Hierarchical Collapse Research Articles

Unveiling two expanding stellar groups formed through violent relaxation in the lagoon nebula cluster

ABSTRACT The current kinematic state of young stellar clusters can give clues on their actual dynamical state and origin. In this contribution, we use Gaia DR3 data of the Lagoon Nebula Cluster (LNC) to show that the cluster is composed of two expanding groups, likely formed from different molecular cloud clumps. We find no evidence of massive stars having larger velocity dispersion than low-mass stars or being spatially segregated across the LNC, as a whole, or within the Primary group. However, the Secondary group, with 1/5th of the stars, exhibits intriguing features. On the one hand, it shows a bipolar nature, with an aspect ratio of ∼3:1. In addition, the massive stars in this group exhibit larger velocity dispersion than the low-mass stars, although they are not concentrated towards the centre of the group. This suggests that this group may have undergone dynamical relaxation, first, and some explosive event afterward. However, further observations and numerical work have to be performed to confirm this hypothesis. The results of this work suggest that, although stellar clusters may form by the global and hierarchical collapse of their parent clump, still some dynamical relaxation may take place.

Star Formation in the H ii Region Sh2-87: Evidence of Global Hierarchical Collapse

We present a detailed study of the Sh2-87 H ii region using a multiwavelength data set in optical to radio bands. A Herschel column density map revealed that the host cloud is filamentary in nature, and together they formed a central dense hub. The extinction map generated using near-infrared photometric data also signifies the nonuniform distribution of the cloud and reveals its filamentary nature. We estimated a sizable variable extinction over the region up to A V = 34.4 mag, with an average value of A V = 3.4 mag. Using the various infrared color–color criteria, we identified 13 Class I and 202 Class II young stellar objects (YSOs) and 22 Hα-emitting sources toward this region. Further analysis showed that the cluster is mainly composed of low-mass YSOs with a typical age of ∼3 Myr having masses in the range of 0.1–6.0 M ⊙. The identified evolved YSOs (i.e., Class II YSOs) are primarily distributed along the filaments and in the outer parts of the cloud, while the recent star formation, inferred by the presence of Class I YSOs, ionized gas, and star-forming clumps, is observed in the hub region. The overall star formation scenario in the Sh2-87 region resembles the global hierarchical collapse model of star formation, where younger massive star formation activity is expected at the central hub along with the distribution of evolved low-mass YSOs in the filaments and the outer parts of the cloud.

Massive star formation in the hub–filament system of RCW 117

ABSTRACT We present a multiwavelength investigation of the hub–filament system RCW 117 (IRAS 17059-4132), which shows intricate filamentary features in the far-infrared, mapped using Herschel images. We obtain the column density and dust temperature maps for the region using the Herschel images and identify 88 cores and 12 filaments from the column density map of the region (18 arcmin × 18 arcmin). We employ the ThrUMMS 13CO (J = 1–0) data for probing the kinematics in RCW 117, and find velocity gradients (∼0.3–1 km s−1 pc−1) with hints of matter inflow along the filamentary structures. Ionized gas emission from the associated H ii region is examined using the Giant Metrewave Radio Telescope (GMRT) at 610 and 1280 MHz and is found to be of extent 5 × 3 pc2 with intensity being brightest towards the hub. We estimate the peak electron density towards the hub to be ∼750 cm−3. Thirty-four Class 0/I young stellar objects (YSOs) have been identified in the region using the Spitzer GLIMPSE colour–colour diagram, with many lying along the filamentary structures. Based on the (i) presence of filamentary structures, (ii) distribution of cores across the region, with ∼39 per cent found along the filamentary structures, (iii) massive star formation tracers in the hub, and (iv) the kinematics, we believe that global hierarchical collapse can plausibly explain the observed features in RCW 117.

