The light curved in the CM field

In this paper we introduce the CM field in Sections 2 and 3 based on the paper by Wang and Peng (1985), and calculate the light curved in the CM field in Section 4. The result shows thatP makes _CM larger than _C at , and smaller at . Under a special circumstance which source, CM lens, and observer are in the same line, if we get | ₀₌₀ , and | _=/2 , we can determine theP(M) andQ(M) of the CM lens,M is the mass of the CM lens.     

Charged particle diffusion in the presence of Alfvén waves: The perpendicular particle flux

The diffusion of charged particles through a weak stochastic electro-magnetic field which is superimposed on a constant background magnetic field is considered. The stochastic electromagnetic fields are assumed to consist of unpolarized Alfvén waves propagating at arbitrary angles to the direction . When the Alfvén waves are propagating in directions other than and the particle gyro-radius,r _g, is sufficiently large (but may be smaller than the correlation length of the stochastic fields) it is shown that the particle flux perpendicular to the direction is , wherev is the particle speed andf the particle density. The expression forK differs from those calculated by previous authors. For small particle gyro-radii the flux S has a different functional form and is identical to that found by Urch (1977) to describe particle diffusion when the Alfvén waves only propagate in the direction .     

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.     

YO molecule in the sun

Vibrational transition probabilities, namely Franck—Condon factors and -centroids have been evaluated by an approximate analytical method for the (A–X), (A–X), and (A–X) system of YO molecule. Morse potential energy curves forX ²⁺,A ²₂,A²₂, andA²₂, states of YO have been constructed using the latest spectroscopic data. The value of -centroids for the band have been found to decrease linearly with the corresponding wavelengths. We show results for two new transitions of (A–X) and (A–X) and five new bands of (A–X) of YO in the umbral spectrum of the Sun.     

Invariant properties of families of moon-to-earth trajectories

Analytical techniques are employed to demonstrate certain invariant properties of families of moon-to-earth trajectories. The analytical expressions which demonstrate these properties have been derived from an earlier analytical solution of the restricted three-body problem which was developed by the method of matched asymptotic expansions. These expressions are given explicitly to orderµ ^1/2 where is the dimensionless mass of the moon. It is also shown that the inclusion of higher order corrections does not affect the nature of the invariant properties but only increases the accuracy of the analytic expressions.The results are compared with the work of Hoelker, Braud, and Herring who first discovered invariant properties of earth-to-moon trajectories by exact numerical integration of the equations of motion. (Similar properties for moon-to-earth trajectories follow from the principle of reflection). In each instance the analytical expressions result in properties which are equivalent, to orderµ ^1/2, with those found by numerical integration. Some quantitative comparisons are presented which show the analytical expressions to be quite accurate for calculating particular geometrical characteristics.

Nomenclature

Orbital Elements near the Moon energy - angular momentum - semi-major axis - eccentricity - inclination - argument of node - argument of pericynthion Orbital Elements near the Earth h _e energy - l _e angular momentum - i inclination - argument of node - argument of perigee - t _f time of flight Other symbols parameters used in matehing - U a function of the energy near the earth - a function of the angular momentum near the earth - r _p perigee radius - perincynthion radius - radius at node near moon - true anomaly of node near moon - initial angle between node near moon and earth-moon line - a function ofU, , andi - earth phase angle - dimensionless mass of the moon - U ₀, U₁ U=U ₀+U ₁ - i ₀, i_1/2, i₁ i=i ₀+µ ^1/2 i _1/2+µ i ₁ - ₀, _1/2, ₁ = ₀+µ ^1/2 i _1/2+µ i ₁ - _p longitude of vertex line - _n latitude of vertex line - R _o ,S _o ,N _o functions ofU ₀ and - a function ofU ₀, and      

