The boundary conditions for relativistic polytropic fluid spheres

The differential equations governing relativistic polytropic fluid spheres have been integrated numerically for polytropic indexn = 0.0 (0.1) 4.9 and relativity parameter = 0.0 (0.1) 0.9, and the resulting boundary conditions for and other related quantities are presented in this paper.     

On the influence of gradients in the angular velocity on the solar meridional motions

If fluctuations in the density are neglected, the large-scale, axisymmetric azimuthal momentum equation for the solar convection zone (SCZ) contains only the velocity correlations and where u are the turbulent convective velocities and the brackets denote a large-scale average. The angular velocity, , and meridional motions are expanded in Legendre polynomials and in these expansions only the two leading terms are retained (for example, where is the polar angle). Per hemisphere, the meridional circulation is, in consequence, the superposition of two flows, characterized by one, and two cells in latitude respectively. Two equations can be derived from the azimuthal momentum equation. The first one expresses the conservation of angular momentum and essentially determines the stream function of the one-cell flow in terms of : the convective motions feed angular momentum to the inner regions of the SCZ and in the steady state a meridional flow must be present to remove this angular momentum. The second equation contains also the integral indicative of a transport of angular momentum towards the equator.With the help of a formalism developed earlier we evaluate, for solid body rotation, the velocity correlations and for several values of an arbitrary parameter, D, left unspecified by the theory. The most striking result of these calculations is the increase of with D. Next we calculate the turbulent viscosity coefficients defined by whereC _ro ⁰ and C _o ⁰ are the velocity correlations for solid body rotation. In these calculations it was assumed that ₂ was a linear function of r. The arbitrary parameter D was chosen so that the meridional flow vanishes at the surface for the rotation laws specified below. The coefficients v _ro ⁱ and v _0o ⁱ that allow for the calculation of C _ro and C _0o for any specified rotation law (with the proviso that ₂ be linear) are the turbulent viscosity coefficients. These coefficients comply well with intuitive expectations: v _ro ¹ and –v _0o ³ are the largest in each group, and v _0o ³ is negative.The equations for the meridional flow were first solved with ₀ and ₂ two linear functions of r ( ₀ ¹ = – 2 × 10 ^–12 cm ^–1) and ( ₂ ¹ = – 6 × 10 ¹² cm ^–1). The corresponding angular velocity increases slightly inwards at the poles and decreases at the equator in broad agreement with heliosismic observations. The computed meridional motions are far too large ( 150m s^–1). Reasonable values for the meridional motions can only be obtained if _o (and in consequence ), increase sharply with depth below the surface. The calculated meridional motion at the surface consists of a weak equatorward flow for gq < 29° and of a stronger poleward flow for > 29°.In the Sun, the Taylor-Proudman balance (the Coriolis force is balanced by the pressure gradient), must be altered to include the buoyancy force. The consequences of this modification are far reaching: is not required, now, to be constant along cylinders. Instead, the latitudinal dependence of the superadiabatic gradient is determined by the rotation law. For the above rotation laws, the corresponding latitudinal variations of the convective flux are of the order of 7% in the lower SCZ.     

Identification of Emissions in the Spectrum of Comet C/1989 VI (Skoritchenko–George) Obtained with the 6-m SAO RAS Reflector

The emission spectrum of comet Skoritchenko–George (C/1989 VI), unusual in its information content, was obtained on February 26.7 UT, 1990, with the use of a TV scanner installed on the 6-m BTA reflector of the Special Astronomical Observatory of the Russian Academy of Sciences (SAO RAS) in Nizhni Arkhyz. Detailed identification of the emission lines of this comet was made. The observed spectrum contains 311 emission lines, including those of the molecules. Among others, the lines of the negative carbon C ₂ ^- ion and the lines corresponding to the electron transition in the neutral CO molecule are discovered. The presence of a large number of lines of the neutral CO molecule (the Asundi bands and the triplet bands) in the visible region is one of the uncommon features of the emission spectrum of this comet. The triplet lines : 15–3, 13–2, 11–2, 9–1, 8–1, 7–1, 7–0, 5–0, 4–0; : 7–0, 6–0, 5–0; and a" : 11–1 (K = 3, 4); 16–4 (K= 0, 1, 2, 4); 9-0 (K= 0, 1, 2); 8–0 (K= 0) were identified for the first time. Prior to this work, the lines of CO in the visible range were observed only in the spectrum of comet C/1979 VI (Bradfield) in 1989.     