Self-similar solutions for fuzzy dark matter

Self-similar solutions for fuzzy dark matter are very different from their counterparts in the standard cold dark matter model. In contrast to the familiar hierarchical collapse of the current model for structure formation, they correspond to an inverse hierarchical blow-up. This fact highlights the gravitational cooling process, which in the absence of dissipation, allows the system to eject excess energy through the intermittent expulsion of clumps of matter. These surprising behaviours are due to the wave properties of the Schrödinger equation and to the quantum pressure. These features are printed in the Eulerian density-velocity representation of the nonrelativistic scalar field, or in the Lagrangian representation in the mass-shell trajectories.

The kinetic and magnetic energy budget of hub-filament systems during the gravitational fragmentation of molecular clouds

ABSTRACT We present a numerical study of the balance between the gravitational (Eg), kinetic (Ek), and magnetic (Em) energies of structures within a hub-filament system in a simulation of the formation and global hierarchical collapse (GHC) of a giant molecular cloud. For structures defined by various density thresholds, and at different evolutionary stages, we investigate the scaling of the virial parameter, α, with mass M, and of the Larson ratio, ${\cal {L}}_{\rm v}\equiv \sigma _{\rm v}/R^{1/2}$, with column density Σ, where σv is the 1D velocity dispersion, and R is an effective radius. We also investigate these scalings for the corresponding magnetic parameters αm and ${\cal {L}}_{\rm {m}}$. Finally, we compare our numerical results with an observational sample of massive clumps. We find that: 1) αm and ${\cal {L}}_{\rm {m}}$ follow similar α–M and ${\cal {L}}$–Σ scalings as their kinetic counterparts, although the ratio Em/Ek decreases as |Eg| increases. 2) The largest objects, defined by the lowest thresholds, tend to appear gravitationally bound (and magnetically supercritical), while their internal substructures tend to appear unbound (and subcritical). This suggests that the latter are being compressed by the infall of their parent structures, and supports earlier suggestions that the measured mass-to-magnetic flux ratio μ decreases inwards in a centrally-peaked cloud under ideal MHD. 3) The scatter in the α–M and ${\cal {L}}$–Σ plots is reduced when Ek and Em are plotted directly against Eg, suggesting that the scatter is due to an ambiguity between mass and size. 4) The clumps in our GHC simulation follow the same trends as the observational sample of massive clumps in the ${\cal {L}}$–Σ and α–M diagrams. We conclude that the main controlling parameter of the energy budget in the structures is Eg, with the kinetic and magnetic energies being derived from it.

Membership Analysis and 3D Kinematics of the Star-forming Complex around Trumpler 37 Using Gaia-DR3

Identifying and characterizing young populations of star-forming regions are crucial to unraveling their properties. In this regard, Gaia-DR3 data and machine-learning tools are very useful for studying large star-forming complexes. In this work, we analyze the ∼7.1 deg2 area of one of our Galaxy’s dominant feedback-driven star-forming complexes, i.e., the region around Trumpler 37. Using the Gaussian mixture and random-forest classifier methods, we identify 1243 highly probable members in the complex, of which ∼60% are new members and are complete down to the mass limit of ∼0.1–0.2 M ⊙. The spatial distribution of the stars reveals multiple clusters toward the complex, where the central cluster around the massive star HD 206267 reveals two subclusters. Of the 1243 stars, 152 have radial velocity, with a mean value of −16.41 ± 0.72 km s−1. We investigate stars’ internal and relative movement within the central cluster. The kinematic analysis shows that the cluster’s expansion is relatively slow compared to the whole complex. This slow expansion is possibly due to newly formed young stars within the cluster. We discuss these results in the context of hierarchical collapse and feedback-induced collapse mode of star formation in the complex.

