Optical properties of spheroidal particles

A new exact solution of the diffraction problem for the homogeneous spheroid on the basis of the method of separation of variables is given. This solution is considerably more efficient than the one of Asano and Yamamoto from the computational point of view. The expressions for various characteristics of the scattered radiation are obtained. The radiation pressure on spheroidal particles is considered taking into account the radial and transversal components. The method of calculations and various tests, which were used to control the computer programs, are described. Numerical results for forward and arbitrary angles scattering by prolate and oblate spheroids with the refractive indices typical for ice and silicates are presented. The dependence of the results on the propagation direction and the polarization of the incident radiation, size of particle and its aspect ratio are examined. The asymptotics for the characteristics of the scattered radiation for the extremely prolate and extremely oblate spheroids are derived. The range of the validity of these approximations is studied. Astrophysical applications include: a) the calculations of the interstellar extinction, interstellar linear and circular polarization curves for the ensemble of partially oriented spheroidal grains, and b) the consideration of the profiles and polarization of the interstellar feature 2200 for the partially oriented graphite spheroids. Appendices contain the expressions for integrals of products of the angular spheroidal functions and the asymptotics for the oblate spheroidal functions.     

Similarity solutions for an axially symmetric explosion model in magnetogasdynamics. I

Similarity solutions, for a point explosion in a spheroid with axially symmetric density distribution obeying power laws in the presence of magnetic field, are obtained. A new technique suggested by Bhowmick (1978) has been utilised to study the character of flow variables behind the shock front in an axisymmetric model. The total energy of the wave is constant.     

Emden-Chandrasekhar axisymmetric,rigid-body rotating polytropes

In connection with the basic theory reported in a previous paper (Paper I) for EC1 (rigidly rotating) polytropes, we define exact configurations as configurations for which the equilibrium equation has solutions which are infinitely close to some analytical function and the related gravitational potential coincides, in fact, with the gravitational potential due to mass distribution, at any point not outside the system. Then we restrict to the special casen=5 and divide the related polytropes into two components, a massive body where each mass element has a finite (polytropic) distance from the centre, and a massless atmosphere where each mass element has an infinite (polytropic) distance from te centre. It is found a single exact configuration exists, which under some assumptions may be related to Roche systems. In the special casen=0 it is shown a particular configuration, the spheroidal one, is an exact configuration and evidence is given that spheroidal configurations are the stablest among all the allowed (axisymmetric) configurations. It is also pointed out that EC1 polytropes withn=0 and incompressible MacLaurin spheroids belong to different sequences, even if they exhibit some common features. In the special casen=1 it is shown each allowed configuration is expressible by a convenient series development, which reduces to the relatedn=0 configuration by maintaining only the first two or the first one terms of the sum. It is also deduced, by analogy with the casen=0, that pseudospheroidal configurations are exact and the stablest among all the allowed (axisymmetric) configurations.     

Propagation of a strong cylindrical shock wave in a rotational axisymmetric dusty gas with exponentially varying density

Non-similarity solutions are obtained for one-dimensional isothermal and adiabatic flow behind strong cylindrical shock wave propagation in a rotational axisymmetric dusty gas,which has a variable azimuthal and axial fluid velocity.The dusty gas is assumed to be a mixture of small solid particles and perfect gas.The equilibrium flow conditions are assumed to be maintained,and the density of the mixture is assumed to be varying and obeying an exponential law.The fluid velocities in the ambient medium are assumed to obey exponential laws.The shock wave moves with variable velocity.The effects of variation of the mass concentration of solid particles in the mixture,and the ratio of the density of solid particles to the initial density of the gas on the flow variables in the region behind the shock are investigated at given times.Also,a comparison between the solutions in the cases of isothermal and adiabatic flows is made.     

Propagation of cylindrical relativistic shock waves in the presence of a magnetic field

In this paper we obtain similarity solutions for the propagation of cylindrical relativistic shock waves in the presence of a constant azimuthal magnetic field or in its absence for the medium, where the nucleon number density is uniform. The shock surface moves with constant velocity and the total energy of the disturbance is dependent on time. The solutions are applicable only to an isothermal medium or a cold gas.     