Inertia coefficient considerations and the structure of Jovian planets

For Jupiter, an overall density model of the form= ₀(1–x ⁿ ), withn1/3 and , is consistent with information presently at hand; for Saturn, however, such a density law would lead to unacceptably high densities in the vicinity of the centre. The limiting cases of the previous law are shown to ben=+, corresponding to a homogeneous sphere, andn=–3, corresponding to a particular central particle model, investigated by a number of astronomers over the last hundred years. Forn0, the central density becomes +. Another possible representation, valid both for Jupiter and Saturn, is the density law= ₀(1–x) ^m ), with in the case of Jupiter, and in the case of Saturn. Graber's density law based on a maximum entropy principle leads to unacceptably high surface densities, both for Jupiter and Saturn. Finally, the paper investigates the problems involved in fitting two-layered parametrically simple density laws to theoretically derived much more elaborate models of the Jovian planets.     

Linear waves in a radiating and scattering grey medium

The problem on linear waves in a radiating and scattering grey medium is studied using Whitham's method. Analysis of the basic equations distinguishes two limiting cases: the one is theequilibrium case in which the energy exchange between the gas and radiation plays an essential role, and the other is theScattering case in which the effect of energy exchange is negligible. A new type ofradiation acoustic wave with the speed is found in the scattering case. The governing equations for linearized one-dimensional flow are reduced to one equation of radiative acoustics valid to order 1/c, and the criterion for the two limiting cases is derived from studying this equation. The harmonic solution is analytically studied to show that theeffective optical depth corresponding to the wavelength of perturbation gives the measure of the interaction between the gas and radiation. When _eff1, the sound speeda _g ² =P _g / and the propagating speed of radiative disturbancea _f ² =fc ² appear as the modified classical and radiation-induced modes respectively, wheref is the Eddington factor. When _eff1, the isentropic sound speeda _s ² =(P _g +P _r / appears in the equilibrium case, and the radiation acoustic speeda _A ² appears in the scattering case. The dispersion relation of the harmonic solution is numerically calculated. The result shows that the wave pattern depends critically on the ratio=P _g /(P _g +P _r ). When , the speeda _S anda _A arise from the modified classical mode, and when , they originate from the radiation-induced mode.     

Der offene sternhaufen NGC 2362

The very young open star cluster NGC 2362 was investigated by the strip method on charts of two photographs taken with the 1-m Schmidt telescope of the European Southern Observatory. Up to the limiting magnitudeM _v ^* =5 _. ^m 8 the cluster contains 100 stars and can be described by the Gaussian density law (6). Further results are: Mass = 246 , central mass density ₀ = 43.1 = 246 pc^-3 , radiusR2.6 pc, mean velocity of the stars = 0.64 km s^–1.

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Mitteilungen Serie A.     

Some controversial issues in theories of the solar differential rotation and dynamo

The following points are discussed:

Fourier analysis of the light curves of eclipsing variable stars

The aim of the present paper is to find the eclipse perturbations, in the frequency-domain, of close eclipsing systems exhibiting partial eclipses.After a brief introduction, in Section 2 we shall deal with the evaluation of thea _n ^(l) integrals for partial eclipses and give them in terms ofa ₀ ⁰ ,a ₀ ⁰ (of the associated -functions) and integrals; while Section 3 gives the eclipse perturbations arising from the tidal and rotational distortion of the two components. The are given for uniformly bright discs (h=1) as well as for linear and quadratic limb-darkening (h=2 and 3, respectively).Finally, Section 4 gives a brief discussion of the results and the way in which they can be applied to practical cases.     

Optically thick accretion onto black holes

Spherically symmetric, steady-state, optically thick accretion onto a nonrotating black hole with the mass of is studied. The gas accreting onto the black hole is assumed to be a fully ionized hydrogen plasma withn ₀=10⁸ cm^–3 andT ₀=10⁴ K far from the black hole, and a new approximate expression for the Eddington factor is introduced. The luminosity is estimated to beL=1.875×10³³ erg s^–1, which primarily arises from the optical surface (1) ofT10⁴ K. The accretion flow is characterized by 1 and (v/c)10. In the optically thin region, the flow remains isothermal, and the increase of temperature occurs at 1. The radiative equilibrium is strictly realized at (v/c)10.     

