The distribution law of stellarp-modes

We show that the overall densityg() of asymptotic acoustic frequencies of a star obeys a Weyl lawg() ^D–1, whereD is the dimensionality of the oscillating stellar configuration. For realistic stars with a finite non-zero surface sound speed,D is equal to the actual dimensionality of the star,D=3. For formal models with a vanishing sound velocity at the surface, heuristic arguments lead to a dimensionality parameterD=4.5. The empirical frequencies of Eddington's standard model are found to be consistent with the latter distribution, with reasonable agreement already occurring in the low-frequency range > _i 2× fundamental radial mode. We argue that real stars obey this 3.5-power law in some finite frequency interval _i << _f , _f being a very high frequency critically depending on the surface sound velocity, while the full asymptotic law, withD=3, holds for > _f .     

Almost stable regions for an area-preserving mapping

The area preserving mapping x = x + a(y – y ³), y = y – a(x – x³), for 0.3 a 2.0 has been studied to locate approximately the x-axis points bounding almost stable regions. For each value of a, these are fixed points with variational trace just greater than 2.0. Transition to chaos can occur rapidly as a increases (with n/k fixed).     

The effect of luminosity-redshift correlation on the analysis of the angular size-redshift relation for radio galaxies

The variation of radio luminosity with redshift and its effect on the analysis of the angular size-redshift ( –z) relation for a bright radio source sample (s ₁₇₈ 10Jy) has been investigated. By assuming a power law dependence of luminosity on redshift of the formP (1 +z), it was found that 4.4 (with correlation coefficientr 0.99) for at leastz 0.3. Correction for such a strongP – (1 +z) correlation when considering the –z data for the sample led to a steeper –z slope. This could be explained by assuming linear size evolution of the formD (1 +z)^–n withn = 2.8 – 3.3 consistent with both theoretical results and those obtained for more homogeneous source samples.     

Einstein-like field equations with conserved source and decreasing ∧ term; Their cosmological consequences

The consequences of a cosmological term varying asS ^–2 in a spatially isotropic universe with scale factorS and conserved matter tensor are investigated. One finds a perpetually expanding universe with positive and gravitational constantG that increases with time. The hard equation of state 3P>U (U mass-energy density,P scalar pressure) applied to the early universe leads to the expansion lawSt (t cosmic time) which solves the horizon problem with no need of inflation. Also the flatness problem is resolved without inflation. The model does not affect the well known predictions on the cosmic light elements abundance which come from standard big bang cosmology.In the present, matter dominated universe one findsdG/dt=2H/U (H is the Hubble parameter) which is consistent with observations provided <10^–57 cm^–2. Asymptotically (S) the term equalsGU/2, in agreement with other studies.     

The propagation of solar cosmic-ray bursts

A model is presented in which we show analytically the three phases of anisotropy which occur during solar cosmic-ray events observed in the 7.5 MeV to 21 MeV kinetic-energy interval and reported by McCracken et al. (1971): (i) a highly anisotropic, near field-aligned, initial phase, (ii) a convective phase, and (iii) a late-time phase in which the anisotropy is approximately perpendicular to the mean interplanetary magnetic field. The model is based on the cosmic-ray particles being convectively transported out from the Sun, undergoing anisotropic diffusion along the interplanetary magnetic-field lines, and losing energy by adiabatic deceleration or by collision processes. The event is seen simply as a pulse moving outward from the Sun after a cosmic-ray burst with a negative density-gradient in front of it and a positive gradient behind. The convective phase (ii) occurs as the spatial peak moves past the observer and has a propagation speed V _d associated with it; the anisotropy vector late in the decay phase (iii) is the result of a residual balance between the radial outward convection and the inward radial component of the anisotropic diffusion. The mathematical solutions are based upon a diffusion coefficient proportional to heliocentric radius and independent of energy and are thus rather special. However they yield formulae for the propagation speed of the convective phase and the direction in space of the long-time anisotropy which are useful as a guide to the dependence of these quantities on the solar wind speed V, the diffusion coefficient and the spectral index . In this model V _d increases with V, , and ; and , the angle between the anisotropy vector at infinite time and the outward radial direction increases with /V and decreases as is increased. These predictions of the dependence of and V _d upon V, , and are open to observational verification.     

