From Clouds to Cores to Protostars and Disks New Insights from Numerical Simulations Shantanu Basu The University of Western Ontario Collaborators: Glenn E. Ciolek (RPI), Takahiro Kudoh (NAO, Japan), Eduard I. Vorobyov (UWO) University of Massachusetts, April 13, 2006

Onishi et al. (2002) Taurus Molecular Cloud 5 pc velocity dispersion sound speed distance = 140 pc protostar T Tauri star

Goodman et al. (1990) – polarization map of Taurus region Goldsmith et al. (2005) – high resolution 12 CO map of Taurus (FCRAO) Correlation of Magnetic Field with Gas Structure

Velocity dispersion-Size relations [Size scale of the cloud] [Velocity dispersion] Self-gravitational equilibrium with turbulence for an ensemble of clouds. Solomon et al. (1987) Heyer and Brunt (2004) Internal cloud correlations (open circles) and global correlations (filled circles).

Magnetic Field Strength Data since averaged over all possible viewing angles. Correlation previously noted by Myers & Goodman (1988); also Bertoldi & McKee (1992), Mouschovias & Psaltis (1995). From Basu (2000), based on Zeeman data compiled by Crutcher (1999). Best fit sub-Alfvénic motions.

magnetic force gravity MHD wave pressure Turbulence Magnetic field line Cloud Magnetized Interstellar Cloud Schematic Picture

Protostellar-driven turbulence Quillen et al. (2006) NGC 1333 Green circles = outflow driven cavity locations, from velocity channel map Diamonds = Herbig-Haro objects Triangles = compact submm sources Stars = protostars

Key Questions How is interstellar molecular cloud turbulence driven? external: galactic shear, supernova shells? internal: outflows, OB stars? How is turbulence maintained? dissipation time < lifetime of cloud? What controls fragmentation of molecular clouds? gravity: ambipolar diffusion: turbulence: How is the stellar mass accumulated? Nature of disk accretion? How does the accretion terminate? depends on power spectrum depends on B field, ionization

3D Cloud Model with Protostellar Formation and Feedback Li & Nakamura (2006) – 128 x 128 x 128 simulation 1.5 pc t g =6 x 10 5 yr

Magnetic field line High resolution global 1.5D MHD simulation A sinusoidal driving force is input into the molecular cloud. Self-gravity Magnetic field line Driving force Molecular cloud Hot medium Our simulation box Most of the previous simulations model a local region. Periodic boundary box If we want to study the global structure of the cloud, this is NOT a good setting for the problem. Low density and hot gas Molecular cloud Kudoh & Basu (2003)

3D Periodic Box Simulations Stone, Gammie, & Ostriker (1998). Similar results from Mac Low et al. (1998) and many subsequent studies. Main Results: Turbulent dissipation rate   = fixed mean density of the periodic box  = one-dimensional velocity dispersion   turbulent driving scale Energy dissipation time scale i.e., the crossing time across the driving scale. A local region of a molecular cloud.

Problems with Application of Local Simulations Consider a cloud of size L If driving scale and then can such rapidly decaying turbulence be maintained by equally strong stirring? Basu & Murali (2001) – “Luminosity problem” if Model prediction of CO luminosity would far exceed that actually observed. Resolution: Characteristic scale for dissipation is L for each cloud, not some inner scale   L 

Dissipation time = global crossing time. Is that OK? Compare to estimated lifetime for molecular clouds. would be better!

1-D Magnetohydrodynamic (MHD) equations (mass) (z-momentum) (y-momentum) (magnetic field) (self-gravity) (gas) (isothermality) Ideal MHD

Movie: Time evolution of density and wave component of the magnetic field (z)(z) 0.25pc Interface between cold cloud and hot low-density gas Kudoh & Basu (2003) Note: 1D simulation allows exceptional resolution (50 points per scale length). grid  z = pc

Snapshot of density 0.25pc Shock waves The density structure is complicated and has many shock waves. Kudoh & Basu (2003)

Time averaged density Time averaged quantities and are for Lagrangian particles. Initial condition Time averaged density The scale height is about 3 times larger than that of the initial condition. 0.25pc The time averaged density shows a smooth distribution.

Input constant amplitude disturbance during this period. The density plots at various times are stacked with time increasing upward. Turbulent driving amplitude increases linearly with time between t=0 and t=10t 0. Driving is terminated at t =40 t 0. a Density Evolution Large scale oscillations survive longest after internal driving discontinued.

Dark Cloud Barnard 68 Lada et al. (2003) Thermal linewidths and near-equilibrium, but evidence for large scale oscillatory motion. Angle-averaged profile can be fit by a thermally supported Bonnor-Ebert sphere model. Alves, Lada, & Lada (2001)

Data from an Ensemble of Cloud Models Velocity dispersion (  ) vs. Scale of the clouds Consistent with observations Time-averaged gravitational equilibrium Filled circles = half-mass position, open circles = full-mass position for a variety of driving amplitudes. Best fit to data is for  = 1  ≈ 0.5 V A )  Cloud self-regulation => highly super-Alfvenic motions not possible! Kudoh & Basu (2006)

Result: Power spectrum of a time snap shot Power spectrum as a function of wave number (k) at t =30t 0. Note that there is significant power on scales larger than the driving scale ( ). This is different from power spectra in uniform media. Power spectrum of B y Power spectrum of v y Kudoh & Basu (2006) driving source

Dissipation time of energy Magnetic energy Kinetic energy (vertical) Kinetic energy (lateral) The sum of all The time we stop driving force Dissipation time Note that the energy in transverse modes remains much greater than that in generated longitudinal modes. A few crossing times of the expanded cloud.

