Tomoaki Matsumoto (Hosei Univ. / NAOJ) Masahiro N. Machida (NAOJ) Self-Gravitational Collapse of a Magnetized Cloud Core: High Resolution Simulation with Thee-dimensional MHD Nested Grid Tomoaki Matsumoto (Hosei Univ. / NAOJ) Masahiro N. Machida (NAOJ) Koji Tomisaka (NAOJ) Tomoyuki Hanawa (Chiba Univ.) Introduction and Overview of Nested Grid Application to Astronomical Problems Outflow Formation in collapsing cloud (single star formation) Fragmentation of collapsing cloud (binary star formation) Methods MHD Self-gravity

Introduction: Scenario of Single Star Formation H13CO+　core Orion molecular cloud Optical + radio image Molecular cloud core Radio image Young star and outflow Near-infrared image Molecular Cloud Core 0.1pc Mass Accretion, and Outfow Single Star 5×10-5 1AU This work High dynamic range! Important physics Self-gravity of gas cloud Thermal pressure of gas Interstellar magnetic field

Introduction: Scenario of Binary Star Formation Molecular Cloud Core Mass Accretion and Outflow Fragmentation 0.1pc Binary Stars 5×10-5 This work High dynamic range! 1AU

Introduction of Nested Grid Nested grid is a cheap edition of AMR B-) AMR is somewhat expensive for nearly spherically collapsing gas cloud. Cell width Dxl = 1/2 Dxl-1 Subgrids are formed automatically according to physical conditions. Location of the subgrid is fixed. Schemes： Roe method for HD and MHD Multigrid method for selfgravity. Numerical fluxes are conserved in grid-interfaces, surfaces between fine and coase grids.

Application 1 Outflow from a Rotating Collapsing Cloud (1) Model Construction Matsumoto & Tomisaka (2003) Rigid rotation Gravitational collapse of magnetized rotating cloud Initial condition Spherical cloud Uniform magnetic field Rigid rotation Isothermal gas (γ=1) is assumed for low density, and polytrope gas (γ=5/3) for high density. This mimics formation of a seed of a stellar core. Ideal MHD Ambipolar diffusion (skiped in this talk) q Uniform Magnetic Field

Outflow from a Collapsing Cloud (2) Mechanism of Outflow Magneto-centrifugal driven outflow (Blandford & Peyne 1982) standard mechanism of outflow Velocity of outflow M=10 Outflow Centrifugal Force Fc ・ Bp/B Gas disk rotates faster than ambient gas because central region spins up by collapse. Magnetic field is twisted. Centrifugal force launch gas

Application 2 Fragmentation of a Collapsing Cloud (1) This is a very traditional but important problem for binary formation. Almost stars (~70%) are formed as binary stars. We show the first MHD numerical simulation in this problem. In almost simulations, magnetic field has been neglected. The initial condition is similar to the previous model of q=0, but includes the bar-mode perturbation. Bar-mode perturbation promotes fragmentation of a cloud core.

Fragmentation of a Collapsing Cloud (2) non MHD model Matsumoto & Hanawa(2003) Collapse of the cloud Fragmentation Fragments rotate around each other. Circum binary disk forms in late stage. No outflow forms in non magnetized model. Disk-bar model

Fragmentation of a Collapsing Cloud (3) MHD model Each fragment ejects outflow. Stream lines: magnetic field Arrows: velocity

Implementation of Nested Grid Code MHD part Numerical Flux is obtained by Roe’s method (Fukuda & Hanawa 2000). Linearlized Reman solver. 2nd order accuracy in space by MUSCL 2nd order accuracy in time by predictor-corrector method. Multi-timestep (c.f., standard AMR) Numerical flux is conserved in grid interface by means of flux correction. Two approach to div B: div B free using staggered mesh (Balsara 2001) div B cleaning (Dender et al. 2002) With either methods, low b regions of b = 10-3 - 10-6 are solved stably.

MHD part Multi timestep (1) dt dt/2 dt/4 l High resolution in time in sub-grid Dtl = Min( Dtl-1 , CFL Dx / v) The fine grid takes two or more steps while the coarse grid does one step. This ensure that CFL condition is satisfied in every grid.

Self-gravity part Overview of multigrid method Poisson equation is solved by multigrid iteration in nested grid (Matsumoto & Hanawa 2003 ApJ) Scalability Computation time ∝ Number of cell (=Nx×Ny×Nz×Nl) Fast convergence Residual is reduces by factor 10-3 - 10-2 in each iteration. Good solution Numerical flux (gravity) is conserved at grid interfaces. Solution satisfies Gauss’s and Stokes’ laws. Important feature as mass conservation in astrophysics. High accuracy 2nd order accuracy except for cells in grid interaces. 1st order accuracy for these cells.

Introduction of Multigrid method Multigrid method @ uniform grid Fine Ｓ =Gauss-Seidel ／= Interpolation ＼= Restriction coarse interpolation: Good initial guess A few times of GS iterations converge the sol. Number of cells Real grid Working grids Number of operatons ∝ Number of cells in real grid

Multigrid method @ Nested Grid Straight forward extension Real grid Working grids Number of operations ∝ Number of cell in a grid×Number of level

Multigrid method: Flux Conservation This may be common sense in plasma physics, but not in astrophysics. Poisson equation should be solved in the every grid level simultaneously and consistently. Solve every grid simultaneously  Solve each grid separately  + - coarse grid g≠0 fine grid + - coarse grid g=0 fine grid

Multigrid method: Flux Conservation (cont.) Gauss-Seidel iteration We pay special attention to Gauss-Seidel iteration (smoothing operator). Flux = gravity, g = ∇y Flux in coarse grid is obtained by sum of fluxes in fine grid at grid interface similar to HD method. This ensures Continuity of field lines of g Gauss’s and Stokes’ laws Important in astronomical simulation + =

Multigrid method: Fast convergence： Residual reduces by factor 10-2 – 10-3 in each FMG iteration. Residuals are evaluated in every level. log (Maximum residual) Round off error Number of FMG

Multigrid method: Scalability: computation cost ∝ number of cell in comp domain Computation time is in proportion to total number of cell in the computation domain, and independent of number of grid level.

Summary High resolution, MHD, self-gravitational simulations are performed by using nested grid. The nested grid is a powerful method for astrophysical simulations. The nested grid achieves high dynamic range. MHD code solves even low beta region, b=10-6, stably. Multigrid method for self-gravity shows high perfomance (fast convergence, scalably, and flux conservation). Especially, the solution satisfies Gauss’s and Stokes’ laws.