arXiv subjects

Enrique Vázquez-Semadeni

Publications and source records attributed to Enrique Vázquez-Semadeni.

At least 37 records · Page 2Linked to original sources

arXiv2022

Filamentary collapse flow in molecular clouds

We present idealized numerical simulations of prestellar gravitational collapse of a moderate initial filamentary perturbation with an additional central ellipsoidal enhancement (a core) considering a uniform, and a stratified background, the latter representing flattened clouds. Both simulations maintain the filamentary structure during the collapse, developing a hierarchical accretion flow from the cloud to the plane; from there to the filament, and from the filament to the core. The flow changes direction smoothly at every step of the hierarchy, with no density divergence nor a shock developing at the filament's axis during the studied prestellar evolution. The flow drives accretion onto the central core and drains material from the filament, slowing down the growth of the latter. As a consequence, the ratio of the central density of the core to the filament density increases in time, diverging at the time of singularity formation in the core. The stratified simulation produces the best match for observed Plummer-like radial column density profiles of filaments, while the uniform simulation does not produce a flat central density profile. This result supports recent suggestions that MCs may be preferentially flattened structures. We examine the possibility that the filamentary flow might approach a quasi-stationary regime in which the radial accretion onto the filament is balanced by the longitudinal accretion onto the core. A simple argument suggests that such a stationary state may be an attractor for the system. Our simulations, do not attain this stationary stage, but appear to be approaching it during the prestellar stage.

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arXiv2021

Density profile evolution during prestellar core collapse: Collapse starts at the large scale

We study the gravitationally-dominated, accretion-driven evolution of a prestellar core. In our model, as the core's density increases, it remains immersed in a constant-density environment and so it accretes from this environment, increasing its mass and reducing its Jeans length. Assuming a power-law density profile $ρ\propto r^{-p}$, we compute the rate of change of the slope $p$, and show that the value $p=2$ is stationary, and furthermore, an attractor. The radial profile of the Jeans length scales as $r^{p/2}$, implying that, for $p<2$, there is a radius below which the region is smaller than its Jeans length, thus appearing gravitationally stable and in need of pressure confinement, while, in reality, it is part of a larger-scale collapse and is undergoing compression by the infalling material. In this region, the infall speed decreases towards the center, eventually becoming subsonic, thus appearing "coherent", without the need for turbulence dissipation. We present a compilation of observational determinations of density profiles in dense cores and show that the distribution of their slopes peaks at $p \sim 1.7$--1.9, supporting the notion that the profile steepens over time. Finally, we discuss the case of magnetic support in a core in which the field scales as $B \propto ρ^β$. For the expected value of $β= 2/3$, this implies that the mass to magnetic flux ratio also decreases towards the central parts of the cores, making them appear magnetically supported, while in reality they may be part of larger collapsing supercritical region. We conclude that local signatures of either thermal or magnetic support are not conclusive evidence of stability, that the gravitational instability of a region must be established at the large scales, and that the prestellar stage of collapse is dynamic rather than quasistatic.

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arXiv2020

Non-adiabatic turbulence driving during gravitational collapse

We investigate the generation of turbulence during the prestellar gravitational contraction of a turbulent spherical core. We define the ratio $g$ of the one-dimensional turbulent velocity dispersion, $σ_\mathrm{1D}$ to the gravitational velocity $v_g$, to then analytically estimate $g$ under the assumptions of a) equipartition or virial equilibrium between the gravitational ($E_g$) and turbulent kinetic ($E_\mathrm{turb}$) energies and b) stationarity of transfer from gravitational to turbulent energy (implying $E_\mathrm{turb}/E_g=$cst). In the equipartition and virial cases, we find $g=\sqrt{1/3}\approx0.58$ and $g=\sqrt{1/6}\approx0.41$, respectively; in the stationary case we find $g=\langle v_\mathrm{rad}\rangle L_d/(4π\sqrt{3}ηRv_g)$, where $η$ is an efficiency factor, $L_d$ is the energy injection scale of the turbulence, and $R$ is the core's radius. Next, we perform AMR simulations of the prestellar collapse of an isothermal, transonic turbulent core at two different resolutions, and a non-turbulent control simulation. We find that the turbulent simulations collapse at the same rate as the non-turbulent one, so that the turbulence generation does not significantly slow down the collapse. We also find that a) the simulations approach near balance between the rates of energy injection from the collapse and of turbulence dissipation; b) $g\approx0.395\pm0.035$, close to the "virial" value (turbulence is $\sim35-40\%$ of non-thermal linewidth); c) the injection scale is $L_d\lesssim R$, and d) the "turbulent pressure" $ρσ_\mathrm{1D}^2$ scales as $\simρ^{1.64}$, an apparently nearly-adiabatic scaling. We propose that this scaling and the nearly virial values of the turbulent velocity dispersion may be reconciled with the non-delayed collapse rate if the turbulence is dissipated as soon as it is generated.

