Compressible Fluid Model for the Seismic Waves Generated by A Sunquake

The solar convection zone is modeled as a horizontally stratified atmosphere with a constant gravitational field and an adiabatic temperature gradient (a neutrally stratified polytrope). At equilibrium, the gas pressure and density decreases to zero at the solar surface so that the solar surface is treated as a free surface which is bounded by vacuum. The evolution of small amplitude perturbations about the equilibrium state is described by the linearized Euler equations for an inviscid compressible fluid with an adiabatic equation of state. A sunquake is initiated at time zero by means of an initial perturbation with a Gaussian velocity profile and the exact solution of the initial value problem is obtained in terms of a Fourier integral. Comparisons between theory and observations indicate that this highly simplified model is able to predict the propagation of sunquake waves across the solar surface with an error of approximately 10% or 20%.     

Effect of Moon perturbation on the energy curves and equilibrium points in the Sun–Earth–Moon system

In this paper, we have considered that the Moon motion around the Earth is a source of a perturbation for the infinitesimal body motion in the Sun–Earth system. The perturbation effect is analyzed by using the Sun–Earth–Moon bi–circular model (BCM). We have determined the effect of this perturbation on the Lagrangian points and zero velocity curves. We have obtained the motion of infinitesimal body in the neighborhood of the equivalent equilibria of the triangular equilibrium points. Moreover, to know the nature of the trajectory, we have estimated the first order Lyapunov characteristic exponents of the trajectory emanating from the vicinity of the triangular equilibrium point in the proposed system. It is noticed that due to the generated perturbation by the Moon motion, the results are affected significantly, and the Jacobian constant is fluctuated periodically as the Moon is moving around the Earth. Finally, we emphasize that this model could be applicable to send either satellite or telescope for deep space exploration.     

Average velocity components in a rotating infinitely flattened self-gravitating system near an equilibrium state

The average radial and angular velocity components are obtained for a rotating two-dimensional self-gravitating system near an equilibrium state. First-order perturbation configurations of flaring straight bars emanating from the center provide examples of such systems. In these systems the average velocity field to first order is incompressible and irrotational. The second-order effects on the product of the average velocity components with the spatial density are essentially independent of the angular coordinate.     

Unstable modes of non-axisymmetric gaseous discs

We present a perturbation theory for studying the instabilities of non-axisymmetric gaseous discs. We perturb the dynamical equations of self-gravitating fluids in the vicinity of a non-axisymmetric equilibrium, and expand the perturbed physical quantities in terms of a complete basis set and a small non-axisymmetry parameter ε. We then derive a linear eigenvalue problem in matrix form, and determine the pattern speed, growth rate and mode shapes of the first three unstable modes. In non-axisymmetric discs, the amplitude and the phase angle of travelling waves are functions of both the radius R and the azimuthal angle φ. This is due to the interaction of different wave components in the response spectrum. We demonstrate that wave interaction in unstable discs, with small initial asymmetries, can develop dense clumps during the phase of exponential growth. Local clumps, which occur on the major spiral arms, can constitute seeds of gas giant planets in accretion discs.     

Phase transitions in the ISM a source of dissipative behaviour

By considering a simple fluid model, we investigate the role of phase transitions in the ISM on the galaxy- scale gas dynamics. Cooling and heating timescales in the ISM are typically shorter than typical galactic rotation timescales, so the individual phases in the ISM can be assumed to be in temperature equilibrium with the radiation field. Using this approximation we can construct an equation of state which depends upon the average density and mass fractions in the individual phases. Previous studies suggest that there is an equilibrium phase fraction as a function of pressure. We incorporate evolution towards this equilibrium state as a relaxation term with a time to obtain equilibrium . We derive a condition in terms of a critical Mach number when one dimensional shocks should be continuous. For small values of the relaxation time we show that the relaxation term acts like a viscosity. We show with one dimensional simulations that increasing causes shocks to become smoother. This work suggests that phase changes can strongly effect the gas dynamics of the ISM across spiral arms and bars.     

Resonant behaviour of MHD waves on magnetic flux tubes

The resonances that appear in the linear compressible MHD formulation of waves are studied for equilibrium states with flow. The conservation laws and the jump conditions across the resonance point are determined for 1D cylindrical plasmas. For equilibrium states with straight magnetic field lines and flow along the field lines the conserved quantity is the Eulerian perturbation of total pressure. Curvature of the magnetic field lines and/or velocity field lines leads to more complicated conservation laws. Rewritten in terms of the displacement components in the magnetic surfaces parallel and perpendicular to the magnetic field lines, the conservation laws simply state that the waves are dominated by the parallel motions for the modified slow resonance and by the perpendicular motions for the modified Alfvén resonance.The conservation laws and the jump conditions are then used for studying surface waves in cylindrical plasmas. These waves are characterized by resonances and have complex eigenfrequencies when the classic true discontinuity is replaced by a nonuniform layer. A thin non-uniform layer is considered here in an attempt to obtain analytical results. An important result related to earlier work by Hollweg et al. (1990) for incompressible planar plasmas is found for equilibrium states with straight magnetic field lines and straight velocity field lines. For these equilibrium states the incompressible and compressible surface waves have the same frequencies at least in the long wavelength limit and there is an exact correspondence with the planar case. As a consequence, the conclusions formulated by Hollweg et al. still hold for the straight cylindrical case. The effects of curvature are subsequently considered.     

