A CLUSTER IN THE MAKING: ALMA REVEALS THE INITIAL CONDITIONS FOR HIGH-MASS CLUSTER FORMATION

Figure 1 shows a two-color image of G0.253+0.016 combining infrared images from the Spitzer Space Telescope (24 μm, ∼6'' angular resolution, shown in cyan) with the new millimeter image from ALMA (3 mm, ∼17 angular resolution, shown in red). Contours show single dish dust continuum emission (James Clerk Maxwell telescope 450 μm, ∼75 angular resolution). Identified as an infrared dark cloud, G0.253+0.016 is clearly seen as a prominent extinction feature in mid-IR images. Because it is so cold, its thermal dust emission peaks at millimeter/sub-millimeter wavelengths. Moreover, since the 3 mm dust continuum emission is optically thin, internal structure is easily revealed by the ALMA images. While the overall morphology of the dust continuum emission revealed by ALMA matches well the mid-IR extinction and the large-scale dust continuum emission provided by the single dish image, the ALMA image shows significant structure on the small-scale. The emission is neither smooth nor evenly distributed across the clump; it is highly fragmented and there is a clear concentration of emission toward its center. These small-scale fragments are presumably the cores from which stars may form.

Because the 3 mm continuum emission is optically thin, given standard assumptions about the dust emissivity, it can be used to determine the column density and, hence, mass of its small-scale fragments. Recent observations from Herschel of the 170–500 μm dust continuum emission show that the dust temperature (T_dust) within G0.253+0.016 decreases smoothly from 27 K on its outer edges to 19 K in its interior (Longmore et al. 2012). External to G0.253+0.016, the dust temperatures are considerably higher, >35 K. Within the cloud and on the spatial scales probed by the Herschel data (∼33'', ∼1 pc), no small volumes of heated dust are apparent. All of the millimeter emission detected by ALMA falls within the region for which T_dust was measured to be $\lt 22$ K. Thus, we assume that the temperature of the dust detected by ALMA ranges from 19–22 K and use a value of 20 ± 1 K for all calculations.

Assuming this dust temperature, a standard value of the gas to dust mass ratio of 100, a dust absorption coefficient (${{\kappa }_{3\,{\rm mm}}}$) of 0.27 cm² g⁻¹ (using ${{\kappa }_{1.2\,{\rm mm}}}$ = 0.8 cm² g⁻¹, and $\kappa \propto {{\nu }^{\beta }}$ Chen et al. 2008; Ossenkopf & Henning 1994,) and a β of 1.2 ± 0.1, the intensity of the 3 mm emission can be converted into the H₂ column density, N(H₂), by multiplying the measured fluxes by a single conversion factor of 1.9 ×10²³ ${\rm c}{{{\rm m}}^{-2}}$/mJy. We assume that T_dust is constant across the cloud. For the small range of measured dust temperatures in G0.253+0.016, this approximation is well justified. Given these assumptions, the uncertainties for T_dust and β introduce an uncertainty of 10% for the column density, volume density, mass, and virial ratio.

The column densities across G0.253+0.016 measured from the ALMA-only image range from 0.5–82 × 10²² ${\rm c}{{{\rm m}}^{-2}}$, with a mean value of 1.0 × 10²² ${\rm c}{{{\rm m}}^{-2}}$. Since this image detects only the highest contrast peaks within the cloud and does not include the large-scale emission, we also determine the column density distribution using the combined (ALMA+SD) image which is more representative of the total column of material associated with G0.253+0.016. When including the emission on all spatial scales, we find that the mean column density is ∼8.6 × 10²² ${\rm c}{{{\rm m}}^{-2}}$ and an order of magnitude higher compared to molecular clouds in the solar neighborhood (see Rathborne et al. 2014b for a detailed discussion of the column density distribution across G0.253+0.016 and its comparison to solar neighborhood clouds).

The 3 mm (90 GHz) spectrum is rich in molecular lines that span a range in their excitation energies and critical densities (see Table 1). As such, these observations also provide data cubes from many molecular species and reveal a wealth of information about the gas kinematics and chemistry within G0.253+0.016. Figure 3 shows, along with the dust continuum image, the integrated intensity images for each of the detected molecular transitions. Clearly seen are the differences in the morphology of the emission from the various tracers, indicating that the chemistry and excitation conditions vary considerably within the clump.