Probing the global dust properties and cluster formation potential of the giant molecular cloud G148.24+00.41

ABSTRACT Clouds more massive than about 105 M⊙ are potential sites of massive cluster formation. Studying the properties of such clouds in the early stages of their evolution offers an opportunity to test various cluster formation processes. We make use of CO, Herschel, and UKIDSS observations to study one such cloud, G148.24+00.41. Our results show the cloud to be of high mass ($\sim \, 1.1\times 10^5$ M⊙), low dust temperature (∼ 14.5 K), nearly circular (projected radius ∼ 26 pc), and gravitationally bound with a dense gas fraction of ∼18 per cent and a density profile with a power-law index of ∼−1.5. Comparing its properties with those of nearby molecular clouds, we find that G148.24+00.41 is comparable to the Orion-A molecular cloud in terms of mass, size, and dense gas fraction. From our analyses, we find that the central area of the cloud is actively forming protostars and is moderately fractal with a Q-value of ∼ 0.66. We also find evidence of global mass-segregation with a degree of mass-segregation (ΛMSR) ≈ 3.2. We discuss these results along with the structure and compactness of the cloud, the spatial and temporal distribution of embedded stellar population and their correlation with the cold dust distribution, in the context of high-mass cluster formation. We compare our results with models of star cluster formation, and conclude that the cloud has the potential to form a cluster in the mass range ∼ 2000–3000 M⊙ through dynamical hierarchical collapse and assembly of both gas and stars.

Star cluster formation in clouds with externally driven turbulence

ABSTRACT Star clusters are known to be formed in turbulent molecular clouds. How turbulence is driven in molecular clouds and what effect this has on star formation is still unclear. We compare a simulation setup with turbulent driving everywhere in a periodic box with a setup where turbulence is only driven around the outside of the box. We analyse the resulting gas distribution, kinematics, and the population of stars that are formed from the cloud. Both setups successfully produce a turbulent velocity field with a power-law structure function, the externally driven cloud has a more central, monolithic, clump, while the fully driven cloud has many smaller, more dispersed, clumps. The star formation follows the cloud morphology producing large clusters, with high star-forming efficiency in the externally driven simulations and sparse individual star formation with much lower star formation efficiency in the fully driven case. We conclude that the externally driven method, which resembles a Global Hierarchical Collapse (GHC) scenario, produces star clusters that more closely match with observations.

The Internal Line-of-Sight Kinematics of NGC 346: The Rotation of the Core Region

We present the stellar radial velocity analysis of the central of the young massive Small Magellanic Cloud star cluster NGC 346. Using VLT/MUSE integral field spectroscopy in combination with Hubble Space Telescope photometry, we extract 103 spectra of cluster member stars suited to measure accurate line-of-sight kinematics. The cluster member stars show two distinct velocity groups at and , relative to the systemic velocity of (165.5 ± 0.2) km s−1, and hint at a third group at . We show that there is neither a correlation between the velocity groups and the spatial location of the stars, nor their locus on optical color–magnitude diagrams, which makes the stellar velocity a key parameter to separate individual stellar components in such a young star cluster. Velocity group 2 shows clear rotation with Ω2 =(−0.4 ± 0.1) Myr−1, corresponding to (−4.9 ± 0.7) km s−1 at radial distance of 10 pc from the center, a possible remnant of the formation process of NGC 346 through the hierarchical collapse of the giant molecular cloud. The ionizing gas has lost any natal kinematic imprint and shows clear expansion, driven by far-ultraviolet fluxes and stellar winds of the numerous OB stars in the cluster center. The size of this expanding bubble and its expansion velocity of 7.9 km s−1 are in excellent agreement with the estimate that the latest star formation episode occurred about two million years ago.

