Expansions of (r/a)m cos jv and (r/a)m sin jv to high eccentricities

Expansions of the functions (r/a)^cos jv and (r/a)m ^sin jv of the elliptic motion 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 of these expansions are expressed in terms of the derivatives of Hansen's coefficients with respect to the eccentricity. The new expansions are convergent for values of the eccentricity such that |e–e*|<(e*), where the radius of convergence (e*) is the same of the extended solution of Kepler's equation. The new expansions are intrinsically related to Lagrange's series.     

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.     

Some applications of Jacobi dynamics to the stellar evolution

The stars in the Main Sequence are seen as a hierarchy of objects with different massesM and effective dynamical radiiR _eff=R/ given by the stellar radii and the coefficients for the inner structure of the stars.As seen in a previous work (Paper I), during the lifetime in the Main SequenceR _eff(t) remains a near invariant when compared to the variation in the time ofR(t) and (t).With such an effectiveR _eff one obtains the amounts of actionA _c(M), the effective densities _eff(M)=(M)³(M), the densities of action and of energy (or mean presures in the stellar interior)a _c(M),e _c(M), and the potential energiesE _p(M).The amounts of action areA _cM ^k withk1.87 for the M stars,k5/3 for the KGF stars, andk1.83 for the A and earlier stars, representing very simples conditions for the other dynamical parameters. For instancek5/3 means a near invariant effective density _eff for the KGF stars, while for such stars the mean densities and coefficients present the strongest variations with masses (M)M ^–1.81, (M)M^0.6.The cases for the M stars (e _c(M)M ^–1) and for the A and earlier stars (betweena _c(M)=constant and _eff(M)M ^–1) and also discussed. These conditions for the earlier stars also represent reasonable mean values for the whole stellar hierarchy in the range of masses 0.2M M25M .With all this, one can build dynamical HR diagrams withA _c(M), E_p(M), _eff M ^–p , etc., whose characteristics are analogous to these in the photometrical HR diagram. A comparison is made betweenA _c(M) from the models here and the HR diagram with the best known stars of luminosity classes IV, V, and white dwarfs.The comparison of the potential energiesE _p(M)M ^–p according to the stellar models used here and the observed frequency function (MM _–q (number of stars in a given interval of masses) from different authors suggests the possibility that the productE _p(M)(M) is a constant, but this must be confirmed with further studies of the function (M) and its fine structure.There are analogies between the formulation used here for the stellar hierarchy and other physical processes, for instance, in modified forms of the Kolmogorov law of turbulence and in the formulation used for the hierarchy of molecular clouds in gravitational equilibrium. Besides, the function of actionA _c(M) for the stars has analogous properties to the relations of angular momenta and massesJ(M) for different types of objects. The cosmological implications of all this are discussed.     

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 .     

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.     

On eccentricity functions for eccentric orbits

Situations arise in celestial mechanics where orbital eccentricities are large and yet it is desirable to maintain the Darwin-Kaula Fourier decomposition of the perturbing function. Evaluation of the appropriate eccentricity functionsG _lpq (e) requires a double summation which, for practical purposes, must be truncated. In this note criteria have been established for truncation of the expansion for eccentricities 0.75.     

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=–.     

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     

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.     

A general algorithm for the solution of Kepler's equation for elliptic orbits

An efficient algorithm is presented for the solution of Kepler's equationf(E)=E–M–e sinE=0, wheree is the eccentricity,M the mean anomaly andE the eccentric anomaly. This algorithm is based on simple initial approximations that are cubics inM, and an iterative scheme that is a slight generalization of the Newton-Raphson method. Extensive testing of this algorithm has been performed on the UNIVAC 1108 computer. Solutions for 20 000 pairs of values ofe andM show that for single precision (10^–8) 42.0% of the cases require one iteration, 57.8% two and 0.2% three. For double precision (10^–18) one additional iteration is required. Single- and double-precision FORTRAN subroutines are available from the author.     

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.     

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.     

Similarity between the structure and stability of isothermal and polytropic gas spheres

As shown by Chiéze, polytropic gas spheres imbedded in an ambient medium with a given pressure exhibit a relation between the mass and radius similar to isothermal gas spheres, providedn(polytropic index)>3 or <–1. Here, further investigation is made of the polytropic spheres. Not only the relation between radius and external pressure, but the relation between the central density ( _c ) and the mass (M) is similar to isothermal spheres and neutron star models. Dynamical calculation shows that the stability of polytropic spheres is lost at the configuration which corresponds to the first peak in the _c —M relation, as is the case for neutron star models in general relativity.     

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.     

Hydromagnetic Ekman layer on free convection past an infinite porous flat plate

Free convection in a conducting liquid past an infinite porous vertical flat plate in a rotating frame of reference when the Hall current is present is considered. Exact solutions for the velocity and temperature fields have been derived. The effects ofM (Hartmann number),m (Hall parameter), andE (Ekman number) on the velocity field are discussed.Nomenclature C _p specific heat at constant pressure - g acceleration due to gravity - E Ekman number - G Grashof number - H ₀ applied magnetic field - j _x ,j _y ,j _z components of the current densityJ - k thermal conductivity - M Hartmann number - m Hall parameter - P Prandtl number - Q heat flux per unit area - T temperature of the fluid near the plate - T _w temperature of the plate - T temperature of the fluid in the free-stream - u, v, w components of the velocity fieldq - U uniform free-stream velocity - w ₀ suction velocity - x, y, z Cartesian coordinates - z dimensionless coordinate normal to the plate Greek symbols coefficient of volume expansion - _e cyclotron frequency - _e electron collision time - _u skin friction in the direction ofu - _v skin friction in the direction ofv - dimensionless temperature - density of the fluid - kinematic viscosity - _e magnetic permeability - electrical conductivity of the fluid - angular velocity     

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.     

On some results of calculations of the chromospheric flare initial phase model according to the strong gasodynamic explosion theory

It is shown that while comparing the observed dependences of radius, front and plasma velocities at the time of the very initial flare phase with the family of the theoretical curves of these functions obtained according to strong explosion theory with the set of the initial density ₀ and estimated from the observations of the total flare energy E ₀, one can determine ₀ in the flare core. Knowing ₀ and E ₀, densities in the filaments responsible for the flare radiation and the effective and spectroscopic radiation volumes as well, one can determine the effective mass and the density between the filaments. The estimated mass, electron temperature, front and plasma velocities and densities both in the filaments and in the intervals are in satisfactory agreement with the modern concept of flares.     

Solution of canonical equations which result from elimination of short-periodic terms in second-order planetary theory

We solve the first order non-linear differential equation and we calculate the two quadratures to which are reduced the canonical differential equations resulting from the elimination of the short period terms in a second order planetary theory carried out through Hori's method and slow Delaunay canonical variables when powers of eccentricities and the sines of semi-inclinations which are >3 are neglected and the eccentricity of the disturbing planet is identically equal to zero. The procedure can be extended to the case when the eccentricity of the disturbing planet is not identically equal to zero. In this latter general case, we calculatedthe two quadratures expressing angular slow Delaunay canonical variable ₁ of the disturbed planet and angular slow Delaunay canonical variable ₂ of the disturbing planet in terms of timet.