Possible semi-annual variation of the Newtonian constant of gravitation

A possible semi-annual variation of the Newtonian constant of gravitationG is established. For the aphelion and perihelion points of the Earth's orbit we find, respectively,

Gas,dust and molecules in the Galaxy

The relative abundances of cool neutral hydrogen, carbon monoxide and formaldehyde are studied using all the available observational data in the literature. The obtained mean valuesN _{H 1}/ ,N _{H 1}/N _CO,N _CO/ are approximately constant in the dark clouds of the solar neighbourhood and in the distant molecular clouds.The observed correlationsN _CO,A _v and ,A _v show that formaldehyde can also be used as an indicator of molecular hydrogen. The ratioN H₁/A _v depends on densities and decays considerably in the ranges of visual absorptions in which the molecules become detectable (A _v 2 mg); an average of /N _{H 1}10 is calculated for the dense dark clouds.Indications of systematic temperature gradiens T/A _v are found for formaldehyde and neutral hydrogen inside the dark clouds, and qualitative comparisons are made with theoretical quantum mechanics calculations.The observed carbon monoxide and formaldehyde abundances, the free electron layer in the Galaxy, the distribution of neutral hydrogen in different states are only compatible if an ionization rate of 10^–16 is accepted, provided presumably by 2 MeV protons of cosmic radiation.Three main states for neutral hydrogen and dust are identified from different kinds of observational data (21 cm line in emission, absorption in galactic radio sources and self-absorption in the hot gas background): (1) a homogeneous intercloud stratum of tenuous gas and dust with a galactic halfwidth of 350 pc and mean parametersn _H=0.2 atom cm^–3, spin temperatureT _s 10000 K andn _d 0.3 mg kpc^–1; (2) cool gas and dust concentrated in spiral features with a galactic half-width of less than 100 pc, probably forming clouds with diffuse and indefinite limits, with mean parametersn _H2 atom cm^–3,T _s <1100 K (probable average,T _s =135 K) andn _d 3 mg kpc^–1; (3) dense gas and dust clouds with a mean diameter of 7 pc and mean parametersn _H700 atom cm^–3 (90% in a molecular state),T _s 63 K andn _d 1 mg pc^–1 on which molecules as CO and H₂CO are formed.The application of the Jeans criteria for gravitational instability shows that the dense clouds are gravitationally bound while the gas in the intermediate state (2) can be protected against collapse by the total internal energy in the medium increasing due to cosmic rays and the magnetic field in the Galaxy.The observed velocity halfwidths and galacticZ-halfwidths in states (1) and (2) are compatible with a total mass density in the galactic layer of 90M pc^–2 (gas plus stars) according to the barometric equation.The relative abundancesN _{H 1}/N _CO, calculated from C¹²O and C¹³O data and comparisons with studies in the 21 cm emission line, show that the antenna temperatureT _A ⁺ in the 2.6 mm line of C¹²O is a good indicator of the cool gas densities in the Galaxy. The possible application of this for studies in galactic structure is discussed and hypothetical distributions of carbon monoxide in the zones outside the galactic planeB=0° are presented.From a synthesis based on the results obtained, a cycle is postulated for the neutral hydrogen in the Galaxy: condensation and cooling of gas molecular formation gravitational collapse and star formation gas dissipation and heating by cosmic rays and UV radiation.     

