Christoph Federrath

THE STAR FORMATION RATE OF TURBULENT MAGNETIZED CLOUDS: COMPARING THEORY, SIMULATIONS, AND OBSERVATIONS

The role of turbulence and magnetic fields is studied for star formation in molecular clouds. We derive and compare six theoretical models for the star formation rate (SFR)—the Krumholz & McKee (KM), Padoan & Nordlund (PN), and Hennebelle & Chabrier (HC) models, and three multi-freefall versions of these, suggested by HC—all based on integrals over the log-normal distribution of turbulent gas. We extend all theories to include magnetic fields and show that the SFR depends on four basic parameters: (1) virial parameter αvir; (2) sonic Mach number ; (3) turbulent forcing parameter b, which is a measure for the fraction of energy driven in compressive modes; and (4) plasma with the Alfven Mach number . We compare all six theories with MHD simulations, covering cloud masses of 300 to 4 × 106 M ☉ and Mach numbers -50 and -∞, with solenoidal (b = 1/3), mixed (b = 0.4), and compressive turbulent (b = 1) forcings. We find that the SFR increases by a factor of four between and 50 for compressive turbulent forcing and αvir ~ 1. Comparing forcing parameters, we see that the SFR is more than 10 times higher with compressive than solenoidal forcing for simulations. The SFR and fragmentation are both reduced by a factor of two in strongly magnetized, trans-Alfvenic turbulence compared to hydrodynamic turbulence. All simulations are fit simultaneously by the multi-freefall KM and multi-freefall PN theories within a factor of two over two orders of magnitude in SFR. The simulated SFRs cover the range and correlation of SFR column density with gas column density observed in Galactic clouds, and agree well for star formation efficiencies SFE = 1%-10% and local efficiencies = 0.3-0.7 due to feedback. We conclude that the SFR is primarily controlled by interstellar turbulence, with a secondary effect coming from magnetic fields.

The Density Probability Distribution in Compressible Isothermal Turbulence: Solenoidal versus Compressive Forcing

The probability density function (PDF) of the gas density in turbulent supersonic flows is investigated with high-resolution numerical simulations. In a systematic study, we compare the density statistics of compressible turbulence driven by the usually adopted solenoidal forcing (divergence-free) and by compressive forcing (curl-free). Our results are in agreement with studies using solenoidal forcing. However, compressive forcing yields a significantly broader density distribution with standard deviation ~3 times larger at the same rms Mach number. The standard deviation-Mach number relation used in analytical models of star formation is reviewed and a modification of the existing expression is proposed, which takes into account the ratio of solenoidal and compressive modes of the turbulence forcing.

MODELING COLLAPSE AND ACCRETION IN TURBULENT GAS CLOUDS: IMPLEMENTATION AND COMPARISON OF SINK PARTICLES IN AMR AND SPH

Star formation is such a complex process that accurate numerical tools are needed to quantitatively examine the mass distribution and accretion of fragments in collapsing, turbulent, magnetized gas clouds. To enable a numerical treatment of this regime, we implemented sink particles in the adaptive mesh refinement (AMR) hydrodynamics code FLASH. Sink particles are created in regions of local gravitational collapse, and their trajectories and accretion can be followed over many dynamical times. We perform a series of tests including the time integration of circular and elliptical orbits, the collapse of a Bonnor-Ebert sphere, and a rotating, fragmenting cloud core. We compare the collapse of a highly unstable singular isothermal sphere to the theory by Shu and show that the sink particle accretion rate is in excellent agreement with the theoretical prediction. To model eccentric orbits and close encounters of sink particles accurately, we show that a very small time step is often required, for which we implemented subcycling of the N-body system. We emphasize that a sole density threshold for sink particle creation is insufficient in supersonic flows, if the density threshold is below the opacity limit. In that case, the density can exceed the threshold in strong shocks that do not necessarily lead to local collapse. Additional checks for bound state, gravitational potential minimum, Jeans instability, and converging flows are absolutely necessary for meaningful creation of sink particles. We apply our new sink particle module for FLASH to the formation of a stellar cluster, and compare to a smoothed particle hydrodynamics (SPH) code with sink particles. Our comparison shows encouraging agreement of gas properties, indicated by column density distributions and radial profiles, and of sink particle formation times and positions. We find excellent agreement in the number of sink particles formed, and in their accretion and mass distributions.

