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We explore the Schmidt-Kennicutt (SK) relations and the star formation efficiency per free-fall time ($\eff$), mirroring observational studies, in numerical simulations of filamentary molecular clouds undergoing gravitational contraction. We find that {\it a)} collapsing clouds accurately replicate the observed SK relations for galactic clouds and {\it b)} the so-called efficiency per free-fall time ($\eff$) is small and constant in space and in time, with values similar to those found in local clouds. This constancy is a consequence of the similar radial scaling of the free-fall time and the internal mass in density structures with spherically-averaged density profiles near $r^{-2}$. We additionally show that {\it c)} the star formation rate (SFR) increases rapidly in time; {\it d)} the low values of $\eff$ are due to the different time periods over which $\tauff$ and $\tausf$ are evaluated, together with the fast increasing SFR, and {\it e)} the fact that star clusters are significantly denser than the gas clumps from which they form is a natural consequence of the fast increasing SFR, the continuous replenishment of the star-forming gas by the accretion flow, and the near $r^{-2}$ density profile generated by the collapse Finally, we argue that the interpretation of $\eff$ as an efficiency is problematic because its maximum value is not bounded by unity, and because the total gas mass in the clouds is not fixed, but rather depends on the environment where clouds are embedded. In summary, our results show that the SK relation, the typical observed values of $\eff$, and the mass density of clusters arise as a natural consequence of gravitational contraction., Comment: Submitted to MNRAS

We explain the large variety of star formation laws in terms of one single, simple law that can be inferred from the definition of the star formation rate and basic algebra. The resulting equation, $\SFR = \eff\ \Mcollapsing/\tauff$, although it has been presented elsewhere, is interpreted in terms of clouds undergoing collapse { rather than being turbulence-supported, an idea that different groups have pursued this century}. Under such assumption, one can explain the constancy of $\eff$, the different intra-cloud correlations observed in Milky Way's molecular clouds, as well as the resolved and unresolved extragalactic relationships between SFR and a measurement of the mass in CO, HCN, and CO+HI. We also explain why the slope of the correlation changes when the orbital time $\tauorb$ is considered instead of the free-fall time, and why estimations of the free-fall time from extragalactic observations skew the correlation, providing a false sublinear correlation. We furthermore show that the apparent nearly linear correlation between the star formation rate and the dynamical equilibrium pressure in the midplane of the galaxies, $\PDE$, is just a consequence of $\PDE$ values being dominated by the variation of the column density of molecular gas. All in all, we argue that the star formation law is driven by the collapse of cold, dense gas, which happens to be primarily molecular in the present Universe, and that the role of stellar feedback is just to shut down the star formation process, not to shape the star formation law., Comment: 17 pages, one figure. Accepted by MNRAS

The direct comparison between hydrodynamical simulations and observations is needed to improve the physics included in the former and test biases in the latter. Post-processing radiative transfer and synthetic observations are now the standard way to do this. We report on the first application of the \texttt{SKIRT} radiative transfer code to simulations of a star-forming cloud. The synthetic observations are then analyzed following traditional observational workflows. We find that in the early stages of the simulation, stellar radiation is inefficient in heating dust to the temperatures observed in Galactic clouds, thus the addition of an interstellar radiation field is necessary. The spectral energy distribution of the cloud settles rather quickly after $\sim3$ Myr of evolution from the onset of star formation, but its morphology continues to evolve for $\sim8$ Myr due to the expansion of \textsc{Hii} regions and the respective creation of cavities, filaments, and ridges. Modeling synthetic \textit{Herschel} fluxes with 1- or 2-component modified black bodies underestimates total dust masses by a factor of $\sim2$. Spatially-resolved fitting recovers up to about $70\%$ of the intrinsic value. This ``missing mass'' is located in a very cold dust component with temperatures below $10$ K, which does not contribute appreciably to the far-infrared flux. This effect could bias real observations if such dust exists in large amounts. Finally, we tested observational calibrations of the SFR based on infrared fluxes and concluded that they are in agreement when compared to the intrinsic SFR of the simulation averaged over $\sim100$ Myr.

