Approximate solutions for isothermal flows behind strong spherical shocks with variable energy

Self-similar unsteady flows with zero temperature gradient behind strong spherical shocks propagating in non-uniform perfect gas at rest are investigated. The total energy of the flow is assumed to be varying with the shock radius obeying a power law. Approximate solutions in a closed analytical form are obtained using the integral method. Also these solutions are shown to be useful to calculate easily and quickly the shock temperature, X-ray surface brightness and luminosity which are important in astrophysical problems. It is found that these approximate solutions are in close agreement with numerical solutions.     

Neutrino transport with isoenergetic and neutrino-electron scattering

Solution to steady-state, bigroup neutrino transport with isoenergetic and neutrino-electron scattering have been obtained for idealized conditions representative of the outer core or envelope of a star whose inner core is undergoing gravitational collapse. The solutions are presented in a form suitable for comparison with other numerical schemes. The incident neutrino distribution was chosen to be nonequilibrium, and the solutions clearly show the tendency of the neutrinos to equilibrate via neutrino-electron scattering as they propagate through the material. Care is taken to compare solutions obtained by two completely different techniques to ensure that the solutions presented here are accurate and error free. The solutions are compared with those obtained by a simple diffusion scheme, and by a flux limited diffusion scheme. The solutions given by the latter agree well with the solutions derived here, except when the neutrino angular distribution becomes oblate.     

Exact solutions for the gravitational potential of a family of heterogeneous spheroids

An infinite family of heterogeneous spheroids has been found for which exact or closed-form solutions for the Newtonian gravitational potential can be given. The family includes both axisymmetric spheroids and spheroids where the matter density varies harmonically with the azimuthal angle. For the axisymmetric family of spheroids, which have no azimuthal dependence of the density, the potential external to the spheroid is of the same form as the potential exterior to a spheroidal homoeoid. It is therefore constant on the surface of the spheroid and on all external spheroidal surfaces confocal with it. The potential is also constant on all internal confocal spheroidal surfaces, with the value on each confocal surface dependent on the density distribution chosen. However, the density is not constant on either concentric or confocal spheroids. These solutions can be considered to be generalizations of analogous spherical solutions given in a companion paper by Conway. For the classical solutions for homogeneous spheroids, the surface is not equipotential, and these are not included within the new family of solutions, except in the special case of a homogeneous sphere.     

Spherical radiation-driven shock waves in a non-uniform atmosphere

Similarity solutions are obtained for spherical radiation-driven shock waves propagating in a non-uniform atmosphere at rest obeying a density power law. Approximate analytical solutions are also obtained and found to be in good agreement with the numerical solutions. The effect of the parameter characterizing the initial density distribution of the gas on solutions of the flow field is studied in detail. It is also shown analytically that the shock wave propagates as an overdriven detonation.     

A new shock locus for similarity solutions in one-dimensional unsteady gas dynamics

This paper deals with shock conditions for the progressing wave (or similarity) solutions of one-dimensional, unsteady gas dynamics. These solutions have hitherto been used to deal with the flow behind shocks moving into stationary atmospheres. By generalising the shock conditions to the case of moving atmospheres, it is shown that the progressing wave solutions can be used to describe a certain class of flows, and a new shock locus can be constructed in the phase plane of the solutions. It is hoped that such solutions will be of use in describing the unsteady flow behind shocks propagating into the ambient solar wind.     

A new class of relativistic stellar models

Einstein field equations for a static and spherically symmetric perfect fluid are considered. A formulation given by Patiño and Rago is used to obtain a class of nine solutions, two of them are Tolman solutions I, IV and the remaining seven are new. The solutions are the correct ones corresponding to expressions derived by Patiño and Rago which have been shown by Knutsen to be incorrect. Similar to Tolman solution IV each of the new solutions satisfies energy conditions inside a sphere in some range of two independent parameters. Besides, each solution could be matched to the exterior Schwarzschild solution at a boundary where the pressure vanishes and thus the solutions constitute a class of new physically reasonable stellar models.     

Pseudo-Newtonian models for the equilibrium structures of rotating relativistic stars

We obtain equilibrium solutions for rotating compact stars, including special relativistic effects. The gravity is assumed to be Newtonian, but we use the active mass density, which takes into account all energies such as the motion of the fluid, internal energy and pressure energy in addition to the rest-mass energy, in computing the gravitational potential using Poisson's equation. Such a treatment could be applicable to neutron stars with relativistic motions or a relativistic equation of state. We applied Hachisu's self-consistent field (SCF) method to find spheroidal as well as toroidal sequences of equilibrium solutions. Our solutions show better agreement with general relativistic solutions than the Newtonian relativistic hydrodynamic approach, which does not take into account the active mass. Physical quantities such as the peak density and equatorial radii in our solutions agree with the general relativistic ones to within 5 per cent. Therefore our approach can be used as a simple alternative to the fully relativistic one when a large number of model calculations is necessary, as it requires much fewer computational resources.     

火星非球形引力位田谐项联合摄动分析解

火星非球形引力场模型与地球有明显差别,其非球形引力位中的田谐项系数基本都要比地球的相应值大一个量级,尤其是J_2,2项（赤道椭率项）的大小接近它的动力学扁率项J₂.对于低轨探测器,若要使轨道外推1 d弧段的精度达到500 m(相当于标准单位10^-4量级),在构造环火探测器的轨道分析解时,田谐项与J₂项以及田谐项与田谐项之间的联合摄动不容忽视.根据摄动量级分析和构造的摄动分析解证实,上述联合摄动对轨道沿迹方向的影响可超过10^-4,并给出了数值验证.结果表明,与地球低轨卫星不同,在类似的问题中,构造环火卫星摄动分析解时,必须考虑这些联合摄动项的影响.     