K emission-line widths and the solar chromosphere

Closely spaced microphotometer tracings parallel to the dispersion of one excellent frame of a K-line time sequence have been utilized for a study of the nature of the K_2v , K_2R intensities in the case of the solar chromosphere. The frequency of occurrence of the categories of intensity ratio are as follows: per cent; per cent; per cent; per cent; per cent. Two types of absorbing components are postulated to explain the pattern of observed K_2v , k_2R intensity ratios. One component with minor Doppler displacements acting on the normal K₂₃₂ profile, where K_2V >K_2R , produces the cases K_2v K_2R , K_2v = K_2R , K_2v <K_2R . The other component arises from dark condensations which are of size 3500 kms as seen in K_2R . They have principally large down flowing velocities in the range 5–8 km/sec and are seen on K₃ spectroheliograms with sizes of about 5000 kms, within the coarse network of emission. These dark condensations give rise to the situation K_2R = 0.K₂-line widths are measured for all tracings where K_2v , K_2R are measurable simultaneously. The distribution curve of these widths is extremely sharp. The K₂ emission source is identified with the bright fine mottles visible on the surface. Evidence for this interpretation comes from the study of auto-correlation functions of K₂ intensity variations and the spacing between the bright fine mottles from both spectrograms and spectroheliograms. The life time of the fine mottling is 200 sec.The supergranular boundaries which constitute the coarse network come in two intensity classes. A low intensity network has the fine mottles as its principal contributor to the K emission. When the network is bright, the enhancement is caused by increased K emission due to the accumulation of magnetic fields at the supergranule boundary. The K₂ widths of the low intensity supergranular boundary agree with the value found for the bright mottles. Those for the brighter network are lower than this value, similar to the K₂ widths as seen in the active regions.It is concluded that bright fine mottling is responsible for the relation, found by Wilson and Bappu, between K emission line widths and absolute magnitudes of the stars.The paper discusses the solar cycle equivalents that stellar chromospheres can demonstrate and indicates a possible line of approach for successful detection of cyclic activity in stellar chromospheres.     

Periodic solutions for resonance 4/3: Application to the construction of an intermediary orbit for Hyperion’s motion

The motion of Hyperion is an almost perfect application of second kind and second genius orbit, according to Poincaré’s classification. In order to construct such an orbit, we suppose that Titan’s motion is an elliptical one and that the observed frequencies are such that 4n _H−3n _T+3n _ω=0, where n _H, n _T are the mean motions of Hyperion and Titan, n _ω is the rate of rotation of Hyperion’s pericenter. We admit that the observed motion of Hyperion is a periodic motion such as . Then, .N _H, N _T, k∈ N ⁺. With that hypothesis we show that Hyperion’s orbit tends to a particular periodic solution among the periodic solutions of the Keplerian problem, when Titan’s mass tends to zero. The condition of periodicity allows us to construct this orbit which represents the real motion with a very good approximation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Models of galactic mass distribution

Three groups of galactic mass models, each consisting of nine inhomogeneous spheroids of two kinds are described, according to three adopted values of the total density near the Sun: 0.10, 0.15 and 0.20 M pc^–3. Approximately 20% of the total mass of each model is in the halo, constructed to adequately fit recent RR Lyrae star observations. It is shown that the maxima found in the RR Lyrae star densities towards the galactic axis (Plaut, 1970) should not be interpreted as being associated with the galactic nucleus, but as the result of the greater decrease in density with increasingz over the increase in density as the galactic axis is approached. Even at the low galactic latitude of 5° (l=0°), this effect causes a 0.5 kpc correction to the distance to the galactic centre. A basic model for kpc, km s^–1, M pc^–3 is first constructed, mainly to satisfy structural conditions near the sun and in the halo. An attempt to optimize the basic model is made by scaling it so as to retain constant density and angular velocity near the sun, and to best fit kinematic data, including the recent re-examination of the 21-cm data of Simonson and Mader (1972). No unknown matter is required in the models, in accordance with the results of Weistrop (1972b), and, as pointed out earlier (Innanen, 1966b) the faintM-stars must be in a highly flattened spheroid. The optimizing indicates that an adequate fit to kinematics can be achieved for km s^–1. More detailed results are tabulated for a representative model for which . Two new galactic density functions are discussed in the Appendix.     

