will have a kurtosis of zero, peaky data will have positive kurtosis, and flatter data will exhibit negative kurtosis. For all positions in an integrated intensity map, we compute each higher-order moment within a small, circular region with a radius of five pixels. Figure 2 shows PDFs of the kurtosis and skewness. They are essentially histograms of the moment arrays (integrated intensity maps).

The kurtosis PDFs exhibit similar behavior: both are centered at zero and sharply decrease with increasing kurtosis magnitude. However, the T2t0 distribution falls off more quickly than that of W1T2t0.2. This likely occurs because the winds generate a more extreme range of high intensity values; the intensity distribution deviates further from a normal distribution.

The skewness PDFs have similar shapes, and both have a small tail at negative skewness. However, the W1T2t0.2 distribution center is shifted to positive skewness, while the T2t0 distribution is centered at zero. This makes sense since the winds in W1T2t0.2 create an excess of high-intensity values.

Simulations of pure turbulence find that as the Mach number increases, the skewness and kurtosis of the column density PDF also increase (Kowal et al., 2007; Burkhart et al., 2009). Higher Mach number flows have stronger shocks, which increase the fraction of high-density, and hence high-column density, material. This is consistent with our results, since the winds create density enhancements and the CO intensity is a proxy for underlying column density.

Fig 2—Kurtosis (left) and Skewness (right) PDFs for run W1T2t0.2 (Blue), a wind launching model at 0.2 Myr, and run T2t0 (green), the same model prior to wind launching at 0 Myr.

Principal Component Analysis

Principle component analysis (PCA) determines a set of orthogonal axes that maximize the variance of the data. As applied to spectral data cubes, it identifies differences between the line profiles, and thus, is a useful tool for distinguishing between kinematic changes and noise (Heyer et al., 1997). Subsequent work established an empirical and analytic formalism connecting PCA to the underlying turbulent velocity fluctuations, including the spectral slope (Brunt et al., 2002; Brunt et al., 2002a; Brunt et al., 2013). In PCA analysis, the first step involves constructing a 2-D covariance matrix from the spectra. Next, the eigenvalues and eigenvectors of this matrix are determined. Here, we use the magnitude of the eigenvalues to assess the degree of difference between two datasets.

Figure 3 shows the velocity channel covariance matrices of runs W1T2t0.2 and T2t0. Both runs show a signal for velocities \(|v| \lesssim 2 \kms\), which roughly encompasses the range of turbulent gas velocities. However, W1T2t0.2 exhibits multiple strong covariance peaks around a few \(\kms\). These features exist to a lesser degree for T2t0, but feedback augments and further separates the peaks. The strongest covariance corresponds to the typical expansion rate of the wind shells.

Because the eigenvectors provide a measure of the strength of different eigenvectors, they also serve as a proxy of the amount of power on different scales (citation not found: brunt08). Figure \ref{eigenvalue} shows the relative sizes of the largest eigenvalues. Our algorithm calculates the first 50 and uses these to determine the degree of difference between two datacubes. However, only the first ten are significant. For observations, eigenvalues beyond the first 10 are dominated by noise. We find a clear difference in the eigenvalues for the cases with and without feedback. The case with feedback has more significant second, third and fourth eigenvalues. This is likely due to the additional emission structure created by the winds.

Figure 3—Covariance matrices of the velocity channels for runs W1T2t0.2 (left) and T2t0 (right). The indices mark the two velocity channels in which we calculate the total covariance summed over all positions. The colorbar denotes the covariance magnitude.

Figure 4— The first 50 covariance matrix eigenvalues for runs W1T2t0.2 (left) and T2t0 (right). For each plot, the eigenvalues are normalized with respect to the maximum eigenvalue. Their magnitudes denote the relative variance described by that principle component.

Spectral Correlation Function

the SCF is generally plotted as a single power-law. That might be more informative than the intermediate step that is being plotted here.

The spectral correlation function (SCF) is the normalized root-mean-square difference between two spectra as a function of their projected separation (Rosolowsky et al., 1999). The SCF manifests as a power-law, where flatter slopes indicate more kinematic correlation across spatial scales (large hierarchical emission structures), while steeper slopes indicates less correlation between large and small scales (smaller discrete emission structures). The SCF serves as a useful comparison metric for both simulations and observations (Padoan et al., 1999; Yeremi et al., 2014; Gaches et al., 2015), however, no direct link between the SCF and turbulent properties has been formulated.