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To quantify the differences in the morphology of the integrated intensity for each molecular transition compared to the others and the dust continuum emission, we have determined the 2D cross-correlation function for all combinations of image pairs. Figure 4 shows the derived correlation coefficients for each transition with respect to all others (the lower half of the figure shows the correlation coefficients via a color scale, the upper half of the figure shows the derived values). For each particular tracer (labeled along the diagonal) the molecular transition for which it has the highest correlation coefficient can be found by searching for the darkest square either in the row to its left or in the column below (the corresponding values are mirrored in the upper portion of the figure).

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We find that the morphology of the dust continuum emission is not highly correlated with any gas tracer, but is most correlated with the morphology of the emission from the molecules with high excitation energies (NH₂CHO in particular). In addition, we find that the morphology of the emission from HCN and HCO⁺ are highly correlated with each other and little else, that the shock tracers (SiO and SO) are correlated with each other, and that the dense gas tracers with high excitation energies are well correlated with each other.

Figure 5 shows example spectra from the detected molecular transitions at several locations across the clump (spectra from a single pixel were extracted at the positions marked on Figure 3 and listed in Table 2). The spectra clearly show the complexity of the emission and the differences in intensity and velocity detected from the various tracers. Molecular species with lower excitation energies (E_u/k ∼ 4 K, e.g., HCN, HCO⁺, C₂H) show emission over a wider range in velocity (∼10–15 km s⁻¹) compared to the isotolopogues (e.g., H¹³CN, H¹³CO⁺, HN¹³C, and HCC¹³CN; ∼3–10 km s⁻¹) and those with higher excitation energies (E_u/k > 10 K, e.g., CH₃SH, NH₂CHO, CH₃CHO, H₂CS, NH₂CN, and HC₅N; ∼3 km s⁻¹). The shocked gas (traced by SiO and SO) also shows emission over a wide range in velocity toward most locations in the clump (∼3–20 km s⁻¹).

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For each molecular transition we characterize the emission via a moment analysis using the definitions of Sault et al. (1995). All moments were calculated over the velocity range −10 km s⁻¹ < V_LSR < 70 km s⁻¹ (this range is delineated by dotted vertical lines on the spectra). The moment analysis allows us to characterize the emission at every point across the map and to easily compare the emission from the various molecular transitions to each other and to the dust continuum emission.

We compute the zeroth moment (M₀, or integrated intensity), the first moment (M₁, the intensity weighted velocity field), and the second moment (M₂, the intensity weighted velocity dispersion, σ). These are defined as

$$\begin{eqnarray*}\begin{array}{rcl} {{M}_{0}} & = & \int T_{A}^{*}(v)dv,{{M}_{1}}=\frac{\int T_{A}^{*}(v)v\;dv}{\int T_{A}^{*}(v)dv}, \\ {{M}_{2}} & = & \sqrt{\frac{\int T_{A}^{*}(v){{(v-{{M}_{1}})}^{2}}dv}{\int T_{A}^{*}(v)dv}}, \\ \end{array}\end{eqnarray*}$$

where T $_{A}^{*}$ $(v)$ is the measured brightness at a given velocity v. The peak intensity of the emission was also determined via a three-point quadratic fit to each spectrum (defined as moment −2; Sault et al. 1995). Note that C₂H has well separated hyperfine components which will affect the first and second moment determinations (HCN also has hyperfine components, but their separation is less than the observed linewidth for HCN toward G0.253+0.016).

To include only the highest S/N emission in the moment analysis, the peak intensity, intensity weighted velocity field, and the intensity weighted velocity dispersion were only computed at positions across the clump where the emission was above an intensity threshold. For the emission from the molecular species that are the brightest and include the large-scale emission from the single dish data (i.e., HCN, HCO⁺, C₂H, SiO, and HNCO) we used a threshold in the integrated intensity of $\gt 20$% of the peak. Since the noise level is different and the extended emission is included in these cases, this threshold was chosen so that the moments were calculated only when the emission was significantly detected above the noise. The choice of this threshold was visually checked (the region over which the moments were calculated are clearly outlined by the edges of the moment maps) to ensure that it was appropriate for the moment calculation. For the fainter lines with a lower S/N that do not include the large-scale emission, we used a threshold in the peak intensity of $\gt 3.8$ mJy beam⁻¹ (∼5σ).