Solutions of Einstein-Maxwell field equations for a static charged perfect fluid shphere

The object of this paper is to give a new mathematical and physical method of finding explicit analytical interior solutions of the Einstein-Maxwell field equations of a static perfect fluid sphere with charge. In spite of many successful efforts in solving the field equations, the importance of finding meaningful general analytic solutions remains. Our purpose is to obtain the interior solutions of the field equations that they complete the results, which they have been already published in an earlier paper (Dionysiou, 1982; this paper will be referred to hereafter as Paper I). Using our new formulae, we then rederive some known results as particular solutions.     

Stationary axisymmetric fields in Generalized Theory of Gravitation

The problem of stationary axisymmetric gravitational fields is formulated within the framework of Generalized Theory of Gravitation. It is shown that solutions of the problem mentioned above may be found, if analogous solutions in General Relativity are obtained. As an illustration a Kerr-like solution is offered. A generation theorem for finding magnetostatic solution from stationary vacuum solutions is proposed.     

Comments on the comment on Jordan-Brans-Dicke universes

The principal objections to our paper on Jordan-Brans-Dicke Bianchi-type universes are misleading. The intended solutions assume a power-law form, only in the limit of very large times. Although this is the only one for which we wrote down explicit solutions. The physical point of view behind our paper was to extend the re-scaling method, developed by one of us, to anisotropic universes in general and not to make a detailed study of each of the possible solutions that could be obtained through this process.     

The spectacular Supernova SN1987A: Grain evaporation and condensation

Non-similarity solutions of the equations governing the motion of a perfect gas behind a cylindrical shock wave of variable strength have been obtained. These solutions are applicable to both the weak and the strong shocks. The nature of flow and field variables are illustrated through graphs. The total energy of the wave is taken to be constant.     

Integral surfaces with space-time coordinates,in the gravitational field of a rotating system

A new integration theory is formulated for dynamical systems with two degrees of freedom, in the gravitational field of a rotating system. Four integrals of motion may be determined from complete solutions of a system of three first-order, partial differential equations in three independent variables. The solutions of this system define two integral surfaces with space-time coordinates. These surfaces represent two independent solutions of a second-order kinematic system to which the original fourth-order system has been reduced. An integral curve may be represented as the locus of intersection points of the integral surfaces. The new theory is the theoretical basis for a method of analytic continuation of periodic orbits of the circular restricted problem.     

Linear stability and shape analysis of spinning three-craft Coulomb formations

This paper describes the discovery of families of multiple invariant shape solutions for collinear three-craft Coulomb formations with set charges, as well as the results of linear stability analysis on such formations. The charged spacecraft are assumed to be spinning about each other in deep space without relevant gravitational forces present. Up to three invariant shape solutions are possible for a single set of craft charges. This behavior, only speculated in previous work, is confirmed through analysis and numerical simulation examples. In fact, distinct regions are analytically described where two or three invariant shape solutions exist for a single charge set. These regions are analyzed to determine what range of trajectories are possible. Linear stability analysis for circular trajectories yields the first examples of marginally stable three-craft invariant shape formations. Linearly stable behavior is only observed when two invariant shape solutions result for one set of charges, where one shape will be unstable and the other marginally stable. Numerical simulation illustrates stability for ten orbital periods when perturbations are confined to the orbital plane. When out of plane motion is considered the shapes are found to be weakly unstable, though the out of plane motion appears to be decoupled from in plane motion to first order.     

Relativistic models of magnetars: the twisted torus magnetic field configuration

We find general relativistic solutions of equilibrium magnetic field configurations in magnetars, extending previous results of Colaiuda et al. Our method is based on the solution of the relativistic Grad–Shafranov equation, to which Maxwell's equations can be reduced. We obtain equilibrium solutions with the toroidal magnetic field component confined into a finite region inside the star, and the poloidal component extending to the exterior. These so-called twisted torus configurations have been found to be the final outcome of dynamical simulations in the framework of Newtonian gravity, and appear to be more stable than other configurations. The solutions include higher-order multipoles, which are coupled to the dominant dipolar field. We use arguments of minimal energy to constrain the ratio of the toroidal to the poloidal field.     

Equilibrium solutions of the restricted problem of 2+2 bodies

Fourteen equilibrium solutions of the restricted problem of 2+2 bodies are shown to exist. Six of these solutions are located about the collinear Lagrangian points of the classical restricted problem of three bodies. Eight solutions are found in the neighborhood of the triangular Lagrangian points. Linear stability analysis reveals that all of the equilibrium solutions are unstable with the exception of four solutions; two in the vicinity of each of the triangular Lagrangian points. These four solutions are found to be stable provided the mass parameter of the primary masses is less than a critical value which depends also on the mass of the minor bodies.     

Analytical solutions for the equations of motion of a space vehicle during the atmospheric re-entry phase on a 2-D trajectory

A practical and important problem encountered during the atmospheric re-entry phase is to determine analytical solutions for the space vehicle dynamical equations of motion. The author proposes new solutions for the equations of trajectory and flight-path angle of the space vehicle during the re-entry phase in Earth’s atmosphere. Explicit analytical solutions for the aerodynamic equations of motion can be effectively applied to investigate and control the rocket flight characteristics. Setting the initial conditions for the speed, re-entering flight-path angle, altitude, atmosphere density, lift and drag coefficients, the nonlinear differential equations of motion are linearized by a proper choice of the re-entry range angles. After integration, the solutions are expressed with the Exponential Integral, and Generalized Exponential Integral functions. Theoretical frameworks for proposed solutions as well as, several numerical examples, are presented.