Protostar Formation in Magnetic Molecular Clouds beyond Ion Detachment: I. Formulation of the Problem and Method of Solution

We formulate the problem of the formation of magnetically supercritical cores in magnetically subcritical parent molecular clouds, and the subsequent collapse of the cores to high densities, past the detachment of ions from magnetic field lines and into the opaque regime. We employ the six-fluid MHD equations, accounting for the effects of grains (negative, positive and neutral) including their inelastic collisions with other species. We do not assume that the magnetic flux is frozen in any of the charged species. We derive a generalized Ohm's law that explicitly distinguishes between flux advection (and the associated process of ambipolar diffusion) and Ohmic dissipation, in order to assess the contribution of each mechanism to the increase of the mass-to-flux ratio of the central parts of a collapsing core and possibly to the resolution of the magnetic flux problem of star formation. We show how our formulation is related to and can be transformed into the traditional, directional formulation of the generalized Ohm's law, and we derive formulae for the perpendicular, parallel and Hall conductivities entering the latter, which include, for the first time, the effect of inelastic collisions between grains. In addition, we present a general (valid in any geometry) solution for the velocities of charged species as functions of the velocity of the neutrals and of the effective flux velocity (which can in turn be calculated from the dynamics of the system and Faraday's law). The last two sets of formulae can be adapted for use in any general non-ideal MHD code to study phenomena beyond star formation in magnetic clouds. The results, including a detailed parameter study, are presented in two accompanying papers.Comment: 17 pages, emulateapj; accepted for publication in the Astrophysical Journa

A Comparative Study of the Parker Instability under Three Models of the Galactic Gravity

To examine how non-uniform nature of the Galactic gravity might affect length and time scales of the Parker instability, we took three models of gravity, uniform, linear and realistic ones. To make comparisons of the three gravity models on a common basis, we first fixed the ratio of magnetic pressure to gas pressure at $\alpha$ = 0.25, that of cosmic-ray pressure at $\beta$ = 0.4, and the rms velocity of interstellar clouds at $a_s$ = 6.4 km s$^{-1}$, and then adjusted parameters of the gravity models in such a way that the resulting density scale heights for the three models may all have the same value of 160 pc. Performing linear stability analyses onto equilibrium states under the three models with the typical ISM conditions, we calculate the maximum growth rate and corresponding length scale for each of the gravity models. Under the uniform gravity the Parker instability has the growth time of 1.2$\times10^{8}$ years and the length scale of 1.6 kpc for symmetric mode. Under the realistic gravity it grows in 1.8$\times10^{7}$ years for both symmetric and antisymmetric modes, and develops density condensations at intervals of 400 pc for the symmetric mode and 200 pc for the antisymmetric one. A simple change of the gravity model has thus reduced the growth time by almost an order of magnitude and its length scale by factors of four to eight. These results suggest that an onset of the Parker instability in the ISM may not necessarily be confined to the regions of high $\alpha$ and $\beta$.Comment: Accepted for publication in ApJ, using aaspp4.sty, 18 text pages with 9 figure

Turbulent Control of the Star Formation Efficiency

Supersonic turbulence plays a dual role in molecular clouds: On one hand, it contributes to the global support of the clouds, while on the other it promotes the formation of small-scale density fluctuations, identifiable with clumps and cores. Within these, the local Jeans length \Ljc is reduced, and collapse ensues if \Ljc becomes smaller than the clump size and the magnetic support is insufficient (i.e., the core is ``magnetically supercritical''); otherwise, the clumps do not collapse and are expected to re-expand and disperse on a few free-fall times. This case may correspond to a fraction of the observed starless cores. The star formation efficiency (SFE, the fraction of the cloud's mass that ends up in collapsed objects) is smaller than unity because the mass contained in collapsing clumps is smaller than the total cloud mass. However, in non-magnetic numerical simulations with realistic Mach numbers and turbulence driving scales, the SFE is still larger than observational estimates. The presence of a magnetic field, even if magnetically supercritical, appears to further reduce the SFE, but by reducing the probability of core formation rather than by delaying the collapse of individual cores, as was formerly thought. Precise quantification of these effects as a function of global cloud parameters is still needed.Comment: Invited review for the conference "IMF@50: the Initial Mass Function 50 Years Later", to be published by Kluwer Academic Publishers, eds. E. Corbelli, F. Palla, and H. Zinnecke

