A generic dynamical model of gamma-ray burst remnants

The conventional generic model is considered to explain the dynamics of gamma-ray burst remnants very well, no matter whether they are adiabatic or highly radiative. However, we find that, for adiabatic expansion, the model cannot reproduce the Sedov solution in the non-relativistic phase, and thus it needs to be revised. In this paper a new differential equation is derived. The generic model based on this equation is shown to be correct for both radiative and adiabatic fireballs, and in both ultrarelativistic and non-relativistic phases.     

Sound speed in extended Chaplygin fluid

We consider an extended Chaplygin gas equation of state which is driven from D-brane action and construct a cosmological model based on this equation of state. In this regard, we compute the scale factor of the model under a certain approximation. The conservation equation of this case is a non-linear differential equation which should solve using the special conditions. We also analyze the stability of the model by using sound speed as well as adiabatic index and discuss certain special cases of the model. We find special equation of state in this model which yields to dynamical and thermodynamical stability. Furthermore, we study the cosmological consequences of this model under certain conditions.     

Nonlinear ion-acoustic solitons in a warm plasma with adiabatic positive and negative ions and hot non-isothermal electrons

The propagation of an ion-acoustic soliton in a collisionless plasma with adiabatic positive and negative ions (with equal ion temperature) and hot non-isothermal electrons is studied by use of the renormalization method introduced by Kodama and Taniuti in the reductive perturbation method. The basic set of fluid equations describing the system is reduced to a Korteweg-de Vries (K-dV)-type equation for the first-order perturbed potential and to a linear inhomogeneous differential equation to the second-order of the perturbed potential. A stationary solution of the coupled equations is obtained.     

The formation of primordial stars triggered by the evolution of large-scale density perturbations

This paper summarizes the basic idea of coupling of density perturbation modes in Newtonian cosmology and the formation of primordial stars. The way of deriving the second-order differential equation governing the growth of two coupled adiabatic density perturbation modes in the matter era is briefly discussed. A Jeans criterion for the growth of short wavelength perturbations is given. A mechanism is proposed for the cosmological origin of Population III stars.     

Linear waves in a radiating and scattering grey medium

The equation of radiative acoustics is derived by taking into account the effect of a transverse magnetic field, which is quite similar to the acoustic equation derived in Paper I. The only difference is that theadiabatic, isothermal, andisentropic speeds of sound and theradiation-acoustic speed are replaced by theadiabatic, isothermal, andisentropic magnetoacoustic speeds and theradiation-magnetoacoustic speed, respectively. The main results shown in Paper I are valid even in the presence of a transverse magnetic field.     

Influence of a uniform coronal magnetic field on solar p modes: coupling to slow resonant MHD waves

The influence of a constant coronal magnetic field on solar global oscillations is investigated for a simple planar equilibrium model. The model consists of an atmosphere with a constant horizontal magnetic field and a constant sound speed, on top of an adiabatic interior having a linear temperature profile. The focus is on the possible resonant coupling of global solar oscillation modes to local slow continuum modes of the atmosphere and the consequent damping of the global oscillations. In order to avoid Alfvén resonances, the analysis is restricted to propagation parallel to the coronal magnetic field. Parallel propagating oscillation modes in this equilibrium model have already been studied by Evans and Roberts (1990). However, they avoided the resonant coupling to slow continuum modes by a special choice of the temperature profile. The physical process of resonant absorption of the acoustic modes with frequency in the cusp continuum is mathematically completely described by the ideal MHD differential equations which for this particular equilibrium model reduce to the hypergeometric differential equation. The resonant layer is correctly dealt with in ideal MHD by a proper treatment of the logarithmical branch cut of the hypergeometric function. The result of the resonant coupling with cusp waves is twofold. The eigenfrequencies become complex and the real part of the frequency is shifted. The shift of the real part of the frequency is not negligible and within the limit of observational accuracy. This indicates that resonant interactions should definitely be taken into account when calculating the frequencies of the global solar oscillations.     

Ekman层风速随高度分布方程的数值解试验

根据大气水平运动方程推导出Ekman层风速随高度分布方程,该分布可用二阶线性微分方程表示,用有限差分法求解该二阶线性微分方程的边值问题的数值解,并给出计算程序,输出结果。将结果与经典解析解比较,讨论了经典解的正确性和适用范围。     

Non-stationary atmospheres of supergiants

A model of 5M hydrogen-helium star with no metals has been considered. The properties of the adiabatic oscillations were obtained, and the vibrational instability of the model has been investigated. The model is a double-shell burning star. Calculations were done for the fundamental mode as well as the first and the second harmonics. The model has been found vibrationally unstable against radial adiabatic pulsations.     