We calculate multiple SCFs of outputs W1T2t0.2 and T2t0 using an array projected separations ranging from 0” to 113" Eric, I kept your input of “size” 11, which is the projected pixel separation. Correct me if my interpretation is wrong.. Figure 5 depicts the SCFs of our two fiducial outputs. With the exception of zero spatial offset, the SCF outputs (not sure what to call the output) of run W1T2t0.2 are smaller than the SCF outputs of run W2T2t0 are. Yet, both runs also exhibit a similar rate of change in SCF value with increasing offset.

Figure 5—Spectral Correlation Functions (SCFs) colorplots for runs W1T2t0.2 and W2T2t0. We show a portion of our SCFs as a colorplot matrix. The indicies denote the amount of horizontal and vertical offset used in an SCF computation, and the colorbar denotes the SCF value. A value of 1 indicates total correlation while a value of 0 denotes complete lack of correlation.

\def\rsun{\ifmmode {\rm R}_{\mathord\odot}\else $R_{\mathord\odot}$\fi} \def\msun{\ifmmode {\rm M}_{\mathord\odot}\else $M_{\mathord\odot}$\fi} \def\lsun{\ifmmode {\rm L}_{\mathord\odot}\else $L_{\mathord\odot}$\fi} \def\kms{\ifmmode {\rm km s}^{-1} \else kms$^{-1}$\fi} \subsection{Fourier Statistics} In this section we present statistics based on a Fourier analysis of the spectral cube: velocity channel analysis (VCA), velocity coordinate spectrum (VCS), power spectrum, bicoherence, and wavelet analysis. Since K16 find that the modified velocity centroid (MVC) method does not does reliably discriminate between the models, we do not consider it here. %VCA/VCS, delta variance, MVC, SPS/Bicoherence, wavelet %SSRO Just use the +Text to create a new separated text box for each category (here \subsubsection{}) into it. \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., \verb|\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 \verb|\cite[e.g.,][]{lazarianp00,lazarianp04}|. % Heyer \& Brunt 2002 doesn't exist 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 outputs W1T2t0.2 and T2t0, we first compute the three-dimensional power 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 resultant 1D power spectrum is then fit to a power law. For VCS, we reduce each 3D power spectrum to one dimension by averaging over the spatial frequencies. This yields two distinct power 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 \verb|\citep{chepurnov09}|. \verb|\citet{kowal07}| find that the density-dominated regime is sensitive to the magnetic field strength, where stronger fields correspond to steeper slopes. %To quantity the VCS, we fit the 1D power spectrum to the segmented linear model used Koch et al. (2015). Figures 6 and 7 show the VCA and VCS power law fits, respectively, for outputs W1T2t0.2 and T2t0. VCA produces similarly sloped power laws for both runs, but there is a constant horizontal offset. This implies that at all spatial scales output W1T2t0.2, our run with feedback, has more energy than that of output T2t0. The winds produce a marginally flatter VCA slope, however, since the curves are otherwise very similar, we conclude VCA is not useful for characterizing feedback properties or comparing the turbulent properties of different clouds. In Figure 7, VCS also shows a horizontal offset between the two curves. 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). \verb|\citet{lazarianp08}| define this break as $k_{cr} = \Delta V_{r_0}^{-1} \simeq \sigma(L)^{-1} (r_0/L)^{-1/4}$, where $\sigma(L)$% = D_z(L)^{1/2}$ $D_z(L)$ is the variance is the velocity dispersion on the cloud scale $L$, and the scaling exponent assumes supersonic turbulence. For T2t0 $k_{\rm cr} \simeq 0.16$ and the velocity dispersion is $\sigma =1.4 \kms$\footnote{The velocity dispersion is defined as the second moment of the spectral cube: $\sigma = (\Sigma_i I(v_i)(v_i - \bar v)^2 dv/\Sigma I(v_i) dv})^{1/2}$, where $I$ is the intensity. Note that the values obtained from the spectral cube are slightly higher than the 1D mass-weighted velocity dispersions, which are $1.15 \kms$ and $1.25 \kms$ for the non-wind and wind outputs, respectively. } at $L=5$pc, which gives $V_{r_0}= 1.0 \kms$ and $r_0\sim 1.2$ pc. {\bf Note that I've assumed kv to be in units of dv=40km/s/256.} For W2T2t0.2, $k_{\rm cr} \simeq 0.1$, so $V_{r_0}= 1.6 \kms$. The velocity dispersion with feedback is slightly higher ($\sigma_{\rm 1D}=2.1\kms$), which corresponds to a slightly larger critical scale of $r_0=1.5$ pc. The output without feedback %SSRO: 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 ($k_v\simeq 0.01-0.16$). The velocity-dominated regime is smaller for the run with feedback ($k_v\simeq 0.01-0.1$), such that changes in gas density affect a greater portion of the structure apparent in the cloud emission. It makes sense that feedback extends the density-dominated regime since the winds create extra density enhancements by sweeping up material. Irrespective of the break-point, the velocity-dominated regime should follow a power-law set by the underlying velocity structure function. For supersonic shocks, we expect $P(k_v) \propto k_v^{-4}$ with the slope steepening for $k_v> \Delta V_{r_0}^{-1}$ depending upon the shape of the line profile \verb|\citep{lazarianp08}|. %&pogosyan Indeed, we find that the slopes are statistically similar above and below the break. Thus, variation in the breakpoint location could provide insight into the underlying turbulent driving scale. % (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.