Alfvén node-free vibrations of white dwarf in the model of solid star with toroidal magnetic field

In the context of white dwarf asteroseismology, we investigate the vibrational properties of a non-convective solid star with an axisymmetric purely toroidal intrinsic magnetic field of two different shapes. Focus is laid on the regime of node-free global Lorentz-force-driven vibrations about the symmetry axis at which material displacements have one and the same form as those for nodeless spheroidal and torsional vibrations restored by Hooke’s force of elastic shear stresses. Particular attention is given to the even-parity poloidal Alfvén modes whose frequency spectra are computed in analytic form, showing how the purely toroidal magnetic fields completely buried beneath the star surface can manifest itself in seismic vibrations of non-magnetic white dwarfs. The spectral formulae obtained are discussed in juxtaposition with those for Alfvén modes in the solid star model with the poloidal, homogeneous internal and dipolar external, magnetic field whose inferences are relevant to Alfvén vibrations in magnetic white dwarfs.     

Self-similar propagation of axisymmetric radiative magnetogasdynamic shocks with increasing energy,II

In this paper self-similar solutions have been investigated for the propagation of axisymmetric radiative gasdynamic shocks caused by an explosion into an inhomogeneous ideal gas permeated by a current free azimuthal magnetic field. The effects of radiation flux and magnetic field together have been seen in the region of interest on the other flow variables. The total energy of the flow between the inner expanding surface and the shock is taken to be dependent on shock radius obeying a power law. The radiative pressure and energy have been neglected.     

The temperature of nonspherical circumstellar dust grains

The temperatures of prolate and oblate spheroidal dust grains in the envelopes of stars of various spectral types are calculated. Homogeneous particles with aspect ratios a/b≤10 composed of amorphous carbon, iron, dirty ice, various silicates, and other materials are considered. The temperatures of spherical and spheroidal particles were found to vary similarly with particle size, distance to the star, and stellar temperature. The temperature ratio T _d(spheroid)/T _d(sphere) depends most strongly on the grain chemical composition and shape. Spheroidal grains are generally colder than spherical particles of the same volume; only iron spheroids can be slightly hotter than iron spheres. At a/b≈2, the temperature differences do not exceed 10%. If a/b≥4, the temperatures can differ by 30–40%. For a fixed dust mass in the medium, the fluxes at wavelengths λ≥100 are higher if the grains are nonspherical, which gives overestimated dust masses from millimeter observations. The effect of grain shape should also be taken into account when modeling Galactic-dust emission properties, which are calculated when searching for fluctuations of the cosmic microwave background radiation in its Wien wing.     

Analytical solutions for the Newtonian gravitational field induced by matter within axisymmetric boundaries

An analytical method originally applied to the problem of the actuator disc in fluid mechanics has been applied to the closely analogous problem of constructing the classical Newtonian potential and attractions. The method can treat axisymmetric problems and also non-axisymmetric cases where matter is confined within axisymmetric boundaries. The potential and attractions for the generalized thin finite disc can be given in closed form in terms of elliptic integrals and elementary functions. For the general case of matter within an axisymmetric boundary, the potentials and attractions can be evaluated as one-dimensional integrals of albeit complex analytical expressions. These expressions represent the fields induced by matter in an extended region as a distribution of gravitating discs. For certain special cases, such as matter bounded by a circular cylinder and also for matter distributed in a spherical region, closed-form solutions can be given that appear to be new. Some non-axisymmetric results are also given for the thin disc of infinite radial extent.     

Undamped oscillations of collisionless stellar systems: Spheres,spheroids and discs

The collisionless Boltzmann equation governing self-gravitating systems such as galaxies has recently been shown to admit exact oscillating solutions with planar and spherical symmetry. The relation of the spherically symmetric solutions to the Virial theorem, as well as generalizations to non-uniform spheres, uniform spheroids and discs form the subject of this paper. These models generalize known families of static solutions. The case of the spheroid is worked out in some detail. Quasiperiodic as well as chaotic time variation of the two axes is demonstrated by studying the surface of section for the associated Hamiltonian system with two degrees of freedom. The relation to earlier work and possible implications for the general problem of collisionless relaxation in self gravitating systems are also discussed     

An axially symmetric explosion model in magnetogasdynamics: II

A method using Eulerian coordinate system is developed under a local radiality assumption to study a point explosion in a spheroid with axially symmetric exponential density distribution, including the effect of azimuthal magnetic field.     