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arXiv2020

Simultaneous evolution of the virial parameter and star formation rate in molecular clumps undergoing global hierarchical collapse

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 G14.225 cloud after $\sim 1.6$ Myr. Within this region, we follow the evolution of the Larson ratio $\mathcal{L} \equiv σ_{\rm v}/R^{1/2}$, where $σ_{\rm v}$ 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 $\mathcal{L}$ 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.

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arXiv2020

From diffuse gas to dense molecular cloud cores

Molecular clouds are a fundamental ingredient of galaxies: they are the channels that transform the diffuse gas into stars. The detailed process of how they do it is not completely understood. We review the current knowledge of molecular clouds and their substructure from scales $\sim~$1~kpc down to the filament and core scale. We first review the mechanisms of cloud formation from the warm diffuse interstellar medium down to the cold and dense molecular clouds, the process of molecule formation and the role of the thermal and gravitational instabilities. We also discuss the main physical mechanisms through which clouds gather their mass, and note that all of them may have a role at various stages of the process. In order to understand the dynamics of clouds we then give a critical review of the widely used virial theorem, and its relation to the measurable properties of molecular clouds. Since these properties are the tools we have for understanding the dynamical state of clouds, we critically analyse them. We finally discuss the ubiquitous filamentary structure of molecular clouds and its connection to prestellar cores and star formation.

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arXiv2020

Zooming in on Individual Star Formation: Low- and High-mass Stars

Star formation is a multi-scale, multi-physics problem ranging from the size scale of molecular clouds ($\sim$10s pc) down to the size scales of dense prestellar cores ($\sim$0.1 pc) that are the birth sites of stars. Several physical processes like turbulence, magnetic fields and stellar feedback, such as radiation pressure and outflows, are more or less important for different stellar masses and size scales. During the last decade a variety of technological and computing advances have transformed our understanding of star formation through the use of multi-wavelength observations, large scale observational surveys, and multi-physics multi-dimensional numerical simulations. Additionally, the use of synthetic observations of simulations have provided a useful tool to interpret observational data and evaluate the importance of various physical processes on different scales in star formation. Here, we review these recent advancements in both high- ($M \gtrsim 8 \, M_{\rm \odot}$) and low-mass star formation.

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arXiv2019

Global Hierarchical Collapse In Molecular Clouds. Towards a Comprehensive Scenario

We present a unified description of the scenario of Global Hierarchical Collapse and fragmentation (GHC) in molecular clouds (MCs), owing to the continuous decrease of the average Jeans mass in the contracting cloud. GHC constitutes a regime of collapses within collapses, in which small-scale collapses begin at later times, but occur on shorter timescales than large-scale ones. The difference in timescales allows for most of the clouds' mass to be dispersed by feedback from the first massive stars, maintaining the global star formation rate low. All scales accrete from their parent structures. The main features of GHC are: star-forming MCs are in an essentially pressureless regime, which produces filaments that accrete onto clumps and cores ("hubs"). The filaments constitute the collapse flow from cloud to hub scales and may approach a quasi-stationary state; the molecular and dense mass fractions of the clouds increase over time; the first (low-mass) stars appear several Myr after global contraction began; more massive stars appear after a few Myr in massive hubs resulting from the collapse of larger scales; the minimum fragment mass may extend well into the brown-dwarf regime; Bondi-Hoyle-Lyttleton accretion occurs at the protostellar and core scales, accounting for a near-Salpeter IMF; the extreme anisotropy of the filamentary network explains the difficulty in detecting large-scale infall signatures; the balance between inertial and gravitationally-driven motions in clumps evolves during the contraction; prestellar cores adopt Bonnor-Ebert-like profiles, but are contracting ever since early times when they may appear to be unbound and to require pressure confinement; stellar clusters develop radial age and mass segregation gradients. Finally, we discuss the incompatibility between supersonic turbulence and the observed scalings in the molecular hierarchy.

