Core Formation due to Magnetic Fields, Ambipolar Diffusion, and Turbulence

1. Core Formation due to Magnetic Fields, Ambipolar Diffusion, and Turbulence Shantanu Basu The University of Western Ontario Collaborators: Glenn Ciolek (RPI), Wolfgang Dapp (UWO), Takahiro Kudoh (NAOJ), James Wurster (UWO) The Cosmic Agitator University of Kentucky March 29, 2008

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

3. Large Scale Molecular Gas Structure of Taurus 12CO emission Goldsmith et al. (2008) Striations of gas emission consistent with magnetically-dominated envelope.

4. Layer Instability Consider a layer of surface density S. Linear perturbation analysis yields dispersion relation Gravitational instability if H is the vertical scale height of the layer. Moreover, there is a preferred fragmentation scale. at which the growth time is a minimum.

5. Effect of Magnetic Field Critical magnetic field if self-gravitational pressure magnetic pressure Where magnetic flux-freezing applies: Subcritical cloud No fragmentation occurs Supercritical cloud Fragmentation occurs

6. Ambipolar Diffusion In a weakly ionized gas, the mean velocity of neutral atoms or molecules will not generally equal the mean velocity of ions and electrons. Neutrals do not feel the Lorentz force directly, but only through collisions arising from a drift relative to ions. neutral-ion collision time ion density vs. neutral density relation, primarily due to cosmic ray ionization Even SUBCRITICAL clouds can undergo fragmentation instability due to ambipolar diffusion, i.e. ion-neutral slip.

7. 1D simulation box (Kudoh & Basu) 2D simulation box Dense core Gravitational fragmentation occurs. MHD simulation: 2-dimensional Integrate thorugh structure of the z-direction near the midplane  2D approximation. Magnetic field line Low density and hot gas Molecular cloud Kudoh & Basu (2003,2006) – dense midplane of stratified turbulent cloud has transonic/subsonic motions.

8. Modes of Fragmentation • Gravitational Fragmentation • Turbulent Fragmentation (Linear perturbations) dynamic (supercritical mass-to-flux ratio) quasistatic ambipolar-diffusion (subcritical mass-to-flux ratio) (Highly nonlinear perturbations) supercritical subcritical We can test all of these scenarios including the effects of magnetic fields and ambipolar diffusion.

9. MHD Model of Gravitational Fragmentation Added small (few %) initial random white noise perturbations to column density, magnetic field. In all images, but magnetic field strength varies. Thin disk approximation

10. Linear Perturbation Analysis for Magnetic Disk with AD CR ionization Ciolek & Basu (2006) growth time of instability flux freezing imperfect coupling

11. Fragmentation Scales Solid lines = linear fragmentation theory. Symbols = result of 2D numerical simulations Converges to Case of low external pressure on disk High external pressure case This curve first derived by Morton & Mouschovias (1991) Ciolek & Basu (2006); Basu, Ciolek, & Wurster (2008)

12. Basu, Ciolek & Wurster (2008) MHD Model of Gravitational Instability in all images 1282 cells in each model Strong Very weak Weak Critical Subcritical Highly supercritical Supercritical • - t = 200 • - |v|max=0.4 cs • mildest core • elongation - t = 100 - |v|max=0.7 cs - largest spacing • t = 20 • |v|max=1.1 cs • moderate elongation • - large spacing - t = 10 - |v|max=1.2 cs - most elongated box size ~ 2 pc, time unit ~ 2 x 105 yr if nn,0= 3 x 10-3 cm-3, scales as nn,0-1/2.

13. MHD Model of Gravitational Instability Basu, Ciolek, & Wurster (2008)

14. INITIAL Core Mass Function (Grav. Fragmentation) Narrow lognormal-like. High-mass slope much steeper than observed CMF/IMF. “Core” = enclosed region with Basu, Ciolek, & Wurster (2008)

15. Observed Core Mass Fcn and Initial Mass Fcn Lognormal Lognormal/power-law Data from Muench, Lada, & Lada (2002) Data from Nutter & Ward-Thompson (2007)

16. INITIAL Core Mass Function from Multiple Models Add results from a range of models from m0=0.5 to m0 = 2.0. Cumulative histogram of 1524 cores from over 400 separate simulations

17. Turbulent Fragmentation with B and Ambipolar Diffusion Thin disk approximation Will this work in 3D? Li & Nakamura (2004) time unit = 2 Myr; box width = 3.7 pc (a)-(e) subcritical (m0 = 0.83) model, (f)-(h) supercritical (m0 = 1.25) model. vk2~ k -4 spectrum – has a large-scale flow note filamentarity and velocity vectors

18. MHD simulation: (1+2 =) 3-dimensional We have a new explicit 3D non-ideal MHD code. Magnetic field line Top view Low density and hot gas Low density and hot gas Magnetic field Side view 3D simulation box dense sheet z Molecular cloud y MHD with ion-neutral slip X Input large perturbation perpendicular to magnetic field at t=0 Kudoh, Basu, Ogata, & Yabe (2007), Kudoh & Basu (2008)

19. 3D Turbulent Fragmentation with B and AD Nonlinear IC Linear IC Nonlinear initial velocity field allowed to decay rms amplitude trans-Alfvenic Gas density in midplane (z=0) A vertical slice of gas density Kudoh & Basu (2008) using 64 x 64 x 40 cells box width = 2.5 pc

20. 3D Turbulent Fragmentation with B and AD What’s really happening? b is a proxy for m. Early turbulent compression Then, higher density region evolves with near vertical force balance Rapid contraction when/where Kudoh & Basu (2008)

21. Core Mass Spectrum for 2D Turbulent Model Thin disk (2D) allows many runs in order to compile statistics BROAD tail, ROUGHLY consistent with IMF. But, final fate still undetermined. Basu, Ciolek, Dapp, & Wurster (2008)

22. Conclusions • Transcritical gravitational fragmentation has a maximum (>> Jeans) fragment scale. Subcritical and supercritical fragmentation both occur at ~ Jeans scale • Nonlinear gravitational fragmentation yields expected fragment spacing and observationally testable kinematics for different m’s • Initial CMF is narrow for gravitational fragmentation with any single mass-to-flux ratio, but a variety of m’s can yield an overall broad distribution. • Model of 3D turbulent fragmentation reveals accelerated fragmentation for subcritical clouds with trans-Alfvenic turbulence. • Initial CMF from turbulent fragmentation is also broad but clear mapping to IMF via monolithic collapse not assured A wide range of results are available for comparison to observations.