Nonlinear Parker Instability with the Effect of Cosmic-Ray Diffusion

We present the results of linear analysis and two-dimensional local magnetohydrodynamic (MHD) simulations of the Parker instability, including the effects of cosmic rays (CRs), in magnetized gas disks (galactic disks). As an unperturbed state for both the linear analysis and the MHD simulations, we adopted an equilibrium model of a magnetized two-temperature layered disk with constant gravitational acceleration parallel to the normal of the disk. The disk comprises a thermal gas, cosmic rays and a magnetic field perpendicular to the gravitational accelerartion. Cosmic ray diffusion along the magnetic field is considered; cross field-line diffusion is supposed to be small and is ignored. We investigated two cases in our simulations. In the mechanical perturbation case we add a velocity perturbation parallel to the magnetic field lines, while in the explosional perturbation case we add cosmic ray energy into a sphere where the cosmic rays are injected. Linear analysis shows that the growth rate of the Parker instability becomes smaller if the coupling between the CR and the gas is stronger (i.e., the CR diffusion coefficient is smaller). Our MHD simulations of the mechanical perturbation confirm this result. We show that the falling matter is impeded by the CR pressure gradient, this causes a decrease in the growth rate. In the explosional perturbation case, the growth of the magnetic loop is faster when the coupling is stronger in the early stage. However, in the later stage the behavior of the growth rate becomes similar to the mechanical perturbation case.Comment: 18 pages, 11 figures, accepted in Ap

Three-dimensional simulations of the Parker instability in a uniformly rotating disk

We investigate the effects of rotation on the evolution of the Parker instability by carrying out three-dimensional numerical simulations with an isothermal magnetohydrodynamic code. These simulations extend our previous work on the nonlinear evolution of the Parker instability by J. Kim and coworkers. The initial equilibrium system is composed of exponentially stratified gas and a field (along the azimuthal direction) in a uniform gravity (along the downward vertical direction). The computational box, placed at the solar neighborhood, is set to rotate uniformly around the Galactic center with a constant angular speed. The instability has been initialized by random velocity perturbations. In the linear stage, the evolution is not much different from that without rotation, and the mixed (undular + interchange) mode regulates the system. The interchange mode induces alternating dense and rarefied regions with small radial wavelengths, while the undular mode bends the magnetic field lines in the plane of the azimuthal and vertical directions. In the nonlinear stage, flow motion overall becomes chaotic, as in the case without rotation. However, as the gas in higher positions slides down along field lines forming supersonic flows, the Coriolis force becomes important. As oppositely directed flows fall into valleys along both sides of the magnetic field lines, they experience the Coriolis force toward opposite directions, which twists the magnetic field lines there. Hence, we suggest that the Coriolis force plays a role in randomizing the magnetic field. The three-dimensional density structure formed by the instability is still sheetlike with the short dimension along the radial direction, as in the case without rotation. However, the long dimension is now slightly tilted with respect to the mean field direction. The shape of high-density regions is a bit rounder. The maximum enhancement factor of the vertical column density relative to its initial value is about 1.5, which is smaller than that in the case without rotation. We conclude that uniform rotation does not change our point of view that the Parker instability alone is not a viable mechanism for the formation of giant molecular cloudsopen252

The alignment of molecular cloud magnetic fields with the spiral arms in M33

The formation of molecular clouds, which serve as stellar nurseries in galaxies, is poorly understood. A class of cloud formation models suggests that a large-scale galactic magnetic field is irrelevant at the scale of individual clouds, because the turbulence and rotation of a cloud may randomize the orientation of its magnetic field. Alternatively, galactic fields could be strong enough to impose their direction upon individual clouds, thereby regulating cloud accumulation and fragmentation, and affecting the rate and efficiency of star formation. Our location in the disk of the Galaxy makes an assessment of the situation difficult. Here we report observations of the magnetic field orientation of six giant molecular cloud complexes in the nearby, almost face-on, galaxy M33. The fields are aligned with the spiral arms, suggesting that the large-scale field in M33 anchors the clouds.Comment: to appear in Natur