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arXiv2019

Molecular Cloud Evolution VI. Measuring cloud ages

In previous contributions, we have presented an analytical model describing the evolution of molecular clouds (MCs) undergoing hierarchical gravitational contraction. The cloud's evolution is characterized by an initial increase in its mass, density, and star formation rate (SFR) and efficiency (SFE) as it contracts, followed by a decrease of these quantities as newly formed massive stars begin to disrupt the cloud. The main parameter of the model is the maximum mass reached by the cloud during its evolution. Thus, specifying the instantaneous mass and some other variable completely determines the cloud's evolutionary stage. We apply the model to interpret the observed scatter in SFEs of the cloud sample compiled by Lada et al.\ as an evolutionary effect so that, although clouds such as California and Orion A have similar masses, they are in very different evolutionary stages, causing their very different observed SFRs and SFEs. The model predicts that the California cloud will eventually reach a significantly larger total mass than the Orion A cloud. Next, we apply the model to derive estimated ages of the clouds since the time when approximately 25\% of their mass had become molecular. We find ages from $\sim 1.5$ to 27 Myr, with the most inactive clouds being the youngest. Further predictions of the model are that clouds with very low SFEs should have massive atomic envelopes constituting the majority of their gravitational mass, and that low-mass clouds ($M \sim 10^3$-$10^4 \, M_\odot$) end their lives with a mini-burst of star formation, reaching SFRs $\sim 300$-$500\, M_\odot$ Myr$^{-1}$. By this time, they have contracted to become compact ($\sim 1$ pc) massive star-forming clumps, in general embedded within larger GMCs.

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arXiv2019

Structure and Expansion Law of HII Regions in structured Molecular Clouds

We present radiation-magnetohydrodynamic simulations aimed at studying evolutionary properties of H\,{\normalsize II} regions in turbulent, magnetised, and collapsing molecular clouds formed by converging flows in the warm neutral medium. We focus on the structure, dynamics and expansion laws of these regions. Once a massive star forms in our highly structured clouds, its ionising radiation eventually stops the accretion (through filaments) toward the massive star-forming regions. The new over-pressured H\,{\normalsize II} regions push away the dense gas, thus disrupting the more massive collapse centres. Also, because of the complex density structure in the cloud, the H\,{\normalsize II} regions expand in a hybrid manner: they virtually do not expand toward the densest regions (cores), while they expand according to the classical analytical result towards the rest of the cloud, and in an accelerated way, as a blister region, towards the diffuse medium. Thus, the ionised regions grow anisotropically, and the ionising stars generally appear off-centre of the regions. Finally, we find that the hypotheses assumed in standard H\,{\normalsize II}-region expansion models (fully embedded region, blister-type, or expansion in a density gradient) apply simultaneously in different parts of our simulated H\,{\normalsize II} regions, producing a net expansion law ($R \propto t^α$, with $α$ in the range of 0.93-1.47 and a mean value of $1.2 \pm 0.17$) that differs from any of those of the standard models.

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arXiv2018

Challenges and Techniques for Simulating Line Emission

Modeling emission lines from the millimeter to the UV and producing synthetic spectra is crucial for a good understanding of observations, yet it is an art filled with hazards. This is the proceedings of "Walking the Line", a 3-day conference held in 2018 that brought together scientists working on different aspects of emission line simulations, in order to share knowledge and discuss the methodology. Emission lines across the spectrum from the millimeter to the UV were discussed, with most of the focus on the interstellar medium, but also some topics on the circumgalactic medium. The most important quality of a useful model is a good synergy with observations and experiments. Challenges in simulating line emission are identified, some of which are already being worked upon, and others that must be addressed in the future for models to agree with observations. Recent advances in several areas aiming at achieving that synergy are summarized here, from micro-physical to galactic and circum-galactic scale.