Figure 6—VCA as a function of spatial frequency \(k\) for outputs W1T2t0.2 (top, blue) and T2t0 (bottom, green), each fitted by a single power law. We report a slope of -2.07\(\pm\)0.02 for W1T2t0 and -2.18\(\pm\)0.02 for T2t0.

Figure 7—VCS as a function of velocity-frequency for outputs W1T2t0.2 (top, blue) and T2t0 (bottom, red). The segmented power law fits are overlaid. For W2T2t0.2, we report slopes of -1.59\(\pm\)0.03 and -4.16\(\pm\)0.04 for the velocity-dominated region and density dominated region, respectively. Similarly, we report slopes of -1.78\(\pm\)0.02 and -3.98\(\pm\)0.04 for T2t0. The break points between power-law fits are -1.01\(\pm\)0.02 for W1T1t0.2, and -0.82\(\pm\)0.01 for T2t0. Eric, the slopes for density and velocity-dominated regions seem swapped

Spatial Power Spectrum

\label{ps} Switch order with VCA

The Fourier power spectrum is one of the most widely computed turbulent statistics. Numerical simulations over the last decade have confirmed that the velocity power spectral slope in one dimension is \(P_v(k) \propto k^{-2}\) for supersonically turbulent gas (Mac Low et al., 2004; McKee et al., 2007, and references therein). The slope is similar or slightly flatter for a magnetized gas where the gas and field are well-coupled. The power spectrum of the 3D density distribution of turbulent gas is \(P_\rho(k) \propto k^{-1.5}-k^{-2.3}\) for solenoidal and compressive driving, respectively (Federrath et al., 2010). Observationally, the situation is more complex since the intensity distribution in a spectral line cube is a product of both density and velocity fluctuations, which are inextricably entangled. For lower density tracers, like \(^{12}\)CO, the gas becomes optically thick and emission saturates along high-density sight-lines through the cloud. (Lazarian et al., 2004) predicted that the intensity power spectrum intensity field \(P(k) \propto k^{-11/3}\) and saturates to \(P(k) \propto k^{-3}\) in the optically thick limit. This was confirmed in numerical simulations by (Burkhart et al., 2013), who post-processed the simulations to produce synthetic CO maps in different optical depth regimes. Because the emission behaves differently in different optical depth limits, it is possible to probe the underlying density and velocity slopes by analyzing the spectrum of different slices within the spectral cube (Lazarian et al., 2000), a technique that we discuss further in §\ref{VCA}.

To obtain the spatial power spectrum (SPS), we compute the Fourier transform of the integrated intensity map and calculate the 2D power spectra of the two-point autocorrelation functions. We then radially average the power spectra over bins in spatial frequency. Fitted power laws for each 1D power spectrum are shown in Figure 8. We find similar results to that of the VCA: a constant horizontal offset with the winds exhibiting more power overall and a slightly flatter slope. The slope of the pure turbulence run, \(-2.79 \pm 0.06\), is close to \(k^{-3}\), which is predicted as the limiting slope for a turbulent optically thick gas (Lazarian et al., 2004; Burkhart et al., 2013). The optical depth of the gas may wash out any break in the spectrum. A different result could be expected for an optically thin tracer such as \(^{13}\)CO (e.g., Swift et al., 2008), but OA15 do not find a quantitatively different result for \(^{13}\)CO for these simulations. However, the flatter slope for the output with winds suggests that feedback and optical depth are somewhat degenerate.