Self-similar solutions for fuzzy dark matter

Fuzzy dark matter (FDM) models admit self-similar solutions which are very different from the standard cold dark matter (CDM) self-similar solutions and do not converge to the latter in the semiclassical limit. In contrast with the familiar CDM hierarchical collapse, they correspond to an inverse-hierarchy blowup. Constant-mass shells start in the nonlinear regime, at early times, with small radii and high densities, and expand to reach at late times the Hubble flow, up to small linear perturbations. Thus, larger masses become linear first. This blowup approximately follows the Hubble expansion, so that the central density contrast remains constant with time, although the width of the self-similar profile shrinks in comoving coordinates. As in a gravitational cooling process, matter is ejected from the central peaks through successive clumps. As in wave systems, the velocities of the geometrical structures and of the matter do not coincide, and matter slowly moves from one clump to the next, with intermittent velocity bursts at the transitions. These features are best observed using the density-velocity representation of the nonrelativistic scalar field, or the mass-shell trajectories, than with the Husimi phase-space distribution, where an analog of the Heisenberg uncertainty principle blurs the resolution in the position or velocity direction. These behaviors are due to the quantum pressure and the wavelike properties of the Schr\"odinger equation. Although the latter has been used as an alternative to $N$-body simulations for CDM, these self-similar solutions show that the semiclassical limit needs to be handled with care.

Redshifted methanol absorption tracing infall motions of high-mass star formation regions

Context. Gravitational collapse is one of the most important processes in high-mass star formation. Compared with the classic blue-skewed profiles, redshifted absorption against continuum emission is a more reliable method to detect inward motions within high-mass star formation regions. Aims. We aim to test if methanol transitions can be used to trace infall motions within high-mass star formation regions. Methods. Using the Effelsberg-100 m, IRAM-30 m, and APEX-12 m telescopes, we carried out observations of 37 and 16 methanol transitions towards two well-known collapsing dense clumps, W31C (G10.6−0.4) and W3(OH), to search for redshifted absorption features or inverse P-Cygni profiles. Results. Redshifted absorption is observed in 14 and 11 methanol transitions towards W31C and W3(OH), respectively. The infall velocities fitted from a simple two-layer model agree with previously reported values derived from other tracers, suggesting that redshifted methanol absorption is a reliable tracer of infall motions within high-mass star formation regions. Our observations indicate the presence of large-scale inward motions, and the mass infall rates are roughly estimated to be ≳10−3 M⊙ yr−1, which supports the global hierarchical collapse and clump-fed scenario. Conclusions. With the aid of bright continuum sources and the overcooling of methanol transitions leading to enhanced absorption, redshifted methanol absorption can trace infall motions within high-mass star formation regions hosting bright H II regions.

Unraveling the inner substructure of new candidate hub-filament system in the H ii region G25.4NW

ABSTRACT We present multiscale and multiwavelength data of the Galactic H ii region G25.4−0.14 (hereafter G25.4NW, distance ∼5.7 kpc). The SHARC-II 350-μm continuum map displays a hub-filament configuration containing five parsec-scale filaments and a central compact hub. Through the 5-GHz radio continuum map, four ionized clumps (i.e. Ia–Id) are identified towards the central hub, and are powered by massive OB-stars. The Herschel temperature map depicts the warm dust emission (i.e. T$_{\rm d}\, \sim$23–39 K) towards the hub. High-resolution Atacama Large Millimeter/submillimeter Array (ALMA) 1.3-mm continuum map (resolution ∼0.82 × 0.58 arcsec2) reveals three cores (c1–c3; mass ∼80–130 M⊙) towards the ionized clumps Ia, and another one (c4; mass ∼70 M⊙) towards the ionized clump Ib. A compact near-infrared (NIR) emission feature (extent ∼0.2 pc) is investigated towards the ionized clump Ia excited by an O8V-type star, and contains at least three embedded K-band stars. In the direction of the ionized clump Ia, the ALMA map also shows an elongated feature (extent ∼0.2 pc) hosting the cores c1–c3. All these findings together illustrate the existence of a small cluster of massive stars in the central hub. Considering the detection of the hub-filament morphology and the spatial locations of the mm cores, a global non-isotropic collapse (GNIC) scenario appears to be applicable in G25.4NW, which includes the basic ingredients of the global hierarchical collapse and clump-fed accretion models. Overall, the GNIC scenario explains the birth of massive stars in G25.4NW.