The Energy Equation in the Lower Solar Convection Zone

As a consequence of the Taylor–Proudman balance, a balance between the pressure, Coriolis and buoyancy forces in the radial and latitudinal momentum equations (that is expected to be amply satisfied in the lower solar convection zone), the superadiabatic gradient is determined by the rotation law and by an unspecified function of r, say, S(r), where r is the radial coordinate. If the rotation law and S(r) are known, then the solution of the energy equation, performed in this paper in the framework of the ML formalism, leads to a knowledge of the Reynolds stresses, convective fluxes, and meridional motions. The ML-formalism is an extension of the mixing length theory to rotating convection zones, and the calculations also involve the azimuthal momentum equation, from which an expression for the meridional motions in terms of the Reynolds stresses can be derived. The meridional motions are expanded as U _r(r,)=P ₂(cos)₂(r)/r ²+P ₄(cos)₄(r)/r ² +..., and a corresponding equation for U (r,). Here is the polar angle, is the density, and P ₂(cos), P ₄(cos) are Legendre polynomials. A good approximation to the meridional motion is obtained by setting ₄(r)=–H₂(r) with H–1.6, a constant. The value of ₂(r) is negative, i.e., the P ₂ flow rises at the equator and sinks at the poles. For the value of H obtained in the numerical calculations, the meridional motions have a narrow countercell at the poles, and the convective flux has a relative maximum at the poles, a minimum at mid latitudes and a larger maximum at the equator. Both results are in agreement with the observations.     

The existence of long-lived rays of the coronal streamer belt – Radial density and velocity distributions of the solar wind flowing in them

A technique is proposed for separating the rays of the streamer belt with quasi-stationary and non-stationary solar wind (SW) flows. It is shown that the lifetime of rays with a quasi-stationary SW can exceed 20 days. A new method has been developed for measuring the relative density distribution of a quasi-stationary slow SW flowing along the streamer belt's ray of increased brightness, based on the LASCO/SOHO data. It is shown that the density n for such SW flows varies with the radius R according to the relationship nR ^–, where =₁3.3–3.9 within 4 R ₀ R 6 R ₀ (here R ₀ is the solar radius), and decreases gradually further away. It is also shown that the V(R)-profiles in some rays of the streamer belt differ little from each other, although the value of the mass flow density, j _E, at the Earth's orbit in them can vary more than by a factor of 4. This distinguishes in a crucial respect a slow SW in the streamer belt's rays from a fast SW originating in coronal holes, for which j _Econstant and the dependences V(R) in different fast flows can differ greatly.     

Masses of macroscopic quark configurations in metric and dynamic theories of gravitation

Within the bounds of the general relativity and in gravidynamics, spherically-symmetric configurations are considered with the limit equation of state (P = ( - 4B)/3) and with the density increasing to the center. It is shown that unlike GR, where the existence of strange stars only is permissible (u-, d-, s-quarks), in the consistent dynamic theory of gravitation the existence ofstable configuration withr ^–2 (quark star) is possible with a bag out of quark-gluon plasma which includes all possible quark flavors (u, d, s, c, b, t, .. .). The total mass of such a compact object with the bag of the radius of 10 km (whose surface consists of the strange self-bound matter) must be 6 - 7M .     

