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Register a new userOn the extraction of power-law parts of the probability density functions in star-forming clouds
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We present a new approach to extract the power-law part of a density/column-density probability density function (rho-pdf/N-pdf) in star-forming clouds. It is based on the mathematical method bPLFIT of Virkar & Clauset (2014) and assesses the power-law part of an arbitrary distribution, without any assumptions about the other part of this distribution. The slope and deviation point are derived as averaged values as the number of bins is varied. Neither parameter is sensitive to spikes and other local features of the tail. This adapted bPLFIT method is applied to two different sets of data from numerical simulations of star-forming clouds at scales 0.5 and 500 pc and displays rho-pdf and N-pdf evolution in agreement with a number of numerical and theoretical studies. Applied to Herschel data on the regions Aquila and Rosette, the method extracts pronounced power-law tails, consistent with those seen in simulations of evolved clouds.
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We report the novel detection of complex high-column density tails in the probability distribution functions (PDFs) for three high-mass star-forming regions (CepOB3, MonR2, NGC6334), obtained from dust emission observed with Herschel. The low column density range can be fit with a lognormal distribution. A first power-law tail starts above an extinction (Av) of ~6-14. It has a slope of alpha=1.3-2 for the rho~r^-alpha profile for an equivalent density distribution (spherical or cylindrical geometry), and is thus consistent with free-fall gravitational collapse. Above Av~40, 60, and 140, we detect an excess that can be fitted by a flatter power law tail with alpha>2. It correlates with the central regions of the cloud (ridges/hubs) of size ~1 pc and densities above 10^4 cm^-3. This excess may be caused by physical processes that slow down collapse and reduce the flow of mass towards higher densities. Possible are: 1. rotation, which introduces an angular momentum barrier, 2. increasing optical depth and weaker cooling, 3. magnetic fields, 4. geometrical effects, and 5. protostellar feedback. The excess/second power-law tail is closely linked to high-mass star-formation though it does not imply a universal column density threshold for the formation of (high-mass) stars.
We study the star formation (SF) law in 12 Galactic molecular clouds with ongoing high-mass star formation (HMSF) activity, as traced by the presence of a bright IRAS source and other HMSF tracers. We define the molecular cloud (MC) associated to each IRAS source using 13CO line emission, and count the young stellar objects (YSOs) within these clouds using GLIMPSE and MIPSGAL 24 micron Spitzer databases.The masses for high luminosity YSOs (Lbol>10~Lsun) are determined individually using Pre Main Sequence evolutionary tracks and the evolutionary stages of the sources, whereas a mean mass of 0.5 Msun was adopted to determine the masses in the low luminosity YSO population. The star formation rate surface density (sigsfr) corresponding to a gas surface density (siggas) in each MC is obtained by counting the number of the YSOs within successive contours of 13CO line emission. We find a break in the relation between sigsfr and siggas, with the relation being power-law (sigsfr ~ siggas^N) with the index N varying between 1.4 and 3.6 above the break. The siggas at the break is between 150-360 Msun/pc^2 for the sample clouds, which compares well with the threshold gas density found in recent studies of Galactic star-forming regions. Our clouds treated as a whole lie between the Kennicutt (1998) relation and the linear relation for Galactic and extra-galactic dense star-forming regions. We find a tendency for the high-mass YSOs to be found preferentially in dense regions at densities higher than 1200 Msun/pc^2 (~0.25 g/cm^2).
We derive an analytical theory of the PDF of density fluctuations in supersonic turbulence in the presence of gravity in star-forming clouds. The theory is based on a rigorous derivation of a combination of the Navier-Stokes continuity equations for the fluid motions and the Poisson equation for the gravity. It extends upon previous approaches first by including gravity, second by considering the PDF as a dynamical system, not a stationary one. We derive the transport equations of the density PDF, characterize its evolution and determine the density threshold above which gravity strongly affects and eventually dominates the dynamics of turbulence. We demonstrate the occurence of {it two} power law tails in the PDF, with two characteristic exponents, corresponding to two different stages in the balance between turbulence and gravity. Another important result of this study is to provide a procedure to relate the observed {it column density} PDFs to the corresponding {it volume density} PDFs. This allows to infer, from the observation of column-densities, various physical parameters characterizing molecular clouds, notably the virial parameter. Furthermore, the theory offers the possibility to date the clouds in units of ${t}_{rm coll}$, the time since a statistically significant fraction of the cloud started to collapse. The theoretical results and diagnostics reproduce very well numerical simulations and observations of star-forming clouds. The theory provides a sound theoretical foundation and quantitative diagnostics to analyze observations or numerical simulations of star-forming regions and to characterize the evolution of the density PDF in various regions of molecular clouds.
Numerical simulations of star formation have found that a power-law mass function can develop at high masses. In a previous paper, we employed isothermal simulations which created large numbers of sinks over a large range in masses to show that the power law exponent of the mass function, $dN/dlog M propto M^{Gamma}$, asymptotically and accurately approaches $Gamma = -1.$ Simple analytic models show that such a power law can develop if the mass accretion rate $dot{M} propto M^2$, as in Bondi-Hoyle accretion; however, the sink mass accretion rates in the simulations show significant departures from this relation. In this paper we show that the expected accretion rate dependence is more closely realized provided the gravitating mass is taken to be the sum of the sink mass and the mass in the near environment. This reconciles the observed mass functions with the accretion rate dependencies, and demonstrates that power-law upper mass functions are essentially the result of gravitational focusing, a mechanism present in, for example, the competitive accretion model.
Both observational and theoretical research over the past decade has demonstrated that the probability distribution function (PDF) of the gas density in turbulent molecular clouds is a key ingredient for understanding star formation. It has recently been argued that the PDF of molecular clouds is a pure power-law distribution. It has been claimed that the log-normal part is ruled out when using only the part of the PDF up/down to which it is complete, that is where the column density contours are still closed. By using the results from high-resolution magnetohydrodynamical simulations of molecular cloud formation and evolution, we find that the column density PDF is indeed composed of a log-normal and, if including self-gravity, a power-law part. We show that insufficient sampling of a molecular cloud results in closed contours that cut off the log-normal part. In contrast, systematically increasing the field of view and sampling the entire cloud yields a completeness limit at the lower column densities, which also recovers the log-normal part. This demonstrates that the field of view must be sufficiently large for the PDF to be complete down to its log-normal part, which has important implications for predictions of star-formation activity based on the PDF.
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