Self-gravity plays a decisive role in the final stages of star formation, where dense cores (size ∼0.1 parsecs) inside molecular clouds collapse to form star-plus-disk systems1. But self-gravity’s role at earlier times (and on larger length scales, such as ∼1 parsec) is unclear; some molecular cloud simulations that do not include self-gravity suggest that ‘turbulent fragmentation’ alone is sufficient to create a mass distribution of dense cores that resembles, and sets, the stellar initial mass function2. Here we report a ‘dendrogram’ (hierarchical tree-diagram) analysis that reveals that self-gravity plays a significant role over the full range of possible scales traced by 13CO observations in the L1448 molecular cloud, but not everywhere in the observed region. In particular, more than 90 per cent of the compact ‘pre-stellar cores’ traced by peaks of dust emission3 are projected on the sky within one of the dendrogram’s self-gravitating ‘leaves’. As these peaks mark the locations of already-forming stars, or of those probably about to form, a self-gravitating cocoon seems a critical condition for their existence. Turbulent fragmentation simulations without self-gravity—even of unmagnetized isothermal material—can yield mass and velocity power spectra very similar to what is observed in clouds like L1448. But a dendrogram of such a simulation4 shows that nearly all the gas in it (much more than in the observations) appears to be self-gravitating. A potentially significant role for gravity in ‘non-self-gravitating’ simulations suggests inconsistency in simulation assumptions and output, and that it is necessary to include self-gravity in any realistic simulation of the star-formation process on subparsec scales.
Spectral-line mapping shows whole molecular clouds (typically tens to hundreds of parsecs across, and surrounded by atomic gas) to be marginally self-gravitating5. When attempts are made to further break down clouds into pieces using ‘segmentation’ routines, some self-gravitating structures are always found on whatever scale is sampled6,7. But no observational study to date has successfully used one spectral-line data cube to study how the role of self-gravity varies as a function of scale and conditions, within an individual region.
Most past structure identification in molecular clouds has been explicitly non-hierarchical, which makes difficult the quantification of physical conditions on multiple scales using a single data set. Consider, for example, the often-used algorithm CLUMPFIND7. In three-dimensional (3D) spectral-line data cubes, CLUMPFIND operates as a watershed segmentation algorithm, identifying local maxima in the position–position–velocity (p–p–v) cube and assigning nearby emission to each local maximum. Figure 1 gives a two-dimensional (2D) view of L1448, our sample star-forming region, and Fig. 2 includes a CLUMPFIND decomposition of it based on 13CO observations. As with any algorithm that does not offer hierchically nested or overlapping features as an option, significant emission found between prominent clumps is typically either appended to the nearest clump or turned into a small, usually ‘pathological’, feature needed to encompass all the emission being modelled. When applied to molecular-line data, CLUMPFIND typically finds features on a limited range of scales, above but close to the physical resolution of the data, and its results can be overly dependent on input parameters. By tuning CLUMPFIND’s two free parameters, the same molecular-line data set8 can be used to show either that the frequency distribution of clump mass is the same as the initial mass function of stars or that it follows the much shallower mass function associated with large-scale molecular clouds (Supplementary Fig. 1).
Four years before the advent of CLUMPFIND, ‘structure trees’9 were proposed as a way to characterize clouds’ hierarchical structure using 2D maps of column density. With this early 2D work as inspiration, we have developed a structure-identification algorithm that abstracts the hierarchical structure of a 3D (p–p–v) data cube into an easily visualized representation called a ‘dendrogram’10. Although well developed in other data-intensive fields11,12, it is curious that the application of tree methodologies so far in astrophysics has been rare, and almost exclusively within the area of galaxy evolution, where ‘merger trees’ are being used with increasing frequency13.
Figure 3 and its legend explain the construction of dendrograms schematically. The dendrogram quantifies how and where local maxima of emission merge with each other, and its implementation is explained in Supplementary Methods. Critically, the dendrogram is determined almost entirely by the data itself, and it has negligible sensitivity to algorithm parameters. To make graphical presentation possible on paper and 2D screens, we ‘flatten’ the dendrograms of 3D data (see Fig. 3 and its legend), by sorting their ‘branches’ to not cross, which eliminates dimensional information on the x axis while preserving all information about connectivity and hierarchy. Numbered ‘billiard ball’ labels in the figures let the reader match features between a 2D map (Fig. 1), an interactive 3D map (Fig. 2a online) and a sorted dendrogram (Fig. 2c).