Existence of the continuations in theN-body problem

The method of obtaining the estimates of the maximalt-interval ( _–, ₊) on which the solution of theN-body problem exists and which is such that some fixed mutual distance (e. g. ₁₂) exceeds some fixed non-negative lower bound, for allt contained in ( _–, ₊), is considered. For given masses and initial data, the increasing sequences of the numbers _k , each of which provides the estimate ₊ > _k , are constructed. It appears that if ₊ = +, then .     

Secondary production of antiprotons in cosmic radiation

A reliable representation to the invariant cross-section for the production of antiprotons ( ) in inclusive reactions has been obtained, which fits the data extremely well from threshold to ISR energies. Using this, the production spectrum of by cosmic ray interaction with interstellar gas is calculated and is compared with other existing calculations. The equilibrium spectrum in the Galaxy has been derived from about 100 MeV to a few hundred GeV in kinetic energy using Leaky Box Model for the propagation of cosmic rays, by taking into account all energy loss processes. It is found that /P ratio calculated here is very much smaller than the observed ratio. In view of the fact that this excess of observed /P ratio is much larger than the measured upper limits on the fraction of antimatter in the form of antinuclei, the excess /P ratio has not been attributed to the existence of antimatter. Instead, it is shown that Closed Galaxy Model for the propagation of cosmic rays predicts more antiprotons than observed. Therefore, it is suggested that if cosmic rays contain about 50% new component of local origin, the resulting Closed Galaxy Model explains well the observations. The present calculations also predict too small a flux of below a few hundred MeV to make this energy region ideally suited to look for antiprotons of primordial origin.     

Gravitational orbit-attitude coupling for very large spacecraft

Motion equations for the gravitationally coupled orbit-attitude motion of a spacecraft are presented. The gravitational force and torque are expanded in a Taylor series in the small ratio (spacecraft size/orbital radius). A recursive definition for higher moments of inertia is introduced which permits terms up tofourth order to be retained. The expressions are fully nonlinear in the attitude variables. A quasi-sunpointing (QSP) passive attitude-control mode is used to assess the effects of higher moments of inertia and gravitational coupling. The attitude motion is detectably coupled to the orbital motion. However, the higher moments of inertia influence only the attitude motion.Nomenclature f _G ,g _G ,f _Gi ,g _Gi total gravitational force and torque and their components of orderi in =/r ₀ - angular momentum of spacecraft about 0 and the spacecraft mass center - J ⁱ ,I ⁱ general moment of inertia about 0 and the spacecraft mass center - second (dyadic), third (triadic), and fourth (tetradic) moment of inertia about 0 and the spacecraft mass center - A andB (and related components) of the second, third and fourth moments of inertia about 0, see Equation (9) - M, m Earth's mass, spacecraft mass - Q ^ba rotation matrix taking _a into _b - position vector from attracting body's mass center to a general mass element, to 0 and to the spacecraft mass center - ₁, ₂, ₃ basis vectors of reference frame - , , _N misalignment angle betweenb ₃ and the (projected) true position of the Sun, its oscillatory component and nominal value - unit dyadic (-identity matrix) - ratio of characteristic spacecraft dimension to orbital radius - pitch angle (aboutb ₂ axis) - Earth's gravitational parameter - , position vector from 0 to a general mass element and the spacecraft mass center - , the (projected) true longitude of the Sun and the true longitude of the spacecraft - _/ angular velocity of reference frame with respect to - (·), (^*), (^o) d()/dt with respect to inertial space _I , and orbiting frame _O and a body-fixed spacecraft frame _b Presented at AAS/AIAA Astrodynamics Conference, Aug. 9–11, 1982.     

