The motion of the line of apsides of ζ phoenicis

An estimate of the period of the rotation of the line of apsides of the double-star system Phe is obtained by representing the density function as a product of a normal Gaussian distribution and an associated Legendre polynomial .The asymptotic behaviour of this function coincides with the results obtained by Zeldovichet al. (1981).The period of motion of the line of apsides of Phe (about 63 years) obtained in this way comes close to the period determined by an empirical formula for of Batten (1973).     

Distribution of Flare Energies Based on Independent Reconnecting Structures

A model is presented to explain the observed frequency distribution of flare energies, based on independent flaring at a number of distinct topological structures (separators) within active-region magnetic fields. The model is a modification and generalization of a recent model due to Craig (2001), and reconciles that model with the observed flare waiting-time distribution, and the observed absence of a flare waiting-time versus energy relationship. The basic assumptions of the model are that flares of energy E ² occur at separators of length , and that the frequency of flaring at a separator is defined by the Alfvén transit time of the structure. To reproduce the observed distribution of flare energies the model requires a probability distribution P( ) ^–1 of separator lengths within active regions. This prediction of the model is in principle testable. A theoretical origin for this distribution is also discussed.     

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.     

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     

A Reconnection Model for the Distribution of Flare Energies

A physically based explanation is given for the distribution of flare energies N(E)E ^– where 1.5. In contrast to previous approaches, the present treatment is based on a physical theory of the flare reconnection site. The central assumption is that topological flare energy, although released explosively, is slowly accumulated over several hundred Alfvén timescales. When coupled to the geometric properties of the reconnective flare source, this assumption is shown to lead naturally to a deduction of the flare energy distribution. Current sheet models yield the exponent whereas more compact current structures imply steeper spectra .     

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      

Global modes constituting the solar magnetic cycle

We show that the axisymmetric odd degree SHF modes of 21.4-yr periodicity and degrees l 29 in the solar magnetic field (as inferred from sunspot data during 1874–1976), are at least approximately stationary. Among the sine and cosine components of these SHF modes we find four groups, each defining the geometry of a coherent global oscillation characterized by a distinct power hump and its own level of variation. The first two of these geometrical eigenmodes (viz., B ₁ and B ₂), define the large-scale structure of the butterfly diagrams. Remaining SHF modes define the orderliness of the field distribution even within the wings of the butterflies down to scales l 29. These include the geometrical eigenmodes B ₃ and B ₄, which are not present in simulated data sets in which the latitudes of the sunspot groups are randomly redistributed within the wings of the butterflies.Superposition of B ₁, B ₂, B ₃, and B ₄ is necessary and sufficient to reproduce important observed properties of the latitude-time distribution of the real field, not only in the sunspot zone, but also in the middle (35°–75°) and the high (75°) latitudes, with appropriate relative orders of magnitude and phases. Thus, B ₁, B ₂, B ₃, and B ₄ seem to represent really existing global oscillations in the Sun's internal magnetic field. The geometrical form of B ₁ may also be the form of the forcing oscillation.     

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.     

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.     

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.     

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 .     

Rediscussion of balmer decrement of broad-line radio-galaxies

On the basis of the erenkov line emission theory in the optically thick case, a new interpretation of intensity ratios H/H and H/H in broad-line radio-galaxies (BLRGs) is reported. Calculation shows that if the theoretical ratio H/H is just taken to be the mean observed value 0.21, equivalently, the parameterX(H)=3.0, then the expected ratio H/H=6.70 is almost the same as the observations. By comparing these values with the previous investigations of QSOs (X(H)=20.9), we conclude that the number density of neutral hydrogen gas in BLRGs is almost one order-of-magnitude smaller than that in the QSOs.Preliminary verification of the erenkov line emission has been obtained by Xuet al. (1981) in the laboratory.     

Effects of metal abundances on the evolutionary period changes of classical Cepheids

Some peculiarities in the behaviour of a model self-gravitating system described by hydrodynamical equations and isothermal equation of state connected with the presence of thermodynamical fluctuations in real systems were investigated in numerical experiment. The values of density and velocity , , respectively, were computed by numerical code perturbed on each time-step and in each computational cell by random values , for modeling such fluctuations. Perturbed values _i = _i + _i ,v _i = _i + v _i were used to initiate the next step of computations. This procedure is equivalent to an introduction into original hydrodynamical equations of Langevin sources which are random functions. It is shown that these small fluctuations (= v =0,² =v ² = 10^–8) grow many times in marginally-stable state.     

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 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.     

О лагранжевых и зйлеровых решеиях обощенной задачи трех тел

(, 1969). ( ), ( ), , , . , (=), , , .. , . , , - ( ), ( ). , .

