花輪 知幸

<title>Abstract</title>Filamentary molecular clouds are thought to fragment to form clumps and cores. However, the fragmentation may be suppressed by magnetic force if the magnetic fields run perpendicularly to the cloud axis. We evaluate the effect using a simple model. Our model cloud is assumed to have a Plummer like radial density distribution, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1743921319003600_inline1.png" /><tex-math> $\rho = {\rho _{\rm{c } } }{\left[ {1 + {r^2}/(2p{H^2})} \right]^{2p } }$ </tex-math></alternatives></inline-formula>, where <italic>r</italic> and <italic>H</italic> denote the radial distance from the cloud axis and the scale length, respectively. The symbols, <italic>ρ</italic>_c and <italic>p</italic> denote the density on the axis and radial density index, respectively. The initial magnetic field is assumed to be uniform and perpendicular to the cloud axis. The model cloud is assumed to be supported against the self gravity by gas pressure and turbulence. We have obtained the growth rate of the fragmentation instability as a function of the wavelength, according to the method of Hanawa, Kudoh &amp; Tomisaka (2017). The instability depends crucially on the outer boundary. If the displacement vanishes in regions very far from the cloud axis, cloud fragmentation is suppressed by a moderate magnetic field. If the displacement is constant along the magnetic field in regions very far from the cloud, the cloud is unstable even when the magnetic field is infinitely strong. The wavelength of the most unstable mode is longer for smaller index, <italic>p</italic>.

Studies of the structure and evolution of protoplanetary disks are important for understanding star and planet formation. Here, we present the direct image of an interacting binary protoplanetary system. Both circumprimary and circumsecondary disks are resolved in the near-infrared. There is a bridge of infrared emission connecting the two disks and a long spiral arm extending from the circumprimary disk. Numerical simulations show that the bridge corresponds to gas flow and a shock wave caused by the collision of gas rotating around the primary and secondary stars. Fresh material streams along the spiral arm, consistent with the theoretical scenarios where gas is replenished from a circummultiple reservoir.

We studied the collapse of rotating molecular cloud cores with inclined magnetic fields, based on three-dimensional numerical simulations. The numerical simulations start from a rotating Bonnor-Ebert isothermal cloud in a uniform magnetic field. The magnetic field is initially taken to be inclined from the rotation axis. As the cloud collapses, the magnetic field and rotation axis change their directions. When the rotation is slow and the magnetic field is relatively strong, the direction of the rotation axis changes to align with the magnetic field, as shown earlier by Matsumoto & Tomisaka. When the magnetic field is weak and the rotation is relatively fast, the magnetic field inclines to become perpendicular to the rotation axis. In other words, the evolution of the magnetic field and rotation axis depends on the relative strength of the rotation and magnetic field. Magnetic braking acts to align the rotation axis and magnetic field, while the rotation causes the magnetic field to incline through dynamo action. The latter effect dominates the former when the ratio of the angular velocity to the magnetic field is larger than a critical value Omega(0)/B-0 &gt; 0.39G(1/2)c(s)(-1) where B-0, Omega(0), G, and c(s) denote the initial magnetic field, initial angular velocity, gravitational constant, and sound speed, respectively. When the rotation is relatively strong, the collapsing cloud forms a disk perpendicular to the rotation axis and the magnetic field becomes nearly parallel to the disk surface in the high-density region. A spiral structure appears due to the rotation and the wound up magnetic field in the disk.

We show three-dimensional numerical simulations on the core collapse of a rotating massive star threaded by strong magnetic fields. The initial magnetic field is nearly uniform in the core and inclined by 60° with respect to the rotation axis. A typical model shows four outflows after the bounce. The first outflow is nearly spherical and is launched just after the core bounce. The last outflow is bipolar and parallel to the initial rotation axis. It is driven by the toroidal magnetic field induced by the rotation. Its launch is later when the initial magnetic field is weaker. The features of the outflow and remnant magnetic field depend on the inclination angle of the initial magnetic field.

Subsequent to Paper I, the evolution and fragmentation of a rotating magnetized cloud are studied with use of three-dimensional magnetohydrodynamic nested grid simulations. After the isothermal runaway collapse, an adiabatic gas forms a protostellar first core at the centre of the cloud. When the isothermal gas is stable for fragmentation in a contracting disc, the adiabatic core often breaks into several fragments. Conditions for fragmentation and binary formation are studied. All the cores which show fragmentation are geometrically thin, as the diameter-to-thickness ratio is larger than 3. Two patterns of fragmentation are found. (1) When a thin disc is supported by centrifugal force, the disc fragments into a ring configuration (ring fragmentation). This is realized in a rapidly rotating adiabatic core as Omega &gt; 0.2 tau(ff)(-1), where Omega and tau(ff) represent the angular rotation speed and the free-fall time of the core, respectively. (2) On the other hand, the disc is deformed to an elongated bar in the isothermal stage for a strongly magnetized or rapidly rotating cloud. The bar breaks into 2-4 fragments (bar fragmentation). Even if a disc is thin, the disc dominated by the magnetic force or thermal pressure is stable and forms a single compact body. In either ring or bar fragmentation mode, the fragments contract and a pair of outflows is ejected from the vicinities of the compact cores. The orbital angular momentum is larger than the spin angular momentum in the ring fragmentation. On the other hand, fragments often quickly merge in the bar fragmentation, since the orbital angular momentum is smaller than the spin angular momentum in this case. Comparison with observations is also shown.