On the role of plasma effects in the cosmic ray propagation and isotropization in the Galaxy

Cosmic ray (c. r.) propagation in interstellar magnetic fields is often considered in the diffusion approximation, i.e. by the diffusion equation in the coordinate space. Cosmic ray momentum distribution in this case is considered isotropic when the space gradients of c.r density are absent. This approach, with the use of an unfixed effective diffusion coefficientD independent of the energyE enables one to describe all the data available However, neither the diffusion mechanism nor the limits of applicability of the diffusion approximation is clear particularly ifD is independent ofE. Furthermore, the diffusion coefficientD must be expressed through the characteristics of the interstellar medium and possibly through the flux velocity and density of c.r. etc. One of the possible approaches for the analysis of the mechanism and characteristic features of c.r. distribution and isotropization is the account taken of the plasma effects and specifically, the study of c.r. flux instability arising when c.r. are moving in the interstellar plasma. As a result of such instability c.r. may generate waves of different types (magnetohydrodynamic, high-frequency plasma and other waves). Generation of waves and scattering on them result in isotropization of cosmic rays while their propagation under certain conditions turns out similar to that under diffusion.An attempt is made here to systematically analyse the avove mentioned plasma effects and to find out to what extent they are responsible for the behaviour of c.r. in the Galaxy. It turns out that c.r. In any case this is true if this mechanism is regarded as the only c.r. isotropization mechanizm within a wide energy range from 1 to 1000 GeV. Isotropization and spatial diffusion of c.r. up toE100–1000 GeV on the waves from external sources (for example, on the waves from the supernova shells) also proved impossible if the diffusion coefficient is assumed to be independent of c.r. energy. Some new possibilities of c.r. isotropization are also considered.A List of Notations D cosmic ray (c.r.) space diffusion coefficient - degree of c.r. anyisotropy - E,E _kin total and kinetic particle energy - p,p particle momentum and its absolute value - angle between the particle momentum direction and the magnetic field direction (z-axis) - cos - v, particle velocity and its absolute value - c light velocity - f(p),f(E) momentum and energy particle distribution function - N( > E) = N( > p) = f(p) dp/(2)³ = _E f dE c.r. particle density - c.r. spectrum index,N(>E)=KE ^–+1 - n _H neutral particle density - n=n _e=n _i ion and electron density - H niagnetic field - T temperature - thermal velocities of electrons and ions - Boltzmann constant - Alfén velocity - M, m proton and electron masses - e electron charge - wave frequency - _H =eH/Mc, = _H (Mc ²/E) gyrofrequency of a plasma proton and relativistic particle - _H =eH/mc gyrofrequency of an electron - plasma frequency - v _ii,v _ei,v _en,v _in collision frequencies between ions, electrons and ions, electrons and neutrals, ions and neutrals - growth rate of wave amplitude - k,k wave vector and its absolute value - angle between the directions of the vectorsk andH - wave energy density     

Specificity of Parker instability in polytropic disks

The specificity of the Parker instability in the disks in the presence of the polytropic connection between the equilibrium pressure and density is investigated.It is shown that, in contrary to the caseV _A=const., the even and odd (with reference to the disk perpendicular coordinate) modes consist of non-finite sum of the harmonics, and that the zero (fundamental) harmonic of the even mode, which has the definite wave numberK=K _* will mainly increase. Consequently, this very harmonic will determine the structure, obtained as a result of the Parker instability development.     