Figure 8—SPS for outputs W1T2t0.2 (blue, top) and T2t0 (green, bottom). The solid lines indicate the power-law fits. We report slopes of -2.39\(\pm\)0.04 for W1T2t0.2 and -2.79\(\pm\)0.06 for T2t0.


The bispectrum measures both the magnitude and phase correlation between Fourier signals. This gives it a distinct advantage over two-point correlation methods such as VCA and VCS, which do not preserve phase information. Consequently, the bispectrum is useful to quantify nonlinear wave-wave interactions, which may be prevalent in turbulent magnetized gas (Burkhart et al., 2009).

The bispectrum is obtained by computing the Fourier transform of the three-point correlation function. In our analysis, we use the bispectrum to calculate the bicoherence, a real-valued, normalized equivalent. (Explain why this is more efficient? Eric discuss why in his paper). Following the analysis of Koch et al. (2015), we generate sets of randomly sampled spatial frequencies that are sampled up to half of the image size (i.e., 127 pixels). For each run, we compute the bicoherence of our integrated intensity maps using the randomly samples sets.

Figure 9 depicts the bicoherence matrices for outputs W1T2t0.2 and T2t0. The bicoherence matrix of W1T2t0.2 exhibits a clear signal on the diagonal; this is the trivial case of \(k_1=k_2\). However, it exhibits little correlation elsewhere. In contrast, bicoherence maxtrix of T2t0 shows enhanced correlation for large wavenumbers (small scales). In general it contains a significant fraction of pixels above 0.5, i.e., there is fairly widespread correlation. If magnetic waves enhance correlation across scales, the wind shells may break up the volume and, thus, reduce correlation. Although shell expansion may perturb the magnetic field and excite magnetosonic waves, it is difficult to see any direct evidence of this against the initial turbulence (OA14). The comparison of the two bicohence matricies in fact seems to suggest that the shells reduce correlation perhaps by disrupting the propagation of MHD waves.

Burkhart et al. (2010) compute the bispectrum of HI maps of the SMC. They found that the maps of HI column density exhibit more correlation compared to a turbulent Gaussian random field. They also discovered a break around \(\sim 160\) parsecs, where the correlation decreases, an signature which they attribute to expanding shells. Burkhart et al. (2010) also demonstrate that correlation is much higher for super-Alfvenic turbulence (\(\mathcal{M}_A =\sqrt{12\pi \rho} \sigma/B> 1\)). The Alfven Mach numbers of our outputs range from \(\sim 1-5.5\). Since the velocity dispersion, and hence the Alfvenic Mach number, increases for the strong feedback case, we would a priori expect more correlation. However, we see the opposite. This supports the conclusion that the shells suppress the free propagation of MHD waves and reduce scale coupling.

Figure 9—Bicoherence matrices for output W1T2t0.2 and T2t0. We calculate the bicoherence over 100 randomly sampled spatial frequencies, denoted by k1 and k2. Is it just the angle between the frequencies that is randomly sampled, or is it also the spatial frequencies?The colorscale denotes the bicoherence magnitude and the degree of correlation between wavenumbers \(k_1\) and \(k_2\): a value of 0 indicates random phases, i.e., no correlation, while a value of 1 indicates strong phase coupling. Eric, this seems opposite to the data in Burkhart et al 2010; they find more correlation on large scales and less correlation on small scales - see their §8.3. We have more correlation for large k = small scales and almost no correlation for small k = large scales.


The \(\Delta\)-variance is a filtered average over the Fourier power spectrum (Stutzki et al., 1998). It has been used to characterize the structure distribution and turbulent power spectra of molecular cloud maps. The revised method presented by Ossenkopf et al. (2008); Ossenkopf et al. (2008)a takes into account noise variation and provides a means to discriminate between small-scale map structure and noise. We adopt this method for our analysis (K16). To compute the \(\Delta-\) variance, we generate a series of Mexican hat wavelets that vary in width. We approximate each wavelet as the difference between two Gaussians with a diameter ratio of 1.5. For each output, we weight the integrated intensity map by its inverse variance, convolve it with a Mexican hat wavelet, and calculate the \(\Delta\)-variance in Fourier space. Figure 10 shows the \(\Delta\)-variance as a function of wavelet width, which is by convention denoted as the “lag" (Stutzki et al., 1998). Similar to the Fourier statistics discussed previously, we find a common horizontal offset between the case with feedback and the purely turbulent case. The \(\Delta\)-variance curve of T2t0 also declines more at scales near 0.1 arcminutes than the curve for output W1T2t0.2.