At locations within the clump where the emission arises from well separated velocity features within the range −10 km s⁻¹ < V_LSR < 70 km s⁻¹, the first moment (intensity weighted velocity field) will result in an intensity weighted average of their individual velocities. In addition, the second moment (intensity weighted velocity dispersion) will clearly be larger than the velocity dispersion of the individual velocity components. Thus, while this analysis may not describe well every individual velocity component, it does provide a method by which to easily characterize and compare the emission from the various tracers and the dust continuum across the clump.

Figure 6, panel 6.1, shows the integrated intensity image (zeroth moment, M₀), peak intensity image, intensity weighted velocity field (first moment, M₁), and also the intensity weighted velocity dispersion (second moment, M₂) derived from the HCN emission toward G0.253+0.016 (all other lines are included in the extended version of Figure 6, panels 6.2–6.17). Included on the latter three images are contours of the peak intensity: these highlight the small regions within the clump that are brightest and are included to aid in the comparison to the velocity field and dispersion images. All images showing the velocity field and dispersions use the same range of values for the color coding (i.e., from −20 to 60 km s⁻¹ and from 0 to 20 km s⁻¹ respectively). This common color coding for the different molecular transitions allows easy comparison and readily highlights differences between them.

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The images of the velocity field clearly show the well known large-scale velocity gradient across the clump; most tracers of the dense gas show blueshifted emission toward more positive declination and redshifted emission toward more negative declination (i.e., SiO, SO, HNCO, and H₂CS). Some of the more complex molecules (NH₂CHO, CH₃CHO, and H₂CS) are also consistent with this trend. In contrast to the molecular transitions with lower excitation energies, the transitions from the more complex molecules also reveal emission toward the clump's center at ∼20 km s⁻¹ (colored green in these images).

The obvious exception to these trends is seen in the emission from both HCN and HCO⁺, which are clearly redshifted across the whole clump and with respect to all other tracers. The trend that these optically thick gas tracers are systematically redshifted with respect to the optically thin and hot gas tracers, indicates that G0.253+0.016 exhibits radial motions (either expanding or contracting, depending on the relative excitation energies of the inner and outer portions of the cloud, see Rathborne et al. 2014a for an in-depth discussion). The trend of redshifted HCN and HCO⁺ emission seen in the single-dish data remains in the interferometric data even when the single dish data are not included. Thus, radial motions are evident on all spatial scales.

Figure 7, panel 7.1, shows the channel maps for HCN (all other channel maps are included in the extended version of Figure 7, panels 7.2–7.17). The panels show the emission from each spectral channel (the corresponding velocity is labeled in the top left corner of each panel) toward positions within the image where the emission was above the previously defined intensity threshold. The channel maps also show well the clump's large-scale velocity gradient. On the smaller scales, they reveal a complex network of emission features with a complicated velocity structure: there is emission on all spatial scales, the morphology of which ranges from small, compact regions to extended, filamentary structures. Toward many of the filamentary features we find homogeneous velocity fields, indicating that they are in fact physically coherent structures.

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Surprisingly, we find filamentary features in both emission and in absorption. The absorption filaments are evident in the HCN and HCO⁺ integrated intensity images and channel maps (i.e., Figure 6, panel 6.1 and 6.2, and Figure 7, panels 7.1 and 7.2). The most intriguing absorption filaments are those seen in HCO⁺ that extend for more than 20 km s⁻¹ in velocity. A detailed analysis of these features is discussed in Bally et al. (2014).

Position–velocity diagrams were also generated for each of the detected molecular transitions (Figure 8). The major axis of the clump is defined along the 0-offset position in R.A. The position–velocity diagrams were made by averaging the emission over the clump's minor axis (i.e., in R.A.) and show the emission along the major axis (Declination) as a function of velocity. These diagrams also show the complexity in the emission and the systematic redshift of the HCO⁺ and HCN emission.