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This paper is a continuation and a generalization of one published earlier (Duboshin, 1969): it discusses the problem whether there exist the Lagrangian and the Eulerian solutions of the generalized three-body (material points) problem. Every point in this generalized problem acts on another, one with a force (attractive or repulsive) directed along the straight line passing through these points, and in an arbitrary manner depending on time, mutual distance and its derivatives, the first and the second. Here, generally speaking, the third axiom of dynamics (law of action and reaction) is not presupposed as fulfilled, that is, it is supposed that every two material points interact in a different way.This most general assumption being made, we establish the conditions which must dictate the laws of the interactions, so that the three points can always remain at the apexes of the equilateral triangle (Langrangian solution), or remain always on a straight line (Eulerian solution).The author believes that such general treatment of the three-body problem can be useful for theoretical studies in celestial mechanics and also for practical applications in the study of isolated stellar systems.

     

Doubly averaged effect of the Moon and Sun on a high altitude Earth satellite orbit

Infinite series expansions are obtained for the doubly averaged effects of the Moon and Sun on a high altitude Earth satellite, and the results used to interpret numerically integrated examples. New in this paper are: (1) both sublunar and translunar satellites are considered; (2) analytic expansions include all powers in the satellite and perturbing body semi-major axes; (3) the fact that retrograde orbits have more benign eccentricity behavior than direct orbits should be exploited for high altitude satellite systems; and (4) near circular orbits can be maintained with small expenditures of fuel in the face of an exponential driving force one forI _ab, whereI _b=180°–I _a andI _a is somewhat less than 39.2° for sublunar orbits and somewhat greater than 39.2° for translunar orbits.Nomenclature a semi-major axis - A _lk coefficient defined in Equation (11) - B _lk coefficient defined in Equation (24) - C _km coefficient defined in Equation (25) - D, E, F coefficients in Equations (38), (39) - e eccentricity - H _k expression defined in Equation (34) - expression defined in Equation (35) - I inclination of satellite orbit on lunar (or solar) ring plane - J ₂ coefficient of second harmonic of Earth's gravitational potential (1082.637×10^–6 R _E ² ) - K _k, L_k, M_k expressions in Section 4 - expressions in Section 4 - p=a(1–e ²) semi-latus rectum - P _l Legendre polynomial of degreel - q argument of Legendre polynomial - radial distance of satellite - R _E Earth equatorial radius (6378.16 km) - R, S, W perturbing accelerations in the radial, tangential and orbit normal directions - syn synchronous orbit radius (42 164.2 km=6.6107R _E) - t time - T satellite orbital period - T orbital period of perturbing body (Moon) - T _e period of long periodic oscillations ine for |I|<I _a - T _s synodic period - U gravitational potential of lunar (or solar) ring - x, y, z Cartesian coordinates of a satellite with (x, y) being the ring plane - coefficient defined in Equation (20) - average change in orbital element over one orbit (=a, e, I, , ) - ₁,₂₃ unit vectors in thex, y, z coordinate directions - _r , _s , _w unit vectors in the radial, tangential and orbit normal directions - =+ angle along the orbital plane from the ascending node on the ring plane to the true position of the satellite - angle around the ring - gravitational constant times mass of Earth (3.986 013×10⁵ km s^–2) - gravitational constant times mass of Moon (or Sun) - _m gravitational constant times mass of Moon (/81.301) - _s gravitational constant time mass of Sun (332 946 ) - ratio of the circumference of a circle to its diameter - radius of lunar (or solar) ring - _m radius of lunar ring (60.2665R _E) - _s radius of solar ring (23455R _E) - true anomaly - argument of perigee - ₀ initial value of - _i critical value of in quadranti(i=1, 2, 3, 4) - longitude of ascending node on ring plane This work was sponsored by the Department of the Air Force.     

Emden-chandrasekhar axisymmetric,rigidly rotating polytropes

An approximate analytical method of solving the polytropic equilibrium equations, first developed by Seidov and Kuzakhmedov (1978), has been extended and generalized to equilibrium configurations of axisymmetric systems in rigid rotation, with polytropic index,n =n _p + _n , nearn _p =0, 1, and 5. Though the details of the method depend on the value ofn _p , acceptable results are obtained for | _n | 0.5 to describe slowly rotating configurations in the range 0n1.5, 4.5n5. In the limit of rotational equilibrium configurations, when the distorsion may be large enough, a satisfactory approximation holds only in the range 0n, 1n1.5, 4.5n5.     

Effect of central condensation on the modes of nonradial oscillations of stellar models

Modes of nonradial oscillations of six composite polytropic models have been investigated numerically to study the effect of central condensation parameter being the density at the centre and the mean density of a stellar model) on the modes of nonradial oscillations of stellar models.