In noisy observations, the \(\Delta\)-variance increases towards small lags, indicating enhanced structure. Here, the difference between the curves indicates that the wind output has slightly more structure on smaller scales, probably caused by the wind shells, which have a thickness of \(\sim\) pixels (0.1 pc). However, the \(\Delta\)-variance of W1T20.2 does not exhibit any break, which would indicate a preferred structure scale. In fact, it more directly resembles a pure power law than the non-wind \(\Delta\)-variance curve. Ossenkopf et al. (2008)a also found a smooth power-law \(\Delta\)-variance for rho Ophiuchus, even though clump-finding on the same map produced a mass distribution with a break (Motte et al., 1998). These results suggests that the \(\Delta-\)variance statistic as applied to integrated intensity maps is not especially sensitive to feedback signatures.

Figure 10—\(\Delta\)-variance spectra for outputs W1T2t0.2 (blue, top) and T2t0 (green, bottom). We convolve the Fourier transform of our integrated intensity maps with Mexican Hat wavelets of different widths (“Lag”) and calculate an average over each filtered power spectra.

Wavelet Transform

Wavelet transforms offer an alternative data decomposition to Fourier transforms for studying intermittency and nonlinear scale coupling. Wavelet transforms have been utilized to study MHD and plasma turbulence for more than two decades (Farge et al., 2015). They are less frequently applied in studies of astrophysical turbulence, although the first application of the wavelet transform was presented for \(^{13}\)CO molecular emission of L1551 by Gill et al. (1990). Here, we define the wavelet transform as the average value of the positive regions of a convolved image (K16); it is essentially an intensity average computed over a range of size scales. We convolve the integrated intensity maps of the outputs with a Mexican hat kernel, a process similar to that of the \(\Delta\)-variance technique described in §3.2.4.

Figure 10 shows the wavelet transform for the fiducial outputs. Following K16, we fit a portion of the transform to a power-law, where the range is informed by the results of Gill et al. (1990). Although the resultant slopes are similar, output T2t0, the purely turbulent model, diverges more from power-law behavior than output W1T2t0.2. We also note that the wavelet transforms are higher for output W1T2t0.2 than than T2t0, which is consistent with the increased molecular excitation produced by the higher density and temperature in the wind shells. I’d like to compare the slope with Gill & Henriksen: 2.35 (all gas), 2.42 (off outflow), 2.29 (on outflow), however their result is totally different. Is this because they use 13CO or because we’re defining the transform differently?

While the shape of the Wavelet transform may provide insight into underlying turbulent properties, neither the offset nor the slope appear to exhibit sufficiently different behavior to serve as a diagnostic for embedded feedback. Indeed, the first astrophysical application of the Wavelet transform by Gill et al. (1990) compared the wavelet transform both “on" and “off" the outflow region of L1551; however, they found little difference in slope between the two regions. Their data did exhibit a turnover around \(log(a) \sim -0.6\), which they postulated was a transition between two competing physical processes. The turnover here is more subtle, but it occurs in a similar point in both outputs, so it more likely represents edge effects. Given the similarity between the two curves, we tentatively conclude that this formulation of wavelet analysis is not a good indicator of feedback.

Eric, do you define the Mexican Hat Wavelet here differently than you define it in the Delta Variance section?

Figure 11—Wavelet Transform for outputs W1T2t0.2 (blue) and T2t0 (green) as a function of the Mexican hat wavelet width \(a\). The lines indicate the best-fit power-law for the range \(-2.03<log(a)<-0.43\). We report slopes of 1.43\(\pm\)0.03 for W1T2t0.2 and 1.37\(\pm\)0.04 for T2t0.

Morphology Statistics

In this section we present statistics quantifying the morphology of the emission distribution, namely, the genus statistic and dendrograms.

Genus Statistic

Genus statistics characterize spatial information in a data map by identifying and counting minima and maxima. They essentially compute the difference between the number of isolated features, or peaks, and holes, or voids, above and below a given threshold, respectively. Beginning with the work of \citet{gott86}, genus statistics have been frequently used in cosmological studies to characterize the distribution of mass in the universe. \citet{kowal07} were the first to apply them to interstellar turbulence. They analyzed density and column density maps produced by MHD simulations and found that the shape of the distribution correlates with the sonic Mach number. This analysis was extended to observational data of the Small Magellenic Cloud (SMC) by \citet{chepurnov08}, who noted the statistic could be sensitive to the presence of shells.