Meridional flow in thick accretion disks

We have studied the effect of the flow in the accretion disk. The specific angular momentum of the disk is assumed to be constant and the polytropic relation is used. We have solved the structure of the disk and the flow patterns of the irrotational perfect fluid.As far as the obtained results are concerned, the flow does not affect the shape of the configuration in the bulk of the disk, although the flow velocity reaches even a half of the sound velocity at the inner edge of the disk. Therefore, in order to study accretion disk models with the moderate mass accretion rate—i.e.,

Cosmic-ray flow lines and energy changes

In this paper we discuss the particle flow or streaming and energy changes of cosmic rays in the interplanetary region via flow lines in momentum-position space. We consider the steady-state case where particles are released monoenergetically from the Sun or from infinity and study the cosmic-ray traffic pattern in momentum and position arising from monoenergetic sources. The analysis makes extensive use of the result (wherep is the particle momentum,V the solar wind velocity andG the cosmic-ray density gradient) for the mean time rate of change of momentum of cosmic rays reckoned for a fixed volume in a reference frame fixed in the solar system, developed by us in several recent papers.Deceased.     

Epstein weight functions for non-radial oscillations of polytropes

Weight functions for the determination of the periods of linear adiabatic non-radial oscillations have been calculated in the same manner as Epstein's classic treatment of purely radial oscillations. Quadrupole (l=2) oscillations for thef and lower orderp andg-modes were considered. One group of static models were polytropes in the range 1.0n4.0 with ; thus included were configurations that were convectively stable, unstable and neutrally stable throughout. Another group consisted ofn=3.0 polytropes with convective shells or convective cores; ₁ was set at different values in each region in order to produce stability ( ) or instability ( ). The weight function provides a pictorial means for assessing the relative importance of each region of a given static model with respect to generating a given non-radial mode.     

On numerical evaluation of theH-functions of transport problems by kernel approximation for the albedo 0<ω≤1

Das Gupta represented theH-functions of transport problems for the albedo [0, 1] in the formH(z)=R(z)–S(z) (see Das Gupta, 1977) whereR(z) is a rational function ofz andS(z) is regular on [–1, 0] ^c . In this paper we have representedS(z) through a Fredholm integral equation of the second kind with a symmetric real kernelL(y, z) as . The problem is then solved as an eigenvalue problem. The kernel is converted into a degenerate kernel through finite Taylor's expansion and the integral equation forS(z) takes the form: (which is solved by the usual procedure) where _r 's are the discrete eigenvalues andF _r 's the corresponding eigenfunctions of the real symmetric kernelL(y, z).     

Apsidal motion in the eclipsing binary system of BW AQR

The first photoelectric light curve of the eclipsing binary system BW Aqr (F71V+F81V;P=6^d.7;V=10 ^m .31), discovered by Miss Leavitt at the beginning of the century, was obtained. The photometric elements were detemined. The components of this system are considerably evolved stars: the age of the system is about 2×10⁹ yr. It follows from the photometric data that the secondary component should have greater mass than the primary one The zero-age spectral classes of components were F2V and F1V. The system has an elliptical orbit with the eccentricitye=0.18. The angular rate of the apsidal motion (_obs = 0.070 deg yr^–1) and the corresponding value of the apsidal parameterk ₂=0.0090 (the relativistic term included) were found. The derived valuek ₂ exceeds by more than a factor of 2 the theoretical coefficient obtained from the modern theory of internal structure of stars with moderate masses .     

The spherically-symmetrical trajectories of spin particles in the Schwarzschild field

The motion of spin particles in the Schwarzschild field is examined in this paper. It is shown that Mathisson-Papapetrou equations under additional conditions , where is the Lie derivative of the Killing vector _j , permit only radial motion, motion in the equatorial plane and in the equilibrium points. All these types of motion are considered more in detail.     