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arXiv2017

Gravity or turbulence? IV. Collapsing cores in out-of-virial disguise

We study the dynamical state of cores by using a simple analytical model, a sample of observational massive cores, and numerical simulations of collapsing massive cores. From the model, we find that, if cores are formed from turbulent compressions, they evolve from small to large column densities, increasing their velocity dispersion as they collapse, while they tend to equipartition between gravity and kinetic energy. From the observational sample, we find that: (a) cores with substantially different column densities in the sample do not follow a Larson-like linewidth-size relation. Instead, cores with higher column densities tend to be located in the upper-left corner of the Larson velocity dispersion-size diagram, a result predicted previously (Ballesteros-Paredes et al. 2011a). (b) The data exhibit cores with overvirial values. Finally, in the simulations we reproduce the behavior depicted by the model and observational sample: cores evolve towards larger velocity dispersions and smaller sizes as they collapse and increase their column density. However, collapsing cores appear to approach overvirial states within a free-fall time. The cause of this apparent excess of kinetic energy is an underestimation of the gravitational energy, due to the assumption that the gravitational energy is given by the energy of an isolated sphere of constant column density. This excess disappears when the gravitational energy is correctly calculated from the actual spatial mass distribution, where inhomogeneities, as well as the potential due to the mass outside of the core, also contribute to the gravitational energy. We conclude that the observed energy budget of cores in surveys is consistent with their non-thermal motions being driven by their self-gravity and in a hierarchical and chaotic collapse.

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arXiv2016

Energy budget of forming clumps in numerical simulations of collapsing clouds

We analyze the physical properties and energy balance of density enhancements in two SPH simulations of the formation, evolution, and collapse of giant molecular clouds. In the simulations, no feedback is included, so all motions are due either to the initial, decaying turbulence, or to gravitational contraction. We define clumps as connected regions above a series of density thresholds. The resultingfull set of clumps follows the generalized energy-equipartition relation $σ_{v}/R^{1/2} \propto Σ^{1/2}$, where $σ_{v}$ is the velocity dispersion, $R$ is the "radius", and $Σ$ is the column density. We interpret this as a natural consequence of gravitational contraction at all scales, rather than virial equilibrium. Nevertheless, clumps with low $Σ$ tend to show a large scatter around equipartition. In more than half of the cases, this scatter is dominated by external turbulent compressions that {\it assemble} the clumps, rather than by small-scale random motions that would disperse them. The other half does actually disperse. Moreover, clump sub-samples selected by means of different criteria exhibit different scalings. Sub-samples with narrow $Σ$ ranges follow Larson-like relations, although characterized by their respective value of $Σ$. Finally, we find that: i) clumps lying in filaments tend to appear sub-virial; ii) high-density cores ($n \ge 10^5$ cm$^3$) that exhibit moderate kinetic energy excesses often contain sink ("stellar") particles, and the excess disappears when the stellar mass is taken into account in the energy balance; iii) cores with kinetic energy excess but no stellar particles are truly in a state of dispersal.

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arXiv2016

Supernova Feedback in Molecular Clouds: Global Evolution and Dynamics

We use magnetohydrodynamical simulations of converging warm neutral medium flows to analyse the formation and global evolution of magnetised and turbulent molecular clouds subject to supernova feedback from massive stars. We show that supernova feedback alone fails to disrupt entire, gravitationally bound, molecular clouds, but is able to disperse small--sized (~10 pc) regions on timescales of less than 1 Myr. Efficient radiative cooling of the supernova remnant as well as strong compression of the surrounding gas result in non-persistent energy and momentum input from the supernovae. However, if the time between subsequent supernovae is short and they are clustered, large hot bubbles form that disperse larger regions of the parental cloud. On longer timescales, supernova feedback increases the amount of gas with moderate temperatures (T~300-3000 K). Despite its inability to disrupt molecular clouds, supernova feedback leaves a strong imprint on the star formation process. We find an overall reduction of the star formation efficiency by a factor of 2 and of the star formation rate by roughly factors of 2-4.