To compute the genus statistic for each output, we normalize the integrated intensity map and convolve it with a 2D Gaussian kernel with width of 1 pixel. This smooths the map so that small scale variations and noise do not contribute to the number of features. We then divide the intensity range \([I_{\rm min}, I_{\rm max}]\) into 100 evenly spaced threshold values and compute the genus for those values above 20 percent of the minimum intensity. We fit the distribution with cubic splines of equal domain size not sure what ’equal domain size’ means for intensities \(<4\)K\(\kms\), which is the maximum threshold for the purely turbulent case.

Figure 11 shows the genus for our two fiducial outputs. Positive values indicate a relative excess of peaks (a “clump-dominated" topology), while a negative genus indicates an excess of voids. We find both curves exhibit similar behavior for intensities below 4 \(\kms\). As expected, output W1T2t0.2 has a broader range of intensity values due to the higher velocities and excitation in the wind shells, and thus, exhibits some structure for higher integrated intensities. Between thresholds of \(-1\kms\) and \(1\kms\), the genus is smaller for the purely turbulent model, which indicates that there are more voids in the emission compared to the case with feedback. The genus for W1T2t0.2 is higher at low-intensities, but this may be because the voids created by winds are larger than those created by pure turbulence, such that the total number of minima is reduced. This effect would likely be enhanced for real clouds, where winds can break out and create sight-lines nearly empty of molecular emission (A11).

Our analysis highlights one advantage of the genus statistic: it is sensitive to both over-densities and voids. \citet{chepurnov08} analyzed HI data of the SMC, which visually displays a large number of expanding shells with sizes of \(\sim 100\) pc. The shells were not apparent at small scales (\(< 100\) pc), but at intermediate scales (120-200 pc) the genus had a neutral or slightly positive value, which they attributed to shells. In practice, however, the thickness and morphology of clouds varies significantly between different star forming regions. In our comparison, the difference between the two curves is relatively subtle. Thus, it may be most informative when employed to compare sub-regions within clouds.

Figure 11—Genus curves and spline fits for outputs W1T2t0.2 (blue) and T2t0 (green). The curves are constructed using a one-dimensional interpolating spline of degree 3.


Dendrograms are hierarchical structure trees, which may be created for both 2D and 3D data (Rosolowsky et al., 2008). The number of peaks (“leaves") and the number of hierarchical levels (“branches") are useful metrics that prior studies have shown to be sensitive to underlying physics, including gravity and magnetic field strength, as well as emission properties (Goodman et al., 2009; Burkhart et al., 2013a; Beaumont et al., 2013). Here, we use dendrograms to characterize the hierarchical structure of our intensity maps. To create the dendrogram, we first identify the peak intensity value in the data and then proceed to smaller intensities and successively catalog local maxima. The leaves on the same level of hierarchy are connected by a branch. To account for simulated noise in our data maps, we set a minimum distance between two local maxima, \(\delta_{\rm min}\). Increasing \(\delta_{\rm min}\) decreases the total number of features, i.e., it “prunes" the tree (e.g., Burkhart et al., 2013a).

K16 consider two dendrogram statistics: the number of features or leaves and the histogram of leaf intensities. To compute the first statistic, we generate multiple dendrograms per output by varying \(\delta_{\rm min}\) from \(10^{-2.5}\) K\(-\)10\(^{0.5}\) K in 100 logarithmic steps. We then count the total number of leaves associated with each \(\delta_{\rm min}\). To compute the second statistic, we create a series of dendrograms for the outputs using the same range of \(\delta\), but instead of counting features we produce histograms of the leaf intensities for each value. We renormalize the intensities so that the mean of the histogram falls at zero.

Figure 12 displays the number of leaves as a function of \(\delta\) for the two fiducial outputs. Output W2T1t0.2 follows a pure power-law, while output T2t0 deviates from a power-law at \(\delta_{\rm min}\sim\)1 K. The latter trend agrees with the results of Burkhart et al. (2013), who analyzed for MHD simulations of turbulence and found that the number of leaves significantly declines as \(\delta\) increases. They also demonstrated that the power-law index for larger \(\delta\) values steepens from -1.1 to -3.9 as sonic Mach number declines from 7 to 0.7. The appearance of a pure power-law for output W2T1t0.2 suggests an interesting signature of feedback; it increases the hierarchy significantly for large \(\delta\). As a result, we also find that the output with feedback contains more structure than the purely turbulent output at all scales.