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, $\alpha$, with mass $M$, and of the Larson ratio, ${\cal{L}}\equiv\sigma_v/R^{1/2}$, with column density $\Sigma$, where $\sigma_v$ is the 1D velocity dispersion, and $R$ is an effective radius. We also investigate these scalings for the corresponding magnetic parameters $\alpha_m$ and ${\cal{L}}_m$. Finally, we compare our numerical results with an observational sample of massive clumps. We find that: 1) $\alpha_m$ and ${\cal{L}}_m$ follow similar 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 $\mu$ decreases inwards in a centrally-peaked cloud under ideal MHD. 3)~The scatter in the $\alpha$-$M$ and ${\cal{L}}$-$\Sigma$ 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 $\alpha$-$M$ and ${\cal{L}}$-$\Sigma$ 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., Comment: Accepted for publication in MNRAS

Observational and theoretical evidence suggests that a substantial population of molecular clouds (MCs) appear to be unbound, dominated by turbulent motions. However, these estimations are made typically via the so-called viral parameter $\alpha_{\rm vir}^{\rm class}$, which is an observational proxy to the virial ratio between the kinetic and the gravitational energy. This parameter intrinsically assumes that MCs are isolated, spherical, and with constant density. However, MCs are embedded in their parent galaxy and thus are subject to compressive and disruptive tidal forces from their galaxy, exhibit irregular shapes, and show substantial substructure. We, therefore, compare the typical estimations of $\alpha_{\rm vir}^{\rm class}$ to a more precise definition of the virial parameter, $\alpha_{\rm vir}^{\rm full}$, which accounts not only for the self-gravity (as $\alpha_{\rm vir}^{\rm class}$), but also for the tidal stresses, and thus, it can take negative (self-gravity) and positive (tides) values. While we recover the classical result that most of the clouds appear to be unbound, having $\alpha_{\rm vir}^{\rm class} > 2$, we show that, with the more detailed definition considering the full gravitational energy, (i) 50\%\ of the total population is gravitationally bound, however, (ii) another 20\%\ is gravitationally dominated, but with tides tearing them apart; (iii) the source of those tides does not come from the galactic structure (bulge, halo, spiral arms), but from the molecular cloud complexes in which clouds reside, and probably (iv) from massive young stellar complexes, if they were present. (v) Finally, our results also suggest that, interstellar turbulence can have, at least partially, a gravitational origin., Comment: 17 pages, 11 figures, submitted to MNRAS

We investigate the effect of numerical magnetic diffusion in magnetohydrodynamic simulations of magnetically supported molecular clouds. To this end, we have performed numerical studies on adaptive mesh isothermal simulations of marginally sub-critical molecular clouds. We find that simulations with low and intermediate resolutions collapse, contrary to what is theoretically expected. However, the simulation with the highest numerical resolution oscillates around an equilibrium state without collapsing. In order to quantify the numerical diffusion of the magnetic field, we ran a second suit of current-sheet simulations in which the numerical magnetic diffusion coefficient can be directly measured, and computed the corresponding diffusion times at various numerical resolutions. On this basis, we propose a criterion for the resolution of magnetic fields in MHD simulations based on requiring that the diffusion time to be larger than the characteristic timescale of the physical process responsible for the dynamic evolution of the structure., Comment: 7 pages, 5 figures, 2 tables. Accepted to MNRAS

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 $\mathcal{L} \equiv \sigma_{\rm v}/R^{1/2}$, where $\sigma_{\rm v}$ is the velocity dispersion and $R$ is the size, the virial parameter $\alpha$, 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 $\alpha$ 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 $\alpha$ followed by an increase when star formation begins; $iii)$ collectively, the ensemble of cores/clumps reproduces the observed trend of lower $\alpha$ 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., Comment: Accepted for publication in ApJ

Different studies have reported a power-law mass-size relation $M \propto R^q$ for ensembles of molecular clouds. In the case of nearby clouds, the index of the power-law $q$ is close to 2. However, for clouds spread all over the Galaxy, indexes larger than 2 are reported. We show that indexes larger than 2 could be the result of line-of-sight superposition of emission that does not belong to the cloud itself. We found that a random factor of gas contamination, between 0.001\%\ and 10\%\ of the line-of-sight, allows to reproduce the mass-size relation with $q \sim 2.2-2.3$ observed in Galactic CO surveys. Furthermore, for dense cores within a single cloud, or molecular clouds within a single galaxy, we argue that, even in these cases, there is observational and theoretical evidence that some degree of superposition may be occurring. However, additional effects may be present in each case, and are briefly discussed. We also argue that defining the fractal dimension of clouds via the mass-size relation is not adequate, since the mass is not {necessarily} a proxy to the area, and the size reported in $M-R$ relations is typically obtained from the square root of the area, rather than from an estimation of the size independent from the area. Finally, we argue that the statistical analysis of finding clouds satisfying the Larson's relations does not mean that each individual cloud is in virial equilibrium., Comment: 8 pages, 4 figures. Accepted for publication in MNRAS