Modelling CO formation in the turbulent interstellar medium

We present results from high-resolution three-dimensional simulations of turbulent interstellar gas that self-consistently follow its coupled thermal, chemical and dynamical evolution, with a particular focus on the formation and destruction of H 2 and CO. We quantify the formation time-scales for H 2 and CO in physical conditions corresponding to those found in nearby giant molecular clouds, and show that both species form rapidly, with chemical time-scales that are comparable to the dynamical time-scale of the gas. We also investigate the spatial distributions of H 2 and CO, and how they relate to the underlying gas distribution. We show that H 2 is a good tracer of the gas distribution, but that the relationship between CO abundance and gas density is more complex. The CO abundance is not well-correlated with either the gas number density n or the visual extinction Av : both have a large influence on the CO abundance, but the inhomogeneous nature of the density field produced by the turbulence means that n and A v are only poorly correlated. There is a large scatter in A v , and hence CO abundance, for gas with any particular density, and similarly a large scatter in density and CO abundance for gas with any particular visual extinction. This will have important consequences for the interpretation of the CO emission observed from real molecular clouds. Finally, we also examine the temperature structure of the simulated gas. We show that the molecular gas is not isothermal. Most of it has a temperature in the range of 10-20 K, but there is also a significant fraction of warmer gas, located in low-extinction regions where photoelectric heating remains effective.

A NEW JEANS RESOLUTION CRITERION FOR (M)HD SIMULATIONS OF SELF-GRAVITATING GAS: APPLICATION TO MAGNETIC FIELD AMPLIFICATION BY GRAVITY-DRIVEN TURBULENCE

Cosmic structure formation is characterized by the complex interplay between gravity, turbulence, and magnetic fields. The processes by which gravitational energy is converted into turbulent and magnetic energies, however, remain poorly understood. Here, we show with high-resolution, adaptive-mesh simulations that MHD turbulence is efficiently driven by extracting energy from the gravitational potential during the collapse of a dense gas cloud. Compressible motions generated during the contraction are converted into solenoidal, turbulent motions, leading to a natural energy ratio of E sol/E tot 2/3. We find that the energy injection scale of gravity-driven turbulence is close to the local Jeans scale. If small seeds of the magnetic field are present, they are amplified exponentially fast via the small-scale dynamo process. The magnetic field grows most efficiently on the smallest scales, for which the stretching, twisting, and folding of field lines, and the turbulent vortices are sufficiently resolved. We find that this scale corresponds to about 30 grid cells in the simulations. We thus suggest a new minimum resolution criterion of 30 cells per Jeans length in (magneto)hydrodynamical simulations of self-gravitating gas, in order to resolve turbulence on the Jeans scale, and to capture minimum dynamo amplification of the magnetic field. Due to numerical diffusion, however, any existing simulation today can at best provide lower limits on the physical growth rates. We conclude that a small, initial magnetic field can grow to dynamically important strength on timescales significantly shorter than the free-fall time of the cloud.

Cluster-formation in the Rosette molecular cloud at the junctions of filaments (Corrigendum)

For many years feedback processes generated by OB-stars in molecular clouds, including expanding ionization fronts, stellar winds, or UV-radiation, have been proposed to trigger subsequent star formation. However, hydrodynamic models including radiation and gravity show that UV-illumination has little or no impact on the global dynamical evolution of the cloud. The Rosette molecular cloud, irradiated by the NGC2244 cluster, is a template region for triggered star-formation, and we investigated its spatial and density structure by applying a curvelet analysis, a filament-tracing algorithm (DisPerSE), and probability density functions (PDFs) on Herschel column density maps, obtained within the HOBYS key program. The analysis reveals not only the filamentary structure of the cloud but also that all known infrared clusters except one lie at junctions of filaments, as predicted by turbulence simulations. The PDFs of sub-regions in the cloud show systematic differences. The two UV-exposed regions have a double-peaked PDF we interprete as caused by shock compression. The deviations of the PDF from the log-normal shape typically associated with low- and high-mass star-forming regions at Av~3-4m and 8-10m, respectively, are found here within the very same cloud. This shows that there is no fundamental difference in the density structure of low- and high-mass star-forming regions. We conclude that star-formation in Rosette - and probably in high-mass star-forming clouds in general - is not globally triggered by the impact of UV-radiation. Moreover, star formation takes place in filaments that arose from the primordial turbulent structure built up during the formation of the cloud. Clusters form at filament mergers, but star formation can be locally induced in the direct interaction zone between an expanding HII--region and the molecular cloud.

