DEVELOPMENT OF SELF-GRAVITATIONAL ADAPTIVE MESH REFINEMENT FOR SIMULATING BINARY STARFORMATION. Tomoaki Matsumoto, Department of Humanity and Environment, Hosei University, Fujimi, Chiyoda-ku, Tokyo,102-8160, Japan, (matsu@i.hosei.ac.jp).

Introduction: A self-gravitational adaptive mesh refinement (AMR) code is newly developed for simulating protostellar collapse and fragmentation of a cloud core.
It is widely accepted that binary and multiple stars form
as a result of fragmentation in a collapsing molecular cloudcore [1]. Fragmentation of a molecular cloud core has beeninvestigated through numerical simulations by many authors
in the last two decades.
In order to follow the protostellar collapse and fragmentation, extremely wide dynamic range of spatial resolution isrequired of a numerical scheme. An initial condition of protostellar collapse is a molecular cloud core, of which size is∼ 0.01 − 0.1 pc. On the other hand, a fragment, a seed of abinary star, has a radius of only ∼ 1 AU. Wide spatial dynamic
range of ∼ 103−4 is therefore required to resolve both the entire cloud core and the small fragments. Truelove et al. [2]indicates that sufficient resolution is necessary for avoiding artificial fragmentation during protostellar collapse calculations(the Jeans condition). The AMR is one of the most popular
technique to obtain the resolution with wide dynamic range inthe recent simulations [2,3,4,5].
In this paper, implementation of a new AMR code is presented, and preliminary results with the code is also shown.
Implementation: The AMR code is vectorized and parallized on the supercomputer Fujitsu VPP5000. The MPI library
is used for parallization. The entire code is written in FORTRAN90. The AMR code consists of three parts: (1) gridgeneration, (2) hydrodynamics, and (3) self-gravity.
Grid Generation: Block-structured grids are adopted forgrid configuration. Each block has Nx × Ny × Nz cubic cells,
where Nx , Ny , and Nz denote the number of cells in the x,y, and z directions, and they are constant over the blocks.
The results with Nx = Ny = Nz = 8 are shown here. The
number of cells within a block is fixed, but the cell width differs
depending on the grid-level; the cell width of each grid-level
decreases successively by a factor of 2. The coarsest grid-levelis labeled ` = 0 (the base grid), and the `-th grid-level has 2`times higher spatial resolution than the coarsest grid-level.
Data is stored by octree structure; the parent (coarse) block
is linked with eight fine (children) blocks. Moreover, a block
is linked with its neighbor blocks.
As a cloud collapses, small structures form in the dense
region. When a block capturing the dense region satisfies a criterion for grid generation, eight finer blocks are automaticallygenerated from the parent block, obtaining finer resolution.
In the present calculations, the criterion of λJ /8 < h is employed, where λJ and h denote the Jeans length and the cellwidth of the parent block. This criterion satisfies the Jeanscondition introduced by Truelove et al. [2].
Hydrodynamics: The hydrodynamics method for theAMR has been developed by extending the method for a nested
grid [6]. The solver is based on Roe method [7], but has