The birth of very massive stars is not well understood(1-3), in contrast to the formation process of low-mass stars like our Sun(4,5). It is not even clear that massive stars can form as single entities; rather, they might form through the mergers of smaller ones born in tight groups(6,7). The recent claim of the discovery of a massive protostar in M17 (a nearby giant ionized region) forming through the same mechanism as low-mass stars(8) has therefore generated considerable interest. Here we show that this protostar has an intermediate mass of only 2.5 to 8 solar masses (M.), contrary to the earlier claim of 20. (ref. 8). The surrounding circumstellar envelope contains only 0.09M. and a much more extended local molecular cloud has 4-9M..

We reexamine accretion onto a protobinary based on two-dimensional numerical simulations with high spatial resolution. We focus our attention on the ratio of the primary and secondary accretion rates. Fifty-eight models are made for studying the dependence of the accretion rates on the specific angular momentum of infalling gas j(inf), the mass ratio of the binary q, and the sound speed c(s). When j(inf) is small, the binary accretes the gas mainly through two channels ( type I): one through the Lagrange point L2 and the other through L3. When j(inf) is large, the binary accretes the gas only through the L2 point ( type II). The primary accretes more than the secondary in both the cases, although the L2 point is closer to the secondary. After flowing through the L2 point, the gas flows halfway around the secondary and through the L1 point to the primary. Only a small amount of gas flows back to the secondary, and the rest forms a circumstellar ring around the primary. The boundary between types I and II depends on q. When j(inf) is very large, the accretion begins after several rotations ( type III). The beginning of the accretion is later when j(inf) is larger and c(s) is smaller. Our result that the primary accretion rate is higher for a large j(inf) is qualitatively different from results of earlier simulations. The difference is mainly due to limited spatial resolution and large numerical viscosity in the numerical simulations thus far.

The decay of Alfven waves in a filamentary molecular cloud is investigated through three-dimensional numerical simulations. We have considered a filamentary molecular cloud supported in part by the Alfven wave against the self-gravity. Our attention has been focused on the basic physical mechanism for the decay. The decay rate is obtained as a function of the wavelength and amplitude. It is found that when the wave is circularly polarized, the decay e-folding timescale is several times the fast wave crossing timescale for the filament and independent of the wavelength, whereas when the wave is linearly polarized, the amplitude of the wave decreases inversely proportional to time. It is also found that the decay of Alfven waves induces rotation and shear flow in the filamentary cloud. The propagation of two Alfven waves in the medium results in the excitation of daughter waves due to nonlinear coupling between mother waves. The wavenumber of the daughter waves is the sum or difference between those of the mother waves, and below a critical wavenumber of the daughter wave, the filamentary cloud fragments as a result of Jeans instability. The fragments collapse to form high-density rotating magnetized disks. In contrast, below a critical wavenumber of the mother wave, the cloud becomes a dense helical filament after the decay of the Alfven waves. The present models are compared with previous simulations and observations with regard to the rotation, fragmentation, and helical structure of filamentary clouds.

A new self-similar solution describing spherical implosions of a gaseous sphere under both self-gravity and radiative diffusion is investigated in detail, where the diffusivity is modeled by a power law with respect to density and temperature. The reduced two-dimensional eigenvalue problem is solved to show that there is a unique quantitative relation between the two physical effects for the self-similar dynamics. The resultant spatial and temporal behaviors are also determined uniquely, once the opacity is specified. For a reference case of an inverse bremsstrahlung opacity in a fully ionized hydrogen plasma, the solution predicts that the system evolves within the applicable parameter ranges: 10-11 to 10-8 g cm-3 for the central density, a few to tens of 103 K for the central temperature, a few to tens of years for the collapse period, and a few to a dozen times the solar mass for the core mass. Persistent entropy emission via radiation plays an important role in the core formation with mass accretion, which is contrary to the predictions of implosion models under isothermal or adiabatic assumptions. The mass accretion rate is found to increase with time in a power-law form. The present solution turns out to be convectively stable.

We present We three-dimensional numerical simulations on binary formation through fragmentation. The simulations follow gravitational collapse of a molecular cloud core up to growth of the first core by accretion. At the initial stage, the gravity is only slightly dominant over the gas pressure. We made various models by changing initial velocity distribution ( rotation speed, rotation law, and bar-mode perturbation). The cloud fragments whenever the cloud rotates sufficiently slowly to allow collapse but faster enough to form a disk before first-core formation. The latter condition is equivalent to Omega(0)t(ff) greater than or similar to 0.05, where Omega(0) and t(ff) denote the initial central angular velocity and the freefall time measured from the central density, and the condition is independent of the initial rotation law and bar-mode perturbation. Fragmentation is classified into six types. When the initial cloud rotates rigidly the cloud collapses to form a adiabatic disk supported by rotation. When the bar-mode perturbation is very minor, the disk deforms to a rotating bar, and the bar fragments. Otherwise, the adiabatic disk evolves into a central core surrounded by a circumstellar disk, and the the circumstellar disk fragments. When the initial cloud rotates differentially, the cloud deforms to a ring or bar in the isothermal collapse phase. The ring fragments into free or more cores, while the bar fragments into only two cores. In the latter case, the core merges due to low orbital angular momentum and new satellite cores form in the later stages.