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The combination of molecular transitions covered by these observations trace distinct chemical and physical conditions. For comparison, we group the molecules into four "families": (1) those that have "low" excitation energies E_u/k ∼ 4 K (i.e., HCN, HCO⁺, C₂H), (2) those that are tracers of shocked gas (i.e., SiO and SO), (3) the isotopologues (i.e., H¹³CN, H¹³CO⁺, HN¹³C, and HCC¹³CN), and (4) those that have "high" excitation energies E_u/k ≳ 10 K (i.e., HNCO, CH₃SH, NH₂CHO, CH₃CHO, H₂CS, NH₂CN, and HC₅N).

Figures 9 –11 compare the derived parameters from the moment analysis to the dust column density. These plots were generated by extracting the values from the moment maps and column density image at every pixel where there were non-zero values in both images. Because the number of plotted points within each panel is high, the data were binned so that a contour map of the distribution could be made. Contours show the overall distribution and location of the most common values (the contour levels are shown at 10–90% in steps of 10% of the peak number; the dotted contour outlines the 1% level). From top to bottom the rows show the normalized integrated intensities, normalized peak intensities, intensity weighted velocity field, and intensity weighted velocity dispersion as a function of dust column density. For ease of comparison, the molecular transitions are grouped according to the molecular "families." Within each group the molecular transitions are ordered (from left to right) by increasing excitation energies (see Table 1 for the specific values).

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Figure 9 shows the plots for HCN and HCO⁺. For comparison, we show the same plots made including and excluding the large-scale emission (ALMA+SD and ALMA only; left and right respectively) to show the effect of its inclusion when calculating the intensity-weighted moments. As expected, when including the large-scale emission, we find a clear correspondence between HCN and HCO⁺ and the lower dust column density material (i.e., at column densities $\lt 5$ × 10²² ${\rm c}{{{\rm m}}^{-2}}$). These molecules clearly show an absence of blueshifted emission, compared to most others (compare to Figures 10 and 11). When the large-scale emission is included, the measured intensity weighted velocity dispersion is more centrally peaked (because the emission is widespread, with a broad line profile, the "intensity weighting" will tend to result in a similar calculated value for the intensity weighted velocity despite the emission being detected over a broad range). Moreover, a clear trend is evident: the measured velocity dispersions decrease as the dust column density increases. This trend is not evident in the plots made with the ALMA-only data because, in this case, there is no emission from the molecular transitions at the same location as the low column density emission, which is required to see the trend.

The shock tracers (SiO and SO) show bright and widespread emission in both position and velocity across G0.253+0.016 (Figure 10). Moreover, their measured velocity dispersions extend to ∼20 km s⁻¹, which is higher than most other tracers and expected since SiO and SO trace gas that is stirred up in regions of shocks. When including the large-scale emission, SiO shows the opposite trend in its velocity dispersion compared to HCN and HCO⁺: its velocity dispersion increases as the dust column density increases, consistent with increased shocks in the higher density material. For SO, while its typical measured dispersion is low (∼2 km s⁻¹), there is a significant tail in this distribution up to dispersions of ∼20 km s⁻¹. In contrast to SiO, there is no clear trend for an increasing SO velocity dispersion with increased dust column density: SO shows low velocity dispersions regardless of the column density. The differences in the velocity dispersions measured for SiO and SO could be understood in terms of the differences in their excitation energies and critical densities: while SO has a higher excitation energy compared to SiO, it has a critical density that is an order of magnitude lower. Thus these species may in fact be tracing different material within the clump.

There are many filamentary features found within G0.253+0.016 that are detected in SiO and SO, that have high velocity dispersions, and large steps in velocity over small spatial regions (i.e., linear features seen in the moment maps in Figures 6.5 and 6.6). Several of these are also evident in the channel maps of H¹³CN (at LSR velocities of ∼10–17 km s⁻¹; see Figures 7.5, 7.6, 7.7). They likely correspond to shock fronts within the clump: their details will be presented in a future paper.