The distribution of peak intensities provides additional insight into the emission structure. Figure 13 shows superimposed histograms of the two fiducial outputs for all \(\delta\). These histograms don’t look very different for 100 steps in delta. The two outputs yield significantly different distributions. The histograms of T2t0 contain a wider range of leaf values than those of W1T2t0.2, whose histograms are all strongly peaked on the mean value. Output W1T2t0.2 also produces a long tail of high intensity values. Like the previous statistic, this indicates that feedback increases the amount of hierarchy in the emission.

Figure 13- Number of dendrogram features as a function of intensity spacing, \(\delta\), for output W1T2t0.2 (top, blue) and T2t0 (bottom, green). The lines indicate power-law fits to each curve, having slopes -0.673\(\pm\)0.004 and -1.58\(\pm\)0.07 for runs W1T2t0.2 and T2to, respectively. Note Burkhart et al. 2013 only show (they don’t fit) reference lines for the top part of the curve, say log delta = 0.2 - 0.5.

Figure 14–Histograms of the renormalized dendrogram peak leaf intensities. The distributions for all \(\delta_{\rm min}\) values are stacked.

Statistical Analysis

\label{distance} RB & SO (e.g. implications for observations)

We use pseudo-distance metrics to efficiently study differences between all synthetic observations. As described in (sec. 2/tab. 1), a pseudo-distance is a single value that encapsulates the degree of difference between two metrics, and they can be used to compare the statistical outputs of various spectral cubes. Section 3 identifies qualitative differences between a simulation with strong feedback and one that is purely turbulent. Expanding upon this, we now quantify all simulation differences and determine the sensitivities of the statistics to stellar mass-loss rate, magnetic field strength, and evolutionary time. This allows us to check if the previously identified features actually pinpoint signatures of feedback, rather than specifically corresponding to any combination of simulation parameters.

For each statistic, we produce a color-plot showing distances between all simulation pairs. Figures ???, ???, and ??? display all of the color-plots. Here, we provide a brief method for analyzing one. Each colored square represents the distance between one simulation pair, denoted by the horizontal and vertical indices. The colorbar denotes the distance values, whose range depends on the statistic. We arrange the simulations in order to easily compare strong wind models (W1) with weaker wind models (W2) or purely-turbulent models (TXto). Table ??? provides a summary of our findings, which we discuss in sec 4.1, 4.2, and 4.3.

Intensity statistics

We show the colorplots for all intensity statistics in Figure ???. With the exception of the Cramer statistic, we find that these statistics exhibit strong sensitivities to changes in stellar mass-loss rate. As seen in their colorplots, the largest distances appear when any strong wind model (W1) is compared to either a weak wind model (W2) or a purely turbulent model (TXt0).The Kurtosis, Skewness, and SCF are clear examples of this, as they display a sensitivity trend among pairings. These statistics yield the largest distances between pairs of W1 and TXt0, followed by pairs of W1 and W2. And, they capture similar, weaker sensitivities between pairs of W2 and TXt0.

The sensitivity trend with wind strength is less clear for the PDF and PCA. We find that time evolution randomly impacts the magnitude of these statistics’ strong wind distances. Weaker wind comparisons among the PDF appear to be correlated to magnetic field strength, given their structure. This does not occur in the PCA and SCF, since their distances for W2 are quite small. Thus, it is only weakly sensitive to magnetic field strength.

The Cramer statistic is by definition a distance metric, so it we include its discussion and analysis here. As described in Yeremi et. al (2014), this statistic compares the interpoint differences between two data sets with the point differences between each individual data set. Following K16, we compute the Cramer statistic using only the top 20% of the integrated-intensity values. The statistic exhibits a behavior different from that of the other intensity statistics. As figure ??? shows, the Cramer statistic displays very large distances between purely turbulent runs and runs with any degree of feedback. Wind strength appears less important to the statistic than wind presence does, which indicates a binary sensitivity to stellar-mass loss rates. The Cramer statistic is also sensitive to magnetic field strength, as seen in the varying distances between the purely turbulent models.