On the peculiarities in the rotational frequency evolution of isolated neutron stars

The measurements of pulsar frequency second derivatives have shown that they are 10²−10⁶ times larger than expected for standard pulsar spin-down law, and are even negative for about half of pulsars. We explain these paradoxical results on the basis of the statistical analysis of the rotational parameters ν, and of the subset of 295 pulsars taken mostly from the ATNF database. We have found a strong correlation between and for both and , as well as between ν and . We interpret these dependencies as evolutionary ones due to being nearly proportional to the pulsars’ age. The derived statistical relations as well as “anomalous” values of are well described by assuming the long-time variations of the spin-down rate. The pulsar frequency evolution, therefore, consists of secular change of ν _ev(t), and according to the power law with n≈5, the irregularities, observed within a timespan as a timing noise, and the variations on the timescale larger than that—several decades. This work has been supported by the Russian Foundation for Basic Research (grant No 04-02-17555), Russian Academy of Sciences (program “Evolution of Stars and Galaxies”), and by the Russian Science Support Foundation. The authors would also like to thank the anonymous referee for valuable comments.     

On the uniqueness of the determination of the coronal potential magnetic field from line-of-sight boundary conditions

We consider a simple model in which the coronal magnetic field B is assumed to be potential in the region between the solar surface _o and an exterior source-surface ₁ of arbitrary shape. We prove that the boundary value problem that determines B from the value B _lof its component on ₀ along either (orthoradial direction) or (fixed direction) has at most one solution. On the other hand, we show that a solution can exist only if B _lsatisfies some solubility conditions.     

Re-analysis of close binary systems

Line-forming regions around close binaries with strong winds ( /4r _* v 10^–4 g cm^–2) are large in extent compared with the stars, large enough to screen them. Their orbitally-modulated Doppler shifts can overestimate the mass function, because of a larger rotational lever arm. In particular, most of the black-hole candidates need not involve companions more massive than a neutron star.The solar-wind problem is reconsidered. An extrapolation to Wolf-Rayet stars suggests that their winds are centrifugally driven. Their mass-loss rates tend to have been overestimated.Seemingly single (massive) stars can hide a (compact) companion.     

Graphisches Konstruktionsschema zur Bestimmung der Bewegungsparameter eines drallstabilisierten Flugkörpers mit Nutationsdämpfung unter dem Einflusz von Lagekorrekturimpulsen

Zusammenfassung Es wird gezeigt, daß die unter der Einwirkung einer Momentenimpulsserie entstehende Bewegung eines rotierenden Flugkörpers mit Nutationsdämpfung sich vollständig einem regelmäßigen Polygon entnehmen läßt, das durch das Trägheitsmomentenverhältnis, den Integralwert eines Einzelimpulses, den Drall und eine die Dämpfung charakterisierende KonstanteK ₀ bestimmt ist.Die Bewegung setzt sich aus logarithmischen Spiralen zusammen, derenn-ten Anfangsradius man erhält, indem man den Teilungspunkt des im VerhältnisK ₀:1 geteilten (n–1)-ten Radius mit der (n+1)-ten Polygonecke verbindet.Es wird bewiesen, daß das Konstruktionsnetz zu einem im äußeren Polygon liegenden ähnlichen inneren Polygon konvergiert, das gegenüber ersterem gedreht ist.Einfache Beziehungen zur Bewegungsbestimmung mit dem Polygonschema werden für Pulsfrequenzen angegeben, die ganzzahlige Vielfache oder Bruchteile der Spinfrequenz sind.

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It is shown that the motion of a spinning body with nutation damping due to a series of torque pulses can be completely derived from a regular polygon determined by the ratio of inertias, the integral of one pulse, the momentum and a constantK ₀ characterizing damping.The motion is composed of spirals thenth initial radius of which is obtained by connecting the dividing point of the (n–1)th radius with the (n+1)th polygon corner. Each dividing point divides the respective radius in the ratioK ₀:1. The net of construction lines converges into an inner polygon turned against the outer one and having the same shape.Simple rules are shown for the application of the scheme on pulse frequencies which are multiples or fractions of spin frequency.