Stability of charged thin shell wormhole supported by polytropic gas

In the framework of Darmois-Israel formalism, the general equations describing the motion of thin shell wormhole with a general form of equation of state of a polytropic gas are derived. The mechanical stability analysis of thin shell wormhole with charge in Reissner–Nordstrom (RN) to linearized spherically symmetric perturbation about static equilibrium solution is carried out.     

Motion in a modified Chermnykh’s restricted three-body problem with oblateness

In this paper, the restricted problem of three bodies is generalized to include a case when the passively gravitating test particle is an oblate spheroid under effect of small perturbations in the Coriolis and centrifugal forces when the first primary is a source of radiation and the second one an oblate spheroid, coupled with the influence of the gravitational potential from the belt. The equilibrium points are found and it is seen that, in addition to the usual three collinear equilibrium points, there appear two new ones due to the potential from the belt and the mass ratio. Two triangular equilibrium points exist. These equilibria are affected by radiation of the first primary, small perturbation in the centrifugal force, oblateness of both the test particle and second primary and the effect arising from the mass of the belt. The linear stability of the equilibrium points is explored and the stability outcome of the collinear equilibrium points remains unstable. In the case of the triangular points, motion is stable with respect to some conditions which depend on the critical mass parameter; influenced by the small perturbations, radiating effect of the first primary, oblateness of the test body and second primary and the gravitational potential from the belt. The effects of each of the imposed free parameters are analyzed. The potential from the belt and small perturbation in the Coriolis force are stabilizing parameters while radiation, small perturbation in the centrifugal force and oblateness reduce the stable regions. The overall effect is that the region of stable motion increases under the combine action of these parameters. We have also found the frequencies of the long and short periodic motion around stable triangular points. Illustrative numerical exploration is rendered in the Sun–Jupiter and Sun–Earth systems where we show that in reality, for some values of the system parameters, the additional equilibrium points do not in general exist even when there is a belt to interact with.     

Free-convection and mass transfer effects on the flow past a vertical plate

In sequel to an earlier work (Mitra, 1984), I set out to find the appropriate equation of state of an isothermal (self-gravitating) cosmic gas sphere. It will be shown that because of self-gravitational interaction, a cosmic gas which is perfect otherwise, is destined to behave like an imperfect gas. It is found that for a cosmic gas in hydrostatic equilibrium the only mode of free expansion (or contraction) is by a homologous evolution. This theorem enables one to construct the equation of state in terms of the local thermodynamical parameters of the inhomogeneous medium in such a way that it is consistent with the boundary conditions at the centre and at the natural boundary of the system. This method can be extended even when there is a temperature gradient in the system. This analysis predicts that the density profile should be the same at least for all nondegenerate Newtonian stars having the same temperature profile and which can be considered to be in hydrostatic equilibrium. And the density profile will be that of a polytrope of indexn=3, modulated by a temperature profile.     

Spiral structure formed in a pair of interacting galaxies

The formation of spiral structure in a galaxy, as a result of the gravitational perturbation caused by a permanent companion, is studied. It is found that spiral structure appears only when a resonance exists between the rotational frequency of the stars in the galaxy and the rotational frequency of the companion galaxy. The number of spiral arms depends strongly on the particular resonance. In the case where the companion moves in an elliptic orbit, spiral arms are formed when a resonance, inside the galactic body, exists in almost all the parts of the orbit or, at least, in the largest part of it.     

Radiation Induced Dynamics of the Galactic Halo

We show results of numerical simulations of a three component plasma consisting of electrons, ions and dust with external gravitation and radiation fields. We perform simulation runs, starting from an analytic halo equilibrium, balancing pressure, gravitational, and radiative forces. Within these the equilibrium is perturbed by the radiation of a typical OB-star association. The perturbation has a total energy input of 10⁷ L _⊙ and a duration of 30 Myrs. After switching off the perturbation, the simulations are continued to further investigate the dynamics induced. We start with a self consistent one-fluid MHD model without background magnetic field and show for an asymmetric case that the system approaches a new equilibrium after switching on the perturbation. Later it relaxes into the starting configuration again, when the additional radiation is turned off. We then show, first by including a disk-parallel magnetic field and then by redoing the simulations with a full three-fluid code, the influence of magnetic fields and species separation on the plasma dynamics. With our computations we demonstrate that these features can be important for the explanation of the structures of galactic halos and large scale mass flows. This revised version was published online in July 2006 with corrections to the Cover Date.     

