Fragmentation of a dynamically condensing radiative layer

In this paper, the stability of a dynamically condensing radiative gas layer is investigated by linear analysis. Our own time-dependent, self-similar solutions describing a dynamical condensing radiative gas layer are used as an unperturbed state. We consider perturbations that are both perpendicular and parallel to the direction of condensation. The transverse wave number of the perturbation is defined by $k$. For $k=0$, it is found that the condensing gas layer is unstable. However, the growth rate is too low to become nonlinear during dynamical condensation. For $k\ne0$, in general, perturbation equations for constant wave number cannot be reduced to an eigenvalue problem due to the unsteady unperturbed state. Therefore, direct numerical integration of the perturbation equations is performed. For comparison, an eigenvalue problem neglecting the time evolution of the unperturbed state is also solved and both results agree well. The gas layer is unstable for all wave numbers, and the growth rate depends a little on wave number. The behaviour of the perturbation is specified by $kL_\mathrm{cool}$ at the centre, where the cooling length, $L_\mathrm{cool}$, represents the length that a sound wave can travel during the cooling time. For $kL_\mathrm{cool}\gg1$, the perturbation grows isobarically. For $kL_\mathrm{cool}\ll1$, the perturbation grows because each part has a different collapse time without interaction. Since the growth rate is sufficiently high, it is not long before the perturbations become nonlinear during the dynamical condensation. Therefore, according to the linear analysis, the cooling layer is expected to split into fragments with various scales.Comment: 12 pages, 10 figures, accepted for publication in Astronomy & Astrophysic

Collapse of Primordial Filamentary Clouds under Far-Ultraviolet Radiation

Collapse and fragmentation of primordial filamentary clouds under isotropic dissociation radiation is investigated with one-dimensional hydrodynamical calculations. We investigate the effect of dissociation photon on the filamentary clouds with calculating non-equilibrium chemical reactions. With the external radiation assumed to turn on when the filamentary cloud forms, the filamentary cloud with low initial density ($n_0 \le 10^2 \mathrm{cm^{-3}}$) suffers photodissociation of hydrogen molecules. In such a case, since main coolant is lost, temperature increases adiabatically enough to suppress collapse. As a result, the filamentary cloud fragments into very massive clouds ($\sim 10^5 M_\odot$). On the other hand, the evolution of the filamentary clouds with high initial density ($n_0>10^2 \mathrm{cm^{-3}}$) is hardly affected by the external radiation. This is because the filamentary cloud with high initial density shields itself from the external radiation. It is found that the external radiation increases fragment mass. This result is consistent with previous results with one-zone models. It is also found that fragment mass decreases owing to the external dissociation radiation in the case with sufficiently large line mass.Comment: 26 pages, 15 figures, accepted by PAS

The effect of dust cooling on low-metallicity star-forming clouds

The theory for the formation of the first population of stars (Pop III) predicts a IMF composed predominantly of high-mass stars, in contrast to the present-day IMF, which tends to yield stars with masses less than 1 M_Solar. The leading theory for the transition in the characteristic stellar mass predicts that the cause is the extra cooling provided by increasing metallicity and in particular the cooling provided at high densities by dust. The aim of this work is to test whether dust cooling can lead to fragmentation and be responsible for this transition. To investigate this, we make use of high-resolution hydrodynamic simulations. We follow the thermodynamic evolution of the gas by solving the full thermal energy equation, and also track the evolution of the dust temperature and the chemical evolution of the gas. We model clouds with different metallicities, and determine the properties of the cloud at the point at which it undergoes gravitational fragmentation. We follow the further collapse to scales of an AU when we replace very dense, gravitationally bound, and collapsing regions by a simple and nongaseous object, a sink particle. Our results suggest that for metallicities as small as 10^{-5}Z_Solar, dust cooling produces low-mass fragments and hence can potentially enable the formation of low mass stars. We conclude that dust cooling affects the fragmentation of low-metallicity gas clouds and plays an important role in shaping the stellar IMF even at these very low metallicities. We find that the characteristic fragment mass increases with decreasing metallicity, but find no evidence for a sudden transition in the behaviour of the IMF within the range of metallicites examined in our present study.Comment: 5 pages, 4 figure