Formation and Fragmentation of Gaseous Spurs in Spiral Galaxies

Intermediate-scale spurs are common in spiral galaxies, but perhaps most distinctively evident in a recent HST image of M51 (Scoville & Rector 2001). We investigate, using time-dependent numerical MHD simulations, how such spurs could form (and subsequently fragment) from the interaction of a gaseous ISM with a stellar spiral arm. We model the gaseous medium as a self-gravitating, magnetized, differentially-rotating, razor-thin disk. The basic flow shocks and compresses as it passes through a local segment of a tightly-wound, trailing stellar spiral arm, modeled as a rigidly-rotating gravitational potential. We first construct 1D profiles for flows with spiral shocks. When the post-shock Toomre parameter Q_sp is sufficiently small, self-gravity is too large for one-dimensional steady solutions to exist. The critical values of Q_sp are 0.8, 0.5, and 0.4 for our models with zero, sub-equipartition, and equipartition magnetic fields, respectively. We then study the growth of self-gravitating perturbations in fully 2D flows, and find that spur-like structures rapidly emerge in our magnetized models. We associate this gravitational instability with the magneto-Jeans mechanism, in which magnetic tension forces oppose the Coriolis forces. The shearing and expanding velocity field shapes the condensed material into spurs as it flows downstream from the arms. Although we find swing amplification can help form spurs when the arm-interarm contrast is moderate, unmagnetized systems that are quasi-axisymmetrically stable are generally also stable to nonaxisymmetric perturbations, suggesting that magnetic effects are essential. In nonlinear stages, the spurs in our models undergo fragmentation to form 4\times 10^6 solar mass clumps, which we suggest could evolve into bright arm/interarm HII regions as seen in spiral galaxies.Comment: 32 pages, 14 figures, Accepted for publication in ApJ; better postscript figures available from http://www.astro.umd.edu/~kimwt/FIGURE2/ ; for associated Animated GIF movies, see http://www.astro.umd.edu/~kimwt/MOVIES

Magnetic Reconnection Triggered by the Parker Instability in the Galaxy: Two-Dimensional Numerical Magnetohydrodynamic Simulations and Application to the Origin of X-Ray Gas in the Galactic Halo

We propose the Galactic flare model for the origin of the X-ray gas in the Galactic halo. For this purpose, we examine the magnetic reconnection triggered by Parker instability (magnetic buoyancy instability), by performing the two-dimensional resistive numerical magnetohydrodynamic simulations. As a result of numerical simulations, the system evolves as following phases: Parker instability occurs in the Galactic disk. In the nonlinear phase of Parker instability, the magnetic loop inflates from the Galactic disk into the Galactic halo, and collides with the anti-parallel magnetic field, so that the current sheets are created in the Galactic halo. The tearing instability occurs, and creates the plasmoids (magnetic islands). Just after the plasmoid ejection, further current-sheet thinning occurs in the sheet, and the anomalous resistivity sets in. Petschek reconnection starts, and heats the gas quickly in the Galactic halo. It also creates the slow and fast shock regions in the Galactic halo. The magnetic field ($B\sim 3 \mu$G), for example, can heat the gas ($n\sim 10^{-3}$ cm$^{-3}$) to temperature of $\sim 10^6$ K via the reconnection in the Galactic halo. The gas is accelerated to Alfv\'en velocity ($\sim 300$ km s$^{-1}$). Such high velocity jets are the evidence of the Galactic flare model we present in this paper, if the Doppler shift of the bipolar jet is detected in the Galactic halo. Full size figures are available at http://www.kwasan.kyoto-u.ac.jp/~tanuma/study/ApJ2002/ApJ2002.htmlComment: 13 pages, 12 figures, uses emulateapj.sty, accepted by Ap

The Parker instability under a linear gravity

A linear stability analysis has been done to a magnetized disk under a linear gravity. We have reduced the linearized perturbation equations to a second-order differential equation that resembles the Schrodinger equation with the potential of a harmonic oscillator. Depending on the signs of energy and potential terms, eigensolutions can be classified into ''continuum'' and ''discrete'' families. When the magnetic field is ignored, the continuum family is identified as the convective mode, while the discrete family is identified as acoustic-gravity waves. If the effective adiabatic index gamma is less than unity, the former develops into the convective instability. When a magnetic field is included, the continuum and discrete families further branch into several solutions with different characters. The continuum family is divided into two modes: one is the original Parker mode, which is a slow MHD mode modulated by the gravity, and the other is a stable Alfven mode. The Parker modes can be either stable or unstable depending on gamma. When gamma is smaller than a critical value gamma(cr), the Parker mode becomes unstable. The discrete family is divided into three modes: a stable fast MHD mode modulated by the gravity, a stable slow MHD mode modulated by the gravity, and an unstable mode that is also attributed to a slow MHD mode. The unstable discrete mode does not always exist. Even though the unstable discrete mode exists, the Parker mode dominates it if the Parker mode is unstable. However, if gamma greater than or equal to gamma(cr), then the discrete mode could be the only unstable one. When gamma is equal gamma(cr), the minimum growth time of the unstable discrete mode is 1.3 x 10(8) yr, with a corresponding length scale of 2.4 kpc. It is suggestive that the corrugated features seen in the Galaxy and external galaxies are related to the unstable discrete modeopen131