Gravitational Instability in the Dust Layer of a Protoplanetary Disk with Interaction between the Layer and the Surrounding Gas

We consider gravitational instability of the dust layer in the midplane of a protoplanetary disk with turbulence and shear stresses between the gas in the disk and that in the dust layer. We solve a linearized system of hydrodynamic equations for perturbations of dust (monodisperse) and gas phases in the incompressible gas approximation. We take into account the gas drag of solid particles (dust aggregates), turbulent diffusion and the velocity dispersion of particles, and the perturbation of the azimuthal velocity of gas in the layer upon the transfer of angular momentum from solid particles to it and from this gas to the surrounding gas in the disk. We obtain and solve the dispersion equation for the layer with the ratio of surface densities of the dust phase and gas being well above unity. The following parameters of gravitational instability in the dust layer are calculated: the critical surface density of solid matter and the Stokes number of particles corresponding to the onset of instability, the wavelength range in which instability occurs, and the rate of its growth as a function of the perturbation wavelength in the circumsolar disk at radial distances of 1 and 10 AU. We show that at 10 AU, the maximum instability growth rate increases due to the transfer of angular momentum of gas in the layer to gas outside it, a new maximum emerges at a longer wavelength, a long-wavelength instability “tail” forms, and the critical surface density initiating instability decreases relative to that determined without the transfer of angular momentum to gas outside the layer. None of these effects are observed at 1 AU, since instability in this region probably develops faster than the transfer of angular momentum to the surrounding gаs of a protoplanetary disk occurs.     

The effects of the tidal force on shear instabilities in the dust layer of the solar nebula

The linear analysis of the instability due to vertical shear in the dust layer of the solar nebula is performed. The following assumptions are adopted throughout this paper: (1) The self-gravity of the dust layer is neglected. (2) One fluid model is adopted, where the dust aggregates have the same velocity with the gas due to strong coupling by the drag force. (3) The gas is incompressible. The calculations with both the Coriolis and the tidal forces show that the tidal force has a stabilizing effect. The tidal force causes the radial shear in the disk. This radial shear changes the wave number of the mode which is at first unstable, and the mode is eventually stabilized. Thus the behavior of the mode is divided into two stages: (1) the first growth of the unstable mode which is similar to the results without the tidal force, and (2) the subsequent stabilization due to an increase of the wave number by the radial shear. If the midplane dust/gas density ratio is smaller than 2, the stabilization occurs before the unstable mode grows largely. On the other hand, the mode grows faster by one hundred orders of magnitude, if this ratio is larger than 20. Because the critical density of the gravitational instability is a few hundreds times as large as the gas density, the hydrodynamic instability investigated in this paper grows largely before the onset of the gravitational instability. It is expected that the hydrodynamic instability develops turbulence in the dust layer and the dust aggregates are stirred up to prevent from settling further. The formation of planetesimals through the gravitational instabilities is difficult to occur as long as the dust/gas surface density ratio is equal to that for the solar abundance. On the other hand, the shear instability is suppressed and the planetesimal formation through the gravitational instability may occur, if dust/gas surface density ratio is hundreds times as large as that for the solar abundance.     

Gravitational instability in the dust layer of a protoplanetary disk: Interaction of solid particles with turbulent gas in the layer

