Paolo Padoan

The Statistics of Supersonic Isothermal Turbulence

We present results of large-scale three-dimensional simulations of supersonic Euler turbulence with the piecewise parabolic method and multiple grid resolutions up to 2048 3 points. Our numerical experiments describe non-magnetized driven turbulent o ws with an isothermal equation of state and an rms Mach number of 6. We discuss numerical resolution issues and demonstrate convergence, in a statistical sense, of the inertial range dynamics in simulations on grids larger than 512 3 points. The simulations allowed us to measure the absolute velocity scaling exponents for the rst time. The inertial range velocity scaling in this strongly compressible regime deviates substantially from the incompressible Kolmogorov laws. The slope of the velocity power spectrum, for instance, is -1:95 compared to -5=3 in the incompressible case. The exponent of the third-order velocity structure function is 1:28, while in incompressible turbulence it is known to be unity. We propose a natural extension of Kolmogorov’s phenomenology that takes into account compressibility by mixing the velocity and density statistics and preserves the Kolmogorov scaling of the power spectrum and structure functions of the density-weighted velocity v 1=3 u. The low-order statistics of v appear to be invariant with respect to changes in the Mach number. For instance, at Mach 6 the slope of the power spectrum of v v is -1:69, and the exponent of the third-order structure function of v v is unity. We also directly measure the mass dimension of the ifractali density distribution in the inertial subrange, Dm 2:4, which is similar to the observed fractal dimension of molecular clouds and agrees well with the cascade phenomenology. Subject headings: hydrodynamics o instabilities o ISM: structure o methods: numerical o turbulence

A Super-Alfvénic Model of Dark Clouds

Supersonic random motions are observed in dark clouds and are traditionally interpreted as Alfven waves, but the possibility that these motions are super-Alfvenic has not been ruled out. In this work we report the results of numerical experiments in two opposite regimes: A ~ 1 and A 1, where A is the initial Alfvenic Mach number—the ratio of the rms velocity to the Alfven speed. Our results show that models with A 1 are consistent with the observed properties of molecular clouds that we have tested (statistics of extinction measurements, distribution of integrated antenna temperature, Zeeman-splitting measurements of magnetic field strength, line width versus integrated antenna temperature of molecular emission-line spectra, statistical B-n relation, and scatter in that relation), while models withA ~ 1 have properties that are in conflict with the observations. We find that both the density and the magnetic field in molecular clouds may be very intermittent. The statistical distributions of the magnetic field and gas density are related by a power law, with an index that decreases with time in experiments with decaying turbulence. After about one dynamical time it stabilizes at B ∝ n0.4. Magnetically dominated cores form early in the evolution, while later on the intermittency in the density field wins out, and also cores with a weak field can be generated by mass accretion along magnetic field lines.

THE STAR FORMATION RATE OF SUPERSONIC MAGNETOHYDRODYNAMIC TURBULENCE

This work presents a new physical model of the star formation rate (SFR), which is verified with an unprecedented set of large numerical simulations of driven, supersonic, self-gravitating, magneto-hydrodynamic (MHD) turbulence, where collapsing cores are captured with accreting sink particles. The model depends on the relative importance of gravitational, turbulent, magnetic, and thermal energies, expressed through the virial parameter, αvir, the rms sonic Mach number, , and the ratio of mean gas pressure to mean magnetic pressure, β0. The SFR is predicted to decrease with increasing αvir (stronger turbulence relative to gravity), to increase with increasing (for constant values of αvir), and to depend weakly on β0 for values typical of star forming regions (-20 and β0 1-20). In the unrealistic limit of β0 → ∞, that is, in the complete absence of a magnetic field, the SFR increases approximately by a factor of three, which shows the importance of magnetic fields in the star formation process, even when they are relatively weak (super-Alfvenic turbulence). The star-formation simulations used to test the model result in an approximately constant SFR, after an initial transient phase. The dependence of the SFR on the virial parameter is shown to agree very well with the theoretical predictions.