The relativistic quasilinear theory of particle acceleration by hydromagnetic turbulence

The quasilinear theory of acceleration of relativistic particles by hydromagnetic turbulence is treated in the adiabatic limit of small gyration radius. The theory is based on the relativistic Vlasov equation; however, a given pitch-angle scattering rate by microturbulence is postulated and is added to this equation. The resulting acceleration is found to be given by a diffusion coefficient in total momentum, which is proportional to the spectrum of turbulence with a rate coefficient . is a frequency that represents the efficiency of each wave component of the turbulence in producing acceleration. It is given as an integral over the solution of a differential equation in pitch angle. is evaluated in various limiting cases and is shown to lead to familiar forms of acceleration, such as Fermi acceleration and magnetic pumping. Thus, a comprehensive theory of these forms of heating is achieved.     

Linear waves in a radiating and scattering grey medium

The equation of radiative acoustics is derived by taking into account the effect of a non-transverse magnetic field, and the solutions are schematically represented. The main results shown in Paper I and Paper II are valid even in the presence of a non-transverse magnetic field, and the only difference is that theadiabatic, isothermal, andisentropic speeds of sound and theradiation-acoustic speed in Paper I which respectively correspond to theadiabatic, isothermal, andisentropic magnetoacoustic speeds and theradiation-magnetoacoustic speed in Paper II are replaced by the sets of speeds ofadiabatic, isothermal, isentropic, andradiation-acoustic fast andslow waves, respectively.     

The convection zone in stars with radiation pressure in the presence of magnetic field

The effect of a uniform magnetic field on the envelope convection zone of an 8.8M star has been studied. The adiabatic exponents _i (i=1, 2, 3), adiabatic temperature gradient and specific heat of stellar matter has been computed. It is shown that the magnetic field tends to increase the values of adiabatic temperature gradient and specific heats of stellar matter in the envelope convection zone.     

Vibrational stability of stars in thermal imbalance: A solution in terms of asymptotic expansions

The solution of the partial differential equation describing the ‘non-isentropic’ oscillations of a star in thermal imbalance has been obtained in terms of asymptotic expansions up to the first order in the parameterII/t _s, whereII is the adiabatic pulsation period for the fundamental mode andt _s , a secular time scale of the order of the Kelvin-Helmholtz time. Use has been made of the zeroth order ‘isentopic’ solution derived in I. The solution obtained allows one to derive unambiguously a general integral expression for the coefficient of vibrational stability for arbitrary stellar models in thermal imbalance. The physical interpretation of this stability coefficient is discussed and its generality and its simplicity are stressed. Application to some simple analytic stellar models in homologous and nonhomologous contraction enables one to recover, in a more straightforward manner, results obtained by Coxet al. (1973). Aizenman and Cox (1974) and Davey (1974). Finally, we emphasize that the inclusion of the effects of thermal imbalance in the stability calculations of realistic evolutionary sequences of stellar models, not considered up to now by the other authors, is quite easy and straightforward with the simple formula derived here.     

Electrostatic solitary structures in a four-component adiabatic dusty plasma

A theoretical investigation has been made of propagating electrostatic waves in a four-component adiabatic dusty plasma, whose constituents are adiabatic electrons, adiabatic ions, adiabatic positively and as well as negatively charged warm dust. The basic features of the solitary structures in such a four-component adiabatic dusty plasma are studied by the reductive perturbation method. It is found that the presence of the positive dust component does not only significantly modify the basic properties of the solitary waves, but also causes the existence of the positive solitary potential structures, which is an interesting feature shown in an adiabatic dusty plasma with the dust of opposite polarity. It is also observed that the basic properties (polarity, speed, amplitude and width) of the DA SWs are significantly modified by the effects of adiabaticity (γ>1) of electrons, ions, negatively as well as positively charged warm dust. The present investigation can be of relevance to the electrostatic solitary structures observed in various dusty space plasma environments (viz. cometary tails, upper mesosphere, Jupiter’s magnetosphere, etc.).     

The growth of density perturbations in the Lemaître universe

The rate of growth of density perturbations in certain Lemaître universes has been investigated using the differential equation derived by Bonnor. The perturbations that must be postulated at decoupling are not significantly different from those required in the conventional zero-pressure Friedmann-Lemaître models.     