The isotopologues and the molecular transitions with higher excitation energies (e.g., H¹³CN, H¹³CO⁺, HN¹³C, HCC¹³CN, HNCO, CH₃SH, NH₂CHO, CH₃CHO, H₂CS, NH₂CN, and HC₅N; Figures 10 and 11) are detected over a much smaller range in dust column densities and are more sharply peaked at, or slightly above, the mean value of the dust column density (∼8.6 × 10²² ${\rm c}{{{\rm m}}^{-3}}$, shown as the vertical dotted line) compared to the molecules with lower excitation energies.¹⁴ Their integrated intensity distribution appears bi-modal, indicating the presence of two distinct integrated intensity values. Their measured velocities span a range from 0–50 km s⁻¹; this velocity distribution is broader compared to that derived from HCN and HCO⁺.¹⁵

While the peak intensities for the isotopologues and the molecules with the higher excitation energies are smoothly distributed over a similar range in column densities to their integrated intensities, their velocity dispersion distributions are typically concentrated at low values of the velocity dispersion. With the exception of HNCO, which is far more abundant than these other molecules, the emission predominantly arises from very narrow velocity features ($\sigma \;\lesssim \;$ 2 km s⁻¹). Such narrow velocity dispersions have also been observed toward G0.253+0.016 on small spatial scales from other molecular tracers (e.g., N₂H⁺(3–2); Kauffmann et al. 2013). The tail in these distributions to considerably higher velocity dispersions (dotted contours) indicates positions within the clump where the moment analysis has overestimated the velocity dispersion due to multiple velocity components within the spectrum. This occurs in fewer than 1% of the spectra.

The narrow range in dust column density over which the higher excitation energy molecular transitions are detected can be understood in terms of their critical densities. These molecules are typically detected when the gas has either a high temperature (T $_{{\rm gas}}\gt 100$ K) or a high density ($n\;\gt \;$n_crit), or both. Because these conditions are needed for both their formation and excitation, the presence of these molecules can be used to pinpoint gas with these physical conditions. These molecular transitions all have critical densities of $\sim {{10}^{5}}-{{10}^{7}}$ ${\rm c}{{{\rm m}}^{-3}}$ (see Table 1). Thus, while the gas temperature may be low toward the clump's center, the volume density may be very high. If the volume densities are sufficiently higher than their critical densities, the molecules may be excited regardless of the gas temperature.

We see a clear trend where molecules with higher excitation energies and critical densities peak toward the center of the clump (compared to those with lower excitation energies and critical densities). The position–velocity diagrams also reveal that the molecular transitions with higher excitation energies and critical densities peak toward the center of the cloud in both position and velocity (decl. offset of ∼0, and velocity of ∼20 km s⁻¹; e.g., CH₃CHO and NH₂CN in Figure 8). These trends are exactly what is expected for a clump with a denser interior.

While the molecular transitions detected by these observations all probe dense gas, they span a range in their critical densities and excitation energies, trace distinct chemical conditions, and often suffer from high optical depths. As such, they may often trace different material within G0.253+0.016 and may not necessarily correlate well with the optically thin dust continuum emission.

One of our key goals is to identify star-forming dust cores within the clump, to measure their sizes, masses, and virial state. Determining these properties will allow us to ascertain the potential of G0.253+0.016 to form a cluster and the distribution of masses for its cores. In order to identify bound star-forming cores, we need to reliably associate molecular line emission with each dust continuum core in order derive their velocity dispersions. Ideally, we need to identify a molecular transition that is optically thin and is bright enough to be detected in most cores.

Given the limited velocity resolution of our data, a detailed analysis of the cores' virial states is challenging with the current data.¹⁶ However, we can gauge, overall, which molecular transitions correlate with the dust emission. We find that emission from the molecular transitions with higher excitation energies typically provide a better morphological match to the dust column density compared to the transitions with lower excitation energies, the isotopologues, or the shock tracers (compare the morphology of the dust emission to NH₂CHO, CH₃CHO, HNCO and H₂CS in Figure 3 and also their 2D correlation coefficients shown in Figure 4). Moreover, since their velocity dispersions are typically narrow (∼2 km s⁻¹), these molecules may in fact be tracing individual high column density cores within the clump with very narrow velocity dispersions.