Gravitational instability of the dust layer formed after the aggregates of dust particles settle toward the midplane of a protoplanetary disk under turbulence is considered. A linearized system of hydrodynamic equations for perturbations of dust (monodisperse) and gas phases in the incompressible gas approximation is solved. Turbulent diffusion and the velocity dispersion of solid particles and the perturbation of gas azimuthal velocity in the layer upon the transfer of angular momentum from the dust phase due to gas drag are taken into account. Such an interaction of the particles and the gas establishes upper and lower bounds on the perturbation wavelength that renders the instability possible. The dispersion equation for the layer in the case when the ratio of surface densities of the dust phase and the gas in the layer is well above unity is obtained and solved. An approximate gravitational instability criterion, which takes the size-dependent stopping time of a particle (aggregate) in the gas into account, is derived. The following parameters of the layer instability are calculated: the wavelength range of its subsistence and the dependence of the perturbation growth rate on the perturbation wavelength in the circumsolar disk at a radial distance of 1 and 10 AU. It is demonstrated that at a distance of 1 AU, the gas–dust disk should be enriched with solids by a factor of 5–10 relative to the initial abundance as well as the particle aggregates should grow to the sizes higher than about 0.3 m in order for the instability to emerge in the layer in the available turbulence models. Such high disk enrichment and aggregate growth is not needed at a distance of 10 AU. The conditions under which this gravitational instability in the layer may be examined with no allowance made for the transfer of angular momentum from the gas in the layer to the gas in a protoplanetary disk outside the layer are discussed.     

The gravitational instability in the dust layer of a protoplanetary disk:: axisymmetric solutions for nonuniform dust density distributions in the direction vertical to the midplane

The gravitational instability in the dust layer of a protoplanetary disk with nonuniform dust density distributions in the direction vertical to the midplane is investigated. The linear analysis of the gravitational instability is performed. The following assumptions are used: (1) One fluid model is adopted, that is, difference of velocities between dust and gas are neglected. (2) The gas is incompressible. (3) Models are axisymmetric with respect to the rotation axis of the disk. Numerical results show that the critical density at the midplane is higher than the one for the uniform dust density distribution by Sekiya (1983, Prog. Theor. Phys. 69, 1116-1130). For the Gaussian dust density distribution, the critical density is 1.3 times higher, although we do not consider this dust density distribution to be realistic because of the shear instability in the dust layer. For the dust density distribution with a constant Richardson number, which is considered to be realized due to the shear instability, the critical density is 2.85 times higher and is independent of the value of the Richardson number. Further, if a constant Richardson number could decrease to the order of 0.001, the gravitational instability would be realized even for the dust to gas surface density ratio with the solar abundance. Our results give a new restriction on planetesimal formation by the gravitational instability.     

The role of Kelvin–Helmholtz instability in dusty and partially ionized outflows

We investigate the linear theory of Kelvin–Helmholtz instability at the interface between a partially ionized dusty outflow and the ambient material analytically. We model the interaction as a multifluid system in a planar geometry. The unstable modes are independent from the charge polarity of the dust particles. Although our results show a stabilizing effect for charged dust particles, the growth time-scale of the growing modes gradually becomes independent of the mass or charge of the dust particles when the magnetic-field strength increases. We show that growth time-scale decreases with increasing the magnetic field. Also, as the mass of the dust particles increases, the growth time-scale of the unstable mode increases.     

Planetesimal growth in turbulent discs before the onset of gravitational instability

It is difficult to imagine a planet formation model that does not at some stage include a gravitationally unstable disc. Initially unstable gas–dust discs may form planets directly, but the high surface density required has motivated the alternative that gravitational instability occurs in a dust sub-layer only after grains have grown large enough by electrostatic sticking. Although such growth up to the instability stage is efficient for laminar discs, previous research concluded that realistic disc turbulence catastrophically increases the settling time, thereby requiring additional processes to facilitate planet formation on the needed time scales. We develop a different model for the influence of turbulence on the collisional velocity of grains and on the scale height of the dust layer and find that the earlier conclusions must be revisited. The model produces a disc-radius dependent time scale to reach a gravitationally unstable phase of planet formation. For a range of dust sticking and disc parameters, we find that for viscosity parameters α ⩽ 10⁻³, this time scale is short enough over a significant range in radii R that turbulence does not catastrophically slow the early phases of planet formation, even in the absence of agglomeration enhancement agents like vortices.     

