---
abstract: 'In the standard scenario of isolated low-mass star formation, strongly magnetized molecular clouds are envisioned to condense gradually into cores, driven by ambipolar diffusion. Once the cores become magnetically supercritical, they collapse to form stars. Most previous studies based on this scenario are limited to axisymmetric calculations leading to single supercritical core formation. The assumption of axisymmetry has precluded a detailed investigation of cloud fragmentation, generally thought to be a necessary step in the formation of binary and multiple stars. In this contribution, we describe the non-axisymmetric evolution of initially magnetically subcritical clouds using a newly-developed MHD code. It is shown that non-axisymmetric perturbations of modest fractional amplitude ($\sim 5\%$) can grow nonlinearly in such clouds during the supercritical phase of cloud evolution, leading to the production of either a highly elongated bar or a set of multiple dense cores.'
author:
- Fumitaka Nakamura
- 'Zhi-Yun Li'
title: On the Formation of Binary Stars and Small Stellar Groups in Magnetically Subcritical Clouds
---

\#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} =

\#1 1.25in .125in .25in

Introduction
============

Over the last few decades, a basic framework has been developed for the formation of low-mass stars in relative isolation (Shu, Adams, & Lizano 1987). In this by now “standard” picture, a molecular cloud, which is initially supported by strong magnetic field against its self-gravity, gradually contracts as the magnetic support weakens by ambipolar diffusion. Magnetically supercritical cores are formed, which collapse to produce stars. Quantitative studies based on this scenario have been carried out by many authors. In most of such studies, axisymmetry has been adopted. However, observations have shown that binary and multiple stars are common product of star formation. We need to understand how such (non-axisymmetric) stellar systems are formed in magnetically supported clouds. To elucidate the formation mechanism of binary stars and stellar groups, we have begun a systematic numerical study of the non-axisymmetric evolution initially magnetically subcritical clouds, by removing the restriction of axisymmetry. In this contribution, we present some of our recent results on this investigation.

Model and Numerical Method
==========================

As a first step, we adopted the thin-disk approximation often used in axisymmetric calculations (e.g., Basu & Mouschovias 1994; Li 2001). The disk is assumed in hydrostatic equilibrium in the vertical direction. The vertically-integrated MHD equations are solved numerically for the cloud evolution in the disk plane, with a 2D MHD code (see Li & Nakamura 2002 for code description). The magnetic structure is solved in 3D space.

The initial conditions for star formation are not well determined either observationally and theoretically. Following Basu & Mouschovias (1994), we prescribe an axisymmetric reference state. See Nakamura & Li and Li & Nakamura (2002) for the details of the reference cloud model. The reference cloud is allowed to evolve into an equilibrium configuration, with the magnetic field frozen-in. Once the equilibrium state is obtained, we reset the time to $t=0$ and add a non-axisymmetric perturbation to the surface density distribution. Then, the cloud evolution is followed with the ambipolar diffusion turned on.

Numerical Results
=================

From axisymmetric calculations, Li (2001) classified the evolution of magnetically subcritical clouds into two cases, depending mainly on the initial cloud mass and the initial density distribution. When the initial cloud is not so massive and/or has a centrally-condensed density distribution, it collapses to form a single supercritical core ([core-forming cloud]{}). On the other hand, when the initial cloud has many thermal Jeans masses and/or a relatively flat density distribution near the center, it collapses to form a ring after the central region becomes magnetically supercritical ([ring-forming cloud]{}). In the following, we show that the core-forming cloud doesn’t fragment during the dynamic collapse phase, but becomes unstable to the bar mode ([*bar growth*]{}), whereas the ring-forming cloud can break up into several blobs ([*multiple fragmentation*]{}).

Bar growth: Implication for Binary Formation
--------------------------------------------

In Fig. 1 we show an example of the bar growth models. In this model, we adopted the reference density distribution of Basu & Mouschovias (1994), which is more centrally-condensed than the model to be shown in the next subsection, and the rotation profile of Nakamura & Hawana (1997). It has a characteristic radius of $r_0=7.5\pi c_s^2/(2\pi G\Sigma_{0,\rm ref})$ (where $c_s$ is the effective isothermal sound speed and $\Sigma_{0,
\rm ref}$ the central cloud surface density in the reference state), initial flux-to-mass ratio of $\Gamma _0 = 1.5 B_{\infty}/(2\pi G^{1/2}\Sigma_{0,\rm ref})$ (where $B_\infty$ is the strength of the initially uniform background field), and a dimensionless rotation rate of $\omega=0.1$. We added to the equilibrium state an $m=2$ perturbation of surface density, with a fractional amplitude of merely 5%. During the initial quasi-static contraction phase, a central core condenses gradually out of the magnetically subcritical cloud, with no apparent tendency for the mode to grow. Rather, the iso-density contours appear to oscillate, changing the direction of elongation along $x$-axis in the disk plane to $y$-axis. After a supercritical core develops, the contraction becomes dynamic and the bar mode grows significantly. During the intermediate stages \[panels (c) and (d)\], the aspect ratio of the bar remains more or less frozen at $R\sim 2$. As the collapse continues, the growth rate of the bar increases dramatically by the very end of the starless collapse. The density distribution along the minor axis of the bar is well reproduced by a power-law profile of $r^{-2}$, which is different from that of an isothermal equilibrium filament ($\propto r^{-4}$). When the volume density exceeds a critical value of $10^{12}$ cm$^{-3}$, we changed the equation of state from isothermal to adiabatic, to mimic the transition to the optically thick regime. The bar is surrounded by an accretion shock, which is analogous to the first core of spherical calculations \[panel (f)\]. The aspect ratio of this “first” bar continues to increase during the early optically thick regime. The highly elongated first bar is expected to break up into two or more pieces. We suspect that bar fragmentation is an important, perhaps the dominant, route for binary and small multiple-star formation.