Numerical simulations of rotating axisymmetric sunspots

A numerical model of axisymmetric convection in the presence of a vertical magnetic flux bundle and rotation about the axis is presented. The model contains a compressible plasma described by the non-linear MHD equations, with density and temperature gradients simulating the upper layer of the Sun's convection zone. The solutions exhibit a central magnetic flux tube in a cylindrical numerical domain, with convection cells forming collar flows around the tube. When the numerical domain is rotated with a constant angular velocity, the plasma forms a Rankine vortex, with the plasma rotating as a rigid body where the magnetic field is strong, as in the flux tube, while experiencing sheared azimuthal flow in the surrounding convection cells, forming a free vortex. As a result, the azimuthal velocity component has its maximum value close to the outer edge of the flux tube. The azimuthal flow inside the magnetic flux tube and the vortex flow is prograde relative to the rotating cylindrical reference frame. A retrograde flow appears at the outer wall. The most significant convection cell outside the flux tube is the location for the maximum value of the azimuthal magnetic field component. The azimuthal flow and magnetic structure are not generated spontaneously, but decay exponentially in the absence of any imposed rotation of the cylindrical domain.     

Tidal effects of a massive spheroidal halo on an inner spheroidal contracting visible body: Application to an evolutionary disk-galaxy model

The effects of the tidal interactions between two coaxial, homogeneous spheroids, one (the “Brigt Component”: B) completely embedded in the other (the “Dark Halo”: D), along a quasi-static contraction, are considered. The aim is to look how the dynamical properties and the final morphology of the B subsystem may be affected by the presence of the D component. Three initial configurations are considered: the quasi-spherical “Dark Halo” D coincides with the “Visible Component” B (case C); D is flatter than B but the two spheroids have the same semiminor axis (case N); no D component is assumed; the “visible” spheroid is single (case S). The application to an evolutionary disk-galaxy model is considered under some simple assumptions: i) in cases C and N the spheroidal halo is massive (mass ratio “dark”/“bright” about ten) and dissipationless. For a mass ratio like this, the tidal interaction of the B component over D turns to be negligible in the course of the contraction; adding to that the lack of dissipation, it appears plausible to take the D component as frozen along the evolution; ii) the degree of anisotropy and the angular momentum of B, J_B, are conserved. The conservation of J_B provides us the time-independent relationship among the key physical quantities and gives the possibility to draw the evolutionary tracks on the plane (axis ratio σ_B, semimajor axis a_B) without any explicit time-scale; iii) the global “star formation rate” is parametrized according to a simple “Schmidt power law” proportional to the square of gas density. At every step of the quasi-static contraction, the structure is determined by the tensor virial theorem extended to a double configuration. The model is very idealized, particularly because there are no available tidal gravitational terms other than for the case of two homogeneous. Nevertheless, the method based on the tensor virial appears powerful to gain insight into the correlations among the physical quantities involved and into their trend along the evolution. One of the main result is a clear indication of a leading role played by the axis ratio of the “Dark-Halo” component which might be, to the extent that this simple picture can be compared with a real galactic system, a possible new physical parameter to be added to mass and angular momentum for separating spirals from S0 galaxies.     

External and internal solutions for the twisted,flux-tube,prominence model

In this paper the twisted flux-tube model for the support of a prominence sheet with constant axial current density, given by Ridgway, Priest, and Amari (1991), is considered.The model is extended in Section 2 to incorporate a current sheet of finite height. The sheet is supported in a constant current density force-free field in the configuration of a twisted flux tube. The mass of the prominence sheet, using a typical height and field strength, is computed. Outside the flux tube the background magnetic field is assumed to be potential but the matching of the flux tube onto this background field is not considered here.Instead our attention is focussed, in Section 3, on the interior of the prominence. An expanded scale is used to stretch the prominence sheet to a finite width. We analytically select solutions for the internal magnetic field in this region which match smoothly onto the external force-free solutions at the prominence edge.The force balance equation applied inside the prominence then yields expressions for the pressure and density and a corresponding temperature may be computed.     