Supersonic turbulence is an important mechanism by which overdensities within molecular clouds may be generated (Vázquez-Semadeni 1994; Padoan et al. 1997). As such, understanding the degree to which turbulence sets the initial structure within molecular clouds is an important part of understanding core and, ultimately, star formation. Recent work using the ALMA dust continuum emission presented here shows that the log-normal shape, dispersion, and mean of its column density probability distribution function (N-PDF) very closely matches the predictions of theoretical models of supersonic turbulence in gas given the high-pressure environment in which G0.253+0.016 is immersed (Rathborne et al. 2014b).

These results also show that while, overall, the N-PDF shows a log-normal distribution, there is a small deviation from log-normal at the highest column densities, which indicates self-gravitating gas. This small deviation is coincident with the location of a known water maser (Lis & Carlstrom 1994) and, thus, pinpoints where star formation is beginning in the cloud. Assuming a dust temperature of 20 K, this small core (radius of 0.04 pc) has a mass of ∼72 M $_{\odot }$ and virial ratio, ${{\alpha }_{{\rm vir}}}$, of ≲1 (this is an upper limit since the associated molecular line emission is unresolved in velocity, $\sigma \lt 1.4$ km s⁻¹; see Rathborne et al. 2014b for details). Since ${{\alpha }_{{\rm vir}}}$ is close to unity, this core is gravitationally bound and unstable to collapse.

The fact that the N-PDF for G0.253+0.016 matches well predictions from turbulent theory suggests that our current understanding of molecular cloud structure derived from the solar neighborhood also holds in high-pressure environments (Rathborne et al. 2014b). This has important implications for understanding the initial gas structure for star formation in a range of different environments across the universe (Kruijssen & Longmore 2013).

To further determine the extent to which turbulence shapes the observed gas structure within G0.253+0.016, we have calculated the fractal dimension and the turbulent power spectrum. These statistical methods are commonly used to describe the structure of molecular clouds in both simulations and observations.

5.3.1. Fractal Structure on Small Spatial Scales

An important characteristic of a protocluster clump, and one that is easily revealed by these data, is the distribution of its material on small spatial scales. If G0.253+0.016 represents a clump that will give rise to a high-mass cluster, then the distribution of the material within it before star formation has commenced can provide important constraints for theories that describe cluster formation. As discussed previously, there have been a wide range of theoretical and numerical studies showing that the formation of high-mass stellar clusters should proceed hierarchically, but thus far the degree of hierarchal structure within high-mass protoclusters has not been estimated.

The fractal dimension (D_f) can be used to characterize the structure within a molecular cloud. It is a simple tool that determines the degree to which the emission is smooth and centrally condensed or structured and hierarchical. Since observations of dust or molecular line emission reveal only the two-dimensional projection of a three-dimensional structure, care needs to be taken to accurately convert the measured projected fractal dimension (D_p) to the real three-dimensional fractal dimension (D_f). This conversion is often assumed to be D_f = D_p + 1 (Beech 1992); however, using simulations of cloud structure, Sánchez et al. (2005) derive an empirical relation to convert between D_f and D_p (see their Figure 8). Rather than a single conversion factor of unity for all values of D_p, their simulations suggest that D_f may actually increase linearly as D_p decreases. Previous observations derive projected fractal dimensions ${{D}_{p}}\sim 1.35$ (see Federrath et al. 2009 and references therein) leading to a quoted D_f for molecular clouds of ∼2.3 (Elmegreen 1997).

To calculate the projected fractal dimension (D_p) for G0.253+0.016, we employ the perimeter-area method (Mandelbrot 1977) which is often used to measure the fractal dimension of interstellar gas clouds (e.g., Falgarone et al. 1991; Sánchez et al. 2005; Federrath et al. 2009). Structures were defined as connected regions within a contour at a given intensity level. The area (A) was calculated as the sum of the number of pixels within the contour; the perimeter (P) was calculated as the number of pixels externally adjacent to the contour level. By varying the contour levels from the image minimum to its maximum, values for P and A were determined on many spatial scales. With a power-law fit of the form P ∝ A $^{{{D}_{p}}/2}$, we can determine the perimeter-area fractal dimension D_p. A smooth structure will have D_p = 1, while a very convoluted structure will have D_p = 2 (Federrath et al. 2009).