The Turbulent Shock Origin of Proto-Stellar Cores

The fragmentation of molecular clouds (MC) into proto-stellar cores is a central aspect of the process of star formation. Because of the turbulent nature of supersonic motions in MCs, it has been suggested that dense structures such as filaments and clumps are formed by shocks in a turbulent flow. In this work we present strong evidence in favor of the turbulent origin of the fragmentation of MCs. The most generic result of turbulent fragmentation is that dense postshock gas traces a gas component with a smaller velocity dispersion than lower density gas, since shocks correspond to regions of converging flows, where the kinetic energy of the turbulent motion is dissipated. Using synthetic maps of spectra of molecular transitions, computed from the results of numerical simulations of supersonic turbulence, we show that the dependence of velocity dispersion on gas density generates an observable relation between the rms velocity centroid and the integrated intensity (column density), σ(V0)-I, which is indeed found in the observational data. The comparison between the theoretical model (maps of synthetic 13CO spectra) with 13CO maps from the Perseus, Rosette, and Taurus MC complexes shows excellent agreement in the σ(V0)-I relation. The σ(V0)-I relation of different observational maps with the same total rms velocity are remarkably similar, which is a strong indication of their origin from a very general property of the fluid equations, such as the turbulent fragmentation process.

Supersonic Turbulence in the Interstellar Medium: Stellar Extinction Determinations as Probes of the Structure and Dynamics of Dark Clouds

Lada et al. have described a method for studying the distribution of dust in dark clouds using infrared imaging surveys. In particular, they show that the method provides some information about the structure of the gas (dust) on scales smaller than their resolution. In the present work we clarify the nature of the information provided by their method. We show that: 1. The three-dimensional density field of the gas is well described by a lognormal distribution down to very small scales. 2. The power spectrum and the standard deviation of the three-dimensional density field can be constrained. 3. Such a structure of the density field is likely to be the effect of random supersonic motions present in the gas. In fact, we find a qualitative and quantitative agreement between the predictions based on recent numerical simulations of randomly forced supersonic flows by Nordlund & Padoan and by Padoan, Nordlund, & Jones and the constraints given by the infrared dust extinction measurements.

Two regimes of turbulent fragmentation and the stellar initial mass function from primordial to present-day star formation

The Padoan and Nordlund model of the stellar initial mass function (IMF) is derived from low-order statistics of supersonic turbulence, neglecting gravity (e.g., gravitational fragmentation, accretion, and merging). In this work, the predictions of that model are tested using the largest numerical experiments of supersonic hydrodynamic (HD) and magnetohydrodynamic (MHD) turbulence to date (~10003 computational zones) and three different codes (Enzo, Zeus, and the Stagger code). The model predicts a power-law distribution for large masses, related to the turbulence-energy power-spectrum slope and the shock-jump conditions. This power-law mass distribution is confirmed by the numerical experiments. The model also predicts a sharp difference between the HD and MHD regimes, which is recovered in the experiments as well, implying that the magnetic field, even below energy equipartition on the large scale, is a crucial component of the process of turbulent fragmentation. These results suggest that the stellar IMF of primordial stars may differ from that in later epochs of star formation, due to differences in both gas temperature and magnetic field strength. In particular, we find that the IMF of primordial stars born in turbulent clouds may be narrowly peaked around a mass of order 10 M☉, as long as the column density of such clouds is not much in excess of 1022 cm-2.

Galaxy Formation and Evolution: Low Surface Brightness Galaxies

ABSTRACT We investigatein detail the hypothesisthat low surfacebrightnessgalaxies(LSB) differfrom ordinary galaxies simply because they form in halos with large spin parameters.We compute star formation rates using the Schmidt law, assuming the same gas infalldependence on surface density as used in models of the Milky Way. We build stellarpopulation models, predicting colours, spectra, and chemical abundances. We compareour predictions with observed values of metallicity and colours for LSB galaxies andfind excellent agreement with all observables. In particular, integrated colours, colourgradients, surface brightness and metallicity match very well to the observed valuesof LSBs for models with ages larger than 7 Gyr and high values (λ > 0.05) for thespin parameter of the halos. We also compute the global star formation rate (SFR)in the Universe due to LSBs and show that it has a flatter evolution with redshiftthan the corresponding SFR for normal discs. We furthermore compare the evolutionin redshift of [Zn/H] for our models to those observed in Damped Lyman α systemsby Pettini et al. (1997) and show that Damped Lyman α systems abundances areconsistent with the predicted abundances at different radii for LSBs. Finally, we showhow the required late redshift of collapse of the halo may constrain the power spectrumof fluctuations.Key words: galaxies: formation – galaxies: evolution – galaxies: spiral – galaxies:stellar content.