Conditions for Gravitational Instability in Protoplanetary Disks

Gravitational instability is one of considerable mechanisms to explain the formation of giant planets. We study the gravitational stability for the protoplanetary disks around a protostar. The temperature and Toomre's Q-value are calculated by assuming local equilibrium between viscous heating and radiative cooling (local thermal equilibrium). We assume constant $\alpha$ viscosity and use a cooling function with realistic opacity. Then, we derive the critical surface density $\Sigma_{\rm{c}}$ that is necessary for a disk to become gravitationally unstable as a function of $r$. This critical surface density $\Sigma_{\rm c}$ is strongly affected by the temperature dependence of the opacity. At the radius $r_{\rm c}\sim 20$AU, where ices form, the value of $\Sigma_{\rm c}$ changes discontinuously by one order of magnitude. This $\Sigma_{\rm c}$ is determined only by local thermal process and criterion of gravitational instability. By comparing a given surface density profile to $\Sigma_{\rm c}$, one can discuss the gravitational instability of protoplanetary disks. As an example, we discuss the gravitational instability of two semi-analytic models for protoplanetary disks. One is the steady state accretion disk, which is realized after the viscous evolution. The other is the disk that has the same angular momentum distribution with its parent cloud core, which corresponds to the disk that has just formed. As a result, it is found that the disks tend to become gravitationally unstable for $r\ge r_{\rm c}$ because ices enable the disks to become low temperature. In the region closer to the protostar than $r_{\rm c}$, it is difficult for a typical protoplanetary disk to fragment because of the high temperature and the large Coriolis force. From this result, we conclude that the fragmentation near the central star is possible but difficult.Comment: accepted for publication in PASJ. Draft version with 26 pages, 8 figures, 1 tabl

Gravitational Fragmentation of Expanding Shells. I. Linear Analysis

We perform a linear perturbation analysis of expanding shells driven by expansions of HII regions. The ambient gas is assumed to be uniform. As an unperturbed state, we develop a semi-analytic method for deriving the time evolution of the density profile across the thickness. It is found that the time evolution of the density profile can be divided into three evolutionary phases, deceleration-dominated, intermediate, and self-gravity-dominated phases. The density peak moves relatively from the shock front to the contact discontinuity as the shell expands. We perform a linear analysis taking into account the asymmetric density profile obtained by the semi-analytic method, and imposing the boundary conditions for the shock front and the contact discontinuity while the evolutionary effect of the shell is neglected. It is found that the growth rate is enhanced compared with the previous studies based on the thin-shell approximation. This is due to the boundary effect of the contact discontinuity and asymmetric density profile that were not taken into account in previous works.Comment: 13 pages, 13 figures, to be published in the Astrophysical Journa

Binary Formation in Star-Forming Clouds with Various Metallicities

Cloud evolution for various metallicities is investigated by three-dimensional nested grid simulations, in which the initial ratio of rotational to gravitational energy of the host cloud \beta_0 (=10^-1 - 10^-6) and cloud metallicity Z (=0 - Z_\odot) are parameters. Starting from a central number density of n = 10^4 cm^-3, cloud evolution for 48 models is calculated until the protostar is formed (n \simeq 10^23 cm^-3) or fragmentation occurs. The fragmentation condition depends both on the initial rotational energy and cloud metallicity. Cloud rotation promotes fragmentation, while fragmentation tends to be suppressed in clouds with higher metallicity. Fragmentation occurs when \beta_0 > 10^-3 in clouds with solar metallicity, while fragmentation occurs when \beta_0 > 10^-5 in the primordial gas cloud. Clouds with lower metallicity have larger probability of fragmentation, which indicates that the binary frequency is a decreasing function of cloud metallicity. Thus, the binary frequency at the early universe (or lower metallicity environment) is higher than at present day (or higher metallicity environment). In addition, binary stars born from low-metallicity clouds have shorter orbital periods than those from high-metallicity clouds. These trends are explained in terms of the thermal history of the collapsing cloud.Comment: 11 pages, 2 figures, Submitted to ApJL, For high resolution figures see http://astro3.sci.hokudai.ac.jp/~machida/binary-metal.pd

Gravitational Instability of Shocked Interstellar Gas Layers

In this paper we investigate gravitational instability of shocked gas layers using linear analysis. An unperturbed state is a self-gravitating isothermal layer which grows with time by the accretion of gas through shock fronts due to a cloud-cloud collision. Since the unperturbed state is not static, and cannot be described by a self-similar solution, we numerically solved the perturbation equations and directly integrated them over time. We took account of the distribution of physical quantities across the thickness. Linearized Rankine-Hugoniot relations were imposed at shock fronts as boundary conditions. The following results are found from our unsteady linear analysis: the perturbation initially evolves in oscillatory mode, and begins to grow at a certain epoch. The wavenumber of the fastest growing mode is given by k=2\sqrt{2\pi G\rho_\mathrm{E} {\cal M\mit}}/c_\mathrm{s}, where $\rho_\mathrm{E}, c_\mathrm{s}$ and \cal M\mit are the density of parent clouds, the sound velocity and the Mach number of the collision velocity, respectively. For this mode, the transition epoch from oscillatory to growing mode is given by t_g = 1.2/\sqrt{2\pi G\rho_\mathrm{E} {\cal M\mit}}. The epoch at which the fastest growing mode becomes non-linear is given by 2.4\delta_0^{-0.1}/\sqrt{2\pi G \rho_\mathrm{E}{\cal M\mit}}, where $\delta_0$ is the initial amplitude of the perturbation of the column density. As an application of our linear analysis, we investigate criteria for collision-induced fragmentation. Collision-induced fragmentation will occur only when parent clouds are cold, or $\alpha_0=5c_\mathrm{s}^2 R/2G M < 1$, where $R$ and $M$ are the radius and the mass of parent clouds, respectively.Comment: 12 pages, 21 figures, accepted for publication in PAS