Rapidly rotating polytropes in the post-Newtonian approximation to general relativity

The study of uniformly polytropes with axial symmetry is extended to include all rotational terms of order ⁴, where is the angular velocity, consistently within the first post-Newtonian approximation to general relativity. The equilibrium structure is determined by treating the effects of rotation and post-Newtonian gravitation as independent perturbations on the classical polytropic structure. The perturbation effects are characterized by a rotation parameter = ²/2G _c and a relativity parameter, =p _c / _c C ² , wherep _c and _c are the central pressure and density respectively. The solution to the structural problem is obtained by following Chandrasekhar's series expansion technique and is complete to the post-Newtonian rotation terms of order ². The critical rotation parameterv _c , which characterizes the configuration with maximum uniform rotation, is accurately evaluated as a function of . Numerical values for all the structural parameters needed to determine the equilibrium configurations are presented for polytropes with indicesn=1, 1.5, 2, 2 5, 3, and 3.5.     

Period distributions in pulsating and binary stars

We analyze the hypothesis of quantization in bands for the angular momenta of binary systems and for the maount of actionA _c in stable and pulsating stars. This parameter isA _c=Mv _eff R _eff, where the effective velocity corresponds to the kinetic energy in the stellar interior and the effective radius corresponds to the potential energyGM ²/R _eff. Analogous parameters can be defined for a pulsating star withm=M where is the rate of the massm participating in the oscillation to the total massM andv _osc,R _osc the effective velocity and oscillation radius.From an elementary dimensional analysis one has thetA _c (energy x time) (period)^1/3 independently ifA _c corresponds to the angular momentum in a binary system, or to the oscillation in a pulsating star or the inner energy and its time-scaleP _eff in a stable star.From evolving stellar models one has that P _effP _eff(solar)1.22 hr a near-invariant for the Main Sequence and for the range of masses 0.6M <M<1.6M .With this one can give scalesn _k=kn ₁ withk integers andn ₁=(P/P ₁)^1/3 withP ₁=P _eff1.22 hr. In these scales proportional toA _c, one sees that the periods in binary and pulsating stars are clustered in discrete unitsn ₁,n ₂,n ₃, etc.This can be seen in pulsating Scuti, Cephei, RR Lyrae, W Virginis, Cephei, semi-regular variables, and Miras and in binary stars as cataclysmic binaries, W Ursa Majoris, Algols, and Lyrae with the corresponding subgroups in all these materials. Phase functions (n _k) in RR Lyrae and Cephei are also associated with discrete levelsn _k.the suggested scenario is that the potential energies and the amounts of actionE _p(t), A_c(t) are indeed time-dependent, but the stars remain more time in determinated most proble states. The Main Sequence itself is an example of this. These most probable states in binary systems, or pulsating or stable stars, must be associated with velocities sub-multiplesc/ _F , given by the velocity of light and the fine structure constant.Additional tests for such a hypothesis are suggested when the sufficient amount of observational data are available. They can made with oscillation velocities in pulsating stars and velocity differences of pairs of galaxies.     

Velocity dispersion profiles ofN-body simulations

The time evolution of the velocity dispersion as a function of radius, called v-profiles, of threeN-body simulations of Wielen are presented in units ofr/R _G (whereR _G is the gravitational or virial radius) and discussed as a function of mass sample. The evolution of the radial and tangential components of the velocity dispersion is discussed, and each v-profile is fitted to a simple power law in the halo (0.15r/R _G2.0). Several structural features appear at late time intervals: (a) an upturn in the radial component of v which occurs in a decreasing shell (closer to the core) in time, (b) the v-profile of the massive particles mimics that of the total sample, since equipartition of kinetic energy does not obtain, and (c) a local minimum atr0.3–0.5R _G appears in one model which coincides with the local minimum in the number density profiles and possibly with feature (a).The line-of-sight v-profile, called LS-profile, of each model as a function of time and mass sample are also presented and discussed. They contain the same structural features as the v-profiles. Projection factors at small radii are also discussed. The LS-profiles of the models can be compared with the observed velocity dispersion profiles of clusters of galaxies in Struble (1979a).     