A dendrogram of a spectral-line data cube allows for the estimation of key physical properties associated with volumes bounded by isosurfaces, such as radius (R), velocity dispersion (σv ) and luminosity (L). The volumes can have any shape, and in other work14 we focus on the significance of the especially elongated features seen in L1448 (Fig. 2a). The luminosity is an approximate proxy for mass, such that Mlum = X13COL13CO, where X13CO = 8.0 × 1020 cm2 K-1 km-1 s (ref. 15; see Supplementary Methods and Supplementary Fig. 2). The derived values for size, mass and velocity dispersion can then be used to estimate the role of self-gravity at each point in the hierarchy, via calculation of an ‘observed’ virial parameter, αobs = 5σv2R/GMlum. In principle, extended portions of the tree (Fig. 2, yellow highlighting) where αobs < 2 (where gravitational energy is comparable to or larger than kinetic energy) correspond to regions of p–p–v space where self-gravity is significant. As αobs only represents the ratio of kinetic energy to gravitational energy at one point in time, and does not explicitly capture external over-pressure and/or magnetic fields16, its measured value should only be used as a guide to the longevity (boundedness) of any particular feature.
In calculating αobs, we are implicitly assuming that there is a one-to-one relationship (known as a ‘bijection’) between a volume in p–p–v space and a volume of physical (position–position–position, p–p–p) space. This bijection paradigm is fine for regions which are dominated by a single structure, but the complexities of relating p–p–v space to physical space in regions with multiple features along a line of sight does mean that this treatment can only ever give an approximate measure of the true dynamical state of the cloud17. Alternatives to bijection are considered in the Supplementary Information. The bijection assumption comes into play when measuring physical properties of individual features, but it does not influence the characterization of hierarchical structure.
In Fig. 2c, we show the dendrogram for the same L1448 13CO spectral-line map shown using contours in Fig. 1. All of the portions shaded yellow have αobs < 2, meaning that they are (most) likely to be self-gravitating. The four most compact p–p–v structures (leaves) where αobs < 2 are numbered in Figs 1 and 2, and they are not as apparent in the projected (2D) view (Fig. 1) as they are in p–p–v (3D) space (Fig. 2a). In the CLUMPFIND decomposition of the cloud (Fig. 2b), these features are not apparent as special.
Overall, the pattern of yellow highlighting in Fig. 2 suggests the importance of gravity on all possible scales, but not within the full possible volume, in a cloud like L1448. With the exception of the gas around region 4, which appears not to be bound to the rest of L1448, the tree shows a fully yellow-highlighted ‘trunk’ and only sporadic highlighting on the dendrogram’s tallest branches and leaves. So for the material traced by 13CO observations, it appears that self-gravitating structures are more prevalent on larger scales than on smaller. At densities surpassing 5 × 103 cm-3, 13CO becomes an increasingly poor tracer of mass18, so it can only give upper limits for the ‘true’ virial parameters of the densest, most compact, structures seen in the dendrogram. Thus, the highest-density non-yellow leaves in Fig. 2c may harbour bound structures only visible with thinner or less-depleted molecular lines. On the other hand, lower-density non-yellow leaves in Fig. 2c probably represent actual low-mass unbound structures in the gas, similar to the ‘pressure-confined’ low-mass clumps found in clump-based segmentations. Importantly, the full pattern of highlighting explicitly indicates that core-like leaves often reside within structures where the mutual gravity between the cores (leaves) and/or their environs (branches) is significant enough to cause meaningful interactions between cores—possibly even, in the most extreme cases, competitive accretion. Recent work18 has shown that the overall (column) density distribution of material traced by 13CO in a 10-pc-scale molecular cloud is roughly log-normal, and our result here implies that some of the high-density fluctuations in that statistical distribution are bound within themselves and/or to each other, and some not.
Tree hierarchies can be used to intercompare the topology and physical properties (for example boundedness) of structures within star-forming regions, and such intercomparison can be profitably extended to simulations as well. In Fig. 4, we summarize such a comparison (see Supplementary Information) with a plot showing the fraction of ‘self-gravitating’ (αobs < 2) material as a function of spatial scale for both our L1448 data and for a synthetic data cube4. The simulation used to produce the synthetic data is purely hydrodynamic, meaning that the effects of magnetic fields, heating and cooling, and self-gravity are not included. The power-law exponent characterizing the power spectrum of turbulence in these synthetic 13CO data and in the COMPLETE Perseus data8 (from which our L1448 example is drawn) is ∼1.8, to within small uncertainties (∼0.2; ref. 4). However, inspection of Fig. 4 (and of Supplementary Fig. 4) clearly shows that the data and simulation appear quite different in the context of dendrogram analysis: in the simulation, nearly all material (much more than in the observations) is self-gravitating, on all spatial scales. Critically, the analysis of the synthetic 13CO cube4 (Supplementary Fig. 4) is done on a simulated observation of it where we have deliberately matched resolution, noise properties and region extent to the L1448 cube (Supplementary Methods). The (constant) abundance of 13CO used for the synthetic map (Supplementary Information) is set to match the known column densities in the simulation, and because abundance is simply a multiplicative constant, changing it cannot reproduce the scale dependence of gravity found in the L1448 data.