We have also followed the evolution of this model cloud perturbed by other (higher) $m$ modes ($m\ge3$), and found no significant mode growth. The reason why the cloud is unstable only to the bar mode appears to be the following. In the absence of nonaxisymmetric perturbations, the supercritical collapse approaches a self-similar solution derived approximately by Nakamura & Hanawa (1997). In the self-similar solution, the effective radius of the central plateau is at most 3-4 times the effective Jeans length, making the cloud unstable to dynamic contraction but not to multiple fragmentation. Indeed, Nakamura & Hanawa (1997) showed that the self-similar solution is unstable only to the $m=2$ mode, consistent with our result. The tendency for the supercritical collapse to approach the self-similar solution is responsible for the bar formation during the dynamic collapse. Detailed numerical results on bar formation will appear elsewhere (Nakamura & Li 2002, in preparation).

Multiple Fragmentation and Formation of Small Stellar Groups
------------------------------------------------------------

In Fig. 2 we show an example of the multiple fragmentation models. In this model, we adopted the reference density profile of Li (2001) with $n=8$, which is less centrally-condensed than the model shown in the previous subsection. The model has a characteristic radius of $r_0=7\pi c_s^2/(2\pi G\Sigma_{0,\rm ref})$, initial flux-to-mass ratio of $\Gamma _0 = 1.5B_{\infty}/(2\pi G^{1/2}\Sigma_{0,\rm ref})$, and rotation rate of $\omega=0.1$. Random density perturbations are added to the axisymmetric equilibrium state. The maximum fractional amplitude of the perturbations is set to 10%. During the quasi-static contraction phase, the infall motions are subsonic, and there is no sign of fragmentation. Once the flux-to-mass ratio in the central high-density region drops below the critical value, the contraction is accelerated near the center. As the collapse continues, the central supercritical region begins to fragment into five blobs. By the time shown in panel (f), the blobs are well separated from the background material and are significantly elongated. Subsequent dynamic collapse of each blob is similar to that of the bar growth case. Individually, we expect each core to produce a highly elongated bar, which could further break up into pieces, producing perhaps binary or multiple stars. Together, the formation of a small stellar group is the most likely outcome. Detailed numerical results on multiple fragmentation are given in Li & Nakamura (2002).

Summary
=======

Our main conclusion is that despite (indeed because of) the presence of the strong magnetic field, the initially magnetically subcritical clouds are unstable to non-axisymmetric perturbations during the supercritical phase of cloud evolution. The cloud evolution is classified into two cases, depending mainly on the initial cloud mass and density distribution. When the initial cloud is not so massive and has a centrally condensed density distribution, it doesn’t break into pieces but becomes unstable to a bar mode ([*bar growth*]{}). This bar is expected to fragment into two or more pieces to form binary or small multiple stars, when the bar becomes opaque to dust emission and is surrounded by an accretion shock. On the other hand, when the initial cloud has many Jeans masses and a relatively flat density distribution near the center, it can fragment into several or many cores after a supercritical region develops near the center ([*multiple fragmentation*]{}). This fragmentation may be responsible for small cluster formation in relatively isolated regions.

Boss (2000) showed the fragmentation of 3D magnetic clouds numerically, treating the magnetic forces and ambipolar diffusion in an approximate way (see also the contribution by Boss). He concluded that magnetic fields (magnetic tension force) can enhance cloud fragmentation by reducing the tendency for the development of a central singularity, which would make fragmentation more difficult. We also find that magnetic fields can have beneficial effects on fragmentation. Strong magnetic fields can support clouds with many Jeans masses and flatten mass distribution, both of which are conducive to fragmentation once the magnetic support weakens through ambipolar diffusion.

Numerical computations in this work were carried out at the Yukawa Institute Computer Facilities, Kyoto University. F.N. gratefully acknowledges the support of the JSPS Postdoctoral Fellowships for Research Abroad.
