Extension of the solution of Kepler's equation to high eccentricities

The classic Lagrange's expansion of the solutionE(e, M) of Kepler's equation in powers of eccentricity is extended to highly eccentric orbits, 0.6627 ... <e<1. The solutionE(e, M) is developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients of the expansion are given in terms of the derivatives of the Bessel functionsJ _n (ne). The expansion is convergent for values of the eccentricity such that |e–e*|<(e*), where the radius of convergence (e*) is a positive real number, which is calculated numerically.     

Some expansions of the elliptic motion to high eccentricities

Some classic expansions of the elliptic motion — cosmE and sinmE — in powers of the eccentricity are extended to highly eccentric orbits, 0.6627...<e<1. The new expansions are developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients are given in terms of the derivatives of Bessel functions with respect to the eccentricity. The expansions have the same radius of convergence (e*) of the extended solution of Kepler's equation, previously derived by the author. Some other simple expansions — (a/r), (r/a), (r/a) sinv, ..., — derived straightforward from the expansions ofE, cosE and sinE are also presented.     

Models of Strange Stars with a Crust and Strange Dwarfs

Strange quark stars with a crust and strange dwarfs consisting of a compact strange quark core and an extended crust are investigated in terms of a bag model. The crust, which consists of atomic nuclei and degenerate electrons, has a limiting density of _cr=_drip=4.3·10¹¹ g/cm³. A series of configurations are calculated for two sets of bag model parameters and three different values of _cr (10⁹ g/cm³ _{cr drip}) to find the dependence of a star's mass M and radius R on the central density. Sequences of stars ranging from compact strange stars to extended strange dwarfs are constructed out of strange quark matter with a crust. The effect of the bag model parameters and limiting crust density _cr on the parameters of the strange stars and strange dwarfs is examined. The strange dwarfs are compared with ordinary white dwarfs and observational differences between the two are pointed out.     

On Low-Mass, Strange Quark Stars with a Crust

Low-mass strange stars with a crust are investigated within the framework of the bag model. The crust, which consists of degenerate electrons and atomic nuclei, has a limiting boundary density _cr , which is determined by the mass of the crust, and it cannot exceed the value _drip = 4.3·10¹¹ g/cm³, corresponding to the density at which neutrons drip from nuclei. For different values of _cr in the low-mass range (M 0.1 M) we calculate several series of configurations: we find the dependence of the stellar mass M on the central density _c for _cr = const, with 10⁹ g/cm³ _{cr drip} , and for each series we determine the parameters of the configuration for which the condition dM/d _c > 0 is violated. When the boundary density of the crust decreases to 10⁹ g/cm³, the minimum mass of a strange star decreases to M _min 10^-3 M, while the radius reaches 600 km.     

Three-component fluid model of the universe of critical density

A three-component fluid model of the Universe during the recombination era is analysed for = _c ( _c is the critical density). In addition to the well-known instability of the Jeans mode at 10⁹ M , we find two more unstable modes at 10¹² M .     

Kelvin-Helmholtz instability of composite plasma in an oblique magnetic field with resistivity

Analytic structure of high-density steady isothermal spheres is discussed using the TOV equation of hydrostatic equilibrium which satisfies an equation of state of the kind:P = K _g , = _g c ².Approximate analytical solutions to the Tolman-Oppenheimer-Volkoff (TOV) equations of hydrostatic equilibrium in (, ), (,U) and (u, v) phase planes in concise and simple form useful for short computer programmes or on small calculator, have been given. In Figures 1, 2, and 3, respectively, we display the qualitative behaviours of the ratio of gas density _g to the central density _gc , _g / _gc ; pressureP to the _gc ,P/ _gc ; and the metric componente ^–, for three representative general relativistic (GR) isothermal configurations =0.1, 0.2, and 0.3. Figure 4 shows the solution curve (, ) for =0.1, 0.2, and 0.3 (=0 represents the classical (Newtonian) curve). Numerical values of physical quantitiesv (=4r ² P ^*(r)), in steps ofu (=M(r)/r)=0.03, and the mass functionU, in steps of =0.2 (dimensionless radial distance), are given, respectively, in Tables I and II. Other interesting features of the configurations, such as ratio of gravitational radius 2GM/c ² to the coordinate radiusR, mass distributionM(r)/M, pressure (or density) distributionP/P _c , binding energy (B.E.), etc., have also been incorporated in the text. It has further been shown that velocity of sound inside the configurations is always less than the velocity of light.Part of the work done at Azerbaijan State University, Baku, U.S.S.R., and Mosul University, Mosul, Iraq, 1985-1986     