Key Conclusions of High-resolution 1.5D Turbulence Model Turbulence from local sources quickly propagate to fill the cloud. Outer regions of low density have high Alfvén speed  leads to large amplitude motions and generation of long wavelength modes in outer cloud.   Z 1/2 and  V A relations naturally satisfied by time-averaged quantities. Highly super-Alfvénic motions not possible. Power spectrum contains most power on the largest scales, in spite of driving on a smaller inner scale (unlike periodic models). Large scale oscillations survive longest after internal turbulence dissipates. Dissipation time is a few crossing times of the expanded cloud, less than but within reach of estimated cloud lifetimes. It is longer than in periodic 3D models. But, will this result survive in a 3D global model?

Magnetic field line MHD simulation: 2-dimensional Self-gravity Magnetic field line Driving force Molecular cloud Hot medium 1D simulation box Low density and hot gas Molecular cloud Structure of the z-direction is integrated into the plane  2D approximation. 2D simulation box Indebetouw & Zweibel (2000) Basu & Ciolek (2004) Li & Nakamura (2004) Gravitational collapse occurs. Dense cloud

Two-Fluid Thin-Disk MHD Equations (some higher order terms dropped) Magnetic thin-disk approximation. Basu & Ciolek (2004) (mass) (momentum) (vertical magnetic field) (ion-neutral drift) (vertical equilibrium) (ionization balance) (planar gravity) (planar magnetic field)

Supercritical ( B = ½ B critical )Transcritical ( B = B critical ) Gravitational collapse happens quickly: sound crossing time ~ 10 6 year Infall velocity supersonic on ~ 0.1 pc scales Gravity wins again, but slowly: magnetic diffusion time ~ 10 7 year Infall velocity is subsonic Core spacing is larger Column density MHD Model of Gravitational Instability Basu & Ciolek (2004) - Two-dimensional grid (128 x 128), normal to mean B field. Small random perturbations added to initially uniform state.

Comparison of models to observations Is core formation driven by gravity, ambipolar diffusion, or turbulence? Or what combination of these? Gravity (i.e., highly supercritical fragmentation)  accounts for YSO spacings in dense regions, Ambipolar diffusion (i.e., transcritical or subcritical fragmentation)  accounts for large-scale subsonic infall and possibly for the low star formation efficiency (SFE). Transcritical fragmentation may be related to “isolated star formation”. Turbulence  is strong in cloud common envelope, and may also enforce low SFE if not dissipated quickly. Taurus Onishi et al. (2002) Still an early stage of comparison between theory and observation.

Zoom in to simulate the collapse of a nonaxisymmetric supercritical core Basu & Ciolek (2004) A self-consistent model of core collapse leading to protostar and disk formation Vorobyov & Basu (2005) A disk that forms naturally from the collapse of the core. Previous models have usually studied isolated disks. 0.5 pc 100 AU

Disk Formation and Protostellar Accretion Vorobyov & Basu (2005) Ideal MHD 2-D (r,  simulation of rotating supercritical core. Logarithmically spaced grid. Finest spacing (0.3 AU) near center. Sink cell introduced after protostar formation. Disk evolution driven by infalling envelope.

Spiral Structure and Episodic Accretion Vorobyov & Basu (2005) FU Ori events smooth mode burst mode 2D logarithmically spaced grid follows collapse to late accretion phase.

Hartmann (1998) – empirical inference, based on ideas advocated by Kenyon et al. (1990). YSO Accretion History Vorobyov & Basu (2006) – theoretical calculation of disk formation and evolution

Spiral Structure Sharp spiral structure and embedded clumps (denoted by arrows) just before a burst occurs. Diffuse, flocculent spiral structure during the quiescent phase between bursts. Vorobyov & Basu (2006)

Summary: From Clouds to Cores to Disks One-dimensional simulations of turbulence: - largest (supersonic) speeds in outermost parts of stratified cloud - significant power generated on largest scales even with driving on smaller scales, due to stratification effect - dissipation time is related to cloud size, not internal driving scale: provides a way out of “luminosity problem” if this result holds for the large cloud complexes. Two-dimensional simulations of magnetically-regulated fragmentation: - infall speed subsonic or supersonic depending on magnetic field strength. Core spacing can also depend on B field. - transcritical fragmentation may be important for understanding isolated low mass star formation and low SFE. Detailed collapse of nonaxisymmetric rotating cores: - newly discovered “burst mode” of accretion - envelope accretion onto disk  disk instability  clump formation  clumps driven onto protostar  repeats until envelope accretion declines sufficiently - explains FU Ori bursts, low disk luminosity. Protoplanet formation?