Figure 12 shows the perimeter-area plots measured from contours of the dust and line integrated intensity emission. For this analysis we use, where possible, the data that include the large-scale emission. Since this analysis is based on contours within an image, the effect of including the large-scale emission simply increases the number of contours to lower levels of the intensity which translates to an increase in the number of points plotted at large perimeters and areas. Their inclusion has little effect on the derived slope.

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In all panels the solid lines show two extreme values for D_p (1 for smooth, centrally condensed emission, 2 for a highly fractal cloud). The dotted lines show linear fits to the data points: the derived values for D_p range from 1.4–1.7, depending on the molecular tracer. For the dust continuum that includes the large-scale emission, we find a D_p of 1.6 (plotted in the top left panel, marked as ALMA+SD). For comparison we also calculate the fractal dimension derived from only the large-scale dust continuum emission (in the same panel marked as SD), and find D_p to be ∼1.2, consistent with its smooth distribution.

Since the fractal dimension is best indicated by emission that is optically thin, i.e., from the dust emission and the molecular transitions with higher excitation energies, we select the derived D_p from these molecules to characterize the emission from within G0.253+0.016 (i.e., D_p ∼ 1.6). Using the empirical conversion from the two-dimensional fractal dimension D_p to the three-dimensional fractal dimension D_f from Sánchez et al. (2005), we find that D_f ∼ 1.8 for G0.253+0.016. Simulations of compressively driven turbulence in the ISM (Federrath et al. 2009) with Mach numbers, $\mathcal{M}$, of ∼5.5 reveal typical values for D_f of 2.3–2.6 (which correspond to D_p ∼ 1.3–1.5 using the empirical relation derived by Sánchez et al. 2005). Since $\mathcal{M}$ is ∼30 for G0.253+0.016, it has a much higher level of turbulence compared to most other molecular clouds. Given this higher level of turbulence, it may be expected that the sonic length may be lower and the degree of fragmentation might be higher in G0.253+0.016 (i.e., a higher D_p, and thus, lower D_f for higher $\mathcal{M}$). While detailed simulations are necessary to determine the effect of $\mathcal{M}$ on the fractal dimension, our results are consistent with a marginally higher value for D_p for G0.253+0.016 when compared to typical molecular clouds. This analysis suggests that the clump is highly fragmented on small spatial scales and its structure consistent with that produced by compressively driven turbulence.

5.3.2. Turbulent Power Spectrum

The power spectrum provides a measure of how turbulence is propagated within a molecular cloud and it identifies the preferential spatial scale, if any, at which the structure changes. On large scales a break in the power spectrum indicates the spatial scale at which turbulent energy is injected, on the small scales it can indicate the spatial scales at which either turbulence is dissipated or injected. Measuring the small-scale break point of the power spectrum can therefore reveal the spatial scale at which gravity overcomes the turbulent pressure or the spatial scale at which energy may be injected (i.e., from protostellar outflows).

We use the dust continuum image and several of the integrated intensity images to compute the power spectra. Because the turbulent power spectrum is a statistical measure of the energy on various spatial scales, it is important that the emission is well detected and well sampled across the clump. As such, we perform this analysis using HCN, HCO⁺, HNCO, and SiO because their data include the zero-spacing information and the emission is detected with high S/N at all spatial scales across G0.253+0.016.

Figure 13 shows the annular averages of each of the power spectra, P(k), as a function of the spatial frequency, k. The power spectra were generated using a Fast Fourier Transform of the images. In all panels the dotted vertical line marks the spatial frequency corresponding to the angular resolution of the data: the power spectra at spatial frequencies greater than this are not well sampled by these data.

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In cases where the power spectrum is close to a power law (i.e., for HCN, HCO⁺, HNCO, and SiO emission) we fit it with a single power law to derive α where P(k) ∝ k^−α. In these cases, we find that $\alpha \sim 3$. A power-law slope of 3 is predicted for a line intensity power spectrum of an optically thick medium (Lazarian & Pogosyan 2004; Burkhart et al. 2013). Thus, the derived power spectrum slope of ∼3 for these molecular transitions are consistent with them being optically thick toward G0.253+0.016.