Physically realizable relativistic stellar structures

Relativistic stellar structures can be obtained both analytically and by computation, but all these models do not stand the tests of physical reality. It is shown that for a physically reasonable solution d/dr -r and d(e ^v /dr r near the centre, d(P/)/dr<-0 and (dP/d)>-(P/). If we change the variabler tox=Cr ², whereC is a constant, the field equations are reduced to a form which is easier to solve. A new set of exact solutions is obtained by consideringe ^v (1-x) ⁿ . Also, a method has been given to obtain generalized solution.It is shown that the solution discussed by Durgapal and Rawat (1980) is the only exact solution which in its most generalized form for a given density distribution stands all the tests of physical reality and for which both (P/) and (dP/d) decrase with increasing value ofr.Neutron star model is proposed by assuming >-2×10¹⁴g cm^–3. Two specific cases are considered, viz.,P<-1/3 and dP/d<-1, respectively. The structures are found to be bound and stable under radial perturbations. The models have been studied for slow rotation and the mass of the Crab pulsar has been estimated for different mechanisms.     

A note on the equation of state of a scalar field

The energy momentum tensor of a scalar field is considered as being that of a perfect fluid with equation of statep=p(). In the extreme case that the field energy is purely kinetic,p=p, whereas if it is purely potential,p=–.     

Electrical conductivity of neutron star cores in the presence of a magnetic field

General theory of electrical conductivity of a multicomponent mixture of degenerate fermions in a magnetic fieldB, developed in the preceding article (this volume), is applied to a matter in neutron star interiors at densities ₀, where ₀ = 2.8×10¹⁴ g cm^–3 is the standard nuclear matter density. A model of free-particle mixture ofn, p, e is used, with account for appearance of ^–-hyperons at > _c , where _c 4₀. The electric resistivities along and acrossB, and , and the Hall resistivity _H are calculated and fitted by simple analytical formulae at _c and > _c for the cases of normal or superfluid neutrons provided other particles are normal. Charge transport alongB is produced by electrons, due to their Coulombic collisions with other charged particles; is independent ofB and almost independent of the neutron superfluidity. Charge transport acrossB at largeB may be essentially determined by other charged particles. If _c , one has = [1 + (B/B ₀)²] for the normal neutrons, and for the superfluid neutrons, while _H = B/B _e for both cases. HereB _e 10⁹ T ₈ ² G,B ₀10¹¹ T ₈ ² G, andT ₈ is temperature in units of 10⁸ K. Accordingly for the normal neutrons atBB ₀, the transverse resistivity suffers an enhancement, _{1/4 1}. When 5₀ andB varies from 0 toBB _p 10¹³ T ₈ ² G, increases by a factor of about 10³–10⁴ and _H changes sign. WhenBB _p , remains constant for the superfluid neutrons, and _H B ² for the normal neutrons, while _H B for any neutron state. Strong dependence of resistivity onB, T, and may affect evolution of magnetic fields in neutron star cores. In particular, the enhancement of at highB may noticeably speed up the Ohmic decay of those electric currents which are perpendicular toB.     

Analytical solutions of cylindrical shock waves in magnetogasdynamics,I

The C.C.W. method has been used to investigate the propagation of converging and diverging cylindrical shock waves in a non-uniform medium under the influence of a magnetic field of constant strength. A comparison has also been made between the two types of cylindrical shock waves, simultaneously for both weak and strong cases of the magnetic field. Density distribution is assumed to be _o = r ^–, where is the density at the axis of symmetry and a constant. The analytical expressions for shock velocity and shock strength as well as the pressure, the density, and the particle velocity just behind the shock front have been derived for both the cases.     