The fate of sub-micron circumplanetary dust grains I: Aligned dipolar magnetic fields

We study the stability of charged dust grains orbiting a planet and subject to gravity and the electromagnetic force. Our numerical models cover a broad range of launch distances from the planetary surface to beyond synchronous orbit, and the full range of charge-to-mass ratios from ions to rocks. Treating the spinning planetary magnetic field as an aligned dipole, we map regions of radial and vertical instability where dust grains are driven to escape or crash into the planet. We derive the boundaries between stable and unstable trajectories analytically, and apply our models to Jupiter, Saturn and the Earth, whose magnetic fields are reasonably well represented by aligned dipoles.     

The effect of finite electrical and thermal conductivities on magnetic buoyancy in a rotating gas

A horizontal magnetic field if increasing in strength downwards can cause a horizontal layer of electrically conducting fluid to become unstable, a phenomenon known as ‘magnetic buoyancy’, and sometimes thought to have relevance to magnetic A stars, and to sunspot creation. Analyses that assume infinite thermal and electrical conductivities (and zero viscosity) predict that modes of zero horizontal wave-length, in the direction perpendidular to the field, are maximally unstable but are stabilised by even small Coriolis forces. It is shown here, however, that when proper allowance is made for the finite (though large) conductivities of the fluid the layer may experience a ‘conductive instability’ that grows on the ohmic time-scale and is maximally unstable to a mode of non-zero horizontal extent.     

On the Formation of Planetesimals: Radial Contraction of the Dust Layer Interacting with the Protoplanetary Disk Gas

Radial contraction of the dust layer in the midplane of a gas–dust protoplanetary disk that consists of large dust aggregates is modeled. Sizes of aggregates vary from centimeters to meters assuming the monodispersion of the layer. The highly nonlinear continuity equation for the solid phase of the dust layer is solved numerically. The purpose of the study is to identify the conditions under which the solid matter is accumulated in the layer, which contributes to the formation of planetesimals as a result of gravitational instability of the dust phase of the layer. We consider the collective interaction of the layer with the surrounding gas of the protoplanetary disk: shear stresses act on the gas in the dust layer that has a higher orbital velocity than the gas outside the layer, this leads to a loss of angular momentum and a radial drift of the layer. The stress magnitude is determined by the turbulent viscosity, which is represented as the sum of the α-viscosity associated with global turbulence in the disk and the viscosity associated with turbulence that is localized in a thin equatorial region comprising the dust layer and is caused by the Kelvin–Helmholtz instability. The evaporation of water ice and the continuity of the mass flux of the nonvolatile component on the ice line is also taken into account. It is shown that the accumulation of solid matter on either side of the ice line and in other regions of the disk is determined primarily by the ratio of the radii of dust aggregates on either side of the ice line. If after the ice evaporation the sizes (or density) of dust aggregates decrease by an order of magnitude or more, the density of the solid phase of the layer’s matter in the annular zone adjacent to the ice line from the inside increases sharply. If, however, the sizes of the aggregates on the inner side of the ice line are only a few times smaller than behind the ice line, then in the same zone there is a deficit of mass at the place of the modern asteroid belt. We have obtained constraints on the parameters at which the layer compaction is possible: the global turbulence viscosity parameter (α < 10^?5), the initial radial distribution of the surface density of the dust layer, and the distribution of the gas surface density in the disk. Restrictions on the surface density depend on the size of dust aggregates. It is shown that the timescale of radial contraction of a dust layer consisting of meter-sized bodies is two orders of magnitude and that of decimeter ones, an order of magnitude greater than the timescale of the radial drift of individual particles if there is no dust layer.     