Simultaneous Evolution of the Virial Parameter and Star Formation Rate in Molecular Clumps Undergoing Global Hierarchical Collapse

Abstract We compare dense clumps and cores in a numerical simulation of molecular clouds (MCs) undergoing global hierarchical collapse (GHC) to observations in two MCs at different evolutionary stages, the Pipe and the G14.225 clouds, to test the ability of the GHC scenario to follow the early evolution of the energy budget and star formation activity of these structures. In the simulation, we select a region that contains cores of sizes and densities similar to the Pipe cores and find that it evolves through accretion, developing substructure similar to that of the G14.225 cloud after ∼1.6 Myr. Within this region, we follow the evolution of the Larson ratio , where is the velocity dispersion and R is the size; the virial parameter α; and the star formation activity of the cores/clumps. In the simulation, we find that as the region evolves, (i) its clumps have and α values first consistent with those of the Pipe substructures and later with those of G14.225; (ii) the individual cores first exhibit a decrease in α followed by an increase when star formation begins; (iii) collectively, the ensemble of cores/clumps reproduces the observed trend of lower α for higher-mass objects; and (iv) the star formation rate and star formation efficiency increase monotonically. We suggest that this evolution is due to the simultaneous loss of externally driven compressive kinetic energy and increase of the self-gravity-driven motions. We conclude that the GHC scenario provides a realistic description of the evolution of the energy budget of the clouds’ substructure at early times, which occurs simultaneously with an evolution of the star formation activity.

The effect of photoionizing feedback on the shaping of hierarchically-forming stellar clusters

ABSTRACT We use two hydrodynamical simulations (with and without photoionizing feedback) of the self-consistent evolution of molecular clouds (MCs) undergoing global hierarchical collapse (GHC), to study the effect of the feedback on the structural and kinematic properties of the gas and the stellar clusters formed in the clouds. During this early stage, the evolution of the two simulations is very similar (implying that the feedback from low-mass stars does not affect the cloud-scale evolution significantly) and the star-forming region accretes faster than it can convert gas into stars, causing the instantaneous measured star formation efficiency (SFE) to remain low even in the absence of significant feedback. Afterwards, the ionizing feedback first destroys the filamentary supply to star-forming hubs and ultimately removes the gas from it, thus first reducing the star formation (SF) and finally halting it. The ionizing feedback also affects the initial kinematics and spatial distribution of the forming stars because the gas being dispersed continues to form stars, which inherit its motion. In the non-feedback simulation, the groups remain highly compact and do not mix, while in the run with feedback, the gas dispersal causes each group to expand, and the cluster expansion thus consists of the combined expansion of the groups. Most secondary star-forming sites around the main hub are also present in the non-feedback run, implying a primordial rather than triggered nature. We do find one example of a peripheral star-forming site that appears only in the feedback run, thus having a triggered origin. However, this appears to be the exception rather than the rule, although this may be an artefact of our simplified radiative transfer scheme.

Nearby Quintuple Systems κ Tucanae and ξ Scorpii

Abstract Architecture and parameters of two wide, nearby hierarchical systems containing five solar-type stars each, κ Tuc and ξ Sco, are studied. Using Gaia astrometry and photometry, masses are determined from visual orbits and isochrones, and effective temperatures from spectra or colors. Both systems are ∼2 Gyr old. Their spatial motion corresponds to a young disk but does not match any known kinematic group. Internal proper motions relative to the center of mass and radial velocities show that wide ∼8 kau outer pairs are bound. No correlation between orbit orientations in the inner subsystems is observed. All masses except one are confined to the narrow range from 0.8 to 1.5 solar. Strongly correlated masses and wide orbits can be explained if those systems formed by fragmentation in relative isolation and their components accreted gas from a common source, as expected in a hierarchical collapse. Young moving groups could be formed in similar environments, and many of them contain high-order hierarchies.

