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PROTOSTELLAR COLLAPSE OF MAGNETO-TURBULENT CLOUD CORES: FORMATION OF
PROTOPLANETARY DISKS AND OUTFLOWS
Tomoaki Matsumoto1 , Masahiro M. Machida2 , and Shuichiro Inutsuka3

ABSTRACT
We investigate formation of protoplanetary disks and outflows inside collapsing turbulent molecular
cloud cores by resistive MHD simulations. By using a self-gravitational adaptive mesh refinement(AMR) code, SFUMATO, the collapse of the cloud core is followed in a wide dynamic range; we resolve
both a cloud cores scale and a protoplanetary disk scale. A protostar is modeled with a Lagrangian
sink particle. The ohmic dissipation is solved by an implicit scheme. We followed collapse of theturbulent molecular cloud cores and accretion onto the protostars (sink particles) up to ∼ 1000 yr
after the protostar formation. The simulations show that weak magnetic fields and strong turbulence
promote a relatively rapid growth of a protoplanetary disk around the sink particle. The strong
magnetic field models exhibit the cavities in infalling envelopes around the protostars. The protostar
accrates the gas while the magnetic fields are decoupled from the gas around the protostar because of
the resistivity. The decoupled magnetic fields generate strong filed region, of which magnetic pressurecauses the cavity in the infalling envelope. All the models exhibit protostellar outflows irrespective of
the initial field strength and the initial turbulent velocity.
Subject headings:
HD Roe1. INTRODUCTION

2. MODELS OF CLOUD CORESAs an initial model of a molecular cloud core, we consider a turbulent, spherical cloud threaded by a uniform
magnetic field. The cloud is confined by a uniform ambient gas. The initial central density is set equal toρ0 = 10−18 g cm−3 , which corresponds to a number density of n0 = 2.61 × 105 cm−3 for an assumed mean molecular weight of 2.3.
Electronic address: matsu@hosei.ac.jp
1 Faculty of Humanity and Environment, Hosei University, Fujimi, Chiyoda-ku, Tokyo 102-8160, Japan
2 Department of Earth and Planetary Sciences, Faculty of Sciences, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka
812-8581, Japan
3 Department of Physics, Nagoya University, Furo-cho, Chikusaku Nagoya, Aichi 464-8602, JapanMHD HLLD

SfumatoMagnetic fields and interstellar turbulence are believedto play important roles in the gravitational collapse ofmolecular cloud cores. The measured magnetic fields ofmolecular clouds and molecular cloud cores are strong
and the magnetic energy is approximately equal to thekinetic energy (Crutcher 1999). Moreover, Burkert &Bodenheimer (2000) suggests that the rotation of cloud
cores originates in turbulence.
There have been very few theoretical studies of collapseof magnetized turbulent cloud cores in protostars, despite
the importance of turbulence and magnetic fields. Although self-gravitational turbulent simulations have beenperformed by many researchers (e.g., Gammie et al. 2003;Li et al. 2004; Offner et al. 2008; Bate 2009), most investigated large-scale turbulence and focused on cloud coreformation.
In this paper, we investigate formation of protoplanetary disks and outflows inside magnetized turbulent
cloud cores. In order to achieve a wide spatial dynamic
range of the protostellar collapse, a self-gravitational
MHD-AMR code is adopted.

MHD Roehydrosalar

vectormultigridlinearsegravity

Scalar, linear

heat cond.

Scalar, linear

Ohmic dissp.

Vector, linear

OD+HE+AD

Vector, nonlinear

nonlinearsink particle

Figure 1. Composition of a self-gravitational MHD-AMR code,
SFUMATO.

The initial velocity field is an incompressible turbulent flow with a power spectrum of P (k) ∝ k −4 , generated according to Dubinski et al. (1995), where k is thewavenumber. The models are constructed by changing
the mean Mach number of the initial velocity field in therange M = 0.5 − 3.
The initial magnetic field is uniform in the z-direction.
The field strength is given by Bz = αBcr , where α denotes the non-dimensional flux-to-mass ratio, Bcr denotes the critical field strength given by Bcr = 2πG1/2 Σ(Nakano & Nakamura 1978; Tomisaka et al. 1988), andΣ denotes the column density at the cloud center.
The barotropic equation of state is assumed wherethe gas temperature is 10 K below the critical densityρcr = 2 × 10−13 g cm−3 (ncr = 5.24 × 1010 cm−3 ), andit increases with the adiabatic index γ = 7/5 above ρcr .
The model of the resistivity is taken from Machida et al.
(2008).
3. NUMERICAL METHODS

We calculated the collapse of the cloud cores using
the AMR code, SFUMATO (Matsumoto 2007), of which
the module composition is shown in Figure 1. The code
adopts a block-structured grid as the grid of the AMR hierarchy. The HLLD scheme of Miyoshi & Kusano (2005)

Matsumoto et al.

Figure 2. Column density distributions in the x − z planes on thescale of the infalling envelopes for all the models. The regions of(200 AU)3 are plotted for the last stages. The white lines denotethe loci of the sink particles.

is adopted as the MHD solver, with the hyperbolic divergence cleaning method of Dedner et al. (2002). TheMHD solver achieves second-order accuracy in space andtime due to the TVD and predictor-corrector methods.
The self-gravity is solved by the multigrid method, exhibiting spatial second-order accuracy. The numericalfluxes are conserved by using a refluxing procedure inboth the MHD and self-gravity solvers.
The dissipation term due to resistivity (the ohmic dissipation) in the MHD equations is solved by an operator
splitting approach. The dissipation term is solved by animplicit scheme with the multigrid method (Matsumoto2011). We also adopt a correction proposed by Graves
et al. (2008) for acceleration of convergence.
The Lagrangian sink particle is adopted as a sub-gridmodeling of a protostar (e.g., Krumholz et al. 2004; Federrath et al. 2010).
4. RESULTS

Figure 2 shows infalling envelopes on a 200 AU scale.
The strong turbulent models (M = 3) have filamentaryenvelopes while the weak turbulent models (M = 0.5)
have disk envelopes. The shapes of the envelopes reflect
the masses of the cloud cores at the initial conditions; thestrong turbulent models have higher mass than the weakturbulent models in order to promote the gravitational
collapse.
The strong field modes (B = 0.25Bcr ) exhibit cavities in the envelopes. We confirmed that the cavities are
created by the magnetic pressure. The strong magnetic
pressure is caused by decoupling of the magnetic fieldfrom the accreted gas.
Figure 3 shows the protoplanetary disks on a 20 AU
scale. The strong turbulent models and weak magnetic
field models tend to have large disks. This implies thatthe angular momenta of the disks originate in the initial
turbulent flow.
Figure 4 shows the growth of the disks qualitatively.
The centrifugal radii of disks increase for ∼ 1000 yr for
all the models except for the weak turbulent models with
M = 0.5. The continuous growth of the disk radii implies
a larger disks are obtained in the further stages.
All the models exhibit protostellar outflows. Note thatthe outflows are not responsible to the cavities of theenvelopes in the strong field models.

Figure 3. Column density distributions in the x − y planes on
the scale of the protoplanetary disks. The regions of (20 AU)3 areplotted for the last stages. The white lines denote the loci of thesink particles.
4(M, α ) =(0.5, 0.1)
(1, 0.1)(3, 0.1)(0.5, 0.25)(1, 0.25)(3, 0.25)

3
rkeplar [AU]

22

1

0
0

200

400
600
time [yr]

800

1000

Figure 4. Mean centrifugal radii of the disks as a function ofelapse time after the protostar formation. Thick and thin lines are
for resistive MHD models and ideal MHD models, respectively.
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