been extended for solving the ideal, barotropic equations ofstate. A MUSCL approach and predictor-corrector method
are adopted here for integration over time in order to achieve
second-order accuracy in space and time. A multi-timestep
scheme is adopted, and all the numerical fluxes are conservedat the interfaces between the adjacent blocks, and even between
coarse and fine blocks [8].
Self-Gravity: Self-gravity is updated in every time step
across all grid-levels by using the multigrid iteration for thePoisson equation. The method shows fast convergence ofthe solution; a residual reduces by a factor ∼ 10−3 by eachiteration. The numerical flux of gravity, which is equivalent
to the gravitational force, is also conserved at the interfaces
between the adjacent blocks, similarly to the hydrodynamics
method described above. This flux conservation means thata field line of gravity never ends at the interface, leading toaccurate evaluation of gravitational torque [11].
The method consists of three parts: (1) a full multigrid(FMG) scheme on the AMR grid, (2) V-cycle full approximation scheme (FAS) on the AMR grid, and (3) FMG for thebase grid. In the first FMG scheme, the solution is converged
simultaneously over the fine and coarse grids (composite grids
corresponding to the hierarchy of grids). The composite grids
are coarsen during the FMG cycle, so that the number of cells
per block decreases from Nx × Ny × Nz to 23 . The solution
on these 23 cells is converged further by the V-cycle FAS, andthe FMG for the base grid.
Application to Binary Formation: Using the present
AMR code, fragmentation during protostellar collapse is followed for the same model as Matsumoto & Hanawa [6]. Figure 1 shows the initial condition by cross-section. At theinitial stage, the cloud core has density distribution factor of
1.1 denser than that of the critical Bonner-Ebert sphere [9, 10],and rotates rigidly at angular velocity of Ω = 0.1t−1
ﬀ , where tﬀ
denotes the initial freefall timescale at the cloud center and defined as tﬀ = (3π/32Gρc )1/2 . The cloud also includes a velocity perturbation of m = 3 mode with a very small amplitude(10−3 ) to break the point symmetry. The initial central densityis ρc = 1 × 10−19 g cm−3 , the temperature is T = 10 K, theradius of the cloud core is Rc = 0.144 pc, and the mass of thecloud is Mc = 3.24 M¯ . The parameter α(= Eth /|Egrav |)has been discussed in terms of instability against fragmentation, where Eth and Egrav are thermal energy and gravitationalenergy. The model cloud has α = 0.765.
A barotropic equation of state is adopted: P = c2s ρ forρ < ρcr , and P = c2s ρcr (ρ/ρcr )7/5 for ρ ≥ ρcr , where P ,cs , and ρcr denote pressure, isothermal sound speed, and thecritical density for switching the equation of state. We set
cs = 0.19 km s−1 , and ρcr = 2 × 10−13 g cm−3 .
The cloud core undergoes the isothermal runaway collapse
in the early stage, and an adiabatic core forms at the cloud center after the maximum density exceeds the critical density ρcr .

Self-gravitational AMR for Binary Star Formation: T. Matsumotolog ρ [g cm-3]

t = 0.0000000E+00 [yr]log ρ [g cm-3]

t = 1.1985403E+06 [yr]3043•10-92•104-19.2

201•104-19.410-10-19.6-1•104

y [AU]y [AU]-1100-12-10-19.8-13

4-2•10-20-20.0-144-3•10-3•104 -2•104 -1•104

0x [AU]

1•104

2•104-30

3•104

Figure 1: Density distribution in the mid-plane for the initial condition. The vertical and horizontal lines represent theblock structure (boundaries of the blocks), and each blockcontains 83 cells. The cloud core is a factor 1.1 denser thanthe critical Bonner-Ebert sphere, and rotates at the angular velocity of Ω = 0.1t−1ﬀ .

y [AU]

30-920-1010-110-12-10-13-20-100x [AU]102030Figure 3: Same as Fig. 1 but for the stage of binary formation.
This figure shows grid-level of ` = 10 − 13.
log ρ [g cm-3]

t = 1.1987082E+06 [yr]3020log ρ [g cm-3]

t = 1.1984447E+06 [yr]-15-30-10y [AU]100-12-10-20-14-30-30-20-14-30-30-20-100x [AU]102030Figure 2: Same as Fig. 1 but for the stage of fragmentation.
The oblate adiabatic core deforms to a ring-shape and then toa bar-shape. This figure shows grid-level of ` = 10−13. Notethat the spatial scale is different form Fig. 1 by magnifying bya factor of 103 .
The adiabatic core has an oblate shape; its radius and thickness
are 6 AU and 3 AU, respectively. The oblate adiabatic core
deforms to a ring shape temporarily, and then to a bar shape.
The bar-shaped adiabatic core breaks into two fragments asshown in Figure 2. The fragments evolve to a proto-binary
system 100 yr after the fragmentation, as shown in Figure 3.
The binary system is surrounded by a circumbinary disk having spiral arms. One of the large spiral arms extending out
in the y-direction evolves to an additional adiabatic core because of self-gravity. The new adiabatic core rotates aroundthe central binary system, and the whole system becomes a hierarchical triple system as shown in Figure 4. The separation-20-100x [AU]102030Figure 4: Same as Fig. 1 but for the stage of hierarchical triple
system formation. This figure shows grid-level of ` = 10−14.
between the central adiabatic cores is ∼ 10 AU, while the newadiabatic core is separated by ∼ 30 AU from the central closesystem. The orbits of the adiabatic cores seem to evolve in thefurther stages as indicated by Matsumoto & Hanawa [6]. The
overall evolution is also consistent with that of Matsumoto &Hanawa [6].
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