Thus it appears that the synthetic data cube created from the simulation4 contains much material that would be significantly affected by gravity, if gravity were actually included in the simulation.
The accuracy with which dendrograms can offer estimates of αobs is at or below the 25% level (Supplementary Information). The uncertainty results primarily from the need to glean a 3D geometry and density based on 2D size and column density (mass/area), and any analysis of p–p–v data will be subject to the same limitations. More analysis, using simulations, of the translation from p–p–v to p–p–p space17 should be, and is being, carried out to quantify these uncertainties more finely. Comparative measurements (for example Fig. 4) are far more certain as these biases should affect all data sets similarly. Thus, the apparent disagreement between observations and simulation in Fig. 4 can be explained by claiming that either, or both, of the following are true: (1) the assumptions/calculations leading to the creation of the synthetic 13CO observations are faulty; or (2) there is missing physics in the simulation (for example gravity, thermal effects), making it an insufficient approximation to real star-forming regions.
Finally, we turn to the relationship between the apparently ‘self-gravitating’ regions in L1448 and the star-formation process itself. Compact millimetre-wavelength emission peaks caused by dust emission (marked by yellow circles in Fig. 1) are typically taken as markers of cores that are forming, or are able to form, stars. Within the region of L1448 considered here, more than 90% of the compact millimetre-dust peaks traced in bolometer observations3 are found projected on the sky within one of the dendrogram’s ‘self-gravitating’ leaves, and none is found outside a self-gravitating branch. Recent NH3 observations19 suggest that all, or all but one, of these ‘pre-stellar cores’ lie within self-gravitating structures along the velocity dimension as well14. As young sources get a little older, they can be detected in the mid-infrared (IRAC) bands of the Spitzer Space Telescope. Four out of the five sources identified by such IRAC imaging as protostar candidates20 also lie within a leaf, and each of those four is associated with a millimetre-dust peak, suggesting they are embedded in dense natal cocoons. Interestingly, the one IRAC protostar candidate in the region not associated with a self-gravitating leaf is also not associated with a millimetre-dust peak, suggesting it is a more evolved source. All told, these associations suggest that a self-gravitating home is critical to the earliest phases of star formation.
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We thank A. Munshi for putting us in touch with M. Thomas and colleagues at Right Hemisphere, whose software and assistance enabled the interactive PDF in this paper; P. Padoan for providing the simulated data cube; R. Shetty for comments on the paper; F. Shu for suggesting we extend our analysis to measure boundedness of p–p–v ‘bound’ objects in p–p–p space using simulations; and S. Hyman, Provost of Harvard University, for supporting the start-up of the Initiative in Innovative Computing at Harvard, which substantially enabled the creation of this work. 3D Slicer is developed by the National Alliance for Medical Image Computing and funded by the National Institutes of Health grant U54-EB005149. The COMPLETE group is supported in part by the National Science Foundation. E.W.R. is supported by the NSF AST-0502605.
Author Contributions The dendrogram algorithm and software was created by E.W.R. The interactive figures were assembled by M.A.B., J.K. and M.H. using software from Right Hemisphere and Adobe. J.K. and M.H. worked to allow 3D Slicer to plot the surfaces relevant to the dendrograms shown in the 3D figures. J.B.F. produced Fig. 1, and J.E.P. carried out the ‘CLUMPFINDing’ analysis shown in Fig. 2 and Supplementary Fig. 1. A.A.G. wrote most of the text, and all authors contributed their thoughts to the discussions and analysis that led to this work.
The 3D Slicer software used to create the surface renderings is available at http://am.iic.harvard.edu/.
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Goodman, A., Rosolowsky, E., Borkin, M. et al. A role for self-gravity at multiple length scales in the process of star formation. Nature 457, 63–66 (2009). https://doi.org/10.1038/nature07609
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