The ALMA Survey of 70 μm Dark High-mass Clumps in Early Stages (ASHES). I. Pilot Survey: Clump Fragmentation

Abstract The ALMA Survey of 70 μm dark High-mass clumps in Early Stages (ASHES) is designed to systematically characterize the earliest stages and constrain theories of high-mass star formation. Twelve massive (>500 ), cold (≤15 K), 3.6–70 μm dark prestellar clump candidates, embedded in infrared dark clouds, were carefully selected in the pilot survey to be observed with the Atacama Large Millimeter/submillimeter Array (ALMA). We have mosaicked each clump (∼1 arcmin2) in continuum and line emission with the 12 m, 7 m, and Total Power (TP) arrays at 224 GHz (1.34 mm), resulting in ∼1.″2 resolution (∼4800 au, at the average source distance). As the first paper in the series, we concentrate on the continuum emission to reveal clump fragmentation. We detect 294 cores, from which 84 (29%) are categorized as protostellar based on outflow activity or “warm core” line emission. The remaining 210 (71%) are considered prestellar core candidates. The number of detected cores is independent of the mass sensitivity range of the observations and, on average, more massive clumps tend to form more cores. We find a large population of low-mass (<1 ) cores and no high-mass (>30 ) prestellar cores (maximum mass 11 ). From the prestellar core mass function, we derive a power-law index of 1.17 ± 0.10, which is slightly shallower than Salpeter. We used the minimum spanning tree (MST) technique to characterize the separation between cores and their spatial distribution, and to derive mass segregation ratios. While there is a range of core masses and separations detected in the sample, the mean separation and mass per clump are well explained by thermal Jeans fragmentation and are inconsistent with turbulent Jeans fragmentation. Core spatial distribution is well described by hierarchical subclustering rather than centrally peaked clustering. There is no conclusive evidence of mass segregation. We test several theoretical conditions and conclude that overall, competitive accretion and global hierarchical collapse scenarios are favored over the turbulent core accretion scenario.

Global hierarchical collapse in molecular clouds. Towards a comprehensive scenario

AbstractWe present a unified description of the scenario of global hierarchical collapse (GHC). GHC constitutes a flow regime of (non-homologous) collapses within collapses, in which all scales accrete from their parent structures, and small, dense regions begin to contract at later times, but on shorter time-scales than large, diffuse ones. The different time-scales allow for most of the clouds’ mass to be dispersed by the feedback from the first massive stars, maintaining the cloud-scale star formation rate low. Molecular clouds (MCs), clumps, and cores are not in equilibrium, but rather are either undergoing contraction or dispersal. The main features of GHC are as follows: (1) The gravitational contraction is initially very slow, and begins when the cloud still consists of mostly atomic gas. (2) Star-forming MCs are in an essentially pressureless regime, causing filamentary accretion flows from the cloud to the core scale to arise spontaneously. (3) Accreting objects have longer lifetimes than their own free-fall time, due to the continuous replenishment of material. (4) The clouds’ total mass and its molecular and dense mass fractions increase over time. (5) The clouds’ masses stop growing when feedback becomes important. (6) The first stars appear several megayears after global contraction began, and are of low mass; massive stars appear a few megayears later, in massive hubs. (7) The minimum fragment mass may well extend into the brown-dwarf regime. (8) Bondi–Hoyle–Lyttleton-like accretion occurs at both the protostellar and the core scales, accounting for an IMF with slope dN/dM ∝ M−2. (9) The extreme anisotropy of the filamentary network explains the difficulty in detecting large-scale infall signatures. (10) The balance between inertial and gravitationally driven motions in clumps evolves during the contraction, explaining the approach to apparent virial equilibrium, from supervirial states in low-column density clumps and from subvirial states in dense cores. (11) Prestellar cores adopt Bonnor–Ebert-like profiles, but are contracting ever since when they may appear to be unbound. (12) Stellar clusters develop radial age and mass segregation gradients. We also discuss the incompatibility between supersonic turbulence and the observed scalings in the molecular hierarchy. Since gravitationally formed filaments do not develop shocks at their axes, we suggest that a diagnostic for the GHC scenario should be the absence of strong shocks in them. Finally, we critically discuss some recent objections to the GHC mechanism.