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arXiv2015

Hierarchical gravitational fragmentation. I. Collapsing cores within collapsing clouds

We investigate the Hierarchical Gravitational Fragmentation scenario through numerical simulations of the prestellar stages of the collapse of a marginally gravitationally unstable isothermal sphere immersed in a strongly gravitationally unstable, uniform background medium. The core developes a Bonnor-Ebert (BE)-like density profile, while at the time of singularity (the protostar) formation the envelope approaches a singular-isothermal-sphere (SIS)-like $r^-2$ density profile. However, these structures are never hydrostatic. In this case, the central flat region is characterized by an infall speed, while the envelope is characterized by a uniform speed. This implies that the hydrostatic SIS initial condition leading to Shu's classical inside-out solution is not expected to occur, and therefore neither should the inside-out solution. Instead, the solution collapses from the outside-in, naturally explaining the observation of extended infall velocities. The core, defined by the radius at which it merges with the background, has a time-variable mass, and evolves along the locus of the ensemble of observed prestellar cores in a plot of $M/M_{BE}$ vs. $M$, where $M$ is the core's mass and $M_{BE}$ is the critical Bonnor-Ebert mass, spanning the range from the "stable" to the "unstable" regimes, even though it is collapsing at all times. We conclude that the presence of an unstable background allows a core to evolve dynamically from the time when it first appears, even when it resembles a pressure-confined, stable BE-sphere. The core can be thought of as a ram-pressure confined BE-sphere, with an increasing mass due to the accretion from the unstable background.

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arXiv2015

Inverse Hubble Flows in Molecular Clouds

Motivated by recent numerical simulations of molecular cloud (MC)evolution, in which the clouds engage in global gravitational contraction, and local collapse events culminate significantly earlier than the global collapse, we investigate the growth of density perturbations embedded in a collapsing background, to which we refer as an Inverse Hubble Flow (IHF). We use the standard procedure for the growth of perturbations in a universe that first expands (the usual Hubble Flow) and then recollapses (the IHF). We find that linear density perturbations immersed in an IHF grow faster than perturbations evolving in a static background (the standard Jeans analysis). A fundamental distinction between the two regimes is that, in the Jeans case, the time $τ_\mathrm{nl}$ for a density fluctuation to become nonlinear increases without limit as its initial value approaches zero, while in the IHF case $τ_\mathrm{nl} \le τ_\mathrm{ff}$ always, where $τ_\mathrm{ff}$ is the free-fall time of the background density. We suggest that this effect, although moderate, implies that small-scale density fluctuations embedded in globally-collapsing clouds must collapse earlier than their parent cloud, regardless of whether the initial amplitude of the fluctuations is moderate or strongly nonlinear, thus allowing the classical mechanism of Hoyle fragmentation to operate in multi-Jeans-mass MCs. More fundamentally, our results show that, contrary to the standard paradigm that fluctuations of all scales grow at the same rate in the linear regime, the hierarchical nesting of the fluctuations of different scales does affect their growth even in the linear stage.

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arXiv2014

An Evolutionary Model for Collapsing Molecular Clouds and Their Star Formation Activity. II. Mass Dependence of the Star Formation Rate