Zones of anomalous activity of the magnetorotational instability in protostellar disks

A criterion for the magnetorotational instability of a protostellar disk in which the dust particles are assumed to be well mixed with the gas over the entire disk volume has been obtained within the framework of Hall magnetohydrodynamics. It is shown that the dusty plasma component affects significantly the Hall current and, under certain conditions, can cause its direction to be reversed compared to the case of a weakly ionized electron-ion plasma. A significant expansion of the range of wave numbers for unstable magnetic fluctuations is a consequence of the Hall current reversal. The spatial localization of the regions of protostellar disks in which not only the long-wavelength Alfvén disturbances but also the short-wavelength ones are subject to the magnetorotational instability is investigated. Possible physical consequences of the presence of anomalously active zones in cold disks for their structure and evolution are pointed out.     

Stability and equilibrium of emerged magnetic flux

We analyse stability and equilibrium of a unipolar large-scale magnetic field pervading a plane horizontal subphotospheric layer with the possible implications for sunspots in mind. Eddy diffusivity is applied to account for the effects of the small-scale convective turbulence. Diffusivity quenching by magnetic field results in a secondary large-scale instability. A linear stability analysis is performed to define the marginal stability boundary in parametric space and the unstable mode structure. The nonlinear dynamics of the unstable modes are followed numerically. The original state of a uniform vertical magnetic field is transformed via the instability into the nonlinear dynamical equilibrium with a highly intermittant distribution of the magnetic field. Magnetic flux is concentrated in a relatively small area surrounded by an almost field-free region. The role of the fluid motion in the hydromagnetic equilibrium is emphasized. Although the relevance of the instability to the process of sunspot formation is rather questionable, the resulting equilibrium structures are similar to mature spots in their thermal and magnetic properties. Also, the simulated flow structure agrees with helioseismic tomography results.     

Settling and growth of dust particles in a laminar phase of a low-mass solar nebula

It is considered that some vertical convection as well as possible turbulence in an early phase of solar nebula soon terminates owing to diminution of the temperature dependence of dust opacity due to rapid growth of dust particles. We reexamine settling and growth of dust particles in the subsequent laminar phase of the solar nebula in detail, treating a dust layer as a two-component fluid composed of the dust and the gas. We obtain analytic expressions for the settling path, the growing size, and the settling time. The settling process is divided into two phases, i.e., an early gas-dominant phase and a later dust-dominant phase. So far, only the former phase, where the particle path finally turns from vertical to radial, has been investigated. In the latter phase, dust particles drag the gas, rather than the gas does dust particles. Consequently, the particle path turns from radial to vertical. Dust particles grow most appreciably and rapidly in a radially sweeping phase. The final radii of dust particles at the onset of gravitational instability of the dust layer are 20, 5.9, and 0.60 cm in the Earth's, Jupiter's, and Neptune's zones, respectively. These values are much smaller than those obtained previously by S.J. Weidenschilling [1980, Icarus44, 172–189] and Y. Nakagawa et al. [1981, Icarus45, 517–528]. The total settling times are 1.9 × 10³, 4.6 × 10³, and 2.8 × 10⁴ years in the above-mentioned three zones, respectively. These are somewhat smaller than those obtained by the previous studies. Most of the settling time is spent in the early vertically settling phase.     

Gas flow and dust acceleration in a cometary Knudsen layer

An analytical model of the innermost gas–dust coma region is proposed. The kinetic Knudsen layer adjacent to the surface of the cometary nucleus, where the initially non-equilibrium velocity distribution function of gas molecules relaxes to Maxwell equilibrium distribution function and, as a result, the macro-characteristics of gas and dust flows vary several-fold, is considered. The gas phase model is based on the equations for mass, momentum and energy flux conservation, and is a natural development of the Anisimov, 1968 and Cercignani, 1981 approaches. The analytical relations between the characteristics of the gas flow on the boundaries of the non-equilibrium layer and the characteristics of the returning gas flow adsorbed by the surface are determined. These values form a consistent basis both for hydrodynamic models of the inner coma and for jet force models. Three particular models are presented: (1) sublimation of a polyatomic one-component gas; (2) sublimation of a two-component polyatomic gas mixture, in both cases from a plane surface; and (3) sublimation of water ice through a porous dust mantle. We conclude that the characteristics of the gas flow emerging from the Knudsen layer over a porous dust mantle is not very sensitive to the structure of the mantle.We also treat the expansion of dust into the coma, concentrating on the interaction between a non-equilibrium gas flow and a test particle. The dynamics of a grain of idealized shape is explored by using several simplifying assumptions for the variation of the drag force. The velocity of a particle at the exterior boundary of the Knudsen layer is thus estimated. Examining various model behaviours of the drag force inside the Knudsen layer, we show that the dust velocity is not sensitive to these variations.     