Evolution of galaxies: One-zone model with a birth-rate stellar function depending on gas density and temperature

In this paper closed one-zone models with a birth-rate stellar function depending on gas density and temperature are built up, in order to make a zeroth order approximation of thermo-chemical evolution of galaxies. To get a more realistic picture, the models are allowed first to expand with the universe and then to contract according to three successive time-scales, in order to simulate violent relaxation, rapid contraction, and slow contraction processes.Star formation occurs according to a birth-rate (m, t) _g ⁿ T ^r m ^p ( _g , gas density;T, gas temperature; andm, star mass), which reproduces the usual Schmidt-type law in the limitr0. Further we suppose that synthesized metals and helium are instantaneously mixed when returned to gas component at star death, and that relative metal abundances are time-independent: ( any element heavier than He, ). An energy fraction of each typical supernova event is assumed to heat interstellar medium, which is also though always to maintain a steady state.Under these hypotheses it becomes possible to use the interstellar cooling function of Dalgarno and McCray (1972), which is related to a special cosmic abundance of elements, to express analytically an interstellar cooling function related to any abundance. At this stage, numerical computations are performed up to about 12 Gy and the results analyzed in order to reproduce some morphological features observed in the galaxy, namely: (i) the extreme scarcity of stars with nearly zero metal content; (ii) the metal content and mass fraction of halo stars; (iii) the metal content and mass fraction of disk stars; and (iv) the ratio of the present-day birth-rate to the past birth-rate averaged over the age of the disk. It is found that a whole class of theoretical models exist, with slope of the mass spectrum between –2.5 and –2.6, which range from a Schmidt's law limitn=1.3 andr=0 to the opposite limitn=0 andr=–1.2, and are able to fit more or less at the same extent the observations mentioned above. On the analysis of further computations, it is also shown that an universal birth-rate stellar function is in contradiction with neither a dynamical hystory starting from the initial expansion of proto-galaxies, nor a morphological sequence where the amount of angular momentum is the only free parameter for any given mass, at least for what concerns spiral types. Finally, possible refinements for future work are briefly outlined.     

Simulations of cosmic-ray particle diffusion

The diffusion of charged particles in a stochastic magnetic field (strengthB) which is superimposed on a uniform magnetic fieldB ₀ k is studied. A slab model of the stochastic magnetic field is used. Many particles were released into different realizations of the magnetic field and their subsequent displacements z in the direction of the uniform magnetic field numerically computed. The particle trajectories were calculated over periods of many particle scattering times. The ensemble average was then used to find the parallel diffusion coefficient . The simulations were performed for several types of stochastic magnetic fields and for a wide range of particle gyro-radius and the parameterB/B ₀. The calculations have shown that the theory of charged particle diffusion is a good approximation even when the stochastic magnetic field is of the same strength as the uniform magnetic field.     

Über Den Aufbau Der Stern- Und Galaxienhaufen

In star clusters and in galaxy clusters the exponentn of the generalized law of Schuster statistically decreases with increasing masses of the clusters or increasing numbersL of their members. In most casesn(gal. cl.) is smaller thann(st. cl.). It is confirmed, that in larger galaxy clusters the brighter members are somewhat more concentrated at the center than the fainter members.

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Mitteilungen Serie A.     

Static spherical configurations of cold matter in the Einstein-Cartan theory of gravitation

We investigate static, spherical configurations of cold catalized matter in the Einstein-Cartan theory of gravitation. Assuming that density of spin is proportional to the number density of baryonsn and using an equation of state of a degenerate, relativistic Fermi gas, we numerically integrated the relativistic equation of equilibrium. We have also studied the stability of those configurations. Configurations with central number densityn _c such that where is the effective pressure, are very similar to general relativistic configurations with the same central density. In the Einstein-Cartan theory there exists another disjoint family of equilibrium configurations for which but . Those configurations have very small masses 10^–6 g and raddi 10^–34 cm and are unstable.Supported in part by Research Grant MR-I-7.