We discuss the dependence of various properties of the star formation rate (SFR) and efficiency (SFE) in molecular clouds (MCs) on the maximum mass reached by the clouds, based on a previously-published model for MC and SFR evolution in which the clouds were assumed to be undergoing global collapse, and the SFR was controlled by ioniztion feedback. Because the model neglects various other processes, the results presented are upper limits. We find that clouds with $\Mmax \lesssim 10^4 \Msun$ end their lives with a mini-burst, at which the SFR reaches a peak of $\sim 10^4 ~\Msun \Myr^{-1}$, although its time average is only $\SFRavg \sim \hbox{ a few} \times 10^2 \Msun \Myr^{-1}$. The corresponding efficiencies are $\SFEmax \lesssim $60\%$ and $\SFEavg \lesssim $1\%$. For more massive clouds ($\Mmax \gtrsim 10^5 ~ \Msun$), the SFR first increases and then remains roughly constant for $\sim 10^7$ yr, because the clouds are influenced by the stellar feedback since earlier in their evolution. We find that $\SFRavg$ and $\SFEavg$ are well represented by the fits $\SFRavg \approx 100 (1+\Mmax/2 \times 10^5 ~ \Msun)^{2} ~ \Msun \Myr^{-1}$ and $\SFEavg \approx 0.024 (\Mmax/10^5 ~ \Msun)^{0.28}$, respectively. The massive model clouds follow the SFR-dense gas mass relation obtained by Gao \& Solomon for infrared galaxies, extrapolated down to MCs scales. Low-mass clouds fall above this relation, in agreement with recent observations. An integration of the model-predicted $\SFRavg$ over a Galactic GMC mass spectrum yields a realistic value for the Galactic SFR. Our results reinforce the suggestion that star-forming GMCs may be in global collapse, and still have low net SFRs and SFEs, due to the evaporation of most of the cloud material by the feedback from massive stars.

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arXiv2014

The Milky Way as a Star Formation Engine

The cycling of material from the interstellar medium (ISM) into stars and the return of stellar ejecta into the ISM is the engine that drives the "galactic ecology" in normal spirals, a cornerstone in the formation and evolution of galaxies through cosmic time. Major observational and theoretical challenges need to be addressed in determining the processes responsible for converting the low-density ISM into dense molecular clouds, forming dense filaments and clumps, fragmenting them into stars, OB associations and bound clusters, and characterizing the feedback that limits the rate and efficiency of star formation. This formidable task can be now effectively attacked thanks to the combination of new global-scale surveys of the Milky Way Galactic Plane from infrared to radio wavelengths, offering the possibility of bridging the gap between local and extragalactic star formation studies. The Herschel, Spitzer and WISE mid to far infrared continuum surveys, complemented by analogue surveys from ground-based facilities in the millimetre and radio wavelengths, enables us to measure the Galactic distribution and physical properties of dust on all scales and in all components of the ISM from diffuse clouds to filamentary complexes and tens of thousands of dense clumps. A complementary suite of spectroscopic surveys in various atomic and molecular tracers is providing the chemical fingerprinting of dense clumps and filaments, as well as essential kinematic information to derive distances and thus transform panoramic data into a 3D representation. The latest results emerging from these Galaxy-scale surveys are reviewed. New insights into cloud formation and evolution, filaments and their relationship to channeling gas onto gravitationally-bound clumps, the properties of these clumps, density thresholds for gravitational collapse, and star and cluster formation rates are discussed.

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arXiv2014

Testing assumptions and predictions of star-formation theories

(Abridged). We present numerical simulations of isothermal, MHD, supersonic turbulence, designed to test various hypotheses frequently assumed in star formation(SF) theories. We consider three simulations, each with a different combination of physical size, rms sonic Mach number, and Jeans parameter, but chosen as to give the same value of the virial parameter and to conform with Larson's scaling relations. As in the non-magnetic case: we find no simultaneously subsonic and super-Jeans structures in our MHD simulations. We find that the fraction of small-scale super-Jeans structures increases when self gravity is turned on, and that the production of gravitationally unstable dense cores by turbulence alone is very low. This implies that self-gravity is in general necessary not only to induce the collapse of Jeans-unstable cores, but also to form them. We find that denser regions tend to have more negative values of the velocity divergence, implying a net inwards flow towards the regions' centers. We compare the results from our simulations with the predictions from the recent SF theories by Krumholz & McKee, Padoan & Nordlund, and Hennebelle & Chabrier, using the expressions recently provided by Federrath & Klessen. We find that none of these theories reproduces the dependence of the SFEff with Ms observed in our simulations in the MHD case. The SFEff predicted by the theories ranges between half and one order of magnitude larger than what we observe in the simulations in both the HD and the MHD cases. We conclude that the type of flow used in simulations like the ones presented here and assumed in recent SF theories, may not correctly represent the flow within actual clouds, and that theories that assume it does may be missing a fundamental aspect of the flow. We suggest that a more realistic regime may be that of hierarchical gravitational collapse.

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