Star formation induced by supersonic turbulence in interstellar molecular cloud

We studied fragmentation process of the interstellar molecular cloud which is predominated by supersonic turbulence with special regard to collisions of turbulent gas elements and formation of a shock-compressed layer by receding shock waves. The propagation of the shock waves and the evolution of the compressed layer are followed by one-dimensional gas dynamical simulation until self-gravity becomes significant, taking account of the effects of thermal properties of the molecular gas and magnetic fields. It is shown that the efficient cooling by CO molecules and its sensitive dependence on gas density make the shock-compressed layer so cold and dense that the layer becomes gravitationally unstable and breaks into fragments even if the gas elements are gravitationally stable prior to the collision. The mass of the unstable fragment is estimated to be about two solar masses or less, irrespective of the presence of the magnetic field. The stars formed by collisions of supersonic turbulent gas elements accelerate the surrounding gas in T Tauri stage and replenish the turbulent energy to maintain the mechanical equilibrium of the molecular cloud.     

Evolution and saturation of Autowaves in photodissociation regions

The propagation of plane, cylindrical, and spherical waves in a thermally unstable gas–dust medium has been simulated numerically. As applied to the photodissociation regions near O and B stars, we take into account the interaction of ultraviolet radiation with dust grains and large polycyclic aromatic hydrocarbon molecules as well as the gas cooling through the excitation of СII ions and OI atoms and the deexcitation of rotational levels of CO molecules. The instability regions have been determined. The perturbation growth times corresponding to them are ~10³?10⁵ yr. We show that wave breaking occurs irrespective of the geometry of motion, while a perturbation in the form of a single pulse gives rise to a sequence of shock waves. The post-shock gas velocity is approximately 0.1?0.5 of the sound velocity, so that the autowaves can contribute noticeably to the observed velocity dispersion of the gas near the boundaries of HII regions. Two-dimensional simulations suggest that the presence of multiple shocks in a thermally unstable medium can accelerate significantly the destruction of preexisting isolated condensations.     

Streaming Instability in the Gas-Dust Medium of the Protoplanetary Disc and the Formation of Fractal Dust Clusters

The sequence of evolution of the protoplanetary gas-and-dust disk around the parent star includes, according to modern concepts, its compression in the central plane and decay into separate dust condensations (clusters) due to the occurrence of various types of instabilities. The interaction of dust clusters of a fractal structure during their collisions is considered as a key mechanism for the formation and growth of primary solids, which serve as the basis for the subsequent formation of planetesimals and embryos of planets. Among the mechanisms contributing to the formation of planetesimals, an important place belongs, along with gravitational instability, hydrodynamic instabilities, in particular, the socalled streaming instability of the two-phase gas-dust layer due to its ability to concentrate dispersed particles in dense clots. In contrast to a number of existing models of streaming instability, in which dust particles are considered structurally compact and monodisperse, this paper proposes a more realistic model of polydisperse particles of fractal nature, forming dust clusters as a result of coagulation. The instability of the dust layer in the central plane of the protoplanetary disk under linear axisymmetric perturbations of its parameters is considered. A preliminary conclusion can be drawn that the proposed model of dust fractal aggregates of different scales increases the efficiency of linear growth of hydrodynamic instabilities, including the streaming instabilities associated with the difference between the velocities of the dust and gas phases.