Unsteady MHD flow past a porous plate with step function change in suction at slip flow regime

Unsteady two-dimensional hydromagnetic flow of an electrically conducting viscous incompressible fluid past a semi-infinite porous flat plate with step function change in suction velocity is studied allowing a first order velocity slip at the boundary condition. The solution of the problem is obtained in closed form and the results are discussed with the aid of graphs for various parameters entering in the problem.Notations B intensity of magnetic field - H magnetic field parameter,H=(M+1/4)^1/2–1/2 - h rarefaction parameter - L ₁ slip coefficient; ;I, mean free path of gas molecules;f, Maxwell's reflection coefficient - M magnetic field parameter - r suction parameter - t time - t dimensionless time - u velocity of the fluid - u dimensionless velocity of the fluid - U velocity of the fluid at infinity - v suction velocity - v ₁ suction velocity att<=0 - v ₂ suction velocity att>0 - x distance parallel to the plate - y distance normal to the plate - y nondimensional distance normal to the plate - v kinematic viscosity - electric conductivity of the fluid - density of the fluid - shear stress at the wall - nondimensional shear stress at the wall - erf error function - erfc complementary error function     

Hollow beam distribution of energetic electrons and higher harmonics of electron cyclotron maser

The variations of the growth rates of ECM at first four harmonics in X-, Z-, and O-modes excited by a hollow beam distribution of weakly relativistic electrons with a parameter _p / _e are presented in this paper. The results show that the second harmonic of ECM in X-mode dominates the instability if < 1, and if 1.2 , 2 or 2.2 3 the third or fourth harmonic will dominate. The second and third harmonics of Z-mode waves grow faster only if 2 2.2 and 3 3.2, respectively, so it would not be a competition in most cases. It is suggested that the ECM emission at these harmonics in X-mode is a possible mechanism to produce solar spike emissions with high brightness temperature at shorter and longer decimetric wavelengths.Proceedings of the Workshop on Radio Continua during Solar Flares, held at Duino (Trieste), Italy, 27–31 May, 1985.On leave from the Department of Astronomy, Nanjing University, Nanjing, The People's Republic of China.     

A relation in families of periodic solutions

We show the existence of a general relation between the parameters of periodic solutions in dynamical systems with ignorable coordinates. In particular, for time-independent systems with an axis of symmetry, the relation takes the form T/A=–/E, whereT is the period,A is the angular momentum, is the angle through which the system has rotated after one period, andE is the energy.     

Absolute and relative amplitudes of variations in radius of classical cepheids

On the basis of observational data for the absolute R and relative R/R amplitudes of variations in radius of galactic classical cepheids (55 stars from Balona and Stobie (1979) and 30 stars from Sollazzoet al. (1981)), four kinds of empirical linear relations are obtained: log(P V)–logR, logP–logR, log(P V)–log(R/R), and logP–log(R/R);P, R, and V are the pulsation periods, the mean stellar radii, and the amplitudes of light variations, respectively. Three groups of stars are considered: short-period cepheids (SPC)-with logP1.1; long-period cepheids (LPC)-with logP>1.1; and s-cepheids (sC). Both the R values and the R/R values increase withP andP V, for a given group of variables. A comparison is performed with our results obtained from data in other sources (Kurochkin, 1966; Gieren, 1982; etc.). The investigated relations can be applied for determining R and R/R of galactic classical cepheids, by using their observedP and V. All studied galactic classical cepheids have R/R<0.35, R<10R for SPC and 10R <R60R for LPC. The sC have smaller R and R/R values than other classical cepheids, at the same periods (the difference is about 2 times for R and 1.4–2.8 times for R/R); the studied sC have R/R in the range 0.025–0.075 and R in the range 1–3R (only Y Oph has R8R ).     