Recent analyses of Gaia data have provided direct evidence that most young stellar clusters are in a state of expansion, with velocities of the order of ~0.5 km/s. Traditionally, expanding young clusters have been pictured as entities that became unbound due to the lack of gravitational binding once the gas from the parental cloud that formed the cluster has been expelled by the stellar radiation of the massive stars in the cluster. In the present contribution, we used radiation-magnetohydrodynamic numerical simulations of molecular cloud formation and evolution to understand how stellar clusters form and disperse. We found that the ionising feedback from the newborn massive stars expels the gas from the collapse centre, flipping-up the gravitational potential as a consequence of the mass removal from the inside-out. Since neither the parental clouds nor the formed shells are distributed symmetrically around the HII region, net forces pulling out the stars are present, accelerating them towards the edges of the cavity. We call this mechanism ``gravitational feedback", in which the gravity from the expelled gas appears to be the crucial mechanism producing unbound clusters that expand away from their formation centre in an accelerated way in young stellar clusters. This mechanism naturally explains the "Hubble flow-like" expansion observed in several young clusters., Comment: Accepted for publication in MNRAS

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., Comment: Accepted by MNRAS. Sec. 6. 4 has been modified to provide an evolutionary expression for the viral parameter, including turbulent and self-gravitating components. New table comparing various SF scenarios. Abstract abridged

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 ($R \propto t^\alpha$, with $\alpha$ 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., Comment: Accepted for publication in MNRAS

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., Comment: 11 pages, 3 figures, and 2 tables. Published in MNRAS. Journal-ref: MNRAS, 2018, Volume 479, p.3254-3263. ADS link: https://ui.adsabs.harvard.edu/abs/2018MNRAS.479.3254V/abstract

We explore the structure of magnetic field lines in and around filaments in simulations of molecular clouds undergoing global, multi-scale gravitational collapse. In these simulations, filaments are not in a static equilibrium, but are long-lived flow structures that accrete gas from their environment and direct it toward clumps embedded in the filament or at the nodes at the conjunction with other filaments. In this context, the magnetic field is dragged by the collapsing gas, so its structure must reflect the flow that generates the filament. Around the filament, the gas is accreted onto it, and the magnetic lines must then be perpendicular to the filament. As the gas density increases, the gas flow changes direction, becoming almost parallel to the filament, and magnetic lines also tend to align with it. At the spine of the filament, however, magnetic lines become perpendicular again since they must connect to lines on the opposite side of the filament, resulting in "U"-shaped magnetic structures, which tend to be stretched by the longitudinal flow along the filament. Magnetic diffusive processes, however, allow the gas to continue to flow. Assuming a stationary state in which the ram pressure of the flow balances the magnetic tension, the curvature of the field lines is determined by the diffusion rate. We derive an expression relating the curvature of the field lines to the diffusive coefficient, which may be used to observationally determine the nature of the diffusive process., Comment: Submitted to MNRAS

We analyse the morphology and kinematics of dense filamentary structures produced in a numerical simulation of a star--forming cloud of $1.4 \times 10^4 \, \Msun$ evolving under their own self--gravity in a magnetized medium. This study is motivated by recent observations of velocity--coherent substructures ("fibres") in star-forming filaments. We find such "fibres" ubiquitously in our simulated filament. We found that a fibre in one projection is not necessarily a fibre in another projection, and thus, caution should be taken into account when considering them as real objects. We found that only the densest parts of the filament ($\sim$30\% of the densest volume, which contains $\sim$70\% of the mass) belong to fibres in 2 projections. Moreover, it is quite common that they are formed by separated density enhancements superposed along the line of sight. Observations of fibres can yield insight into the level of turbulent substructure driven by gravity, but care should be taken in interpreting the results given the problem of line of sight superposition. We also studied the morphology and kinematics of the 3D skeleton (spine), finding that subfilaments accrete structured material mainly along the magnetic field lines, which are preferentially perpendicular to the skeleton. The magnetic field is at the same time dragged by the velocity field due to the gravitational collapse., Comment: Accepted for publication in MNRAS (a minor correction of the second author affiliations)