The fragmentation of molecular cloud cores a factor of 1.1 denser than the critical Bonnor-Ebert sphere is examined though three-dimensional numerical simulations. A nested grid is employed to resolve. ne structure down to 1 AU while following the entire structure of the molecular cloud core of radius 0.14 pc. A barotropic equation of state is assumed to take account of the change in temperature during collapse, allowing simulation of the formation of the first core. A total of 225 models are shown to survey the effects of initial rotation speed, rotation law, and amplitude of bar mode perturbation. The simulations show that the cloud fragments whenever the cloud rotates sufficiently slowly to allow collapse but fast enough to form a disk before first-core formation. The latter condition is equivalent to Omega(0)t(ff) greater than or similar to 0.05, where Omega(0) and t(ff) denote the initial central angular velocity and the freefall time measured from the central density, respectively. Fragmentation is classified into six types: disk-bar, ring-bar, satellite, bar, ring, and dumbbell types according to the morphology of collapse and fragmentation. When the outward decrease in initial angular velocity is more steep, the cloud deforms from spherical at an early stage. The cloud deforms into a ring only when the bar mode (m = 2) perturbation is very minor. The ring fragments into two or three fragments via ring-bar type fragmentation and into at least three fragments via ring type fragmentation. When the bar mode is significant, the cloud fragments into two fragments via either bar or dumbbell type fragmentation. These fragments eventually merge because of their low angular momenta, after which several new fragments form around the merged fragment via satellite type fragmentation. This satellite type fragmentation may be responsible for the observed wide range of binary separation.

We present a numerical method for solving the Poisson equation on a nested grid. A nested grid consists of uniform grids having different grid spacing and is designed to cover the space closer to the center with a finer grid. Thus, our numerical method is suitable for computing the gravity of a centrally condensed object. It consists of two parts: the difference scheme for the Poisson equation on the nested grid and the multigrid iteration algorithm. It has three advantages: accuracy, fast convergence, and scalability. First, it computes the gravitational potential of a close binary accurately up to the quadrupole moment, even when the binary is resolved only in the ne grids. Second, the residual decreases by a factor of 300 or more with each iteration. We confirmed experimentally that the iteration always converges to the exact solution of the difference equation. Third, the computation load of the iteration is proportional to the total number of the cells in the nested grid. Thus, our method gives a good solution at a minimum expense when the nested grid is large. The difference scheme is applicable also to adaptive mesh refinement, in which cells of different sizes are used to cover a domain of computation.

We present three-dimensional numerical simulations for the formation and evolution of an accreting protoplanetary disk. The disk was assumed to be formed by the gravitational collapse of a rotating gas cloud. It was assumed to be nearly axisymmetric but to have a nonaxisymmetric perturbation. We constructed 11 models, changing the angular momentum distribution and initial perturbation. In a typical model, the growth of the bar mode (m = 2) begins 300 yr after the formation of the protoplanetary disk. The growth is exponential in the linear regime and has an e-folding timescale of 100 yr. The bar mode changes the protoplanetary disk into a fast-rotating bar. The angular momentum is transferred from the bar to the outer infalling envelope, and spiral density waves are emitted outward. The bar shrinks into a smaller round disk after the angular momentum transfer. This increases the accretion rate temporarily. The disk grows through accretion and becomes unstable against the bar mode again. The recurrence period was about 800 yr. Our simulations indicate that accretion is dynamical and that its rate is highly variable in the early protoplanetary disk.

We investigate the growth of the bar mode during the main accretion phase of protostar formation. Our stability analysis shows that the bar mode grows both in the pre-protostellar phase, and in the main accretion phase. The perturbation has a larger amplitude at a smaller radius in the main accretion phase. The perturbation propagates outward to have a large amplitude at a large distance at a later stage. The growth during the main accretion phase makes the envelope nonspherical. Thus, the accretion onto the protostar becomes nonspherical at a later stage. The nonspherical accretion may enhance formation of multiple stars. We also show the application of our stability analysis to the spin-up of the protostar and to its infalling envelope.

We developed a numerical simulation code using a nested grid method. The code allows us to follow a collapse and fragmentation of molecular cloud cores with a fine resolution. Our simulations show that a molecular cloud core, of which size is 0.1 pc, gravitationally collapses to form a opaque core, of which size is 10-100 AU. The core fragments into seeds of binary stars, of which size is 1-10 AU. The fragmentation is classified by the initial perturbations. When the initial cloud has a large perturbation of barmode, the opaque core is elongated enough to fragment into two clumps. When the initial cloud has small perturbation of barmode, a central clump surrounded by spiral arms is formed. The spiral arms fragment into the dense clumps. These clumps are gravitationally bounded, which correspond to seeds of binary stars.

We show a fast algorithm for solving the Poisson equation on a nested grid. The nested grid consists of uniform grids having potentials us to compute the gravitational potential of compact bodies embedded in a less dense cloud. The computation load is scalable, i.e., proportional to the total number of grid points. The algorithm is partly applicable to adaptive mesh refinement. We used this algorithm for computing the gravitational potential in a simulation of star formation.

We investigate the stability of a gravitationally collapsing iron core against nonspherical perturbation. The gravitationally collapsing iron core is approximated by a similarity solution for a dynamically collapsing polytropic gas sphere. We find that the similarity solution is unstable against nonspherical perturbations. The perturbation grows in proportion to (t - t(0))(-sigma) while the central density increases in proportion to (t - t(0))(-2). The growth rate is sigma = 1/3 + l(gamma - 4/3), where gamma and l denote the polytropic index and the parameter l of the spherical harmonics, Y-l(m)(theta, phi), respectively. The growing perturbation is dominated by vortex motion. Thus, it excites global convection during the collapse and may contribute to material mixing in a Type II supernova.

We show three-dimensional numerical simulations in which stars form sequentially in a filamentary molecular cloud. The star formation is triggered by expansion of an H II region. The H II region is distant from the filamentary cloud at the initial stage. As it expands, it interacts with the filamentary cloud. The cloud is pinched and separated into two parts. Subsequently, the gravitational instability is induced to form two cores of the first generation along the filament axis in a typical model. The separation of the two cores is several times larger than the filament diameter. It is comparable to the wavelength of the fastest growing fragmentation mode. The first-generation cores become isolated, and filamentary clouds shorten to widen the separation. New cores of second generation form at the edges of the shortened filamentary clouds. This core formation is recursive, and our model shows sequential star formation triggered by an expanding H II region. The age difference is several times that of the dynamical timescale between the first- and second-generation cores. This sequential star formation is similar to that observed in the filamentary cloud associated with the H II region NGC 2024. Our first-generation cores correspond to FIR 4 and FIR 5, while the second-generation cores correspond to FIR 3 and FIR 6.

We discuss the stability of dynamically collapsing gas spheres. We use a similarity solution for a dynamically collapsing sphere as the unperturbed state. In the similarity solution the gas pressure is approximated by a polytrope of P = K rho(gamma). We examined three types of perturbations: bar (l = 2) mode, spin-up mode, and Ori-Piran mode. When gamma &lt; 1.097, it is unstable against the bar mode. It is unstable against the spin-up mode for any gamma. When gamma &lt; 0.961, the similarity solution is unstable against the Ori-Piran mode. The unstable mode grows in proportion to \t - t(0)\(-sigma), while the central density increases in proportion to rho(c) proportional to (t - t(0))(-2) in the similarity solution. The growth rate, sigma, is obtained numerically as a function of gamma for the bar mode and the Ori-Piran mode. The growth rate of the bar mode is larger for a smaller gamma. The spin-up mode has a growth rate of sigma = 1/3 for any gamma.

We show two-dimensional numerical simulations of the gravitational collapse of rotating gas clouds. We assume the polytropic equation of state, P = K rho(gamma), to take account of the temperature change during the collapse. Our numerical simulations have two model parameters, beta and gamma, which specify the initial rotation velocity and polytropic index, respectively. We show three models, beta = 1.0, 0.5, and 0.2, for each gamma, which is taken to be 0.8, 0.9, 0.95, 1.05, 1.1, or 1.2. These 18 models are compared with previously reported isothermal models (gamma = 1). In each model a rotating cylindrical cloud initially in equilibrium fragments periodically because of the growth of a velocity perturbation and forms cloud cores. The cloud core becomes a dynamically collapsing gaseous disk whose central density (rho(c)) increases with time (t) in proportion to rho(c) proportional to (t - t(0))(-2). This collapse is qualitatively similar in density and velocity distributions to the runaway collapse of a rotating isothermal cloud. The surface density of the disk, Sigma, is proportional to the power of the radial distance, Sigma(r) proportional to r(1-2 gamma), in the envelope. Models with gamma &gt; 1 have geometrically thick disks (aspect ratio r(d)/z(d) similar or equal to 2), while those with gamma &lt; 1 have very thin disks (r(d)/z(d) &gt; 10). While the former disks are stable, the latter disks are unstable against fragmentation if we adopt the Toomre stability criterion for a thin gaseous disk. Our numerical simulations also show the growth of a rotationally supported disk by radial accretion in a period t &gt; t(0) for models with gamma &gt; 1. The accretion phase starts at a stage in which the central density is still finite. The central density at the beginning of the accretion phase is lower when beta and gamma are larger. Our models with gamma &gt; 1 are applicable to star formation in turbulent gas clouds in which the effective sound speed decreases with increase in the density. Our models with gamma &gt; 1 are applicable to star formation in primordial clouds in which the temperature increase during the collapse is due to less efficient cooling.

We investigate the stability of a similarity solution for a gravitationally collapsing isothermal sphere by means of a normal-mode analysis. When the central density increases in proportion to rho(c) proportional to (t(0) - t)(-2), the bar mode grows in proportion to (t(0) - t)(-0.354). Here the symbol t denotes the time and the symbol t(0) denotes an epoch. When the bar mode grows, the dense part of the collapsing cloud becomes either a thin filament or a disk. Since a thin filament is likely to fragment, the bar-mode instability will probably result in multiple stars at the center of the collapsing cloud.

We investigated the dynamical collapse of a molecular cloud core with three-dimensional numerical simulations. Our simulations show that an initially spherical core produces a bar or disk during its dynamical collapse. The disk and bar formation is due to instability. Velocity perturbations grow in proportion to rho(c)(1/6), where rho(c) denotes the central density. The growing velocity perturbations are due to two effects, rotation and shear. Rotation makes the core spin faster to produce a disk at the center. On the other hand, velocity shear elongates or shortens the core in one direction to form a bar or nonrotating disk. When the core rotates nonuniformly, the collapse produces a disk containing a bar at its center by the growth of rotation and shear. This bar is much longer than the Jeans length and is likely to be unstable against fragmentation. We expect that the bar will evolve into a binary or multiple stars. The binary or multiple stars will be surrounded by a common disk. We also demonstrate the growth of an eigenmode that leads to bar and disk formation. On the basis of our numerical simulations, we give a condition for formation of a disk and bar during the isothermal collapse phase of a molecular cloud core. If the core has an oblateness of 10% or an equivalent velocity shear at n(H2) similar or equal to 10(4) cm(-3), the core produces a bar by the end of its isothermal collapse phase.

Taking self-gravity into account, we investigated the stability of the interstellar medium in which the circularly polarized Alfven wave travels with large amplitude. The interstellar medium suffers from two types of instabilities: the Jeans instability and the parametric instability. Using a linear stability analysis, we show both the effect of the Alfven wave on the Jeans instability and the effect of self-gravity on the parametric instability. The Alfven wave suppresses the Jeans instability, having effective pressure which acts against gravity. The effective pressure is proportional to the wave energy density and depends on the wavelength of the Alfven wave and Alfven speed. The parametric instability transforms the Alfven wave into other magnetohydrodynamical waves. We followed the nonlinear growth of the instabilities with a newly developed computation code. The development of the computation code is summarized in Appendix A. Based on a linear stability analysis and numerical simulation, we discuss the magnetohydrodynamical evolution of the interstellar medium.

We investigate time-dependent inviscid hydrodynamical accretion hows onto a black hole using numerical simulations. We consider accretion that consists of hot tenuous gas with low specific angular momentum and cold dense gas with high specific angular momentum. The former accretes continuously and the latter highly intermittently as blobs. The high specific angular momentum gas blobs bounce at the centrifugal barrier and create shock waves. The low specific angular momentum gas is heated at the shock fronts and escapes along the rotation axis. The outgoing gas evolves into pressure-driven jets. Jet acceleration lasts until the shack waves fade out. The total amount of the mass ejection is about 1%-11% of the mass of the blobs. The jet mass increases when the gas blobs are more massive or have larger specific angular momentum. We get narrower, well-collimated jets when the hot continuous Bow has a lower temperature. In the numerical simulations we used a finite difference code based on the total variation diminishing scheme. It is extended to include the blackbody radiation and to apply a multitime-step scheme for time marching.

We study the dynamical collapse of isothermal magnetized clouds with two-dimensional axisymmetric numerical simulations. As a model of a cloud, we consider an infinitely long filamentary cloud with a longitudinal magnetic field. An initial model is constructed by adding an axisymmetric perturbation to an equilibrium model. Because of gravitational instability, it fragments into magnetically supercritical cloud cores, which collapse to form dynamically contracting disks keeping nearly quasi-static equilibrium in the vertical direction. The disk contraction is followed until the central density increases by a factor of more than 10(7). The disk collapses self-similarly while oscillating with appreciable amplitude; the structure of the disk at the different times is similar, except for the scale. In each cycle of the oscillation, MHD fast and slow shock waves form. This oscillation is essentially the same as that during the collapse of an isothermal rotating cloud. We also follow the evolution of various models, changing the cloud mass and magnetic field strength. The disk evolution depends only weakly on the initial condition. Taking account of the magnetic pressure and tension, we refine the similarity solution for a magnetized thin disk obtained by Nakamura, Hanawa, & Nakano. We find that the refined similarity solution can reproduce the main features of our simulations. We also apply the similarity solution to the collapse of a magnetically subcritical cloud. We confirm that the dynamical collapse phase of the subcritical cloud can also be well approximated by the similarity solution. The structure of a dynamically collapsing magnetized disk is essentially similar irrespective of whether the initial cloud is supercritical or subcritical. It indicates that the similarity collapse is a universal characteristic of the dynamical collapse of magnetized clouds.

We consider gravitational collapse of a filamentary cloud under the assumption that it is axisymmetric and uniform along the axis. The pressure is approximated by a polytrope of P = K rho(gamma). We found a similarity solution for the collapse when the polytropic index lies in the range 0 &lt; gamma &lt; 1. According to the similarity solution, the collapse consits of two phases. In the first phase the filament becomes denser and thiner. The density at the center (rho(c)) increases in proportion to (t(0) - t)(-2), where t(0) denotes the epoch at the end of the first phase. Meanwhile, the filament diameter (FWHM) decreases in proportion to (t(0) - t)((2 - gamma)). Th, line density of the central filament of rho &gt; 0.1 rho(c) vanishes at t = t(0), since it is proportional to the central density and the square of the diameter [alpha (t(0) - t)(2(1 - gamma))]. In the second phase the central filament grows in mass by accretion. The collapse in the second phase is similar to the inside-out collapse of a spherical gas cloud. When the polytropic index gamma is closer to unity, the collapse is slower. When gamma = 0.999, the collapse is 12-times slower than the dynamical one. We also found from numerical simulations having various initial conditions that the collapse of a filamentary cloud approaches asymptotically the similarity solution irrespectively of the initial condition.

We study clustering of pre-main-sequence stars in the Orion, Ophiuchus, Chamaeleon, Vela, and Lupus star-forming regions. We calculate the average surface density of companions, Sigma(theta), as a function of angular distance, theta, from each star. We employ the method developed by Larson in a 1995 study for the calculation. In most of the regions studied, the function can be fitted by two power laws (Sigma proportional to theta(gamma)) with a break as found by Larson for the Taurus star-forming region. The power index, gamma, is smaller at small separations than at large separations. The power index at large separations shows significant variation from region to region (-0.8 &lt; gamma &lt; -0.1), while the power index at small separations does not (gamma similar to-2). The power index at large separations relates to the distribution of the nearest-neighbor distance. When the latter can be fitted by the Poisson distribution, the power index is close to 0. When the latter is broader than the Poisson distribution, the power index is negatively large. This correlation can be interpreted as the result of the variation in the surface density within the region. At large separations, the power-law fit may indicate star formation history in the region and not the spatial structure like the self-similar hierarchical, or fractal, one. Because of the velocity dispersion, stars move from their birthplaces, and the surface density of coeval stars decreases with their age. When a star-forming region contains several groups of stars with different ages, a power law may fit the average surface density of companions for it. The break of the power law is located around 0.01-0.1 pc. There is a clear correlation between the break position and the mean nearest-neighbor distance. The break position may reflect dispersal of newly formed stars.

We present similarity solutions that describe the runaway collapse of a rotating isothermal disk and its subsequent inside-out collapse. The similarity solutions contain the sound speed c(s) and the ratio omega of the specific angular momentum to the mass as model parameters. During the runaway collapse, the surface density of the disk is nearly constant in the central part and inversely proportional to the radius in the tail. As the central surface density increases by collapse, the central high surface density part shrinks its radius. Thus the surface density becomes a 1/r power law at the end of runaway collapse. In the subsequent inside-out collapse phase, the disk has two parts: an inner rotating disk in quasiequilibrium and an outer dynamically infalling envelope. The inner disk and outer envelope are bounded by a shock wave. The mass and outer edge of the inner disk grow at a constant rate. The accretion rate is proportional to the cube of the sound speed, i.e., M over dot proportional to c(s)(3)/G, where G denotes the gravitational constant. The similarity solution of the runaway collapse can reproduce numerical simulations of dynamical collapse of either rotating or magnetized disks. The similarity solution of the inside-out collapse denotes growth of a centrifugally supported circumstellar disk. This solution can apply to a protostar that accretes gas substantially through the disk.

We study clustering of pre-main-sequence stars in the Orion, Ophiuchus, Chamaeleon, Vela, and Lupus star-forming regions. We calculate the average surface density of companions, ∑(θ), as a function of angular distance, θ, from each star. We employ the method developed by Larson in a 1995 study for the calculation. In most of the regions studied, the function can be fitted by two power laws (∑ ∝ θγ) with a break as found by Larson for the Taurus star-forming region. The power index, γ, is smaller at small separations than at large separations. The power index at large separations shows significant variation from region to region (-0.8 &lt γ &lt -0.1), while the power index at small separations does not (γ∼ -2). The power index at large separations relates to the distribution of the nearest-neighbor distance. When the latter can be fitted by the Poisson distribution, the power index is close to 0. When the latter is broader than the Poisson distribution, the power index is negatively large. This correlation can be interpreted as the result of the variation in the surface density within the region. At large separations, the power-law fit may indicate star formation history in the region and not the spatial structure like the self-similar hierarchical, or fractal, one. Because of the velocity dispersion, stars move from their birthplaces, and the surface density of coeval stars decreases with their age. When a star-forming region contains several groups of stars with different ages, a power law may fit the average surface density of companions for it. The break of the power law is located around 0.01-0.1 pc. There is a clear correlation between the break position and the mean nearest-neighbor distance. The break position may reflect dispersal of newly formed stars. © 1998. The American Astronomical Society. All rights reserved.

Magnetized gas layers in gravitational fields (e.g., accretion disks and galactic disks) can be subject to the Parker instability, an undular mode of the magnetic buoyancy instabilities. By means of a linear stability analysis, we examined the effects of hot, tenuous regions (''coronae'') on the growth of the Parker instability in the underlying magnetized gas layers. As an unperturbed state, Ne consider the magnetized gas layers in static equilibrium. The stratified gas layers are threaded by horizontal magnetic fields in the x-direction. The temperature varies almost discontinuously at the coronal base in the z-direction. The ratio of magnetic pressure to gas pressure, alpha, is assumed to be constant. Our analysis has confirmed that the presence of a corona reduces the growth rate of the Parker instability and increases the critical wavelength. It is found that the growth of the Parker instability is more sensitive to the height of the coronal base than the temperature ratio between the disk and the corona is. In particular, the Parker instability is stabilized substantially when the coronal base lies below the height of maximum gravitational acceleration. When the wavenumber vector of the perturbation is parallel to the magnetic field (k(y) = 0), the growth rates of all modes in the disk are reduced considerably in the limit of the vanishing coronal base height. The first harmonic mode (Ih-mode with odd symmetric velocity eigen-functions with respect to the equatorial plane) is more easily stabilized by coronae than the fundamental mode is (f-mode with even symmetric velocity eigenfunctions). This is because global convective motion across the equatorial plane is allowed for the f-mode even when k(y) = 0, whereas it is not allowed for the 1h-mode. For the f-mode, furthermore, we find that the smallest possible gamma (critical gamma) against the instability is gamma(crit) = 1 + alpha, regardless of the value of k(y). The reason for this is discussed briefly.

We investigated the stability of similarity solutions for a gravitationally contracting isothermal sphere by means of a normal mode analysis. We found that a normal mode grows in proportion to (t(0) - t)(-sigma), where t denotes the time. The symbol, t(0), denotes an epoch at which the central density increases infinitely [rho proportional to (t(0) - t)(-2)]. The Hunter-b and -d solutions are very unstable against spherical perturbations; the growth rates of the unstable perturbations are as large as sigma = 56.2 and 6.48 x 10(3) for the Hunter-b and -d solutions, respectively. The Hunter solutions are unlikely to be realized in astrophysical situations and even in numerical simulations. The Larson-Penston solution is less unstable. Even if an unstable spherical perturbation exists, the growth rate should be lower than sigma less than or equal to 1. We have found an unstable mode in which the angular velocity increases as Omega infinity(t(0) - t)(-4/3). Since this mode grows slowly, the Larson-Penston solution will be realized approximately when the initial rotation is very small.

Using the thin disk approximation we study nonaxisymmetric evolution of a rotating magnetized disk during its dynamic collapse. Since the disk shrinks in size and mass during the collapse, we introduce the zooming coordinates in which the length and timescales shorten with the time in accordance with the collapse. These coordinates enable us to follow the nonaxisymmetric evolution of the disk with high spatial resolution even for the late stage. Numerical simulations show the nonlinear evolution of a perturbation superimposed on a dynamically contracting disk. When a perturbation is absent, the disk is apparently stationary in the zooming coordinates and fits a similarity solution for a dynamically collapsing disk. The disk is unstable against the bar (m = 2) mode; a perturbation grows to change the disk into a bar. In the early phase, the growth is proportional to (t(0)-t)(-0.6), where t(0) denotes an epoch. The length of the bar is about a hundredth of the initial disk diameter. The width of the bar is much shorter than the length. Then the bar is likely to be unstable against fragmentation and to fragment into many small, denser cores. We expect that the small denser cores evolve into binary or multiple stars after merger, scatter, and accretion.

We investigate the isothermal gravitational collapse of rotating interstellar clouds with axisymmetric numerical simulations. The simulations show that a filamentary cloud fragments owing to the gravitational instability and the fragment evolves into a dynamically contracting disk. The disk contraction is followed until the central density increases by a factor of 10(16) at most. The disk evolution shows similarity: the disk structure at a given time is similar to that at another time except for the scale. We construct various models, in which we change the wavelength of the perturbation and the initial rotation velocity, and study the dependences of the disk evolution on these model parameters. The surface density of the disk is proportional to the square of the sound speed, Sigma proportional to c(s)(2) and almost independent of the wavelength of the perturbation imposed, i.e., the mass contained in the fragment. It indicates that the mass of the gravitationally contracting disk is independent of the parent cloud mass. When the initial cloud rotates slowly, the dense part of the fragment is nearly spherical in the early contraction phase and evolves into a disk. When the initial cloud rotates fast, the fragment has a disk shape from the early contraction phase. In the late contraction phase, the surface density and the rotation velocity do not depend strongly on the initial rotation velocity and depend weakly on it when it is small. Although the disk evolution is well understand by similarity collapse, it shows an oscillation around similarity collapse. A new shock wave forms each cycle of oscillation.

Using two-dimensional numerical simulations, we have constructed a model for formation of a filamentary molecular cloud permeated with an almost perpendicular magnetic field. The model filamentary cloud is formed by fragmentation of a magnetized sheetlike cloud. It stays in a quasi-static equilibrium for a period that is much longer than its free-fall time. In the quasi-static equilibrium, the magnetic field runs parallel to the cloud axis in the central part of the cloud and almost perpendicular to the axis in the less dense part of the cloud. The inner parallel magnetic field supports the cloud in part against the cloud gravity. The parallel magnetic field escapes from the cloud through the Alfven wave and the quasistatic equilibrium state ends. During the quasi-static equilibrium, the cloud is expected to fragment in the direction of the axis by gravitational instability. We discuss the application of our model to the filamentary clouds in Taurus.

We followed the fragmentation of magnetized filamentary clouds and the formation of disks with numerical simulations and a semianalytical approach. Two-dimensional magnetohydrodynamical simulations showed that a filamentary cloud with longitudinal magnetic fields fragments to form geometrically thin disks perpendicular to the magnetic field. Each disk contracts dynamically toward the symmetry axis keeping quasi-static equilibrium in the direction parallel to the axis. We followed the late-stage evolution of the disk with a one-dimensional numerical simulation assuming that the disk is infinitesimally thin and found that the disk approaches a state in which the surface density is inversely proportional to the distance from the center, suggesting the existence of a similarity solution. With some simplification we obtained numerically some similarity solutions for one-dimensional dynamically contracting disks.

We have studied the gravitational instability of a gaseous slab formed by high-velocity collisions between massive gas clouds. When the collision is oblique, the compressed gaseous slab rotates end-over-end with internal velocity shear. We obtained the growth rates of the instability for gaseous slabs with various, but realistic, parameters. We also depict the density and velocity distributions of the perturbed gaseous slabs. Two types of unstable modes are found: one is mainly due to the self-gravity of the gaseous slab; the other is due to velocity shear. The former instability leads to the formation of gaseous clumps, whose masses are comparable to those of stars and star clusters. The velocity shear reduces the growth rate of the former instability, and enhances that of the latter instability. The rotation also decreases the growth of the former instability.

The dynamical instability of a self-gravitating magnetized filamentary cloud was investigated while taking account of rotation around its axis. The filamentary cloud of our model is supported against self-gravity in part by both a magnetic field and rotation. The density distribution in equilibrium was assumed to be a function of the radial distance from the axis, rho0(r) = rho(c)(1 + r2/8H2)-2, where rho(c) and H are model parameters specifying the density on the axis and the length scale, respectively; the magnetic field was assumed to have both longitudinal (z-) and azimuthal (phi-) components with a strength of B0(r) is-proportional-to square-root rho0(r). The rotation velocity was assumed to be upsilon0phi = OMEGA(c)r(1 + r2/8H2)-1/2. We obtained the growth rate and eigenfunction numerically for (1) axisymmetric (m = 0) perturbations imposed on a rotating cloud with a longitudinal magnetic field, (2) non-axisymmetric (m = 1) perturbations imposed on a rotating cloud with a longitudinal magnetic field, and (3) axisymmetric perturbations imposed on a rotating cloud with a helical magnetic field. The fastest growing perturbation is an axisymmetric one for all of the model clouds studied. Its wavelength is lambda(max) less-than-or-equal-to 11.14H for a non-rotating cloud without a magnetic field, and is shorter when the magnetic field is stronger and/or the rotation is faster. For a rotating cloud without a magnetic field the most unstable axisymmetric mode is excited mainly by self-gravity (the Jeans instability), while the unstable non-axisymmetric mode is excited mainly by non-uniform rotation (the Kelvin-Helmholtz instability). The unstable non-axisymmetric perturbation corotates with a fluid at r = (2-4)H and grows in time. When the equilibrium magnetic field is helical, the unstable perturbation grows in time and propagates along the axis. A rotating cloud with a helical magnetic field is less unstable than that with a longitudinal magnetic field.

We discuss the fragmentation of a filamentary cloud on the basis of a one-dimensional hydrodynamical simulation of a self-gravitating gas cloud. The simulation shows that dense cores are produced with a semiregular interval in space and time from one edge to the other. At the initial stage the gas near one of the edges is attracted inwards by gravity and the accumulation of the gas makes a dense core near the edge. When the dense core grows in mass up to a certain amount, it gathers gas from the other direction. Accordingly the dense core becomes isolated from the main cloud and the parent filamentary cloud has a new edge. This cycle repeats and the fragmentation process propagates toward the other edge. The propagation speed is a few tens of percent larger than the sound speed. According to the theory, the age difference for the northwest-most and southeast-most cores in TMC-1 is estimated to be 0.68 pc/0.6 km s-1 = 10(6) yr. The estimated age difference is consistent with that obtained from the chemical chronology.

The dynamical instability of accretion flow onto a black hole with a standing shock wave is investigated. When gas possessing angular momentum accretes onto a black hole, the flow can be associated with a standing shock wave. If the flow is decelerated at the pre-shock side of the front, it is unstable and the shock front leaves its equilibrium position. Assuming an isothermal accreting gas flow and a pseudo-Newtonian potential, we obtained the eigenvalues and eigenfunctions of unstable perturbations. The nonlinear evolution of the instability was also studied by a numerical simulation. The physical mechanism of the instability is explained in terms of shock wave propagation. It is shown that the motion of the shock front depends on the balance between the total pressures of the pre- and post-shock flows.

We discuss the fragmentation of a filamentary molecular cloud on the basis of a magnetohydrodynamical stability analysis and the observations of the Orion A cloud with the Nobeyama 45 m telescope. Our model cloud has axial and helical magnetic fields and rotates around the axis. The dispersion relation for this cloud shows that the presence of a magnetic field and/or rotation shortens the wavelength of the most unstable mode, i.e., the mean distance between the adjacent fragments, measured in units of the filament diameter. We compare the theoretical results with the observed clumpy structure of the filamentary Orion A cloud and find that the sum of magnetic and centrifugal forces in the parent cloud was comparable with the pressure force (thermal and turbulent) and has had significant effects on the fragmentation. The rotation velocity of the filament measured in the (CO)-C-13 emission line is consistent with this result.

The effect of the magnetic skew on the Parker instability is investigated by means of the linear stability analysis for a gravitationally stratified gas layer permeated by a horizontal magnetic field. When the magnetic field is skewed (i.e., the field line direction is a function of the height), the wavelength of the most unstable mode is lambda approximately 10H where H is the pressure scale height. The growth rate of the short-wavelength modes is greatly reduced when the gradient in the magnetic field direction exceeds 0.5 rad per scale height. Our results indicate that the Parker instability in a skewed magnetic field preferentially forms large-scale structures like giant molecular clouds.

The fragmentation of the Galactic gaseous disk was investigated by means of a linear stability analysis. We approximated the equilibrium Galactic disk to be a uniformly rotating gaseous sheet in which the magnetic field runs in a horizontal direction. The gravitational acceleration during equilibrium was set to be equal to Oort's value. The perturbation equation was solved numerically while taking account of the self-gravity of the gas, the interstellar magnetic field, and the rotation. The Parker and Jeans instabilities take place almost independently, although they slightly interfere with each other. At a gas density of n(c) approximately 1 cm-3 the Jeans instability proceeds at the rate of the angular momentum extraction from a fragment due to the magnetic torque. The growth rate of not only the Parker instability, but also the Jeans instability, is thus proportional to the magnetic field strength. We derived semi-analytical dispersion relations which reproduce the numerical results. For a standard Galaxy model the Parker instability dominates over the Jeans instability.