Coherence analysis of granular intensity

A high resolution spectrogram of the Mg b₂ line from the quiet Sun disc centre is subjected to a coherence analysis. We find that the coherence between intensity fluctuations in the continuum and the wings of the line breaks down at a distance = 0.35 Å from line centre. From this and the r.m.s. intensity contrast as a function of we are led to the following simple model of temperature fluctuation T in the solar photosphere: A lower part (below 50 km, or ₅₀₀₀ > 0.25) with strongly inward increasing T and an upper part (above 50 km) with constant T = 75 K. The two parts are supposed to fluctuate incoherently.Mitteilungen aus dem Kiepenheuer Institut Nr. 166.     

Lunar properties from transient and steady magnetic field measurements

The electrical conductivity of the lunar interior has been determined from magnetic field step transients measured on the lunar dark side. The simplest model which best fits the data is a spherically symmetric three layer model having a nonconducting outer crust of radial thickness 0.03R _moon; an intermediate layer of thicknessR0.37R _moon, with electrical conductivity ₁ 3.5 × 10^–4 mhos/m; and an inner core of radiusR ₂ 0.6R _m with conductivity ₂ 10^–2 mhos/m. Temperatures calculated from these conductivities in the three regions for an example of an olivine Moon are as follows: crust, < 440 K; intermediate layer, 890 K; and core, 1240 K. The whole-moon relative permeability has been calculated from the measurements to be/ ₀ = 1.03 ± 0.13. Remanent magnetic fields at the landing sites are 38 ± 3 at Apollo 12, 43 ± 6 and 103 ± 5 at two Apollo 14 sites separated by 1.1 km, and 6 ± 4 at the Apollo 15 site. Measurements show that the 38 remanent field at the Apollo 12 site is compressed to 54 by a solar wind pressure increase of 7 × 10^–8 dynes/cm².National Research Council Postdoctoral Associate.     

Search for the matter clumps in scalesZ ~ 1

The diagramV - log(1 +z _e ) as function of (, ) is considered for the quasars. HereV is the apparent visual magnitude,z _e is the emission line redshift, and are the equatorial coordinates. Two opposite extreme spots NE and SE are observed on the sky, where the inclination of the straight line fitting the dependenceV - log(1 +z _e ) is maximum and minimum. The coordinates of the centres of these extreme spots are ( _NE, _NE) = (282°, +42°) and ( _SE, _SE) = (70°, -38°) with errors 5°. A hypothesis of the Superattractor (SA) is proposed to explain such an effect. Two independent tests of this hypothesis are realized. First, the dependence or the frequency a of the absorbers in QSO spectra on (, ) is investigated. A region of the larger a is found. The coordinates of its centre are (, ) = (82°, - 10°) with error 5°. Second, the cases ofz _a >z _e are plotted in the Mercatorial projection (, ). The most of the casesz -z _e > 0.02 are concentrated within the circle with radiusR = 34° and centre (, ) = (50°, - 15°). The both anomalous regions overlap the Southern extreme spot around SE. The SA direction is (, ) = (67°, -21°) with errors about 12°. The redshift of SA isz _SA = 1.7 ± 0.3 that corresponds to the distancer _SA = (3100 ± 300)h ^–1 Mpc for the Hubble constantH ₀ = 75h kms^–1 Mpc^–1. The SA mass isM _SA ~ 10¹⁸-10²⁰ M . The orientation of the normal to the quasiperiodical large-scale sheet structure on the sky occurs near SA.     

Genericity of the non-periodic solutions of the central force problem

We study the classical problem of two-dimensional motion of a particle in the field of a central force proportional to a real power of the distancer. for negative energy and (0, 2), each energy levelI _h is foliated by the invariant toriI _hc of constant angular momentumc and, by Liouville-Arnold's theorem, the flow on eachI _hc is conjugated to a linear flow of rotation number _h (c).A well-known result asserts that if we require _h (c) to be rational for every value ofh andc, the, must be equal to one (Kepler's problem). In this paper we prove that for almost every (0, 2) _h (c) is a non-constant continuous function ofc, for everyh<0. In particular, we deduce that motion under central potentials is generically non-periodic.Partially supported by CIRIT under grant No. EE88/2.     

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

The following points are discussed: