1. Magnetic Fields and Protostellar Cores Shantanu Basu University of Western Ontario YLU Meeting, La Thuile, Italy, March 24, 2004

2. Magnetic Field Strength Data Crutcher (1999) and Basu (2000) A better correlation Best fit slope = 0.47 Best fit slope = 1.00 1-D velocity dispersion

3. Magnetic Field Strength Data Two separate correlations Best fit => (1) However, (2) Dimensionless mass-to-flux ratio e.g., Myers & Goodman (1988) Pressure of self-gravityTurbulent pressure

4. Magnetic Field Strength Data Using B los, best fit implies i.e., Alfvenic motions in molecular clouds? e.g., Myers & Goodman (1988), Bertoldi & McKee (1992), Mouschovias & Psaltis (1995). (3) Basu (2000)

5. self-gravity perturbation Molecular cloud Magnetic field line Schematic picture of our simulation A sinusoidal perturbation is input into the molecular cloud. Magnetic field line Low-density and hot medium Simulation box Molecular cloud Hot medium Kudoh & Basu (2003)

6. Basic MHD equations in 1.5 dimensions mass continuity z -momentum y -momentum isothermality magnetic induction self-gravity (Poisson’s eqn.) ideal gas law

7. A Model for Turbulent Molecular Clouds Numerical solution of MHD equations in 1-D. Start with Spitzer 1-D equilibrium state Cloud has a moving boundary Density stratification due to gravity Add nonlinear forcing near z = 0 => nonzero Kudoh & Basu (2003) Molecular cloud Hot medium

8. A Model for MHD Turbulence in Molecular Clouds Kudoh & Basu (2003) Highlights: Cloud expands due to turbulent pressure, achieves “steady state” between t = 10 and t = 40; later contracts when forcing discontinued at t = 40. Outer cloud undergoes largest amplitude oscillations. Resolution: 50 points per length H 0. in this model. Parameters:

9. Snapshots of density 0.25pc Shock waves The density structure is complicated and has many shock waves.

10. Time averaged density Time averaged quantities and are for Lagrangian particles. Initial condition 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.

11. A Model for MHD Turbulence Transverse standing wave => boundary is a node for B y, antinode for v y. sub-Alfvenic motions

12. Results for an ensemble of clouds with different turbulent driving strengths: Solid circles => half-mass position Open circles => edge of cloud Correlations of Global Properties

13. Ideal MHD Turbulence in a Stratified Cloud Clouds are in a time-averaged balance between turbulent support and gravity. Inner cloud obeys equipartition of transverse wave energy, Transverse modes dominate, Outer low density part of cloud undergoes large longitudinal oscillations, and exhibits transverse (Alfvenic) standing wave modes. Correlations and naturally satisfied.

14. MHD Model of Gravitational Instability Courtesy of Nakamura & Hanawa (1997) Complementary to previous model. Solve for dynamics in plane perpendicular to mean magnetic field. No driven turbulence. Ion-neutral friction allowed => non-ideal MHD. Basu & Ciolek (2004) A sub-region of a cloud in which turbulence has largely dissipated.

15. Two-Fluid MHD Equations (some higher order terms dropped) Magnetic thin-disk approximation.

16. MHD Model of Gravitational Instability Basu & Ciolek (2004) Small perturbations added to periodic initially uniform state. Column densityMass-to-flux ratio Triaxial but more nearly oblate cores.

17. MHD Model of Gravitational Instability Infall motions are subsonic. Maximum e.g., observations of L1544, Tafalla et al. (1998) Note merger of column density into background, e.g., mid-infrared maps of Bacmann et al. (2000). Horizontal slice through a core.

18. MHD Model of Gravitational Instability supercritical cloud. All other parameters identical. Supersonic infall in cores and extended near-sonic infall. Observationally distinguishable! Basu & Ciolek (2004)

19. Two-Fluid Non-ideal MHD Gravitational Instability Ambipolar diffusion leads naturally to a non-uniform distribution of mass-to-flux ratio. Stars form preferentially in the most supercritical regions. Supercritical cores and subcritical envelopes created simultaneously by flux redistribution if Initially critical model => subsonic infall. Initially significantly supercritical model => supersonic infall. Neutral speeds typically greater than ion speeds – gravitationally driven motions. Core densities merge into background near-uniform value.

20. MHD Model of Gravitational Instability Subcritical sheet

21. The coefficient of Chandrasekhar-Fermi formula Surface of the cloud  =1 (for linear wave)  =0.23  <1 at the surface of the cloud 0.25pc  B y is small near the surface but v y is not – a standing wave effect!

22. Dissipation time of energy Magnetic energy Kinetic energy (vertical) Kinetic energy (lateral) The sum of the 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.