     

Dynamical and secular instability of a self-gravitating rotating cylinder in a toroidal magnetic field

The oscillations and stability of a homogeneous self-gravitating rotating cylinder in a toroidal magnetic field are investigated. It is assumed that the field is proportional to the distance to the axis of the cylinder. We show the existence of four infinite discreta spectra of magnetic (or rotational) modes. Rotation stabilizes the magneticm=1 instability. The magnetic field decreases the growth rate of rotational instability and reduces the interval of unstable wavenumbers. Ifm=1, instability always occurs with the exception of the equipartition state. Ifm>1, the instability can be suppressed by a sufficiently large magnetic field. Resistivity decreases the growth rate of magnetic instability, but increases the growth rate of rotational instability. For zero wavenumber perturbations secular instability occurs due to the action of resistivity before a neutral point is attained where a second secular instabiliity initiates due to the action of resistivity.     

Dust to planetesimals: Settling and coagulation in the solar nebula

The behavior of solid particles in a low-mass solar nebula during settling to the central plane and the formation of planetesimals is examined. Gravitational instability in a dust layer and collisional accretion are considered as possible mechanisms of planetesimal formation. Non-Keplerian rotation of the nebula results in shear between the gas and a dust layer. This shear produces turbulence within the layer which inhibits gravitational instability, unless the mean particle size exceeds a critical value, ～1 cm at 1 AU. The size requirement is less stringent at larger heliocentric distances, suggesting a possible difference in planetesimal formation mechanisms between the inner and outer nebula. Coagulation of grains during settling is expected in the solar nebula environment. Van der Waals forces appear adequate to produce centimeter-sized aggregates. Growth is primarily due to sweepup of small particles by larger ones due to size-dependent settling velocities. A numerical model for computing simultaneous coagulation and settling is described. Relative velocities are determined by gas drag and the non-Keplerian rotation of the nebula. The settling is very nonhomologous. Most of the solid matter reaches the central plane as centimeter-sized aggregates in a few times 10³ revolutions, but some remains suspended in the form of fine dust. Drag-induced relative velocities result in collisions. The growth of bodies in the central plane is initially rapid. After sizes reach ～10³ cm, relative velocities decrease and the growth rate declines. Gas drag rapidly damps the out-of-plane motions of these intermediate-sized bodies. They settle into a thin layer which is subject to gravitational instability. Kilometer-sized planetesimals are formed by this composite process.     

Stability of Electrostatic Electron Cyclotron Waves in a Multi-Ion Plasma

We have studied the stability of the electrostatic electron cyclotron wave in a plasma composed of hydrogen, oxygen and electrons. To conform to satellite observations in the low latitude boundary layer we model both the ionic components as drifting perpendicular to the magnetic field. Expressions for the frequency and the growth rate of the wave have been derived. We find that the plasma can support electron cyclotron waves with a frequency slightly greater than the electron cyclotron frequency ω _ce ; these waves can be driven unstable when the drift velocities of both the ions are greater than the phase velocity of the wave. We thus introduce another source of instability for these waves namely multiple ion beams drifting perpendicular to the magnetic field.     

Rayleigh-Taylor instability of a two-fluid layer under a general rotation field and a horizontal magnetic field

In this paper the Rayleigh-Taylor instability (RTI) of a two-fluid layer system under the simultaneous action of a general rotation field and a horizontal magnetic field is presented. An approximate and an exact solution of the eigenvalue equation are calculated. These solutions are important not only to understand more deeply the physical problem but also to determine the correct numerical solutions. Numerical calculations are done for an unstable density stratification in the cases of horizontal magnetic field parallel and perpendicular to the horizontal component of the angular velocity. For an adverse density stratification, it is shown that in comparison to previous works, the horizontal magnetic field creates new angular areas (of the angle of propagation of the perturbation) at which the perturbation is stable and propagates as traveling waves. It is also shown that the vertical component of the angular velocity has a destabilizing effect because it works to eliminate the stable angular areas.