Families of planar orbits generated by homogeneous potentials

The second order partial differential equation which relates the potentialV(x,y) to a family of planar orbitsf(x,y)=c generated by this potential is applied for the case of homogeneousV(x,y) of any degreem. It is shown that, if the functionf(x,y) is also homogeneous, there exists, for eachm, a monoparametric set of homogeneous potentials which are the solutions of an ordinary, linear differential equation of the second order. Iff(x,y) is not homogeneous, in general, there is not a homogeneous potential which can create the given family; only if =f _y /f _x satisfies two conditions, a homogeneous potential does exist and can be determined uniquely, apart from a multiplicative constant. Examples are offered for all cases.     

Structures in a class of magnetized scale-free discs

We construct analytically stationary global configurations for both aligned and logarithmic spiral coplanar magnetohydrodynamics (MHD) perturbations in an axisymmetric background MHD disc with a power-law surface mass density  Σ₀∝ r ^−α  , a coplanar azimuthal magnetic field   B ₀∝ r ^−γ  , a consistent self-gravity and a power-law rotation curve   v ₀∝ r ^−β  , where v ₀ is the linear azimuthal gas rotation speed. The barotropic equation of state  Π∝Σ ⁿ   is adopted for both MHD background equilibrium and coplanar MHD perturbations where Π is the vertically integrated pressure and n is the barotropic index. For a scale-free background MHD equilibrium, a relation exists among  α, β, γ  and n such that only one parameter (e.g. β) is independent. For a linear axisymmetric stability analysis, we provide global criteria in various parameter regimes. For non-axisymmetric aligned and logarithmic spiral cases, two branches of perturbation modes (i.e. fast and slow MHD density waves) can be derived once β is specified. To complement the magnetized singular isothermal disc analysis of Lou, we extend the analysis to a wider range of  −1/4 < β < 1/2  . As an illustrative example, we discuss specifically the  β= 1/4  case when the background magnetic field is force-free. Angular momentum conservation for coplanar MHD perturbations and other relevant aspects of our approach are discussed.     

A theory for the gravitational potentials of spheroidal stellar systems and its application to the Galaxy

A new formula for the gravitational potential for spheroidal systems is proposed, and is applied to the galactic system. The applied model consists of a disk, a nucleus and a massive halo. Although this model is not quite so accurate as those produced by the superposition of large numbers of simple spheroids, it has the important advantage of retaining analytic simplicity for its acceleration formulae, thereby producing significant economies for orbit computations. As better observational data become available, the method can be readily expanded.     

ULF pulsation mode coupling in the ionosphere and magnetosphere

The effect of realistic ionospheric Hall conductances on axisymmetric toroidal mode hydromagnetic wave resonances is investigated. The toroidal modes couple to evanescent poloidal modes near the ionospheres such that the composite modes resonate at the constant frequencies of the corresponding single-field-shell resonances for zero Hall conductance. A model for these composite modes is developed which has narrow but finite latitudinal resonance widths such as to make the modes valid solutions of the hydromagnetic equations. The modes also suggest that “shell” solutions can realistically describe such properties of real pulsations as frequency, damping, phase variation along the field-line and node-antinode behaviour at the ionospheres. Estimates of ionospheric coupling strength are obtained and compared with magnetospheric coupling strength. It is found that magnetospheric coupling dominates ionospheric coupling for any single non-axisymmetric mode. However, ionospherically coupled axisymmetric modes should be necessary components of the Fourier sum of modes required to model any real pulsation of low to moderate apparent azimuthal wave number.Estimates of the range of magnetospheric coupling strength are obtained for pulsations under a variety of conditions.     

On the gravitational fields of Pandora and Prometheus

Utilizing topographic models of Saturn's F-ring shepherd satellites Prometheus (S16 1980S27) and Pandora (S15 1980S26), derived by Stooke (1994), and supposing that their mass density is constant, we derived basic geometrical and dynamical characteristics of the moons. They include the volume and mass, the mean radii, the tensor of inertia, and Stokes coefficients of the harmonic expansions of external gravitational potential. The best fitting ellipsoid approximations of the topography were calculated. A simple method of determining the gravitational potential on the surface of an irregular satellite is presented. Examples of equipotential surfaces of the satellites are shown     

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.