A further study of the solution of Poisson's equation for three-dimensional spiral galaxies

We consider a three-dimensional model of spiral arms, which are in the form of logarithmic spirals in the galactic plane and falling off exponentially in the perpendicular direction. We derive an analytic solution of the gravitational potential corresponding to this density perturbation. We discuss the effect of the finite thickness on the spiral pattern and the question of stability of the spiral arms.     

Non-equilibrium of a cylindrical magnetic arcade

A cylindrically-symmetric magnetic arcade with its axis on the photosphere is perturbed by means of an alteration in the pressure along the base. The perturbation is examined with a view to finding equilibrium configurations close to the original equilibrium. It is found that equilibria can only be found when the integral of the excess pressure along the base is zero. In other cases no equilibria can be found and the arcade is likely either to collapse or, in the case of a coronal mass ejection, to erupt. For an initial arcade whose field increases linearly with radial distance from the axis, the neighbouring equilibria have been found.     

The possible origin of the spiral-arm instability

Physical arguments suggest the spiral arms may be manifestations of the galaxy not being in dynamical equilibrium — in the sense that the kinetic energy of tis stars and gas is less relative to its binding energy than that dictated by the virial theorem. Without constant cooling of the galactic disk (i.e., a progressive increase in the binding energy of the galaxy) such a departure from dynamical equilibrium would be corrected and the spiral arms destroyed in about 10⁹ yr due to an increase in the velocity dispersion of the stars in the disk resulting from their interacting with the spiral arms. The rate of cooling required to maintain the spiral arms, about 6×10⁴ L , may be provided by mass loss from stars in the disk population. The cooling arises from the average scale-heights and velocities of these stars being larger than that of the gas in the disk, so that there is a net loss of kinetic energy and an increase in the binding energy of the galaxy due to the ejected gas settling down to a lower terminal velocity and scale-height in the galactic disk.     

Influence of the resonance in a gravity-gradient stabilized satellite

Using Hamiltonian formalism the translational-rotational motion of a satellite is studied near a resonance considering the orbital and rotational motions. A first order perturbation theory is derived by Hori's transformation in order to eliminate short and long periodic terms, preserving in the new Hamiltonian secular and resonant terms. This theory is again applied to study the resonant system whose analysis lead us to a system of equations equivalent to the equations of a simple pendulum which is integrable in terms of elliptical integrals.     

Stellar oscillations and magnetic field perturbations of the boundary conditions

The effect of a weak magnetic field on the adiabatic radial and non-radial oscillations of a stellar configuration is studied by means of a perturbation method. Special attention is devoted to the perturbation of the oscillation frequencies resulting from the change of the boundary conditions caused by the magnetic field. This change is related to the fact that the introduction of a magnetic field removes the singularity at the surface of the equilibrium configuration. The perturbation method is applied to Ferraro's model and the influence of a magnetic field on the frequencies of the different types of oscillation modes is discussed.     

滞弹地球自转速率的潮汐变化

本文根据勒夫数和负荷载夫数的数值扰动原理，利用ｈａｎｄｌｅｒ摆动的理论周期作为滞弹吸收带模型参数估计的约束条件，讨论了地幔滞弹性对有效勒夫数ｋ的直接影响，以及滞弹地球对平均海潮的响应所产生的间接扰动，分析了它们对带谐潮尺度因子ｋ／Ｃ的影响，由此定义了一个具有动力学海潮、滞弹地幔和液核地球的世界时ＵＴ１潮汐变化序列，与天文新技术观测结果相比较表明，高频带谐潮变化的理论值与实测值是一致的。频散效应对低     

Analytic characterization of measurement uncertainty and initial orbit determination on orbital element representations

This paper presents an approach to characterize the uncertainty associated with the state vector obtained from the Herrick-Gibbs orbit determination approach using transformation of variables. The approach is applied to estimate the state vector and its probability density function for objects in low Earth orbit using sparse observations. The state vector and associated uncertainty estimates are computed in Cartesian coordinates and Keplerian elements. The approach is then extended to accommodate the $J_2$ perturbation where the state vector is written in terms of mean orbital elements. The results obtained from the analytical approach presented in this paper are validated using Monte Carlo simulations and compared with the often utilized similarity transformation for Kepler, mean, and nonsingular elements. The measurement uncertainty characterization obtained is used to initialize conventional nonlinear filters as well as operate a Bayesian approach for orbit determination and object tracking.     

Stability of a class of non-static axial self-gravitating systems in f(R) gravity

In this paper, we analyze stability regions of a non-static restricted class of axially symmetric spacetime with anisotropic matter distribution. We consider f(R)=R+?R ² model and assume hydrostatic equilibrium of the axial self-gravitating system at large past time. Considering perturbation from hydrostatic phase, we develop dynamical as well as collapse equations and explore dynamical instabilities at Newtonian and post-Newtonian regimes. It is concluded with the help of stiffness parameter, Γ ₁, that radial profile of physical parameters like pressure anisotropy, energy density and higher curvature terms of the f(R) model affect the instability ranges.