Fast Recursive Reconnection and the Hall Effect: Hall-MHD Simulations

Abstract Magnetohydrodynamic (MHD) theory and simulations have shown that reconnection is triggered via a fast “ideal” tearing instability in current sheets whose inverse aspect ratio decreases to , with S as the Lundquist number defined by the half-length L of the current sheet (of a thickness of 2a). Ideal tearing, in 2D sheets, triggers a hierarchical collapse via stretching of X-points and recursive instability. At each step, the local Lundquist number decreases, until the subsequent sheet thickness either approaches kinetic scales or the Lundquist number becomes sufficiently small. Here we carry out a series of Hall-MHD simulations to show how the Hall effect modifies recursive reconnection once the ion inertial scale is approached. We show that as the ion inertial length becomes of the order of the inner, singular layer thickness at some step of the recursive collapse, reconnection transits from the plasmoid-dominant regime to an intermediate plasmoid+Hall regime and then to the Hall-dominant regime. The structure around the X-point, the reconnection rate, the dissipation property, and the power spectra are also modified significantly by the Hall effect.

Filamentary Accretion Flows in the Infrared Dark Cloud G14.225–0.506 Revealed by ALMA

Abstract Filaments are ubiquitous structures in molecular clouds and play an important role in the mass assembly of stars. We present results of dynamical stability analyses for filaments in the infrared dark cloud G14.225−0.506, where a delayed onset of massive star formation was reported in the two hubs at the convergence of multiple filaments of parsec length. Full-synthesis imaging is performed with the Atacama Large Millimeter/submillimeter Array to map the emission in two hub-filament systems with a spatial resolution of ∼0.034 pc. Kinematics are derived from a sophisticated spectral fitting algorithm that accounts for line blending, large optical depth, and multiple velocity components. We identify five velocity coherent filaments and derive their velocity gradients with principal component analysis. The mass accretion rates along the filaments are up to and are significant enough to affect the hub dynamics within one freefall time (∼105 yr). The filaments are in equilibrium with virial parameter α vir ∼ 1.2. We compare α vir measured in the filaments, filaments, 870 μm dense clumps, and 3 mm dense cores. The decreasing trend in α vir with decreasing spatial scales persists, suggesting an increasingly important role of gravity at small scales. Meanwhile, α vir also decreases with decreasing nonthermal motions. In combination with the absence of high-mass protostars and massive cores, our results are consistent with the global hierarchical collapse scenario.

Hierarchical collapse of regular islands via dissipation

In this work we investigate how regular islands localized in a mixed phase-space of generic area-preserving Hamiltonian systems are affected by a small amount of dissipation. Mainly we search for a universality (hierarchy) in the convergence of higher-order resonances and their periods when dissipation increases. One very simple scenario is already known: when subjected to small dissipation, stable periodic points become sinks attracting almost all the surrounding orbits, destroying all invariant curves which divide the phase-space in chaotic and regular domains. However, performing numerical experiments with the paradigmatic Chirikov–Taylor standard mapping we show that this presumably simple scenario can be rather complicated. The first, not trivial, scenario is what happens to chaotic trajectories, since they can be attracted by the sinks or by chaotic attractors, in cases when they exist. We show that this depends very much on how basins of attraction are formed as dissipation increases. In addition, we demonstrate that higher-order resonances are usually first affected by small dissipation when compared to lower-order resonances from the conservative case. Nevertheless, this is not a generic behaviour. We show that a local hierarchical collapse of resonances, as dissipation increases, is related to the area of the islands from the conservative case surrounding the periodic orbits. All observed resonance destructions occur via the bifurcation phenomena and are quantified here by determining the largest finite-time Lyapunov exponent.