Scaling Relations of Supersonic Turbulence in Star-forming Molecular Clouds

We present a direct numerical and analytical study of driven supersonic magnetohydrodynamic turbulence that is believed to govern the dynamics of star-forming molecular clouds. We describe statistical properties of the turbulence by measuring the velocity difference structure functions up to the fifth order. In particular, the velocity power spectrum in the inertial range is found to be close to Ek ~ k-1.74, and the velocity difference scales as |Δu| ~ L0.42. The results agree well with the Kolmogorov-Burgers analytical model suggested for supersonic turbulence.We then generalize the model to more realistic, fractal structure of molecular clouds and show that depending on the fractal dimension of a given molecular cloud, the theoretical value for the velocity spectrum spans the interval [-1.74, -1.89], while the corresponding window for the velocity difference scaling exponent is [0.42, 0.78].

Interstellar Turbulence: The Density PDFs of Supersonic Random Flows

The question of the shape of the density PDF for supersonic turbulence is addressed, using both analytical and numerical methods. For isothermal supersonic turbulence, the PDF is Log-Normal, with a width that scales approximately linearly with the Mach number. For a polytropic equation of state, with an eeective gamma smaller than one, the PDF becomes skewed and becomes reminiscent of (but not identical to) a power law on the high density side.

Theoretical Models of Polarized Dust Emission from Protostellar Cores

We model the polarized thermal dust emission from protostellar cores that are assembled by supersonic turbulent flows in molecular clouds. Self-gravitating cores are selected from a three-dimensional simulation of supersonic and super-Alfvenic magnetohydrodynamic (MHD) turbulence. The polarization is computed in two ways. In model A it is assumed that dust properties and grain alignment efficiency are uniform; in model B it is assumed that grains are not aligned at visual extinction larger than AV,0 = 3 mag, consistent with theoretical expectations for grain alignment mechanisms. Instead of using a specific set of grain properties, we adopt a maximum degree of polarization Pmax = 15%. Results are therefore sensitive mainly to the topology of the magnetic field (model A) and to the gas distribution that determines the distribution of AV (model B). Furthermore, the radiative transfer in the MHD model is solved with a non-LTE Monte Carlo method, to compute spectral maps of the J = 1-0 transition of CS. The CS spectral maps are used to estimate the turbulent velocity, as in the observations. The main results of this work are the following: (1) Values of P between 1% and 10% (up to almost Pmax) are typical, despite the super-Alfvenic nature of the turbulence. (2) A steep decrease of P with increasing values of the submillimeter dust continuum intensity I is always found in self-gravitating cores selected from the MHD simulations if grains are not aligned above a certain value of visual extinction AV,0 (model B). (3) The same behavior is hard to reproduce if grains are aligned independently of AV (model A). (4) The Chandrasekhar-Fermi formula, corrected by a factor f ≈ 0.4, provides an approximate estimate of the average magnetic field strength in the cores. Submillimeter dust continuum polarization maps of quiescent protostellar cores and Bok globules have recently been obtained. They always show a decrease in P with increasing value of I consistent with the predictions of our model B. We therefore conclude that submillimeter polarization maps of quiescent cores do not map the magnetic field inside the cores at visual extinction larger than AV,0 ≈ 3 mag. The use of such maps to constrain models of protostellar core formation and evolution is questionable. This conclusion suggests that there is no inconsistency between the results from optical and near-IR polarized absorption of background stars and the observed polarization of submillimeter dust continuum from quiescent cores. In both cases, grains at large visual extinction appear to be virtually unaligned.