The effect of chemical abundance oscillations on the pulsational properties of A 5M ⊙ hydrogen-helium star

A model of a first generation intermediate star of 5M , with Z=0 has been considered. The model is at an advanced stage of its evolution and has a double shell burning. It burns helium in the inner shell, and hydrogen, via CNO cycle, in the outer shell. =(log/log) _T and _T =(log/logT) were computed allowing for the oscillations of the relative mass abundance of the reagents in nuclear reactions. Including =(log/log) _T and =(log/logT) of mean molecular weight and the effect of the oscillations of abundances due to nuclear reactions, stability was studied. Contrary to the results of the static calculations, we found that instability due to the excitation mechanism provided by the high temperature sensitivity of energy generation rate propagates up to the surface. Thus the model in question was found to be unstable against radial adiabatic pulsations, in its fundamental mode.     

Neutron-phonon interaction in neutron stars: Phonon spectrum of Coulomb lattice

The phonon excitation spectrum of Coulomb lattice in the neutron star crusts is studied by solving Dyson's equation for phonons. It is shown that a strong renormalization of the phonon spectrum occurs at densities _s /4 for the crustal matter compositions with spherical nuclei, which imply relatively small nuclear mass numbers and charges. It is shown that, the lattice becomes unstable against density fluctuations above a critical density of the order of _s /3, where _s 2.6x10¹⁴ g cm^–3 is the nuclear saturation density. The neutron quasiparticle spectrum and the virtual mass of a nucleus are briefly discussed.     

A new kind of 3-body problem

A new kind of restricted 3-body problem is considered. One body,m ₁, is a rigid spherical shell filled with an homogeneous incompressible fluid of density ₁. The second one,m ₂, is a mass point outside the shell andm ₃ a small solid sphere of density ₃ supposed movinginside the shell and subjected to the attraction ofm ₂ and the buoyancy force due to the fluid ₁. There exists a solution withm ₃ at the center of the shell whilem ₂ describes a Keplerian orbit around it. The linear stability of this configuration is studied assuming the mass ofm ₃ to beinfinitesimal. Explicitly two cases are considered. In the first case, the orbit ofm ₂ aroundm ₁ is circular. In the second case, this orbit is elliptic but the shell is empty (i.e. no fluid inside it) or the densities ₁ and ₃ are equal. In each case, the domain of stability is investigated for the whole range of the parameters characterizing the problem.     

The jeans instability criterion for a compressible fluid including viscosity and heat conduction

For the region after the recombination era of the Universe the hydrodynamical density waves are analyzed including shear viscosity and heat conduction for =_c as well as for <_c(_c is the critical density of the Universe). Very near to the end of the recombination era (z=1200) we find the well-known Jeans instability. It is shown that the influence of the shear viscosity on the instabilities in negligible, however, a visible influence of the bulk viscosity is present.     

The maximum mass of a neutron star

We consider the possibility that gravitational energy may play a local as well as global role in the behavior of matter in strong gravitational fields. A particular idealized equation, suggested as representing uniform energy density in general relativity, is examined, and its stability with respect to oscillatory and convective perturbations shown to be consistent with general relativistic hydrodynamics, subject to a new physical effect predicted for the behavior of fluids moving in strong fields. We calculate from this idealized equation the mass of a non-rotating neutron star, obtaining a maximum surface redshift ofz=2.48 and a maximum core mass of 9.79 ₁₄ ^–1/2 M. This compares withz=2.00 and 11.4 ₁₄ ^–1/2 M for a Schwarzschild star (=const.) and 6.8 ₁₄ ^–1/2 M for a causal star (dP/d1).     