Considering the various degrees of sensitivity, we find the PCA to be strong a candidate for constraining feedback signatures. As Figure 3 shows, this statistic displays sharp, distinct features for a strong wind model, and its color-plot only shows strong sensitivity to changes in stellar-mass loss rate. The other intensity statistics either exhibit less-distinct features, or react to multiple physical changes. Because of this, we recommend using these statistics in concert with the PCA. Of the remaining intensity statistics, the SCF is the second most promising candidate, as its color-plot behaves similar to the PCA.

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Figure 15--Intensity statistics color-plot tables. Each statistic utilizes a different distance metric to quantify the difference in the outputs of two simulations. The colored squares represent the distance between simulations denoted by the horizontal and vertical indices.

Fourier statistics

All Fourier statistic color plots are shown in Fig ???. Unlike the Intensity statistics, the Fourier statistics do not share a common behavior, and their color plots appear more heterogeneous. As a whole, we note various sensitivities to changes in stellar mass-loss rate, magnetic field strength, and evolutionary time. The Delta-Variance and Wavelet transform color plots closely resemble those of the intensity statistics, as their greatest sensitivities correspond to changes in stellar-mass loss rates. The Delta-Variance’s strong wind comparisons also appear slightly impacted by magnetic field strength, as seen in their distance magnitudes. The Wavelet transform displays a similar trend that is further augmented by time evolution.

The VCS statistic demonstrates roughly equal sensitivities to both stellar mass-loss rate and magnetic field strength. As its color-plot shows, distances solely quantifying changes in stellar mass-loss rate tend to resemble those explicitly comparing changes in magnetic field. In fact, some of the largest distances involve T4, the run with the strongest magnetic field. We also note large distances between the turbulent clouds T1 and T2 in the presence of strong winds. These clouds have the same magnetic field strength, indicating that the VCS is sensitive to the initial turbulence conditions. We also find the SPS to be sensitive to all simulation parameters, but unlike the VCS, it’s sensitivities are not structured.

We find the Bicoherence to exhibit strong sensitivities to stellar mass-loss rate, magnetic field strength, and evolutionary time. As time evolves, wind models do become more alike, but they remain largely different from turbulent models. Turbulent models also appear relatively similar to each other.

Out of all of the statistics, the VCA demonstrates the weakest sensitivity towards magnetic field strength. The behavior of its color-plot suggests an insensitivity to turbulent structure, as distances only change with wind model and evolution time. This allows the VCA to clearly detect changes in stellar mass-loss rate. The color plot shows the distances for strong wind models to be different from those of all other models. But as time evolves, the weak wind distances more closely resemble the strong wind distances. This trend is clear because of the magnetic field’s weak impact on the statistic.

Despite the various degrees of sensitivities, many of the Fourier statistics fail to produce distinct visual differences corresponding to feedback. As discussed in section 3.2, the most common difference is horizontal offset/power spectra/characteristic energy scale (one of these), which is relatively minor. The VCS does produce distinct responses, but its color plot suggests sensitivities to parameters other than stellar mass-loss rate.

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Figure 16--Fourier statistic color-plot tables.

Morphology statistics

The morphology statistic color plots are shown in Fig. ???. Although the Genus statistic produces a wide range of distances, we find that it fails to display any clear trends, given its color-plot’s shape. Both Dendrogram statistics show clear sensitivities to stellar mass-loss rate. The histogram statistic yields the largest distances for strong wind and purely turbulent pairings, followed by strong wind and weak wind pairings. The behavior is similar to that of many other statistics, but the histogram statistic’s trend continues down onto weak wind model comparisons, indicating a very clear sensitivity towards wind activity.

The number of features statistic is also sensitive to winds, but it shows different correlations between strong wind model pairings, opposite of the histogram statistic’s. By a significant amount, the largest distances occur for strong wind and weak wind pairings, as opposed to those of strong wind and purely turbulent models. The behavior does not occur in weaker wind model comparisons, as pairings between these and purely turbulent ones are larger than pairings between weaker wind models.

Although the histogram statistic produces cleaner trends, we find both Dendrogram statistics to effectively highlight feedback signatures. They are most sensitive to changes in winds, meaning their statistical outputs produce distinct signatures corresponding to feedback.

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Figure 17--Morphology statistic color-plot tables



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 (Lazarian et al., 2004; Burkhart et al., 2013) and tracers with different optical depth and chemistry may yield different results (e.g, Swift et al., 2008; Gaches et al., 2015).


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