On the integrability cases of the equation of motion for a satellite in an axially symmetric gravitational field

The projection of an axially symmetric satellite's orbit on a plane perpendicular to the rotation axis (z=const.) is given by the second-order differential equation. $$\frac{{y''}}{{1 + y'^2 }} = \bar \Psi _y - y'\bar \Psi _{x,}$$ where the prime denotes the derivative with respect tox and \(\bar \Psi (x,y)\) is a known function. Two integrability cases have been investigated and it has been shown that for these two cases the integration can be carried out either by quadratures or reduced to a first-order differential equation. Analytical and physical properties are expressed, and it is shown that the equation can be derived from the calssical plane eikonal equation of geometric optics.     

On the stability of a class of radiating viscous self-gravitating stars with axial symmetry

This study investigates the stability of a class of radiating viscous self-gravitating stars with axial symmetry having anisotropic pressure. We use perturbation technique to establish the perturbed form of the Einstein field equations and dynamical equations. The instability range in the Newtonian and post-Newtonian eras has been analyzed by constructing the collapse equation. It is found that the adiabatic index has a key role in the discussion of instability ranges which depends upon the physical parameters, i.e., energy density, anisotropic pressure and shear viscosity of the fluid and heat flux. We conclude that the shear viscosity decreases the instability range and makes the system more stable.     

Motion in the interiors and atmospheres of Jupiter and Saturn: scale analysis,anelastic equations,barotropic stability criterion

If Jupiter's and Saturn's fluid interiors were inviscid and adiabatic, any steady zonal motion would take the form of differentially rotating cylinders concentric about the planetary axis of rotation. B. A. Smith et al. [Science215, 504–537 (1982)] showed that Saturn's observed zonal wind profile extends a significant distance below cloud base. Further extension into the interior occurs if the values of the eddy viscosity and superadiabaticity are small. We estimate these values using a scaling analysis of deep convection in the presence of differential rotation. The differential rotation inhibits the convection and reduces the effective eddy viscosity. Viscous dissipation of zonal mean kinetic energy is then within the bounds set by the internal heat source. The differential rotation increases the superadiabaticity, but not so much as to eliminate the cylindrical structure of the flow. Very large departures from adiabaticity, necessary for decoupling the atmosphere and interior, do not occur. Using our scaling analysis we develop the anelastic equations that describe motions in Jupiter's and Saturn's interiors. A simple problem is solved, that of an adiabatic fluid with a steady zonal wind varying as a function of cylindrical radius. Low zonal wavenumber perturbations are two dimensional (independent of the axial coordinate) and obey a modified barotropic stability equation. The parameter analogous to β is negative and is three to four times larger than the β for thin atmospheres. Jupiter's and Saturn's observed zonal wind profiles are close to marginal stability according to this deep sphere criterion, but are several times supercritical according to the thin atmosphere criterion.     

Force-free fields in the vicinity of a Reissner-Nordstrøm black hole

The behavioyr of a force-free field has been studied in a Reissner-Nordström metric. An expansion in tensor harmonics of even-odd parity reduced the radial equations in a differential equation of the Sturm-Liouville system which was solved asymptotically in a conveniently defined space coordinate. Further, it has been possible to regularize the singular behaviour of the Reissner-Nordström metric at the event horizon and the modified metric to be given explicitly.     

Quasi-viscous accretion flow – I. Equilibrium conditions and asymptotic behaviour

In a novel approach to studying viscous accretion flows, viscosity has been introduced as a perturbative effect, involving a first-order correction in the α-viscosity parameter. This method reduces the problem of solving a second-order non-linear differential equation (Navier–Stokes equation) to that of an effective first-order equation. Viscosity breaks down the invariance of the equilibrium conditions for stationary inflow and outflow solutions, and distinguishes accretion from wind. Under a dynamical systems classification, the only feasible critical points of this 'quasi-viscous' flow are saddle points and spirals. On large spatial scales of the disc, where a linearized and radially propagating time-dependent perturbation is known to cause a secular instability, the velocity evolution equation of the quasi-viscous flow has been transformed to bear a formal closeness with Schrödinger's equation with a repulsive potential. Compatible with the transport of angular momentum to the outer regions of the disc, a viscosity-limited length-scale has been defined for the full spatial extent over which the accretion process would be viable.     

Dynamical Evolution of Gamma-Ray Burst Remnants with Evolving Radiative Efficiency

In previous works, a generic dynamical model has been suggested by Huang et al., which is shown to be correct for both adiabatic and radiative blast-waves, in both ultra-relativistic and non-relativistic phases. In deriving their equations, Huang et al. have assumed that the radiative efficiency of the fireball is constant. They then applied their model directly to realistic cases where the radiative efficiency evolves with time. In this paper, we abandon the above assumption and re-derive a more accurate dynamical equation for gamma-ray burst remnants. Numerical results show that the model presented by Huang et al. is accurate enough in general cases.