The angular momentum problem and magnetic braking during star formation: Exact solutions for an aligned and a perpendicular rotator

The problems of fragmentation, angular momentum, and magnetic flux during star formation are reviewed briefly. Then the resolution of the angular momentum problem through magnetic braking is studied rigorously.A disk-like interstellar cloud of uniform density _cl is given an initial angular velocity _o about its axis of symmetry, which isaligned with an initially uniform, frozen-in magnetic field. Torsional Alfvén waves transport angular momentum from the cloud to the external medium, which has a uniform density _ext . The angular velocity of the cloud ( _cl ) is determined analytically as a function of space and time for different ratios _cl / _ext (the only free parameter in the equations), representing different stages of contraction. Despite dissimilar transient response of the cloud (or fragment) structure to different initial conditions, the characteristic time for magnetic braking of the rotation of the cloud (or fragment) as a whole is remarkably insensitive to the initial conditions and independent of the stage of contraction. The latter conclusion is in agreement with an approximate result obtained recently (Mouschovias, 1978; 1979a).A cylindrical cloud (or fragment) of uniform density is also imparted an initial angular velocity about its axis of symmetry with respect to the external medium. The frozen-in magnetic field is now initially radial andperpendicular to the axis of symmetry. In this case magnetic braking becomes more efficient upon contraction. It is more efficient than the aligned rotator case typically by one order of magnitude. The angular momentum problem can be resolved in about 10⁶ yr during the early stages of cloud contraction. Planetary systems, such as the Sun-Jupiter pair, become dynamically possible. A stage exists in which a cloud (or fragment) is in retrograde rotation with respect to its surroundings. This provides the first and only observable prediction of magnetic braking in action. It also constitutes a natural explantation of retrograde rotation in stellar and planetary systems.This work was supported in part by the National Science Foundation under grant NSF AST-77-23568.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

Cosmological mass model in general relativity

Relativistic cosmological field equations are obtained for a non-static stationary Bertotti-Robinson-type space-time for interacting perfect fluid and electromagnetic field. The cosmological solution to the field equations are obtained and the nature of the electromagnetic field as well the perfect fluid are studied. The electromagnetic field generated here corresponds to a special generic case and the perfect fluid distribution degenerates into a barotropic perfect fluid with equation of statep+=0, >0. It is shown here that the interacting barotropic fluid can generate gravitation only when the cosmological constant being a function ofx in a dynamic field.     

Evolution of Wide Star Pairs Owing to Irregular Forces

The dynamic evolution of wide star pairs (separated by 10⁴ AU for typical galactic conditions) under the influence of random interactions with background stars during transits is studied theoretically. Transits which form a roughly equilateral triangle at some time are considered. The combined effect of many transits can be separated into systematic heating of the pair plus random diffusion over the Kepler elements a and e. The corresponding Fokker-Planck equation is derived and its range of validity indicated. A steady state distribution over a and e is obtained.     

Distribution of minima of the current size of the triple systems and their disruption

The dynamical evolution of triple systems has been studied by computer simulations. A function (t) has been defined, where p is the maximum distance of the components from their centre of inertia, and t is the time. The value of is used to indicate the current size of the triple system. The minima of have been followed during the course of evolution of the triples. A distribution of f(_min) has been obtained, which is described by the following statistical parameters: the mode is equal to 0.65d, the mean value _min= 0.750d, r.m.s. is 0.477d, the asymmetry is 0.218, the excess is 2.04 where d is the mean harmonic distance between the bodies in the equilibrium state of the triple system. As a rule, escapes from triples occur only after close three-body approaches.     

The physical properties of an analytic model for a relativistic star

An exact solution of Einstein's equations for a static isentropic perfect fluid sphere is examined in detail. The analysis yields a strong indication that the model isstable with respect to infinitesimal radial pulsations. This means that the temperature is decreasing outwards. We prove that the adiabatic speed of sound is everywhere less than the speed of light if and only if the radius of the sphere is larger than 1.61 times its Schwarzschild radius. We further show that the strong energy condition is fulfilled everywhere if and only if the radius is larger than 1.76 times the Schwarzschild radius. The necessary and sufficient condition for the speed of sound to be decreasing outwards is given, and we find that this criterion is fulfilled if the fluid is causal. Taking the values of the pressure and the density to be somewhere given by the maximum values from Baymet al.'s equation of state, i.e., ₀=5.1×10¹⁴ g cm^–3 andp ₀=7.4×10³³ dyne cm^–2, we find the maximum mass of the fluid sphere to be 2.5 solar masses.Dedicated to the memory of the late George Cunliffe McVittie (1904–1988).