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Symbole 1-2-3 Achsen des flugkörperfesten Koordinatensystems - a,b,c Hilfsgrößen zur Bestimmung der Iterationsgrößen - E _i i-te Polygonecke - H Drall des Flugkörpers - K _i Verhältnis deri-ten Drehzeigerlängen zu Beginn und am Ende eines Impulses - M Iterationsmatrix - Integralwert des Momentenimpulses - P ₀ Äußeres Polygon - P ₁ Spitze des Drehzeigersr _00e - P Drehpunkt des Drehzeigersr ₀₀ - P Konvergierendes Polygon - P _i Teilungspunkt des [i–1]-ten Zeigers - r _0i Drehzeiger aufgrund desi-ten Impulses allein - r _0ia Zeigerr _0i in Anfangslage - r _0ie Zeigerr _0i in Endlage - r _i i-ter Summenzeiger - r _ia Zeigerr _i in Anfangslage - r _ie Zeigerr _i in Endlage - T Dauer einer Flugkörperumdrehung - t,t, Zeitargumente - x-y-z Achsen eines raumfesten Koordinatensystems - x _i ,y _i Iterationskoordinaten - _n Phase desn-ten Radius gegenüber der anliegenden Polygonseite - Drehung des inneren Polygons gegenüber dem äußeren - Abklingkonstante - Phasenänderung des Drehzeigers innerhalb einer Flugkörperumdrehung - ₀ Anteil der über 2 hinausgehenden Phasenänderung des Drehzeigers - ₃ Trägheitsmoment um die Spinachse - ₁₂ Trägheitsmoment um die Querachsen - Zahl der Ecken des Konstruktionspolygons - _1,2 Eigenwerte der Iterationsmatrix - Zahl der vollen Umläufe des Konstruktionspolygons - Fortbewegungsachse des Drallvektors - ₀ Ausgangsphasenwinkel - _i Phasenlage desi-ten Summenzeigers - _x, _y Drehwinkel nach Einzelimpuls fürt - , Funktionen der Iterationsgrößen - , Drehwinkel umx-bzw.y-Achse - Drehgeschwindigkeit der Spinachse um den Drallvektor - Fiktive Größen bei Pulsfrequenzen kleiner als Spinfrequenz - Fiktive Größen bei Pulsfrequenzen größer als Spinfrequenz     

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.     

Supersymetric Taub model,the micro-superspace sector

Since there are reasons for expecting supersymmetry in an underlying quantum theory of gravity, one is led to study quantum and classical cosmology with supergravity. In particular, classical solutions corresponding to these models could also be used to generate the quantization of supersymmetric minisuperspaces. In generating these solutions, the solution to the Rarita-Schwinger field in the cosmological background is also obtained. In this paper the supercosmological equations of Einstein-Rarita-Schwinger are solved for the micro-superspace sector of the Taub model, under the assumption =₁₁*₂₂ and . The solution for the parameters of the metric and are proportional to each other in each order, the zeroth-order and also the second-order terms. The zeroth-order terms correspond to the solution in general relativity and are logarithmic in time, the ₁₂ terms have an hyperbolic time-dependence. The Rarita-Schwinger field has the form cos((2/D ³)ln |t–t ₀|) and oscillates an infinite number of times astt ₀. This oscillating behaviour of the solution for is not only present when spinor fields are treated in a curved background, but also some cosmological wave functions behave in this manner. This solution is at the same time the supercosmological solution for the microsuperspace sector of the Taub model and also the Rarita-Schwinger field in this background.This work was supported in part by CONACYT grant P228CCOX891723, and DGICSA SEP grant C90-03-0347.