A comparison between grid and particle methods on the statistics of driven, supersonic, isothermal turbulence

We compare the statistics of driven, supersonic turbulence at a high Mach number using FLASH, a widely used Eulerian grid-based code, and PHANTOM, a Lagrangian smoothed particle hydrodynamics (SPH) code at resolutions of up to 512 3 in both grid cells and SPH particles. We find excellent agreement between codes on the basic statistical properties: a slope of k −1.95 in the velocity power spectrum for hydrodynamic, Mach 10 turbulence, evidence in both codes for a Kolmogorov-like slope of k −5/3 in the variable ρ 1/3 v as suggested by Kritsuk et al. and a lognormal probability distribution function (PDF) with a width that scales with the Mach number and proportionality constant b = 0.33–0.5 in the density variance–Mach number relation. The measured structure function slopes are not converged in either code at 512 3 elements. We find that for measuring volumetric statistics such as the power spectrum slope and structure function scaling, SPH and grid codes give roughly comparable results when the number of SPH particles is approximately equal to the number of grid cells. In particular, to accurately measure the power spectrum slope in the inertial range, in the absence of sub-grid turbulence models, requires at least 512 3 computational elements in either code. On the other hand the SPH code was found to be better at resolving dense structures, giving maximum densities at a resolution of 128 3 particles that were similar to the maximum densities resolved in the grid code at 512 3 cells, reflected also in the high density tail of the PDF. We find SPH to be more dissipative at comparable numbers of computational elements in statistics of the velocity field, but correspondingly less dissipative than the grid code in the statistics of density-weighted quantities such as ρ 1/3 v. For SPH simulations of high Mach number turbulence, we find it important to use sufficient non-linear β-viscosity in order to prevent particle interpenetration in shocks (we require βvisc = 4 instead of the widely used default value, βvisc = 2).

Dynamic star formation in the massive DR21 filament

The formation of massive stars is a highly complex process in which it is not clear whether the star-forming gas is in global gravitational collapse or in an equilibrium state, supported by turbulence. By studying one of the most massive and dense star-forming regions in the Galaxy at a distance of less than 3 kpc, the filament containing the well-known sources DR21 and DR21(OH), we expect to find observational signatures that allow to discriminate between the two views. We use molecular line data from our 13CO 1-0, CS 2-1, and N2H+ 1-0 survey of the Cygnus X region obtained with the FCRAO and high-angular resolution observations of CO, CS, HCO+, N2H+, and H2CO, obtained with the IRAM 30m telescope. We observe a complex velocity field and velocity dispersion in the DR21 filament in which regions of highest column-density, i.e. dense cores, have a lower velocity dispersion than the surrounding gas and velocity gradients that are not (only) due to rotation. Infall signatures in optically thick line profiles of HCO+ and 12CO are observed along and across the whole DR21 filament. From modelling the observed spectra, we obtain a typical infall speed of 0.6 km/s and mass accretion rates of the order of a few 10^-3 Msun/yr for the two main clumps constituting the filament. These massive (4900 and 3300 Msun) clumps are both gravitationally contracting. All observed kinematic features in the DR21 filament can be explained if it is formed by the convergence of flows at large scales and is now in a state of global gravitational collapse. Whether this convergence of flows originated from self-gravity at larger scales or from other processes can not be settled with the present study. The observed velocity field and velocity dispersion are consistent with results from (magneto)-hydrodynamic simulations where the cores lie at the stagnation points of convergent turbulent flows.