In contrast, for the dust continuum emission that is optically thin, the power spectrum shows a clear break. While this break occurs at large spatial frequencies (i.e., small spatial scales), it is lies above the spatial frequencies corresponding to the angular resolution of these data (marked with the vertical dotted line). For this power spectrum we fit two distinct power laws and derive their indicies and the break point. For small values of k (i.e., large spatial scales) the derived power-law index is $\alpha \sim 2.5$ compared to $\alpha \sim 8.0$ for large values of k (i.e., small spatial scales). Values for α of ∼2.8 are observed toward other star-forming regions such as Perseus, Taurus, and Rosetta (Padoan et al. 2004).

The trend of a steeper spectrum on smaller spatial scales implies that the higher column density material is concentrated on smaller spatial scales, while the lower column density material is distributed on larger spatial scales. This is consistent with the distribution of the emission in the dust continuum image. Since the effect of gravity on small scales will cause a steepening of the power-law slope at large spatial frequencies, the spatial scale at which gravity may dominate can be determined from the break point in the power spectrum. From the dust continuum emission power spectrum, we find that the break occurs at spatial frequencies corresponding to size scales of 0.12 pc. This size scale is comparable to the size of the small region of self-gravitating gas that was identified as the deviation at high column densities via the N-PDF (Rathborne et al. 2014b). As such, this lends support to the idea that gravity has overcome the internal pressure on these small spatial scales.

For cores within molecular clouds to collapse into stars, they need to be sufficiently dense that their self-gravity overcomes the thermal pressure. The Jeans length, defined as the minimum length scale for gravitational fragmentation to occur, was determined for G0.253+0.016 using the relation

$$\begin{eqnarray}&&{{\lambda }_{J}}=0.4\;{\rm pc}\left( \frac{{{c}_{s}}}{0.2\;{\rm km}\;{{{\rm s}}^{-1}}} \right){{\left( \frac{n({{{\rm H}}_{2}})}{{{10}^{3}}\;{\rm c}{{{\rm m}}^{-3}}} \right)}^{-0.5}}\end{eqnarray} \tag{ 1 }$$

where c_s is the sound speed and n(H₂) the volume density. While the measured dust temperature toward G0.253+0.016 is ∼20 K (Longmore et al. 2012), recent observations indicate that the gas temperature is considerably higher, ∼65 K (Ao et al. 2013), leading to the speculation that the gas may be heated by the dissipation of turbulent energy and/or cosmic rays rather than by photon heating. These observations are supported by smoothed particle hydrodynamics modeling of the dust and gas temperature distribution in G0.253+0.016 which suggest that the gas and dust temperatures are not coupled and that the high interstellar radiation field and cosmic ray ionization rate in the CMZ are responsible for the discrepant dust and gas temperatures (Clark et al. 2013).

Assuming a global gas temperature for G0.253+0.016 of 65 K (Ao et al. 2013) the corresponding gas sound speed, c_s, is 0.5 km s⁻¹. Using an average volume density for this material of ∼10⁵ ${\rm c}{{{\rm m}}^{-3}}$, we derive ${{\lambda }_{J}}$ to be 0.10 pc. This value is comparable to the size scale derived from the break point in the power spectrum of the dust continuum emission and of the size of the self-gravitating core identified via the N-PDF. Extrapolating the global size–linewidth relation in Figure 14 (top right panel) to smaller scales suggests that the line-widths are subsonic on spatial scales of ∼0.1 pc (the sonic length is ∼0.2 pc). As such, since the sonic length exceeds the derived break point in the power spectrum, the thermal Jeans length is indeed the relevant quantity for describing the minimum size scale for gravitational fragmentation. An absence of a similar break in the power spectra derived from the molecular line emission is consistent with the transitions being optically thick and the dust continuum optically thin. Thus, the small, high-column density features detected in the optically thin dust continuum emission may have formed because gravity was able to overcome the thermal pressure on these small scales.

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