We present Submillimeter Array (SMA) observations at 345 GHz towards the intermediate/high-mass cluster-forming region NGC 6334 V. From the dust emission we spatially resolve three dense condensations, the brightest one presenting the typical chemistry of a hot core. The magnetic field (derived from the dust polarized emission) shows a bimodal converging pattern towards the hot core. The molecular emission traces two filamentary structures at two different velocities, separated by 2 km/s, converging to the hot core and following the magnetic field distribution. We compare the velocity field and the magnetic field derived from the SMA observations with MHD simulations of star-forming regions dominated by gravity. This comparison allows us to show how the gas falls in from the larger-scale extended dense core (~0.1 pc) of NGC 6334 V towards the higher-density hot core region (~0.02 pc) through two distinctive converging flows dragging the magnetic field, whose strength seems to have been overcome by gravity., Comment: 13 pages, 11 figures, accepted for publication in ApJ

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., Comment: 16 pages, 12 figures (2 in appendix), revised version, submitted to MNRAS

We discuss the mechanism of cluster formation in hierarchically collapsing molecular clouds. Recent evidence, both observational and numerical, suggests that molecular clouds (MCs) may be undergoing global, hierarchical gravitational collapse. The "hierarchical" regime consists of small-scale collapses within larger-scale ones. The latter implies that the star formation rate increases systematically during the early stages of evolution, and occurs via filamentary flows onto "hubs" of higher density, mass, and velocity dispersion, and culminates a few Myr after than the small-scale collapses have started to form stars. In turn, the small-scale collapses occur in clumps embedded in the filaments, and are themselves falling into the larger potential well of the still-ongoing large-scale collapse. The stars formed in the early, small-scale collapses share the infall motion of their parent clumps towards the larger potential trough, so that the filaments feed both gaseous and stellar material to the hubs. This leads to the presence of older stars in a region where new protostars are still forming, to a scale-free or fractal structure of the clusters, in which each unit is composed of smaller-scale ones, and to the eventual merging of the subunits, explaining the observed structural features of open clusters., Comment: 6 pages, 7 figures, to appear in the proceedings of IAU Symposium 316: Formation, Evolution, and Survival of Massive Star Clusters

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., Comment: Accepted for publication in the ApJ

We present an idealized, semi-empirical model for the evolution of gravitationally contracting molecular clouds (MCs) and their star formation rate (SFR) and efficiency (SFE). The model assumes that the instantaneous SFR is given by the mass above a certain density threshold divided by its free-fall time. The instantaneous number of massive stars is computed assuming a Kroupa IMF. These stars feed back on the cloud through ionizing radiation, eroding it. The main controlling parameter of the evolution turns out to be the maximum cloud mass, $\Mmax$. This allows us to compare various properties of the model clouds against their observational counterparts. A giant molecular cloud (GMC) model ($\Mmax \sim 10^5 \Msun$) adheres very well to the evolutionary scenario recently inferred by Kawamura et al. (2009) for GMCs in the Large Magellanic Cloud. A model cloud with $\Mmax \approx 2000 \Msun$ evolves in the Kennicutt-Schmidt diagram first passing through the locus of typical low- to-intermediate mass star-forming clouds, and then moving towards the locus of high-mass star-forming ones over the course of $\sim 10$ Myr. Also, the stellar age histograms for this cloud a few Myr before its destruction agree very well with those observed in the $\rho$-Oph stellar association, whose parent cloud has a similar mass, and imply that the SFR of the clouds increases with time. Our model thus agrees well with various observed properties of star-forming MCs, suggesting that the scenario of gravitationally collapsing MCs, with their SFR regulated by stellar feedback, is entirely feasible and in agreement with key observed properties of molecular clouds., Comment: Version accepted for publication in ApJ. At referee's suggestion, includes comparison with numerical models in addition to comparison with observational data

We discuss the nature of the velocity dispersion vs. size relation for molecular clouds. In particular, we add to previous observational results showing that the velocity dispersions in molecular clouds and cores are not purely functions of spatial scale but involve surface gas densities as well. We emphasize that hydrodynamic turbulence is required to produce the first condensations in the progenitor medium. However, as the cloud is forming, it also becomes bound, and gravitational accelerations dominate the motions. Energy conservation in this case implies $|E_g| \sim E_k$, in agreement with observational data, and providing an interpretation for two recent observational results: the scatter in the $\delta v-R$ plane, and the dependence of the velocity dispersion on the surface density ${\delta v^2/ R} \propto \Sigma$. We argue that the observational data are consistent with molecular clouds in a state of hierarchical gravitational collapse, i.e., developing local centers of collapse throughout the whole cloud while the cloud itself is collapsing, and making equilibrium unnecessary at all stages prior to the formation of actual stars. Finally, we discuss how this mechanism need not be in conflict with the observed star formation rate., Comment: Accepted by MNRAS. 7 pages, 3 figures