Numerical simulations of compressively driven interstellar turbulence - I. Isothermal gas

Context. Supersonic turbulence is ubiquitous in the interstellar me dium and plays an important role in contemporary star formation. Aims. To perform a high-resolution numerical simulation of supersonic isothermal turbulence driven by compressive large-scale forcing and to analyse various statistical properties. Methods. The compressible Euler equations with an external stochastic force field dominated by rotation-free modes are solved wi th the piecewise parabolic method. Both a static grid and adaptive mesh refinement is used with an effective resolution N = 768 3 . Results. After a transient phase dominated by shocks, turbulence evolves into a steady state with root mean square Mach number ≈ 2.2... 2.5, in which cloud-like structures of over-dense gas are surr ounded by highly rarefied gas. The index of the turbulence ene rgy spectrum functionβ≈ 2.0 in the shock-dominated phase. As the flow approaches statis tical equilibrium, the spectrum flattens, with β≈ 1.9. For the scaling exponent of the root mean square velocity fl uctuation, we obtainγ≈ 0.43 from the velocity structure functions of second order. These results are well within the range of observed scaling properties for the velocity dispersion in molecular clouds. Calculating structure functions of order p = 1,..., 5, we find for all scaling exponents significant deviations from the Kolmogor ovBurgers model proposed by Boldyrev. Our results are very well described by a general log-Poisson model with a higher degree of intermittency, which implies an influence of the forcing on t he scaling properties. The spectral index of the quadratic l ogarithmic density fluctuation βδ≈ 1.8. Contrary to previous numerical results for isothermal turbulence, we obtain a skewed probability density function of the mass density fluctuations that is not consist ent with log-normal statistics and entails a substantially higher fraction of mass in the density peaks than implied by the Padoan-Nordlund relation between the variance of the density fluctuation s and the Mach number. Conclusions. Even putting aside further complexity due to magnetic fields , gravity or thermal processes, we question the notion that Larson-type relations are a consequence of universal supersonic turbulence scaling. For a genuine understanding, it seems necessary to account for the production mechanism of turbulence in the ISM.

WHAT DETERMINES THE DENSITY STRUCTURE OF MOLECULAR CLOUDS? A CASE STUDY OF ORION B WITH HERSCHEL ∗

A key parameter to the description of all star formation processes is the density structure of the gas. In this letter, we make use of probability distribution functions (PDFs) of Herschel column density maps of Orion B, Aquila, and Polaris, obtained with the Herschel Gould Belt survey (HGBS). We aim to understand which physical processes influence the PDF shape, and with which signatures. The PDFs of Orion B (Aquila) show a lognormal distribution for low column densities until Av 3 (6), and a power-law tail for high column densities, consistent with a rho r^-2 profile for the equivalent spherical density distribution. The PDF of Orion B is broadened by external compression due to the nearby OB stellar aggregates. The PDF of a quiescent subregion of the non-star-forming Polaris cloud is nearly lognormal, indicating that supersonic turbulence governs the density distribution. But we also observe a deviation from the lognormal shape at Av>1 for a subregion in Polaris that includes a prominent filament. We conclude that (i) the point where the PDF deviates from the lognormal form does not trace a universal Av-threshold for star formation, (ii) statistical density fluctuations, intermittency and magnetic fields can cause excess from the lognormal PDF at an early cloud formation stage, (iii) core formation and/or global collapse of filaments and a non-isothermal gas distribution lead to a power-law tail, and (iv) external compression broadens the column density PDF, consistent with numerical simulations.A key parameter to the description of all star formation processes is the density structure of the gas. In this Letter, we make use of probability distribution functions (PDFs) of Herschel column density maps of Orion B, Aquila, and Polaris, obtained with the Herschel Gould Belt survey (HGBS). We aim to understand which physical processes influence the PDF shape, and with which signatures. The PDFs of Orion B (Aquila) show a lognormal distribution for low column densities until A V 3 (6), and a power-law tail for high column densities, consistent with a ρr -2 profile for the equivalent spherical density distribution. The PDF of Orion B is broadened by external compression due to the nearby OB stellar aggregates. The PDF of a quiescent subregion of the non-star-forming Polaris cloud is nearly lognormal, indicating that supersonic turbulence governs the density distribution. But we also observe a deviation from the lognormal shape at A V > 1 for a subregion in Polaris that includes a prominent filament. We conclude that (1) the point where the PDF deviates from the lognormal form does not trace a universal A V -threshold for star formation, (2) statistical density fluctuations, intermittency, and magnetic fields can cause excess from the lognormal PDF at an early cloud formation stage, (3) core formation and/or global collapse of filaments and a non-isothermal gas distribution lead to a power-law tail, and (4) external compression broadens the column density PDF, consistent with numerical simulations.